%------------------------------------------------------------------------------
% File     : ITP210_4 : TPTP v8.2.0. Released v8.0.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem Assertions 00353_010205
% 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    : 0024_Assertions_00353_010205 [Des22]

% Status   : Theorem
% Rating   : 1.00 v8.1.0
% Syntax   : Number of formulae    : 12253 (4535 unt;2321 typ;   0 def)
%            Number of atoms       : 15293 (8141 equ)
%            Maximal formula atoms :   38 (   1 avg)
%            Number of connectives : 16781 (2007   ~; 242   |;1370   &)
%                                         (1958 <=>;11204  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   24 (   6 avg)
%            Maximal term depth    :   37 (   2 avg)
%            Number of FOOLs       :  770 ( 524 fml; 246 var)
%            Number of X terms     :  498 (   0  []; 464 ite;  34 let)
%            Number of types       :   14 (  13 usr)
%            Number of type conns  : 2127 (1670   >; 457   *;   0   +;   0  <<)
%            Number of predicates  :  299 ( 296 usr;   2 prp; 0-9 aty)
%            Number of functors    : 2030 (2030 usr; 102 con; 0-8 aty)
%            Number of variables   : 37965 (33670   !; 568   ?;37965   :)
%                                         (3727  !>;   0  ?*;   0  @-;   0  @+)
% SPC      : TX1_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            from the van Emde Boas Trees session in the Archive of Formal
%            proofs - 
%            www.isa-afp.org/browser_info/current/AFP/Van_Emde_Boas_Trees
%            2022-02-17 14:35:53.877
%------------------------------------------------------------------------------
% Could-be-implicit typings (26)
tff(ty_t_Code__Numeral_Onatural,type,
    code_natural: $tType ).

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

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

tff(ty_t_Heap_Oheap_Oheap__ext,type,
    heap_ext: $tType > $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_Multiset_Omultiset,type,
    multiset: $tType > $tType ).

tff(ty_t_Assertions_Oassn,type,
    assn: $tType ).

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

tff(ty_t_Predicate_Opred,type,
    pred: $tType > $tType ).

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

tff(ty_t_Predicate_Oseq,type,
    seq: $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_List_Olist,type,
    list: $tType > $tType ).

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

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

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

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

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

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

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

tff(ty_tf_b,type,
    b: $tType ).

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

% Explicit typings (2295)
tff(sy_cl_Typerep_Otyperep,type,
    typerep: 
      !>[A: $tType] : $o ).

tff(sy_cl_Enum_Oenum,type,
    enum: 
      !>[A: $tType] : $o ).

tff(sy_cl_Code__Evaluation_Oterm__of,type,
    code_term_of: 
      !>[A: $tType] : $o ).

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

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

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

tff(sy_cl_Groups_Otimes,type,
    times: 
      !>[A: $tType] : $o ).

tff(sy_cl_Lattices_Oinf,type,
    inf: 
      !>[A: $tType] : $o ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_cl_Enum_Ofinite__lattice,type,
    finite_lattice: 
      !>[A: $tType] : $o ).

tff(sy_cl_Fields_Ofield__char__0,type,
    field_char_0: 
      !>[A: $tType] : $o ).

tff(sy_cl_Groups_Oab__group__add,type,
    ab_group_add: 
      !>[A: $tType] : $o ).

tff(sy_cl_Nat_Osemiring__char__0,type,
    semiring_char_0: 
      !>[A: $tType] : $o ).

tff(sy_cl_Orderings_Oorder__bot,type,
    order_bot: 
      !>[A: $tType] : $o ).

tff(sy_cl_Orderings_Oorder__top,type,
    order_top: 
      !>[A: $tType] : $o ).

tff(sy_cl_Orderings_Owellorder,type,
    wellorder: 
      !>[A: $tType] : $o ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_cl_Quickcheck__Random_Orandom,type,
    quickcheck_random: 
      !>[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_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_Bit__Operations_Osemiring__bits,type,
    bit_semiring_bits: 
      !>[A: $tType] : $o ).

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

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

tff(sy_cl_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_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_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_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_Quickcheck__Exhaustive_Oexhaustive,type,
    quickc658316121487927005ustive: 
      !>[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_Rings_Olinordered__nonzero__semiring,type,
    linord181362715937106298miring: 
      !>[A: $tType] : $o ).

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

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

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

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

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

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

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

tff(sy_cl_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_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_Quickcheck__Exhaustive_Ofull__exhaustive,type,
    quickc3360725361186068524ustive: 
      !>[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_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_Divides_Ounique__euclidean__semiring__numeral,type,
    unique1627219031080169319umeral: 
      !>[A: $tType] : $o ).

tff(sy_cl_Complete__Lattices_Ocomplete__boolean__algebra,type,
    comple489889107523837845lgebra: 
      !>[A: $tType] : $o ).

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

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

tff(sy_cl_Euclidean__Division_Ounique__euclidean__ring__with__nat,type,
    euclid8789492081693882211th_nat: 
      !>[A: $tType] : $o ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_ATP_058Lamp__ama____,type,
    aTP_Lamp_ama: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_ATP_058Lamp__ara____,type,
    aTP_Lamp_ara: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_ATP_058Lamp__arw____,type,
    aTP_Lamp_arw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__arx____,type,
    aTP_Lamp_arx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ary____,type,
    aTP_Lamp_ary: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__arz____,type,
    aTP_Lamp_arz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

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

tff(sy_c_ATP_058Lamp__asa____,type,
    aTP_Lamp_asa: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__asb____,type,
    aTP_Lamp_asb: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__asc____,type,
    aTP_Lamp_asc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__asd____,type,
    aTP_Lamp_asd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ase____,type,
    aTP_Lamp_ase: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__asf____,type,
    aTP_Lamp_asf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__asg____,type,
    aTP_Lamp_asg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ash____,type,
    aTP_Lamp_ash: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__asi____,type,
    aTP_Lamp_asi: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__asj____,type,
    aTP_Lamp_asj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ask____,type,
    aTP_Lamp_ask: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__asl____,type,
    aTP_Lamp_asl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__asm____,type,
    aTP_Lamp_asm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__asn____,type,
    aTP_Lamp_asn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aso____,type,
    aTP_Lamp_aso: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__asp____,type,
    aTP_Lamp_asp: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__asq____,type,
    aTP_Lamp_asq: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__asr____,type,
    aTP_Lamp_asr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ass____,type,
    aTP_Lamp_ass: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ast____,type,
    aTP_Lamp_ast: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__asu____,type,
    aTP_Lamp_asu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__asv____,type,
    aTP_Lamp_asv: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__asw____,type,
    aTP_Lamp_asw: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__asx____,type,
    aTP_Lamp_asx: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__asy____,type,
    aTP_Lamp_asy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__asz____,type,
    aTP_Lamp_asz: 
      !>[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__ata____,type,
    aTP_Lamp_ata: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__atb____,type,
    aTP_Lamp_atb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__atc____,type,
    aTP_Lamp_atc: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__atd____,type,
    aTP_Lamp_atd: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__ate____,type,
    aTP_Lamp_ate: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__atf____,type,
    aTP_Lamp_atf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__atg____,type,
    aTP_Lamp_atg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ath____,type,
    aTP_Lamp_ath: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ati____,type,
    aTP_Lamp_ati: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__atj____,type,
    aTP_Lamp_atj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__atk____,type,
    aTP_Lamp_atk: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__atl____,type,
    aTP_Lamp_atl: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__atm____,type,
    aTP_Lamp_atm: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__atn____,type,
    aTP_Lamp_atn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ato____,type,
    aTP_Lamp_ato: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__atp____,type,
    aTP_Lamp_atp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__atq____,type,
    aTP_Lamp_atq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__atr____,type,
    aTP_Lamp_atr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ats____,type,
    aTP_Lamp_ats: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__att____,type,
    aTP_Lamp_att: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__atu____,type,
    aTP_Lamp_atu: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__atv____,type,
    aTP_Lamp_atv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__atw____,type,
    aTP_Lamp_atw: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__atx____,type,
    aTP_Lamp_atx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aty____,type,
    aTP_Lamp_aty: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__atz____,type,
    aTP_Lamp_atz: 
      !>[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__aua____,type,
    aTP_Lamp_aua: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__aub____,type,
    aTP_Lamp_aub: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__auc____,type,
    aTP_Lamp_auc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aud____,type,
    aTP_Lamp_aud: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aue____,type,
    aTP_Lamp_aue: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__auf____,type,
    aTP_Lamp_auf: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__aug____,type,
    aTP_Lamp_aug: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__auh____,type,
    aTP_Lamp_auh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aui____,type,
    aTP_Lamp_aui: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__auj____,type,
    aTP_Lamp_auj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__auk____,type,
    aTP_Lamp_auk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aul____,type,
    aTP_Lamp_aul: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aum____,type,
    aTP_Lamp_aum: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aun____,type,
    aTP_Lamp_aun: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__auo____,type,
    aTP_Lamp_auo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aup____,type,
    aTP_Lamp_aup: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__auq____,type,
    aTP_Lamp_auq: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__aur____,type,
    aTP_Lamp_aur: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__aus____,type,
    aTP_Lamp_aus: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__aut____,type,
    aTP_Lamp_aut: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__auu____,type,
    aTP_Lamp_auu: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__auv____,type,
    aTP_Lamp_auv: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__auw____,type,
    aTP_Lamp_auw: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__aux____,type,
    aTP_Lamp_aux: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__auy____,type,
    aTP_Lamp_auy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__auz____,type,
    aTP_Lamp_auz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

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

tff(sy_c_ATP_058Lamp__ava____,type,
    aTP_Lamp_ava: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__avb____,type,
    aTP_Lamp_avb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__avc____,type,
    aTP_Lamp_avc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__avd____,type,
    aTP_Lamp_avd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ave____,type,
    aTP_Lamp_ave: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__avf____,type,
    aTP_Lamp_avf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__avg____,type,
    aTP_Lamp_avg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__avh____,type,
    aTP_Lamp_avh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__avi____,type,
    aTP_Lamp_avi: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__avj____,type,
    aTP_Lamp_avj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__avk____,type,
    aTP_Lamp_avk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__avl____,type,
    aTP_Lamp_avl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__avm____,type,
    aTP_Lamp_avm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__avn____,type,
    aTP_Lamp_avn: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__avo____,type,
    aTP_Lamp_avo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__avp____,type,
    aTP_Lamp_avp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__avq____,type,
    aTP_Lamp_avq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__avr____,type,
    aTP_Lamp_avr: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__avs____,type,
    aTP_Lamp_avs: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__avt____,type,
    aTP_Lamp_avt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__avu____,type,
    aTP_Lamp_avu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__avv____,type,
    aTP_Lamp_avv: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__avw____,type,
    aTP_Lamp_avw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__avx____,type,
    aTP_Lamp_avx: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__avy____,type,
    aTP_Lamp_avy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__avz____,type,
    aTP_Lamp_avz: 
      !>[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__awa____,type,
    aTP_Lamp_awa: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__awb____,type,
    aTP_Lamp_awb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__awc____,type,
    aTP_Lamp_awc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__awd____,type,
    aTP_Lamp_awd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__awe____,type,
    aTP_Lamp_awe: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__awf____,type,
    aTP_Lamp_awf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__awg____,type,
    aTP_Lamp_awg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__awh____,type,
    aTP_Lamp_awh: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__awi____,type,
    aTP_Lamp_awi: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__awj____,type,
    aTP_Lamp_awj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__awk____,type,
    aTP_Lamp_awk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__awl____,type,
    aTP_Lamp_awl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__awm____,type,
    aTP_Lamp_awm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__awn____,type,
    aTP_Lamp_awn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__awo____,type,
    aTP_Lamp_awo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__awp____,type,
    aTP_Lamp_awp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__awq____,type,
    aTP_Lamp_awq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__awr____,type,
    aTP_Lamp_awr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aws____,type,
    aTP_Lamp_aws: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__awt____,type,
    aTP_Lamp_awt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__awu____,type,
    aTP_Lamp_awu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__awv____,type,
    aTP_Lamp_awv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aww____,type,
    aTP_Lamp_aww: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__awx____,type,
    aTP_Lamp_awx: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__awy____,type,
    aTP_Lamp_awy: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__awz____,type,
    aTP_Lamp_awz: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

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

tff(sy_c_ATP_058Lamp__axa____,type,
    aTP_Lamp_axa: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__axb____,type,
    aTP_Lamp_axb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__axc____,type,
    aTP_Lamp_axc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__axd____,type,
    aTP_Lamp_axd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__axe____,type,
    aTP_Lamp_axe: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__axf____,type,
    aTP_Lamp_axf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__axg____,type,
    aTP_Lamp_axg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__axh____,type,
    aTP_Lamp_axh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__axi____,type,
    aTP_Lamp_axi: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__axj____,type,
    aTP_Lamp_axj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__axk____,type,
    aTP_Lamp_axk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__axl____,type,
    aTP_Lamp_axl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__axm____,type,
    aTP_Lamp_axm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__axn____,type,
    aTP_Lamp_axn: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__axo____,type,
    aTP_Lamp_axo: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__axp____,type,
    aTP_Lamp_axp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__axq____,type,
    aTP_Lamp_axq: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__axr____,type,
    aTP_Lamp_axr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__axs____,type,
    aTP_Lamp_axs: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__axt____,type,
    aTP_Lamp_axt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__axu____,type,
    aTP_Lamp_axu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__axv____,type,
    aTP_Lamp_axv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__axw____,type,
    aTP_Lamp_axw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__axx____,type,
    aTP_Lamp_axx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__axy____,type,
    aTP_Lamp_axy: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__axz____,type,
    aTP_Lamp_axz: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

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

tff(sy_c_ATP_058Lamp__aya____,type,
    aTP_Lamp_aya: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ayb____,type,
    aTP_Lamp_ayb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ayc____,type,
    aTP_Lamp_ayc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ayd____,type,
    aTP_Lamp_ayd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aye____,type,
    aTP_Lamp_aye: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ayf____,type,
    aTP_Lamp_ayf: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__ayg____,type,
    aTP_Lamp_ayg: 
      !>[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] : fun(A,B) ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_ATP_058Lamp__pz____,type,
    aTP_Lamp_pz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qa____,type,
    aTP_Lamp_qa: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qb____,type,
    aTP_Lamp_qb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qc____,type,
    aTP_Lamp_qc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qd____,type,
    aTP_Lamp_qd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qe____,type,
    aTP_Lamp_qe: 
      !>[A: $tType,B: $tType] : fun(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] : fun(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] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__rb____,type,
    aTP_Lamp_rb: 
      !>[A: $tType,B: $tType] : fun(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] : fun(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] : fun(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] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__so____,type,
    aTP_Lamp_so: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__sp____,type,
    aTP_Lamp_sp: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__sq____,type,
    aTP_Lamp_sq: 
      !>[A: $tType,B: $tType] : fun(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] : fun(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] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__ta____,type,
    aTP_Lamp_ta: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__tb____,type,
    aTP_Lamp_tb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__tc____,type,
    aTP_Lamp_tc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__td____,type,
    aTP_Lamp_td: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__te____,type,
    aTP_Lamp_te: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__tf____,type,
    aTP_Lamp_tf: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__tg____,type,
    aTP_Lamp_tg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__th____,type,
    aTP_Lamp_th: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ti____,type,
    aTP_Lamp_ti: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__tj____,type,
    aTP_Lamp_tj: 
      !>[A: $tType,B: $tType] : ( 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] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__tm____,type,
    aTP_Lamp_tm: 
      !>[A: $tType,B: $tType] : fun(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] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__tx____,type,
    aTP_Lamp_tx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ty____,type,
    aTP_Lamp_ty: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__tz____,type,
    aTP_Lamp_tz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ua____,type,
    aTP_Lamp_ua: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ub____,type,
    aTP_Lamp_ub: 
      !>[A: $tType,B: $tType] : fun(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] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__uh____,type,
    aTP_Lamp_uh: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__ui____,type,
    aTP_Lamp_ui: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__uj____,type,
    aTP_Lamp_uj: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__uk____,type,
    aTP_Lamp_uk: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__ul____,type,
    aTP_Lamp_ul: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__um____,type,
    aTP_Lamp_um: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__un____,type,
    aTP_Lamp_un: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__uo____,type,
    aTP_Lamp_uo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__up____,type,
    aTP_Lamp_up: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__uq____,type,
    aTP_Lamp_uq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ur____,type,
    aTP_Lamp_ur: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__us____,type,
    aTP_Lamp_us: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ut____,type,
    aTP_Lamp_ut: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__uu____,type,
    aTP_Lamp_uu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__uv____,type,
    aTP_Lamp_uv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__uw____,type,
    aTP_Lamp_uw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ux____,type,
    aTP_Lamp_ux: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__uy____,type,
    aTP_Lamp_uy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__uz____,type,
    aTP_Lamp_uz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__va____,type,
    aTP_Lamp_va: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vb____,type,
    aTP_Lamp_vb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vc____,type,
    aTP_Lamp_vc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vd____,type,
    aTP_Lamp_vd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ve____,type,
    aTP_Lamp_ve: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vf____,type,
    aTP_Lamp_vf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vg____,type,
    aTP_Lamp_vg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vh____,type,
    aTP_Lamp_vh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vi____,type,
    aTP_Lamp_vi: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vj____,type,
    aTP_Lamp_vj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vk____,type,
    aTP_Lamp_vk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vl____,type,
    aTP_Lamp_vl: 
      !>[A: $tType,B: $tType] : fun(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] : fun(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] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__vu____,type,
    aTP_Lamp_vu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vv____,type,
    aTP_Lamp_vv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vw____,type,
    aTP_Lamp_vw: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__vx____,type,
    aTP_Lamp_vx: 
      !>[A: $tType,B: $tType] : fun(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] : fun(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] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__wk____,type,
    aTP_Lamp_wk: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__wl____,type,
    aTP_Lamp_wl: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__wm____,type,
    aTP_Lamp_wm: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__wn____,type,
    aTP_Lamp_wn: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__wo____,type,
    aTP_Lamp_wo: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__wp____,type,
    aTP_Lamp_wp: 
      !>[A: $tType,B: $tType] : fun(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] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__wt____,type,
    aTP_Lamp_wt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wu____,type,
    aTP_Lamp_wu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wv____,type,
    aTP_Lamp_wv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ww____,type,
    aTP_Lamp_ww: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wx____,type,
    aTP_Lamp_wx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wy____,type,
    aTP_Lamp_wy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wz____,type,
    aTP_Lamp_wz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xa____,type,
    aTP_Lamp_xa: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xb____,type,
    aTP_Lamp_xb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xc____,type,
    aTP_Lamp_xc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xd____,type,
    aTP_Lamp_xd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xe____,type,
    aTP_Lamp_xe: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xf____,type,
    aTP_Lamp_xf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xg____,type,
    aTP_Lamp_xg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xh____,type,
    aTP_Lamp_xh: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__xi____,type,
    aTP_Lamp_xi: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xj____,type,
    aTP_Lamp_xj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xk____,type,
    aTP_Lamp_xk: 
      !>[A: $tType,B: $tType] : ( 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] : fun(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] : fun(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] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__xt____,type,
    aTP_Lamp_xt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xu____,type,
    aTP_Lamp_xu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xv____,type,
    aTP_Lamp_xv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xw____,type,
    aTP_Lamp_xw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xx____,type,
    aTP_Lamp_xx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xy____,type,
    aTP_Lamp_xy: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__xz____,type,
    aTP_Lamp_xz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ya____,type,
    aTP_Lamp_ya: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yb____,type,
    aTP_Lamp_yb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yc____,type,
    aTP_Lamp_yc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yd____,type,
    aTP_Lamp_yd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ye____,type,
    aTP_Lamp_ye: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yf____,type,
    aTP_Lamp_yf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yg____,type,
    aTP_Lamp_yg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yh____,type,
    aTP_Lamp_yh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yi____,type,
    aTP_Lamp_yi: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yj____,type,
    aTP_Lamp_yj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yk____,type,
    aTP_Lamp_yk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yl____,type,
    aTP_Lamp_yl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ym____,type,
    aTP_Lamp_ym: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yn____,type,
    aTP_Lamp_yn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yo____,type,
    aTP_Lamp_yo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yp____,type,
    aTP_Lamp_yp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yq____,type,
    aTP_Lamp_yq: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__yr____,type,
    aTP_Lamp_yr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ys____,type,
    aTP_Lamp_ys: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yt____,type,
    aTP_Lamp_yt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yu____,type,
    aTP_Lamp_yu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yv____,type,
    aTP_Lamp_yv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yw____,type,
    aTP_Lamp_yw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yx____,type,
    aTP_Lamp_yx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yy____,type,
    aTP_Lamp_yy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yz____,type,
    aTP_Lamp_yz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__za____,type,
    aTP_Lamp_za: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__zb____,type,
    aTP_Lamp_zb: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__zc____,type,
    aTP_Lamp_zc: 
      !>[A: $tType,B: $tType] : 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] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__zf____,type,
    aTP_Lamp_zf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zg____,type,
    aTP_Lamp_zg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zh____,type,
    aTP_Lamp_zh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zi____,type,
    aTP_Lamp_zi: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zj____,type,
    aTP_Lamp_zj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zk____,type,
    aTP_Lamp_zk: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__zl____,type,
    aTP_Lamp_zl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zm____,type,
    aTP_Lamp_zm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zn____,type,
    aTP_Lamp_zn: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__zo____,type,
    aTP_Lamp_zo: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__zp____,type,
    aTP_Lamp_zp: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__zq____,type,
    aTP_Lamp_zq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zr____,type,
    aTP_Lamp_zr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zs____,type,
    aTP_Lamp_zs: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__zt____,type,
    aTP_Lamp_zt: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__zu____,type,
    aTP_Lamp_zu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zv____,type,
    aTP_Lamp_zv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zw____,type,
    aTP_Lamp_zw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zx____,type,
    aTP_Lamp_zx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zy____,type,
    aTP_Lamp_zy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zz____,type,
    aTP_Lamp_zz: 
      !>[A: $tType,B: $tType] : ( 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_Assertions_Oassn_OAbs__assn,type,
    abs_assn: fun(fun(product_prod(heap_ext(product_unit),set(nat)),$o),assn) ).

tff(sy_c_Assertions_Oassn_ORep__assn,type,
    rep_assn: fun(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o)) ).

tff(sy_c_Assertions_Oin__range,type,
    in_range: fun(product_prod(heap_ext(product_unit),set(nat)),$o) ).

tff(sy_c_Assertions_Oone__assn__raw,type,
    one_assn_raw: fun(product_prod(heap_ext(product_unit),set(nat)),$o) ).

tff(sy_c_Assertions_Oproper,type,
    proper: fun(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o) ).

tff(sy_c_Assertions_OrelH,type,
    relH: ( set(nat) * heap_ext(product_unit) * heap_ext(product_unit) ) > $o ).

tff(sy_c_Assertions_Otimes__assn__raw,type,
    times_assn_raw: ( fun(product_prod(heap_ext(product_unit),set(nat)),$o) * fun(product_prod(heap_ext(product_unit),set(nat)),$o) ) > fun(product_prod(heap_ext(product_unit),set(nat)),$o) ).

tff(sy_c_Assertions_Otimes__assn__raw__rel,type,
    times_assn_raw_rel: fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),$o)) ).

tff(sy_c_Assertions_Owand__assn,type,
    wand_assn: ( assn * assn ) > assn ).

tff(sy_c_Assertions_Owand__raw,type,
    wand_raw: ( fun(product_prod(heap_ext(product_unit),set(nat)),$o) * fun(product_prod(heap_ext(product_unit),set(nat)),$o) ) > fun(product_prod(heap_ext(product_unit),set(nat)),$o) ).

tff(sy_c_Assertions_Owand__raw__rel,type,
    wand_raw_rel: fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),$o)) ).

tff(sy_c_BNF__Cardinal__Arithmetic_OCsum,type,
    bNF_Cardinal_Csum: 
      !>[A: $tType,B: $tType] : ( ( set(product_prod(A,A)) * fun(A,set(product_prod(B,B))) ) > set(product_prod(product_prod(A,B),product_prod(A,B))) ) ).

tff(sy_c_BNF__Cardinal__Arithmetic_Ocexp,type,
    bNF_Cardinal_cexp: 
      !>[B: $tType,A: $tType] : ( ( set(product_prod(B,B)) * set(product_prod(A,A)) ) > set(product_prod(fun(A,B),fun(A,B))) ) ).

tff(sy_c_BNF__Cardinal__Arithmetic_Ocfinite,type,
    bNF_Cardinal_cfinite: 
      !>[A: $tType] : ( set(product_prod(A,A)) > $o ) ).

tff(sy_c_BNF__Cardinal__Arithmetic_Ocinfinite,type,
    bNF_Ca4139267488887388095finite: 
      !>[A: $tType] : ( set(product_prod(A,A)) > $o ) ).

tff(sy_c_BNF__Cardinal__Arithmetic_Ocone,type,
    bNF_Cardinal_cone: set(product_prod(product_unit,product_unit)) ).

tff(sy_c_BNF__Cardinal__Arithmetic_Ocprod,type,
    bNF_Cardinal_cprod: 
      !>[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_BNF__Cardinal__Arithmetic_Ocsum,type,
    bNF_Cardinal_csum: 
      !>[A: $tType,B: $tType] : ( ( set(product_prod(A,A)) * set(product_prod(B,B)) ) > set(product_prod(sum_sum(A,B),sum_sum(A,B))) ) ).

tff(sy_c_BNF__Cardinal__Arithmetic_Octwo,type,
    bNF_Cardinal_ctwo: set(product_prod($o,$o)) ).

tff(sy_c_BNF__Cardinal__Arithmetic_Oczero,type,
    bNF_Cardinal_czero: 
      !>[A: $tType] : set(product_prod(A,A)) ).

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: $tType] : ( set(A) > fun(set(product_prod(A,A)),$o) ) ).

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: $tType] : ( set(product_prod(A,A)) > fun(set(product_prod(set(A),set(A))),$o) ) ).

tff(sy_c_BNF__Cardinal__Order__Relation_OnatLeq,type,
    bNF_Ca8665028551170535155natLeq: set(product_prod(nat,nat)) ).

tff(sy_c_BNF__Cardinal__Order__Relation_OnatLess,type,
    bNF_Ca8459412986667044542atLess: set(product_prod(nat,nat)) ).

tff(sy_c_BNF__Cardinal__Order__Relation_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__Composition_Oid__bnf,type,
    bNF_id_bnf: 
      !>[A: $tType] : fun(A,A) ).

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: $tType] : ( ( set(A) * fun(A,B) ) > fun(A,fun(B,$o)) ) ).

tff(sy_c_BNF__Def_Ocollect,type,
    bNF_collect: 
      !>[B: $tType,A: $tType] : ( set(fun(B,set(A))) > fun(B,set(A)) ) ).

tff(sy_c_BNF__Def_Oeq__onp,type,
    bNF_eq_onp: 
      !>[A: $tType] : ( fun(A,$o) > fun(A,fun(A,$o)) ) ).

tff(sy_c_BNF__Def_OfstOp,type,
    bNF_fstOp: 
      !>[A: $tType,B: $tType,C: $tType] : ( ( fun(A,fun(B,$o)) * fun(B,fun(C,$o)) * product_prod(A,C) ) > product_prod(A,B) ) ).

tff(sy_c_BNF__Def_Opick__middlep,type,
    bNF_pick_middlep: 
      !>[B: $tType,A: $tType,C: $tType] : ( ( fun(B,fun(A,$o)) * fun(A,fun(C,$o)) * B * C ) > A ) ).

tff(sy_c_BNF__Def_Orel__fun,type,
    bNF_rel_fun: 
      !>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( fun(A,fun(C,$o)) * fun(B,fun(D,$o)) ) > fun(fun(A,B),fun(fun(C,D),$o)) ) ).

tff(sy_c_BNF__Def_Orel__set,type,
    bNF_rel_set: 
      !>[A: $tType,B: $tType] : ( fun(A,fun(B,$o)) > fun(set(A),fun(set(B),$o)) ) ).

tff(sy_c_BNF__Def_Orel__sum,type,
    bNF_rel_sum: 
      !>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( fun(A,fun(C,$o)) * fun(B,fun(D,$o)) ) > fun(sum_sum(A,B),fun(sum_sum(C,D),$o)) ) ).

tff(sy_c_BNF__Def_OsndOp,type,
    bNF_sndOp: 
      !>[C: $tType,A: $tType,B: $tType] : ( ( fun(C,fun(A,$o)) * fun(A,fun(B,$o)) * product_prod(C,B) ) > product_prod(A,B) ) ).

tff(sy_c_BNF__Def_Ovimage2p,type,
    bNF_vimage2p: 
      !>[A: $tType,D: $tType,B: $tType,E: $tType,C: $tType] : ( ( fun(A,D) * fun(B,E) * fun(D,fun(E,C)) ) > fun(A,fun(B,C)) ) ).

tff(sy_c_BNF__Greatest__Fixpoint_OShift,type,
    bNF_Greatest_Shift: 
      !>[A: $tType] : ( ( set(list(A)) * A ) > set(list(A)) ) ).

tff(sy_c_BNF__Greatest__Fixpoint_OSucc,type,
    bNF_Greatest_Succ: 
      !>[A: $tType] : ( ( set(list(A)) * list(A) ) > set(A) ) ).

tff(sy_c_BNF__Greatest__Fixpoint_OfromCard,type,
    bNF_Gr5436034075474128252omCard: 
      !>[A: $tType,B: $tType] : ( ( set(A) * set(product_prod(B,B)) * B ) > A ) ).

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_Oimage2p,type,
    bNF_Greatest_image2p: 
      !>[C: $tType,A: $tType,D: $tType,B: $tType] : ( ( fun(C,A) * fun(D,B) * fun(C,fun(D,$o)) * A * B ) > $o ) ).

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__Greatest__Fixpoint_OtoCard,type,
    bNF_Greatest_toCard: 
      !>[A: $tType,B: $tType] : ( ( set(A) * set(product_prod(B,B)) ) > fun(A,B) ) ).

tff(sy_c_BNF__Greatest__Fixpoint_OtoCard__pred,type,
    bNF_Gr1419584066657907630d_pred: 
      !>[A: $tType,B: $tType] : ( ( set(A) * set(product_prod(B,B)) ) > fun(fun(A,B),$o) ) ).

tff(sy_c_BNF__Greatest__Fixpoint_Ouniv,type,
    bNF_Greatest_univ: 
      !>[B: $tType,A: $tType] : ( ( fun(B,A) * set(B) ) > 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_OFunc__map,type,
    bNF_We4925052301507509544nc_map: 
      !>[B: $tType,C: $tType,A: $tType,D: $tType] : ( ( set(B) * fun(C,A) * fun(B,D) ) > fun(fun(D,C),fun(B,A)) ) ).

tff(sy_c_BNF__Wellorder__Constructions_Obsqr,type,
    bNF_Wellorder_bsqr: 
      !>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(product_prod(A,A),product_prod(A,A))) ) ).

tff(sy_c_BNF__Wellorder__Constructions_Ocurr,type,
    bNF_Wellorder_curr: 
      !>[A: $tType,B: $tType,C: $tType] : ( set(A) > fun(fun(product_prod(A,B),C),fun(A,fun(B,C))) ) ).

tff(sy_c_BNF__Wellorder__Constructions_Odir__image,type,
    bNF_We2720479622203943262_image: 
      !>[A: $tType,A2: $tType] : ( ( set(product_prod(A,A)) * fun(A,A2) ) > set(product_prod(A2,A2)) ) ).

tff(sy_c_BNF__Wellorder__Constructions_OofilterIncl,type,
    bNF_We413866401316099525erIncl: 
      !>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(set(A),set(A))) ) ).

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__Constructions_Oord__to__filter,type,
    bNF_We8469521843155493636filter: 
      !>[A: $tType] : ( set(product_prod(A,A)) > fun(set(product_prod(A,A)),set(A)) ) ).

tff(sy_c_BNF__Wellorder__Embedding_Ocompat,type,
    bNF_Wellorder_compat: 
      !>[A: $tType,A2: $tType] : ( ( set(product_prod(A,A)) * set(product_prod(A2,A2)) * fun(A,A2) ) > $o ) ).

tff(sy_c_BNF__Wellorder__Embedding_Oembed,type,
    bNF_Wellorder_embed: 
      !>[A: $tType,A2: $tType] : ( ( set(product_prod(A,A)) * set(product_prod(A2,A2)) ) > fun(fun(A,A2),$o) ) ).

tff(sy_c_BNF__Wellorder__Embedding_OembedS,type,
    bNF_Wellorder_embedS: 
      !>[A: $tType,A2: $tType] : ( ( set(product_prod(A,A)) * set(product_prod(A2,A2)) * fun(A,A2) ) > $o ) ).

tff(sy_c_BNF__Wellorder__Embedding_Oiso,type,
    bNF_Wellorder_iso: 
      !>[A: $tType,A2: $tType] : ( ( set(product_prod(A,A)) * set(product_prod(A2,A2)) * fun(A,A2) ) > $o ) ).

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: $tType] : ( ( set(product_prod(A,A)) * set(A) ) > fun(A,$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__BNFs_Ofsts,type,
    basic_fsts: 
      !>[A: $tType,B: $tType] : ( product_prod(A,B) > set(A) ) ).

tff(sy_c_Basic__BNFs_Ofstsp,type,
    basic_fstsp: 
      !>[A: $tType,B: $tType] : ( product_prod(A,B) > fun(A,$o) ) ).

tff(sy_c_Basic__BNFs_Opred__fun,type,
    basic_pred_fun: 
      !>[A: $tType,B: $tType] : ( fun(A,$o) > fun(fun(B,$o),fun(fun(A,B),$o)) ) ).

tff(sy_c_Basic__BNFs_Opred__prod,type,
    basic_pred_prod: 
      !>[A: $tType,B: $tType] : ( ( fun(A,$o) * fun(B,$o) ) > fun(product_prod(A,B),$o) ) ).

tff(sy_c_Basic__BNFs_Orel__prod,type,
    basic_rel_prod: 
      !>[A: $tType,B: $tType,C: $tType,D: $tType] : ( ( fun(A,fun(B,$o)) * fun(C,fun(D,$o)) ) > fun(product_prod(A,C),fun(product_prod(B,D),$o)) ) ).

tff(sy_c_Basic__BNFs_Osetl,type,
    basic_setl: 
      !>[A: $tType,B: $tType] : fun(sum_sum(A,B),set(A)) ).

tff(sy_c_Basic__BNFs_Osetlp,type,
    basic_setlp: 
      !>[A: $tType,B: $tType] : ( sum_sum(A,B) > fun(A,$o) ) ).

tff(sy_c_Basic__BNFs_Osetr,type,
    basic_setr: 
      !>[A: $tType,B: $tType] : fun(sum_sum(A,B),set(B)) ).

tff(sy_c_Basic__BNFs_Osetrp,type,
    basic_setrp: 
      !>[A: $tType,B: $tType] : ( sum_sum(A,B) > fun(B,$o) ) ).

tff(sy_c_Basic__BNFs_Osnds,type,
    basic_snds: 
      !>[A: $tType,B: $tType] : ( product_prod(A,B) > set(B) ) ).

tff(sy_c_Basic__BNFs_Osndsp,type,
    basic_sndsp: 
      !>[A: $tType,B: $tType] : ( product_prod(A,B) > fun(B,$o) ) ).

tff(sy_c_Binomial_Obinomial,type,
    binomial: nat > fun(nat,nat) ).

tff(sy_c_Binomial_Ogbinomial,type,
    gbinomial: 
      !>[A: $tType] : ( A > fun(nat,A) ) ).

tff(sy_c_Bit__Operations_Oand__int__rel,type,
    bit_and_int_rel: fun(product_prod(int,int),fun(product_prod(int,int),$o)) ).

tff(sy_c_Bit__Operations_Oand__not__num,type,
    bit_and_not_num: ( num * num ) > option(num) ).

tff(sy_c_Bit__Operations_Oand__not__num__rel,type,
    bit_and_not_num_rel: fun(product_prod(num,num),fun(product_prod(num,num),$o)) ).

tff(sy_c_Bit__Operations_Oconcat__bit,type,
    bit_concat_bit: ( nat * int * int ) > int ).

tff(sy_c_Bit__Operations_Oor__not__num__neg,type,
    bit_or_not_num_neg: ( num * num ) > num ).

tff(sy_c_Bit__Operations_Oor__not__num__neg__rel,type,
    bit_or3848514188828904588eg_rel: fun(product_prod(num,num),fun(product_prod(num,num),$o)) ).

tff(sy_c_Bit__Operations_Oring__bit__operations__class_Onot,type,
    bit_ri4277139882892585799ns_not: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit,type,
    bit_ri4674362597316999326ke_bit: 
      !>[A: $tType] : ( ( nat * A ) > A ) ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand,type,
    bit_se5824344872417868541ns_and: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit,type,
    bit_se4197421643247451524op_bit: 
      !>[A: $tType] : ( ( nat * A ) > A ) ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit,type,
    bit_se8732182000553998342ip_bit: 
      !>[A: $tType] : ( ( nat * A ) > A ) ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask,type,
    bit_se2239418461657761734s_mask: 
      !>[A: $tType] : ( nat > A ) ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor,type,
    bit_se1065995026697491101ons_or: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit,type,
    bit_se4730199178511100633sh_bit: 
      !>[A: $tType] : ( ( nat * A ) > A ) ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit,type,
    bit_se5668285175392031749et_bit: 
      !>[A: $tType] : ( ( nat * A ) > A ) ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit,type,
    bit_se2584673776208193580ke_bit: 
      !>[A: $tType] : ( ( nat * A ) > A ) ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit,type,
    bit_se2638667681897837118et_bit: 
      !>[A: $tType] : ( ( nat * A ) > A ) ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor,type,
    bit_se5824344971392196577ns_xor: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_Bit__Operations_Osemiring__bits__class_Obit,type,
    bit_se5641148757651400278ts_bit: 
      !>[A: $tType] : ( A > fun(nat,$o) ) ).

tff(sy_c_Bit__Operations_Osemiring__bits__class_Opossible__bit,type,
    bit_se6407376104438227557le_bit: 
      !>[A: $tType] : ( ( itself(A) * nat ) > $o ) ).

tff(sy_c_Bit__Operations_Otake__bit__num,type,
    bit_take_bit_num: ( nat * num ) > option(num) ).

tff(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oand__num,type,
    bit_un7362597486090784418nd_num: ( num * num ) > option(num) ).

tff(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oand__num__rel,type,
    bit_un4731106466462545111um_rel: fun(product_prod(num,num),fun(product_prod(num,num),$o)) ).

tff(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oor__num,type,
    bit_un6697907153464112080or_num: ( num * num ) > num ).

tff(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oor__num__rel,type,
    bit_un4773296044027857193um_rel: fun(product_prod(num,num),fun(product_prod(num,num),$o)) ).

tff(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oxor__num,type,
    bit_un2480387367778600638or_num: ( num * num ) > option(num) ).

tff(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oxor__num__rel,type,
    bit_un2901131394128224187um_rel: fun(product_prod(num,num),fun(product_prod(num,num),$o)) ).

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__axioms,type,
    boolea6902313364301356556axioms: 
      !>[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_Boolean__Algebras_Oabstract__boolean__algebra__sym__diff__axioms,type,
    boolea5476839437570043046axioms: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * fun(A,fun(A,A)) * fun(A,A) * fun(A,fun(A,A)) ) > $o ) ).

tff(sy_c_Code__Numeral_OSuc,type,
    code_Suc: code_natural > code_natural ).

tff(sy_c_Code__Numeral_Obit__cut__integer,type,
    code_bit_cut_integer: code_integer > product_prod(code_integer,$o) ).

tff(sy_c_Code__Numeral_Ocr__integer,type,
    code_cr_integer: ( int * code_integer ) > $o ).

tff(sy_c_Code__Numeral_Ocr__natural,type,
    code_cr_natural: ( nat * code_natural ) > $o ).

tff(sy_c_Code__Numeral_Odivmod__abs,type,
    code_divmod_abs: ( code_integer * code_integer ) > product_prod(code_integer,code_integer) ).

tff(sy_c_Code__Numeral_Odivmod__integer,type,
    code_divmod_integer: ( code_integer * code_integer ) > product_prod(code_integer,code_integer) ).

tff(sy_c_Code__Numeral_Odup,type,
    code_dup: fun(code_integer,code_integer) ).

tff(sy_c_Code__Numeral_Ointeger_Oint__of__integer,type,
    code_int_of_integer: fun(code_integer,int) ).

tff(sy_c_Code__Numeral_Ointeger_Ointeger__of__int,type,
    code_integer_of_int: fun(int,code_integer) ).

tff(sy_c_Code__Numeral_Ointeger__of__nat,type,
    code_integer_of_nat: nat > code_integer ).

tff(sy_c_Code__Numeral_Ointeger__of__num,type,
    code_integer_of_num: num > code_integer ).

tff(sy_c_Code__Numeral_Onat__of__integer,type,
    code_nat_of_integer: code_integer > nat ).

tff(sy_c_Code__Numeral_Onatural_Ocase__natural,type,
    code_case_natural: 
      !>[T: $tType] : ( ( T * fun(code_natural,T) ) > fun(code_natural,T) ) ).

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_Code__Numeral_Onatural_Orec__natural,type,
    code_rec_natural: 
      !>[T: $tType] : ( ( T * fun(code_natural,fun(T,T)) ) > fun(code_natural,T) ) ).

tff(sy_c_Code__Numeral_Onatural_Orec__set__natural,type,
    code_rec_set_natural: 
      !>[T: $tType] : ( ( T * fun(code_natural,fun(T,T)) * code_natural ) > fun(T,$o) ) ).

tff(sy_c_Code__Numeral_Onegative,type,
    code_negative: fun(num,code_integer) ).

tff(sy_c_Code__Numeral_Onum__of__integer,type,
    code_num_of_integer: code_integer > num ).

tff(sy_c_Code__Numeral_Opcr__integer,type,
    code_pcr_integer: fun(int,fun(code_integer,$o)) ).

tff(sy_c_Code__Numeral_Osub,type,
    code_sub: fun(num,fun(num,code_integer)) ).

tff(sy_c_Complete__Lattices_OInf__class_OInf,type,
    complete_Inf_Inf: 
      !>[A: $tType] : fun(set(A),A) ).

tff(sy_c_Complete__Lattices_OSup__class_OSup,type,
    complete_Sup_Sup: 
      !>[A: $tType] : fun(set(A),A) ).

tff(sy_c_Complete__Partial__Order_Occpo__class_Oiteratesp,type,
    comple7512665784863727008ratesp: 
      !>[A: $tType] : ( fun(A,A) > fun(A,$o) ) ).

tff(sy_c_Complete__Partial__Order_Ochain,type,
    comple1602240252501008431_chain: 
      !>[A: $tType] : ( ( fun(A,fun(A,$o)) * set(A) ) > $o ) ).

tff(sy_c_Conditionally__Complete__Lattices_Opreorder_Obdd__above,type,
    condit8047198070973881523_above: 
      !>[A: $tType] : ( ( fun(A,fun(A,$o)) * set(A) ) > $o ) ).

tff(sy_c_Conditionally__Complete__Lattices_Opreorder_Obdd__below,type,
    condit8119078960628432327_below: 
      !>[A: $tType] : ( fun(A,fun(A,$o)) > fun(set(A),$o) ) ).

tff(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__above,type,
    condit941137186595557371_above: 
      !>[A: $tType] : ( set(A) > $o ) ).

tff(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__below,type,
    condit1013018076250108175_below: 
      !>[A: $tType] : fun(set(A),$o) ).

tff(sy_c_Conditionally__Complete__Lattices_Opreordering__bdd,type,
    condit622319405099724424ng_bdd: 
      !>[A: $tType] : ( ( fun(A,fun(A,$o)) * fun(A,fun(A,$o)) ) > $o ) ).

tff(sy_c_Conditionally__Complete__Lattices_Opreordering__bdd_Obdd,type,
    condit16957441358409770ng_bdd: 
      !>[A: $tType] : ( fun(A,fun(A,$o)) > fun(set(A),$o) ) ).

tff(sy_c_Countable_Ofrom__nat,type,
    from_nat: 
      !>[A: $tType] : fun(nat,A) ).

tff(sy_c_Countable_Onat__to__rat__surj,type,
    nat_to_rat_surj: fun(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,$o)) ).

tff(sy_c_Countable_Oto__nat,type,
    to_nat: 
      !>[A: $tType] : fun(A,nat) ).

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: $tType] : ( fun(A,fun(A,$o)) > $o ) ).

tff(sy_c_Equiv__Relations_Opart__equivp,type,
    equiv_part_equivp: 
      !>[A: $tType] : ( fun(A,fun(A,$o)) > $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_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_Ocofinite,type,
    cofinite: 
      !>[A: $tType] : filter(A) ).

tff(sy_c_Filter_Oeventually,type,
    eventually: 
      !>[A: $tType] : ( ( fun(A,$o) * filter(A) ) > $o ) ).

tff(sy_c_Filter_Ofilter_OAbs__filter,type,
    abs_filter: 
      !>[A: $tType] : ( fun(fun(A,$o),$o) > filter(A) ) ).

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(fun(A,B),fun(filter(A),filter(B))) ).

tff(sy_c_Filter_Ofinite__subsets__at__top,type,
    finite5375528669736107172at_top: 
      !>[A: $tType] : ( set(A) > filter(set(A)) ) ).

tff(sy_c_Filter_Ofrequently,type,
    frequently: 
      !>[A: $tType] : ( ( fun(A,$o) * filter(A) ) > $o ) ).

tff(sy_c_Filter_Omap__filter__on,type,
    map_filter_on: 
      !>[A: $tType,B: $tType] : ( set(A) > fun(fun(A,B),fun(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_Filter_Orel__filter,type,
    rel_filter: 
      !>[A: $tType,B: $tType] : ( fun(A,fun(B,$o)) > fun(filter(A),fun(filter(B),$o)) ) ).

tff(sy_c_Finite__Set_OFpow,type,
    finite_Fpow: 
      !>[A: $tType] : ( set(A) > set(set(A)) ) ).

tff(sy_c_Finite__Set_Ocard,type,
    finite_card: 
      !>[B: $tType] : fun(set(B),nat) ).

tff(sy_c_Finite__Set_Ocomp__fun__commute,type,
    finite6289374366891150609ommute: 
      !>[A: $tType,B: $tType] : ( fun(A,fun(B,B)) > $o ) ).

tff(sy_c_Finite__Set_Ocomp__fun__commute__on,type,
    finite4664212375090638736ute_on: 
      !>[A: $tType,B: $tType] : ( ( set(A) * fun(A,fun(B,B)) ) > $o ) ).

tff(sy_c_Finite__Set_Ocomp__fun__idem,type,
    finite_comp_fun_idem: 
      !>[A: $tType,B: $tType] : ( fun(A,fun(B,B)) > $o ) ).

tff(sy_c_Finite__Set_Ocomp__fun__idem__on,type,
    finite673082921795544331dem_on: 
      !>[A: $tType,B: $tType] : ( ( set(A) * fun(A,fun(B,B)) ) > $o ) ).

tff(sy_c_Finite__Set_Ofinite,type,
    finite_finite2: 
      !>[A: $tType] : fun(set(A),$o) ).

tff(sy_c_Finite__Set_Ofold,type,
    finite_fold: 
      !>[A: $tType,B: $tType] : ( ( fun(A,fun(B,B)) * B * set(A) ) > B ) ).

tff(sy_c_Finite__Set_Ofold__graph,type,
    finite_fold_graph: 
      !>[A: $tType,B: $tType] : ( ( fun(A,fun(B,B)) * B * set(A) ) > fun(B,$o) ) ).

tff(sy_c_Finite__Set_Ofolding,type,
    finite_folding: 
      !>[A: $tType,B: $tType] : ( fun(A,fun(B,B)) > $o ) ).

tff(sy_c_Finite__Set_Ofolding__on,type,
    finite_folding_on: 
      !>[A: $tType,B: $tType] : ( ( set(A) * fun(A,fun(B,B)) ) > $o ) ).

tff(sy_c_Finite__Set_Ofolding__on_OF,type,
    finite_folding_F: 
      !>[A: $tType,B: $tType] : ( ( fun(A,fun(B,B)) * B ) > fun(set(A),B) ) ).

tff(sy_c_Fun_Obij__betw,type,
    bij_betw: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * set(A) * set(B) ) > $o ) ).

tff(sy_c_Fun_Ocomp,type,
    comp: 
      !>[B: $tType,C: $tType,A: $tType] : ( fun(B,C) > fun(fun(A,B),fun(A,C)) ) ).

tff(sy_c_Fun_Ofun__upd,type,
    fun_upd: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * A * B ) > fun(A,B) ) ).

tff(sy_c_Fun_Oid,type,
    id: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Fun_Oinj__on,type,
    inj_on: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * set(A) ) > $o ) ).

tff(sy_c_Fun_Omap__fun,type,
    map_fun: 
      !>[C: $tType,A: $tType,B: $tType,D: $tType] : ( ( fun(C,A) * fun(B,D) ) > fun(fun(A,B),fun(C,D)) ) ).

tff(sy_c_Fun_Ooverride__on,type,
    override_on: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * fun(A,B) * set(A) ) > fun(A,B) ) ).

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) ) > fun(B,A) ) ).

tff(sy_c_Fun__Def_OTHE__default,type,
    fun_THE_default: 
      !>[A: $tType] : ( ( A * fun(A,$o) ) > A ) ).

tff(sy_c_Fun__Def_Omax__strict,type,
    fun_max_strict: set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))) ).

tff(sy_c_Fun__Def_Omax__weak,type,
    fun_max_weak: set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))) ).

tff(sy_c_Fun__Def_Omin__strict,type,
    fun_min_strict: set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))) ).

tff(sy_c_Fun__Def_Omin__weak,type,
    fun_min_weak: set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))) ).

tff(sy_c_Fun__Def_Opair__leq,type,
    fun_pair_leq: set(product_prod(product_prod(nat,nat),product_prod(nat,nat))) ).

tff(sy_c_Fun__Def_Opair__less,type,
    fun_pair_less: set(product_prod(product_prod(nat,nat),product_prod(nat,nat))) ).

tff(sy_c_Fun__Def_Oreduction__pair,type,
    fun_reduction_pair: 
      !>[A: $tType] : ( product_prod(set(product_prod(A,A)),set(product_prod(A,A))) > $o ) ).

tff(sy_c_Fun__Def_Orp__inv__image,type,
    fun_rp_inv_image: 
      !>[A: $tType,B: $tType] : fun(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),fun(fun(B,A),product_prod(set(product_prod(B,B)),set(product_prod(B,B))))) ).

tff(sy_c_GCD_OGcd__class_OGcd,type,
    gcd_Gcd: 
      !>[A: $tType] : ( set(A) > A ) ).

tff(sy_c_GCD_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),$o)) ).

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),$o)) ).

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_Oabel__semigroup,type,
    abel_semigroup: 
      !>[A: $tType] : ( fun(A,fun(A,A)) > $o ) ).

tff(sy_c_Groups_Oabel__semigroup__axioms,type,
    abel_s757365448890700780axioms: 
      !>[A: $tType] : ( fun(A,fun(A,A)) > $o ) ).

tff(sy_c_Groups_Oabs__class_Oabs,type,
    abs_abs: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Groups_Ocomm__monoid,type,
    comm_monoid: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * A ) > $o ) ).

tff(sy_c_Groups_Ocomm__monoid__axioms,type,
    comm_monoid_axioms: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * A ) > $o ) ).

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_Osum,type,
    groups3894954378712506084id_sum: 
      !>[A: $tType,B: $tType] : ( ( fun(A,fun(A,A)) * A * fun(B,A) * set(B) ) > A ) ).

tff(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum,type,
    groups7311177749621191930dd_sum: 
      !>[B: $tType,A: $tType] : fun(fun(B,A),fun(set(B),A)) ).

tff(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_H,type,
    groups1027152243600224163dd_sum: 
      !>[C: $tType,A: $tType] : ( ( fun(C,A) * set(C) ) > A ) ).

tff(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod,type,
    groups7121269368397514597t_prod: 
      !>[B: $tType,A: $tType] : fun(fun(B,A),fun(set(B),A)) ).

tff(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_H,type,
    groups1962203154675924110t_prod: 
      !>[C: $tType,A: $tType] : fun(fun(C,A),fun(set(C),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] : ( 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: $tType] : ( fun(A,$o) > A ) ).

tff(sy_c_HOL_OUniq,type,
    uniq: 
      !>[A: $tType] : ( fun(A,$o) > $o ) ).

tff(sy_c_HOL_Oundefined,type,
    undefined: 
      !>[A: $tType] : A ).

tff(sy_c_Heap_Oheap_Olim,type,
    lim: 
      !>[Z: $tType] : ( heap_ext(Z) > nat ) ).

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_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) > fun(nat,A) ) ).

tff(sy_c_Int_OAbs__Integ,type,
    abs_Integ: fun(product_prod(nat,nat),int) ).

tff(sy_c_Int_ONeg,type,
    neg: num > int ).

tff(sy_c_Int_ORep__Integ,type,
    rep_Integ: fun(int,product_prod(nat,nat)) ).

tff(sy_c_Int_Ocr__int,type,
    cr_int: product_prod(nat,nat) > fun(int,$o) ).

tff(sy_c_Int_Odup,type,
    dup: int > int ).

tff(sy_c_Int_Oint_OAbs__int,type,
    abs_int: fun(set(product_prod(nat,nat)),int) ).

tff(sy_c_Int_Oint_ORep__int,type,
    rep_int: fun(int,set(product_prod(nat,nat))) ).

tff(sy_c_Int_Oint__ge__less__than,type,
    int_ge_less_than: int > set(product_prod(int,int)) ).

tff(sy_c_Int_Oint__ge__less__than2,type,
    int_ge_less_than2: int > set(product_prod(int,int)) ).

tff(sy_c_Int_Ointrel,type,
    intrel: fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)) ).

tff(sy_c_Int_Onat,type,
    nat2: fun(int,nat) ).

tff(sy_c_Int_Opcr__int,type,
    pcr_int: fun(product_prod(nat,nat),fun(int,$o)) ).

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_Int_Osub,type,
    sub: ( num * num ) > int ).

tff(sy_c_Lattices_Oinf__class_Oinf,type,
    inf_inf: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_Lattices_Osemilattice,type,
    semilattice: 
      !>[A: $tType] : ( fun(A,fun(A,A)) > $o ) ).

tff(sy_c_Lattices_Osemilattice__axioms,type,
    semilattice_axioms: 
      !>[A: $tType] : ( fun(A,fun(A,A)) > $o ) ).

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: $tType] : ( ( fun(A,fun(A,A)) * A * fun(A,fun(A,$o)) * fun(A,fun(A,$o)) ) > $o ) ).

tff(sy_c_Lattices_Osemilattice__order,type,
    semilattice_order: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * fun(A,fun(A,$o)) * fun(A,fun(A,$o)) ) > $o ) ).

tff(sy_c_Lattices_Osemilattice__order__axioms,type,
    semila6385135966242565138axioms: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * fun(A,fun(A,$o)) * fun(A,fun(A,$o)) ) > $o ) ).

tff(sy_c_Lattices_Osup__class_Osup,type,
    sup_sup: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_Lattices__Big_Olinorder_OMax,type,
    lattices_Max: 
      !>[A: $tType] : ( fun(A,fun(A,$o)) > fun(set(A),A) ) ).

tff(sy_c_Lattices__Big_Olinorder_OMin,type,
    lattices_Min: 
      !>[A: $tType] : ( fun(A,fun(A,$o)) > fun(set(A),A) ) ).

tff(sy_c_Lattices__Big_Olinorder__class_OMax,type,
    lattic643756798349783984er_Max: 
      !>[A: $tType] : fun(set(A),A) ).

tff(sy_c_Lattices__Big_Olinorder__class_OMin,type,
    lattic643756798350308766er_Min: 
      !>[A: $tType] : fun(set(A),A) ).

tff(sy_c_Lattices__Big_Oord__class_Oarg__min,type,
    lattices_ord_arg_min: 
      !>[B: $tType,A: $tType] : ( ( fun(B,A) * fun(B,$o) ) > B ) ).

tff(sy_c_Lattices__Big_Oord__class_Oarg__min__on,type,
    lattic7623131987881927897min_on: 
      !>[B: $tType,A: $tType] : ( ( fun(B,A) * set(B) ) > B ) ).

tff(sy_c_Lattices__Big_Oord__class_Ois__arg__min,type,
    lattic501386751177426532rg_min: 
      !>[B: $tType,A: $tType] : ( ( fun(B,A) * fun(B,$o) ) > fun(B,$o) ) ).

tff(sy_c_Lattices__Big_Osemilattice__inf_OInf__fin,type,
    lattic8678736583308907530nf_fin: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * set(A) ) > A ) ).

tff(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin,type,
    lattic7752659483105999362nf_fin: 
      !>[A: $tType] : fun(set(A),A) ).

tff(sy_c_Lattices__Big_Osemilattice__neutr__set,type,
    lattic5652469242046573047tr_set: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * A ) > $o ) ).

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: $tType] : ( ( fun(A,fun(A,A)) * fun(A,fun(A,$o)) * fun(A,fun(A,$o)) ) > $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)) > fun(set(A),A) ) ).

tff(sy_c_Lattices__Big_Osemilattice__sup_OSup__fin,type,
    lattic4630905495605216202up_fin: 
      !>[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] : fun(set(A),A) ).

tff(sy_c_Lifting_OQuotient,type,
    quotient: 
      !>[A: $tType,B: $tType] : ( ( fun(A,fun(A,$o)) * fun(A,B) * fun(B,A) * fun(A,fun(B,$o)) ) > $o ) ).

tff(sy_c_Lifting_Orel__pred__comp,type,
    rel_pred_comp: 
      !>[A: $tType,B: $tType] : ( ( fun(A,fun(B,$o)) * fun(B,$o) * A ) > $o ) ).

tff(sy_c_List_OBleast,type,
    bleast: 
      !>[A: $tType] : ( ( set(A) * fun(A,$o) ) > A ) ).

tff(sy_c_List_Oabort__Bleast,type,
    abort_Bleast: 
      !>[A: $tType] : ( ( set(A) * fun(A,$o) ) > A ) ).

tff(sy_c_List_Oappend,type,
    append: 
      !>[A: $tType] : fun(list(A),fun(list(A),list(A))) ).

tff(sy_c_List_Oarg__min__list,type,
    arg_min_list: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * list(A) ) > A ) ).

tff(sy_c_List_Oarg__min__list__rel,type,
    arg_min_list_rel: 
      !>[A: $tType,B: $tType] : fun(product_prod(fun(A,B),list(A)),fun(product_prod(fun(A,B),list(A)),$o)) ).

tff(sy_c_List_Obind,type,
    bind: 
      !>[A: $tType,B: $tType] : ( ( list(A) * fun(A,list(B)) ) > list(B) ) ).

tff(sy_c_List_Obutlast,type,
    butlast: 
      !>[A: $tType] : ( list(A) > list(A) ) ).

tff(sy_c_List_Oconcat,type,
    concat: 
      !>[A: $tType] : ( list(list(A)) > list(A) ) ).

tff(sy_c_List_Ocoset,type,
    coset: 
      !>[A: $tType] : ( list(A) > set(A) ) ).

tff(sy_c_List_Ocount__list,type,
    count_list: 
      !>[A: $tType] : ( ( list(A) * A ) > nat ) ).

tff(sy_c_List_Odistinct,type,
    distinct: 
      !>[A: $tType] : ( list(A) > $o ) ).

tff(sy_c_List_Odrop,type,
    drop: 
      !>[A: $tType] : ( ( nat * list(A) ) > list(A) ) ).

tff(sy_c_List_OdropWhile,type,
    dropWhile: 
      !>[A: $tType] : ( ( fun(A,$o) * list(A) ) > list(A) ) ).

tff(sy_c_List_Oenumerate,type,
    enumerate: 
      !>[A: $tType] : ( ( nat * list(A) ) > list(product_prod(nat,A)) ) ).

tff(sy_c_List_Oextract,type,
    extract: 
      !>[A: $tType] : ( ( fun(A,$o) * list(A) ) > option(product_prod(list(A),product_prod(A,list(A)))) ) ).

tff(sy_c_List_Ofilter,type,
    filter2: 
      !>[A: $tType] : ( fun(A,$o) > fun(list(A),list(A)) ) ).

tff(sy_c_List_Ofind,type,
    find: 
      !>[A: $tType] : ( ( fun(A,$o) * list(A) ) > option(A) ) ).

tff(sy_c_List_Ofold,type,
    fold: 
      !>[A: $tType,B: $tType] : ( ( fun(A,fun(B,B)) * list(A) ) > fun(B,B) ) ).

tff(sy_c_List_Ofolding__insort__key,type,
    folding_insort_key: 
      !>[A: $tType,B: $tType] : ( ( fun(A,fun(A,$o)) * fun(A,fun(A,$o)) * set(B) * fun(B,A) ) > $o ) ).

tff(sy_c_List_Ofoldl,type,
    foldl: 
      !>[B: $tType,A: $tType] : ( ( fun(B,fun(A,B)) * B * list(A) ) > B ) ).

tff(sy_c_List_Ofoldr,type,
    foldr: 
      !>[A: $tType,B: $tType] : ( ( fun(A,fun(B,B)) * list(A) ) > fun(B,B) ) ).

tff(sy_c_List_Olast,type,
    last: 
      !>[A: $tType] : ( list(A) > A ) ).

tff(sy_c_List_Olenlex,type,
    lenlex: 
      !>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(list(A),list(A))) ) ).

tff(sy_c_List_Olex,type,
    lex: 
      !>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(list(A),list(A))) ) ).

tff(sy_c_List_Olexn,type,
    lexn: 
      !>[A: $tType] : ( set(product_prod(A,A)) > fun(nat,set(product_prod(list(A),list(A)))) ) ).

tff(sy_c_List_Olexord,type,
    lexord: 
      !>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(list(A),list(A))) ) ).

tff(sy_c_List_Olexordp,type,
    lexordp: 
      !>[A: $tType] : ( ( fun(A,fun(A,$o)) * list(A) * list(A) ) > $o ) ).

tff(sy_c_List_Olinorder_Oinsort__key,type,
    insort_key: 
      !>[A: $tType,B: $tType] : ( ( fun(A,fun(A,$o)) * fun(B,A) * B * list(B) ) > list(B) ) ).

tff(sy_c_List_Olinorder_Osorted__key__list__of__set,type,
    sorted8670434370408473282of_set: 
      !>[A: $tType,B: $tType] : ( ( fun(A,fun(A,$o)) * fun(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) > fun(B,fun(list(B),list(B))) ) ).

tff(sy_c_List_Olinorder__class_Osort__key,type,
    linorder_sort_key: 
      !>[B: $tType,A: $tType] : ( fun(B,A) > fun(list(B),list(B)) ) ).

tff(sy_c_List_Olinorder__class_Osorted__key__list__of__set,type,
    linord144544945434240204of_set: 
      !>[B: $tType,A: $tType] : ( fun(B,A) > fun(set(B),list(B)) ) ).

tff(sy_c_List_Olinorder__class_Osorted__list__of__set,type,
    linord4507533701916653071of_set: 
      !>[A: $tType] : fun(set(A),list(A)) ).

tff(sy_c_List_Olinorder__class_Ostable__sort__key,type,
    linord3483353639454293061rt_key: 
      !>[B: $tType,A: $tType] : ( fun(fun(B,A),fun(list(B),list(B))) > $o ) ).

tff(sy_c_List_Olist_OCons,type,
    cons: 
      !>[A: $tType] : fun(A,fun(list(A),list(A))) ).

tff(sy_c_List_Olist_ONil,type,
    nil: 
      !>[A: $tType] : list(A) ).

tff(sy_c_List_Olist_Ocase__list,type,
    case_list: 
      !>[B: $tType,A: $tType] : ( ( B * fun(A,fun(list(A),B)) ) > fun(list(A),B) ) ).

tff(sy_c_List_Olist_Ohd,type,
    hd: 
      !>[A: $tType] : fun(list(A),A) ).

tff(sy_c_List_Olist_Olist__all,type,
    list_all: 
      !>[A: $tType] : ( fun(A,$o) > fun(list(A),$o) ) ).

tff(sy_c_List_Olist_Olist__all2,type,
    list_all2: 
      !>[A: $tType,B: $tType] : ( fun(A,fun(B,$o)) > fun(list(A),fun(list(B),$o)) ) ).

tff(sy_c_List_Olist_Omap,type,
    map: 
      !>[A: $tType,Aa: $tType] : ( fun(A,Aa) > fun(list(A),list(Aa)) ) ).

tff(sy_c_List_Olist_Orec__list,type,
    rec_list: 
      !>[C: $tType,A: $tType] : ( ( C * fun(A,fun(list(A),fun(C,C))) ) > fun(list(A),C) ) ).

tff(sy_c_List_Olist_Oset,type,
    set2: 
      !>[A: $tType] : fun(list(A),set(A)) ).

tff(sy_c_List_Olist_Osize__list,type,
    size_list: 
      !>[A: $tType] : ( ( fun(A,nat) * list(A) ) > nat ) ).

tff(sy_c_List_Olist_Otl,type,
    tl: 
      !>[A: $tType] : fun(list(A),list(A)) ).

tff(sy_c_List_Olist__ex1,type,
    list_ex1: 
      !>[A: $tType] : ( ( fun(A,$o) * 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_Olistrel1p,type,
    listrel1p: 
      !>[A: $tType] : ( ( fun(A,fun(A,$o)) * list(A) * list(A) ) > $o ) ).

tff(sy_c_List_Olistrelp,type,
    listrelp: 
      !>[A: $tType,B: $tType] : ( fun(A,fun(B,$o)) > fun(list(A),fun(list(B),$o)) ) ).

tff(sy_c_List_Olists,type,
    lists: 
      !>[A: $tType] : ( set(A) > set(list(A)) ) ).

tff(sy_c_List_Olistset,type,
    listset: 
      !>[A: $tType] : ( list(set(A)) > set(list(A)) ) ).

tff(sy_c_List_Olistsp,type,
    listsp: 
      !>[A: $tType] : ( fun(A,$o) > fun(list(A),$o) ) ).

tff(sy_c_List_Omap__filter,type,
    map_filter: 
      !>[A: $tType,B: $tType] : ( ( fun(A,option(B)) * list(A) ) > list(B) ) ).

tff(sy_c_List_Omap__project,type,
    map_project: 
      !>[A: $tType,B: $tType] : ( ( fun(A,option(B)) * set(A) ) > set(B) ) ).

tff(sy_c_List_Omap__tailrec__rev,type,
    map_tailrec_rev: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * list(A) * list(B) ) > list(B) ) ).

tff(sy_c_List_Omap__tailrec__rev__rel,type,
    map_tailrec_rev_rel: 
      !>[A: $tType,B: $tType] : fun(product_prod(fun(A,B),product_prod(list(A),list(B))),fun(product_prod(fun(A,B),product_prod(list(A),list(B))),$o)) ).

tff(sy_c_List_Omeasures,type,
    measures: 
      !>[A: $tType] : ( list(fun(A,nat)) > set(product_prod(A,A)) ) ).

tff(sy_c_List_Omin__list,type,
    min_list: 
      !>[A: $tType] : ( list(A) > A ) ).

tff(sy_c_List_Omin__list__rel,type,
    min_list_rel: 
      !>[A: $tType] : fun(list(A),fun(list(A),$o)) ).

tff(sy_c_List_On__lists,type,
    n_lists: 
      !>[A: $tType] : ( ( nat * list(A) ) > list(list(A)) ) ).

tff(sy_c_List_Onth,type,
    nth: 
      !>[A: $tType] : ( list(A) > fun(nat,A) ) ).

tff(sy_c_List_Onths,type,
    nths: 
      !>[A: $tType] : ( ( list(A) * set(nat) ) > list(A) ) ).

tff(sy_c_List_Oord_Olexordp,type,
    lexordp2: 
      !>[A: $tType] : ( fun(A,fun(A,$o)) > fun(list(A),fun(list(A),$o)) ) ).

tff(sy_c_List_Oord_Olexordp__eq,type,
    lexordp_eq: 
      !>[A: $tType] : ( fun(A,fun(A,$o)) > fun(list(A),fun(list(A),$o)) ) ).

tff(sy_c_List_Oord__class_Olexordp,type,
    ord_lexordp: 
      !>[A: $tType] : fun(list(A),fun(list(A),$o)) ).

tff(sy_c_List_Opartition,type,
    partition: 
      !>[A: $tType] : ( fun(A,$o) > fun(list(A),product_prod(list(A),list(A))) ) ).

tff(sy_c_List_Oproduct,type,
    product: 
      !>[A: $tType,B: $tType] : ( ( list(A) * list(B) ) > list(product_prod(A,B)) ) ).

tff(sy_c_List_Oproduct__lists,type,
    product_lists: 
      !>[A: $tType] : ( list(list(A)) > list(list(A)) ) ).

tff(sy_c_List_Oremdups,type,
    remdups: 
      !>[A: $tType] : fun(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 > fun(list(A),list(A)) ) ).

tff(sy_c_List_Oreplicate,type,
    replicate: 
      !>[A: $tType] : ( ( nat * A ) > list(A) ) ).

tff(sy_c_List_Orev,type,
    rev: 
      !>[A: $tType] : ( list(A) > list(A) ) ).

tff(sy_c_List_Orotate,type,
    rotate: 
      !>[A: $tType] : ( ( nat * list(A) ) > list(A) ) ).

tff(sy_c_List_Orotate1,type,
    rotate1: 
      !>[A: $tType] : ( list(A) > list(A) ) ).

tff(sy_c_List_Oset__Cons,type,
    set_Cons: 
      !>[A: $tType] : ( ( set(A) * set(list(A)) ) > set(list(A)) ) ).

tff(sy_c_List_Oshuffles,type,
    shuffles: 
      !>[A: $tType] : ( ( list(A) * list(A) ) > set(list(A)) ) ).

tff(sy_c_List_Oshuffles__rel,type,
    shuffles_rel: 
      !>[A: $tType] : fun(product_prod(list(A),list(A)),fun(product_prod(list(A),list(A)),$o)) ).

tff(sy_c_List_Osorted__wrt,type,
    sorted_wrt: 
      !>[A: $tType] : ( ( fun(A,fun(A,$o)) * list(A) ) > $o ) ).

tff(sy_c_List_Osorted__wrt__rel,type,
    sorted_wrt_rel: 
      !>[A: $tType] : fun(product_prod(fun(A,fun(A,$o)),list(A)),fun(product_prod(fun(A,fun(A,$o)),list(A)),$o)) ).

tff(sy_c_List_Osplice,type,
    splice: 
      !>[A: $tType] : ( ( list(A) * list(A) ) > list(A) ) ).

tff(sy_c_List_Osplice__rel,type,
    splice_rel: 
      !>[A: $tType] : fun(product_prod(list(A),list(A)),fun(product_prod(list(A),list(A)),$o)) ).

tff(sy_c_List_Osubseqs,type,
    subseqs: 
      !>[A: $tType] : ( list(A) > list(list(A)) ) ).

tff(sy_c_List_Otake,type,
    take: 
      !>[A: $tType] : ( ( nat * list(A) ) > list(A) ) ).

tff(sy_c_List_OtakeWhile,type,
    takeWhile: 
      !>[A: $tType] : ( ( fun(A,$o) * list(A) ) > list(A) ) ).

tff(sy_c_List_Othose,type,
    those: 
      !>[A: $tType] : ( list(option(A)) > option(list(A)) ) ).

tff(sy_c_List_Otranspose,type,
    transpose: 
      !>[A: $tType] : ( list(list(A)) > list(list(A)) ) ).

tff(sy_c_List_Otranspose__rel,type,
    transpose_rel: 
      !>[A: $tType] : fun(list(list(A)),fun(list(list(A)),$o)) ).

tff(sy_c_List_Ounion,type,
    union: 
      !>[A: $tType] : ( ( list(A) * list(A) ) > list(A) ) ).

tff(sy_c_List_Oupt,type,
    upt: ( nat * nat ) > list(nat) ).

tff(sy_c_List_Oupto,type,
    upto: ( int * int ) > list(int) ).

tff(sy_c_List_Oupto__aux,type,
    upto_aux: ( int * int * list(int) ) > list(int) ).

tff(sy_c_List_Oupto__rel,type,
    upto_rel: fun(product_prod(int,int),fun(product_prod(int,int),$o)) ).

tff(sy_c_List_Ozip,type,
    zip: 
      !>[A: $tType,B: $tType] : ( ( list(A) * list(B) ) > list(product_prod(A,B)) ) ).

tff(sy_c_Map_Odom,type,
    dom: 
      !>[A: $tType,B: $tType] : ( fun(A,option(B)) > set(A) ) ).

tff(sy_c_Map_Ograph,type,
    graph: 
      !>[A: $tType,B: $tType] : ( fun(A,option(B)) > set(product_prod(A,B)) ) ).

tff(sy_c_Map_Omap__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_Misc_OEps__Opt,type,
    eps_Opt: 
      !>[A: $tType] : ( fun(A,$o) > option(A) ) ).

tff(sy_c_Misc_Obijective,type,
    bijective: 
      !>[A: $tType,B: $tType] : ( set(product_prod(A,B)) > $o ) ).

tff(sy_c_Misc_Obrk__rel,type,
    brk_rel: 
      !>[A: $tType,B: $tType] : ( set(product_prod(A,B)) > set(product_prod(product_prod($o,A),product_prod($o,B))) ) ).

tff(sy_c_Misc_Odflt__None__set,type,
    dflt_None_set: 
      !>[A: $tType] : ( set(A) > option(set(A)) ) ).

tff(sy_c_Misc_Ofun__of__rel,type,
    fun_of_rel: 
      !>[B: $tType,A: $tType] : ( ( set(product_prod(B,A)) * B ) > A ) ).

tff(sy_c_Misc_Oinv__on,type,
    inv_on: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * set(A) * B ) > A ) ).

tff(sy_c_Misc_Olist__all__zip,type,
    list_all_zip: 
      !>[A: $tType,B: $tType] : ( ( fun(A,fun(B,$o)) * list(A) * list(B) ) > $o ) ).

tff(sy_c_Misc_Olist__all__zip__rel,type,
    list_all_zip_rel: 
      !>[A: $tType,B: $tType] : fun(product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),fun(product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),$o)) ).

tff(sy_c_Misc_Olist__collect__set,type,
    list_collect_set: 
      !>[B: $tType,A: $tType] : ( ( fun(B,set(A)) * list(B) ) > set(A) ) ).

tff(sy_c_Misc_Omap__mmupd,type,
    map_mmupd: 
      !>[B: $tType,A: $tType] : ( ( fun(B,option(A)) * set(B) * A ) > fun(B,option(A)) ) ).

tff(sy_c_Misc_Omap__to__set,type,
    map_to_set: 
      !>[A: $tType,B: $tType] : ( fun(A,option(B)) > set(product_prod(A,B)) ) ).

tff(sy_c_Misc_Omerge,type,
    merge: 
      !>[A: $tType] : ( ( list(A) * list(A) ) > list(A) ) ).

tff(sy_c_Misc_Omerge__list,type,
    merge_list: 
      !>[A: $tType] : ( ( list(list(A)) * list(list(A)) ) > list(A) ) ).

tff(sy_c_Misc_Omerge__list__rel,type,
    merge_list_rel: 
      !>[A: $tType] : fun(product_prod(list(list(A)),list(list(A))),fun(product_prod(list(list(A)),list(list(A))),$o)) ).

tff(sy_c_Misc_Omerge__rel,type,
    merge_rel: 
      !>[A: $tType] : fun(product_prod(list(A),list(A)),fun(product_prod(list(A),list(A)),$o)) ).

tff(sy_c_Misc_Omergesort,type,
    mergesort: 
      !>[A: $tType] : fun(list(A),list(A)) ).

tff(sy_c_Misc_Omergesort__by__rel,type,
    mergesort_by_rel: 
      !>[A: $tType] : ( fun(A,fun(A,$o)) > fun(list(A),list(A)) ) ).

tff(sy_c_Misc_Omergesort__by__rel__merge,type,
    merges9089515139780605204_merge: 
      !>[A: $tType] : ( ( fun(A,fun(A,$o)) * list(A) * list(A) ) > list(A) ) ).

tff(sy_c_Misc_Omergesort__by__rel__merge__rel,type,
    merges2244889521215249637ge_rel: 
      !>[A: $tType] : fun(product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),fun(product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),$o)) ).

tff(sy_c_Misc_Omergesort__by__rel__rel,type,
    mergesort_by_rel_rel: 
      !>[A: $tType] : fun(product_prod(fun(A,fun(A,$o)),list(A)),fun(product_prod(fun(A,fun(A,$o)),list(A)),$o)) ).

tff(sy_c_Misc_Omergesort__by__rel__split,type,
    merges295452479951948502_split: 
      !>[A: $tType] : ( ( product_prod(list(A),list(A)) * list(A) ) > product_prod(list(A),list(A)) ) ).

tff(sy_c_Misc_Omergesort__by__rel__split__rel,type,
    merges7066485432131860899it_rel: 
      !>[A: $tType] : fun(product_prod(product_prod(list(A),list(A)),list(A)),fun(product_prod(product_prod(list(A),list(A)),list(A)),$o)) ).

tff(sy_c_Misc_Omergesort__remdups,type,
    mergesort_remdups: 
      !>[A: $tType] : fun(list(A),list(A)) ).

tff(sy_c_Misc_Opairself,type,
    pairself: 
      !>[A: $tType,B: $tType] : ( fun(A,B) > fun(product_prod(A,A),product_prod(B,B)) ) ).

tff(sy_c_Misc_Opairself__rel,type,
    pairself_rel: 
      !>[A: $tType,B: $tType] : fun(product_prod(fun(A,B),product_prod(A,A)),fun(product_prod(fun(A,B),product_prod(A,A)),$o)) ).

tff(sy_c_Misc_Opartition__rev,type,
    partition_rev: 
      !>[A: $tType] : ( ( fun(A,$o) * product_prod(list(A),list(A)) * list(A) ) > product_prod(list(A),list(A)) ) ).

tff(sy_c_Misc_Opartition__rev__rel,type,
    partition_rev_rel: 
      !>[A: $tType] : fun(product_prod(fun(A,$o),product_prod(product_prod(list(A),list(A)),list(A))),fun(product_prod(fun(A,$o),product_prod(product_prod(list(A),list(A)),list(A))),$o)) ).

tff(sy_c_Misc_Oquicksort__by__rel,type,
    quicksort_by_rel: 
      !>[A: $tType] : ( ( fun(A,fun(A,$o)) * list(A) ) > fun(list(A),list(A)) ) ).

tff(sy_c_Misc_Oquicksort__by__rel__rel,type,
    quicksort_by_rel_rel: 
      !>[A: $tType] : fun(product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),fun(product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),$o)) ).

tff(sy_c_Misc_Orel__of,type,
    rel_of: 
      !>[A: $tType,B: $tType] : ( ( fun(A,option(B)) * fun(product_prod(A,B),$o) ) > set(product_prod(A,B)) ) ).

tff(sy_c_Misc_Orel__restrict,type,
    rel_restrict: 
      !>[A: $tType] : ( ( set(product_prod(A,A)) * set(A) ) > set(product_prod(A,A)) ) ).

tff(sy_c_Misc_Oremove__rev,type,
    remove_rev: 
      !>[A: $tType] : ( ( A * list(A) ) > list(A) ) ).

tff(sy_c_Misc_Orevg,type,
    revg: 
      !>[A: $tType] : ( ( list(A) * list(A) ) > list(A) ) ).

tff(sy_c_Misc_Orevg__rel,type,
    revg_rel: 
      !>[A: $tType] : fun(product_prod(list(A),list(A)),fun(product_prod(list(A),list(A)),$o)) ).

tff(sy_c_Misc_Oset__to__map,type,
    set_to_map: 
      !>[B: $tType,A: $tType] : ( set(product_prod(B,A)) > fun(B,option(A)) ) ).

tff(sy_c_Misc_Oslice,type,
    slice: 
      !>[A: $tType] : ( ( nat * nat * list(A) ) > list(A) ) ).

tff(sy_c_Misc_Osu__rel__fun,type,
    su_rel_fun: 
      !>[A: $tType,B: $tType] : ( ( set(product_prod(A,B)) * fun(A,B) ) > $o ) ).

tff(sy_c_Misc_Othe__default,type,
    the_default: 
      !>[A: $tType] : ( ( A * option(A) ) > A ) ).

tff(sy_c_Misc_Ouncurry,type,
    uncurry: 
      !>[A: $tType,B: $tType,C: $tType] : ( fun(A,fun(B,C)) > fun(product_prod(A,B),C) ) ).

tff(sy_c_Misc_Ozipf,type,
    zipf: 
      !>[A: $tType,B: $tType,C: $tType] : ( ( fun(A,fun(B,C)) * list(A) * list(B) ) > list(C) ) ).

tff(sy_c_Misc_Ozipf__rel,type,
    zipf_rel: 
      !>[A: $tType,B: $tType,C: $tType] : fun(product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B))),fun(product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B))),$o)) ).

tff(sy_c_Multiset_Oadd__mset,type,
    add_mset: 
      !>[A: $tType] : fun(A,fun(multiset(A),multiset(A))) ).

tff(sy_c_Multiset_Ocomm__monoid__add__class_Osum__mset,type,
    comm_m7189776963980413722m_mset: 
      !>[A: $tType] : ( multiset(A) > A ) ).

tff(sy_c_Multiset_Ocomm__monoid__mset,type,
    comm_monoid_mset: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * A ) > $o ) ).

tff(sy_c_Multiset_Ocomm__monoid__mset_OF,type,
    comm_monoid_F: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * A ) > fun(multiset(A),A) ) ).

tff(sy_c_Multiset_Ocomm__monoid__mult__class_Oprod__mset,type,
    comm_m9189036328036947845d_mset: 
      !>[A: $tType] : fun(multiset(A),A) ).

tff(sy_c_Multiset_Ocr__multiset,type,
    cr_multiset: 
      !>[A: $tType] : fun(fun(A,nat),fun(multiset(A),$o)) ).

tff(sy_c_Multiset_Ofilter__mset,type,
    filter_mset: 
      !>[A: $tType] : fun(fun(A,$o),fun(multiset(A),multiset(A))) ).

tff(sy_c_Multiset_Ofold__mset,type,
    fold_mset: 
      !>[A: $tType,B: $tType] : ( ( fun(A,fun(B,B)) * B * multiset(A) ) > B ) ).

tff(sy_c_Multiset_Oimage__mset,type,
    image_mset: 
      !>[A: $tType,B: $tType] : ( fun(A,B) > fun(multiset(A),multiset(B)) ) ).

tff(sy_c_Multiset_Ointer__mset,type,
    inter_mset: 
      !>[A: $tType] : fun(multiset(A),fun(multiset(A),multiset(A))) ).

tff(sy_c_Multiset_Olinorder__class_Opart,type,
    linorder_part: 
      !>[B: $tType,A: $tType] : ( ( fun(B,A) * A * list(B) ) > product_prod(list(B),product_prod(list(B),list(B))) ) ).

tff(sy_c_Multiset_Olinorder__class_Osorted__list__of__multiset,type,
    linord6283353356039996273ltiset: 
      !>[A: $tType] : ( multiset(A) > list(A) ) ).

tff(sy_c_Multiset_Oms__strict,type,
    ms_strict: set(product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)))) ).

tff(sy_c_Multiset_Oms__weak,type,
    ms_weak: set(product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)))) ).

tff(sy_c_Multiset_Omset,type,
    mset: 
      !>[A: $tType] : fun(list(A),multiset(A)) ).

tff(sy_c_Multiset_Omset__set,type,
    mset_set: 
      !>[B: $tType] : ( set(B) > multiset(B) ) ).

tff(sy_c_Multiset_Omult,type,
    mult: 
      !>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(multiset(A),multiset(A))) ) ).

tff(sy_c_Multiset_Omult1,type,
    mult1: 
      !>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(multiset(A),multiset(A))) ) ).

tff(sy_c_Multiset_Omulteqp__code,type,
    multeqp_code: 
      !>[A: $tType] : ( fun(A,fun(A,$o)) > fun(multiset(A),fun(multiset(A),$o)) ) ).

tff(sy_c_Multiset_Omultiset_OAbs__multiset,type,
    abs_multiset: 
      !>[A: $tType] : fun(fun(A,nat),multiset(A)) ).

tff(sy_c_Multiset_Omultiset_Ocount,type,
    count: 
      !>[A: $tType] : fun(multiset(A),fun(A,nat)) ).

tff(sy_c_Multiset_Omultp,type,
    multp: 
      !>[A: $tType] : ( fun(A,fun(A,$o)) > fun(multiset(A),fun(multiset(A),$o)) ) ).

tff(sy_c_Multiset_Omultp__code,type,
    multp_code: 
      !>[A: $tType] : ( ( fun(A,fun(A,$o)) * multiset(A) * multiset(A) ) > $o ) ).

tff(sy_c_Multiset_Opcr__multiset,type,
    pcr_multiset: 
      !>[C: $tType,B: $tType] : ( fun(C,fun(B,$o)) > fun(fun(C,nat),fun(multiset(B),$o)) ) ).

tff(sy_c_Multiset_Opw__leq,type,
    pw_leq: ( multiset(product_prod(nat,nat)) * multiset(product_prod(nat,nat)) ) > $o ).

tff(sy_c_Multiset_Orel__mset,type,
    rel_mset: 
      !>[A: $tType,B: $tType] : ( fun(A,fun(B,$o)) > fun(multiset(A),fun(multiset(B),$o)) ) ).

tff(sy_c_Multiset_Orepeat__mset,type,
    repeat_mset: 
      !>[A: $tType] : fun(nat,fun(multiset(A),multiset(A))) ).

tff(sy_c_Multiset_Oreplicate__mset,type,
    replicate_mset: 
      !>[A: $tType] : ( ( nat * A ) > multiset(A) ) ).

tff(sy_c_Multiset_Oset__mset,type,
    set_mset: 
      !>[A: $tType] : fun(multiset(A),set(A)) ).

tff(sy_c_Multiset_Osize__multiset,type,
    size_multiset: 
      !>[A: $tType] : ( fun(A,nat) > fun(multiset(A),nat) ) ).

tff(sy_c_Multiset_Osubset__eq__mset__impl,type,
    subset_eq_mset_impl: 
      !>[A: $tType] : ( ( list(A) * list(A) ) > option($o) ) ).

tff(sy_c_Multiset_Osubset__eq__mset__impl__rel,type,
    subset751672762298770561pl_rel: 
      !>[A: $tType] : fun(product_prod(list(A),list(A)),fun(product_prod(list(A),list(A)),$o)) ).

tff(sy_c_Multiset_Osubset__mset,type,
    subset_mset: 
      !>[A: $tType] : fun(multiset(A),fun(multiset(A),$o)) ).

tff(sy_c_Multiset_Osubseteq__mset,type,
    subseteq_mset: 
      !>[A: $tType] : fun(multiset(A),fun(multiset(A),$o)) ).

tff(sy_c_Multiset_Ounion__mset,type,
    union_mset: 
      !>[A: $tType] : fun(multiset(A),fun(multiset(A),multiset(A))) ).

tff(sy_c_Multiset_Owcount,type,
    wcount: 
      !>[A: $tType] : ( ( fun(A,nat) * multiset(A) ) > fun(A,nat) ) ).

tff(sy_c_Nat_OSuc,type,
    suc: fun(nat,nat) ).

tff(sy_c_Nat_Ocompow,type,
    compow: 
      !>[A: $tType] : ( ( nat * A ) > A ) ).

tff(sy_c_Nat_Onat_Ocase__nat,type,
    case_nat: 
      !>[A: $tType] : fun(A,fun(fun(nat,A),fun(nat,A))) ).

tff(sy_c_Nat_Onat_Opred,type,
    pred2: nat > nat ).

tff(sy_c_Nat_Oold_Onat_Orec__nat,type,
    rec_nat: 
      !>[T: $tType] : ( ( T * fun(nat,fun(T,T)) ) > fun(nat,T) ) ).

tff(sy_c_Nat_Oold_Onat_Orec__set__nat,type,
    rec_set_nat: 
      !>[T: $tType] : ( ( T * fun(nat,fun(T,T)) * nat ) > fun(T,$o) ) ).

tff(sy_c_Nat_Osemiring__1__class_ONats,type,
    semiring_1_Nats: 
      !>[A: $tType] : set(A) ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat,type,
    semiring_1_of_nat: 
      !>[A: $tType] : fun(nat,A) ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat__aux,type,
    semiri8178284476397505188at_aux: 
      !>[A: $tType] : ( ( fun(A,A) * nat * A ) > A ) ).

tff(sy_c_Nat_Osize__class_Osize,type,
    size_size: 
      !>[A: $tType] : fun(A,nat) ).

tff(sy_c_Nat__Bijection_Oint__decode,type,
    nat_int_decode: fun(nat,int) ).

tff(sy_c_Nat__Bijection_Oint__encode,type,
    nat_int_encode: fun(int,nat) ).

tff(sy_c_Nat__Bijection_Olist__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,$o)) ).

tff(sy_c_Nat__Bijection_Olist__encode,type,
    nat_list_encode: fun(list(nat),nat) ).

tff(sy_c_Nat__Bijection_Olist__encode__rel,type,
    nat_list_encode_rel: fun(list(nat),fun(list(nat),$o)) ).

tff(sy_c_Nat__Bijection_Oprod__decode,type,
    nat_prod_decode: fun(nat,product_prod(nat,nat)) ).

tff(sy_c_Nat__Bijection_Oprod__decode__aux,type,
    nat_prod_decode_aux: ( nat * nat ) > product_prod(nat,nat) ).

tff(sy_c_Nat__Bijection_Oprod__decode__aux__rel,type,
    nat_pr5047031295181774490ux_rel: fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)) ).

tff(sy_c_Nat__Bijection_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: 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_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: 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: fun(num,num) ).

tff(sy_c_Num_Onum_OBit1,type,
    bit1: fun(num,num) ).

tff(sy_c_Num_Onum_OOne,type,
    one2: num ).

tff(sy_c_Num_Onum_Ocase__num,type,
    case_num: 
      !>[A: $tType] : ( ( A * fun(num,A) * fun(num,A) * num ) > A ) ).

tff(sy_c_Num_Onum__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] : fun(sum_sum(A,nat),set(old_node(A,B))) ).

tff(sy_c_Old__Datatype_OCase,type,
    old_Case: 
      !>[A: $tType,B: $tType,C: $tType] : ( ( fun(set(old_node(A,B)),C) * fun(set(old_node(A,B)),C) * set(old_node(A,B)) ) > C ) ).

tff(sy_c_Old__Datatype_OIn0,type,
    old_In0: 
      !>[A: $tType,B: $tType] : fun(set(old_node(A,B)),set(old_node(A,B))) ).

tff(sy_c_Old__Datatype_OIn1,type,
    old_In1: 
      !>[A: $tType,B: $tType] : fun(set(old_node(A,B)),set(old_node(A,B))) ).

tff(sy_c_Old__Datatype_OLeaf,type,
    old_Leaf: 
      !>[A: $tType,B: $tType] : fun(A,set(old_node(A,B))) ).

tff(sy_c_Old__Datatype_OLim,type,
    old_Lim: 
      !>[B: $tType,A: $tType] : ( fun(B,set(old_node(A,B))) > 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] : fun(nat,set(old_node(A,B))) ).

tff(sy_c_Old__Datatype_OPush,type,
    old_Push: 
      !>[B: $tType] : fun(sum_sum(B,nat),fun(fun(nat,sum_sum(B,nat)),fun(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_OSplit,type,
    old_Split: 
      !>[A: $tType,B: $tType,C: $tType] : ( fun(set(old_node(A,B)),fun(set(old_node(A,B)),C)) > fun(set(old_node(A,B)),C) ) ).

tff(sy_c_Old__Datatype_Odprod,type,
    old_dprod: 
      !>[A: $tType,B: $tType] : ( ( set(product_prod(set(old_node(A,B)),set(old_node(A,B)))) * set(product_prod(set(old_node(A,B)),set(old_node(A,B)))) ) > set(product_prod(set(old_node(A,B)),set(old_node(A,B)))) ) ).

tff(sy_c_Old__Datatype_Odsum,type,
    old_dsum: 
      !>[A: $tType,B: $tType] : ( ( set(product_prod(set(old_node(A,B)),set(old_node(A,B)))) * set(product_prod(set(old_node(A,B)),set(old_node(A,B)))) ) > set(product_prod(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_Old__Datatype_Ouprod,type,
    old_uprod: 
      !>[A: $tType,B: $tType] : ( ( set(set(old_node(A,B))) * set(set(old_node(A,B))) ) > set(set(old_node(A,B))) ) ).

tff(sy_c_Old__Datatype_Ousum,type,
    old_usum: 
      !>[A: $tType,B: $tType] : ( ( set(set(old_node(A,B))) * set(set(old_node(A,B))) ) > set(set(old_node(A,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_Ooption_ONone,type,
    none: 
      !>[A: $tType] : option(A) ).

tff(sy_c_Option_Ooption_OSome,type,
    some: 
      !>[A: $tType] : fun(A,option(A)) ).

tff(sy_c_Option_Ooption_Ocase__option,type,
    case_option: 
      !>[B: $tType,A: $tType] : ( ( B * fun(A,B) * option(A) ) > B ) ).

tff(sy_c_Option_Ooption_Omap__option,type,
    map_option: 
      !>[A: $tType,Aa: $tType] : ( fun(A,Aa) > fun(option(A),option(Aa)) ) ).

tff(sy_c_Option_Ooption_Orec__option,type,
    rec_option: 
      !>[C: $tType,A: $tType] : ( ( C * fun(A,C) ) > fun(option(A),C) ) ).

tff(sy_c_Option_Ooption_Orel__option,type,
    rel_option: 
      !>[A: $tType,B: $tType] : ( fun(A,fun(B,$o)) > fun(option(A),fun(option(B),$o)) ) ).

tff(sy_c_Option_Ooption_Oset__option,type,
    set_option: 
      !>[A: $tType] : ( option(A) > set(A) ) ).

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__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: $tType] : ( ( fun(A,fun(A,$o)) * set(A) ) > set(product_prod(A,A)) ) ).

tff(sy_c_Order__Relation_Ounder,type,
    order_under: 
      !>[A: $tType] : ( set(product_prod(A,A)) > fun(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: $tType] : ( fun(A,fun(A,$o)) > fun(fun(A,$o),A) ) ).

tff(sy_c_Orderings_Oord_Omax,type,
    max: 
      !>[A: $tType] : ( fun(A,fun(A,$o)) > fun(A,fun(A,A)) ) ).

tff(sy_c_Orderings_Oord_Omin,type,
    min: 
      !>[A: $tType] : ( fun(A,fun(A,$o)) > fun(A,fun(A,A)) ) ).

tff(sy_c_Orderings_Oord__class_OLeast,type,
    ord_Least: 
      !>[A: $tType] : ( fun(A,$o) > A ) ).

tff(sy_c_Orderings_Oord__class_Oless,type,
    ord_less: 
      !>[A: $tType] : fun(A,fun(A,$o)) ).

tff(sy_c_Orderings_Oord__class_Oless__eq,type,
    ord_less_eq: 
      !>[A: $tType] : fun(A,fun(A,$o)) ).

tff(sy_c_Orderings_Oord__class_Omax,type,
    ord_max: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_Orderings_Oord__class_Omin,type,
    ord_min: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_Orderings_Oorder_OGreatest,type,
    greatest: 
      !>[A: $tType] : ( fun(A,fun(A,$o)) > fun(fun(A,$o),A) ) ).

tff(sy_c_Orderings_Oorder_Omono,type,
    mono: 
      !>[A: $tType,B: $tType] : ( fun(A,fun(A,$o)) > fun(fun(A,B),$o) ) ).

tff(sy_c_Orderings_Oorder__class_OGreatest,type,
    order_Greatest: 
      !>[A: $tType] : ( fun(A,$o) > A ) ).

tff(sy_c_Orderings_Oorder__class_Oantimono,type,
    order_antimono: 
      !>[A: $tType,B: $tType] : ( fun(A,B) > $o ) ).

tff(sy_c_Orderings_Oorder__class_Omono,type,
    order_mono: 
      !>[A: $tType,B: $tType] : ( fun(A,B) > $o ) ).

tff(sy_c_Orderings_Oorder__class_Ostrict__mono,type,
    order_strict_mono: 
      !>[A: $tType,B: $tType] : ( fun(A,B) > $o ) ).

tff(sy_c_Orderings_Oordering,type,
    ordering: 
      !>[A: $tType] : ( ( fun(A,fun(A,$o)) * fun(A,fun(A,$o)) ) > $o ) ).

tff(sy_c_Orderings_Oordering__axioms,type,
    ordering_axioms: 
      !>[A: $tType] : ( ( fun(A,fun(A,$o)) * fun(A,fun(A,$o)) ) > $o ) ).

tff(sy_c_Orderings_Oordering__top,type,
    ordering_top: 
      !>[A: $tType] : ( ( fun(A,fun(A,$o)) * fun(A,fun(A,$o)) * A ) > $o ) ).

tff(sy_c_Orderings_Oordering__top__axioms,type,
    ordering_top_axioms: 
      !>[A: $tType] : ( ( fun(A,fun(A,$o)) * A ) > $o ) ).

tff(sy_c_Orderings_Opartial__preordering,type,
    partial_preordering: 
      !>[A: $tType] : ( fun(A,fun(A,$o)) > $o ) ).

tff(sy_c_Orderings_Opreordering,type,
    preordering: 
      !>[A: $tType] : ( ( fun(A,fun(A,$o)) * fun(A,fun(A,$o)) ) > $o ) ).

tff(sy_c_Orderings_Opreordering__axioms,type,
    preordering_axioms: 
      !>[A: $tType] : ( ( fun(A,fun(A,$o)) * fun(A,fun(A,$o)) ) > $o ) ).

tff(sy_c_Orderings_Otop__class_Otop,type,
    top_top: 
      !>[A: $tType] : A ).

tff(sy_c_Partial__Function_Oflat__lub,type,
    partial_flat_lub: 
      !>[A: $tType] : ( ( A * set(A) ) > A ) ).

tff(sy_c_Partial__Function_Oimg__ord,type,
    partial_img_ord: 
      !>[A: $tType,C: $tType,B: $tType] : ( ( fun(A,C) * fun(C,fun(C,B)) * A * A ) > B ) ).

tff(sy_c_Partial__Function_Omk__less,type,
    partial_mk_less: 
      !>[A: $tType] : ( ( fun(A,fun(A,$o)) * A * A ) > $o ) ).

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_Predicate_OSeq,type,
    seq2: 
      !>[A: $tType] : ( fun(product_unit,seq(A)) > pred(A) ) ).

tff(sy_c_Predicate_Oadjunct,type,
    adjunct: 
      !>[A: $tType] : ( ( pred(A) * seq(A) ) > seq(A) ) ).

tff(sy_c_Predicate_Oapply,type,
    apply: 
      !>[A: $tType,B: $tType] : ( ( fun(A,pred(B)) * seq(A) ) > seq(B) ) ).

tff(sy_c_Predicate_Obind,type,
    bind2: 
      !>[A: $tType,B: $tType] : ( ( pred(A) * fun(A,pred(B)) ) > pred(B) ) ).

tff(sy_c_Predicate_Ocontained,type,
    contained: 
      !>[A: $tType] : ( ( seq(A) * pred(A) ) > $o ) ).

tff(sy_c_Predicate_Oif__pred,type,
    if_pred: $o > pred(product_unit) ).

tff(sy_c_Predicate_Ois__empty,type,
    is_empty: 
      !>[A: $tType] : ( pred(A) > $o ) ).

tff(sy_c_Predicate_Oiterate__upto,type,
    iterate_upto: 
      !>[A: $tType] : ( ( fun(code_natural,A) * code_natural * code_natural ) > pred(A) ) ).

tff(sy_c_Predicate_Oiterate__upto__rel,type,
    iterate_upto_rel: 
      !>[A: $tType] : fun(product_prod(fun(code_natural,A),product_prod(code_natural,code_natural)),fun(product_prod(fun(code_natural,A),product_prod(code_natural,code_natural)),$o)) ).

tff(sy_c_Predicate_Omap,type,
    map2: 
      !>[A: $tType,B: $tType] : ( fun(A,B) > fun(pred(A),pred(B)) ) ).

tff(sy_c_Predicate_Onot__pred,type,
    not_pred: pred(product_unit) > pred(product_unit) ).

tff(sy_c_Predicate_Onull,type,
    null: 
      !>[A: $tType] : ( seq(A) > $o ) ).

tff(sy_c_Predicate_Opred_OPred,type,
    pred3: 
      !>[A: $tType] : fun(fun(A,$o),pred(A)) ).

tff(sy_c_Predicate_Opred_Oeval,type,
    eval: 
      !>[A: $tType] : ( pred(A) > fun(A,$o) ) ).

tff(sy_c_Predicate_Opred__of__seq,type,
    pred_of_seq: 
      !>[A: $tType] : ( seq(A) > pred(A) ) ).

tff(sy_c_Predicate_Opred__of__set,type,
    pred_of_set: 
      !>[A: $tType] : fun(set(A),pred(A)) ).

tff(sy_c_Predicate_Oseq_OEmpty,type,
    empty: 
      !>[A: $tType] : seq(A) ).

tff(sy_c_Predicate_Oseq_OInsert,type,
    insert: 
      !>[A: $tType] : ( ( A * pred(A) ) > seq(A) ) ).

tff(sy_c_Predicate_Oseq_OJoin,type,
    join: 
      !>[A: $tType] : ( ( pred(A) * seq(A) ) > seq(A) ) ).

tff(sy_c_Predicate_Oseq_Ocase__seq,type,
    case_seq: 
      !>[B: $tType,A: $tType] : ( ( B * fun(A,fun(pred(A),B)) * fun(pred(A),fun(seq(A),B)) * seq(A) ) > B ) ).

tff(sy_c_Predicate_Oset__of__pred,type,
    set_of_pred: 
      !>[A: $tType] : ( pred(A) > set(A) ) ).

tff(sy_c_Predicate_Oset__of__seq,type,
    set_of_seq: 
      !>[A: $tType] : ( seq(A) > set(A) ) ).

tff(sy_c_Predicate_Osingle,type,
    single: 
      !>[A: $tType] : fun(A,pred(A)) ).

tff(sy_c_Predicate_Osingleton,type,
    singleton: 
      !>[A: $tType] : ( ( fun(product_unit,A) * pred(A) ) > A ) ).

tff(sy_c_Predicate_Othe,type,
    the3: 
      !>[A: $tType] : ( pred(A) > A ) ).

tff(sy_c_Predicate_Othe__only,type,
    the_only: 
      !>[A: $tType] : ( ( fun(product_unit,A) * seq(A) ) > A ) ).

tff(sy_c_Predicate__Compile_Ocontains,type,
    predicate_contains: 
      !>[A: $tType] : ( ( set(A) * A ) > $o ) ).

tff(sy_c_Predicate__Compile_Ocontains__pred,type,
    predic7144156976422707464s_pred: 
      !>[A: $tType] : ( ( set(A) * A ) > pred(product_unit) ) ).

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_OUnity,type,
    product_Unity: product_unit ).

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_Ointernal__case__prod,type,
    produc5280177257484947105e_prod: 
      !>[A: $tType,B: $tType,C: $tType] : ( ( fun(A,fun(B,C)) * 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 * $o ) > T ) ).

tff(sy_c_Product__Type_Oold_Obool_Orec__set__bool,type,
    product_rec_set_bool: 
      !>[T: $tType] : ( ( T * T * $o ) > fun(T,$o) ) ).

tff(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
    product_rec_prod: 
      !>[A: $tType,B: $tType,T: $tType] : ( ( fun(A,fun(B,T)) * product_prod(A,B) ) > T ) ).

tff(sy_c_Product__Type_Oold_Oprod_Orec__set__prod,type,
    product_rec_set_prod: 
      !>[A: $tType,B: $tType,T: $tType] : ( ( fun(A,fun(B,T)) * product_prod(A,B) ) > fun(T,$o) ) ).

tff(sy_c_Product__Type_Oold_Ounit_Orec__set__unit,type,
    product_rec_set_unit: 
      !>[T: $tType] : ( ( T * product_unit ) > fun(T,$o) ) ).

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_Product__Type_Ounit_OAbs__unit,type,
    product_Abs_unit: fun($o,product_unit) ).

tff(sy_c_Product__Type_Ounit_ORep__unit,type,
    product_Rep_unit: fun(product_unit,$o) ).

tff(sy_c_Pure_Otype,type,
    type2: 
      !>[A: $tType] : itself(A) ).

tff(sy_c_Quicksort_Olinorder__class_Oquicksort,type,
    linorder_quicksort: 
      !>[A: $tType] : fun(list(A),list(A)) ).

tff(sy_c_Quicksort_Olinorder__class_Oquicksort__rel,type,
    linord6200660962353139674rt_rel: 
      !>[A: $tType] : fun(list(A),fun(list(A),$o)) ).

tff(sy_c_Random_Oinc__shift,type,
    inc_shift: ( code_natural * code_natural ) > code_natural ).

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: $tType] : fun(product_prod(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B)),fun(product_prod(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B)),$o)) ).

tff(sy_c_Random_Olog,type,
    log: ( code_natural * code_natural ) > code_natural ).

tff(sy_c_Random_Olog__rel,type,
    log_rel: fun(product_prod(code_natural,code_natural),fun(product_prod(code_natural,code_natural),$o)) ).

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)) > fun(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_Random_Oselect__weight,type,
    select_weight: 
      !>[A: $tType] : ( list(product_prod(code_natural,A)) > fun(product_prod(code_natural,code_natural),product_prod(A,product_prod(code_natural,code_natural))) ) ).

tff(sy_c_Random_Osplit__seed,type,
    split_seed: product_prod(code_natural,code_natural) > product_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural)) ).

tff(sy_c_Random__Pred_ORandom,type,
    random_Random: 
      !>[A: $tType] : ( fun(product_prod(code_natural,code_natural),product_prod(product_prod(A,fun(product_unit,code_term)),product_prod(code_natural,code_natural))) > fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))) ) ).

tff(sy_c_Random__Pred_Obind,type,
    random_bind: 
      !>[A: $tType,B: $tType] : ( ( fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))) * fun(A,fun(product_prod(code_natural,code_natural),product_prod(pred(B),product_prod(code_natural,code_natural)))) * product_prod(code_natural,code_natural) ) > product_prod(pred(B),product_prod(code_natural,code_natural)) ) ).

tff(sy_c_Random__Pred_Oempty,type,
    random_empty: 
      !>[A: $tType] : fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))) ).

tff(sy_c_Random__Pred_Oiterate__upto,type,
    random_iterate_upto: 
      !>[A: $tType] : ( ( fun(code_natural,A) * code_natural * code_natural ) > fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))) ) ).

tff(sy_c_Random__Pred_Onot__randompred,type,
    random6974930770145893639ompred: ( fun(product_prod(code_natural,code_natural),product_prod(pred(product_unit),product_prod(code_natural,code_natural))) * product_prod(code_natural,code_natural) ) > product_prod(pred(product_unit),product_prod(code_natural,code_natural)) ).

tff(sy_c_Random__Pred_Osingle,type,
    random_single: 
      !>[A: $tType] : ( A > fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))) ) ).

tff(sy_c_Random__Pred_Ounion,type,
    random_union: 
      !>[A: $tType] : ( ( fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))) * fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))) * product_prod(code_natural,code_natural) ) > product_prod(pred(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: fun(int,fun(int,rat)) ).

tff(sy_c_Rat_OFrct,type,
    frct: product_prod(int,int) > rat ).

tff(sy_c_Rat_ORep__Rat,type,
    rep_Rat: fun(rat,product_prod(int,int)) ).

tff(sy_c_Rat_Ocr__rat,type,
    cr_rat: ( product_prod(int,int) * rat ) > $o ).

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] : fun(rat,A) ).

tff(sy_c_Rat_Onormalize,type,
    normalize: product_prod(int,int) > product_prod(int,int) ).

tff(sy_c_Rat_Oof__int,type,
    of_int: int > rat ).

tff(sy_c_Rat_Opcr__rat,type,
    pcr_rat: fun(product_prod(int,int),fun(rat,$o)) ).

tff(sy_c_Rat_Opositive,type,
    positive: fun(rat,$o) ).

tff(sy_c_Rat_Oquotient__of,type,
    quotient_of: rat > product_prod(int,int) ).

tff(sy_c_Rat_Orat_OAbs__rat,type,
    abs_rat: fun(set(product_prod(int,int)),rat) ).

tff(sy_c_Rat_Orat_ORep__rat,type,
    rep_rat: fun(rat,set(product_prod(int,int))) ).

tff(sy_c_Rat_Oratrel,type,
    ratrel: fun(product_prod(int,int),fun(product_prod(int,int),$o)) ).

tff(sy_c_Relation_ODomain,type,
    domain: 
      !>[A: $tType,B: $tType] : fun(set(product_prod(A,B)),set(A)) ).

tff(sy_c_Relation_ODomainp,type,
    domainp: 
      !>[A: $tType,B: $tType] : fun(fun(A,fun(B,$o)),fun(A,$o)) ).

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)) > fun(set(A),set(B)) ) ).

tff(sy_c_Relation_OPowp,type,
    powp: 
      !>[A: $tType] : ( ( fun(A,$o) * set(A) ) > $o ) ).

tff(sy_c_Relation_ORange,type,
    range2: 
      !>[A: $tType,B: $tType] : fun(set(product_prod(A,B)),set(B)) ).

tff(sy_c_Relation_ORangep,type,
    rangep: 
      !>[A: $tType,B: $tType] : ( fun(A,fun(B,$o)) > fun(B,$o) ) ).

tff(sy_c_Relation_Oantisym,type,
    antisym: 
      !>[A: $tType] : ( set(product_prod(A,A)) > $o ) ).

tff(sy_c_Relation_Oantisymp,type,
    antisymp: 
      !>[A: $tType] : ( fun(A,fun(A,$o)) > $o ) ).

tff(sy_c_Relation_Oasym,type,
    asym: 
      !>[A: $tType] : ( set(product_prod(A,A)) > $o ) ).

tff(sy_c_Relation_Oasymp,type,
    asymp: 
      !>[A: $tType] : ( fun(A,fun(A,$o)) > $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: $tType] : ( fun(A,fun(B,$o)) > fun(B,fun(A,$o)) ) ).

tff(sy_c_Relation_Oinv__image,type,
    inv_image: 
      !>[B: $tType,A: $tType] : ( ( set(product_prod(B,B)) * fun(A,B) ) > set(product_prod(A,A)) ) ).

tff(sy_c_Relation_Oirrefl,type,
    irrefl: 
      !>[A: $tType] : ( set(product_prod(A,A)) > $o ) ).

tff(sy_c_Relation_Oirreflp,type,
    irreflp: 
      !>[A: $tType] : ( fun(A,fun(A,$o)) > $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: $tType] : ( fun(A,fun(A,$o)) > $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: $tType] : fun(fun(A,fun(B,$o)),fun(fun(B,fun(C,$o)),fun(A,fun(C,$o)))) ).

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: $tType] : ( fun(A,fun(B,$o)) > $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: $tType] : ( fun(A,fun(A,$o)) > $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] : fun(A,fun(A,A)) ).

tff(sy_c_Rings_Odvd__class_Odvd,type,
    dvd_dvd: 
      !>[A: $tType] : fun(A,fun(A,$o)) ).

tff(sy_c_Rings_Omodulo__class_Omodulo,type,
    modulo_modulo: 
      !>[A: $tType] : ( ( A * A ) > A ) ).

tff(sy_c_Rings_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: $tType] : fun($o,A) ).

tff(sy_c_Set_OBall,type,
    ball: 
      !>[A: $tType] : fun(set(A),fun(fun(A,$o),$o)) ).

tff(sy_c_Set_OBex,type,
    bex: 
      !>[A: $tType] : fun(set(A),fun(fun(A,$o),$o)) ).

tff(sy_c_Set_OCollect,type,
    collect: 
      !>[A: $tType] : fun(fun(A,$o),set(A)) ).

tff(sy_c_Set_OPow,type,
    pow2: 
      !>[A: $tType] : ( set(A) > set(set(A)) ) ).

tff(sy_c_Set_Obind,type,
    bind3: 
      !>[A: $tType,B: $tType] : ( ( set(A) * fun(A,set(B)) ) > set(B) ) ).

tff(sy_c_Set_Odisjnt,type,
    disjnt: 
      !>[A: $tType] : fun(set(A),fun(set(A),$o)) ).

tff(sy_c_Set_Ofilter,type,
    filter3: 
      !>[A: $tType] : ( ( fun(A,$o) * set(A) ) > set(A) ) ).

tff(sy_c_Set_Oimage,type,
    image2: 
      !>[A: $tType,B: $tType] : ( fun(A,B) > fun(set(A),set(B)) ) ).

tff(sy_c_Set_Oinsert,type,
    insert2: 
      !>[A: $tType] : fun(A,fun(set(A),set(A))) ).

tff(sy_c_Set_Ois__empty,type,
    is_empty2: 
      !>[A: $tType] : ( set(A) > $o ) ).

tff(sy_c_Set_Ois__singleton,type,
    is_singleton: 
      !>[A: $tType] : ( set(A) > $o ) ).

tff(sy_c_Set_Opairwise,type,
    pairwise: 
      !>[A: $tType] : ( ( fun(A,fun(A,$o)) * set(A) ) > $o ) ).

tff(sy_c_Set_Oremove,type,
    remove: 
      !>[A: $tType] : fun(A,fun(set(A),set(A))) ).

tff(sy_c_Set_Othe__elem,type,
    the_elem: 
      !>[A: $tType] : ( set(A) > A ) ).

tff(sy_c_Set_Ovimage,type,
    vimage: 
      !>[A: $tType,B: $tType] : fun(fun(A,B),fun(set(B),set(A))) ).

tff(sy_c_Set__Interval_Ofold__atLeastAtMost__nat,type,
    set_fo6178422350223883121st_nat: 
      !>[A: $tType] : ( ( fun(nat,fun(A,A)) * nat * nat * A ) > A ) ).

tff(sy_c_Set__Interval_Ofold__atLeastAtMost__nat__rel,type,
    set_fo1817059534552279752at_rel: 
      !>[A: $tType] : fun(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),fun(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),$o)) ).

tff(sy_c_Set__Interval_Oord_OatLeast,type,
    set_atLeast: 
      !>[A: $tType] : ( ( fun(A,fun(A,$o)) * A ) > set(A) ) ).

tff(sy_c_Set__Interval_Oord_OatLeastAtMost,type,
    set_atLeastAtMost: 
      !>[A: $tType] : ( ( fun(A,fun(A,$o)) * A * A ) > set(A) ) ).

tff(sy_c_Set__Interval_Oord_OatLeastLessThan,type,
    set_atLeastLessThan: 
      !>[A: $tType] : ( ( fun(A,fun(A,$o)) * fun(A,fun(A,$o)) * A * A ) > set(A) ) ).

tff(sy_c_Set__Interval_Oord_OatMost,type,
    set_atMost: 
      !>[A: $tType] : ( ( fun(A,fun(A,$o)) * A ) > set(A) ) ).

tff(sy_c_Set__Interval_Oord_OgreaterThan,type,
    set_greaterThan: 
      !>[A: $tType] : ( ( fun(A,fun(A,$o)) * A ) > set(A) ) ).

tff(sy_c_Set__Interval_Oord_OgreaterThanAtMost,type,
    set_gr3752724095348155675AtMost: 
      !>[A: $tType] : ( ( fun(A,fun(A,$o)) * fun(A,fun(A,$o)) * A * A ) > set(A) ) ).

tff(sy_c_Set__Interval_Oord_OgreaterThanLessThan,type,
    set_gr287244882034783167ssThan: 
      !>[A: $tType] : ( ( fun(A,fun(A,$o)) * A * A ) > set(A) ) ).

tff(sy_c_Set__Interval_Oord_OlessThan,type,
    set_lessThan: 
      !>[A: $tType] : ( ( fun(A,fun(A,$o)) * A ) > set(A) ) ).

tff(sy_c_Set__Interval_Oord__class_OatLeast,type,
    set_ord_atLeast: 
      !>[A: $tType] : fun(A,set(A)) ).

tff(sy_c_Set__Interval_Oord__class_OatLeastAtMost,type,
    set_or1337092689740270186AtMost: 
      !>[A: $tType] : ( ( A * A ) > set(A) ) ).

tff(sy_c_Set__Interval_Oord__class_OatLeastLessThan,type,
    set_or7035219750837199246ssThan: 
      !>[A: $tType] : ( ( A * A ) > set(A) ) ).

tff(sy_c_Set__Interval_Oord__class_OatMost,type,
    set_ord_atMost: 
      !>[A: $tType] : fun(A,set(A)) ).

tff(sy_c_Set__Interval_Oord__class_OgreaterThan,type,
    set_ord_greaterThan: 
      !>[A: $tType] : fun(A,set(A)) ).

tff(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost,type,
    set_or3652927894154168847AtMost: 
      !>[A: $tType] : ( ( A * A ) > set(A) ) ).

tff(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan,type,
    set_or5935395276787703475ssThan: 
      !>[A: $tType] : ( ( A * A ) > set(A) ) ).

tff(sy_c_Set__Interval_Oord__class_OlessThan,type,
    set_ord_lessThan: 
      !>[A: $tType] : fun(A,set(A)) ).

tff(sy_c_String_OCode_Oabort,type,
    abort: 
      !>[A: $tType] : ( ( literal * fun(product_unit,A) ) > A ) ).

tff(sy_c_String_OLiteral,type,
    literal2: fun($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun(literal,literal)))))))) ).

tff(sy_c_String_Oascii__of,type,
    ascii_of: fun(char,char) ).

tff(sy_c_String_Oasciis__of__literal,type,
    asciis_of_literal: literal > list(code_integer) ).

tff(sy_c_String_Ochar_OChar,type,
    char2: ( $o * $o * $o * $o * $o * $o * $o ) > fun($o,char) ).

tff(sy_c_String_Ochar_Odigit7,type,
    digit7: char > $o ).

tff(sy_c_String_Ochar__of__integer,type,
    char_of_integer: code_integer > char ).

tff(sy_c_String_Ocomm__semiring__1__class_Oof__char,type,
    comm_s6883823935334413003f_char: 
      !>[A: $tType] : fun(char,A) ).

tff(sy_c_String_Ocr__literal,type,
    cr_literal: fun(list(char),fun(literal,$o)) ).

tff(sy_c_String_Oliteral_OAbs__literal,type,
    abs_literal: fun(list(char),literal) ).

tff(sy_c_String_Oliteral_Oexplode,type,
    explode: fun(literal,list(char)) ).

tff(sy_c_String_Opcr__literal,type,
    pcr_literal: fun(list(char),fun(literal,$o)) ).

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_Omap__sum,type,
    sum_map_sum: 
      !>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( fun(A,C) * fun(B,D) ) > fun(sum_sum(A,B),sum_sum(C,D)) ) ).

tff(sy_c_Sum__Type_Oold_Osum_Orec__set__sum,type,
    sum_rec_set_sum: 
      !>[A: $tType,T: $tType,B: $tType] : ( ( fun(A,T) * fun(B,T) * sum_sum(A,B) ) > fun(T,$o) ) ).

tff(sy_c_Sum__Type_Oold_Osum_Orec__sum,type,
    sum_rec_sum: 
      !>[A: $tType,T: $tType,B: $tType] : ( ( fun(A,T) * fun(B,T) * sum_sum(A,B) ) > T ) ).

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_Sum__Type_Osum_Oprojl,type,
    sum_projl: 
      !>[A: $tType,B: $tType] : ( sum_sum(A,B) > A ) ).

tff(sy_c_Sum__Type_Osum_Oprojr,type,
    sum_projr: 
      !>[A: $tType,B: $tType] : ( sum_sum(A,B) > B ) ).

tff(sy_c_Syntax__Match_Osyntax__fo__nomatch,type,
    syntax7388354845996824322omatch: 
      !>[A: $tType,B: $tType] : ( ( A * B ) > $o ) ).

tff(sy_c_Transfer_Obi__total,type,
    bi_total: 
      !>[A: $tType,B: $tType] : ( fun(A,fun(B,$o)) > $o ) ).

tff(sy_c_Transfer_Obi__unique,type,
    bi_unique: 
      !>[A: $tType,B: $tType] : ( fun(A,fun(B,$o)) > $o ) ).

tff(sy_c_Transfer_Oleft__total,type,
    left_total: 
      !>[A: $tType,B: $tType] : ( fun(A,fun(B,$o)) > $o ) ).

tff(sy_c_Transfer_Oleft__unique,type,
    left_unique: 
      !>[A: $tType,B: $tType] : ( fun(A,fun(B,$o)) > $o ) ).

tff(sy_c_Transfer_Oright__total,type,
    right_total: 
      !>[A: $tType,B: $tType] : ( fun(A,fun(B,$o)) > $o ) ).

tff(sy_c_Transfer_Otransfer__bforall,type,
    transfer_bforall: 
      !>[A: $tType] : ( ( fun(A,$o) * fun(A,$o) ) > $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: $tType] : ( fun(A,fun(A,$o)) > fun(A,fun(A,$o)) ) ).

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: $tType] : ( fun(A,fun(A,$o)) > fun(A,fun(A,$o)) ) ).

tff(sy_c_Typedef_Otype__definition,type,
    type_definition: 
      !>[B: $tType,A: $tType] : ( ( fun(B,A) * fun(A,B) * set(A) ) > $o ) ).

tff(sy_c_Wellfounded_Oaccp,type,
    accp: 
      !>[A: $tType] : ( ( fun(A,fun(A,$o)) * A ) > $o ) ).

tff(sy_c_Wellfounded_Ofinite__psubset,type,
    finite_psubset: 
      !>[A: $tType] : set(product_prod(set(A),set(A))) ).

tff(sy_c_Wellfounded_Oless__than,type,
    less_than: set(product_prod(nat,nat)) ).

tff(sy_c_Wellfounded_Olex__prod,type,
    lex_prod: 
      !>[A: $tType,B: $tType] : ( ( set(product_prod(A,A)) * set(product_prod(B,B)) ) > set(product_prod(product_prod(A,B),product_prod(A,B))) ) ).

tff(sy_c_Wellfounded_Omax__ext,type,
    max_ext: 
      !>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(set(A),set(A))) ) ).

tff(sy_c_Wellfounded_Omax__extp,type,
    max_extp: 
      !>[A: $tType] : ( fun(A,fun(A,$o)) > fun(set(A),fun(set(A),$o)) ) ).

tff(sy_c_Wellfounded_Omeasure,type,
    measure: 
      !>[A: $tType] : ( fun(A,nat) > set(product_prod(A,A)) ) ).

tff(sy_c_Wellfounded_Omin__ext,type,
    min_ext: 
      !>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(set(A),set(A))) ) ).

tff(sy_c_Wellfounded_Omlex__prod,type,
    mlex_prod: 
      !>[A: $tType] : ( ( fun(A,nat) * set(product_prod(A,A)) ) > set(product_prod(A,A)) ) ).

tff(sy_c_Wellfounded_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: $tType] : ( fun(A,fun(A,$o)) > $o ) ).

tff(sy_c_Wfrec_Oadm__wf,type,
    adm_wf: 
      !>[A: $tType,B: $tType] : ( ( set(product_prod(A,A)) * fun(fun(A,B),fun(A,B)) ) > $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: $tType] : ( ( fun(A,$o) * fun(A,set(product_prod(B,B))) ) > set(product_prod(product_prod(A,B),product_prod(A,B))) ) ).

tff(sy_c_Zorn_OChains,type,
    chains: 
      !>[A: $tType] : ( set(product_prod(A,A)) > set(set(A)) ) ).

tff(sy_c_Zorn_Ochain__subset,type,
    chain_subset: 
      !>[A: $tType] : ( set(set(A)) > $o ) ).

tff(sy_c_Zorn_Ochains,type,
    chains2: 
      !>[A: $tType] : ( set(set(A)) > set(set(set(A))) ) ).

tff(sy_c_Zorn_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: $tType] : ( ( set(A) * fun(A,fun(A,$o)) ) > fun(set(A),$o) ) ).

tff(sy_c_Zorn_Opred__on_Omaxchain,type,
    pred_maxchain: 
      !>[A: $tType] : ( ( set(A) * fun(A,fun(A,$o)) * set(A) ) > $o ) ).

tff(sy_c_Zorn_Opred__on_Osuc,type,
    pred_suc: 
      !>[A: $tType] : ( ( set(A) * fun(A,fun(A,$o)) * set(A) ) > set(A) ) ).

tff(sy_c_Zorn_Opred__on_Osuc__Union__closed,type,
    pred_s596693808085603175closed: 
      !>[A: $tType] : ( ( set(A) * fun(A,fun(A,$o)) ) > set(set(A)) ) ).

tff(sy_c_Zorn_Opred__on_Osuc__Union__closedp,type,
    pred_s7749564232668923593losedp: 
      !>[A: $tType] : ( ( set(A) * fun(A,fun(A,$o)) ) > fun(set(A),$o) ) ).

tff(sy_c_aa,type,
    aa: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * A ) > B ) ).

tff(sy_c_fAll,type,
    fAll: 
      !>[A: $tType] : fun(fun(A,$o),$o) ).

tff(sy_c_fChoice,type,
    fChoice: 
      !>[A: $tType] : ( fun(A,$o) > A ) ).

tff(sy_c_fNot,type,
    fNot: fun($o,$o) ).

tff(sy_c_fequal,type,
    fequal: 
      !>[A: $tType] : fun(A,fun(A,$o)) ).

tff(sy_c_member,type,
    member: 
      !>[A: $tType] : ( A > fun(set(A),$o) ) ).

tff(sy_v_x,type,
    x: assn ).

tff(sy_v_y,type,
    y: assn ).

% Relevant facts (9088)
tff(fact_0_Rep__assn__inject,axiom,
    ! [X: assn,Y: assn] :
      ( ( aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,X) = aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,Y) )
    <=> ( X = Y ) ) ).

% Rep_assn_inject
tff(fact_1_Rep__assn__inverse,axiom,
    ! [X: assn] : aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),assn,abs_assn,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,X)) = X ).

% Rep_assn_inverse
tff(fact_2_Abs__assn__eqI_I2_J,axiom,
    ! [P: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Pr: assn] :
      ( ! [H: product_prod(heap_ext(product_unit),set(nat))] :
          ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,P,H)
        <=> aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,Pr),H) )
     => ( Pr = aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),assn,abs_assn,P) ) ) ).

% Abs_assn_eqI(2)
tff(fact_3_Abs__assn__eqI_I1_J,axiom,
    ! [P: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Pr: assn] :
      ( ! [H: product_prod(heap_ext(product_unit),set(nat))] :
          ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,P,H)
        <=> aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,Pr),H) )
     => ( aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),assn,abs_assn,P) = Pr ) ) ).

% Abs_assn_eqI(1)
tff(fact_4_wand__assn__def,axiom,
    ! [P: assn,Q: assn] : wand_assn(P,Q) = aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),assn,abs_assn,wand_raw(aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,P),aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,Q))) ).

% wand_assn_def
tff(fact_5_Abs__assn__inverse,axiom,
    ! [Y: fun(product_prod(heap_ext(product_unit),set(nat)),$o)] :
      ( aa(set(fun(product_prod(heap_ext(product_unit),set(nat)),$o)),$o,member(fun(product_prod(heap_ext(product_unit),set(nat)),$o),Y),aa(fun(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o),set(fun(product_prod(heap_ext(product_unit),set(nat)),$o)),collect(fun(product_prod(heap_ext(product_unit),set(nat)),$o)),proper))
     => ( aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),assn,abs_assn,Y)) = Y ) ) ).

% Abs_assn_inverse
tff(fact_6_inf__assn__def,axiom,
    ! [P: assn,Q: assn] : aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),P),Q) = aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),assn,abs_assn,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),aTP_Lamp_ab(assn,fun(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o)),P),Q)) ).

% inf_assn_def
tff(fact_7_sup__assn__def,axiom,
    ! [P: assn,Q: assn] : aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),P),Q) = aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),assn,abs_assn,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),aTP_Lamp_ac(assn,fun(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o)),P),Q)) ).

% sup_assn_def
tff(fact_8_bot__assn__def,axiom,
    bot_bot(assn) = aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),assn,abs_assn,aTP_Lamp_ad(product_prod(heap_ext(product_unit),set(nat)),$o)) ).

% bot_assn_def
tff(fact_9_Abs__assn__cases,axiom,
    ! [X: assn] :
      ~ ! [Y2: fun(product_prod(heap_ext(product_unit),set(nat)),$o)] :
          ( ( X = aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),assn,abs_assn,Y2) )
         => ~ aa(set(fun(product_prod(heap_ext(product_unit),set(nat)),$o)),$o,member(fun(product_prod(heap_ext(product_unit),set(nat)),$o),Y2),aa(fun(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o),set(fun(product_prod(heap_ext(product_unit),set(nat)),$o)),collect(fun(product_prod(heap_ext(product_unit),set(nat)),$o)),proper)) ) ).

% Abs_assn_cases
tff(fact_10_Abs__assn__induct,axiom,
    ! [P: fun(assn,$o),X: assn] :
      ( ! [Y2: fun(product_prod(heap_ext(product_unit),set(nat)),$o)] :
          ( aa(set(fun(product_prod(heap_ext(product_unit),set(nat)),$o)),$o,member(fun(product_prod(heap_ext(product_unit),set(nat)),$o),Y2),aa(fun(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o),set(fun(product_prod(heap_ext(product_unit),set(nat)),$o)),collect(fun(product_prod(heap_ext(product_unit),set(nat)),$o)),proper))
         => aa(assn,$o,P,aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),assn,abs_assn,Y2)) )
     => aa(assn,$o,P,X) ) ).

% Abs_assn_induct
tff(fact_11_Abs__assn__inject,axiom,
    ! [X: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Y: fun(product_prod(heap_ext(product_unit),set(nat)),$o)] :
      ( aa(set(fun(product_prod(heap_ext(product_unit),set(nat)),$o)),$o,member(fun(product_prod(heap_ext(product_unit),set(nat)),$o),X),aa(fun(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o),set(fun(product_prod(heap_ext(product_unit),set(nat)),$o)),collect(fun(product_prod(heap_ext(product_unit),set(nat)),$o)),proper))
     => ( aa(set(fun(product_prod(heap_ext(product_unit),set(nat)),$o)),$o,member(fun(product_prod(heap_ext(product_unit),set(nat)),$o),Y),aa(fun(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o),set(fun(product_prod(heap_ext(product_unit),set(nat)),$o)),collect(fun(product_prod(heap_ext(product_unit),set(nat)),$o)),proper))
       => ( ( aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),assn,abs_assn,X) = aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),assn,abs_assn,Y) )
        <=> ( X = Y ) ) ) ) ).

% Abs_assn_inject
tff(fact_12_Rep__assn,axiom,
    ! [X: assn] : aa(set(fun(product_prod(heap_ext(product_unit),set(nat)),$o)),$o,member(fun(product_prod(heap_ext(product_unit),set(nat)),$o),aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,X)),aa(fun(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o),set(fun(product_prod(heap_ext(product_unit),set(nat)),$o)),collect(fun(product_prod(heap_ext(product_unit),set(nat)),$o)),proper)) ).

% Rep_assn
tff(fact_13_Rep__assn__cases,axiom,
    ! [Y: fun(product_prod(heap_ext(product_unit),set(nat)),$o)] :
      ( aa(set(fun(product_prod(heap_ext(product_unit),set(nat)),$o)),$o,member(fun(product_prod(heap_ext(product_unit),set(nat)),$o),Y),aa(fun(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o),set(fun(product_prod(heap_ext(product_unit),set(nat)),$o)),collect(fun(product_prod(heap_ext(product_unit),set(nat)),$o)),proper))
     => ~ ! [X2: assn] : Y != aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,X2) ) ).

% Rep_assn_cases
tff(fact_14_Rep__assn__induct,axiom,
    ! [Y: fun(product_prod(heap_ext(product_unit),set(nat)),$o),P: fun(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o)] :
      ( aa(set(fun(product_prod(heap_ext(product_unit),set(nat)),$o)),$o,member(fun(product_prod(heap_ext(product_unit),set(nat)),$o),Y),aa(fun(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o),set(fun(product_prod(heap_ext(product_unit),set(nat)),$o)),collect(fun(product_prod(heap_ext(product_unit),set(nat)),$o)),proper))
     => ( ! [X2: assn] : aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o,P,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,X2))
       => aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o,P,Y) ) ) ).

% Rep_assn_induct
tff(fact_15_times__assn__def,axiom,
    ! [P: assn,Q: assn] : aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),Q) = aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),assn,abs_assn,times_assn_raw(aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,P),aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,Q))) ).

% times_assn_def
tff(fact_16_bool__assn__proper_I2_J,axiom,
    aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o,proper,aTP_Lamp_ad(product_prod(heap_ext(product_unit),set(nat)),$o)) ).

% bool_assn_proper(2)
tff(fact_17_bool__assn__proper_I3_J,axiom,
    ! [P: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Q: fun(product_prod(heap_ext(product_unit),set(nat)),$o)] :
      ( aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o,proper,P)
     => ( aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o,proper,Q)
       => aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o,proper,aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(heap_ext(product_unit),set(nat)),$o),aTP_Lamp_ae(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(heap_ext(product_unit),set(nat)),$o)),P),Q)) ) ) ).

% bool_assn_proper(3)
tff(fact_18_bool__assn__proper_I4_J,axiom,
    ! [P: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Q: fun(product_prod(heap_ext(product_unit),set(nat)),$o)] :
      ( aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o,proper,P)
     => ( aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o,proper,Q)
       => aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o,proper,aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(heap_ext(product_unit),set(nat)),$o),aTP_Lamp_af(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(heap_ext(product_unit),set(nat)),$o)),P),Q)) ) ) ).

% bool_assn_proper(4)
tff(fact_19_wand__proper,axiom,
    ! [P: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Q: fun(product_prod(heap_ext(product_unit),set(nat)),$o)] : aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o,proper,wand_raw(P,Q)) ).

% wand_proper
tff(fact_20_times__assn__proper,axiom,
    ! [P: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Q: fun(product_prod(heap_ext(product_unit),set(nat)),$o)] :
      ( aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o,proper,P)
     => ( aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o,proper,Q)
       => aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o,proper,times_assn_raw(P,Q)) ) ) ).

% times_assn_proper
tff(fact_21_assn__times__comm,axiom,
    ! [P: assn,Q: assn] : aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),Q) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Q),P) ).

% assn_times_comm
tff(fact_22_assn__times__assoc,axiom,
    ! [P: assn,Q: assn,R: assn] : aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),Q)),R) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Q),R)) ).

% assn_times_assoc
tff(fact_23_mod__starE,axiom,
    ! [A3: assn,B2: assn,H2: product_prod(heap_ext(product_unit),set(nat))] :
      ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),A3),B2)),H2)
     => ~ ( ? [X_1: product_prod(heap_ext(product_unit),set(nat))] : aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,A3),X_1)
         => ! [H_2: product_prod(heap_ext(product_unit),set(nat))] : ~ aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,B2),H_2) ) ) ).

% mod_starE
tff(fact_24_mod__starD,axiom,
    ! [A4: assn,B3: assn,H2: product_prod(heap_ext(product_unit),set(nat))] :
      ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),A4),B3)),H2)
     => ? [H1: product_prod(heap_ext(product_unit),set(nat)),H22: product_prod(heap_ext(product_unit),set(nat))] :
          ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,A4),H1)
          & aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,B3),H22) ) ) ).

% mod_starD
tff(fact_25_inf__sup__absorb,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y)) = X ) ).

% inf_sup_absorb
tff(fact_26_sup__inf__absorb,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)) = X ) ).

% sup_inf_absorb
tff(fact_27_sup__bot__left,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),bot_bot(A)),X) = X ) ).

% sup_bot_left
tff(fact_28_sup__bot__right,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),bot_bot(A)) = X ) ).

% sup_bot_right
tff(fact_29_bot__eq__sup__iff,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [X: A,Y: A] :
          ( ( bot_bot(A) = aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y) )
        <=> ( ( X = bot_bot(A) )
            & ( Y = bot_bot(A) ) ) ) ) ).

% bot_eq_sup_iff
tff(fact_30_sup__eq__bot__iff,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [X: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y) = bot_bot(A) )
        <=> ( ( X = bot_bot(A) )
            & ( Y = bot_bot(A) ) ) ) ) ).

% sup_eq_bot_iff
tff(fact_31_sup__bot_Oeq__neutr__iff,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [A3: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2) = bot_bot(A) )
        <=> ( ( A3 = bot_bot(A) )
            & ( B2 = bot_bot(A) ) ) ) ) ).

% sup_bot.eq_neutr_iff
tff(fact_32_sup__bot_Oleft__neutral,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),bot_bot(A)),A3) = A3 ) ).

% sup_bot.left_neutral
tff(fact_33_sup__bot_Oneutr__eq__iff,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [A3: A,B2: A] :
          ( ( bot_bot(A) = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2) )
        <=> ( ( A3 = bot_bot(A) )
            & ( B2 = bot_bot(A) ) ) ) ) ).

% sup_bot.neutr_eq_iff
tff(fact_34_sup__bot_Oright__neutral,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),bot_bot(A)) = A3 ) ).

% sup_bot.right_neutral
tff(fact_35_inf__bot__left,axiom,
    ! [A: $tType] :
      ( bounded_lattice_bot(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),bot_bot(A)),X) = bot_bot(A) ) ).

% inf_bot_left
tff(fact_36_inf__bot__right,axiom,
    ! [A: $tType] :
      ( bounded_lattice_bot(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),X),bot_bot(A)) = bot_bot(A) ) ).

% inf_bot_right
tff(fact_37_inf__apply,axiom,
    ! [A: $tType,B: $tType] :
      ( semilattice_inf(A)
     => ! [F: fun(B,A),G: fun(B,A),X: B] : aa(B,A,aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),inf_inf(fun(B,A)),F),G),X) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(B,A,F,X)),aa(B,A,G,X)) ) ).

% inf_apply
tff(fact_38_inf__right__idem,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)),Y) = aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y) ) ).

% inf_right_idem
tff(fact_39_inf_Oright__idem,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)),B2) = aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2) ) ).

% inf.right_idem
tff(fact_40_inf__left__idem,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y) ) ).

% inf_left_idem
tff(fact_41_inf_Oleft__idem,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2) ) ).

% inf.left_idem
tff(fact_42_inf__idem,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),X),X) = X ) ).

% inf_idem
tff(fact_43_inf_Oidem,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),A3) = A3 ) ).

% inf.idem
tff(fact_44_sup__apply,axiom,
    ! [A: $tType,B: $tType] :
      ( semilattice_sup(A)
     => ! [F: fun(B,A),G: fun(B,A),X: B] : aa(B,A,aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),sup_sup(fun(B,A)),F),G),X) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(B,A,F,X)),aa(B,A,G,X)) ) ).

% sup_apply
tff(fact_45_mem__Collect__eq,axiom,
    ! [A: $tType,A3: A,P: fun(A,$o)] :
      ( aa(set(A),$o,member(A,A3),aa(fun(A,$o),set(A),collect(A),P))
    <=> aa(A,$o,P,A3) ) ).

% mem_Collect_eq
tff(fact_46_Collect__mem__eq,axiom,
    ! [A: $tType,A4: set(A)] : aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_ag(set(A),fun(A,$o)),A4)) = A4 ).

% Collect_mem_eq
tff(fact_47_Collect__cong,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] :
      ( ! [X2: A] :
          ( aa(A,$o,P,X2)
        <=> aa(A,$o,Q,X2) )
     => ( aa(fun(A,$o),set(A),collect(A),P) = aa(fun(A,$o),set(A),collect(A),Q) ) ) ).

% Collect_cong
tff(fact_48_ext,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),G: fun(A,B)] :
      ( ! [X2: A] : aa(A,B,F,X2) = aa(A,B,G,X2)
     => ( F = G ) ) ).

% ext
tff(fact_49_sup_Oright__idem,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2)),B2) = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2) ) ).

% sup.right_idem
tff(fact_50_sup__left__idem,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),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),Y) ) ).

% sup_left_idem
tff(fact_51_sup_Oleft__idem,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2) ) ).

% sup.left_idem
tff(fact_52_sup__idem,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),X) = X ) ).

% sup_idem
tff(fact_53_sup_Oidem,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),A3) = A3 ) ).

% sup.idem
tff(fact_54_inf__fun__def,axiom,
    ! [B: $tType,A: $tType] :
      ( semilattice_inf(B)
     => ! [F: fun(A,B),G: fun(A,B),X3: A] : aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),inf_inf(fun(A,B)),F),G),X3) = aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(A,B,F,X3)),aa(A,B,G,X3)) ) ).

% inf_fun_def
tff(fact_55_inf__left__commute,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [X: A,Y: A,Z2: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Z2)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Z2)) ) ).

% inf_left_commute
tff(fact_56_inf_Oleft__commute,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,A3: A,C2: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),B2),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),C2)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),aa(A,A,aa(A,fun(A,A),inf_inf(A),B2),C2)) ) ).

% inf.left_commute
tff(fact_57_inf__commute,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [X: A,Y: 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),Y),X) ) ).

% inf_commute
tff(fact_58_inf_Ocommute,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B2: 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),B2),A3) ) ).

% inf.commute
tff(fact_59_inf__assoc,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [X: A,Y: A,Z2: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)),Z2) = aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Z2)) ) ).

% inf_assoc
tff(fact_60_inf_Oassoc,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),aa(A,A,aa(A,fun(A,A),inf_inf(A),B2),C2)) ) ).

% inf.assoc
tff(fact_61_inf__sup__aci_I1_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [X: A,Y: 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),Y),X) ) ).

% inf_sup_aci(1)
tff(fact_62_inf__sup__aci_I2_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [X: A,Y: A,Z2: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)),Z2) = aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Z2)) ) ).

% inf_sup_aci(2)
tff(fact_63_inf__sup__aci_I3_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [X: A,Y: A,Z2: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Z2)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Z2)) ) ).

% inf_sup_aci(3)
tff(fact_64_inf__sup__aci_I4_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y) ) ).

% inf_sup_aci(4)
tff(fact_65_sup__fun__def,axiom,
    ! [B: $tType,A: $tType] :
      ( semilattice_sup(B)
     => ! [F: fun(A,B),G: fun(A,B),X3: A] : aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),sup_sup(fun(A,B)),F),G),X3) = aa(B,B,aa(B,fun(B,B),sup_sup(B),aa(A,B,F,X3)),aa(A,B,G,X3)) ) ).

% sup_fun_def
tff(fact_66_sup__left__commute,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,Y: A,Z2: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z2)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Z2)) ) ).

% sup_left_commute
tff(fact_67_sup_Oleft__commute,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A3: A,C2: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),B2),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),C2)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),aa(A,A,aa(A,fun(A,A),sup_sup(A),B2),C2)) ) ).

% sup.left_commute
tff(fact_68_sup__commute,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,Y: 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),Y),X) ) ).

% sup_commute
tff(fact_69_sup_Ocommute,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: A,B2: 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),B2),A3) ) ).

% sup.commute
tff(fact_70_sup__assoc,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,Y: A,Z2: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y)),Z2) = aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z2)) ) ).

% sup_assoc
tff(fact_71_sup_Oassoc,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),aa(A,A,aa(A,fun(A,A),sup_sup(A),B2),C2)) ) ).

% sup.assoc
tff(fact_72_inf__sup__aci_I5_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [X: A,Y: 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),Y),X) ) ).

% inf_sup_aci(5)
tff(fact_73_inf__sup__aci_I6_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [X: A,Y: A,Z2: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y)),Z2) = aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z2)) ) ).

% inf_sup_aci(6)
tff(fact_74_inf__sup__aci_I7_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [X: A,Y: A,Z2: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z2)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Z2)) ) ).

% inf_sup_aci(7)
tff(fact_75_inf__sup__aci_I8_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),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),Y) ) ).

% inf_sup_aci(8)
tff(fact_76_sup__inf__distrib2,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [Y: A,Z2: A,X: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Z2)),X) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),X)),aa(A,A,aa(A,fun(A,A),sup_sup(A),Z2),X)) ) ).

% sup_inf_distrib2
tff(fact_77_sup__inf__distrib1,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [X: A,Y: A,Z2: 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)) ) ).

% sup_inf_distrib1
tff(fact_78_inf__sup__distrib2,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [Y: A,Z2: A,X: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z2)),X) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),X)),aa(A,A,aa(A,fun(A,A),inf_inf(A),Z2),X)) ) ).

% inf_sup_distrib2
tff(fact_79_inf__sup__distrib1,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [X: A,Y: A,Z2: A] : 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)) = 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)) ) ).

% inf_sup_distrib1
tff(fact_80_distrib__imp2,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [X: A,Y: A,Z2: A] :
          ( ! [X2: A,Y2: A,Z3: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X2),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y2),Z3)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),X2),Y2)),aa(A,A,aa(A,fun(A,A),sup_sup(A),X2),Z3))
         => ( 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)) = 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)) ) ) ) ).

% distrib_imp2
tff(fact_81_distrib__imp1,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [X: A,Y: A,Z2: A] :
          ( ! [X2: A,Y2: A,Z3: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),X2),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y2),Z3)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X2),Y2)),aa(A,A,aa(A,fun(A,A),inf_inf(A),X2),Z3))
         => ( 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_imp1
tff(fact_82_boolean__algebra_Oconj__zero__right,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),X),bot_bot(A)) = bot_bot(A) ) ).

% boolean_algebra.conj_zero_right
tff(fact_83_boolean__algebra_Oconj__zero__left,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),bot_bot(A)),X) = bot_bot(A) ) ).

% boolean_algebra.conj_zero_left
tff(fact_84_bot__apply,axiom,
    ! [B: $tType,A: $tType] :
      ( bot(A)
     => ! [X: B] : aa(B,A,bot_bot(fun(B,A)),X) = bot_bot(A) ) ).

% bot_apply
tff(fact_85_boolean__algebra_Odisj__conj__distrib2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Y: A,Z2: A,X: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Z2)),X) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),X)),aa(A,A,aa(A,fun(A,A),sup_sup(A),Z2),X)) ) ).

% boolean_algebra.disj_conj_distrib2
tff(fact_86_boolean__algebra_Oconj__disj__distrib2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Y: A,Z2: A,X: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z2)),X) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),X)),aa(A,A,aa(A,fun(A,A),inf_inf(A),Z2),X)) ) ).

% boolean_algebra.conj_disj_distrib2
tff(fact_87_boolean__algebra_Odisj__conj__distrib,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y: A,Z2: 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)) ) ).

% boolean_algebra.disj_conj_distrib
tff(fact_88_boolean__algebra_Oconj__disj__distrib,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y: A,Z2: A] : 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)) = 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)) ) ).

% boolean_algebra.conj_disj_distrib
tff(fact_89_boolean__algebra_Odisj__zero__right,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),bot_bot(A)) = X ) ).

% boolean_algebra.disj_zero_right
tff(fact_90_one__assn__proper,axiom,
    aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o,proper,one_assn_raw) ).

% one_assn_proper
tff(fact_91_type__definition__assn,axiom,
    type_definition(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,abs_assn,aa(fun(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o),set(fun(product_prod(heap_ext(product_unit),set(nat)),$o)),collect(fun(product_prod(heap_ext(product_unit),set(nat)),$o)),proper)) ).

% type_definition_assn
tff(fact_92_boolean__algebra__cancel_Osup2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B3: A,K: A,B2: A,A3: A] :
          ( ( B3 = aa(A,A,aa(A,fun(A,A),sup_sup(A),K),B2) )
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3) = aa(A,A,aa(A,fun(A,A),sup_sup(A),K),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2)) ) ) ) ).

% boolean_algebra_cancel.sup2
tff(fact_93_boolean__algebra__cancel_Osup1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A4: A,K: A,A3: A,B2: A] :
          ( ( A4 = aa(A,A,aa(A,fun(A,A),sup_sup(A),K),A3) )
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A4),B2) = aa(A,A,aa(A,fun(A,A),sup_sup(A),K),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2)) ) ) ) ).

% boolean_algebra_cancel.sup1
tff(fact_94_bot__fun__def,axiom,
    ! [A: $tType,B: $tType] :
      ( bot(B)
     => ! [X3: A] : aa(A,B,bot_bot(fun(A,B)),X3) = bot_bot(B) ) ).

% bot_fun_def
tff(fact_95_boolean__algebra__cancel_Oinf1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A4: A,K: A,A3: A,B2: A] :
          ( ( A4 = aa(A,A,aa(A,fun(A,A),inf_inf(A),K),A3) )
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A4),B2) = aa(A,A,aa(A,fun(A,A),inf_inf(A),K),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)) ) ) ) ).

% boolean_algebra_cancel.inf1
tff(fact_96_boolean__algebra__cancel_Oinf2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B3: A,K: A,B2: A,A3: A] :
          ( ( B3 = aa(A,A,aa(A,fun(A,A),inf_inf(A),K),B2) )
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B3) = aa(A,A,aa(A,fun(A,A),inf_inf(A),K),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)) ) ) ) ).

% boolean_algebra_cancel.inf2
tff(fact_97_type__definition_ORep,axiom,
    ! [A: $tType,B: $tType,Rep: fun(A,B),Abs: fun(B,A),A4: set(B),X: A] :
      ( type_definition(A,B,Rep,Abs,A4)
     => aa(set(B),$o,member(B,aa(A,B,Rep,X)),A4) ) ).

% type_definition.Rep
tff(fact_98_type__definition_Ointro,axiom,
    ! [A: $tType,B: $tType,Rep: fun(A,B),A4: set(B),Abs: fun(B,A)] :
      ( ! [X2: A] : aa(set(B),$o,member(B,aa(A,B,Rep,X2)),A4)
     => ( ! [X2: A] : aa(B,A,Abs,aa(A,B,Rep,X2)) = X2
       => ( ! [Y2: B] :
              ( aa(set(B),$o,member(B,Y2),A4)
             => ( aa(A,B,Rep,aa(B,A,Abs,Y2)) = Y2 ) )
         => type_definition(A,B,Rep,Abs,A4) ) ) ) ).

% type_definition.intro
tff(fact_99_type__definition_OAbs__cases,axiom,
    ! [A: $tType,B: $tType,Rep: fun(A,B),Abs: fun(B,A),A4: set(B),X: A] :
      ( type_definition(A,B,Rep,Abs,A4)
     => ~ ! [Y2: B] :
            ( ( X = aa(B,A,Abs,Y2) )
           => ~ aa(set(B),$o,member(B,Y2),A4) ) ) ).

% type_definition.Abs_cases
tff(fact_100_type__definition_ORep__cases,axiom,
    ! [B: $tType,A: $tType,Rep: fun(A,B),Abs: fun(B,A),A4: set(B),Y: B] :
      ( type_definition(A,B,Rep,Abs,A4)
     => ( aa(set(B),$o,member(B,Y),A4)
       => ~ ! [X2: A] : Y != aa(A,B,Rep,X2) ) ) ).

% type_definition.Rep_cases
tff(fact_101_type__definition_OAbs__induct,axiom,
    ! [B: $tType,A: $tType,Rep: fun(A,B),Abs: fun(B,A),A4: set(B),P: fun(A,$o),X: A] :
      ( type_definition(A,B,Rep,Abs,A4)
     => ( ! [Y2: B] :
            ( aa(set(B),$o,member(B,Y2),A4)
           => aa(A,$o,P,aa(B,A,Abs,Y2)) )
       => aa(A,$o,P,X) ) ) ).

% type_definition.Abs_induct
tff(fact_102_type__definition_OAbs__inject,axiom,
    ! [A: $tType,B: $tType,Rep: fun(A,B),Abs: fun(B,A),A4: set(B),X: B,Y: B] :
      ( type_definition(A,B,Rep,Abs,A4)
     => ( aa(set(B),$o,member(B,X),A4)
       => ( aa(set(B),$o,member(B,Y),A4)
         => ( ( aa(B,A,Abs,X) = aa(B,A,Abs,Y) )
          <=> ( X = Y ) ) ) ) ) ).

% type_definition.Abs_inject
tff(fact_103_type__definition_ORep__induct,axiom,
    ! [A: $tType,B: $tType,Rep: fun(A,B),Abs: fun(B,A),A4: set(B),Y: B,P: fun(B,$o)] :
      ( type_definition(A,B,Rep,Abs,A4)
     => ( aa(set(B),$o,member(B,Y),A4)
       => ( ! [X2: A] : aa(B,$o,P,aa(A,B,Rep,X2))
         => aa(B,$o,P,Y) ) ) ) ).

% type_definition.Rep_induct
tff(fact_104_type__definition_ORep__inject,axiom,
    ! [B: $tType,A: $tType,Rep: fun(A,B),Abs: fun(B,A),A4: set(B),X: A,Y: A] :
      ( type_definition(A,B,Rep,Abs,A4)
     => ( ( aa(A,B,Rep,X) = aa(A,B,Rep,Y) )
      <=> ( X = Y ) ) ) ).

% type_definition.Rep_inject
tff(fact_105_type__definition_OAbs__inverse,axiom,
    ! [A: $tType,B: $tType,Rep: fun(A,B),Abs: fun(B,A),A4: set(B),Y: B] :
      ( type_definition(A,B,Rep,Abs,A4)
     => ( aa(set(B),$o,member(B,Y),A4)
       => ( aa(A,B,Rep,aa(B,A,Abs,Y)) = Y ) ) ) ).

% type_definition.Abs_inverse
tff(fact_106_type__definition_ORep__inverse,axiom,
    ! [B: $tType,A: $tType,Rep: fun(A,B),Abs: fun(B,A),A4: set(B),X: A] :
      ( type_definition(A,B,Rep,Abs,A4)
     => ( aa(B,A,Abs,aa(A,B,Rep,X)) = X ) ) ).

% type_definition.Rep_inverse
tff(fact_107_type__definition__def,axiom,
    ! [A: $tType,B: $tType,Rep: fun(A,B),Abs: fun(B,A),A4: set(B)] :
      ( type_definition(A,B,Rep,Abs,A4)
    <=> ( ! [X4: A] : aa(set(B),$o,member(B,aa(A,B,Rep,X4)),A4)
        & ! [X4: A] : aa(B,A,Abs,aa(A,B,Rep,X4)) = X4
        & ! [Y3: B] :
            ( aa(set(B),$o,member(B,Y3),A4)
           => ( aa(A,B,Rep,aa(B,A,Abs,Y3)) = Y3 ) ) ) ) ).

% type_definition_def
tff(fact_108_mult_Oright__assoc,axiom,
    ! [A: $tType] :
      ( ab_semigroup_mult(A)
     => ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ).

% mult.right_assoc
tff(fact_109_mult_Oright__commute,axiom,
    ! [A: $tType] :
      ( ab_semigroup_mult(A)
     => ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),B2) ) ).

% mult.right_commute
tff(fact_110_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [A: $tType] :
      ( ab_semigroup_mult(A)
     => ! [B2: A,A3: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ).

% ab_semigroup_mult_class.mult.left_commute
tff(fact_111_ab__semigroup__mult__class_Omult_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) ) ).

% ab_semigroup_mult_class.mult.commute
tff(fact_112_mult_Oassoc,axiom,
    ! [A: $tType] :
      ( semigroup_mult(A)
     => ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ).

% mult.assoc
tff(fact_113_mult_Osafe__commute,axiom,
    ! [A: $tType] :
      ( ab_semigroup_mult(A)
     => ! [X: A,Y: A,A3: A,B2: A] :
          ( syntax7388354845996824322omatch(A,A,aa(A,A,aa(A,fun(A,A),times_times(A),X),Y),A3)
         => ( 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.safe_commute
tff(fact_114_sup__bot_Osemilattice__neutr__axioms,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => semilattice_neutr(A,sup_sup(A),bot_bot(A)) ) ).

% sup_bot.semilattice_neutr_axioms
tff(fact_115_sup__bot_Omonoid__axioms,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => monoid(A,sup_sup(A),bot_bot(A)) ) ).

% sup_bot.monoid_axioms
tff(fact_116_one__assn__def,axiom,
    one_one(assn) = aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),assn,abs_assn,one_assn_raw) ).

% one_assn_def
tff(fact_117_boolean__algebra_Ocomplement__unique,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [A3: A,X: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),X) = bot_bot(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),X) = top_top(A) )
           => ( ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),Y) = bot_bot(A) )
             => ( ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),Y) = top_top(A) )
               => ( X = Y ) ) ) ) ) ) ).

% boolean_algebra.complement_unique
tff(fact_118_mod__star__conv,axiom,
    ! [A4: assn,B3: assn,H2: product_prod(heap_ext(product_unit),set(nat))] :
      ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),A4),B3)),H2)
    <=> ? [Hr: heap_ext(product_unit),As1: set(nat),As2: set(nat)] :
          ( ( H2 = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),Hr),aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As1),As2)) )
          & ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As1),As2) = bot_bot(set(nat)) )
          & aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,A4),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),Hr),As1))
          & aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,B3),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),Hr),As2)) ) ) ).

% mod_star_conv
tff(fact_119_star__assnI,axiom,
    ! [P: assn,H2: heap_ext(product_unit),As: set(nat),Q: assn,As3: set(nat)] :
      ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,P),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As))
     => ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,Q),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As3))
       => ( ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As),As3) = bot_bot(set(nat)) )
         => aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),Q)),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As),As3))) ) ) ) ).

% star_assnI
tff(fact_120_sup__bot_Ocomm__monoid__axioms,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => comm_monoid(A,sup_sup(A),bot_bot(A)) ) ).

% sup_bot.comm_monoid_axioms
tff(fact_121_boolean__algebra_Ode__Morgan__disj,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,uminus_uminus(A),X)),aa(A,A,uminus_uminus(A),Y)) ) ).

% boolean_algebra.de_Morgan_disj
tff(fact_122_uminus__apply,axiom,
    ! [A: $tType,B: $tType] :
      ( uminus(A)
     => ! [A4: fun(B,A),X: B] : aa(B,A,aa(fun(B,A),fun(B,A),uminus_uminus(fun(B,A)),A4),X) = aa(A,A,uminus_uminus(A),aa(B,A,A4,X)) ) ).

% uminus_apply
tff(fact_123_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_124_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_125_boolean__algebra__class_Oboolean__algebra_Ocompl__eq__compl__iff,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y: A] :
          ( ( aa(A,A,uminus_uminus(A),X) = aa(A,A,uminus_uminus(A),Y) )
        <=> ( X = Y ) ) ) ).

% boolean_algebra_class.boolean_algebra.compl_eq_compl_iff
tff(fact_126_boolean__algebra__class_Oboolean__algebra_Odouble__compl,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A] : aa(A,A,uminus_uminus(A),aa(A,A,uminus_uminus(A),X)) = X ) ).

% boolean_algebra_class.boolean_algebra.double_compl
tff(fact_127_top__apply,axiom,
    ! [B: $tType,A: $tType] :
      ( top(A)
     => ! [X: B] : aa(B,A,top_top(fun(B,A)),X) = top_top(A) ) ).

% top_apply
tff(fact_128_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_129_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_130_inf__top__left,axiom,
    ! [A: $tType] :
      ( bounde4346867609351753570nf_top(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),top_top(A)),X) = X ) ).

% inf_top_left
tff(fact_131_inf__top__right,axiom,
    ! [A: $tType] :
      ( bounde4346867609351753570nf_top(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),X),top_top(A)) = X ) ).

% inf_top_right
tff(fact_132_inf__eq__top__iff,axiom,
    ! [A: $tType] :
      ( bounde4346867609351753570nf_top(A)
     => ! [X: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y) = top_top(A) )
        <=> ( ( X = top_top(A) )
            & ( Y = top_top(A) ) ) ) ) ).

% inf_eq_top_iff
tff(fact_133_top__eq__inf__iff,axiom,
    ! [A: $tType] :
      ( bounde4346867609351753570nf_top(A)
     => ! [X: A,Y: A] :
          ( ( top_top(A) = aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y) )
        <=> ( ( X = top_top(A) )
            & ( Y = top_top(A) ) ) ) ) ).

% top_eq_inf_iff
tff(fact_134_inf__top_Oeq__neutr__iff,axiom,
    ! [A: $tType] :
      ( bounde4346867609351753570nf_top(A)
     => ! [A3: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2) = top_top(A) )
        <=> ( ( A3 = top_top(A) )
            & ( B2 = top_top(A) ) ) ) ) ).

% inf_top.eq_neutr_iff
tff(fact_135_inf__top_Oleft__neutral,axiom,
    ! [A: $tType] :
      ( bounde4346867609351753570nf_top(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),top_top(A)),A3) = A3 ) ).

% inf_top.left_neutral
tff(fact_136_inf__top_Oneutr__eq__iff,axiom,
    ! [A: $tType] :
      ( bounde4346867609351753570nf_top(A)
     => ! [A3: A,B2: A] :
          ( ( top_top(A) = aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2) )
        <=> ( ( A3 = top_top(A) )
            & ( B2 = top_top(A) ) ) ) ) ).

% inf_top.neutr_eq_iff
tff(fact_137_inf__top_Oright__neutral,axiom,
    ! [A: $tType] :
      ( bounde4346867609351753570nf_top(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),top_top(A)) = A3 ) ).

% inf_top.right_neutral
tff(fact_138_boolean__algebra_Odisj__one__left,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),top_top(A)),X) = top_top(A) ) ).

% boolean_algebra.disj_one_left
tff(fact_139_boolean__algebra_Odisj__one__right,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),top_top(A)) = top_top(A) ) ).

% boolean_algebra.disj_one_right
tff(fact_140_sup__top__left,axiom,
    ! [A: $tType] :
      ( bounded_lattice_top(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),top_top(A)),X) = top_top(A) ) ).

% sup_top_left
tff(fact_141_sup__top__right,axiom,
    ! [A: $tType] :
      ( bounded_lattice_top(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),top_top(A)) = top_top(A) ) ).

% sup_top_right
tff(fact_142_boolean__algebra_Ocompl__one,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ( aa(A,A,uminus_uminus(A),top_top(A)) = bot_bot(A) ) ) ).

% boolean_algebra.compl_one
tff(fact_143_boolean__algebra_Ocompl__zero,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ( aa(A,A,uminus_uminus(A),bot_bot(A)) = top_top(A) ) ) ).

% boolean_algebra.compl_zero
tff(fact_144_inf__compl__bot__left1,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,uminus_uminus(A),X)),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)) = bot_bot(A) ) ).

% inf_compl_bot_left1
tff(fact_145_inf__compl__bot__left2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,uminus_uminus(A),X)),Y)) = bot_bot(A) ) ).

% inf_compl_bot_left2
tff(fact_146_inf__compl__bot__right,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),aa(A,A,uminus_uminus(A),X))) = bot_bot(A) ) ).

% inf_compl_bot_right
tff(fact_147_boolean__algebra_Oconj__cancel__left,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,uminus_uminus(A),X)),X) = bot_bot(A) ) ).

% boolean_algebra.conj_cancel_left
tff(fact_148_boolean__algebra_Oconj__cancel__right,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(A,A,uminus_uminus(A),X)) = bot_bot(A) ) ).

% boolean_algebra.conj_cancel_right
tff(fact_149_sup__compl__top__left1,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,uminus_uminus(A),X)),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y)) = top_top(A) ) ).

% sup_compl_top_left1
tff(fact_150_sup__compl__top__left2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,uminus_uminus(A),X)),Y)) = top_top(A) ) ).

% sup_compl_top_left2
tff(fact_151_boolean__algebra_Odisj__cancel__left,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,uminus_uminus(A),X)),X) = top_top(A) ) ).

% boolean_algebra.disj_cancel_left
tff(fact_152_boolean__algebra_Odisj__cancel__right,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(A,A,uminus_uminus(A),X)) = top_top(A) ) ).

% boolean_algebra.disj_cancel_right
tff(fact_153_boolean__algebra_Ode__Morgan__conj,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,uminus_uminus(A),X)),aa(A,A,uminus_uminus(A),Y)) ) ).

% boolean_algebra.de_Morgan_conj
tff(fact_154_mult_Omonoid__axioms,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => monoid(A,times_times(A),one_one(A)) ) ).

% mult.monoid_axioms
tff(fact_155_mult_Ocomm__monoid__axioms,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => comm_monoid(A,times_times(A),one_one(A)) ) ).

% mult.comm_monoid_axioms
tff(fact_156_fun__Compl__def,axiom,
    ! [B: $tType,A: $tType] :
      ( uminus(B)
     => ! [A4: fun(A,B),X3: A] : aa(A,B,aa(fun(A,B),fun(A,B),uminus_uminus(fun(A,B)),A4),X3) = aa(B,B,uminus_uminus(B),aa(A,B,A4,X3)) ) ).

% fun_Compl_def
tff(fact_157_semilattice__neutr_Oaxioms_I2_J,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
      ( semilattice_neutr(A,F,Z2)
     => comm_monoid(A,F,Z2) ) ).

% semilattice_neutr.axioms(2)
tff(fact_158_monoid_Oleft__neutral,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z2: A,A3: A] :
      ( monoid(A,F,Z2)
     => ( aa(A,A,aa(A,fun(A,A),F,Z2),A3) = A3 ) ) ).

% monoid.left_neutral
tff(fact_159_monoid_Oright__neutral,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z2: A,A3: A] :
      ( monoid(A,F,Z2)
     => ( aa(A,A,aa(A,fun(A,A),F,A3),Z2) = A3 ) ) ).

% monoid.right_neutral
tff(fact_160_one__assn__raw_Ocases,axiom,
    ! [X: product_prod(heap_ext(product_unit),set(nat))] :
      ~ ! [H: heap_ext(product_unit),As4: set(nat)] : X != aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As4) ).

% one_assn_raw.cases
tff(fact_161_comm__monoid_Ocomm__neutral,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z2: A,A3: A] :
      ( comm_monoid(A,F,Z2)
     => ( aa(A,A,aa(A,fun(A,A),F,A3),Z2) = A3 ) ) ).

% comm_monoid.comm_neutral
tff(fact_162_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_163_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_164_one__reorient,axiom,
    ! [A: $tType] :
      ( one(A)
     => ! [X: A] :
          ( ( one_one(A) = X )
        <=> ( X = one_one(A) ) ) ) ).

% one_reorient
tff(fact_165_sup__cancel__left1,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),A3)),aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,uminus_uminus(A),X)),B2)) = top_top(A) ) ).

% sup_cancel_left1
tff(fact_166_sup__cancel__left2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,uminus_uminus(A),X)),A3)),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),B2)) = top_top(A) ) ).

% sup_cancel_left2
tff(fact_167_inf__top_Ocomm__monoid__axioms,axiom,
    ! [A: $tType] :
      ( bounde4346867609351753570nf_top(A)
     => comm_monoid(A,inf_inf(A),top_top(A)) ) ).

% inf_top.comm_monoid_axioms
tff(fact_168_inf__top_Omonoid__axioms,axiom,
    ! [A: $tType] :
      ( bounde4346867609351753570nf_top(A)
     => monoid(A,inf_inf(A),top_top(A)) ) ).

% inf_top.monoid_axioms
tff(fact_169_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_170_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_171_inf__top_Osemilattice__neutr__axioms,axiom,
    ! [A: $tType] :
      ( bounde4346867609351753570nf_top(A)
     => semilattice_neutr(A,inf_inf(A),top_top(A)) ) ).

% inf_top.semilattice_neutr_axioms
tff(fact_172_syntax__fo__nomatch__def,axiom,
    ! [A: $tType,B: $tType,Pat: A,Obj: B] : syntax7388354845996824322omatch(A,B,Pat,Obj) ).

% syntax_fo_nomatch_def
tff(fact_173_boolean__algebra_Oconj__one__right,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),X),top_top(A)) = X ) ).

% boolean_algebra.conj_one_right
tff(fact_174_mod__h__bot__indep,axiom,
    ! [P: assn,H2: heap_ext(product_unit),H3: heap_ext(product_unit)] :
      ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,P),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),bot_bot(set(nat))))
    <=> aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,P),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H3),bot_bot(set(nat)))) ) ).

% mod_h_bot_indep
tff(fact_175_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y) = bot_bot(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y) = top_top(A) )
           => ( aa(A,A,uminus_uminus(A),X) = Y ) ) ) ) ).

% boolean_algebra_class.boolean_algebra.compl_unique
tff(fact_176_one__assn__raw_Oelims_I3_J,axiom,
    ! [X: product_prod(heap_ext(product_unit),set(nat))] :
      ( ~ aa(product_prod(heap_ext(product_unit),set(nat)),$o,one_assn_raw,X)
     => ~ ! [H: heap_ext(product_unit),As4: set(nat)] :
            ( ( X = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As4) )
           => ( As4 = bot_bot(set(nat)) ) ) ) ).

% one_assn_raw.elims(3)
tff(fact_177_one__assn__raw_Oelims_I2_J,axiom,
    ! [X: product_prod(heap_ext(product_unit),set(nat))] :
      ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,one_assn_raw,X)
     => ~ ! [H: heap_ext(product_unit),As4: set(nat)] :
            ( ( X = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As4) )
           => ( As4 != bot_bot(set(nat)) ) ) ) ).

% one_assn_raw.elims(2)
tff(fact_178_one__assn__raw_Oelims_I1_J,axiom,
    ! [X: product_prod(heap_ext(product_unit),set(nat)),Y: $o] :
      ( ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,one_assn_raw,X)
      <=> (Y) )
     => ~ ! [H: heap_ext(product_unit),As4: set(nat)] :
            ( ( X = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As4) )
           => ( (Y)
            <=> ( As4 != bot_bot(set(nat)) ) ) ) ) ).

% one_assn_raw.elims(1)
tff(fact_179_one__assn__raw_Osimps,axiom,
    ! [H2: heap_ext(product_unit),As: set(nat)] :
      ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,one_assn_raw,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As))
    <=> ( As = bot_bot(set(nat)) ) ) ).

% one_assn_raw.simps
tff(fact_180_times__assn__raw_Osimps,axiom,
    ! [P: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Q: fun(product_prod(heap_ext(product_unit),set(nat)),$o),H2: heap_ext(product_unit),As: set(nat)] :
      ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,times_assn_raw(P,Q),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As))
    <=> ? [As1: set(nat),As2: set(nat)] :
          ( ( As = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As1),As2) )
          & ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As1),As2) = bot_bot(set(nat)) )
          & aa(product_prod(heap_ext(product_unit),set(nat)),$o,P,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As1))
          & aa(product_prod(heap_ext(product_unit),set(nat)),$o,Q,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As2)) ) ) ).

% times_assn_raw.simps
tff(fact_181_times__assn__raw_Oelims_I1_J,axiom,
    ! [X: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xa: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xb: product_prod(heap_ext(product_unit),set(nat)),Y: $o] :
      ( ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,times_assn_raw(X,Xa),Xb)
      <=> (Y) )
     => ~ ! [H: heap_ext(product_unit),As4: set(nat)] :
            ( ( Xb = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As4) )
           => ( (Y)
            <=> ~ ? [As1: set(nat),As2: set(nat)] :
                    ( ( As4 = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As1),As2) )
                    & ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As1),As2) = bot_bot(set(nat)) )
                    & aa(product_prod(heap_ext(product_unit),set(nat)),$o,X,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As1))
                    & aa(product_prod(heap_ext(product_unit),set(nat)),$o,Xa,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2)) ) ) ) ) ).

% times_assn_raw.elims(1)
tff(fact_182_times__assn__raw_Oelims_I2_J,axiom,
    ! [X: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xa: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xb: product_prod(heap_ext(product_unit),set(nat))] :
      ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,times_assn_raw(X,Xa),Xb)
     => ~ ! [H: heap_ext(product_unit),As4: set(nat)] :
            ( ( Xb = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As4) )
           => ~ ? [As12: set(nat),As22: set(nat)] :
                  ( ( As4 = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As12),As22) )
                  & ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As12),As22) = bot_bot(set(nat)) )
                  & aa(product_prod(heap_ext(product_unit),set(nat)),$o,X,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As12))
                  & aa(product_prod(heap_ext(product_unit),set(nat)),$o,Xa,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As22)) ) ) ) ).

% times_assn_raw.elims(2)
tff(fact_183_times__assn__raw_Oelims_I3_J,axiom,
    ! [X: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xa: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xb: product_prod(heap_ext(product_unit),set(nat))] :
      ( ~ aa(product_prod(heap_ext(product_unit),set(nat)),$o,times_assn_raw(X,Xa),Xb)
     => ~ ! [H: heap_ext(product_unit),As4: set(nat)] :
            ( ( Xb = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As4) )
           => ? [As13: set(nat),As23: set(nat)] :
                ( ( As4 = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As13),As23) )
                & ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As13),As23) = bot_bot(set(nat)) )
                & aa(product_prod(heap_ext(product_unit),set(nat)),$o,X,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As13))
                & aa(product_prod(heap_ext(product_unit),set(nat)),$o,Xa,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As23)) ) ) ) ).

% times_assn_raw.elims(3)
tff(fact_184_assn__one__left,axiom,
    ! [P: assn] : aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),one_one(assn)),P) = P ).

% assn_one_left
tff(fact_185_inf__cancel__left1,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),A3)),aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,uminus_uminus(A),X)),B2)) = bot_bot(A) ) ).

% inf_cancel_left1
tff(fact_186_inf__cancel__left2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,uminus_uminus(A),X)),A3)),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),B2)) = bot_bot(A) ) ).

% inf_cancel_left2
tff(fact_187_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_188_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_189_Un__Int__eq_I1_J,axiom,
    ! [A: $tType,S: set(A),T2: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),S),T2)),S) = S ).

% Un_Int_eq(1)
tff(fact_190_Un__Int__eq_I2_J,axiom,
    ! [A: $tType,S: set(A),T2: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),S),T2)),T2) = T2 ).

% Un_Int_eq(2)
tff(fact_191_Un__Int__eq_I3_J,axiom,
    ! [A: $tType,S: set(A),T2: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),S),T2)) = S ).

% Un_Int_eq(3)
tff(fact_192_Un__Int__eq_I4_J,axiom,
    ! [A: $tType,T2: set(A),S: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),T2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),S),T2)) = T2 ).

% Un_Int_eq(4)
tff(fact_193_Int__Un__eq_I1_J,axiom,
    ! [A: $tType,S: set(A),T2: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S),T2)),S) = S ).

% Int_Un_eq(1)
tff(fact_194_Int__Un__eq_I2_J,axiom,
    ! [A: $tType,S: set(A),T2: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S),T2)),T2) = T2 ).

% Int_Un_eq(2)
tff(fact_195_Int__Un__eq_I3_J,axiom,
    ! [A: $tType,S: set(A),T2: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),S),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S),T2)) = S ).

% Int_Un_eq(3)
tff(fact_196_Int__Un__eq_I4_J,axiom,
    ! [A: $tType,T2: set(A),S: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),T2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S),T2)) = T2 ).

% Int_Un_eq(4)
tff(fact_197_Un__empty,axiom,
    ! [A: $tType,A4: set(A),B3: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3) = bot_bot(set(A)) )
    <=> ( ( A4 = bot_bot(set(A)) )
        & ( B3 = bot_bot(set(A)) ) ) ) ).

% Un_empty
tff(fact_198_Compl__disjoint2,axiom,
    ! [A: $tType,A4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),A4)),A4) = bot_bot(set(A)) ).

% Compl_disjoint2
tff(fact_199_empty__iff,axiom,
    ! [A: $tType,C2: A] : ~ aa(set(A),$o,member(A,C2),bot_bot(set(A))) ).

% empty_iff
tff(fact_200_all__not__in__conv,axiom,
    ! [A: $tType,A4: set(A)] :
      ( ! [X4: A] : ~ aa(set(A),$o,member(A,X4),A4)
    <=> ( A4 = bot_bot(set(A)) ) ) ).

% all_not_in_conv
tff(fact_201_Collect__empty__eq,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( ( aa(fun(A,$o),set(A),collect(A),P) = bot_bot(set(A)) )
    <=> ! [X4: A] : ~ aa(A,$o,P,X4) ) ).

% Collect_empty_eq
tff(fact_202_empty__Collect__eq,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( ( bot_bot(set(A)) = aa(fun(A,$o),set(A),collect(A),P) )
    <=> ! [X4: A] : ~ aa(A,$o,P,X4) ) ).

% empty_Collect_eq
tff(fact_203_IntI,axiom,
    ! [A: $tType,C2: A,A4: set(A),B3: set(A)] :
      ( aa(set(A),$o,member(A,C2),A4)
     => ( aa(set(A),$o,member(A,C2),B3)
       => aa(set(A),$o,member(A,C2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3)) ) ) ).

% IntI
tff(fact_204_Int__iff,axiom,
    ! [A: $tType,C2: A,A4: set(A),B3: set(A)] :
      ( aa(set(A),$o,member(A,C2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3))
    <=> ( aa(set(A),$o,member(A,C2),A4)
        & aa(set(A),$o,member(A,C2),B3) ) ) ).

% Int_iff
tff(fact_205_Int__UNIV,axiom,
    ! [A: $tType,A4: set(A),B3: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3) = top_top(set(A)) )
    <=> ( ( A4 = top_top(set(A)) )
        & ( B3 = top_top(set(A)) ) ) ) ).

% Int_UNIV
tff(fact_206_UnCI,axiom,
    ! [A: $tType,C2: A,B3: set(A),A4: set(A)] :
      ( ( ~ aa(set(A),$o,member(A,C2),B3)
       => aa(set(A),$o,member(A,C2),A4) )
     => aa(set(A),$o,member(A,C2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3)) ) ).

% UnCI
tff(fact_207_Un__iff,axiom,
    ! [A: $tType,C2: A,A4: set(A),B3: set(A)] :
      ( aa(set(A),$o,member(A,C2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3))
    <=> ( aa(set(A),$o,member(A,C2),A4)
        | aa(set(A),$o,member(A,C2),B3) ) ) ).

% Un_iff
tff(fact_208_ComplI,axiom,
    ! [A: $tType,C2: A,A4: set(A)] :
      ( ~ aa(set(A),$o,member(A,C2),A4)
     => aa(set(A),$o,member(A,C2),aa(set(A),set(A),uminus_uminus(set(A)),A4)) ) ).

% ComplI
tff(fact_209_Compl__iff,axiom,
    ! [A: $tType,C2: A,A4: set(A)] :
      ( aa(set(A),$o,member(A,C2),aa(set(A),set(A),uminus_uminus(set(A)),A4))
    <=> ~ aa(set(A),$o,member(A,C2),A4) ) ).

% Compl_iff
tff(fact_210_Compl__eq__Compl__iff,axiom,
    ! [A: $tType,A4: set(A),B3: set(A)] :
      ( ( aa(set(A),set(A),uminus_uminus(set(A)),A4) = aa(set(A),set(A),uminus_uminus(set(A)),B3) )
    <=> ( A4 = B3 ) ) ).

% Compl_eq_Compl_iff
tff(fact_211_Collect__const,axiom,
    ! [A: $tType,P: $o] :
      aa(fun(A,$o),set(A),collect(A),aTP_Lamp_ah($o,fun(A,$o),(P))) = $ite((P),top_top(set(A)),bot_bot(set(A))) ).

% Collect_const
tff(fact_212_Compl__disjoint,axiom,
    ! [A: $tType,A4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),aa(set(A),set(A),uminus_uminus(set(A)),A4)) = bot_bot(set(A)) ).

% Compl_disjoint
tff(fact_213_ComplD,axiom,
    ! [A: $tType,C2: A,A4: set(A)] :
      ( aa(set(A),$o,member(A,C2),aa(set(A),set(A),uminus_uminus(set(A)),A4))
     => ~ aa(set(A),$o,member(A,C2),A4) ) ).

% ComplD
tff(fact_214_Compl__eq,axiom,
    ! [A: $tType,A4: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),A4) = aa(fun(A,$o),set(A),collect(A),aTP_Lamp_ai(set(A),fun(A,$o),A4)) ).

% Compl_eq
tff(fact_215_Collect__neg__eq,axiom,
    ! [A: $tType,P: fun(A,$o)] : aa(fun(A,$o),set(A),collect(A),aTP_Lamp_aj(fun(A,$o),fun(A,$o),P)) = aa(set(A),set(A),uminus_uminus(set(A)),aa(fun(A,$o),set(A),collect(A),P)) ).

% Collect_neg_eq
tff(fact_216_uminus__set__def,axiom,
    ! [A: $tType,A4: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),A4) = aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),uminus_uminus(fun(A,$o)),aa(set(A),fun(A,$o),aTP_Lamp_ag(set(A),fun(A,$o)),A4))) ).

% uminus_set_def
tff(fact_217_double__complement,axiom,
    ! [A: $tType,A4: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),A4)) = A4 ).

% double_complement
tff(fact_218_emptyE,axiom,
    ! [A: $tType,A3: A] : ~ aa(set(A),$o,member(A,A3),bot_bot(set(A))) ).

% emptyE
tff(fact_219_equals0D,axiom,
    ! [A: $tType,A4: set(A),A3: A] :
      ( ( A4 = bot_bot(set(A)) )
     => ~ aa(set(A),$o,member(A,A3),A4) ) ).

% equals0D
tff(fact_220_equals0I,axiom,
    ! [A: $tType,A4: set(A)] :
      ( ! [Y2: A] : ~ aa(set(A),$o,member(A,Y2),A4)
     => ( A4 = bot_bot(set(A)) ) ) ).

% equals0I
tff(fact_221_ex__in__conv,axiom,
    ! [A: $tType,A4: set(A)] :
      ( ? [X4: A] : aa(set(A),$o,member(A,X4),A4)
    <=> ( A4 != bot_bot(set(A)) ) ) ).

% ex_in_conv
tff(fact_222_bot__set__def,axiom,
    ! [A: $tType] : bot_bot(set(A)) = aa(fun(A,$o),set(A),collect(A),bot_bot(fun(A,$o))) ).

% bot_set_def
tff(fact_223_Compl__UNIV__eq,axiom,
    ! [A: $tType] : aa(set(A),set(A),uminus_uminus(set(A)),top_top(set(A))) = bot_bot(set(A)) ).

% Compl_UNIV_eq
tff(fact_224_Compl__empty__eq,axiom,
    ! [A: $tType] : aa(set(A),set(A),uminus_uminus(set(A)),bot_bot(set(A))) = top_top(set(A)) ).

% Compl_empty_eq
tff(fact_225_empty__not__UNIV,axiom,
    ! [A: $tType] : bot_bot(set(A)) != top_top(set(A)) ).

% empty_not_UNIV
tff(fact_226_IntE,axiom,
    ! [A: $tType,C2: A,A4: set(A),B3: set(A)] :
      ( aa(set(A),$o,member(A,C2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3))
     => ~ ( aa(set(A),$o,member(A,C2),A4)
         => ~ aa(set(A),$o,member(A,C2),B3) ) ) ).

% IntE
tff(fact_227_IntD1,axiom,
    ! [A: $tType,C2: A,A4: set(A),B3: set(A)] :
      ( aa(set(A),$o,member(A,C2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3))
     => aa(set(A),$o,member(A,C2),A4) ) ).

% IntD1
tff(fact_228_IntD2,axiom,
    ! [A: $tType,C2: A,A4: set(A),B3: set(A)] :
      ( aa(set(A),$o,member(A,C2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3))
     => aa(set(A),$o,member(A,C2),B3) ) ).

% IntD2
tff(fact_229_Int__assoc,axiom,
    ! [A: $tType,A4: set(A),B3: set(A),C3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3)),C3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),C3)) ).

% Int_assoc
tff(fact_230_Int__absorb,axiom,
    ! [A: $tType,A4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),A4) = A4 ).

% Int_absorb
tff(fact_231_Int__commute,axiom,
    ! [A: $tType,A4: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),A4) ).

% Int_commute
tff(fact_232_Int__UNIV__left,axiom,
    ! [A: $tType,B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),top_top(set(A))),B3) = B3 ).

% Int_UNIV_left
tff(fact_233_Int__UNIV__right,axiom,
    ! [A: $tType,A4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),top_top(set(A))) = A4 ).

% Int_UNIV_right
tff(fact_234_Int__left__absorb,axiom,
    ! [A: $tType,A4: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3) ).

% Int_left_absorb
tff(fact_235_Int__left__commute,axiom,
    ! [A: $tType,A4: set(A),B3: set(A),C3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),C3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),C3)) ).

% Int_left_commute
tff(fact_236_UnE,axiom,
    ! [A: $tType,C2: A,A4: set(A),B3: set(A)] :
      ( aa(set(A),$o,member(A,C2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3))
     => ( ~ aa(set(A),$o,member(A,C2),A4)
       => aa(set(A),$o,member(A,C2),B3) ) ) ).

% UnE
tff(fact_237_UnI1,axiom,
    ! [A: $tType,C2: A,A4: set(A),B3: set(A)] :
      ( aa(set(A),$o,member(A,C2),A4)
     => aa(set(A),$o,member(A,C2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3)) ) ).

% UnI1
tff(fact_238_UnI2,axiom,
    ! [A: $tType,C2: A,B3: set(A),A4: set(A)] :
      ( aa(set(A),$o,member(A,C2),B3)
     => aa(set(A),$o,member(A,C2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3)) ) ).

% UnI2
tff(fact_239_bex__Un,axiom,
    ! [A: $tType,A4: set(A),B3: set(A),P: fun(A,$o)] :
      ( ? [X4: A] :
          ( aa(set(A),$o,member(A,X4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3))
          & aa(A,$o,P,X4) )
    <=> ( ? [X4: A] :
            ( aa(set(A),$o,member(A,X4),A4)
            & aa(A,$o,P,X4) )
        | ? [X4: A] :
            ( aa(set(A),$o,member(A,X4),B3)
            & aa(A,$o,P,X4) ) ) ) ).

% bex_Un
tff(fact_240_ball__Un,axiom,
    ! [A: $tType,A4: set(A),B3: set(A),P: fun(A,$o)] :
      ( ! [X4: A] :
          ( aa(set(A),$o,member(A,X4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3))
         => aa(A,$o,P,X4) )
    <=> ( ! [X4: A] :
            ( aa(set(A),$o,member(A,X4),A4)
           => aa(A,$o,P,X4) )
        & ! [X4: A] :
            ( aa(set(A),$o,member(A,X4),B3)
           => aa(A,$o,P,X4) ) ) ) ).

% ball_Un
tff(fact_241_Un__assoc,axiom,
    ! [A: $tType,A4: set(A),B3: set(A),C3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3)),C3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B3),C3)) ).

% Un_assoc
tff(fact_242_Un__absorb,axiom,
    ! [A: $tType,A4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),A4) = A4 ).

% Un_absorb
tff(fact_243_Un__commute,axiom,
    ! [A: $tType,A4: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B3),A4) ).

% Un_commute
tff(fact_244_Un__UNIV__left,axiom,
    ! [A: $tType,B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),top_top(set(A))),B3) = top_top(set(A)) ).

% Un_UNIV_left
tff(fact_245_Un__UNIV__right,axiom,
    ! [A: $tType,A4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),top_top(set(A))) = top_top(set(A)) ).

% Un_UNIV_right
tff(fact_246_Un__left__absorb,axiom,
    ! [A: $tType,A4: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3) ).

% Un_left_absorb
tff(fact_247_Compl__partition,axiom,
    ! [A: $tType,A4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),aa(set(A),set(A),uminus_uminus(set(A)),A4)) = top_top(set(A)) ).

% Compl_partition
tff(fact_248_Un__left__commute,axiom,
    ! [A: $tType,A4: set(A),B3: set(A),C3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B3),C3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),C3)) ).

% Un_left_commute
tff(fact_249_Compl__partition2,axiom,
    ! [A: $tType,A4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),A4)),A4) = top_top(set(A)) ).

% Compl_partition2
tff(fact_250_Set_Oempty__def,axiom,
    ! [A: $tType] : bot_bot(set(A)) = aa(fun(A,$o),set(A),collect(A),aTP_Lamp_ak(A,$o)) ).

% Set.empty_def
tff(fact_251_Int__def,axiom,
    ! [A: $tType,A4: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3) = aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_al(set(A),fun(set(A),fun(A,$o)),A4),B3)) ).

% Int_def
tff(fact_252_Int__Collect,axiom,
    ! [A: $tType,X: A,A4: set(A),P: fun(A,$o)] :
      ( aa(set(A),$o,member(A,X),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),aa(fun(A,$o),set(A),collect(A),P)))
    <=> ( aa(set(A),$o,member(A,X),A4)
        & aa(A,$o,P,X) ) ) ).

% Int_Collect
tff(fact_253_inf__set__def,axiom,
    ! [A: $tType,A4: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3) = aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aa(fun(A,$o),fun(fun(A,$o),fun(A,$o)),inf_inf(fun(A,$o)),aa(set(A),fun(A,$o),aTP_Lamp_ag(set(A),fun(A,$o)),A4)),aa(set(A),fun(A,$o),aTP_Lamp_ag(set(A),fun(A,$o)),B3))) ).

% inf_set_def
tff(fact_254_Collect__conj__eq,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] : aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_am(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(fun(A,$o),set(A),collect(A),P)),aa(fun(A,$o),set(A),collect(A),Q)) ).

% Collect_conj_eq
tff(fact_255_Un__def,axiom,
    ! [A: $tType,A4: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3) = aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_an(set(A),fun(set(A),fun(A,$o)),A4),B3)) ).

% Un_def
tff(fact_256_sup__set__def,axiom,
    ! [A: $tType,A4: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3) = aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aa(fun(A,$o),fun(fun(A,$o),fun(A,$o)),sup_sup(fun(A,$o)),aa(set(A),fun(A,$o),aTP_Lamp_ag(set(A),fun(A,$o)),A4)),aa(set(A),fun(A,$o),aTP_Lamp_ag(set(A),fun(A,$o)),B3))) ).

% sup_set_def
tff(fact_257_Collect__imp__eq,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] : aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_ao(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),aa(fun(A,$o),set(A),collect(A),P))),aa(fun(A,$o),set(A),collect(A),Q)) ).

% Collect_imp_eq
tff(fact_258_Collect__disj__eq,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] : aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_ap(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(fun(A,$o),set(A),collect(A),P)),aa(fun(A,$o),set(A),collect(A),Q)) ).

% Collect_disj_eq
tff(fact_259_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_260_Int__emptyI,axiom,
    ! [A: $tType,A4: set(A),B3: set(A)] :
      ( ! [X2: A] :
          ( aa(set(A),$o,member(A,X2),A4)
         => ~ aa(set(A),$o,member(A,X2),B3) )
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3) = bot_bot(set(A)) ) ) ).

% Int_emptyI
tff(fact_261_disjoint__iff,axiom,
    ! [A: $tType,A4: set(A),B3: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3) = bot_bot(set(A)) )
    <=> ! [X4: A] :
          ( aa(set(A),$o,member(A,X4),A4)
         => ~ aa(set(A),$o,member(A,X4),B3) ) ) ).

% disjoint_iff
tff(fact_262_Int__empty__left,axiom,
    ! [A: $tType,B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),bot_bot(set(A))),B3) = bot_bot(set(A)) ).

% Int_empty_left
tff(fact_263_Int__empty__right,axiom,
    ! [A: $tType,A4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),bot_bot(set(A))) = bot_bot(set(A)) ).

% Int_empty_right
tff(fact_264_disjoint__iff__not__equal,axiom,
    ! [A: $tType,A4: set(A),B3: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3) = bot_bot(set(A)) )
    <=> ! [X4: A] :
          ( aa(set(A),$o,member(A,X4),A4)
         => ! [Xa2: A] :
              ( aa(set(A),$o,member(A,Xa2),B3)
             => ( X4 != Xa2 ) ) ) ) ).

% disjoint_iff_not_equal
tff(fact_265_Un__empty__left,axiom,
    ! [A: $tType,B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),bot_bot(set(A))),B3) = B3 ).

% Un_empty_left
tff(fact_266_Un__empty__right,axiom,
    ! [A: $tType,A4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),bot_bot(set(A))) = A4 ).

% Un_empty_right
tff(fact_267_Compl__Un,axiom,
    ! [A: $tType,A4: set(A),B3: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),A4)),aa(set(A),set(A),uminus_uminus(set(A)),B3)) ).

% Compl_Un
tff(fact_268_Compl__Int,axiom,
    ! [A: $tType,A4: set(A),B3: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),A4)),aa(set(A),set(A),uminus_uminus(set(A)),B3)) ).

% Compl_Int
tff(fact_269_Un__Int__crazy,axiom,
    ! [A: $tType,A4: set(A),B3: set(A),C3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),C3))),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),C3),A4)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B3),C3))),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),C3),A4)) ).

% Un_Int_crazy
tff(fact_270_Int__Un__distrib,axiom,
    ! [A: $tType,A4: set(A),B3: set(A),C3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B3),C3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),C3)) ).

% Int_Un_distrib
tff(fact_271_Un__Int__distrib,axiom,
    ! [A: $tType,A4: set(A),B3: set(A),C3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),C3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),C3)) ).

% Un_Int_distrib
tff(fact_272_Int__Un__distrib2,axiom,
    ! [A: $tType,B3: set(A),C3: set(A),A4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B3),C3)),A4) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),A4)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),C3),A4)) ).

% Int_Un_distrib2
tff(fact_273_Un__Int__distrib2,axiom,
    ! [A: $tType,B3: set(A),C3: set(A),A4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),C3)),A4) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B3),A4)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),C3),A4)) ).

% Un_Int_distrib2
tff(fact_274_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_275_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_276_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_277_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_278_verit__minus__simplify_I4_J,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [B2: A] : aa(A,A,uminus_uminus(A),aa(A,A,uminus_uminus(A),B2)) = B2 ) ).

% verit_minus_simplify(4)
tff(fact_279_old_Oprod_Oinject,axiom,
    ! [A: $tType,B: $tType,A3: A,B2: B,A5: A,B4: 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),A5),B4) )
    <=> ( ( A3 = A5 )
        & ( B2 = B4 ) ) ) ).

% old.prod.inject
tff(fact_280_prod_Oinject,axiom,
    ! [A: $tType,B: $tType,X1: A,X22: B,Y1: A,Y22: B] :
      ( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X1),X22) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Y1),Y22) )
    <=> ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
tff(fact_281_lambda__one,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ( aTP_Lamp_aq(A,A) = aa(A,fun(A,A),times_times(A),one_one(A)) ) ) ).

% lambda_one
tff(fact_282_disjointI,axiom,
    ! [A: $tType,A3: set(A),B2: set(A)] :
      ( ! [X2: A] :
          ( aa(set(A),$o,member(A,X2),A3)
         => ~ aa(set(A),$o,member(A,X2),B2) )
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B2) = bot_bot(set(A)) ) ) ).

% disjointI
tff(fact_283_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_284_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_285_UNIV__I,axiom,
    ! [A: $tType,X: A] : aa(set(A),$o,member(A,X),top_top(set(A))) ).

% UNIV_I
tff(fact_286_inf1I,axiom,
    ! [A: $tType,A4: fun(A,$o),X: A,B3: fun(A,$o)] :
      ( aa(A,$o,A4,X)
     => ( aa(A,$o,B3,X)
       => aa(A,$o,aa(fun(A,$o),fun(A,$o),aa(fun(A,$o),fun(fun(A,$o),fun(A,$o)),inf_inf(fun(A,$o)),A4),B3),X) ) ) ).

% inf1I
tff(fact_287_sup1CI,axiom,
    ! [A: $tType,B3: fun(A,$o),X: A,A4: fun(A,$o)] :
      ( ( ~ aa(A,$o,B3,X)
       => aa(A,$o,A4,X) )
     => aa(A,$o,aa(fun(A,$o),fun(A,$o),aa(fun(A,$o),fun(fun(A,$o),fun(A,$o)),sup_sup(fun(A,$o)),A4),B3),X) ) ).

% sup1CI
tff(fact_288_UNIV__def,axiom,
    ! [A: $tType] : top_top(set(A)) = aa(fun(A,$o),set(A),collect(A),aTP_Lamp_ar(A,$o)) ).

% UNIV_def
tff(fact_289_UNIV__eq__I,axiom,
    ! [A: $tType,A4: set(A)] :
      ( ! [X2: A] : aa(set(A),$o,member(A,X2),A4)
     => ( top_top(set(A)) = A4 ) ) ).

% UNIV_eq_I
tff(fact_290_top__set__def,axiom,
    ! [A: $tType] : top_top(set(A)) = aa(fun(A,$o),set(A),collect(A),top_top(fun(A,$o))) ).

% top_set_def
tff(fact_291_UNIV__witness,axiom,
    ! [A: $tType] :
    ? [X2: A] : aa(set(A),$o,member(A,X2),top_top(set(A))) ).

% UNIV_witness
tff(fact_292_sup1I2,axiom,
    ! [A: $tType,B3: fun(A,$o),X: A,A4: fun(A,$o)] :
      ( aa(A,$o,B3,X)
     => aa(A,$o,aa(fun(A,$o),fun(A,$o),aa(fun(A,$o),fun(fun(A,$o),fun(A,$o)),sup_sup(fun(A,$o)),A4),B3),X) ) ).

% sup1I2
tff(fact_293_sup1I1,axiom,
    ! [A: $tType,A4: fun(A,$o),X: A,B3: fun(A,$o)] :
      ( aa(A,$o,A4,X)
     => aa(A,$o,aa(fun(A,$o),fun(A,$o),aa(fun(A,$o),fun(fun(A,$o),fun(A,$o)),sup_sup(fun(A,$o)),A4),B3),X) ) ).

% sup1I1
tff(fact_294_inf1D2,axiom,
    ! [A: $tType,A4: fun(A,$o),B3: fun(A,$o),X: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aa(fun(A,$o),fun(fun(A,$o),fun(A,$o)),inf_inf(fun(A,$o)),A4),B3),X)
     => aa(A,$o,B3,X) ) ).

% inf1D2
tff(fact_295_inf1D1,axiom,
    ! [A: $tType,A4: fun(A,$o),B3: fun(A,$o),X: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aa(fun(A,$o),fun(fun(A,$o),fun(A,$o)),inf_inf(fun(A,$o)),A4),B3),X)
     => aa(A,$o,A4,X) ) ).

% inf1D1
tff(fact_296_sup1E,axiom,
    ! [A: $tType,A4: fun(A,$o),B3: fun(A,$o),X: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aa(fun(A,$o),fun(fun(A,$o),fun(A,$o)),sup_sup(fun(A,$o)),A4),B3),X)
     => ( ~ aa(A,$o,A4,X)
       => aa(A,$o,B3,X) ) ) ).

% sup1E
tff(fact_297_inf1E,axiom,
    ! [A: $tType,A4: fun(A,$o),B3: fun(A,$o),X: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aa(fun(A,$o),fun(fun(A,$o),fun(A,$o)),inf_inf(fun(A,$o)),A4),B3),X)
     => ~ ( aa(A,$o,A4,X)
         => ~ aa(A,$o,B3,X) ) ) ).

% inf1E
tff(fact_298_times__assn__raw_Ocases,axiom,
    ! [X: product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))))] :
      ~ ! [P2: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Q2: fun(product_prod(heap_ext(product_unit),set(nat)),$o),H: heap_ext(product_unit),As4: set(nat)] : X != aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),P2),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),Q2),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As4))) ).

% times_assn_raw.cases
tff(fact_299_pairself_Ocases,axiom,
    ! [B: $tType,A: $tType,X: product_prod(fun(A,B),product_prod(A,A))] :
      ~ ! [F2: fun(A,B),A6: A,B5: A] : X != aa(product_prod(A,A),product_prod(fun(A,B),product_prod(A,A)),aa(fun(A,B),fun(product_prod(A,A),product_prod(fun(A,B),product_prod(A,A))),product_Pair(fun(A,B),product_prod(A,A)),F2),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A6),B5)) ).

% pairself.cases
tff(fact_300_old_Oprod_Oexhaust,axiom,
    ! [A: $tType,B: $tType,Y: product_prod(A,B)] :
      ~ ! [A6: A,B5: B] : Y != aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A6),B5) ).

% old.prod.exhaust
tff(fact_301_bex2I,axiom,
    ! [A: $tType,B: $tType,A3: A,B2: B,S: set(product_prod(A,B)),P: fun(A,fun(B,$o))] :
      ( aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),S)
     => ( ( aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),S)
         => aa(B,$o,aa(A,fun(B,$o),P,A3),B2) )
       => ? [A6: A,B5: B] :
            ( aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A6),B5)),S)
            & aa(B,$o,aa(A,fun(B,$o),P,A6),B5) ) ) ) ).

% bex2I
tff(fact_302_surj__pair,axiom,
    ! [A: $tType,B: $tType,P3: product_prod(A,B)] :
    ? [X2: A,Y2: B] : P3 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Y2) ).

% surj_pair
tff(fact_303_prod__cases,axiom,
    ! [B: $tType,A: $tType,P: fun(product_prod(A,B),$o),P3: product_prod(A,B)] :
      ( ! [A6: A,B5: B] : aa(product_prod(A,B),$o,P,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A6),B5))
     => aa(product_prod(A,B),$o,P,P3) ) ).

% prod_cases
tff(fact_304_Pair__inject,axiom,
    ! [A: $tType,B: $tType,A3: A,B2: B,A5: A,B4: 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),A5),B4) )
     => ~ ( ( A3 = A5 )
         => ( B2 != B4 ) ) ) ).

% Pair_inject
tff(fact_305_prod__cases3,axiom,
    ! [A: $tType,B: $tType,C: $tType,Y: product_prod(A,product_prod(B,C))] :
      ~ ! [A6: A,B5: B,C4: C] : Y != aa(product_prod(B,C),product_prod(A,product_prod(B,C)),aa(A,fun(product_prod(B,C),product_prod(A,product_prod(B,C))),product_Pair(A,product_prod(B,C)),A6),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),B5),C4)) ).

% prod_cases3
tff(fact_306_prod__cases4,axiom,
    ! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod(A,product_prod(B,product_prod(C,D)))] :
      ~ ! [A6: A,B5: B,C4: 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))),A6),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)),B5),aa(D,product_prod(C,D),aa(C,fun(D,product_prod(C,D)),product_Pair(C,D),C4),D2))) ).

% prod_cases4
tff(fact_307_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))))] :
      ~ ! [A6: A,B5: B,C4: 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)))),A6),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))),B5),aa(product_prod(D,E),product_prod(C,product_prod(D,E)),aa(C,fun(product_prod(D,E),product_prod(C,product_prod(D,E))),product_Pair(C,product_prod(D,E)),C4),aa(E,product_prod(D,E),aa(D,fun(E,product_prod(D,E)),product_Pair(D,E),D2),E2)))) ).

% prod_cases5
tff(fact_308_prod__cases6,axiom,
    ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F3: $tType,Y: product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F3)))))] :
      ~ ! [A6: A,B5: B,C4: C,D2: D,E2: E,F2: F3] : Y != aa(product_prod(B,product_prod(C,product_prod(D,product_prod(E,F3)))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F3))))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,product_prod(E,F3)))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F3)))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F3))))),A6),aa(product_prod(C,product_prod(D,product_prod(E,F3))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,F3)))),aa(B,fun(product_prod(C,product_prod(D,product_prod(E,F3))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,F3))))),product_Pair(B,product_prod(C,product_prod(D,product_prod(E,F3)))),B5),aa(product_prod(D,product_prod(E,F3)),product_prod(C,product_prod(D,product_prod(E,F3))),aa(C,fun(product_prod(D,product_prod(E,F3)),product_prod(C,product_prod(D,product_prod(E,F3)))),product_Pair(C,product_prod(D,product_prod(E,F3))),C4),aa(product_prod(E,F3),product_prod(D,product_prod(E,F3)),aa(D,fun(product_prod(E,F3),product_prod(D,product_prod(E,F3))),product_Pair(D,product_prod(E,F3)),D2),aa(F3,product_prod(E,F3),aa(E,fun(F3,product_prod(E,F3)),product_Pair(E,F3),E2),F2))))) ).

% prod_cases6
tff(fact_309_prod__cases7,axiom,
    ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F3: $tType,G2: $tType,Y: product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F3,G2))))))] :
      ~ ! [A6: A,B5: B,C4: C,D2: D,E2: E,F2: F3,G3: G2] : Y != aa(product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F3,G2))))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F3,G2)))))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F3,G2))))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F3,G2))))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F3,G2)))))),A6),aa(product_prod(C,product_prod(D,product_prod(E,product_prod(F3,G2)))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F3,G2))))),aa(B,fun(product_prod(C,product_prod(D,product_prod(E,product_prod(F3,G2)))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F3,G2)))))),product_Pair(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F3,G2))))),B5),aa(product_prod(D,product_prod(E,product_prod(F3,G2))),product_prod(C,product_prod(D,product_prod(E,product_prod(F3,G2)))),aa(C,fun(product_prod(D,product_prod(E,product_prod(F3,G2))),product_prod(C,product_prod(D,product_prod(E,product_prod(F3,G2))))),product_Pair(C,product_prod(D,product_prod(E,product_prod(F3,G2)))),C4),aa(product_prod(E,product_prod(F3,G2)),product_prod(D,product_prod(E,product_prod(F3,G2))),aa(D,fun(product_prod(E,product_prod(F3,G2)),product_prod(D,product_prod(E,product_prod(F3,G2)))),product_Pair(D,product_prod(E,product_prod(F3,G2))),D2),aa(product_prod(F3,G2),product_prod(E,product_prod(F3,G2)),aa(E,fun(product_prod(F3,G2),product_prod(E,product_prod(F3,G2))),product_Pair(E,product_prod(F3,G2)),E2),aa(G2,product_prod(F3,G2),aa(F3,fun(G2,product_prod(F3,G2)),product_Pair(F3,G2),F2),G3)))))) ).

% prod_cases7
tff(fact_310_prod__induct3,axiom,
    ! [C: $tType,B: $tType,A: $tType,P: fun(product_prod(A,product_prod(B,C)),$o),X: product_prod(A,product_prod(B,C))] :
      ( ! [A6: A,B5: B,C4: C] : aa(product_prod(A,product_prod(B,C)),$o,P,aa(product_prod(B,C),product_prod(A,product_prod(B,C)),aa(A,fun(product_prod(B,C),product_prod(A,product_prod(B,C))),product_Pair(A,product_prod(B,C)),A6),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),B5),C4)))
     => aa(product_prod(A,product_prod(B,C)),$o,P,X) ) ).

% prod_induct3
tff(fact_311_prod__induct4,axiom,
    ! [D: $tType,C: $tType,B: $tType,A: $tType,P: fun(product_prod(A,product_prod(B,product_prod(C,D))),$o),X: product_prod(A,product_prod(B,product_prod(C,D)))] :
      ( ! [A6: A,B5: B,C4: C,D2: D] : aa(product_prod(A,product_prod(B,product_prod(C,D))),$o,P,aa(product_prod(B,product_prod(C,D)),product_prod(A,product_prod(B,product_prod(C,D))),aa(A,fun(product_prod(B,product_prod(C,D)),product_prod(A,product_prod(B,product_prod(C,D)))),product_Pair(A,product_prod(B,product_prod(C,D))),A6),aa(product_prod(C,D),product_prod(B,product_prod(C,D)),aa(B,fun(product_prod(C,D),product_prod(B,product_prod(C,D))),product_Pair(B,product_prod(C,D)),B5),aa(D,product_prod(C,D),aa(C,fun(D,product_prod(C,D)),product_Pair(C,D),C4),D2))))
     => aa(product_prod(A,product_prod(B,product_prod(C,D))),$o,P,X) ) ).

% prod_induct4
tff(fact_312_prod__induct5,axiom,
    ! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: fun(product_prod(A,product_prod(B,product_prod(C,product_prod(D,E)))),$o),X: product_prod(A,product_prod(B,product_prod(C,product_prod(D,E))))] :
      ( ! [A6: A,B5: B,C4: C,D2: D,E2: E] : aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,E)))),$o,P,aa(product_prod(B,product_prod(C,product_prod(D,E))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,E)))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,E))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,E))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,E)))),A6),aa(product_prod(C,product_prod(D,E)),product_prod(B,product_prod(C,product_prod(D,E))),aa(B,fun(product_prod(C,product_prod(D,E)),product_prod(B,product_prod(C,product_prod(D,E)))),product_Pair(B,product_prod(C,product_prod(D,E))),B5),aa(product_prod(D,E),product_prod(C,product_prod(D,E)),aa(C,fun(product_prod(D,E),product_prod(C,product_prod(D,E))),product_Pair(C,product_prod(D,E)),C4),aa(E,product_prod(D,E),aa(D,fun(E,product_prod(D,E)),product_Pair(D,E),D2),E2)))))
     => aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,E)))),$o,P,X) ) ).

% prod_induct5
tff(fact_313_prod__induct6,axiom,
    ! [F3: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: fun(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F3))))),$o),X: product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F3)))))] :
      ( ! [A6: A,B5: B,C4: C,D2: D,E2: E,F2: F3] : aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F3))))),$o,P,aa(product_prod(B,product_prod(C,product_prod(D,product_prod(E,F3)))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F3))))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,product_prod(E,F3)))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F3)))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F3))))),A6),aa(product_prod(C,product_prod(D,product_prod(E,F3))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,F3)))),aa(B,fun(product_prod(C,product_prod(D,product_prod(E,F3))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,F3))))),product_Pair(B,product_prod(C,product_prod(D,product_prod(E,F3)))),B5),aa(product_prod(D,product_prod(E,F3)),product_prod(C,product_prod(D,product_prod(E,F3))),aa(C,fun(product_prod(D,product_prod(E,F3)),product_prod(C,product_prod(D,product_prod(E,F3)))),product_Pair(C,product_prod(D,product_prod(E,F3))),C4),aa(product_prod(E,F3),product_prod(D,product_prod(E,F3)),aa(D,fun(product_prod(E,F3),product_prod(D,product_prod(E,F3))),product_Pair(D,product_prod(E,F3)),D2),aa(F3,product_prod(E,F3),aa(E,fun(F3,product_prod(E,F3)),product_Pair(E,F3),E2),F2))))))
     => aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F3))))),$o,P,X) ) ).

% prod_induct6
tff(fact_314_prod__induct7,axiom,
    ! [G2: $tType,F3: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: fun(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F3,G2)))))),$o),X: product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F3,G2))))))] :
      ( ! [A6: A,B5: B,C4: C,D2: D,E2: E,F2: F3,G3: G2] : aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F3,G2)))))),$o,P,aa(product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F3,G2))))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F3,G2)))))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F3,G2))))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F3,G2))))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F3,G2)))))),A6),aa(product_prod(C,product_prod(D,product_prod(E,product_prod(F3,G2)))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F3,G2))))),aa(B,fun(product_prod(C,product_prod(D,product_prod(E,product_prod(F3,G2)))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F3,G2)))))),product_Pair(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F3,G2))))),B5),aa(product_prod(D,product_prod(E,product_prod(F3,G2))),product_prod(C,product_prod(D,product_prod(E,product_prod(F3,G2)))),aa(C,fun(product_prod(D,product_prod(E,product_prod(F3,G2))),product_prod(C,product_prod(D,product_prod(E,product_prod(F3,G2))))),product_Pair(C,product_prod(D,product_prod(E,product_prod(F3,G2)))),C4),aa(product_prod(E,product_prod(F3,G2)),product_prod(D,product_prod(E,product_prod(F3,G2))),aa(D,fun(product_prod(E,product_prod(F3,G2)),product_prod(D,product_prod(E,product_prod(F3,G2)))),product_Pair(D,product_prod(E,product_prod(F3,G2))),D2),aa(product_prod(F3,G2),product_prod(E,product_prod(F3,G2)),aa(E,fun(product_prod(F3,G2),product_prod(E,product_prod(F3,G2))),product_Pair(E,product_prod(F3,G2)),E2),aa(G2,product_prod(F3,G2),aa(F3,fun(G2,product_prod(F3,G2)),product_Pair(F3,G2),F2),G3)))))))
     => aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F3,G2)))))),$o,P,X) ) ).

% prod_induct7
tff(fact_315_memb__imp__not__empty,axiom,
    ! [A: $tType,X: A,S: set(A)] :
      ( aa(set(A),$o,member(A,X),S)
     => ( S != bot_bot(set(A)) ) ) ).

% memb_imp_not_empty
tff(fact_316_set__notEmptyE,axiom,
    ! [A: $tType,S: set(A)] :
      ( ( S != bot_bot(set(A)) )
     => ~ ! [X2: A] : ~ aa(set(A),$o,member(A,X2),S) ) ).

% set_notEmptyE
tff(fact_317_verit__negate__coefficient_I3_J,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A] :
          ( ( A3 = B2 )
         => ( aa(A,A,uminus_uminus(A),A3) = aa(A,A,uminus_uminus(A),B2) ) ) ) ).

% verit_negate_coefficient(3)
tff(fact_318_old_Oprod_Orec,axiom,
    ! [B: $tType,A: $tType,C: $tType,F1: fun(B,fun(C,A)),A3: B,B2: C] : product_rec_prod(B,C,A,F1,aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A3),B2)) = aa(C,A,aa(B,fun(C,A),F1,A3),B2) ).

% old.prod.rec
tff(fact_319_sup__Un__eq,axiom,
    ! [A: $tType,R: set(A),S: set(A),X3: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aa(fun(A,$o),fun(fun(A,$o),fun(A,$o)),sup_sup(fun(A,$o)),aa(set(A),fun(A,$o),aTP_Lamp_ag(set(A),fun(A,$o)),R)),aa(set(A),fun(A,$o),aTP_Lamp_ag(set(A),fun(A,$o)),S)),X3)
    <=> aa(set(A),$o,member(A,X3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),R),S)) ) ).

% sup_Un_eq
tff(fact_320_inf__Int__eq,axiom,
    ! [A: $tType,R: set(A),S: set(A),X3: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aa(fun(A,$o),fun(fun(A,$o),fun(A,$o)),inf_inf(fun(A,$o)),aa(set(A),fun(A,$o),aTP_Lamp_ag(set(A),fun(A,$o)),R)),aa(set(A),fun(A,$o),aTP_Lamp_ag(set(A),fun(A,$o)),S)),X3)
    <=> aa(set(A),$o,member(A,X3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),R),S)) ) ).

% inf_Int_eq
tff(fact_321_iso__tuple__UNIV__I,axiom,
    ! [A: $tType,X: A] : aa(set(A),$o,member(A,X),top_top(set(A))) ).

% iso_tuple_UNIV_I
tff(fact_322_Collect__empty__eq__bot,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( ( aa(fun(A,$o),set(A),collect(A),P) = bot_bot(set(A)) )
    <=> ( P = bot_bot(fun(A,$o)) ) ) ).

% Collect_empty_eq_bot
tff(fact_323_bot__empty__eq,axiom,
    ! [A: $tType,X3: A] :
      ( aa(A,$o,bot_bot(fun(A,$o)),X3)
    <=> aa(set(A),$o,member(A,X3),bot_bot(set(A))) ) ).

% bot_empty_eq
tff(fact_324_type__copy__ex__RepI,axiom,
    ! [B: $tType,A: $tType,Rep: fun(A,B),Abs: fun(B,A),F4: fun(B,$o)] :
      ( type_definition(A,B,Rep,Abs,top_top(set(B)))
     => ( ? [X_12: B] : aa(B,$o,F4,X_12)
      <=> ? [B6: A] : aa(B,$o,F4,aa(A,B,Rep,B6)) ) ) ).

% type_copy_ex_RepI
tff(fact_325_type__copy__obj__one__point__absE,axiom,
    ! [A: $tType,B: $tType,Rep: fun(A,B),Abs: fun(B,A),S2: A] :
      ( type_definition(A,B,Rep,Abs,top_top(set(B)))
     => ~ ! [X2: B] : S2 != aa(B,A,Abs,X2) ) ).

% type_copy_obj_one_point_absE
tff(fact_326_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_327_boolean__algebra_Oabstract__boolean__algebra__axioms,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => boolea2506097494486148201lgebra(A,inf_inf(A),sup_sup(A),uminus_uminus(A),bot_bot(A),top_top(A)) ) ).

% boolean_algebra.abstract_boolean_algebra_axioms
tff(fact_328_uncurry__apply,axiom,
    ! [B: $tType,A: $tType,C: $tType,F: fun(B,fun(C,A)),A3: B,B2: C] : aa(product_prod(B,C),A,uncurry(B,C,A,F),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),F,A3),B2) ).

% uncurry_apply
tff(fact_329_top1I,axiom,
    ! [A: $tType,X: A] : aa(A,$o,top_top(fun(A,$o)),X) ).

% top1I
tff(fact_330_top__conj_I2_J,axiom,
    ! [A: $tType,P: $o,X: A] :
      ( ( (P)
        & aa(A,$o,top_top(fun(A,$o)),X) )
    <=> (P) ) ).

% top_conj(2)
tff(fact_331_top__conj_I1_J,axiom,
    ! [A: $tType,X: A,P: $o] :
      ( ( aa(A,$o,top_top(fun(A,$o)),X)
        & (P) )
    <=> (P) ) ).

% top_conj(1)
tff(fact_332_abstract__boolean__algebra_Ocompl__one,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A] :
      ( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
     => ( aa(A,A,Compl,One) = Zero ) ) ).

% abstract_boolean_algebra.compl_one
tff(fact_333_abstract__boolean__algebra_Ocompl__zero,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A] :
      ( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
     => ( aa(A,A,Compl,Zero) = One ) ) ).

% abstract_boolean_algebra.compl_zero
tff(fact_334_abstract__boolean__algebra_Ocompl__unique,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,X: A,Y: A] :
      ( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
     => ( ( aa(A,A,aa(A,fun(A,A),Conj,X),Y) = Zero )
       => ( ( aa(A,A,aa(A,fun(A,A),Disj,X),Y) = One )
         => ( aa(A,A,Compl,X) = Y ) ) ) ) ).

% abstract_boolean_algebra.compl_unique
tff(fact_335_abstract__boolean__algebra_Odouble__compl,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,X: A] :
      ( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
     => ( aa(A,A,Compl,aa(A,A,Compl,X)) = X ) ) ).

% abstract_boolean_algebra.double_compl
tff(fact_336_abstract__boolean__algebra_Odisj__one__left,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,X: A] :
      ( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
     => ( aa(A,A,aa(A,fun(A,A),Disj,One),X) = One ) ) ).

% abstract_boolean_algebra.disj_one_left
tff(fact_337_abstract__boolean__algebra_Oconj__one__right,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,X: A] :
      ( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
     => ( aa(A,A,aa(A,fun(A,A),Conj,X),One) = X ) ) ).

% abstract_boolean_algebra.conj_one_right
tff(fact_338_abstract__boolean__algebra_Oconj__zero__left,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,X: A] :
      ( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
     => ( aa(A,A,aa(A,fun(A,A),Conj,Zero),X) = Zero ) ) ).

% abstract_boolean_algebra.conj_zero_left
tff(fact_339_abstract__boolean__algebra_Ode__Morgan__conj,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,X: A,Y: A] :
      ( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
     => ( aa(A,A,Compl,aa(A,A,aa(A,fun(A,A),Conj,X),Y)) = aa(A,A,aa(A,fun(A,A),Disj,aa(A,A,Compl,X)),aa(A,A,Compl,Y)) ) ) ).

% abstract_boolean_algebra.de_Morgan_conj
tff(fact_340_abstract__boolean__algebra_Ode__Morgan__disj,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,X: A,Y: A] :
      ( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
     => ( aa(A,A,Compl,aa(A,A,aa(A,fun(A,A),Disj,X),Y)) = aa(A,A,aa(A,fun(A,A),Conj,aa(A,A,Compl,X)),aa(A,A,Compl,Y)) ) ) ).

% abstract_boolean_algebra.de_Morgan_disj
tff(fact_341_abstract__boolean__algebra_Odisj__one__right,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,X: A] :
      ( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
     => ( aa(A,A,aa(A,fun(A,A),Disj,X),One) = One ) ) ).

% abstract_boolean_algebra.disj_one_right
tff(fact_342_abstract__boolean__algebra_Oconj__zero__right,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,X: A] :
      ( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
     => ( aa(A,A,aa(A,fun(A,A),Conj,X),Zero) = Zero ) ) ).

% abstract_boolean_algebra.conj_zero_right
tff(fact_343_abstract__boolean__algebra_Odisj__zero__right,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,X: A] :
      ( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
     => ( aa(A,A,aa(A,fun(A,A),Disj,X),Zero) = X ) ) ).

% abstract_boolean_algebra.disj_zero_right
tff(fact_344_abstract__boolean__algebra_Oconj__cancel__left,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,X: A] :
      ( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
     => ( aa(A,A,aa(A,fun(A,A),Conj,aa(A,A,Compl,X)),X) = Zero ) ) ).

% abstract_boolean_algebra.conj_cancel_left
tff(fact_345_abstract__boolean__algebra_Odisj__cancel__left,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,X: A] :
      ( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
     => ( aa(A,A,aa(A,fun(A,A),Disj,aa(A,A,Compl,X)),X) = One ) ) ).

% abstract_boolean_algebra.disj_cancel_left
tff(fact_346_abstract__boolean__algebra_Ocomplement__unique,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,A3: A,X: A,Y: A] :
      ( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
     => ( ( aa(A,A,aa(A,fun(A,A),Conj,A3),X) = Zero )
       => ( ( aa(A,A,aa(A,fun(A,A),Disj,A3),X) = One )
         => ( ( aa(A,A,aa(A,fun(A,A),Conj,A3),Y) = Zero )
           => ( ( aa(A,A,aa(A,fun(A,A),Disj,A3),Y) = One )
             => ( X = Y ) ) ) ) ) ) ).

% abstract_boolean_algebra.complement_unique
tff(fact_347_abstract__boolean__algebra_Oconj__cancel__right,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,X: A] :
      ( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
     => ( aa(A,A,aa(A,fun(A,A),Conj,X),aa(A,A,Compl,X)) = Zero ) ) ).

% abstract_boolean_algebra.conj_cancel_right
tff(fact_348_abstract__boolean__algebra_Oconj__disj__distrib,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,X: A,Y: A,Z2: A] :
      ( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
     => ( aa(A,A,aa(A,fun(A,A),Conj,X),aa(A,A,aa(A,fun(A,A),Disj,Y),Z2)) = aa(A,A,aa(A,fun(A,A),Disj,aa(A,A,aa(A,fun(A,A),Conj,X),Y)),aa(A,A,aa(A,fun(A,A),Conj,X),Z2)) ) ) ).

% abstract_boolean_algebra.conj_disj_distrib
tff(fact_349_abstract__boolean__algebra_Odisj__cancel__right,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,X: A] :
      ( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
     => ( aa(A,A,aa(A,fun(A,A),Disj,X),aa(A,A,Compl,X)) = One ) ) ).

% abstract_boolean_algebra.disj_cancel_right
tff(fact_350_abstract__boolean__algebra_Odisj__conj__distrib,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,X: A,Y: A,Z2: A] :
      ( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
     => ( aa(A,A,aa(A,fun(A,A),Disj,X),aa(A,A,aa(A,fun(A,A),Conj,Y),Z2)) = aa(A,A,aa(A,fun(A,A),Conj,aa(A,A,aa(A,fun(A,A),Disj,X),Y)),aa(A,A,aa(A,fun(A,A),Disj,X),Z2)) ) ) ).

% abstract_boolean_algebra.disj_conj_distrib
tff(fact_351_abstract__boolean__algebra_Ocompl__eq__compl__iff,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,X: A,Y: A] :
      ( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
     => ( ( aa(A,A,Compl,X) = aa(A,A,Compl,Y) )
      <=> ( X = Y ) ) ) ).

% abstract_boolean_algebra.compl_eq_compl_iff
tff(fact_352_abstract__boolean__algebra_Oconj__disj__distrib2,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Y: A,Z2: A,X: A] :
      ( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
     => ( aa(A,A,aa(A,fun(A,A),Conj,aa(A,A,aa(A,fun(A,A),Disj,Y),Z2)),X) = aa(A,A,aa(A,fun(A,A),Disj,aa(A,A,aa(A,fun(A,A),Conj,Y),X)),aa(A,A,aa(A,fun(A,A),Conj,Z2),X)) ) ) ).

% abstract_boolean_algebra.conj_disj_distrib2
tff(fact_353_abstract__boolean__algebra_Odisj__conj__distrib2,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Y: A,Z2: A,X: A] :
      ( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
     => ( aa(A,A,aa(A,fun(A,A),Disj,aa(A,A,aa(A,fun(A,A),Conj,Y),Z2)),X) = aa(A,A,aa(A,fun(A,A),Conj,aa(A,A,aa(A,fun(A,A),Disj,Y),X)),aa(A,A,aa(A,fun(A,A),Disj,Z2),X)) ) ) ).

% abstract_boolean_algebra.disj_conj_distrib2
tff(fact_354_top__empty__eq,axiom,
    ! [A: $tType,X3: A] :
      ( aa(A,$o,top_top(fun(A,$o)),X3)
    <=> aa(set(A),$o,member(A,X3),top_top(set(A))) ) ).

% top_empty_eq
tff(fact_355_sup__Un__eq2,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,B)),S: set(product_prod(A,B)),X3: A,Xa3: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),sup_sup(fun(A,fun(B,$o))),aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_as(set(product_prod(A,B)),fun(A,fun(B,$o))),R)),aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_as(set(product_prod(A,B)),fun(A,fun(B,$o))),S)),X3),Xa3)
    <=> aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Xa3)),aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),sup_sup(set(product_prod(A,B))),R),S)) ) ).

% sup_Un_eq2
tff(fact_356_inf__Int__eq2,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,B)),S: set(product_prod(A,B)),X3: A,Xa3: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),inf_inf(fun(A,fun(B,$o))),aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_as(set(product_prod(A,B)),fun(A,fun(B,$o))),R)),aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_as(set(product_prod(A,B)),fun(A,fun(B,$o))),S)),X3),Xa3)
    <=> aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Xa3)),aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),inf_inf(set(product_prod(A,B))),R),S)) ) ).

% inf_Int_eq2
tff(fact_357_bot__empty__eq2,axiom,
    ! [B: $tType,A: $tType,X3: A,Xa3: B] :
      ( aa(B,$o,aa(A,fun(B,$o),bot_bot(fun(A,fun(B,$o))),X3),Xa3)
    <=> aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Xa3)),bot_bot(set(product_prod(A,B)))) ) ).

% bot_empty_eq2
tff(fact_358_pred__equals__eq2,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,B)),S: set(product_prod(A,B))] :
      ( ! [X4: A,Xa2: B] :
          ( aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Xa2)),R)
        <=> aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Xa2)),S) )
    <=> ( R = S ) ) ).

% pred_equals_eq2
tff(fact_359_top__empty__eq2,axiom,
    ! [B: $tType,A: $tType,X3: A,Xa3: B] :
      ( aa(B,$o,aa(A,fun(B,$o),top_top(fun(A,fun(B,$o))),X3),Xa3)
    <=> aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Xa3)),top_top(set(product_prod(A,B)))) ) ).

% top_empty_eq2
tff(fact_360_times__assn__raw_Opelims_I3_J,axiom,
    ! [X: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xa: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xb: product_prod(heap_ext(product_unit),set(nat))] :
      ( ~ aa(product_prod(heap_ext(product_unit),set(nat)),$o,times_assn_raw(X,Xa),Xb)
     => ( accp(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),times_assn_raw_rel,aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),Xa),Xb)))
       => ~ ! [H: heap_ext(product_unit),As4: set(nat)] :
              ( ( Xb = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As4) )
             => ( accp(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),times_assn_raw_rel,aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),Xa),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As4))))
               => ? [As13: set(nat),As23: set(nat)] :
                    ( ( As4 = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As13),As23) )
                    & ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As13),As23) = bot_bot(set(nat)) )
                    & aa(product_prod(heap_ext(product_unit),set(nat)),$o,X,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As13))
                    & aa(product_prod(heap_ext(product_unit),set(nat)),$o,Xa,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As23)) ) ) ) ) ) ).

% times_assn_raw.pelims(3)
tff(fact_361_times__assn__raw_Opelims_I2_J,axiom,
    ! [X: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xa: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xb: product_prod(heap_ext(product_unit),set(nat))] :
      ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,times_assn_raw(X,Xa),Xb)
     => ( accp(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),times_assn_raw_rel,aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),Xa),Xb)))
       => ~ ! [H: heap_ext(product_unit),As4: set(nat)] :
              ( ( Xb = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As4) )
             => ( accp(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),times_assn_raw_rel,aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),Xa),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As4))))
               => ~ ? [As12: set(nat),As22: set(nat)] :
                      ( ( As4 = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As12),As22) )
                      & ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As12),As22) = bot_bot(set(nat)) )
                      & aa(product_prod(heap_ext(product_unit),set(nat)),$o,X,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As12))
                      & aa(product_prod(heap_ext(product_unit),set(nat)),$o,Xa,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As22)) ) ) ) ) ) ).

% times_assn_raw.pelims(2)
tff(fact_362_times__assn__raw_Opelims_I1_J,axiom,
    ! [X: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xa: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xb: product_prod(heap_ext(product_unit),set(nat)),Y: $o] :
      ( ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,times_assn_raw(X,Xa),Xb)
      <=> (Y) )
     => ( accp(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),times_assn_raw_rel,aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),Xa),Xb)))
       => ~ ! [H: heap_ext(product_unit),As4: set(nat)] :
              ( ( Xb = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As4) )
             => ( ( (Y)
                <=> ? [As1: set(nat),As2: set(nat)] :
                      ( ( As4 = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As1),As2) )
                      & ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As1),As2) = bot_bot(set(nat)) )
                      & aa(product_prod(heap_ext(product_unit),set(nat)),$o,X,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As1))
                      & aa(product_prod(heap_ext(product_unit),set(nat)),$o,Xa,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2)) ) )
               => ~ accp(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),times_assn_raw_rel,aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),Xa),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As4)))) ) ) ) ) ).

% times_assn_raw.pelims(1)
tff(fact_363_Set_Ois__empty__def,axiom,
    ! [A: $tType,A4: set(A)] :
      ( is_empty2(A,A4)
    <=> ( A4 = bot_bot(set(A)) ) ) ).

% Set.is_empty_def
tff(fact_364_internal__case__prod__conv,axiom,
    ! [B: $tType,A: $tType,C: $tType,C2: fun(B,fun(C,A)),A3: B,B2: C] : produc5280177257484947105e_prod(B,C,A,C2,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),C2,A3),B2) ).

% internal_case_prod_conv
tff(fact_365_wand__assnI,axiom,
    ! [H2: heap_ext(product_unit),As: set(nat),Q: assn,R: assn] :
      ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As))
     => ( ! [H4: heap_ext(product_unit),As5: set(nat)] :
            ( ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As),As5) = bot_bot(set(nat)) )
           => ( relH(As,H2,H4)
             => ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),As))
               => ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,Q),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),As5))
                 => aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,R),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As),As5))) ) ) ) )
       => aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,wand_assn(Q,R)),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As)) ) ) ).

% wand_assnI
tff(fact_366_vimage__if,axiom,
    ! [B: $tType,A: $tType,B3: set(A),C2: B,D3: B,A4: set(B)] :
      aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),aa(B,fun(A,B),aa(B,fun(B,fun(A,B)),aTP_Lamp_at(set(A),fun(B,fun(B,fun(A,B))),B3),C2),D3)),A4) = $ite(
        aa(set(B),$o,member(B,C2),A4),
        $ite(aa(set(B),$o,member(B,D3),A4),top_top(set(A)),B3),
        $ite(aa(set(B),$o,member(B,D3),A4),aa(set(A),set(A),uminus_uminus(set(A)),B3),bot_bot(set(A))) ) ).

% vimage_if
tff(fact_367_BNF__Composition_Otype__definition__id__bnf__UNIV,axiom,
    ! [A: $tType] : type_definition(A,A,bNF_id_bnf(A),bNF_id_bnf(A),top_top(set(A))) ).

% BNF_Composition.type_definition_id_bnf_UNIV
tff(fact_368_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_369_wand__raw_Oelims_I3_J,axiom,
    ! [X: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xa: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xb: product_prod(heap_ext(product_unit),set(nat))] :
      ( ~ aa(product_prod(heap_ext(product_unit),set(nat)),$o,wand_raw(X,Xa),Xb)
     => ~ ! [H: heap_ext(product_unit),As4: set(nat)] :
            ( ( Xb = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As4) )
           => ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As4))
              & ! [H4: heap_ext(product_unit),As5: set(nat)] :
                  ( ( ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As4),As5) = bot_bot(set(nat)) )
                    & relH(As4,H,H4)
                    & aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),As4))
                    & aa(product_prod(heap_ext(product_unit),set(nat)),$o,X,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),As5)) )
                 => aa(product_prod(heap_ext(product_unit),set(nat)),$o,Xa,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As4),As5))) ) ) ) ) ).

% wand_raw.elims(3)
tff(fact_370_wand__raw_Oelims_I2_J,axiom,
    ! [X: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xa: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xb: product_prod(heap_ext(product_unit),set(nat))] :
      ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,wand_raw(X,Xa),Xb)
     => ~ ! [H: heap_ext(product_unit),As4: set(nat)] :
            ( ( Xb = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As4) )
           => ~ ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As4))
                & ! [H5: heap_ext(product_unit),As6: set(nat)] :
                    ( ( ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As4),As6) = bot_bot(set(nat)) )
                      & relH(As4,H,H5)
                      & aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H5),As4))
                      & aa(product_prod(heap_ext(product_unit),set(nat)),$o,X,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H5),As6)) )
                   => aa(product_prod(heap_ext(product_unit),set(nat)),$o,Xa,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H5),aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As4),As6))) ) ) ) ) ).

% wand_raw.elims(2)
tff(fact_371_sup2CI,axiom,
    ! [A: $tType,B: $tType,B3: fun(A,fun(B,$o)),X: A,Y: B,A4: fun(A,fun(B,$o))] :
      ( ( ~ aa(B,$o,aa(A,fun(B,$o),B3,X),Y)
       => aa(B,$o,aa(A,fun(B,$o),A4,X),Y) )
     => aa(B,$o,aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),sup_sup(fun(A,fun(B,$o))),A4),B3),X),Y) ) ).

% sup2CI
tff(fact_372_inf2I,axiom,
    ! [A: $tType,B: $tType,A4: fun(A,fun(B,$o)),X: A,Y: B,B3: fun(A,fun(B,$o))] :
      ( aa(B,$o,aa(A,fun(B,$o),A4,X),Y)
     => ( aa(B,$o,aa(A,fun(B,$o),B3,X),Y)
       => aa(B,$o,aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),inf_inf(fun(A,fun(B,$o))),A4),B3),X),Y) ) ) ).

% inf2I
tff(fact_373_vimage__ident,axiom,
    ! [A: $tType,Y4: set(A)] : aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),vimage(A,A),aTP_Lamp_au(A,A)),Y4) = Y4 ).

% vimage_ident
tff(fact_374_vimage__Collect__eq,axiom,
    ! [A: $tType,B: $tType,F: fun(A,B),P: fun(B,$o)] : aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F),aa(fun(B,$o),set(B),collect(B),P)) = aa(fun(A,$o),set(A),collect(A),aa(fun(B,$o),fun(A,$o),aTP_Lamp_av(fun(A,B),fun(fun(B,$o),fun(A,$o)),F),P)) ).

% vimage_Collect_eq
tff(fact_375_vimage__empty,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B)] : aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F),bot_bot(set(B))) = bot_bot(set(A)) ).

% vimage_empty
tff(fact_376_vimage__UNIV,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B)] : aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F),top_top(set(B))) = top_top(set(A)) ).

% vimage_UNIV
tff(fact_377_vimage__Int,axiom,
    ! [A: $tType,B: $tType,F: fun(A,B),A4: set(B),B3: set(B)] : aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),B3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F),A4)),aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F),B3)) ).

% vimage_Int
tff(fact_378_vimage__Un,axiom,
    ! [A: $tType,B: $tType,F: fun(A,B),A4: set(B),B3: set(B)] : aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A4),B3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F),A4)),aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F),B3)) ).

% vimage_Un
tff(fact_379_top2I,axiom,
    ! [A: $tType,B: $tType,X: A,Y: B] : aa(B,$o,aa(A,fun(B,$o),top_top(fun(A,fun(B,$o))),X),Y) ).

% top2I
tff(fact_380_bool__assn__proper_I1_J,axiom,
    aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o,proper,in_range) ).

% bool_assn_proper(1)
tff(fact_381_relH__dist__union,axiom,
    ! [As: set(nat),As3: set(nat),H2: heap_ext(product_unit),H3: heap_ext(product_unit)] :
      ( relH(aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As),As3),H2,H3)
    <=> ( relH(As,H2,H3)
        & relH(As3,H2,H3) ) ) ).

% relH_dist_union
tff(fact_382_bool__assn__proper_I5_J,axiom,
    ! [P: fun(product_prod(heap_ext(product_unit),set(nat)),$o)] :
      ( aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o,proper,P)
     => aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o,proper,aTP_Lamp_aw(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(heap_ext(product_unit),set(nat)),$o),P)) ) ).

% bool_assn_proper(5)
tff(fact_383_in__range__empty,axiom,
    ! [H2: heap_ext(product_unit)] : aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),bot_bot(set(nat)))) ).

% in_range_empty
tff(fact_384_vimage__const,axiom,
    ! [B: $tType,A: $tType,C2: B,A4: set(B)] :
      aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),aTP_Lamp_ax(B,fun(A,B),C2)),A4) = $ite(aa(set(B),$o,member(B,C2),A4),top_top(set(A)),bot_bot(set(A))) ).

% vimage_const
tff(fact_385_in__range__dist__union,axiom,
    ! [H2: heap_ext(product_unit),As: set(nat),As3: set(nat)] :
      ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As),As3)))
    <=> ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As))
        & aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As3)) ) ) ).

% in_range_dist_union
tff(fact_386_relH__sym,axiom,
    ! [As: set(nat),H2: heap_ext(product_unit),H3: heap_ext(product_unit)] :
      ( relH(As,H2,H3)
     => relH(As,H3,H2) ) ).

% relH_sym
tff(fact_387_relH__trans,axiom,
    ! [As: set(nat),H12: heap_ext(product_unit),H23: heap_ext(product_unit),H32: heap_ext(product_unit)] :
      ( relH(As,H12,H23)
     => ( relH(As,H23,H32)
       => relH(As,H12,H32) ) ) ).

% relH_trans
tff(fact_388_bot2E,axiom,
    ! [A: $tType,B: $tType,X: A,Y: B] : ~ aa(B,$o,aa(A,fun(B,$o),bot_bot(fun(A,fun(B,$o))),X),Y) ).

% bot2E
tff(fact_389_inf2E,axiom,
    ! [A: $tType,B: $tType,A4: fun(A,fun(B,$o)),B3: fun(A,fun(B,$o)),X: A,Y: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),inf_inf(fun(A,fun(B,$o))),A4),B3),X),Y)
     => ~ ( aa(B,$o,aa(A,fun(B,$o),A4,X),Y)
         => ~ aa(B,$o,aa(A,fun(B,$o),B3,X),Y) ) ) ).

% inf2E
tff(fact_390_sup2E,axiom,
    ! [A: $tType,B: $tType,A4: fun(A,fun(B,$o)),B3: fun(A,fun(B,$o)),X: A,Y: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),sup_sup(fun(A,fun(B,$o))),A4),B3),X),Y)
     => ( ~ aa(B,$o,aa(A,fun(B,$o),A4,X),Y)
       => aa(B,$o,aa(A,fun(B,$o),B3,X),Y) ) ) ).

% sup2E
tff(fact_391_inf2D1,axiom,
    ! [A: $tType,B: $tType,A4: fun(A,fun(B,$o)),B3: fun(A,fun(B,$o)),X: A,Y: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),inf_inf(fun(A,fun(B,$o))),A4),B3),X),Y)
     => aa(B,$o,aa(A,fun(B,$o),A4,X),Y) ) ).

% inf2D1
tff(fact_392_inf2D2,axiom,
    ! [A: $tType,B: $tType,A4: fun(A,fun(B,$o)),B3: fun(A,fun(B,$o)),X: A,Y: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),inf_inf(fun(A,fun(B,$o))),A4),B3),X),Y)
     => aa(B,$o,aa(A,fun(B,$o),B3,X),Y) ) ).

% inf2D2
tff(fact_393_sup2I1,axiom,
    ! [A: $tType,B: $tType,A4: fun(A,fun(B,$o)),X: A,Y: B,B3: fun(A,fun(B,$o))] :
      ( aa(B,$o,aa(A,fun(B,$o),A4,X),Y)
     => aa(B,$o,aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),sup_sup(fun(A,fun(B,$o))),A4),B3),X),Y) ) ).

% sup2I1
tff(fact_394_sup2I2,axiom,
    ! [A: $tType,B: $tType,B3: fun(A,fun(B,$o)),X: A,Y: B,A4: fun(A,fun(B,$o))] :
      ( aa(B,$o,aa(A,fun(B,$o),B3,X),Y)
     => aa(B,$o,aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),sup_sup(fun(A,fun(B,$o))),A4),B3),X),Y) ) ).

% sup2I2
tff(fact_395_vimage__def,axiom,
    ! [A: $tType,B: $tType,F: fun(A,B),B3: set(B)] : aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F),B3) = aa(fun(A,$o),set(A),collect(A),aa(set(B),fun(A,$o),aTP_Lamp_ay(fun(A,B),fun(set(B),fun(A,$o)),F),B3)) ).

% vimage_def
tff(fact_396_relH__in__rangeI_I2_J,axiom,
    ! [As: set(nat),H2: heap_ext(product_unit),H3: heap_ext(product_unit)] :
      ( relH(As,H2,H3)
     => aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H3),As)) ) ).

% relH_in_rangeI(2)
tff(fact_397_relH__in__rangeI_I1_J,axiom,
    ! [As: set(nat),H2: heap_ext(product_unit),H3: heap_ext(product_unit)] :
      ( relH(As,H2,H3)
     => aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As)) ) ).

% relH_in_rangeI(1)
tff(fact_398_relH__refl,axiom,
    ! [H2: heap_ext(product_unit),As: set(nat)] :
      ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As))
     => relH(As,H2,H2) ) ).

% relH_refl
tff(fact_399_vimage__inter__cong,axiom,
    ! [B: $tType,A: $tType,S: set(A),F: fun(A,B),G: fun(A,B),Y: set(B)] :
      ( ! [W: A] :
          ( aa(set(A),$o,member(A,W),S)
         => ( aa(A,B,F,W) = aa(A,B,G,W) ) )
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F),Y)),S) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),G),Y)),S) ) ) ).

% vimage_inter_cong
tff(fact_400_vimage__Compl,axiom,
    ! [A: $tType,B: $tType,F: fun(A,B),A4: set(B)] : aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F),aa(set(B),set(B),uminus_uminus(set(B)),A4)) = aa(set(A),set(A),uminus_uminus(set(A)),aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F),A4)) ).

% vimage_Compl
tff(fact_401_proper__iff,axiom,
    ! [P: fun(product_prod(heap_ext(product_unit),set(nat)),$o),As: set(nat),H2: heap_ext(product_unit),H3: heap_ext(product_unit)] :
      ( aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o,proper,P)
     => ( relH(As,H2,H3)
       => ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H3),As))
         => ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,P,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As))
          <=> aa(product_prod(heap_ext(product_unit),set(nat)),$o,P,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H3),As)) ) ) ) ) ).

% proper_iff
tff(fact_402_proper__def,axiom,
    ! [P: fun(product_prod(heap_ext(product_unit),set(nat)),$o)] :
      ( aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o,proper,P)
    <=> ! [H6: heap_ext(product_unit),H7: heap_ext(product_unit),As7: set(nat)] :
          ( ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,P,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H6),As7))
           => aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H6),As7)) )
          & ( ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,P,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H6),As7))
              & relH(As7,H6,H7)
              & aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H7),As7)) )
           => aa(product_prod(heap_ext(product_unit),set(nat)),$o,P,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H7),As7)) ) ) ) ).

% proper_def
tff(fact_403_properD2,axiom,
    ! [P: fun(product_prod(heap_ext(product_unit),set(nat)),$o),H2: heap_ext(product_unit),As: set(nat),H3: heap_ext(product_unit)] :
      ( aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o,proper,P)
     => ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,P,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As))
       => ( relH(As,H2,H3)
         => ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H3),As))
           => aa(product_prod(heap_ext(product_unit),set(nat)),$o,P,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H3),As)) ) ) ) ) ).

% properD2
tff(fact_404_properI,axiom,
    ! [P: fun(product_prod(heap_ext(product_unit),set(nat)),$o)] :
      ( ! [As4: set(nat),H: heap_ext(product_unit)] :
          ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,P,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As4))
         => aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As4)) )
     => ( ! [As4: set(nat),H: heap_ext(product_unit),H4: heap_ext(product_unit)] :
            ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,P,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As4))
           => ( relH(As4,H,H4)
             => ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),As4))
               => aa(product_prod(heap_ext(product_unit),set(nat)),$o,P,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),As4)) ) ) )
       => aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o,proper,P) ) ) ).

% properI
tff(fact_405_models__in__range,axiom,
    ! [P: assn,H2: product_prod(heap_ext(product_unit),set(nat))] :
      ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,P),H2)
     => aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,H2) ) ).

% models_in_range
tff(fact_406_top__assn__def,axiom,
    top_top(assn) = aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),assn,abs_assn,in_range) ).

% top_assn_def
tff(fact_407_properD1,axiom,
    ! [P: fun(product_prod(heap_ext(product_unit),set(nat)),$o),H2: heap_ext(product_unit),As: set(nat)] :
      ( aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o,proper,P)
     => ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,P,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As))
       => aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As)) ) ) ).

% properD1
tff(fact_408_mod__relH,axiom,
    ! [As: set(nat),H2: heap_ext(product_unit),H3: heap_ext(product_unit),P: assn] :
      ( relH(As,H2,H3)
     => ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,P),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As))
      <=> aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,P),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H3),As)) ) ) ).

% mod_relH
tff(fact_409_uminus__assn__def,axiom,
    ! [P: assn] : aa(assn,assn,uminus_uminus(assn),P) = aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),assn,abs_assn,aTP_Lamp_az(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),P)) ).

% uminus_assn_def
tff(fact_410_wand__raw_Osimps,axiom,
    ! [P: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Q: fun(product_prod(heap_ext(product_unit),set(nat)),$o),H2: heap_ext(product_unit),As: set(nat)] :
      ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,wand_raw(P,Q),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As))
    <=> ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As))
        & ! [H7: heap_ext(product_unit),As8: set(nat)] :
            ( ( ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As),As8) = bot_bot(set(nat)) )
              & relH(As,H2,H7)
              & aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H7),As))
              & aa(product_prod(heap_ext(product_unit),set(nat)),$o,P,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H7),As8)) )
           => aa(product_prod(heap_ext(product_unit),set(nat)),$o,Q,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H7),aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As),As8))) ) ) ) ).

% wand_raw.simps
tff(fact_411_wand__raw_Oelims_I1_J,axiom,
    ! [X: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xa: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xb: product_prod(heap_ext(product_unit),set(nat)),Y: $o] :
      ( ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,wand_raw(X,Xa),Xb)
      <=> (Y) )
     => ~ ! [H: heap_ext(product_unit),As4: set(nat)] :
            ( ( Xb = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As4) )
           => ( (Y)
            <=> ~ ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As4))
                  & ! [H7: heap_ext(product_unit),As8: set(nat)] :
                      ( ( ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As4),As8) = bot_bot(set(nat)) )
                        & relH(As4,H,H7)
                        & aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H7),As4))
                        & aa(product_prod(heap_ext(product_unit),set(nat)),$o,X,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H7),As8)) )
                     => aa(product_prod(heap_ext(product_unit),set(nat)),$o,Xa,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H7),aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As4),As8))) ) ) ) ) ) ).

% wand_raw.elims(1)
tff(fact_412_wand__raw_Opelims_I1_J,axiom,
    ! [X: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xa: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xb: product_prod(heap_ext(product_unit),set(nat)),Y: $o] :
      ( ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,wand_raw(X,Xa),Xb)
      <=> (Y) )
     => ( accp(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),wand_raw_rel,aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),Xa),Xb)))
       => ~ ! [H: heap_ext(product_unit),As4: set(nat)] :
              ( ( Xb = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As4) )
             => ( ( (Y)
                <=> ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As4))
                    & ! [H7: heap_ext(product_unit),As8: set(nat)] :
                        ( ( ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As4),As8) = bot_bot(set(nat)) )
                          & relH(As4,H,H7)
                          & aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H7),As4))
                          & aa(product_prod(heap_ext(product_unit),set(nat)),$o,X,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H7),As8)) )
                       => aa(product_prod(heap_ext(product_unit),set(nat)),$o,Xa,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H7),aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As4),As8))) ) ) )
               => ~ accp(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),wand_raw_rel,aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),Xa),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As4)))) ) ) ) ) ).

% wand_raw.pelims(1)
tff(fact_413_wand__raw_Opelims_I2_J,axiom,
    ! [X: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xa: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xb: product_prod(heap_ext(product_unit),set(nat))] :
      ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,wand_raw(X,Xa),Xb)
     => ( accp(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),wand_raw_rel,aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),Xa),Xb)))
       => ~ ! [H: heap_ext(product_unit),As4: set(nat)] :
              ( ( Xb = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As4) )
             => ( accp(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),wand_raw_rel,aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),Xa),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As4))))
               => ~ ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As4))
                    & ! [H5: heap_ext(product_unit),As6: set(nat)] :
                        ( ( ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As4),As6) = bot_bot(set(nat)) )
                          & relH(As4,H,H5)
                          & aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H5),As4))
                          & aa(product_prod(heap_ext(product_unit),set(nat)),$o,X,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H5),As6)) )
                       => aa(product_prod(heap_ext(product_unit),set(nat)),$o,Xa,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H5),aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As4),As6))) ) ) ) ) ) ) ).

% wand_raw.pelims(2)
tff(fact_414_wand__raw_Opelims_I3_J,axiom,
    ! [X: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xa: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xb: product_prod(heap_ext(product_unit),set(nat))] :
      ( ~ aa(product_prod(heap_ext(product_unit),set(nat)),$o,wand_raw(X,Xa),Xb)
     => ( accp(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),wand_raw_rel,aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),Xa),Xb)))
       => ~ ! [H: heap_ext(product_unit),As4: set(nat)] :
              ( ( Xb = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As4) )
             => ( accp(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),wand_raw_rel,aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),Xa),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As4))))
               => ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As4))
                  & ! [H4: heap_ext(product_unit),As5: set(nat)] :
                      ( ( ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As4),As5) = bot_bot(set(nat)) )
                        & relH(As4,H,H4)
                        & aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),As4))
                        & aa(product_prod(heap_ext(product_unit),set(nat)),$o,X,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),As5)) )
                     => aa(product_prod(heap_ext(product_unit),set(nat)),$o,Xa,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As4),As5))) ) ) ) ) ) ) ).

% wand_raw.pelims(3)
tff(fact_415_reflclp__idemp,axiom,
    ! [A: $tType,P: fun(A,fun(A,$o))] : aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(fun(A,fun(A,$o)),fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),sup_sup(fun(A,fun(A,$o))),aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(fun(A,fun(A,$o)),fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),sup_sup(fun(A,fun(A,$o))),P),fequal(A))),fequal(A)) = aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(fun(A,fun(A,$o)),fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),sup_sup(fun(A,fun(A,$o))),P),fequal(A)) ).

% reflclp_idemp
tff(fact_416_bijective__Empty,axiom,
    ! [B: $tType,A: $tType] : bijective(A,B,bot_bot(set(product_prod(A,B)))) ).

% bijective_Empty
tff(fact_417_dbl__inc__simps_I1_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [K: num] : neg_numeral_dbl_inc(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),K))) = aa(A,A,uminus_uminus(A),neg_numeral_dbl_dec(A,aa(num,A,numeral_numeral(A),K))) ) ).

% dbl_inc_simps(1)
tff(fact_418_dbl__dec__simps_I1_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [K: num] : neg_numeral_dbl_dec(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),K))) = aa(A,A,uminus_uminus(A),neg_numeral_dbl_inc(A,aa(num,A,numeral_numeral(A),K))) ) ).

% dbl_dec_simps(1)
tff(fact_419_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_420_pairself_Opelims,axiom,
    ! [A: $tType,B: $tType,X: fun(B,A),Xa: product_prod(B,B),Y: product_prod(A,A)] :
      ( ( aa(product_prod(B,B),product_prod(A,A),pairself(B,A,X),Xa) = Y )
     => ( accp(product_prod(fun(B,A),product_prod(B,B)),pairself_rel(B,A),aa(product_prod(B,B),product_prod(fun(B,A),product_prod(B,B)),aa(fun(B,A),fun(product_prod(B,B),product_prod(fun(B,A),product_prod(B,B))),product_Pair(fun(B,A),product_prod(B,B)),X),Xa))
       => ~ ! [A6: B,B5: B] :
              ( ( Xa = aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),A6),B5) )
             => ( ( Y = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(B,A,X,A6)),aa(B,A,X,B5)) )
               => ~ accp(product_prod(fun(B,A),product_prod(B,B)),pairself_rel(B,A),aa(product_prod(B,B),product_prod(fun(B,A),product_prod(B,B)),aa(fun(B,A),fun(product_prod(B,B),product_prod(fun(B,A),product_prod(B,B))),product_Pair(fun(B,A),product_prod(B,B)),X),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),A6),B5))) ) ) ) ) ).

% pairself.pelims
tff(fact_421_rel__restrict__compl,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A4: set(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))),rel_restrict(A,R,A4)),rel_restrict(A,R,aa(set(A),set(A),uminus_uminus(set(A)),A4))) = bot_bot(set(product_prod(A,A))) ).

% rel_restrict_compl
tff(fact_422_Id__on__empty,axiom,
    ! [A: $tType] : id_on(A,bot_bot(set(A))) = bot_bot(set(product_prod(A,A))) ).

% Id_on_empty
tff(fact_423_mult__cancel__right,axiom,
    ! [A: $tType] :
      ( semiri6575147826004484403cancel(A)
     => ! [A3: A,C2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2) = aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2) )
        <=> ( ( C2 = zero_zero(A) )
            | ( A3 = B2 ) ) ) ) ).

% mult_cancel_right
tff(fact_424_mult__cancel__left,axiom,
    ! [A: $tType] :
      ( semiri6575147826004484403cancel(A)
     => ! [C2: A,A3: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3) = aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2) )
        <=> ( ( C2 = zero_zero(A) )
            | ( A3 = B2 ) ) ) ) ).

% mult_cancel_left
tff(fact_425_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_426_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_427_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_428_mult__numeral__left__semiring__numeral,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [V: num,W2: num,Z2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V)),aa(A,A,aa(A,fun(A,A),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),V),W2))),Z2) ) ).

% mult_numeral_left_semiring_numeral
tff(fact_429_numeral__times__numeral,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [M: num,N: num] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),N)) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),M),N)) ) ).

% numeral_times_numeral
tff(fact_430_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_431_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_432_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_433_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_434_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_435_neg__numeral__eq__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [M: num,N: num] :
          ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N)) )
        <=> ( M = N ) ) ) ).

% neg_numeral_eq_iff
tff(fact_436_rel__restrict__empty,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] : rel_restrict(A,R,bot_bot(set(A))) = R ).

% rel_restrict_empty
tff(fact_437_Id__onI,axiom,
    ! [A: $tType,A3: A,A4: set(A)] :
      ( aa(set(A),$o,member(A,A3),A4)
     => aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),A3)),id_on(A,A4)) ) ).

% Id_onI
tff(fact_438_mult__cancel__right2,axiom,
    ! [A: $tType] :
      ( ring_15535105094025558882visors(A)
     => ! [A3: A,C2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2) = C2 )
        <=> ( ( C2 = zero_zero(A) )
            | ( A3 = one_one(A) ) ) ) ) ).

% mult_cancel_right2
tff(fact_439_mult__cancel__right1,axiom,
    ! [A: $tType] :
      ( ring_15535105094025558882visors(A)
     => ! [C2: A,B2: A] :
          ( ( C2 = aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2) )
        <=> ( ( C2 = zero_zero(A) )
            | ( B2 = one_one(A) ) ) ) ) ).

% mult_cancel_right1
tff(fact_440_mult__cancel__left2,axiom,
    ! [A: $tType] :
      ( ring_15535105094025558882visors(A)
     => ! [C2: A,A3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3) = C2 )
        <=> ( ( C2 = zero_zero(A) )
            | ( A3 = one_one(A) ) ) ) ) ).

% mult_cancel_left2
tff(fact_441_mult__cancel__left1,axiom,
    ! [A: $tType] :
      ( ring_15535105094025558882visors(A)
     => ! [C2: A,B2: A] :
          ( ( C2 = aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2) )
        <=> ( ( C2 = zero_zero(A) )
            | ( B2 = one_one(A) ) ) ) ) ).

% mult_cancel_left1
tff(fact_442_mult__neg__numeral__simps_I1_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [M: 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),M))),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),M),N)) ) ).

% mult_neg_numeral_simps(1)
tff(fact_443_mult__neg__numeral__simps_I2_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [M: 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),M))),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),M),N))) ) ).

% mult_neg_numeral_simps(2)
tff(fact_444_mult__neg__numeral__simps_I3_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [M: num,N: num] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),M)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),M),N))) ) ).

% mult_neg_numeral_simps(3)
tff(fact_445_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_446_zero__reorient,axiom,
    ! [A: $tType] :
      ( zero(A)
     => ! [X: A] :
          ( ( zero_zero(A) = X )
        <=> ( X = zero_zero(A) ) ) ) ).

% zero_reorient
tff(fact_447_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_448_fun__cong__unused__0,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( zero(B)
     => ! [F: fun(fun(A,B),C),G: C] :
          ( ! [X2: fun(A,B)] : aa(fun(A,B),C,F,X2) = G
         => ( aa(fun(A,B),C,F,aTP_Lamp_ba(A,B)) = G ) ) ) ).

% fun_cong_unused_0
tff(fact_449_numeral__neq__neg__numeral,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [M: num,N: num] : aa(num,A,numeral_numeral(A),M) != aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N)) ) ).

% numeral_neq_neg_numeral
tff(fact_450_neg__numeral__neq__numeral,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [M: num,N: num] : aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M)) != aa(num,A,numeral_numeral(A),N) ) ).

% neg_numeral_neq_numeral
tff(fact_451_mult__right__cancel,axiom,
    ! [A: $tType] :
      ( semiri6575147826004484403cancel(A)
     => ! [C2: A,A3: A,B2: A] :
          ( ( C2 != zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2) = aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2) )
          <=> ( A3 = B2 ) ) ) ) ).

% mult_right_cancel
tff(fact_452_mult__left__cancel,axiom,
    ! [A: $tType] :
      ( semiri6575147826004484403cancel(A)
     => ! [C2: A,A3: A,B2: A] :
          ( ( C2 != zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3) = aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2) )
          <=> ( A3 = B2 ) ) ) ) ).

% mult_left_cancel
tff(fact_453_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_454_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_455_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_456_zero__neq__one,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ( zero_zero(A) != one_one(A) ) ) ).

% zero_neq_one
tff(fact_457_rel__restrict__lift,axiom,
    ! [A: $tType,X: A,Y: A,E3: set(product_prod(A,A)),R: set(A)] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),rel_restrict(A,E3,R))
     => aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),E3) ) ).

% rel_restrict_lift
tff(fact_458_rel__restrictI,axiom,
    ! [A: $tType,X: A,R: set(A),Y: A,E3: set(product_prod(A,A))] :
      ( ~ aa(set(A),$o,member(A,X),R)
     => ( ~ aa(set(A),$o,member(A,Y),R)
       => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),E3)
         => aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),rel_restrict(A,E3,R)) ) ) ) ).

% rel_restrictI
tff(fact_459_rel__restrict__notR_I1_J,axiom,
    ! [A: $tType,X: A,Y: A,A4: set(product_prod(A,A)),R: set(A)] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),rel_restrict(A,A4,R))
     => ~ aa(set(A),$o,member(A,X),R) ) ).

% rel_restrict_notR(1)
tff(fact_460_rel__restrict__notR_I2_J,axiom,
    ! [A: $tType,X: A,Y: A,A4: set(product_prod(A,A)),R: set(A)] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),rel_restrict(A,A4,R))
     => ~ aa(set(A),$o,member(A,Y),R) ) ).

% rel_restrict_notR(2)
tff(fact_461_rel__restrict__union,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A4: set(A),B3: set(A)] : rel_restrict(A,R,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3)) = rel_restrict(A,rel_restrict(A,R,A4),B3) ).

% rel_restrict_union
tff(fact_462_Id__on__iff,axiom,
    ! [A: $tType,X: A,Y: A,A4: set(A)] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),id_on(A,A4))
    <=> ( ( X = Y )
        & aa(set(A),$o,member(A,X),A4) ) ) ).

% Id_on_iff
tff(fact_463_Id__on__eqI,axiom,
    ! [A: $tType,A3: A,B2: A,A4: set(A)] :
      ( ( A3 = B2 )
     => ( aa(set(A),$o,member(A,A3),A4)
       => aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),id_on(A,A4)) ) ) ).

% Id_on_eqI
tff(fact_464_Id__onE,axiom,
    ! [A: $tType,C2: product_prod(A,A),A4: set(A)] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),C2),id_on(A,A4))
     => ~ ! [X2: A] :
            ( aa(set(A),$o,member(A,X2),A4)
           => ( C2 != aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),X2) ) ) ) ).

% Id_onE
tff(fact_465_lambda__zero,axiom,
    ! [A: $tType] :
      ( mult_zero(A)
     => ( aTP_Lamp_bb(A,A) = aa(A,fun(A,A),times_times(A),zero_zero(A)) ) ) ).

% lambda_zero
tff(fact_466_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_467_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_468_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_469_pairself_Osimps,axiom,
    ! [A: $tType,B: $tType,F: fun(B,A),A3: B,B2: B] : aa(product_prod(B,B),product_prod(A,A),pairself(B,A,F),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),A3),B2)) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(B,A,F,A3)),aa(B,A,F,B2)) ).

% pairself.simps
tff(fact_470_pairself_Oelims,axiom,
    ! [A: $tType,B: $tType,X: fun(B,A),Xa: product_prod(B,B),Y: product_prod(A,A)] :
      ( ( aa(product_prod(B,B),product_prod(A,A),pairself(B,A,X),Xa) = Y )
     => ~ ! [A6: B,B5: B] :
            ( ( Xa = aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),A6),B5) )
           => ( Y != aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(B,A,X,A6)),aa(B,A,X,B5)) ) ) ) ).

% pairself.elims
tff(fact_471_bijective__def,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,B))] :
      ( bijective(A,B,R)
    <=> ( ! [X4: A,Y3: B,Z4: B] :
            ( ( aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Y3)),R)
              & aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Z4)),R) )
           => ( Y3 = Z4 ) )
        & ! [X4: A,Y3: A,Z4: B] :
            ( ( aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Z4)),R)
              & aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Y3),Z4)),R) )
           => ( X4 = Y3 ) ) ) ) ).

% bijective_def
tff(fact_472_semiring__norm_I170_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [V: 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),V))),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),V),W2)))),Y) ) ).

% semiring_norm(170)
tff(fact_473_semiring__norm_I171_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [V: num,W2: num,Y: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V)),aa(A,A,aa(A,fun(A,A),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),V),W2)))),Y) ) ).

% semiring_norm(171)
tff(fact_474_semiring__norm_I172_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [V: 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),V))),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),V),W2))),Y) ) ).

% semiring_norm(172)
tff(fact_475_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_476_dbl__simps_I1_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [K: num] : neg_numeral_dbl(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),K))) = aa(A,A,uminus_uminus(A),neg_numeral_dbl(A,aa(num,A,numeral_numeral(A),K))) ) ).

% dbl_simps(1)
tff(fact_477_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B2: A,W2: num] :
          ( ( A3 = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))) )
        <=> $ite(aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)) != zero_zero(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))) = B2,A3 = zero_zero(A)) ) ) ).

% eq_divide_eq_numeral1(2)
tff(fact_478_divide__eq__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,W2: num,A3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))) = A3 )
        <=> $ite(aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)) != zero_zero(A),B2 = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),A3 = zero_zero(A)) ) ) ).

% divide_eq_eq_numeral1(2)
tff(fact_479_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_480_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_481_divides__aux__eq,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Q3: A,R2: A] :
          ( unique5940410009612947441es_aux(A,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q3),R2))
        <=> ( R2 = zero_zero(A) ) ) ) ).

% divides_aux_eq
tff(fact_482_minus__assn__def,axiom,
    ! [A3: assn,B2: assn] : aa(assn,assn,aa(assn,fun(assn,assn),minus_minus(assn),A3),B2) = aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),A3),aa(assn,assn,uminus_uminus(assn),B2)) ).

% minus_assn_def
tff(fact_483_minus__apply,axiom,
    ! [A: $tType,B: $tType] :
      ( minus(A)
     => ! [A4: fun(B,A),B3: fun(B,A),X: B] : aa(B,A,aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),minus_minus(fun(B,A)),A4),B3),X) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(B,A,A4,X)),aa(B,A,B3,X)) ) ).

% minus_apply
tff(fact_484_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_485_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_486_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_487_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_488_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_489_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_490_div__by__1,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),one_one(A)) = A3 ) ).

% div_by_1
tff(fact_491_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_492_left__diff__distrib__numeral,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & ring(A) )
     => ! [A3: A,B2: A,V: 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),V)) = 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),V))),aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(num,A,numeral_numeral(A),V))) ) ).

% left_diff_distrib_numeral
tff(fact_493_right__diff__distrib__numeral,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & ring(A) )
     => ! [V: num,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V)),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V)),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V)),C2)) ) ).

% right_diff_distrib_numeral
tff(fact_494_verit__minus__simplify_I3_J,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),zero_zero(A)),B2) = aa(A,A,uminus_uminus(A),B2) ) ).

% verit_minus_simplify(3)
tff(fact_495_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_496_nonzero__mult__div__cancel__left,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A3: A,B2: A] :
          ( ( A3 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),A3) = B2 ) ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_497_nonzero__mult__div__cancel__right,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [B2: A,A3: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),B2) = A3 ) ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_498_div__self,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A3: A] :
          ( ( A3 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),A3) = one_one(A) ) ) ) ).

% div_self
tff(fact_499_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_500_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B2: A,W2: num] :
          ( ( A3 = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(num,A,numeral_numeral(A),W2)) )
        <=> $ite(aa(num,A,numeral_numeral(A),W2) != zero_zero(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),W2)) = B2,A3 = zero_zero(A)) ) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_501_divide__eq__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,W2: num,A3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(num,A,numeral_numeral(A),W2)) = A3 )
        <=> $ite(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)),A3 = zero_zero(A)) ) ) ).

% divide_eq_eq_numeral1(1)
tff(fact_502_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_503_fun__diff__def,axiom,
    ! [B: $tType,A: $tType] :
      ( minus(B)
     => ! [A4: fun(A,B),B3: fun(A,B),X3: A] : aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),minus_minus(fun(A,B)),A4),B3),X3) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,A4,X3)),aa(A,B,B3,X3)) ) ).

% fun_diff_def
tff(fact_504_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [A3: A,C2: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),C2)),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)),C2) ) ).

% cancel_ab_semigroup_add_class.diff_right_commute
tff(fact_505_diff__eq__diff__eq,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B2: A,C2: 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),C2),D3) )
         => ( ( A3 = B2 )
          <=> ( C2 = D3 ) ) ) ) ).

% diff_eq_diff_eq
tff(fact_506_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_507_left__diff__distrib,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ).

% left_diff_distrib
tff(fact_508_right__diff__distrib,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A3: A,B2: A,C2: 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),C2)) = 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),C2)) ) ).

% right_diff_distrib
tff(fact_509_left__diff__distrib_H,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [B2: A,C2: 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),C2)),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),C2),A3)) ) ).

% left_diff_distrib'
tff(fact_510_right__diff__distrib_H,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A3: A,B2: A,C2: 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),C2)) = 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),C2)) ) ).

% right_diff_distrib'
tff(fact_511_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_512_not__iszero__1,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ~ ring_1_iszero(A,one_one(A)) ) ).

% not_iszero_1
tff(fact_513_diff__eq,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Y) = aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(A,A,uminus_uminus(A),Y)) ) ).

% diff_eq
tff(fact_514_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_515_eq__divide__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [W2: num,B2: A,C2: A] :
          ( ( aa(num,A,numeral_numeral(A),W2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) )
        <=> $ite(C2 != zero_zero(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),C2) = B2,aa(num,A,numeral_numeral(A),W2) = zero_zero(A)) ) ) ).

% eq_divide_eq_numeral(1)
tff(fact_516_divide__eq__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,C2: A,W2: num] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) = aa(num,A,numeral_numeral(A),W2) )
        <=> $ite(C2 != zero_zero(A),B2 = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),C2),aa(num,A,numeral_numeral(A),W2) = zero_zero(A)) ) ) ).

% divide_eq_eq_numeral(1)
tff(fact_517_divide__eq__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,C2: A,W2: num] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)) )
        <=> $ite(C2 != zero_zero(A),B2 = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),C2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)) = zero_zero(A)) ) ) ).

% divide_eq_eq_numeral(2)
tff(fact_518_eq__divide__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [W2: num,B2: A,C2: A] :
          ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) )
        <=> $ite(C2 != zero_zero(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),C2) = B2,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)) = zero_zero(A)) ) ) ).

% eq_divide_eq_numeral(2)
tff(fact_519_nonzero__divide__mult__cancel__left,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A,B2: A] :
          ( ( A3 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),B2) ) ) ) ).

% nonzero_divide_mult_cancel_left
tff(fact_520_nonzero__divide__mult__cancel__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [B2: A,A3: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A3) ) ) ) ).

% nonzero_divide_mult_cancel_right
tff(fact_521_divide__minus1,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),X),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),X) ) ).

% divide_minus1
tff(fact_522_div__minus1__right,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),A3) ) ).

% div_minus1_right
tff(fact_523_divide__eq__1__iff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) = one_one(A) )
        <=> ( ( B2 != zero_zero(A) )
            & ( A3 = B2 ) ) ) ) ).

% divide_eq_1_iff
tff(fact_524_one__eq__divide__iff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A,B2: A] :
          ( ( one_one(A) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) )
        <=> ( ( B2 != zero_zero(A) )
            & ( A3 = B2 ) ) ) ) ).

% one_eq_divide_iff
tff(fact_525_divide__self,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A] :
          ( ( A3 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),A3) = one_one(A) ) ) ) ).

% divide_self
tff(fact_526_divide__self__if,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A] :
          aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),A3) = $ite(A3 = zero_zero(A),zero_zero(A),one_one(A)) ) ).

% divide_self_if
tff(fact_527_divide__eq__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3) = one_one(A) )
        <=> ( ( A3 != zero_zero(A) )
            & ( A3 = B2 ) ) ) ) ).

% divide_eq_eq_1
tff(fact_528_eq__divide__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A3: A] :
          ( ( one_one(A) = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3) )
        <=> ( ( A3 != zero_zero(A) )
            & ( A3 = B2 ) ) ) ) ).

% eq_divide_eq_1
tff(fact_529_Diff__empty,axiom,
    ! [A: $tType,A4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),bot_bot(set(A))) = A4 ).

% Diff_empty
tff(fact_530_empty__Diff,axiom,
    ! [A: $tType,A4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),bot_bot(set(A))),A4) = bot_bot(set(A)) ).

% empty_Diff
tff(fact_531_Diff__cancel,axiom,
    ! [A: $tType,A4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),A4) = bot_bot(set(A)) ).

% Diff_cancel
tff(fact_532_Un__Diff__cancel,axiom,
    ! [A: $tType,A4: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B3),A4)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3) ).

% Un_Diff_cancel
tff(fact_533_Un__Diff__cancel2,axiom,
    ! [A: $tType,B3: set(A),A4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B3),A4)),A4) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B3),A4) ).

% Un_Diff_cancel2
tff(fact_534_times__divide__eq__right,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),C2) ) ).

% times_divide_eq_right
tff(fact_535_divide__divide__eq__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),B2) ) ).

% divide_divide_eq_right
tff(fact_536_divide__divide__eq__left,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ).

% divide_divide_eq_left
tff(fact_537_times__divide__eq__left,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [B2: A,C2: A,A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),A3) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A3)),C2) ) ).

% times_divide_eq_left
tff(fact_538_Diff__UNIV,axiom,
    ! [A: $tType,A4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),top_top(set(A))) = bot_bot(set(A)) ).

% Diff_UNIV
tff(fact_539_div__minus__minus,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,uminus_uminus(A),A3)),aa(A,A,uminus_uminus(A),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) ) ).

% div_minus_minus
tff(fact_540_Diff__disjoint,axiom,
    ! [A: $tType,A4: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B3),A4)) = bot_bot(set(A)) ).

% Diff_disjoint
tff(fact_541_Diff__Compl,axiom,
    ! [A: $tType,A4: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(set(A),set(A),uminus_uminus(set(A)),B3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3) ).

% Diff_Compl
tff(fact_542_inter__compl__diff__conv,axiom,
    ! [A: $tType,A4: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),aa(set(A),set(A),uminus_uminus(set(A)),B3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),B3) ).

% inter_compl_diff_conv
tff(fact_543_Compl__Diff__eq,axiom,
    ! [A: $tType,A4: set(A),B3: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),B3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),A4)),B3) ).

% Compl_Diff_eq
tff(fact_544_div__mult__mult1__if,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,A3: A,B2: A] :
          aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) = $ite(C2 = zero_zero(A),zero_zero(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)) ) ).

% div_mult_mult1_if
tff(fact_545_div__mult__mult2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,A3: A,B2: A] :
          ( ( C2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) ) ) ) ).

% div_mult_mult2
tff(fact_546_div__mult__mult1,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,A3: A,B2: A] :
          ( ( C2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) ) ) ) ).

% div_mult_mult1
tff(fact_547_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: A,A3: A,B2: A] :
          ( ( C2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) ) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
tff(fact_548_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: A,A3: A,B2: A] :
          ( ( C2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) ) ) ) ).

% nonzero_mult_divide_mult_cancel_right
tff(fact_549_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: A,A3: A,B2: A] :
          ( ( C2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) ) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
tff(fact_550_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: A,A3: A,B2: A] :
          ( ( C2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) ) ) ) ).

% nonzero_mult_divide_mult_cancel_left
tff(fact_551_mult__divide__mult__cancel__left__if,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: A,A3: A,B2: A] :
          aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) = $ite(C2 = zero_zero(A),zero_zero(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)) ) ).

% mult_divide_mult_cancel_left_if
tff(fact_552_zero__eq__1__divide__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( ( zero_zero(A) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A3) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% zero_eq_1_divide_iff
tff(fact_553_one__divide__eq__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A3) = zero_zero(A) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% one_divide_eq_0_iff
tff(fact_554_Int__Diff,axiom,
    ! [A: $tType,A4: set(A),B3: set(A),C3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3)),C3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B3),C3)) ).

% Int_Diff
tff(fact_555_Diff__Int2,axiom,
    ! [A: $tType,A4: set(A),C3: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),C3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),C3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),C3)),B3) ).

% Diff_Int2
tff(fact_556_Diff__Diff__Int,axiom,
    ! [A: $tType,A4: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),B3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3) ).

% Diff_Diff_Int
tff(fact_557_Diff__Int__distrib,axiom,
    ! [A: $tType,C3: set(A),A4: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),C3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),B3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),C3),A4)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),C3),B3)) ).

% Diff_Int_distrib
tff(fact_558_Diff__Int__distrib2,axiom,
    ! [A: $tType,A4: set(A),B3: set(A),C3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),B3)),C3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),C3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),C3)) ).

% Diff_Int_distrib2
tff(fact_559_Un__Diff,axiom,
    ! [A: $tType,A4: set(A),B3: set(A),C3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3)),C3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),C3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B3),C3)) ).

% Un_Diff
tff(fact_560_set__diff__diff__left,axiom,
    ! [A: $tType,A4: set(A),B3: set(A),C3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),B3)),C3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B3),C3)) ).

% set_diff_diff_left
tff(fact_561_Diff__triv,axiom,
    ! [A: $tType,A4: set(A),B3: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3) = bot_bot(set(A)) )
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),B3) = A4 ) ) ).

% Diff_triv
tff(fact_562_Int__Diff__disjoint,axiom,
    ! [A: $tType,A4: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),B3)) = bot_bot(set(A)) ).

% Int_Diff_disjoint
tff(fact_563_disjoint__alt__simp1,axiom,
    ! [A: $tType,A4: set(A),B3: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),B3) = A4 )
    <=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3) = bot_bot(set(A)) ) ) ).

% disjoint_alt_simp1
tff(fact_564_disjoint__alt__simp2,axiom,
    ! [A: $tType,A4: set(A),B3: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),B3) != A4 )
    <=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3) != bot_bot(set(A)) ) ) ).

% disjoint_alt_simp2
tff(fact_565_Diff__Un,axiom,
    ! [A: $tType,A4: set(A),B3: set(A),C3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B3),C3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),B3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),C3)) ).

% Diff_Un
tff(fact_566_Diff__Int,axiom,
    ! [A: $tType,A4: set(A),B3: set(A),C3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),C3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),B3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),C3)) ).

% Diff_Int
tff(fact_567_Int__Diff__Un,axiom,
    ! [A: $tType,A4: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),B3)) = A4 ).

% Int_Diff_Un
tff(fact_568_Un__Diff__Int,axiom,
    ! [A: $tType,A4: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),B3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3)) = A4 ).

% Un_Diff_Int
tff(fact_569_Compl__eq__Diff__UNIV,axiom,
    ! [A: $tType,A4: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),A4) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),top_top(set(A))),A4) ).

% Compl_eq_Diff_UNIV
tff(fact_570_divide__divide__eq__left_H,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) ) ).

% divide_divide_eq_left'
tff(fact_571_divide__divide__times__eq,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [X: A,Y: A,Z2: A,W2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Z2),W2)) = aa(A,A,aa(A,fun(A,A),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_572_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),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Z2),W2)) = aa(A,A,aa(A,fun(A,A),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_573_div__minus__right,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,uminus_uminus(A),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,uminus_uminus(A),A3)),B2) ) ).

% div_minus_right
tff(fact_574_minus__divide__left,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B2: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,uminus_uminus(A),A3)),B2) ) ).

% minus_divide_left
tff(fact_575_minus__divide__divide,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,uminus_uminus(A),A3)),aa(A,A,uminus_uminus(A),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) ) ).

% minus_divide_divide
tff(fact_576_minus__divide__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A,B2: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,uminus_uminus(A),B2)) ) ).

% minus_divide_right
tff(fact_577_nonzero__eq__divide__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [C2: A,A3: A,B2: A] :
          ( ( C2 != zero_zero(A) )
         => ( ( A3 = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) )
          <=> ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2) = B2 ) ) ) ) ).

% nonzero_eq_divide_eq
tff(fact_578_nonzero__divide__eq__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [C2: A,B2: A,A3: A] :
          ( ( C2 != zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) = A3 )
          <=> ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2) ) ) ) ) ).

% nonzero_divide_eq_eq
tff(fact_579_eq__divide__imp,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [C2: A,A3: A,B2: A] :
          ( ( C2 != zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2) = B2 )
           => ( A3 = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) ) ) ) ) ).

% eq_divide_imp
tff(fact_580_divide__eq__imp,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [C2: A,B2: A,A3: A] :
          ( ( C2 != zero_zero(A) )
         => ( ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2) )
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) = A3 ) ) ) ) ).

% divide_eq_imp
tff(fact_581_eq__divide__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B2: A,C2: A] :
          ( ( A3 = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) )
        <=> $ite(C2 != zero_zero(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2) = B2,A3 = zero_zero(A)) ) ) ).

% eq_divide_eq
tff(fact_582_divide__eq__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,C2: A,A3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) = A3 )
        <=> $ite(C2 != zero_zero(A),B2 = aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2),A3 = zero_zero(A)) ) ) ).

% divide_eq_eq
tff(fact_583_frac__eq__eq,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y: A,Z2: A,X: A,W2: A] :
          ( ( Y != zero_zero(A) )
         => ( ( Z2 != zero_zero(A) )
           => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y) = aa(A,A,aa(A,fun(A,A),divide_divide(A),W2),Z2) )
            <=> ( aa(A,A,aa(A,fun(A,A),times_times(A),X),Z2) = aa(A,A,aa(A,fun(A,A),times_times(A),W2),Y) ) ) ) ) ) ).

% frac_eq_eq
tff(fact_584_right__inverse__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,A3: A] :
          ( ( B2 != zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) = one_one(A) )
          <=> ( A3 = B2 ) ) ) ) ).

% right_inverse_eq
tff(fact_585_nonzero__minus__divide__right,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,A3: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,uminus_uminus(A),B2)) ) ) ) ).

% nonzero_minus_divide_right
tff(fact_586_nonzero__minus__divide__divide,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,A3: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,uminus_uminus(A),A3)),aa(A,A,uminus_uminus(A),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) ) ) ) ).

% nonzero_minus_divide_divide
tff(fact_587_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,aa(A,fun(A,A),divide_divide(A),X),Z2)),Y) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z2))),Z2) ) ) ) ).

% divide_diff_eq_iff
tff(fact_588_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),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y),Z2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),Z2)),Y)),Z2) ) ) ) ).

% diff_divide_eq_iff
tff(fact_589_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),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)),aa(A,A,aa(A,fun(A,A),divide_divide(A),W2),Z2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),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_590_add__divide__eq__if__simps_I4_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B2: A,Z2: A] :
          aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),Z2)) = $ite(Z2 = zero_zero(A),A3,aa(A,A,aa(A,fun(A,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),A3),Z2)),B2)),Z2)) ) ).

% add_divide_eq_if_simps(4)
tff(fact_591_nonzero__neg__divide__eq__eq2,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,C2: A,A3: A] :
          ( ( B2 != zero_zero(A) )
         => ( ( C2 = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)) )
          <=> ( aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2) = aa(A,A,uminus_uminus(A),A3) ) ) ) ) ).

% nonzero_neg_divide_eq_eq2
tff(fact_592_nonzero__neg__divide__eq__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,A3: A,C2: A] :
          ( ( B2 != zero_zero(A) )
         => ( ( aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)) = C2 )
          <=> ( aa(A,A,uminus_uminus(A),A3) = aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2) ) ) ) ) ).

% nonzero_neg_divide_eq_eq
tff(fact_593_minus__divide__eq__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,C2: A,A3: A] :
          ( ( aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) = A3 )
        <=> $ite(C2 != zero_zero(A),aa(A,A,uminus_uminus(A),B2) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2),A3 = zero_zero(A)) ) ) ).

% minus_divide_eq_eq
tff(fact_594_eq__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B2: A,C2: A] :
          ( ( A3 = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) )
        <=> $ite(C2 != zero_zero(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2) = aa(A,A,uminus_uminus(A),B2),A3 = zero_zero(A)) ) ) ).

% eq_minus_divide_eq
tff(fact_595_divide__eq__minus__1__iff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,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_596_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),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Z2))),Y) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),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_597_add__divide__eq__if__simps_I5_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,Z2: A,B2: A] :
          aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),Z2)),B2) = $ite(Z2 = zero_zero(A),aa(A,A,uminus_uminus(A),B2),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,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_598_add__divide__eq__if__simps_I6_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,Z2: A,B2: A] :
          aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),Z2))),B2) = $ite(Z2 = zero_zero(A),aa(A,A,uminus_uminus(A),B2),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,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_599_bits__div__by__1,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),one_one(A)) = A3 ) ).

% bits_div_by_1
tff(fact_600_inf__period_I1_J,axiom,
    ! [A: $tType] :
      ( ( comm_ring(A)
        & dvd(A) )
     => ! [P: fun(A,$o),D4: A,Q: fun(A,$o)] :
          ( ! [X2: A,K2: A] :
              ( aa(A,$o,P,X2)
            <=> aa(A,$o,P,aa(A,A,aa(A,fun(A,A),minus_minus(A),X2),aa(A,A,aa(A,fun(A,A),times_times(A),K2),D4))) )
         => ( ! [X2: A,K2: A] :
                ( aa(A,$o,Q,X2)
              <=> aa(A,$o,Q,aa(A,A,aa(A,fun(A,A),minus_minus(A),X2),aa(A,A,aa(A,fun(A,A),times_times(A),K2),D4))) )
           => ! [X3: A,K3: A] :
                ( ( aa(A,$o,P,X3)
                  & aa(A,$o,Q,X3) )
              <=> ( aa(A,$o,P,aa(A,A,aa(A,fun(A,A),minus_minus(A),X3),aa(A,A,aa(A,fun(A,A),times_times(A),K3),D4)))
                  & aa(A,$o,Q,aa(A,A,aa(A,fun(A,A),minus_minus(A),X3),aa(A,A,aa(A,fun(A,A),times_times(A),K3),D4))) ) ) ) ) ) ).

% inf_period(1)
tff(fact_601_inf__period_I2_J,axiom,
    ! [A: $tType] :
      ( ( comm_ring(A)
        & dvd(A) )
     => ! [P: fun(A,$o),D4: A,Q: fun(A,$o)] :
          ( ! [X2: A,K2: A] :
              ( aa(A,$o,P,X2)
            <=> aa(A,$o,P,aa(A,A,aa(A,fun(A,A),minus_minus(A),X2),aa(A,A,aa(A,fun(A,A),times_times(A),K2),D4))) )
         => ( ! [X2: A,K2: A] :
                ( aa(A,$o,Q,X2)
              <=> aa(A,$o,Q,aa(A,A,aa(A,fun(A,A),minus_minus(A),X2),aa(A,A,aa(A,fun(A,A),times_times(A),K2),D4))) )
           => ! [X3: A,K3: A] :
                ( ( aa(A,$o,P,X3)
                  | aa(A,$o,Q,X3) )
              <=> ( aa(A,$o,P,aa(A,A,aa(A,fun(A,A),minus_minus(A),X3),aa(A,A,aa(A,fun(A,A),times_times(A),K3),D4)))
                  | aa(A,$o,Q,aa(A,A,aa(A,fun(A,A),minus_minus(A),X3),aa(A,A,aa(A,fun(A,A),times_times(A),K3),D4))) ) ) ) ) ) ).

% inf_period(2)
tff(fact_602_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_603_diff__numeral__special_I6_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: num] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(num,A,numeral_numeral(A),M)),aa(A,A,uminus_uminus(A),one_one(A))) = aa(num,A,numeral_numeral(A),inc(M)) ) ).

% diff_numeral_special(6)
tff(fact_604_left__diff__distrib__NO__MATCH,axiom,
    ! [A: $tType,B: $tType] :
      ( ring(B)
     => ! [X: A,Y: A,C2: B,A3: B,B2: B] :
          ( nO_MATCH(A,B,aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y),C2)
         => ( aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),A3),B2)),C2) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),times_times(B),A3),C2)),aa(B,B,aa(B,fun(B,B),times_times(B),B2),C2)) ) ) ) ).

% left_diff_distrib_NO_MATCH
tff(fact_605_right__diff__distrib__NO__MATCH,axiom,
    ! [A: $tType,B: $tType] :
      ( ring(B)
     => ! [X: A,Y: A,A3: B,B2: B,C2: B] :
          ( nO_MATCH(A,B,aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y),A3)
         => ( aa(B,B,aa(B,fun(B,B),times_times(B),A3),aa(B,B,aa(B,fun(B,B),minus_minus(B),B2),C2)) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),times_times(B),A3),B2)),aa(B,B,aa(B,fun(B,B),times_times(B),A3),C2)) ) ) ) ).

% right_diff_distrib_NO_MATCH
tff(fact_606_divide__less__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,W2: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),C2)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),C2)),B2),aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)))) ) ) ) ).

% divide_less_eq_numeral(2)
tff(fact_607_less__divide__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [W2: num,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),C2)),B2),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),C2)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),zero_zero(A))) ) ) ) ).

% less_divide_eq_numeral(2)
tff(fact_608_le__divide__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,W2: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)))) ) ) ).

% le_divide_eq_numeral1(2)
tff(fact_609_dual__order_Orefl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),A3) ) ).

% dual_order.refl
tff(fact_610_order__refl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),X) ) ).

% order_refl
tff(fact_611_empty__subsetI,axiom,
    ! [A: $tType,A4: set(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),bot_bot(set(A))),A4) ).

% empty_subsetI
tff(fact_612_subset__empty,axiom,
    ! [A: $tType,A4: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),bot_bot(set(A)))
    <=> ( A4 = bot_bot(set(A)) ) ) ).

% subset_empty
tff(fact_613_Int__subset__iff,axiom,
    ! [A: $tType,C3: set(A),A4: set(A),B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3))
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C3),A4)
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C3),B3) ) ) ).

% Int_subset_iff
tff(fact_614_Un__subset__iff,axiom,
    ! [A: $tType,A4: set(A),B3: set(A),C3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3)),C3)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),C3)
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),C3) ) ) ).

% Un_subset_iff
tff(fact_615_Compl__subset__Compl__iff,axiom,
    ! [A: $tType,A4: set(A),B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),A4)),aa(set(A),set(A),uminus_uminus(set(A)),B3))
    <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),A4) ) ).

% Compl_subset_Compl_iff
tff(fact_616_Compl__anti__mono,axiom,
    ! [A: $tType,A4: set(A),B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B3)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),B3)),aa(set(A),set(A),uminus_uminus(set(A)),A4)) ) ).

% Compl_anti_mono
tff(fact_617_le__zero__eq,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [N: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),N),zero_zero(A))
        <=> ( N = zero_zero(A) ) ) ) ).

% le_zero_eq
tff(fact_618_not__gr__zero,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [N: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),N)
        <=> ( N = zero_zero(A) ) ) ) ).

% not_gr_zero
tff(fact_619_neg__le__iff__le,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,uminus_uminus(A),A3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2) ) ) ).

% neg_le_iff_le
tff(fact_620_compl__le__compl__iff,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),X)),aa(A,A,uminus_uminus(A),Y))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X) ) ) ).

% compl_le_compl_iff
tff(fact_621_neg__less__iff__less,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,uminus_uminus(A),A3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2) ) ) ).

% neg_less_iff_less
tff(fact_622_compl__less__compl__iff,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),X)),aa(A,A,uminus_uminus(A),Y))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),X) ) ) ).

% compl_less_compl_iff
tff(fact_623_le__inf__iff,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [X: A,Y: A,Z2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Z2))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Z2) ) ) ) ).

% le_inf_iff
tff(fact_624_inf_Obounded__iff,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),inf_inf(A),B2),C2))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),C2) ) ) ) ).

% inf.bounded_iff
tff(fact_625_le__sup__iff,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,Y: A,Z2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y)),Z2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Z2)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z2) ) ) ) ).

% le_sup_iff
tff(fact_626_sup_Obounded__iff,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,C2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),B2),C2)),A3)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A3) ) ) ) ).

% sup.bounded_iff
tff(fact_627_Diff__eq__empty__iff,axiom,
    ! [A: $tType,A4: set(A),B3: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),B3) = bot_bot(set(A)) )
    <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B3) ) ).

% Diff_eq_empty_iff
tff(fact_628_diff__ge__0__iff__ge,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3) ) ) ).

% diff_ge_0_iff_ge
tff(fact_629_diff__gt__0__iff__gt,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3) ) ) ).

% diff_gt_0_iff_gt
tff(fact_630_neg__0__le__iff__le,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,uminus_uminus(A),A3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),zero_zero(A)) ) ) ).

% neg_0_le_iff_le
tff(fact_631_neg__le__0__iff__le,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),A3)),zero_zero(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3) ) ) ).

% neg_le_0_iff_le
tff(fact_632_less__eq__neg__nonpos,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,uminus_uminus(A),A3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),zero_zero(A)) ) ) ).

% less_eq_neg_nonpos
tff(fact_633_neg__less__eq__nonneg,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),A3)),A3)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3) ) ) ).

% neg_less_eq_nonneg
tff(fact_634_less__neg__neg,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(A,A,uminus_uminus(A),A3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A)) ) ) ).

% less_neg_neg
tff(fact_635_neg__less__pos,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),A3)),A3)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3) ) ) ).

% neg_less_pos
tff(fact_636_neg__0__less__iff__less,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,uminus_uminus(A),A3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A)) ) ) ).

% neg_0_less_iff_less
tff(fact_637_neg__less__0__iff__less,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),A3)),zero_zero(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3) ) ) ).

% neg_less_0_iff_less
tff(fact_638_neg__numeral__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num,N: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N)))
        <=> aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),N),M) ) ) ).

% neg_numeral_le_iff
tff(fact_639_neg__numeral__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num,N: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N)))
        <=> aa(num,$o,aa(num,fun(num,$o),ord_less(num),N),M) ) ) ).

% neg_numeral_less_iff
tff(fact_640_divide__le__0__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A3)),zero_zero(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),zero_zero(A)) ) ) ).

% divide_le_0_1_iff
tff(fact_641_zero__le__divide__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3) ) ) ).

% zero_le_divide_1_iff
tff(fact_642_divide__less__0__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A3)),zero_zero(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A)) ) ) ).

% divide_less_0_1_iff
tff(fact_643_divide__less__eq__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3)),one_one(A))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2) ) ) ) ).

% divide_less_eq_1_neg
tff(fact_644_divide__less__eq__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3)),one_one(A))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3) ) ) ) ).

% divide_less_eq_1_pos
tff(fact_645_less__divide__eq__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3) ) ) ) ).

% less_divide_eq_1_neg
tff(fact_646_less__divide__eq__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2) ) ) ) ).

% less_divide_eq_1_pos
tff(fact_647_zero__less__divide__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3) ) ) ).

% zero_less_divide_1_iff
tff(fact_648_le__divide__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,W2: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(num,A,numeral_numeral(A),W2)))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),W2))),B2) ) ) ).

% le_divide_eq_numeral1(1)
tff(fact_649_divide__le__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,W2: num,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(num,A,numeral_numeral(A),W2))),A3)
        <=> aa(A,$o,aa(A,fun(A,$o),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_650_less__divide__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,W2: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(num,A,numeral_numeral(A),W2)))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),W2))),B2) ) ) ).

% less_divide_eq_numeral1(1)
tff(fact_651_divide__less__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,W2: num,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(num,A,numeral_numeral(A),W2))),A3)
        <=> aa(A,$o,aa(A,fun(A,$o),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_652_le__divide__eq__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2) ) ) ) ).

% le_divide_eq_1_pos
tff(fact_653_le__divide__eq__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3) ) ) ) ).

% le_divide_eq_1_neg
tff(fact_654_divide__le__eq__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3)),one_one(A))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3) ) ) ) ).

% divide_le_eq_1_pos
tff(fact_655_divide__le__eq__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3)),one_one(A))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2) ) ) ) ).

% divide_le_eq_1_neg
tff(fact_656_divide__le__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,W2: num,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)))),A3)
        <=> aa(A,$o,aa(A,fun(A,$o),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_657_divide__less__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,W2: num,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)))),A3)
        <=> aa(A,$o,aa(A,fun(A,$o),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_658_less__divide__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,W2: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)))) ) ) ).

% less_divide_eq_numeral1(2)
tff(fact_659_minus__set__def,axiom,
    ! [A: $tType,A4: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),B3) = aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aa(fun(A,$o),fun(fun(A,$o),fun(A,$o)),minus_minus(fun(A,$o)),aa(set(A),fun(A,$o),aTP_Lamp_ag(set(A),fun(A,$o)),A4)),aa(set(A),fun(A,$o),aTP_Lamp_ag(set(A),fun(A,$o)),B3))) ).

% minus_set_def
tff(fact_660_set__diff__eq,axiom,
    ! [A: $tType,A4: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),B3) = aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_bc(set(A),fun(set(A),fun(A,$o)),A4),B3)) ).

% set_diff_eq
tff(fact_661_order__less__imp__not__less,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),X) ) ) ).

% order_less_imp_not_less
tff(fact_662_order__le__imp__less__or__eq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
            | ( X = Y ) ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_663_linorder__le__less__linear,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
          | aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),X) ) ) ).

% linorder_le_less_linear
tff(fact_664_order__less__imp__not__eq2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
         => ( Y != X ) ) ) ).

% order_less_imp_not_eq2
tff(fact_665_order__less__imp__not__eq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
         => ( X != Y ) ) ) ).

% order_less_imp_not_eq
tff(fact_666_order__less__le__subst2,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A3: A,B2: A,F: fun(A,B),C2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,B2)),C2)
           => ( ! [X2: A,Y2: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X2),Y2)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F,X2)),aa(A,B,F,Y2)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F,A3)),C2) ) ) ) ) ).

% order_less_le_subst2
tff(fact_667_order__less__le__subst1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A3: A,F: fun(B,A),B2: B,C2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(B,A,F,B2))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),C2)
           => ( ! [X2: B,Y2: B] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X2),Y2)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,F,X2)),aa(B,A,F,Y2)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(B,A,F,C2)) ) ) ) ) ).

% order_less_le_subst1
tff(fact_668_order__le__less__subst2,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A3: A,B2: A,F: fun(A,B),C2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F,B2)),C2)
           => ( ! [X2: A,Y2: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),Y2)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,X2)),aa(A,B,F,Y2)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F,A3)),C2) ) ) ) ) ).

% order_le_less_subst2
tff(fact_669_order__le__less__subst1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A3: A,F: fun(B,A),B2: B,C2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(B,A,F,B2))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),C2)
           => ( ! [X2: B,Y2: B] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X2),Y2)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,F,X2)),aa(B,A,F,Y2)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(B,A,F,C2)) ) ) ) ) ).

% order_le_less_subst1
tff(fact_670_linorder__less__linear,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
          | ( X = Y )
          | aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),X) ) ) ).

% linorder_less_linear
tff(fact_671_order__less__le__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A,Z2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Z2) ) ) ) ).

% order_less_le_trans
tff(fact_672_order__less__imp__triv,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A,P: $o] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),X)
           => (P) ) ) ) ).

% order_less_imp_triv
tff(fact_673_order__le__less__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A,Z2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Z2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Z2) ) ) ) ).

% order_le_less_trans
tff(fact_674_order__neq__le__trans,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B2: A] :
          ( ( A3 != B2 )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2) ) ) ) ).

% order_neq_le_trans
tff(fact_675_order__less__not__sym,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),X) ) ) ).

% order_less_not_sym
tff(fact_676_order__le__neq__trans,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
         => ( ( A3 != B2 )
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2) ) ) ) ).

% order_le_neq_trans
tff(fact_677_order__antisym__conv,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Y: A,X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
          <=> ( X = Y ) ) ) ) ).

% order_antisym_conv
tff(fact_678_order__less__subst2,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A3: A,B2: A,F: fun(A,B),C2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F,B2)),C2)
           => ( ! [X2: A,Y2: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X2),Y2)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F,X2)),aa(A,B,F,Y2)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F,A3)),C2) ) ) ) ) ).

% order_less_subst2
tff(fact_679_order__less__subst1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A3: A,F: fun(B,A),B2: B,C2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(B,A,F,B2))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),C2)
           => ( ! [X2: B,Y2: B] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X2),Y2)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,F,X2)),aa(B,A,F,Y2)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(B,A,F,C2)) ) ) ) ) ).

% order_less_subst1
tff(fact_680_order__less__irrefl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),X) ) ).

% order_less_irrefl
tff(fact_681_order__less__imp__le,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y) ) ) ).

% order_less_imp_le
tff(fact_682_ord__less__eq__subst,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & ord(A) )
     => ! [A3: A,B2: A,F: fun(A,B),C2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
         => ( ( aa(A,B,F,B2) = C2 )
           => ( ! [X2: A,Y2: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X2),Y2)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F,X2)),aa(A,B,F,Y2)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F,A3)),C2) ) ) ) ) ).

% ord_less_eq_subst
tff(fact_683_ord__eq__less__subst,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & ord(A) )
     => ! [A3: A,F: fun(B,A),B2: B,C2: B] :
          ( ( A3 = aa(B,A,F,B2) )
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),C2)
           => ( ! [X2: B,Y2: B] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X2),Y2)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,F,X2)),aa(B,A,F,Y2)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(B,A,F,C2)) ) ) ) ) ).

% ord_eq_less_subst
tff(fact_684_linorder__not__less,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X) ) ) ).

% linorder_not_less
tff(fact_685_linorder__le__cases,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X) ) ) ).

% linorder_le_cases
tff(fact_686_order__less__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A,Z2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Z2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Z2) ) ) ) ).

% order_less_trans
tff(fact_687_order__less__asym_H,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3) ) ) ).

% order_less_asym'
tff(fact_688_linorder__neq__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( ( X != Y )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
            | aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),X) ) ) ) ).

% linorder_neq_iff
tff(fact_689_order__less__asym,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),X) ) ) ).

% order_less_asym
tff(fact_690_ord__le__eq__subst,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & ord(A) )
     => ! [A3: A,B2: A,F: fun(A,B),C2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
         => ( ( aa(A,B,F,B2) = C2 )
           => ( ! [X2: A,Y2: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),Y2)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,X2)),aa(A,B,F,Y2)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,A3)),C2) ) ) ) ) ).

% ord_le_eq_subst
tff(fact_691_ord__eq__le__subst,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & ord(A) )
     => ! [A3: A,F: fun(B,A),B2: B,C2: B] :
          ( ( A3 = aa(B,A,F,B2) )
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),C2)
           => ( ! [X2: B,Y2: B] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X2),Y2)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,F,X2)),aa(B,A,F,Y2)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(B,A,F,C2)) ) ) ) ) ).

% ord_eq_le_subst
tff(fact_692_linorder__not__le,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),X) ) ) ).

% linorder_not_le
tff(fact_693_linorder__linear,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
          | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X) ) ) ).

% linorder_linear
tff(fact_694_order__less__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
            & ( X != Y ) ) ) ) ).

% order_less_le
tff(fact_695_order__le__less,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
            | ( X = Y ) ) ) ) ).

% order_le_less
tff(fact_696_order__eq__refl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A] :
          ( ( X = Y )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y) ) ) ).

% order_eq_refl
tff(fact_697_linorder__neqE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( ( X != Y )
         => ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),X) ) ) ) ).

% linorder_neqE
tff(fact_698_order__subst2,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A3: A,B2: A,F: fun(A,B),C2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,B2)),C2)
           => ( ! [X2: A,Y2: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),Y2)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,X2)),aa(A,B,F,Y2)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,A3)),C2) ) ) ) ) ).

% order_subst2
tff(fact_699_order__subst1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A3: A,F: fun(B,A),B2: B,C2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(B,A,F,B2))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),C2)
           => ( ! [X2: B,Y2: B] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X2),Y2)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,F,X2)),aa(B,A,F,Y2)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(B,A,F,C2)) ) ) ) ) ).

% order_subst1
tff(fact_700_Orderings_Oorder__eq__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B2: A] :
          ( ( A3 = B2 )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3) ) ) ) ).

% Orderings.order_eq_iff
tff(fact_701_le__fun__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F: fun(A,B),G: fun(A,B)] :
          ( aa(fun(A,B),$o,aa(fun(A,B),fun(fun(A,B),$o),ord_less_eq(fun(A,B)),F),G)
        <=> ! [X4: A] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,X4)),aa(A,B,G,X4)) ) ) ).

% le_fun_def
tff(fact_702_le__funI,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F: fun(A,B),G: fun(A,B)] :
          ( ! [X2: A] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,X2)),aa(A,B,G,X2))
         => aa(fun(A,B),$o,aa(fun(A,B),fun(fun(A,B),$o),ord_less_eq(fun(A,B)),F),G) ) ) ).

% le_funI
tff(fact_703_le__funE,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F: fun(A,B),G: fun(A,B),X: A] :
          ( aa(fun(A,B),$o,aa(fun(A,B),fun(fun(A,B),$o),ord_less_eq(fun(A,B)),F),G)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,X)),aa(A,B,G,X)) ) ) ).

% le_funE
tff(fact_704_le__funD,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F: fun(A,B),G: fun(A,B),X: A] :
          ( aa(fun(A,B),$o,aa(fun(A,B),fun(fun(A,B),$o),ord_less_eq(fun(A,B)),F),G)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,X)),aa(A,B,G,X)) ) ) ).

% le_funD
tff(fact_705_antisym,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3)
           => ( A3 = B2 ) ) ) ) ).

% antisym
tff(fact_706_dual__order_Ostrict__implies__order,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3) ) ) ).

% dual_order.strict_implies_order
tff(fact_707_dual__order_Ostrict__implies__not__eq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3)
         => ( A3 != B2 ) ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_708_order_Ostrict__implies__order,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2) ) ) ).

% order.strict_implies_order
tff(fact_709_dual__order_Ostrict__iff__not,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3)
            & ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2) ) ) ) ).

% dual_order.strict_iff_not
tff(fact_710_dual__order_Ostrict__trans2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),A3) ) ) ) ).

% dual_order.strict_trans2
tff(fact_711_dual__order_Ostrict__trans1,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),A3) ) ) ) ).

% dual_order.strict_trans1
tff(fact_712_order_Ostrict__implies__not__eq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
         => ( A3 != B2 ) ) ) ).

% order.strict_implies_not_eq
tff(fact_713_dual__order_Ostrict__iff__order,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3)
            & ( A3 != B2 ) ) ) ) ).

% dual_order.strict_iff_order
tff(fact_714_dual__order_Oorder__iff__strict,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3)
            | ( A3 = B2 ) ) ) ) ).

% dual_order.order_iff_strict
tff(fact_715_dual__order_Ostrict__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),A3) ) ) ) ).

% dual_order.strict_trans
tff(fact_716_dense__le__bounded,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [X: A,Y: A,Z2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
         => ( ! [W: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),W)
               => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),W),Y)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),W),Z2) ) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z2) ) ) ) ).

% dense_le_bounded
tff(fact_717_dense__ge__bounded,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [Z2: A,X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z2),X)
         => ( ! [W: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z2),W)
               => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),W),X)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),W) ) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z2) ) ) ) ).

% dense_ge_bounded
tff(fact_718_not__less__iff__gr__or__eq,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),X)
            | ( X = Y ) ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_719_order_Ostrict__iff__not,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
            & ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3) ) ) ) ).

% order.strict_iff_not
tff(fact_720_order_Ostrict__trans2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),C2) ) ) ) ).

% order.strict_trans2
tff(fact_721_order_Ostrict__trans1,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),C2) ) ) ) ).

% order.strict_trans1
tff(fact_722_order_Ostrict__iff__order,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
            & ( A3 != B2 ) ) ) ) ).

% order.strict_iff_order
tff(fact_723_order_Oorder__iff__strict,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
            | ( A3 = B2 ) ) ) ) ).

% order.order_iff_strict
tff(fact_724_order_Ostrict__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),C2) ) ) ) ).

% order.strict_trans
tff(fact_725_linorder__less__wlog,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,fun(A,$o)),A3: A,B2: A] :
          ( ! [A6: A,B5: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A6),B5)
             => aa(A,$o,aa(A,fun(A,$o),P,A6),B5) )
         => ( ! [A6: A] : aa(A,$o,aa(A,fun(A,$o),P,A6),A6)
           => ( ! [A6: A,B5: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),P,B5),A6)
                 => aa(A,$o,aa(A,fun(A,$o),P,A6),B5) )
             => aa(A,$o,aa(A,fun(A,$o),P,A3),B2) ) ) ) ) ).

% linorder_less_wlog
tff(fact_726_exists__least__iff,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P: fun(A,$o)] :
          ( ? [X_12: A] : aa(A,$o,P,X_12)
        <=> ? [N2: A] :
              ( aa(A,$o,P,N2)
              & ! [M2: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),M2),N2)
                 => ~ aa(A,$o,P,M2) ) ) ) ) ).

% exists_least_iff
tff(fact_727_dual__order_Oirrefl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),A3) ) ).

% dual_order.irrefl
tff(fact_728_dual__order_Otrans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A3) ) ) ) ).

% dual_order.trans
tff(fact_729_dual__order_Oasym,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2) ) ) ).

% dual_order.asym
tff(fact_730_dual__order_Oantisym,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
           => ( A3 = B2 ) ) ) ) ).

% dual_order.antisym
tff(fact_731_not__le__imp__less,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Y: A,X: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y) ) ) ).

% not_le_imp_less
tff(fact_732_less__le__not__le,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
            & ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X) ) ) ) ).

% less_le_not_le
tff(fact_733_dual__order_Oeq__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B2: A] :
          ( ( A3 = B2 )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2) ) ) ) ).

% dual_order.eq_iff
tff(fact_734_linorder__cases,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
         => ( ( X != Y )
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),X) ) ) ) ).

% linorder_cases
tff(fact_735_dense__le,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [Y: A,Z2: A] :
          ( ! [X2: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X2),Y)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),Z2) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z2) ) ) ).

% dense_le
tff(fact_736_dense__ge,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [Z2: A,Y: A] :
          ( ! [X2: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z2),X2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X2) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z2) ) ) ).

% dense_ge
tff(fact_737_linorder__wlog,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,fun(A,$o)),A3: A,B2: A] :
          ( ! [A6: A,B5: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A6),B5)
             => aa(A,$o,aa(A,fun(A,$o),P,A6),B5) )
         => ( ! [A6: A,B5: A] :
                ( aa(A,$o,aa(A,fun(A,$o),P,B5),A6)
               => aa(A,$o,aa(A,fun(A,$o),P,A6),B5) )
           => aa(A,$o,aa(A,fun(A,$o),P,A3),B2) ) ) ) ).

% linorder_wlog
tff(fact_738_antisym__conv3,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Y: A,X: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),X)
         => ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
          <=> ( X = Y ) ) ) ) ).

% antisym_conv3
tff(fact_739_less__induct,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P: fun(A,$o),A3: A] :
          ( ! [X2: A] :
              ( ! [Y5: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y5),X2)
                 => aa(A,$o,P,Y5) )
             => aa(A,$o,P,X2) )
         => aa(A,$o,P,A3) ) ) ).

% less_induct
tff(fact_740_ord__less__eq__trans,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
         => ( ( B2 = C2 )
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),C2) ) ) ) ).

% ord_less_eq_trans
tff(fact_741_ord__eq__less__trans,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A3: A,B2: A,C2: A] :
          ( ( A3 = B2 )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),C2) ) ) ) ).

% ord_eq_less_trans
tff(fact_742_order__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A,Z2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Z2) ) ) ) ).

% order_trans
tff(fact_743_order_Otrans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),C2) ) ) ) ).

% order.trans
tff(fact_744_order_Oasym,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3) ) ) ).

% order.asym
tff(fact_745_order__antisym,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X)
           => ( X = Y ) ) ) ) ).

% order_antisym
tff(fact_746_antisym__conv2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
         => ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
          <=> ( X = Y ) ) ) ) ).

% antisym_conv2
tff(fact_747_antisym__conv1,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
          <=> ( X = Y ) ) ) ) ).

% antisym_conv1
tff(fact_748_ord__le__eq__trans,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
         => ( ( B2 = C2 )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),C2) ) ) ) ).

% ord_le_eq_trans
tff(fact_749_ord__eq__le__trans,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A3: A,B2: A,C2: A] :
          ( ( A3 = B2 )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),C2) ) ) ) ).

% ord_eq_le_trans
tff(fact_750_order__class_Oorder__eq__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y: A] :
          ( ( X = Y )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X) ) ) ) ).

% order_class.order_eq_iff
tff(fact_751_less__imp__neq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
         => ( X != Y ) ) ) ).

% less_imp_neq
tff(fact_752_le__cases3,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A,Z2: A] :
          ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
           => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z2) )
         => ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Z2) )
           => ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Z2)
               => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z2),Y) )
             => ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z2),Y)
                 => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X) )
               => ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z2)
                   => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z2),X) )
                 => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z2),X)
                     => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y) ) ) ) ) ) ) ) ).

% le_cases3
tff(fact_753_dense,axiom,
    ! [A: $tType] :
      ( dense_order(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
         => ? [Z3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Z3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z3),Y) ) ) ) ).

% dense
tff(fact_754_nle__le,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3)
            & ( B2 != A3 ) ) ) ) ).

% nle_le
tff(fact_755_nless__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B2: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
        <=> ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
            | ( A3 = B2 ) ) ) ) ).

% nless_le
tff(fact_756_gt__ex,axiom,
    ! [A: $tType] :
      ( no_top(A)
     => ! [X: A] :
        ? [X_1: A] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),X_1) ) ).

% gt_ex
tff(fact_757_lt__ex,axiom,
    ! [A: $tType] :
      ( no_bot(A)
     => ! [X: A] :
        ? [Y2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y2),X) ) ).

% lt_ex
tff(fact_758_leI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X) ) ) ).

% leI
tff(fact_759_leD,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Y: A,X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y) ) ) ).

% leD
tff(fact_760_pred__subset__eq2,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,B)),S: set(product_prod(A,B))] :
      ( aa(fun(A,fun(B,$o)),$o,aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),$o),ord_less_eq(fun(A,fun(B,$o))),aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_as(set(product_prod(A,B)),fun(A,fun(B,$o))),R)),aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_as(set(product_prod(A,B)),fun(A,fun(B,$o))),S))
    <=> aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),R),S) ) ).

% pred_subset_eq2
tff(fact_761_mult__less__le__imp__less,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B2: A,C2: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),D3)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D3)) ) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_762_mult__le__less__imp__less,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B2: A,C2: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),D3)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D3)) ) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_763_mult__right__le__imp__le,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2) ) ) ) ).

% mult_right_le_imp_le
tff(fact_764_mult__left__le__imp__le,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [C2: A,A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2) ) ) ) ).

% mult_left_le_imp_le
tff(fact_765_mult__le__cancel__left__pos,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2) ) ) ) ).

% mult_le_cancel_left_pos
tff(fact_766_mult__le__cancel__left__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3) ) ) ) ).

% mult_le_cancel_left_neg
tff(fact_767_mult__less__cancel__right,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A3: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3) ) ) ) ) ).

% mult_less_cancel_right
tff(fact_768_mult__strict__mono_H,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B2: A,C2: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),D3)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D3)) ) ) ) ) ) ).

% mult_strict_mono'
tff(fact_769_mult__right__less__imp__less,axiom,
    ! [A: $tType] :
      ( linordered_semiring(A)
     => ! [A3: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2) ) ) ) ).

% mult_right_less_imp_less
tff(fact_770_mult__less__cancel__left,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3) ) ) ) ) ).

% mult_less_cancel_left
tff(fact_771_mult__strict__mono,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B2: A,C2: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),D3)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D3)) ) ) ) ) ) ).

% mult_strict_mono
tff(fact_772_mult__left__less__imp__less,axiom,
    ! [A: $tType] :
      ( linordered_semiring(A)
     => ! [C2: A,A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2) ) ) ) ).

% mult_left_less_imp_less
tff(fact_773_mult__le__cancel__right,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A3: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3) ) ) ) ) ).

% mult_le_cancel_right
tff(fact_774_mult__le__cancel__left,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3) ) ) ) ) ).

% mult_le_cancel_left
tff(fact_775_zero__less__iff__neq__zero,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [N: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),N)
        <=> ( N != zero_zero(A) ) ) ) ).

% zero_less_iff_neq_zero
tff(fact_776_gr__implies__not__zero,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [M: A,N: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),M),N)
         => ( N != zero_zero(A) ) ) ) ).

% gr_implies_not_zero
tff(fact_777_not__less__zero,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [N: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),N),zero_zero(A)) ) ).

% not_less_zero
tff(fact_778_gr__zeroI,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [N: A] :
          ( ( N != zero_zero(A) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),N) ) ) ).

% gr_zeroI
tff(fact_779_zero__le,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [X: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),X) ) ).

% zero_le
tff(fact_780_less__numeral__extra_I4_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),one_one(A)) ) ).

% less_numeral_extra(4)
tff(fact_781_diff__strict__right__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),C2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),C2)) ) ) ).

% diff_strict_right_mono
tff(fact_782_diff__strict__left__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [B2: A,A3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),A3)),aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),B2)) ) ) ).

% diff_strict_left_mono
tff(fact_783_diff__eq__diff__less,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A,C2: A,D3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),D3) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),D3) ) ) ) ).

% diff_eq_diff_less
tff(fact_784_diff__strict__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A,D3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),D3),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),C2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),D3)) ) ) ) ).

% diff_strict_mono
tff(fact_785_bot_Oextremum__strict,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [A3: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),bot_bot(A)) ) ).

% bot.extremum_strict
tff(fact_786_bot_Onot__eq__extremum,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [A3: A] :
          ( ( A3 != bot_bot(A) )
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),bot_bot(A)),A3) ) ) ).

% bot.not_eq_extremum
tff(fact_787_not__psubset__empty,axiom,
    ! [A: $tType,A4: set(A)] : ~ aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A4),bot_bot(set(A))) ).

% not_psubset_empty
tff(fact_788_verit__negate__coefficient_I2_J,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,uminus_uminus(A),A3)) ) ) ).

% verit_negate_coefficient(2)
tff(fact_789_less__minus__iff,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(A,A,uminus_uminus(A),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,uminus_uminus(A),A3)) ) ) ).

% less_minus_iff
tff(fact_790_minus__less__iff,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),A3)),B2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),B2)),A3) ) ) ).

% minus_less_iff
tff(fact_791_compl__less__swap2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Y: A,X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),Y)),X)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),X)),Y) ) ) ).

% compl_less_swap2
tff(fact_792_compl__less__swap1,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Y: A,X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),aa(A,A,uminus_uminus(A),X))
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),aa(A,A,uminus_uminus(A),Y)) ) ) ).

% compl_less_swap1
tff(fact_793_le__numeral__extra_I4_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),one_one(A)) ) ).

% le_numeral_extra(4)
tff(fact_794_diff__eq__diff__less__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A,C2: A,D3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),D3) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),D3) ) ) ) ).

% diff_eq_diff_less_eq
tff(fact_795_diff__right__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),C2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),C2)) ) ) ).

% diff_right_mono
tff(fact_796_diff__left__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [B2: A,A3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),A3)),aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),B2)) ) ) ).

% diff_left_mono
tff(fact_797_diff__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A,D3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),D3),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),C2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),D3)) ) ) ) ).

% diff_mono
tff(fact_798_subrelI,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,B)),S2: set(product_prod(A,B))] :
      ( ! [X2: A,Y2: B] :
          ( aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Y2)),R2)
         => aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Y2)),S2) )
     => aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),R2),S2) ) ).

% subrelI
tff(fact_799_bot_Oextremum,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [A3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),bot_bot(A)),A3) ) ).

% bot.extremum
tff(fact_800_bot_Oextremum__unique,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),bot_bot(A))
        <=> ( A3 = bot_bot(A) ) ) ) ).

% bot.extremum_unique
tff(fact_801_bot_Oextremum__uniqueI,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),bot_bot(A))
         => ( A3 = bot_bot(A) ) ) ) ).

% bot.extremum_uniqueI
tff(fact_802_top_Onot__eq__extremum,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [A3: A] :
          ( ( A3 != top_top(A) )
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),top_top(A)) ) ) ).

% top.not_eq_extremum
tff(fact_803_top_Oextremum__strict,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [A3: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),top_top(A)),A3) ) ).

% top.extremum_strict
tff(fact_804_less__infI1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,X: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),X)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)),X) ) ) ).

% less_infI1
tff(fact_805_less__infI2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,X: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),X)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)),X) ) ) ).

% less_infI2
tff(fact_806_inf_Oabsorb3,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2) = A3 ) ) ) ).

% inf.absorb3
tff(fact_807_inf_Oabsorb4,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3)
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2) = B2 ) ) ) ).

% inf.absorb4
tff(fact_808_inf_Ostrict__boundedE,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),inf_inf(A),B2),C2))
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),C2) ) ) ) ).

% inf.strict_boundedE
tff(fact_809_inf_Ostrict__order__iff,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
        <=> ( ( A3 = aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2) )
            & ( A3 != B2 ) ) ) ) ).

% inf.strict_order_iff
tff(fact_810_inf_Ostrict__coboundedI1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),C2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)),C2) ) ) ).

% inf.strict_coboundedI1
tff(fact_811_inf_Ostrict__coboundedI2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,C2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),C2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)),C2) ) ) ).

% inf.strict_coboundedI2
tff(fact_812_less__supI1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),A3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2)) ) ) ).

% less_supI1
tff(fact_813_less__supI2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2)) ) ) ).

% less_supI2
tff(fact_814_sup_Oabsorb3,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3)
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2) = A3 ) ) ) ).

% sup.absorb3
tff(fact_815_sup_Oabsorb4,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2) = B2 ) ) ) ).

% sup.absorb4
tff(fact_816_sup_Ostrict__boundedE,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,C2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),B2),C2)),A3)
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),A3) ) ) ) ).

% sup.strict_boundedE
tff(fact_817_sup_Ostrict__order__iff,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3)
        <=> ( ( A3 = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2) )
            & ( A3 != B2 ) ) ) ) ).

% sup.strict_order_iff
tff(fact_818_sup_Ostrict__coboundedI1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [C2: A,A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),A3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2)) ) ) ).

% sup.strict_coboundedI1
tff(fact_819_sup_Ostrict__coboundedI2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [C2: A,B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2)) ) ) ).

% sup.strict_coboundedI2
tff(fact_820_le__minus__iff,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,uminus_uminus(A),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,uminus_uminus(A),A3)) ) ) ).

% le_minus_iff
tff(fact_821_minus__le__iff,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),A3)),B2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),A3) ) ) ).

% minus_le_iff
tff(fact_822_le__imp__neg__le,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,uminus_uminus(A),A3)) ) ) ).

% le_imp_neg_le
tff(fact_823_compl__le__swap2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Y: A,X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),Y)),X)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),X)),Y) ) ) ).

% compl_le_swap2
tff(fact_824_compl__le__swap1,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Y: A,X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),aa(A,A,uminus_uminus(A),X))
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(A,A,uminus_uminus(A),Y)) ) ) ).

% compl_le_swap1
tff(fact_825_compl__mono,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),Y)),aa(A,A,uminus_uminus(A),X)) ) ) ).

% compl_mono
tff(fact_826_subset__minus__empty,axiom,
    ! [A: $tType,A4: set(A),B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B3)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),B3) = bot_bot(set(A)) ) ) ).

% subset_minus_empty
tff(fact_827_top_Oextremum__uniqueI,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),top_top(A)),A3)
         => ( A3 = top_top(A) ) ) ) ).

% top.extremum_uniqueI
tff(fact_828_top_Oextremum__unique,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),top_top(A)),A3)
        <=> ( A3 = top_top(A) ) ) ) ).

% top.extremum_unique
tff(fact_829_top__greatest,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [A3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),top_top(A)) ) ).

% top_greatest
tff(fact_830_inf__sup__ord_I2_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [X: A,Y: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)),Y) ) ).

% inf_sup_ord(2)
tff(fact_831_inf__sup__ord_I1_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [X: A,Y: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)),X) ) ).

% inf_sup_ord(1)
tff(fact_832_inf__le1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [X: A,Y: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)),X) ) ).

% inf_le1
tff(fact_833_inf__le2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [X: A,Y: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)),Y) ) ).

% inf_le2
tff(fact_834_le__infE,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [X: A,A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2))
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),A3)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),B2) ) ) ) ).

% le_infE
tff(fact_835_le__infI,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [X: A,A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)) ) ) ) ).

% le_infI
tff(fact_836_inf__mono,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,C2: A,B2: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)),aa(A,A,aa(A,fun(A,A),inf_inf(A),C2),D3)) ) ) ) ).

% inf_mono
tff(fact_837_le__infI1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,X: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),X)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)),X) ) ) ).

% le_infI1
tff(fact_838_le__infI2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,X: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),X)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)),X) ) ) ).

% le_infI2
tff(fact_839_inf_OorderE,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
         => ( A3 = aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2) ) ) ) ).

% inf.orderE
tff(fact_840_inf_OorderI,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B2: A] :
          ( ( A3 = aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2) ) ) ).

% inf.orderI
tff(fact_841_inf__unique,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [F: fun(A,fun(A,A)),X: A,Y: A] :
          ( ! [X2: A,Y2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),F,X2),Y2)),X2)
         => ( ! [X2: A,Y2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),F,X2),Y2)),Y2)
           => ( ! [X2: A,Y2: A,Z3: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),Y2)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),Z3)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),aa(A,A,aa(A,fun(A,A),F,Y2),Z3)) ) )
             => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y) = aa(A,A,aa(A,fun(A,A),F,X),Y) ) ) ) ) ) ).

% inf_unique
tff(fact_842_le__iff__inf,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
        <=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y) = X ) ) ) ).

% le_iff_inf
tff(fact_843_inf_Oabsorb1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2) = A3 ) ) ) ).

% inf.absorb1
tff(fact_844_inf_Oabsorb2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3)
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2) = B2 ) ) ) ).

% inf.absorb2
tff(fact_845_inf__absorb1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y) = X ) ) ) ).

% inf_absorb1
tff(fact_846_inf__absorb2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Y: A,X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X)
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y) = Y ) ) ) ).

% inf_absorb2
tff(fact_847_inf_OboundedE,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),inf_inf(A),B2),C2))
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),C2) ) ) ) ).

% inf.boundedE
tff(fact_848_inf_OboundedI,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),inf_inf(A),B2),C2)) ) ) ) ).

% inf.boundedI
tff(fact_849_inf__greatest,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [X: A,Y: A,Z2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Z2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Z2)) ) ) ) ).

% inf_greatest
tff(fact_850_inf_Oorder__iff,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),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_851_inf_Ocobounded1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)),A3) ) ).

% inf.cobounded1
tff(fact_852_inf_Ocobounded2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)),B2) ) ).

% inf.cobounded2
tff(fact_853_inf_Oabsorb__iff1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),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_854_inf_Oabsorb__iff2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),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_855_inf_OcoboundedI1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),C2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)),C2) ) ) ).

% inf.coboundedI1
tff(fact_856_inf_OcoboundedI2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,C2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),C2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)),C2) ) ) ).

% inf.coboundedI2
tff(fact_857_inf__sup__ord_I4_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Y: A,X: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y)) ) ).

% inf_sup_ord(4)
tff(fact_858_inf__sup__ord_I3_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [X: A,Y: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y)) ) ).

% inf_sup_ord(3)
tff(fact_859_le__supE,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: A,B2: A,X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2)),X)
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),X)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),X) ) ) ) ).

% le_supE
tff(fact_860_le__supI,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: A,X: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),X)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),X)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2)),X) ) ) ) ).

% le_supI
tff(fact_861_sup__ge1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,Y: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y)) ) ).

% sup_ge1
tff(fact_862_sup__ge2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Y: A,X: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y)) ) ).

% sup_ge2
tff(fact_863_le__supI1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),A3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2)) ) ) ).

% le_supI1
tff(fact_864_le__supI2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2)) ) ) ).

% le_supI2
tff(fact_865_sup_Omono,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [C2: A,A3: A,D3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),D3),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),C2),D3)),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2)) ) ) ) ).

% sup.mono
tff(fact_866_sup__mono,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: A,C2: A,B2: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2)),aa(A,A,aa(A,fun(A,A),sup_sup(A),C2),D3)) ) ) ) ).

% sup_mono
tff(fact_867_sup__least,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Y: A,X: A,Z2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z2),X)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z2)),X) ) ) ) ).

% sup_least
tff(fact_868_le__iff__sup,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
        <=> ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y) = Y ) ) ) ).

% le_iff_sup
tff(fact_869_sup_OorderE,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3)
         => ( A3 = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2) ) ) ) ).

% sup.orderE
tff(fact_870_sup_OorderI,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: A,B2: A] :
          ( ( A3 = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3) ) ) ).

% sup.orderI
tff(fact_871_sup__unique,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [F: fun(A,fun(A,A)),X: A,Y: A] :
          ( ! [X2: A,Y2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),aa(A,A,aa(A,fun(A,A),F,X2),Y2))
         => ( ! [X2: A,Y2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y2),aa(A,A,aa(A,fun(A,A),F,X2),Y2))
           => ( ! [X2: A,Y2: A,Z3: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y2),X2)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z3),X2)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),F,Y2),Z3)),X2) ) )
             => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y) = aa(A,A,aa(A,fun(A,A),F,X),Y) ) ) ) ) ) ).

% sup_unique
tff(fact_872_sup_Oabsorb1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3)
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2) = A3 ) ) ) ).

% sup.absorb1
tff(fact_873_sup_Oabsorb2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2) = B2 ) ) ) ).

% sup.absorb2
tff(fact_874_sup__absorb1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Y: A,X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X)
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y) = X ) ) ) ).

% sup_absorb1
tff(fact_875_sup__absorb2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y) = Y ) ) ) ).

% sup_absorb2
tff(fact_876_sup_OboundedE,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,C2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),B2),C2)),A3)
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A3) ) ) ) ).

% sup.boundedE
tff(fact_877_sup_OboundedI,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),B2),C2)),A3) ) ) ) ).

% sup.boundedI
tff(fact_878_sup_Oorder__iff,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),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_879_sup_Ocobounded1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: A,B2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2)) ) ).

% sup.cobounded1
tff(fact_880_sup_Ocobounded2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2)) ) ).

% sup.cobounded2
tff(fact_881_sup_Oabsorb__iff1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),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_882_sup_Oabsorb__iff2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),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_883_sup_OcoboundedI1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [C2: A,A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2)) ) ) ).

% sup.coboundedI1
tff(fact_884_sup_OcoboundedI2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [C2: A,B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2)) ) ) ).

% sup.coboundedI2
tff(fact_885_subset__UNIV,axiom,
    ! [A: $tType,A4: set(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),top_top(set(A))) ).

% subset_UNIV
tff(fact_886_inter__eq__subsetI,axiom,
    ! [A: $tType,S: set(A),S3: set(A),A4: set(A),B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),S3)
     => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),S3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),S3) )
       => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),S) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),S) ) ) ) ).

% inter_eq_subsetI
tff(fact_887_Int__Collect__mono,axiom,
    ! [A: $tType,A4: set(A),B3: set(A),P: fun(A,$o),Q: fun(A,$o)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B3)
     => ( ! [X2: A] :
            ( aa(set(A),$o,member(A,X2),A4)
           => ( aa(A,$o,P,X2)
             => aa(A,$o,Q,X2) ) )
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),aa(fun(A,$o),set(A),collect(A),P))),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),aa(fun(A,$o),set(A),collect(A),Q))) ) ) ).

% Int_Collect_mono
tff(fact_888_Int__greatest,axiom,
    ! [A: $tType,C3: set(A),A4: set(A),B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C3),A4)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C3),B3)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3)) ) ) ).

% Int_greatest
tff(fact_889_Int__absorb2,axiom,
    ! [A: $tType,A4: set(A),B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B3)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3) = A4 ) ) ).

% Int_absorb2
tff(fact_890_Int__absorb1,axiom,
    ! [A: $tType,B3: set(A),A4: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),A4)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3) = B3 ) ) ).

% Int_absorb1
tff(fact_891_Int__lower2,axiom,
    ! [A: $tType,A4: set(A),B3: set(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3)),B3) ).

% Int_lower2
tff(fact_892_Int__lower1,axiom,
    ! [A: $tType,A4: set(A),B3: set(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3)),A4) ).

% Int_lower1
tff(fact_893_Int__mono,axiom,
    ! [A: $tType,A4: set(A),C3: set(A),B3: set(A),D4: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),C3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),D4)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),C3),D4)) ) ) ).

% Int_mono
tff(fact_894_subset__Un__eq,axiom,
    ! [A: $tType,A4: set(A),B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B3)
    <=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3) = B3 ) ) ).

% subset_Un_eq
tff(fact_895_subset__UnE,axiom,
    ! [A: $tType,C3: set(A),A4: set(A),B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3))
     => ~ ! [A7: set(A)] :
            ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A7),A4)
           => ! [B7: set(A)] :
                ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B7),B3)
               => ( C3 != aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A7),B7) ) ) ) ) ).

% subset_UnE
tff(fact_896_Un__absorb2,axiom,
    ! [A: $tType,B3: set(A),A4: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),A4)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3) = A4 ) ) ).

% Un_absorb2
tff(fact_897_Un__absorb1,axiom,
    ! [A: $tType,A4: set(A),B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B3)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3) = B3 ) ) ).

% Un_absorb1
tff(fact_898_Un__upper2,axiom,
    ! [A: $tType,B3: set(A),A4: set(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3)) ).

% Un_upper2
tff(fact_899_Un__upper1,axiom,
    ! [A: $tType,A4: set(A),B3: set(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3)) ).

% Un_upper1
tff(fact_900_Un__least,axiom,
    ! [A: $tType,A4: set(A),C3: set(A),B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),C3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),C3)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3)),C3) ) ) ).

% Un_least
tff(fact_901_Un__mono,axiom,
    ! [A: $tType,A4: set(A),C3: set(A),B3: set(A),D4: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),C3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),D4)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),C3),D4)) ) ) ).

% Un_mono
tff(fact_902_Diff__partition,axiom,
    ! [A: $tType,A4: set(A),B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B3)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B3),A4)) = B3 ) ) ).

% Diff_partition
tff(fact_903_Diff__subset__conv,axiom,
    ! [A: $tType,A4: set(A),B3: set(A),C3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),B3)),C3)
    <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B3),C3)) ) ).

% Diff_subset_conv
tff(fact_904_relH__subset,axiom,
    ! [Bs: set(nat),H2: heap_ext(product_unit),H3: heap_ext(product_unit),As: set(nat)] :
      ( relH(Bs,H2,H3)
     => ( aa(set(nat),$o,aa(set(nat),fun(set(nat),$o),ord_less_eq(set(nat)),As),Bs)
       => relH(As,H2,H3) ) ) ).

% relH_subset
tff(fact_905_mult__le__cancel__left1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),B2) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),one_one(A)) ) ) ) ) ).

% mult_le_cancel_left1
tff(fact_906_mult__le__cancel__left2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),C2)
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),one_one(A)) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),A3) ) ) ) ) ).

% mult_le_cancel_left2
tff(fact_907_mult__le__cancel__right1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),B2) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),one_one(A)) ) ) ) ) ).

% mult_le_cancel_right1
tff(fact_908_mult__le__cancel__right2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),C2)
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),one_one(A)) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),A3) ) ) ) ) ).

% mult_le_cancel_right2
tff(fact_909_mult__less__cancel__left1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),B2) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),one_one(A)) ) ) ) ) ).

% mult_less_cancel_left1
tff(fact_910_mult__less__cancel__left2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),C2)
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),one_one(A)) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A3) ) ) ) ) ).

% mult_less_cancel_left2
tff(fact_911_mult__less__cancel__right1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),B2) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),one_one(A)) ) ) ) ) ).

% mult_less_cancel_right1
tff(fact_912_mult__less__cancel__right2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),C2)
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),one_one(A)) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A3) ) ) ) ) ).

% mult_less_cancel_right2
tff(fact_913_field__le__mult__one__interval,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( ! [Z3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Z3)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z3),one_one(A))
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z3),X)),Y) ) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y) ) ) ).

% field_le_mult_one_interval
tff(fact_914_divide__left__mono__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),A3)),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),B2)) ) ) ) ) ).

% divide_left_mono_neg
tff(fact_915_mult__imp__le__div__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Z2: A,X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z2),Y)),X)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z2),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)) ) ) ) ).

% mult_imp_le_div_pos
tff(fact_916_mult__imp__div__pos__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,X: A,Z2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),times_times(A),Z2),Y))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)),Z2) ) ) ) ).

% mult_imp_div_pos_le
tff(fact_917_pos__le__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),B2) ) ) ) ).

% pos_le_divide_eq
tff(fact_918_pos__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),A3)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)) ) ) ) ).

% pos_divide_le_eq
tff(fact_919_neg__le__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)) ) ) ) ).

% neg_le_divide_eq
tff(fact_920_neg__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),A3)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),B2) ) ) ) ).

% neg_divide_le_eq
tff(fact_921_divide__left__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),A3)),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),B2)) ) ) ) ) ).

% divide_left_mono
tff(fact_922_le__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
              aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),B2),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),zero_zero(A))) ) ) ) ).

% le_divide_eq
tff(fact_923_divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),A3)
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
              aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),B2),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)) ) ) ) ).

% divide_le_eq
tff(fact_924_le__divide__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3) ) ) ) ) ).

% le_divide_eq_1
tff(fact_925_divide__le__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3)),one_one(A))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2) )
            | ( A3 = zero_zero(A) ) ) ) ) ).

% divide_le_eq_1
tff(fact_926_le__divide__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [W2: num,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),W2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
              aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),C2)),B2),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),C2)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),W2)),zero_zero(A))) ) ) ) ).

% le_divide_eq_numeral(1)
tff(fact_927_divide__le__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,W2: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),aa(num,A,numeral_numeral(A),W2))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
              aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),C2)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),C2)),B2),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(num,A,numeral_numeral(A),W2))) ) ) ) ).

% divide_le_eq_numeral(1)
tff(fact_928_pos__minus__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))),A3)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)) ) ) ) ).

% pos_minus_divide_le_eq
tff(fact_929_pos__le__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,uminus_uminus(A),B2)) ) ) ) ).

% pos_le_minus_divide_eq
tff(fact_930_neg__minus__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))),A3)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,uminus_uminus(A),B2)) ) ) ) ).

% neg_minus_divide_le_eq
tff(fact_931_neg__le__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)) ) ) ) ).

% neg_le_minus_divide_eq
tff(fact_932_minus__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))),A3)
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
              aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,uminus_uminus(A),B2)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)) ) ) ) ).

% minus_divide_le_eq
tff(fact_933_le__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
              aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,uminus_uminus(A),B2)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),zero_zero(A))) ) ) ) ).

% le_minus_divide_eq
tff(fact_934_less__eq__assn__def,axiom,
    ! [A3: assn,B2: assn] :
      ( aa(assn,$o,aa(assn,fun(assn,$o),ord_less_eq(assn),A3),B2)
    <=> ( A3 = aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),A3),B2) ) ) ).

% less_eq_assn_def
tff(fact_935_mult__neg__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) ) ) ) ).

% mult_neg_neg
tff(fact_936_not__square__less__zero,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [A3: A] : ~ aa(A,$o,aa(A,fun(A,$o),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_937_mult__less__0__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),zero_zero(A))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),zero_zero(A)) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2) ) ) ) ) ).

% mult_less_0_iff
tff(fact_938_mult__neg__pos,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),zero_zero(A)) ) ) ) ).

% mult_neg_pos
tff(fact_939_mult__pos__neg,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),zero_zero(A)) ) ) ) ).

% mult_pos_neg
tff(fact_940_mult__pos__pos,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) ) ) ) ).

% mult_pos_pos
tff(fact_941_mult__pos__neg2,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A3)),zero_zero(A)) ) ) ) ).

% mult_pos_neg2
tff(fact_942_zero__less__mult__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),zero_zero(A)) ) ) ) ) ).

% zero_less_mult_iff
tff(fact_943_zero__less__mult__pos,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2) ) ) ) ).

% zero_less_mult_pos
tff(fact_944_zero__less__mult__pos2,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A3))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2) ) ) ) ).

% zero_less_mult_pos2
tff(fact_945_mult__less__cancel__left__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3) ) ) ) ).

% mult_less_cancel_left_neg
tff(fact_946_mult__less__cancel__left__pos,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2) ) ) ) ).

% mult_less_cancel_left_pos
tff(fact_947_mult__strict__left__mono__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [B2: A,A3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) ) ) ) ).

% mult_strict_left_mono_neg
tff(fact_948_mult__strict__left__mono,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) ) ) ) ).

% mult_strict_left_mono
tff(fact_949_mult__less__cancel__left__disj,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3) ) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_950_mult__strict__right__mono__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [B2: A,A3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ) ) ).

% mult_strict_right_mono_neg
tff(fact_951_mult__strict__right__mono,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ) ) ).

% mult_strict_right_mono
tff(fact_952_mult__less__cancel__right__disj,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A3: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3) ) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_953_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: $tType] :
      ( linord2810124833399127020strict(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_954_zero__less__one,axiom,
    ! [A: $tType] :
      ( zero_less_one(A)
     => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),one_one(A)) ) ).

% zero_less_one
tff(fact_955_not__one__less__zero,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),zero_zero(A)) ) ).

% not_one_less_zero
tff(fact_956_less__numeral__extra_I1_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),one_one(A)) ) ).

% less_numeral_extra(1)
tff(fact_957_less__iff__diff__less__0,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
        <=> aa(A,$o,aa(A,fun(A,$o),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_958_not__numeral__less__one,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [N: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(num,A,numeral_numeral(A),N)),one_one(A)) ) ).

% not_numeral_less_one
tff(fact_959_less__1__mult,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [M: A,N: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),M)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),N)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),M),N)) ) ) ) ).

% less_1_mult
tff(fact_960_mult__mono,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A3: A,B2: A,C2: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),D3)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D3)) ) ) ) ) ) ).

% mult_mono
tff(fact_961_mult__mono_H,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A3: A,B2: A,C2: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),D3)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D3)) ) ) ) ) ) ).

% mult_mono'
tff(fact_962_zero__le__square,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [A3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),A3)) ) ).

% zero_le_square
tff(fact_963_split__mult__pos__le,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A3: A,B2: A] :
          ( ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A)) ) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) ) ) ).

% split_mult_pos_le
tff(fact_964_mult__left__mono__neg,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [B2: A,A3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) ) ) ) ).

% mult_left_mono_neg
tff(fact_965_mult__nonpos__nonpos,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) ) ) ) ).

% mult_nonpos_nonpos
tff(fact_966_mult__left__mono,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) ) ) ) ).

% mult_left_mono
tff(fact_967_mult__right__mono__neg,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [B2: A,A3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ) ) ).

% mult_right_mono_neg
tff(fact_968_mult__right__mono,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ) ) ).

% mult_right_mono
tff(fact_969_mult__le__0__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),zero_zero(A))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A)) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2) ) ) ) ) ).

% mult_le_0_iff
tff(fact_970_split__mult__neg__le,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A3: A,B2: A] :
          ( ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A)) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2) ) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),zero_zero(A)) ) ) ).

% split_mult_neg_le
tff(fact_971_mult__nonneg__nonneg,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) ) ) ) ).

% mult_nonneg_nonneg
tff(fact_972_mult__nonneg__nonpos,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),zero_zero(A)) ) ) ) ).

% mult_nonneg_nonpos
tff(fact_973_mult__nonpos__nonneg,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),zero_zero(A)) ) ) ) ).

% mult_nonpos_nonneg
tff(fact_974_mult__nonneg__nonpos2,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A3)),zero_zero(A)) ) ) ) ).

% mult_nonneg_nonpos2
tff(fact_975_zero__le__mult__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A)) ) ) ) ) ).

% zero_le_mult_iff
tff(fact_976_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: $tType] :
      ( ordere2520102378445227354miring(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_977_zero__less__one__class_Ozero__le__one,axiom,
    ! [A: $tType] :
      ( zero_less_one(A)
     => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),one_one(A)) ) ).

% zero_less_one_class.zero_le_one
tff(fact_978_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),one_one(A)) ) ).

% linordered_nonzero_semiring_class.zero_le_one
tff(fact_979_not__one__le__zero,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),zero_zero(A)) ) ).

% not_one_le_zero
tff(fact_980_le__iff__diff__le__0,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
        <=> aa(A,$o,aa(A,fun(A,$o),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_981_not__numeral__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num,N: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(num,A,numeral_numeral(A),M)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))) ) ).

% not_numeral_less_neg_numeral
tff(fact_982_neg__numeral__less__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num,N: num] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(num,A,numeral_numeral(A),N)) ) ).

% neg_numeral_less_numeral
tff(fact_983_less__minus__one__simps_I4_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(A,A,uminus_uminus(A),one_one(A))) ) ).

% less_minus_one_simps(4)
tff(fact_984_less__minus__one__simps_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),one_one(A))),one_one(A)) ) ).

% less_minus_one_simps(2)
tff(fact_985_one__le__numeral,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [N: num] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(num,A,numeral_numeral(A),N)) ) ).

% one_le_numeral
tff(fact_986_diff__shunt__var,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Y) = bot_bot(A) )
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y) ) ) ).

% diff_shunt_var
tff(fact_987_not__numeral__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num,N: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),M)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))) ) ).

% not_numeral_le_neg_numeral
tff(fact_988_neg__numeral__le__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num,N: num] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(num,A,numeral_numeral(A),N)) ) ).

% neg_numeral_le_numeral
tff(fact_989_le__minus__one__simps_I4_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(A,A,uminus_uminus(A),one_one(A))) ) ).

% le_minus_one_simps(4)
tff(fact_990_le__minus__one__simps_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),one_one(A))),one_one(A)) ) ).

% le_minus_one_simps(2)
tff(fact_991_disjoint__alt__simp3,axiom,
    ! [A: $tType,A4: set(A),B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),B3)),A4)
    <=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3) != bot_bot(set(A)) ) ) ).

% disjoint_alt_simp3
tff(fact_992_distrib__inf__le,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [X: A,Y: A,Z2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Z2))),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z2))) ) ).

% distrib_inf_le
tff(fact_993_distrib__sup__le,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [X: A,Y: A,Z2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Z2))),aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y)),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Z2))) ) ).

% distrib_sup_le
tff(fact_994_disjoint__mono,axiom,
    ! [A: $tType,A3: set(A),A5: set(A),B2: set(A),B4: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),A5)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),B4)
       => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B4) = bot_bot(set(A)) )
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B2) = bot_bot(set(A)) ) ) ) ) ).

% disjoint_mono
tff(fact_995_Un__Int__assoc__eq,axiom,
    ! [A: $tType,A4: set(A),B3: set(A),C3: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3)),C3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B3),C3)) )
    <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C3),A4) ) ).

% Un_Int_assoc_eq
tff(fact_996_subset__Compl__self__eq,axiom,
    ! [A: $tType,A4: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),aa(set(A),set(A),uminus_uminus(set(A)),A4))
    <=> ( A4 = bot_bot(set(A)) ) ) ).

% subset_Compl_self_eq
tff(fact_997_le__divide__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [W2: num,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
              aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),C2)),B2),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),C2)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),zero_zero(A))) ) ) ) ).

% le_divide_eq_numeral(2)
tff(fact_998_divide__le__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,W2: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
              aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),C2)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),C2)),B2),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)))) ) ) ) ).

% divide_le_eq_numeral(2)
tff(fact_999_in__range__subset,axiom,
    ! [As: set(nat),As3: set(nat),H2: heap_ext(product_unit)] :
      ( aa(set(nat),$o,aa(set(nat),fun(set(nat),$o),ord_less_eq(set(nat)),As),As3)
     => ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As3))
       => aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As)) ) ) ).

% in_range_subset
tff(fact_1000_not__zero__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))) ) ).

% not_zero_less_neg_numeral
tff(fact_1001_neg__numeral__less__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: num] : aa(A,$o,aa(A,fun(A,$o),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_1002_less__minus__one__simps_I3_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,uminus_uminus(A),one_one(A))) ) ).

% less_minus_one_simps(3)
tff(fact_1003_less__minus__one__simps_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),one_one(A))),zero_zero(A)) ) ).

% less_minus_one_simps(1)
tff(fact_1004_divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),A3)
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),B2),aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)) ) ) ) ).

% divide_less_eq
tff(fact_1005_less__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),B2),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A))) ) ) ) ).

% less_divide_eq
tff(fact_1006_neg__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),A3)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),B2) ) ) ) ).

% neg_divide_less_eq
tff(fact_1007_neg__less__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)) ) ) ) ).

% neg_less_divide_eq
tff(fact_1008_pos__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),A3)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)) ) ) ) ).

% pos_divide_less_eq
tff(fact_1009_pos__less__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),B2) ) ) ) ).

% pos_less_divide_eq
tff(fact_1010_mult__imp__div__pos__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,X: A,Z2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),aa(A,A,aa(A,fun(A,A),times_times(A),Z2),Y))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)),Z2) ) ) ) ).

% mult_imp_div_pos_less
tff(fact_1011_mult__imp__less__div__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Z2: A,X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z2),Y)),X)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z2),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)) ) ) ) ).

% mult_imp_less_div_pos
tff(fact_1012_divide__strict__left__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),A3)),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),B2)) ) ) ) ) ).

% divide_strict_left_mono
tff(fact_1013_divide__strict__left__mono__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),A3)),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),B2)) ) ) ) ) ).

% divide_strict_left_mono_neg
tff(fact_1014_divide__less__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3)),one_one(A))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2) )
            | ( A3 = zero_zero(A) ) ) ) ) ).

% divide_less_eq_1
tff(fact_1015_less__divide__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3) ) ) ) ) ).

% less_divide_eq_1
tff(fact_1016_neg__numeral__less__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),one_one(A)) ) ).

% neg_numeral_less_one
tff(fact_1017_neg__one__less__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(num,A,numeral_numeral(A),M)) ) ).

% neg_one_less_numeral
tff(fact_1018_not__numeral__less__neg__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(num,A,numeral_numeral(A),M)),aa(A,A,uminus_uminus(A),one_one(A))) ) ).

% not_numeral_less_neg_one
tff(fact_1019_not__one__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))) ) ).

% not_one_less_neg_numeral
tff(fact_1020_not__neg__one__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))) ) ).

% not_neg_one_less_neg_numeral
tff(fact_1021_mult__left__le__one__le,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),X)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Y)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),one_one(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Y),X)),X) ) ) ) ) ).

% mult_left_le_one_le
tff(fact_1022_mult__right__le__one__le,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),X)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Y)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),one_one(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),Y)),X) ) ) ) ) ).

% mult_right_le_one_le
tff(fact_1023_mult__le__one,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),one_one(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),one_one(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),one_one(A)) ) ) ) ) ).

% mult_le_one
tff(fact_1024_mult__left__le,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [C2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),one_one(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),A3) ) ) ) ).

% mult_left_le
tff(fact_1025_not__zero__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))) ) ).

% not_zero_le_neg_numeral
tff(fact_1026_neg__numeral__le__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: num] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))),zero_zero(A)) ) ).

% neg_numeral_le_zero
tff(fact_1027_le__minus__one__simps_I3_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,uminus_uminus(A),one_one(A))) ) ).

% le_minus_one_simps(3)
tff(fact_1028_le__minus__one__simps_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),one_one(A))),zero_zero(A)) ) ).

% le_minus_one_simps(1)
tff(fact_1029_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [C2: A,A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),C2) ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_1030_neg__numeral__le__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),one_one(A)) ) ).

% neg_numeral_le_one
tff(fact_1031_neg__one__le__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(num,A,numeral_numeral(A),M)) ) ).

% neg_one_le_numeral
tff(fact_1032_neg__numeral__le__neg__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(A,A,uminus_uminus(A),one_one(A))) ) ).

% neg_numeral_le_neg_one
tff(fact_1033_not__numeral__le__neg__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),M)),aa(A,A,uminus_uminus(A),one_one(A))) ) ).

% not_numeral_le_neg_one
tff(fact_1034_not__one__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))) ) ).

% not_one_le_neg_numeral
tff(fact_1035_inf__shunt,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y) = bot_bot(A) )
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(A,A,uminus_uminus(A),Y)) ) ) ).

% inf_shunt
tff(fact_1036_sup__shunt,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y) = top_top(A) )
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),X)),Y) ) ) ).

% sup_shunt
tff(fact_1037_shunt1,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y: A,Z2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)),Z2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,uminus_uminus(A),Y)),Z2)) ) ) ).

% shunt1
tff(fact_1038_shunt2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y: A,Z2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(A,A,uminus_uminus(A),Y))),Z2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z2)) ) ) ).

% shunt2
tff(fact_1039_sup__neg__inf,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [P3: A,Q3: A,R2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),P3),aa(A,A,aa(A,fun(A,A),sup_sup(A),Q3),R2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),P3),aa(A,A,uminus_uminus(A),Q3))),R2) ) ) ).

% sup_neg_inf
tff(fact_1040_disjoint__eq__subset__Compl,axiom,
    ! [A: $tType,A4: set(A),B3: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3) = bot_bot(set(A)) )
    <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),aa(set(A),set(A),uminus_uminus(set(A)),B3)) ) ).

% disjoint_eq_subset_Compl
tff(fact_1041_less__divide__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [W2: num,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(num,A,numeral_numeral(A),W2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),C2)),B2),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),C2)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(num,A,numeral_numeral(A),W2)),zero_zero(A))) ) ) ) ).

% less_divide_eq_numeral(1)
tff(fact_1042_divide__less__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,W2: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),aa(num,A,numeral_numeral(A),W2))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),C2)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),C2)),B2),aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(num,A,numeral_numeral(A),W2))) ) ) ) ).

% divide_less_eq_numeral(1)
tff(fact_1043_frac__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Z2: A,X: A,W2: A] :
          ( ( Y != zero_zero(A) )
         => ( ( Z2 != zero_zero(A) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)),aa(A,A,aa(A,fun(A,A),divide_divide(A),W2),Z2))
            <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,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_1044_pos__minus__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))),A3)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)) ) ) ) ).

% pos_minus_divide_less_eq
tff(fact_1045_pos__less__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,uminus_uminus(A),B2)) ) ) ) ).

% pos_less_minus_divide_eq
tff(fact_1046_neg__minus__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))),A3)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,uminus_uminus(A),B2)) ) ) ) ).

% neg_minus_divide_less_eq
tff(fact_1047_neg__less__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)) ) ) ) ).

% neg_less_minus_divide_eq
tff(fact_1048_minus__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))),A3)
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,uminus_uminus(A),B2)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)) ) ) ) ).

% minus_divide_less_eq
tff(fact_1049_less__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,uminus_uminus(A),B2)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A))) ) ) ) ).

% less_minus_divide_eq
tff(fact_1050_frac__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Z2: A,X: A,W2: A] :
          ( ( Y != zero_zero(A) )
         => ( ( Z2 != zero_zero(A) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)),aa(A,A,aa(A,fun(A,A),divide_divide(A),W2),Z2))
            <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,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_1051_mult__le__cancel__iff1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z2: A,X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Z2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),Z2)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y) ) ) ) ).

% mult_le_cancel_iff1
tff(fact_1052_mult__le__cancel__iff2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z2: A,X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Z2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z2),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Z2),Y))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y) ) ) ) ).

% mult_le_cancel_iff2
tff(fact_1053_mult__less__iff1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z2: A,X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Z2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),Z2)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y) ) ) ) ).

% mult_less_iff1
tff(fact_1054_sup__bot_Osemilattice__neutr__order__axioms,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => semila1105856199041335345_order(A,sup_sup(A),bot_bot(A),aTP_Lamp_bd(A,fun(A,$o)),aTP_Lamp_be(A,fun(A,$o))) ) ).

% sup_bot.semilattice_neutr_order_axioms
tff(fact_1055_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_1056_bot_Oordering__top__axioms,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ordering_top(A,aTP_Lamp_bf(A,fun(A,$o)),aTP_Lamp_bg(A,fun(A,$o)),bot_bot(A)) ) ).

% bot.ordering_top_axioms
tff(fact_1057_neg__numeral__less__neg__one__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(A,A,uminus_uminus(A),one_one(A)))
        <=> ( M != one2 ) ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_1058_not__neg__one__le__neg__numeral__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M)))
        <=> ( M != one2 ) ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_1059_add__neg__numeral__special_I6_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),inc(M))) ) ).

% add_neg_numeral_special(6)
tff(fact_1060_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_1061_add__left__cancel,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [A3: A,B2: A,C2: 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),C2) )
        <=> ( B2 = C2 ) ) ) ).

% add_left_cancel
tff(fact_1062_add__right__cancel,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [B2: A,A3: A,C2: 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),C2),A3) )
        <=> ( B2 = C2 ) ) ) ).

% add_right_cancel
tff(fact_1063_predicate2I,axiom,
    ! [B: $tType,A: $tType,P: fun(A,fun(B,$o)),Q: fun(A,fun(B,$o))] :
      ( ! [X2: A,Y2: B] :
          ( aa(B,$o,aa(A,fun(B,$o),P,X2),Y2)
         => aa(B,$o,aa(A,fun(B,$o),Q,X2),Y2) )
     => aa(fun(A,fun(B,$o)),$o,aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),$o),ord_less_eq(fun(A,fun(B,$o))),P),Q) ) ).

% predicate2I
tff(fact_1064_add__le__cancel__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [C2: A,A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2) ) ) ).

% add_le_cancel_left
tff(fact_1065_add__le__cancel__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [A3: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2) ) ) ).

% add_le_cancel_right
tff(fact_1066_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_1067_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_1068_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_1069_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_1070_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_1071_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_1072_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_1073_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_1074_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_1075_add__less__cancel__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [C2: A,A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2) ) ) ).

% add_less_cancel_left
tff(fact_1076_add__less__cancel__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [A3: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2) ) ) ).

% add_less_cancel_right
tff(fact_1077_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_1078_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_1079_add__diff__cancel__left,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [C2: 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),C2),A3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2) ) ).

% add_diff_cancel_left
tff(fact_1080_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_1081_add__diff__cancel__right,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [A3: A,C2: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2) ) ).

% add_diff_cancel_right
tff(fact_1082_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_1083_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_1084_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_1085_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_1086_add__le__same__cancel1,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A3)),B2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),zero_zero(A)) ) ) ).

% add_le_same_cancel1
tff(fact_1087_add__le__same__cancel2,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),B2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),zero_zero(A)) ) ) ).

% add_le_same_cancel2
tff(fact_1088_le__add__same__cancel1,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2) ) ) ).

% le_add_same_cancel1
tff(fact_1089_le__add__same__cancel2,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2) ) ) ).

% le_add_same_cancel2
tff(fact_1090_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),A3)),zero_zero(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),zero_zero(A)) ) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_1091_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),A3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3) ) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_1092_add__less__same__cancel1,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A3)),B2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A)) ) ) ).

% add_less_same_cancel1
tff(fact_1093_add__less__same__cancel2,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),B2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A)) ) ) ).

% add_less_same_cancel2
tff(fact_1094_less__add__same__cancel1,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2) ) ) ).

% less_add_same_cancel1
tff(fact_1095_less__add__same__cancel2,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2) ) ) ).

% less_add_same_cancel2
tff(fact_1096_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),A3)),zero_zero(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A)) ) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_1097_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),A3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3) ) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_1098_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_1099_distrib__right__numeral,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & semiring(A) )
     => ! [A3: A,B2: A,V: num] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),aa(num,A,numeral_numeral(A),V)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),V))),aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(num,A,numeral_numeral(A),V))) ) ).

% distrib_right_numeral
tff(fact_1100_distrib__left__numeral,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & semiring(A) )
     => ! [V: num,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V)),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V)),C2)) ) ).

% distrib_left_numeral
tff(fact_1101_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_1102_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_1103_semiring__norm_I168_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [V: num,W2: num,Y: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),Y)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),V),W2)))),Y) ) ).

% semiring_norm(168)
tff(fact_1104_add__neg__numeral__simps_I3_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: 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),M))),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),M)),aa(num,A,numeral_numeral(A),N))) ) ).

% add_neg_numeral_simps(3)
tff(fact_1105_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_1106_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_1107_diff__numeral__simps_I3_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: 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),M))),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),M),N))) ) ).

% diff_numeral_simps(3)
tff(fact_1108_diff__numeral__simps_I2_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: num,N: num] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(num,A,numeral_numeral(A),M)),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),M),N)) ) ).

% diff_numeral_simps(2)
tff(fact_1109_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_1110_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_1111_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_1112_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_1113_div__mult__self4,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,C2: A,A3: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)),A3)),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)) ) ) ) ).

% div_mult_self4
tff(fact_1114_div__mult__self3,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,C2: A,A3: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)),A3)),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)) ) ) ) ).

% div_mult_self3
tff(fact_1115_div__mult__self2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,A3: A,C2: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)) ) ) ) ).

% div_mult_self2
tff(fact_1116_div__mult__self1,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,A3: A,C2: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)) ) ) ) ).

% div_mult_self1
tff(fact_1117_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_1118_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_1119_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_1120_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_1121_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_1122_diff__numeral__special_I4_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: num] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),one_one(A)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),M),one2))) ) ).

% diff_numeral_special(4)
tff(fact_1123_numeral__le__one__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [N: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),N)),one_one(A))
        <=> aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),N),one2) ) ) ).

% numeral_le_one_iff
tff(fact_1124_one__less__numeral__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [N: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(num,A,numeral_numeral(A),N))
        <=> aa(num,$o,aa(num,fun(num,$o),ord_less(num),one2),N) ) ) ).

% one_less_numeral_iff
tff(fact_1125_pred__subset__eq,axiom,
    ! [A: $tType,R: set(A),S: set(A)] :
      ( aa(fun(A,$o),$o,aa(fun(A,$o),fun(fun(A,$o),$o),ord_less_eq(fun(A,$o)),aa(set(A),fun(A,$o),aTP_Lamp_ag(set(A),fun(A,$o)),R)),aa(set(A),fun(A,$o),aTP_Lamp_ag(set(A),fun(A,$o)),S))
    <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),R),S) ) ).

% pred_subset_eq
tff(fact_1126_less__eq__set__def,axiom,
    ! [A: $tType,A4: set(A),B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B3)
    <=> aa(fun(A,$o),$o,aa(fun(A,$o),fun(fun(A,$o),$o),ord_less_eq(fun(A,$o)),aa(set(A),fun(A,$o),aTP_Lamp_ag(set(A),fun(A,$o)),A4)),aa(set(A),fun(A,$o),aTP_Lamp_ag(set(A),fun(A,$o)),B3)) ) ).

% less_eq_set_def
tff(fact_1127_less__set__def,axiom,
    ! [A: $tType,A4: set(A),B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A4),B3)
    <=> aa(fun(A,$o),$o,aa(fun(A,$o),fun(fun(A,$o),$o),ord_less(fun(A,$o)),aa(set(A),fun(A,$o),aTP_Lamp_ag(set(A),fun(A,$o)),A4)),aa(set(A),fun(A,$o),aTP_Lamp_ag(set(A),fun(A,$o)),B3)) ) ).

% less_set_def
tff(fact_1128_less__assn__def,axiom,
    ! [A3: assn,B2: assn] :
      ( aa(assn,$o,aa(assn,fun(assn,$o),ord_less(assn),A3),B2)
    <=> ( aa(assn,$o,aa(assn,fun(assn,$o),ord_less_eq(assn),A3),B2)
        & ( A3 != B2 ) ) ) ).

% less_assn_def
tff(fact_1129_rev__predicate2D,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,$o)),X: A,Y: B,Q: fun(A,fun(B,$o))] :
      ( aa(B,$o,aa(A,fun(B,$o),P,X),Y)
     => ( aa(fun(A,fun(B,$o)),$o,aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),$o),ord_less_eq(fun(A,fun(B,$o))),P),Q)
       => aa(B,$o,aa(A,fun(B,$o),Q,X),Y) ) ) ).

% rev_predicate2D
tff(fact_1130_less__fun__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F: fun(A,B),G: fun(A,B)] :
          ( aa(fun(A,B),$o,aa(fun(A,B),fun(fun(A,B),$o),ord_less(fun(A,B)),F),G)
        <=> ( aa(fun(A,B),$o,aa(fun(A,B),fun(fun(A,B),$o),ord_less_eq(fun(A,B)),F),G)
            & ~ aa(fun(A,B),$o,aa(fun(A,B),fun(fun(A,B),$o),ord_less_eq(fun(A,B)),G),F) ) ) ) ).

% less_fun_def
tff(fact_1131_predicate2D,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,$o)),Q: fun(A,fun(B,$o)),X: A,Y: B] :
      ( aa(fun(A,fun(B,$o)),$o,aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),$o),ord_less_eq(fun(A,fun(B,$o))),P),Q)
     => ( aa(B,$o,aa(A,fun(B,$o),P,X),Y)
       => aa(B,$o,aa(A,fun(B,$o),Q,X),Y) ) ) ).

% predicate2D
tff(fact_1132_eq__subset,axiom,
    ! [A: $tType,P: fun(A,fun(A,$o))] : aa(fun(A,fun(A,$o)),$o,aa(fun(A,fun(A,$o)),fun(fun(A,fun(A,$o)),$o),ord_less_eq(fun(A,fun(A,$o))),fequal(A)),aTP_Lamp_bh(fun(A,fun(A,$o)),fun(A,fun(A,$o)),P)) ).

% eq_subset
tff(fact_1133_Collect__subset,axiom,
    ! [A: $tType,A4: set(A),P: fun(A,$o)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_bi(set(A),fun(fun(A,$o),fun(A,$o)),A4),P))),A4) ).

% Collect_subset
tff(fact_1134_conj__subset__def,axiom,
    ! [A: $tType,A4: set(A),P: fun(A,$o),Q: fun(A,$o)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_am(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q)))
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),aa(fun(A,$o),set(A),collect(A),P))
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),aa(fun(A,$o),set(A),collect(A),Q)) ) ) ).

% conj_subset_def
tff(fact_1135_ordering__top_Oextremum,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),Top: A,A3: A] :
      ( ordering_top(A,Less_eq,Less,Top)
     => aa(A,$o,aa(A,fun(A,$o),Less_eq,A3),Top) ) ).

% ordering_top.extremum
tff(fact_1136_ordering__top_Oextremum__strict,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),Top: A,A3: A] :
      ( ordering_top(A,Less_eq,Less,Top)
     => ~ aa(A,$o,aa(A,fun(A,$o),Less,Top),A3) ) ).

% ordering_top.extremum_strict
tff(fact_1137_ordering__top_Oextremum__unique,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),Top: A,A3: A] :
      ( ordering_top(A,Less_eq,Less,Top)
     => ( aa(A,$o,aa(A,fun(A,$o),Less_eq,Top),A3)
      <=> ( A3 = Top ) ) ) ).

% ordering_top.extremum_unique
tff(fact_1138_ordering__top_Onot__eq__extremum,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),Top: A,A3: A] :
      ( ordering_top(A,Less_eq,Less,Top)
     => ( ( A3 != Top )
      <=> aa(A,$o,aa(A,fun(A,$o),Less,A3),Top) ) ) ).

% ordering_top.not_eq_extremum
tff(fact_1139_ordering__top_Oextremum__uniqueI,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),Top: A,A3: A] :
      ( ordering_top(A,Less_eq,Less,Top)
     => ( aa(A,$o,aa(A,fun(A,$o),Less_eq,Top),A3)
       => ( A3 = Top ) ) ) ).

% ordering_top.extremum_uniqueI
tff(fact_1140_semilattice__neutr__order_Oeq__neutr__iff,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z2: A,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A3: A,B2: A] :
      ( semila1105856199041335345_order(A,F,Z2,Less_eq,Less)
     => ( ( aa(A,A,aa(A,fun(A,A),F,A3),B2) = Z2 )
      <=> ( ( A3 = Z2 )
          & ( B2 = Z2 ) ) ) ) ).

% semilattice_neutr_order.eq_neutr_iff
tff(fact_1141_semilattice__neutr__order_Oneutr__eq__iff,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z2: A,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A3: A,B2: A] :
      ( semila1105856199041335345_order(A,F,Z2,Less_eq,Less)
     => ( ( Z2 = aa(A,A,aa(A,fun(A,A),F,A3),B2) )
      <=> ( ( A3 = Z2 )
          & ( B2 = Z2 ) ) ) ) ).

% semilattice_neutr_order.neutr_eq_iff
tff(fact_1142_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( ( I = J )
            & ( K = L ) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K) = aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L) ) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_1143_group__cancel_Oadd1,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A4: A,K: A,A3: A,B2: A] :
          ( ( A4 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),A3) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)) ) ) ) ).

% group_cancel.add1
tff(fact_1144_group__cancel_Oadd2,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [B3: A,K: A,B2: A,A3: A] :
          ( ( B3 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),B2) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)) ) ) ) ).

% group_cancel.add2
tff(fact_1145_add_Oassoc,axiom,
    ! [A: $tType] :
      ( semigroup_add(A)
     => ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) ) ).

% add.assoc
tff(fact_1146_add_Oleft__cancel,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B2: A,C2: 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),C2) )
        <=> ( B2 = C2 ) ) ) ).

% add.left_cancel
tff(fact_1147_add_Oright__cancel,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [B2: A,A3: A,C2: 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),C2),A3) )
        <=> ( B2 = C2 ) ) ) ).

% add.right_cancel
tff(fact_1148_ab__semigroup__add__class_Oadd_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) ) ).

% ab_semigroup_add_class.add.commute
tff(fact_1149_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [A: $tType] :
      ( ab_semigroup_add(A)
     => ! [B2: A,A3: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) ) ).

% ab_semigroup_add_class.add.left_commute
tff(fact_1150_add__left__imp__eq,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [A3: A,B2: A,C2: 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),C2) )
         => ( B2 = C2 ) ) ) ).

% add_left_imp_eq
tff(fact_1151_add__right__imp__eq,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [B2: A,A3: A,C2: 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),C2),A3) )
         => ( B2 = C2 ) ) ) ).

% add_right_imp_eq
tff(fact_1152_add_Oright__assoc,axiom,
    ! [A: $tType] :
      ( ab_semigroup_add(A)
     => ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) ) ).

% add.right_assoc
tff(fact_1153_add_Oright__commute,axiom,
    ! [A: $tType] :
      ( ab_semigroup_add(A)
     => ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)),B2) ) ).

% add.right_commute
tff(fact_1154_numerals_I1_J,axiom,
    aa(num,nat,numeral_numeral(nat),one2) = one_one(nat) ).

% numerals(1)
tff(fact_1155_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),I),J)
            & ( K = L ) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_1156_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( ( I = J )
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),K),L) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_1157_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),I),J)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),K),L) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_1158_add__mono,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [A3: A,B2: A,C2: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),D3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),D3)) ) ) ) ).

% add_mono
tff(fact_1159_add__left__mono,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2)) ) ) ).

% add_left_mono
tff(fact_1160_less__eqE,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
         => ~ ! [C4: A] : B2 != aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C4) ) ) ).

% less_eqE
tff(fact_1161_add__right__mono,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) ) ) ).

% add_right_mono
tff(fact_1162_le__iff__add,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
        <=> ? [C5: A] : B2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C5) ) ) ).

% le_iff_add
tff(fact_1163_add__le__imp__le__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [C2: A,A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2))
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2) ) ) ).

% add_le_imp_le_left
tff(fact_1164_add__le__imp__le__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [A3: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2))
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2) ) ) ).

% add_le_imp_le_right
tff(fact_1165_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_1166_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_1167_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_1168_add__mono__thms__linordered__field_I5_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),I),J)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),K),L) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_1169_add__mono__thms__linordered__field_I2_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( ( I = J )
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),K),L) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_1170_add__mono__thms__linordered__field_I1_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),I),J)
            & ( K = L ) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_1171_add__strict__mono,axiom,
    ! [A: $tType] :
      ( strict9044650504122735259up_add(A)
     => ! [A3: A,B2: A,C2: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),D3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),D3)) ) ) ) ).

% add_strict_mono
tff(fact_1172_add__strict__left__mono,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2)) ) ) ).

% add_strict_left_mono
tff(fact_1173_add__strict__right__mono,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) ) ) ).

% add_strict_right_mono
tff(fact_1174_add__less__imp__less__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [C2: A,A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2))
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2) ) ) ).

% add_less_imp_less_left
tff(fact_1175_add__less__imp__less__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [A3: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2))
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2) ) ) ).

% add_less_imp_less_right
tff(fact_1176_combine__common__factor,axiom,
    ! [A: $tType] :
      ( semiring(A)
     => ! [A3: A,E4: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),E4)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E4)),C2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),E4)),C2) ) ).

% combine_common_factor
tff(fact_1177_distrib__right,axiom,
    ! [A: $tType] :
      ( semiring(A)
     => ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ).

% distrib_right
tff(fact_1178_distrib__left,axiom,
    ! [A: $tType] :
      ( semiring(A)
     => ! [A3: A,B2: A,C2: 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),C2)) = 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),C2)) ) ).

% distrib_left
tff(fact_1179_comm__semiring__class_Odistrib,axiom,
    ! [A: $tType] :
      ( comm_semiring(A)
     => ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ).

% comm_semiring_class.distrib
tff(fact_1180_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A3: A,B2: A,C2: 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),C2)) = 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),C2)) ) ).

% ring_class.ring_distribs(1)
tff(fact_1181_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ).

% ring_class.ring_distribs(2)
tff(fact_1182_group__cancel_Osub1,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A4: A,K: A,A3: A,B2: A] :
          ( ( A4 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),A3) )
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A4),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)) ) ) ) ).

% group_cancel.sub1
tff(fact_1183_diff__eq__eq,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B2: A,C2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2) = C2 )
        <=> ( A3 = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2) ) ) ) ).

% diff_eq_eq
tff(fact_1184_eq__diff__eq,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,C2: A,B2: A] :
          ( ( A3 = aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),B2) )
        <=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2) = C2 ) ) ) ).

% eq_diff_eq
tff(fact_1185_add__diff__eq,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B2: A,C2: 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),C2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),C2) ) ).

% add_diff_eq
tff(fact_1186_diff__diff__eq2,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B2: A,C2: 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),C2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)),B2) ) ).

% diff_diff_eq2
tff(fact_1187_diff__add__eq,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)),B2) ) ).

% diff_add_eq
tff(fact_1188_diff__add__eq__diff__diff__swap,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B2: A,C2: 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),C2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),C2)),B2) ) ).

% diff_add_eq_diff_diff_swap
tff(fact_1189_add__implies__diff,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [C2: A,B2: A,A3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2) = A3 )
         => ( C2 = aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2) ) ) ) ).

% add_implies_diff
tff(fact_1190_diff__diff__eq,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) ) ).

% diff_diff_eq
tff(fact_1191_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_1192_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_1193_group__cancel_Oneg1,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A4: A,K: A,A3: A] :
          ( ( A4 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),A3) )
         => ( aa(A,A,uminus_uminus(A),A4) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),K)),aa(A,A,uminus_uminus(A),A3)) ) ) ) ).

% group_cancel.neg1
tff(fact_1194_add_Osafe__commute,axiom,
    ! [A: $tType] :
      ( ab_semigroup_add(A)
     => ! [X: A,Y: A,A3: A,B2: A] :
          ( syntax7388354845996824322omatch(A,A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y),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),B2),A3) ) ) ) ).

% add.safe_commute
tff(fact_1195_add__nonpos__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),zero_zero(A))
           => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y) = zero_zero(A) )
            <=> ( ( X = zero_zero(A) )
                & ( Y = zero_zero(A) ) ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_1196_add__nonneg__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),X)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Y)
           => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y) = zero_zero(A) )
            <=> ( ( X = zero_zero(A) )
                & ( Y = zero_zero(A) ) ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_1197_add__nonpos__nonpos,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),zero_zero(A)) ) ) ) ).

% add_nonpos_nonpos
tff(fact_1198_add__nonneg__nonneg,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)) ) ) ) ).

% add_nonneg_nonneg
tff(fact_1199_add__increasing2,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [C2: A,B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)) ) ) ) ).

% add_increasing2
tff(fact_1200_add__decreasing2,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [C2: A,A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)),B2) ) ) ) ).

% add_decreasing2
tff(fact_1201_add__increasing,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)) ) ) ) ).

% add_increasing
tff(fact_1202_add__decreasing,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)),B2) ) ) ) ).

% add_decreasing
tff(fact_1203_add__mono__thms__linordered__field_I4_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),I),J)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),K),L) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_1204_add__mono__thms__linordered__field_I3_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),I),J)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),K),L) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_1205_add__le__less__mono,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [A3: A,B2: A,C2: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),D3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),D3)) ) ) ) ).

% add_le_less_mono
tff(fact_1206_add__less__le__mono,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [A3: A,B2: A,C2: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),D3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),D3)) ) ) ) ).

% add_less_le_mono
tff(fact_1207_pos__add__strict,axiom,
    ! [A: $tType] :
      ( strict7427464778891057005id_add(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)) ) ) ) ).

% pos_add_strict
tff(fact_1208_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
         => ~ ! [C4: A] :
                ( ( B2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C4) )
               => ( C4 = zero_zero(A) ) ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
tff(fact_1209_add__pos__pos,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)) ) ) ) ).

% add_pos_pos
tff(fact_1210_add__neg__neg,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),zero_zero(A)) ) ) ) ).

% add_neg_neg
tff(fact_1211_diff__le__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)),C2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2)) ) ) ).

% diff_le_eq
tff(fact_1212_le__diff__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),C2) ) ) ).

% le_diff_eq
tff(fact_1213_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),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 ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add
tff(fact_1214_le__add__diff,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)),A3)) ) ) ).

% le_add_diff
tff(fact_1215_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A3)),B2) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
tff(fact_1216_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2)),A3) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
tff(fact_1217_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),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),C2),B2)),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3)) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
tff(fact_1218_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),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)),C2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)),A3) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
tff(fact_1219_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),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),C2)),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)),C2) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
tff(fact_1220_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A3)),B2) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_diff_right
tff(fact_1221_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),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_1222_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
           => ( ( aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3) = C2 )
            <=> ( B2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A3) ) ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
tff(fact_1223_add__mono1,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
         => aa(A,$o,aa(A,fun(A,$o),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_1224_less__add__one,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A))) ) ).

% less_add_one
tff(fact_1225_diff__less__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)),C2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2)) ) ) ).

% diff_less_eq
tff(fact_1226_less__diff__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),C2) ) ) ).

% less_diff_eq
tff(fact_1227_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_1228_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_1229_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_1230_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_1231_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_1232_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_1233_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_1234_eq__add__iff2,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A3: A,E4: A,C2: A,B2: A,D3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),E4)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E4)),D3) )
        <=> ( C2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3)),E4)),D3) ) ) ) ).

% eq_add_iff2
tff(fact_1235_eq__add__iff1,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A3: A,E4: A,C2: A,B2: A,D3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),E4)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E4)),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)),E4)),C2) = D3 ) ) ) ).

% eq_add_iff1
tff(fact_1236_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_1237_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_1238_group__cancel_Osub2,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [B3: A,K: A,B2: A,A3: A] :
          ( ( B3 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),B2) )
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),K)),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)) ) ) ) ).

% group_cancel.sub2
tff(fact_1239_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_1240_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_1241_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_1242_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_1243_numeral__One,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ( aa(num,A,numeral_numeral(A),one2) = one_one(A) ) ) ).

% numeral_One
tff(fact_1244_add_Ocomm__monoid__axioms,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => comm_monoid(A,plus_plus(A),zero_zero(A)) ) ).

% add.comm_monoid_axioms
tff(fact_1245_add_Omonoid__axioms,axiom,
    ! [A: $tType] :
      ( monoid_add(A)
     => monoid(A,plus_plus(A),zero_zero(A)) ) ).

% add.monoid_axioms
tff(fact_1246_semilattice__neutr__order_Oaxioms_I1_J,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z2: A,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o))] :
      ( semila1105856199041335345_order(A,F,Z2,Less_eq,Less)
     => semilattice_neutr(A,F,Z2) ) ).

% semilattice_neutr_order.axioms(1)
tff(fact_1247_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_1248_add__strict__increasing2,axiom,
    ! [A: $tType] :
      ( ordere8940638589300402666id_add(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)) ) ) ) ).

% add_strict_increasing2
tff(fact_1249_add__strict__increasing,axiom,
    ! [A: $tType] :
      ( ordere8940638589300402666id_add(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)) ) ) ) ).

% add_strict_increasing
tff(fact_1250_add__pos__nonneg,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)) ) ) ) ).

% add_pos_nonneg
tff(fact_1251_add__nonpos__neg,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),zero_zero(A)) ) ) ) ).

% add_nonpos_neg
tff(fact_1252_add__nonneg__pos,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)) ) ) ) ).

% add_nonneg_pos
tff(fact_1253_add__neg__nonpos,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),zero_zero(A)) ) ) ) ).

% add_neg_nonpos
tff(fact_1254_sum__squares__ge__zero,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [X: A,Y: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y))) ) ).

% sum_squares_ge_zero
tff(fact_1255_not__sum__squares__lt__zero,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [X: A,Y: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y))),zero_zero(A)) ) ).

% not_sum_squares_lt_zero
tff(fact_1256_discrete,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A))),B2) ) ) ).

% discrete
tff(fact_1257_zero__less__two,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),one_one(A))) ) ).

% zero_less_two
tff(fact_1258_le__add__iff1,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A3: A,E4: A,C2: A,B2: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),E4)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E4)),D3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)),E4)),C2)),D3) ) ) ).

% le_add_iff1
tff(fact_1259_le__add__iff2,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A3: A,E4: A,C2: A,B2: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),E4)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E4)),D3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3)),E4)),D3)) ) ) ).

% le_add_iff2
tff(fact_1260_less__add__iff1,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A3: A,E4: A,C2: A,B2: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),E4)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E4)),D3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)),E4)),C2)),D3) ) ) ).

% less_add_iff1
tff(fact_1261_less__add__iff2,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A3: A,E4: A,C2: A,B2: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),E4)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E4)),D3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3)),E4)),D3)) ) ) ).

% less_add_iff2
tff(fact_1262_divide__add__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z2: A,X: A,Y: A] :
          ( ( Z2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Z2)),Y) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z2))),Z2) ) ) ) ).

% divide_add_eq_iff
tff(fact_1263_add__divide__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z2: A,X: A,Y: A] :
          ( ( Z2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y),Z2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),Z2)),Y)),Z2) ) ) ) ).

% add_divide_eq_iff
tff(fact_1264_add__num__frac,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y: A,Z2: A,X: A] :
          ( ( Y != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),Z2),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(A,A,aa(A,fun(A,A),times_times(A),Z2),Y))),Y) ) ) ) ).

% add_num_frac
tff(fact_1265_add__frac__num,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y: A,X: A,Z2: A] :
          ( ( Y != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)),Z2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(A,A,aa(A,fun(A,A),times_times(A),Z2),Y))),Y) ) ) ) ).

% add_frac_num
tff(fact_1266_add__frac__eq,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y: A,Z2: A,X: A,W2: A] :
          ( ( Y != zero_zero(A) )
         => ( ( Z2 != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)),aa(A,A,aa(A,fun(A,A),divide_divide(A),W2),Z2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),Z2)),aa(A,A,aa(A,fun(A,A),times_times(A),W2),Y))),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z2)) ) ) ) ) ).

% add_frac_eq
tff(fact_1267_add__divide__eq__if__simps_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B2: A,Z2: A] :
          aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),Z2)) = $ite(Z2 = zero_zero(A),A3,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),Z2)),B2)),Z2)) ) ).

% add_divide_eq_if_simps(1)
tff(fact_1268_add__divide__eq__if__simps_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,Z2: A,B2: A] :
          aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),Z2)),B2) = $ite(Z2 = zero_zero(A),B2,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),Z2))),Z2)) ) ).

% add_divide_eq_if_simps(2)
tff(fact_1269_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_1270_div__add__self2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,A3: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),one_one(A)) ) ) ) ).

% div_add_self2
tff(fact_1271_div__add__self1,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,A3: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A3)),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),one_one(A)) ) ) ) ).

% div_add_self1
tff(fact_1272_less__half__sum,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(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),one_one(A)),one_one(A)))) ) ) ).

% less_half_sum
tff(fact_1273_gt__half__sum,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),one_one(A)))),B2) ) ) ).

% gt_half_sum
tff(fact_1274_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_1275_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_1276_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_1277_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_1278_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_1279_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_1280_distrib__left__NO__MATCH,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring(B)
     => ! [X: A,Y: A,A3: B,B2: B,C2: B] :
          ( nO_MATCH(A,B,aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y),A3)
         => ( aa(B,B,aa(B,fun(B,B),times_times(B),A3),aa(B,B,aa(B,fun(B,B),plus_plus(B),B2),C2)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),A3),B2)),aa(B,B,aa(B,fun(B,B),times_times(B),A3),C2)) ) ) ) ).

% distrib_left_NO_MATCH
tff(fact_1281_distrib__right__NO__MATCH,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring(B)
     => ! [X: A,Y: A,C2: B,A3: B,B2: B] :
          ( nO_MATCH(A,B,aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y),C2)
         => ( aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),A3),B2)),C2) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),A3),C2)),aa(B,B,aa(B,fun(B,B),times_times(B),B2),C2)) ) ) ) ).

% distrib_right_NO_MATCH
tff(fact_1282_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_1283_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_1284_convex__bound__le,axiom,
    ! [A: $tType] :
      ( linord6961819062388156250ring_1(A)
     => ! [X: A,A3: A,Y: A,U: A,V: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),A3)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),U)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),V)
               => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),U),V) = one_one(A) )
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),U),X)),aa(A,A,aa(A,fun(A,A),times_times(A),V),Y))),A3) ) ) ) ) ) ) ).

% convex_bound_le
tff(fact_1285_minus__divide__add__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z2: A,X: A,Y: A] :
          ( ( Z2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Z2))),Y) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z2))),Z2) ) ) ) ).

% minus_divide_add_eq_iff
tff(fact_1286_add__divide__eq__if__simps_I3_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,Z2: A,B2: A] :
          aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),Z2))),B2) = $ite(Z2 = zero_zero(A),B2,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),Z2))),Z2)) ) ).

% add_divide_eq_if_simps(3)
tff(fact_1287_convex__bound__lt,axiom,
    ! [A: $tType] :
      ( linord715952674999750819strict(A)
     => ! [X: A,A3: A,Y: A,U: A,V: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),A3)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),U)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),V)
               => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),U),V) = one_one(A) )
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),U),X)),aa(A,A,aa(A,fun(A,A),times_times(A),V),Y))),A3) ) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_1288_scaling__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [U: A,V: A,R2: A,S2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),U),V)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),R2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),R2),S2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),U),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),R2),aa(A,A,aa(A,fun(A,A),minus_minus(A),V),U))),S2))),V) ) ) ) ) ).

% scaling_mono
tff(fact_1289_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_1290_div__add__self2__no__field,axiom,
    ! [A: $tType,B: $tType] :
      ( ( euclid4440199948858584721cancel(B)
        & field(A) )
     => ! [X: A,B2: B,A3: B] :
          ( nO_MATCH(A,B,X,B2)
         => ( ( B2 != zero_zero(B) )
           => ( aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),A3),B2)),B2) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),A3),B2)),one_one(B)) ) ) ) ) ).

% div_add_self2_no_field
tff(fact_1291_div__add__self1__no__field,axiom,
    ! [A: $tType,B: $tType] :
      ( ( euclid4440199948858584721cancel(B)
        & field(A) )
     => ! [X: A,B2: B,A3: B] :
          ( nO_MATCH(A,B,X,B2)
         => ( ( B2 != zero_zero(B) )
           => ( aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),B2),A3)),B2) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),A3),B2)),one_one(B)) ) ) ) ) ).

% div_add_self1_no_field
tff(fact_1292_sum__squares__gt__zero__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y)))
        <=> ( ( X != zero_zero(A) )
            | ( Y != zero_zero(A) ) ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_1293_sum__squares__le__zero__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y))),zero_zero(A))
        <=> ( ( X = zero_zero(A) )
            & ( Y = zero_zero(A) ) ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_1294_add__scale__eq__noteq,axiom,
    ! [A: $tType] :
      ( semiri1453513574482234551roduct(A)
     => ! [R2: A,A3: A,B2: A,C2: A,D3: A] :
          ( ( R2 != zero_zero(A) )
         => ( ( ( A3 = B2 )
              & ( C2 != D3 ) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),R2),C2)) != aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),R2),D3)) ) ) ) ) ).

% add_scale_eq_noteq
tff(fact_1295_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_1296_less__by__empty,axiom,
    ! [A: $tType,A4: set(product_prod(A,A)),B3: set(product_prod(A,A))] :
      ( ( A4 = bot_bot(set(product_prod(A,A))) )
     => aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),A4),B3) ) ).

% less_by_empty
tff(fact_1297_divmod__algorithm__code_I2_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num] : unique8689654367752047608divmod(A,M,one2) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(num,A,numeral_numeral(A),M)),zero_zero(A)) ) ).

% divmod_algorithm_code(2)
tff(fact_1298_minus__sub__one__diff__one,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [M: num] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),neg_numeral_sub(A,M,one2))),one_one(A)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M)) ) ).

% minus_sub_one_diff_one
tff(fact_1299_predicate1I,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] :
      ( ! [X2: A] :
          ( aa(A,$o,P,X2)
         => aa(A,$o,Q,X2) )
     => aa(fun(A,$o),$o,aa(fun(A,$o),fun(fun(A,$o),$o),ord_less_eq(fun(A,$o)),P),Q) ) ).

% predicate1I
tff(fact_1300_semiring__norm_I166_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [V: num,W2: num,Y: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),V)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),Y)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),neg_numeral_sub(A,V,W2)),Y) ) ).

% semiring_norm(166)
tff(fact_1301_semiring__norm_I167_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [V: num,W2: num,Y: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),W2)),Y)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),neg_numeral_sub(A,W2,V)),Y) ) ).

% semiring_norm(167)
tff(fact_1302_add__neg__numeral__simps_I1_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: num,N: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),M)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))) = neg_numeral_sub(A,M,N) ) ).

% add_neg_numeral_simps(1)
tff(fact_1303_add__neg__numeral__simps_I2_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: 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),M))),aa(num,A,numeral_numeral(A),N)) = neg_numeral_sub(A,N,M) ) ).

% add_neg_numeral_simps(2)
tff(fact_1304_diff__numeral__simps_I4_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: 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),M))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))) = neg_numeral_sub(A,N,M) ) ).

% diff_numeral_simps(4)
tff(fact_1305_diff__numeral__special_I2_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: num] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(num,A,numeral_numeral(A),M)),one_one(A)) = neg_numeral_sub(A,M,one2) ) ).

% diff_numeral_special(2)
tff(fact_1306_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_1307_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_1308_add__neg__numeral__special_I3_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),M)),aa(A,A,uminus_uminus(A),one_one(A))) = neg_numeral_sub(A,M,one2) ) ).

% add_neg_numeral_special(3)
tff(fact_1309_add__neg__numeral__special_I2_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),one_one(A)) = neg_numeral_sub(A,one2,M) ) ).

% add_neg_numeral_special(2)
tff(fact_1310_add__neg__numeral__special_I1_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: 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),M))) = neg_numeral_sub(A,one2,M) ) ).

% add_neg_numeral_special(1)
tff(fact_1311_diff__numeral__special_I8_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: num] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(A,A,uminus_uminus(A),one_one(A))) = neg_numeral_sub(A,one2,M) ) ).

% diff_numeral_special(8)
tff(fact_1312_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_1313_div__eq__dividend__iff,axiom,
    ! [M: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N) = M )
      <=> ( N = one_one(nat) ) ) ) ).

% div_eq_dividend_iff
tff(fact_1314_div__less__dividend,axiom,
    ! [N: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),one_one(nat)),N)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N)),M) ) ) ).

% div_less_dividend
tff(fact_1315_nat__geq__1__eq__neqz,axiom,
    ! [X: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),X)
    <=> ( X != zero_zero(nat) ) ) ).

% nat_geq_1_eq_neqz
tff(fact_1316_rev__predicate1D,axiom,
    ! [A: $tType,P: fun(A,$o),X: A,Q: fun(A,$o)] :
      ( aa(A,$o,P,X)
     => ( aa(fun(A,$o),$o,aa(fun(A,$o),fun(fun(A,$o),$o),ord_less_eq(fun(A,$o)),P),Q)
       => aa(A,$o,Q,X) ) ) ).

% rev_predicate1D
tff(fact_1317_predicate1D,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o),X: A] :
      ( aa(fun(A,$o),$o,aa(fun(A,$o),fun(fun(A,$o),$o),ord_less_eq(fun(A,$o)),P),Q)
     => ( aa(A,$o,P,X)
       => aa(A,$o,Q,X) ) ) ).

% predicate1D
tff(fact_1318_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_1319_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_1320_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_1321_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_1322_crossproduct__noteq,axiom,
    ! [A: $tType] :
      ( semiri1453513574482234551roduct(A)
     => ! [A3: A,B2: A,C2: A,D3: A] :
          ( ( ( A3 != B2 )
            & ( C2 != D3 ) )
        <=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),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),C2)) ) ) ) ).

% crossproduct_noteq
tff(fact_1323_less__one,axiom,
    ! [N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),one_one(nat))
    <=> ( N = zero_zero(nat) ) ) ).

% less_one
tff(fact_1324_nat__mult__eq__1__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N) = one_one(nat) )
    <=> ( ( M = one_one(nat) )
        & ( N = one_one(nat) ) ) ) ).

% nat_mult_eq_1_iff
tff(fact_1325_nat__1__eq__mult__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( one_one(nat) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N) )
    <=> ( ( M = one_one(nat) )
        & ( N = one_one(nat) ) ) ) ).

% nat_1_eq_mult_iff
tff(fact_1326_subset__emptyI,axiom,
    ! [A: $tType,A4: set(A)] :
      ( ! [X2: A] : ~ aa(set(A),$o,member(A,X2),A4)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),bot_bot(set(A))) ) ).

% subset_emptyI
tff(fact_1327_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),aa(num,num,bit0,one2))) ) ) ).

% dbl_simps(4)
tff(fact_1328_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_1329_divmod__algorithm__code_I3_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [N: num] : unique8689654367752047608divmod(A,one2,aa(num,num,bit0,N)) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),zero_zero(A)),aa(num,A,numeral_numeral(A),one2)) ) ).

% divmod_algorithm_code(3)
tff(fact_1330_divmod__step__eq,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [L: num,Q3: A,R2: A] :
          unique1321980374590559556d_step(A,L,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q3),R2)) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),L)),R2),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Q3)),one_one(A))),aa(A,A,aa(A,fun(A,A),minus_minus(A),R2),aa(num,A,numeral_numeral(A),L))),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Q3)),R2)) ) ).

% divmod_step_eq
tff(fact_1331_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),aa(num,num,bit0,one2)) ) ) ).

% one_add_one
tff(fact_1332_dbl__simps_I3_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( neg_numeral_dbl(A,one_one(A)) = aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)) ) ) ).

% dbl_simps(3)
tff(fact_1333_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_1334_one__div__two__eq__zero,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = zero_zero(A) ) ) ).

% one_div_two_eq_zero
tff(fact_1335_bits__1__div__2,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = zero_zero(A) ) ) ).

% bits_1_div_2
tff(fact_1336_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),aa(num,num,bit0,one2))) ) ) ).

% add_neg_numeral_special(9)
tff(fact_1337_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),aa(num,num,bit0,one2))) ) ) ).

% diff_numeral_special(10)
tff(fact_1338_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),aa(num,num,bit0,one2)) ) ) ).

% diff_numeral_special(11)
tff(fact_1339_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_1340_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),aa(num,num,bit0,L))) ) ).

% sub_num_simps(3)
tff(fact_1341_divmod__algorithm__code_I7_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,N: num] :
          unique8689654367752047608divmod(A,aa(num,num,bit0,M),aa(num,num,bit1,N)) = $ite(aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),M),N),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),zero_zero(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,M))),unique1321980374590559556d_step(A,aa(num,num,bit1,N),unique8689654367752047608divmod(A,aa(num,num,bit0,M),aa(num,num,bit0,aa(num,num,bit1,N))))) ) ).

% divmod_algorithm_code(7)
tff(fact_1342_divmod__algorithm__code_I8_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,N: num] :
          unique8689654367752047608divmod(A,aa(num,num,bit1,M),aa(num,num,bit1,N)) = $ite(aa(num,$o,aa(num,fun(num,$o),ord_less(num),M),N),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),zero_zero(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit1,M))),unique1321980374590559556d_step(A,aa(num,num,bit1,N),unique8689654367752047608divmod(A,aa(num,num,bit1,M),aa(num,num,bit0,aa(num,num,bit1,N))))) ) ).

% divmod_algorithm_code(8)
tff(fact_1343_bot__nat__0_Oordering__top__axioms,axiom,
    ordering_top(nat,aTP_Lamp_bj(nat,fun(nat,$o)),aTP_Lamp_bk(nat,fun(nat,$o)),zero_zero(nat)) ).

% bot_nat_0.ordering_top_axioms
tff(fact_1344_bot__nat__def,axiom,
    bot_bot(nat) = zero_zero(nat) ).

% bot_nat_def
tff(fact_1345_xor__num_Ocases,axiom,
    ! [X: product_prod(num,num)] :
      ( ( X != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),one2) )
     => ( ! [N3: num] : X != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),aa(num,num,bit0,N3))
       => ( ! [N3: num] : X != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),aa(num,num,bit1,N3))
         => ( ! [M3: num] : X != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit0,M3)),one2)
           => ( ! [M3: num,N3: num] : X != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit0,M3)),aa(num,num,bit0,N3))
             => ( ! [M3: num,N3: num] : X != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit0,M3)),aa(num,num,bit1,N3))
               => ( ! [M3: num] : X != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M3)),one2)
                 => ( ! [M3: num,N3: num] : X != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M3)),aa(num,num,bit0,N3))
                   => ~ ! [M3: num,N3: num] : X != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M3)),aa(num,num,bit1,N3)) ) ) ) ) ) ) ) ) ).

% xor_num.cases
tff(fact_1346_neg__zdiv__mult__2,axiom,
    ! [A3: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A3),zero_zero(int))
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),B2))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),A3)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),B2),one_one(int))),A3) ) ) ).

% neg_zdiv_mult_2
tff(fact_1347_pos__zdiv__mult__2,axiom,
    ! [A3: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),A3)
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),B2))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),A3)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),B2),A3) ) ) ).

% pos_zdiv_mult_2
tff(fact_1348_int__bit__induct,axiom,
    ! [P: fun(int,$o),K: int] :
      ( aa(int,$o,P,zero_zero(int))
     => ( aa(int,$o,P,aa(int,int,uminus_uminus(int),one_one(int)))
       => ( ! [K2: int] :
              ( aa(int,$o,P,K2)
             => ( ( K2 != zero_zero(int) )
               => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),times_times(int),K2),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))) ) )
         => ( ! [K2: int] :
                ( aa(int,$o,P,K2)
               => ( ( K2 != aa(int,int,uminus_uminus(int),one_one(int)) )
                 => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),K2),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))))) ) )
           => aa(int,$o,P,K) ) ) ) ) ).

% int_bit_induct
tff(fact_1349_numeral__code_I2_J,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [N: num] :
          aa(num,A,numeral_numeral(A),aa(num,num,bit0,N)) = $let(
            m: A,
            m:= aa(num,A,numeral_numeral(A),N),
            aa(A,A,aa(A,fun(A,A),plus_plus(A),m),m) ) ) ).

% numeral_code(2)
tff(fact_1350_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_1351_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),aa(num,num,bit0,one2)) ).

% nat_1_add_1
tff(fact_1352_ssubst__Pair__rhs,axiom,
    ! [B: $tType,A: $tType,R2: A,S2: B,R: set(product_prod(A,B)),S4: B] :
      ( aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),R2),S2)),R)
     => ( ( S4 = S2 )
       => aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),R2),S4)),R) ) ) ).

% ssubst_Pair_rhs
tff(fact_1353_numeral__code_I3_J,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [N: num] :
          aa(num,A,numeral_numeral(A),aa(num,num,bit1,N)) = $let(
            m: A,
            m:= aa(num,A,numeral_numeral(A),N),
            aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),m),m)),one_one(A)) ) ) ).

% numeral_code(3)
tff(fact_1354_mult__eq__self__implies__10,axiom,
    ! [M: nat,N: nat] :
      ( ( M = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N) )
     => ( ( N = one_one(nat) )
        | ( M = zero_zero(nat) ) ) ) ).

% mult_eq_self_implies_10
tff(fact_1355_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_1356_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_1357_mult__eq__if,axiom,
    ! [M: nat,N: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N) = $ite(M = zero_zero(nat),zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),one_one(nat))),N))) ).

% mult_eq_if
tff(fact_1358_divmod__divmod__step,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,N: num] :
          unique8689654367752047608divmod(A,M,N) = $ite(aa(num,$o,aa(num,fun(num,$o),ord_less(num),M),N),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),zero_zero(A)),aa(num,A,numeral_numeral(A),M)),unique1321980374590559556d_step(A,N,unique8689654367752047608divmod(A,M,aa(num,num,bit0,N)))) ) ).

% divmod_divmod_step
tff(fact_1359_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),aa(num,num,bit0,one2))),Z2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Z2),Z2) ) ).

% mult_2
tff(fact_1360_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),aa(num,num,bit0,one2))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Z2),Z2) ) ).

% mult_2_right
tff(fact_1361_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),aa(num,num,bit0,one2))),A3)),B2) ) ).

% left_add_twice
tff(fact_1362_prop__restrict,axiom,
    ! [A: $tType,X: A,Z5: set(A),X5: set(A),P: fun(A,$o)] :
      ( aa(set(A),$o,member(A,X),Z5)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Z5),aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_bi(set(A),fun(fun(A,$o),fun(A,$o)),X5),P)))
       => aa(A,$o,P,X) ) ) ).

% prop_restrict
tff(fact_1363_Collect__restrict,axiom,
    ! [A: $tType,X5: set(A),P: fun(A,$o)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_bi(set(A),fun(fun(A,$o),fun(A,$o)),X5),P))),X5) ).

% Collect_restrict
tff(fact_1364_minus__1__div__2__eq,axiom,
    ! [A: $tType] :
      ( euclid8789492081693882211th_nat(A)
     => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ).

% minus_1_div_2_eq
tff(fact_1365_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),aa(num,num,bit0,one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))) ) ).

% set_bit_0
tff(fact_1366_nat__induct2,axiom,
    ! [P: fun(nat,$o),N: nat] :
      ( aa(nat,$o,P,zero_zero(nat))
     => ( aa(nat,$o,P,one_one(nat))
       => ( ! [N3: nat] :
              ( aa(nat,$o,P,N3)
             => aa(nat,$o,P,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) )
         => aa(nat,$o,P,N) ) ) ) ).

% nat_induct2
tff(fact_1367_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),aa(num,num,bit0,one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))) ) ).

% unset_bit_0
tff(fact_1368_divmod__algorithm__code_I6_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,N: num] : unique8689654367752047608divmod(A,aa(num,num,bit1,M),aa(num,num,bit0,N)) = aa(product_prod(A,A),product_prod(A,A),aa(fun(A,fun(A,product_prod(A,A))),fun(product_prod(A,A),product_prod(A,A)),product_case_prod(A,A,product_prod(A,A)),aTP_Lamp_bl(A,fun(A,product_prod(A,A)))),unique8689654367752047608divmod(A,M,N)) ) ).

% divmod_algorithm_code(6)
tff(fact_1369_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_bm(num,fun(A,fun(A,product_prod(A,A))),L)),Qr) ) ).

% divmod_step_def
tff(fact_1370_divmod__digit__1_I1_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),modulo_modulo(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),B2)))
             => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),B2)))),one_one(A)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) ) ) ) ) ) ).

% divmod_digit_1(1)
tff(fact_1371_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),aa(num,num,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),aa(num,num,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_1372_mod__minus__minus,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A3: A,B2: A] : modulo_modulo(A,aa(A,A,uminus_uminus(A),A3),aa(A,A,uminus_uminus(A),B2)) = aa(A,A,uminus_uminus(A),modulo_modulo(A,A3,B2)) ) ).

% mod_minus_minus
tff(fact_1373_case__prod__conv,axiom,
    ! [B: $tType,A: $tType,C: $tType,F: 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),F),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),F,A3),B2) ).

% case_prod_conv
tff(fact_1374_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_1375_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_1376_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_1377_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_1378_mod__mult__self4,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,C2: 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),C2)),A3),B2) = modulo_modulo(A,A3,B2) ) ).

% mod_mult_self4
tff(fact_1379_mod__mult__self3,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: 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),C2),B2)),A3),B2) = modulo_modulo(A,A3,B2) ) ).

% mod_mult_self3
tff(fact_1380_mod__mult__self2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A3: A,B2: A,C2: 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),C2)),B2) = modulo_modulo(A,A3,B2) ) ).

% mod_mult_self2
tff(fact_1381_mod__mult__self1,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A3: A,C2: 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),C2),B2)),B2) = modulo_modulo(A,A3,B2) ) ).

% mod_mult_self1
tff(fact_1382_minus__mod__self1,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [B2: A,A3: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3),B2) = modulo_modulo(A,aa(A,A,uminus_uminus(A),A3),B2) ) ).

% minus_mod_self1
tff(fact_1383_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_1384_bits__one__mod__two__eq__one,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ( modulo_modulo(A,one_one(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = one_one(A) ) ) ).

% bits_one_mod_two_eq_one
tff(fact_1385_one__mod__two__eq__one,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ( modulo_modulo(A,one_one(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = one_one(A) ) ) ).

% one_mod_two_eq_one
tff(fact_1386_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),aa(num,num,bit0,X))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Y))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,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_1387_xor__numerals_I8_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,X))),one_one(A)) = aa(num,A,numeral_numeral(A),aa(num,num,bit0,X)) ) ).

% xor_numerals(8)
tff(fact_1388_xor__numerals_I5_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,X))),one_one(A)) = aa(num,A,numeral_numeral(A),aa(num,num,bit1,X)) ) ).

% xor_numerals(5)
tff(fact_1389_xor__numerals_I2_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Y: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Y))) = aa(num,A,numeral_numeral(A),aa(num,num,bit0,Y)) ) ).

% xor_numerals(2)
tff(fact_1390_xor__numerals_I1_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Y: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Y))) = aa(num,A,numeral_numeral(A),aa(num,num,bit1,Y)) ) ).

% xor_numerals(1)
tff(fact_1391_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),aa(num,num,bit0,one2))) != zero_zero(A) )
        <=> ( modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = one_one(A) ) ) ) ).

% not_mod_2_eq_0_eq_1
tff(fact_1392_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),aa(num,num,bit0,one2))) != one_one(A) )
        <=> ( modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = zero_zero(A) ) ) ) ).

% not_mod_2_eq_1_eq_0
tff(fact_1393_minus__1__mod__2__eq,axiom,
    ! [A: $tType] :
      ( euclid8789492081693882211th_nat(A)
     => ( modulo_modulo(A,aa(A,A,uminus_uminus(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = one_one(A) ) ) ).

% minus_1_mod_2_eq
tff(fact_1394_bits__minus__1__mod__2__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ( modulo_modulo(A,aa(A,A,uminus_uminus(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = one_one(A) ) ) ).

% bits_minus_1_mod_2_eq
tff(fact_1395_mod2__gr__0,axiom,
    ! [M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),modulo_modulo(nat,M,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))
    <=> ( modulo_modulo(nat,M,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = one_one(nat) ) ) ).

% mod2_gr_0
tff(fact_1396_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),aa(num,num,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_1397_zmod__numeral__Bit1,axiom,
    ! [V: num,W2: num] : modulo_modulo(int,aa(num,int,numeral_numeral(int),aa(num,num,bit1,V)),aa(num,int,numeral_numeral(int),aa(num,num,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),aa(num,num,bit0,one2))),modulo_modulo(int,aa(num,int,numeral_numeral(int),V),aa(num,int,numeral_numeral(int),W2)))),one_one(int)) ).

% zmod_numeral_Bit1
tff(fact_1398_divmod__algorithm__code_I5_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,N: num] : unique8689654367752047608divmod(A,aa(num,num,bit0,M),aa(num,num,bit0,N)) = aa(product_prod(A,A),product_prod(A,A),aa(fun(A,fun(A,product_prod(A,A))),fun(product_prod(A,A),product_prod(A,A)),product_case_prod(A,A,product_prod(A,A)),aTP_Lamp_bn(A,fun(A,product_prod(A,A)))),unique8689654367752047608divmod(A,M,N)) ) ).

% divmod_algorithm_code(5)
tff(fact_1399_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),aa(num,num,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),aa(num,num,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_1400_zmod__minus1,axiom,
    ! [B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
     => ( modulo_modulo(int,aa(int,int,uminus_uminus(int),one_one(int)),B2) = aa(int,int,aa(int,fun(int,int),minus_minus(int),B2),one_one(int)) ) ) ).

% zmod_minus1
tff(fact_1401_minus__mod__int__eq,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),L)
     => ( modulo_modulo(int,aa(int,int,uminus_uminus(int),K),L) = aa(int,int,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),K),one_one(int)),L)) ) ) ).

% minus_mod_int_eq
tff(fact_1402_zdiv__zminus1__eq__if,axiom,
    ! [B2: int,A3: int] :
      ( ( B2 != zero_zero(int) )
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,uminus_uminus(int),A3)),B2) = $ite(modulo_modulo(int,A3,B2) = zero_zero(int),aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B2)),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B2))),one_one(int))) ) ) ).

% zdiv_zminus1_eq_if
tff(fact_1403_zdiv__zminus2__eq__if,axiom,
    ! [B2: int,A3: int] :
      ( ( B2 != zero_zero(int) )
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),aa(int,int,uminus_uminus(int),B2)) = $ite(modulo_modulo(int,A3,B2) = zero_zero(int),aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B2)),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B2))),one_one(int))) ) ) ).

% zdiv_zminus2_eq_if
tff(fact_1404_div__pos__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),L)),zero_zero(int))
       => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ) ).

% div_pos_neg_trivial
tff(fact_1405_div__pos__geq,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),L)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),L),K)
       => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),K),L)),L)),one_one(int)) ) ) ) ).

% div_pos_geq
tff(fact_1406_int__div__less__self,axiom,
    ! [X: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),X)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),one_one(int)),K)
       => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),X),K)),X) ) ) ).

% int_div_less_self
tff(fact_1407_verit__less__mono__div__int2,axiom,
    ! [A4: int,B3: int,N: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A4),B3)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),aa(int,int,uminus_uminus(int),N))
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),B3),N)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A4),N)) ) ) ).

% verit_less_mono_div_int2
tff(fact_1408_verit__le__mono__div__int,axiom,
    ! [A4: int,B3: int,N: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),A4),B3)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),N)
       => aa(int,$o,
            aa(int,fun(int,$o),ord_less_eq(int),
              aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A4),N)),
                $ite(modulo_modulo(int,B3,N) = zero_zero(int),one_one(int),zero_zero(int)))),
            aa(int,int,aa(int,fun(int,int),divide_divide(int),B3),N)) ) ) ).

% verit_le_mono_div_int
tff(fact_1409_div__eq__minus1,axiom,
    ! [B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,uminus_uminus(int),one_one(int))),B2) = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ).

% div_eq_minus1
tff(fact_1410_divmod__int__def,axiom,
    ! [M: num,N: num] : unique8689654367752047608divmod(int,M,N) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(num,int,numeral_numeral(int),M)),aa(num,int,numeral_numeral(int),N))),modulo_modulo(int,aa(num,int,numeral_numeral(int),M),aa(num,int,numeral_numeral(int),N))) ).

% divmod_int_def
tff(fact_1411_zmod__zminus1__not__zero,axiom,
    ! [K: int,L: int] :
      ( ( modulo_modulo(int,aa(int,int,uminus_uminus(int),K),L) != zero_zero(int) )
     => ( modulo_modulo(int,K,L) != zero_zero(int) ) ) ).

% zmod_zminus1_not_zero
tff(fact_1412_zmod__zminus2__not__zero,axiom,
    ! [K: int,L: int] :
      ( ( modulo_modulo(int,K,aa(int,int,uminus_uminus(int),L)) != zero_zero(int) )
     => ( modulo_modulo(int,K,L) != zero_zero(int) ) ) ).

% zmod_zminus2_not_zero
tff(fact_1413_zmod__zminus2__eq__if,axiom,
    ! [A3: int,B2: int] :
      modulo_modulo(int,A3,aa(int,int,uminus_uminus(int),B2)) = $ite(modulo_modulo(int,A3,B2) = zero_zero(int),zero_zero(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),modulo_modulo(int,A3,B2)),B2)) ).

% zmod_zminus2_eq_if
tff(fact_1414_zmod__zminus1__eq__if,axiom,
    ! [A3: int,B2: int] :
      modulo_modulo(int,aa(int,int,uminus_uminus(int),A3),B2) = $ite(modulo_modulo(int,A3,B2) = zero_zero(int),zero_zero(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),B2),modulo_modulo(int,A3,B2))) ).

% zmod_zminus1_eq_if
tff(fact_1415_prod_Ocase__distrib,axiom,
    ! [B: $tType,A: $tType,D: $tType,C: $tType,H2: fun(B,A),F: fun(C,fun(D,B)),Prod: product_prod(C,D)] : aa(B,A,H2,aa(product_prod(C,D),B,aa(fun(C,fun(D,B)),fun(product_prod(C,D),B),product_case_prod(C,D,B),F),Prod)) = aa(product_prod(C,D),A,aa(fun(C,fun(D,A)),fun(product_prod(C,D),A),product_case_prod(C,D,A),aa(fun(C,fun(D,B)),fun(C,fun(D,A)),aTP_Lamp_bo(fun(B,A),fun(fun(C,fun(D,B)),fun(C,fun(D,A))),H2),F)),Prod) ).

% prod.case_distrib
tff(fact_1416_case__prod__app,axiom,
    ! [D: $tType,A: $tType,C: $tType,B: $tType,F: fun(B,fun(C,fun(D,A))),X: product_prod(B,C),Y: D] : aa(D,A,aa(product_prod(B,C),fun(D,A),aa(fun(B,fun(C,fun(D,A))),fun(product_prod(B,C),fun(D,A)),product_case_prod(B,C,fun(D,A)),F),X),Y) = aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),aa(D,fun(B,fun(C,A)),aTP_Lamp_bp(fun(B,fun(C,fun(D,A))),fun(D,fun(B,fun(C,A))),F),Y)),X) ).

% case_prod_app
tff(fact_1417_nested__case__prod__simp,axiom,
    ! [D: $tType,A: $tType,C: $tType,B: $tType,F: fun(B,fun(C,fun(D,A))),X: product_prod(B,C),Y: D] : aa(D,A,aa(product_prod(B,C),fun(D,A),aa(fun(B,fun(C,fun(D,A))),fun(product_prod(B,C),fun(D,A)),product_case_prod(B,C,fun(D,A)),F),X),Y) = aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),aa(D,fun(B,fun(C,A)),aTP_Lamp_bp(fun(B,fun(C,fun(D,A))),fun(D,fun(B,fun(C,A))),F),Y)),X) ).

% nested_case_prod_simp
tff(fact_1418_mod__mult__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A3: A,C2: A,B2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),modulo_modulo(A,A3,C2)),modulo_modulo(A,B2,C2)),C2) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2),C2) ) ).

% mod_mult_eq
tff(fact_1419_mod__mult__cong,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A3: A,C2: A,A5: A,B2: A,B4: A] :
          ( ( modulo_modulo(A,A3,C2) = modulo_modulo(A,A5,C2) )
         => ( ( modulo_modulo(A,B2,C2) = modulo_modulo(A,B4,C2) )
           => ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2),C2) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A5),B4),C2) ) ) ) ) ).

% mod_mult_cong
tff(fact_1420_mod__mult__mult2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A3: A,C2: A,B2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),times_times(A),modulo_modulo(A,A3,B2)),C2) ) ).

% mod_mult_mult2
tff(fact_1421_mult__mod__right,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),C2),modulo_modulo(A,A3,B2)) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) ) ).

% mult_mod_right
tff(fact_1422_mod__mult__left__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A3: A,C2: A,B2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),modulo_modulo(A,A3,C2)),B2),C2) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2),C2) ) ).

% mod_mult_left_eq
tff(fact_1423_mod__mult__right__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A3: A,B2: A,C2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),modulo_modulo(A,B2,C2)),C2) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2),C2) ) ).

% mod_mult_right_eq
tff(fact_1424_mod__minus__eq,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A3: A,B2: A] : modulo_modulo(A,aa(A,A,uminus_uminus(A),modulo_modulo(A,A3,B2)),B2) = modulo_modulo(A,aa(A,A,uminus_uminus(A),A3),B2) ) ).

% mod_minus_eq
tff(fact_1425_mod__minus__cong,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A3: A,B2: A,A5: A] :
          ( ( modulo_modulo(A,A3,B2) = modulo_modulo(A,A5,B2) )
         => ( modulo_modulo(A,aa(A,A,uminus_uminus(A),A3),B2) = modulo_modulo(A,aa(A,A,uminus_uminus(A),A5),B2) ) ) ) ).

% mod_minus_cong
tff(fact_1426_mod__minus__right,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A3: A,B2: A] : modulo_modulo(A,A3,aa(A,A,uminus_uminus(A),B2)) = aa(A,A,uminus_uminus(A),modulo_modulo(A,aa(A,A,uminus_uminus(A),A3),B2)) ) ).

% mod_minus_right
tff(fact_1427_split__cong,axiom,
    ! [C: $tType,B: $tType,A: $tType,Q3: product_prod(A,B),F: fun(A,fun(B,C)),G: fun(A,fun(B,C)),P3: product_prod(A,B)] :
      ( ! [X2: A,Y2: B] :
          ( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Y2) = Q3 )
         => ( aa(B,C,aa(A,fun(B,C),F,X2),Y2) = aa(B,C,aa(A,fun(B,C),G,X2),Y2) ) )
     => ( ( P3 = Q3 )
       => ( aa(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F),P3) = aa(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),G),Q3) ) ) ) ).

% split_cong
tff(fact_1428_old_Oprod_Ocase,axiom,
    ! [B: $tType,A: $tType,C: $tType,F: fun(B,fun(C,A)),X1: B,X22: 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),F),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),X1),X22)) = aa(C,A,aa(B,fun(C,A),F,X1),X22) ).

% old.prod.case
tff(fact_1429_case__prod__Pair__iden,axiom,
    ! [B: $tType,A: $tType,P3: product_prod(A,B)] : aa(product_prod(A,B),product_prod(A,B),aa(fun(A,fun(B,product_prod(A,B))),fun(product_prod(A,B),product_prod(A,B)),product_case_prod(A,B,product_prod(A,B)),product_Pair(A,B)),P3) = P3 ).

% case_prod_Pair_iden
tff(fact_1430_case__prodE2,axiom,
    ! [B: $tType,A: $tType,C: $tType,Q: fun(A,$o),P: fun(B,fun(C,A)),Z2: product_prod(B,C)] :
      ( aa(A,$o,Q,aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),P),Z2))
     => ~ ! [X2: B,Y2: C] :
            ( ( Z2 = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),X2),Y2) )
           => ~ aa(A,$o,Q,aa(C,A,aa(B,fun(C,A),P,X2),Y2)) ) ) ).

% case_prodE2
tff(fact_1431_case__prod__eta,axiom,
    ! [C: $tType,B: $tType,A: $tType,F: 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_bq(fun(product_prod(A,B),C),fun(A,fun(B,C)),F)) = F ).

% case_prod_eta
tff(fact_1432_cond__case__prod__eta,axiom,
    ! [C: $tType,B: $tType,A: $tType,F: fun(A,fun(B,C)),G: fun(product_prod(A,B),C)] :
      ( ! [X2: A,Y2: B] : aa(B,C,aa(A,fun(B,C),F,X2),Y2) = aa(product_prod(A,B),C,G,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Y2))
     => ( aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F) = G ) ) ).

% cond_case_prod_eta
tff(fact_1433_mod__eqE,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A3: A,C2: A,B2: A] :
          ( ( modulo_modulo(A,A3,C2) = modulo_modulo(A,B2,C2) )
         => ~ ! [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),C2),D2)) ) ) ).

% mod_eqE
tff(fact_1434_pos__zmod__mult__2,axiom,
    ! [A3: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),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),aa(num,num,bit0,one2))),B2)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,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),aa(num,num,bit0,one2))),modulo_modulo(int,B2,A3))) ) ) ).

% pos_zmod_mult_2
tff(fact_1435_neg__zmod__mult__2,axiom,
    ! [A3: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),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),aa(num,num,bit0,one2))),B2)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,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),aa(num,num,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_1436_divmod_H__nat__def,axiom,
    ! [M: num,N: num] : unique8689654367752047608divmod(nat,M,N) = aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(num,nat,numeral_numeral(nat),M)),aa(num,nat,numeral_numeral(nat),N))),modulo_modulo(nat,aa(num,nat,numeral_numeral(nat),M),aa(num,nat,numeral_numeral(nat),N))) ).

% divmod'_nat_def
tff(fact_1437_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_br(num,fun(int,fun(int,product_prod(int,int))),L)),Qr) ).

% divmod_step_int_def
tff(fact_1438_uncurry__def,axiom,
    ! [C: $tType,B: $tType,A: $tType,F: fun(A,fun(B,C))] : uncurry(A,B,C,F) = aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F) ).

% uncurry_def
tff(fact_1439_xor_Ocomm__monoid__axioms,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => comm_monoid(A,bit_se5824344971392196577ns_xor(A),zero_zero(A)) ) ).

% xor.comm_monoid_axioms
tff(fact_1440_xor_Omonoid__axioms,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => monoid(A,bit_se5824344971392196577ns_xor(A),zero_zero(A)) ) ).

% xor.monoid_axioms
tff(fact_1441_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),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2))),modulo_modulo(A,A3,B2)) = A3 ) ).

% mult_div_mod_eq
tff(fact_1442_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),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2))) = A3 ) ).

% mod_mult_div_eq
tff(fact_1443_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),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),B2)) = A3 ) ).

% mod_div_mult_eq
tff(fact_1444_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),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),B2)),modulo_modulo(A,A3,B2)) = A3 ) ).

% div_mult_mod_eq
tff(fact_1445_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),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),B2)),modulo_modulo(A,A3,B2)) ) ).

% mod_div_decomp
tff(fact_1446_cancel__div__mod__rules_I1_J,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),B2)),modulo_modulo(A,A3,B2))),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2) ) ).

% cancel_div_mod_rules(1)
tff(fact_1447_cancel__div__mod__rules_I2_J,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [B2: A,A3: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2))),modulo_modulo(A,A3,B2))),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2) ) ).

% cancel_div_mod_rules(2)
tff(fact_1448_div__mult1__eq,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),modulo_modulo(A,B2,C2))),C2)) ) ).

% div_mult1_eq
tff(fact_1449_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),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2))) = modulo_modulo(A,A3,B2) ) ).

% minus_mult_div_eq_mod
tff(fact_1450_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),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)) ) ).

% minus_mod_eq_mult_div
tff(fact_1451_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),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),B2) ) ).

% minus_mod_eq_div_mult
tff(fact_1452_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),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),B2)) = modulo_modulo(A,A3,B2) ) ).

% minus_div_mult_eq_mod
tff(fact_1453_divmod__def,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,N: num] : unique8689654367752047608divmod(A,M,N) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),N))),modulo_modulo(A,aa(num,A,numeral_numeral(A),M),aa(num,A,numeral_numeral(A),N))) ) ).

% divmod_def
tff(fact_1454_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [C2: A,A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
         => ( modulo_modulo(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),modulo_modulo(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2),C2))),modulo_modulo(A,A3,B2)) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_1455_verit__le__mono__div,axiom,
    ! [A4: nat,B3: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A4),B3)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
       => aa(nat,$o,
            aa(nat,fun(nat,$o),ord_less_eq(nat),
              aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),A4),N)),
                $ite(modulo_modulo(nat,B3,N) = zero_zero(nat),one_one(nat),zero_zero(nat)))),
            aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),B3),N)) ) ) ).

% verit_le_mono_div
tff(fact_1456_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_bs(num,fun(nat,fun(nat,product_prod(nat,nat))),L)),Qr) ).

% divmod_step_nat_def
tff(fact_1457_divmod__digit__0_I2_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),modulo_modulo(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),B2))),B2)
           => ( modulo_modulo(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),B2)) = modulo_modulo(A,A3,B2) ) ) ) ) ).

% divmod_digit_0(2)
tff(fact_1458_divmod__digit__0_I1_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),modulo_modulo(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),B2))),B2)
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),B2))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) ) ) ) ) ).

% divmod_digit_0(1)
tff(fact_1459_mod__double__modulus,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: A,X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),M)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),X)
           => ( ( modulo_modulo(A,X,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),M)) = modulo_modulo(A,X,M) )
              | ( modulo_modulo(A,X,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),M)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,X,M)),M) ) ) ) ) ) ).

% mod_double_modulus
tff(fact_1460_divmod__digit__1_I2_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),modulo_modulo(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),B2)))
             => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),modulo_modulo(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),B2))),B2) = modulo_modulo(A,A3,B2) ) ) ) ) ) ).

% divmod_digit_1(2)
tff(fact_1461_zle__add1__eq__le,axiom,
    ! [W2: int,Z2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),W2),aa(int,int,aa(int,fun(int,int),plus_plus(int),Z2),one_one(int)))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),W2),Z2) ) ).

% zle_add1_eq_le
tff(fact_1462_zle__diff1__eq,axiom,
    ! [W2: int,Z2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),W2),aa(int,int,aa(int,fun(int,int),minus_minus(int),Z2),one_one(int)))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),W2),Z2) ) ).

% zle_diff1_eq
tff(fact_1463_one__div__minus__numeral,axiom,
    ! [N: num] : aa(int,int,aa(int,fun(int,int),divide_divide(int),one_one(int)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = aa(int,int,uminus_uminus(int),adjust_div(unique8689654367752047608divmod(int,one2,N))) ).

% one_div_minus_numeral
tff(fact_1464_minus__one__div__numeral,axiom,
    ! [N: num] : aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,uminus_uminus(int),one_one(int))),aa(num,int,numeral_numeral(int),N)) = aa(int,int,uminus_uminus(int),adjust_div(unique8689654367752047608divmod(int,one2,N))) ).

% minus_one_div_numeral
tff(fact_1465_le__imp__0__less,axiom,
    ! [Z2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z2)
     => aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),Z2)) ) ).

% le_imp_0_less
tff(fact_1466_signed__take__bit__rec,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat,A3: A] :
          bit_ri4674362597316999326ke_bit(A,N,A3) = $ite(N = zero_zero(nat),aa(A,A,uminus_uminus(A),modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),bit_ri4674362597316999326ke_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))))) ) ).

% signed_take_bit_rec
tff(fact_1467_numeral__div__minus__numeral,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = aa(int,int,uminus_uminus(int),adjust_div(unique8689654367752047608divmod(int,M,N))) ).

% numeral_div_minus_numeral
tff(fact_1468_minus__numeral__div__numeral,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),aa(num,int,numeral_numeral(int),N)) = aa(int,int,uminus_uminus(int),adjust_div(unique8689654367752047608divmod(int,M,N))) ).

% minus_numeral_div_numeral
tff(fact_1469_case__prodI2,axiom,
    ! [B: $tType,A: $tType,P3: product_prod(A,B),C2: fun(A,fun(B,$o))] :
      ( ! [A6: A,B5: B] :
          ( ( P3 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A6),B5) )
         => aa(B,$o,aa(A,fun(B,$o),C2,A6),B5) )
     => aa(product_prod(A,B),$o,aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),C2),P3) ) ).

% case_prodI2
tff(fact_1470_case__prodI,axiom,
    ! [A: $tType,B: $tType,F: fun(A,fun(B,$o)),A3: A,B2: B] :
      ( aa(B,$o,aa(A,fun(B,$o),F,A3),B2)
     => aa(product_prod(A,B),$o,aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),F),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)) ) ).

% case_prodI
tff(fact_1471_mem__case__prodI2,axiom,
    ! [C: $tType,B: $tType,A: $tType,P3: product_prod(A,B),Z2: C,C2: fun(A,fun(B,set(C)))] :
      ( ! [A6: A,B5: B] :
          ( ( P3 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A6),B5) )
         => aa(set(C),$o,member(C,Z2),aa(B,set(C),aa(A,fun(B,set(C)),C2,A6),B5)) )
     => aa(set(C),$o,member(C,Z2),aa(product_prod(A,B),set(C),aa(fun(A,fun(B,set(C))),fun(product_prod(A,B),set(C)),product_case_prod(A,B,set(C)),C2),P3)) ) ).

% mem_case_prodI2
tff(fact_1472_mem__case__prodI,axiom,
    ! [A: $tType,B: $tType,C: $tType,Z2: A,C2: fun(B,fun(C,set(A))),A3: B,B2: C] :
      ( aa(set(A),$o,member(A,Z2),aa(C,set(A),aa(B,fun(C,set(A)),C2,A3),B2))
     => aa(set(A),$o,member(A,Z2),aa(product_prod(B,C),set(A),aa(fun(B,fun(C,set(A))),fun(product_prod(B,C),set(A)),product_case_prod(B,C,set(A)),C2),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A3),B2))) ) ).

% mem_case_prodI
tff(fact_1473_case__prodI2_H,axiom,
    ! [A: $tType,B: $tType,C: $tType,P3: product_prod(A,B),C2: fun(A,fun(B,fun(C,$o))),X: C] :
      ( ! [A6: A,B5: B] :
          ( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A6),B5) = P3 )
         => aa(C,$o,aa(B,fun(C,$o),aa(A,fun(B,fun(C,$o)),C2,A6),B5),X) )
     => aa(C,$o,aa(product_prod(A,B),fun(C,$o),aa(fun(A,fun(B,fun(C,$o))),fun(product_prod(A,B),fun(C,$o)),product_case_prod(A,B,fun(C,$o)),C2),P3),X) ) ).

% case_prodI2'
tff(fact_1474_Collect__const__case__prod,axiom,
    ! [B: $tType,A: $tType,P: $o] :
      aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aTP_Lamp_bt($o,fun(A,fun(B,$o)),(P)))) = $ite((P),top_top(set(product_prod(A,B))),bot_bot(set(product_prod(A,B)))) ).

% Collect_const_case_prod
tff(fact_1475_signed__take__bit__of__minus__1,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat] : bit_ri4674362597316999326ke_bit(A,N,aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).

% signed_take_bit_of_minus_1
tff(fact_1476_signed__take__bit__numeral__of__1,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [K: num] : bit_ri4674362597316999326ke_bit(A,aa(num,nat,numeral_numeral(nat),K),one_one(A)) = one_one(A) ) ).

% signed_take_bit_numeral_of_1
tff(fact_1477_signed__take__bit__0,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A3: 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),aa(num,num,bit0,one2)))) ) ).

% signed_take_bit_0
tff(fact_1478_mem__case__prodE,axiom,
    ! [B: $tType,A: $tType,C: $tType,Z2: A,C2: fun(B,fun(C,set(A))),P3: product_prod(B,C)] :
      ( aa(set(A),$o,member(A,Z2),aa(product_prod(B,C),set(A),aa(fun(B,fun(C,set(A))),fun(product_prod(B,C),set(A)),product_case_prod(B,C,set(A)),C2),P3))
     => ~ ! [X2: B,Y2: C] :
            ( ( P3 = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),X2),Y2) )
           => ~ aa(set(A),$o,member(A,Z2),aa(C,set(A),aa(B,fun(C,set(A)),C2,X2),Y2)) ) ) ).

% mem_case_prodE
tff(fact_1479_signed__take__bit__minus,axiom,
    ! [N: nat,K: int] : bit_ri4674362597316999326ke_bit(int,N,aa(int,int,uminus_uminus(int),bit_ri4674362597316999326ke_bit(int,N,K))) = bit_ri4674362597316999326ke_bit(int,N,aa(int,int,uminus_uminus(int),K)) ).

% signed_take_bit_minus
tff(fact_1480_case__prodE,axiom,
    ! [A: $tType,B: $tType,C2: fun(A,fun(B,$o)),P3: product_prod(A,B)] :
      ( aa(product_prod(A,B),$o,aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),C2),P3)
     => ~ ! [X2: A,Y2: B] :
            ( ( P3 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Y2) )
           => ~ aa(B,$o,aa(A,fun(B,$o),C2,X2),Y2) ) ) ).

% case_prodE
tff(fact_1481_case__prodD,axiom,
    ! [A: $tType,B: $tType,F: fun(A,fun(B,$o)),A3: A,B2: B] :
      ( aa(product_prod(A,B),$o,aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),F),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2))
     => aa(B,$o,aa(A,fun(B,$o),F,A3),B2) ) ).

% case_prodD
tff(fact_1482_case__prodE_H,axiom,
    ! [B: $tType,A: $tType,C: $tType,C2: fun(A,fun(B,fun(C,$o))),P3: product_prod(A,B),Z2: C] :
      ( aa(C,$o,aa(product_prod(A,B),fun(C,$o),aa(fun(A,fun(B,fun(C,$o))),fun(product_prod(A,B),fun(C,$o)),product_case_prod(A,B,fun(C,$o)),C2),P3),Z2)
     => ~ ! [X2: A,Y2: B] :
            ( ( P3 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Y2) )
           => ~ aa(C,$o,aa(B,fun(C,$o),aa(A,fun(B,fun(C,$o)),C2,X2),Y2),Z2) ) ) ).

% case_prodE'
tff(fact_1483_case__prodD_H,axiom,
    ! [B: $tType,A: $tType,C: $tType,R: fun(A,fun(B,fun(C,$o))),A3: A,B2: B,C2: C] :
      ( aa(C,$o,aa(product_prod(A,B),fun(C,$o),aa(fun(A,fun(B,fun(C,$o))),fun(product_prod(A,B),fun(C,$o)),product_case_prod(A,B,fun(C,$o)),R),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),C2)
     => aa(C,$o,aa(B,fun(C,$o),aa(A,fun(B,fun(C,$o)),R,A3),B2),C2) ) ).

% case_prodD'
tff(fact_1484_Id__on__def_H,axiom,
    ! [A: $tType,A4: fun(A,$o)] : id_on(A,aa(fun(A,$o),set(A),collect(A),A4)) = aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),aTP_Lamp_bu(fun(A,$o),fun(A,fun(A,$o)),A4))) ).

% Id_on_def'
tff(fact_1485_rel__restrict__def,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A4: set(A)] : rel_restrict(A,R,A4) = aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),aa(set(A),fun(A,fun(A,$o)),aTP_Lamp_bv(set(product_prod(A,A)),fun(set(A),fun(A,fun(A,$o))),R),A4))) ).

% rel_restrict_def
tff(fact_1486_int__less__induct,axiom,
    ! [I: int,K: int,P: fun(int,$o)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),I),K)
     => ( aa(int,$o,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),K),one_one(int)))
       => ( ! [I2: int] :
              ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),I2),K)
             => ( aa(int,$o,P,I2)
               => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),I2),one_one(int))) ) )
         => aa(int,$o,P,I) ) ) ) ).

% int_less_induct
tff(fact_1487_minus__int__code_I2_J,axiom,
    ! [L: int] : aa(int,int,aa(int,fun(int,int),minus_minus(int),zero_zero(int)),L) = aa(int,int,uminus_uminus(int),L) ).

% minus_int_code(2)
tff(fact_1488_uminus__int__code_I1_J,axiom,
    aa(int,int,uminus_uminus(int),zero_zero(int)) = zero_zero(int) ).

% uminus_int_code(1)
tff(fact_1489_int__le__induct,axiom,
    ! [I: int,K: int,P: fun(int,$o)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I),K)
     => ( aa(int,$o,P,K)
       => ( ! [I2: int] :
              ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I2),K)
             => ( aa(int,$o,P,I2)
               => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),I2),one_one(int))) ) )
         => aa(int,$o,P,I) ) ) ) ).

% int_le_induct
tff(fact_1490_zless__add1__eq,axiom,
    ! [W2: int,Z2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),W2),aa(int,int,aa(int,fun(int,int),plus_plus(int),Z2),one_one(int)))
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),W2),Z2)
        | ( W2 = Z2 ) ) ) ).

% zless_add1_eq
tff(fact_1491_int__gr__induct,axiom,
    ! [K: int,I: int,P: fun(int,$o)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),I)
     => ( aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),K),one_one(int)))
       => ( ! [I2: int] :
              ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),I2)
             => ( aa(int,$o,P,I2)
               => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),I2),one_one(int))) ) )
         => aa(int,$o,P,I) ) ) ) ).

% int_gr_induct
tff(fact_1492_int__one__le__iff__zero__less,axiom,
    ! [Z2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),one_one(int)),Z2)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Z2) ) ).

% int_one_le_iff_zero_less
tff(fact_1493_pos__zmult__eq__1__iff,axiom,
    ! [M: int,N: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),M)
     => ( ( aa(int,int,aa(int,fun(int,int),times_times(int),M),N) = one_one(int) )
      <=> ( ( M = one_one(int) )
          & ( N = one_one(int) ) ) ) ) ).

% pos_zmult_eq_1_iff
tff(fact_1494_odd__nonzero,axiom,
    ! [Z2: int] : aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),Z2)),Z2) != zero_zero(int) ).

% odd_nonzero
tff(fact_1495_odd__less__0__iff,axiom,
    ! [Z2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),Z2)),Z2)),zero_zero(int))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z2),zero_zero(int)) ) ).

% odd_less_0_iff
tff(fact_1496_zless__imp__add1__zle,axiom,
    ! [W2: int,Z2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),W2),Z2)
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),W2),one_one(int))),Z2) ) ).

% zless_imp_add1_zle
tff(fact_1497_int__ge__induct,axiom,
    ! [K: int,I: int,P: fun(int,$o)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),I)
     => ( aa(int,$o,P,K)
       => ( ! [I2: int] :
              ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),I2)
             => ( aa(int,$o,P,I2)
               => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),I2),one_one(int))) ) )
         => aa(int,$o,P,I) ) ) ) ).

% int_ge_induct
tff(fact_1498_add1__zle__eq,axiom,
    ! [W2: int,Z2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),W2),one_one(int))),Z2)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),W2),Z2) ) ).

% add1_zle_eq
tff(fact_1499_int__induct,axiom,
    ! [P: fun(int,$o),K: int,I: int] :
      ( aa(int,$o,P,K)
     => ( ! [I2: int] :
            ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),I2)
           => ( aa(int,$o,P,I2)
             => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),I2),one_one(int))) ) )
       => ( ! [I2: int] :
              ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I2),K)
             => ( aa(int,$o,P,I2)
               => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),I2),one_one(int))) ) )
         => aa(int,$o,P,I) ) ) ) ).

% int_induct
tff(fact_1500_zmult__eq__1__iff,axiom,
    ! [M: int,N: int] :
      ( ( aa(int,int,aa(int,fun(int,int),times_times(int),M),N) = one_one(int) )
    <=> ( ( ( M = one_one(int) )
          & ( N = one_one(int) ) )
        | ( ( M = 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_1501_pos__zmult__eq__1__iff__lemma,axiom,
    ! [M: int,N: int] :
      ( ( aa(int,int,aa(int,fun(int,int),times_times(int),M),N) = one_one(int) )
     => ( ( M = one_one(int) )
        | ( M = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ) ).

% pos_zmult_eq_1_iff_lemma
tff(fact_1502_neg__eucl__rel__int__mult__2,axiom,
    ! [B2: int,A3: int,Q3: int,R2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),B2),zero_zero(int))
     => ( eucl_rel_int(aa(int,int,aa(int,fun(int,int),plus_plus(int),A3),one_one(int)),B2,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),R2))
       => eucl_rel_int(aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),A3)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),B2),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),R2)),one_one(int)))) ) ) ).

% neg_eucl_rel_int_mult_2
tff(fact_1503_signed__take__bit__numeral__minus__bit1,axiom,
    ! [L: num,K: num] : bit_ri4674362597316999326ke_bit(int,aa(num,nat,numeral_numeral(nat),L),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,K)))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),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),K))),one_one(int)))),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),one_one(int)) ).

% signed_take_bit_numeral_minus_bit1
tff(fact_1504_pos__eucl__rel__int__mult__2,axiom,
    ! [B2: int,A3: int,Q3: int,R2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),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),Q3),R2))
       => eucl_rel_int(aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),A3)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),B2),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),R2)))) ) ) ).

% pos_eucl_rel_int_mult_2
tff(fact_1505_signed__take__bit__Suc__minus__bit1,axiom,
    ! [N: nat,K: num] : 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,K)))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(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),K))),one_one(int)))),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),one_one(int)) ).

% signed_take_bit_Suc_minus_bit1
tff(fact_1506_divmod__BitM__2__eq,axiom,
    ! [M: num] : unique8689654367752047608divmod(int,bitM(M),aa(num,num,bit0,one2)) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(num,int,numeral_numeral(int),M)),one_one(int))),one_one(int)) ).

% divmod_BitM_2_eq
tff(fact_1507_split__part,axiom,
    ! [B: $tType,A: $tType,P: $o,Q: fun(A,fun(B,$o)),X3: product_prod(A,B)] :
      ( aa(product_prod(A,B),$o,aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aTP_Lamp_bw($o,fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),(P)),Q)),X3)
    <=> ( (P)
        & aa(product_prod(A,B),$o,aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),Q),X3) ) ) ).

% split_part
tff(fact_1508_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_1509_signed__take__bit__Suc__1,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat] : bit_ri4674362597316999326ke_bit(A,aa(nat,nat,suc,N),one_one(A)) = one_one(A) ) ).

% signed_take_bit_Suc_1
tff(fact_1510_Suc__diff,axiom,
    ! [M: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),N)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),M)
       => ( aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),one_one(nat))) ) ) ) ).

% Suc_diff
tff(fact_1511_Suc__1,axiom,
    aa(nat,nat,suc,one_one(nat)) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)) ).

% Suc_1
tff(fact_1512_Suc__diff__1,axiom,
    ! [N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),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_1513_Suc__times__numeral__mod__eq,axiom,
    ! [K: num,N: nat] :
      ( ( aa(num,nat,numeral_numeral(nat),K) != one_one(nat) )
     => ( modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),K)),N)),aa(num,nat,numeral_numeral(nat),K)) = one_one(nat) ) ) ).

% Suc_times_numeral_mod_eq
tff(fact_1514_sub__num__simps_I2_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [L: num] : neg_numeral_sub(A,one2,aa(num,num,bit0,L)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),bitM(L))) ) ).

% sub_num_simps(2)
tff(fact_1515_signed__take__bit__Suc__minus__bit0,axiom,
    ! [N: nat,K: num] : bit_ri4674362597316999326ke_bit(int,aa(nat,nat,suc,N),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,K)))) = aa(int,int,aa(int,fun(int,int),times_times(int),bit_ri4674362597316999326ke_bit(int,N,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K)))),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))) ).

% signed_take_bit_Suc_minus_bit0
tff(fact_1516_signed__take__bit__numeral__minus__bit0,axiom,
    ! [L: num,K: num] : 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,bit0,K)))) = aa(int,int,aa(int,fun(int,int),times_times(int),bit_ri4674362597316999326ke_bit(int,pred_numeral(L),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K)))),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))) ).

% signed_take_bit_numeral_minus_bit0
tff(fact_1517_signed__take__bit__Suc__bit1,axiom,
    ! [N: nat,K: num] : bit_ri4674362597316999326ke_bit(int,aa(nat,nat,suc,N),aa(num,int,numeral_numeral(int),aa(num,num,bit1,K))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),bit_ri4674362597316999326ke_bit(int,N,aa(num,int,numeral_numeral(int),K))),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),one_one(int)) ).

% signed_take_bit_Suc_bit1
tff(fact_1518_signed__take__bit__numeral__bit1,axiom,
    ! [L: num,K: num] : bit_ri4674362597316999326ke_bit(int,aa(num,nat,numeral_numeral(nat),L),aa(num,int,numeral_numeral(int),aa(num,num,bit1,K))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),bit_ri4674362597316999326ke_bit(int,pred_numeral(L),aa(num,int,numeral_numeral(int),K))),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),one_one(int)) ).

% signed_take_bit_numeral_bit1
tff(fact_1519_prod_Odisc__eq__case,axiom,
    ! [B: $tType,A: $tType,Prod: product_prod(A,B)] : aa(product_prod(A,B),$o,aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aTP_Lamp_bx(A,fun(B,$o))),Prod) ).

% prod.disc_eq_case
tff(fact_1520_unique__remainder,axiom,
    ! [A3: int,B2: int,Q3: int,R2: int,Q4: int,R3: int] :
      ( eucl_rel_int(A3,B2,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),R2))
     => ( eucl_rel_int(A3,B2,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q4),R3))
       => ( R2 = R3 ) ) ) ).

% unique_remainder
tff(fact_1521_unique__quotient,axiom,
    ! [A3: int,B2: int,Q3: int,R2: int,Q4: int,R3: int] :
      ( eucl_rel_int(A3,B2,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),R2))
     => ( eucl_rel_int(A3,B2,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q4),R3))
       => ( Q3 = Q4 ) ) ) ).

% unique_quotient
tff(fact_1522_One__nat__def,axiom,
    one_one(nat) = aa(nat,nat,suc,zero_zero(nat)) ).

% One_nat_def
tff(fact_1523_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_1524_plus__1__eq__Suc,axiom,
    aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)) = suc ).

% plus_1_eq_Suc
tff(fact_1525_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_1526_diff__Suc__eq__diff__pred,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),aa(nat,nat,suc,N)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),one_one(nat))),N) ).

% diff_Suc_eq_diff_pred
tff(fact_1527_eucl__rel__int__by0,axiom,
    ! [K: int] : eucl_rel_int(K,zero_zero(int),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),K)) ).

% eucl_rel_int_by0
tff(fact_1528_nat__induct__non__zero,axiom,
    ! [N: nat,P: fun(nat,$o)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
     => ( aa(nat,$o,P,one_one(nat))
       => ( ! [N3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N3)
             => ( aa(nat,$o,P,N3)
               => aa(nat,$o,P,aa(nat,nat,suc,N3)) ) )
         => aa(nat,$o,P,N) ) ) ) ).

% nat_induct_non_zero
tff(fact_1529_div__int__unique,axiom,
    ! [K: int,L: int,Q3: int,R2: int] :
      ( eucl_rel_int(K,L,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),R2))
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) = Q3 ) ) ).

% div_int_unique
tff(fact_1530_mod__int__unique,axiom,
    ! [K: int,L: int,Q3: int,R2: int] :
      ( eucl_rel_int(K,L,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),R2))
     => ( modulo_modulo(int,K,L) = R2 ) ) ).

% mod_int_unique
tff(fact_1531_pred__numeral__def,axiom,
    ! [K: num] : pred_numeral(K) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(num,nat,numeral_numeral(nat),K)),one_one(nat)) ).

% pred_numeral_def
tff(fact_1532_small__lazy_H_Ocases,axiom,
    ! [X: product_prod(int,int)] :
      ~ ! [D2: int,I2: int] : X != aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),D2),I2) ).

% small_lazy'.cases
tff(fact_1533_exhaustive__int_H_Ocases,axiom,
    ! [X: product_prod(fun(int,option(product_prod($o,list(code_term)))),product_prod(int,int))] :
      ~ ! [F2: fun(int,option(product_prod($o,list(code_term)))),D2: int,I2: int] : X != aa(product_prod(int,int),product_prod(fun(int,option(product_prod($o,list(code_term)))),product_prod(int,int)),aa(fun(int,option(product_prod($o,list(code_term)))),fun(product_prod(int,int),product_prod(fun(int,option(product_prod($o,list(code_term)))),product_prod(int,int))),product_Pair(fun(int,option(product_prod($o,list(code_term)))),product_prod(int,int)),F2),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),D2),I2)) ).

% exhaustive_int'.cases
tff(fact_1534_full__exhaustive__int_H_Ocases,axiom,
    ! [X: product_prod(fun(product_prod(int,fun(product_unit,code_term)),option(product_prod($o,list(code_term)))),product_prod(int,int))] :
      ~ ! [F2: fun(product_prod(int,fun(product_unit,code_term)),option(product_prod($o,list(code_term)))),D2: int,I2: int] : X != aa(product_prod(int,int),product_prod(fun(product_prod(int,fun(product_unit,code_term)),option(product_prod($o,list(code_term)))),product_prod(int,int)),aa(fun(product_prod(int,fun(product_unit,code_term)),option(product_prod($o,list(code_term)))),fun(product_prod(int,int),product_prod(fun(product_prod(int,fun(product_unit,code_term)),option(product_prod($o,list(code_term)))),product_prod(int,int))),product_Pair(fun(product_prod(int,fun(product_unit,code_term)),option(product_prod($o,list(code_term)))),product_prod(int,int)),F2),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),D2),I2)) ).

% full_exhaustive_int'.cases
tff(fact_1535_Suc__pred_H,axiom,
    ! [N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),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_1536_Suc__diff__eq__diff__pred,axiom,
    ! [N: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,M)),N) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))) ) ) ).

% Suc_diff_eq_diff_pred
tff(fact_1537_add__eq__if,axiom,
    ! [M: nat,N: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N) = $ite(M = zero_zero(nat),N,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),one_one(nat))),N))) ).

% add_eq_if
tff(fact_1538_Suc__n__minus__m__eq,axiom,
    ! [M: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),N)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),one_one(nat)),M)
       => ( aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),one_one(nat))) ) ) ) ).

% Suc_n_minus_m_eq
tff(fact_1539_eucl__rel__int__dividesI,axiom,
    ! [L: int,K: int,Q3: int] :
      ( ( L != zero_zero(int) )
     => ( ( K = aa(int,int,aa(int,fun(int,int),times_times(int),Q3),L) )
       => eucl_rel_int(K,L,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),zero_zero(int))) ) ) ).

% eucl_rel_int_dividesI
tff(fact_1540_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),aa(num,num,bit0,N))),one_one(A)) ) ).

% numeral_BitM
tff(fact_1541_eucl__rel__int,axiom,
    ! [K: int,L: int] : eucl_rel_int(K,L,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L)),modulo_modulo(int,K,L))) ).

% eucl_rel_int
tff(fact_1542_Suc__times__mod__eq,axiom,
    ! [M: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),M)
     => ( modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N)),M) = one_one(nat) ) ) ).

% Suc_times_mod_eq
tff(fact_1543_zminus1__lemma,axiom,
    ! [A3: int,B2: int,Q3: int,R2: int] :
      ( eucl_rel_int(A3,B2,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),R2))
     => ( ( 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),
                $ite(R2 = zero_zero(int),aa(int,int,uminus_uminus(int),Q3),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,uminus_uminus(int),Q3)),one_one(int)))),
              $ite(R2 = zero_zero(int),zero_zero(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),B2),R2)))) ) ) ).

% zminus1_lemma
tff(fact_1544_signed__take__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat,A3: 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),aa(num,num,bit0,one2)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),bit_ri4674362597316999326ke_bit(A,N,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))))) ) ).

% signed_take_bit_Suc
tff(fact_1545_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),aa(num,num,bit0,one2)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),bit_se2638667681897837118et_bit(A,N,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))))) ) ).

% unset_bit_Suc
tff(fact_1546_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),aa(num,num,bit0,one2)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),bit_se5668285175392031749et_bit(A,N,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))))) ) ).

% set_bit_Suc
tff(fact_1547_eucl__rel__int__iff,axiom,
    ! [K: int,L: int,Q3: int,R2: int] :
      ( eucl_rel_int(K,L,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),R2))
    <=> ( ( K = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),L),Q3)),R2) )
        & $ite(
            aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),L),
            ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),R2)
            & aa(int,$o,aa(int,fun(int,$o),ord_less(int),R2),L) ),
            $ite(
              aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),zero_zero(int)),
              ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),R2)
              & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),R2),zero_zero(int)) ),
              Q3 = zero_zero(int) ) ) ) ) ).

% eucl_rel_int_iff
tff(fact_1548_divmod__nat__if,axiom,
    ! [M: nat,N: nat] :
      divmod_nat(M,N) = $ite(
        ( ( N = zero_zero(nat) )
        | aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),N) ),
        aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),zero_zero(nat)),M),
        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_by(nat,fun(nat,product_prod(nat,nat)))),divmod_nat(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N),N)) ) ).

% divmod_nat_if
tff(fact_1549_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),aa(num,num,bit0,one2)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),bit_se8732182000553998342ip_bit(A,N,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))))) ) ).

% flip_bit_Suc
tff(fact_1550_take__bit__numeral__minus__bit1,axiom,
    ! [L: num,K: num] : bit_se2584673776208193580ke_bit(int,aa(num,nat,numeral_numeral(nat),L),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,K)))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),bit_se2584673776208193580ke_bit(int,pred_numeral(L),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),inc(K))))),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),one_one(int)) ).

% take_bit_numeral_minus_bit1
tff(fact_1551_int__ge__less__than2__def,axiom,
    ! [D3: int] : int_ge_less_than2(D3) = aa(fun(product_prod(int,int),$o),set(product_prod(int,int)),collect(product_prod(int,int)),aa(fun(int,fun(int,$o)),fun(product_prod(int,int),$o),product_case_prod(int,int,$o),aTP_Lamp_bz(int,fun(int,fun(int,$o)),D3))) ).

% int_ge_less_than2_def
tff(fact_1552_int__ge__less__than__def,axiom,
    ! [D3: int] : int_ge_less_than(D3) = aa(fun(product_prod(int,int),$o),set(product_prod(int,int)),collect(product_prod(int,int)),aa(fun(int,fun(int,$o)),fun(product_prod(int,int),$o),product_case_prod(int,int,$o),aTP_Lamp_ca(int,fun(int,fun(int,$o)),D3))) ).

% int_ge_less_than_def
tff(fact_1553_take__bit__Suc__1,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: nat] : bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,N),one_one(A)) = one_one(A) ) ).

% take_bit_Suc_1
tff(fact_1554_take__bit__numeral__1,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [L: num] : bit_se2584673776208193580ke_bit(A,aa(num,nat,numeral_numeral(nat),L),one_one(A)) = one_one(A) ) ).

% take_bit_numeral_1
tff(fact_1555_take__bit__of__1__eq__0__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: nat] :
          ( ( bit_se2584673776208193580ke_bit(A,N,one_one(A)) = zero_zero(A) )
        <=> ( N = zero_zero(nat) ) ) ) ).

% take_bit_of_1_eq_0_iff
tff(fact_1556_take__bit__minus,axiom,
    ! [N: nat,K: int] : bit_se2584673776208193580ke_bit(int,N,aa(int,int,uminus_uminus(int),bit_se2584673776208193580ke_bit(int,N,K))) = bit_se2584673776208193580ke_bit(int,N,aa(int,int,uminus_uminus(int),K)) ).

% take_bit_minus
tff(fact_1557_take__bit__decr__eq,axiom,
    ! [N: nat,K: int] :
      ( ( bit_se2584673776208193580ke_bit(int,N,K) != zero_zero(int) )
     => ( bit_se2584673776208193580ke_bit(int,N,aa(int,int,aa(int,fun(int,int),minus_minus(int),K),one_one(int))) = aa(int,int,aa(int,fun(int,int),minus_minus(int),bit_se2584673776208193580ke_bit(int,N,K)),one_one(int)) ) ) ).

% take_bit_decr_eq
tff(fact_1558_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)),A6: nat,B5: 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)),A6),aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),B5),Acc))) ).

% fold_atLeastAtMost_nat.cases
tff(fact_1559_take__bit__Suc__bit0,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: nat,K: num] : bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,N),aa(num,A,numeral_numeral(A),aa(num,num,bit0,K))) = aa(A,A,aa(A,fun(A,A),times_times(A),bit_se2584673776208193580ke_bit(A,N,aa(num,A,numeral_numeral(A),K))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ).

% take_bit_Suc_bit0
tff(fact_1560_take__bit__numeral__bit0,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [L: num,K: num] : bit_se2584673776208193580ke_bit(A,aa(num,nat,numeral_numeral(nat),L),aa(num,A,numeral_numeral(A),aa(num,num,bit0,K))) = aa(A,A,aa(A,fun(A,A),times_times(A),bit_se2584673776208193580ke_bit(A,pred_numeral(L),aa(num,A,numeral_numeral(A),K))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ).

% take_bit_numeral_bit0
tff(fact_1561_take__bit__Suc__minus__bit0,axiom,
    ! [N: nat,K: num] : bit_se2584673776208193580ke_bit(int,aa(nat,nat,suc,N),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,K)))) = aa(int,int,aa(int,fun(int,int),times_times(int),bit_se2584673776208193580ke_bit(int,N,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K)))),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))) ).

% take_bit_Suc_minus_bit0
tff(fact_1562_take__bit__numeral__minus__bit0,axiom,
    ! [L: num,K: num] : 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,bit0,K)))) = aa(int,int,aa(int,fun(int,int),times_times(int),bit_se2584673776208193580ke_bit(int,pred_numeral(L),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K)))),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))) ).

% take_bit_numeral_minus_bit0
tff(fact_1563_take__bit__Suc__bit1,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: nat,K: num] : bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,N),aa(num,A,numeral_numeral(A),aa(num,num,bit1,K))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),bit_se2584673776208193580ke_bit(A,N,aa(num,A,numeral_numeral(A),K))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),one_one(A)) ) ).

% take_bit_Suc_bit1
tff(fact_1564_take__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A3: 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),bit_se2584673776208193580ke_bit(A,N,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))) ) ).

% take_bit_Suc
tff(fact_1565_take__bit__numeral__bit1,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [L: num,K: num] : bit_se2584673776208193580ke_bit(A,aa(num,nat,numeral_numeral(nat),L),aa(num,A,numeral_numeral(A),aa(num,num,bit1,K))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),bit_se2584673776208193580ke_bit(A,pred_numeral(L),aa(num,A,numeral_numeral(A),K))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),one_one(A)) ) ).

% take_bit_numeral_bit1
tff(fact_1566_divmod__nat__def,axiom,
    ! [M: nat,N: nat] : divmod_nat(M,N) = aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N)),modulo_modulo(nat,M,N)) ).

% divmod_nat_def
tff(fact_1567_take__bit__Suc__minus__bit1,axiom,
    ! [N: nat,K: num] : 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,K)))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),bit_se2584673776208193580ke_bit(int,N,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),inc(K))))),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),one_one(int)) ).

% take_bit_Suc_minus_bit1
tff(fact_1568_take__bit__rec,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A3: A] :
          bit_se2584673776208193580ke_bit(A,N,A3) = $ite(N = zero_zero(nat),zero_zero(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),bit_se2584673776208193580ke_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))) ) ).

% take_bit_rec
tff(fact_1569_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),aa(num,num,bit0,one2))),bit_se2239418461657761734s_mask(A,pred_numeral(N)))) ) ).

% mask_numeral
tff(fact_1570_fold__atLeastAtMost__nat_Opinduct,axiom,
    ! [A: $tType,A0: fun(nat,fun(A,A)),A1: nat,A22: nat,A32: A,P: fun(fun(nat,fun(A,A)),fun(nat,fun(nat,fun(A,$o))))] :
      ( accp(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),set_fo1817059534552279752at_rel(A),aa(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),aa(fun(nat,fun(A,A)),fun(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A)))),product_Pair(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),A0),aa(product_prod(nat,A),product_prod(nat,product_prod(nat,A)),aa(nat,fun(product_prod(nat,A),product_prod(nat,product_prod(nat,A))),product_Pair(nat,product_prod(nat,A)),A1),aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),A22),A32))))
     => ( ! [F2: fun(nat,fun(A,A)),A6: nat,B5: 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)),A6),aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),B5),Acc))))
           => ( ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),B5),A6)
               => aa(A,$o,aa(nat,fun(A,$o),aa(nat,fun(nat,fun(A,$o)),aa(fun(nat,fun(A,A)),fun(nat,fun(nat,fun(A,$o))),P,F2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A6),one_one(nat))),B5),aa(A,A,aa(nat,fun(A,A),F2,A6),Acc)) )
             => aa(A,$o,aa(nat,fun(A,$o),aa(nat,fun(nat,fun(A,$o)),aa(fun(nat,fun(A,A)),fun(nat,fun(nat,fun(A,$o))),P,F2),A6),B5),Acc) ) )
       => aa(A,$o,aa(nat,fun(A,$o),aa(nat,fun(nat,fun(A,$o)),aa(fun(nat,fun(A,A)),fun(nat,fun(nat,fun(A,$o))),P,A0),A1),A22),A32) ) ) ).

% fold_atLeastAtMost_nat.pinduct
tff(fact_1571_and__int__unfold,axiom,
    ! [K: int,L: int] :
      aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L) = $ite(
        ( ( K = zero_zero(int) )
        | ( L = zero_zero(int) ) ),
        zero_zero(int),
        $ite(
          K = aa(int,int,uminus_uminus(int),one_one(int)),
          L,
          $ite(L = aa(int,int,uminus_uminus(int),one_one(int)),K,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),modulo_modulo(int,K,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),modulo_modulo(int,L,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))))))) ) ) ).

% and_int_unfold
tff(fact_1572_take__bit__minus__small__eq,axiom,
    ! [K: int,N: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N))
       => ( bit_se2584673776208193580ke_bit(int,N,aa(int,int,uminus_uminus(int),K)) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N)),K) ) ) ) ).

% take_bit_minus_small_eq
tff(fact_1573_signed__take__bit__int__greater__eq,axiom,
    ! [K: int,N: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N)))
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(nat,nat,suc,N)))),bit_ri4674362597316999326ke_bit(int,N,K)) ) ).

% signed_take_bit_int_greater_eq
tff(fact_1574_normalize__negative,axiom,
    ! [Q3: int,P3: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Q3),zero_zero(int))
     => ( normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P3),Q3)) = normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,uminus_uminus(int),P3)),aa(int,int,uminus_uminus(int),Q3))) ) ) ).

% normalize_negative
tff(fact_1575_odd__two__times__div__two__nat,axiom,
    ! [N: nat] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),aa(num,num,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_1576_odd__two__times__div__two__succ,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A3: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,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),aa(num,num,bit0,one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))),one_one(A)) = A3 ) ) ) ).

% odd_two_times_div_two_succ
tff(fact_1577_dvd__minus__iff,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),X),aa(A,A,uminus_uminus(A),Y))
        <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),X),Y) ) ) ).

% dvd_minus_iff
tff(fact_1578_minus__dvd__iff,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,uminus_uminus(A),X)),Y)
        <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),X),Y) ) ) ).

% minus_dvd_iff
tff(fact_1579_power__one,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [N: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),one_one(A)),N) = one_one(A) ) ).

% power_one
tff(fact_1580_power__one__right,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A3: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),one_one(nat)) = A3 ) ).

% power_one_right
tff(fact_1581_nat__dvd__1__iff__1,axiom,
    ! [M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),M),one_one(nat))
    <=> ( M = one_one(nat) ) ) ).

% nat_dvd_1_iff_1
tff(fact_1582_dvd__mult__cancel__left,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [C2: A,A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
        <=> ( ( C2 = zero_zero(A) )
            | aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A3),B2) ) ) ) ).

% dvd_mult_cancel_left
tff(fact_1583_dvd__mult__cancel__right,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [A3: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))
        <=> ( ( C2 = zero_zero(A) )
            | aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A3),B2) ) ) ) ).

% dvd_mult_cancel_right
tff(fact_1584_dvd__times__left__cancel__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A,C2: A] :
          ( ( A3 != zero_zero(A) )
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2))
          <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),C2) ) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_1585_dvd__times__right__cancel__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A,C2: A] :
          ( ( A3 != zero_zero(A) )
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3))
          <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),C2) ) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_1586_dvd__add__times__triv__left__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A3: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A3),B2) ) ) ).

% dvd_add_times_triv_left_iff
tff(fact_1587_dvd__add__times__triv__right__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)))
        <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A3),B2) ) ) ).

% dvd_add_times_triv_right_iff
tff(fact_1588_unit__prod,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A3),one_one(A))
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),one_one(A))
           => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),one_one(A)) ) ) ) ).

% unit_prod
tff(fact_1589_power__inject__exp,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,M: nat,N: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A3)
         => ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),M) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N) )
          <=> ( M = N ) ) ) ) ).

% power_inject_exp
tff(fact_1590_dvd__div__mult__self,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A3),B2)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3)),A3) = B2 ) ) ) ).

% dvd_div_mult_self
tff(fact_1591_dvd__mult__div__cancel,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A3),B2)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3)) = B2 ) ) ) ).

% dvd_mult_div_cancel
tff(fact_1592_unit__div,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A3),one_one(A))
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),one_one(A))
           => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),one_one(A)) ) ) ) ).

% unit_div
tff(fact_1593_unit__div__1__unit,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A3),one_one(A))
         => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A3)),one_one(A)) ) ) ).

% unit_div_1_unit
tff(fact_1594_unit__div__1__div__1,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A3),one_one(A))
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A3)) = A3 ) ) ) ).

% unit_div_1_div_1
tff(fact_1595_and_Oleft__neutral,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,uminus_uminus(A),one_one(A))),A3) = A3 ) ).

% and.left_neutral
tff(fact_1596_and_Oright__neutral,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),aa(A,A,uminus_uminus(A),one_one(A))) = A3 ) ).

% and.right_neutral
tff(fact_1597_bit_Oconj__one__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),X),aa(A,A,uminus_uminus(A),one_one(A))) = X ) ).

% bit.conj_one_right
tff(fact_1598_unit__mult__div__div,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A3),one_one(A))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A3)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3) ) ) ) ).

% unit_mult_div_div
tff(fact_1599_unit__div__mult__self,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A3),one_one(A))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3)),A3) = B2 ) ) ) ).

% unit_div_mult_self
tff(fact_1600_power__strict__increasing__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,X: nat,Y: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),X)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Y))
          <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),Y) ) ) ) ).

% power_strict_increasing_iff
tff(fact_1601_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_1602_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_1603_and__numerals_I8_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,X))),one_one(A)) = one_one(A) ) ).

% and_numerals(8)
tff(fact_1604_and__numerals_I2_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Y: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Y))) = one_one(A) ) ).

% and_numerals(2)
tff(fact_1605_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_1606_power__add__numeral,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A3: A,M: 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),M))),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),M),N))) ) ).

% power_add_numeral
tff(fact_1607_power__add__numeral2,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A3: A,M: 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),M))),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),M),N)))),B2) ) ).

% power_add_numeral2
tff(fact_1608_take__bit__minus__one__eq__mask,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat] : bit_se2584673776208193580ke_bit(A,N,aa(A,A,uminus_uminus(A),one_one(A))) = bit_se2239418461657761734s_mask(A,N) ) ).

% take_bit_minus_one_eq_mask
tff(fact_1609_normalize__denom__zero,axiom,
    ! [P3: int] : normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P3),zero_zero(int))) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int)) ).

% normalize_denom_zero
tff(fact_1610_power__strict__decreasing__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,M: nat,N: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),one_one(A))
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),N))
            <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),M) ) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_1611_even__mult__iff,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2))
        <=> ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A3)
            | aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),B2) ) ) ) ).

% even_mult_iff
tff(fact_1612_power__increasing__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,X: nat,Y: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),X)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Y))
          <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X),Y) ) ) ) ).

% power_increasing_iff
tff(fact_1613_and__numerals_I5_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,X))),one_one(A)) = zero_zero(A) ) ).

% and_numerals(5)
tff(fact_1614_and__numerals_I1_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Y: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Y))) = zero_zero(A) ) ).

% and_numerals(1)
tff(fact_1615_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),aa(num,num,bit0,X))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Y))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,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_1616_power2__minus,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A3: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),A3)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ) ).

% power2_minus
tff(fact_1617_and__minus__numerals_I6_J,axiom,
    ! [N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N)))),one_one(int)) = one_one(int) ).

% and_minus_numerals(6)
tff(fact_1618_and__minus__numerals_I2_J,axiom,
    ! [N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),one_one(int)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N)))) = one_one(int) ).

% and_minus_numerals(2)
tff(fact_1619_power__decreasing__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,M: nat,N: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),one_one(A))
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),N))
            <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),M) ) ) ) ) ).

% power_decreasing_iff
tff(fact_1620_even__plus__one__iff,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A)))
        <=> ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A3) ) ) ).

% even_plus_one_iff
tff(fact_1621_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),aa(num,num,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),aa(num,num,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_1622_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),aa(num,num,bit0,Y))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,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_1623_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),aa(num,num,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),aa(num,num,bit0,one2))),N)) ) ).

% Power.ring_1_class.power_minus_even
tff(fact_1624_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [N: nat,A3: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),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),N) ) ) ) ).

% Parity.ring_1_class.power_minus_even
tff(fact_1625_power__minus__odd,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [N: nat,A3: A] :
          ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),A3)),N) = aa(A,A,uminus_uminus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)) ) ) ) ).

% power_minus_odd
tff(fact_1626_and__minus__numerals_I5_J,axiom,
    ! [N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,N)))),one_one(int)) = zero_zero(int) ).

% and_minus_numerals(5)
tff(fact_1627_and__minus__numerals_I1_J,axiom,
    ! [N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),one_one(int)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,N)))) = zero_zero(int) ).

% and_minus_numerals(1)
tff(fact_1628_even__succ__div__two,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A3)
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ) ) ).

% even_succ_div_two
tff(fact_1629_odd__succ__div__two,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A3: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A3)
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),one_one(A)) ) ) ) ).

% odd_succ_div_two
tff(fact_1630_even__succ__div__2,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A3)
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),A3)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ) ) ).

% even_succ_div_2
tff(fact_1631_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),aa(num,num,bit0,one2))),N)) = one_one(A) ) ).

% power_minus1_even
tff(fact_1632_neg__one__odd__power,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [N: nat] :
          ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),N) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ).

% neg_one_odd_power
tff(fact_1633_neg__one__even__power,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [N: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),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) ) ) ) ).

% neg_one_even_power
tff(fact_1634_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [N: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)),one_one(A)))
        <=> ( N = zero_zero(nat) ) ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_1635_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),aa(num,num,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_1636_even__succ__div__exp,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,N: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A3)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),A3)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)) ) ) ) ) ).

% even_succ_div_exp
tff(fact_1637_even__succ__mod__exp,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,N: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A3)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),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),aa(num,num,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),aa(num,num,bit0,one2))),N))) ) ) ) ) ).

% even_succ_mod_exp
tff(fact_1638_is__unit__power__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,N: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)),one_one(A))
        <=> ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A3),one_one(A))
            | ( N = zero_zero(nat) ) ) ) ) ).

% is_unit_power_iff
tff(fact_1639_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_1640_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_1641_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_1642_dvdE,axiom,
    ! [A: $tType] :
      ( dvd(A)
     => ! [B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),A3)
         => ~ ! [K2: A] : A3 != aa(A,A,aa(A,fun(A,A),times_times(A),B2),K2) ) ) ).

% dvdE
tff(fact_1643_dvdI,axiom,
    ! [A: $tType] :
      ( dvd(A)
     => ! [A3: A,B2: A,K: A] :
          ( ( A3 = aa(A,A,aa(A,fun(A,A),times_times(A),B2),K) )
         => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),A3) ) ) ).

% dvdI
tff(fact_1644_dvd__def,axiom,
    ! [A: $tType] :
      ( dvd(A)
     => ! [B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),A3)
        <=> ? [K4: A] : A3 = aa(A,A,aa(A,fun(A,A),times_times(A),B2),K4) ) ) ).

% dvd_def
tff(fact_1645_dvd__mult,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A3),C2)
         => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ) ).

% dvd_mult
tff(fact_1646_dvd__mult2,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A3),B2)
         => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ) ).

% dvd_mult2
tff(fact_1647_dvd__mult__left,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),C2)
         => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A3),C2) ) ) ).

% dvd_mult_left
tff(fact_1648_dvd__triv__left,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A,B2: A] : aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) ) ).

% dvd_triv_left
tff(fact_1649_mult__dvd__mono,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A,B2: A,C2: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A3),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C2),D3)
           => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D3)) ) ) ) ).

% mult_dvd_mono
tff(fact_1650_dvd__mult__right,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),C2)
         => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),C2) ) ) ).

% dvd_mult_right
tff(fact_1651_dvd__triv__right,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A,B2: A] : aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A3)) ) ).

% dvd_triv_right
tff(fact_1652_one__dvd,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A] : aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),one_one(A)),A3) ) ).

% one_dvd
tff(fact_1653_unit__imp__dvd,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),one_one(A))
         => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),A3) ) ) ).

% unit_imp_dvd
tff(fact_1654_dvd__unit__imp__unit,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A3),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),one_one(A))
           => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A3),one_one(A)) ) ) ) ).

% dvd_unit_imp_unit
tff(fact_1655_uminus__dvd__conv_I1_J,axiom,
    ! [D3: int,T3: int] :
      ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),D3),T3)
    <=> aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(int,int,uminus_uminus(int),D3)),T3) ) ).

% uminus_dvd_conv(1)
tff(fact_1656_uminus__dvd__conv_I2_J,axiom,
    ! [D3: int,T3: int] :
      ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),D3),T3)
    <=> aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),D3),aa(int,int,uminus_uminus(int),T3)) ) ).

% uminus_dvd_conv(2)
tff(fact_1657_dvd__power__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [X: A,M: nat,N: nat] :
          ( ( X != zero_zero(A) )
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N))
          <=> ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),X),one_one(A))
              | aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),N) ) ) ) ) ).

% dvd_power_iff
tff(fact_1658_dvd__power,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [N: nat,X: A] :
          ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
            | ( X = one_one(A) ) )
         => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),X),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N)) ) ) ).

% dvd_power
tff(fact_1659_subset__divisors__dvd,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A,B2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_cb(A,fun(A,$o),A3))),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_cb(A,fun(A,$o),B2)))
        <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A3),B2) ) ) ).

% subset_divisors_dvd
tff(fact_1660_strict__subset__divisors__dvd,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A,B2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_cb(A,fun(A,$o),A3))),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_cb(A,fun(A,$o),B2)))
        <=> ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A3),B2)
            & ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),A3) ) ) ) ).

% strict_subset_divisors_dvd
tff(fact_1661_and__eq__minus__1__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A3: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),B2) = aa(A,A,uminus_uminus(A),one_one(A)) )
        <=> ( ( A3 = aa(A,A,uminus_uminus(A),one_one(A)) )
            & ( B2 = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ) ).

% and_eq_minus_1_iff
tff(fact_1662_uminus__power__if,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) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N),aa(A,A,uminus_uminus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N))) ) ).

% uminus_power_if
tff(fact_1663_take__bit__eq__mask__iff__exp__dvd,axiom,
    ! [N: nat,K: int] :
      ( ( bit_se2584673776208193580ke_bit(int,N,K) = bit_se2239418461657761734s_mask(int,N) )
    <=> aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N)),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),one_one(int))) ) ).

% take_bit_eq_mask_iff_exp_dvd
tff(fact_1664_one__le__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),A3)
         => aa(A,$o,aa(A,fun(A,$o),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_1665_not__is__unit__0,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),zero_zero(A)),one_one(A)) ) ).

% not_is_unit_0
tff(fact_1666_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_1667_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_1668_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_1669_is__unit__mult__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),one_one(A))
        <=> ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A3),one_one(A))
            & aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),one_one(A)) ) ) ) ).

% is_unit_mult_iff
tff(fact_1670_dvd__mult__unit__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),one_one(A))
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
          <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A3),C2) ) ) ) ).

% dvd_mult_unit_iff
tff(fact_1671_mult__unit__dvd__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),one_one(A))
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),C2)
          <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A3),C2) ) ) ) ).

% mult_unit_dvd_iff
tff(fact_1672_dvd__mult__unit__iff_H,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),one_one(A))
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))
          <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A3),C2) ) ) ) ).

% dvd_mult_unit_iff'
tff(fact_1673_mult__unit__dvd__iff_H,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A3),one_one(A))
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),C2)
          <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),C2) ) ) ) ).

% mult_unit_dvd_iff'
tff(fact_1674_unit__mult__left__cancel,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),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),C2) )
          <=> ( B2 = C2 ) ) ) ) ).

% unit_mult_left_cancel
tff(fact_1675_unit__mult__right__cancel,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),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),C2),A3) )
          <=> ( B2 = C2 ) ) ) ) ).

% unit_mult_right_cancel
tff(fact_1676_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_1677_power__one__over,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,N: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A3)),N) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)) ) ).

% power_one_over
tff(fact_1678_dvd__div__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C2),B2)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),A3) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A3)),C2) ) ) ) ).

% dvd_div_mult
tff(fact_1679_div__mult__swap,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C2),B2)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),C2) ) ) ) ).

% div_mult_swap
tff(fact_1680_div__div__eq__right,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),A3)
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),C2) ) ) ) ) ).

% div_div_eq_right
tff(fact_1681_dvd__div__mult2__eq,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,C2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)),A3)
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),C2) ) ) ) ).

% dvd_div_mult2_eq
tff(fact_1682_dvd__mult__imp__div,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),B2)
         => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) ) ) ).

% dvd_mult_imp_div
tff(fact_1683_div__mult__div__if__dvd,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A3: A,D3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),D3),C2)
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),D3)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D3)) ) ) ) ) ).

% div_mult_div_if_dvd
tff(fact_1684_power__add,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A3: A,M: 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),M),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)) ) ).

% power_add
tff(fact_1685_unit__div__cancel,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A3),one_one(A))
         => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3) = aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),A3) )
          <=> ( B2 = C2 ) ) ) ) ).

% unit_div_cancel
tff(fact_1686_div__unit__dvd__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),one_one(A))
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),C2)
          <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A3),C2) ) ) ) ).

% div_unit_dvd_iff
tff(fact_1687_dvd__div__unit__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),one_one(A))
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),B2))
          <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A3),C2) ) ) ) ).

% dvd_div_unit_iff
tff(fact_1688_dvd__div__neg,axiom,
    ! [A: $tType] :
      ( idom_divide(A)
     => ! [B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),A3)
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,uminus_uminus(A),B2)) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)) ) ) ) ).

% dvd_div_neg
tff(fact_1689_dvd__neg__div,axiom,
    ! [A: $tType] :
      ( idom_divide(A)
     => ! [B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),A3)
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,uminus_uminus(A),A3)),B2) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)) ) ) ) ).

% dvd_neg_div
tff(fact_1690_minus__one__power__iff,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))),N) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N),one_one(A),aa(A,A,uminus_uminus(A),one_one(A))) ) ).

% minus_one_power_iff
tff(fact_1691_mask__int__def,axiom,
    ! [N: nat] : bit_se2239418461657761734s_mask(int,N) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N)),one_one(int)) ).

% mask_int_def
tff(fact_1692_mask__eq__exp__minus__1,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat] : bit_se2239418461657761734s_mask(A,N) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)),one_one(A)) ) ).

% mask_eq_exp_minus_1
tff(fact_1693_power__le__one,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),one_one(A))
           => aa(A,$o,aa(A,fun(A,$o),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_1694_power__gt1__lemma,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N))) ) ) ).

% power_gt1_lemma
tff(fact_1695_power__less__power__Suc,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A3)
         => aa(A,$o,aa(A,fun(A,$o),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_1696_unity__coeff__ex,axiom,
    ! [A: $tType] :
      ( ( dvd(A)
        & semiring_0(A) )
     => ! [P: fun(A,$o),L: A] :
          ( ? [X4: A] : aa(A,$o,P,aa(A,A,aa(A,fun(A,A),times_times(A),L),X4))
        <=> ? [X4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),L),aa(A,A,aa(A,fun(A,A),plus_plus(A),X4),zero_zero(A)))
              & aa(A,$o,P,X4) ) ) ) ).

% unity_coeff_ex
tff(fact_1697_power__0__left,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [N: nat] :
          aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),N) = $ite(N = zero_zero(nat),one_one(A),zero_zero(A)) ) ).

% power_0_left
tff(fact_1698_unit__dvdE,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A3),one_one(A))
         => ~ ( ( A3 != zero_zero(A) )
             => ! [C4: A] : B2 != aa(A,A,aa(A,fun(A,A),times_times(A),A3),C4) ) ) ) ).

% unit_dvdE
tff(fact_1699_power__gt1,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A3)
         => aa(A,$o,aa(A,fun(A,$o),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_1700_and_Ocomm__monoid__axioms,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => comm_monoid(A,bit_se5824344872417868541ns_and(A),aa(A,A,uminus_uminus(A),one_one(A))) ) ).

% and.comm_monoid_axioms
tff(fact_1701_power__less__imp__less__exp,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,M: nat,N: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N))
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),N) ) ) ) ).

% power_less_imp_less_exp
tff(fact_1702_power__strict__increasing,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat,N4: nat,A3: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),N4)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A3)
           => aa(A,$o,aa(A,fun(A,$o),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),A3),N4)) ) ) ) ).

% power_strict_increasing
tff(fact_1703_power__increasing,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat,N4: nat,A3: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),N4)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),A3)
           => aa(A,$o,aa(A,fun(A,$o),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),N4)) ) ) ) ).

% power_increasing
tff(fact_1704_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_1705_inf__period_I4_J,axiom,
    ! [A: $tType] :
      ( ( comm_ring(A)
        & dvd(A) )
     => ! [D3: A,D4: A,T3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),D3),D4)
         => ! [X3: A,K3: A] :
              ( ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),D3),aa(A,A,aa(A,fun(A,A),plus_plus(A),X3),T3))
            <=> ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),D3),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X3),aa(A,A,aa(A,fun(A,A),times_times(A),K3),D4))),T3)) ) ) ) ).

% inf_period(4)
tff(fact_1706_inf__period_I3_J,axiom,
    ! [A: $tType] :
      ( ( comm_ring(A)
        & dvd(A) )
     => ! [D3: A,D4: A,T3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),D3),D4)
         => ! [X3: A,K3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),D3),aa(A,A,aa(A,fun(A,A),plus_plus(A),X3),T3))
            <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),D3),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X3),aa(A,A,aa(A,fun(A,A),times_times(A),K3),D4))),T3)) ) ) ) ).

% inf_period(3)
tff(fact_1707_dvd__div__div__eq__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,C2: A,B2: A,D3: A] :
          ( ( A3 != zero_zero(A) )
         => ( ( C2 != zero_zero(A) )
           => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A3),B2)
             => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C2),D3)
               => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3) = aa(A,A,aa(A,fun(A,A),divide_divide(A),D3),C2) )
                <=> ( aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),D3) ) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
tff(fact_1708_dvd__div__iff__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,B2: A,A3: A] :
          ( ( C2 != zero_zero(A) )
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C2),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))
            <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),B2) ) ) ) ) ).

% dvd_div_iff_mult
tff(fact_1709_div__dvd__iff__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A3: A,C2: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),A3)
           => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),C2)
            <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) ) ) ) ) ).

% div_dvd_iff_mult
tff(fact_1710_dvd__div__eq__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A,C2: A] :
          ( ( A3 != zero_zero(A) )
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A3),B2)
           => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3) = C2 )
            <=> ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3) ) ) ) ) ) ).

% dvd_div_eq_mult
tff(fact_1711_unit__div__eq__0__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),one_one(A))
         => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) = zero_zero(A) )
          <=> ( A3 = zero_zero(A) ) ) ) ) ).

% unit_div_eq_0_iff
tff(fact_1712_and_Omonoid__axioms,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => monoid(A,bit_se5824344872417868541ns_and(A),aa(A,A,uminus_uminus(A),one_one(A))) ) ).

% and.monoid_axioms
tff(fact_1713_unit__eq__div1,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),one_one(A))
         => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) = C2 )
          <=> ( A3 = aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2) ) ) ) ) ).

% unit_eq_div1
tff(fact_1714_unit__eq__div2,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),one_one(A))
         => ( ( A3 = aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),B2) )
          <=> ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2) = C2 ) ) ) ) ).

% unit_eq_div2
tff(fact_1715_div__mult__unit2,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C2),one_one(A))
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),A3)
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),C2) ) ) ) ) ).

% div_mult_unit2
tff(fact_1716_unit__div__commute,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),one_one(A))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),B2) ) ) ) ).

% unit_div_commute
tff(fact_1717_unit__div__mult__swap,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C2),one_one(A))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),C2) ) ) ) ).

% unit_div_mult_swap
tff(fact_1718_is__unit__div__mult2__eq,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,C2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),one_one(A))
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C2),one_one(A))
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),C2) ) ) ) ) ).

% is_unit_div_mult2_eq
tff(fact_1719_unit__imp__mod__eq__0,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),one_one(A))
         => ( modulo_modulo(A,A3,B2) = zero_zero(A) ) ) ) ).

% unit_imp_mod_eq_0
tff(fact_1720_power__minus__Bit0,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [X: A,K: num] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,K))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,K))) ) ).

% power_minus_Bit0
tff(fact_1721_and_Osemilattice__neutr__axioms,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => semilattice_neutr(A,bit_se5824344872417868541ns_and(A),aa(A,A,uminus_uminus(A),one_one(A))) ) ).

% and.semilattice_neutr_axioms
tff(fact_1722_power__minus__Bit1,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [X: A,K: num] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,K))) = aa(A,A,uminus_uminus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,K)))) ) ).

% power_minus_Bit1
tff(fact_1723_even__mask__div__iff_H,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [M: nat,N: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),M)),one_one(A))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),N) ) ) ).

% even_mask_div_iff'
tff(fact_1724_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),aa(num,num,bit0,W2))) = $let(
            w: A,
            w:= aa(nat,A,aa(A,fun(nat,A),power_power(A),Z2),aa(num,nat,numeral_numeral(nat),W2)),
            aa(A,A,aa(A,fun(A,A),times_times(A),w),w) ) ) ).

% power_numeral_even
tff(fact_1725_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))) = $let(
            w: A,
            w:= aa(nat,A,aa(A,fun(nat,A),power_power(A),Z2),aa(num,nat,numeral_numeral(nat),W2)),
            aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z2),w)),w) ) ) ).

% power_numeral_odd
tff(fact_1726_even__mask__div__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M: nat,N: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),M)),one_one(A))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)))
        <=> ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N) = zero_zero(A) )
            | aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),N) ) ) ) ).

% even_mask_div_iff
tff(fact_1727_stable__imp__take__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A,N: nat] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = A3 )
         => ( bit_se2584673776208193580ke_bit(A,N,A3) = $ite(aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A3),zero_zero(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)),one_one(A))) ) ) ) ).

% stable_imp_take_bit_eq
tff(fact_1728_power__Suc__less,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),one_one(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),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_1729_power__Suc__le__self,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),one_one(A))
           => aa(A,$o,aa(A,fun(A,$o),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_1730_power__Suc__less__one,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),one_one(A))
           => aa(A,$o,aa(A,fun(A,$o),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_1731_power__strict__decreasing,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat,N4: nat,A3: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),N4)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),one_one(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N4)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)) ) ) ) ) ).

% power_strict_decreasing
tff(fact_1732_power__decreasing,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat,N4: nat,A3: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),N4)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),one_one(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N4)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)) ) ) ) ) ).

% power_decreasing
tff(fact_1733_power__le__imp__le__exp,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,M: nat,N: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N))
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),N) ) ) ) ).

% power_le_imp_le_exp
tff(fact_1734_self__le__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),A3)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)) ) ) ) ).

% self_le_power
tff(fact_1735_one__less__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A3)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)) ) ) ) ).

% one_less_power
tff(fact_1736_is__unit__div__mult__cancel__right,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A] :
          ( ( A3 != zero_zero(A) )
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),one_one(A))
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A3)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),B2) ) ) ) ) ).

% is_unit_div_mult_cancel_right
tff(fact_1737_is__unit__div__mult__cancel__left,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A] :
          ( ( A3 != zero_zero(A) )
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),one_one(A))
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),B2) ) ) ) ) ).

% is_unit_div_mult_cancel_left
tff(fact_1738_is__unitE,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A3),one_one(A))
         => ~ ( ( A3 != zero_zero(A) )
             => ! [B5: A] :
                  ( ( B5 != zero_zero(A) )
                 => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B5),one_one(A))
                   => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A3) = B5 )
                     => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),B5) = A3 )
                       => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),B5) = one_one(A) )
                         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),A3) != aa(A,A,aa(A,fun(A,A),times_times(A),C2),B5) ) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_1739_evenE,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A3)
         => ~ ! [B5: A] : A3 != aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),B5) ) ) ).

% evenE
tff(fact_1740_odd__one,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),one_one(A)) ) ).

% odd_one
tff(fact_1741_even__minus,axiom,
    ! [A: $tType] :
      ( ring_parity(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,uminus_uminus(A),A3))
        <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A3) ) ) ).

% even_minus
tff(fact_1742_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),aa(num,num,bit0,aa(num,num,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_1743_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),aa(num,num,bit0,one2))) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),A3) ) ).

% power2_eq_square
tff(fact_1744_one__power2,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),one_one(A)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = one_one(A) ) ) ).

% one_power2
tff(fact_1745_power2__eq__iff,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [X: A,Y: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) )
        <=> ( ( X = Y )
            | ( X = aa(A,A,uminus_uminus(A),Y) ) ) ) ) ).

% power2_eq_iff
tff(fact_1746_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_1747_dvd__mult__cancel2,axiom,
    ! [M: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),M)),M)
      <=> ( N = one_one(nat) ) ) ) ).

% dvd_mult_cancel2
tff(fact_1748_dvd__mult__cancel1,axiom,
    ! [M: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N)),M)
      <=> ( N = one_one(nat) ) ) ) ).

% dvd_mult_cancel1
tff(fact_1749_even__mult__exp__div__exp__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,M: nat,N: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),M))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)))
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),M)
            | ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N) = zero_zero(A) )
            | ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),N)
              & aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M)))) ) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_1750_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_1751_and__one__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),one_one(A)) = modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ).

% and_one_eq
tff(fact_1752_one__and__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),one_one(A)),A3) = modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ).

% one_and_eq
tff(fact_1753_take__bit__eq__mask__iff,axiom,
    ! [N: nat,K: int] :
      ( ( bit_se2584673776208193580ke_bit(int,N,K) = bit_se2239418461657761734s_mask(int,N) )
    <=> ( bit_se2584673776208193580ke_bit(int,N,aa(int,int,aa(int,fun(int,int),plus_plus(int),K),one_one(int))) = zero_zero(int) ) ) ).

% take_bit_eq_mask_iff
tff(fact_1754_even__two__times__div__two,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A3)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))) = A3 ) ) ) ).

% even_two_times_div_two
tff(fact_1755_power2__eq__1__iff,axiom,
    ! [A: $tType] :
      ( ring_15535105094025558882visors(A)
     => ! [A3: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = one_one(A) )
        <=> ( ( A3 = one_one(A) )
            | ( A3 = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ) ).

% power2_eq_1_iff
tff(fact_1756_power__diff__power__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A3: A,M: nat,N: nat] :
          ( ( A3 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),M),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N)),aa(A,A,aa(A,fun(A,A),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),M)))) ) ) ) ).

% power_diff_power_eq
tff(fact_1757_odd__iff__mod__2__eq__one,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A3: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A3)
        <=> ( modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = one_one(A) ) ) ) ).

% odd_iff_mod_2_eq_one
tff(fact_1758_normalize__denom__pos,axiom,
    ! [R2: product_prod(int,int),P3: int,Q3: int] :
      ( ( normalize(R2) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P3),Q3) )
     => aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Q3) ) ).

% normalize_denom_pos
tff(fact_1759_power__eq__if,axiom,
    ! [A: $tType] :
      ( power(A)
     => ! [P3: A,M: nat] :
          aa(nat,A,aa(A,fun(nat,A),power_power(A),P3),M) = $ite(M = zero_zero(nat),one_one(A),aa(A,A,aa(A,fun(A,A),times_times(A),P3),aa(nat,A,aa(A,fun(nat,A),power_power(A),P3),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),one_one(nat))))) ) ).

% power_eq_if
tff(fact_1760_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [K: nat,N: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),N)
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K)) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_1761_normalize__crossproduct,axiom,
    ! [Q3: int,S2: int,P3: int,R2: int] :
      ( ( Q3 != 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),P3),Q3)) = normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),R2),S2)) )
         => ( aa(int,int,aa(int,fun(int,int),times_times(int),P3),S2) = aa(int,int,aa(int,fun(int,int),times_times(int),R2),Q3) ) ) ) ) ).

% normalize_crossproduct
tff(fact_1762_power__minus__mult,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [N: nat,A3: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),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_1763_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),aa(num,num,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),aa(num,num,bit0,one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),X)),Y)) ) ).

% power2_sum
tff(fact_1764_oddE,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A3: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A3)
         => ~ ! [B5: 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),aa(num,num,bit0,one2))),B5)),one_one(A)) ) ) ).

% oddE
tff(fact_1765_square__le__1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),one_one(A))),X)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),one_one(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),one_one(A)) ) ) ) ).

% square_le_1
tff(fact_1766_parity__cases,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A3: A] :
          ( ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A3)
           => ( modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) != zero_zero(A) ) )
         => ~ ( ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A3)
             => ( modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) != one_one(A) ) ) ) ) ).

% parity_cases
tff(fact_1767_mod2__eq__if,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A3: A] :
          modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = $ite(aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A3),zero_zero(A),one_one(A)) ) ).

% mod2_eq_if
tff(fact_1768_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),aa(num,num,bit0,one2))),N)) ) ).

% minus_power_mult_self
tff(fact_1769_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),aa(num,num,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),aa(num,num,bit0,one2)))) ) ).

% power_odd_eq
tff(fact_1770_minus__1__div__exp__eq__int,axiom,
    ! [N: nat] : aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,uminus_uminus(int),one_one(int))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N)) = aa(int,int,uminus_uminus(int),one_one(int)) ).

% minus_1_div_exp_eq_int
tff(fact_1771_signed__take__bit__int__greater__eq__minus__exp,axiom,
    ! [N: nat,K: int] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N))),bit_ri4674362597316999326ke_bit(int,N,K)) ).

% signed_take_bit_int_greater_eq_minus_exp
tff(fact_1772_signed__take__bit__int__less__eq__self__iff,axiom,
    ! [N: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),bit_ri4674362597316999326ke_bit(int,N,K)),K)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N))),K) ) ).

% signed_take_bit_int_less_eq_self_iff
tff(fact_1773_signed__take__bit__int__greater__self__iff,axiom,
    ! [K: int,N: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),bit_ri4674362597316999326ke_bit(int,N,K))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N))) ) ).

% signed_take_bit_int_greater_self_iff
tff(fact_1774_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),aa(num,num,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),aa(num,num,bit0,one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),X)),Y)) ) ).

% power2_diff
tff(fact_1775_mask__half__int,axiom,
    ! [N: nat] : aa(int,int,aa(int,fun(int,int),divide_divide(int),bit_se2239418461657761734s_mask(int,N)),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))) = bit_se2239418461657761734s_mask(int,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))) ).

% mask_half_int
tff(fact_1776_mult__exp__mod__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M: nat,N: nat,A3: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),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),aa(num,num,bit0,one2))),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),modulo_modulo(A,A3,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),M)) ) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_1777_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),aa(num,num,bit0,one2))),N))) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).

% power_minus1_odd
tff(fact_1778_signed__take__bit__int__eq__self__iff,axiom,
    ! [N: nat,K: int] :
      ( ( bit_ri4674362597316999326ke_bit(int,N,K) = K )
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N))),K)
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N)) ) ) ).

% signed_take_bit_int_eq_self_iff
tff(fact_1779_signed__take__bit__int__eq__self,axiom,
    ! [N: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N))),K)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N))
       => ( bit_ri4674362597316999326ke_bit(int,N,K) = K ) ) ) ).

% signed_take_bit_int_eq_self
tff(fact_1780_take__bit__incr__eq,axiom,
    ! [N: nat,K: int] :
      ( ( bit_se2584673776208193580ke_bit(int,N,K) != aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N)),one_one(int)) )
     => ( bit_se2584673776208193580ke_bit(int,N,aa(int,int,aa(int,fun(int,int),plus_plus(int),K),one_one(int))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),bit_se2584673776208193580ke_bit(int,N,K)) ) ) ).

% take_bit_incr_eq
tff(fact_1781_take__bit__Suc__minus__1__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat] : 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),aa(num,num,bit0,one2))),aa(nat,nat,suc,N))),one_one(A)) ) ).

% take_bit_Suc_minus_1_eq
tff(fact_1782_take__bit__numeral__minus__1__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [K: num] : bit_se2584673776208193580ke_bit(A,aa(num,nat,numeral_numeral(nat),K),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),aa(num,num,bit0,one2))),aa(num,nat,numeral_numeral(nat),K))),one_one(A)) ) ).

% take_bit_numeral_minus_1_eq
tff(fact_1783_gcd__nat_Oordering__top__axioms,axiom,
    ordering_top(nat,dvd_dvd(nat),aTP_Lamp_cc(nat,fun(nat,$o)),zero_zero(nat)) ).

% gcd_nat.ordering_top_axioms
tff(fact_1784_fold__atLeastAtMost__nat_Opsimps,axiom,
    ! [A: $tType,F: 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))),F),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))))
     => ( set_fo6178422350223883121st_nat(A,F,A3,B2,Acc2) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),B2),A3),Acc2,set_fo6178422350223883121st_nat(A,F,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A3),one_one(nat)),B2,aa(A,A,aa(nat,fun(A,A),F,A3),Acc2))) ) ) ).

% fold_atLeastAtMost_nat.psimps
tff(fact_1785_fold__atLeastAtMost__nat_Opelims,axiom,
    ! [A: $tType,X: fun(nat,fun(A,A)),Xa: nat,Xb: nat,Xc: A,Y: A] :
      ( ( set_fo6178422350223883121st_nat(A,X,Xa,Xb,Xc) = Y )
     => ( accp(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),set_fo1817059534552279752at_rel(A),aa(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),aa(fun(nat,fun(A,A)),fun(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A)))),product_Pair(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),X),aa(product_prod(nat,A),product_prod(nat,product_prod(nat,A)),aa(nat,fun(product_prod(nat,A),product_prod(nat,product_prod(nat,A))),product_Pair(nat,product_prod(nat,A)),Xa),aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),Xb),Xc))))
       => ~ ( ( Y = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),Xa),Xc,set_fo6178422350223883121st_nat(A,X,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Xa),one_one(nat)),Xb,aa(A,A,aa(nat,fun(A,A),X,Xa),Xc))) )
           => ~ accp(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),set_fo1817059534552279752at_rel(A),aa(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),aa(fun(nat,fun(A,A)),fun(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A)))),product_Pair(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),X),aa(product_prod(nat,A),product_prod(nat,product_prod(nat,A)),aa(nat,fun(product_prod(nat,A),product_prod(nat,product_prod(nat,A))),product_Pair(nat,product_prod(nat,A)),Xa),aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),Xb),Xc)))) ) ) ) ).

% fold_atLeastAtMost_nat.pelims
tff(fact_1786_flip__bit__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : bit_se8732182000553998342ip_bit(A,zero_zero(nat),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa($o,A,zero_neq_one_of_bool(A),aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A3))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))) ) ).

% flip_bit_0
tff(fact_1787_one__mod__2__pow__eq,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [N: nat] : modulo_modulo(A,one_one(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)) = aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)) ) ).

% one_mod_2_pow_eq
tff(fact_1788_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: num,N: nat,A3: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),X))),N)),aa(int,A,ring_1_of_int(A),A3))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),X))),N)),A3) ) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_1789_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: int,X: num,N: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(int,A,ring_1_of_int(A),A3)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),X))),N))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(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_less_neg_numeral_power_cancel_iff
tff(fact_1790_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: num,N: nat,A3: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),X))),N)),aa(int,A,ring_1_of_int(A),A3))
        <=> aa(int,$o,aa(int,fun(int,$o),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_1791_of__bool__eq__1__iff,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ! [P: $o] :
          ( ( aa($o,A,zero_neq_one_of_bool(A),(P)) = one_one(A) )
        <=> (P) ) ) ).

% of_bool_eq_1_iff
tff(fact_1792_of__bool__eq_I2_J,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ( aa($o,A,zero_neq_one_of_bool(A),$true) = one_one(A) ) ) ).

% of_bool_eq(2)
tff(fact_1793_of__bool__less__one__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [P: $o] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa($o,A,zero_neq_one_of_bool(A),(P))),one_one(A))
        <=> ~ (P) ) ) ).

% of_bool_less_one_iff
tff(fact_1794_of__bool__not__iff,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [P: $o] : aa($o,A,zero_neq_one_of_bool(A),~ (P)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa($o,A,zero_neq_one_of_bool(A),(P))) ) ).

% of_bool_not_iff
tff(fact_1795_of__int__1,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ( aa(int,A,ring_1_of_int(A),one_one(int)) = one_one(A) ) ) ).

% of_int_1
tff(fact_1796_of__int__eq__1__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Z2: int] :
          ( ( aa(int,A,ring_1_of_int(A),Z2) = one_one(A) )
        <=> ( Z2 = one_one(int) ) ) ) ).

% of_int_eq_1_iff
tff(fact_1797_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_1798_of__int__minus,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Z2: int] : aa(int,A,ring_1_of_int(A),aa(int,int,uminus_uminus(int),Z2)) = aa(A,A,uminus_uminus(A),aa(int,A,ring_1_of_int(A),Z2)) ) ).

% of_int_minus
tff(fact_1799_Divides_Oadjust__div__eq,axiom,
    ! [Q3: int,R2: int] : adjust_div(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),R2)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Q3),aa($o,int,zero_neq_one_of_bool(int),R2 != zero_zero(int))) ).

% Divides.adjust_div_eq
tff(fact_1800_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_1801_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_1802_of__int__le__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z2: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z2)),one_one(A))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Z2),one_one(int)) ) ) ).

% of_int_le_1_iff
tff(fact_1803_of__int__1__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z2: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(int,A,ring_1_of_int(A),Z2))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),one_one(int)),Z2) ) ) ).

% of_int_1_le_iff
tff(fact_1804_of__int__less__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z2: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(int,A,ring_1_of_int(A),Z2)),one_one(A))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z2),one_one(int)) ) ) ).

% of_int_less_1_iff
tff(fact_1805_of__int__1__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z2: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(int,A,ring_1_of_int(A),Z2))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),one_one(int)),Z2) ) ) ).

% of_int_1_less_iff
tff(fact_1806_take__bit__of__1,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat] : bit_se2584673776208193580ke_bit(A,N,one_one(A)) = aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)) ) ).

% take_bit_of_1
tff(fact_1807_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [X: num,N: nat,Y: int] :
          ( ( 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),Y) )
        <=> ( 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) = Y ) ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_1808_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Y: int,X: num,N: nat] :
          ( ( aa(int,A,ring_1_of_int(A),Y) = 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) )
        <=> ( Y = 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_eq_neg_numeral_power_cancel_iff
tff(fact_1809_one__div__2__pow__eq,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [N: nat] : aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)) = aa($o,A,zero_neq_one_of_bool(A),N = zero_zero(nat)) ) ).

% one_div_2_pow_eq
tff(fact_1810_bits__1__div__exp,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [N: nat] : aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)) = aa($o,A,zero_neq_one_of_bool(A),N = zero_zero(nat)) ) ).

% bits_1_div_exp
tff(fact_1811_take__bit__of__exp,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M: nat,N: nat] : bit_se2584673776208193580ke_bit(A,M,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),M))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)) ) ).

% take_bit_of_exp
tff(fact_1812_take__bit__of__2,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: nat] : bit_se2584673776208193580ke_bit(A,N,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ).

% take_bit_of_2
tff(fact_1813_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: int,X: num,N: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),A3)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),X))),N))
        <=> aa(int,$o,aa(int,fun(int,$o),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_1814_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_1815_of__bool__conj,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [P: $o,Q: $o] :
          aa($o,A,zero_neq_one_of_bool(A),
            ( (P)
            & (Q) )) = aa(A,A,aa(A,fun(A,A),times_times(A),aa($o,A,zero_neq_one_of_bool(A),(P))),aa($o,A,zero_neq_one_of_bool(A),(Q))) ) ).

% of_bool_conj
tff(fact_1816_of__bool__less__eq__one,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [P: $o] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa($o,A,zero_neq_one_of_bool(A),(P))),one_one(A)) ) ).

% of_bool_less_eq_one
tff(fact_1817_split__of__bool__asm,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ! [P: fun(A,$o),P3: $o] :
          ( aa(A,$o,P,aa($o,A,zero_neq_one_of_bool(A),(P3)))
        <=> ~ ( ( (P3)
                & ~ aa(A,$o,P,one_one(A)) )
              | ( ~ (P3)
                & ~ aa(A,$o,P,zero_zero(A)) ) ) ) ) ).

% split_of_bool_asm
tff(fact_1818_split__of__bool,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ! [P: fun(A,$o),P3: $o] :
          ( aa(A,$o,P,aa($o,A,zero_neq_one_of_bool(A),(P3)))
        <=> ( ( (P3)
             => aa(A,$o,P,one_one(A)) )
            & ( ~ (P3)
             => aa(A,$o,P,zero_zero(A)) ) ) ) ) ).

% split_of_bool
tff(fact_1819_of__bool__def,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ! [P3: $o] :
          aa($o,A,zero_neq_one_of_bool(A),(P3)) = $ite((P3),one_one(A),zero_zero(A)) ) ).

% of_bool_def
tff(fact_1820_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_cd(int,fun(int,int))),Qr) ).

% Divides.adjust_div_def
tff(fact_1821_of__int__neg__numeral,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [K: num] : aa(int,A,ring_1_of_int(A),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),K)) ) ).

% of_int_neg_numeral
tff(fact_1822_power__dvd__imp__le,axiom,
    ! [I: nat,M: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),I),M)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),I),N))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),one_one(nat)),I)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),N) ) ) ).

% power_dvd_imp_le
tff(fact_1823_fold__atLeastAtMost__nat_Osimps,axiom,
    ! [A: $tType,F: fun(nat,fun(A,A)),A3: nat,B2: nat,Acc2: A] :
      set_fo6178422350223883121st_nat(A,F,A3,B2,Acc2) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),B2),A3),Acc2,set_fo6178422350223883121st_nat(A,F,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A3),one_one(nat)),B2,aa(A,A,aa(nat,fun(A,A),F,A3),Acc2))) ).

% fold_atLeastAtMost_nat.simps
tff(fact_1824_fold__atLeastAtMost__nat_Oelims,axiom,
    ! [A: $tType,X: fun(nat,fun(A,A)),Xa: nat,Xb: nat,Xc: A,Y: A] :
      ( ( set_fo6178422350223883121st_nat(A,X,Xa,Xb,Xc) = Y )
     => ( Y = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),Xa),Xc,set_fo6178422350223883121st_nat(A,X,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Xa),one_one(nat)),Xb,aa(A,A,aa(nat,fun(A,A),X,Xa),Xc))) ) ) ).

% fold_atLeastAtMost_nat.elims
tff(fact_1825_mask__nat__def,axiom,
    ! [N: nat] : bit_se2239418461657761734s_mask(nat,N) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)),one_one(nat)) ).

% mask_nat_def
tff(fact_1826_bits__induct,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [P: fun(A,$o),A3: A] :
          ( ! [A6: A] :
              ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A6),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = A6 )
             => aa(A,$o,P,A6) )
         => ( ! [A6: A,B5: $o] :
                ( aa(A,$o,P,A6)
               => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa($o,A,zero_neq_one_of_bool(A),(B5))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A6))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = A6 )
                 => aa(A,$o,P,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa($o,A,zero_neq_one_of_bool(A),(B5))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A6))) ) )
           => aa(A,$o,P,A3) ) ) ) ).

% bits_induct
tff(fact_1827_exp__mod__exp,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [M: nat,N: nat] : modulo_modulo(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),M),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),N))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),M)) ) ).

% exp_mod_exp
tff(fact_1828_dvd__productE,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [P3: A,A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),P3),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2))
         => ~ ! [X2: A,Y2: A] :
                ( ( P3 = aa(A,A,aa(A,fun(A,A),times_times(A),X2),Y2) )
               => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),X2),A3)
                 => ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),Y2),B2) ) ) ) ) ).

% dvd_productE
tff(fact_1829_division__decomp,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))
         => ? [B8: A,C6: A] :
              ( ( A3 = aa(A,A,aa(A,fun(A,A),times_times(A),B8),C6) )
              & aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B8),B2)
              & aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C6),C2) ) ) ) ).

% division_decomp
tff(fact_1830_ex__power__ivl1,axiom,
    ! [B2: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),B2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),K)
       => ? [N3: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),N3)),K)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N3),one_one(nat)))) ) ) ) ).

% ex_power_ivl1
tff(fact_1831_ex__power__ivl2,axiom,
    ! [B2: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),B2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),K)
       => ? [N3: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),N3)),K)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N3),one_one(nat)))) ) ) ) ).

% ex_power_ivl2
tff(fact_1832_xor__one__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A3),one_one(A)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa($o,A,zero_neq_one_of_bool(A),aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A3)))),aa($o,A,zero_neq_one_of_bool(A),~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A3))) ) ).

% xor_one_eq
tff(fact_1833_one__xor__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),one_one(A)),A3) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa($o,A,zero_neq_one_of_bool(A),aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A3)))),aa($o,A,zero_neq_one_of_bool(A),~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A3))) ) ).

% one_xor_eq
tff(fact_1834_exp__div__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M: nat,N: nat] :
          aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)) = aa(A,A,
            aa(A,fun(A,A),times_times(A),
              aa($o,A,zero_neq_one_of_bool(A),
                ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),M) != zero_zero(A) )
                & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),M) ))),
            aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N))) ) ).

% exp_div_exp_eq
tff(fact_1835_floor__exists1,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A] :
        ? [X2: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),X2)),X)
          & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),plus_plus(int),X2),one_one(int))))
          & ! [Y5: int] :
              ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Y5)),X)
                & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),plus_plus(int),Y5),one_one(int)))) )
             => ( Y5 = X2 ) ) ) ) ).

% floor_exists1
tff(fact_1836_floor__exists,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A] :
        ? [Z3: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z3)),X)
          & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),plus_plus(int),Z3),one_one(int)))) ) ) ).

% floor_exists
tff(fact_1837_and__int_Osimps,axiom,
    ! [K: int,L: int] :
      aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L) = $ite(
        ( aa(set(int),$o,member(int,K),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))
        & aa(set(int),$o,member(int,L),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))) ),
        aa(int,int,uminus_uminus(int),
          aa($o,int,zero_neq_one_of_bool(int),
            ( ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),K)
            & ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),L) ))),
        aa(int,int,
          aa(int,fun(int,int),plus_plus(int),
            aa($o,int,zero_neq_one_of_bool(int),
              ( ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),K)
              & ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),L) ))),
          aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))))) ) ).

% and_int.simps
tff(fact_1838_and__int_Oelims,axiom,
    ! [X: int,Xa: int,Y: int] :
      ( ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),X),Xa) = Y )
     => ( Y = $ite(
            ( aa(set(int),$o,member(int,X),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))
            & aa(set(int),$o,member(int,Xa),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))) ),
            aa(int,int,uminus_uminus(int),
              aa($o,int,zero_neq_one_of_bool(int),
                ( ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),X)
                & ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Xa) ))),
            aa(int,int,
              aa(int,fun(int,int),plus_plus(int),
                aa($o,int,zero_neq_one_of_bool(int),
                  ( ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),X)
                  & ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Xa) ))),
              aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),X),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),Xa),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))))) ) ) ) ).

% and_int.elims
tff(fact_1839_round__unique,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))),aa(int,A,ring_1_of_int(A),Y))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Y)),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))))
           => ( archimedean_round(A,X) = Y ) ) ) ) ).

% round_unique
tff(fact_1840_push__bit__numeral__minus__1,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: num] : bit_se4730199178511100633sh_bit(A,aa(num,nat,numeral_numeral(nat),N),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(num,nat,numeral_numeral(nat),N))) ) ).

% push_bit_numeral_minus_1
tff(fact_1841_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_ce(num,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),L)),Qr) ).

% divmod_step_integer_def
tff(fact_1842_singletonI,axiom,
    ! [A: $tType,A3: A] : aa(set(A),$o,member(A,A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A)))) ).

% singletonI
tff(fact_1843_Int__insert__right__if1,axiom,
    ! [A: $tType,A3: A,A4: set(A),B3: set(A)] :
      ( aa(set(A),$o,member(A,A3),A4)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),B3)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3)) ) ) ).

% Int_insert_right_if1
tff(fact_1844_Int__insert__right__if0,axiom,
    ! [A: $tType,A3: A,A4: set(A),B3: set(A)] :
      ( ~ aa(set(A),$o,member(A,A3),A4)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),B3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3) ) ) ).

% Int_insert_right_if0
tff(fact_1845_insert__inter__insert,axiom,
    ! [A: $tType,A3: A,A4: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),A4)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),B3)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3)) ).

% insert_inter_insert
tff(fact_1846_Int__insert__left__if1,axiom,
    ! [A: $tType,A3: A,C3: set(A),B3: set(A)] :
      ( aa(set(A),$o,member(A,A3),C3)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),B3)),C3) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),C3)) ) ) ).

% Int_insert_left_if1
tff(fact_1847_Int__insert__left__if0,axiom,
    ! [A: $tType,A3: A,C3: set(A),B3: set(A)] :
      ( ~ aa(set(A),$o,member(A,A3),C3)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),B3)),C3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),C3) ) ) ).

% Int_insert_left_if0
tff(fact_1848_Un__insert__right,axiom,
    ! [A: $tType,A4: set(A),A3: A,B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),B3)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3)) ).

% Un_insert_right
tff(fact_1849_Un__insert__left,axiom,
    ! [A: $tType,A3: A,B3: set(A),C3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),B3)),C3) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B3),C3)) ).

% Un_insert_left
tff(fact_1850_singleton__conv2,axiom,
    ! [A: $tType,A3: A] : aa(fun(A,$o),set(A),collect(A),aa(A,fun(A,$o),fequal(A),A3)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A))) ).

% singleton_conv2
tff(fact_1851_singleton__conv,axiom,
    ! [A: $tType,A3: A] : aa(fun(A,$o),set(A),collect(A),aTP_Lamp_cf(A,fun(A,$o),A3)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A))) ).

% singleton_conv
tff(fact_1852_singleton__insert__inj__eq,axiom,
    ! [A: $tType,B2: A,A3: A,A4: set(A)] :
      ( ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),A4) )
    <=> ( ( A3 = B2 )
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A)))) ) ) ).

% singleton_insert_inj_eq
tff(fact_1853_singleton__insert__inj__eq_H,axiom,
    ! [A: $tType,A3: A,A4: set(A),B2: A] :
      ( ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),A4) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A))) )
    <=> ( ( A3 = B2 )
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A)))) ) ) ).

% singleton_insert_inj_eq'
tff(fact_1854_insert__disjoint_I1_J,axiom,
    ! [A: $tType,A3: A,A4: set(A),B3: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),A4)),B3) = bot_bot(set(A)) )
    <=> ( ~ aa(set(A),$o,member(A,A3),B3)
        & ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3) = bot_bot(set(A)) ) ) ) ).

% insert_disjoint(1)
tff(fact_1855_insert__disjoint_I2_J,axiom,
    ! [A: $tType,A3: A,A4: set(A),B3: set(A)] :
      ( ( bot_bot(set(A)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),A4)),B3) )
    <=> ( ~ aa(set(A),$o,member(A,A3),B3)
        & ( bot_bot(set(A)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3) ) ) ) ).

% insert_disjoint(2)
tff(fact_1856_disjoint__insert_I1_J,axiom,
    ! [A: $tType,B3: set(A),A3: A,A4: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),A4)) = bot_bot(set(A)) )
    <=> ( ~ aa(set(A),$o,member(A,A3),B3)
        & ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),A4) = bot_bot(set(A)) ) ) ) ).

% disjoint_insert(1)
tff(fact_1857_disjoint__insert_I2_J,axiom,
    ! [A: $tType,A4: set(A),B2: A,B3: set(A)] :
      ( ( bot_bot(set(A)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),B3)) )
    <=> ( ~ aa(set(A),$o,member(A,B2),A4)
        & ( bot_bot(set(A)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3) ) ) ) ).

% disjoint_insert(2)
tff(fact_1858_insert__Diff__single,axiom,
    ! [A: $tType,A3: A,A4: set(A)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A))))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),A4) ).

% insert_Diff_single
tff(fact_1859_round__1,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( archimedean_round(A,one_one(A)) = one_one(int) ) ) ).

% round_1
tff(fact_1860_subset__Compl__singleton,axiom,
    ! [A: $tType,A4: set(A),B2: A] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A)))))
    <=> ~ aa(set(A),$o,member(A,B2),A4) ) ).

% subset_Compl_singleton
tff(fact_1861_push__bit__Suc__minus__numeral,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat,K: num] : bit_se4730199178511100633sh_bit(A,aa(nat,nat,suc,N),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),K))) = bit_se4730199178511100633sh_bit(A,N,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,K)))) ) ).

% push_bit_Suc_minus_numeral
tff(fact_1862_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_1863_push__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A3: A] : bit_se4730199178511100633sh_bit(A,aa(nat,nat,suc,N),A3) = bit_se4730199178511100633sh_bit(A,N,aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))) ) ).

% push_bit_Suc
tff(fact_1864_push__bit__of__1,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat] : bit_se4730199178511100633sh_bit(A,N,one_one(A)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N) ) ).

% push_bit_of_1
tff(fact_1865_push__bit__minus__numeral,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [L: num,K: num] : bit_se4730199178511100633sh_bit(A,aa(num,nat,numeral_numeral(nat),L),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),K))) = bit_se4730199178511100633sh_bit(A,pred_numeral(L),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,K)))) ) ).

% push_bit_minus_numeral
tff(fact_1866_minus__integer__code_I2_J,axiom,
    ! [L: code_integer] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),zero_zero(code_integer)),L) = aa(code_integer,code_integer,uminus_uminus(code_integer),L) ).

% minus_integer_code(2)
tff(fact_1867_insert__Collect,axiom,
    ! [A: $tType,A3: A,P: fun(A,$o)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),aa(fun(A,$o),set(A),collect(A),P)) = aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_cg(A,fun(fun(A,$o),fun(A,$o)),A3),P)) ).

% insert_Collect
tff(fact_1868_insert__compr,axiom,
    ! [A: $tType,A3: A,B3: set(A)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),B3) = aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_ch(A,fun(set(A),fun(A,$o)),A3),B3)) ).

% insert_compr
tff(fact_1869_divmod__integer_H__def,axiom,
    ! [M: num,N: num] : unique8689654367752047608divmod(code_integer,M,N) = aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),divide_divide(code_integer),aa(num,code_integer,numeral_numeral(code_integer),M)),aa(num,code_integer,numeral_numeral(code_integer),N))),modulo_modulo(code_integer,aa(num,code_integer,numeral_numeral(code_integer),M),aa(num,code_integer,numeral_numeral(code_integer),N))) ).

% divmod_integer'_def
tff(fact_1870_exhaustive__integer_H_Ocases,axiom,
    ! [X: product_prod(fun(code_integer,option(product_prod($o,list(code_term)))),product_prod(code_integer,code_integer))] :
      ~ ! [F2: fun(code_integer,option(product_prod($o,list(code_term)))),D2: code_integer,I2: code_integer] : X != aa(product_prod(code_integer,code_integer),product_prod(fun(code_integer,option(product_prod($o,list(code_term)))),product_prod(code_integer,code_integer)),aa(fun(code_integer,option(product_prod($o,list(code_term)))),fun(product_prod(code_integer,code_integer),product_prod(fun(code_integer,option(product_prod($o,list(code_term)))),product_prod(code_integer,code_integer))),product_Pair(fun(code_integer,option(product_prod($o,list(code_term)))),product_prod(code_integer,code_integer)),F2),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),D2),I2)) ).

% exhaustive_integer'.cases
tff(fact_1871_full__exhaustive__integer_H_Ocases,axiom,
    ! [X: product_prod(fun(product_prod(code_integer,fun(product_unit,code_term)),option(product_prod($o,list(code_term)))),product_prod(code_integer,code_integer))] :
      ~ ! [F2: fun(product_prod(code_integer,fun(product_unit,code_term)),option(product_prod($o,list(code_term)))),D2: code_integer,I2: code_integer] : X != aa(product_prod(code_integer,code_integer),product_prod(fun(product_prod(code_integer,fun(product_unit,code_term)),option(product_prod($o,list(code_term)))),product_prod(code_integer,code_integer)),aa(fun(product_prod(code_integer,fun(product_unit,code_term)),option(product_prod($o,list(code_term)))),fun(product_prod(code_integer,code_integer),product_prod(fun(product_prod(code_integer,fun(product_unit,code_term)),option(product_prod($o,list(code_term)))),product_prod(code_integer,code_integer))),product_Pair(fun(product_prod(code_integer,fun(product_unit,code_term)),option(product_prod($o,list(code_term)))),product_prod(code_integer,code_integer)),F2),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),D2),I2)) ).

% full_exhaustive_integer'.cases
tff(fact_1872_ID_Opred__cong,axiom,
    ! [A: $tType,X: A,Ya: A,P: fun(A,$o),Pa: fun(A,$o)] :
      ( ( X = Ya )
     => ( ! [Z3: A] :
            ( aa(set(A),$o,member(A,Z3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Ya),bot_bot(set(A))))
           => ( aa(A,$o,P,Z3)
            <=> aa(A,$o,Pa,Z3) ) )
       => ( aa(A,$o,aa(fun(A,$o),fun(A,$o),bNF_id_bnf(fun(A,$o)),P),X)
        <=> aa(A,$o,aa(fun(A,$o),fun(A,$o),bNF_id_bnf(fun(A,$o)),Pa),Ya) ) ) ) ).

% ID.pred_cong
tff(fact_1873_ID_Opred__mono__strong,axiom,
    ! [A: $tType,P: fun(A,$o),X: A,Pa: fun(A,$o)] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),bNF_id_bnf(fun(A,$o)),P),X)
     => ( ! [Z3: A] :
            ( aa(set(A),$o,member(A,Z3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))))
           => ( aa(A,$o,P,Z3)
             => aa(A,$o,Pa,Z3) ) )
       => aa(A,$o,aa(fun(A,$o),fun(A,$o),bNF_id_bnf(fun(A,$o)),Pa),X) ) ) ).

% ID.pred_mono_strong
tff(fact_1874_ID_Orel__cong,axiom,
    ! [A: $tType,B: $tType,X: A,Ya: A,Y: B,Xa: B,R: fun(A,fun(B,$o)),Ra: fun(A,fun(B,$o))] :
      ( ( X = Ya )
     => ( ( Y = Xa )
       => ( ! [Z3: A,Yb: B] :
              ( aa(set(A),$o,member(A,Z3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Ya),bot_bot(set(A))))
             => ( aa(set(B),$o,member(B,Yb),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),Xa),bot_bot(set(B))))
               => ( aa(B,$o,aa(A,fun(B,$o),R,Z3),Yb)
                <=> aa(B,$o,aa(A,fun(B,$o),Ra,Z3),Yb) ) ) )
         => ( aa(B,$o,aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),bNF_id_bnf(fun(A,fun(B,$o))),R),X),Y)
          <=> aa(B,$o,aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),bNF_id_bnf(fun(A,fun(B,$o))),Ra),Ya),Xa) ) ) ) ) ).

% ID.rel_cong
tff(fact_1875_ID_Orel__mono__strong,axiom,
    ! [A: $tType,B: $tType,R: fun(A,fun(B,$o)),X: A,Y: B,Ra: fun(A,fun(B,$o))] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),bNF_id_bnf(fun(A,fun(B,$o))),R),X),Y)
     => ( ! [Z3: A,Yb: B] :
            ( aa(set(A),$o,member(A,Z3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))))
           => ( aa(set(B),$o,member(B,Yb),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),Y),bot_bot(set(B))))
             => ( aa(B,$o,aa(A,fun(B,$o),R,Z3),Yb)
               => aa(B,$o,aa(A,fun(B,$o),Ra,Z3),Yb) ) ) )
       => aa(B,$o,aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),bNF_id_bnf(fun(A,fun(B,$o))),Ra),X),Y) ) ) ).

% ID.rel_mono_strong
tff(fact_1876_ID_Orel__refl__strong,axiom,
    ! [A: $tType,X: A,Ra: fun(A,fun(A,$o))] :
      ( ! [Z3: A] :
          ( aa(set(A),$o,member(A,Z3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))))
         => aa(A,$o,aa(A,fun(A,$o),Ra,Z3),Z3) )
     => aa(A,$o,aa(A,fun(A,$o),aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),bNF_id_bnf(fun(A,fun(A,$o))),Ra),X),X) ) ).

% ID.rel_refl_strong
tff(fact_1877_singleton__inject,axiom,
    ! [A: $tType,A3: A,B2: A] :
      ( ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A))) )
     => ( A3 = B2 ) ) ).

% singleton_inject
tff(fact_1878_insert__not__empty,axiom,
    ! [A: $tType,A3: A,A4: set(A)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),A4) != bot_bot(set(A)) ).

% insert_not_empty
tff(fact_1879_doubleton__eq__iff,axiom,
    ! [A: $tType,A3: A,B2: A,C2: A,D3: A] :
      ( ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A)))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),C2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),D3),bot_bot(set(A)))) )
    <=> ( ( ( A3 = C2 )
          & ( B2 = D3 ) )
        | ( ( A3 = D3 )
          & ( B2 = C2 ) ) ) ) ).

% doubleton_eq_iff
tff(fact_1880_singleton__iff,axiom,
    ! [A: $tType,B2: A,A3: A] :
      ( aa(set(A),$o,member(A,B2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A))))
    <=> ( B2 = A3 ) ) ).

% singleton_iff
tff(fact_1881_singletonD,axiom,
    ! [A: $tType,B2: A,A3: A] :
      ( aa(set(A),$o,member(A,B2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A))))
     => ( B2 = A3 ) ) ).

% singletonD
tff(fact_1882_flip__bit__int__def,axiom,
    ! [N: nat,K: int] : bit_se8732182000553998342ip_bit(int,N,K) = aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),bit_se4730199178511100633sh_bit(int,N,one_one(int))) ).

% flip_bit_int_def
tff(fact_1883_insert__UNIV,axiom,
    ! [A: $tType,X: A] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),top_top(set(A))) = top_top(set(A)) ).

% insert_UNIV
tff(fact_1884_Int__insert__right,axiom,
    ! [A: $tType,A4: set(A),A3: A,B3: set(A)] :
      aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),B3)) = $ite(aa(set(A),$o,member(A,A3),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3)) ).

% Int_insert_right
tff(fact_1885_Int__insert__left,axiom,
    ! [A: $tType,A3: A,B3: set(A),C3: set(A)] :
      aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),B3)),C3) = $ite(aa(set(A),$o,member(A,A3),C3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),C3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),C3)) ).

% Int_insert_left
tff(fact_1886_flip__bit__nat__def,axiom,
    ! [M: nat,N: nat] : bit_se8732182000553998342ip_bit(nat,M,N) = aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),N),bit_se4730199178511100633sh_bit(nat,M,one_one(nat))) ).

% flip_bit_nat_def
tff(fact_1887_push__bit__minus,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat,A3: A] : bit_se4730199178511100633sh_bit(A,N,aa(A,A,uminus_uminus(A),A3)) = aa(A,A,uminus_uminus(A),bit_se4730199178511100633sh_bit(A,N,A3)) ) ).

% push_bit_minus
tff(fact_1888_Collect__conv__if2,axiom,
    ! [A: $tType,A3: A,P: fun(A,$o)] :
      aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_ci(A,fun(fun(A,$o),fun(A,$o)),A3),P)) = $ite(aa(A,$o,P,A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A))),bot_bot(set(A))) ).

% Collect_conv_if2
tff(fact_1889_Collect__conv__if,axiom,
    ! [A: $tType,A3: A,P: fun(A,$o)] :
      aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_cj(A,fun(fun(A,$o),fun(A,$o)),A3),P)) = $ite(aa(A,$o,P,A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A))),bot_bot(set(A))) ).

% Collect_conv_if
tff(fact_1890_insert__def,axiom,
    ! [A: $tType,A3: A,B3: set(A)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),B3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_cf(A,fun(A,$o),A3))),B3) ).

% insert_def
tff(fact_1891_subset__singletonD,axiom,
    ! [A: $tType,A4: set(A),X: A] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))))
     => ( ( A4 = bot_bot(set(A)) )
        | ( A4 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))) ) ) ) ).

% subset_singletonD
tff(fact_1892_subset__singleton__iff,axiom,
    ! [A: $tType,X5: set(A),A3: A] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A))))
    <=> ( ( X5 = bot_bot(set(A)) )
        | ( X5 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A))) ) ) ) ).

% subset_singleton_iff
tff(fact_1893_singleton__Un__iff,axiom,
    ! [A: $tType,X: A,A4: set(A),B3: set(A)] :
      ( ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3) )
    <=> ( ( ( A4 = bot_bot(set(A)) )
          & ( B3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))) ) )
        | ( ( A4 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))) )
          & ( B3 = bot_bot(set(A)) ) )
        | ( ( A4 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))) )
          & ( B3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))) ) ) ) ) ).

% singleton_Un_iff
tff(fact_1894_Un__singleton__iff,axiom,
    ! [A: $tType,A4: set(A),B3: set(A),X: A] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))) )
    <=> ( ( ( A4 = bot_bot(set(A)) )
          & ( B3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))) ) )
        | ( ( A4 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))) )
          & ( B3 = bot_bot(set(A)) ) )
        | ( ( A4 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))) )
          & ( B3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))) ) ) ) ) ).

% Un_singleton_iff
tff(fact_1895_insert__is__Un,axiom,
    ! [A: $tType,A3: A,A4: set(A)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),A4) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A)))),A4) ).

% insert_is_Un
tff(fact_1896_set__minus__singleton__eq,axiom,
    ! [A: $tType,X: A,X5: set(A)] :
      ( ~ aa(set(A),$o,member(A,X),X5)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),X5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) = X5 ) ) ).

% set_minus_singleton_eq
tff(fact_1897_insert__minus__eq,axiom,
    ! [A: $tType,X: A,Y: A,A4: set(A)] :
      ( ( X != Y )
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A4)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y),bot_bot(set(A)))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y),bot_bot(set(A))))) ) ) ).

% insert_minus_eq
tff(fact_1898_Diff__insert,axiom,
    ! [A: $tType,A4: set(A),A3: A,B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),B3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),B3)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A)))) ).

% Diff_insert
tff(fact_1899_insert__Diff,axiom,
    ! [A: $tType,A3: A,A4: set(A)] :
      ( aa(set(A),$o,member(A,A3),A4)
     => ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A))))) = A4 ) ) ).

% insert_Diff
tff(fact_1900_Diff__insert2,axiom,
    ! [A: $tType,A4: set(A),A3: A,B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),B3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A))))),B3) ).

% Diff_insert2
tff(fact_1901_Diff__insert__absorb,axiom,
    ! [A: $tType,X: A,A4: set(A)] :
      ( ~ aa(set(A),$o,member(A,X),A4)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A4)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) = A4 ) ) ).

% Diff_insert_absorb
tff(fact_1902_vimage__singleton__eq,axiom,
    ! [A: $tType,B: $tType,A3: A,F: fun(A,B),B2: B] :
      ( aa(set(A),$o,member(A,A3),aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),B2),bot_bot(set(B)))))
    <=> ( aa(A,B,F,A3) = B2 ) ) ).

% vimage_singleton_eq
tff(fact_1903_subset__insert__iff,axiom,
    ! [A: $tType,A4: set(A),X: A,B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),B3))
    <=> $ite(aa(set(A),$o,member(A,X),A4),aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))))),B3),aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B3)) ) ).

% subset_insert_iff
tff(fact_1904_Diff__single__insert,axiom,
    ! [A: $tType,A4: set(A),X: A,B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))))),B3)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),B3)) ) ).

% Diff_single_insert
tff(fact_1905_remove__subset,axiom,
    ! [A: $tType,X: A,S: set(A)] :
      ( aa(set(A),$o,member(A,X),S)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))))),S) ) ).

% remove_subset
tff(fact_1906_vimage__insert,axiom,
    ! [A: $tType,B: $tType,F: fun(A,B),A3: B,B3: set(B)] : aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),A3),B3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),A3),bot_bot(set(B))))),aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F),B3)) ).

% vimage_insert
tff(fact_1907_Compl__insert,axiom,
    ! [A: $tType,X: A,A4: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A4)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),A4)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) ).

% Compl_insert
tff(fact_1908_one__integer_Orsp,axiom,
    one_one(int) = one_one(int) ).

% one_integer.rsp
tff(fact_1909_one__natural_Orsp,axiom,
    one_one(nat) = one_one(nat) ).

% one_natural.rsp
tff(fact_1910_flip__bit__eq__xor,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A3: A] : bit_se8732182000553998342ip_bit(A,N,A3) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A3),bit_se4730199178511100633sh_bit(A,N,one_one(A))) ) ).

% flip_bit_eq_xor
tff(fact_1911_psubset__insert__iff,axiom,
    ! [A: $tType,A4: set(A),X: A,B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),B3))
    <=> $ite(
          aa(set(A),$o,member(A,X),B3),
          aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A4),B3),
          $ite(aa(set(A),$o,member(A,X),A4),aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))))),B3),aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B3)) ) ) ).

% psubset_insert_iff
tff(fact_1912_push__bit__double,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A3: A] : bit_se4730199178511100633sh_bit(A,N,aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))) = aa(A,A,aa(A,fun(A,A),times_times(A),bit_se4730199178511100633sh_bit(A,N,A3)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ).

% push_bit_double
tff(fact_1913_push__bit__eq__mult,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A3: 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),aa(num,num,bit0,one2))),N)) ) ).

% push_bit_eq_mult
tff(fact_1914_push__bit__minus__one,axiom,
    ! [N: nat] : bit_se4730199178511100633sh_bit(int,N,aa(int,int,uminus_uminus(int),one_one(int))) = aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N)) ).

% push_bit_minus_one
tff(fact_1915_of__int__round__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),archimedean_round(A,X))),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))) ) ).

% of_int_round_le
tff(fact_1916_of__int__round__ge,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))),aa(int,A,ring_1_of_int(A),archimedean_round(A,X))) ) ).

% of_int_round_ge
tff(fact_1917_of__int__round__gt,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))),aa(int,A,ring_1_of_int(A),archimedean_round(A,X))) ) ).

% of_int_round_gt
tff(fact_1918_integer__of__int__code,axiom,
    ! [K: int] :
      aa(int,code_integer,code_integer_of_int,K) = $ite(
        aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int)),
        aa(code_integer,code_integer,uminus_uminus(code_integer),aa(int,code_integer,code_integer_of_int,aa(int,int,uminus_uminus(int),K))),
        $ite(
          K = zero_zero(int),
          zero_zero(code_integer),
          $let(
            l: code_integer,
            l:= aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2))),aa(int,code_integer,code_integer_of_int,aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))))),
            $ite(modulo_modulo(int,K,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))) = zero_zero(int),l,aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),l),one_one(code_integer))) ) ) ) ).

% integer_of_int_code
tff(fact_1919_and__int_Opsimps,axiom,
    ! [K: int,L: int] :
      ( accp(product_prod(int,int),bit_and_int_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),K),L))
     => ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L) = $ite(
            ( aa(set(int),$o,member(int,K),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))
            & aa(set(int),$o,member(int,L),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))) ),
            aa(int,int,uminus_uminus(int),
              aa($o,int,zero_neq_one_of_bool(int),
                ( ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),K)
                & ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),L) ))),
            aa(int,int,
              aa(int,fun(int,int),plus_plus(int),
                aa($o,int,zero_neq_one_of_bool(int),
                  ( ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),K)
                  & ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),L) ))),
              aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))))) ) ) ) ).

% and_int.psimps
tff(fact_1920_and__int_Opelims,axiom,
    ! [X: int,Xa: int,Y: int] :
      ( ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),X),Xa) = Y )
     => ( accp(product_prod(int,int),bit_and_int_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),X),Xa))
       => ~ ( ( Y = $ite(
                  ( aa(set(int),$o,member(int,X),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))
                  & aa(set(int),$o,member(int,Xa),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))) ),
                  aa(int,int,uminus_uminus(int),
                    aa($o,int,zero_neq_one_of_bool(int),
                      ( ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),X)
                      & ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Xa) ))),
                  aa(int,int,
                    aa(int,fun(int,int),plus_plus(int),
                      aa($o,int,zero_neq_one_of_bool(int),
                        ( ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),X)
                        & ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Xa) ))),
                    aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),X),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),Xa),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))))) ) )
           => ~ accp(product_prod(int,int),bit_and_int_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),X),Xa)) ) ) ) ).

% and_int.pelims
tff(fact_1921_neg__numeral__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: num,X: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V))),archimedean_ceiling(A,X))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),one_one(A))),X) ) ) ).

% neg_numeral_le_ceiling
tff(fact_1922_ceiling__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V: num] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archimedean_ceiling(A,X)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V)))
        <=> aa(A,$o,aa(A,fun(A,$o),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),V))),one_one(A))) ) ) ).

% ceiling_less_neg_numeral
tff(fact_1923_floor__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V: num] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archim6421214686448440834_floor(A,X)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V)))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),one_one(A))) ) ) ).

% floor_le_neg_numeral
tff(fact_1924_neg__numeral__less__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: num,X: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V))),archim6421214686448440834_floor(A,X))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),one_one(A))),X) ) ) ).

% neg_numeral_less_floor
tff(fact_1925_floor__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( archim6421214686448440834_floor(A,one_one(A)) = one_one(int) ) ) ).

% floor_one
tff(fact_1926_ceiling__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( archimedean_ceiling(A,one_one(A)) = one_one(int) ) ) ).

% ceiling_one
tff(fact_1927_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_1928_zero__less__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),archim6421214686448440834_floor(A,X))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),X) ) ) ).

% zero_less_floor
tff(fact_1929_floor__le__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archim6421214686448440834_floor(A,X)),zero_zero(int))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),one_one(A)) ) ) ).

% floor_le_zero
tff(fact_1930_one__le__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),one_one(int)),archim6421214686448440834_floor(A,X))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),X) ) ) ).

% one_le_floor
tff(fact_1931_ceiling__less__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archimedean_ceiling(A,X)),one_one(int))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),zero_zero(A)) ) ) ).

% ceiling_less_one
tff(fact_1932_one__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),one_one(int)),archimedean_ceiling(A,X))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),X) ) ) ).

% one_le_ceiling
tff(fact_1933_floor__less__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archim6421214686448440834_floor(A,X)),one_one(int))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),one_one(A)) ) ) ).

% floor_less_one
tff(fact_1934_ceiling__le__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(A,X)),one_one(int))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),one_one(A)) ) ) ).

% ceiling_le_one
tff(fact_1935_one__less__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),one_one(int)),archimedean_ceiling(A,X))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),X) ) ) ).

% one_less_ceiling
tff(fact_1936_floor__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: num] : archim6421214686448440834_floor(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V)) ) ).

% floor_neg_numeral
tff(fact_1937_ceiling__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: num] : archimedean_ceiling(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V)) ) ).

% ceiling_neg_numeral
tff(fact_1938_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_1939_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_1940_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_1941_ceiling__less__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archimedean_ceiling(A,X)),zero_zero(int))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(A,A,uminus_uminus(A),one_one(A))) ) ) ).

% ceiling_less_zero
tff(fact_1942_zero__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),archimedean_ceiling(A,X))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),one_one(A))),X) ) ) ).

% zero_le_ceiling
tff(fact_1943_numeral__less__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: num,X: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(num,int,numeral_numeral(int),V)),archim6421214686448440834_floor(A,X))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),V)),one_one(A))),X) ) ) ).

% numeral_less_floor
tff(fact_1944_floor__le__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V: num] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archim6421214686448440834_floor(A,X)),aa(num,int,numeral_numeral(int),V))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),V)),one_one(A))) ) ) ).

% floor_le_numeral
tff(fact_1945_ceiling__less__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V: num] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archimedean_ceiling(A,X)),aa(num,int,numeral_numeral(int),V))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(num,A,numeral_numeral(A),V)),one_one(A))) ) ) ).

% ceiling_less_numeral
tff(fact_1946_numeral__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: num,X: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(num,int,numeral_numeral(int),V)),archimedean_ceiling(A,X))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(num,A,numeral_numeral(A),V)),one_one(A))),X) ) ) ).

% numeral_le_ceiling
tff(fact_1947_one__less__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),one_one(int)),archim6421214686448440834_floor(A,X))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),X) ) ) ).

% one_less_floor
tff(fact_1948_floor__le__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archim6421214686448440834_floor(A,X)),one_one(int))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ) ).

% floor_le_one
tff(fact_1949_neg__numeral__le__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: num,X: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V))),archim6421214686448440834_floor(A,X))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),X) ) ) ).

% neg_numeral_le_floor
tff(fact_1950_floor__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V: num] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archim6421214686448440834_floor(A,X)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V)))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))) ) ) ).

% floor_less_neg_numeral
tff(fact_1951_ceiling__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V: num] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(A,X)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V)))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))) ) ) ).

% ceiling_le_neg_numeral
tff(fact_1952_neg__numeral__less__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: num,X: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V))),archimedean_ceiling(A,X))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),X) ) ) ).

% neg_numeral_less_ceiling
tff(fact_1953_integer__of__int__cases,axiom,
    ! [X: code_integer] :
      ~ ! [Y2: int] :
          ( ( X = aa(int,code_integer,code_integer_of_int,Y2) )
         => ~ aa(set(int),$o,member(int,Y2),top_top(set(int))) ) ).

% integer_of_int_cases
tff(fact_1954_integer__of__int__induct,axiom,
    ! [P: fun(code_integer,$o),X: code_integer] :
      ( ! [Y2: int] :
          ( aa(set(int),$o,member(int,Y2),top_top(set(int)))
         => aa(code_integer,$o,P,aa(int,code_integer,code_integer_of_int,Y2)) )
     => aa(code_integer,$o,P,X) ) ).

% integer_of_int_induct
tff(fact_1955_integer__of__int__inject,axiom,
    ! [X: int,Y: int] :
      ( aa(set(int),$o,member(int,X),top_top(set(int)))
     => ( aa(set(int),$o,member(int,Y),top_top(set(int)))
       => ( ( aa(int,code_integer,code_integer_of_int,X) = aa(int,code_integer,code_integer_of_int,Y) )
        <=> ( X = Y ) ) ) ) ).

% integer_of_int_inject
tff(fact_1956_uminus__integer__code_I1_J,axiom,
    aa(code_integer,code_integer,uminus_uminus(code_integer),zero_zero(code_integer)) = zero_zero(code_integer) ).

% uminus_integer_code(1)
tff(fact_1957_ID_Omap__cong__pred,axiom,
    ! [B: $tType,A: $tType,X: A,Ya: A,F: fun(A,B),G: fun(A,B)] :
      ( ( X = Ya )
     => ( aa(A,$o,aa(fun(A,$o),fun(A,$o),bNF_id_bnf(fun(A,$o)),aa(fun(A,B),fun(A,$o),aTP_Lamp_ck(fun(A,B),fun(fun(A,B),fun(A,$o)),F),G)),Ya)
       => ( aa(A,B,aa(fun(A,B),fun(A,B),bNF_id_bnf(fun(A,B)),F),X) = aa(A,B,aa(fun(A,B),fun(A,B),bNF_id_bnf(fun(A,B)),G),Ya) ) ) ) ).

% ID.map_cong_pred
tff(fact_1958_ID_Opred__True,axiom,
    ! [A: $tType,X3: A] : aa(A,$o,aa(fun(A,$o),fun(A,$o),bNF_id_bnf(fun(A,$o)),aTP_Lamp_ar(A,$o)),X3) ).

% ID.pred_True
tff(fact_1959_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_1960_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_1961_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_1962_uminus__integer_Oabs__eq,axiom,
    ! [X: int] : aa(code_integer,code_integer,uminus_uminus(code_integer),aa(int,code_integer,code_integer_of_int,X)) = aa(int,code_integer,code_integer_of_int,aa(int,int,uminus_uminus(int),X)) ).

% uminus_integer.abs_eq
tff(fact_1963_ceiling__diff__floor__le__1,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,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_1964_ceiling__altdef,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          archimedean_ceiling(A,X) = $ite(X = aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,X)),archim6421214686448440834_floor(A,X),aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,X)),one_one(int))) ) ).

% ceiling_altdef
tff(fact_1965_one__integer__def,axiom,
    one_one(code_integer) = aa(int,code_integer,code_integer_of_int,one_one(int)) ).

% one_integer_def
tff(fact_1966_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_1967_floor__unique,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z2: int,X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z2)),X)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),Z2)),one_one(A)))
           => ( archim6421214686448440834_floor(A,X) = Z2 ) ) ) ) ).

% floor_unique
tff(fact_1968_floor__eq__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,A3: int] :
          ( ( archim6421214686448440834_floor(A,X) = A3 )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),A3)),X)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),A3)),one_one(A))) ) ) ) ).

% floor_eq_iff
tff(fact_1969_floor__split,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [P: fun(int,$o),T3: A] :
          ( aa(int,$o,P,archim6421214686448440834_floor(A,T3))
        <=> ! [I3: int] :
              ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),I3)),T3)
                & aa(A,$o,aa(A,fun(A,$o),ord_less(A),T3),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),I3)),one_one(A))) )
             => aa(int,$o,P,I3) ) ) ) ).

% floor_split
tff(fact_1970_le__mult__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
           => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),times_times(int),archim6421214686448440834_floor(A,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_1971_less__floor__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z2: int,X: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z2),archim6421214686448440834_floor(A,X))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),Z2)),one_one(A))),X) ) ) ).

% less_floor_iff
tff(fact_1972_floor__le__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Z2: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archim6421214686448440834_floor(A,X)),Z2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),Z2)),one_one(A))) ) ) ).

% floor_le_iff
tff(fact_1973_floor__correct,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,X))),X)
          & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,X)),one_one(int)))) ) ) ).

% floor_correct
tff(fact_1974_ceiling__correct,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,X))),one_one(A))),X)
          & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,X))) ) ) ).

% ceiling_correct
tff(fact_1975_ceiling__unique,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z2: int,X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),Z2)),one_one(A))),X)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(int,A,ring_1_of_int(A),Z2))
           => ( archimedean_ceiling(A,X) = Z2 ) ) ) ) ).

% ceiling_unique
tff(fact_1976_ceiling__eq__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,A3: int] :
          ( ( archimedean_ceiling(A,X) = A3 )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),A3)),one_one(A))),X)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(int,A,ring_1_of_int(A),A3)) ) ) ) ).

% ceiling_eq_iff
tff(fact_1977_ceiling__split,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [P: fun(int,$o),T3: A] :
          ( aa(int,$o,P,archimedean_ceiling(A,T3))
        <=> ! [I3: int] :
              ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),I3)),one_one(A))),T3)
                & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),T3),aa(int,A,ring_1_of_int(A),I3)) )
             => aa(int,$o,P,I3) ) ) ) ).

% ceiling_split
tff(fact_1978_mult__ceiling__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
           => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),times_times(A),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_1979_ceiling__less__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Z2: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archimedean_ceiling(A,X)),Z2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(A,A,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_1980_le__ceiling__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z2: int,X: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Z2),archimedean_ceiling(A,X))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,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_1981_floor__divide__lower,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q3: A,P3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Q3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),P3),Q3)))),Q3)),P3) ) ) ).

% floor_divide_lower
tff(fact_1982_ceiling__divide__upper,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q3: A,P3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Q3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),P3),aa(A,A,aa(A,fun(A,A),times_times(A),aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),P3),Q3)))),Q3)) ) ) ).

% ceiling_divide_upper
tff(fact_1983_floor__divide__upper,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q3: A,P3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Q3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),P3),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),P3),Q3)))),one_one(A))),Q3)) ) ) ).

% floor_divide_upper
tff(fact_1984_round__def,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : archimedean_round(A,X) = archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))) ) ).

% round_def
tff(fact_1985_ceiling__divide__lower,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q3: A,P3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Q3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),P3),Q3)))),one_one(A))),Q3)),P3) ) ) ).

% ceiling_divide_lower
tff(fact_1986_and__int_Opinduct,axiom,
    ! [A0: int,A1: int,P: fun(int,fun(int,$o))] :
      ( accp(product_prod(int,int),bit_and_int_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),A0),A1))
     => ( ! [K2: int,L2: int] :
            ( accp(product_prod(int,int),bit_and_int_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),K2),L2))
           => ( ( ~ ( aa(set(int),$o,member(int,K2),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))
                    & aa(set(int),$o,member(int,L2),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))) )
               => aa(int,$o,aa(int,fun(int,$o),P,aa(int,int,aa(int,fun(int,int),divide_divide(int),K2),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L2),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))) )
             => aa(int,$o,aa(int,fun(int,$o),P,K2),L2) ) )
       => aa(int,$o,aa(int,fun(int,$o),P,A0),A1) ) ) ).

% and_int.pinduct
tff(fact_1987_upto_Opinduct,axiom,
    ! [A0: int,A1: int,P: fun(int,fun(int,$o))] :
      ( accp(product_prod(int,int),upto_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),A0),A1))
     => ( ! [I2: int,J2: int] :
            ( accp(product_prod(int,int),upto_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),I2),J2))
           => ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I2),J2)
               => aa(int,$o,aa(int,fun(int,$o),P,aa(int,int,aa(int,fun(int,int),plus_plus(int),I2),one_one(int))),J2) )
             => aa(int,$o,aa(int,fun(int,$o),P,I2),J2) ) )
       => aa(int,$o,aa(int,fun(int,$o),P,A0),A1) ) ) ).

% upto.pinduct
tff(fact_1988_round__altdef,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          archimedean_round(A,X) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),archimedean_frac(A,X)),archimedean_ceiling(A,X),archim6421214686448440834_floor(A,X)) ) ).

% round_altdef
tff(fact_1989_the__elem__eq,axiom,
    ! [A: $tType,X: A] : the_elem(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) = X ).

% the_elem_eq
tff(fact_1990_round__unique_H,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,N: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),aa(int,A,ring_1_of_int(A),N)))),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))
         => ( archimedean_round(A,X) = N ) ) ) ).

% round_unique'
tff(fact_1991_of__int__round__abs__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),archimedean_round(A,X))),X))),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))) ) ).

% of_int_round_abs_le
tff(fact_1992_num__of__integer__code,axiom,
    ! [K: code_integer] :
      code_num_of_integer(K) = $ite(aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less_eq(code_integer),K),one_one(code_integer)),one2,aa(product_prod(code_integer,code_integer),num,aa(fun(code_integer,fun(code_integer,num)),fun(product_prod(code_integer,code_integer),num),product_case_prod(code_integer,code_integer,num),aTP_Lamp_cl(code_integer,fun(code_integer,num))),code_divmod_integer(K,aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2))))) ).

% num_of_integer_code
tff(fact_1993_bit__cut__integer__def,axiom,
    ! [K: code_integer] : code_bit_cut_integer(K) = aa($o,product_prod(code_integer,$o),aa(code_integer,fun($o,product_prod(code_integer,$o)),product_Pair(code_integer,$o),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),divide_divide(code_integer),K),aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2)))),~ aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),dvd_dvd(code_integer),aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2))),K)) ).

% bit_cut_integer_def
tff(fact_1994_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_1995_abs__zero,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ( aa(A,A,abs_abs(A),zero_zero(A)) = zero_zero(A) ) ) ).

% abs_zero
tff(fact_1996_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_1997_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_1998_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_1999_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_2000_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_2001_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_2002_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_2003_zdvd1__eq,axiom,
    ! [X: int] :
      ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),X),one_one(int))
    <=> ( aa(int,int,abs_abs(int),X) = one_one(int) ) ) ).

% zdvd1_eq
tff(fact_2004_abs__le__zero__iff,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),A3)),zero_zero(A))
        <=> ( A3 = zero_zero(A) ) ) ) ).

% abs_le_zero_iff
tff(fact_2005_abs__le__self__iff,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),A3)),A3)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3) ) ) ).

% abs_le_self_iff
tff(fact_2006_abs__of__nonneg,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
         => ( aa(A,A,abs_abs(A),A3) = A3 ) ) ) ).

% abs_of_nonneg
tff(fact_2007_zero__less__abs__iff,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,abs_abs(A),A3))
        <=> ( A3 != zero_zero(A) ) ) ) ).

% zero_less_abs_iff
tff(fact_2008_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_2009_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_2010_abs__power__minus,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,N: nat] : aa(A,A,abs_abs(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),A3)),N)) = aa(A,A,abs_abs(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)) ) ).

% abs_power_minus
tff(fact_2011_zabs__less__one__iff,axiom,
    ! [Z2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,abs_abs(int),Z2)),one_one(int))
    <=> ( Z2 = zero_zero(int) ) ) ).

% zabs_less_one_iff
tff(fact_2012_abs__of__nonpos,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),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_2013_abs__ge__self,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,abs_abs(A),A3)) ) ).

% abs_ge_self
tff(fact_2014_abs__le__D1,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),A3)),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2) ) ) ).

% abs_le_D1
tff(fact_2015_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_2016_abs__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ( aa(A,A,abs_abs(A),one_one(A)) = one_one(A) ) ) ).

% abs_one
tff(fact_2017_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_2018_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_2019_abs__ge__zero,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,abs_abs(A),A3)) ) ).

% abs_ge_zero
tff(fact_2020_abs__of__pos,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
         => ( aa(A,A,abs_abs(A),A3) = A3 ) ) ) ).

% abs_of_pos
tff(fact_2021_abs__not__less__zero,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),A3)),zero_zero(A)) ) ).

% abs_not_less_zero
tff(fact_2022_abs__triangle__ineq,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),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_2023_abs__triangle__ineq2__sym,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,abs_abs(A),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_2024_abs__triangle__ineq3,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,abs_abs(A),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_2025_abs__triangle__ineq2,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,abs_abs(A),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_2026_abs__mult__less,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,C2: A,B2: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),A3)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),B2)),D3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(A),B2))),aa(A,A,aa(A,fun(A,A),times_times(A),C2),D3)) ) ) ) ).

% abs_mult_less
tff(fact_2027_abs__ge__minus__self,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),A3)),aa(A,A,abs_abs(A),A3)) ) ).

% abs_ge_minus_self
tff(fact_2028_abs__le__iff,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),A3)),B2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),A3)),B2) ) ) ) ).

% abs_le_iff
tff(fact_2029_abs__le__D2,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),A3)),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),A3)),B2) ) ) ).

% abs_le_D2
tff(fact_2030_abs__leI,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),A3)),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),A3)),B2) ) ) ) ).

% abs_leI
tff(fact_2031_abs__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),A3)),B2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),A3)),B2) ) ) ) ).

% abs_less_iff
tff(fact_2032_abs__zmult__eq__1,axiom,
    ! [M: int,N: int] :
      ( ( aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),times_times(int),M),N)) = one_one(int) )
     => ( aa(int,int,abs_abs(int),M) = one_one(int) ) ) ).

% abs_zmult_eq_1
tff(fact_2033_frac__lt__1,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),archimedean_frac(A,X)),one_one(A)) ) ).

% frac_lt_1
tff(fact_2034_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_2035_dense__eq0__I,axiom,
    ! [A: $tType] :
      ( ( ordere166539214618696060dd_abs(A)
        & dense_linorder(A) )
     => ! [X: A] :
          ( ! [E2: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),E2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),X)),E2) )
         => ( X = zero_zero(A) ) ) ) ).

% dense_eq0_I
tff(fact_2036_abs__eq__mult,axiom,
    ! [A: $tType] :
      ( ordered_ring_abs(A)
     => ! [A3: A,B2: A] :
          ( ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
              | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),zero_zero(A)) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
              | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A)) ) )
         => ( aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),times_times(A),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_2037_abs__mult__pos,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),X)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,abs_abs(A),Y)),X) = aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),times_times(A),Y),X)) ) ) ) ).

% abs_mult_pos
tff(fact_2038_abs__minus__le__zero,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(A,A,abs_abs(A),A3))),zero_zero(A)) ) ).

% abs_minus_le_zero
tff(fact_2039_eq__abs__iff_H,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [A3: A,B2: A] :
          ( ( A3 = aa(A,A,abs_abs(A),B2) )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
            & ( ( B2 = A3 )
              | ( B2 = aa(A,A,uminus_uminus(A),A3) ) ) ) ) ) ).

% eq_abs_iff'
tff(fact_2040_abs__eq__iff_H,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [A3: A,B2: A] :
          ( ( aa(A,A,abs_abs(A),A3) = B2 )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
            & ( ( A3 = B2 )
              | ( A3 = aa(A,A,uminus_uminus(A),B2) ) ) ) ) ) ).

% abs_eq_iff'
tff(fact_2041_abs__diff__triangle__ineq,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B2: A,C2: A,D3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,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),C2),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),C2))),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_2042_abs__triangle__ineq4,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),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_2043_abs__if__raw,axiom,
    ! [A: $tType] :
      ( abs_if(A)
     => ! [X3: A] :
          aa(A,A,abs_abs(A),X3) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),zero_zero(A)),aa(A,A,uminus_uminus(A),X3),X3) ) ).

% abs_if_raw
tff(fact_2044_abs__of__neg,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),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_2045_abs__if,axiom,
    ! [A: $tType] :
      ( abs_if(A)
     => ! [A3: A] :
          aa(A,A,abs_abs(A),A3) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A)),aa(A,A,uminus_uminus(A),A3),A3) ) ).

% abs_if
tff(fact_2046_zabs__def,axiom,
    ! [I: int] :
      aa(int,int,abs_abs(int),I) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less(int),I),zero_zero(int)),aa(int,int,uminus_uminus(int),I),I) ).

% zabs_def
tff(fact_2047_abs__integer__code,axiom,
    ! [K: code_integer] :
      aa(code_integer,code_integer,abs_abs(code_integer),K) = $ite(aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),K),zero_zero(code_integer)),aa(code_integer,code_integer,uminus_uminus(code_integer),K),K) ).

% abs_integer_code
tff(fact_2048_abs__add__one__gt__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,abs_abs(A),X))) ) ).

% abs_add_one_gt_zero
tff(fact_2049_of__int__leD,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: int,X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(int,A,ring_1_of_int(A),N))),X)
         => ( ( N = zero_zero(int) )
            | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),X) ) ) ) ).

% of_int_leD
tff(fact_2050_of__int__lessD,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: int,X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),aa(int,A,ring_1_of_int(A),N))),X)
         => ( ( N = zero_zero(int) )
            | aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),X) ) ) ) ).

% of_int_lessD
tff(fact_2051_zdvd__mult__cancel1,axiom,
    ! [M: int,N: int] :
      ( ( M != zero_zero(int) )
     => ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(int,int,aa(int,fun(int,int),times_times(int),M),N)),M)
      <=> ( aa(int,int,abs_abs(int),N) = one_one(int) ) ) ) ).

% zdvd_mult_cancel1
tff(fact_2052_abs__square__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = one_one(A) )
        <=> ( aa(A,A,abs_abs(A),X) = one_one(A) ) ) ) ).

% abs_square_eq_1
tff(fact_2053_frac__eq,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( ( archimedean_frac(A,X) = X )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),X)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),one_one(A)) ) ) ) ).

% frac_eq
tff(fact_2054_frac__add,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y: A] :
          archimedean_frac(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,X)),archimedean_frac(A,Y))),one_one(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,X)),archimedean_frac(A,Y)),aa(A,A,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_2055_abs__square__le__1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),one_one(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),X)),one_one(A)) ) ) ).

% abs_square_le_1
tff(fact_2056_abs__square__less__1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),one_one(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),X)),one_one(A)) ) ) ).

% abs_square_less_1
tff(fact_2057_nat__intermed__int__val,axiom,
    ! [M: nat,N: nat,F: fun(nat,int),K: int] :
      ( ! [I2: nat] :
          ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),I2)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),N) )
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,F,aa(nat,nat,suc,I2))),aa(nat,int,F,I2)))),one_one(int)) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),N)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,F,M)),K)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),aa(nat,int,F,N))
           => ? [I2: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),I2)
                & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),N)
                & ( aa(nat,int,F,I2) = K ) ) ) ) ) ) ).

% nat_intermed_int_val
tff(fact_2058_incr__lemma,axiom,
    ! [D3: int,Z2: int,X: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D3)
     => aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z2),aa(int,int,aa(int,fun(int,int),plus_plus(int),X),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),X),Z2))),one_one(int))),D3))) ) ).

% incr_lemma
tff(fact_2059_decr__lemma,axiom,
    ! [D3: int,X: int,Z2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D3)
     => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,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_2060_nat__ivt__aux,axiom,
    ! [N: nat,F: fun(nat,int),K: int] :
      ( ! [I2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),N)
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,F,aa(nat,nat,suc,I2))),aa(nat,int,F,I2)))),one_one(int)) )
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,F,zero_zero(nat))),K)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),aa(nat,int,F,N))
         => ? [I2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),N)
              & ( aa(nat,int,F,I2) = K ) ) ) ) ) ).

% nat_ivt_aux
tff(fact_2061_floor__add,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y: A] :
          archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,X)),archimedean_frac(A,Y))),one_one(A)),aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,X)),archim6421214686448440834_floor(A,Y)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,X)),archim6421214686448440834_floor(A,Y))),one_one(int))) ) ).

% floor_add
tff(fact_2062_nat0__intermed__int__val,axiom,
    ! [N: nat,F: fun(nat,int),K: int] :
      ( ! [I2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),N)
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,F,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),one_one(nat)))),aa(nat,int,F,I2)))),one_one(int)) )
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,F,zero_zero(nat))),K)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),aa(nat,int,F,N))
         => ? [I2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),N)
              & ( aa(nat,int,F,I2) = K ) ) ) ) ) ).

% nat0_intermed_int_val
tff(fact_2063_divmod__integer__def,axiom,
    ! [K: code_integer,L: code_integer] : code_divmod_integer(K,L) = aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),divide_divide(code_integer),K),L)),modulo_modulo(code_integer,K,L)) ).

% divmod_integer_def
tff(fact_2064_bit__cut__integer__code,axiom,
    ! [K: code_integer] :
      code_bit_cut_integer(K) = $ite(K = zero_zero(code_integer),aa($o,product_prod(code_integer,$o),aa(code_integer,fun($o,product_prod(code_integer,$o)),product_Pair(code_integer,$o),zero_zero(code_integer)),$false),aa(product_prod(code_integer,code_integer),product_prod(code_integer,$o),aa(fun(code_integer,fun(code_integer,product_prod(code_integer,$o))),fun(product_prod(code_integer,code_integer),product_prod(code_integer,$o)),product_case_prod(code_integer,code_integer,product_prod(code_integer,$o)),aTP_Lamp_cm(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,$o))),K)),code_divmod_abs(K,aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2))))) ).

% bit_cut_integer_code
tff(fact_2065_nat__of__integer__code,axiom,
    ! [K: code_integer] :
      code_nat_of_integer(K) = $ite(aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less_eq(code_integer),K),zero_zero(code_integer)),zero_zero(nat),aa(product_prod(code_integer,code_integer),nat,aa(fun(code_integer,fun(code_integer,nat)),fun(product_prod(code_integer,code_integer),nat),product_case_prod(code_integer,code_integer,nat),aTP_Lamp_cn(code_integer,fun(code_integer,nat))),code_divmod_integer(K,aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2))))) ).

% nat_of_integer_code
tff(fact_2066_int__of__integer__code,axiom,
    ! [K: code_integer] :
      aa(code_integer,int,code_int_of_integer,K) = $ite(
        aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),K),zero_zero(code_integer)),
        aa(int,int,uminus_uminus(int),aa(code_integer,int,code_int_of_integer,aa(code_integer,code_integer,uminus_uminus(code_integer),K))),
        $ite(K = zero_zero(code_integer),zero_zero(int),aa(product_prod(code_integer,code_integer),int,aa(fun(code_integer,fun(code_integer,int)),fun(product_prod(code_integer,code_integer),int),product_case_prod(code_integer,code_integer,int),aTP_Lamp_co(code_integer,fun(code_integer,int))),code_divmod_integer(K,aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2))))) ) ).

% int_of_integer_code
tff(fact_2067_xor__int__unfold,axiom,
    ! [K: int,L: int] :
      aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),L) = $ite(
        K = aa(int,int,uminus_uminus(int),one_one(int)),
        aa(int,int,bit_ri4277139882892585799ns_not(int),L),
        $ite(
          L = aa(int,int,uminus_uminus(int),one_one(int)),
          aa(int,int,bit_ri4277139882892585799ns_not(int),K),
          $ite(
            K = zero_zero(int),
            L,
            $ite(L = zero_zero(int),K,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),modulo_modulo(int,K,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),modulo_modulo(int,L,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))))))) ) ) ) ).

% xor_int_unfold
tff(fact_2068_is__singleton__the__elem,axiom,
    ! [A: $tType,A4: set(A)] :
      ( is_singleton(A,A4)
    <=> ( A4 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),the_elem(A,A4)),bot_bot(set(A))) ) ) ).

% is_singleton_the_elem
tff(fact_2069_integer__of__num_I3_J,axiom,
    ! [N: num] :
      code_integer_of_num(aa(num,num,bit1,N)) = $let(
        k: code_integer,
        k:= code_integer_of_num(N),
        aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),k),k)),one_one(code_integer)) ) ).

% integer_of_num(3)
tff(fact_2070_mask__eq__sum__exp__nat,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)),aa(nat,nat,suc,zero_zero(nat))) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_bk(nat,fun(nat,$o)),N))) ).

% mask_eq_sum_exp_nat
tff(fact_2071_of__int__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( ring_1(A)
     => ! [F: fun(B,int),A4: set(B)] : aa(int,A,ring_1_of_int(A),aa(set(B),int,aa(fun(B,int),fun(set(B),int),groups7311177749621191930dd_sum(B,int),F),A4)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aTP_Lamp_cp(fun(B,int),fun(B,A),F)),A4) ) ).

% of_int_sum
tff(fact_2072_one__integer_Orep__eq,axiom,
    aa(code_integer,int,code_int_of_integer,one_one(code_integer)) = one_one(int) ).

% one_integer.rep_eq
tff(fact_2073_uminus__integer_Orep__eq,axiom,
    ! [X: code_integer] : aa(code_integer,int,code_int_of_integer,aa(code_integer,code_integer,uminus_uminus(code_integer),X)) = aa(int,int,uminus_uminus(int),aa(code_integer,int,code_int_of_integer,X)) ).

% uminus_integer.rep_eq
tff(fact_2074_is__singletonI,axiom,
    ! [A: $tType,X: A] : is_singleton(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) ).

% is_singletonI
tff(fact_2075_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_2076_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_2077_bit_Oxor__cancel__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),X),aa(A,A,bit_ri4277139882892585799ns_not(A),X)) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).

% bit.xor_cancel_right
tff(fact_2078_bit_Oxor__cancel__left,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(A,A,bit_ri4277139882892585799ns_not(A),X)),X) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).

% bit.xor_cancel_left
tff(fact_2079_bit_Oxor__one__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),X),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,bit_ri4277139882892585799ns_not(A),X) ) ).

% bit.xor_one_right
tff(fact_2080_bit_Oxor__one__left,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(A,A,uminus_uminus(A),one_one(A))),X) = aa(A,A,bit_ri4277139882892585799ns_not(A),X) ) ).

% bit.xor_one_left
tff(fact_2081_minus__not__numeral__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: num] : aa(A,A,uminus_uminus(A),aa(A,A,bit_ri4277139882892585799ns_not(A),aa(num,A,numeral_numeral(A),N))) = aa(num,A,numeral_numeral(A),inc(N)) ) ).

% minus_not_numeral_eq
tff(fact_2082_not__minus__numeral__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: num] : aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))) = neg_numeral_sub(A,N,one2) ) ).

% not_minus_numeral_eq
tff(fact_2083_push__bit__minus__one__eq__not__mask,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat] : bit_se4730199178511100633sh_bit(A,N,aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,bit_ri4277139882892585799ns_not(A),bit_se2239418461657761734s_mask(A,N)) ) ).

% push_bit_minus_one_eq_not_mask
tff(fact_2084_xor__minus__numerals_I2_J,axiom,
    ! [K: int,N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = aa(int,int,bit_ri4277139882892585799ns_not(int),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),neg_numeral_sub(int,N,one2))) ).

% xor_minus_numerals(2)
tff(fact_2085_xor__minus__numerals_I1_J,axiom,
    ! [N: num,K: int] : aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))),K) = aa(int,int,bit_ri4277139882892585799ns_not(int),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),neg_numeral_sub(int,N,one2)),K)) ).

% xor_minus_numerals(1)
tff(fact_2086_not__one__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ( aa(A,A,bit_ri4277139882892585799ns_not(A),one_one(A)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ) ).

% not_one_eq
tff(fact_2087_int__of__integer,axiom,
    ! [X: code_integer] : aa(set(int),$o,member(int,aa(code_integer,int,code_int_of_integer,X)),top_top(set(int))) ).

% int_of_integer
tff(fact_2088_int__of__integer__cases,axiom,
    ! [Y: int] :
      ( aa(set(int),$o,member(int,Y),top_top(set(int)))
     => ~ ! [X2: code_integer] : Y != aa(code_integer,int,code_int_of_integer,X2) ) ).

% int_of_integer_cases
tff(fact_2089_int__of__integer__induct,axiom,
    ! [Y: int,P: fun(int,$o)] :
      ( aa(set(int),$o,member(int,Y),top_top(set(int)))
     => ( ! [X2: code_integer] : aa(int,$o,P,aa(code_integer,int,code_int_of_integer,X2))
       => aa(int,$o,P,Y) ) ) ).

% int_of_integer_induct
tff(fact_2090_mod__sum__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [F: fun(B,A),A3: A,A4: set(B)] : modulo_modulo(A,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(A,fun(B,A),aTP_Lamp_cq(fun(B,A),fun(A,fun(B,A)),F),A3)),A4),A3) = modulo_modulo(A,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F),A4),A3) ) ).

% mod_sum_eq
tff(fact_2091_type__definition__integer,axiom,
    type_definition(code_integer,int,code_int_of_integer,code_integer_of_int,top_top(set(int))) ).

% type_definition_integer
tff(fact_2092_integer__of__int__inverse,axiom,
    ! [Y: int] :
      ( aa(set(int),$o,member(int,Y),top_top(set(int)))
     => ( aa(code_integer,int,code_int_of_integer,aa(int,code_integer,code_integer_of_int,Y)) = Y ) ) ).

% integer_of_int_inverse
tff(fact_2093_is__singletonI_H,axiom,
    ! [A: $tType,A4: set(A)] :
      ( ( A4 != bot_bot(set(A)) )
     => ( ! [X2: A,Y2: A] :
            ( aa(set(A),$o,member(A,X2),A4)
           => ( aa(set(A),$o,member(A,Y2),A4)
             => ( X2 = Y2 ) ) )
       => is_singleton(A,A4) ) ) ).

% is_singletonI'
tff(fact_2094_minus__eq__not__plus__1,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A3: A] : aa(A,A,uminus_uminus(A),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,bit_ri4277139882892585799ns_not(A),A3)),one_one(A)) ) ).

% minus_eq_not_plus_1
tff(fact_2095_minus__eq__not__minus__1,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A3: A] : aa(A,A,uminus_uminus(A),A3) = aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),one_one(A))) ) ).

% minus_eq_not_minus_1
tff(fact_2096_not__eq__complement,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A3: A] : aa(A,A,bit_ri4277139882892585799ns_not(A),A3) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),A3)),one_one(A)) ) ).

% not_eq_complement
tff(fact_2097_not__int__def,axiom,
    ! [K: int] : aa(int,int,bit_ri4277139882892585799ns_not(int),K) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,uminus_uminus(int),K)),one_one(int)) ).

% not_int_def
tff(fact_2098_and__not__numerals_I1_J,axiom,
    aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),one_one(int)),aa(int,int,bit_ri4277139882892585799ns_not(int),one_one(int))) = zero_zero(int) ).

% and_not_numerals(1)
tff(fact_2099_sum__power__add,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [X: A,M: nat,I4: 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_cr(A,fun(nat,fun(nat,A)),X),M)),I4) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),M)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X)),I4)) ) ).

% sum_power_add
tff(fact_2100_minus__numeral__inc__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: num] : aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),inc(N))) = aa(A,A,bit_ri4277139882892585799ns_not(A),aa(num,A,numeral_numeral(A),N)) ) ).

% minus_numeral_inc_eq
tff(fact_2101_unset__bit__int__def,axiom,
    ! [N: nat,K: int] : bit_se2638667681897837118et_bit(int,N,K) = aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),aa(int,int,bit_ri4277139882892585799ns_not(int),bit_se4730199178511100633sh_bit(int,N,one_one(int)))) ).

% unset_bit_int_def
tff(fact_2102_divmod__abs__code_I6_J,axiom,
    ! [J: code_integer] : code_divmod_abs(zero_zero(code_integer),J) = aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),zero_zero(code_integer)),zero_zero(code_integer)) ).

% divmod_abs_code(6)
tff(fact_2103_divmod__abs__code_I5_J,axiom,
    ! [J: code_integer] : code_divmod_abs(J,zero_zero(code_integer)) = aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),zero_zero(code_integer)),aa(code_integer,code_integer,abs_abs(code_integer),J)) ).

% divmod_abs_code(5)
tff(fact_2104_and__not__numerals_I2_J,axiom,
    ! [N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),one_one(int)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,N)))) = one_one(int) ).

% and_not_numerals(2)
tff(fact_2105_and__not__numerals_I4_J,axiom,
    ! [M: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,M))),aa(int,int,bit_ri4277139882892585799ns_not(int),one_one(int))) = aa(num,int,numeral_numeral(int),aa(num,num,bit0,M)) ).

% and_not_numerals(4)
tff(fact_2106_not__numeral__Bit0__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: num] : aa(A,A,bit_ri4277139882892585799ns_not(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,N))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,N))) ) ).

% not_numeral_Bit0_eq
tff(fact_2107_minus__numeral__eq__not__sub__one,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: num] : aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N)) = aa(A,A,bit_ri4277139882892585799ns_not(A),neg_numeral_sub(A,N,one2)) ) ).

% minus_numeral_eq_not_sub_one
tff(fact_2108_not__numeral__BitM__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: num] : aa(A,A,bit_ri4277139882892585799ns_not(A),aa(num,A,numeral_numeral(A),bitM(N))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,N))) ) ).

% not_numeral_BitM_eq
tff(fact_2109_unset__bit__eq__and__not,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat,A3: A] : bit_se2638667681897837118et_bit(A,N,A3) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),aa(A,A,bit_ri4277139882892585799ns_not(A),bit_se4730199178511100633sh_bit(A,N,one_one(A)))) ) ).

% unset_bit_eq_and_not
tff(fact_2110_is__singletonE,axiom,
    ! [A: $tType,A4: set(A)] :
      ( is_singleton(A,A4)
     => ~ ! [X2: A] : A4 != aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X2),bot_bot(set(A))) ) ).

% is_singletonE
tff(fact_2111_is__singleton__def,axiom,
    ! [A: $tType,A4: set(A)] :
      ( is_singleton(A,A4)
    <=> ? [X4: A] : A4 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X4),bot_bot(set(A))) ) ).

% is_singleton_def
tff(fact_2112_and__not__numerals_I7_J,axiom,
    ! [M: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,M))),aa(int,int,bit_ri4277139882892585799ns_not(int),one_one(int))) = aa(num,int,numeral_numeral(int),aa(num,num,bit0,M)) ).

% and_not_numerals(7)
tff(fact_2113_and__not__numerals_I3_J,axiom,
    ! [N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),one_one(int)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N)))) = zero_zero(int) ).

% and_not_numerals(3)
tff(fact_2114_integer__of__num__triv_I1_J,axiom,
    code_integer_of_num(one2) = one_one(code_integer) ).

% integer_of_num_triv(1)
tff(fact_2115_integer__of__num_I2_J,axiom,
    ! [N: num] :
      code_integer_of_num(aa(num,num,bit0,N)) = $let(
        k: code_integer,
        k:= code_integer_of_num(N),
        aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),k),k) ) ).

% integer_of_num(2)
tff(fact_2116_minus__exp__eq__not__mask,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat] : aa(A,A,uminus_uminus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)) = aa(A,A,bit_ri4277139882892585799ns_not(A),bit_se2239418461657761734s_mask(A,N)) ) ).

% minus_exp_eq_not_mask
tff(fact_2117_divmod__abs__def,axiom,
    ! [K: code_integer,L: code_integer] : code_divmod_abs(K,L) = aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),divide_divide(code_integer),aa(code_integer,code_integer,abs_abs(code_integer),K)),aa(code_integer,code_integer,abs_abs(code_integer),L))),modulo_modulo(code_integer,aa(code_integer,code_integer,abs_abs(code_integer),K),aa(code_integer,code_integer,abs_abs(code_integer),L))) ).

% divmod_abs_def
tff(fact_2118_mask__eq__sum__exp,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [N: nat] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)),one_one(A)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_bk(nat,fun(nat,$o)),N))) ) ).

% mask_eq_sum_exp
tff(fact_2119_and__not__numerals_I8_J,axiom,
    ! [M: 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,M))),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),aa(num,num,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),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),N))))) ).

% and_not_numerals(8)
tff(fact_2120_sum__abs__ge__zero,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [F: fun(B,A),A4: set(B)] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aTP_Lamp_cs(fun(B,A),fun(B,A),F)),A4)) ) ).

% sum_abs_ge_zero
tff(fact_2121_sum__abs,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [F: fun(B,A),A4: set(B)] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F),A4))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aTP_Lamp_cs(fun(B,A),fun(B,A),F)),A4)) ) ).

% sum_abs
tff(fact_2122_sum_Oempty,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(B,A)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),bot_bot(set(B))) = zero_zero(A) ) ).

% sum.empty
tff(fact_2123_convex__sum__bound__le,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_idom(B)
     => ! [I4: set(A),X: fun(A,B),A3: fun(A,B),B2: B,Delta: B] :
          ( ! [I2: A] :
              ( aa(set(A),$o,member(A,I2),I4)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,X,I2)) )
         => ( ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),X),I4) = one_one(B) )
           => ( ! [I2: A] :
                  ( aa(set(A),$o,member(A,I2),I4)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,abs_abs(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,A3,I2)),B2))),Delta) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,abs_abs(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aa(fun(A,B),fun(A,B),aTP_Lamp_ct(fun(A,B),fun(fun(A,B),fun(A,B)),X),A3)),I4)),B2))),Delta) ) ) ) ) ).

% convex_sum_bound_le
tff(fact_2124_abs__sum__abs,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [F: fun(B,A),A4: set(B)] : aa(A,A,abs_abs(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aTP_Lamp_cs(fun(B,A),fun(B,A),F)),A4)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aTP_Lamp_cs(fun(B,A),fun(B,A),F)),A4) ) ).

% abs_sum_abs
tff(fact_2125_sum_Oneutral__const,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [A4: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aTP_Lamp_cu(B,A)),A4) = zero_zero(A) ) ).

% sum.neutral_const
tff(fact_2126_sum__diff1__nat,axiom,
    ! [A: $tType,F: fun(A,nat),A4: set(A),A3: A] :
      aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A))))) = $ite(aa(set(A),$o,member(A,A3),A4),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),F),A4)),aa(A,nat,F,A3)),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F),A4)) ).

% sum_diff1_nat
tff(fact_2127_sum_Oswap,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(B,fun(C,A)),B3: set(C),A4: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(set(C),fun(B,A),aTP_Lamp_cv(fun(B,fun(C,A)),fun(set(C),fun(B,A)),G),B3)),A4) = aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7311177749621191930dd_sum(C,A),aa(set(B),fun(C,A),aTP_Lamp_cx(fun(B,fun(C,A)),fun(set(B),fun(C,A)),G),A4)),B3) ) ).

% sum.swap
tff(fact_2128_sum__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [K5: set(A),F: fun(A,B),G: fun(A,B)] :
          ( ! [I2: A] :
              ( aa(set(A),$o,member(A,I2),K5)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,I2)),aa(A,B,G,I2)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F),K5)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),K5)) ) ) ).

% sum_mono
tff(fact_2129_sum_Odistrib,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(B,A),H2: fun(B,A),A4: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(fun(B,A),fun(B,A),aTP_Lamp_cy(fun(B,A),fun(fun(B,A),fun(B,A)),G),H2)),A4) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),A4)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),H2),A4)) ) ).

% sum.distrib
tff(fact_2130_sum__product,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( semiring_0(A)
     => ! [F: fun(B,A),A4: set(B),G: fun(C,A),B3: set(C)] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F),A4)),aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7311177749621191930dd_sum(C,A),G),B3)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(set(C),fun(B,A),aa(fun(C,A),fun(set(C),fun(B,A)),aTP_Lamp_da(fun(B,A),fun(fun(C,A),fun(set(C),fun(B,A))),F),G),B3)),A4) ) ).

% sum_product
tff(fact_2131_sum__distrib__right,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_0(A)
     => ! [F: fun(B,A),A4: set(B),R2: 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),F),A4)),R2) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(A,fun(B,A),aTP_Lamp_db(fun(B,A),fun(A,fun(B,A)),F),R2)),A4) ) ).

% sum_distrib_right
tff(fact_2132_sum__distrib__left,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_0(A)
     => ! [R2: A,F: fun(B,A),A4: set(B)] : aa(A,A,aa(A,fun(A,A),times_times(A),R2),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F),A4)) = 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_dc(A,fun(fun(B,A),fun(B,A)),R2),F)),A4) ) ).

% sum_distrib_left
tff(fact_2133_sum__subtractf,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_add(A)
     => ! [F: fun(B,A),G: fun(B,A),A4: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(fun(B,A),fun(B,A),aTP_Lamp_dd(fun(B,A),fun(fun(B,A),fun(B,A)),F),G)),A4) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F),A4)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),A4)) ) ).

% sum_subtractf
tff(fact_2134_sum__negf,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_add(A)
     => ! [F: fun(B,A),A4: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aTP_Lamp_de(fun(B,A),fun(B,A),F)),A4) = aa(A,A,uminus_uminus(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F),A4)) ) ).

% sum_negf
tff(fact_2135_sum__divide__distrib,axiom,
    ! [A: $tType,B: $tType] :
      ( field(A)
     => ! [F: fun(B,A),A4: set(B),R2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F),A4)),R2) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(A,fun(B,A),aTP_Lamp_df(fun(B,A),fun(A,fun(B,A)),F),R2)),A4) ) ).

% sum_divide_distrib
tff(fact_2136_sum__subtractf__nat,axiom,
    ! [A: $tType,A4: set(A),G: fun(A,nat),F: fun(A,nat)] :
      ( ! [X2: A] :
          ( aa(set(A),$o,member(A,X2),A4)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,G,X2)),aa(A,nat,F,X2)) )
     => ( 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_dg(fun(A,nat),fun(fun(A,nat),fun(A,nat)),G),F)),A4) = 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),F),A4)),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),G),A4)) ) ) ).

% sum_subtractf_nat
tff(fact_2137_divmod__integer__code,axiom,
    ! [K: code_integer,L: code_integer] :
      code_divmod_integer(K,L) = $ite(
        K = zero_zero(code_integer),
        aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),zero_zero(code_integer)),zero_zero(code_integer)),
        $ite(
          aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),zero_zero(code_integer)),L),
          $ite(aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),zero_zero(code_integer)),K),code_divmod_abs(K,L),aa(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer),aa(fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)),product_case_prod(code_integer,code_integer,product_prod(code_integer,code_integer)),aTP_Lamp_dh(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),L)),code_divmod_abs(K,L))),
          $ite(
            L = zero_zero(code_integer),
            aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),zero_zero(code_integer)),K),
            aa(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer),aa(fun(code_integer,code_integer),fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)),product_apsnd(code_integer,code_integer,code_integer),uminus_uminus(code_integer)),
              $ite(aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),K),zero_zero(code_integer)),code_divmod_abs(K,L),aa(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer),aa(fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)),product_case_prod(code_integer,code_integer,product_prod(code_integer,code_integer)),aTP_Lamp_di(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),L)),code_divmod_abs(K,L)))) ) ) ) ).

% divmod_integer_code
tff(fact_2138_Sum__Icc__nat,axiom,
    ! [M: nat,N: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_dj(nat,nat)),set_or1337092689740270186AtMost(nat,M,N)) = aa(nat,nat,aa(nat,fun(nat,nat),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),M),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),one_one(nat))))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ).

% Sum_Icc_nat
tff(fact_2139_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_dk(nat,fun(nat,fun(nat,nat)),A3),D3)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,N)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),A3)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),D3)))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ).

% arith_series_nat
tff(fact_2140_Sum__Icc__int,axiom,
    ! [M: int,N: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),M),N)
     => ( aa(set(int),int,aa(fun(int,int),fun(set(int),int),groups7311177749621191930dd_sum(int,int),aTP_Lamp_dl(int,int)),set_or1337092689740270186AtMost(int,M,N)) = aa(int,int,aa(int,fun(int,int),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),M),aa(int,int,aa(int,fun(int,int),minus_minus(int),M),one_one(int))))),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))) ) ) ).

% Sum_Icc_int
tff(fact_2141_Sum__Ico__nat,axiom,
    ! [M: nat,N: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_dj(nat,nat)),set_or7035219750837199246ssThan(nat,M,N)) = aa(nat,nat,aa(nat,fun(nat,nat),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),M),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),one_one(nat))))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ).

% Sum_Ico_nat
tff(fact_2142_sum__power2,axiom,
    ! [K: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),K)),one_one(nat)) ).

% sum_power2
tff(fact_2143_gauss__sum__nat,axiom,
    ! [N: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_dj(nat,nat)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,suc,N))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ).

% gauss_sum_nat
tff(fact_2144_apsnd__conv,axiom,
    ! [A: $tType,B: $tType,C: $tType,F: 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),F),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,F,Y)) ).

% apsnd_conv
tff(fact_2145_atLeastatMost__empty__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B2: A] :
          ( ( set_or1337092689740270186AtMost(A,A3,B2) = bot_bot(set(A)) )
        <=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2) ) ) ).

% atLeastatMost_empty_iff
tff(fact_2146_atLeastatMost__empty__iff2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B2: A] :
          ( ( bot_bot(set(A)) = set_or1337092689740270186AtMost(A,A3,B2) )
        <=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2) ) ) ).

% atLeastatMost_empty_iff2
tff(fact_2147_atLeastatMost__empty,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3)
         => ( set_or1337092689740270186AtMost(A,A3,B2) = bot_bot(set(A)) ) ) ) ).

% atLeastatMost_empty
tff(fact_2148_atLeastLessThan__empty,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3)
         => ( set_or7035219750837199246ssThan(A,A3,B2) = bot_bot(set(A)) ) ) ) ).

% atLeastLessThan_empty
tff(fact_2149_atLeastLessThan__empty__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B2: A] :
          ( ( set_or7035219750837199246ssThan(A,A3,B2) = bot_bot(set(A)) )
        <=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2) ) ) ).

% atLeastLessThan_empty_iff
tff(fact_2150_atLeastLessThan__empty__iff2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B2: A] :
          ( ( bot_bot(set(A)) = set_or7035219750837199246ssThan(A,A3,B2) )
        <=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2) ) ) ).

% atLeastLessThan_empty_iff2
tff(fact_2151_atLeastAtMost__singleton,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A] : set_or1337092689740270186AtMost(A,A3,A3) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A))) ) ).

% atLeastAtMost_singleton
tff(fact_2152_atLeastAtMost__singleton__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B2: A,C2: A] :
          ( ( set_or1337092689740270186AtMost(A,A3,B2) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),C2),bot_bot(set(A))) )
        <=> ( ( A3 = B2 )
            & ( B2 = C2 ) ) ) ) ).

% atLeastAtMost_singleton_iff
tff(fact_2153_atLeastLessThan__singleton,axiom,
    ! [M: nat] : set_or7035219750837199246ssThan(nat,M,aa(nat,nat,suc,M)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),M),bot_bot(set(nat))) ).

% atLeastLessThan_singleton
tff(fact_2154_ivl__disj__un__two_I7_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),M)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),M),U)
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or7035219750837199246ssThan(A,L,M)),set_or1337092689740270186AtMost(A,M,U)) = set_or1337092689740270186AtMost(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(7)
tff(fact_2155_ivl__disj__int__two_I7_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,M: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or7035219750837199246ssThan(A,L,M)),set_or1337092689740270186AtMost(A,M,U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_two(7)
tff(fact_2156_not__UNIV__eq__Icc,axiom,
    ! [A: $tType] :
      ( no_top(A)
     => ! [L3: A,H3: A] : top_top(set(A)) != set_or1337092689740270186AtMost(A,L3,H3) ) ).

% not_UNIV_eq_Icc
tff(fact_2157_ivl__disj__un__two__touch_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),M)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),M),U)
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or1337092689740270186AtMost(A,L,M)),set_or7035219750837199246ssThan(A,M,U)) = set_or7035219750837199246ssThan(A,L,U) ) ) ) ) ).

% ivl_disj_un_two_touch(2)
tff(fact_2158_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),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A)))) ) ).

% atLeastLessThan_eq_atLeastAtMost_diff
tff(fact_2159_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_dm(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_do(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),A3),N)),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)) ) ).

% sum.nested_swap
tff(fact_2160_ivl__disj__un__singleton_I6_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),U)
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or7035219750837199246ssThan(A,L,U)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),U),bot_bot(set(A)))) = set_or1337092689740270186AtMost(A,L,U) ) ) ) ).

% ivl_disj_un_singleton(6)
tff(fact_2161_ivl__disj__un__two__touch_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),M)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),M),U)
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or1337092689740270186AtMost(A,L,M)),set_or1337092689740270186AtMost(A,M,U)) = set_or1337092689740270186AtMost(A,L,U) ) ) ) ) ).

% ivl_disj_un_two_touch(4)
tff(fact_2162_atLeastAtMost__singleton_H,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B2: A] :
          ( ( A3 = B2 )
         => ( set_or1337092689740270186AtMost(A,A3,B2) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A))) ) ) ) ).

% atLeastAtMost_singleton'
tff(fact_2163_not__UNIV__le__Icc,axiom,
    ! [A: $tType] :
      ( no_top(A)
     => ! [L: A,H2: A] : ~ aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),top_top(set(A))),set_or1337092689740270186AtMost(A,L,H2)) ) ).

% not_UNIV_le_Icc
tff(fact_2164_ivl__disj__un__two_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),M)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),M),U)
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or7035219750837199246ssThan(A,L,M)),set_or7035219750837199246ssThan(A,M,U)) = set_or7035219750837199246ssThan(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(3)
tff(fact_2165_ivl__disj__int__two_I3_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,M: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or7035219750837199246ssThan(A,L,M)),set_or7035219750837199246ssThan(A,M,U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_two(3)
tff(fact_2166_sum_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),N: nat,M: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,N,M)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_dp(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),N),M)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,N),M)) ) ).

% sum.atLeastLessThan_rev_at_least_Suc_atMost
tff(fact_2167_atLeastLessThan0,axiom,
    ! [M: nat] : set_or7035219750837199246ssThan(nat,M,zero_zero(nat)) = bot_bot(set(nat)) ).

% atLeastLessThan0
tff(fact_2168_sum_Oshift__bounds__Suc__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_dq(fun(nat,A),fun(nat,A),G)),set_or7035219750837199246ssThan(nat,M,N)) ) ).

% sum.shift_bounds_Suc_ivl
tff(fact_2169_sum_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_dq(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,M,N)) ) ).

% sum.shift_bounds_cl_Suc_ivl
tff(fact_2170_sum_Oshift__bounds__nat__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,K: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aTP_Lamp_dr(fun(nat,A),fun(nat,fun(nat,A)),G),K)),set_or7035219750837199246ssThan(nat,M,N)) ) ).

% sum.shift_bounds_nat_ivl
tff(fact_2171_sum_Oshift__bounds__cl__nat__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,K: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aTP_Lamp_dr(fun(nat,A),fun(nat,fun(nat,A)),G),K)),set_or1337092689740270186AtMost(nat,M,N)) ) ).

% sum.shift_bounds_cl_nat_ivl
tff(fact_2172_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_2173_aset_I2_J,axiom,
    ! [D4: int,A4: set(int),P: fun(int,$o),Q: fun(int,$o)] :
      ( ! [X2: int] :
          ( ! [Xa3: int] :
              ( aa(set(int),$o,member(int,Xa3),set_or1337092689740270186AtMost(int,one_one(int),D4))
             => ! [Xb2: int] :
                  ( aa(set(int),$o,member(int,Xb2),A4)
                 => ( X2 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb2),Xa3) ) ) )
         => ( aa(int,$o,P,X2)
           => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X2),D4)) ) )
     => ( ! [X2: int] :
            ( ! [Xa3: int] :
                ( aa(set(int),$o,member(int,Xa3),set_or1337092689740270186AtMost(int,one_one(int),D4))
               => ! [Xb2: int] :
                    ( aa(set(int),$o,member(int,Xb2),A4)
                   => ( X2 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb2),Xa3) ) ) )
           => ( aa(int,$o,Q,X2)
             => aa(int,$o,Q,aa(int,int,aa(int,fun(int,int),plus_plus(int),X2),D4)) ) )
       => ! [X3: int] :
            ( ! [Xa4: int] :
                ( aa(set(int),$o,member(int,Xa4),set_or1337092689740270186AtMost(int,one_one(int),D4))
               => ! [Xb3: int] :
                    ( aa(set(int),$o,member(int,Xb3),A4)
                   => ( X3 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa4) ) ) )
           => ( ( aa(int,$o,P,X3)
                | aa(int,$o,Q,X3) )
             => ( aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D4))
                | aa(int,$o,Q,aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D4)) ) ) ) ) ) ).

% aset(2)
tff(fact_2174_aset_I1_J,axiom,
    ! [D4: int,A4: set(int),P: fun(int,$o),Q: fun(int,$o)] :
      ( ! [X2: int] :
          ( ! [Xa3: int] :
              ( aa(set(int),$o,member(int,Xa3),set_or1337092689740270186AtMost(int,one_one(int),D4))
             => ! [Xb2: int] :
                  ( aa(set(int),$o,member(int,Xb2),A4)
                 => ( X2 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb2),Xa3) ) ) )
         => ( aa(int,$o,P,X2)
           => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X2),D4)) ) )
     => ( ! [X2: int] :
            ( ! [Xa3: int] :
                ( aa(set(int),$o,member(int,Xa3),set_or1337092689740270186AtMost(int,one_one(int),D4))
               => ! [Xb2: int] :
                    ( aa(set(int),$o,member(int,Xb2),A4)
                   => ( X2 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb2),Xa3) ) ) )
           => ( aa(int,$o,Q,X2)
             => aa(int,$o,Q,aa(int,int,aa(int,fun(int,int),plus_plus(int),X2),D4)) ) )
       => ! [X3: int] :
            ( ! [Xa4: int] :
                ( aa(set(int),$o,member(int,Xa4),set_or1337092689740270186AtMost(int,one_one(int),D4))
               => ! [Xb3: int] :
                    ( aa(set(int),$o,member(int,Xb3),A4)
                   => ( X3 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa4) ) ) )
           => ( ( aa(int,$o,P,X3)
                & aa(int,$o,Q,X3) )
             => ( aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D4))
                & aa(int,$o,Q,aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D4)) ) ) ) ) ) ).

% aset(1)
tff(fact_2175_bset_I2_J,axiom,
    ! [D4: int,B3: set(int),P: fun(int,$o),Q: fun(int,$o)] :
      ( ! [X2: int] :
          ( ! [Xa3: int] :
              ( aa(set(int),$o,member(int,Xa3),set_or1337092689740270186AtMost(int,one_one(int),D4))
             => ! [Xb2: int] :
                  ( aa(set(int),$o,member(int,Xb2),B3)
                 => ( X2 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa3) ) ) )
         => ( aa(int,$o,P,X2)
           => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),X2),D4)) ) )
     => ( ! [X2: int] :
            ( ! [Xa3: int] :
                ( aa(set(int),$o,member(int,Xa3),set_or1337092689740270186AtMost(int,one_one(int),D4))
               => ! [Xb2: int] :
                    ( aa(set(int),$o,member(int,Xb2),B3)
                   => ( X2 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa3) ) ) )
           => ( aa(int,$o,Q,X2)
             => aa(int,$o,Q,aa(int,int,aa(int,fun(int,int),minus_minus(int),X2),D4)) ) )
       => ! [X3: int] :
            ( ! [Xa4: int] :
                ( aa(set(int),$o,member(int,Xa4),set_or1337092689740270186AtMost(int,one_one(int),D4))
               => ! [Xb3: int] :
                    ( aa(set(int),$o,member(int,Xb3),B3)
                   => ( X3 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa4) ) ) )
           => ( ( aa(int,$o,P,X3)
                | aa(int,$o,Q,X3) )
             => ( aa(int,$o,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),D4))
                | aa(int,$o,Q,aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),D4)) ) ) ) ) ) ).

% bset(2)
tff(fact_2176_bset_I1_J,axiom,
    ! [D4: int,B3: set(int),P: fun(int,$o),Q: fun(int,$o)] :
      ( ! [X2: int] :
          ( ! [Xa3: int] :
              ( aa(set(int),$o,member(int,Xa3),set_or1337092689740270186AtMost(int,one_one(int),D4))
             => ! [Xb2: int] :
                  ( aa(set(int),$o,member(int,Xb2),B3)
                 => ( X2 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa3) ) ) )
         => ( aa(int,$o,P,X2)
           => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),X2),D4)) ) )
     => ( ! [X2: int] :
            ( ! [Xa3: int] :
                ( aa(set(int),$o,member(int,Xa3),set_or1337092689740270186AtMost(int,one_one(int),D4))
               => ! [Xb2: int] :
                    ( aa(set(int),$o,member(int,Xb2),B3)
                   => ( X2 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa3) ) ) )
           => ( aa(int,$o,Q,X2)
             => aa(int,$o,Q,aa(int,int,aa(int,fun(int,int),minus_minus(int),X2),D4)) ) )
       => ! [X3: int] :
            ( ! [Xa4: int] :
                ( aa(set(int),$o,member(int,Xa4),set_or1337092689740270186AtMost(int,one_one(int),D4))
               => ! [Xb3: int] :
                    ( aa(set(int),$o,member(int,Xb3),B3)
                   => ( X3 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa4) ) ) )
           => ( ( aa(int,$o,P,X3)
                & aa(int,$o,Q,X3) )
             => ( aa(int,$o,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),D4))
                & aa(int,$o,Q,aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),D4)) ) ) ) ) ) ).

% bset(1)
tff(fact_2177_atLeastLessThanSuc,axiom,
    ! [M: nat,N: nat] :
      set_or7035219750837199246ssThan(nat,M,aa(nat,nat,suc,N)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),N),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),N),set_or7035219750837199246ssThan(nat,M,N)),bot_bot(set(nat))) ).

% atLeastLessThanSuc
tff(fact_2178_atLeastLessThan__add__Un,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( set_or7035219750837199246ssThan(nat,I,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K)) = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),set_or7035219750837199246ssThan(nat,I,J)),set_or7035219750837199246ssThan(nat,J,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K))) ) ) ).

% atLeastLessThan_add_Un
tff(fact_2179_sum_OatLeastAtMost__rev,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),N: nat,M: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,N,M)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_dp(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),N),M)),set_or1337092689740270186AtMost(nat,N,M)) ) ).

% sum.atLeastAtMost_rev
tff(fact_2180_aset_I10_J,axiom,
    ! [D3: int,D4: int,A4: set(int),T3: int] :
      ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),D3),D4)
     => ! [X3: int] :
          ( ! [Xa4: int] :
              ( aa(set(int),$o,member(int,Xa4),set_or1337092689740270186AtMost(int,one_one(int),D4))
             => ! [Xb3: int] :
                  ( aa(set(int),$o,member(int,Xb3),A4)
                 => ( X3 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa4) ) ) )
         => ( ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),D3),aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),T3))
           => ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),D3),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D4)),T3)) ) ) ) ).

% aset(10)
tff(fact_2181_aset_I9_J,axiom,
    ! [D3: int,D4: int,A4: set(int),T3: int] :
      ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),D3),D4)
     => ! [X3: int] :
          ( ! [Xa4: int] :
              ( aa(set(int),$o,member(int,Xa4),set_or1337092689740270186AtMost(int,one_one(int),D4))
             => ! [Xb3: int] :
                  ( aa(set(int),$o,member(int,Xb3),A4)
                 => ( X3 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa4) ) ) )
         => ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),D3),aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),T3))
           => aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),D3),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D4)),T3)) ) ) ) ).

% aset(9)
tff(fact_2182_bset_I10_J,axiom,
    ! [D3: int,D4: int,B3: set(int),T3: int] :
      ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),D3),D4)
     => ! [X3: int] :
          ( ! [Xa4: int] :
              ( aa(set(int),$o,member(int,Xa4),set_or1337092689740270186AtMost(int,one_one(int),D4))
             => ! [Xb3: int] :
                  ( aa(set(int),$o,member(int,Xb3),B3)
                 => ( X3 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa4) ) ) )
         => ( ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),D3),aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),T3))
           => ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),D3),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),D4)),T3)) ) ) ) ).

% bset(10)
tff(fact_2183_bset_I9_J,axiom,
    ! [D3: int,D4: int,B3: set(int),T3: int] :
      ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),D3),D4)
     => ! [X3: int] :
          ( ! [Xa4: int] :
              ( aa(set(int),$o,member(int,Xa4),set_or1337092689740270186AtMost(int,one_one(int),D4))
             => ! [Xb3: int] :
                  ( aa(set(int),$o,member(int,Xb3),B3)
                 => ( X3 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa4) ) ) )
         => ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),D3),aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),T3))
           => aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),D3),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),D4)),T3)) ) ) ) ).

% bset(9)
tff(fact_2184_atLeastAtMostPlus1__int__conv,axiom,
    ! [M: int,N: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),M),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),N))
     => ( set_or1337092689740270186AtMost(int,M,aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),N)) = aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),N)),set_or1337092689740270186AtMost(int,M,N)) ) ) ).

% atLeastAtMostPlus1_int_conv
tff(fact_2185_atLeastLessThan__nat__numeral,axiom,
    ! [M: nat,K: num] :
      set_or7035219750837199246ssThan(nat,M,aa(num,nat,numeral_numeral(nat),K)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),pred_numeral(K)),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),pred_numeral(K)),set_or7035219750837199246ssThan(nat,M,pred_numeral(K))),bot_bot(set(nat))) ).

% atLeastLessThan_nat_numeral
tff(fact_2186_sum__Suc__diff_H,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [M: nat,N: nat,F: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),N)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_ds(fun(nat,A),fun(nat,A),F)),set_or7035219750837199246ssThan(nat,M,N)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F,N)),aa(nat,A,F,M)) ) ) ) ).

% sum_Suc_diff'
tff(fact_2187_sum_OSuc__reindex__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,N: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),N)
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M,N))),aa(nat,A,G,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,M)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_dq(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,M,N))) ) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_2188_sum__Suc__diff,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [M: nat,N: nat,F: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),aa(nat,nat,suc,N))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_ds(fun(nat,A),fun(nat,A),F)),set_or1337092689740270186AtMost(nat,M,N)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F,aa(nat,nat,suc,N))),aa(nat,A,F,M)) ) ) ) ).

% sum_Suc_diff
tff(fact_2189_sum_OatLeastLessThan__rev,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),N: nat,M: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,N,M)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_dt(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),N),M)),set_or7035219750837199246ssThan(nat,N,M)) ) ).

% sum.atLeastLessThan_rev
tff(fact_2190_sum__atLeastAtMost__code,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [F: fun(nat,A),A3: nat,B2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F),set_or1337092689740270186AtMost(nat,A3,B2)) = set_fo6178422350223883121st_nat(A,aTP_Lamp_du(fun(nat,A),fun(nat,fun(A,A)),F),A3,B2,zero_zero(A)) ) ).

% sum_atLeastAtMost_code
tff(fact_2191_periodic__finite__ex,axiom,
    ! [D3: int,P: fun(int,$o)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D3)
     => ( ! [X2: int,K2: int] :
            ( aa(int,$o,P,X2)
          <=> aa(int,$o,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),X2),aa(int,int,aa(int,fun(int,int),times_times(int),K2),D3))) )
       => ( ? [X_12: int] : aa(int,$o,P,X_12)
        <=> ? [X4: int] :
              ( aa(set(int),$o,member(int,X4),set_or1337092689740270186AtMost(int,one_one(int),D3))
              & aa(int,$o,P,X4) ) ) ) ) ).

% periodic_finite_ex
tff(fact_2192_aset_I7_J,axiom,
    ! [D4: int,A4: set(int),T3: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D4)
     => ! [X3: int] :
          ( ! [Xa4: int] :
              ( aa(set(int),$o,member(int,Xa4),set_or1337092689740270186AtMost(int,one_one(int),D4))
             => ! [Xb3: int] :
                  ( aa(set(int),$o,member(int,Xb3),A4)
                 => ( X3 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa4) ) ) )
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),T3),X3)
           => aa(int,$o,aa(int,fun(int,$o),ord_less(int),T3),aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D4)) ) ) ) ).

% aset(7)
tff(fact_2193_aset_I5_J,axiom,
    ! [D4: int,T3: int,A4: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D4)
     => ( aa(set(int),$o,member(int,T3),A4)
       => ! [X3: int] :
            ( ! [Xa4: int] :
                ( aa(set(int),$o,member(int,Xa4),set_or1337092689740270186AtMost(int,one_one(int),D4))
               => ! [Xb3: int] :
                    ( aa(set(int),$o,member(int,Xb3),A4)
                   => ( X3 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa4) ) ) )
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),X3),T3)
             => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D4)),T3) ) ) ) ) ).

% aset(5)
tff(fact_2194_aset_I4_J,axiom,
    ! [D4: int,T3: int,A4: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D4)
     => ( aa(set(int),$o,member(int,T3),A4)
       => ! [X3: int] :
            ( ! [Xa4: int] :
                ( aa(set(int),$o,member(int,Xa4),set_or1337092689740270186AtMost(int,one_one(int),D4))
               => ! [Xb3: int] :
                    ( aa(set(int),$o,member(int,Xb3),A4)
                   => ( X3 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa4) ) ) )
           => ( ( X3 != T3 )
             => ( aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D4) != T3 ) ) ) ) ) ).

% aset(4)
tff(fact_2195_aset_I3_J,axiom,
    ! [D4: int,T3: int,A4: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D4)
     => ( aa(set(int),$o,member(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),T3),one_one(int))),A4)
       => ! [X3: int] :
            ( ! [Xa4: int] :
                ( aa(set(int),$o,member(int,Xa4),set_or1337092689740270186AtMost(int,one_one(int),D4))
               => ! [Xb3: int] :
                    ( aa(set(int),$o,member(int,Xb3),A4)
                   => ( X3 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa4) ) ) )
           => ( ( X3 = T3 )
             => ( aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D4) = T3 ) ) ) ) ) ).

% aset(3)
tff(fact_2196_bset_I7_J,axiom,
    ! [D4: int,T3: int,B3: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D4)
     => ( aa(set(int),$o,member(int,T3),B3)
       => ! [X3: int] :
            ( ! [Xa4: int] :
                ( aa(set(int),$o,member(int,Xa4),set_or1337092689740270186AtMost(int,one_one(int),D4))
               => ! [Xb3: int] :
                    ( aa(set(int),$o,member(int,Xb3),B3)
                   => ( X3 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa4) ) ) )
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),T3),X3)
             => aa(int,$o,aa(int,fun(int,$o),ord_less(int),T3),aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),D4)) ) ) ) ) ).

% bset(7)
tff(fact_2197_bset_I5_J,axiom,
    ! [D4: int,B3: set(int),T3: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D4)
     => ! [X3: int] :
          ( ! [Xa4: int] :
              ( aa(set(int),$o,member(int,Xa4),set_or1337092689740270186AtMost(int,one_one(int),D4))
             => ! [Xb3: int] :
                  ( aa(set(int),$o,member(int,Xb3),B3)
                 => ( X3 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa4) ) ) )
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),X3),T3)
           => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),D4)),T3) ) ) ) ).

% bset(5)
tff(fact_2198_bset_I4_J,axiom,
    ! [D4: int,T3: int,B3: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D4)
     => ( aa(set(int),$o,member(int,T3),B3)
       => ! [X3: int] :
            ( ! [Xa4: int] :
                ( aa(set(int),$o,member(int,Xa4),set_or1337092689740270186AtMost(int,one_one(int),D4))
               => ! [Xb3: int] :
                    ( aa(set(int),$o,member(int,Xb3),B3)
                   => ( X3 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa4) ) ) )
           => ( ( X3 != T3 )
             => ( aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),D4) != T3 ) ) ) ) ) ).

% bset(4)
tff(fact_2199_bset_I3_J,axiom,
    ! [D4: int,T3: int,B3: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D4)
     => ( aa(set(int),$o,member(int,aa(int,int,aa(int,fun(int,int),minus_minus(int),T3),one_one(int))),B3)
       => ! [X3: int] :
            ( ! [Xa4: int] :
                ( aa(set(int),$o,member(int,Xa4),set_or1337092689740270186AtMost(int,one_one(int),D4))
               => ! [Xb3: int] :
                    ( aa(set(int),$o,member(int,Xb3),B3)
                   => ( X3 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa4) ) ) )
           => ( ( X3 = T3 )
             => ( aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),D4) = T3 ) ) ) ) ) ).

% bset(3)
tff(fact_2200_sum_Oub__add__nat,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,N: nat,G: fun(nat,A),P3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),P3))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M,N))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),P3)))) ) ) ) ).

% sum.ub_add_nat
tff(fact_2201_simp__from__to,axiom,
    ! [I: int,J: int] :
      set_or1337092689740270186AtMost(int,I,J) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less(int),J),I),bot_bot(set(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),I),set_or1337092689740270186AtMost(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),I),one_one(int)),J))) ).

% simp_from_to
tff(fact_2202_aset_I8_J,axiom,
    ! [D4: int,A4: set(int),T3: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D4)
     => ! [X3: int] :
          ( ! [Xa4: int] :
              ( aa(set(int),$o,member(int,Xa4),set_or1337092689740270186AtMost(int,one_one(int),D4))
             => ! [Xb3: int] :
                  ( aa(set(int),$o,member(int,Xb3),A4)
                 => ( X3 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa4) ) ) )
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),T3),X3)
           => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),T3),aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D4)) ) ) ) ).

% aset(8)
tff(fact_2203_aset_I6_J,axiom,
    ! [D4: int,T3: int,A4: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D4)
     => ( aa(set(int),$o,member(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),T3),one_one(int))),A4)
       => ! [X3: int] :
            ( ! [Xa4: int] :
                ( aa(set(int),$o,member(int,Xa4),set_or1337092689740270186AtMost(int,one_one(int),D4))
               => ! [Xb3: int] :
                    ( aa(set(int),$o,member(int,Xb3),A4)
                   => ( X3 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa4) ) ) )
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),X3),T3)
             => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D4)),T3) ) ) ) ) ).

% aset(6)
tff(fact_2204_bset_I8_J,axiom,
    ! [D4: int,T3: int,B3: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D4)
     => ( aa(set(int),$o,member(int,aa(int,int,aa(int,fun(int,int),minus_minus(int),T3),one_one(int))),B3)
       => ! [X3: int] :
            ( ! [Xa4: int] :
                ( aa(set(int),$o,member(int,Xa4),set_or1337092689740270186AtMost(int,one_one(int),D4))
               => ! [Xb3: int] :
                    ( aa(set(int),$o,member(int,Xb3),B3)
                   => ( X3 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa4) ) ) )
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),T3),X3)
             => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),T3),aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),D4)) ) ) ) ) ).

% bset(8)
tff(fact_2205_bset_I6_J,axiom,
    ! [D4: int,B3: set(int),T3: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D4)
     => ! [X3: int] :
          ( ! [Xa4: int] :
              ( aa(set(int),$o,member(int,Xa4),set_or1337092689740270186AtMost(int,one_one(int),D4))
             => ! [Xb3: int] :
                  ( aa(set(int),$o,member(int,Xb3),B3)
                 => ( X3 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa4) ) ) )
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),X3),T3)
           => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),D4)),T3) ) ) ) ).

% bset(6)
tff(fact_2206_cpmi,axiom,
    ! [D4: int,P: fun(int,$o),P4: fun(int,$o),B3: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D4)
     => ( ? [Z6: int] :
          ! [X2: int] :
            ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),X2),Z6)
           => ( aa(int,$o,P,X2)
            <=> aa(int,$o,P4,X2) ) )
       => ( ! [X2: int] :
              ( ! [Xa3: int] :
                  ( aa(set(int),$o,member(int,Xa3),set_or1337092689740270186AtMost(int,one_one(int),D4))
                 => ! [Xb2: int] :
                      ( aa(set(int),$o,member(int,Xb2),B3)
                     => ( X2 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa3) ) ) )
             => ( aa(int,$o,P,X2)
               => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),X2),D4)) ) )
         => ( ! [X2: int,K2: int] :
                ( aa(int,$o,P4,X2)
              <=> aa(int,$o,P4,aa(int,int,aa(int,fun(int,int),minus_minus(int),X2),aa(int,int,aa(int,fun(int,int),times_times(int),K2),D4))) )
           => ( ? [X_12: int] : aa(int,$o,P,X_12)
            <=> ( ? [X4: int] :
                    ( aa(set(int),$o,member(int,X4),set_or1337092689740270186AtMost(int,one_one(int),D4))
                    & aa(int,$o,P4,X4) )
                | ? [X4: int] :
                    ( aa(set(int),$o,member(int,X4),set_or1337092689740270186AtMost(int,one_one(int),D4))
                    & ? [Xa2: int] :
                        ( aa(set(int),$o,member(int,Xa2),B3)
                        & aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),Xa2),X4)) ) ) ) ) ) ) ) ) ).

% cpmi
tff(fact_2207_cppi,axiom,
    ! [D4: int,P: fun(int,$o),P4: fun(int,$o),A4: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D4)
     => ( ? [Z6: int] :
          ! [X2: int] :
            ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z6),X2)
           => ( aa(int,$o,P,X2)
            <=> aa(int,$o,P4,X2) ) )
       => ( ! [X2: int] :
              ( ! [Xa3: int] :
                  ( aa(set(int),$o,member(int,Xa3),set_or1337092689740270186AtMost(int,one_one(int),D4))
                 => ! [Xb2: int] :
                      ( aa(set(int),$o,member(int,Xb2),A4)
                     => ( X2 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb2),Xa3) ) ) )
             => ( aa(int,$o,P,X2)
               => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X2),D4)) ) )
         => ( ! [X2: int,K2: int] :
                ( aa(int,$o,P4,X2)
              <=> aa(int,$o,P4,aa(int,int,aa(int,fun(int,int),minus_minus(int),X2),aa(int,int,aa(int,fun(int,int),times_times(int),K2),D4))) )
           => ( ? [X_12: int] : aa(int,$o,P,X_12)
            <=> ( ? [X4: int] :
                    ( aa(set(int),$o,member(int,X4),set_or1337092689740270186AtMost(int,one_one(int),D4))
                    & aa(int,$o,P4,X4) )
                | ? [X4: int] :
                    ( aa(set(int),$o,member(int,X4),set_or1337092689740270186AtMost(int,one_one(int),D4))
                    & ? [Xa2: int] :
                        ( aa(set(int),$o,member(int,Xa2),A4)
                        & aa(int,$o,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),Xa2),X4)) ) ) ) ) ) ) ) ) ).

% cppi
tff(fact_2208_sum__natinterval__diff,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [F: fun(nat,A),M: nat,N: nat] :
          aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_dv(fun(nat,A),fun(nat,A),F)),set_or1337092689740270186AtMost(nat,M,N)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),N),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F,M)),aa(nat,A,F,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)))),zero_zero(A)) ) ).

% sum_natinterval_diff
tff(fact_2209_sum__telescope_H_H,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [M: nat,N: nat,F: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),N)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_dw(fun(nat,A),fun(nat,A),F)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),N)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F,N)),aa(nat,A,F,M)) ) ) ) ).

% sum_telescope''
tff(fact_2210_sum__gp__multiplied,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [M: nat,N: nat,X: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),N)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),X)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X)),set_or1337092689740270186AtMost(nat,M,N))) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(nat,nat,suc,N))) ) ) ) ).

% sum_gp_multiplied
tff(fact_2211_sum_Oin__pairs,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),M),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_dx(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,M,N)) ) ).

% sum.in_pairs
tff(fact_2212_sum__gp,axiom,
    ! [A: $tType] :
      ( ( division_ring(A)
        & comm_ring(A) )
     => ! [X: A,M: nat,N: nat] :
          aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X)),set_or1337092689740270186AtMost(nat,M,N)) = $ite(
            aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),M),
            zero_zero(A),
            $ite(X = one_one(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))),M)),aa(A,A,aa(A,fun(A,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),X),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(nat,nat,suc,N)))),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),X))) ) ) ).

% sum_gp
tff(fact_2213_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)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N)),one_one(A)))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ).

% gauss_sum_from_Suc_0
tff(fact_2214_sum__gp__offset,axiom,
    ! [A: $tType] :
      ( ( division_ring(A)
        & comm_ring(A) )
     => ! [X: A,M: nat,N: nat] :
          aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X)),set_or1337092689740270186AtMost(nat,M,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N))) = $ite(X = one_one(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),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),M)),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_2215_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_dy(A,fun(A,fun(nat,A)),A3),D3)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N)),one_one(A))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),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),aa(num,num,bit0,one2))) ) ).

% arith_series
tff(fact_2216_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)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N)),one_one(A)))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ).

% gauss_sum
tff(fact_2217_double__gauss__sum__from__Suc__0,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [N: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),semiring_1_of_nat(A)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N)),one_one(A))) ) ).

% double_gauss_sum_from_Suc_0
tff(fact_2218_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),aa(num,num,bit0,one2))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_dz(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),aa(num,num,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_2219_negative__eq__positive,axiom,
    ! [N: nat,M: nat] :
      ( ( aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N)) = aa(nat,int,semiring_1_of_nat(int),M) )
    <=> ( ( N = zero_zero(nat) )
        & ( M = zero_zero(nat) ) ) ) ).

% negative_eq_positive
tff(fact_2220_negative__zle,axiom,
    ! [N: nat,M: nat] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N))),aa(nat,int,semiring_1_of_nat(int),M)) ).

% negative_zle
tff(fact_2221_of__nat__mult,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [M: nat,N: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N)) ) ).

% of_nat_mult
tff(fact_2222_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_2223_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_2224_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_2225_negative__zless,axiom,
    ! [N: nat,M: nat] : aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N)))),aa(nat,int,semiring_1_of_nat(int),M)) ).

% negative_zless
tff(fact_2226_of__nat__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_1(A)
     => ! [F: fun(B,nat),A4: set(B)] : aa(nat,A,semiring_1_of_nat(A),aa(set(B),nat,aa(fun(B,nat),fun(set(B),nat),groups7311177749621191930dd_sum(B,nat),F),A4)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aTP_Lamp_ea(fun(B,nat),fun(B,A),F)),A4) ) ).

% of_nat_sum
tff(fact_2227_of__nat__Suc,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [M: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,M)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(nat,A,semiring_1_of_nat(A),M)) ) ).

% of_nat_Suc
tff(fact_2228_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_2229_atLeastLessThanPlusOne__atLeastAtMost__integer,axiom,
    ! [L: code_integer,U: code_integer] : set_or7035219750837199246ssThan(code_integer,L,aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),U),one_one(code_integer))) = set_or1337092689740270186AtMost(code_integer,L,U) ).

% atLeastLessThanPlusOne_atLeastAtMost_integer
tff(fact_2230_int__cases2,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),N3)) ) ).

% int_cases2
tff(fact_2231_div__mult2__eq_H,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A3: A,M: nat,N: nat] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(nat,A,semiring_1_of_nat(A),M))),aa(nat,A,semiring_1_of_nat(A),N)) ) ).

% div_mult2_eq'
tff(fact_2232_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_2233_int__of__nat__induct,axiom,
    ! [P: fun(int,$o),Z2: int] :
      ( ! [N3: nat] : aa(int,$o,P,aa(nat,int,semiring_1_of_nat(int),N3))
     => ( ! [N3: nat] : aa(int,$o,P,aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N3))))
       => aa(int,$o,P,Z2) ) ) ).

% int_of_nat_induct
tff(fact_2234_int__ops_I2_J,axiom,
    aa(nat,int,semiring_1_of_nat(int),one_one(nat)) = one_one(int) ).

% int_ops(2)
tff(fact_2235_not__int__zless__negative,axiom,
    ! [N: nat,M: nat] : ~ aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(nat,int,semiring_1_of_nat(int),N)),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),M))) ).

% not_int_zless_negative
tff(fact_2236_nat__less__as__int,axiom,
    ! [X3: nat,Xa3: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X3),Xa3)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(nat,int,semiring_1_of_nat(int),X3)),aa(nat,int,semiring_1_of_nat(int),Xa3)) ) ).

% nat_less_as_int
tff(fact_2237_nat__leq__as__int,axiom,
    ! [X3: nat,Xa3: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X3),Xa3)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,semiring_1_of_nat(int),X3)),aa(nat,int,semiring_1_of_nat(int),Xa3)) ) ).

% nat_leq_as_int
tff(fact_2238_ex__less__of__nat__mult,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),X)
         => ? [N3: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N3)),X)) ) ) ).

% ex_less_of_nat_mult
tff(fact_2239_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_2240_int__cases4,axiom,
    ! [M: int] :
      ( ! [N3: nat] : M != aa(nat,int,semiring_1_of_nat(int),N3)
     => ~ ! [N3: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N3)
           => ( M != aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N3)) ) ) ) ).

% int_cases4
tff(fact_2241_int__zle__neg,axiom,
    ! [N: nat,M: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,semiring_1_of_nat(int),N)),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),M)))
    <=> ( ( N = zero_zero(nat) )
        & ( M = zero_zero(nat) ) ) ) ).

% int_zle_neg
tff(fact_2242_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_2243_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_2244_negative__zle__0,axiom,
    ! [N: nat] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N))),zero_zero(int)) ).

% negative_zle_0
tff(fact_2245_nonpos__int__cases,axiom,
    ! [K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),zero_zero(int))
     => ~ ! [N3: nat] : K != aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N3)) ) ).

% nonpos_int_cases
tff(fact_2246_int__sum,axiom,
    ! [A: $tType,F: fun(A,nat),A4: set(A)] : aa(nat,int,semiring_1_of_nat(int),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F),A4)) = aa(set(A),int,aa(fun(A,int),fun(set(A),int),groups7311177749621191930dd_sum(A,int),aTP_Lamp_eb(fun(A,nat),fun(A,int),F)),A4) ).

% int_sum
tff(fact_2247_mod__mult2__eq_H,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A3: A,M: 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),M)),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),M)),modulo_modulo(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N)))),modulo_modulo(A,A3,aa(nat,A,semiring_1_of_nat(A),M))) ) ).

% mod_mult2_eq'
tff(fact_2248_int__cases3,axiom,
    ! [K: int] :
      ( ( K != zero_zero(int) )
     => ( ! [N3: nat] :
            ( ( K = aa(nat,int,semiring_1_of_nat(int),N3) )
           => ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N3) )
       => ~ ! [N3: nat] :
              ( ( K = aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N3)) )
             => ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N3) ) ) ) ).

% int_cases3
tff(fact_2249_not__zle__0__negative,axiom,
    ! [N: nat] : ~ aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N)))) ).

% not_zle_0_negative
tff(fact_2250_negD,axiom,
    ! [X: int] :
      ( aa(int,$o,aa(int,fun(int,$o),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_2251_negative__zless__0,axiom,
    ! [N: nat] : aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N)))),zero_zero(int)) ).

% negative_zless_0
tff(fact_2252_nat__approx__posE,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [E4: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),E4)
         => ~ ! [N3: nat] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,N3)))),E4) ) ) ).

% nat_approx_posE
tff(fact_2253_neg__int__cases,axiom,
    ! [K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int))
     => ~ ! [N3: nat] :
            ( ( K = aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N3)) )
           => ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N3) ) ) ).

% neg_int_cases
tff(fact_2254_double__gauss__sum,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [N: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),semiring_1_of_nat(A)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N)),one_one(A))) ) ).

% double_gauss_sum
tff(fact_2255_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),aa(num,num,bit0,one2))),Z2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,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),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)))),comm_s3205402744901411588hammer(A,Z2,N))),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z2),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),N)) ) ).

% pochhammer_double
tff(fact_2256_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_ec(A,A),N,zero_zero(A)) ) ).

% of_nat_code
tff(fact_2257_pochhammer__code,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A,N: nat] :
          comm_s3205402744901411588hammer(A,A3,N) = $ite(N = zero_zero(nat),one_one(A),set_fo6178422350223883121st_nat(A,aTP_Lamp_ed(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_2258_sum__gp0,axiom,
    ! [A: $tType] :
      ( ( division_ring(A)
        & comm_ring(A) )
     => ! [X: A,N: nat] :
          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)) = $ite(X = one_one(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))),aa(A,A,aa(A,fun(A,A),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_2259_gchoose__row__sum__weighted,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [R2: A,M: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_ee(A,fun(nat,A),R2)),set_or1337092689740270186AtMost(nat,zero_zero(nat),M)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,M))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),aa(nat,A,gbinomial(A,R2),aa(nat,nat,suc,M))) ) ).

% gchoose_row_sum_weighted
tff(fact_2260_case__nat__add__eq__if,axiom,
    ! [A: $tType,A3: A,F: fun(nat,A),V: num,N: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aa(A,fun(fun(nat,A),fun(nat,A)),case_nat(A),A3),F),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),V)),N)) = aa(nat,A,F,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),pred_numeral(V)),N)) ).

% case_nat_add_eq_if
tff(fact_2261_gbinomial__1,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [A3: A] : aa(nat,A,gbinomial(A,A3),one_one(nat)) = A3 ) ).

% gbinomial_1
tff(fact_2262_pochhammer__1,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A] : comm_s3205402744901411588hammer(A,A3,one_one(nat)) = A3 ) ).

% pochhammer_1
tff(fact_2263_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_2264_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_2265_case__nat__numeral,axiom,
    ! [A: $tType,A3: A,F: fun(nat,A),V: num] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aa(A,fun(fun(nat,A),fun(nat,A)),case_nat(A),A3),F),aa(num,nat,numeral_numeral(nat),V)) = aa(nat,A,F,pred_numeral(V)) ).

% case_nat_numeral
tff(fact_2266_atMost__0,axiom,
    aa(nat,set(nat),set_ord_atMost(nat),zero_zero(nat)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),zero_zero(nat)),bot_bot(set(nat))) ).

% atMost_0
tff(fact_2267_nat_Ocase__distrib,axiom,
    ! [B: $tType,A: $tType,H2: fun(B,A),F1: B,F22: fun(nat,B),Nat: nat] : aa(B,A,H2,aa(nat,B,aa(fun(nat,B),fun(nat,B),aa(B,fun(fun(nat,B),fun(nat,B)),case_nat(B),F1),F22),Nat)) = aa(nat,A,aa(fun(nat,A),fun(nat,A),aa(A,fun(fun(nat,A),fun(nat,A)),case_nat(A),aa(B,A,H2,F1)),aa(fun(nat,B),fun(nat,A),aTP_Lamp_ef(fun(B,A),fun(fun(nat,B),fun(nat,A)),H2),F22)),Nat) ).

% nat.case_distrib
tff(fact_2268_not__empty__eq__Iic__eq__empty,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [H2: A] : bot_bot(set(A)) != aa(A,set(A),set_ord_atMost(A),H2) ) ).

% not_empty_eq_Iic_eq_empty
tff(fact_2269_not__UNIV__eq__Iic,axiom,
    ! [A: $tType] :
      ( no_top(A)
     => ! [H3: A] : top_top(set(A)) != aa(A,set(A),set_ord_atMost(A),H3) ) ).

% not_UNIV_eq_Iic
tff(fact_2270_atMost__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [U: A] : aa(A,set(A),set_ord_atMost(A),U) = aa(fun(A,$o),set(A),collect(A),aTP_Lamp_eg(A,fun(A,$o),U)) ) ).

% atMost_def
tff(fact_2271_gbinomial__parallel__sum,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_eh(A,fun(nat,A),A3)),aa(nat,set(nat),set_ord_atMost(nat),N)) = aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(nat,A,semiring_1_of_nat(A),N))),one_one(A))),N) ) ).

% gbinomial_parallel_sum
tff(fact_2272_gbinomial__Suc__Suc,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K: 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,K)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,gbinomial(A,A3),K)),aa(nat,A,gbinomial(A,A3),aa(nat,nat,suc,K))) ) ).

% gbinomial_Suc_Suc
tff(fact_2273_nat_Odisc__eq__case_I2_J,axiom,
    ! [Nat: nat] :
      ( ( Nat != zero_zero(nat) )
    <=> aa(nat,$o,aa(fun(nat,$o),fun(nat,$o),aa($o,fun(fun(nat,$o),fun(nat,$o)),case_nat($o),$false),aTP_Lamp_ei(nat,$o)),Nat) ) ).

% nat.disc_eq_case(2)
tff(fact_2274_nat_Odisc__eq__case_I1_J,axiom,
    ! [Nat: nat] :
      ( ( Nat = zero_zero(nat) )
    <=> aa(nat,$o,aa(fun(nat,$o),fun(nat,$o),aa($o,fun(fun(nat,$o),fun(nat,$o)),case_nat($o),$true),aTP_Lamp_ej(nat,$o)),Nat) ) ).

% nat.disc_eq_case(1)
tff(fact_2275_not__UNIV__le__Iic,axiom,
    ! [A: $tType] :
      ( no_top(A)
     => ! [H2: A] : ~ aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),top_top(set(A))),aa(A,set(A),set_ord_atMost(A),H2)) ) ).

% not_UNIV_le_Iic
tff(fact_2276_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_2277_gbinomial__sum__lower__neg,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,M: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_ek(A,fun(nat,A),A3)),aa(nat,set(nat),set_ord_atMost(nat),M)) = 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))),M)),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),one_one(A))),M)) ) ).

% gbinomial_sum_lower_neg
tff(fact_2278_gbinomial__addition__formula,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K: nat] : aa(nat,A,gbinomial(A,A3),aa(nat,nat,suc,K)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),one_one(A))),aa(nat,nat,suc,K))),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),one_one(A))),K)) ) ).

% gbinomial_addition_formula
tff(fact_2279_gbinomial__absorb__comp,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),aa(nat,A,semiring_1_of_nat(A),K))),aa(nat,A,gbinomial(A,A3),K)) = 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))),K)) ) ).

% gbinomial_absorb_comp
tff(fact_2280_gbinomial__mult__1,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(nat,A,gbinomial(A,A3),K)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),K)),aa(nat,A,gbinomial(A,A3),K))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,K))),aa(nat,A,gbinomial(A,A3),aa(nat,nat,suc,K)))) ) ).

% gbinomial_mult_1
tff(fact_2281_gbinomial__mult__1_H,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,A3),K)),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),K)),aa(nat,A,gbinomial(A,A3),K))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,K))),aa(nat,A,gbinomial(A,A3),aa(nat,nat,suc,K)))) ) ).

% gbinomial_mult_1'
tff(fact_2282_gbinomial__partial__sum__poly,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [M: 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_el(nat,fun(A,fun(A,fun(A,fun(nat,A)))),M),A3),X),Y)),aa(nat,set(nat),set_ord_atMost(nat),M)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aa(A,fun(A,fun(A,fun(nat,A))),aTP_Lamp_em(nat,fun(A,fun(A,fun(A,fun(nat,A)))),M),A3),X),Y)),aa(nat,set(nat),set_ord_atMost(nat),M)) ) ).

% gbinomial_partial_sum_poly
tff(fact_2283_pochhammer__0__left,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [N: nat] :
          comm_s3205402744901411588hammer(A,zero_zero(A),N) = $ite(N = zero_zero(nat),one_one(A),zero_zero(A)) ) ).

% pochhammer_0_left
tff(fact_2284_gbinomial__partial__sum__poly__xpos,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [M: 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_el(nat,fun(A,fun(A,fun(A,fun(nat,A)))),M),A3),X),Y)),aa(nat,set(nat),set_ord_atMost(nat),M)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aa(A,fun(A,fun(A,fun(nat,A))),aTP_Lamp_en(nat,fun(A,fun(A,fun(A,fun(nat,A)))),M),A3),X),Y)),aa(nat,set(nat),set_ord_atMost(nat),M)) ) ).

% gbinomial_partial_sum_poly_xpos
tff(fact_2285_gbinomial__sum__nat__pow2,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [M: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_eo(nat,fun(nat,A),M)),aa(nat,set(nat),set_ord_atMost(nat),M)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),M) ) ).

% gbinomial_sum_nat_pow2
tff(fact_2286_Suc__times__gbinomial,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: 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,K))),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A))),aa(nat,nat,suc,K))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A))),aa(nat,A,gbinomial(A,A3),K)) ) ).

% Suc_times_gbinomial
tff(fact_2287_gbinomial__absorption,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: 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,K))),aa(nat,A,gbinomial(A,A3),aa(nat,nat,suc,K))) = 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))),K)) ) ).

% gbinomial_absorption
tff(fact_2288_less__eq__nat_Osimps_I2_J,axiom,
    ! [M: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,M)),N)
    <=> aa(nat,$o,aa(fun(nat,$o),fun(nat,$o),aa($o,fun(fun(nat,$o),fun(nat,$o)),case_nat($o),$false),aa(nat,fun(nat,$o),ord_less_eq(nat),M)),N) ) ).

% less_eq_nat.simps(2)
tff(fact_2289_gbinomial__trinomial__revision,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,M: nat,A3: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),M)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,A3),M)),aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),M)),K)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,A3),K)),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),K))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),K))) ) ) ) ).

% gbinomial_trinomial_revision
tff(fact_2290_sum_OatMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),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,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_dq(fun(nat,A),fun(nat,A),G)),aa(nat,set(nat),set_ord_atMost(nat),N))) ) ).

% sum.atMost_Suc_shift
tff(fact_2291_sum__telescope,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [F: fun(nat,A),I: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_ep(fun(nat,A),fun(nat,A),F)),aa(nat,set(nat),set_ord_atMost(nat),I)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F,zero_zero(nat))),aa(nat,A,F,aa(nat,nat,suc,I))) ) ).

% sum_telescope
tff(fact_2292_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_2293_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_2294_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_2295_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [N: nat,K: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),K)
         => ( comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),N)),K) = zero_zero(A) ) ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_2296_pochhammer__of__nat__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ( ring_char_0(A)
        & idom(A) )
     => ! [N: nat,K: nat] :
          ( ( comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),N)),K) = zero_zero(A) )
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),K) ) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_2297_pochhammer__eq__0__iff,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,N: nat] :
          ( ( comm_s3205402744901411588hammer(A,A3,N) = zero_zero(A) )
        <=> ? [K4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K4),N)
              & ( A3 = aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),K4)) ) ) ) ) ).

% pochhammer_eq_0_iff
tff(fact_2298_gbinomial__rec,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K: 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,K)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,A3),K)),aa(A,A,aa(A,fun(A,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,K)))) ) ).

% gbinomial_rec
tff(fact_2299_gbinomial__factors,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K: 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,K)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A))),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,K)))),aa(nat,A,gbinomial(A,A3),K)) ) ).

% gbinomial_factors
tff(fact_2300_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [A: $tType] :
      ( ( ring_char_0(A)
        & idom(A) )
     => ! [K: nat,N: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),N)
         => ( comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),N)),K) != zero_zero(A) ) ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_2301_pochhammer__product_H,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Z2: A,N: nat,M: nat] : comm_s3205402744901411588hammer(A,Z2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M)) = 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)),M)) ) ).

% pochhammer_product'
tff(fact_2302_gbinomial__negated__upper,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K: nat] : aa(nat,A,gbinomial(A,A3),K) = 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))),K)),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,semiring_1_of_nat(A),K)),A3)),one_one(A))),K)) ) ).

% gbinomial_negated_upper
tff(fact_2303_gbinomial__index__swap,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: 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))),K)),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))),K)) = 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),K))),one_one(A))),N)) ) ).

% gbinomial_index_swap
tff(fact_2304_gbinomial__r__part__sum,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [M: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(nat,A,semiring_1_of_nat(A),M))),one_one(A)))),aa(nat,set(nat),set_ord_atMost(nat),M)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),M)) ) ).

% gbinomial_r_part_sum
tff(fact_2305_diff__Suc,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),aa(nat,nat,suc,N)) = aa(nat,nat,aa(fun(nat,nat),fun(nat,nat),aa(nat,fun(fun(nat,nat),fun(nat,nat)),case_nat(nat),zero_zero(nat)),aTP_Lamp_dj(nat,nat)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N)) ).

% diff_Suc
tff(fact_2306_gbinomial__partial__row__sum,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,M: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_ee(A,fun(nat,A),A3)),aa(nat,set(nat),set_ord_atMost(nat),M)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),M)),one_one(A))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),aa(nat,A,gbinomial(A,A3),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),one_one(nat)))) ) ).

% gbinomial_partial_row_sum
tff(fact_2307_Nitpick_Ocase__nat__unfold,axiom,
    ! [A: $tType,X: A,F: fun(nat,A),N: nat] :
      aa(nat,A,aa(fun(nat,A),fun(nat,A),aa(A,fun(fun(nat,A),fun(nat,A)),case_nat(A),X),F),N) = $ite(N = zero_zero(nat),X,aa(nat,A,F,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)))) ).

% Nitpick.case_nat_unfold
tff(fact_2308_gbinomial__minus,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K: nat] : aa(nat,A,gbinomial(A,aa(A,A,uminus_uminus(A),A3)),K) = 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))),K)),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(nat,A,semiring_1_of_nat(A),K))),one_one(A))),K)) ) ).

% gbinomial_minus
tff(fact_2309_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),aa(nat,fun(set(nat),set(nat)),insert2(nat),zero_zero(nat)),bot_bot(set(nat)))) ).

% atLeast1_atMost_eq_remove0
tff(fact_2310_gbinomial__reduce__nat,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,A3: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
         => ( aa(nat,A,gbinomial(A,A3),K) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),one_one(A))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K),one_one(nat)))),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),one_one(A))),K)) ) ) ) ).

% gbinomial_reduce_nat
tff(fact_2311_sum_Otriangle__reindex__eq,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,fun(nat,A)),N: nat] : aa(set(product_prod(nat,nat)),A,aa(fun(product_prod(nat,nat),A),fun(set(product_prod(nat,nat)),A),groups7311177749621191930dd_sum(product_prod(nat,nat),A),aa(fun(nat,fun(nat,A)),fun(product_prod(nat,nat),A),product_case_prod(nat,nat,A),G)),aa(fun(product_prod(nat,nat),$o),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aTP_Lamp_eq(nat,fun(nat,fun(nat,$o)),N)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_es(fun(nat,fun(nat,A)),fun(nat,A),G)),aa(nat,set(nat),set_ord_atMost(nat),N)) ) ).

% sum.triangle_reindex_eq
tff(fact_2312_pochhammer__product,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [M: nat,N: nat,Z2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),N)
         => ( comm_s3205402744901411588hammer(A,Z2,N) = aa(A,A,aa(A,fun(A,A),times_times(A),comm_s3205402744901411588hammer(A,Z2,M)),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z2),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M))) ) ) ) ).

% pochhammer_product
tff(fact_2313_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_2314_sum__power__shift,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [M: nat,N: nat,X: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),N)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X)),set_or1337092689740270186AtMost(nat,M,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),M)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X)),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M)))) ) ) ) ).

% sum_power_shift
tff(fact_2315_gbinomial__sum__up__index,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_et(nat,fun(nat,A),K)),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),K),one_one(nat))) ) ).

% gbinomial_sum_up_index
tff(fact_2316_pochhammer__absorb__comp,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [R2: A,K: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),R2),aa(nat,A,semiring_1_of_nat(A),K))),comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),R2),K)) = aa(A,A,aa(A,fun(A,A),times_times(A),R2),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),R2)),one_one(A)),K)) ) ).

% pochhammer_absorb_comp
tff(fact_2317_gbinomial__absorption_H,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,A3: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
         => ( aa(nat,A,gbinomial(A,A3),K) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(nat,A,semiring_1_of_nat(A),K))),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),one_one(A))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K),one_one(nat)))) ) ) ) ).

% gbinomial_absorption'
tff(fact_2318_sum_Oin__pairs__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),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),aa(num,num,bit0,one2))),N)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_dx(fun(nat,A),fun(nat,A),G)),aa(nat,set(nat),set_ord_atMost(nat),N)) ) ).

% sum.in_pairs_0
tff(fact_2319_pochhammer__minus_H,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [B2: A,K: nat] : comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),aa(nat,A,semiring_1_of_nat(A),K))),one_one(A)),K) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),K)),comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),B2),K)) ) ).

% pochhammer_minus'
tff(fact_2320_pochhammer__minus,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [B2: A,K: nat] : comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),B2),K) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),K)),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),aa(nat,A,semiring_1_of_nat(A),K))),one_one(A)),K)) ) ).

% pochhammer_minus
tff(fact_2321_sum_Ozero__middle,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [P3: nat,K: nat,G: fun(nat,A),H2: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),P3)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),P3)
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_eu(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K),G),H2)),aa(nat,set(nat),set_ord_atMost(nat),P3)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_ev(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K),G),H2)),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),P3),aa(nat,nat,suc,zero_zero(nat))))) ) ) ) ) ).

% sum.zero_middle
tff(fact_2322_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),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),aa(nat,nat,suc,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_ew(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),aa(num,num,bit0,one2))),N)),one_one(nat)))) ) ).

% pochhammer_times_pochhammer_half
tff(fact_2323_choose__odd__sum,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [N: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_ex(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),aa(num,num,bit0,one2))),N) ) ) ) ).

% choose_odd_sum
tff(fact_2324_choose__even__sum,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [N: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_ey(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),aa(num,num,bit0,one2))),N) ) ) ) ).

% choose_even_sum
tff(fact_2325_gbinomial__code,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K: nat] :
          aa(nat,A,gbinomial(A,A3),K) = $ite(K = zero_zero(nat),one_one(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),set_fo6178422350223883121st_nat(A,aTP_Lamp_ez(A,fun(nat,fun(A,A)),A3),zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K),one_one(nat)),one_one(A))),semiring_char_0_fact(A,K))) ) ).

% gbinomial_code
tff(fact_2326_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),aa(num,num,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),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N))),semiring_char_0_fact(A,N)) ) ).

% fact_double
tff(fact_2327_choose__alternating__sum,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [N: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),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_fa(nat,fun(nat,A),N)),aa(nat,set(nat),set_ord_atMost(nat),N)) = zero_zero(A) ) ) ) ).

% choose_alternating_sum
tff(fact_2328_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_2329_binomial__n__n,axiom,
    ! [N: nat] : aa(nat,nat,binomial(N),N) = one_one(nat) ).

% binomial_n_n
tff(fact_2330_prod_Oneutral__const,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A4: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aTP_Lamp_fb(B,A)),A4) = one_one(A) ) ).

% prod.neutral_const
tff(fact_2331_of__nat__prod,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(A)
     => ! [F: fun(B,nat),A4: set(B)] : aa(nat,A,semiring_1_of_nat(A),aa(set(B),nat,aa(fun(B,nat),fun(set(B),nat),groups7121269368397514597t_prod(B,nat),F),A4)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aTP_Lamp_fc(fun(B,nat),fun(B,A),F)),A4) ) ).

% of_nat_prod
tff(fact_2332_of__int__prod,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_ring_1(A)
     => ! [F: fun(B,int),A4: set(B)] : aa(int,A,ring_1_of_int(A),aa(set(B),int,aa(fun(B,int),fun(set(B),int),groups7121269368397514597t_prod(B,int),F),A4)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aTP_Lamp_fd(fun(B,int),fun(B,A),F)),A4) ) ).

% of_int_prod
tff(fact_2333_prod_Oempty,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(B,A)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),bot_bot(set(B))) = one_one(A) ) ).

% prod.empty
tff(fact_2334_fact__0,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ( semiring_char_0_fact(A,zero_zero(nat)) = one_one(A) ) ) ).

% fact_0
tff(fact_2335_fact__1,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ( semiring_char_0_fact(A,one_one(nat)) = one_one(A) ) ) ).

% fact_1
tff(fact_2336_binomial__n__0,axiom,
    ! [N: nat] : aa(nat,nat,binomial(N),zero_zero(nat)) = one_one(nat) ).

% binomial_n_0
tff(fact_2337_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_2338_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_2339_prod_OatMost__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),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),G),aa(nat,set(nat),set_ord_atMost(nat),N))),aa(nat,A,G,aa(nat,nat,suc,N))) ) ).

% prod.atMost_Suc
tff(fact_2340_prod_Ocl__ivl__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,N: nat] :
          aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M,aa(nat,nat,suc,N))) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,N)),M),one_one(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M,N))),aa(nat,A,G,aa(nat,nat,suc,N)))) ) ).

% prod.cl_ivl_Suc
tff(fact_2341_prod_Oop__ivl__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,N: nat] :
          aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,M,aa(nat,nat,suc,N))) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),M),one_one(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,M,N))),aa(nat,A,G,N))) ) ).

% prod.op_ivl_Suc
tff(fact_2342_prod_Oswap,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(B,fun(C,A)),B3: set(C),A4: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(set(C),fun(B,A),aTP_Lamp_fe(fun(B,fun(C,A)),fun(set(C),fun(B,A)),G),B3)),A4) = aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7121269368397514597t_prod(C,A),aa(set(B),fun(C,A),aTP_Lamp_fg(fun(B,fun(C,A)),fun(set(B),fun(C,A)),G),A4)),B3) ) ).

% prod.swap
tff(fact_2343_prod_Oneutral,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [A4: set(A),G: fun(A,B)] :
          ( ! [X2: A] :
              ( aa(set(A),$o,member(A,X2),A4)
             => ( aa(A,B,G,X2) = one_one(B) ) )
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A4) = one_one(B) ) ) ) ).

% prod.neutral
tff(fact_2344_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(B,A),A4: set(B)] :
          ( ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),A4) != one_one(A) )
         => ~ ! [A6: B] :
                ( aa(set(B),$o,member(B,A6),A4)
               => ( aa(B,A,G,A6) = one_one(A) ) ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_2345_choose__one,axiom,
    ! [N: nat] : aa(nat,nat,binomial(N),one_one(nat)) = N ).

% choose_one
tff(fact_2346_prod_Odistrib,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(B,A),H2: fun(B,A),A4: 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_fh(fun(B,A),fun(fun(B,A),fun(B,A)),G),H2)),A4) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),A4)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),H2),A4)) ) ).

% prod.distrib
tff(fact_2347_prod__dividef,axiom,
    ! [A: $tType,B: $tType] :
      ( field(A)
     => ! [F: fun(B,A),G: fun(B,A),A4: 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_fi(fun(B,A),fun(fun(B,A),fun(B,A)),F),G)),A4) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F),A4)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),A4)) ) ).

% prod_dividef
tff(fact_2348_prod__power__distrib,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(A)
     => ! [F: fun(B,A),A4: set(B),N: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F),A4)),N) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(nat,fun(B,A),aTP_Lamp_fj(fun(B,A),fun(nat,fun(B,A)),F),N)),A4) ) ).

% prod_power_distrib
tff(fact_2349_mod__prod__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [F: fun(B,A),A3: A,A4: set(B)] : modulo_modulo(A,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(A,fun(B,A),aTP_Lamp_cq(fun(B,A),fun(A,fun(B,A)),F),A3)),A4),A3) = modulo_modulo(A,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F),A4),A3) ) ).

% mod_prod_eq
tff(fact_2350_abs__prod,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_field(A)
     => ! [F: fun(B,A),A4: set(B)] : aa(A,A,abs_abs(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F),A4)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aTP_Lamp_fk(fun(B,A),fun(B,A),F)),A4) ) ).

% abs_prod
tff(fact_2351_fact__prod,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),aTP_Lamp_dj(nat,nat)),set_or1337092689740270186AtMost(nat,one_one(nat),N))) ) ).

% fact_prod
tff(fact_2352_prod__ge__1,axiom,
    ! [B: $tType,A: $tType] :
      ( linord181362715937106298miring(B)
     => ! [A4: set(A),F: fun(A,B)] :
          ( ! [X2: A] :
              ( aa(set(A),$o,member(A,X2),A4)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),one_one(B)),aa(A,B,F,X2)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),one_one(B)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F),A4)) ) ) ).

% prod_ge_1
tff(fact_2353_fact__eq__fact__times,axiom,
    ! [N: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),M)
     => ( semiring_char_0_fact(nat,M) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),semiring_char_0_fact(nat,N)),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),aTP_Lamp_dj(nat,nat)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,N),M))) ) ) ).

% fact_eq_fact_times
tff(fact_2354_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_2355_fact__ge__1,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),semiring_char_0_fact(A,N)) ) ).

% fact_ge_1
tff(fact_2356_pochhammer__fact,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & comm_semiring_1(A) )
     => ! [N: nat] : semiring_char_0_fact(A,N) = comm_s3205402744901411588hammer(A,one_one(A),N) ) ).

% pochhammer_fact
tff(fact_2357_prod_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_fl(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,M,N)) ) ).

% prod.shift_bounds_cl_Suc_ivl
tff(fact_2358_prod_Oshift__bounds__Suc__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_fl(fun(nat,A),fun(nat,A),G)),set_or7035219750837199246ssThan(nat,M,N)) ) ).

% prod.shift_bounds_Suc_ivl
tff(fact_2359_power__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [C2: A,F: fun(B,nat),A4: set(B)] : aa(nat,A,aa(A,fun(nat,A),power_power(A),C2),aa(set(B),nat,aa(fun(B,nat),fun(set(B),nat),groups7311177749621191930dd_sum(B,nat),F),A4)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(fun(B,nat),fun(B,A),aTP_Lamp_fm(A,fun(fun(B,nat),fun(B,A)),C2),F)),A4) ) ).

% power_sum
tff(fact_2360_prod_Oshift__bounds__cl__nat__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,K: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_fn(fun(nat,A),fun(nat,fun(nat,A)),G),K)),set_or1337092689740270186AtMost(nat,M,N)) ) ).

% prod.shift_bounds_cl_nat_ivl
tff(fact_2361_prod_Oshift__bounds__nat__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,K: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_fn(fun(nat,A),fun(nat,fun(nat,A)),G),K)),set_or7035219750837199246ssThan(nat,M,N)) ) ).

% prod.shift_bounds_nat_ivl
tff(fact_2362_fact__binomial,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,N: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),N)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),semiring_char_0_fact(A,K)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(N),K))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),semiring_char_0_fact(A,N)),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K))) ) ) ) ).

% fact_binomial
tff(fact_2363_binomial__fact,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,N: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),N)
         => ( aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(N),K)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),semiring_char_0_fact(A,N)),aa(A,A,aa(A,fun(A,A),times_times(A),semiring_char_0_fact(A,K)),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K)))) ) ) ) ).

% binomial_fact
tff(fact_2364_fact__div__fact,axiom,
    ! [N: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),M)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),semiring_char_0_fact(nat,M)),semiring_char_0_fact(nat,N)) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),aTP_Lamp_dj(nat,nat)),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)),M)) ) ) ).

% fact_div_fact
tff(fact_2365_prod__le__1,axiom,
    ! [A: $tType,B: $tType] :
      ( linord181362715937106298miring(B)
     => ! [A4: set(A),F: fun(A,B)] :
          ( ! [X2: A] :
              ( aa(set(A),$o,member(A,X2),A4)
             => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F,X2))
                & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,X2)),one_one(B)) ) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F),A4)),one_one(B)) ) ) ).

% prod_le_1
tff(fact_2366_fact__split,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [K: nat,N: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),N)
         => ( semiring_char_0_fact(A,N) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),suc),set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K),N)))),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K))) ) ) ) ).

% fact_split
tff(fact_2367_prod_OatLeastLessThan__concat,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,N: nat,P3: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),N)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),P3)
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,M,N))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,N,P3))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,M,P3)) ) ) ) ) ).

% prod.atLeastLessThan_concat
tff(fact_2368_binomial__altdef__of__nat,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,N: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),N)
         => ( aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(N),K)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_fo(nat,fun(nat,fun(nat,A)),K),N)),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ) ) ).

% binomial_altdef_of_nat
tff(fact_2369_binomial__absorb__comp,axiom,
    ! [N: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K)),aa(nat,nat,binomial(N),K)) = 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))),K)) ).

% binomial_absorb_comp
tff(fact_2370_dvd__fact,axiom,
    ! [M: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),M)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),N)
       => aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),M),semiring_char_0_fact(nat,N)) ) ) ).

% dvd_fact
tff(fact_2371_gbinomial__mult__fact_H,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,A3),K)),semiring_char_0_fact(A,K)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_fp(A,fun(nat,A),A3)),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ).

% gbinomial_mult_fact'
tff(fact_2372_gbinomial__mult__fact,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),semiring_char_0_fact(A,K)),aa(nat,A,gbinomial(A,A3),K)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_fp(A,fun(nat,A),A3)),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ).

% gbinomial_mult_fact
tff(fact_2373_gbinomial__prod__rev,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [A3: A,K: nat] : aa(nat,A,gbinomial(A,A3),K) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_fq(A,fun(nat,A),A3)),set_or7035219750837199246ssThan(nat,zero_zero(nat),K))),semiring_char_0_fact(A,K)) ) ).

% gbinomial_prod_rev
tff(fact_2374_fact__fact__dvd__fact,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K: nat,N: nat] : aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),semiring_char_0_fact(A,K)),semiring_char_0_fact(A,N))),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),N))) ) ).

% fact_fact_dvd_fact
tff(fact_2375_prod_OatLeastAtMost__rev,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: nat,M: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,N,M)) = 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_fr(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),N),M)),set_or1337092689740270186AtMost(nat,N,M)) ) ).

% prod.atLeastAtMost_rev
tff(fact_2376_gbinomial__Suc,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [A3: A,K: nat] : aa(nat,A,gbinomial(A,A3),aa(nat,nat,suc,K)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_fq(A,fun(nat,A),A3)),set_or1337092689740270186AtMost(nat,zero_zero(nat),K))),semiring_char_0_fact(A,aa(nat,nat,suc,K))) ) ).

% gbinomial_Suc
tff(fact_2377_sum__choose__upper,axiom,
    ! [M: nat,N: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_fs(nat,fun(nat,nat),M)),aa(nat,set(nat),set_ord_atMost(nat),N)) = aa(nat,nat,binomial(aa(nat,nat,suc,N)),aa(nat,nat,suc,M)) ).

% sum_choose_upper
tff(fact_2378_prod_OatLeast0__atMost__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,zero_zero(nat),N))),aa(nat,A,G,aa(nat,nat,suc,N))) ) ).

% prod.atLeast0_atMost_Suc
tff(fact_2379_prod_OatLeast0__lessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,zero_zero(nat),N))),aa(nat,A,G,N)) ) ).

% prod.atLeast0_lessThan_Suc
tff(fact_2380_prod_OatLeast__Suc__lessThan,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,N: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),N)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,M,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,M)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),N))) ) ) ) ).

% prod.atLeast_Suc_lessThan
tff(fact_2381_prod_Onat__ivl__Suc_H,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,N: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),aa(nat,nat,suc,N))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,aa(nat,nat,suc,N))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M,N))) ) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_2382_prod_OatLeast__Suc__atMost,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,N: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),N)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,M)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),N))) ) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_2383_prod_OatLeastLessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: nat,B2: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A3),B2)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),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),G),set_or7035219750837199246ssThan(nat,A3,B2))),aa(nat,A,G,B2)) ) ) ) ).

% prod.atLeastLessThan_Suc
tff(fact_2384_prod_Olast__plus,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,N: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),N)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,N)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,M,N))) ) ) ) ).

% prod.last_plus
tff(fact_2385_binomial__absorption,axiom,
    ! [K: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),aa(nat,nat,binomial(N),aa(nat,nat,suc,K))) = 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))),K)) ).

% binomial_absorption
tff(fact_2386_prod_OSuc__reindex__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,N: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),N)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M,N))),aa(nat,A,G,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,M)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_fl(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,M,N))) ) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_2387_prod_OatMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),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,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_fl(fun(nat,A),fun(nat,A),G)),aa(nat,set(nat),set_ord_atMost(nat),N))) ) ).

% prod.atMost_Suc_shift
tff(fact_2388_prod_OatLeastLessThan__rev,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: nat,M: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,N,M)) = 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_ft(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),N),M)),set_or7035219750837199246ssThan(nat,N,M)) ) ).

% prod.atLeastLessThan_rev
tff(fact_2389_choose__dvd,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K: nat,N: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),N)
         => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),semiring_char_0_fact(A,K)),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K)))),semiring_char_0_fact(A,N)) ) ) ).

% choose_dvd
tff(fact_2390_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_fu(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_fw(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),A3),N)),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)) ) ).

% prod.nested_swap
tff(fact_2391_fact__numeral,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [K: num] : semiring_char_0_fact(A,aa(num,nat,numeral_numeral(nat),K)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),K)),semiring_char_0_fact(A,pred_numeral(K))) ) ).

% fact_numeral
tff(fact_2392_prod__atLeastAtMost__code,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [F: fun(nat,A),A3: nat,B2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),F),set_or1337092689740270186AtMost(nat,A3,B2)) = set_fo6178422350223883121st_nat(A,aTP_Lamp_fx(fun(nat,A),fun(nat,fun(A,A)),F),A3,B2,one_one(A)) ) ).

% prod_atLeastAtMost_code
tff(fact_2393_sum__choose__lower,axiom,
    ! [R2: nat,N: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_fy(nat,fun(nat,nat),R2)),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),R2),N))),N) ).

% sum_choose_lower
tff(fact_2394_choose__rising__sum_I2_J,axiom,
    ! [N: nat,M: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_fz(nat,fun(nat,nat),N)),aa(nat,set(nat),set_ord_atMost(nat),M)) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M)),one_one(nat))),M) ).

% choose_rising_sum(2)
tff(fact_2395_choose__rising__sum_I1_J,axiom,
    ! [N: nat,M: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_fz(nat,fun(nat,nat),N)),aa(nat,set(nat),set_ord_atMost(nat),M)) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M)),one_one(nat))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))) ).

% choose_rising_sum(1)
tff(fact_2396_binomial__code,axiom,
    ! [N: nat,K: nat] :
      aa(nat,nat,binomial(N),K) = $ite(
        aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),K),
        zero_zero(nat),
        $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),K)),aa(nat,nat,binomial(N),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),set_fo6178422350223883121st_nat(nat,times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K)),one_one(nat)),N,one_one(nat))),semiring_char_0_fact(nat,K))) ) ).

% binomial_code
tff(fact_2397_prod_Ohead__if,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,N: nat] :
          aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M,N)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),M),one_one(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,M,N))),aa(nat,A,G,N))) ) ).

% prod.head_if
tff(fact_2398_prod_Oub__add__nat,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,N: nat,G: fun(nat,A),P3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),P3))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M,N))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),P3)))) ) ) ) ).

% prod.ub_add_nat
tff(fact_2399_choose__reduce__nat,axiom,
    ! [N: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
       => ( aa(nat,nat,binomial(N),K) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K),one_one(nat)))),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))),K)) ) ) ) ).

% choose_reduce_nat
tff(fact_2400_times__binomial__minus1__eq,axiom,
    ! [K: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),aa(nat,nat,binomial(N),K)) = 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),K),one_one(nat)))) ) ) ).

% times_binomial_minus1_eq
tff(fact_2401_prod_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: nat,M: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,N,M)) = 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_fr(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),N),M)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,N),M)) ) ).

% prod.atLeastLessThan_rev_at_least_Suc_atMost
tff(fact_2402_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_ga(A,fun(nat,A),A3)),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)) ) ).

% pochhammer_prod
tff(fact_2403_sum__choose__diagonal,axiom,
    ! [M: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),N)
     => ( aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(nat,fun(nat,nat),aTP_Lamp_gb(nat,fun(nat,fun(nat,nat)),M),N)),aa(nat,set(nat),set_ord_atMost(nat),M)) = aa(nat,nat,binomial(aa(nat,nat,suc,N)),M) ) ) ).

% sum_choose_diagonal
tff(fact_2404_vandermonde,axiom,
    ! [M: nat,N: nat,R2: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(nat,fun(nat,nat),aa(nat,fun(nat,fun(nat,nat)),aTP_Lamp_gc(nat,fun(nat,fun(nat,fun(nat,nat))),M),N),R2)),aa(nat,set(nat),set_ord_atMost(nat),R2)) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)),R2) ).

% vandermonde
tff(fact_2405_binomial__addition__formula,axiom,
    ! [N: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
     => ( aa(nat,nat,binomial(N),aa(nat,nat,suc,K)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))),aa(nat,nat,suc,K))),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))),K)) ) ) ).

% binomial_addition_formula
tff(fact_2406_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_ga(A,fun(nat,A),A3)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) ) ).

% pochhammer_Suc_prod
tff(fact_2407_pochhammer__prod__rev,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),aa(nat,fun(nat,A),aTP_Lamp_gd(A,fun(nat,fun(nat,A)),A3),N)),set_or1337092689740270186AtMost(nat,one_one(nat),N)) ) ).

% pochhammer_prod_rev
tff(fact_2408_fact__num__eq__if,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [M: nat] :
          semiring_char_0_fact(A,M) = $ite(M = zero_zero(nat),one_one(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),M)),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),one_one(nat))))) ) ).

% fact_num_eq_if
tff(fact_2409_fact__reduce,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),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_2410_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),aa(num,num,bit0,one2)),N,one_one(nat))) ) ).

% fact_code
tff(fact_2411_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_2412_choose__row__sum,axiom,
    ! [N: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),binomial(N)),aa(nat,set(nat),set_ord_atMost(nat),N)) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N) ).

% choose_row_sum
tff(fact_2413_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_ge(nat,fun(nat,fun(nat,fun(nat,nat))),A3),B2),N)),aa(nat,set(nat),set_ord_atMost(nat),N)) ).

% binomial
tff(fact_2414_choose__two,axiom,
    ! [N: nat] : aa(nat,nat,binomial(N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = aa(nat,nat,aa(nat,fun(nat,nat),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),aa(num,num,bit0,one2))) ).

% choose_two
tff(fact_2415_prod_Oin__pairs,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),M),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_gf(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,M,N)) ) ).

% prod.in_pairs
tff(fact_2416_prod_Oin__pairs__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),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),aa(num,num,bit0,one2))),N)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_gf(fun(nat,A),fun(nat,A),G)),aa(nat,set(nat),set_ord_atMost(nat),N)) ) ).

% prod.in_pairs_0
tff(fact_2417_gbinomial__altdef__of__nat,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K: nat] : aa(nat,A,gbinomial(A,A3),K) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_gg(A,fun(nat,fun(nat,A)),A3),K)),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ).

% gbinomial_altdef_of_nat
tff(fact_2418_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_gd(A,fun(nat,fun(nat,A)),A3),N)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) ) ).

% pochhammer_Suc_prod_rev
tff(fact_2419_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_gh(A,fun(A,fun(nat,fun(nat,A))),A3),B2),N)),aa(nat,set(nat),set_ord_atMost(nat),N)) ) ).

% binomial_ring
tff(fact_2420_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_gi(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_2421_gbinomial__pochhammer_H,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K: nat] : aa(nat,A,gbinomial(A,A3),K) = aa(A,A,aa(A,fun(A,A),divide_divide(A),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),aa(nat,A,semiring_1_of_nat(A),K))),one_one(A)),K)),semiring_char_0_fact(A,K)) ) ).

% gbinomial_pochhammer'
tff(fact_2422_choose__square__sum,axiom,
    ! [N: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_gj(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),aa(num,num,bit0,one2))),N)),N) ).

% choose_square_sum
tff(fact_2423_gbinomial__pochhammer,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K: nat] : aa(nat,A,gbinomial(A,A3),K) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),K)),comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),A3),K))),semiring_char_0_fact(A,K)) ) ).

% gbinomial_pochhammer
tff(fact_2424_prod_Ozero__middle,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [P3: nat,K: nat,G: fun(nat,A),H2: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),P3)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),P3)
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_gk(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K),G),H2)),aa(nat,set(nat),set_ord_atMost(nat),P3)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_gl(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K),G),H2)),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),P3),aa(nat,nat,suc,zero_zero(nat))))) ) ) ) ) ).

% prod.zero_middle
tff(fact_2425_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_gm(nat,fun(nat,A),N)),aa(nat,set(nat),set_ord_atMost(nat),N)) = zero_zero(A) ) ) ) ).

% choose_alternating_linear_sum
tff(fact_2426_binomial__r__part__sum,axiom,
    ! [M: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),M)),one_one(nat)))),aa(nat,set(nat),set_ord_atMost(nat),M)) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),M)) ).

% 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_gn(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),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)))) ).

% choose_linear_sum
tff(fact_2428_sum_Otriangle__reindex,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,fun(nat,A)),N: nat] : aa(set(product_prod(nat,nat)),A,aa(fun(product_prod(nat,nat),A),fun(set(product_prod(nat,nat)),A),groups7311177749621191930dd_sum(product_prod(nat,nat),A),aa(fun(nat,fun(nat,A)),fun(product_prod(nat,nat),A),product_case_prod(nat,nat,A),G)),aa(fun(product_prod(nat,nat),$o),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aTP_Lamp_go(nat,fun(nat,fun(nat,$o)),N)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_es(fun(nat,fun(nat,A)),fun(nat,A),G)),aa(nat,set(nat),set_ord_lessThan(nat),N)) ) ).

% sum.triangle_reindex
tff(fact_2429_divide__int__unfold,axiom,
    ! [K: int,M: nat,L: int,N: nat] :
      aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),K)),aa(nat,int,semiring_1_of_nat(int),M))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L)),aa(nat,int,semiring_1_of_nat(int),N))) = $ite(
        ( ( aa(int,int,sgn_sgn(int),L) = zero_zero(int) )
        | ( aa(int,int,sgn_sgn(int),K) = zero_zero(int) )
        | ( N = zero_zero(nat) ) ),
        zero_zero(int),
        $ite(aa(int,int,sgn_sgn(int),K) = aa(int,int,sgn_sgn(int),L),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N)),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N)),aa($o,nat,zero_neq_one_of_bool(nat),~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),N),M)))))) ) ).

% divide_int_unfold
tff(fact_2430_rec__nat__add__eq__if,axiom,
    ! [A: $tType,A3: A,F: fun(nat,fun(A,A)),V: num,N: nat] :
      aa(nat,A,rec_nat(A,A3,F),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),V)),N)) = $let(
        pv: nat,
        pv:= pred_numeral(V),
        aa(A,A,aa(nat,fun(A,A),F,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),pv),N)),aa(nat,A,rec_nat(A,A3,F),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),pv),N))) ) ).

% rec_nat_add_eq_if
tff(fact_2431_sum__zero__power_H,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: fun(nat,A),D3: fun(nat,A),A4: set(nat)] :
          aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,A),fun(nat,A),aTP_Lamp_gp(fun(nat,A),fun(fun(nat,A),fun(nat,A)),C2),D3)),A4) = $ite(
            ( aa(set(nat),$o,finite_finite2(nat),A4)
            & aa(set(nat),$o,member(nat,zero_zero(nat)),A4) ),
            aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,C2,zero_zero(nat))),aa(nat,A,D3,zero_zero(nat))),
            zero_zero(A) ) ) ).

% sum_zero_power'
tff(fact_2432_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_2433_drop__bit__numeral__minus__bit1,axiom,
    ! [L: num,K: num] : bit_se4197421643247451524op_bit(int,aa(num,nat,numeral_numeral(nat),L),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,K)))) = bit_se4197421643247451524op_bit(int,pred_numeral(L),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),inc(K)))) ).

% drop_bit_numeral_minus_bit1
tff(fact_2434_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_2435_sgn__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)) ) ).

% sgn_minus
tff(fact_2436_prod_Oinfinite,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [A4: set(A),G: fun(A,B)] :
          ( ~ aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A4) = one_one(B) ) ) ) ).

% prod.infinite
tff(fact_2437_divide__sgn,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,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_2438_lessThan__0,axiom,
    aa(nat,set(nat),set_ord_lessThan(nat),zero_zero(nat)) = bot_bot(set(nat)) ).

% lessThan_0
tff(fact_2439_drop__bit__minus__one,axiom,
    ! [N: nat] : bit_se4197421643247451524op_bit(int,N,aa(int,int,uminus_uminus(int),one_one(int))) = aa(int,int,uminus_uminus(int),one_one(int)) ).

% drop_bit_minus_one
tff(fact_2440_sum_Odelta_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [S: set(A),A3: A,B2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),S)
         => ( 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_gq(A,fun(fun(A,B),fun(A,B)),A3),B2)),S) = $ite(aa(set(A),$o,member(A,A3),S),aa(A,B,B2,A3),zero_zero(B)) ) ) ) ).

% sum.delta'
tff(fact_2441_sum_Odelta,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [S: set(A),A3: A,B2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),S)
         => ( 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_gr(A,fun(fun(A,B),fun(A,B)),A3),B2)),S) = $ite(aa(set(A),$o,member(A,A3),S),aa(A,B,B2,A3),zero_zero(B)) ) ) ) ).

% sum.delta
tff(fact_2442_prod__eq__1__iff,axiom,
    ! [A: $tType,A4: set(A),F: fun(A,nat)] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7121269368397514597t_prod(A,nat),F),A4) = one_one(nat) )
      <=> ! [X4: A] :
            ( aa(set(A),$o,member(A,X4),A4)
           => ( aa(A,nat,F,X4) = one_one(nat) ) ) ) ) ).

% prod_eq_1_iff
tff(fact_2443_prod_Odelta,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [S: set(A),A3: A,B2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),S)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),aa(fun(A,B),fun(A,B),aTP_Lamp_gs(A,fun(fun(A,B),fun(A,B)),A3),B2)),S) = $ite(aa(set(A),$o,member(A,A3),S),aa(A,B,B2,A3),one_one(B)) ) ) ) ).

% prod.delta
tff(fact_2444_prod_Odelta_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [S: set(A),A3: A,B2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),S)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),aa(fun(A,B),fun(A,B),aTP_Lamp_gt(A,fun(fun(A,B),fun(A,B)),A3),B2)),S) = $ite(aa(set(A),$o,member(A,A3),S),aa(A,B,B2,A3),one_one(B)) ) ) ) ).

% prod.delta'
tff(fact_2445_rec__nat__numeral,axiom,
    ! [A: $tType,A3: A,F: fun(nat,fun(A,A)),V: num] :
      aa(nat,A,rec_nat(A,A3,F),aa(num,nat,numeral_numeral(nat),V)) = $let(
        pv: nat,
        pv:= pred_numeral(V),
        aa(A,A,aa(nat,fun(A,A),F,pv),aa(nat,A,rec_nat(A,A3,F),pv)) ) ).

% rec_nat_numeral
tff(fact_2446_sgn__pos,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
         => ( aa(A,A,sgn_sgn(A),A3) = one_one(A) ) ) ) ).

% sgn_pos
tff(fact_2447_prod_Oinsert,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A4: set(A),X: A,G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ~ aa(set(A),$o,member(A,X),A4)
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A4)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,G,X)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A4)) ) ) ) ) ).

% prod.insert
tff(fact_2448_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_2449_single__Diff__lessThan,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [K: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),K),bot_bot(set(A)))),aa(A,set(A),set_ord_lessThan(A),K)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),K),bot_bot(set(A))) ) ).

% single_Diff_lessThan
tff(fact_2450_sgn__mult__self__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,sgn_sgn(A),A3)),aa(A,A,sgn_sgn(A),A3)) = aa($o,A,zero_neq_one_of_bool(A),A3 != zero_zero(A)) ) ).

% sgn_mult_self_eq
tff(fact_2451_drop__bit__of__1,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat] : bit_se4197421643247451524op_bit(A,N,one_one(A)) = aa($o,A,zero_neq_one_of_bool(A),N = zero_zero(nat)) ) ).

% drop_bit_of_1
tff(fact_2452_prod_OlessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),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),G),aa(nat,set(nat),set_ord_lessThan(nat),N))),aa(nat,A,G,N)) ) ).

% prod.lessThan_Suc
tff(fact_2453_sgn__neg,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),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_2454_drop__bit__Suc__minus__bit0,axiom,
    ! [N: nat,K: num] : bit_se4197421643247451524op_bit(int,aa(nat,nat,suc,N),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,K)))) = bit_se4197421643247451524op_bit(int,N,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K))) ).

% drop_bit_Suc_minus_bit0
tff(fact_2455_sum__mult__of__bool__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(B)
     => ! [A4: set(A),F: fun(A,B),P: fun(A,$o)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aa(fun(A,$o),fun(A,B),aTP_Lamp_gu(fun(A,B),fun(fun(A,$o),fun(A,B)),F),P)),A4) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),aa(fun(A,$o),set(A),collect(A),P))) ) ) ) ).

% sum_mult_of_bool_eq
tff(fact_2456_sum__of__bool__mult__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(B)
     => ! [A4: set(A),P: fun(A,$o),F: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aa(fun(A,B),fun(A,B),aTP_Lamp_gv(fun(A,$o),fun(fun(A,B),fun(A,B)),P),F)),A4) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),aa(fun(A,$o),set(A),collect(A),P))) ) ) ) ).

% sum_of_bool_mult_eq
tff(fact_2457_drop__bit__numeral__minus__bit0,axiom,
    ! [L: num,K: num] : bit_se4197421643247451524op_bit(int,aa(num,nat,numeral_numeral(nat),L),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,K)))) = bit_se4197421643247451524op_bit(int,pred_numeral(L),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K))) ).

% drop_bit_numeral_minus_bit0
tff(fact_2458_sum__zero__power,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [C2: fun(nat,A),A4: set(nat)] :
          aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_gw(fun(nat,A),fun(nat,A),C2)),A4) = $ite(
            ( aa(set(nat),$o,finite_finite2(nat),A4)
            & aa(set(nat),$o,member(nat,zero_zero(nat)),A4) ),
            aa(nat,A,C2,zero_zero(nat)),
            zero_zero(A) ) ) ).

% sum_zero_power
tff(fact_2459_drop__bit__Suc__minus__bit1,axiom,
    ! [N: nat,K: num] : 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,K)))) = bit_se4197421643247451524op_bit(int,N,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),inc(K)))) ).

% drop_bit_Suc_minus_bit1
tff(fact_2460_int__prod,axiom,
    ! [A: $tType,F: fun(A,nat),A4: set(A)] : aa(nat,int,semiring_1_of_nat(int),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7121269368397514597t_prod(A,nat),F),A4)) = aa(set(A),int,aa(fun(A,int),fun(set(A),int),groups7121269368397514597t_prod(A,int),aTP_Lamp_eb(fun(A,nat),fun(A,int),F)),A4) ).

% int_prod
tff(fact_2461_finite__if__eq__beyond__finite,axiom,
    ! [A: $tType,S: set(A),S4: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),S)
     => aa(set(set(A)),$o,finite_finite2(set(A)),aa(fun(set(A),$o),set(set(A)),collect(set(A)),aa(set(A),fun(set(A),$o),aTP_Lamp_gx(set(A),fun(set(A),fun(set(A),$o)),S),S4))) ) ).

% finite_if_eq_beyond_finite
tff(fact_2462_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_2463_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_2464_finite__M__bounded__by__nat,axiom,
    ! [P: fun(nat,$o),I: nat] : aa(set(nat),$o,finite_finite2(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_gy(fun(nat,$o),fun(nat,fun(nat,$o)),P),I))) ).

% finite_M_bounded_by_nat
tff(fact_2465_finite__less__ub,axiom,
    ! [F: fun(nat,nat),U: nat] :
      ( ! [N3: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N3),aa(nat,nat,F,N3))
     => aa(set(nat),$o,finite_finite2(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_gz(fun(nat,nat),fun(nat,fun(nat,$o)),F),U))) ) ).

% finite_less_ub
tff(fact_2466_sum_Oswap__restrict,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [A4: set(A),B3: set(B),G: fun(A,fun(B,C)),R: fun(A,fun(B,$o))] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(B),$o,finite_finite2(B),B3)
           => ( aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7311177749621191930dd_sum(A,C),aa(fun(A,fun(B,$o)),fun(A,C),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(A,C)),aTP_Lamp_hb(set(B),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(A,C))),B3),G),R)),A4) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),aa(fun(A,fun(B,$o)),fun(B,C),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(B,C)),aTP_Lamp_he(set(A),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(B,C))),A4),G),R)),B3) ) ) ) ) ).

% sum.swap_restrict
tff(fact_2467_prod_Oswap__restrict,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [A4: set(A),B3: set(B),G: fun(A,fun(B,C)),R: fun(A,fun(B,$o))] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(B),$o,finite_finite2(B),B3)
           => ( aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7121269368397514597t_prod(A,C),aa(fun(A,fun(B,$o)),fun(A,C),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(A,C)),aTP_Lamp_hf(set(B),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(A,C))),B3),G),R)),A4) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),aa(fun(A,fun(B,$o)),fun(B,C),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(B,C)),aTP_Lamp_hh(set(A),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(B,C))),A4),G),R)),B3) ) ) ) ) ).

% prod.swap_restrict
tff(fact_2468_lessThan__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [U: A] : aa(A,set(A),set_ord_lessThan(A),U) = aa(fun(A,$o),set(A),collect(A),aTP_Lamp_hi(A,fun(A,$o),U)) ) ).

% lessThan_def
tff(fact_2469_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_2470_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_2471_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_2472_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_2473_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_2474_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_2475_linordered__idom__class_Oabs__sgn,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [K: A] : aa(A,A,abs_abs(A),K) = aa(A,A,aa(A,fun(A,A),times_times(A),K),aa(A,A,sgn_sgn(A),K)) ) ).

% linordered_idom_class.abs_sgn
tff(fact_2476_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_2477_sum_Ofinite__Collect__op,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [I4: set(A),X: fun(A,B),Y: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_hj(set(A),fun(fun(A,B),fun(A,$o)),I4),X)))
         => ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_hj(set(A),fun(fun(A,B),fun(A,$o)),I4),Y)))
           => aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aa(fun(A,B),fun(fun(A,B),fun(A,$o)),aTP_Lamp_hk(set(A),fun(fun(A,B),fun(fun(A,B),fun(A,$o))),I4),X),Y))) ) ) ) ).

% sum.finite_Collect_op
tff(fact_2478_prod_Ofinite__Collect__op,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [I4: set(A),X: fun(A,B),Y: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_hl(set(A),fun(fun(A,B),fun(A,$o)),I4),X)))
         => ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_hl(set(A),fun(fun(A,B),fun(A,$o)),I4),Y)))
           => aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aa(fun(A,B),fun(fun(A,B),fun(A,$o)),aTP_Lamp_hm(set(A),fun(fun(A,B),fun(fun(A,B),fun(A,$o))),I4),X),Y))) ) ) ) ).

% prod.finite_Collect_op
tff(fact_2479_sum_Ointer__filter,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A4: set(A),G: fun(A,B),P: fun(A,$o)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_bi(set(A),fun(fun(A,$o),fun(A,$o)),A4),P))) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aa(fun(A,$o),fun(A,B),aTP_Lamp_hn(fun(A,B),fun(fun(A,$o),fun(A,B)),G),P)),A4) ) ) ) ).

% sum.inter_filter
tff(fact_2480_prod_Ointer__filter,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A4: set(A),G: fun(A,B),P: fun(A,$o)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_bi(set(A),fun(fun(A,$o),fun(A,$o)),A4),P))) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),aa(fun(A,$o),fun(A,B),aTP_Lamp_ho(fun(A,B),fun(fun(A,$o),fun(A,B)),G),P)),A4) ) ) ) ).

% prod.inter_filter
tff(fact_2481_sum__strict__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( strict7427464778891057005id_add(B)
     => ! [A4: set(A),F: fun(A,B),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( ! [X2: A] :
                  ( aa(set(A),$o,member(A,X2),A4)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F,X2)),aa(A,B,G,X2)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F),A4)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),A4)) ) ) ) ) ).

% sum_strict_mono
tff(fact_2482_sgn__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( ( aa(A,A,sgn_sgn(A),A3) = one_one(A) )
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3) ) ) ).

% sgn_1_pos
tff(fact_2483_prod_Orelated,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [R: fun(A,fun(A,$o)),S: set(B),H2: fun(B,A),G: fun(B,A)] :
          ( aa(A,$o,aa(A,fun(A,$o),R,one_one(A)),one_one(A))
         => ( ! [X12: A,Y12: A,X23: A,Y23: A] :
                ( ( aa(A,$o,aa(A,fun(A,$o),R,X12),X23)
                  & aa(A,$o,aa(A,fun(A,$o),R,Y12),Y23) )
               => aa(A,$o,aa(A,fun(A,$o),R,aa(A,A,aa(A,fun(A,A),times_times(A),X12),Y12)),aa(A,A,aa(A,fun(A,A),times_times(A),X23),Y23)) )
           => ( aa(set(B),$o,finite_finite2(B),S)
             => ( ! [X2: B] :
                    ( aa(set(B),$o,member(B,X2),S)
                   => aa(A,$o,aa(A,fun(A,$o),R,aa(B,A,H2,X2)),aa(B,A,G,X2)) )
               => aa(A,$o,aa(A,fun(A,$o),R,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),H2),S)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),S)) ) ) ) ) ) ).

% prod.related
tff(fact_2484_prod_Oinsert__if,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A4: set(A),G: fun(A,B),X: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A4)) = $ite(aa(set(A),$o,member(A,X),A4),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A4),aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,G,X)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A4))) ) ) ) ).

% prod.insert_if
tff(fact_2485_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [S3: set(A),T4: set(B),S: set(A),I: fun(B,A),J: fun(A,B),T2: set(B),G: fun(A,C),H2: fun(B,C)] :
          ( aa(set(A),$o,finite_finite2(A),S3)
         => ( aa(set(B),$o,finite_finite2(B),T4)
           => ( ! [A6: A] :
                  ( aa(set(A),$o,member(A,A6),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S),S3))
                 => ( aa(B,A,I,aa(A,B,J,A6)) = A6 ) )
             => ( ! [A6: A] :
                    ( aa(set(A),$o,member(A,A6),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S),S3))
                   => aa(set(B),$o,member(B,aa(A,B,J,A6)),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T2),T4)) )
               => ( ! [B5: B] :
                      ( aa(set(B),$o,member(B,B5),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T2),T4))
                     => ( aa(A,B,J,aa(B,A,I,B5)) = B5 ) )
                 => ( ! [B5: B] :
                        ( aa(set(B),$o,member(B,B5),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T2),T4))
                       => aa(set(A),$o,member(A,aa(B,A,I,B5)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S),S3)) )
                   => ( ! [A6: A] :
                          ( aa(set(A),$o,member(A,A6),S3)
                         => ( aa(A,C,G,A6) = one_one(C) ) )
                     => ( ! [B5: B] :
                            ( aa(set(B),$o,member(B,B5),T4)
                           => ( aa(B,C,H2,B5) = one_one(C) ) )
                       => ( ! [A6: A] :
                              ( aa(set(A),$o,member(A,A6),S)
                             => ( aa(B,C,H2,aa(A,B,J,A6)) = aa(A,C,G,A6) ) )
                         => ( aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7121269368397514597t_prod(A,C),G),S) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),H2),T2) ) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_2486_abs__sgn__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          aa(A,A,abs_abs(A),aa(A,A,sgn_sgn(A),A3)) = $ite(A3 = zero_zero(A),zero_zero(A),one_one(A)) ) ).

% abs_sgn_eq
tff(fact_2487_sum_Onat__diff__reindex,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aTP_Lamp_hp(fun(nat,A),fun(nat,fun(nat,A)),G),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),G),aa(nat,set(nat),set_ord_lessThan(nat),N)) ) ).

% sum.nat_diff_reindex
tff(fact_2488_prod_Onat__diff__reindex,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_hq(fun(nat,A),fun(nat,fun(nat,A)),G),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),G),aa(nat,set(nat),set_ord_lessThan(nat),N)) ) ).

% prod.nat_diff_reindex
tff(fact_2489_sum__eq__1__iff,axiom,
    ! [A: $tType,A4: set(A),F: fun(A,nat)] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F),A4) = one_one(nat) )
      <=> ? [X4: A] :
            ( aa(set(A),$o,member(A,X4),A4)
            & ( aa(A,nat,F,X4) = one_one(nat) )
            & ! [Xa2: A] :
                ( aa(set(A),$o,member(A,Xa2),A4)
               => ( ( X4 != Xa2 )
                 => ( aa(A,nat,F,Xa2) = zero_zero(nat) ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_2490_prod__int__eq,axiom,
    ! [I: nat,J: nat] : aa(set(nat),int,aa(fun(nat,int),fun(set(nat),int),groups7121269368397514597t_prod(nat,int),semiring_1_of_nat(int)),set_or1337092689740270186AtMost(nat,I,J)) = aa(set(int),int,aa(fun(int,int),fun(set(int),int),groups7121269368397514597t_prod(int,int),aTP_Lamp_dl(int,int)),set_or1337092689740270186AtMost(int,aa(nat,int,semiring_1_of_nat(int),I),aa(nat,int,semiring_1_of_nat(int),J))) ).

% prod_int_eq
tff(fact_2491_div__push__bit__of__1__eq__drop__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A,N: nat] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),bit_se4730199178511100633sh_bit(A,N,one_one(A))) = bit_se4197421643247451524op_bit(A,N,A3) ) ).

% div_push_bit_of_1_eq_drop_bit
tff(fact_2492_sum__nonneg__leq__bound,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [S2: set(A),F: fun(A,B),B3: B,I: A] :
          ( aa(set(A),$o,finite_finite2(A),S2)
         => ( ! [I2: A] :
                ( aa(set(A),$o,member(A,I2),S2)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F,I2)) )
           => ( ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F),S2) = B3 )
             => ( aa(set(A),$o,member(A,I),S2)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,I)),B3) ) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_2493_sum__nonneg__0,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [S2: set(A),F: fun(A,B),I: A] :
          ( aa(set(A),$o,finite_finite2(A),S2)
         => ( ! [I2: A] :
                ( aa(set(A),$o,member(A,I2),S2)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F,I2)) )
           => ( ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F),S2) = zero_zero(B) )
             => ( aa(set(A),$o,member(A,I),S2)
               => ( aa(A,B,F,I) = zero_zero(B) ) ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_2494_ivl__disj__un__one_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),U)
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(A,set(A),set_ord_lessThan(A),L)),set_or7035219750837199246ssThan(A,L,U)) = aa(A,set(A),set_ord_lessThan(A),U) ) ) ) ).

% ivl_disj_un_one(2)
tff(fact_2495_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_2496_prod_Otriangle__reindex,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,fun(nat,A)),N: nat] : aa(set(product_prod(nat,nat)),A,aa(fun(product_prod(nat,nat),A),fun(set(product_prod(nat,nat)),A),groups7121269368397514597t_prod(product_prod(nat,nat),A),aa(fun(nat,fun(nat,A)),fun(product_prod(nat,nat),A),product_case_prod(nat,nat,A),G)),aa(fun(product_prod(nat,nat),$o),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aTP_Lamp_go(nat,fun(nat,fun(nat,$o)),N)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_hs(fun(nat,fun(nat,A)),fun(nat,A),G)),aa(nat,set(nat),set_ord_lessThan(nat),N)) ) ).

% prod.triangle_reindex
tff(fact_2497_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_2498_sum_Ointer__restrict,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A4: set(A),G: fun(A,B),B3: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3)) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aa(set(A),fun(A,B),aTP_Lamp_ht(fun(A,B),fun(set(A),fun(A,B)),G),B3)),A4) ) ) ) ).

% sum.inter_restrict
tff(fact_2499_sum_Osetdiff__irrelevant,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A4: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_hu(fun(A,B),fun(A,$o),G)))) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),A4) ) ) ) ).

% sum.setdiff_irrelevant
tff(fact_2500_prod_Ointer__restrict,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A4: set(A),G: fun(A,B),B3: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3)) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),aa(set(A),fun(A,B),aTP_Lamp_hv(fun(A,B),fun(set(A),fun(A,B)),G),B3)),A4) ) ) ) ).

% prod.inter_restrict
tff(fact_2501_prod_Osetdiff__irrelevant,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A4: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_hw(fun(A,B),fun(A,$o),G)))) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A4) ) ) ) ).

% prod.setdiff_irrelevant
tff(fact_2502_finite__divisors__nat,axiom,
    ! [M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
     => aa(set(nat),$o,finite_finite2(nat),aa(fun(nat,$o),set(nat),collect(nat),aTP_Lamp_hx(nat,fun(nat,$o),M))) ) ).

% finite_divisors_nat
tff(fact_2503_zsgn__def,axiom,
    ! [I: int] :
      aa(int,int,sgn_sgn(int),I) = $ite(
        I = zero_zero(int),
        zero_zero(int),
        $ite(aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),I),one_one(int),aa(int,int,uminus_uminus(int),one_one(int))) ) ).

% zsgn_def
tff(fact_2504_sum__diff__distrib,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Q: fun(A,nat),P: fun(A,nat),N: A] :
          ( ! [X2: A] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,Q,X2)),aa(A,nat,P,X2))
         => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),P),aa(A,set(A),set_ord_lessThan(A),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_hy(fun(A,nat),fun(fun(A,nat),fun(A,nat)),Q),P)),aa(A,set(A),set_ord_lessThan(A),N)) ) ) ) ).

% sum_diff_distrib
tff(fact_2505_sum__pos,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [I4: set(A),F: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),I4)
         => ( ( I4 != bot_bot(set(A)) )
           => ( ! [I2: A] :
                  ( aa(set(A),$o,member(A,I2),I4)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(A,B,F,I2)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F),I4)) ) ) ) ) ).

% sum_pos
tff(fact_2506_less__1__prod2,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_idom(B)
     => ! [I4: set(A),I: A,F: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),I4)
         => ( aa(set(A),$o,member(A,I),I4)
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),one_one(B)),aa(A,B,F,I))
             => ( ! [I2: A] :
                    ( aa(set(A),$o,member(A,I2),I4)
                   => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),one_one(B)),aa(A,B,F,I2)) )
               => aa(B,$o,aa(B,fun(B,$o),ord_less(B),one_one(B)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F),I4)) ) ) ) ) ) ).

% less_1_prod2
tff(fact_2507_less__1__prod,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_idom(B)
     => ! [I4: set(A),F: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),I4)
         => ( ( I4 != bot_bot(set(A)) )
           => ( ! [I2: A] :
                  ( aa(set(A),$o,member(A,I2),I4)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less(B),one_one(B)),aa(A,B,F,I2)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),one_one(B)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F),I4)) ) ) ) ) ).

% less_1_prod
tff(fact_2508_sum_OlessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),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,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_dq(fun(nat,A),fun(nat,A),G)),aa(nat,set(nat),set_ord_lessThan(nat),N))) ) ).

% sum.lessThan_Suc_shift
tff(fact_2509_sum__lessThan__telescope_H,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [F: fun(nat,A),M: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_ep(fun(nat,A),fun(nat,A),F)),aa(nat,set(nat),set_ord_lessThan(nat),M)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F,zero_zero(nat))),aa(nat,A,F,M)) ) ).

% sum_lessThan_telescope'
tff(fact_2510_sum__lessThan__telescope,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [F: fun(nat,A),M: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_ds(fun(nat,A),fun(nat,A),F)),aa(nat,set(nat),set_ord_lessThan(nat),M)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F,M)),aa(nat,A,F,zero_zero(nat))) ) ).

% sum_lessThan_telescope
tff(fact_2511_sgn__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( ( aa(A,A,sgn_sgn(A),A3) = aa(A,A,uminus_uminus(A),one_one(A)) )
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A)) ) ) ).

% sgn_1_neg
tff(fact_2512_sgn__if,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A] :
          aa(A,A,sgn_sgn(A),X) = $ite(
            X = zero_zero(A),
            zero_zero(A),
            $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),X),one_one(A),aa(A,A,uminus_uminus(A),one_one(A))) ) ) ).

% sgn_if
tff(fact_2513_prod_OlessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),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,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_fl(fun(nat,A),fun(nat,A),G)),aa(nat,set(nat),set_ord_lessThan(nat),N))) ) ).

% prod.lessThan_Suc_shift
tff(fact_2514_sum_Omono__neutral__cong,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [T2: set(A),S: set(A),H2: fun(A,B),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),T2)
         => ( aa(set(A),$o,finite_finite2(A),S)
           => ( ! [I2: A] :
                  ( aa(set(A),$o,member(A,I2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T2),S))
                 => ( aa(A,B,H2,I2) = zero_zero(B) ) )
             => ( ! [I2: A] :
                    ( aa(set(A),$o,member(A,I2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S),T2))
                   => ( aa(A,B,G,I2) = zero_zero(B) ) )
               => ( ! [X2: A] :
                      ( aa(set(A),$o,member(A,X2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S),T2))
                     => ( aa(A,B,G,X2) = aa(A,B,H2,X2) ) )
                 => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),S) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),H2),T2) ) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
tff(fact_2515_prod_Osubset__diff,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [B3: set(A),A4: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),A4)
         => ( aa(set(A),$o,finite_finite2(A),A4)
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A4) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),B3))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),B3)) ) ) ) ) ).

% prod.subset_diff
tff(fact_2516_prod_Omono__neutral__cong__right,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [T2: set(A),S: set(A),G: fun(A,B),H2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),T2)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),T2)
           => ( ! [X2: A] :
                  ( aa(set(A),$o,member(A,X2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T2),S))
                 => ( aa(A,B,G,X2) = one_one(B) ) )
             => ( ! [X2: A] :
                    ( aa(set(A),$o,member(A,X2),S)
                   => ( aa(A,B,G,X2) = aa(A,B,H2,X2) ) )
               => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),T2) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),H2),S) ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_2517_prod_Omono__neutral__cong__left,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [T2: set(A),S: set(A),H2: fun(A,B),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),T2)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),T2)
           => ( ! [X2: A] :
                  ( aa(set(A),$o,member(A,X2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T2),S))
                 => ( aa(A,B,H2,X2) = one_one(B) ) )
             => ( ! [X2: A] :
                    ( aa(set(A),$o,member(A,X2),S)
                   => ( aa(A,B,G,X2) = aa(A,B,H2,X2) ) )
               => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),S) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),H2),T2) ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_2518_prod_Omono__neutral__right,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [T2: set(A),S: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),T2)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),T2)
           => ( ! [X2: A] :
                  ( aa(set(A),$o,member(A,X2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T2),S))
                 => ( aa(A,B,G,X2) = one_one(B) ) )
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),T2) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),S) ) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_2519_prod_Omono__neutral__left,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [T2: set(A),S: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),T2)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),T2)
           => ( ! [X2: A] :
                  ( aa(set(A),$o,member(A,X2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T2),S))
                 => ( aa(A,B,G,X2) = one_one(B) ) )
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),S) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),T2) ) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_2520_prod_Osame__carrierI,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [C3: set(A),A4: set(A),B3: set(A),G: fun(A,B),H2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),C3)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),C3)
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),C3)
             => ( ! [A6: A] :
                    ( aa(set(A),$o,member(A,A6),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),C3),A4))
                   => ( aa(A,B,G,A6) = one_one(B) ) )
               => ( ! [B5: A] :
                      ( aa(set(A),$o,member(A,B5),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),C3),B3))
                     => ( aa(A,B,H2,B5) = one_one(B) ) )
                 => ( ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),C3) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),H2),C3) )
                   => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A4) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),H2),B3) ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_2521_prod_Osame__carrier,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [C3: set(A),A4: set(A),B3: set(A),G: fun(A,B),H2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),C3)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),C3)
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),C3)
             => ( ! [A6: A] :
                    ( aa(set(A),$o,member(A,A6),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),C3),A4))
                   => ( aa(A,B,G,A6) = one_one(B) ) )
               => ( ! [B5: A] :
                      ( aa(set(A),$o,member(A,B5),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),C3),B3))
                     => ( aa(A,B,H2,B5) = one_one(B) ) )
                 => ( ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A4) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),H2),B3) )
                  <=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),C3) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),H2),C3) ) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_2522_sum_Ounion__inter,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A4: set(A),B3: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),$o,finite_finite2(A),B3)
           => ( aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3))) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),A4)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),B3)) ) ) ) ) ).

% sum.union_inter
tff(fact_2523_sum_OInt__Diff,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A4: set(A),G: fun(A,B),B3: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),A4) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),B3))) ) ) ) ).

% sum.Int_Diff
tff(fact_2524_sum_OatLeast1__atMost__eq,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_dq(fun(nat,A),fun(nat,A),G)),aa(nat,set(nat),set_ord_lessThan(nat),N)) ) ).

% sum.atLeast1_atMost_eq
tff(fact_2525_prod_Ounion__inter,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A4: set(A),B3: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),$o,finite_finite2(A),B3)
           => ( aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3))) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A4)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),B3)) ) ) ) ) ).

% prod.union_inter
tff(fact_2526_prod_OInt__Diff,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A4: set(A),G: fun(A,B),B3: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A4) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),B3))) ) ) ) ).

% prod.Int_Diff
tff(fact_2527_prod_Omono__neutral__cong,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [T2: set(A),S: set(A),H2: fun(A,B),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),T2)
         => ( aa(set(A),$o,finite_finite2(A),S)
           => ( ! [I2: A] :
                  ( aa(set(A),$o,member(A,I2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T2),S))
                 => ( aa(A,B,H2,I2) = one_one(B) ) )
             => ( ! [I2: A] :
                    ( aa(set(A),$o,member(A,I2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S),T2))
                   => ( aa(A,B,G,I2) = one_one(B) ) )
               => ( ! [X2: A] :
                      ( aa(set(A),$o,member(A,X2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S),T2))
                     => ( aa(A,B,G,X2) = aa(A,B,H2,X2) ) )
                 => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),S) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),H2),T2) ) ) ) ) ) ) ) ).

% prod.mono_neutral_cong
tff(fact_2528_prod_OatLeast1__atMost__eq,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_fl(fun(nat,A),fun(nat,A),G)),aa(nat,set(nat),set_ord_lessThan(nat),N)) ) ).

% prod.atLeast1_atMost_eq
tff(fact_2529_sum__bounds__lt__plus1,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [F: fun(nat,A),Mm: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_dq(fun(nat,A),fun(nat,A),F)),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),F),set_or1337092689740270186AtMost(nat,one_one(nat),Mm)) ) ).

% sum_bounds_lt_plus1
tff(fact_2530_prod__int__plus__eq,axiom,
    ! [I: nat,J: nat] : aa(set(nat),int,aa(fun(nat,int),fun(set(nat),int),groups7121269368397514597t_prod(nat,int),semiring_1_of_nat(int)),set_or1337092689740270186AtMost(nat,I,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),J))) = aa(set(int),int,aa(fun(int,int),fun(set(int),int),groups7121269368397514597t_prod(int,int),aTP_Lamp_dl(int,int)),set_or1337092689740270186AtMost(int,aa(nat,int,semiring_1_of_nat(int),I),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),J)))) ).

% prod_int_plus_eq
tff(fact_2531_sum_Onat__group,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),K: 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_hz(fun(nat,A),fun(nat,fun(nat,A)),G),K)),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),G),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),K))) ) ).

% sum.nat_group
tff(fact_2532_prod_Onat__group,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),K: 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_ia(fun(nat,A),fun(nat,fun(nat,A)),G),K)),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),G),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),K))) ) ).

% prod.nat_group
tff(fact_2533_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_ib(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_do(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_2534_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_ic(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_fw(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_2535_Iio__Int__singleton,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [K: A,X: A] :
          aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(A,set(A),set_ord_lessThan(A),K)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),K),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))),bot_bot(set(A))) ) ).

% Iio_Int_singleton
tff(fact_2536_ivl__disj__un__singleton_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(A,set(A),set_ord_lessThan(A),U)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),U),bot_bot(set(A)))) = aa(A,set(A),set_ord_atMost(A),U) ) ).

% ivl_disj_un_singleton(2)
tff(fact_2537_ivl__disj__un__one_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),U)
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(A,set(A),set_ord_lessThan(A),L)),set_or1337092689740270186AtMost(A,L,U)) = aa(A,set(A),set_ord_atMost(A),U) ) ) ) ).

% ivl_disj_un_one(4)
tff(fact_2538_sum_OIf__cases,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A4: set(A),P: fun(A,$o),H2: fun(A,B),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_id(fun(A,$o),fun(fun(A,B),fun(fun(A,B),fun(A,B))),P),H2),G)),A4) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),H2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),aa(fun(A,$o),set(A),collect(A),P)))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),aa(set(A),set(A),uminus_uminus(set(A)),aa(fun(A,$o),set(A),collect(A),P))))) ) ) ) ).

% sum.If_cases
tff(fact_2539_prod_OIf__cases,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A4: set(A),P: fun(A,$o),H2: fun(A,B),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_ie(fun(A,$o),fun(fun(A,B),fun(fun(A,B),fun(A,B))),P),H2),G)),A4) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),H2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),aa(fun(A,$o),set(A),collect(A),P)))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),aa(set(A),set(A),uminus_uminus(set(A)),aa(fun(A,$o),set(A),collect(A),P))))) ) ) ) ).

% prod.If_cases
tff(fact_2540_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_2541_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_2542_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)) = aa(A,A,aa(A,fun(A,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),X),N)),one_one(A))),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),one_one(A))) ) ) ) ).

% geometric_sum
tff(fact_2543_prod__mono__strict,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_semidom(B)
     => ! [A4: set(A),F: fun(A,B),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ! [I2: A] :
                ( aa(set(A),$o,member(A,I2),A4)
               => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F,I2))
                  & aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F,I2)),aa(A,B,G,I2)) ) )
           => ( ( A4 != bot_bot(set(A)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F),A4)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A4)) ) ) ) ) ).

% prod_mono_strict
tff(fact_2544_sum_OatMost__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),aa(nat,set(nat),set_ord_atMost(nat),N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_dq(fun(nat,A),fun(nat,A),G)),aa(nat,set(nat),set_ord_lessThan(nat),N))) ) ).

% sum.atMost_shift
tff(fact_2545_sum_Ounion__inter__neutral,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A4: set(A),B3: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),$o,finite_finite2(A),B3)
           => ( ! [X2: A] :
                  ( aa(set(A),$o,member(A,X2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3))
                 => ( aa(A,B,G,X2) = zero_zero(B) ) )
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),A4)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),B3)) ) ) ) ) ) ).

% sum.union_inter_neutral
tff(fact_2546_sum_Oremove,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A4: set(A),X: A,G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),$o,member(A,X),A4)
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),A4) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,G,X)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))))) ) ) ) ) ).

% sum.remove
tff(fact_2547_sum_Oinsert__remove,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A4: set(A),G: fun(A,B),X: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A4)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,G,X)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))))) ) ) ) ).

% sum.insert_remove
tff(fact_2548_prod_OatMost__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),aa(nat,set(nat),set_ord_atMost(nat),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_fl(fun(nat,A),fun(nat,A),G)),aa(nat,set(nat),set_ord_lessThan(nat),N))) ) ).

% prod.atMost_shift
tff(fact_2549_sum__diff1,axiom,
    ! [B: $tType,A: $tType] :
      ( ab_group_add(B)
     => ! [A4: set(A),F: fun(A,B),A3: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A))))) = $ite(aa(set(A),$o,member(A,A3),A4),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),F),A4)),aa(A,B,F,A3)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F),A4)) ) ) ) ).

% sum_diff1
tff(fact_2550_sum__Un,axiom,
    ! [B: $tType,A: $tType] :
      ( ab_group_add(B)
     => ! [A4: set(A),B3: set(A),F: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),$o,finite_finite2(A),B3)
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3)) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F),A4)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F),B3))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3))) ) ) ) ) ).

% sum_Un
tff(fact_2551_sum_Ounion__disjoint,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A4: set(A),B3: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),$o,finite_finite2(A),B3)
           => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3) = bot_bot(set(A)) )
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),A4)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),B3)) ) ) ) ) ) ).

% sum.union_disjoint
tff(fact_2552_prod_Oremove,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A4: set(A),X: A,G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),$o,member(A,X),A4)
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A4) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,G,X)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))))) ) ) ) ) ).

% prod.remove
tff(fact_2553_prod_Oinsert__remove,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A4: set(A),G: fun(A,B),X: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A4)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,G,X)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))))) ) ) ) ).

% prod.insert_remove
tff(fact_2554_prod_Ounion__inter__neutral,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A4: set(A),B3: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),$o,finite_finite2(A),B3)
           => ( ! [X2: A] :
                  ( aa(set(A),$o,member(A,X2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3))
                 => ( aa(A,B,G,X2) = one_one(B) ) )
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A4)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),B3)) ) ) ) ) ) ).

% prod.union_inter_neutral
tff(fact_2555_prod_Ounion__disjoint,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A4: set(A),B3: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),$o,finite_finite2(A),B3)
           => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3) = bot_bot(set(A)) )
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A4)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),B3)) ) ) ) ) ) ).

% prod.union_disjoint
tff(fact_2556_sum_Ounion__diff2,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A4: set(A),B3: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),$o,finite_finite2(A),B3)
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),B3))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B3),A4)))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3))) ) ) ) ) ).

% sum.union_diff2
tff(fact_2557_sum__Un2,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A4: set(A),B3: set(A),F: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3))
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),B3))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B3),A4)))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3))) ) ) ) ).

% sum_Un2
tff(fact_2558_prod_Ounion__diff2,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A4: set(A),B3: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),$o,finite_finite2(A),B3)
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),B3))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B3),A4)))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3))) ) ) ) ) ).

% prod.union_diff2
tff(fact_2559_sum__Un__nat,axiom,
    ! [A: $tType,A4: set(A),B3: set(A),F: fun(A,nat)] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( aa(set(A),$o,finite_finite2(A),B3)
       => ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F),A4)),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F),B3))),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3))) ) ) ) ).

% sum_Un_nat
tff(fact_2560_sum_Odelta__remove,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [S: set(A),A3: A,B2: fun(A,B),C2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),S)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_if(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),A3),B2),C2)),S) = $ite(aa(set(A),$o,member(A,A3),S),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,B2,A3)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),C2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A)))))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),C2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A)))))) ) ) ) ).

% sum.delta_remove
tff(fact_2561_sum__div__partition,axiom,
    ! [A: $tType,B: $tType] :
      ( euclid4440199948858584721cancel(B)
     => ! [A4: set(A),F: fun(A,B),B2: B] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F),A4)),B2) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aa(B,fun(A,B),aTP_Lamp_ig(fun(A,B),fun(B,fun(A,B)),F),B2)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),aa(fun(A,$o),set(A),collect(A),aa(B,fun(A,$o),aTP_Lamp_ih(fun(A,B),fun(B,fun(A,$o)),F),B2))))),aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),aa(fun(A,$o),set(A),collect(A),aa(B,fun(A,$o),aTP_Lamp_ii(fun(A,B),fun(B,fun(A,$o)),F),B2))))),B2)) ) ) ) ).

% sum_div_partition
tff(fact_2562_prod_Odelta__remove,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [S: set(A),A3: A,B2: fun(A,B),C2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),S)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_ij(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),A3),B2),C2)),S) = $ite(aa(set(A),$o,member(A,A3),S),aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,B2,A3)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),C2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A)))))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),C2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A)))))) ) ) ) ).

% prod.delta_remove
tff(fact_2563_sum__gp__strict,axiom,
    ! [A: $tType] :
      ( ( division_ring(A)
        & comm_ring(A) )
     => ! [X: A,N: nat] :
          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)) = $ite(X = one_one(A),aa(nat,A,semiring_1_of_nat(A),N),aa(A,A,aa(A,fun(A,A),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_2564_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_ik(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_2565_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_il(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_2566_eucl__rel__int__remainderI,axiom,
    ! [R2: int,L: int,K: int,Q3: int] :
      ( ( aa(int,int,sgn_sgn(int),R2) = aa(int,int,sgn_sgn(int),L) )
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,abs_abs(int),R2)),aa(int,int,abs_abs(int),L))
       => ( ( K = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Q3),L)),R2) )
         => eucl_rel_int(K,L,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),R2)) ) ) ) ).

% eucl_rel_int_remainderI
tff(fact_2567_member__le__sum,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ordere6911136660526730532id_add(B)
        & semiring_1(B) )
     => ! [I: A,A4: set(A),F: fun(A,B)] :
          ( aa(set(A),$o,member(A,I),A4)
         => ( ! [X2: A] :
                ( aa(set(A),$o,member(A,X2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),I),bot_bot(set(A)))))
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F,X2)) )
           => ( aa(set(A),$o,finite_finite2(A),A4)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,I)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F),A4)) ) ) ) ) ).

% member_le_sum
tff(fact_2568_prod__mono2,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_idom(B)
     => ! [B3: set(A),A4: set(A),F: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),B3)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B3)
           => ( ! [B5: A] :
                  ( aa(set(A),$o,member(A,B5),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B3),A4))
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),one_one(B)),aa(A,B,F,B5)) )
             => ( ! [A6: A] :
                    ( aa(set(A),$o,member(A,A6),A4)
                   => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F,A6)) )
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F),A4)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F),B3)) ) ) ) ) ) ).

% prod_mono2
tff(fact_2569_prod__diff1,axiom,
    ! [B: $tType,A: $tType] :
      ( semidom_divide(B)
     => ! [A4: set(A),F: fun(A,B),A3: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( aa(A,B,F,A3) != zero_zero(B) )
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A))))) = $ite(aa(set(A),$o,member(A,A3),A4),aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F),A4)),aa(A,B,F,A3)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F),A4)) ) ) ) ) ).

% prod_diff1
tff(fact_2570_prod__Un,axiom,
    ! [B: $tType,A: $tType] :
      ( field(B)
     => ! [A4: set(A),B3: set(A),F: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),$o,finite_finite2(A),B3)
           => ( ! [X2: A] :
                  ( aa(set(A),$o,member(A,X2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3))
                 => ( aa(A,B,F,X2) != zero_zero(B) ) )
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3)) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F),A4)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F),B3))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3))) ) ) ) ) ) ).

% prod_Un
tff(fact_2571_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),aa(nat,fun(set(nat),set(nat)),insert2(nat),zero_zero(nat)),bot_bot(set(nat)))) ).

% atLeast1_lessThan_eq_remove0
tff(fact_2572_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_im(A,fun(nat,fun(nat,A)),X),N)),aa(nat,set(nat),set_ord_lessThan(nat),N))) ) ).

% one_diff_power_eq'
tff(fact_2573_eucl__rel__int_Ocases,axiom,
    ! [A1: int,A22: int,A32: product_prod(int,int)] :
      ( eucl_rel_int(A1,A22,A32)
     => ( ( ( A22 = zero_zero(int) )
         => ( A32 != aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),A1) ) )
       => ( ! [Q5: int] :
              ( ( A32 = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q5),zero_zero(int)) )
             => ( ( A22 != zero_zero(int) )
               => ( A1 != aa(int,int,aa(int,fun(int,int),times_times(int),Q5),A22) ) ) )
         => ~ ! [R4: int,Q5: int] :
                ( ( A32 = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q5),R4) )
               => ( ( aa(int,int,sgn_sgn(int),R4) = aa(int,int,sgn_sgn(int),A22) )
                 => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,abs_abs(int),R4)),aa(int,int,abs_abs(int),A22))
                   => ( A1 != aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Q5),A22)),R4) ) ) ) ) ) ) ) ).

% eucl_rel_int.cases
tff(fact_2574_eucl__rel__int_Osimps,axiom,
    ! [A1: int,A22: int,A32: product_prod(int,int)] :
      ( eucl_rel_int(A1,A22,A32)
    <=> ( ? [K4: int] :
            ( ( A1 = K4 )
            & ( A22 = zero_zero(int) )
            & ( A32 = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),K4) ) )
        | ? [L4: int,K4: int,Q6: int] :
            ( ( A1 = K4 )
            & ( A22 = L4 )
            & ( A32 = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q6),zero_zero(int)) )
            & ( L4 != zero_zero(int) )
            & ( K4 = aa(int,int,aa(int,fun(int,int),times_times(int),Q6),L4) ) )
        | ? [R5: int,L4: int,K4: int,Q6: int] :
            ( ( A1 = K4 )
            & ( A22 = L4 )
            & ( A32 = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q6),R5) )
            & ( aa(int,int,sgn_sgn(int),R5) = aa(int,int,sgn_sgn(int),L4) )
            & aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,abs_abs(int),R5)),aa(int,int,abs_abs(int),L4))
            & ( K4 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Q6),L4)),R5) ) ) ) ) ).

% eucl_rel_int.simps
tff(fact_2575_div__noneq__sgn__abs,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero(int) )
     => ( ( aa(int,int,sgn_sgn(int),K) != aa(int,int,sgn_sgn(int),L) )
       => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,abs_abs(int),K)),aa(int,int,abs_abs(int),L)))),aa($o,int,zero_neq_one_of_bool(int),~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),L),K))) ) ) ) ).

% div_noneq_sgn_abs
tff(fact_2576_prod_Otriangle__reindex__eq,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,fun(nat,A)),N: nat] : aa(set(product_prod(nat,nat)),A,aa(fun(product_prod(nat,nat),A),fun(set(product_prod(nat,nat)),A),groups7121269368397514597t_prod(product_prod(nat,nat),A),aa(fun(nat,fun(nat,A)),fun(product_prod(nat,nat),A),product_case_prod(nat,nat,A),G)),aa(fun(product_prod(nat,nat),$o),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aTP_Lamp_eq(nat,fun(nat,fun(nat,$o)),N)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_hs(fun(nat,fun(nat,A)),fun(nat,A),G)),aa(nat,set(nat),set_ord_atMost(nat),N)) ) ).

% prod.triangle_reindex_eq
tff(fact_2577_drop__bit__rec,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A3: A] :
          bit_se4197421643247451524op_bit(A,N,A3) = $ite(N = zero_zero(nat),A3,bit_se4197421643247451524op_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))) ) ).

% drop_bit_rec
tff(fact_2578_finite__compl,axiom,
    ! [A: $tType,A4: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( aa(set(A),$o,finite_finite2(A),aa(set(A),set(A),uminus_uminus(set(A)),A4))
      <=> aa(set(A),$o,finite_finite2(A),top_top(set(A))) ) ) ).

% finite_compl
tff(fact_2579_finite__Collect__le__nat,axiom,
    ! [K: nat] : aa(set(nat),$o,finite_finite2(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_bj(nat,fun(nat,$o)),K))) ).

% finite_Collect_le_nat
tff(fact_2580_finite__Collect__less__nat,axiom,
    ! [K: nat] : aa(set(nat),$o,finite_finite2(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_bk(nat,fun(nat,$o)),K))) ).

% finite_Collect_less_nat
tff(fact_2581_finite__Collect__not,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),P))
     => ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_aj(fun(A,$o),fun(A,$o),P)))
      <=> aa(set(A),$o,finite_finite2(A),top_top(set(A))) ) ) ).

% finite_Collect_not
tff(fact_2582_finite__Collect__subsets,axiom,
    ! [A: $tType,A4: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => aa(set(set(A)),$o,finite_finite2(set(A)),aa(fun(set(A),$o),set(set(A)),collect(set(A)),aTP_Lamp_in(set(A),fun(set(A),$o),A4))) ) ).

% finite_Collect_subsets
tff(fact_2583_finite__finite__vimage__IntI,axiom,
    ! [A: $tType,B: $tType,F4: set(A),H2: fun(B,A),A4: set(B)] :
      ( aa(set(A),$o,finite_finite2(A),F4)
     => ( ! [Y2: A] :
            ( aa(set(A),$o,member(A,Y2),F4)
           => aa(set(B),$o,finite_finite2(B),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(set(A),set(B),aa(fun(B,A),fun(set(A),set(B)),vimage(B,A),H2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y2),bot_bot(set(A))))),A4)) )
       => aa(set(B),$o,finite_finite2(B),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(set(A),set(B),aa(fun(B,A),fun(set(A),set(B)),vimage(B,A),H2),F4)),A4)) ) ) ).

% finite_finite_vimage_IntI
tff(fact_2584_finite__Collect__conjI,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] :
      ( ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),P))
        | aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),Q)) )
     => aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_am(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q))) ) ).

% finite_Collect_conjI
tff(fact_2585_finite__Collect__disjI,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] :
      ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_ap(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q)))
    <=> ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),P))
        & aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),Q)) ) ) ).

% finite_Collect_disjI
tff(fact_2586_finite__interval__int1,axiom,
    ! [A3: int,B2: int] : aa(set(int),$o,finite_finite2(int),aa(fun(int,$o),set(int),collect(int),aa(int,fun(int,$o),aTP_Lamp_io(int,fun(int,fun(int,$o)),A3),B2))) ).

% finite_interval_int1
tff(fact_2587_finite__interval__int4,axiom,
    ! [A3: int,B2: int] : aa(set(int),$o,finite_finite2(int),aa(fun(int,$o),set(int),collect(int),aa(int,fun(int,$o),aTP_Lamp_ip(int,fun(int,fun(int,$o)),A3),B2))) ).

% finite_interval_int4
tff(fact_2588_finite__Plus__UNIV__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( aa(set(sum_sum(A,B)),$o,finite_finite2(sum_sum(A,B)),top_top(set(sum_sum(A,B))))
    <=> ( aa(set(A),$o,finite_finite2(A),top_top(set(A)))
        & aa(set(B),$o,finite_finite2(B),top_top(set(B))) ) ) ).

% finite_Plus_UNIV_iff
tff(fact_2589_finite__Int,axiom,
    ! [A: $tType,F4: set(A),G4: set(A)] :
      ( ( aa(set(A),$o,finite_finite2(A),F4)
        | aa(set(A),$o,finite_finite2(A),G4) )
     => aa(set(A),$o,finite_finite2(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),F4),G4)) ) ).

% finite_Int
tff(fact_2590_finite__Un,axiom,
    ! [A: $tType,F4: set(A),G4: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),F4),G4))
    <=> ( aa(set(A),$o,finite_finite2(A),F4)
        & aa(set(A),$o,finite_finite2(A),G4) ) ) ).

% finite_Un
tff(fact_2591_finite__interval__int2,axiom,
    ! [A3: int,B2: int] : aa(set(int),$o,finite_finite2(int),aa(fun(int,$o),set(int),collect(int),aa(int,fun(int,$o),aTP_Lamp_iq(int,fun(int,fun(int,$o)),A3),B2))) ).

% finite_interval_int2
tff(fact_2592_finite__interval__int3,axiom,
    ! [A3: int,B2: int] : aa(set(int),$o,finite_finite2(int),aa(fun(int,$o),set(int),collect(int),aa(int,fun(int,$o),aTP_Lamp_ir(int,fun(int,fun(int,$o)),A3),B2))) ).

% finite_interval_int3
tff(fact_2593_infinite__UNIV__int,axiom,
    ~ aa(set(int),$o,finite_finite2(int),top_top(set(int))) ).

% infinite_UNIV_int
tff(fact_2594_finite__divisors__int,axiom,
    ! [I: int] :
      ( ( I != zero_zero(int) )
     => aa(set(int),$o,finite_finite2(int),aa(fun(int,$o),set(int),collect(int),aTP_Lamp_is(int,fun(int,$o),I))) ) ).

% finite_divisors_int
tff(fact_2595_infinite__UNIV__nat,axiom,
    ~ aa(set(nat),$o,finite_finite2(nat),top_top(set(nat))) ).

% infinite_UNIV_nat
tff(fact_2596_not__finite__existsD,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( ~ aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),P))
     => ? [X_1: A] : aa(A,$o,P,X_1) ) ).

% not_finite_existsD
tff(fact_2597_pigeonhole__infinite__rel,axiom,
    ! [A: $tType,B: $tType,A4: set(A),B3: set(B),R: fun(A,fun(B,$o))] :
      ( ~ aa(set(A),$o,finite_finite2(A),A4)
     => ( aa(set(B),$o,finite_finite2(B),B3)
       => ( ! [X2: A] :
              ( aa(set(A),$o,member(A,X2),A4)
             => ? [Xa3: B] :
                  ( aa(set(B),$o,member(B,Xa3),B3)
                  & aa(B,$o,aa(A,fun(B,$o),R,X2),Xa3) ) )
         => ? [X2: B] :
              ( aa(set(B),$o,member(B,X2),B3)
              & ~ aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(B,fun(A,$o),aa(fun(A,fun(B,$o)),fun(B,fun(A,$o)),aTP_Lamp_hd(set(A),fun(fun(A,fun(B,$o)),fun(B,fun(A,$o))),A4),R),X2))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_2598_sgn__integer__code,axiom,
    ! [K: code_integer] :
      aa(code_integer,code_integer,sgn_sgn(code_integer),K) = $ite(
        K = zero_zero(code_integer),
        zero_zero(code_integer),
        $ite(aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),K),zero_zero(code_integer)),aa(code_integer,code_integer,uminus_uminus(code_integer),one_one(code_integer)),one_one(code_integer)) ) ).

% sgn_integer_code
tff(fact_2599_finite_OemptyI,axiom,
    ! [A: $tType] : aa(set(A),$o,finite_finite2(A),bot_bot(set(A))) ).

% finite.emptyI
tff(fact_2600_infinite__imp__nonempty,axiom,
    ! [A: $tType,S: set(A)] :
      ( ~ aa(set(A),$o,finite_finite2(A),S)
     => ( S != bot_bot(set(A)) ) ) ).

% infinite_imp_nonempty
tff(fact_2601_finite__fun__UNIVD2,axiom,
    ! [A: $tType,B: $tType] :
      ( aa(set(fun(A,B)),$o,finite_finite2(fun(A,B)),top_top(set(fun(A,B))))
     => aa(set(B),$o,finite_finite2(B),top_top(set(B))) ) ).

% finite_fun_UNIVD2
tff(fact_2602_finite__UNIV,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => aa(set(A),$o,finite_finite2(A),top_top(set(A))) ) ).

% finite_UNIV
tff(fact_2603_ex__new__if__finite,axiom,
    ! [A: $tType,A4: set(A)] :
      ( ~ aa(set(A),$o,finite_finite2(A),top_top(set(A)))
     => ( aa(set(A),$o,finite_finite2(A),A4)
       => ? [A6: A] : ~ aa(set(A),$o,member(A,A6),A4) ) ) ).

% ex_new_if_finite
tff(fact_2604_infinite__UNIV__char__0,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ~ aa(set(A),$o,finite_finite2(A),top_top(set(A))) ) ).

% infinite_UNIV_char_0
tff(fact_2605_finite__UnI,axiom,
    ! [A: $tType,F4: set(A),G4: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),F4)
     => ( aa(set(A),$o,finite_finite2(A),G4)
       => aa(set(A),$o,finite_finite2(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),F4),G4)) ) ) ).

% finite_UnI
tff(fact_2606_Un__infinite,axiom,
    ! [A: $tType,S: set(A),T2: set(A)] :
      ( ~ aa(set(A),$o,finite_finite2(A),S)
     => ~ aa(set(A),$o,finite_finite2(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),S),T2)) ) ).

% Un_infinite
tff(fact_2607_infinite__Un,axiom,
    ! [A: $tType,S: set(A),T2: set(A)] :
      ( ~ aa(set(A),$o,finite_finite2(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),S),T2))
    <=> ( ~ aa(set(A),$o,finite_finite2(A),S)
        | ~ aa(set(A),$o,finite_finite2(A),T2) ) ) ).

% infinite_Un
tff(fact_2608_finite__has__minimal,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A4: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ? [X2: A] :
                ( aa(set(A),$o,member(A,X2),A4)
                & ! [Xa3: A] :
                    ( aa(set(A),$o,member(A,Xa3),A4)
                   => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa3),X2)
                     => ( X2 = Xa3 ) ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_2609_finite__has__maximal,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A4: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ? [X2: A] :
                ( aa(set(A),$o,member(A,X2),A4)
                & ! [Xa3: A] :
                    ( aa(set(A),$o,member(A,Xa3),A4)
                   => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),Xa3)
                     => ( X2 = Xa3 ) ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_2610_finite_Ocases,axiom,
    ! [A: $tType,A3: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( ( A3 != bot_bot(set(A)) )
       => ~ ! [A8: set(A)] :
              ( ? [A6: A] : A3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A6),A8)
             => ~ aa(set(A),$o,finite_finite2(A),A8) ) ) ) ).

% finite.cases
tff(fact_2611_finite_Osimps,axiom,
    ! [A: $tType,A3: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A3)
    <=> ( ( A3 = bot_bot(set(A)) )
        | ? [A9: set(A),A10: A] :
            ( ( A3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A10),A9) )
            & aa(set(A),$o,finite_finite2(A),A9) ) ) ) ).

% finite.simps
tff(fact_2612_finite__induct,axiom,
    ! [A: $tType,F4: set(A),P: fun(set(A),$o)] :
      ( aa(set(A),$o,finite_finite2(A),F4)
     => ( aa(set(A),$o,P,bot_bot(set(A)))
       => ( ! [X2: A,F5: set(A)] :
              ( aa(set(A),$o,finite_finite2(A),F5)
             => ( ~ aa(set(A),$o,member(A,X2),F5)
               => ( aa(set(A),$o,P,F5)
                 => aa(set(A),$o,P,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X2),F5)) ) ) )
         => aa(set(A),$o,P,F4) ) ) ) ).

% finite_induct
tff(fact_2613_finite__ne__induct,axiom,
    ! [A: $tType,F4: set(A),P: fun(set(A),$o)] :
      ( aa(set(A),$o,finite_finite2(A),F4)
     => ( ( F4 != bot_bot(set(A)) )
       => ( ! [X2: A] : aa(set(A),$o,P,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X2),bot_bot(set(A))))
         => ( ! [X2: A,F5: set(A)] :
                ( aa(set(A),$o,finite_finite2(A),F5)
               => ( ( F5 != bot_bot(set(A)) )
                 => ( ~ aa(set(A),$o,member(A,X2),F5)
                   => ( aa(set(A),$o,P,F5)
                     => aa(set(A),$o,P,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X2),F5)) ) ) ) )
           => aa(set(A),$o,P,F4) ) ) ) ) ).

% finite_ne_induct
tff(fact_2614_infinite__finite__induct,axiom,
    ! [A: $tType,P: fun(set(A),$o),A4: set(A)] :
      ( ! [A8: set(A)] :
          ( ~ aa(set(A),$o,finite_finite2(A),A8)
         => aa(set(A),$o,P,A8) )
     => ( aa(set(A),$o,P,bot_bot(set(A)))
       => ( ! [X2: A,F5: set(A)] :
              ( aa(set(A),$o,finite_finite2(A),F5)
             => ( ~ aa(set(A),$o,member(A,X2),F5)
               => ( aa(set(A),$o,P,F5)
                 => aa(set(A),$o,P,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X2),F5)) ) ) )
         => aa(set(A),$o,P,A4) ) ) ) ).

% infinite_finite_induct
tff(fact_2615_finite__prod,axiom,
    ! [A: $tType,B: $tType] :
      ( aa(set(product_prod(A,B)),$o,finite_finite2(product_prod(A,B)),top_top(set(product_prod(A,B))))
    <=> ( aa(set(A),$o,finite_finite2(A),top_top(set(A)))
        & aa(set(B),$o,finite_finite2(B),top_top(set(B))) ) ) ).

% finite_prod
tff(fact_2616_finite__Prod__UNIV,axiom,
    ! [B: $tType,A: $tType] :
      ( aa(set(A),$o,finite_finite2(A),top_top(set(A)))
     => ( aa(set(B),$o,finite_finite2(B),top_top(set(B)))
       => aa(set(product_prod(A,B)),$o,finite_finite2(product_prod(A,B)),top_top(set(product_prod(A,B)))) ) ) ).

% finite_Prod_UNIV
tff(fact_2617_Finite__Set_Ofinite__set,axiom,
    ! [A: $tType] :
      ( aa(set(set(A)),$o,finite_finite2(set(A)),top_top(set(set(A))))
    <=> aa(set(A),$o,finite_finite2(A),top_top(set(A))) ) ).

% Finite_Set.finite_set
tff(fact_2618_finite__subset__induct_H,axiom,
    ! [A: $tType,F4: set(A),A4: set(A),P: fun(set(A),$o)] :
      ( aa(set(A),$o,finite_finite2(A),F4)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),F4),A4)
       => ( aa(set(A),$o,P,bot_bot(set(A)))
         => ( ! [A6: A,F5: set(A)] :
                ( aa(set(A),$o,finite_finite2(A),F5)
               => ( aa(set(A),$o,member(A,A6),A4)
                 => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),F5),A4)
                   => ( ~ aa(set(A),$o,member(A,A6),F5)
                     => ( aa(set(A),$o,P,F5)
                       => aa(set(A),$o,P,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A6),F5)) ) ) ) ) )
           => aa(set(A),$o,P,F4) ) ) ) ) ).

% finite_subset_induct'
tff(fact_2619_finite__subset__induct,axiom,
    ! [A: $tType,F4: set(A),A4: set(A),P: fun(set(A),$o)] :
      ( aa(set(A),$o,finite_finite2(A),F4)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),F4),A4)
       => ( aa(set(A),$o,P,bot_bot(set(A)))
         => ( ! [A6: A,F5: set(A)] :
                ( aa(set(A),$o,finite_finite2(A),F5)
               => ( aa(set(A),$o,member(A,A6),A4)
                 => ( ~ aa(set(A),$o,member(A,A6),F5)
                   => ( aa(set(A),$o,P,F5)
                     => aa(set(A),$o,P,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A6),F5)) ) ) ) )
           => aa(set(A),$o,P,F4) ) ) ) ) ).

% finite_subset_induct
tff(fact_2620_infinite__remove,axiom,
    ! [A: $tType,S: set(A),A3: A] :
      ( ~ aa(set(A),$o,finite_finite2(A),S)
     => ~ aa(set(A),$o,finite_finite2(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A))))) ) ).

% infinite_remove
tff(fact_2621_infinite__coinduct,axiom,
    ! [A: $tType,X5: fun(set(A),$o),A4: set(A)] :
      ( aa(set(A),$o,X5,A4)
     => ( ! [A8: set(A)] :
            ( aa(set(A),$o,X5,A8)
           => ? [X3: A] :
                ( aa(set(A),$o,member(A,X3),A8)
                & ( aa(set(A),$o,X5,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A8),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X3),bot_bot(set(A)))))
                  | ~ aa(set(A),$o,finite_finite2(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A8),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X3),bot_bot(set(A))))) ) ) )
       => ~ aa(set(A),$o,finite_finite2(A),A4) ) ) ).

% infinite_coinduct
tff(fact_2622_finite__empty__induct,axiom,
    ! [A: $tType,A4: set(A),P: fun(set(A),$o)] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( aa(set(A),$o,P,A4)
       => ( ! [A6: A,A8: set(A)] :
              ( aa(set(A),$o,finite_finite2(A),A8)
             => ( aa(set(A),$o,member(A,A6),A8)
               => ( aa(set(A),$o,P,A8)
                 => aa(set(A),$o,P,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A8),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A6),bot_bot(set(A))))) ) ) )
         => aa(set(A),$o,P,bot_bot(set(A))) ) ) ) ).

% finite_empty_induct
tff(fact_2623_finite__remove__induct,axiom,
    ! [A: $tType,B3: set(A),P: fun(set(A),$o)] :
      ( aa(set(A),$o,finite_finite2(A),B3)
     => ( aa(set(A),$o,P,bot_bot(set(A)))
       => ( ! [A8: set(A)] :
              ( aa(set(A),$o,finite_finite2(A),A8)
             => ( ( A8 != bot_bot(set(A)) )
               => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A8),B3)
                 => ( ! [X3: A] :
                        ( aa(set(A),$o,member(A,X3),A8)
                       => aa(set(A),$o,P,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A8),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X3),bot_bot(set(A))))) )
                   => aa(set(A),$o,P,A8) ) ) ) )
         => aa(set(A),$o,P,B3) ) ) ) ).

% finite_remove_induct
tff(fact_2624_remove__induct,axiom,
    ! [A: $tType,P: fun(set(A),$o),B3: set(A)] :
      ( aa(set(A),$o,P,bot_bot(set(A)))
     => ( ( ~ aa(set(A),$o,finite_finite2(A),B3)
         => aa(set(A),$o,P,B3) )
       => ( ! [A8: set(A)] :
              ( aa(set(A),$o,finite_finite2(A),A8)
             => ( ( A8 != bot_bot(set(A)) )
               => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A8),B3)
                 => ( ! [X3: A] :
                        ( aa(set(A),$o,member(A,X3),A8)
                       => aa(set(A),$o,P,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A8),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X3),bot_bot(set(A))))) )
                   => aa(set(A),$o,P,A8) ) ) ) )
         => aa(set(A),$o,P,B3) ) ) ) ).

% remove_induct
tff(fact_2625_finite__induct__select,axiom,
    ! [A: $tType,S: set(A),P: fun(set(A),$o)] :
      ( aa(set(A),$o,finite_finite2(A),S)
     => ( aa(set(A),$o,P,bot_bot(set(A)))
       => ( ! [T5: set(A)] :
              ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),T5),S)
             => ( aa(set(A),$o,P,T5)
               => ? [X3: A] :
                    ( aa(set(A),$o,member(A,X3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S),T5))
                    & aa(set(A),$o,P,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X3),T5)) ) ) )
         => aa(set(A),$o,P,S) ) ) ) ).

% finite_induct_select
tff(fact_2626_finite__option__UNIV,axiom,
    ! [A: $tType] :
      ( aa(set(option(A)),$o,finite_finite2(option(A)),top_top(set(option(A))))
    <=> aa(set(A),$o,finite_finite2(A),top_top(set(A))) ) ).

% finite_option_UNIV
tff(fact_2627_diff__preserves__multiset,axiom,
    ! [A: $tType,M4: fun(A,nat),N4: fun(A,nat)] :
      ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_it(fun(A,nat),fun(A,$o),M4)))
     => aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,nat),fun(A,$o),aTP_Lamp_iu(fun(A,nat),fun(fun(A,nat),fun(A,$o)),M4),N4))) ) ).

% diff_preserves_multiset
tff(fact_2628_add__mset__in__multiset,axiom,
    ! [A: $tType,M4: fun(A,nat),A3: A] :
      ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_it(fun(A,nat),fun(A,$o),M4)))
     => aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(A,fun(A,$o),aTP_Lamp_iv(fun(A,nat),fun(A,fun(A,$o)),M4),A3))) ) ).

% add_mset_in_multiset
tff(fact_2629_finite__linorder__min__induct,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),P: fun(set(A),$o)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),$o,P,bot_bot(set(A)))
           => ( ! [B5: A,A8: set(A)] :
                  ( aa(set(A),$o,finite_finite2(A),A8)
                 => ( ! [X3: A] :
                        ( aa(set(A),$o,member(A,X3),A8)
                       => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B5),X3) )
                   => ( aa(set(A),$o,P,A8)
                     => aa(set(A),$o,P,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B5),A8)) ) ) )
             => aa(set(A),$o,P,A4) ) ) ) ) ).

% finite_linorder_min_induct
tff(fact_2630_finite__linorder__max__induct,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),P: fun(set(A),$o)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),$o,P,bot_bot(set(A)))
           => ( ! [B5: A,A8: set(A)] :
                  ( aa(set(A),$o,finite_finite2(A),A8)
                 => ( ! [X3: A] :
                        ( aa(set(A),$o,member(A,X3),A8)
                       => aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),B5) )
                   => ( aa(set(A),$o,P,A8)
                     => aa(set(A),$o,P,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B5),A8)) ) ) )
             => aa(set(A),$o,P,A4) ) ) ) ) ).

% finite_linorder_max_induct
tff(fact_2631_finite__ranking__induct,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [S: set(A),P: fun(set(A),$o),F: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),S)
         => ( aa(set(A),$o,P,bot_bot(set(A)))
           => ( ! [X2: A,S5: set(A)] :
                  ( aa(set(A),$o,finite_finite2(A),S5)
                 => ( ! [Y5: A] :
                        ( aa(set(A),$o,member(A,Y5),S5)
                       => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,Y5)),aa(A,B,F,X2)) )
                   => ( aa(set(A),$o,P,S5)
                     => aa(set(A),$o,P,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X2),S5)) ) ) )
             => aa(set(A),$o,P,S) ) ) ) ) ).

% finite_ranking_induct
tff(fact_2632_infinite__growing,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X5: set(A)] :
          ( ( X5 != bot_bot(set(A)) )
         => ( ! [X2: A] :
                ( aa(set(A),$o,member(A,X2),X5)
               => ? [Xa3: A] :
                    ( aa(set(A),$o,member(A,Xa3),X5)
                    & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X2),Xa3) ) )
           => ~ aa(set(A),$o,finite_finite2(A),X5) ) ) ) ).

% infinite_growing
tff(fact_2633_ex__min__if__finite,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [S: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),S)
         => ( ( S != bot_bot(set(A)) )
           => ? [X2: A] :
                ( aa(set(A),$o,member(A,X2),S)
                & ~ ? [Xa3: A] :
                      ( aa(set(A),$o,member(A,Xa3),S)
                      & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa3),X2) ) ) ) ) ) ).

% ex_min_if_finite
tff(fact_2634_filter__preserves__multiset,axiom,
    ! [A: $tType,M4: fun(A,nat),P: fun(A,$o)] :
      ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_it(fun(A,nat),fun(A,$o),M4)))
     => aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_iw(fun(A,nat),fun(fun(A,$o),fun(A,$o)),M4),P))) ) ).

% filter_preserves_multiset
tff(fact_2635_divide__int__def,axiom,
    ! [K: int,L: int] :
      aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) = $ite(
        L = zero_zero(int),
        zero_zero(int),
        $ite(aa(int,int,sgn_sgn(int),K) = aa(int,int,sgn_sgn(int),L),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(int,nat,nat2,aa(int,int,abs_abs(int),K))),aa(int,nat,nat2,aa(int,int,abs_abs(int),L)))),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(int,nat,nat2,aa(int,int,abs_abs(int),K))),aa(int,nat,nat2,aa(int,int,abs_abs(int),L)))),aa($o,nat,zero_neq_one_of_bool(nat),~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),L),K)))))) ) ).

% divide_int_def
tff(fact_2636_rec__nat__Suc__imp,axiom,
    ! [A: $tType,F: fun(nat,A),F1: A,F22: fun(nat,fun(A,A)),N: nat] :
      ( ( F = rec_nat(A,F1,F22) )
     => ( aa(nat,A,F,aa(nat,nat,suc,N)) = aa(A,A,aa(nat,fun(A,A),F22,N),aa(nat,A,F,N)) ) ) ).

% rec_nat_Suc_imp
tff(fact_2637_rec__nat__0__imp,axiom,
    ! [A: $tType,F: fun(nat,A),F1: A,F22: fun(nat,fun(A,A))] :
      ( ( F = rec_nat(A,F1,F22) )
     => ( aa(nat,A,F,zero_zero(nat)) = F1 ) ) ).

% rec_nat_0_imp
tff(fact_2638_sum__diff1_H__aux,axiom,
    ! [B: $tType,A: $tType] :
      ( ab_group_add(B)
     => ! [F4: set(A),I4: set(A),F: fun(A,B),I: A] :
          ( aa(set(A),$o,finite_finite2(A),F4)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_ix(set(A),fun(fun(A,B),fun(A,$o)),I4),F))),F4)
           => ( groups1027152243600224163dd_sum(A,B,F,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),I4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),I),bot_bot(set(A))))) = $ite(aa(set(A),$o,member(A,I),I4),aa(B,B,aa(B,fun(B,B),minus_minus(B),groups1027152243600224163dd_sum(A,B,F,I4)),aa(A,B,F,I)),groups1027152243600224163dd_sum(A,B,F,I4)) ) ) ) ) ).

% sum_diff1'_aux
tff(fact_2639_even__sum__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_parity(B)
     => ! [A4: set(A),F: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F),A4))
          <=> aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(set(A),nat,finite_card(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_iy(set(A),fun(fun(A,B),fun(A,$o)),A4),F)))) ) ) ) ).

% even_sum_iff
tff(fact_2640_nat__not__finite,axiom,
    ~ aa(set(nat),$o,finite_finite2(nat),top_top(set(nat))) ).

% nat_not_finite
tff(fact_2641_card__UNIV__unit,axiom,
    aa(set(product_unit),nat,finite_card(product_unit),top_top(set(product_unit))) = one_one(nat) ).

% card_UNIV_unit
tff(fact_2642_card__Collect__less__nat,axiom,
    ! [N: nat] : aa(set(nat),nat,finite_card(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_bk(nat,fun(nat,$o)),N))) = N ).

% card_Collect_less_nat
tff(fact_2643_card__eq__UNIV,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [S: set(A)] :
          ( ( aa(set(A),nat,finite_card(A),S) = aa(set(A),nat,finite_card(A),top_top(set(A))) )
        <=> ( S = top_top(set(A)) ) ) ) ).

% card_eq_UNIV
tff(fact_2644_card__eq__UNIV2,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [S: set(A)] :
          ( ( aa(set(A),nat,finite_card(A),top_top(set(A))) = aa(set(A),nat,finite_card(A),S) )
        <=> ( S = top_top(set(A)) ) ) ) ).

% card_eq_UNIV2
tff(fact_2645_card__Collect__le__nat,axiom,
    ! [N: nat] : aa(set(nat),nat,finite_card(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_bj(nat,fun(nat,$o)),N))) = aa(nat,nat,suc,N) ).

% card_Collect_le_nat
tff(fact_2646_card__UNIV__bool,axiom,
    aa(set($o),nat,finite_card($o),top_top(set($o))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)) ).

% card_UNIV_bool
tff(fact_2647_card_Oempty,axiom,
    ! [A: $tType] : aa(set(A),nat,finite_card(A),bot_bot(set(A))) = zero_zero(nat) ).

% card.empty
tff(fact_2648_card__ge__UNIV,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [S: set(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),top_top(set(A)))),aa(set(A),nat,finite_card(A),S))
        <=> ( S = top_top(set(A)) ) ) ) ).

% card_ge_UNIV
tff(fact_2649_sum_Oempty_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [P3: fun(B,A)] : groups1027152243600224163dd_sum(B,A,P3,bot_bot(set(B))) = zero_zero(A) ) ).

% sum.empty'
tff(fact_2650_prod__constant,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [Y: A,A4: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aTP_Lamp_iz(A,fun(B,A),Y)),A4) = aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(set(B),nat,finite_card(B),A4)) ) ).

% prod_constant
tff(fact_2651_card__0__eq,axiom,
    ! [A: $tType,A4: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( ( aa(set(A),nat,finite_card(A),A4) = zero_zero(nat) )
      <=> ( A4 = bot_bot(set(A)) ) ) ) ).

% card_0_eq
tff(fact_2652_nat__1,axiom,
    aa(int,nat,nat2,one_one(int)) = aa(nat,nat,suc,zero_zero(nat)) ).

% nat_1
tff(fact_2653_nat__neg__numeral,axiom,
    ! [K: num] : aa(int,nat,nat2,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K))) = zero_zero(nat) ).

% nat_neg_numeral
tff(fact_2654_nat__zminus__int,axiom,
    ! [N: nat] : aa(int,nat,nat2,aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N))) = zero_zero(nat) ).

% nat_zminus_int
tff(fact_2655_sum__constant,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(A)
     => ! [Y: A,A4: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aTP_Lamp_ja(A,fun(B,A),Y)),A4) = 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),A4))),Y) ) ).

% sum_constant
tff(fact_2656_card__Diff__insert,axiom,
    ! [A: $tType,A3: A,A4: set(A),B3: set(A)] :
      ( aa(set(A),$o,member(A,A3),A4)
     => ( ~ aa(set(A),$o,member(A,A3),B3)
       => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),B3))) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),B3))),one_one(nat)) ) ) ) ).

% card_Diff_insert
tff(fact_2657_card__atLeastAtMost__int,axiom,
    ! [L: int,U: int] : aa(set(int),nat,finite_card(int),set_or1337092689740270186AtMost(int,L,U)) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),U),L)),one_one(int))) ).

% card_atLeastAtMost_int
tff(fact_2658_sum_Oinsert_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [I4: set(A),P3: fun(A,B),I: A] :
          ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_hj(set(A),fun(fun(A,B),fun(A,$o)),I4),P3)))
         => ( groups1027152243600224163dd_sum(A,B,P3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),I),I4)) = $ite(aa(set(A),$o,member(A,I),I4),groups1027152243600224163dd_sum(A,B,P3,I4),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,P3,I)),groups1027152243600224163dd_sum(A,B,P3,I4))) ) ) ) ).

% sum.insert'
tff(fact_2659_card__doubleton__eq__2__iff,axiom,
    ! [A: $tType,A3: A,B2: A] :
      ( ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A))))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)) )
    <=> ( A3 != B2 ) ) ).

% card_doubleton_eq_2_iff
tff(fact_2660_one__less__nat__eq,axiom,
    ! [Z2: int] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),aa(int,nat,nat2,Z2))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),one_one(int)),Z2) ) ).

% one_less_nat_eq
tff(fact_2661_sum__of__bool__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(B)
     => ! [A4: set(A),P: fun(A,$o)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),$o,finite_finite2(A),A4)
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aTP_Lamp_jb(fun(A,$o),fun(A,B),P)),A4) = aa(nat,B,semiring_1_of_nat(B),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),aa(fun(A,$o),set(A),collect(A),P)))) ) ) ) ) ).

% sum_of_bool_eq
tff(fact_2662_nat__numeral__diff__1,axiom,
    ! [V: num] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(num,nat,numeral_numeral(nat),V)),one_one(nat)) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(num,int,numeral_numeral(int),V)),one_one(int))) ).

% nat_numeral_diff_1
tff(fact_2663_n__subsets,axiom,
    ! [A: $tType,A4: set(A),K: nat] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( aa(set(set(A)),nat,finite_card(set(A)),aa(fun(set(A),$o),set(set(A)),collect(set(A)),aa(nat,fun(set(A),$o),aTP_Lamp_jc(set(A),fun(nat,fun(set(A),$o)),A4),K))) = aa(nat,nat,binomial(aa(set(A),nat,finite_card(A),A4)),K) ) ) ).

% n_subsets
tff(fact_2664_sum_Onon__neutral_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(B,A),I4: set(B)] : groups1027152243600224163dd_sum(B,A,G,aa(fun(B,$o),set(B),collect(B),aa(set(B),fun(B,$o),aTP_Lamp_jd(fun(B,A),fun(set(B),fun(B,$o)),G),I4))) = groups1027152243600224163dd_sum(B,A,G,I4) ) ).

% sum.non_neutral'
tff(fact_2665_card__eq__UNIV__imp__eq__UNIV,axiom,
    ! [A: $tType,A4: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),top_top(set(A)))
     => ( ( aa(set(A),nat,finite_card(A),A4) = aa(set(A),nat,finite_card(A),top_top(set(A))) )
       => ( A4 = top_top(set(A)) ) ) ) ).

% card_eq_UNIV_imp_eq_UNIV
tff(fact_2666_nat__one__as__int,axiom,
    one_one(nat) = aa(int,nat,nat2,one_one(int)) ).

% nat_one_as_int
tff(fact_2667_sum__multicount__gen,axiom,
    ! [A: $tType,B: $tType,S2: set(A),T3: set(B),R: fun(A,fun(B,$o)),K: fun(B,nat)] :
      ( aa(set(A),$o,finite_finite2(A),S2)
     => ( aa(set(B),$o,finite_finite2(B),T3)
       => ( ! [X2: B] :
              ( aa(set(B),$o,member(B,X2),T3)
             => ( aa(set(A),nat,finite_card(A),aa(fun(A,$o),set(A),collect(A),aa(B,fun(A,$o),aa(fun(A,fun(B,$o)),fun(B,fun(A,$o)),aTP_Lamp_hd(set(A),fun(fun(A,fun(B,$o)),fun(B,fun(A,$o))),S2),R),X2))) = aa(B,nat,K,X2) ) )
         => ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aa(fun(A,fun(B,$o)),fun(A,nat),aTP_Lamp_je(set(B),fun(fun(A,fun(B,$o)),fun(A,nat)),T3),R)),S2) = aa(set(B),nat,aa(fun(B,nat),fun(set(B),nat),groups7311177749621191930dd_sum(B,nat),K),T3) ) ) ) ) ).

% sum_multicount_gen
tff(fact_2668_card__eq__sum,axiom,
    ! [A: $tType,A4: set(A)] : aa(set(A),nat,finite_card(A),A4) = aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aTP_Lamp_jf(A,nat)),A4) ).

% card_eq_sum
tff(fact_2669_is__singleton__altdef,axiom,
    ! [A: $tType,A4: set(A)] :
      ( is_singleton(A,A4)
    <=> ( aa(set(A),nat,finite_card(A),A4) = one_one(nat) ) ) ).

% is_singleton_altdef
tff(fact_2670_sum_Odistrib__triv_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [I4: set(A),G: fun(A,B),H2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),I4)
         => ( groups1027152243600224163dd_sum(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_jg(fun(A,B),fun(fun(A,B),fun(A,B)),G),H2),I4) = aa(B,B,aa(B,fun(B,B),plus_plus(B),groups1027152243600224163dd_sum(A,B,G,I4)),groups1027152243600224163dd_sum(A,B,H2,I4)) ) ) ) ).

% sum.distrib_triv'
tff(fact_2671_card__eq__0__iff,axiom,
    ! [A: $tType,A4: set(A)] :
      ( ( aa(set(A),nat,finite_card(A),A4) = zero_zero(nat) )
    <=> ( ( A4 = bot_bot(set(A)) )
        | ~ aa(set(A),$o,finite_finite2(A),A4) ) ) ).

% card_eq_0_iff
tff(fact_2672_card__1__singletonE,axiom,
    ! [A: $tType,A4: set(A)] :
      ( ( aa(set(A),nat,finite_card(A),A4) = one_one(nat) )
     => ~ ! [X2: A] : A4 != aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X2),bot_bot(set(A))) ) ).

% card_1_singletonE
tff(fact_2673_card__Un__le,axiom,
    ! [A: $tType,A4: set(A),B3: set(A)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(A),nat,finite_card(A),A4)),aa(set(A),nat,finite_card(A),B3))) ).

% card_Un_le
tff(fact_2674_card__less,axiom,
    ! [M4: set(nat),I: nat] :
      ( aa(set(nat),$o,member(nat,zero_zero(nat)),M4)
     => ( aa(set(nat),nat,finite_card(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_jh(set(nat),fun(nat,fun(nat,$o)),M4),I))) != zero_zero(nat) ) ) ).

% card_less
tff(fact_2675_card__less__Suc,axiom,
    ! [M4: set(nat),I: nat] :
      ( aa(set(nat),$o,member(nat,zero_zero(nat)),M4)
     => ( aa(nat,nat,suc,aa(set(nat),nat,finite_card(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_ji(set(nat),fun(nat,fun(nat,$o)),M4),I)))) = aa(set(nat),nat,finite_card(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_jh(set(nat),fun(nat,fun(nat,$o)),M4),I))) ) ) ).

% card_less_Suc
tff(fact_2676_card__less__Suc2,axiom,
    ! [M4: set(nat),I: nat] :
      ( ~ aa(set(nat),$o,member(nat,zero_zero(nat)),M4)
     => ( aa(set(nat),nat,finite_card(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_ji(set(nat),fun(nat,fun(nat,$o)),M4),I))) = aa(set(nat),nat,finite_card(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_jh(set(nat),fun(nat,fun(nat,$o)),M4),I))) ) ) ).

% card_less_Suc2
tff(fact_2677_nat__plus__as__int,axiom,
    ! [X3: nat,Xa3: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),X3),Xa3) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),X3)),aa(nat,int,semiring_1_of_nat(int),Xa3))) ).

% nat_plus_as_int
tff(fact_2678_nat__times__as__int,axiom,
    ! [X3: nat,Xa3: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),X3),Xa3) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,semiring_1_of_nat(int),X3)),aa(nat,int,semiring_1_of_nat(int),Xa3))) ).

% nat_times_as_int
tff(fact_2679_nat__minus__as__int,axiom,
    ! [X3: nat,Xa3: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),X3),Xa3) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,semiring_1_of_nat(int),X3)),aa(nat,int,semiring_1_of_nat(int),Xa3))) ).

% nat_minus_as_int
tff(fact_2680_sum__Suc,axiom,
    ! [A: $tType,F: fun(A,nat),A4: set(A)] : aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aTP_Lamp_jj(fun(A,nat),fun(A,nat),F)),A4) = 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),F),A4)),aa(set(A),nat,finite_card(A),A4)) ).

% sum_Suc
tff(fact_2681_sum__multicount,axiom,
    ! [A: $tType,B: $tType,S: set(A),T2: set(B),R: fun(A,fun(B,$o)),K: nat] :
      ( aa(set(A),$o,finite_finite2(A),S)
     => ( aa(set(B),$o,finite_finite2(B),T2)
       => ( ! [X2: B] :
              ( aa(set(B),$o,member(B,X2),T2)
             => ( aa(set(A),nat,finite_card(A),aa(fun(A,$o),set(A),collect(A),aa(B,fun(A,$o),aa(fun(A,fun(B,$o)),fun(B,fun(A,$o)),aTP_Lamp_hd(set(A),fun(fun(A,fun(B,$o)),fun(B,fun(A,$o))),S),R),X2))) = K ) )
         => ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aa(fun(A,fun(B,$o)),fun(A,nat),aTP_Lamp_je(set(B),fun(fun(A,fun(B,$o)),fun(A,nat)),T2),R)),S) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),aa(set(B),nat,finite_card(B),T2)) ) ) ) ) ).

% sum_multicount
tff(fact_2682_nat__div__as__int,axiom,
    ! [X3: nat,Xa3: nat] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),X3),Xa3) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(nat,int,semiring_1_of_nat(int),X3)),aa(nat,int,semiring_1_of_nat(int),Xa3))) ).

% nat_div_as_int
tff(fact_2683_nat__mod__as__int,axiom,
    ! [X3: nat,Xa3: nat] : modulo_modulo(nat,X3,Xa3) = aa(int,nat,nat2,modulo_modulo(int,aa(nat,int,semiring_1_of_nat(int),X3),aa(nat,int,semiring_1_of_nat(int),Xa3))) ).

% nat_mod_as_int
tff(fact_2684_card__gt__0__iff,axiom,
    ! [A: $tType,A4: set(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(set(A),nat,finite_card(A),A4))
    <=> ( ( A4 != bot_bot(set(A)) )
        & aa(set(A),$o,finite_finite2(A),A4) ) ) ).

% card_gt_0_iff
tff(fact_2685_card__Suc__eq,axiom,
    ! [A: $tType,A4: set(A),K: nat] :
      ( ( aa(set(A),nat,finite_card(A),A4) = aa(nat,nat,suc,K) )
    <=> ? [B6: A,B9: set(A)] :
          ( ( A4 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B6),B9) )
          & ~ aa(set(A),$o,member(A,B6),B9)
          & ( aa(set(A),nat,finite_card(A),B9) = K )
          & ( ( K = zero_zero(nat) )
           => ( B9 = bot_bot(set(A)) ) ) ) ) ).

% card_Suc_eq
tff(fact_2686_card__eq__SucD,axiom,
    ! [A: $tType,A4: set(A),K: nat] :
      ( ( aa(set(A),nat,finite_card(A),A4) = aa(nat,nat,suc,K) )
     => ? [B5: A,B10: set(A)] :
          ( ( A4 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B5),B10) )
          & ~ aa(set(A),$o,member(A,B5),B10)
          & ( aa(set(A),nat,finite_card(A),B10) = K )
          & ( ( K = zero_zero(nat) )
           => ( B10 = bot_bot(set(A)) ) ) ) ) ).

% card_eq_SucD
tff(fact_2687_card__1__singleton__iff,axiom,
    ! [A: $tType,A4: set(A)] :
      ( ( aa(set(A),nat,finite_card(A),A4) = aa(nat,nat,suc,zero_zero(nat)) )
    <=> ? [X4: A] : A4 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X4),bot_bot(set(A))) ) ).

% card_1_singleton_iff
tff(fact_2688_sum__bounded__below,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ordere6911136660526730532id_add(B)
        & semiring_1(B) )
     => ! [A4: set(A),K5: B,F: fun(A,B)] :
          ( ! [I2: A] :
              ( aa(set(A),$o,member(A,I2),A4)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),K5),aa(A,B,F,I2)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,semiring_1_of_nat(B),aa(set(A),nat,finite_card(A),A4))),K5)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F),A4)) ) ) ).

% sum_bounded_below
tff(fact_2689_sum__bounded__above,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ordere6911136660526730532id_add(B)
        & semiring_1(B) )
     => ! [A4: set(A),F: fun(A,B),K5: B] :
          ( ! [I2: A] :
              ( aa(set(A),$o,member(A,I2),A4)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,I2)),K5) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F),A4)),aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,semiring_1_of_nat(B),aa(set(A),nat,finite_card(A),A4))),K5)) ) ) ).

% sum_bounded_above
tff(fact_2690_finite__UNIV__card__ge__0,axiom,
    ! [A: $tType] :
      ( aa(set(A),$o,finite_finite2(A),top_top(set(A)))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(set(A),nat,finite_card(A),top_top(set(A)))) ) ).

% finite_UNIV_card_ge_0
tff(fact_2691_card__1__singletonI,axiom,
    ! [A: $tType,S: set(A),X: A] :
      ( aa(set(A),$o,finite_finite2(A),S)
     => ( ( aa(set(A),nat,finite_card(A),S) = one_one(nat) )
       => ( aa(set(A),$o,member(A,X),S)
         => ( S = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))) ) ) ) ) ).

% card_1_singletonI
tff(fact_2692_card__Diff1__le,axiom,
    ! [A: $tType,A4: set(A),X: A] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))))),aa(set(A),nat,finite_card(A),A4)) ).

% card_Diff1_le
tff(fact_2693_card__Un__Int,axiom,
    ! [A: $tType,A4: set(A),B3: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( aa(set(A),$o,finite_finite2(A),B3)
       => ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(A),nat,finite_card(A),A4)),aa(set(A),nat,finite_card(A),B3)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3))),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3))) ) ) ) ).

% card_Un_Int
tff(fact_2694_sum_Odistrib_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [I4: set(A),G: fun(A,B),H2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_hj(set(A),fun(fun(A,B),fun(A,$o)),I4),G)))
         => ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_hj(set(A),fun(fun(A,B),fun(A,$o)),I4),H2)))
           => ( groups1027152243600224163dd_sum(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_jg(fun(A,B),fun(fun(A,B),fun(A,B)),G),H2),I4) = aa(B,B,aa(B,fun(B,B),plus_plus(B),groups1027152243600224163dd_sum(A,B,G,I4)),groups1027152243600224163dd_sum(A,B,H2,I4)) ) ) ) ) ).

% sum.distrib'
tff(fact_2695_card__Diff__subset__Int,axiom,
    ! [A: $tType,A4: set(A),B3: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3))
     => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),B3)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,finite_card(A),A4)),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3))) ) ) ).

% card_Diff_subset_Int
tff(fact_2696_Suc__as__int,axiom,
    ! [X3: nat] : aa(nat,nat,suc,X3) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),X3)),one_one(int))) ).

% Suc_as_int
tff(fact_2697_sum_OG__def,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [P3: fun(B,A),I4: set(B)] :
          groups1027152243600224163dd_sum(B,A,P3,I4) = $ite(aa(set(B),$o,finite_finite2(B),aa(fun(B,$o),set(B),collect(B),aa(set(B),fun(B,$o),aTP_Lamp_jd(fun(B,A),fun(set(B),fun(B,$o)),P3),I4))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),P3),aa(fun(B,$o),set(B),collect(B),aa(set(B),fun(B,$o),aTP_Lamp_jd(fun(B,A),fun(set(B),fun(B,$o)),P3),I4))),zero_zero(A)) ) ).

% sum.G_def
tff(fact_2698_card__sum__le__nat__sum,axiom,
    ! [S: set(nat)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_dj(nat,nat)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(set(nat),nat,finite_card(nat),S)))),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_dj(nat,nat)),S)) ).

% card_sum_le_nat_sum
tff(fact_2699_subset__Collect__iff,axiom,
    ! [A: $tType,B3: set(A),A4: set(A),P: fun(A,$o)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),A4)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_bi(set(A),fun(fun(A,$o),fun(A,$o)),A4),P)))
      <=> ! [X4: A] :
            ( aa(set(A),$o,member(A,X4),B3)
           => aa(A,$o,P,X4) ) ) ) ).

% subset_Collect_iff
tff(fact_2700_subset__CollectI,axiom,
    ! [A: $tType,B3: set(A),A4: set(A),Q: fun(A,$o),P: fun(A,$o)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),A4)
     => ( ! [X2: A] :
            ( aa(set(A),$o,member(A,X2),B3)
           => ( aa(A,$o,Q,X2)
             => aa(A,$o,P,X2) ) )
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_bi(set(A),fun(fun(A,$o),fun(A,$o)),B3),Q))),aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_bi(set(A),fun(fun(A,$o),fun(A,$o)),A4),P))) ) ) ).

% subset_CollectI
tff(fact_2701_card__2__iff,axiom,
    ! [A: $tType,S: set(A)] :
      ( ( aa(set(A),nat,finite_card(A),S) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)) )
    <=> ? [X4: A,Y3: A] :
          ( ( S = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y3),bot_bot(set(A)))) )
          & ( X4 != Y3 ) ) ) ).

% card_2_iff
tff(fact_2702_card_Oremove,axiom,
    ! [A: $tType,A4: set(A),X: A] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( aa(set(A),$o,member(A,X),A4)
       => ( aa(set(A),nat,finite_card(A),A4) = 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)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))))) ) ) ) ).

% card.remove
tff(fact_2703_card_Oinsert__remove,axiom,
    ! [A: $tType,A4: set(A),X: A] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A4)) = 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)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))))) ) ) ).

% card.insert_remove
tff(fact_2704_card__Suc__Diff1,axiom,
    ! [A: $tType,A4: set(A),X: A] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( aa(set(A),$o,member(A,X),A4)
       => ( 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)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))))) = aa(set(A),nat,finite_card(A),A4) ) ) ) ).

% card_Suc_Diff1
tff(fact_2705_card__Diff1__less__iff,axiom,
    ! [A: $tType,A4: set(A),X: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))))),aa(set(A),nat,finite_card(A),A4))
    <=> ( aa(set(A),$o,finite_finite2(A),A4)
        & aa(set(A),$o,member(A,X),A4) ) ) ).

% card_Diff1_less_iff
tff(fact_2706_card__Diff2__less,axiom,
    ! [A: $tType,A4: set(A),X: A,Y: A] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( aa(set(A),$o,member(A,X),A4)
       => ( aa(set(A),$o,member(A,Y),A4)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))))),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y),bot_bot(set(A)))))),aa(set(A),nat,finite_card(A),A4)) ) ) ) ).

% card_Diff2_less
tff(fact_2707_card__Diff1__less,axiom,
    ! [A: $tType,A4: set(A),X: A] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( aa(set(A),$o,member(A,X),A4)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))))),aa(set(A),nat,finite_card(A),A4)) ) ) ).

% card_Diff1_less
tff(fact_2708_card__3__iff,axiom,
    ! [A: $tType,S: set(A)] :
      ( ( aa(set(A),nat,finite_card(A),S) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) )
    <=> ? [X4: A,Y3: A,Z4: A] :
          ( ( S = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Z4),bot_bot(set(A))))) )
          & ( X4 != Y3 )
          & ( Y3 != Z4 )
          & ( X4 != Z4 ) ) ) ).

% card_3_iff
tff(fact_2709_odd__card__imp__not__empty,axiom,
    ! [A: $tType,A4: set(A)] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(set(A),nat,finite_card(A),A4))
     => ( A4 != bot_bot(set(A)) ) ) ).

% odd_card_imp_not_empty
tff(fact_2710_card__Un__disjoint,axiom,
    ! [A: $tType,A4: set(A),B3: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( aa(set(A),$o,finite_finite2(A),B3)
       => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3) = bot_bot(set(A)) )
         => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(A),nat,finite_card(A),A4)),aa(set(A),nat,finite_card(A),B3)) ) ) ) ) ).

% card_Un_disjoint
tff(fact_2711_Suc__nat__eq__nat__zadd1,axiom,
    ! [Z2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z2)
     => ( aa(nat,nat,suc,aa(int,nat,nat2,Z2)) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),Z2)) ) ) ).

% Suc_nat_eq_nat_zadd1
tff(fact_2712_card__Diff__singleton__if,axiom,
    ! [A: $tType,A4: set(A),X: A] :
      aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))))) = $ite(aa(set(A),$o,member(A,X),A4),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,finite_card(A),A4)),one_one(nat)),aa(set(A),nat,finite_card(A),A4)) ).

% card_Diff_singleton_if
tff(fact_2713_card__Diff__singleton,axiom,
    ! [A: $tType,X: A,A4: set(A)] :
      ( aa(set(A),$o,member(A,X),A4)
     => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))))) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,finite_card(A),A4)),one_one(nat)) ) ) ).

% card_Diff_singleton
tff(fact_2714_nat__mult__distrib__neg,axiom,
    ! [Z2: int,Z7: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Z2),zero_zero(int))
     => ( aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),times_times(int),Z2),Z7)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(int,nat,nat2,aa(int,int,uminus_uminus(int),Z2))),aa(int,nat,nat2,aa(int,int,uminus_uminus(int),Z7))) ) ) ).

% nat_mult_distrib_neg
tff(fact_2715_diff__nat__eq__if,axiom,
    ! [Z2: int,Z7: int] :
      aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(int,nat,nat2,Z2)),aa(int,nat,nat2,Z7)) = $ite(
        aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z7),zero_zero(int)),
        aa(int,nat,nat2,Z2),
        $let(
          d: int,
          d:= aa(int,int,aa(int,fun(int,int),minus_minus(int),Z2),Z7),
          $ite(aa(int,$o,aa(int,fun(int,$o),ord_less(int),d),zero_zero(int)),zero_zero(nat),aa(int,nat,nat2,d)) ) ) ).

% diff_nat_eq_if
tff(fact_2716_prod__le__power,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_semidom(B)
     => ! [A4: set(A),F: fun(A,B),N: B,K: nat] :
          ( ! [I2: A] :
              ( aa(set(A),$o,member(A,I2),A4)
             => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F,I2))
                & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,I2)),N) ) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A4)),K)
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),one_one(B)),N)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F),A4)),aa(nat,B,aa(B,fun(nat,B),power_power(B),N),K)) ) ) ) ) ).

% prod_le_power
tff(fact_2717_sum__bounded__above__divide,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_field(B)
     => ! [A4: set(A),F: fun(A,B),K5: B] :
          ( ! [I2: A] :
              ( aa(set(A),$o,member(A,I2),A4)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,I2)),aa(B,B,aa(B,fun(B,B),divide_divide(B),K5),aa(nat,B,semiring_1_of_nat(B),aa(set(A),nat,finite_card(A),A4)))) )
         => ( aa(set(A),$o,finite_finite2(A),A4)
           => ( ( A4 != bot_bot(set(A)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F),A4)),K5) ) ) ) ) ).

% sum_bounded_above_divide
tff(fact_2718_sum__bounded__above__strict,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ordere8940638589300402666id_add(B)
        & semiring_1(B) )
     => ! [A4: set(A),F: fun(A,B),K5: B] :
          ( ! [I2: A] :
              ( aa(set(A),$o,member(A,I2),A4)
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F,I2)),K5) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(set(A),nat,finite_card(A),A4))
           => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F),A4)),aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,semiring_1_of_nat(B),aa(set(A),nat,finite_card(A),A4))),K5)) ) ) ) ).

% sum_bounded_above_strict
tff(fact_2719_of__int__of__nat,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [K: int] :
          aa(int,A,ring_1_of_int(A),K) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int)),aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),aa(int,nat,nat2,aa(int,int,uminus_uminus(int),K)))),aa(nat,A,semiring_1_of_nat(A),aa(int,nat,nat2,K))) ) ).

% of_int_of_nat
tff(fact_2720_card__insert__le__m1,axiom,
    ! [A: $tType,N: nat,Y: set(A),X: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),Y)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)))
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),Y))),N) ) ) ).

% card_insert_le_m1
tff(fact_2721_prod__gen__delta,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [S: set(A),A3: A,B2: fun(A,B),C2: B] :
          ( aa(set(A),$o,finite_finite2(A),S)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),aa(B,fun(A,B),aa(fun(A,B),fun(B,fun(A,B)),aTP_Lamp_jk(A,fun(fun(A,B),fun(B,fun(A,B))),A3),B2),C2)),S) = $ite(aa(set(A),$o,member(A,A3),S),aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,B2,A3)),aa(nat,B,aa(B,fun(nat,B),power_power(B),C2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,finite_card(A),S)),one_one(nat)))),aa(nat,B,aa(B,fun(nat,B),power_power(B),C2),aa(set(A),nat,finite_card(A),S))) ) ) ) ).

% prod_gen_delta
tff(fact_2722_finite__transitivity__chain,axiom,
    ! [A: $tType,A4: set(A),R: fun(A,fun(A,$o))] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( ! [X2: A] : ~ aa(A,$o,aa(A,fun(A,$o),R,X2),X2)
       => ( ! [X2: A,Y2: A,Z3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),R,X2),Y2)
             => ( aa(A,$o,aa(A,fun(A,$o),R,Y2),Z3)
               => aa(A,$o,aa(A,fun(A,$o),R,X2),Z3) ) )
         => ( ! [X2: A] :
                ( aa(set(A),$o,member(A,X2),A4)
               => ? [Y5: A] :
                    ( aa(set(A),$o,member(A,Y5),A4)
                    & aa(A,$o,aa(A,fun(A,$o),R,X2),Y5) ) )
           => ( A4 = bot_bot(set(A)) ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_2723_sum__diff1_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ab_group_add(B)
     => ! [I4: set(A),F: fun(A,B),I: A] :
          ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_ix(set(A),fun(fun(A,B),fun(A,$o)),I4),F)))
         => ( groups1027152243600224163dd_sum(A,B,F,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),I4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),I),bot_bot(set(A))))) = $ite(aa(set(A),$o,member(A,I),I4),aa(B,B,aa(B,fun(B,B),minus_minus(B),groups1027152243600224163dd_sum(A,B,F,I4)),aa(A,B,F,I)),groups1027152243600224163dd_sum(A,B,F,I4)) ) ) ) ).

% sum_diff1'
tff(fact_2724_finite__fun__UNIVD1,axiom,
    ! [B: $tType,A: $tType] :
      ( aa(set(fun(A,B)),$o,finite_finite2(fun(A,B)),top_top(set(fun(A,B))))
     => ( ( aa(set(B),nat,finite_card(B),top_top(set(B))) != aa(nat,nat,suc,zero_zero(nat)) )
       => aa(set(A),$o,finite_finite2(A),top_top(set(A))) ) ) ).

% finite_fun_UNIVD1
tff(fact_2725_card__UNIV__char,axiom,
    aa(set(char),nat,finite_card(char),top_top(set(char))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,one2))))))))) ).

% card_UNIV_char
tff(fact_2726_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),aa(num,num,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),aa(num,num,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_2727_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),aa(num,num,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),aa(num,num,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_2728_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),aa(num,num,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_2729_divmod__integer__eq__cases,axiom,
    ! [K: code_integer,L: code_integer] :
      code_divmod_integer(K,L) = $ite(
        K = zero_zero(code_integer),
        aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),zero_zero(code_integer)),zero_zero(code_integer)),
        $ite(
          L = zero_zero(code_integer),
          aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),zero_zero(code_integer)),K),
          aa(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer),aa(code_integer,fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)),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),
            $ite(aa(code_integer,code_integer,sgn_sgn(code_integer),K) = aa(code_integer,code_integer,sgn_sgn(code_integer),L),code_divmod_abs(K,L),aa(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer),aa(fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)),product_case_prod(code_integer,code_integer,product_prod(code_integer,code_integer)),aTP_Lamp_jl(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),L)),code_divmod_abs(K,L)))) ) ) ).

% divmod_integer_eq_cases
tff(fact_2730_bit_Odisj__one__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),X),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).

% bit.disj_one_right
tff(fact_2731_bit_Odisj__one__left,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,uminus_uminus(A),one_one(A))),X) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).

% bit.disj_one_left
tff(fact_2732_or__numerals_I2_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Y: num] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Y))) = aa(num,A,numeral_numeral(A),aa(num,num,bit1,Y)) ) ).

% or_numerals(2)
tff(fact_2733_or__numerals_I8_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,X))),one_one(A)) = aa(num,A,numeral_numeral(A),aa(num,num,bit1,X)) ) ).

% or_numerals(8)
tff(fact_2734_or__minus__numerals_I2_J,axiom,
    ! [N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),one_one(int)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N)))) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N))) ).

% or_minus_numerals(2)
tff(fact_2735_or__minus__numerals_I6_J,axiom,
    ! [N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N)))),one_one(int)) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N))) ).

% or_minus_numerals(6)
tff(fact_2736_bit_Odisj__cancel__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),X),aa(A,A,bit_ri4277139882892585799ns_not(A),X)) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).

% bit.disj_cancel_right
tff(fact_2737_bit_Odisj__cancel__left,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,bit_ri4277139882892585799ns_not(A),X)),X) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).

% bit.disj_cancel_left
tff(fact_2738_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),aa(num,num,bit0,X))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Y))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,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_2739_or__numerals_I1_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Y: num] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Y))) = aa(num,A,numeral_numeral(A),aa(num,num,bit1,Y)) ) ).

% or_numerals(1)
tff(fact_2740_or__numerals_I5_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,X))),one_one(A)) = aa(num,A,numeral_numeral(A),aa(num,num,bit1,X)) ) ).

% or_numerals(5)
tff(fact_2741_or__minus__minus__numerals,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = aa(int,int,bit_ri4277139882892585799ns_not(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(num,int,numeral_numeral(int),M)),one_one(int))),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(num,int,numeral_numeral(int),N)),one_one(int)))) ).

% or_minus_minus_numerals
tff(fact_2742_and__minus__minus__numerals,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = aa(int,int,bit_ri4277139882892585799ns_not(int),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(num,int,numeral_numeral(int),M)),one_one(int))),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(num,int,numeral_numeral(int),N)),one_one(int)))) ).

% and_minus_minus_numerals
tff(fact_2743_K__record__comp,axiom,
    ! [A: $tType,C: $tType,B: $tType,C2: B,F: fun(A,C),X3: A] : aa(A,B,aa(fun(A,C),fun(A,B),comp(C,B,A,aTP_Lamp_jm(B,fun(C,B),C2)),F),X3) = C2 ).

% K_record_comp
tff(fact_2744_UNIV__bool,axiom,
    top_top(set($o)) = aa(set($o),set($o),aa($o,fun(set($o),set($o)),insert2($o),$false),aa(set($o),set($o),aa($o,fun(set($o),set($o)),insert2($o),$true),bot_bot(set($o)))) ).

% UNIV_bool
tff(fact_2745_type__copy__map__comp0__undo,axiom,
    ! [E: $tType,A: $tType,C: $tType,B: $tType,D: $tType,F3: $tType,Rep: fun(A,B),Abs: fun(B,A),Rep2: fun(C,D),Abs2: fun(D,C),Rep3: fun(E,F3),Abs3: fun(F3,E),M4: fun(F3,D),M1: fun(B,D),M22: fun(F3,B)] :
      ( type_definition(A,B,Rep,Abs,top_top(set(B)))
     => ( type_definition(C,D,Rep2,Abs2,top_top(set(D)))
       => ( type_definition(E,F3,Rep3,Abs3,top_top(set(F3)))
         => ( ( aa(fun(E,F3),fun(E,C),comp(F3,C,E,aa(fun(F3,D),fun(F3,C),comp(D,C,F3,Abs2),M4)),Rep3) = aa(fun(E,A),fun(E,C),comp(A,C,E,aa(fun(A,B),fun(A,C),comp(B,C,A,aa(fun(B,D),fun(B,C),comp(D,C,B,Abs2),M1)),Rep)),aa(fun(E,F3),fun(E,A),comp(F3,A,E,aa(fun(F3,B),fun(F3,A),comp(B,A,F3,Abs),M22)),Rep3)) )
           => ( aa(fun(F3,B),fun(F3,D),comp(B,D,F3,M1),M22) = M4 ) ) ) ) ) ).

% type_copy_map_comp0_undo
tff(fact_2746_type__copy__map__comp0,axiom,
    ! [F3: $tType,D: $tType,B: $tType,A: $tType,C: $tType,E: $tType,Rep: fun(A,B),Abs: fun(B,A),M4: fun(C,D),M1: fun(B,D),M22: fun(C,B),F: fun(D,F3),G: fun(E,C)] :
      ( type_definition(A,B,Rep,Abs,top_top(set(B)))
     => ( ( M4 = aa(fun(C,B),fun(C,D),comp(B,D,C,M1),M22) )
       => ( aa(fun(E,C),fun(E,F3),comp(C,F3,E,aa(fun(C,D),fun(C,F3),comp(D,F3,C,F),M4)),G) = aa(fun(E,A),fun(E,F3),comp(A,F3,E,aa(fun(A,B),fun(A,F3),comp(B,F3,A,aa(fun(B,D),fun(B,F3),comp(D,F3,B,F),M1)),Rep)),aa(fun(E,C),fun(E,A),comp(C,A,E,aa(fun(C,B),fun(C,A),comp(B,A,C,Abs),M22)),G)) ) ) ) ).

% type_copy_map_comp0
tff(fact_2747_or_Ocomm__monoid__axioms,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => comm_monoid(A,bit_se1065995026697491101ons_or(A),zero_zero(A)) ) ).

% or.comm_monoid_axioms
tff(fact_2748_or_Omonoid__axioms,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => monoid(A,bit_se1065995026697491101ons_or(A),zero_zero(A)) ) ).

% or.monoid_axioms
tff(fact_2749_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_2750_set__bit__int__def,axiom,
    ! [N: nat,K: int] : bit_se5668285175392031749et_bit(int,N,K) = aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),bit_se4730199178511100633sh_bit(int,N,one_one(int))) ).

% set_bit_int_def
tff(fact_2751_set__bit__nat__def,axiom,
    ! [M: nat,N: nat] : bit_se5668285175392031749et_bit(nat,M,N) = aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),N),bit_se4730199178511100633sh_bit(nat,M,one_one(nat))) ).

% set_bit_nat_def
tff(fact_2752_or__not__numerals_I1_J,axiom,
    aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),one_one(int)),aa(int,int,bit_ri4277139882892585799ns_not(int),one_one(int))) = aa(int,int,bit_ri4277139882892585799ns_not(int),zero_zero(int)) ).

% or_not_numerals(1)
tff(fact_2753_set__bit__eq__or,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A3: A] : bit_se5668285175392031749et_bit(A,N,A3) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A3),bit_se4730199178511100633sh_bit(A,N,one_one(A))) ) ).

% set_bit_eq_or
tff(fact_2754_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_2755_or__not__numerals_I2_J,axiom,
    ! [N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),one_one(int)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,N)))) = aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,N))) ).

% or_not_numerals(2)
tff(fact_2756_or__not__numerals_I4_J,axiom,
    ! [M: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,M))),aa(int,int,bit_ri4277139882892585799ns_not(int),one_one(int))) = aa(int,int,bit_ri4277139882892585799ns_not(int),one_one(int)) ).

% or_not_numerals(4)
tff(fact_2757_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_2758_or__not__numerals_I3_J,axiom,
    ! [N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),one_one(int)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N)))) = aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,N))) ).

% or_not_numerals(3)
tff(fact_2759_or__not__numerals_I7_J,axiom,
    ! [M: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,M))),aa(int,int,bit_ri4277139882892585799ns_not(int),one_one(int))) = aa(int,int,bit_ri4277139882892585799ns_not(int),zero_zero(int)) ).

% or_not_numerals(7)
tff(fact_2760_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_2761_or__one__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A3),one_one(A)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa($o,A,zero_neq_one_of_bool(A),aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A3))) ) ).

% or_one_eq
tff(fact_2762_one__or__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),one_one(A)),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa($o,A,zero_neq_one_of_bool(A),aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A3))) ) ).

% one_or_eq
tff(fact_2763_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),aa(num,num,bit0,one2))),bit_se2239418461657761734s_mask(A,N))) ) ).

% mask_Suc_double
tff(fact_2764_or__not__numerals_I5_J,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,M))),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),aa(num,num,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),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),N))))) ).

% or_not_numerals(5)
tff(fact_2765_or__not__numerals_I9_J,axiom,
    ! [M: 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,M))),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),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),N))))) ).

% or_not_numerals(9)
tff(fact_2766_or__not__numerals_I8_J,axiom,
    ! [M: 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,M))),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),aa(num,num,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),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),N))))) ).

% or_not_numerals(8)
tff(fact_2767_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_2768_or__int__unfold,axiom,
    ! [K: int,L: int] :
      aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),L) = $ite(
        ( ( K = aa(int,int,uminus_uminus(int),one_one(int)) )
        | ( L = aa(int,int,uminus_uminus(int),one_one(int)) ) ),
        aa(int,int,uminus_uminus(int),one_one(int)),
        $ite(
          K = zero_zero(int),
          L,
          $ite(L = zero_zero(int),K,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),ord_max(int),modulo_modulo(int,K,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),modulo_modulo(int,L,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))))))) ) ) ).

% or_int_unfold
tff(fact_2769_or__minus__numerals_I5_J,axiom,
    ! [N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,N)))),one_one(int)) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit_or_not_num_neg(one2,bitM(N)))) ).

% or_minus_numerals(5)
tff(fact_2770_or__minus__numerals_I1_J,axiom,
    ! [N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),one_one(int)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,N)))) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit_or_not_num_neg(one2,bitM(N)))) ).

% or_minus_numerals(1)
tff(fact_2771_card__greaterThanLessThan__int,axiom,
    ! [L: int,U: int] : aa(set(int),nat,finite_card(int),set_or5935395276787703475ssThan(int,L,U)) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),minus_minus(int),U),aa(int,int,aa(int,fun(int,int),plus_plus(int),L),one_one(int)))) ).

% card_greaterThanLessThan_int
tff(fact_2772_card__partition,axiom,
    ! [A: $tType,C3: set(set(A)),K: nat] :
      ( aa(set(set(A)),$o,finite_finite2(set(A)),C3)
     => ( aa(set(A),$o,finite_finite2(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C3))
       => ( ! [C4: set(A)] :
              ( aa(set(set(A)),$o,member(set(A),C4),C3)
             => ( aa(set(A),nat,finite_card(A),C4) = K ) )
         => ( ! [C1: set(A),C22: set(A)] :
                ( aa(set(set(A)),$o,member(set(A),C1),C3)
               => ( aa(set(set(A)),$o,member(set(A),C22),C3)
                 => ( ( C1 != C22 )
                   => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),C1),C22) = bot_bot(set(A)) ) ) ) )
           => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),aa(set(set(A)),nat,finite_card(set(A)),C3)) = aa(set(A),nat,finite_card(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C3)) ) ) ) ) ) ).

% card_partition
tff(fact_2773_max_Oright__idem,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2)),B2) = aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2) ) ).

% max.right_idem
tff(fact_2774_max_Oleft__idem,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),A3),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2) ) ).

% max.left_idem
tff(fact_2775_max_Oidem,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),A3),A3) = A3 ) ).

% max.idem
tff(fact_2776_max_Obounded__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,C2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),C2)),A3)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A3) ) ) ) ).

% max.bounded_iff
tff(fact_2777_max_Oabsorb2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
         => ( aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2) = B2 ) ) ) ).

% max.absorb2
tff(fact_2778_max_Oabsorb1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3)
         => ( aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2) = A3 ) ) ) ).

% max.absorb1
tff(fact_2779_max__less__iff__conj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A,Z2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y)),Z2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Z2)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Z2) ) ) ) ).

% max_less_iff_conj
tff(fact_2780_max_Oabsorb4,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
         => ( aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2) = B2 ) ) ) ).

% max.absorb4
tff(fact_2781_max_Oabsorb3,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3)
         => ( aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2) = A3 ) ) ) ).

% max.absorb3
tff(fact_2782_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_2783_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_2784_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_2785_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_2786_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_2787_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_2788_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_2789_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_2790_greaterThanLessThan__empty__iff2,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B2: A] :
          ( ( bot_bot(set(A)) = set_or5935395276787703475ssThan(A,A3,B2) )
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3) ) ) ).

% greaterThanLessThan_empty_iff2
tff(fact_2791_greaterThanLessThan__empty__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B2: A] :
          ( ( set_or5935395276787703475ssThan(A,A3,B2) = bot_bot(set(A)) )
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3) ) ) ).

% greaterThanLessThan_empty_iff
tff(fact_2792_greaterThanLessThan__empty,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,K: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),K)
         => ( set_or5935395276787703475ssThan(A,K,L) = bot_bot(set(A)) ) ) ) ).

% greaterThanLessThan_empty
tff(fact_2793_max__number__of_I2_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V: num] :
          aa(A,A,aa(A,fun(A,A),ord_max(A),aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V)),aa(num,A,numeral_numeral(A),U)) ) ).

% max_number_of(2)
tff(fact_2794_max__number__of_I3_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V: num] :
          aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(num,A,numeral_numeral(A),V)) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(num,A,numeral_numeral(A),V)),aa(num,A,numeral_numeral(A),V),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))) ) ).

% max_number_of(3)
tff(fact_2795_max__number__of_I4_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V: num] :
          aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))) ) ).

% max_number_of(4)
tff(fact_2796_or__minus__numerals_I8_J,axiom,
    ! [N: num,M: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N)))),aa(num,int,numeral_numeral(int),M)) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit_or_not_num_neg(M,aa(num,num,bit0,N)))) ).

% or_minus_numerals(8)
tff(fact_2797_or__minus__numerals_I4_J,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N)))) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit_or_not_num_neg(M,aa(num,num,bit0,N)))) ).

% or_minus_numerals(4)
tff(fact_2798_or__minus__numerals_I3_J,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,N)))) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit_or_not_num_neg(M,bitM(N)))) ).

% or_minus_numerals(3)
tff(fact_2799_or__minus__numerals_I7_J,axiom,
    ! [N: num,M: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,N)))),aa(num,int,numeral_numeral(int),M)) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit_or_not_num_neg(M,bitM(N)))) ).

% or_minus_numerals(7)
tff(fact_2800_in__Union__o__assoc,axiom,
    ! [B: $tType,A: $tType,C: $tType,X: A,Gset: fun(B,set(set(A))),Gmap: fun(C,B),A4: C] :
      ( aa(set(A),$o,member(A,X),aa(C,set(A),aa(fun(C,B),fun(C,set(A)),comp(B,set(A),C,aa(fun(B,set(set(A))),fun(B,set(A)),comp(set(set(A)),set(A),B,complete_Sup_Sup(set(A))),Gset)),Gmap),A4))
     => aa(set(A),$o,member(A,X),aa(C,set(A),aa(fun(C,set(set(A))),fun(C,set(A)),comp(set(set(A)),set(A),C,complete_Sup_Sup(set(A))),aa(fun(C,B),fun(C,set(set(A))),comp(B,set(set(A)),C,Gset),Gmap)),A4)) ) ).

% in_Union_o_assoc
tff(fact_2801_max__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A3: A,B2: A] :
          aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2),B2,A3) ) ).

% max_def
tff(fact_2802_max__absorb1,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Y: A,X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X)
         => ( aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y) = X ) ) ) ).

% max_absorb1
tff(fact_2803_max__absorb2,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
         => ( aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y) = Y ) ) ) ).

% max_absorb2
tff(fact_2804_max_Omono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A,A3: A,D3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),D3),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_max(A),C2),D3)),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2)) ) ) ) ).

% max.mono
tff(fact_2805_max_OorderE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3)
         => ( A3 = aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2) ) ) ) ).

% max.orderE
tff(fact_2806_max_OorderI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] :
          ( ( A3 = aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3) ) ) ).

% max.orderI
tff(fact_2807_max_OboundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,C2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),C2)),A3)
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A3) ) ) ) ).

% max.boundedE
tff(fact_2808_max_OboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),C2)),A3) ) ) ) ).

% max.boundedI
tff(fact_2809_max_Oorder__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),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_2810_max_Ocobounded1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2)) ) ).

% max.cobounded1
tff(fact_2811_max_Ocobounded2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2)) ) ).

% max.cobounded2
tff(fact_2812_le__max__iff__disj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Z2: A,X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z2),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z2),X)
            | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z2),Y) ) ) ) ).

% le_max_iff_disj
tff(fact_2813_max_Oabsorb__iff1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),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_2814_max_Oabsorb__iff2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),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_2815_max_OcoboundedI1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A,A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2)) ) ) ).

% max.coboundedI1
tff(fact_2816_max_OcoboundedI2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A,B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2)) ) ) ).

% max.coboundedI2
tff(fact_2817_less__max__iff__disj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Z2: A,X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z2),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z2),X)
            | aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z2),Y) ) ) ) ).

% less_max_iff_disj
tff(fact_2818_max_Ostrict__boundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,C2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),C2)),A3)
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),A3) ) ) ) ).

% max.strict_boundedE
tff(fact_2819_max_Ostrict__order__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3)
        <=> ( ( A3 = aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2) )
            & ( A3 != B2 ) ) ) ) ).

% max.strict_order_iff
tff(fact_2820_max_Ostrict__coboundedI1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A,A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),A3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2)) ) ) ).

% max.strict_coboundedI1
tff(fact_2821_max_Ostrict__coboundedI2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A,B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2)) ) ) ).

% max.strict_coboundedI2
tff(fact_2822_sup__int__def,axiom,
    sup_sup(int) = ord_max(int) ).

% sup_int_def
tff(fact_2823_max_Oleft__commute,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A3: A,C2: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),B2),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),C2)) = aa(A,A,aa(A,fun(A,A),ord_max(A),A3),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),C2)) ) ).

% max.left_commute
tff(fact_2824_max_Ocommute,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2) = aa(A,A,aa(A,fun(A,A),ord_max(A),B2),A3) ) ).

% max.commute
tff(fact_2825_max_Oassoc,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),ord_max(A),A3),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),C2)) ) ).

% max.assoc
tff(fact_2826_abstract__boolean__algebra__sym__diff_Oconj__xor__distrib2,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xor: fun(A,fun(A,A)),Y: A,Z2: A,X: A] :
      ( boolea3799213064322606851m_diff(A,Conj,Disj,Compl,Zero,One,Xor)
     => ( aa(A,A,aa(A,fun(A,A),Conj,aa(A,A,aa(A,fun(A,A),Xor,Y),Z2)),X) = aa(A,A,aa(A,fun(A,A),Xor,aa(A,A,aa(A,fun(A,A),Conj,Y),X)),aa(A,A,aa(A,fun(A,A),Conj,Z2),X)) ) ) ).

% abstract_boolean_algebra_sym_diff.conj_xor_distrib2
tff(fact_2827_abstract__boolean__algebra__sym__diff_Oxor__cancel__right,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xor: fun(A,fun(A,A)),X: A] :
      ( boolea3799213064322606851m_diff(A,Conj,Disj,Compl,Zero,One,Xor)
     => ( aa(A,A,aa(A,fun(A,A),Xor,X),aa(A,A,Compl,X)) = One ) ) ).

% abstract_boolean_algebra_sym_diff.xor_cancel_right
tff(fact_2828_abstract__boolean__algebra__sym__diff_Oconj__xor__distrib,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xor: fun(A,fun(A,A)),X: A,Y: A,Z2: A] :
      ( boolea3799213064322606851m_diff(A,Conj,Disj,Compl,Zero,One,Xor)
     => ( aa(A,A,aa(A,fun(A,A),Conj,X),aa(A,A,aa(A,fun(A,A),Xor,Y),Z2)) = aa(A,A,aa(A,fun(A,A),Xor,aa(A,A,aa(A,fun(A,A),Conj,X),Y)),aa(A,A,aa(A,fun(A,A),Conj,X),Z2)) ) ) ).

% abstract_boolean_algebra_sym_diff.conj_xor_distrib
tff(fact_2829_abstract__boolean__algebra__sym__diff_Oxor__compl__right,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xor: fun(A,fun(A,A)),X: A,Y: A] :
      ( boolea3799213064322606851m_diff(A,Conj,Disj,Compl,Zero,One,Xor)
     => ( aa(A,A,aa(A,fun(A,A),Xor,X),aa(A,A,Compl,Y)) = aa(A,A,Compl,aa(A,A,aa(A,fun(A,A),Xor,X),Y)) ) ) ).

% abstract_boolean_algebra_sym_diff.xor_compl_right
tff(fact_2830_abstract__boolean__algebra__sym__diff_Oxor__cancel__left,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xor: fun(A,fun(A,A)),X: A] :
      ( boolea3799213064322606851m_diff(A,Conj,Disj,Compl,Zero,One,Xor)
     => ( aa(A,A,aa(A,fun(A,A),Xor,aa(A,A,Compl,X)),X) = One ) ) ).

% abstract_boolean_algebra_sym_diff.xor_cancel_left
tff(fact_2831_abstract__boolean__algebra__sym__diff_Oxor__compl__left,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xor: fun(A,fun(A,A)),X: A,Y: A] :
      ( boolea3799213064322606851m_diff(A,Conj,Disj,Compl,Zero,One,Xor)
     => ( aa(A,A,aa(A,fun(A,A),Xor,aa(A,A,Compl,X)),Y) = aa(A,A,Compl,aa(A,A,aa(A,fun(A,A),Xor,X),Y)) ) ) ).

% abstract_boolean_algebra_sym_diff.xor_compl_left
tff(fact_2832_abstract__boolean__algebra__sym__diff_Oxor__one__right,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xor: fun(A,fun(A,A)),X: A] :
      ( boolea3799213064322606851m_diff(A,Conj,Disj,Compl,Zero,One,Xor)
     => ( aa(A,A,aa(A,fun(A,A),Xor,X),One) = aa(A,A,Compl,X) ) ) ).

% abstract_boolean_algebra_sym_diff.xor_one_right
tff(fact_2833_abstract__boolean__algebra__sym__diff_Oxor__left__self,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xor: fun(A,fun(A,A)),X: A,Y: A] :
      ( boolea3799213064322606851m_diff(A,Conj,Disj,Compl,Zero,One,Xor)
     => ( aa(A,A,aa(A,fun(A,A),Xor,X),aa(A,A,aa(A,fun(A,A),Xor,X),Y)) = Y ) ) ).

% abstract_boolean_algebra_sym_diff.xor_left_self
tff(fact_2834_abstract__boolean__algebra__sym__diff_Oxor__one__left,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xor: fun(A,fun(A,A)),X: A] :
      ( boolea3799213064322606851m_diff(A,Conj,Disj,Compl,Zero,One,Xor)
     => ( aa(A,A,aa(A,fun(A,A),Xor,One),X) = aa(A,A,Compl,X) ) ) ).

% abstract_boolean_algebra_sym_diff.xor_one_left
tff(fact_2835_abstract__boolean__algebra__sym__diff_Oxor__self,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xor: fun(A,fun(A,A)),X: A] :
      ( boolea3799213064322606851m_diff(A,Conj,Disj,Compl,Zero,One,Xor)
     => ( aa(A,A,aa(A,fun(A,A),Xor,X),X) = Zero ) ) ).

% abstract_boolean_algebra_sym_diff.xor_self
tff(fact_2836_abstract__boolean__algebra__sym__diff_Oxor__def2,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xor: fun(A,fun(A,A)),X: A,Y: A] :
      ( boolea3799213064322606851m_diff(A,Conj,Disj,Compl,Zero,One,Xor)
     => ( aa(A,A,aa(A,fun(A,A),Xor,X),Y) = aa(A,A,aa(A,fun(A,A),Conj,aa(A,A,aa(A,fun(A,A),Disj,X),Y)),aa(A,A,aa(A,fun(A,A),Disj,aa(A,A,Compl,X)),aa(A,A,Compl,Y))) ) ) ).

% abstract_boolean_algebra_sym_diff.xor_def2
tff(fact_2837_abstract__boolean__algebra__sym__diff_Oxor__def,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xor: fun(A,fun(A,A)),X: A,Y: A] :
      ( boolea3799213064322606851m_diff(A,Conj,Disj,Compl,Zero,One,Xor)
     => ( aa(A,A,aa(A,fun(A,A),Xor,X),Y) = aa(A,A,aa(A,fun(A,A),Disj,aa(A,A,aa(A,fun(A,A),Conj,X),aa(A,A,Compl,Y))),aa(A,A,aa(A,fun(A,A),Conj,aa(A,A,Compl,X)),Y)) ) ) ).

% abstract_boolean_algebra_sym_diff.xor_def
tff(fact_2838_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_2839_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_2840_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_2841_sup__max,axiom,
    ! [A: $tType] :
      ( ( semilattice_sup(A)
        & linorder(A) )
     => ( sup_sup(A) = ord_max(A) ) ) ).

% sup_max
tff(fact_2842_prod_OUnion__comp,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [B3: set(set(A)),G: fun(A,B)] :
          ( ! [X2: set(A)] :
              ( aa(set(set(A)),$o,member(set(A),X2),B3)
             => aa(set(A),$o,finite_finite2(A),X2) )
         => ( ! [A12: set(A)] :
                ( aa(set(set(A)),$o,member(set(A),A12),B3)
               => ! [A23: set(A)] :
                    ( aa(set(set(A)),$o,member(set(A),A23),B3)
                   => ( ( A12 != A23 )
                     => ! [X2: A] :
                          ( aa(set(A),$o,member(A,X2),A12)
                         => ( aa(set(A),$o,member(A,X2),A23)
                           => ( aa(A,B,G,X2) = one_one(B) ) ) ) ) ) )
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),B3)) = aa(set(set(A)),B,aa(fun(A,B),fun(set(set(A)),B),aa(fun(fun(A,B),fun(set(A),B)),fun(fun(A,B),fun(set(set(A)),B)),comp(fun(set(A),B),fun(set(set(A)),B),fun(A,B),groups7121269368397514597t_prod(set(A),B)),groups7121269368397514597t_prod(A,B)),G),B3) ) ) ) ) ).

% prod.Union_comp
tff(fact_2843_max__def__raw,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X3: A,Xa3: A] :
          aa(A,A,aa(A,fun(A,A),ord_max(A),X3),Xa3) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Xa3),Xa3,X3) ) ).

% max_def_raw
tff(fact_2844_sum_OUnion__disjoint,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [C3: set(set(A)),G: fun(A,B)] :
          ( ! [X2: set(A)] :
              ( aa(set(set(A)),$o,member(set(A),X2),C3)
             => aa(set(A),$o,finite_finite2(A),X2) )
         => ( ! [X2: set(A)] :
                ( aa(set(set(A)),$o,member(set(A),X2),C3)
               => ! [Xa4: set(A)] :
                    ( aa(set(set(A)),$o,member(set(A),Xa4),C3)
                   => ( ( X2 != Xa4 )
                     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),X2),Xa4) = bot_bot(set(A)) ) ) ) )
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C3)) = aa(set(set(A)),B,aa(fun(A,B),fun(set(set(A)),B),aa(fun(fun(A,B),fun(set(A),B)),fun(fun(A,B),fun(set(set(A)),B)),comp(fun(set(A),B),fun(set(set(A)),B),fun(A,B),groups7311177749621191930dd_sum(set(A),B)),groups7311177749621191930dd_sum(A,B)),G),C3) ) ) ) ) ).

% sum.Union_disjoint
tff(fact_2845_prod_OUnion__disjoint,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [C3: set(set(A)),G: fun(A,B)] :
          ( ! [X2: set(A)] :
              ( aa(set(set(A)),$o,member(set(A),X2),C3)
             => aa(set(A),$o,finite_finite2(A),X2) )
         => ( ! [X2: set(A)] :
                ( aa(set(set(A)),$o,member(set(A),X2),C3)
               => ! [Xa4: set(A)] :
                    ( aa(set(set(A)),$o,member(set(A),Xa4),C3)
                   => ( ( X2 != Xa4 )
                     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),X2),Xa4) = bot_bot(set(A)) ) ) ) )
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C3)) = aa(set(set(A)),B,aa(fun(A,B),fun(set(set(A)),B),aa(fun(fun(A,B),fun(set(A),B)),fun(fun(A,B),fun(set(set(A)),B)),comp(fun(set(A),B),fun(set(set(A)),B),fun(A,B),groups7121269368397514597t_prod(set(A),B)),groups7121269368397514597t_prod(A,B)),G),C3) ) ) ) ) ).

% prod.Union_disjoint
tff(fact_2846_abstract__boolean__algebra__sym__diff_Oaxioms_I1_J,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xor: fun(A,fun(A,A))] :
      ( boolea3799213064322606851m_diff(A,Conj,Disj,Compl,Zero,One,Xor)
     => boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One) ) ).

% abstract_boolean_algebra_sym_diff.axioms(1)
tff(fact_2847_Sup__inf__eq__bot__iff,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [B3: set(A),A3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Sup_Sup(A),B3)),A3) = bot_bot(A) )
        <=> ! [X4: A] :
              ( aa(set(A),$o,member(A,X4),B3)
             => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X4),A3) = bot_bot(A) ) ) ) ) ).

% Sup_inf_eq_bot_iff
tff(fact_2848_type__copy__wit,axiom,
    ! [A: $tType,C: $tType,B: $tType,Rep: fun(A,B),Abs: fun(B,A),X: C,S: fun(B,set(C)),Y: B] :
      ( type_definition(A,B,Rep,Abs,top_top(set(B)))
     => ( aa(set(C),$o,member(C,X),aa(A,set(C),aa(fun(A,B),fun(A,set(C)),comp(B,set(C),A,S),Rep),aa(B,A,Abs,Y)))
       => aa(set(C),$o,member(C,X),aa(B,set(C),S,Y)) ) ) ).

% type_copy_wit
tff(fact_2849_insert__partition,axiom,
    ! [A: $tType,X: set(A),F4: set(set(A))] :
      ( ~ aa(set(set(A)),$o,member(set(A),X),F4)
     => ( ! [X2: set(A)] :
            ( aa(set(set(A)),$o,member(set(A),X2),aa(set(set(A)),set(set(A)),aa(set(A),fun(set(set(A)),set(set(A))),insert2(set(A)),X),F4))
           => ! [Xa4: set(A)] :
                ( aa(set(set(A)),$o,member(set(A),Xa4),aa(set(set(A)),set(set(A)),aa(set(A),fun(set(set(A)),set(set(A))),insert2(set(A)),X),F4))
               => ( ( X2 != Xa4 )
                 => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),X2),Xa4) = bot_bot(set(A)) ) ) ) )
       => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),X),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),F4)) = bot_bot(set(A)) ) ) ) ).

% insert_partition
tff(fact_2850_finite__Sup__in,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A4: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( ! [X2: A,Y2: A] :
                  ( aa(set(A),$o,member(A,X2),A4)
                 => ( aa(set(A),$o,member(A,Y2),A4)
                   => aa(set(A),$o,member(A,aa(A,A,aa(A,fun(A,A),sup_sup(A),X2),Y2)),A4) ) )
             => aa(set(A),$o,member(A,aa(set(A),A,complete_Sup_Sup(A),A4)),A4) ) ) ) ) ).

% finite_Sup_in
tff(fact_2851_ivl__disj__int__two_I5_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,M: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or5935395276787703475ssThan(A,L,M)),set_or1337092689740270186AtMost(A,M,U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_two(5)
tff(fact_2852_ivl__disj__int__two_I4_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,M: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or1337092689740270186AtMost(A,L,M)),set_or5935395276787703475ssThan(A,M,U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_two(4)
tff(fact_2853_ivl__disj__int__two_I1_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,M: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or5935395276787703475ssThan(A,L,M)),set_or7035219750837199246ssThan(A,M,U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_two(1)
tff(fact_2854_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_2855_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_2856_cSup__asclose,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & linordered_idom(A) )
     => ! [S: set(A),L: A,E4: A] :
          ( ( S != bot_bot(set(A)) )
         => ( ! [X2: A] :
                ( aa(set(A),$o,member(A,X2),S)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X2),L))),E4) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(set(A),A,complete_Sup_Sup(A),S)),L))),E4) ) ) ) ).

% cSup_asclose
tff(fact_2857_int__numeral__or__not__num__neg,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),N))) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit_or_not_num_neg(M,N))) ).

% int_numeral_or_not_num_neg
tff(fact_2858_int__numeral__not__or__num__neg,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),M))),aa(num,int,numeral_numeral(int),N)) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit_or_not_num_neg(N,M))) ).

% int_numeral_not_or_num_neg
tff(fact_2859_numeral__or__not__num__eq,axiom,
    ! [M: num,N: num] : aa(num,int,numeral_numeral(int),bit_or_not_num_neg(M,N)) = aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),N)))) ).

% numeral_or_not_num_eq
tff(fact_2860_ivl__disj__un__two_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),L),M)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),M),U)
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or5935395276787703475ssThan(A,L,M)),set_or7035219750837199246ssThan(A,M,U)) = set_or5935395276787703475ssThan(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(1)
tff(fact_2861_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),aa(A,fun(set(A),set(A)),insert2(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A))))) = set_or5935395276787703475ssThan(A,A3,B2) ) ).

% atLeastAtMost_diff_ends
tff(fact_2862_ivl__disj__un__one_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),L),U)
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(A,set(A),set_ord_atMost(A),L)),set_or5935395276787703475ssThan(A,L,U)) = aa(A,set(A),set_ord_lessThan(A),U) ) ) ) ).

% ivl_disj_un_one(1)
tff(fact_2863_prod_OatLeast0__atMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),suc)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N))) ) ).

% prod.atLeast0_atMost_Suc_shift
tff(fact_2864_prod_OatLeast0__lessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),suc)),set_or7035219750837199246ssThan(nat,zero_zero(nat),N))) ) ).

% prod.atLeast0_lessThan_Suc_shift
tff(fact_2865_dvd__partition,axiom,
    ! [A: $tType,C3: set(set(A)),K: nat] :
      ( aa(set(A),$o,finite_finite2(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C3))
     => ( ! [X2: set(A)] :
            ( aa(set(set(A)),$o,member(set(A),X2),C3)
           => aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),K),aa(set(A),nat,finite_card(A),X2)) )
       => ( ! [X2: set(A)] :
              ( aa(set(set(A)),$o,member(set(A),X2),C3)
             => ! [Xa4: set(A)] :
                  ( aa(set(set(A)),$o,member(set(A),Xa4),C3)
                 => ( ( X2 != Xa4 )
                   => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),X2),Xa4) = bot_bot(set(A)) ) ) ) )
         => aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),K),aa(set(A),nat,finite_card(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C3))) ) ) ) ).

% dvd_partition
tff(fact_2866_sum_OatLeast__atMost__pred__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),aTP_Lamp_jn(nat,nat))),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M,N)) ) ).

% sum.atLeast_atMost_pred_shift
tff(fact_2867_sum_OatLeast__lessThan__pred__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),aTP_Lamp_jn(nat,nat))),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,M,N)) ) ).

% sum.atLeast_lessThan_pred_shift
tff(fact_2868_prod_OatLeast__atMost__pred__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),aTP_Lamp_jn(nat,nat))),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M,N)) ) ).

% prod.atLeast_atMost_pred_shift
tff(fact_2869_prod_OatLeast__lessThan__pred__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),aTP_Lamp_jn(nat,nat))),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,M,N)) ) ).

% prod.atLeast_lessThan_pred_shift
tff(fact_2870_ivl__disj__un__singleton_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),L),U)
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),L),bot_bot(set(A)))),set_or5935395276787703475ssThan(A,L,U)) = set_or7035219750837199246ssThan(A,L,U) ) ) ) ).

% ivl_disj_un_singleton(3)
tff(fact_2871_ivl__disj__un__two_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),M)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),M),U)
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or1337092689740270186AtMost(A,L,M)),set_or5935395276787703475ssThan(A,M,U)) = set_or7035219750837199246ssThan(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(4)
tff(fact_2872_Sup__insert,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: A,A4: set(A)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),A4)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),aa(set(A),A,complete_Sup_Sup(A),A4)) ) ).

% Sup_insert
tff(fact_2873_Sup__UNIV,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ( aa(set(A),A,complete_Sup_Sup(A),top_top(set(A))) = top_top(A) ) ) ).

% Sup_UNIV
tff(fact_2874_ccpo__Sup__singleton,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [X: A] : aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) = X ) ).

% ccpo_Sup_singleton
tff(fact_2875_cSup__singleton,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X: A] : aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) = X ) ).

% cSup_singleton
tff(fact_2876_Sup__empty,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ( aa(set(A),A,complete_Sup_Sup(A),bot_bot(set(A))) = bot_bot(A) ) ) ).

% Sup_empty
tff(fact_2877_Sup__eq__top__iff,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [A4: set(A)] :
          ( ( aa(set(A),A,complete_Sup_Sup(A),A4) = top_top(A) )
        <=> ! [X4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),top_top(A))
             => ? [Xa2: A] :
                  ( aa(set(A),$o,member(A,Xa2),A4)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Xa2) ) ) ) ) ).

% Sup_eq_top_iff
tff(fact_2878_Sup__bot__conv_I2_J,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A4: set(A)] :
          ( ( bot_bot(A) = aa(set(A),A,complete_Sup_Sup(A),A4) )
        <=> ! [X4: A] :
              ( aa(set(A),$o,member(A,X4),A4)
             => ( X4 = bot_bot(A) ) ) ) ) ).

% Sup_bot_conv(2)
tff(fact_2879_Sup__bot__conv_I1_J,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A4: set(A)] :
          ( ( aa(set(A),A,complete_Sup_Sup(A),A4) = bot_bot(A) )
        <=> ! [X4: A] :
              ( aa(set(A),$o,member(A,X4),A4)
             => ( X4 = bot_bot(A) ) ) ) ) ).

% Sup_bot_conv(1)
tff(fact_2880_Union__Un__distrib,axiom,
    ! [A: $tType,A4: set(set(A)),B3: set(set(A))] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(set(A)),set(set(A)),aa(set(set(A)),fun(set(set(A)),set(set(A))),sup_sup(set(set(A))),A4),B3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A4)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),B3)) ).

% Union_Un_distrib
tff(fact_2881_Sup__nat__empty,axiom,
    aa(set(nat),nat,complete_Sup_Sup(nat),bot_bot(set(nat))) = zero_zero(nat) ).

% Sup_nat_empty
tff(fact_2882_max__nat_Osemilattice__neutr__axioms,axiom,
    semilattice_neutr(nat,ord_max(nat),zero_zero(nat)) ).

% max_nat.semilattice_neutr_axioms
tff(fact_2883_max__nat_Ocomm__monoid__axioms,axiom,
    comm_monoid(nat,ord_max(nat),zero_zero(nat)) ).

% max_nat.comm_monoid_axioms
tff(fact_2884_max__nat_Omonoid__axioms,axiom,
    monoid(nat,ord_max(nat),zero_zero(nat)) ).

% max_nat.monoid_axioms
tff(fact_2885_max__Suc2,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),M),aa(nat,nat,suc,N)) = aa(nat,nat,aa(fun(nat,nat),fun(nat,nat),aa(nat,fun(fun(nat,nat),fun(nat,nat)),case_nat(nat),aa(nat,nat,suc,N)),aTP_Lamp_jo(nat,fun(nat,nat),N)),M) ).

% max_Suc2
tff(fact_2886_max__Suc1,axiom,
    ! [N: nat,M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,suc,N)),M) = aa(nat,nat,aa(fun(nat,nat),fun(nat,nat),aa(nat,fun(fun(nat,nat),fun(nat,nat)),case_nat(nat),aa(nat,nat,suc,N)),aTP_Lamp_jp(nat,fun(nat,nat),N)),M) ).

% max_Suc1
tff(fact_2887_max__nat_Osemilattice__neutr__order__axioms,axiom,
    semila1105856199041335345_order(nat,ord_max(nat),zero_zero(nat),aTP_Lamp_bj(nat,fun(nat,$o)),aTP_Lamp_bk(nat,fun(nat,$o))) ).

% max_nat.semilattice_neutr_order_axioms
tff(fact_2888_atLeastPlusOneLessThan__greaterThanLessThan__integer,axiom,
    ! [L: code_integer,U: code_integer] : set_or7035219750837199246ssThan(code_integer,aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),L),one_one(code_integer)),U) = set_or5935395276787703475ssThan(code_integer,L,U) ).

% atLeastPlusOneLessThan_greaterThanLessThan_integer
tff(fact_2889_empty__Union__conv,axiom,
    ! [A: $tType,A4: set(set(A))] :
      ( ( bot_bot(set(A)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A4) )
    <=> ! [X4: set(A)] :
          ( aa(set(set(A)),$o,member(set(A),X4),A4)
         => ( X4 = bot_bot(set(A)) ) ) ) ).

% empty_Union_conv
tff(fact_2890_Union__empty__conv,axiom,
    ! [A: $tType,A4: set(set(A))] :
      ( ( aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A4) = bot_bot(set(A)) )
    <=> ! [X4: set(A)] :
          ( aa(set(set(A)),$o,member(set(A),X4),A4)
         => ( X4 = bot_bot(set(A)) ) ) ) ).

% Union_empty_conv
tff(fact_2891_Union__empty,axiom,
    ! [A: $tType] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),bot_bot(set(set(A)))) = bot_bot(set(A)) ).

% Union_empty
tff(fact_2892_Union__insert,axiom,
    ! [A: $tType,A3: set(A),B3: set(set(A))] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(set(A)),set(set(A)),aa(set(A),fun(set(set(A)),set(set(A))),insert2(set(A)),A3),B3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),B3)) ).

% Union_insert
tff(fact_2893_complete__linorder__sup__max,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ( sup_sup(A) = ord_max(A) ) ) ).

% complete_linorder_sup_max
tff(fact_2894_less__eq__Sup,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A4: set(A),U: A] :
          ( ! [V2: A] :
              ( aa(set(A),$o,member(A,V2),A4)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),U),V2) )
         => ( ( A4 != bot_bot(set(A)) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),U),aa(set(A),A,complete_Sup_Sup(A),A4)) ) ) ) ).

% less_eq_Sup
tff(fact_2895_cSup__least,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X5: set(A),Z2: A] :
          ( ( X5 != bot_bot(set(A)) )
         => ( ! [X2: A] :
                ( aa(set(A),$o,member(A,X2),X5)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),Z2) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),X5)),Z2) ) ) ) ).

% cSup_least
tff(fact_2896_cSup__eq__non__empty,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X5: set(A),A3: A] :
          ( ( X5 != bot_bot(set(A)) )
         => ( ! [X2: A] :
                ( aa(set(A),$o,member(A,X2),X5)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),A3) )
           => ( ! [Y2: A] :
                  ( ! [X3: A] :
                      ( aa(set(A),$o,member(A,X3),X5)
                     => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Y2) )
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),Y2) )
             => ( aa(set(A),A,complete_Sup_Sup(A),X5) = A3 ) ) ) ) ) ).

% cSup_eq_non_empty
tff(fact_2897_less__cSupD,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X5: set(A),Z2: A] :
          ( ( X5 != bot_bot(set(A)) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z2),aa(set(A),A,complete_Sup_Sup(A),X5))
           => ? [X2: A] :
                ( aa(set(A),$o,member(A,X2),X5)
                & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z2),X2) ) ) ) ) ).

% less_cSupD
tff(fact_2898_less__cSupE,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [Y: A,X5: set(A)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),aa(set(A),A,complete_Sup_Sup(A),X5))
         => ( ( X5 != bot_bot(set(A)) )
           => ~ ! [X2: A] :
                  ( aa(set(A),$o,member(A,X2),X5)
                 => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),X2) ) ) ) ) ).

% less_cSupE
tff(fact_2899_Sup__union__distrib,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A4: set(A),B3: set(A)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Sup_Sup(A),A4)),aa(set(A),A,complete_Sup_Sup(A),B3)) ) ).

% Sup_union_distrib
tff(fact_2900_Union__disjoint,axiom,
    ! [A: $tType,C3: set(set(A)),A4: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C3)),A4) = bot_bot(set(A)) )
    <=> ! [X4: set(A)] :
          ( aa(set(set(A)),$o,member(set(A),X4),C3)
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),X4),A4) = bot_bot(set(A)) ) ) ) ).

% Union_disjoint
tff(fact_2901_Union__Int__subset,axiom,
    ! [A: $tType,A4: set(set(A)),B3: set(set(A))] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(set(A)),set(set(A)),aa(set(set(A)),fun(set(set(A)),set(set(A))),inf_inf(set(set(A))),A4),B3))),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A4)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),B3))) ).

% Union_Int_subset
tff(fact_2902_Union__UNIV,axiom,
    ! [A: $tType] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),top_top(set(set(A)))) = top_top(set(A)) ).

% Union_UNIV
tff(fact_2903_finite__Sup__less__iff,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X5: set(A),A3: A] :
          ( aa(set(A),$o,finite_finite2(A),X5)
         => ( ( X5 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(A),A,complete_Sup_Sup(A),X5)),A3)
            <=> ! [X4: A] :
                  ( aa(set(A),$o,member(A,X4),X5)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),A3) ) ) ) ) ) ).

% finite_Sup_less_iff
tff(fact_2904_cSup__abs__le,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & linordered_idom(A) )
     => ! [S: set(A),A3: A] :
          ( ( S != bot_bot(set(A)) )
         => ( ! [X2: A] :
                ( aa(set(A),$o,member(A,X2),S)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),X2)),A3) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(set(A),A,complete_Sup_Sup(A),S))),A3) ) ) ) ).

% cSup_abs_le
tff(fact_2905_Sup__inter__less__eq,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A4: set(A),B3: set(A)] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3))),aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Sup_Sup(A),A4)),aa(set(A),A,complete_Sup_Sup(A),B3))) ) ).

% Sup_inter_less_eq
tff(fact_2906_Sup__insert__finite,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [S: set(A),X: A] :
          ( aa(set(A),$o,finite_finite2(A),S)
         => ( aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),S)) = $ite(S = bot_bot(set(A)),X,aa(A,A,aa(A,fun(A,A),ord_max(A),X),aa(set(A),A,complete_Sup_Sup(A),S))) ) ) ) ).

% Sup_insert_finite
tff(fact_2907_Code__Numeral_Onegative__def,axiom,
    code_negative = aa(fun(num,code_integer),fun(num,code_integer),comp(code_integer,code_integer,num,uminus_uminus(code_integer)),numeral_numeral(code_integer)) ).

% Code_Numeral.negative_def
tff(fact_2908_card__UNION,axiom,
    ! [A: $tType,A4: set(set(A))] :
      ( aa(set(set(A)),$o,finite_finite2(set(A)),A4)
     => ( ! [X2: set(A)] :
            ( aa(set(set(A)),$o,member(set(A),X2),A4)
           => aa(set(A),$o,finite_finite2(A),X2) )
       => ( aa(set(A),nat,finite_card(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A4)) = aa(int,nat,nat2,aa(set(set(set(A))),int,aa(fun(set(set(A)),int),fun(set(set(set(A))),int),groups7311177749621191930dd_sum(set(set(A)),int),aTP_Lamp_jq(set(set(A)),int)),aa(fun(set(set(A)),$o),set(set(set(A))),collect(set(set(A))),aTP_Lamp_jr(set(set(A)),fun(set(set(A)),$o),A4)))) ) ) ) ).

% card_UNION
tff(fact_2909_Sup__finite__insert,axiom,
    ! [A: $tType] :
      ( finite_lattice(A)
     => ! [A3: A,A4: set(A)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),A4)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),aa(set(A),A,complete_Sup_Sup(A),A4)) ) ).

% Sup_finite_insert
tff(fact_2910_top__finite__def,axiom,
    ! [A: $tType] :
      ( finite_lattice(A)
     => ( top_top(A) = aa(set(A),A,complete_Sup_Sup(A),top_top(set(A))) ) ) ).

% top_finite_def
tff(fact_2911_finite__mono__strict__prefix__implies__finite__fixpoint,axiom,
    ! [A: $tType,F: fun(nat,set(A)),S: set(A)] :
      ( ! [I2: nat] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(nat,set(A),F,I2)),S)
     => ( aa(set(A),$o,finite_finite2(A),S)
       => ( ? [N5: nat] :
              ( ! [N3: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N3),N5)
                 => ! [M3: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M3),N5)
                     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M3),N3)
                       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),aa(nat,set(A),F,M3)),aa(nat,set(A),F,N3)) ) ) )
              & ! [N3: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N5),N3)
                 => ( aa(nat,set(A),F,N5) = aa(nat,set(A),F,N3) ) ) )
         => ( aa(nat,set(A),F,aa(set(A),nat,finite_card(A),S)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),F),top_top(set(nat)))) ) ) ) ) ).

% finite_mono_strict_prefix_implies_finite_fixpoint
tff(fact_2912_image__ident,axiom,
    ! [A: $tType,Y4: set(A)] : aa(set(A),set(A),image2(A,A,aTP_Lamp_au(A,A)),Y4) = Y4 ).

% image_ident
tff(fact_2913_image__is__empty,axiom,
    ! [A: $tType,B: $tType,F: fun(B,A),A4: set(B)] :
      ( ( aa(set(B),set(A),image2(B,A,F),A4) = bot_bot(set(A)) )
    <=> ( A4 = bot_bot(set(B)) ) ) ).

% image_is_empty
tff(fact_2914_empty__is__image,axiom,
    ! [A: $tType,B: $tType,F: fun(B,A),A4: set(B)] :
      ( ( bot_bot(set(A)) = aa(set(B),set(A),image2(B,A,F),A4) )
    <=> ( A4 = bot_bot(set(B)) ) ) ).

% empty_is_image
tff(fact_2915_image__empty,axiom,
    ! [B: $tType,A: $tType,F: fun(B,A)] : aa(set(B),set(A),image2(B,A,F),bot_bot(set(B))) = bot_bot(set(A)) ).

% image_empty
tff(fact_2916_Inf__top__conv_I1_J,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A4: set(A)] :
          ( ( aa(set(A),A,complete_Inf_Inf(A),A4) = top_top(A) )
        <=> ! [X4: A] :
              ( aa(set(A),$o,member(A,X4),A4)
             => ( X4 = top_top(A) ) ) ) ) ).

% Inf_top_conv(1)
tff(fact_2917_Inf__top__conv_I2_J,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A4: set(A)] :
          ( ( top_top(A) = aa(set(A),A,complete_Inf_Inf(A),A4) )
        <=> ! [X4: A] :
              ( aa(set(A),$o,member(A,X4),A4)
             => ( X4 = top_top(A) ) ) ) ) ).

% Inf_top_conv(2)
tff(fact_2918_Inter__UNIV__conv_I1_J,axiom,
    ! [A: $tType,A4: set(set(A))] :
      ( ( aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),A4) = top_top(set(A)) )
    <=> ! [X4: set(A)] :
          ( aa(set(set(A)),$o,member(set(A),X4),A4)
         => ( X4 = top_top(set(A)) ) ) ) ).

% Inter_UNIV_conv(1)
tff(fact_2919_Inter__UNIV__conv_I2_J,axiom,
    ! [A: $tType,A4: set(set(A))] :
      ( ( top_top(set(A)) = aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),A4) )
    <=> ! [X4: set(A)] :
          ( aa(set(set(A)),$o,member(set(A),X4),A4)
         => ( X4 = top_top(set(A)) ) ) ) ).

% Inter_UNIV_conv(2)
tff(fact_2920_SUP__apply,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( complete_Sup(A)
     => ! [F: fun(C,fun(B,A)),A4: set(C),X: B] : aa(B,A,aa(set(fun(B,A)),fun(B,A),complete_Sup_Sup(fun(B,A)),aa(set(C),set(fun(B,A)),image2(C,fun(B,A),F),A4)),X) = aa(set(A),A,complete_Sup_Sup(A),aa(set(C),set(A),image2(C,A,aa(B,fun(C,A),aTP_Lamp_js(fun(C,fun(B,A)),fun(B,fun(C,A)),F),X)),A4)) ) ).

% SUP_apply
tff(fact_2921_SUP__identity__eq,axiom,
    ! [A: $tType] :
      ( complete_Sup(A)
     => ! [A4: set(A)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),image2(A,A,aTP_Lamp_jt(A,A)),A4)) = aa(set(A),A,complete_Sup_Sup(A),A4) ) ).

% SUP_identity_eq
tff(fact_2922_INF__apply,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( complete_Inf(A)
     => ! [F: fun(C,fun(B,A)),A4: set(C),X: B] : aa(B,A,aa(set(fun(B,A)),fun(B,A),complete_Inf_Inf(fun(B,A)),aa(set(C),set(fun(B,A)),image2(C,fun(B,A),F),A4)),X) = aa(set(A),A,complete_Inf_Inf(A),aa(set(C),set(A),image2(C,A,aa(B,fun(C,A),aTP_Lamp_ju(fun(C,fun(B,A)),fun(B,fun(C,A)),F),X)),A4)) ) ).

% INF_apply
tff(fact_2923_INF__identity__eq,axiom,
    ! [A: $tType] :
      ( complete_Inf(A)
     => ! [A4: set(A)] : aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),image2(A,A,aTP_Lamp_jv(A,A)),A4)) = aa(set(A),A,complete_Inf_Inf(A),A4) ) ).

% INF_identity_eq
tff(fact_2924_UN__iff,axiom,
    ! [A: $tType,B: $tType,B2: A,B3: fun(B,set(A)),A4: set(B)] :
      ( aa(set(A),$o,member(A,B2),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B3),A4)))
    <=> ? [X4: B] :
          ( aa(set(B),$o,member(B,X4),A4)
          & aa(set(A),$o,member(A,B2),aa(B,set(A),B3,X4)) ) ) ).

% UN_iff
tff(fact_2925_UN__I,axiom,
    ! [B: $tType,A: $tType,A3: A,A4: set(A),B2: B,B3: fun(A,set(B))] :
      ( aa(set(A),$o,member(A,A3),A4)
     => ( aa(set(B),$o,member(B,B2),aa(A,set(B),B3,A3))
       => aa(set(B),$o,member(B,B2),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),B3),A4))) ) ) ).

% UN_I
tff(fact_2926_INT__iff,axiom,
    ! [A: $tType,B: $tType,B2: A,B3: fun(B,set(A)),A4: set(B)] :
      ( aa(set(A),$o,member(A,B2),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B3),A4)))
    <=> ! [X4: B] :
          ( aa(set(B),$o,member(B,X4),A4)
         => aa(set(A),$o,member(A,B2),aa(B,set(A),B3,X4)) ) ) ).

% INT_iff
tff(fact_2927_INT__I,axiom,
    ! [B: $tType,A: $tType,A4: set(A),B2: B,B3: fun(A,set(B))] :
      ( ! [X2: A] :
          ( aa(set(A),$o,member(A,X2),A4)
         => aa(set(B),$o,member(B,B2),aa(A,set(B),B3,X2)) )
     => aa(set(B),$o,member(B,B2),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image2(A,set(B),B3),A4))) ) ).

% INT_I
tff(fact_2928_Inf__eq__bot__iff,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [A4: set(A)] :
          ( ( aa(set(A),A,complete_Inf_Inf(A),A4) = bot_bot(A) )
        <=> ! [X4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),bot_bot(A)),X4)
             => ? [Xa2: A] :
                  ( aa(set(A),$o,member(A,Xa2),A4)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa2),X4) ) ) ) ) ).

% Inf_eq_bot_iff
tff(fact_2929_Inf__empty,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ( aa(set(A),A,complete_Inf_Inf(A),bot_bot(set(A))) = top_top(A) ) ) ).

% Inf_empty
tff(fact_2930_Inf__UNIV,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ( aa(set(A),A,complete_Inf_Inf(A),top_top(set(A))) = bot_bot(A) ) ) ).

% Inf_UNIV
tff(fact_2931_cInf__singleton,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X: A] : aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) = X ) ).

% cInf_singleton
tff(fact_2932_image__uminus__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [X: A,Y: A] : aa(set(A),set(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_2933_Inf__insert,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: A,A4: set(A)] : aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),A4)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),aa(set(A),A,complete_Inf_Inf(A),A4)) ) ).

% Inf_insert
tff(fact_2934_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_2935_image__uminus__greaterThanLessThan,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [X: A,Y: A] : aa(set(A),set(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_2936_SUP__bot__conv_I2_J,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [B3: fun(B,A),A4: set(B)] :
          ( ( bot_bot(A) = aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,B3),A4)) )
        <=> ! [X4: B] :
              ( aa(set(B),$o,member(B,X4),A4)
             => ( aa(B,A,B3,X4) = bot_bot(A) ) ) ) ) ).

% SUP_bot_conv(2)
tff(fact_2937_SUP__bot__conv_I1_J,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [B3: fun(B,A),A4: set(B)] :
          ( ( aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,B3),A4)) = bot_bot(A) )
        <=> ! [X4: B] :
              ( aa(set(B),$o,member(B,X4),A4)
             => ( aa(B,A,B3,X4) = bot_bot(A) ) ) ) ) ).

% SUP_bot_conv(1)
tff(fact_2938_SUP__bot,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A4: set(B)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,aTP_Lamp_jw(B,A)),A4)) = bot_bot(A) ) ).

% SUP_bot
tff(fact_2939_cSUP__const,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [A4: set(A),C2: B] :
          ( ( A4 != bot_bot(set(A)) )
         => ( aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,aTP_Lamp_jx(B,fun(A,B),C2)),A4)) = C2 ) ) ) ).

% cSUP_const
tff(fact_2940_SUP__const,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A4: set(A),F: B] :
          ( ( A4 != bot_bot(set(A)) )
         => ( aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,aTP_Lamp_jy(B,fun(A,B),F)),A4)) = F ) ) ) ).

% SUP_const
tff(fact_2941_cINF__const,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [A4: set(A),C2: B] :
          ( ( A4 != bot_bot(set(A)) )
         => ( aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,aTP_Lamp_jx(B,fun(A,B),C2)),A4)) = C2 ) ) ) ).

% cINF_const
tff(fact_2942_INF__const,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A4: set(A),F: B] :
          ( ( A4 != bot_bot(set(A)) )
         => ( aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,aTP_Lamp_jy(B,fun(A,B),F)),A4)) = F ) ) ) ).

% INF_const
tff(fact_2943_image__add__atLeastAtMost_H,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K: A,I: A,J: A] : aa(set(A),set(A),image2(A,A,aTP_Lamp_jz(A,fun(A,A),K)),set_or1337092689740270186AtMost(A,I,J)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),K)) ) ).

% image_add_atLeastAtMost'
tff(fact_2944_image__minus__const__atLeastAtMost_H,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [D3: A,A3: A,B2: A] : aa(set(A),set(A),image2(A,A,aTP_Lamp_ka(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_2945_INF__top,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A4: set(B)] : aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,aTP_Lamp_kb(B,A)),A4)) = top_top(A) ) ).

% INF_top
tff(fact_2946_INF__top__conv_I1_J,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [B3: fun(B,A),A4: set(B)] :
          ( ( aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,B3),A4)) = top_top(A) )
        <=> ! [X4: B] :
              ( aa(set(B),$o,member(B,X4),A4)
             => ( aa(B,A,B3,X4) = top_top(A) ) ) ) ) ).

% INF_top_conv(1)
tff(fact_2947_INF__top__conv_I2_J,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [B3: fun(B,A),A4: set(B)] :
          ( ( top_top(A) = aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,B3),A4)) )
        <=> ! [X4: B] :
              ( aa(set(B),$o,member(B,X4),A4)
             => ( aa(B,A,B3,X4) = top_top(A) ) ) ) ) ).

% INF_top_conv(2)
tff(fact_2948_image__add__atLeastLessThan_H,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K: A,I: A,J: A] : aa(set(A),set(A),image2(A,A,aTP_Lamp_jz(A,fun(A,A),K)),set_or7035219750837199246ssThan(A,I,J)) = set_or7035219750837199246ssThan(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),K)) ) ).

% image_add_atLeastLessThan'
tff(fact_2949_if__image__distrib,axiom,
    ! [A: $tType,B: $tType,P: fun(B,$o),F: fun(B,A),G: fun(B,A),S: set(B)] : aa(set(B),set(A),image2(B,A,aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_kc(fun(B,$o),fun(fun(B,A),fun(fun(B,A),fun(B,A))),P),F),G)),S) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(B),set(A),image2(B,A,F),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),S),aa(fun(B,$o),set(B),collect(B),P)))),aa(set(B),set(A),image2(B,A,G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),S),aa(fun(B,$o),set(B),collect(B),aTP_Lamp_kd(fun(B,$o),fun(B,$o),P))))) ).

% if_image_distrib
tff(fact_2950_UN__constant,axiom,
    ! [B: $tType,A: $tType,C2: set(A),A4: set(B)] :
      aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aTP_Lamp_ke(set(A),fun(B,set(A)),C2)),A4)) = $ite(A4 = bot_bot(set(B)),bot_bot(set(A)),C2) ).

% UN_constant
tff(fact_2951_finite__UN__I,axiom,
    ! [B: $tType,A: $tType,A4: set(A),B3: fun(A,set(B))] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( ! [A6: A] :
            ( aa(set(A),$o,member(A,A6),A4)
           => aa(set(B),$o,finite_finite2(B),aa(A,set(B),B3,A6)) )
       => aa(set(B),$o,finite_finite2(B),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),B3),A4))) ) ) ).

% finite_UN_I
tff(fact_2952_UN__Un,axiom,
    ! [A: $tType,B: $tType,M4: fun(B,set(A)),A4: set(B),B3: set(B)] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),M4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A4),B3))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),M4),A4))),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),M4),B3))) ).

% UN_Un
tff(fact_2953_finite__INT,axiom,
    ! [B: $tType,A: $tType,I4: set(A),A4: fun(A,set(B))] :
      ( ? [X3: A] :
          ( aa(set(A),$o,member(A,X3),I4)
          & aa(set(B),$o,finite_finite2(B),aa(A,set(B),A4,X3)) )
     => aa(set(B),$o,finite_finite2(B),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image2(A,set(B),A4),I4))) ) ).

% finite_INT
tff(fact_2954_image__vimage__eq,axiom,
    ! [A: $tType,B: $tType,F: fun(B,A),A4: set(A)] : aa(set(B),set(A),image2(B,A,F),aa(set(A),set(B),aa(fun(B,A),fun(set(A),set(B)),vimage(B,A),F),A4)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),aa(set(B),set(A),image2(B,A,F),top_top(set(B)))) ).

% image_vimage_eq
tff(fact_2955_INF__eq__bot__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [F: fun(B,A),A4: set(B)] :
          ( ( aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F),A4)) = bot_bot(A) )
        <=> ! [X4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),bot_bot(A)),X4)
             => ? [Xa2: B] :
                  ( aa(set(B),$o,member(B,Xa2),A4)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,F,Xa2)),X4) ) ) ) ) ).

% INF_eq_bot_iff
tff(fact_2956_SUP__eq__top__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [F: fun(B,A),A4: set(B)] :
          ( ( aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F),A4)) = top_top(A) )
        <=> ! [X4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),top_top(A))
             => ? [Xa2: B] :
                  ( aa(set(B),$o,member(B,Xa2),A4)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),aa(B,A,F,Xa2)) ) ) ) ) ).

% SUP_eq_top_iff
tff(fact_2957_range__constant,axiom,
    ! [B: $tType,A: $tType,X: A] : aa(set(B),set(A),image2(B,A,aa(A,fun(B,A),aTP_Lamp_kf(A,fun(B,A)),X)),top_top(set(B))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))) ).

% range_constant
tff(fact_2958_UN__singleton,axiom,
    ! [A: $tType,A4: set(A)] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(A),set(set(A)),image2(A,set(A),aTP_Lamp_kg(A,set(A))),A4)) = A4 ).

% UN_singleton
tff(fact_2959_UN__simps_I1_J,axiom,
    ! [A: $tType,B: $tType,A3: A,B3: fun(B,set(A)),C3: set(B)] :
      aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_kh(A,fun(fun(B,set(A)),fun(B,set(A))),A3),B3)),C3)) = $ite(C3 = bot_bot(set(B)),bot_bot(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B3),C3)))) ).

% UN_simps(1)
tff(fact_2960_UN__simps_I3_J,axiom,
    ! [A: $tType,B: $tType,A4: set(A),B3: fun(B,set(A)),C3: set(B)] :
      aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_ki(set(A),fun(fun(B,set(A)),fun(B,set(A))),A4),B3)),C3)) = $ite(C3 = bot_bot(set(B)),bot_bot(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B3),C3)))) ).

% UN_simps(3)
tff(fact_2961_UN__simps_I2_J,axiom,
    ! [B: $tType,A: $tType,A4: fun(B,set(A)),B3: set(A),C3: set(B)] :
      aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(set(A),fun(B,set(A)),aTP_Lamp_kj(fun(B,set(A)),fun(set(A),fun(B,set(A))),A4),B3)),C3)) = $ite(C3 = bot_bot(set(B)),bot_bot(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),A4),C3))),B3)) ).

% UN_simps(2)
tff(fact_2962_UN__insert,axiom,
    ! [A: $tType,B: $tType,B3: fun(B,set(A)),A3: B,A4: set(B)] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B3),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),A3),A4))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(B,set(A),B3,A3)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B3),A4))) ).

% UN_insert
tff(fact_2963_INT__constant,axiom,
    ! [B: $tType,A: $tType,C2: set(A),A4: set(B)] :
      aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aTP_Lamp_ke(set(A),fun(B,set(A)),C2)),A4)) = $ite(A4 = bot_bot(set(B)),top_top(set(A)),C2) ).

% INT_constant
tff(fact_2964_INT__insert,axiom,
    ! [A: $tType,B: $tType,B3: fun(B,set(A)),A3: B,A4: set(B)] : aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B3),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),A3),A4))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(B,set(A),B3,A3)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B3),A4))) ).

% INT_insert
tff(fact_2965_Compl__INT,axiom,
    ! [A: $tType,B: $tType,B3: fun(B,set(A)),A4: set(B)] : aa(set(A),set(A),uminus_uminus(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B3),A4))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aTP_Lamp_kk(fun(B,set(A)),fun(B,set(A)),B3)),A4)) ).

% Compl_INT
tff(fact_2966_Compl__UN,axiom,
    ! [A: $tType,B: $tType,B3: fun(B,set(A)),A4: set(B)] : aa(set(A),set(A),uminus_uminus(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B3),A4))) = aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aTP_Lamp_kk(fun(B,set(A)),fun(B,set(A)),B3)),A4)) ).

% Compl_UN
tff(fact_2967_image__mult__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [D3: A,A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),D3)
         => ( aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),times_times(A),D3)),set_or1337092689740270186AtMost(A,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_2968_Inf__atMostLessThan,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),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_2969_image__divide__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [D3: A,A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),D3)
         => ( aa(set(A),set(A),image2(A,A,aTP_Lamp_kl(A,fun(A,A),D3)),set_or1337092689740270186AtMost(A,A3,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),D3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),D3)) ) ) ) ).

% image_divide_atLeastAtMost
tff(fact_2970_INT__simps_I1_J,axiom,
    ! [B: $tType,A: $tType,A4: fun(B,set(A)),B3: set(A),C3: set(B)] :
      aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(set(A),fun(B,set(A)),aTP_Lamp_km(fun(B,set(A)),fun(set(A),fun(B,set(A))),A4),B3)),C3)) = $ite(C3 = bot_bot(set(B)),top_top(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),A4),C3))),B3)) ).

% INT_simps(1)
tff(fact_2971_INT__simps_I2_J,axiom,
    ! [A: $tType,B: $tType,A4: set(A),B3: fun(B,set(A)),C3: set(B)] :
      aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_kn(set(A),fun(fun(B,set(A)),fun(B,set(A))),A4),B3)),C3)) = $ite(C3 = bot_bot(set(B)),top_top(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B3),C3)))) ).

% INT_simps(2)
tff(fact_2972_INT__simps_I3_J,axiom,
    ! [B: $tType,A: $tType,A4: fun(B,set(A)),B3: set(A),C3: set(B)] :
      aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(set(A),fun(B,set(A)),aTP_Lamp_ko(fun(B,set(A)),fun(set(A),fun(B,set(A))),A4),B3)),C3)) = $ite(C3 = bot_bot(set(B)),top_top(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),A4),C3))),B3)) ).

% INT_simps(3)
tff(fact_2973_INT__simps_I4_J,axiom,
    ! [A: $tType,B: $tType,A4: set(A),B3: fun(B,set(A)),C3: set(B)] :
      aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_kp(set(A),fun(fun(B,set(A)),fun(B,set(A))),A4),B3)),C3)) = $ite(C3 = bot_bot(set(B)),top_top(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B3),C3)))) ).

% INT_simps(4)
tff(fact_2974_Union__natural,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B)] : aa(fun(set(set(A)),set(set(B))),fun(set(set(A)),set(B)),comp(set(set(B)),set(B),set(set(A)),complete_Sup_Sup(set(B))),image2(set(A),set(B),image2(A,B,F))) = aa(fun(set(set(A)),set(A)),fun(set(set(A)),set(B)),comp(set(A),set(B),set(set(A)),image2(A,B,F)),complete_Sup_Sup(set(A))) ).

% Union_natural
tff(fact_2975_uminus__Inf,axiom,
    ! [A: $tType] :
      ( comple489889107523837845lgebra(A)
     => ! [A4: set(A)] : aa(A,A,uminus_uminus(A),aa(set(A),A,complete_Inf_Inf(A),A4)) = aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),image2(A,A,uminus_uminus(A)),A4)) ) ).

% uminus_Inf
tff(fact_2976_uminus__Sup,axiom,
    ! [A: $tType] :
      ( comple489889107523837845lgebra(A)
     => ! [A4: set(A)] : aa(A,A,uminus_uminus(A),aa(set(A),A,complete_Sup_Sup(A),A4)) = aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),image2(A,A,uminus_uminus(A)),A4)) ) ).

% uminus_Sup
tff(fact_2977_sup__nat__def,axiom,
    sup_sup(nat) = ord_max(nat) ).

% sup_nat_def
tff(fact_2978_UN__extend__simps_I10_J,axiom,
    ! [A: $tType,C: $tType,B: $tType,B3: fun(C,set(A)),F: fun(B,C),A4: set(B)] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(fun(B,C),fun(B,set(A)),aTP_Lamp_kq(fun(C,set(A)),fun(fun(B,C),fun(B,set(A))),B3),F)),A4)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(C),set(set(A)),image2(C,set(A),B3),aa(set(B),set(C),image2(B,C,F),A4))) ).

% UN_extend_simps(10)
tff(fact_2979_UN__extend__simps_I8_J,axiom,
    ! [A: $tType,B: $tType,B3: fun(B,set(A)),A4: set(set(B))] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(set(B)),set(set(A)),image2(set(B),set(A),aTP_Lamp_kr(fun(B,set(A)),fun(set(B),set(A)),B3)),A4)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B3),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),A4))) ).

% UN_extend_simps(8)
tff(fact_2980_INT__extend__simps_I9_J,axiom,
    ! [A: $tType,C: $tType,B: $tType,C3: fun(C,set(A)),B3: fun(B,set(C)),A4: set(B)] : aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(fun(B,set(C)),fun(B,set(A)),aTP_Lamp_ks(fun(C,set(A)),fun(fun(B,set(C)),fun(B,set(A))),C3),B3)),A4)) = aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(C),set(set(A)),image2(C,set(A),C3),aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(B),set(set(C)),image2(B,set(C),B3),A4)))) ).

% INT_extend_simps(9)
tff(fact_2981_INT__extend__simps_I8_J,axiom,
    ! [A: $tType,B: $tType,B3: fun(B,set(A)),A4: set(set(B))] : aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(set(B)),set(set(A)),image2(set(B),set(A),aTP_Lamp_kt(fun(B,set(A)),fun(set(B),set(A)),B3)),A4)) = aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B3),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),A4))) ).

% INT_extend_simps(8)
tff(fact_2982_image__UN,axiom,
    ! [B: $tType,A: $tType,C: $tType,F: fun(B,A),B3: fun(C,set(B)),A4: set(C)] : aa(set(B),set(A),image2(B,A,F),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),image2(C,set(B),B3),A4))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(C),set(set(A)),image2(C,set(A),aa(fun(C,set(B)),fun(C,set(A)),aTP_Lamp_ku(fun(B,A),fun(fun(C,set(B)),fun(C,set(A))),F),B3)),A4)) ).

% image_UN
tff(fact_2983_image__Union,axiom,
    ! [A: $tType,B: $tType,F: fun(B,A),S: set(set(B))] : aa(set(B),set(A),image2(B,A,F),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),S)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(set(B)),set(set(A)),image2(set(B),set(A),image2(B,A,F)),S)) ).

% image_Union
tff(fact_2984_cINF__greatest,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [A4: set(A),M: B,F: fun(A,B)] :
          ( ( A4 != bot_bot(set(A)) )
         => ( ! [X2: A] :
                ( aa(set(A),$o,member(A,X2),A4)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),M),aa(A,B,F,X2)) )
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),M),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F),A4))) ) ) ) ).

% cINF_greatest
tff(fact_2985_INF__eq__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [I4: set(A),F: fun(A,B),C2: B] :
          ( ( I4 != bot_bot(set(A)) )
         => ( ! [I2: A] :
                ( aa(set(A),$o,member(A,I2),I4)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,I2)),C2) )
           => ( ( aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F),I4)) = C2 )
            <=> ! [X4: A] :
                  ( aa(set(A),$o,member(A,X4),I4)
                 => ( aa(A,B,F,X4) = C2 ) ) ) ) ) ) ).

% INF_eq_iff
tff(fact_2986_INF__eq__const,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [I4: set(A),F: fun(A,B),X: B] :
          ( ( I4 != bot_bot(set(A)) )
         => ( ! [I2: A] :
                ( aa(set(A),$o,member(A,I2),I4)
               => ( aa(A,B,F,I2) = X ) )
           => ( aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F),I4)) = X ) ) ) ) ).

% INF_eq_const
tff(fact_2987_INT__insert__distrib,axiom,
    ! [B: $tType,A: $tType,U: A,A4: set(A),A3: B,B3: fun(A,set(B))] :
      ( aa(set(A),$o,member(A,U),A4)
     => ( aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image2(A,set(B),aa(fun(A,set(B)),fun(A,set(B)),aTP_Lamp_kv(B,fun(fun(A,set(B)),fun(A,set(B))),A3),B3)),A4)) = aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),A3),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image2(A,set(B),B3),A4))) ) ) ).

% INT_insert_distrib
tff(fact_2988_INT__extend__simps_I5_J,axiom,
    ! [A: $tType,B: $tType,A3: A,B3: fun(B,set(A)),C3: set(B)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B3),C3))) = aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_kh(A,fun(fun(B,set(A)),fun(B,set(A))),A3),B3)),C3)) ).

% INT_extend_simps(5)
tff(fact_2989_INF__superset__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [B3: set(A),A4: set(A),F: fun(A,B),G: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),A4)
         => ( ! [X2: A] :
                ( aa(set(A),$o,member(A,X2),B3)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,X2)),aa(A,B,G,X2)) )
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F),A4))),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,G),B3))) ) ) ) ).

% INF_superset_mono
tff(fact_2990_INT__subset__iff,axiom,
    ! [A: $tType,B: $tType,B3: set(A),A4: fun(B,set(A)),I4: set(B)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),A4),I4)))
    <=> ! [X4: B] :
          ( aa(set(B),$o,member(B,X4),I4)
         => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),aa(B,set(A),A4,X4)) ) ) ).

% INT_subset_iff
tff(fact_2991_INT__anti__mono,axiom,
    ! [B: $tType,A: $tType,A4: set(A),B3: set(A),F: fun(A,set(B)),G: fun(A,set(B))] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B3)
     => ( ! [X2: A] :
            ( aa(set(A),$o,member(A,X2),A4)
           => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(A,set(B),F,X2)),aa(A,set(B),G,X2)) )
       => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image2(A,set(B),F),B3))),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image2(A,set(B),G),A4))) ) ) ).

% INT_anti_mono
tff(fact_2992_INT__greatest,axiom,
    ! [B: $tType,A: $tType,A4: set(A),C3: set(B),B3: fun(A,set(B))] :
      ( ! [X2: A] :
          ( aa(set(A),$o,member(A,X2),A4)
         => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),C3),aa(A,set(B),B3,X2)) )
     => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),C3),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image2(A,set(B),B3),A4))) ) ).

% INT_greatest
tff(fact_2993_INT__lower,axiom,
    ! [B: $tType,A: $tType,A3: A,A4: set(A),B3: fun(A,set(B))] :
      ( aa(set(A),$o,member(A,A3),A4)
     => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image2(A,set(B),B3),A4))),aa(A,set(B),B3,A3)) ) ).

% INT_lower
tff(fact_2994_INF__greatest,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A4: set(A),U: B,F: fun(A,B)] :
          ( ! [I2: A] :
              ( aa(set(A),$o,member(A,I2),A4)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),U),aa(A,B,F,I2)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),U),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F),A4))) ) ) ).

% INF_greatest
tff(fact_2995_le__INF__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [U: A,F: fun(B,A),A4: set(B)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),U),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F),A4)))
        <=> ! [X4: B] :
              ( aa(set(B),$o,member(B,X4),A4)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),U),aa(B,A,F,X4)) ) ) ) ).

% le_INF_iff
tff(fact_2996_INF__lower2,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [I: A,A4: set(A),F: fun(A,B),U: B] :
          ( aa(set(A),$o,member(A,I),A4)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,I)),U)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F),A4))),U) ) ) ) ).

% INF_lower2
tff(fact_2997_INF__mono_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [F: fun(A,B),G: fun(A,B),A4: set(A)] :
          ( ! [X2: A] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,X2)),aa(A,B,G,X2))
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F),A4))),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,G),A4))) ) ) ).

% INF_mono'
tff(fact_2998_INF__lower,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [I: A,A4: set(A),F: fun(A,B)] :
          ( aa(set(A),$o,member(A,I),A4)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F),A4))),aa(A,B,F,I)) ) ) ).

% INF_lower
tff(fact_2999_INF__mono,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( comple6319245703460814977attice(C)
     => ! [B3: set(A),A4: set(B),F: fun(B,C),G: fun(A,C)] :
          ( ! [M3: A] :
              ( aa(set(A),$o,member(A,M3),B3)
             => ? [X3: B] :
                  ( aa(set(B),$o,member(B,X3),A4)
                  & aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(B,C,F,X3)),aa(A,C,G,M3)) ) )
         => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(set(C),C,complete_Inf_Inf(C),aa(set(B),set(C),image2(B,C,F),A4))),aa(set(C),C,complete_Inf_Inf(C),aa(set(A),set(C),image2(A,C,G),B3))) ) ) ).

% INF_mono
tff(fact_3000_INF__eqI,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A4: set(A),X: B,F: fun(A,B)] :
          ( ! [I2: A] :
              ( aa(set(A),$o,member(A,I2),A4)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),aa(A,B,F,I2)) )
         => ( ! [Y2: B] :
                ( ! [I5: A] :
                    ( aa(set(A),$o,member(A,I5),A4)
                   => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Y2),aa(A,B,F,I5)) )
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Y2),X) )
           => ( aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F),A4)) = X ) ) ) ) ).

% INF_eqI
tff(fact_3001_vimage__INT,axiom,
    ! [B: $tType,A: $tType,C: $tType,F: fun(A,B),B3: fun(C,set(B)),A4: set(C)] : aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(C),set(set(B)),image2(C,set(B),B3),A4))) = aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(C),set(set(A)),image2(C,set(A),aa(fun(C,set(B)),fun(C,set(A)),aTP_Lamp_kw(fun(A,B),fun(fun(C,set(B)),fun(C,set(A))),F),B3)),A4)) ).

% vimage_INT
tff(fact_3002_Un__INT__distrib2,axiom,
    ! [C: $tType,A: $tType,B: $tType,A4: fun(B,set(A)),I4: set(B),B3: fun(C,set(A)),J3: set(C)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),A4),I4))),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(C),set(set(A)),image2(C,set(A),B3),J3))) = aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(set(C),fun(B,set(A)),aa(fun(C,set(A)),fun(set(C),fun(B,set(A))),aTP_Lamp_ky(fun(B,set(A)),fun(fun(C,set(A)),fun(set(C),fun(B,set(A)))),A4),B3),J3)),I4)) ).

% Un_INT_distrib2
tff(fact_3003_Un__INT__distrib,axiom,
    ! [A: $tType,B: $tType,B3: set(A),A4: fun(B,set(A)),I4: set(B)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B3),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),A4),I4))) = aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_ki(set(A),fun(fun(B,set(A)),fun(B,set(A))),B3),A4)),I4)) ).

% Un_INT_distrib
tff(fact_3004_INT__extend__simps_I6_J,axiom,
    ! [A: $tType,B: $tType,A4: fun(B,set(A)),C3: set(B),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),A4),C3))),B3) = aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(set(A),fun(B,set(A)),aTP_Lamp_kj(fun(B,set(A)),fun(set(A),fun(B,set(A))),A4),B3)),C3)) ).

% INT_extend_simps(6)
tff(fact_3005_INT__extend__simps_I7_J,axiom,
    ! [A: $tType,B: $tType,A4: set(A),B3: fun(B,set(A)),C3: set(B)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B3),C3))) = aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_ki(set(A),fun(fun(B,set(A)),fun(B,set(A))),A4),B3)),C3)) ).

% INT_extend_simps(7)
tff(fact_3006_Int__Inter__image,axiom,
    ! [A: $tType,B: $tType,A4: fun(B,set(A)),B3: fun(B,set(A)),C3: set(B)] : aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_kz(fun(B,set(A)),fun(fun(B,set(A)),fun(B,set(A))),A4),B3)),C3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),A4),C3))),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B3),C3))) ).

% Int_Inter_image
tff(fact_3007_INT__Int__distrib,axiom,
    ! [A: $tType,B: $tType,A4: fun(B,set(A)),B3: fun(B,set(A)),I4: set(B)] : aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_kz(fun(B,set(A)),fun(fun(B,set(A)),fun(B,set(A))),A4),B3)),I4)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),A4),I4))),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B3),I4))) ).

% INT_Int_distrib
tff(fact_3008_INT__absorb,axiom,
    ! [B: $tType,A: $tType,K: A,I4: set(A),A4: fun(A,set(B))] :
      ( aa(set(A),$o,member(A,K),I4)
     => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(A,set(B),A4,K)),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image2(A,set(B),A4),I4))) = aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image2(A,set(B),A4),I4)) ) ) ).

% INT_absorb
tff(fact_3009_INF__absorb,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [K: A,I4: set(A),A4: fun(A,B)] :
          ( aa(set(A),$o,member(A,K),I4)
         => ( aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(A,B,A4,K)),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,A4),I4))) = aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,A4),I4)) ) ) ) ).

% INF_absorb
tff(fact_3010_INF__inf__distrib,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F: fun(B,A),A4: set(B),G: fun(B,A)] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F),A4))),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,G),A4))) = aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_la(fun(B,A),fun(fun(B,A),fun(B,A)),F),G)),A4)) ) ).

% INF_inf_distrib
tff(fact_3011_INT__E,axiom,
    ! [A: $tType,B: $tType,B2: A,B3: fun(B,set(A)),A4: set(B),A3: B] :
      ( aa(set(A),$o,member(A,B2),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B3),A4)))
     => ( ~ aa(set(A),$o,member(A,B2),aa(B,set(A),B3,A3))
       => ~ aa(set(B),$o,member(B,A3),A4) ) ) ).

% INT_E
tff(fact_3012_INT__D,axiom,
    ! [A: $tType,B: $tType,B2: A,B3: fun(B,set(A)),A4: set(B),A3: B] :
      ( aa(set(A),$o,member(A,B2),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B3),A4)))
     => ( aa(set(B),$o,member(B,A3),A4)
       => aa(set(A),$o,member(A,B2),aa(B,set(A),B3,A3)) ) ) ).

% INT_D
tff(fact_3013_INF__commute,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F: fun(B,fun(C,A)),B3: set(C),A4: set(B)] : aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,aa(set(C),fun(B,A),aTP_Lamp_lb(fun(B,fun(C,A)),fun(set(C),fun(B,A)),F),B3)),A4)) = aa(set(A),A,complete_Inf_Inf(A),aa(set(C),set(A),image2(C,A,aa(set(B),fun(C,A),aTP_Lamp_ld(fun(B,fun(C,A)),fun(set(B),fun(C,A)),F),A4)),B3)) ) ).

% INF_commute
tff(fact_3014_Sup_OSUP__identity__eq,axiom,
    ! [A: $tType,Sup: fun(set(A),A),A4: set(A)] : aa(set(A),A,Sup,aa(set(A),set(A),image2(A,A,aTP_Lamp_au(A,A)),A4)) = aa(set(A),A,Sup,A4) ).

% Sup.SUP_identity_eq
tff(fact_3015_Inf_OINF__identity__eq,axiom,
    ! [A: $tType,Inf: fun(set(A),A),A4: set(A)] : aa(set(A),A,Inf,aa(set(A),set(A),image2(A,A,aTP_Lamp_au(A,A)),A4)) = aa(set(A),A,Inf,A4) ).

% Inf.INF_identity_eq
tff(fact_3016_INT__extend__simps_I10_J,axiom,
    ! [A: $tType,C: $tType,B: $tType,B3: fun(C,set(A)),F: fun(B,C),A4: set(B)] : aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(fun(B,C),fun(B,set(A)),aTP_Lamp_kq(fun(C,set(A)),fun(fun(B,C),fun(B,set(A))),B3),F)),A4)) = aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(C),set(set(A)),image2(C,set(A),B3),aa(set(B),set(C),image2(B,C,F),A4))) ).

% INT_extend_simps(10)
tff(fact_3017_INF__less__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [F: fun(B,A),A4: set(B),A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F),A4))),A3)
        <=> ? [X4: B] :
              ( aa(set(B),$o,member(B,X4),A4)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,F,X4)),A3) ) ) ) ).

% INF_less_iff
tff(fact_3018_less__INF__D,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Y: A,F: fun(B,A),A4: set(B),I: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F),A4)))
         => ( aa(set(B),$o,member(B,I),A4)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),aa(B,A,F,I)) ) ) ) ).

% less_INF_D
tff(fact_3019_INTER__UNIV__conv_I1_J,axiom,
    ! [B: $tType,A: $tType,B3: fun(B,set(A)),A4: set(B)] :
      ( ( top_top(set(A)) = aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B3),A4)) )
    <=> ! [X4: B] :
          ( aa(set(B),$o,member(B,X4),A4)
         => ( aa(B,set(A),B3,X4) = top_top(set(A)) ) ) ) ).

% INTER_UNIV_conv(1)
tff(fact_3020_INTER__UNIV__conv_I2_J,axiom,
    ! [B: $tType,A: $tType,B3: fun(B,set(A)),A4: set(B)] :
      ( ( aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B3),A4)) = top_top(set(A)) )
    <=> ! [X4: B] :
          ( aa(set(B),$o,member(B,X4),A4)
         => ( aa(B,set(A),B3,X4) = top_top(set(A)) ) ) ) ).

% INTER_UNIV_conv(2)
tff(fact_3021_INF__sup__distrib2,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [F: fun(B,A),A4: set(B),G: fun(C,A),B3: set(C)] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F),A4))),aa(set(A),A,complete_Inf_Inf(A),aa(set(C),set(A),image2(C,A,G),B3))) = aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,aa(set(C),fun(B,A),aa(fun(C,A),fun(set(C),fun(B,A)),aTP_Lamp_lf(fun(B,A),fun(fun(C,A),fun(set(C),fun(B,A))),F),G),B3)),A4)) ) ).

% INF_sup_distrib2
tff(fact_3022_sup__INF,axiom,
    ! [A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [A3: A,F: fun(B,A),B3: set(B)] : aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F),B3))) = aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_lg(A,fun(fun(B,A),fun(B,A)),A3),F)),B3)) ) ).

% sup_INF
tff(fact_3023_Inf__sup,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [B3: set(A),A3: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Inf_Inf(A),B3)),A3) = aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),image2(A,A,aTP_Lamp_lh(A,fun(A,A),A3)),B3)) ) ).

% Inf_sup
tff(fact_3024_INF__sup,axiom,
    ! [A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [F: fun(B,A),B3: set(B),A3: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F),B3))),A3) = aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,aa(A,fun(B,A),aTP_Lamp_li(fun(B,A),fun(A,fun(B,A)),F),A3)),B3)) ) ).

% INF_sup
tff(fact_3025_sup__Inf,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [A3: A,B3: set(A)] : aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),aa(set(A),A,complete_Inf_Inf(A),B3)) = aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),sup_sup(A),A3)),B3)) ) ).

% sup_Inf
tff(fact_3026_Compr__image__eq,axiom,
    ! [B: $tType,A: $tType,F: fun(B,A),A4: set(B),P: fun(A,$o)] : aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aa(set(B),fun(fun(A,$o),fun(A,$o)),aTP_Lamp_lj(fun(B,A),fun(set(B),fun(fun(A,$o),fun(A,$o))),F),A4),P)) = aa(set(B),set(A),image2(B,A,F),aa(fun(B,$o),set(B),collect(B),aa(fun(A,$o),fun(B,$o),aa(set(B),fun(fun(A,$o),fun(B,$o)),aTP_Lamp_lk(fun(B,A),fun(set(B),fun(fun(A,$o),fun(B,$o))),F),A4),P))) ).

% Compr_image_eq
tff(fact_3027_image__image,axiom,
    ! [B: $tType,A: $tType,C: $tType,F: fun(B,A),G: fun(C,B),A4: set(C)] : aa(set(B),set(A),image2(B,A,F),aa(set(C),set(B),image2(C,B,G),A4)) = aa(set(C),set(A),image2(C,A,aa(fun(C,B),fun(C,A),aTP_Lamp_ll(fun(B,A),fun(fun(C,B),fun(C,A)),F),G)),A4) ).

% image_image
tff(fact_3028_imageE,axiom,
    ! [A: $tType,B: $tType,B2: A,F: fun(B,A),A4: set(B)] :
      ( aa(set(A),$o,member(A,B2),aa(set(B),set(A),image2(B,A,F),A4))
     => ~ ! [X2: B] :
            ( ( B2 = aa(B,A,F,X2) )
           => ~ aa(set(B),$o,member(B,X2),A4) ) ) ).

% imageE
tff(fact_3029_INF__constant,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [C2: A,A4: set(B)] :
          aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,aTP_Lamp_lm(A,fun(B,A),C2)),A4)) = $ite(A4 = bot_bot(set(B)),top_top(A),C2) ) ).

% INF_constant
tff(fact_3030_INF__empty,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F: fun(B,A)] : aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F),bot_bot(set(B)))) = top_top(A) ) ).

% INF_empty
tff(fact_3031_INF__inf__const1,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [I4: set(A),X: B,F: fun(A,B)] :
          ( ( I4 != bot_bot(set(A)) )
         => ( aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ln(B,fun(fun(A,B),fun(A,B)),X),F)),I4)) = aa(B,B,aa(B,fun(B,B),inf_inf(B),X),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F),I4))) ) ) ) ).

% INF_inf_const1
tff(fact_3032_INF__inf__const2,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [I4: set(A),F: fun(A,B),X: B] :
          ( ( I4 != bot_bot(set(A)) )
         => ( aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,aa(B,fun(A,B),aTP_Lamp_lo(fun(A,B),fun(B,fun(A,B)),F),X)),I4)) = aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F),I4))),X) ) ) ) ).

% INF_inf_const2
tff(fact_3033_uminus__SUP,axiom,
    ! [A: $tType,B: $tType] :
      ( comple489889107523837845lgebra(A)
     => ! [B3: fun(B,A),A4: set(B)] : aa(A,A,uminus_uminus(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,B3),A4))) = aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,aTP_Lamp_lp(fun(B,A),fun(B,A),B3)),A4)) ) ).

% uminus_SUP
tff(fact_3034_uminus__INF,axiom,
    ! [A: $tType,B: $tType] :
      ( comple489889107523837845lgebra(A)
     => ! [B3: fun(B,A),A4: set(B)] : aa(A,A,uminus_uminus(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,B3),A4))) = aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,aTP_Lamp_lp(fun(B,A),fun(B,A),B3)),A4)) ) ).

% uminus_INF
tff(fact_3035_INF__insert,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F: fun(B,A),A3: B,A4: set(B)] : aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),A3),A4))) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(B,A,F,A3)),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F),A4))) ) ).

% INF_insert
tff(fact_3036_rangeI,axiom,
    ! [A: $tType,B: $tType,F: fun(B,A),X: B] : aa(set(A),$o,member(A,aa(B,A,F,X)),aa(set(B),set(A),image2(B,A,F),top_top(set(B)))) ).

% rangeI
tff(fact_3037_range__eqI,axiom,
    ! [A: $tType,B: $tType,B2: A,F: fun(B,A),X: B] :
      ( ( B2 = aa(B,A,F,X) )
     => aa(set(A),$o,member(A,B2),aa(set(B),set(A),image2(B,A,F),top_top(set(B)))) ) ).

% range_eqI
tff(fact_3038_INF__union,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [M4: fun(B,A),A4: set(B),B3: set(B)] : aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,M4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A4),B3))) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,M4),A4))),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,M4),B3))) ) ).

% INF_union
tff(fact_3039_INT__empty,axiom,
    ! [B: $tType,A: $tType,B3: fun(B,set(A))] : aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B3),bot_bot(set(B)))) = top_top(set(A)) ).

% INT_empty
tff(fact_3040_Inter__subset,axiom,
    ! [A: $tType,A4: set(set(A)),B3: set(A)] :
      ( ! [X6: set(A)] :
          ( aa(set(set(A)),$o,member(set(A),X6),A4)
         => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X6),B3) )
     => ( ( A4 != bot_bot(set(set(A))) )
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),A4)),B3) ) ) ).

% Inter_subset
tff(fact_3041_INT__extend__simps_I1_J,axiom,
    ! [A: $tType,B: $tType,A4: fun(B,set(A)),C3: set(B),B3: set(A)] :
      aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),A4),C3))),B3) = $ite(C3 = bot_bot(set(B)),B3,aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(set(A),fun(B,set(A)),aTP_Lamp_km(fun(B,set(A)),fun(set(A),fun(B,set(A))),A4),B3)),C3))) ).

% INT_extend_simps(1)
tff(fact_3042_INT__extend__simps_I2_J,axiom,
    ! [A: $tType,B: $tType,A4: set(A),B3: fun(B,set(A)),C3: set(B)] :
      aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B3),C3))) = $ite(C3 = bot_bot(set(B)),A4,aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_kn(set(A),fun(fun(B,set(A)),fun(B,set(A))),A4),B3)),C3))) ).

% INT_extend_simps(2)
tff(fact_3043_SUP__UNION,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F: fun(B,A),G: fun(C,set(B)),A4: set(C)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),image2(C,set(B),G),A4)))) = aa(set(A),A,complete_Sup_Sup(A),aa(set(C),set(A),image2(C,A,aa(fun(C,set(B)),fun(C,A),aTP_Lamp_lq(fun(B,A),fun(fun(C,set(B)),fun(C,A)),F),G)),A4)) ) ).

% SUP_UNION
tff(fact_3044_image__Un,axiom,
    ! [A: $tType,B: $tType,F: fun(B,A),A4: set(B),B3: set(B)] : aa(set(B),set(A),image2(B,A,F),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A4),B3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(B),set(A),image2(B,A,F),A4)),aa(set(B),set(A),image2(B,A,F),B3)) ).

% image_Un
tff(fact_3045_Inter__empty,axiom,
    ! [A: $tType] : aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),bot_bot(set(set(A)))) = top_top(set(A)) ).

% Inter_empty
tff(fact_3046_INT__Un,axiom,
    ! [A: $tType,B: $tType,M4: fun(B,set(A)),A4: set(B),B3: set(B)] : aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),M4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A4),B3))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),M4),A4))),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),M4),B3))) ).

% INT_Un
tff(fact_3047_Inter__Un__distrib,axiom,
    ! [A: $tType,A4: set(set(A)),B3: set(set(A))] : aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(set(A)),set(set(A)),aa(set(set(A)),fun(set(set(A)),set(set(A))),sup_sup(set(set(A))),A4),B3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),A4)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),B3)) ).

% Inter_Un_distrib
tff(fact_3048_UN__extend__simps_I7_J,axiom,
    ! [A: $tType,B: $tType,A4: set(A),B3: fun(B,set(A)),C3: set(B)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B3),C3))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_kp(set(A),fun(fun(B,set(A)),fun(B,set(A))),A4),B3)),C3)) ).

% UN_extend_simps(7)
tff(fact_3049_INF__SUP,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [P: fun(C,fun(B,A))] : aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,aTP_Lamp_ls(fun(C,fun(B,A)),fun(B,A),P)),top_top(set(B)))) = aa(set(A),A,complete_Sup_Sup(A),aa(set(fun(B,C)),set(A),image2(fun(B,C),A,aTP_Lamp_lu(fun(C,fun(B,A)),fun(fun(B,C),A),P)),top_top(set(fun(B,C))))) ) ).

% INF_SUP
tff(fact_3050_SUP__INF,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [P: fun(C,fun(B,A))] : aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,aTP_Lamp_lv(fun(C,fun(B,A)),fun(B,A),P)),top_top(set(B)))) = aa(set(A),A,complete_Inf_Inf(A),aa(set(fun(B,C)),set(A),image2(fun(B,C),A,aTP_Lamp_lw(fun(C,fun(B,A)),fun(fun(B,C),A),P)),top_top(set(fun(B,C))))) ) ).

% SUP_INF
tff(fact_3051_bot__finite__def,axiom,
    ! [A: $tType] :
      ( finite_lattice(A)
     => ( bot_bot(A) = aa(set(A),A,complete_Inf_Inf(A),top_top(set(A))) ) ) ).

% bot_finite_def
tff(fact_3052_Inf__finite__insert,axiom,
    ! [A: $tType] :
      ( finite_lattice(A)
     => ! [A3: A,A4: set(A)] : aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),A4)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),aa(set(A),A,complete_Inf_Inf(A),A4)) ) ).

% Inf_finite_insert
tff(fact_3053_INF__le__SUP,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A4: set(A),F: fun(A,B)] :
          ( ( A4 != bot_bot(set(A)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F),A4))),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F),A4))) ) ) ).

% INF_le_SUP
tff(fact_3054_pigeonhole__infinite,axiom,
    ! [B: $tType,A: $tType,A4: set(A),F: fun(A,B)] :
      ( ~ aa(set(A),$o,finite_finite2(A),A4)
     => ( aa(set(B),$o,finite_finite2(B),aa(set(A),set(B),image2(A,B,F),A4))
       => ? [X2: A] :
            ( aa(set(A),$o,member(A,X2),A4)
            & ~ aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(A,fun(A,$o),aa(fun(A,B),fun(A,fun(A,$o)),aTP_Lamp_lx(set(A),fun(fun(A,B),fun(A,fun(A,$o))),A4),F),X2))) ) ) ) ).

% pigeonhole_infinite
tff(fact_3055_image__Collect__subsetI,axiom,
    ! [A: $tType,B: $tType,P: fun(A,$o),F: fun(A,B),B3: set(B)] :
      ( ! [X2: A] :
          ( aa(A,$o,P,X2)
         => aa(set(B),$o,member(B,aa(A,B,F,X2)),B3) )
     => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F),aa(fun(A,$o),set(A),collect(A),P))),B3) ) ).

% image_Collect_subsetI
tff(fact_3056_range__composition,axiom,
    ! [A: $tType,C: $tType,B: $tType,F: fun(C,A),G: fun(B,C)] : aa(set(B),set(A),image2(B,A,aa(fun(B,C),fun(B,A),aTP_Lamp_ly(fun(C,A),fun(fun(B,C),fun(B,A)),F),G)),top_top(set(B))) = aa(set(C),set(A),image2(C,A,F),aa(set(B),set(C),image2(B,C,G),top_top(set(B)))) ).

% range_composition
tff(fact_3057_rangeE,axiom,
    ! [A: $tType,B: $tType,B2: A,F: fun(B,A)] :
      ( aa(set(A),$o,member(A,B2),aa(set(B),set(A),image2(B,A,F),top_top(set(B))))
     => ~ ! [X2: B] : B2 != aa(B,A,F,X2) ) ).

% rangeE
tff(fact_3058_SUP__commute,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F: fun(B,fun(C,A)),B3: set(C),A4: set(B)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,aa(set(C),fun(B,A),aTP_Lamp_lz(fun(B,fun(C,A)),fun(set(C),fun(B,A)),F),B3)),A4)) = aa(set(A),A,complete_Sup_Sup(A),aa(set(C),set(A),image2(C,A,aa(set(B),fun(C,A),aTP_Lamp_ma(fun(B,fun(C,A)),fun(set(B),fun(C,A)),F),A4)),B3)) ) ).

% SUP_commute
tff(fact_3059_UN__UN__flatten,axiom,
    ! [B: $tType,A: $tType,C: $tType,C3: fun(B,set(A)),B3: fun(C,set(B)),A4: set(C)] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),C3),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),image2(C,set(B),B3),A4)))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(C),set(set(A)),image2(C,set(A),aa(fun(C,set(B)),fun(C,set(A)),aTP_Lamp_mb(fun(B,set(A)),fun(fun(C,set(B)),fun(C,set(A))),C3),B3)),A4)) ).

% UN_UN_flatten
tff(fact_3060_UN__E,axiom,
    ! [A: $tType,B: $tType,B2: A,B3: fun(B,set(A)),A4: set(B)] :
      ( aa(set(A),$o,member(A,B2),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B3),A4)))
     => ~ ! [X2: B] :
            ( aa(set(B),$o,member(B,X2),A4)
           => ~ aa(set(A),$o,member(A,B2),aa(B,set(A),B3,X2)) ) ) ).

% UN_E
tff(fact_3061_UN__extend__simps_I9_J,axiom,
    ! [A: $tType,C: $tType,B: $tType,C3: fun(C,set(A)),B3: fun(B,set(C)),A4: set(B)] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(fun(B,set(C)),fun(B,set(A)),aTP_Lamp_mc(fun(C,set(A)),fun(fun(B,set(C)),fun(B,set(A))),C3),B3)),A4)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(C),set(set(A)),image2(C,set(A),C3),aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(B),set(set(C)),image2(B,set(C),B3),A4)))) ).

% UN_extend_simps(9)
tff(fact_3062_UNION__singleton__eq__range,axiom,
    ! [A: $tType,B: $tType,F: fun(B,A),A4: set(B)] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aTP_Lamp_md(fun(B,A),fun(B,set(A)),F)),A4)) = aa(set(B),set(A),image2(B,A,F),A4) ).

% UNION_singleton_eq_range
tff(fact_3063_INT__extend__simps_I3_J,axiom,
    ! [A: $tType,B: $tType,A4: fun(B,set(A)),C3: set(B),B3: set(A)] :
      aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),A4),C3))),B3) = $ite(C3 = bot_bot(set(B)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),top_top(set(A))),B3),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(set(A),fun(B,set(A)),aTP_Lamp_ko(fun(B,set(A)),fun(set(A),fun(B,set(A))),A4),B3)),C3))) ).

% INT_extend_simps(3)
tff(fact_3064_INT__extend__simps_I4_J,axiom,
    ! [A: $tType,B: $tType,A4: set(A),B3: fun(B,set(A)),C3: set(B)] :
      aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B3),C3))) = $ite(C3 = bot_bot(set(B)),A4,aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_kp(set(A),fun(fun(B,set(A)),fun(B,set(A))),A4),B3)),C3))) ).

% INT_extend_simps(4)
tff(fact_3065_cInf__eq__non__empty,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X5: set(A),A3: A] :
          ( ( X5 != bot_bot(set(A)) )
         => ( ! [X2: A] :
                ( aa(set(A),$o,member(A,X2),X5)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),X2) )
           => ( ! [Y2: A] :
                  ( ! [X3: A] :
                      ( aa(set(A),$o,member(A,X3),X5)
                     => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y2),X3) )
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y2),A3) )
             => ( aa(set(A),A,complete_Inf_Inf(A),X5) = A3 ) ) ) ) ) ).

% cInf_eq_non_empty
tff(fact_3066_cInf__greatest,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X5: set(A),Z2: A] :
          ( ( X5 != bot_bot(set(A)) )
         => ( ! [X2: A] :
                ( aa(set(A),$o,member(A,X2),X5)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z2),X2) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z2),aa(set(A),A,complete_Inf_Inf(A),X5)) ) ) ) ).

% cInf_greatest
tff(fact_3067_Inf__less__eq,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A4: set(A),U: A] :
          ( ! [V2: A] :
              ( aa(set(A),$o,member(A,V2),A4)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),V2),U) )
         => ( ( A4 != bot_bot(set(A)) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A4)),U) ) ) ) ).

% Inf_less_eq
tff(fact_3068_cInf__lessD,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X5: set(A),Z2: A] :
          ( ( X5 != bot_bot(set(A)) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(A),A,complete_Inf_Inf(A),X5)),Z2)
           => ? [X2: A] :
                ( aa(set(A),$o,member(A,X2),X5)
                & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X2),Z2) ) ) ) ) ).

% cInf_lessD
tff(fact_3069_Inf__finite__empty,axiom,
    ! [A: $tType] :
      ( finite_lattice(A)
     => ( aa(set(A),A,complete_Inf_Inf(A),bot_bot(set(A))) = aa(set(A),A,complete_Sup_Sup(A),top_top(set(A))) ) ) ).

% Inf_finite_empty
tff(fact_3070_Sup__finite__empty,axiom,
    ! [A: $tType] :
      ( finite_lattice(A)
     => ( aa(set(A),A,complete_Sup_Sup(A),bot_bot(set(A))) = aa(set(A),A,complete_Inf_Inf(A),top_top(set(A))) ) ) ).

% Sup_finite_empty
tff(fact_3071_SUP__eq__const,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [I4: set(A),F: fun(A,B),X: B] :
          ( ( I4 != bot_bot(set(A)) )
         => ( ! [I2: A] :
                ( aa(set(A),$o,member(A,I2),I4)
               => ( aa(A,B,F,I2) = X ) )
           => ( aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F),I4)) = X ) ) ) ) ).

% SUP_eq_const
tff(fact_3072_Inf__sup__eq__top__iff,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [B3: set(A),A3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Inf_Inf(A),B3)),A3) = top_top(A) )
        <=> ! [X4: A] :
              ( aa(set(A),$o,member(A,X4),B3)
             => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X4),A3) = top_top(A) ) ) ) ) ).

% Inf_sup_eq_top_iff
tff(fact_3073_range__subsetD,axiom,
    ! [B: $tType,A: $tType,F: fun(B,A),B3: set(A),I: B] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(B),set(A),image2(B,A,F),top_top(set(B)))),B3)
     => aa(set(A),$o,member(A,aa(B,A,F,I)),B3) ) ).

% range_subsetD
tff(fact_3074_inf__Sup,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [A3: A,B3: set(A)] : aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),aa(set(A),A,complete_Sup_Sup(A),B3)) = aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),inf_inf(A),A3)),B3)) ) ).

% inf_Sup
tff(fact_3075_surj__fun__eq,axiom,
    ! [B: $tType,C: $tType,A: $tType,F: fun(B,A),X5: set(B),G1: fun(A,C),G22: fun(A,C)] :
      ( ( aa(set(B),set(A),image2(B,A,F),X5) = top_top(set(A)) )
     => ( ! [X2: B] :
            ( aa(set(B),$o,member(B,X2),X5)
           => ( aa(B,C,aa(fun(B,A),fun(B,C),comp(A,C,B,G1),F),X2) = aa(B,C,aa(fun(B,A),fun(B,C),comp(A,C,B,G22),F),X2) ) )
       => ( G1 = G22 ) ) ) ).

% surj_fun_eq
tff(fact_3076_image__Int__subset,axiom,
    ! [A: $tType,B: $tType,F: fun(B,A),A4: set(B),B3: set(B)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(B),set(A),image2(B,A,F),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),B3))),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(B),set(A),image2(B,A,F),A4)),aa(set(B),set(A),image2(B,A,F),B3))) ).

% image_Int_subset
tff(fact_3077_Inf__union__distrib,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A4: set(A),B3: set(A)] : aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Inf_Inf(A),A4)),aa(set(A),A,complete_Inf_Inf(A),B3)) ) ).

% Inf_union_distrib
tff(fact_3078_Inter__Un__subset,axiom,
    ! [A: $tType,A4: set(set(A)),B3: set(set(A))] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),A4)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),B3))),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(set(A)),set(set(A)),aa(set(set(A)),fun(set(set(A)),set(set(A))),inf_inf(set(set(A))),A4),B3))) ).

% Inter_Un_subset
tff(fact_3079_type__definition_ORep__range,axiom,
    ! [A: $tType,B: $tType,Rep: fun(A,B),Abs: fun(B,A),A4: set(B)] :
      ( type_definition(A,B,Rep,Abs,A4)
     => ( aa(set(A),set(B),image2(A,B,Rep),top_top(set(A))) = A4 ) ) ).

% type_definition.Rep_range
tff(fact_3080_type__definition_OAbs__image,axiom,
    ! [B: $tType,A: $tType,Rep: fun(A,B),Abs: fun(B,A),A4: set(B)] :
      ( type_definition(A,B,Rep,Abs,A4)
     => ( aa(set(B),set(A),image2(B,A,Abs),A4) = top_top(set(A)) ) ) ).

% type_definition.Abs_image
tff(fact_3081_SUP__eqI,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A4: set(A),F: fun(A,B),X: B] :
          ( ! [I2: A] :
              ( aa(set(A),$o,member(A,I2),A4)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,I2)),X) )
         => ( ! [Y2: B] :
                ( ! [I5: A] :
                    ( aa(set(A),$o,member(A,I5),A4)
                   => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,I5)),Y2) )
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),Y2) )
           => ( aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F),A4)) = X ) ) ) ) ).

% SUP_eqI
tff(fact_3082_SUP__mono,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comple6319245703460814977attice(C)
     => ! [A4: set(A),B3: set(B),F: fun(A,C),G: fun(B,C)] :
          ( ! [N3: A] :
              ( aa(set(A),$o,member(A,N3),A4)
             => ? [X3: B] :
                  ( aa(set(B),$o,member(B,X3),B3)
                  & aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(A,C,F,N3)),aa(B,C,G,X3)) ) )
         => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(set(C),C,complete_Sup_Sup(C),aa(set(A),set(C),image2(A,C,F),A4))),aa(set(C),C,complete_Sup_Sup(C),aa(set(B),set(C),image2(B,C,G),B3))) ) ) ).

% SUP_mono
tff(fact_3083_SUP__least,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A4: set(A),F: fun(A,B),U: B] :
          ( ! [I2: A] :
              ( aa(set(A),$o,member(A,I2),A4)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,I2)),U) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F),A4))),U) ) ) ).

% SUP_least
tff(fact_3084_SUP__mono_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [F: fun(A,B),G: fun(A,B),A4: set(A)] :
          ( ! [X2: A] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,X2)),aa(A,B,G,X2))
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F),A4))),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,G),A4))) ) ) ).

% SUP_mono'
tff(fact_3085_SUP__upper,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [I: A,A4: set(A),F: fun(A,B)] :
          ( aa(set(A),$o,member(A,I),A4)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,I)),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F),A4))) ) ) ).

% SUP_upper
tff(fact_3086_SUP__le__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F: fun(B,A),A4: set(B),U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F),A4))),U)
        <=> ! [X4: B] :
              ( aa(set(B),$o,member(B,X4),A4)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,F,X4)),U) ) ) ) ).

% SUP_le_iff
tff(fact_3087_SUP__upper2,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [I: A,A4: set(A),U: B,F: fun(A,B)] :
          ( aa(set(A),$o,member(A,I),A4)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),U),aa(A,B,F,I))
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),U),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F),A4))) ) ) ) ).

% SUP_upper2
tff(fact_3088_SUP__lessD,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F: fun(B,A),A4: set(B),Y: A,I: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F),A4))),Y)
         => ( aa(set(B),$o,member(B,I),A4)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,F,I)),Y) ) ) ) ).

% SUP_lessD
tff(fact_3089_less__SUP__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [A3: A,F: fun(B,A),A4: set(B)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F),A4)))
        <=> ? [X4: B] :
              ( aa(set(B),$o,member(B,X4),A4)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(B,A,F,X4)) ) ) ) ).

% less_SUP_iff
tff(fact_3090_image__constant,axiom,
    ! [A: $tType,B: $tType,X: A,A4: set(A),C2: B] :
      ( aa(set(A),$o,member(A,X),A4)
     => ( aa(set(A),set(B),image2(A,B,aTP_Lamp_ax(B,fun(A,B),C2)),A4) = aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),C2),bot_bot(set(B))) ) ) ).

% image_constant
tff(fact_3091_image__constant__conv,axiom,
    ! [B: $tType,A: $tType,C2: A,A4: set(B)] :
      aa(set(B),set(A),image2(B,A,aa(A,fun(B,A),aTP_Lamp_kf(A,fun(B,A)),C2)),A4) = $ite(A4 = bot_bot(set(B)),bot_bot(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),C2),bot_bot(set(A)))) ).

% image_constant_conv
tff(fact_3092_nat__seg__image__imp__finite,axiom,
    ! [A: $tType,A4: set(A),F: fun(nat,A),N: nat] :
      ( ( A4 = aa(set(nat),set(A),image2(nat,A,F),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_bk(nat,fun(nat,$o)),N))) )
     => aa(set(A),$o,finite_finite2(A),A4) ) ).

% nat_seg_image_imp_finite
tff(fact_3093_finite__conv__nat__seg__image,axiom,
    ! [A: $tType,A4: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A4)
    <=> ? [N2: nat,F6: fun(nat,A)] : A4 = aa(set(nat),set(A),image2(nat,A,F6),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_bk(nat,fun(nat,$o)),N2))) ) ).

% finite_conv_nat_seg_image
tff(fact_3094_Inter__insert,axiom,
    ! [A: $tType,A3: set(A),B3: set(set(A))] : aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(set(A)),set(set(A)),aa(set(A),fun(set(set(A)),set(set(A))),insert2(set(A)),A3),B3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),B3)) ).

% Inter_insert
tff(fact_3095_finite__range__imageI,axiom,
    ! [A: $tType,C: $tType,B: $tType,G: fun(B,A),F: fun(A,C)] :
      ( aa(set(A),$o,finite_finite2(A),aa(set(B),set(A),image2(B,A,G),top_top(set(B))))
     => aa(set(C),$o,finite_finite2(C),aa(set(B),set(C),image2(B,C,aa(fun(A,C),fun(B,C),aTP_Lamp_me(fun(B,A),fun(fun(A,C),fun(B,C)),G),F)),top_top(set(B)))) ) ).

% finite_range_imageI
tff(fact_3096_SUP__inf,axiom,
    ! [A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [F: fun(B,A),B3: set(B),A3: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F),B3))),A3) = aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,aa(A,fun(B,A),aTP_Lamp_mf(fun(B,A),fun(A,fun(B,A)),F),A3)),B3)) ) ).

% SUP_inf
tff(fact_3097_Sup__inf,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [B3: set(A),A3: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Sup_Sup(A),B3)),A3) = aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),image2(A,A,aTP_Lamp_mg(A,fun(A,A),A3)),B3)) ) ).

% Sup_inf
tff(fact_3098_inf__SUP,axiom,
    ! [A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [A3: A,F: fun(B,A),B3: set(B)] : aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F),B3))) = aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_mh(A,fun(fun(B,A),fun(B,A)),A3),F)),B3)) ) ).

% inf_SUP
tff(fact_3099_SUP__inf__distrib2,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [F: fun(B,A),A4: set(B),G: fun(C,A),B3: set(C)] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F),A4))),aa(set(A),A,complete_Sup_Sup(A),aa(set(C),set(A),image2(C,A,G),B3))) = aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,aa(set(C),fun(B,A),aa(fun(C,A),fun(set(C),fun(B,A)),aTP_Lamp_mj(fun(B,A),fun(fun(C,A),fun(set(C),fun(B,A))),F),G),B3)),A4)) ) ).

% SUP_inf_distrib2
tff(fact_3100_sum_Oimage__gen,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [S: set(A),H2: fun(A,B),G: fun(A,C)] :
          ( aa(set(A),$o,finite_finite2(A),S)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),H2),S) = aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7311177749621191930dd_sum(C,B),aa(fun(A,C),fun(C,B),aa(fun(A,B),fun(fun(A,C),fun(C,B)),aTP_Lamp_ml(set(A),fun(fun(A,B),fun(fun(A,C),fun(C,B))),S),H2),G)),aa(set(A),set(C),image2(A,C,G),S)) ) ) ) ).

% sum.image_gen
tff(fact_3101_complete__lattice__class_OSUP__sup__distrib,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F: fun(B,A),A4: set(B),G: fun(B,A)] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F),A4))),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,G),A4))) = aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_mm(fun(B,A),fun(fun(B,A),fun(B,A)),F),G)),A4)) ) ).

% complete_lattice_class.SUP_sup_distrib
tff(fact_3102_SUP__absorb,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [K: A,I4: set(A),A4: fun(A,B)] :
          ( aa(set(A),$o,member(A,K),I4)
         => ( aa(B,B,aa(B,fun(B,B),sup_sup(B),aa(A,B,A4,K)),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,A4),I4))) = aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,A4),I4)) ) ) ) ).

% SUP_absorb
tff(fact_3103_prod_Oimage__gen,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [S: set(A),H2: fun(A,B),G: fun(A,C)] :
          ( aa(set(A),$o,finite_finite2(A),S)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),H2),S) = aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7121269368397514597t_prod(C,B),aa(fun(A,C),fun(C,B),aa(fun(A,B),fun(fun(A,C),fun(C,B)),aTP_Lamp_mn(set(A),fun(fun(A,B),fun(fun(A,C),fun(C,B))),S),H2),G)),aa(set(A),set(C),image2(A,C,G),S)) ) ) ) ).

% prod.image_gen
tff(fact_3104_UN__empty2,axiom,
    ! [B: $tType,A: $tType,A4: set(B)] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aTP_Lamp_mo(B,set(A))),A4)) = bot_bot(set(A)) ).

% UN_empty2
tff(fact_3105_UN__empty,axiom,
    ! [B: $tType,A: $tType,B3: fun(B,set(A))] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B3),bot_bot(set(B)))) = bot_bot(set(A)) ).

% UN_empty
tff(fact_3106_UNION__empty__conv_I1_J,axiom,
    ! [B: $tType,A: $tType,B3: fun(B,set(A)),A4: set(B)] :
      ( ( bot_bot(set(A)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B3),A4)) )
    <=> ! [X4: B] :
          ( aa(set(B),$o,member(B,X4),A4)
         => ( aa(B,set(A),B3,X4) = bot_bot(set(A)) ) ) ) ).

% UNION_empty_conv(1)
tff(fact_3107_UNION__empty__conv_I2_J,axiom,
    ! [B: $tType,A: $tType,B3: fun(B,set(A)),A4: set(B)] :
      ( ( aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B3),A4)) = bot_bot(set(A)) )
    <=> ! [X4: B] :
          ( aa(set(B),$o,member(B,X4),A4)
         => ( aa(B,set(A),B3,X4) = bot_bot(set(A)) ) ) ) ).

% UNION_empty_conv(2)
tff(fact_3108_UN__image__subset,axiom,
    ! [C: $tType,B: $tType,A: $tType,F: fun(B,set(A)),G: fun(C,set(B)),X: C,X5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),F),aa(C,set(B),G,X)))),X5)
    <=> aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(C,set(B),G,X)),aa(fun(B,$o),set(B),collect(B),aa(set(A),fun(B,$o),aTP_Lamp_mp(fun(B,set(A)),fun(set(A),fun(B,$o)),F),X5))) ) ).

% UN_image_subset
tff(fact_3109_UN__mono,axiom,
    ! [B: $tType,A: $tType,A4: set(A),B3: set(A),F: fun(A,set(B)),G: fun(A,set(B))] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B3)
     => ( ! [X2: A] :
            ( aa(set(A),$o,member(A,X2),A4)
           => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(A,set(B),F,X2)),aa(A,set(B),G,X2)) )
       => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),F),A4))),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),G),B3))) ) ) ).

% UN_mono
tff(fact_3110_UN__least,axiom,
    ! [A: $tType,B: $tType,A4: set(A),B3: fun(A,set(B)),C3: set(B)] :
      ( ! [X2: A] :
          ( aa(set(A),$o,member(A,X2),A4)
         => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(A,set(B),B3,X2)),C3) )
     => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),B3),A4))),C3) ) ).

% UN_least
tff(fact_3111_UN__upper,axiom,
    ! [B: $tType,A: $tType,A3: A,A4: set(A),B3: fun(A,set(B))] :
      ( aa(set(A),$o,member(A,A3),A4)
     => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(A,set(B),B3,A3)),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),B3),A4))) ) ).

% UN_upper
tff(fact_3112_UN__subset__iff,axiom,
    ! [B: $tType,A: $tType,A4: fun(B,set(A)),I4: set(B),B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),A4),I4))),B3)
    <=> ! [X4: B] :
          ( aa(set(B),$o,member(B,X4),I4)
         => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(B,set(A),A4,X4)),B3) ) ) ).

% UN_subset_iff
tff(fact_3113_UN__insert__distrib,axiom,
    ! [B: $tType,A: $tType,U: A,A4: set(A),A3: B,B3: fun(A,set(B))] :
      ( aa(set(A),$o,member(A,U),A4)
     => ( aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),aa(fun(A,set(B)),fun(A,set(B)),aTP_Lamp_kv(B,fun(fun(A,set(B)),fun(A,set(B))),A3),B3)),A4)) = aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),A3),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),B3),A4))) ) ) ).

% UN_insert_distrib
tff(fact_3114_UN__extend__simps_I5_J,axiom,
    ! [A: $tType,B: $tType,A4: set(A),B3: fun(B,set(A)),C3: set(B)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B3),C3))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_kn(set(A),fun(fun(B,set(A)),fun(B,set(A))),A4),B3)),C3)) ).

% UN_extend_simps(5)
tff(fact_3115_UN__extend__simps_I4_J,axiom,
    ! [A: $tType,B: $tType,A4: fun(B,set(A)),C3: set(B),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),A4),C3))),B3) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(set(A),fun(B,set(A)),aTP_Lamp_km(fun(B,set(A)),fun(set(A),fun(B,set(A))),A4),B3)),C3)) ).

% UN_extend_simps(4)
tff(fact_3116_Int__UN__distrib,axiom,
    ! [A: $tType,B: $tType,B3: set(A),A4: fun(B,set(A)),I4: set(B)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),A4),I4))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_kn(set(A),fun(fun(B,set(A)),fun(B,set(A))),B3),A4)),I4)) ).

% Int_UN_distrib
tff(fact_3117_Int__UN__distrib2,axiom,
    ! [C: $tType,A: $tType,B: $tType,A4: fun(B,set(A)),I4: set(B),B3: fun(C,set(A)),J3: set(C)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),A4),I4))),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(C),set(set(A)),image2(C,set(A),B3),J3))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(set(C),fun(B,set(A)),aa(fun(C,set(A)),fun(set(C),fun(B,set(A))),aTP_Lamp_mr(fun(B,set(A)),fun(fun(C,set(A)),fun(set(C),fun(B,set(A)))),A4),B3),J3)),I4)) ).

% Int_UN_distrib2
tff(fact_3118_UN__absorb,axiom,
    ! [B: $tType,A: $tType,K: A,I4: set(A),A4: fun(A,set(B))] :
      ( aa(set(A),$o,member(A,K),I4)
     => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),aa(A,set(B),A4,K)),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),A4),I4))) = aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),A4),I4)) ) ) ).

% UN_absorb
tff(fact_3119_UN__Un__distrib,axiom,
    ! [A: $tType,B: $tType,A4: fun(B,set(A)),B3: fun(B,set(A)),I4: set(B)] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_ms(fun(B,set(A)),fun(fun(B,set(A)),fun(B,set(A))),A4),B3)),I4)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),A4),I4))),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B3),I4))) ).

% UN_Un_distrib
tff(fact_3120_Un__Union__image,axiom,
    ! [A: $tType,B: $tType,A4: fun(B,set(A)),B3: fun(B,set(A)),C3: set(B)] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_ms(fun(B,set(A)),fun(fun(B,set(A)),fun(B,set(A))),A4),B3)),C3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),A4),C3))),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B3),C3))) ).

% Un_Union_image
tff(fact_3121_UN__extend__simps_I6_J,axiom,
    ! [A: $tType,B: $tType,A4: fun(B,set(A)),C3: set(B),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),A4),C3))),B3) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(set(A),fun(B,set(A)),aTP_Lamp_ko(fun(B,set(A)),fun(set(A),fun(B,set(A))),A4),B3)),C3)) ).

% UN_extend_simps(6)
tff(fact_3122_Inter__UNIV,axiom,
    ! [A: $tType] : aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),top_top(set(set(A)))) = bot_bot(set(A)) ).

% Inter_UNIV
tff(fact_3123_vimage__UN,axiom,
    ! [B: $tType,A: $tType,C: $tType,F: fun(A,B),B3: fun(C,set(B)),A4: set(C)] : aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),image2(C,set(B),B3),A4))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(C),set(set(A)),image2(C,set(A),aa(fun(C,set(B)),fun(C,set(A)),aTP_Lamp_kw(fun(A,B),fun(fun(C,set(B)),fun(C,set(A))),F),B3)),A4)) ).

% vimage_UN
tff(fact_3124_the__elem__image__unique,axiom,
    ! [B: $tType,A: $tType,A4: set(A),F: fun(A,B),X: A] :
      ( ( A4 != bot_bot(set(A)) )
     => ( ! [Y2: A] :
            ( aa(set(A),$o,member(A,Y2),A4)
           => ( aa(A,B,F,Y2) = aa(A,B,F,X) ) )
       => ( the_elem(B,aa(set(A),set(B),image2(A,B,F),A4)) = aa(A,B,F,X) ) ) ) ).

% the_elem_image_unique
tff(fact_3125_UN__atMost__UNIV,axiom,
    aa(set(set(nat)),set(nat),complete_Sup_Sup(set(nat)),aa(set(nat),set(set(nat)),image2(nat,set(nat),set_ord_atMost(nat)),top_top(set(nat)))) = top_top(set(nat)) ).

% UN_atMost_UNIV
tff(fact_3126_UN__lessThan__UNIV,axiom,
    aa(set(set(nat)),set(nat),complete_Sup_Sup(set(nat)),aa(set(nat),set(set(nat)),image2(nat,set(nat),set_ord_lessThan(nat)),top_top(set(nat)))) = top_top(set(nat)) ).

% UN_lessThan_UNIV
tff(fact_3127_finite__less__Inf__iff,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X5: set(A),A3: A] :
          ( aa(set(A),$o,finite_finite2(A),X5)
         => ( ( X5 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(set(A),A,complete_Inf_Inf(A),X5))
            <=> ! [X4: A] :
                  ( aa(set(A),$o,member(A,X4),X5)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),X4) ) ) ) ) ) ).

% finite_less_Inf_iff
tff(fact_3128_Inf__le__Sup,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A4: set(A)] :
          ( ( A4 != bot_bot(set(A)) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A4)),aa(set(A),A,complete_Sup_Sup(A),A4)) ) ) ).

% Inf_le_Sup
tff(fact_3129_cSUP__least,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [A4: set(A),F: fun(A,B),M4: B] :
          ( ( A4 != bot_bot(set(A)) )
         => ( ! [X2: A] :
                ( aa(set(A),$o,member(A,X2),A4)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,X2)),M4) )
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F),A4))),M4) ) ) ) ).

% cSUP_least
tff(fact_3130_SUP__eq__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [I4: set(A),C2: B,F: fun(A,B)] :
          ( ( I4 != bot_bot(set(A)) )
         => ( ! [I2: A] :
                ( aa(set(A),$o,member(A,I2),I4)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),C2),aa(A,B,F,I2)) )
           => ( ( aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F),I4)) = C2 )
            <=> ! [X4: A] :
                  ( aa(set(A),$o,member(A,X4),I4)
                 => ( aa(A,B,F,X4) = C2 ) ) ) ) ) ) ).

% SUP_eq_iff
tff(fact_3131_finite__Inf__in,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A4: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( ! [X2: A,Y2: A] :
                  ( aa(set(A),$o,member(A,X2),A4)
                 => ( aa(set(A),$o,member(A,Y2),A4)
                   => aa(set(A),$o,member(A,aa(A,A,aa(A,fun(A,A),inf_inf(A),X2),Y2)),A4) ) )
             => aa(set(A),$o,member(A,aa(set(A),A,complete_Inf_Inf(A),A4)),A4) ) ) ) ) ).

% finite_Inf_in
tff(fact_3132_cInf__abs__ge,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & linordered_idom(A) )
     => ! [S: set(A),A3: A] :
          ( ( S != bot_bot(set(A)) )
         => ( ! [X2: A] :
                ( aa(set(A),$o,member(A,X2),S)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),X2)),A3) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(set(A),A,complete_Inf_Inf(A),S))),A3) ) ) ) ).

% cInf_abs_ge
tff(fact_3133_less__eq__Inf__inter,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A4: set(A),B3: set(A)] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Inf_Inf(A),A4)),aa(set(A),A,complete_Inf_Inf(A),B3))),aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3))) ) ).

% less_eq_Inf_inter
tff(fact_3134_range__eq__singletonD,axiom,
    ! [B: $tType,A: $tType,F: fun(B,A),A3: A,X: B] :
      ( ( aa(set(B),set(A),image2(B,A,F),top_top(set(B))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A))) )
     => ( aa(B,A,F,X) = A3 ) ) ).

% range_eq_singletonD
tff(fact_3135_Union__image__empty,axiom,
    ! [B: $tType,A: $tType,A4: set(A),F: fun(B,set(A))] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),F),bot_bot(set(B))))) = A4 ).

% Union_image_empty
tff(fact_3136_Union__image__insert,axiom,
    ! [A: $tType,B: $tType,F: fun(B,set(A)),A3: B,B3: set(B)] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),F),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),A3),B3))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(B,set(A),F,A3)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),F),B3))) ).

% Union_image_insert
tff(fact_3137_finite__vimageD,axiom,
    ! [A: $tType,B: $tType,H2: fun(A,B),F4: set(B)] :
      ( aa(set(A),$o,finite_finite2(A),aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),H2),F4))
     => ( ( aa(set(A),set(B),image2(A,B,H2),top_top(set(A))) = top_top(set(B)) )
       => aa(set(B),$o,finite_finite2(B),F4) ) ) ).

% finite_vimageD
tff(fact_3138_SUP__subset__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A4: set(A),B3: set(A),F: fun(A,B),G: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B3)
         => ( ! [X2: A] :
                ( aa(set(A),$o,member(A,X2),A4)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,X2)),aa(A,B,G,X2)) )
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F),A4))),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,G),B3))) ) ) ) ).

% SUP_subset_mono
tff(fact_3139_SUP__empty,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F: fun(B,A)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F),bot_bot(set(B)))) = bot_bot(A) ) ).

% SUP_empty
tff(fact_3140_SUP__constant,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [C2: A,A4: set(B)] :
          aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,aTP_Lamp_lm(A,fun(B,A),C2)),A4)) = $ite(A4 = bot_bot(set(B)),bot_bot(A),C2) ) ).

% SUP_constant
tff(fact_3141_sum_Ogroup,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( comm_monoid_add(C)
     => ! [S: set(A),T2: set(B),G: fun(A,B),H2: fun(A,C)] :
          ( aa(set(A),$o,finite_finite2(A),S)
         => ( aa(set(B),$o,finite_finite2(B),T2)
           => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,G),S)),T2)
             => ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),aa(fun(A,C),fun(B,C),aa(fun(A,B),fun(fun(A,C),fun(B,C)),aTP_Lamp_mu(set(A),fun(fun(A,B),fun(fun(A,C),fun(B,C))),S),G),H2)),T2) = aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7311177749621191930dd_sum(A,C),H2),S) ) ) ) ) ) ).

% sum.group
tff(fact_3142_SUP__insert,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F: fun(B,A),A3: B,A4: set(B)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),A3),A4))) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(B,A,F,A3)),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F),A4))) ) ).

% SUP_insert
tff(fact_3143_prod_Ogroup,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( comm_monoid_mult(C)
     => ! [S: set(A),T2: set(B),G: fun(A,B),H2: fun(A,C)] :
          ( aa(set(A),$o,finite_finite2(A),S)
         => ( aa(set(B),$o,finite_finite2(B),T2)
           => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,G),S)),T2)
             => ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),aa(fun(A,C),fun(B,C),aa(fun(A,B),fun(fun(A,C),fun(B,C)),aTP_Lamp_mv(set(A),fun(fun(A,B),fun(fun(A,C),fun(B,C))),S),G),H2)),T2) = aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7121269368397514597t_prod(A,C),H2),S) ) ) ) ) ) ).

% prod.group
tff(fact_3144_SUP__union,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [M4: fun(B,A),A4: set(B),B3: set(B)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,M4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A4),B3))) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,M4),A4))),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,M4),B3))) ) ).

% SUP_union
tff(fact_3145_UN__extend__simps_I1_J,axiom,
    ! [A: $tType,B: $tType,A3: A,B3: fun(B,set(A)),C3: set(B)] :
      aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B3),C3))) = $ite(C3 = bot_bot(set(B)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A))),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_kh(A,fun(fun(B,set(A)),fun(B,set(A))),A3),B3)),C3))) ).

% UN_extend_simps(1)
tff(fact_3146_UN__extend__simps_I3_J,axiom,
    ! [A: $tType,B: $tType,A4: set(A),B3: fun(B,set(A)),C3: set(B)] :
      aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B3),C3))) = $ite(C3 = bot_bot(set(B)),A4,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_ki(set(A),fun(fun(B,set(A)),fun(B,set(A))),A4),B3)),C3))) ).

% UN_extend_simps(3)
tff(fact_3147_UN__extend__simps_I2_J,axiom,
    ! [A: $tType,B: $tType,A4: fun(B,set(A)),C3: set(B),B3: set(A)] :
      aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),A4),C3))),B3) = $ite(C3 = bot_bot(set(B)),B3,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(set(A),fun(B,set(A)),aTP_Lamp_kj(fun(B,set(A)),fun(set(A),fun(B,set(A))),A4),B3)),C3))) ).

% UN_extend_simps(2)
tff(fact_3148_empty__natural,axiom,
    ! [C: $tType,B: $tType,D: $tType,A: $tType,F: fun(A,C),G: fun(D,B)] : aa(fun(A,C),fun(A,set(B)),comp(C,set(B),A,aTP_Lamp_mw(C,set(B))),F) = aa(fun(A,set(D)),fun(A,set(B)),comp(set(D),set(B),A,image2(D,B,G)),aTP_Lamp_mx(A,set(D))) ).

% empty_natural
tff(fact_3149_cInf__asclose,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & linordered_idom(A) )
     => ! [S: set(A),L: A,E4: A] :
          ( ( S != bot_bot(set(A)) )
         => ( ! [X2: A] :
                ( aa(set(A),$o,member(A,X2),S)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X2),L))),E4) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(set(A),A,complete_Inf_Inf(A),S)),L))),E4) ) ) ) ).

% cInf_asclose
tff(fact_3150_prod_Oreindex__nontrivial,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( comm_monoid_mult(C)
     => ! [A4: set(A),H2: fun(A,B),G: fun(B,C)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ! [X2: A,Y2: A] :
                ( aa(set(A),$o,member(A,X2),A4)
               => ( aa(set(A),$o,member(A,Y2),A4)
                 => ( ( X2 != Y2 )
                   => ( ( aa(A,B,H2,X2) = aa(A,B,H2,Y2) )
                     => ( aa(B,C,G,aa(A,B,H2,X2)) = one_one(C) ) ) ) ) )
           => ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),G),aa(set(A),set(B),image2(A,B,H2),A4)) = aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7121269368397514597t_prod(A,C),aa(fun(A,B),fun(A,C),comp(B,C,A,G),H2)),A4) ) ) ) ) ).

% prod.reindex_nontrivial
tff(fact_3151_inf__img__fin__dom,axiom,
    ! [B: $tType,A: $tType,F: fun(B,A),A4: set(B)] :
      ( aa(set(A),$o,finite_finite2(A),aa(set(B),set(A),image2(B,A,F),A4))
     => ( ~ aa(set(B),$o,finite_finite2(B),A4)
       => ? [X2: A] :
            ( aa(set(A),$o,member(A,X2),aa(set(B),set(A),image2(B,A,F),A4))
            & ~ aa(set(B),$o,finite_finite2(B),aa(set(A),set(B),aa(fun(B,A),fun(set(A),set(B)),vimage(B,A),F),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X2),bot_bot(set(A))))) ) ) ) ).

% inf_img_fin_dom
tff(fact_3152_inf__img__fin__domE,axiom,
    ! [B: $tType,A: $tType,F: fun(B,A),A4: set(B)] :
      ( aa(set(A),$o,finite_finite2(A),aa(set(B),set(A),image2(B,A,F),A4))
     => ( ~ aa(set(B),$o,finite_finite2(B),A4)
       => ~ ! [Y2: A] :
              ( aa(set(A),$o,member(A,Y2),aa(set(B),set(A),image2(B,A,F),A4))
             => aa(set(B),$o,finite_finite2(B),aa(set(A),set(B),aa(fun(B,A),fun(set(A),set(B)),vimage(B,A),F),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y2),bot_bot(set(A))))) ) ) ) ).

% inf_img_fin_domE
tff(fact_3153_finite__vimageD_H,axiom,
    ! [A: $tType,B: $tType,F: fun(A,B),A4: set(B)] :
      ( aa(set(A),$o,finite_finite2(A),aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F),A4))
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A4),aa(set(A),set(B),image2(A,B,F),top_top(set(A))))
       => aa(set(B),$o,finite_finite2(B),A4) ) ) ).

% finite_vimageD'
tff(fact_3154_vimage__eq__UN,axiom,
    ! [A: $tType,B: $tType,F: fun(A,B),B3: set(B)] : aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F),B3) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aTP_Lamp_my(fun(A,B),fun(B,set(A)),F)),B3)) ).

% vimage_eq_UN
tff(fact_3155_UN__UN__finite__eq,axiom,
    ! [A: $tType,A4: fun(nat,set(A))] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),aTP_Lamp_mz(fun(nat,set(A)),fun(nat,set(A)),A4)),top_top(set(nat)))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),A4),top_top(set(nat)))) ).

% UN_UN_finite_eq
tff(fact_3156_UN__le__add__shift,axiom,
    ! [A: $tType,M4: fun(nat,set(A)),K: nat,N: nat] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),aa(nat,fun(nat,set(A)),aTP_Lamp_na(fun(nat,set(A)),fun(nat,fun(nat,set(A))),M4),K)),aa(nat,set(nat),set_ord_atMost(nat),N))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),M4),set_or1337092689740270186AtMost(nat,K,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K)))) ).

% UN_le_add_shift
tff(fact_3157_UN__le__add__shift__strict,axiom,
    ! [A: $tType,M4: fun(nat,set(A)),K: nat,N: nat] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),aa(nat,fun(nat,set(A)),aTP_Lamp_na(fun(nat,set(A)),fun(nat,fun(nat,set(A))),M4),K)),aa(nat,set(nat),set_ord_lessThan(nat),N))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),M4),set_or7035219750837199246ssThan(nat,K,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K)))) ).

% UN_le_add_shift_strict
tff(fact_3158_card__range__greater__zero,axiom,
    ! [A: $tType,B: $tType,F: fun(B,A)] :
      ( aa(set(A),$o,finite_finite2(A),aa(set(B),set(A),image2(B,A,F),top_top(set(B))))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(set(A),nat,finite_card(A),aa(set(B),set(A),image2(B,A,F),top_top(set(B))))) ) ).

% card_range_greater_zero
tff(fact_3159_inf__img__fin__domE_H,axiom,
    ! [A: $tType,B: $tType,F: fun(B,A),A4: set(B)] :
      ( aa(set(A),$o,finite_finite2(A),aa(set(B),set(A),image2(B,A,F),A4))
     => ( ~ aa(set(B),$o,finite_finite2(B),A4)
       => ~ ! [Y2: A] :
              ( aa(set(A),$o,member(A,Y2),aa(set(B),set(A),image2(B,A,F),A4))
             => aa(set(B),$o,finite_finite2(B),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(set(A),set(B),aa(fun(B,A),fun(set(A),set(B)),vimage(B,A),F),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y2),bot_bot(set(A))))),A4)) ) ) ) ).

% inf_img_fin_domE'
tff(fact_3160_inf__img__fin__dom_H,axiom,
    ! [A: $tType,B: $tType,F: fun(B,A),A4: set(B)] :
      ( aa(set(A),$o,finite_finite2(A),aa(set(B),set(A),image2(B,A,F),A4))
     => ( ~ aa(set(B),$o,finite_finite2(B),A4)
       => ? [X2: A] :
            ( aa(set(A),$o,member(A,X2),aa(set(B),set(A),image2(B,A,F),A4))
            & ~ aa(set(B),$o,finite_finite2(B),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(set(A),set(B),aa(fun(B,A),fun(set(A),set(B)),vimage(B,A),F),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X2),bot_bot(set(A))))),A4)) ) ) ) ).

% inf_img_fin_dom'
tff(fact_3161_sum_OUNION__disjoint,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( comm_monoid_add(C)
     => ! [I4: set(A),A4: fun(A,set(B)),G: fun(B,C)] :
          ( aa(set(A),$o,finite_finite2(A),I4)
         => ( ! [X2: A] :
                ( aa(set(A),$o,member(A,X2),I4)
               => aa(set(B),$o,finite_finite2(B),aa(A,set(B),A4,X2)) )
           => ( ! [X2: A] :
                  ( aa(set(A),$o,member(A,X2),I4)
                 => ! [Xa4: A] :
                      ( aa(set(A),$o,member(A,Xa4),I4)
                     => ( ( X2 != Xa4 )
                       => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(A,set(B),A4,X2)),aa(A,set(B),A4,Xa4)) = bot_bot(set(B)) ) ) ) )
             => ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),G),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),A4),I4))) = 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_nb(fun(A,set(B)),fun(fun(B,C),fun(A,C)),A4),G)),I4) ) ) ) ) ) ).

% sum.UNION_disjoint
tff(fact_3162_prod_OUNION__disjoint,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( comm_monoid_mult(C)
     => ! [I4: set(A),A4: fun(A,set(B)),G: fun(B,C)] :
          ( aa(set(A),$o,finite_finite2(A),I4)
         => ( ! [X2: A] :
                ( aa(set(A),$o,member(A,X2),I4)
               => aa(set(B),$o,finite_finite2(B),aa(A,set(B),A4,X2)) )
           => ( ! [X2: A] :
                  ( aa(set(A),$o,member(A,X2),I4)
                 => ! [Xa4: A] :
                      ( aa(set(A),$o,member(A,Xa4),I4)
                     => ( ( X2 != Xa4 )
                       => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(A,set(B),A4,X2)),aa(A,set(B),A4,Xa4)) = bot_bot(set(B)) ) ) ) )
             => ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),G),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),A4),I4))) = aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7121269368397514597t_prod(A,C),aa(fun(B,C),fun(A,C),aTP_Lamp_nc(fun(A,set(B)),fun(fun(B,C),fun(A,C)),A4),G)),I4) ) ) ) ) ) ).

% prod.UNION_disjoint
tff(fact_3163_card__UN__le,axiom,
    ! [B: $tType,A: $tType,I4: set(A),A4: fun(A,set(B))] :
      ( aa(set(A),$o,finite_finite2(A),I4)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(B),nat,finite_card(B),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),A4),I4)))),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aTP_Lamp_nd(fun(A,set(B)),fun(A,nat),A4)),I4)) ) ).

% card_UN_le
tff(fact_3164_UN__finite__subset,axiom,
    ! [A: $tType,A4: fun(nat,set(A)),C3: set(A)] :
      ( ! [N3: nat] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),A4),set_or7035219750837199246ssThan(nat,zero_zero(nat),N3)))),C3)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),A4),top_top(set(nat))))),C3) ) ).

% UN_finite_subset
tff(fact_3165_UN__finite2__eq,axiom,
    ! [A: $tType,A4: fun(nat,set(A)),B3: fun(nat,set(A)),K: nat] :
      ( ! [N3: nat] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),A4),set_or7035219750837199246ssThan(nat,zero_zero(nat),N3))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),B3),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N3),K))))
     => ( aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),A4),top_top(set(nat)))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),B3),top_top(set(nat)))) ) ) ).

% UN_finite2_eq
tff(fact_3166_type__copy__set__map0,axiom,
    ! [A: $tType,B: $tType,D: $tType,E: $tType,C: $tType,F3: $tType,Rep: fun(A,B),Abs: fun(B,A),S: fun(B,set(D)),M4: fun(C,B),F: fun(E,D),S3: fun(C,set(E)),G: fun(F3,C)] :
      ( type_definition(A,B,Rep,Abs,top_top(set(B)))
     => ( ( aa(fun(C,B),fun(C,set(D)),comp(B,set(D),C,S),M4) = aa(fun(C,set(E)),fun(C,set(D)),comp(set(E),set(D),C,image2(E,D,F)),S3) )
       => ( aa(fun(F3,A),fun(F3,set(D)),comp(A,set(D),F3,aa(fun(A,B),fun(A,set(D)),comp(B,set(D),A,S),Rep)),aa(fun(F3,C),fun(F3,A),comp(C,A,F3,aa(fun(C,B),fun(C,A),comp(B,A,C,Abs),M4)),G)) = aa(fun(F3,set(E)),fun(F3,set(D)),comp(set(E),set(D),F3,image2(E,D,F)),aa(fun(F3,C),fun(F3,set(E)),comp(C,set(E),F3,S3),G)) ) ) ) ).

% type_copy_set_map0
tff(fact_3167_image__mult__atLeastAtMost__if,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,X: A,Y: A] :
          aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),times_times(A),C2)),set_or1337092689740270186AtMost(A,X,Y)) = $ite(
            aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
            set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),X),aa(A,A,aa(A,fun(A,A),times_times(A),C2),Y)),
            $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),Y),aa(A,A,aa(A,fun(A,A),times_times(A),C2),X)),bot_bot(set(A))) ) ) ).

% image_mult_atLeastAtMost_if
tff(fact_3168_image__mult__atLeastAtMost__if_H,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,X: A,Y: A] :
          aa(set(A),set(A),image2(A,A,aTP_Lamp_ne(A,fun(A,A),C2)),set_or1337092689740270186AtMost(A,X,Y)) = $ite(
            aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y),
            $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),C2),aa(A,A,aa(A,fun(A,A),times_times(A),Y),C2)),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),Y),C2),aa(A,A,aa(A,fun(A,A),times_times(A),X),C2))),
            bot_bot(set(A)) ) ) ).

% image_mult_atLeastAtMost_if'
tff(fact_3169_image__affinity__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [M: A,C2: A,A3: A,B2: A] :
          aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),aTP_Lamp_nf(A,fun(A,fun(A,A)),M),C2)),set_or1337092689740270186AtMost(A,A3,B2)) = $ite(
            set_or1337092689740270186AtMost(A,A3,B2) = bot_bot(set(A)),
            bot_bot(set(A)),
            $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),M),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M),A3)),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M),B2)),C2)),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M),B2)),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M),A3)),C2))) ) ) ).

% image_affinity_atLeastAtMost
tff(fact_3170_image__affinity__atLeastAtMost__diff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [M: A,C2: A,A3: A,B2: A] :
          aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),aTP_Lamp_ng(A,fun(A,fun(A,A)),M),C2)),set_or1337092689740270186AtMost(A,A3,B2)) = $ite(
            set_or1337092689740270186AtMost(A,A3,B2) = bot_bot(set(A)),
            bot_bot(set(A)),
            $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),M),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M),A3)),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M),B2)),C2)),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M),B2)),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M),A3)),C2))) ) ) ).

% image_affinity_atLeastAtMost_diff
tff(fact_3171_image__affinity__atLeastAtMost__div,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [M: A,C2: A,A3: A,B2: A] :
          aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),aTP_Lamp_nh(A,fun(A,fun(A,A)),M),C2)),set_or1337092689740270186AtMost(A,A3,B2)) = $ite(
            set_or1337092689740270186AtMost(A,A3,B2) = bot_bot(set(A)),
            bot_bot(set(A)),
            $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),M),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),M)),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),M)),C2)),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),M)),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),M)),C2))) ) ) ).

% image_affinity_atLeastAtMost_div
tff(fact_3172_image__affinity__atLeastAtMost__div__diff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [M: A,C2: A,A3: A,B2: A] :
          aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),aTP_Lamp_ni(A,fun(A,fun(A,A)),M),C2)),set_or1337092689740270186AtMost(A,A3,B2)) = $ite(
            set_or1337092689740270186AtMost(A,A3,B2) = bot_bot(set(A)),
            bot_bot(set(A)),
            $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),M),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),M)),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),M)),C2)),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),M)),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),M)),C2))) ) ) ).

% image_affinity_atLeastAtMost_div_diff
tff(fact_3173_sum__fun__comp,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( semiring_1(C)
     => ! [S: set(A),R: set(B),G: fun(A,B),F: fun(B,C)] :
          ( aa(set(A),$o,finite_finite2(A),S)
         => ( aa(set(B),$o,finite_finite2(B),R)
           => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,G),S)),R)
             => ( aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7311177749621191930dd_sum(A,C),aa(fun(B,C),fun(A,C),aTP_Lamp_nj(fun(A,B),fun(fun(B,C),fun(A,C)),G),F)),S) = 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_nk(set(A),fun(fun(A,B),fun(fun(B,C),fun(B,C))),S),G),F)),R) ) ) ) ) ) ).

% sum_fun_comp
tff(fact_3174_card__UN__disjoint,axiom,
    ! [B: $tType,A: $tType,I4: set(A),A4: fun(A,set(B))] :
      ( aa(set(A),$o,finite_finite2(A),I4)
     => ( ! [X2: A] :
            ( aa(set(A),$o,member(A,X2),I4)
           => aa(set(B),$o,finite_finite2(B),aa(A,set(B),A4,X2)) )
       => ( ! [X2: A] :
              ( aa(set(A),$o,member(A,X2),I4)
             => ! [Xa4: A] :
                  ( aa(set(A),$o,member(A,Xa4),I4)
                 => ( ( X2 != Xa4 )
                   => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(A,set(B),A4,X2)),aa(A,set(B),A4,Xa4)) = bot_bot(set(B)) ) ) ) )
         => ( aa(set(B),nat,finite_card(B),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),A4),I4))) = aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aTP_Lamp_nd(fun(A,set(B)),fun(A,nat),A4)),I4) ) ) ) ) ).

% card_UN_disjoint
tff(fact_3175_UN__finite2__subset,axiom,
    ! [A: $tType,A4: fun(nat,set(A)),B3: fun(nat,set(A)),K: nat] :
      ( ! [N3: nat] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),A4),set_or7035219750837199246ssThan(nat,zero_zero(nat),N3)))),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),B3),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N3),K)))))
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),A4),top_top(set(nat))))),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),B3),top_top(set(nat))))) ) ).

% UN_finite2_subset
tff(fact_3176_UN__le__eq__Un0,axiom,
    ! [A: $tType,M4: fun(nat,set(A)),N: nat] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),M4),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)),aa(set(nat),set(set(A)),image2(nat,set(A),M4),set_or1337092689740270186AtMost(nat,one_one(nat),N)))),aa(nat,set(A),M4,zero_zero(nat))) ).

% UN_le_eq_Un0
tff(fact_3177_surj__diff__right,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A] : aa(set(A),set(A),image2(A,A,aTP_Lamp_nl(A,fun(A,A),A3)),top_top(set(A))) = top_top(set(A)) ) ).

% surj_diff_right
tff(fact_3178_surj__plus,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A] : aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),plus_plus(A),A3)),top_top(set(A))) = top_top(set(A)) ) ).

% surj_plus
tff(fact_3179_surj__Compl__image__subset,axiom,
    ! [A: $tType,B: $tType,F: fun(B,A),A4: set(B)] :
      ( ( aa(set(B),set(A),image2(B,A,F),top_top(set(B))) = top_top(set(A)) )
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),aa(set(B),set(A),image2(B,A,F),A4))),aa(set(B),set(A),image2(B,A,F),aa(set(B),set(B),uminus_uminus(set(B)),A4))) ) ).

% surj_Compl_image_subset
tff(fact_3180_vimage__subsetD,axiom,
    ! [A: $tType,B: $tType,F: fun(B,A),B3: set(A),A4: set(B)] :
      ( ( aa(set(B),set(A),image2(B,A,F),top_top(set(B))) = top_top(set(A)) )
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),aa(fun(B,A),fun(set(A),set(B)),vimage(B,A),F),B3)),A4)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),aa(set(B),set(A),image2(B,A,F),A4)) ) ) ).

% vimage_subsetD
tff(fact_3181_surj__vimage__empty,axiom,
    ! [B: $tType,A: $tType,F: fun(B,A),A4: set(A)] :
      ( ( aa(set(B),set(A),image2(B,A,F),top_top(set(B))) = top_top(set(A)) )
     => ( ( aa(set(A),set(B),aa(fun(B,A),fun(set(A),set(B)),vimage(B,A),F),A4) = bot_bot(set(B)) )
      <=> ( A4 = bot_bot(set(A)) ) ) ) ).

% surj_vimage_empty
tff(fact_3182_subset__mset_OcINF__const,axiom,
    ! [A: $tType,B: $tType,A4: set(A),C2: multiset(B)] :
      ( ( A4 != bot_bot(set(A)) )
     => ( aa(set(multiset(B)),multiset(B),complete_Inf_Inf(multiset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),aTP_Lamp_nm(multiset(B),fun(A,multiset(B)),C2)),A4)) = C2 ) ) ).

% subset_mset.cINF_const
tff(fact_3183_subset__mset_OcSUP__const,axiom,
    ! [A: $tType,B: $tType,A4: set(A),C2: multiset(B)] :
      ( ( A4 != bot_bot(set(A)) )
     => ( aa(set(multiset(B)),multiset(B),complete_Sup_Sup(multiset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),aTP_Lamp_nm(multiset(B),fun(A,multiset(B)),C2)),A4)) = C2 ) ) ).

% subset_mset.cSUP_const
tff(fact_3184_pair__imageI,axiom,
    ! [C: $tType,B: $tType,A: $tType,A3: A,B2: B,A4: set(product_prod(A,B)),F: fun(A,fun(B,C))] :
      ( aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),A4)
     => aa(set(C),$o,member(C,aa(B,C,aa(A,fun(B,C),F,A3),B2)),aa(set(product_prod(A,B)),set(C),image2(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F)),A4)) ) ).

% pair_imageI
tff(fact_3185_Sup__apply,axiom,
    ! [A: $tType,B: $tType] :
      ( complete_Sup(A)
     => ! [A4: set(fun(B,A)),X: B] : aa(B,A,aa(set(fun(B,A)),fun(B,A),complete_Sup_Sup(fun(B,A)),A4),X) = aa(set(A),A,complete_Sup_Sup(A),aa(set(fun(B,A)),set(A),image2(fun(B,A),A,aTP_Lamp_nn(B,fun(fun(B,A),A),X)),A4)) ) ).

% Sup_apply
tff(fact_3186_Inf__apply,axiom,
    ! [A: $tType,B: $tType] :
      ( complete_Inf(A)
     => ! [A4: set(fun(B,A)),X: B] : aa(B,A,aa(set(fun(B,A)),fun(B,A),complete_Inf_Inf(fun(B,A)),A4),X) = aa(set(A),A,complete_Inf_Inf(A),aa(set(fun(B,A)),set(A),image2(fun(B,A),A,aTP_Lamp_no(B,fun(fun(B,A),A),X)),A4)) ) ).

% Inf_apply
tff(fact_3187_Inf__int__def,axiom,
    ! [X5: set(int)] : aa(set(int),int,complete_Inf_Inf(int),X5) = aa(int,int,uminus_uminus(int),aa(set(int),int,complete_Sup_Sup(int),aa(set(int),set(int),image2(int,int,uminus_uminus(int)),X5))) ).

% Inf_int_def
tff(fact_3188_SUP__UN__eq2,axiom,
    ! [A: $tType,B: $tType,C: $tType,R2: fun(C,set(product_prod(A,B))),S: set(C),X3: A,Xa3: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(set(fun(A,fun(B,$o))),fun(A,fun(B,$o)),complete_Sup_Sup(fun(A,fun(B,$o))),aa(set(C),set(fun(A,fun(B,$o))),image2(C,fun(A,fun(B,$o)),aTP_Lamp_np(fun(C,set(product_prod(A,B))),fun(C,fun(A,fun(B,$o))),R2)),S)),X3),Xa3)
    <=> aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Xa3)),aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),aa(set(C),set(set(product_prod(A,B))),image2(C,set(product_prod(A,B)),R2),S))) ) ).

% SUP_UN_eq2
tff(fact_3189_Sup__SUP__eq2,axiom,
    ! [B: $tType,A: $tType,S: set(fun(A,fun(B,$o))),X3: A,Xa3: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(set(fun(A,fun(B,$o))),fun(A,fun(B,$o)),complete_Sup_Sup(fun(A,fun(B,$o))),S),X3),Xa3)
    <=> aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Xa3)),aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),aa(set(fun(product_prod(A,B),$o)),set(set(product_prod(A,B))),image2(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B))),aa(set(fun(A,fun(B,$o))),set(fun(product_prod(A,B),$o)),image2(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o)),S)))) ) ).

% Sup_SUP_eq2
tff(fact_3190_Inf__set__def,axiom,
    ! [A: $tType,A4: set(set(A))] : aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),A4) = aa(fun(A,$o),set(A),collect(A),aTP_Lamp_nq(set(set(A)),fun(A,$o),A4)) ).

% Inf_set_def
tff(fact_3191_INF__Int__eq,axiom,
    ! [A: $tType,S: set(set(A)),X3: A] :
      ( aa(A,$o,aa(set(fun(A,$o)),fun(A,$o),complete_Inf_Inf(fun(A,$o)),aa(set(set(A)),set(fun(A,$o)),image2(set(A),fun(A,$o),aTP_Lamp_ag(set(A),fun(A,$o))),S)),X3)
    <=> aa(set(A),$o,member(A,X3),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),S)) ) ).

% INF_Int_eq
tff(fact_3192_INF__INT__eq2,axiom,
    ! [A: $tType,B: $tType,C: $tType,R2: fun(C,set(product_prod(A,B))),S: set(C),X3: A,Xa3: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(set(fun(A,fun(B,$o))),fun(A,fun(B,$o)),complete_Inf_Inf(fun(A,fun(B,$o))),aa(set(C),set(fun(A,fun(B,$o))),image2(C,fun(A,fun(B,$o)),aTP_Lamp_np(fun(C,set(product_prod(A,B))),fun(C,fun(A,fun(B,$o))),R2)),S)),X3),Xa3)
    <=> aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Xa3)),aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Inf_Inf(set(product_prod(A,B))),aa(set(C),set(set(product_prod(A,B))),image2(C,set(product_prod(A,B)),R2),S))) ) ).

% INF_INT_eq2
tff(fact_3193_INF__Int__eq2,axiom,
    ! [B: $tType,A: $tType,S: set(set(product_prod(A,B))),X3: A,Xa3: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(set(fun(A,fun(B,$o))),fun(A,fun(B,$o)),complete_Inf_Inf(fun(A,fun(B,$o))),aa(set(set(product_prod(A,B))),set(fun(A,fun(B,$o))),image2(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_as(set(product_prod(A,B)),fun(A,fun(B,$o)))),S)),X3),Xa3)
    <=> aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Xa3)),aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Inf_Inf(set(product_prod(A,B))),S)) ) ).

% INF_Int_eq2
tff(fact_3194_Inf__INT__eq2,axiom,
    ! [B: $tType,A: $tType,S: set(fun(A,fun(B,$o))),X3: A,Xa3: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(set(fun(A,fun(B,$o))),fun(A,fun(B,$o)),complete_Inf_Inf(fun(A,fun(B,$o))),S),X3),Xa3)
    <=> aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Xa3)),aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Inf_Inf(set(product_prod(A,B))),aa(set(fun(product_prod(A,B),$o)),set(set(product_prod(A,B))),image2(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B))),aa(set(fun(A,fun(B,$o))),set(fun(product_prod(A,B),$o)),image2(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o)),S)))) ) ).

% Inf_INT_eq2
tff(fact_3195_INF__filter__not__bot,axiom,
    ! [A: $tType,B: $tType,B3: set(A),F4: fun(A,filter(B))] :
      ( ! [X6: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X6),B3)
         => ( aa(set(A),$o,finite_finite2(A),X6)
           => ( aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image2(A,filter(B),F4),X6)) != bot_bot(filter(B)) ) ) )
     => ( aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image2(A,filter(B),F4),B3)) != bot_bot(filter(B)) ) ) ).

% INF_filter_not_bot
tff(fact_3196_INF__INT__eq,axiom,
    ! [A: $tType,B: $tType,R2: fun(B,set(A)),S: set(B),X3: A] :
      ( aa(A,$o,aa(set(fun(A,$o)),fun(A,$o),complete_Inf_Inf(fun(A,$o)),aa(set(B),set(fun(A,$o)),image2(B,fun(A,$o),aTP_Lamp_nr(fun(B,set(A)),fun(B,fun(A,$o)),R2)),S)),X3)
    <=> aa(set(A),$o,member(A,X3),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),R2),S))) ) ).

% INF_INT_eq
tff(fact_3197_Inf__nat__def1,axiom,
    ! [K5: set(nat)] :
      ( ( K5 != bot_bot(set(nat)) )
     => aa(set(nat),$o,member(nat,aa(set(nat),nat,complete_Inf_Inf(nat),K5)),K5) ) ).

% Inf_nat_def1
tff(fact_3198_Sup__fun__def,axiom,
    ! [B: $tType,A: $tType] :
      ( complete_Sup(B)
     => ! [A4: set(fun(A,B)),X3: A] : aa(A,B,aa(set(fun(A,B)),fun(A,B),complete_Sup_Sup(fun(A,B)),A4),X3) = aa(set(B),B,complete_Sup_Sup(B),aa(set(fun(A,B)),set(B),image2(fun(A,B),B,aTP_Lamp_ns(A,fun(fun(A,B),B),X3)),A4)) ) ).

% Sup_fun_def
tff(fact_3199_Inf__fun__def,axiom,
    ! [B: $tType,A: $tType] :
      ( complete_Inf(B)
     => ! [A4: set(fun(A,B)),X3: A] : aa(A,B,aa(set(fun(A,B)),fun(A,B),complete_Inf_Inf(fun(A,B)),A4),X3) = aa(set(B),B,complete_Inf_Inf(B),aa(set(fun(A,B)),set(B),image2(fun(A,B),B,aTP_Lamp_nt(A,fun(fun(A,B),B),X3)),A4)) ) ).

% Inf_fun_def
tff(fact_3200_SUP__Sup__eq2,axiom,
    ! [B: $tType,A: $tType,S: set(set(product_prod(A,B))),X3: A,Xa3: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(set(fun(A,fun(B,$o))),fun(A,fun(B,$o)),complete_Sup_Sup(fun(A,fun(B,$o))),aa(set(set(product_prod(A,B))),set(fun(A,fun(B,$o))),image2(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_as(set(product_prod(A,B)),fun(A,fun(B,$o)))),S)),X3),Xa3)
    <=> aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Xa3)),aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),S)) ) ).

% SUP_Sup_eq2
tff(fact_3201_Sup__set__def,axiom,
    ! [A: $tType,A4: set(set(A))] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A4) = aa(fun(A,$o),set(A),collect(A),aTP_Lamp_nu(set(set(A)),fun(A,$o),A4)) ).

% Sup_set_def
tff(fact_3202_SUP__Sup__eq,axiom,
    ! [A: $tType,S: set(set(A)),X3: A] :
      ( aa(A,$o,aa(set(fun(A,$o)),fun(A,$o),complete_Sup_Sup(fun(A,$o)),aa(set(set(A)),set(fun(A,$o)),image2(set(A),fun(A,$o),aTP_Lamp_ag(set(A),fun(A,$o))),S)),X3)
    <=> aa(set(A),$o,member(A,X3),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),S)) ) ).

% SUP_Sup_eq
tff(fact_3203_SUP__UN__eq,axiom,
    ! [A: $tType,B: $tType,R2: fun(B,set(A)),S: set(B),X3: A] :
      ( aa(A,$o,aa(set(fun(A,$o)),fun(A,$o),complete_Sup_Sup(fun(A,$o)),aa(set(B),set(fun(A,$o)),image2(B,fun(A,$o),aTP_Lamp_nr(fun(B,set(A)),fun(B,fun(A,$o)),R2)),S)),X3)
    <=> aa(set(A),$o,member(A,X3),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),R2),S))) ) ).

% SUP_UN_eq
tff(fact_3204_Int__Union2,axiom,
    ! [A: $tType,B3: set(set(A)),A4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),B3)),A4) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(set(A)),set(set(A)),image2(set(A),set(A),aTP_Lamp_nv(set(A),fun(set(A),set(A)),A4)),B3)) ).

% Int_Union2
tff(fact_3205_Int__Union,axiom,
    ! [A: $tType,A4: set(A),B3: set(set(A))] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),B3)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(set(A)),set(set(A)),image2(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4)),B3)) ).

% Int_Union
tff(fact_3206_Un__Inter,axiom,
    ! [A: $tType,A4: set(A),B3: set(set(A))] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),B3)) = aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(set(A)),set(set(A)),image2(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4)),B3)) ).

% Un_Inter
tff(fact_3207_vimage__Union,axiom,
    ! [A: $tType,B: $tType,F: fun(A,B),A4: set(set(B))] : aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),A4)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(set(B)),set(set(A)),image2(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F)),A4)) ).

% vimage_Union
tff(fact_3208_in__image__insert__iff,axiom,
    ! [A: $tType,B3: set(set(A)),X: A,A4: set(A)] :
      ( ! [C7: set(A)] :
          ( aa(set(set(A)),$o,member(set(A),C7),B3)
         => ~ aa(set(A),$o,member(A,X),C7) )
     => ( aa(set(set(A)),$o,member(set(A),A4),aa(set(set(A)),set(set(A)),image2(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X)),B3))
      <=> ( aa(set(A),$o,member(A,X),A4)
          & aa(set(set(A)),$o,member(set(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))))),B3) ) ) ) ).

% in_image_insert_iff
tff(fact_3209_int__in__range__abs,axiom,
    ! [N: nat] : aa(set(int),$o,member(int,aa(nat,int,semiring_1_of_nat(int),N)),aa(set(int),set(int),image2(int,int,abs_abs(int)),top_top(set(int)))) ).

% int_in_range_abs
tff(fact_3210_Int__Inter__eq_I1_J,axiom,
    ! [A: $tType,A4: set(A),B11: set(set(A))] :
      aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),B11)) = $ite(B11 = bot_bot(set(set(A))),A4,aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(set(A)),set(set(A)),image2(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4)),B11))) ).

% Int_Inter_eq(1)
tff(fact_3211_Int__Inter__eq_I2_J,axiom,
    ! [A: $tType,B11: set(set(A)),A4: set(A)] :
      aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),B11)),A4) = $ite(B11 = bot_bot(set(set(A))),A4,aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(set(A)),set(set(A)),image2(set(A),set(A),aTP_Lamp_nv(set(A),fun(set(A),set(A)),A4)),B11))) ).

% Int_Inter_eq(2)
tff(fact_3212_Id__on__def,axiom,
    ! [A: $tType,A4: set(A)] : id_on(A,A4) = aa(set(set(product_prod(A,A))),set(product_prod(A,A)),complete_Sup_Sup(set(product_prod(A,A))),aa(set(A),set(set(product_prod(A,A))),image2(A,set(product_prod(A,A)),aTP_Lamp_nw(A,set(product_prod(A,A)))),A4)) ).

% Id_on_def
tff(fact_3213_SUP__UNIV__bool__expand,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A4: fun($o,A)] : aa(set(A),A,complete_Sup_Sup(A),aa(set($o),set(A),image2($o,A,A4),top_top(set($o)))) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa($o,A,A4,$true)),aa($o,A,A4,$false)) ) ).

% SUP_UNIV_bool_expand
tff(fact_3214_INF__UNIV__bool__expand,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A4: fun($o,A)] : aa(set(A),A,complete_Inf_Inf(A),aa(set($o),set(A),image2($o,A,A4),top_top(set($o)))) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa($o,A,A4,$true)),aa($o,A,A4,$false)) ) ).

% INF_UNIV_bool_expand
tff(fact_3215_UN__bool__eq,axiom,
    ! [A: $tType,A4: fun($o,set(A))] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set($o),set(set(A)),image2($o,set(A),A4),top_top(set($o)))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa($o,set(A),A4,$true)),aa($o,set(A),A4,$false)) ).

% UN_bool_eq
tff(fact_3216_Un__eq__UN,axiom,
    ! [A: $tType,A4: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set($o),set(set(A)),image2($o,set(A),aa(set(A),fun($o,set(A)),aTP_Lamp_nx(set(A),fun(set(A),fun($o,set(A))),A4),B3)),top_top(set($o)))) ).

% Un_eq_UN
tff(fact_3217_INT__bool__eq,axiom,
    ! [A: $tType,A4: fun($o,set(A))] : aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set($o),set(set(A)),image2($o,set(A),A4),top_top(set($o)))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa($o,set(A),A4,$true)),aa($o,set(A),A4,$false)) ).

% INT_bool_eq
tff(fact_3218_image__Suc__atMost,axiom,
    ! [N: nat] : aa(set(nat),set(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_3219_image__Suc__lessThan,axiom,
    ! [N: nat] : aa(set(nat),set(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_3220_image__add__int__atLeastLessThan,axiom,
    ! [L: int,U: int] : aa(set(int),set(int),image2(int,int,aTP_Lamp_ny(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_3221_image__add__integer__atLeastLessThan,axiom,
    ! [L: code_integer,U: code_integer] : aa(set(code_integer),set(code_integer),image2(code_integer,code_integer,aTP_Lamp_nz(code_integer,fun(code_integer,code_integer),L)),set_or7035219750837199246ssThan(code_integer,zero_zero(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),U),L))) = set_or7035219750837199246ssThan(code_integer,L,U) ).

% image_add_integer_atLeastLessThan
tff(fact_3222_range__mod,axiom,
    ! [N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
     => ( aa(set(nat),set(nat),image2(nat,nat,aTP_Lamp_oa(nat,fun(nat,nat),N)),top_top(set(nat))) = set_or7035219750837199246ssThan(nat,zero_zero(nat),N) ) ) ).

% range_mod
tff(fact_3223_image__minus__const__atLeastLessThan__nat,axiom,
    ! [C2: nat,X: nat,Y: nat] :
      aa(set(nat),set(nat),image2(nat,nat,aTP_Lamp_ob(nat,fun(nat,nat),C2)),set_or7035219750837199246ssThan(nat,X,Y)) = $ite(
        aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),C2),Y),
        set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),X),C2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Y),C2)),
        $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),Y),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),zero_zero(nat)),bot_bot(set(nat))),bot_bot(set(nat))) ) ).

% image_minus_const_atLeastLessThan_nat
tff(fact_3224_surjD,axiom,
    ! [A: $tType,B: $tType,F: fun(B,A),Y: A] :
      ( ( aa(set(B),set(A),image2(B,A,F),top_top(set(B))) = top_top(set(A)) )
     => ? [X2: B] : Y = aa(B,A,F,X2) ) ).

% surjD
tff(fact_3225_surjE,axiom,
    ! [A: $tType,B: $tType,F: fun(B,A),Y: A] :
      ( ( aa(set(B),set(A),image2(B,A,F),top_top(set(B))) = top_top(set(A)) )
     => ~ ! [X2: B] : Y != aa(B,A,F,X2) ) ).

% surjE
tff(fact_3226_surjI,axiom,
    ! [B: $tType,A: $tType,G: fun(B,A),F: fun(A,B)] :
      ( ! [X2: A] : aa(B,A,G,aa(A,B,F,X2)) = X2
     => ( aa(set(B),set(A),image2(B,A,G),top_top(set(B))) = top_top(set(A)) ) ) ).

% surjI
tff(fact_3227_surj__def,axiom,
    ! [A: $tType,B: $tType,F: fun(B,A)] :
      ( ( aa(set(B),set(A),image2(B,A,F),top_top(set(B))) = top_top(set(A)) )
    <=> ! [Y3: A] :
        ? [X4: B] : Y3 = aa(B,A,F,X4) ) ).

% surj_def
tff(fact_3228_translation__Int,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A,S2: set(A),T3: set(A)] : aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),plus_plus(A),A3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S2),T3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),plus_plus(A),A3)),S2)),aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),plus_plus(A),A3)),T3)) ) ).

% translation_Int
tff(fact_3229_comp__surj,axiom,
    ! [B: $tType,A: $tType,C: $tType,F: fun(B,A),G: fun(A,C)] :
      ( ( aa(set(B),set(A),image2(B,A,F),top_top(set(B))) = top_top(set(A)) )
     => ( ( aa(set(A),set(C),image2(A,C,G),top_top(set(A))) = top_top(set(C)) )
       => ( aa(set(B),set(C),image2(B,C,aa(fun(B,A),fun(B,C),comp(A,C,B,G),F)),top_top(set(B))) = top_top(set(C)) ) ) ) ).

% comp_surj
tff(fact_3230_translation__Compl,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A,T3: set(A)] : aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),plus_plus(A),A3)),aa(set(A),set(A),uminus_uminus(set(A)),T3)) = aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),plus_plus(A),A3)),T3)) ) ).

% translation_Compl
tff(fact_3231_surj__image__vimage__eq,axiom,
    ! [B: $tType,A: $tType,F: fun(B,A),A4: set(A)] :
      ( ( aa(set(B),set(A),image2(B,A,F),top_top(set(B))) = top_top(set(A)) )
     => ( aa(set(B),set(A),image2(B,A,F),aa(set(A),set(B),aa(fun(B,A),fun(set(A),set(B)),vimage(B,A),F),A4)) = A4 ) ) ).

% surj_image_vimage_eq
tff(fact_3232_translation__subtract__Int,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A,S2: set(A),T3: set(A)] : aa(set(A),set(A),image2(A,A,aTP_Lamp_nl(A,fun(A,A),A3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S2),T3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),image2(A,A,aTP_Lamp_nl(A,fun(A,A),A3)),S2)),aa(set(A),set(A),image2(A,A,aTP_Lamp_nl(A,fun(A,A),A3)),T3)) ) ).

% translation_subtract_Int
tff(fact_3233_translation__subtract__diff,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A,S2: set(A),T3: set(A)] : aa(set(A),set(A),image2(A,A,aTP_Lamp_nl(A,fun(A,A),A3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S2),T3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),image2(A,A,aTP_Lamp_nl(A,fun(A,A),A3)),S2)),aa(set(A),set(A),image2(A,A,aTP_Lamp_nl(A,fun(A,A),A3)),T3)) ) ).

% translation_subtract_diff
tff(fact_3234_translation__subtract__Compl,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A,T3: set(A)] : aa(set(A),set(A),image2(A,A,aTP_Lamp_nl(A,fun(A,A),A3)),aa(set(A),set(A),uminus_uminus(set(A)),T3)) = aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),image2(A,A,aTP_Lamp_nl(A,fun(A,A),A3)),T3)) ) ).

% translation_subtract_Compl
tff(fact_3235_fun_Oset__map,axiom,
    ! [A: $tType,C: $tType,B: $tType,F: fun(C,A),V: fun(B,C)] : aa(set(B),set(A),image2(B,A,aa(fun(B,C),fun(B,A),comp(C,A,B,F),V)),top_top(set(B))) = aa(set(C),set(A),image2(C,A,F),aa(set(B),set(C),image2(B,C,V),top_top(set(B)))) ).

% fun.set_map
tff(fact_3236_fun_Omap__cong,axiom,
    ! [C: $tType,B: $tType,A: $tType,X: fun(A,B),Ya: fun(A,B),F: fun(B,C),G: fun(B,C)] :
      ( ( X = Ya )
     => ( ! [Z3: B] :
            ( aa(set(B),$o,member(B,Z3),aa(set(A),set(B),image2(A,B,Ya),top_top(set(A))))
           => ( aa(B,C,F,Z3) = aa(B,C,G,Z3) ) )
       => ( aa(fun(A,B),fun(A,C),comp(B,C,A,F),X) = aa(fun(A,B),fun(A,C),comp(B,C,A,G),Ya) ) ) ) ).

% fun.map_cong
tff(fact_3237_fun_Omap__cong0,axiom,
    ! [C: $tType,A: $tType,B: $tType,X: fun(B,A),F: fun(A,C),G: fun(A,C)] :
      ( ! [Z3: A] :
          ( aa(set(A),$o,member(A,Z3),aa(set(B),set(A),image2(B,A,X),top_top(set(B))))
         => ( aa(A,C,F,Z3) = aa(A,C,G,Z3) ) )
     => ( aa(fun(B,A),fun(B,C),comp(A,C,B,F),X) = aa(fun(B,A),fun(B,C),comp(A,C,B,G),X) ) ) ).

% fun.map_cong0
tff(fact_3238_fun_Oinj__map__strong,axiom,
    ! [C: $tType,A: $tType,B: $tType,X: fun(B,A),Xa: fun(B,A),F: fun(A,C),Fa: fun(A,C)] :
      ( ! [Z3: A,Za: A] :
          ( aa(set(A),$o,member(A,Z3),aa(set(B),set(A),image2(B,A,X),top_top(set(B))))
         => ( aa(set(A),$o,member(A,Za),aa(set(B),set(A),image2(B,A,Xa),top_top(set(B))))
           => ( ( aa(A,C,F,Z3) = aa(A,C,Fa,Za) )
             => ( Z3 = Za ) ) ) )
     => ( ( aa(fun(B,A),fun(B,C),comp(A,C,B,F),X) = aa(fun(B,A),fun(B,C),comp(A,C,B,Fa),Xa) )
       => ( X = Xa ) ) ) ).

% fun.inj_map_strong
tff(fact_3239_UNIV__char__of__nat,axiom,
    top_top(set(char)) = aa(set(nat),set(char),image2(nat,char,unique5772411509450598832har_of(nat)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,one2))))))))))) ).

% UNIV_char_of_nat
tff(fact_3240_subset__mset_OcInf__singleton,axiom,
    ! [A: $tType,X: multiset(A)] : aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert2(multiset(A)),X),bot_bot(set(multiset(A))))) = X ).

% subset_mset.cInf_singleton
tff(fact_3241_subset__mset_OcSup__singleton,axiom,
    ! [A: $tType,X: multiset(A)] : aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert2(multiset(A)),X),bot_bot(set(multiset(A))))) = X ).

% subset_mset.cSup_singleton
tff(fact_3242_SUP2__I,axiom,
    ! [B: $tType,A: $tType,C: $tType,A3: A,A4: set(A),B3: fun(A,fun(B,fun(C,$o))),B2: B,C2: C] :
      ( aa(set(A),$o,member(A,A3),A4)
     => ( aa(C,$o,aa(B,fun(C,$o),aa(A,fun(B,fun(C,$o)),B3,A3),B2),C2)
       => aa(C,$o,aa(B,fun(C,$o),aa(set(fun(B,fun(C,$o))),fun(B,fun(C,$o)),complete_Sup_Sup(fun(B,fun(C,$o))),aa(set(A),set(fun(B,fun(C,$o))),image2(A,fun(B,fun(C,$o)),B3),A4)),B2),C2) ) ) ).

% SUP2_I
tff(fact_3243_INF1__I,axiom,
    ! [A: $tType,B: $tType,A4: set(A),B3: fun(A,fun(B,$o)),B2: B] :
      ( ! [X2: A] :
          ( aa(set(A),$o,member(A,X2),A4)
         => aa(B,$o,aa(A,fun(B,$o),B3,X2),B2) )
     => aa(B,$o,aa(set(fun(B,$o)),fun(B,$o),complete_Inf_Inf(fun(B,$o)),aa(set(A),set(fun(B,$o)),image2(A,fun(B,$o),B3),A4)),B2) ) ).

% INF1_I
tff(fact_3244_INF2__I,axiom,
    ! [B: $tType,A: $tType,C: $tType,A4: set(A),B3: fun(A,fun(B,fun(C,$o))),B2: B,C2: C] :
      ( ! [X2: A] :
          ( aa(set(A),$o,member(A,X2),A4)
         => aa(C,$o,aa(B,fun(C,$o),aa(A,fun(B,fun(C,$o)),B3,X2),B2),C2) )
     => aa(C,$o,aa(B,fun(C,$o),aa(set(fun(B,fun(C,$o))),fun(B,fun(C,$o)),complete_Inf_Inf(fun(B,fun(C,$o))),aa(set(A),set(fun(B,fun(C,$o))),image2(A,fun(B,fun(C,$o)),B3),A4)),B2),C2) ) ).

% INF2_I
tff(fact_3245_SUP1__I,axiom,
    ! [A: $tType,B: $tType,A3: A,A4: set(A),B3: fun(A,fun(B,$o)),B2: B] :
      ( aa(set(A),$o,member(A,A3),A4)
     => ( aa(B,$o,aa(A,fun(B,$o),B3,A3),B2)
       => aa(B,$o,aa(set(fun(B,$o)),fun(B,$o),complete_Sup_Sup(fun(B,$o)),aa(set(A),set(fun(B,$o)),image2(A,fun(B,$o),B3),A4)),B2) ) ) ).

% SUP1_I
tff(fact_3246_Sup__multiset__empty,axiom,
    ! [A: $tType] : aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),bot_bot(set(multiset(A)))) = zero_zero(multiset(A)) ).

% Sup_multiset_empty
tff(fact_3247_SUP1__E,axiom,
    ! [B: $tType,A: $tType,B3: fun(B,fun(A,$o)),A4: set(B),B2: A] :
      ( aa(A,$o,aa(set(fun(A,$o)),fun(A,$o),complete_Sup_Sup(fun(A,$o)),aa(set(B),set(fun(A,$o)),image2(B,fun(A,$o),B3),A4)),B2)
     => ~ ! [X2: B] :
            ( aa(set(B),$o,member(B,X2),A4)
           => ~ aa(A,$o,aa(B,fun(A,$o),B3,X2),B2) ) ) ).

% SUP1_E
tff(fact_3248_SUP2__E,axiom,
    ! [A: $tType,C: $tType,B: $tType,B3: fun(C,fun(A,fun(B,$o))),A4: set(C),B2: A,C2: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(set(fun(A,fun(B,$o))),fun(A,fun(B,$o)),complete_Sup_Sup(fun(A,fun(B,$o))),aa(set(C),set(fun(A,fun(B,$o))),image2(C,fun(A,fun(B,$o)),B3),A4)),B2),C2)
     => ~ ! [X2: C] :
            ( aa(set(C),$o,member(C,X2),A4)
           => ~ aa(B,$o,aa(A,fun(B,$o),aa(C,fun(A,fun(B,$o)),B3,X2),B2),C2) ) ) ).

% SUP2_E
tff(fact_3249_INF2__E,axiom,
    ! [B: $tType,A: $tType,C: $tType,B3: fun(C,fun(A,fun(B,$o))),A4: set(C),B2: A,C2: B,A3: C] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(set(fun(A,fun(B,$o))),fun(A,fun(B,$o)),complete_Inf_Inf(fun(A,fun(B,$o))),aa(set(C),set(fun(A,fun(B,$o))),image2(C,fun(A,fun(B,$o)),B3),A4)),B2),C2)
     => ( ~ aa(B,$o,aa(A,fun(B,$o),aa(C,fun(A,fun(B,$o)),B3,A3),B2),C2)
       => ~ aa(set(C),$o,member(C,A3),A4) ) ) ).

% INF2_E
tff(fact_3250_INF2__D,axiom,
    ! [A: $tType,C: $tType,B: $tType,B3: fun(C,fun(A,fun(B,$o))),A4: set(C),B2: A,C2: B,A3: C] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(set(fun(A,fun(B,$o))),fun(A,fun(B,$o)),complete_Inf_Inf(fun(A,fun(B,$o))),aa(set(C),set(fun(A,fun(B,$o))),image2(C,fun(A,fun(B,$o)),B3),A4)),B2),C2)
     => ( aa(set(C),$o,member(C,A3),A4)
       => aa(B,$o,aa(A,fun(B,$o),aa(C,fun(A,fun(B,$o)),B3,A3),B2),C2) ) ) ).

% INF2_D
tff(fact_3251_INF1__E,axiom,
    ! [A: $tType,B: $tType,B3: fun(B,fun(A,$o)),A4: set(B),B2: A,A3: B] :
      ( aa(A,$o,aa(set(fun(A,$o)),fun(A,$o),complete_Inf_Inf(fun(A,$o)),aa(set(B),set(fun(A,$o)),image2(B,fun(A,$o),B3),A4)),B2)
     => ( ~ aa(A,$o,aa(B,fun(A,$o),B3,A3),B2)
       => ~ aa(set(B),$o,member(B,A3),A4) ) ) ).

% INF1_E
tff(fact_3252_INF1__D,axiom,
    ! [B: $tType,A: $tType,B3: fun(B,fun(A,$o)),A4: set(B),B2: A,A3: B] :
      ( aa(A,$o,aa(set(fun(A,$o)),fun(A,$o),complete_Inf_Inf(fun(A,$o)),aa(set(B),set(fun(A,$o)),image2(B,fun(A,$o),B3),A4)),B2)
     => ( aa(set(B),$o,member(B,A3),A4)
       => aa(A,$o,aa(B,fun(A,$o),B3,A3),B2) ) ) ).

% INF1_D
tff(fact_3253_INF__filter__bot__base,axiom,
    ! [A: $tType,B: $tType,I4: set(A),F4: fun(A,filter(B))] :
      ( ! [I2: A] :
          ( aa(set(A),$o,member(A,I2),I4)
         => ! [J2: A] :
              ( aa(set(A),$o,member(A,J2),I4)
             => ? [X3: A] :
                  ( aa(set(A),$o,member(A,X3),I4)
                  & aa(filter(B),$o,aa(filter(B),fun(filter(B),$o),ord_less_eq(filter(B)),aa(A,filter(B),F4,X3)),aa(filter(B),filter(B),aa(filter(B),fun(filter(B),filter(B)),inf_inf(filter(B)),aa(A,filter(B),F4,I2)),aa(A,filter(B),F4,J2))) ) ) )
     => ( ( aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image2(A,filter(B),F4),I4)) = bot_bot(filter(B)) )
      <=> ? [X4: A] :
            ( aa(set(A),$o,member(A,X4),I4)
            & ( aa(A,filter(B),F4,X4) = bot_bot(filter(B)) ) ) ) ) ).

% INF_filter_bot_base
tff(fact_3254_Inf__multiset__empty,axiom,
    ! [A: $tType] : aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),bot_bot(set(multiset(A)))) = zero_zero(multiset(A)) ).

% Inf_multiset_empty
tff(fact_3255_Inf__filter__not__bot,axiom,
    ! [A: $tType,B3: set(filter(A))] :
      ( ! [X6: set(filter(A))] :
          ( aa(set(filter(A)),$o,aa(set(filter(A)),fun(set(filter(A)),$o),ord_less_eq(set(filter(A))),X6),B3)
         => ( aa(set(filter(A)),$o,finite_finite2(filter(A)),X6)
           => ( aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),X6) != bot_bot(filter(A)) ) ) )
     => ( aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),B3) != bot_bot(filter(A)) ) ) ).

% Inf_filter_not_bot
tff(fact_3256_fun_Omap__ident,axiom,
    ! [B: $tType,A: $tType,T3: fun(A,B)] : aa(fun(A,B),fun(A,B),comp(B,B,A,aTP_Lamp_oc(B,B)),T3) = T3 ).

% fun.map_ident
tff(fact_3257_UN__UN__split__split__eq,axiom,
    ! [D: $tType,E: $tType,A: $tType,C: $tType,B: $tType,A4: fun(B,fun(C,fun(D,fun(E,set(A))))),Y4: set(product_prod(D,E)),X5: set(product_prod(B,C))] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(product_prod(B,C)),set(set(A)),image2(product_prod(B,C),set(A),aa(fun(B,fun(C,set(A))),fun(product_prod(B,C),set(A)),product_case_prod(B,C,set(A)),aa(set(product_prod(D,E)),fun(B,fun(C,set(A))),aTP_Lamp_od(fun(B,fun(C,fun(D,fun(E,set(A))))),fun(set(product_prod(D,E)),fun(B,fun(C,set(A)))),A4),Y4))),X5)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(product_prod(B,C)),set(set(A)),image2(product_prod(B,C),set(A),aa(set(product_prod(D,E)),fun(product_prod(B,C),set(A)),aTP_Lamp_og(fun(B,fun(C,fun(D,fun(E,set(A))))),fun(set(product_prod(D,E)),fun(product_prod(B,C),set(A))),A4),Y4)),X5)) ).

% UN_UN_split_split_eq
tff(fact_3258_mlex__eq,axiom,
    ! [A: $tType,F: fun(A,nat),R: set(product_prod(A,A))] : mlex_prod(A,F,R) = aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),aa(set(product_prod(A,A)),fun(A,fun(A,$o)),aTP_Lamp_oh(fun(A,nat),fun(set(product_prod(A,A)),fun(A,fun(A,$o))),F),R))) ).

% mlex_eq
tff(fact_3259_UN__constant__eq,axiom,
    ! [A: $tType,B: $tType,A3: A,A4: set(A),F: fun(A,set(B)),C2: set(B)] :
      ( aa(set(A),$o,member(A,A3),A4)
     => ( ! [X2: A] :
            ( aa(set(A),$o,member(A,X2),A4)
           => ( aa(A,set(B),F,X2) = C2 ) )
       => ( aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),F),A4)) = C2 ) ) ) ).

% UN_constant_eq
tff(fact_3260_fold__union__pair,axiom,
    ! [B: $tType,A: $tType,B3: set(A),X: B,A4: set(product_prod(B,A))] :
      ( aa(set(A),$o,finite_finite2(A),B3)
     => ( aa(set(product_prod(B,A)),set(product_prod(B,A)),aa(set(product_prod(B,A)),fun(set(product_prod(B,A)),set(product_prod(B,A))),sup_sup(set(product_prod(B,A))),aa(set(set(product_prod(B,A))),set(product_prod(B,A)),complete_Sup_Sup(set(product_prod(B,A))),aa(set(A),set(set(product_prod(B,A))),image2(A,set(product_prod(B,A)),aTP_Lamp_oi(B,fun(A,set(product_prod(B,A))),X)),B3))),A4) = finite_fold(A,set(product_prod(B,A)),aTP_Lamp_oj(B,fun(A,fun(set(product_prod(B,A)),set(product_prod(B,A)))),X),A4,B3) ) ) ).

% fold_union_pair
tff(fact_3261_range__nat__of__char,axiom,
    aa(set(char),set(nat),image2(char,nat,comm_s6883823935334413003f_char(nat)),top_top(set(char))) = set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,one2)))))))))) ).

% range_nat_of_char
tff(fact_3262_fold__empty,axiom,
    ! [B: $tType,A: $tType,F: fun(B,fun(A,A)),Z2: A] : finite_fold(B,A,F,Z2,bot_bot(set(B))) = Z2 ).

% fold_empty
tff(fact_3263_union__fold__insert,axiom,
    ! [A: $tType,A4: set(A),B3: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3) = finite_fold(A,set(A),insert2(A),B3,A4) ) ) ).

% union_fold_insert
tff(fact_3264_sup__Sup__fold__sup,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A4: set(A),B3: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Sup_Sup(A),A4)),B3) = finite_fold(A,A,sup_sup(A),B3,A4) ) ) ) ).

% sup_Sup_fold_sup
tff(fact_3265_inf__Inf__fold__inf,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A4: set(A),B3: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Inf_Inf(A),A4)),B3) = finite_fold(A,A,inf_inf(A),B3,A4) ) ) ) ).

% inf_Inf_fold_inf
tff(fact_3266_Sup__fold__sup,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A4: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),A,complete_Sup_Sup(A),A4) = finite_fold(A,A,sup_sup(A),bot_bot(A),A4) ) ) ) ).

% Sup_fold_sup
tff(fact_3267_Inf__fold__inf,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A4: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),A,complete_Inf_Inf(A),A4) = finite_fold(A,A,inf_inf(A),top_top(A),A4) ) ) ) ).

% Inf_fold_inf
tff(fact_3268_prod_Oeq__fold,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(B,A),A4: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),A4) = finite_fold(B,A,aa(fun(B,A),fun(B,fun(A,A)),comp(A,fun(A,A),B,times_times(A)),G),one_one(A),A4) ) ).

% prod.eq_fold
tff(fact_3269_image__fold__insert,axiom,
    ! [B: $tType,A: $tType,A4: set(A),F: fun(A,B)] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( aa(set(A),set(B),image2(A,B,F),A4) = finite_fold(A,set(B),aTP_Lamp_ok(fun(A,B),fun(A,fun(set(B),set(B))),F),bot_bot(set(B)),A4) ) ) ).

% image_fold_insert
tff(fact_3270_sup__SUP__fold__sup,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A4: set(A),B3: B,F: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(B,B,aa(B,fun(B,B),sup_sup(B),B3),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F),A4))) = finite_fold(A,B,aa(fun(A,B),fun(A,fun(B,B)),comp(B,fun(B,B),A,sup_sup(B)),F),B3,A4) ) ) ) ).

% sup_SUP_fold_sup
tff(fact_3271_inf__INF__fold__inf,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A4: set(A),B3: B,F: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(B,B,aa(B,fun(B,B),inf_inf(B),B3),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F),A4))) = finite_fold(A,B,aa(fun(A,B),fun(A,fun(B,B)),comp(B,fun(B,B),A,inf_inf(B)),F),B3,A4) ) ) ) ).

% inf_INF_fold_inf
tff(fact_3272_Id__on__fold,axiom,
    ! [A: $tType,A4: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( id_on(A,A4) = finite_fold(A,set(product_prod(A,A)),aTP_Lamp_ol(A,fun(set(product_prod(A,A)),set(product_prod(A,A)))),bot_bot(set(product_prod(A,A))),A4) ) ) ).

% Id_on_fold
tff(fact_3273_SUP__fold__sup,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A4: set(A),F: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F),A4)) = finite_fold(A,B,aa(fun(A,B),fun(A,fun(B,B)),comp(B,fun(B,B),A,sup_sup(B)),F),bot_bot(B),A4) ) ) ) ).

% SUP_fold_sup
tff(fact_3274_INF__fold__inf,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A4: set(A),F: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F),A4)) = finite_fold(A,B,aa(fun(A,B),fun(A,fun(B,B)),comp(B,fun(B,B),A,inf_inf(B)),F),top_top(B),A4) ) ) ) ).

% INF_fold_inf
tff(fact_3275_mlex__leq,axiom,
    ! [A: $tType,F: fun(A,nat),X: A,Y: A,R: set(product_prod(A,A))] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,F,X)),aa(A,nat,F,Y))
     => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R)
       => aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),mlex_prod(A,F,R)) ) ) ).

% mlex_leq
tff(fact_3276_mlex__iff,axiom,
    ! [A: $tType,X: A,Y: A,F: fun(A,nat),R: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),mlex_prod(A,F,R))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,F,X)),aa(A,nat,F,Y))
        | ( ( aa(A,nat,F,X) = aa(A,nat,F,Y) )
          & aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R) ) ) ) ).

% mlex_iff
tff(fact_3277_mlex__less,axiom,
    ! [A: $tType,F: fun(A,nat),X: A,Y: A,R: set(product_prod(A,A))] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,F,X)),aa(A,nat,F,Y))
     => aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),mlex_prod(A,F,R)) ) ).

% mlex_less
tff(fact_3278_Set__filter__fold,axiom,
    ! [A: $tType,A4: set(A),P: fun(A,$o)] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( filter3(A,P,A4) = finite_fold(A,set(A),aTP_Lamp_om(fun(A,$o),fun(A,fun(set(A),set(A))),P),bot_bot(set(A)),A4) ) ) ).

% Set_filter_fold
tff(fact_3279_pred__def,axiom,
    ! [Nat: nat] : pred2(Nat) = aa(nat,nat,aa(fun(nat,nat),fun(nat,nat),aa(nat,fun(fun(nat,nat),fun(nat,nat)),case_nat(nat),zero_zero(nat)),aTP_Lamp_dj(nat,nat)),Nat) ).

% pred_def
tff(fact_3280_Set_Ofilter__def,axiom,
    ! [A: $tType,P: fun(A,$o),A4: set(A)] : filter3(A,P,A4) = aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_on(fun(A,$o),fun(set(A),fun(A,$o)),P),A4)) ).

% Set.filter_def
tff(fact_3281_card_Oeq__fold,axiom,
    ! [A: $tType,A4: set(A)] : aa(set(A),nat,finite_card(A),A4) = finite_fold(A,nat,aTP_Lamp_oo(A,fun(nat,nat)),zero_zero(nat),A4) ).

% card.eq_fold
tff(fact_3282_inter__Set__filter,axiom,
    ! [A: $tType,B3: set(A),A4: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),B3)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3) = filter3(A,aa(set(A),fun(A,$o),aTP_Lamp_ag(set(A),fun(A,$o)),A4),B3) ) ) ).

% inter_Set_filter
tff(fact_3283_in__measure,axiom,
    ! [A: $tType,X: A,Y: A,F: fun(A,nat)] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),measure(A,F))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,F,X)),aa(A,nat,F,Y)) ) ).

% in_measure
tff(fact_3284_in__finite__psubset,axiom,
    ! [A: $tType,A4: set(A),B3: set(A)] :
      ( aa(set(product_prod(set(A),set(A))),$o,member(product_prod(set(A),set(A)),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),A4),B3)),finite_psubset(A))
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A4),B3)
        & aa(set(A),$o,finite_finite2(A),B3) ) ) ).

% in_finite_psubset
tff(fact_3285_Pow__fold,axiom,
    ! [A: $tType,A4: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( pow2(A,A4) = finite_fold(A,set(set(A)),aTP_Lamp_op(A,fun(set(set(A)),set(set(A)))),aa(set(set(A)),set(set(A)),aa(set(A),fun(set(set(A)),set(set(A))),insert2(set(A)),bot_bot(set(A))),bot_bot(set(set(A)))),A4) ) ) ).

% Pow_fold
tff(fact_3286_relpow__finite__bounded1,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),K: nat] :
      ( aa(set(product_prod(A,A)),$o,finite_finite2(product_prod(A,A)),R)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
       => aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),compow(set(product_prod(A,A)),K,R)),aa(set(set(product_prod(A,A))),set(product_prod(A,A)),complete_Sup_Sup(set(product_prod(A,A))),aa(set(nat),set(set(product_prod(A,A))),image2(nat,set(product_prod(A,A)),aTP_Lamp_oq(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),R)),aa(fun(nat,$o),set(nat),collect(nat),aTP_Lamp_or(set(product_prod(A,A)),fun(nat,$o),R))))) ) ) ).

% relpow_finite_bounded1
tff(fact_3287_range__abs__Nats,axiom,
    aa(set(int),set(int),image2(int,int,abs_abs(int)),top_top(set(int))) = semiring_1_Nats(int) ).

% range_abs_Nats
tff(fact_3288_relpow__1,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] : compow(set(product_prod(A,A)),one_one(nat),R) = R ).

% relpow_1
tff(fact_3289_Pow__UNIV,axiom,
    ! [A: $tType] : pow2(A,top_top(set(A))) = top_top(set(set(A))) ).

% Pow_UNIV
tff(fact_3290_Pow__Int__eq,axiom,
    ! [A: $tType,A4: set(A),B3: set(A)] : pow2(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3)) = aa(set(set(A)),set(set(A)),aa(set(set(A)),fun(set(set(A)),set(set(A))),inf_inf(set(set(A))),pow2(A,A4)),pow2(A,B3)) ).

% Pow_Int_eq
tff(fact_3291_Pow__empty,axiom,
    ! [A: $tType] : pow2(A,bot_bot(set(A))) = aa(set(set(A)),set(set(A)),aa(set(A),fun(set(set(A)),set(set(A))),insert2(set(A)),bot_bot(set(A))),bot_bot(set(set(A)))) ).

% Pow_empty
tff(fact_3292_Pow__singleton__iff,axiom,
    ! [A: $tType,X5: set(A),Y4: set(A)] :
      ( ( pow2(A,X5) = aa(set(set(A)),set(set(A)),aa(set(A),fun(set(set(A)),set(set(A))),insert2(set(A)),Y4),bot_bot(set(set(A)))) )
    <=> ( ( X5 = bot_bot(set(A)) )
        & ( Y4 = bot_bot(set(A)) ) ) ) ).

% Pow_singleton_iff
tff(fact_3293_Pow__bottom,axiom,
    ! [A: $tType,B3: set(A)] : aa(set(set(A)),$o,member(set(A),bot_bot(set(A))),pow2(A,B3)) ).

% Pow_bottom
tff(fact_3294_relpow__Suc__D2_H,axiom,
    ! [A: $tType,N: nat,R: set(product_prod(A,A)),X3: A,Y5: A,Z6: A] :
      ( ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y5)),compow(set(product_prod(A,A)),N,R))
        & aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y5),Z6)),R) )
     => ? [W: A] :
          ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),W)),R)
          & aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),W),Z6)),compow(set(product_prod(A,A)),N,R)) ) ) ).

% relpow_Suc_D2'
tff(fact_3295_Pow__not__empty,axiom,
    ! [A: $tType,A4: set(A)] : pow2(A,A4) != bot_bot(set(set(A))) ).

% Pow_not_empty
tff(fact_3296_Pow__def,axiom,
    ! [A: $tType,A4: set(A)] : pow2(A,A4) = aa(fun(set(A),$o),set(set(A)),collect(set(A)),aTP_Lamp_in(set(A),fun(set(A),$o),A4)) ).

% Pow_def
tff(fact_3297_Nats__mult,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [A3: A,B2: A] :
          ( aa(set(A),$o,member(A,A3),semiring_1_Nats(A))
         => ( aa(set(A),$o,member(A,B2),semiring_1_Nats(A))
           => aa(set(A),$o,member(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),semiring_1_Nats(A)) ) ) ) ).

% Nats_mult
tff(fact_3298_Nats__1,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => aa(set(A),$o,member(A,one_one(A)),semiring_1_Nats(A)) ) ).

% Nats_1
tff(fact_3299_relpow__0__E,axiom,
    ! [A: $tType,X: A,Y: A,R: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),compow(set(product_prod(A,A)),zero_zero(nat),R))
     => ( X = Y ) ) ).

% relpow_0_E
tff(fact_3300_relpow__0__I,axiom,
    ! [A: $tType,X: A,R: set(product_prod(A,A))] : aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),X)),compow(set(product_prod(A,A)),zero_zero(nat),R)) ).

% relpow_0_I
tff(fact_3301_relpow__Suc__I2,axiom,
    ! [A: $tType,X: A,Y: A,R: set(product_prod(A,A)),Z2: A,N: nat] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R)
     => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z2)),compow(set(product_prod(A,A)),N,R))
       => aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z2)),compow(set(product_prod(A,A)),aa(nat,nat,suc,N),R)) ) ) ).

% relpow_Suc_I2
tff(fact_3302_relpow__Suc__E2,axiom,
    ! [A: $tType,X: A,Z2: A,N: nat,R: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z2)),compow(set(product_prod(A,A)),aa(nat,nat,suc,N),R))
     => ~ ! [Y2: A] :
            ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y2)),R)
           => ~ aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),Z2)),compow(set(product_prod(A,A)),N,R)) ) ) ).

% relpow_Suc_E2
tff(fact_3303_relpow__Suc__D2,axiom,
    ! [A: $tType,X: A,Z2: A,N: nat,R: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z2)),compow(set(product_prod(A,A)),aa(nat,nat,suc,N),R))
     => ? [Y2: A] :
          ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y2)),R)
          & aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),Z2)),compow(set(product_prod(A,A)),N,R)) ) ) ).

% relpow_Suc_D2
tff(fact_3304_relpow__Suc__I,axiom,
    ! [A: $tType,X: A,Y: A,N: nat,R: set(product_prod(A,A)),Z2: A] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),compow(set(product_prod(A,A)),N,R))
     => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z2)),R)
       => aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z2)),compow(set(product_prod(A,A)),aa(nat,nat,suc,N),R)) ) ) ).

% relpow_Suc_I
tff(fact_3305_relpow__Suc__E,axiom,
    ! [A: $tType,X: A,Z2: A,N: nat,R: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z2)),compow(set(product_prod(A,A)),aa(nat,nat,suc,N),R))
     => ~ ! [Y2: A] :
            ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y2)),compow(set(product_prod(A,A)),N,R))
           => ~ aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),Z2)),R) ) ) ).

% relpow_Suc_E
tff(fact_3306_Pow__INT__eq,axiom,
    ! [A: $tType,B: $tType,B3: fun(B,set(A)),A4: set(B)] : pow2(A,aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B3),A4))) = aa(set(set(set(A))),set(set(A)),complete_Inf_Inf(set(set(A))),aa(set(B),set(set(set(A))),image2(B,set(set(A)),aTP_Lamp_os(fun(B,set(A)),fun(B,set(set(A))),B3)),A4)) ).

% Pow_INT_eq
tff(fact_3307_relpow__E2,axiom,
    ! [A: $tType,X: A,Z2: A,N: nat,R: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z2)),compow(set(product_prod(A,A)),N,R))
     => ( ( ( N = zero_zero(nat) )
         => ( X != Z2 ) )
       => ~ ! [Y2: A,M3: nat] :
              ( ( N = aa(nat,nat,suc,M3) )
             => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y2)),R)
               => ~ aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),Z2)),compow(set(product_prod(A,A)),M3,R)) ) ) ) ) ).

% relpow_E2
tff(fact_3308_relpow__E,axiom,
    ! [A: $tType,X: A,Z2: A,N: nat,R: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z2)),compow(set(product_prod(A,A)),N,R))
     => ( ( ( N = zero_zero(nat) )
         => ( X != Z2 ) )
       => ~ ! [Y2: A,M3: nat] :
              ( ( N = aa(nat,nat,suc,M3) )
             => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y2)),compow(set(product_prod(A,A)),M3,R))
               => ~ aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),Z2)),R) ) ) ) ) ).

% relpow_E
tff(fact_3309_relpow__empty,axiom,
    ! [A: $tType,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
     => ( 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_3310_Un__Pow__subset,axiom,
    ! [A: $tType,A4: set(A),B3: set(A)] : aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),aa(set(set(A)),set(set(A)),aa(set(set(A)),fun(set(set(A)),set(set(A))),sup_sup(set(set(A))),pow2(A,A4)),pow2(A,B3))),pow2(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3))) ).

% Un_Pow_subset
tff(fact_3311_UN__Pow__subset,axiom,
    ! [A: $tType,B: $tType,B3: fun(B,set(A)),A4: set(B)] : aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),aa(set(set(set(A))),set(set(A)),complete_Sup_Sup(set(set(A))),aa(set(B),set(set(set(A))),image2(B,set(set(A)),aTP_Lamp_os(fun(B,set(A)),fun(B,set(set(A))),B3)),A4))),pow2(A,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B3),A4)))) ).

% UN_Pow_subset
tff(fact_3312_Pow__insert,axiom,
    ! [A: $tType,A3: A,A4: set(A)] : pow2(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),A4)) = aa(set(set(A)),set(set(A)),aa(set(set(A)),fun(set(set(A)),set(set(A))),sup_sup(set(set(A))),pow2(A,A4)),aa(set(set(A)),set(set(A)),image2(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3)),pow2(A,A4))) ).

% Pow_insert
tff(fact_3313_relpow__fun__conv,axiom,
    ! [A: $tType,A3: A,B2: A,N: nat,R: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),compow(set(product_prod(A,A)),N,R))
    <=> ? [F6: fun(nat,A)] :
          ( ( aa(nat,A,F6,zero_zero(nat)) = A3 )
          & ( aa(nat,A,F6,N) = B2 )
          & ! [I3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),N)
             => aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(nat,A,F6,I3)),aa(nat,A,F6,aa(nat,nat,suc,I3)))),R) ) ) ) ).

% relpow_fun_conv
tff(fact_3314_Nats__def,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ( semiring_1_Nats(A) = aa(set(nat),set(A),image2(nat,A,semiring_1_of_nat(A)),top_top(set(nat))) ) ) ).

% Nats_def
tff(fact_3315_binomial__def,axiom,
    ! [N: nat,K: nat] : aa(nat,nat,binomial(N),K) = aa(set(set(nat)),nat,finite_card(set(nat)),aa(fun(set(nat),$o),set(set(nat)),collect(set(nat)),aa(nat,fun(set(nat),$o),aTP_Lamp_ot(nat,fun(nat,fun(set(nat),$o)),N),K))) ).

% binomial_def
tff(fact_3316_relpow__finite__bounded,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),K: nat] :
      ( aa(set(product_prod(A,A)),$o,finite_finite2(product_prod(A,A)),R)
     => aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),compow(set(product_prod(A,A)),K,R)),aa(set(set(product_prod(A,A))),set(product_prod(A,A)),complete_Sup_Sup(set(product_prod(A,A))),aa(set(nat),set(set(product_prod(A,A))),image2(nat,set(product_prod(A,A)),aTP_Lamp_oq(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),R)),aa(fun(nat,$o),set(nat),collect(nat),aTP_Lamp_ou(set(product_prod(A,A)),fun(nat,$o),R))))) ) ).

% relpow_finite_bounded
tff(fact_3317_finite__psubset__def,axiom,
    ! [A: $tType] : finite_psubset(A) = aa(fun(product_prod(set(A),set(A)),$o),set(product_prod(set(A),set(A))),collect(product_prod(set(A),set(A))),aa(fun(set(A),fun(set(A),$o)),fun(product_prod(set(A),set(A)),$o),product_case_prod(set(A),set(A),$o),aTP_Lamp_ov(set(A),fun(set(A),$o)))) ).

% finite_psubset_def
tff(fact_3318_ntrancl__def,axiom,
    ! [A: $tType,N: nat,R: set(product_prod(A,A))] : transitive_ntrancl(A,N,R) = aa(set(set(product_prod(A,A))),set(product_prod(A,A)),complete_Sup_Sup(set(product_prod(A,A))),aa(set(nat),set(set(product_prod(A,A))),image2(nat,set(product_prod(A,A)),aTP_Lamp_oq(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),R)),aa(fun(nat,$o),set(nat),collect(nat),aTP_Lamp_ow(nat,fun(nat,$o),N)))) ).

% ntrancl_def
tff(fact_3319_trancl__finite__eq__relpow,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,finite_finite2(product_prod(A,A)),R)
     => ( transitive_trancl(A,R) = aa(set(set(product_prod(A,A))),set(product_prod(A,A)),complete_Sup_Sup(set(product_prod(A,A))),aa(set(nat),set(set(product_prod(A,A))),image2(nat,set(product_prod(A,A)),aTP_Lamp_oq(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),R)),aa(fun(nat,$o),set(nat),collect(nat),aTP_Lamp_or(set(product_prod(A,A)),fun(nat,$o),R)))) ) ) ).

% trancl_finite_eq_relpow
tff(fact_3320_Fpow__Pow__finite,axiom,
    ! [A: $tType,A4: set(A)] : finite_Fpow(A,A4) = aa(set(set(A)),set(set(A)),aa(set(set(A)),fun(set(set(A)),set(set(A))),inf_inf(set(set(A))),pow2(A,A4)),aa(fun(set(A),$o),set(set(A)),collect(set(A)),finite_finite2(A))) ).

% Fpow_Pow_finite
tff(fact_3321_comp__fun__commute__on_Ofold__set__union__disj,axiom,
    ! [B: $tType,A: $tType,S: set(A),F: fun(A,fun(B,B)),A4: set(A),B3: set(A),Z2: B] :
      ( finite4664212375090638736ute_on(A,B,S,F)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),S)
       => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),S)
         => ( aa(set(A),$o,finite_finite2(A),A4)
           => ( aa(set(A),$o,finite_finite2(A),B3)
             => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3) = bot_bot(set(A)) )
               => ( finite_fold(A,B,F,Z2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3)) = finite_fold(A,B,F,finite_fold(A,B,F,Z2,A4),B3) ) ) ) ) ) ) ) ).

% comp_fun_commute_on.fold_set_union_disj
tff(fact_3322_rtrancl__finite__eq__relpow,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,finite_finite2(product_prod(A,A)),R)
     => ( transitive_rtrancl(A,R) = aa(set(set(product_prod(A,A))),set(product_prod(A,A)),complete_Sup_Sup(set(product_prod(A,A))),aa(set(nat),set(set(product_prod(A,A))),image2(nat,set(product_prod(A,A)),aTP_Lamp_oq(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),R)),aa(fun(nat,$o),set(nat),collect(nat),aTP_Lamp_ou(set(product_prod(A,A)),fun(nat,$o),R)))) ) ) ).

% rtrancl_finite_eq_relpow
tff(fact_3323_trancl__empty,axiom,
    ! [A: $tType] : transitive_trancl(A,bot_bot(set(product_prod(A,A)))) = bot_bot(set(product_prod(A,A))) ).

% trancl_empty
tff(fact_3324_trancl__single,axiom,
    ! [A: $tType,A3: A,B2: A] : transitive_trancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(product_prod(A,A),fun(set(product_prod(A,A)),set(product_prod(A,A))),insert2(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),bot_bot(set(product_prod(A,A))))) = aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(product_prod(A,A),fun(set(product_prod(A,A)),set(product_prod(A,A))),insert2(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),bot_bot(set(product_prod(A,A)))) ).

% trancl_single
tff(fact_3325_tranclD,axiom,
    ! [A: $tType,X: A,Y: A,R: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),transitive_trancl(A,R))
     => ? [Z3: A] :
          ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z3)),R)
          & aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z3),Y)),transitive_rtrancl(A,R)) ) ) ).

% tranclD
tff(fact_3326_rtranclD,axiom,
    ! [A: $tType,A3: A,B2: A,R: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_rtrancl(A,R))
     => ( ( A3 = B2 )
        | ( ( A3 != B2 )
          & aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_trancl(A,R)) ) ) ) ).

% rtranclD
tff(fact_3327_tranclD2,axiom,
    ! [A: $tType,X: A,Y: A,R: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),transitive_trancl(A,R))
     => ? [Z3: A] :
          ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z3)),transitive_rtrancl(A,R))
          & aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z3),Y)),R) ) ) ).

% tranclD2
tff(fact_3328_trancl__into__rtrancl,axiom,
    ! [A: $tType,A3: A,B2: A,R2: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_trancl(A,R2))
     => aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_rtrancl(A,R2)) ) ).

% trancl_into_rtrancl
tff(fact_3329_rtrancl__eq__or__trancl,axiom,
    ! [A: $tType,X: A,Y: A,R: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),transitive_rtrancl(A,R))
    <=> ( ( X = Y )
        | ( ( X != Y )
          & aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),transitive_trancl(A,R)) ) ) ) ).

% rtrancl_eq_or_trancl
tff(fact_3330_rtrancl__into__trancl1,axiom,
    ! [A: $tType,A3: A,B2: A,R2: set(product_prod(A,A)),C2: A] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_rtrancl(A,R2))
     => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),C2)),R2)
       => aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),C2)),transitive_trancl(A,R2)) ) ) ).

% rtrancl_into_trancl1
tff(fact_3331_rtrancl__into__trancl2,axiom,
    ! [A: $tType,A3: A,B2: A,R2: set(product_prod(A,A)),C2: A] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R2)
     => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),C2)),transitive_rtrancl(A,R2))
       => aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),C2)),transitive_trancl(A,R2)) ) ) ).

% rtrancl_into_trancl2
tff(fact_3332_rtrancl__trancl__trancl,axiom,
    ! [A: $tType,X: A,Y: A,R2: set(product_prod(A,A)),Z2: A] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),transitive_rtrancl(A,R2))
     => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z2)),transitive_trancl(A,R2))
       => aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z2)),transitive_trancl(A,R2)) ) ) ).

% rtrancl_trancl_trancl
tff(fact_3333_trancl__rtrancl__trancl,axiom,
    ! [A: $tType,A3: A,B2: A,R2: set(product_prod(A,A)),C2: A] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_trancl(A,R2))
     => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),C2)),transitive_rtrancl(A,R2))
       => aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),C2)),transitive_trancl(A,R2)) ) ) ).

% trancl_rtrancl_trancl
tff(fact_3334_trancl__induct2,axiom,
    ! [A: $tType,B: $tType,Ax: A,Ay: B,Bx: A,By: B,R2: set(product_prod(product_prod(A,B),product_prod(A,B))),P: fun(A,fun(B,$o))] :
      ( aa(set(product_prod(product_prod(A,B),product_prod(A,B))),$o,member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Bx),By))),transitive_trancl(product_prod(A,B),R2))
     => ( ! [A6: A,B5: B] :
            ( aa(set(product_prod(product_prod(A,B),product_prod(A,B))),$o,member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A6),B5))),R2)
           => aa(B,$o,aa(A,fun(B,$o),P,A6),B5) )
       => ( ! [A6: A,B5: B,Aa2: A,Ba: B] :
              ( aa(set(product_prod(product_prod(A,B),product_prod(A,B))),$o,member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A6),B5))),transitive_trancl(product_prod(A,B),R2))
             => ( aa(set(product_prod(product_prod(A,B),product_prod(A,B))),$o,member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A6),B5)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa2),Ba))),R2)
               => ( aa(B,$o,aa(A,fun(B,$o),P,A6),B5)
                 => aa(B,$o,aa(A,fun(B,$o),P,Aa2),Ba) ) ) )
         => aa(B,$o,aa(A,fun(B,$o),P,Bx),By) ) ) ) ).

% trancl_induct2
tff(fact_3335_trancl_Ocases,axiom,
    ! [A: $tType,A1: A,A22: A,R2: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A1),A22)),transitive_trancl(A,R2))
     => ( ~ aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A1),A22)),R2)
       => ~ ! [B5: A] :
              ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A1),B5)),transitive_trancl(A,R2))
             => ~ aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B5),A22)),R2) ) ) ) ).

% trancl.cases
tff(fact_3336_trancl_Osimps,axiom,
    ! [A: $tType,A1: A,A22: A,R2: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A1),A22)),transitive_trancl(A,R2))
    <=> ( ? [A10: A,B6: A] :
            ( ( A1 = A10 )
            & ( A22 = B6 )
            & aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A10),B6)),R2) )
        | ? [A10: A,B6: A,C5: A] :
            ( ( A1 = A10 )
            & ( A22 = C5 )
            & aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A10),B6)),transitive_trancl(A,R2))
            & aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B6),C5)),R2) ) ) ) ).

% trancl.simps
tff(fact_3337_trancl_Or__into__trancl,axiom,
    ! [A: $tType,A3: A,B2: A,R2: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R2)
     => aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_trancl(A,R2)) ) ).

% trancl.r_into_trancl
tff(fact_3338_tranclE,axiom,
    ! [A: $tType,A3: A,B2: A,R2: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_trancl(A,R2))
     => ( ~ aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R2)
       => ~ ! [C4: A] :
              ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),C4)),transitive_trancl(A,R2))
             => ~ aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),C4),B2)),R2) ) ) ) ).

% tranclE
tff(fact_3339_trancl__trans,axiom,
    ! [A: $tType,X: A,Y: A,R2: set(product_prod(A,A)),Z2: A] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),transitive_trancl(A,R2))
     => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z2)),transitive_trancl(A,R2))
       => aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z2)),transitive_trancl(A,R2)) ) ) ).

% trancl_trans
tff(fact_3340_trancl__induct,axiom,
    ! [A: $tType,A3: A,B2: A,R2: set(product_prod(A,A)),P: fun(A,$o)] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_trancl(A,R2))
     => ( ! [Y2: A] :
            ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),Y2)),R2)
           => aa(A,$o,P,Y2) )
       => ( ! [Y2: A,Z3: A] :
              ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),Y2)),transitive_trancl(A,R2))
             => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),Z3)),R2)
               => ( aa(A,$o,P,Y2)
                 => aa(A,$o,P,Z3) ) ) )
         => aa(A,$o,P,B2) ) ) ) ).

% trancl_induct
tff(fact_3341_r__r__into__trancl,axiom,
    ! [A: $tType,A3: A,B2: A,R: set(product_prod(A,A)),C2: A] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R)
     => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),C2)),R)
       => aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),C2)),transitive_trancl(A,R)) ) ) ).

% r_r_into_trancl
tff(fact_3342_converse__tranclE,axiom,
    ! [A: $tType,X: A,Z2: A,R2: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z2)),transitive_trancl(A,R2))
     => ( ~ aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z2)),R2)
       => ~ ! [Y2: A] :
              ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y2)),R2)
             => ~ aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),Z2)),transitive_trancl(A,R2)) ) ) ) ).

% converse_tranclE
tff(fact_3343_irrefl__trancl__rD,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),X: A,Y: A] :
      ( ! [X2: A] : ~ aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),X2)),transitive_trancl(A,R2))
     => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R2)
       => ( X != Y ) ) ) ).

% irrefl_trancl_rD
tff(fact_3344_Transitive__Closure_Otrancl__into__trancl,axiom,
    ! [A: $tType,A3: A,B2: A,R2: set(product_prod(A,A)),C2: A] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_trancl(A,R2))
     => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),C2)),R2)
       => aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),C2)),transitive_trancl(A,R2)) ) ) ).

% Transitive_Closure.trancl_into_trancl
tff(fact_3345_trancl__into__trancl2,axiom,
    ! [A: $tType,A3: A,B2: A,R2: set(product_prod(A,A)),C2: A] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R2)
     => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),C2)),transitive_trancl(A,R2))
       => aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),C2)),transitive_trancl(A,R2)) ) ) ).

% trancl_into_trancl2
tff(fact_3346_trancl__trans__induct,axiom,
    ! [A: $tType,X: A,Y: A,R2: set(product_prod(A,A)),P: fun(A,fun(A,$o))] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),transitive_trancl(A,R2))
     => ( ! [X2: A,Y2: A] :
            ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y2)),R2)
           => aa(A,$o,aa(A,fun(A,$o),P,X2),Y2) )
       => ( ! [X2: A,Y2: A,Z3: A] :
              ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y2)),transitive_trancl(A,R2))
             => ( aa(A,$o,aa(A,fun(A,$o),P,X2),Y2)
               => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),Z3)),transitive_trancl(A,R2))
                 => ( aa(A,$o,aa(A,fun(A,$o),P,Y2),Z3)
                   => aa(A,$o,aa(A,fun(A,$o),P,X2),Z3) ) ) ) )
         => aa(A,$o,aa(A,fun(A,$o),P,X),Y) ) ) ) ).

% trancl_trans_induct
tff(fact_3347_converse__trancl__induct,axiom,
    ! [A: $tType,A3: A,B2: A,R2: set(product_prod(A,A)),P: fun(A,$o)] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_trancl(A,R2))
     => ( ! [Y2: A] :
            ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),B2)),R2)
           => aa(A,$o,P,Y2) )
       => ( ! [Y2: A,Z3: A] :
              ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),Z3)),R2)
             => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z3),B2)),transitive_trancl(A,R2))
               => ( aa(A,$o,P,Z3)
                 => aa(A,$o,P,Y2) ) ) )
         => aa(A,$o,P,A3) ) ) ) ).

% converse_trancl_induct
tff(fact_3348_rtrancl__induct2,axiom,
    ! [A: $tType,B: $tType,Ax: A,Ay: B,Bx: A,By: B,R2: set(product_prod(product_prod(A,B),product_prod(A,B))),P: fun(A,fun(B,$o))] :
      ( aa(set(product_prod(product_prod(A,B),product_prod(A,B))),$o,member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Bx),By))),transitive_rtrancl(product_prod(A,B),R2))
     => ( aa(B,$o,aa(A,fun(B,$o),P,Ax),Ay)
       => ( ! [A6: A,B5: B,Aa2: A,Ba: B] :
              ( aa(set(product_prod(product_prod(A,B),product_prod(A,B))),$o,member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A6),B5))),transitive_rtrancl(product_prod(A,B),R2))
             => ( aa(set(product_prod(product_prod(A,B),product_prod(A,B))),$o,member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A6),B5)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa2),Ba))),R2)
               => ( aa(B,$o,aa(A,fun(B,$o),P,A6),B5)
                 => aa(B,$o,aa(A,fun(B,$o),P,Aa2),Ba) ) ) )
         => aa(B,$o,aa(A,fun(B,$o),P,Bx),By) ) ) ) ).

% rtrancl_induct2
tff(fact_3349_converse__rtranclE2,axiom,
    ! [B: $tType,A: $tType,Xa: A,Xb: B,Za2: A,Zb: B,R2: set(product_prod(product_prod(A,B),product_prod(A,B)))] :
      ( aa(set(product_prod(product_prod(A,B),product_prod(A,B))),$o,member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa),Xb)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Za2),Zb))),transitive_rtrancl(product_prod(A,B),R2))
     => ( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa),Xb) != aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Za2),Zb) )
       => ~ ! [A6: A,B5: B] :
              ( aa(set(product_prod(product_prod(A,B),product_prod(A,B))),$o,member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa),Xb)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A6),B5))),R2)
             => ~ aa(set(product_prod(product_prod(A,B),product_prod(A,B))),$o,member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A6),B5)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Za2),Zb))),transitive_rtrancl(product_prod(A,B),R2)) ) ) ) ).

% converse_rtranclE2
tff(fact_3350_converse__rtrancl__induct2,axiom,
    ! [A: $tType,B: $tType,Ax: A,Ay: B,Bx: A,By: B,R2: set(product_prod(product_prod(A,B),product_prod(A,B))),P: fun(A,fun(B,$o))] :
      ( aa(set(product_prod(product_prod(A,B),product_prod(A,B))),$o,member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Bx),By))),transitive_rtrancl(product_prod(A,B),R2))
     => ( aa(B,$o,aa(A,fun(B,$o),P,Bx),By)
       => ( ! [A6: A,B5: B,Aa2: A,Ba: B] :
              ( aa(set(product_prod(product_prod(A,B),product_prod(A,B))),$o,member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A6),B5)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa2),Ba))),R2)
             => ( aa(set(product_prod(product_prod(A,B),product_prod(A,B))),$o,member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa2),Ba)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Bx),By))),transitive_rtrancl(product_prod(A,B),R2))
               => ( aa(B,$o,aa(A,fun(B,$o),P,Aa2),Ba)
                 => aa(B,$o,aa(A,fun(B,$o),P,A6),B5) ) ) )
         => aa(B,$o,aa(A,fun(B,$o),P,Ax),Ay) ) ) ) ).

% converse_rtrancl_induct2
tff(fact_3351_converse__rtranclE_H,axiom,
    ! [A: $tType,U: A,V: A,R: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),U),V)),transitive_rtrancl(A,R))
     => ( ( U != V )
       => ~ ! [Vh: A] :
              ( ( U != Vh )
             => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),U),Vh)),R)
               => ~ aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Vh),V)),transitive_rtrancl(A,R)) ) ) ) ) ).

% converse_rtranclE'
tff(fact_3352_rtrancl_Ocases,axiom,
    ! [A: $tType,A1: A,A22: A,R2: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A1),A22)),transitive_rtrancl(A,R2))
     => ( ( A22 != A1 )
       => ~ ! [B5: A] :
              ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A1),B5)),transitive_rtrancl(A,R2))
             => ~ aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B5),A22)),R2) ) ) ) ).

% rtrancl.cases
tff(fact_3353_rtrancl_Osimps,axiom,
    ! [A: $tType,A1: A,A22: A,R2: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A1),A22)),transitive_rtrancl(A,R2))
    <=> ( ? [A10: A] :
            ( ( A1 = A10 )
            & ( A22 = A10 ) )
        | ? [A10: A,B6: A,C5: A] :
            ( ( A1 = A10 )
            & ( A22 = C5 )
            & aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A10),B6)),transitive_rtrancl(A,R2))
            & aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B6),C5)),R2) ) ) ) ).

% rtrancl.simps
tff(fact_3354_rtrancl_Ortrancl__refl,axiom,
    ! [A: $tType,A3: A,R2: set(product_prod(A,A))] : aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),A3)),transitive_rtrancl(A,R2)) ).

% rtrancl.rtrancl_refl
tff(fact_3355_rtrancl_Ortrancl__into__rtrancl,axiom,
    ! [A: $tType,A3: A,B2: A,R2: set(product_prod(A,A)),C2: A] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_rtrancl(A,R2))
     => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),C2)),R2)
       => aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),C2)),transitive_rtrancl(A,R2)) ) ) ).

% rtrancl.rtrancl_into_rtrancl
tff(fact_3356_rtranclE,axiom,
    ! [A: $tType,A3: A,B2: A,R2: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_rtrancl(A,R2))
     => ( ( A3 != B2 )
       => ~ ! [Y2: A] :
              ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),Y2)),transitive_rtrancl(A,R2))
             => ~ aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),B2)),R2) ) ) ) ).

% rtranclE
tff(fact_3357_rtrancl__trans,axiom,
    ! [A: $tType,X: A,Y: A,R2: set(product_prod(A,A)),Z2: A] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),transitive_rtrancl(A,R2))
     => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z2)),transitive_rtrancl(A,R2))
       => aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z2)),transitive_rtrancl(A,R2)) ) ) ).

% rtrancl_trans
tff(fact_3358_rtrancl__induct,axiom,
    ! [A: $tType,A3: A,B2: A,R2: set(product_prod(A,A)),P: fun(A,$o)] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_rtrancl(A,R2))
     => ( aa(A,$o,P,A3)
       => ( ! [Y2: A,Z3: A] :
              ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),Y2)),transitive_rtrancl(A,R2))
             => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),Z3)),R2)
               => ( aa(A,$o,P,Y2)
                 => aa(A,$o,P,Z3) ) ) )
         => aa(A,$o,P,B2) ) ) ) ).

% rtrancl_induct
tff(fact_3359_converse__rtranclE,axiom,
    ! [A: $tType,X: A,Z2: A,R2: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z2)),transitive_rtrancl(A,R2))
     => ( ( X != Z2 )
       => ~ ! [Y2: A] :
              ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y2)),R2)
             => ~ aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),Z2)),transitive_rtrancl(A,R2)) ) ) ) ).

% converse_rtranclE
tff(fact_3360_converse__rtrancl__induct,axiom,
    ! [A: $tType,A3: A,B2: A,R2: set(product_prod(A,A)),P: fun(A,$o)] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_rtrancl(A,R2))
     => ( aa(A,$o,P,B2)
       => ( ! [Y2: A,Z3: A] :
              ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),Z3)),R2)
             => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z3),B2)),transitive_rtrancl(A,R2))
               => ( aa(A,$o,P,Z3)
                 => aa(A,$o,P,Y2) ) ) )
         => aa(A,$o,P,A3) ) ) ) ).

% converse_rtrancl_induct
tff(fact_3361_converse__rtrancl__into__rtrancl,axiom,
    ! [A: $tType,A3: A,B2: A,R2: set(product_prod(A,A)),C2: A] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R2)
     => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),C2)),transitive_rtrancl(A,R2))
       => aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),C2)),transitive_rtrancl(A,R2)) ) ) ).

% converse_rtrancl_into_rtrancl
tff(fact_3362_trancl__union__outside,axiom,
    ! [A: $tType,V: A,W2: A,E3: set(product_prod(A,A)),U2: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),V),W2)),transitive_trancl(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))),E3),U2)))
     => ( ~ aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),V),W2)),transitive_trancl(A,E3))
       => ? [X2: A,Y2: A] :
            ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),V),X2)),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))),E3),U2)))
            & aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y2)),U2)
            & aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),W2)),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))),E3),U2))) ) ) ) ).

% trancl_union_outside
tff(fact_3363_trancl__over__edgeE,axiom,
    ! [A: $tType,U: A,W2: A,V1: A,V22: A,E3: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),U),W2)),transitive_trancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(product_prod(A,A),fun(set(product_prod(A,A)),set(product_prod(A,A))),insert2(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),V1),V22)),E3)))
     => ( ~ aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),U),W2)),transitive_trancl(A,E3))
       => ~ ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),U),V1)),transitive_rtrancl(A,E3))
           => ~ aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),V22),W2)),transitive_rtrancl(A,E3)) ) ) ) ).

% trancl_over_edgeE
tff(fact_3364_in__rtrancl__UnI,axiom,
    ! [A: $tType,X: product_prod(A,A),R: set(product_prod(A,A)),S: set(product_prod(A,A))] :
      ( ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),X),transitive_rtrancl(A,R))
        | aa(set(product_prod(A,A)),$o,member(product_prod(A,A),X),transitive_rtrancl(A,S)) )
     => aa(set(product_prod(A,A)),$o,member(product_prod(A,A),X),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))),R),S))) ) ).

% in_rtrancl_UnI
tff(fact_3365_rtrancl__Un__rtrancl,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),S: set(product_prod(A,A))] : 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))),transitive_rtrancl(A,R)),transitive_rtrancl(A,S))) = 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))),R),S)) ).

% rtrancl_Un_rtrancl
tff(fact_3366_rtrancl__Un__separator__converseE,axiom,
    ! [A: $tType,A3: A,B2: A,P: set(product_prod(A,A)),Q: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),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))),P),Q)))
     => ( ! [X2: A,Y2: A] :
            ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),B2)),transitive_rtrancl(A,P))
           => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),X2)),Q)
             => ( Y2 = X2 ) ) )
       => aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_rtrancl(A,P)) ) ) ).

% rtrancl_Un_separator_converseE
tff(fact_3367_rtrancl__Un__separatorE,axiom,
    ! [A: $tType,A3: A,B2: A,P: set(product_prod(A,A)),Q: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),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))),P),Q)))
     => ( ! [X2: A,Y2: A] :
            ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),X2)),transitive_rtrancl(A,P))
           => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y2)),Q)
             => ( X2 = Y2 ) ) )
       => aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_rtrancl(A,P)) ) ) ).

% rtrancl_Un_separatorE
tff(fact_3368_rel__restrict__trancl__mem,axiom,
    ! [A: $tType,A3: A,B2: A,A4: set(product_prod(A,A)),R: set(A)] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_trancl(A,rel_restrict(A,A4,R)))
     => aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),rel_restrict(A,transitive_trancl(A,A4),R)) ) ).

% rel_restrict_trancl_mem
tff(fact_3369_rel__restrict__trancl__notR_I1_J,axiom,
    ! [A: $tType,V: A,W2: A,E3: set(product_prod(A,A)),R: set(A)] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),V),W2)),transitive_trancl(A,rel_restrict(A,E3,R)))
     => ~ aa(set(A),$o,member(A,V),R) ) ).

% rel_restrict_trancl_notR(1)
tff(fact_3370_rel__restrict__trancl__notR_I2_J,axiom,
    ! [A: $tType,V: A,W2: A,E3: set(product_prod(A,A)),R: set(A)] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),V),W2)),transitive_trancl(A,rel_restrict(A,E3,R)))
     => ~ aa(set(A),$o,member(A,W2),R) ) ).

% rel_restrict_trancl_notR(2)
tff(fact_3371_trancl__insert,axiom,
    ! [A: $tType,Y: A,X: A,R2: set(product_prod(A,A))] : transitive_trancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(product_prod(A,A),fun(set(product_prod(A,A)),set(product_prod(A,A))),insert2(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),X)),R2)) = aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),transitive_trancl(A,R2)),aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),aa(set(product_prod(A,A)),fun(A,fun(A,$o)),aa(A,fun(set(product_prod(A,A)),fun(A,fun(A,$o))),aTP_Lamp_ox(A,fun(A,fun(set(product_prod(A,A)),fun(A,fun(A,$o)))),Y),X),R2)))) ).

% trancl_insert
tff(fact_3372_empty__in__Fpow,axiom,
    ! [A: $tType,A4: set(A)] : aa(set(set(A)),$o,member(set(A),bot_bot(set(A))),finite_Fpow(A,A4)) ).

% empty_in_Fpow
tff(fact_3373_rtrancl__Un__subset,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),S: set(product_prod(A,A))] : aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),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)),transitive_rtrancl(A,S))),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))),R),S))) ).

% rtrancl_Un_subset
tff(fact_3374_rtrancl__mapI,axiom,
    ! [B: $tType,A: $tType,A3: A,B2: A,E3: set(product_prod(A,A)),F: fun(A,B)] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_rtrancl(A,E3))
     => aa(set(product_prod(B,B)),$o,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,F,A3)),aa(A,B,F,B2))),transitive_rtrancl(B,aa(set(product_prod(A,A)),set(product_prod(B,B)),image2(product_prod(A,A),product_prod(B,B),pairself(A,B,F)),E3))) ) ).

% rtrancl_mapI
tff(fact_3375_Fpow__not__empty,axiom,
    ! [A: $tType,A4: set(A)] : finite_Fpow(A,A4) != bot_bot(set(set(A))) ).

% Fpow_not_empty
tff(fact_3376_finite__trancl__ntranl,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,finite_finite2(product_prod(A,A)),R)
     => ( transitive_trancl(A,R) = transitive_ntrancl(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(product_prod(A,A)),nat,finite_card(product_prod(A,A)),R)),one_one(nat)),R) ) ) ).

% finite_trancl_ntranl
tff(fact_3377_trancl__insert2,axiom,
    ! [A: $tType,A3: A,B2: A,R2: set(product_prod(A,A))] : transitive_trancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(product_prod(A,A),fun(set(product_prod(A,A)),set(product_prod(A,A))),insert2(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R2)) = aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),transitive_trancl(A,R2)),aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),aa(set(product_prod(A,A)),fun(A,fun(A,$o)),aa(A,fun(set(product_prod(A,A)),fun(A,fun(A,$o))),aTP_Lamp_oy(A,fun(A,fun(set(product_prod(A,A)),fun(A,fun(A,$o)))),A3),B2),R2)))) ).

% trancl_insert2
tff(fact_3378_Fpow__def,axiom,
    ! [A: $tType,A4: set(A)] : finite_Fpow(A,A4) = aa(fun(set(A),$o),set(set(A)),collect(set(A)),aTP_Lamp_oz(set(A),fun(set(A),$o),A4)) ).

% Fpow_def
tff(fact_3379_rtrancl__insert,axiom,
    ! [A: $tType,A3: A,B2: A,R2: set(product_prod(A,A))] : transitive_rtrancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(product_prod(A,A),fun(set(product_prod(A,A)),set(product_prod(A,A))),insert2(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R2)) = aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),transitive_rtrancl(A,R2)),aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),aa(set(product_prod(A,A)),fun(A,fun(A,$o)),aa(A,fun(set(product_prod(A,A)),fun(A,fun(A,$o))),aTP_Lamp_ox(A,fun(A,fun(set(product_prod(A,A)),fun(A,fun(A,$o)))),A3),B2),R2)))) ).

% rtrancl_insert
tff(fact_3380_comp__fun__commute__on_Ocomp__comp__fun__commute__on,axiom,
    ! [B: $tType,A: $tType,C: $tType,S: set(A),F: fun(A,fun(B,B)),G: fun(C,A),R: set(C)] :
      ( finite4664212375090638736ute_on(A,B,S,F)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(C),set(A),image2(C,A,G),top_top(set(C)))),S)
       => finite4664212375090638736ute_on(C,B,R,aa(fun(C,A),fun(C,fun(B,B)),comp(A,fun(B,B),C,F),G)) ) ) ).

% comp_fun_commute_on.comp_comp_fun_commute_on
tff(fact_3381_rtrancl__is__UN__relpow,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] : transitive_rtrancl(A,R) = aa(set(set(product_prod(A,A))),set(product_prod(A,A)),complete_Sup_Sup(set(product_prod(A,A))),aa(set(nat),set(set(product_prod(A,A))),image2(nat,set(product_prod(A,A)),aTP_Lamp_oq(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),R)),top_top(set(nat)))) ).

% rtrancl_is_UN_relpow
tff(fact_3382_rtrancl__imp__UN__relpow,axiom,
    ! [A: $tType,P3: product_prod(A,A),R: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),P3),transitive_rtrancl(A,R))
     => aa(set(product_prod(A,A)),$o,member(product_prod(A,A),P3),aa(set(set(product_prod(A,A))),set(product_prod(A,A)),complete_Sup_Sup(set(product_prod(A,A))),aa(set(nat),set(set(product_prod(A,A))),image2(nat,set(product_prod(A,A)),aTP_Lamp_oq(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),R)),top_top(set(nat))))) ) ).

% rtrancl_imp_UN_relpow
tff(fact_3383_comp__fun__commute__on_Ofold__insert__remove,axiom,
    ! [B: $tType,A: $tType,S: set(A),F: fun(A,fun(B,B)),X: A,A4: set(A),Z2: B] :
      ( finite4664212375090638736ute_on(A,B,S,F)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A4)),S)
       => ( aa(set(A),$o,finite_finite2(A),A4)
         => ( finite_fold(A,B,F,Z2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A4)) = aa(B,B,aa(A,fun(B,B),F,X),finite_fold(A,B,F,Z2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))))) ) ) ) ) ).

% comp_fun_commute_on.fold_insert_remove
tff(fact_3384_comp__fun__commute__on_Ofold__rec,axiom,
    ! [B: $tType,A: $tType,S: set(A),F: fun(A,fun(B,B)),A4: set(A),X: A,Z2: B] :
      ( finite4664212375090638736ute_on(A,B,S,F)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),S)
       => ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),$o,member(A,X),A4)
           => ( finite_fold(A,B,F,Z2,A4) = aa(B,B,aa(A,fun(B,B),F,X),finite_fold(A,B,F,Z2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))))) ) ) ) ) ) ).

% comp_fun_commute_on.fold_rec
tff(fact_3385_bit__rec,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,N: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A3),N)
        <=> $ite(N = zero_zero(nat),~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A3),aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)))) ) ) ).

% bit_rec
tff(fact_3386_UNION__fun__upd,axiom,
    ! [B: $tType,A: $tType,A4: fun(B,set(A)),I: B,B3: set(A),J3: set(B)] :
      aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),fun_upd(B,set(A),A4,I,B3)),J3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),A4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),J3),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),I),bot_bot(set(B))))))),
        $ite(aa(set(B),$o,member(B,I),J3),B3,bot_bot(set(A)))) ).

% UNION_fun_upd
tff(fact_3387_bit__sum__mult__2__cases,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A,B2: A,N: nat] :
          ( ! [J2: nat] : ~ aa(nat,$o,bit_se5641148757651400278ts_bit(A,A3),aa(nat,nat,suc,J2))
         => ( aa(nat,$o,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),aa(num,num,bit0,one2))),B2))),N)
          <=> $ite(N = zero_zero(nat),~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A3),aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),B2)),N)) ) ) ) ).

% bit_sum_mult_2_cases
tff(fact_3388_times__int_Oabs__eq,axiom,
    ! [Xa: 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,Xa)),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_pb(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))))),Xa),X)) ).

% times_int.abs_eq
tff(fact_3389_rat__minus__code,axiom,
    ! [P3: rat,Q3: rat] : quotient_of(aa(rat,rat,aa(rat,fun(rat,rat),minus_minus(rat),P3),Q3)) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aTP_Lamp_pd(rat,fun(int,fun(int,product_prod(int,int))),Q3)),quotient_of(P3)) ).

% rat_minus_code
tff(fact_3390_bit__0__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ( bit_se5641148757651400278ts_bit(A,zero_zero(A)) = bot_bot(fun(nat,$o)) ) ) ).

% bit_0_eq
tff(fact_3391_rat__one__code,axiom,
    quotient_of(one_one(rat)) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),one_one(int)) ).

% rat_one_code
tff(fact_3392_bit__minus__numeral__Bit0__Suc__iff,axiom,
    ! [W2: num,N: nat] :
      ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,W2)))),aa(nat,nat,suc,N))
    <=> aa(nat,$o,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_3393_bit__minus__numeral__Bit1__Suc__iff,axiom,
    ! [W2: num,N: nat] :
      ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,W2)))),aa(nat,nat,suc,N))
    <=> ~ aa(nat,$o,bit_se5641148757651400278ts_bit(int,aa(num,int,numeral_numeral(int),W2)),N) ) ).

% bit_minus_numeral_Bit1_Suc_iff
tff(fact_3394_rat__zero__code,axiom,
    quotient_of(zero_zero(rat)) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int)) ).

% rat_zero_code
tff(fact_3395_quotient__of__number_I3_J,axiom,
    ! [K: num] : quotient_of(aa(num,rat,numeral_numeral(rat),K)) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(num,int,numeral_numeral(int),K)),one_one(int)) ).

% quotient_of_number(3)
tff(fact_3396_quotient__of__number_I4_J,axiom,
    quotient_of(aa(rat,rat,uminus_uminus(rat),one_one(rat))) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,uminus_uminus(int),one_one(int))),one_one(int)) ).

% quotient_of_number(4)
tff(fact_3397_bit__minus__numeral__int_I1_J,axiom,
    ! [W2: num,N: num] :
      ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,W2)))),aa(num,nat,numeral_numeral(nat),N))
    <=> aa(nat,$o,bit_se5641148757651400278ts_bit(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),W2))),pred_numeral(N)) ) ).

% bit_minus_numeral_int(1)
tff(fact_3398_bit__minus__numeral__int_I2_J,axiom,
    ! [W2: num,N: num] :
      ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,W2)))),aa(num,nat,numeral_numeral(nat),N))
    <=> ~ aa(nat,$o,bit_se5641148757651400278ts_bit(int,aa(num,int,numeral_numeral(int),W2)),pred_numeral(N)) ) ).

% bit_minus_numeral_int(2)
tff(fact_3399_quotient__of__number_I5_J,axiom,
    ! [K: num] : quotient_of(aa(rat,rat,uminus_uminus(rat),aa(num,rat,numeral_numeral(rat),K))) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K))),one_one(int)) ).

% quotient_of_number(5)
tff(fact_3400_diff__rat__def,axiom,
    ! [Q3: rat,R2: rat] : aa(rat,rat,aa(rat,fun(rat,rat),minus_minus(rat),Q3),R2) = aa(rat,rat,aa(rat,fun(rat,rat),plus_plus(rat),Q3),aa(rat,rat,uminus_uminus(rat),R2)) ).

% diff_rat_def
tff(fact_3401_quotient__of__div,axiom,
    ! [R2: rat,N: int,D3: int] :
      ( ( quotient_of(R2) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),N),D3) )
     => ( R2 = aa(rat,rat,aa(rat,fun(rat,rat),divide_divide(rat),aa(int,rat,ring_1_of_int(rat),N)),aa(int,rat,ring_1_of_int(rat),D3)) ) ) ).

% quotient_of_div
tff(fact_3402_eq__Abs__Integ,axiom,
    ! [Z2: int] :
      ~ ! [X2: nat,Y2: 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),X2),Y2)) ).

% eq_Abs_Integ
tff(fact_3403_finite__update__induct,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),C2: B,P: fun(fun(A,B),$o)] :
      ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(B,fun(A,$o),aTP_Lamp_pe(fun(A,B),fun(B,fun(A,$o)),F),C2)))
     => ( aa(fun(A,B),$o,P,aTP_Lamp_ax(B,fun(A,B),C2))
       => ( ! [A6: A,B5: B,F2: fun(A,B)] :
              ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_pf(B,fun(fun(A,B),fun(A,$o)),C2),F2)))
             => ( ( aa(A,B,F2,A6) = C2 )
               => ( ( B5 != C2 )
                 => ( aa(fun(A,B),$o,P,F2)
                   => aa(fun(A,B),$o,P,fun_upd(A,B,F2,A6,B5)) ) ) ) )
         => aa(fun(A,B),$o,P,F) ) ) ) ).

% finite_update_induct
tff(fact_3404_bit__1__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [N: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,one_one(A)),N)
        <=> ( N = zero_zero(nat) ) ) ) ).

% bit_1_iff
tff(fact_3405_not__bit__1__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat] : ~ aa(nat,$o,bit_se5641148757651400278ts_bit(A,one_one(A)),aa(nat,nat,suc,N)) ) ).

% not_bit_1_Suc
tff(fact_3406_bit__numeral__simps_I1_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: num] : ~ aa(nat,$o,bit_se5641148757651400278ts_bit(A,one_one(A)),aa(num,nat,numeral_numeral(nat),N)) ) ).

% bit_numeral_simps(1)
tff(fact_3407_bit__iff__and__drop__bit__eq__1,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A,N: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A3),N)
        <=> ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),bit_se4197421643247451524op_bit(A,N,A3)),one_one(A)) = one_one(A) ) ) ) ).

% bit_iff_and_drop_bit_eq_1
tff(fact_3408_bit__not__int__iff_H,axiom,
    ! [K: int,N: nat] :
      ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,uminus_uminus(int),K)),one_one(int))),N)
    <=> ~ aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),N) ) ).

% bit_not_int_iff'
tff(fact_3409_quotient__of__denom__pos,axiom,
    ! [R2: rat,P3: int,Q3: int] :
      ( ( quotient_of(R2) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P3),Q3) )
     => aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Q3) ) ).

% quotient_of_denom_pos
tff(fact_3410_bit__numeral__rec_I1_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [W2: num,N: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,W2))),N)
        <=> aa(nat,$o,aa(fun(nat,$o),fun(nat,$o),aa($o,fun(fun(nat,$o),fun(nat,$o)),case_nat($o),$false),bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),W2))),N) ) ) ).

% bit_numeral_rec(1)
tff(fact_3411_bit__numeral__rec_I2_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [W2: num,N: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,W2))),N)
        <=> aa(nat,$o,aa(fun(nat,$o),fun(nat,$o),aa($o,fun(fun(nat,$o),fun(nat,$o)),case_nat($o),$true),bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),W2))),N) ) ) ).

% bit_numeral_rec(2)
tff(fact_3412_bit__iff__and__push__bit__not__eq__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A,N: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A3),N)
        <=> ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),bit_se4730199178511100633sh_bit(A,N,one_one(A))) != zero_zero(A) ) ) ) ).

% bit_iff_and_push_bit_not_eq_0
tff(fact_3413_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_3414_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_3415_bit__minus__int__iff,axiom,
    ! [K: int,N: nat] :
      ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,aa(int,int,uminus_uminus(int),K)),N)
    <=> aa(nat,$o,bit_se5641148757651400278ts_bit(int,aa(int,int,bit_ri4277139882892585799ns_not(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),K),one_one(int)))),N) ) ).

% bit_minus_int_iff
tff(fact_3416_rat__floor__code,axiom,
    ! [P3: rat] : archim6421214686448440834_floor(rat,P3) = aa(product_prod(int,int),int,aa(fun(int,fun(int,int)),fun(product_prod(int,int),int),product_case_prod(int,int,int),divide_divide(int)),quotient_of(P3)) ).

% rat_floor_code
tff(fact_3417_rat__uminus__code,axiom,
    ! [P3: rat] : quotient_of(aa(rat,rat,uminus_uminus(rat),P3)) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aTP_Lamp_pg(int,fun(int,product_prod(int,int)))),quotient_of(P3)) ).

% rat_uminus_code
tff(fact_3418_rat__abs__code,axiom,
    ! [P3: rat] : quotient_of(aa(rat,rat,abs_abs(rat),P3)) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aTP_Lamp_ph(int,fun(int,product_prod(int,int)))),quotient_of(P3)) ).

% rat_abs_code
tff(fact_3419_rat__less__eq__code,axiom,
    ! [P3: rat,Q3: rat] :
      ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),P3),Q3)
    <=> aa(product_prod(int,int),$o,aa(fun(int,fun(int,$o)),fun(product_prod(int,int),$o),product_case_prod(int,int,$o),aTP_Lamp_pj(rat,fun(int,fun(int,$o)),Q3)),quotient_of(P3)) ) ).

% rat_less_eq_code
tff(fact_3420_rat__less__code,axiom,
    ! [P3: rat,Q3: rat] :
      ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),P3),Q3)
    <=> aa(product_prod(int,int),$o,aa(fun(int,fun(int,$o)),fun(product_prod(int,int),$o),product_case_prod(int,int,$o),aTP_Lamp_pl(rat,fun(int,fun(int,$o)),Q3)),quotient_of(P3)) ) ).

% rat_less_code
tff(fact_3421_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_pm(nat,fun(nat,product_prod(nat,nat)))),X)) ).

% uminus_int.abs_eq
tff(fact_3422_int__bit__bound,axiom,
    ! [K: int] :
      ~ ! [N3: nat] :
          ( ! [M5: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N3),M5)
             => ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),M5)
              <=> aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),N3) ) )
         => ~ ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N3)
             => ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N3),one_one(nat)))
              <=> ~ aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),N3) ) ) ) ).

% int_bit_bound
tff(fact_3423_fun__upd__image,axiom,
    ! [A: $tType,B: $tType,F: fun(B,A),X: B,Y: A,A4: set(B)] :
      aa(set(B),set(A),image2(B,A,fun_upd(B,A,F,X,Y)),A4) = $ite(aa(set(B),$o,member(B,X),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y),aa(set(B),set(A),image2(B,A,F),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A4),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),X),bot_bot(set(B)))))),aa(set(B),set(A),image2(B,A,F),A4)) ).

% fun_upd_image
tff(fact_3424_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_3425_of__int_Oabs__eq,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [X: product_prod(nat,nat)] : aa(int,A,ring_1_of_int(A),aa(product_prod(nat,nat),int,abs_Integ,X)) = 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_pn(nat,fun(nat,A))),X) ) ).

% of_int.abs_eq
tff(fact_3426_less__int_Oabs__eq,axiom,
    ! [Xa: product_prod(nat,nat),X: product_prod(nat,nat)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(product_prod(nat,nat),int,abs_Integ,Xa)),aa(product_prod(nat,nat),int,abs_Integ,X))
    <=> aa(product_prod(nat,nat),$o,aa(product_prod(nat,nat),fun(product_prod(nat,nat),$o),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),$o))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)),product_case_prod(nat,nat,fun(product_prod(nat,nat),$o)),aTP_Lamp_pp(nat,fun(nat,fun(product_prod(nat,nat),$o)))),Xa),X) ) ).

% less_int.abs_eq
tff(fact_3427_less__eq__int_Oabs__eq,axiom,
    ! [Xa: product_prod(nat,nat),X: product_prod(nat,nat)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(product_prod(nat,nat),int,abs_Integ,Xa)),aa(product_prod(nat,nat),int,abs_Integ,X))
    <=> aa(product_prod(nat,nat),$o,aa(product_prod(nat,nat),fun(product_prod(nat,nat),$o),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),$o))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)),product_case_prod(nat,nat,fun(product_prod(nat,nat),$o)),aTP_Lamp_pr(nat,fun(nat,fun(product_prod(nat,nat),$o)))),Xa),X) ) ).

% less_eq_int.abs_eq
tff(fact_3428_plus__int_Oabs__eq,axiom,
    ! [Xa: 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,Xa)),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_pt(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))))),Xa),X)) ).

% plus_int.abs_eq
tff(fact_3429_minus__int_Oabs__eq,axiom,
    ! [Xa: 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,Xa)),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_pv(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))))),Xa),X)) ).

% minus_int.abs_eq
tff(fact_3430_rat__divide__code,axiom,
    ! [P3: rat,Q3: rat] : quotient_of(aa(rat,rat,aa(rat,fun(rat,rat),divide_divide(rat),P3),Q3)) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aTP_Lamp_px(rat,fun(int,fun(int,product_prod(int,int))),Q3)),quotient_of(P3)) ).

% rat_divide_code
tff(fact_3431_rat__times__code,axiom,
    ! [P3: rat,Q3: rat] : quotient_of(aa(rat,rat,aa(rat,fun(rat,rat),times_times(rat),P3),Q3)) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aTP_Lamp_pz(rat,fun(int,fun(int,product_prod(int,int))),Q3)),quotient_of(P3)) ).

% rat_times_code
tff(fact_3432_take__bit__sum,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: nat,A3: 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_qa(A,fun(nat,A),A3)),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)) ) ).

% take_bit_sum
tff(fact_3433_even__bit__succ__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,N: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A3)
         => ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),A3)),N)
          <=> ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A3),N)
              | ( N = zero_zero(nat) ) ) ) ) ) ).

% even_bit_succ_iff
tff(fact_3434_odd__bit__iff__bit__pred,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,N: nat] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A3)
         => ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A3),N)
          <=> ( aa(nat,$o,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_3435_signed__take__bit__def,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat,A3: A] : bit_ri4674362597316999326ke_bit(A,N,A3) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),bit_se2584673776208193580ke_bit(A,N,A3)),aa(A,A,aa(A,fun(A,A),times_times(A),aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,bit_se5641148757651400278ts_bit(A,A3),N))),aa(A,A,bit_ri4277139882892585799ns_not(A),bit_se2239418461657761734s_mask(A,N)))) ) ).

% signed_take_bit_def
tff(fact_3436_signed__take__bit__code,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat,A3: A] :
          bit_ri4674362597316999326ke_bit(A,N,A3) = $let(
            l: A,
            l:= bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,N),A3),
            $ite(aa(nat,$o,bit_se5641148757651400278ts_bit(A,l),N),aa(A,A,aa(A,fun(A,A),plus_plus(A),l),bit_se4730199178511100633sh_bit(A,aa(nat,nat,suc,N),aa(A,A,uminus_uminus(A),one_one(A)))),l) ) ) ).

% signed_take_bit_code
tff(fact_3437_rat__plus__code,axiom,
    ! [P3: rat,Q3: rat] : quotient_of(aa(rat,rat,aa(rat,fun(rat,rat),plus_plus(rat),P3),Q3)) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aTP_Lamp_qc(rat,fun(int,fun(int,product_prod(int,int))),Q3)),quotient_of(P3)) ).

% rat_plus_code
tff(fact_3438_quotient__of__int,axiom,
    ! [A3: int] : quotient_of(of_int(A3)) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),A3),one_one(int)) ).

% quotient_of_int
tff(fact_3439_same__fst__trancl,axiom,
    ! [B: $tType,A: $tType,P: fun(A,$o),R: fun(A,set(product_prod(B,B)))] : transitive_trancl(product_prod(A,B),same_fst(A,B,P,R)) = same_fst(A,B,P,aTP_Lamp_qd(fun(A,set(product_prod(B,B))),fun(A,set(product_prod(B,B))),R)) ).

% same_fst_trancl
tff(fact_3440_char__of__def,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: A] : aa(A,char,unique5772411509450598832har_of(A),N) = aa($o,char,char2(~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N),aa(nat,$o,bit_se5641148757651400278ts_bit(A,N),one_one(nat)),aa(nat,$o,bit_se5641148757651400278ts_bit(A,N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,$o,bit_se5641148757651400278ts_bit(A,N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),aa(nat,$o,bit_se5641148757651400278ts_bit(A,N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit0,one2)))),aa(nat,$o,bit_se5641148757651400278ts_bit(A,N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit0,one2)))),aa(nat,$o,bit_se5641148757651400278ts_bit(A,N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit1,one2))))),aa(nat,$o,bit_se5641148757651400278ts_bit(A,N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,one2))))) ) ).

% char_of_def
tff(fact_3441_rat__inverse__code,axiom,
    ! [P3: rat] : quotient_of(aa(rat,rat,inverse_inverse(rat),P3)) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aTP_Lamp_qe(int,fun(int,product_prod(int,int)))),quotient_of(P3)) ).

% rat_inverse_code
tff(fact_3442_in__lex__prod,axiom,
    ! [A: $tType,B: $tType,A3: A,B2: B,A5: A,B4: B,R2: set(product_prod(A,A)),S2: set(product_prod(B,B))] :
      ( aa(set(product_prod(product_prod(A,B),product_prod(A,B))),$o,member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B4))),lex_prod(A,B,R2,S2))
    <=> ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),A5)),R2)
        | ( ( A3 = A5 )
          & aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),B2),B4)),S2) ) ) ) ).

% in_lex_prod
tff(fact_3443_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_3444_inverse__eq__1__iff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [X: A] :
          ( ( aa(A,A,inverse_inverse(A),X) = one_one(A) )
        <=> ( X = one_one(A) ) ) ) ).

% inverse_eq_1_iff
tff(fact_3445_inverse__1,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ( aa(A,A,inverse_inverse(A),one_one(A)) = one_one(A) ) ) ).

% inverse_1
tff(fact_3446_inverse__minus__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: 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)) ) ).

% inverse_minus_eq
tff(fact_3447_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_3448_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_3449_inverse__eq__divide__numeral,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [W2: num] : aa(A,A,inverse_inverse(A),aa(num,A,numeral_numeral(A),W2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),W2)) ) ).

% inverse_eq_divide_numeral
tff(fact_3450_inverse__eq__divide__neg__numeral,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [W2: num] : aa(A,A,inverse_inverse(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))) ) ).

% inverse_eq_divide_neg_numeral
tff(fact_3451_abs__rat__def,axiom,
    ! [A3: rat] :
      aa(rat,rat,abs_abs(rat),A3) = $ite(aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),A3),zero_zero(rat)),aa(rat,rat,uminus_uminus(rat),A3),A3) ).

% abs_rat_def
tff(fact_3452_sgn__rat__def,axiom,
    ! [A3: rat] :
      aa(rat,rat,sgn_sgn(rat),A3) = $ite(
        A3 = zero_zero(rat),
        zero_zero(rat),
        $ite(aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),A3),one_one(rat),aa(rat,rat,uminus_uminus(rat),one_one(rat))) ) ).

% sgn_rat_def
tff(fact_3453_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_3454_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_3455_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_3456_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_3457_field__class_Ofield__divide__inverse,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,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_3458_divide__inverse,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,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_3459_divide__inverse__commute,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,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_3460_inverse__eq__divide,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A] : aa(A,A,inverse_inverse(A),A3) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A3) ) ).

% inverse_eq_divide
tff(fact_3461_power__mult__inverse__distrib,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [X: A,M: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),M)),aa(A,A,inverse_inverse(A),X)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),X)),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),M)) ) ).

% power_mult_inverse_distrib
tff(fact_3462_power__mult__power__inverse__commute,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [X: A,M: 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),M)),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),M)) ) ).

% power_mult_power_inverse_commute
tff(fact_3463_mult__inverse__of__nat__commute,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xa: nat,X: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),aa(nat,A,semiring_1_of_nat(A),Xa))),X) = aa(A,A,aa(A,fun(A,A),times_times(A),X),aa(A,A,inverse_inverse(A),aa(nat,A,semiring_1_of_nat(A),Xa))) ) ).

% mult_inverse_of_nat_commute
tff(fact_3464_mult__inverse__of__int__commute,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xa: int,X: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),aa(int,A,ring_1_of_int(A),Xa))),X) = aa(A,A,aa(A,fun(A,A),times_times(A),X),aa(A,A,inverse_inverse(A),aa(int,A,ring_1_of_int(A),Xa))) ) ).

% mult_inverse_of_int_commute
tff(fact_3465_inverse__le__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,inverse_inverse(A),X)),one_one(A))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),zero_zero(A))
            | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),X) ) ) ) ).

% inverse_le_1_iff
tff(fact_3466_one__less__inverse,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),one_one(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(A,A,inverse_inverse(A),A3)) ) ) ) ).

% one_less_inverse
tff(fact_3467_one__less__inverse__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(A,A,inverse_inverse(A),X))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),X)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),one_one(A)) ) ) ) ).

% one_less_inverse_iff
tff(fact_3468_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_3469_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_3470_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_3471_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_3472_nonzero__inverse__eq__divide,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A] :
          ( ( A3 != zero_zero(A) )
         => ( aa(A,A,inverse_inverse(A),A3) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A3) ) ) ) ).

% nonzero_inverse_eq_divide
tff(fact_3473_inverse__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2) ) ) ) ) ).

% inverse_le_iff
tff(fact_3474_inverse__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2) ) ) ) ) ).

% inverse_less_iff
tff(fact_3475_one__le__inverse__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(A,A,inverse_inverse(A),X))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),X)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),one_one(A)) ) ) ) ).

% one_le_inverse_iff
tff(fact_3476_inverse__less__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),X)),one_one(A))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),zero_zero(A))
            | aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),X) ) ) ) ).

% inverse_less_1_iff
tff(fact_3477_one__le__inverse,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),one_one(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(A,A,inverse_inverse(A),A3)) ) ) ) ).

% one_le_inverse
tff(fact_3478_power__diff__conv__inverse,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [X: A,M: nat,N: nat] :
          ( ( X != zero_zero(A) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),N)
           => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,inverse_inverse(A),X)),M)) ) ) ) ) ).

% power_diff_conv_inverse
tff(fact_3479_lex__prod__def,axiom,
    ! [A: $tType,B: $tType,Ra2: set(product_prod(A,A)),Rb: set(product_prod(B,B))] : lex_prod(A,B,Ra2,Rb) = aa(fun(product_prod(product_prod(A,B),product_prod(A,B)),$o),set(product_prod(product_prod(A,B),product_prod(A,B))),collect(product_prod(product_prod(A,B),product_prod(A,B))),aa(fun(product_prod(A,B),fun(product_prod(A,B),$o)),fun(product_prod(product_prod(A,B),product_prod(A,B)),$o),product_case_prod(product_prod(A,B),product_prod(A,B),$o),aa(fun(A,fun(B,fun(product_prod(A,B),$o))),fun(product_prod(A,B),fun(product_prod(A,B),$o)),product_case_prod(A,B,fun(product_prod(A,B),$o)),aa(set(product_prod(B,B)),fun(A,fun(B,fun(product_prod(A,B),$o))),aTP_Lamp_qg(set(product_prod(A,A)),fun(set(product_prod(B,B)),fun(A,fun(B,fun(product_prod(A,B),$o)))),Ra2),Rb)))) ).

% lex_prod_def
tff(fact_3480_same__fstI,axiom,
    ! [B: $tType,A: $tType,P: fun(A,$o),X: A,Y6: B,Y: B,R: fun(A,set(product_prod(B,B)))] :
      ( aa(A,$o,P,X)
     => ( aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Y6),Y)),aa(A,set(product_prod(B,B)),R,X))
       => aa(set(product_prod(product_prod(A,B),product_prod(A,B))),$o,member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y6)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y))),same_fst(A,B,P,R)) ) ) ).

% same_fstI
tff(fact_3481_same__fst__def,axiom,
    ! [B: $tType,A: $tType,P: fun(A,$o),R: fun(A,set(product_prod(B,B)))] : same_fst(A,B,P,R) = aa(fun(product_prod(product_prod(A,B),product_prod(A,B)),$o),set(product_prod(product_prod(A,B),product_prod(A,B))),collect(product_prod(product_prod(A,B),product_prod(A,B))),aa(fun(product_prod(A,B),fun(product_prod(A,B),$o)),fun(product_prod(product_prod(A,B),product_prod(A,B)),$o),product_case_prod(product_prod(A,B),product_prod(A,B),$o),aa(fun(A,fun(B,fun(product_prod(A,B),$o))),fun(product_prod(A,B),fun(product_prod(A,B),$o)),product_case_prod(A,B,fun(product_prod(A,B),$o)),aa(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(product_prod(A,B),$o))),aTP_Lamp_qi(fun(A,$o),fun(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(product_prod(A,B),$o)))),P),R)))) ).

% same_fst_def
tff(fact_3482_char__of__integer__code,axiom,
    ! [K: code_integer] : char_of_integer(K) = aa(product_prod(code_integer,$o),char,aa(fun(code_integer,fun($o,char)),fun(product_prod(code_integer,$o),char),product_case_prod(code_integer,$o,char),aTP_Lamp_qq(code_integer,fun($o,char))),code_bit_cut_integer(K)) ).

% char_of_integer_code
tff(fact_3483_Frct__code__post_I5_J,axiom,
    ! [K: num] : frct(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),aa(num,int,numeral_numeral(int),K))) = aa(rat,rat,aa(rat,fun(rat,rat),divide_divide(rat),one_one(rat)),aa(num,rat,numeral_numeral(rat),K)) ).

% Frct_code_post(5)
tff(fact_3484_floor__rat__def,axiom,
    ! [X: rat] : archim6421214686448440834_floor(rat,X) = the(int,aTP_Lamp_qr(rat,fun(int,$o),X)) ).

% floor_rat_def
tff(fact_3485_Frct__code__post_I9_J,axiom,
    ! [Q3: product_prod(int,int)] : aa(rat,rat,uminus_uminus(rat),aa(rat,rat,uminus_uminus(rat),frct(Q3))) = frct(Q3) ).

% Frct_code_post(9)
tff(fact_3486_the__elem__def,axiom,
    ! [A: $tType,X5: set(A)] : the_elem(A,X5) = the(A,aTP_Lamp_qs(set(A),fun(A,$o),X5)) ).

% the_elem_def
tff(fact_3487_Frct__code__post_I1_J,axiom,
    ! [A3: int] : frct(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),A3)) = zero_zero(rat) ).

% Frct_code_post(1)
tff(fact_3488_Frct__code__post_I2_J,axiom,
    ! [A3: int] : frct(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),A3),zero_zero(int))) = zero_zero(rat) ).

% Frct_code_post(2)
tff(fact_3489_Frct__code__post_I3_J,axiom,
    frct(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),one_one(int))) = one_one(rat) ).

% Frct_code_post(3)
tff(fact_3490_Frct__code__post_I7_J,axiom,
    ! [A3: int,B2: int] : frct(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,uminus_uminus(int),A3)),B2)) = aa(rat,rat,uminus_uminus(rat),frct(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),A3),B2))) ).

% Frct_code_post(7)
tff(fact_3491_Frct__code__post_I8_J,axiom,
    ! [A3: int,B2: int] : frct(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),A3),aa(int,int,uminus_uminus(int),B2))) = aa(rat,rat,uminus_uminus(rat),frct(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),A3),B2))) ).

% Frct_code_post(8)
tff(fact_3492_Frct__code__post_I4_J,axiom,
    ! [K: num] : frct(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(num,int,numeral_numeral(int),K)),one_one(int))) = aa(num,rat,numeral_numeral(rat),K) ).

% Frct_code_post(4)
tff(fact_3493_Frct__code__post_I6_J,axiom,
    ! [K: num,L: num] : frct(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(num,int,numeral_numeral(int),K)),aa(num,int,numeral_numeral(int),L))) = aa(rat,rat,aa(rat,fun(rat,rat),divide_divide(rat),aa(num,rat,numeral_numeral(rat),K)),aa(num,rat,numeral_numeral(rat),L)) ).

% Frct_code_post(6)
tff(fact_3494_old_Orec__prod__def,axiom,
    ! [C: $tType,B: $tType,A: $tType,X3: fun(A,fun(B,C)),Xa3: product_prod(A,B)] : product_rec_prod(A,B,C,X3,Xa3) = the(C,product_rec_set_prod(A,B,C,X3,Xa3)) ).

% old.rec_prod_def
tff(fact_3495_old_Orec__nat__def,axiom,
    ! [A: $tType,X3: A,Xa3: fun(nat,fun(A,A)),Xb2: nat] : aa(nat,A,rec_nat(A,X3,Xa3),Xb2) = the(A,rec_set_nat(A,X3,Xa3,Xb2)) ).

% old.rec_nat_def
tff(fact_3496_the__equality,axiom,
    ! [A: $tType,P: fun(A,$o),A3: A] :
      ( aa(A,$o,P,A3)
     => ( ! [X2: A] :
            ( aa(A,$o,P,X2)
           => ( X2 = A3 ) )
       => ( the(A,P) = A3 ) ) ) ).

% the_equality
tff(fact_3497_the__eq__trivial,axiom,
    ! [A: $tType,A3: A] : the(A,aTP_Lamp_cf(A,fun(A,$o),A3)) = A3 ).

% the_eq_trivial
tff(fact_3498_the__sym__eq__trivial,axiom,
    ! [A: $tType,X: A] : the(A,aa(A,fun(A,$o),fequal(A),X)) = X ).

% the_sym_eq_trivial
tff(fact_3499_The__split__eq,axiom,
    ! [A: $tType,B: $tType,X: A,Y: B] : the(product_prod(A,B),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aa(B,fun(A,fun(B,$o)),aTP_Lamp_qt(A,fun(B,fun(A,fun(B,$o))),X),Y))) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y) ).

% The_split_eq
tff(fact_3500_the1__equality,axiom,
    ! [A: $tType,P: fun(A,$o),A3: A] :
      ( ? [X3: A] :
          ( aa(A,$o,P,X3)
          & ! [Y2: A] :
              ( aa(A,$o,P,Y2)
             => ( Y2 = X3 ) ) )
     => ( aa(A,$o,P,A3)
       => ( the(A,P) = A3 ) ) ) ).

% the1_equality
tff(fact_3501_the1I2,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] :
      ( ? [X3: A] :
          ( aa(A,$o,P,X3)
          & ! [Y2: A] :
              ( aa(A,$o,P,Y2)
             => ( Y2 = X3 ) ) )
     => ( ! [X2: A] :
            ( aa(A,$o,P,X2)
           => aa(A,$o,Q,X2) )
       => aa(A,$o,Q,the(A,P)) ) ) ).

% the1I2
tff(fact_3502_If__def,axiom,
    ! [A: $tType,P: $o,X: A,Y: A] :
      $ite((P),X,Y) = the(A,aa(A,fun(A,$o),aa(A,fun(A,fun(A,$o)),aTP_Lamp_qu($o,fun(A,fun(A,fun(A,$o))),(P)),X),Y)) ).

% If_def
tff(fact_3503_theI2,axiom,
    ! [A: $tType,P: fun(A,$o),A3: A,Q: fun(A,$o)] :
      ( aa(A,$o,P,A3)
     => ( ! [X2: A] :
            ( aa(A,$o,P,X2)
           => ( X2 = A3 ) )
       => ( ! [X2: A] :
              ( aa(A,$o,P,X2)
             => aa(A,$o,Q,X2) )
         => aa(A,$o,Q,the(A,P)) ) ) ) ).

% theI2
tff(fact_3504_theI_H,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( ? [X3: A] :
          ( aa(A,$o,P,X3)
          & ! [Y2: A] :
              ( aa(A,$o,P,Y2)
             => ( Y2 = X3 ) ) )
     => aa(A,$o,P,the(A,P)) ) ).

% theI'
tff(fact_3505_theI,axiom,
    ! [A: $tType,P: fun(A,$o),A3: A] :
      ( aa(A,$o,P,A3)
     => ( ! [X2: A] :
            ( aa(A,$o,P,X2)
           => ( X2 = A3 ) )
       => aa(A,$o,P,the(A,P)) ) ) ).

% theI
tff(fact_3506_less__eq__int_Orep__eq,axiom,
    ! [X: int,Xa: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),X),Xa)
    <=> aa(product_prod(nat,nat),$o,aa(product_prod(nat,nat),fun(product_prod(nat,nat),$o),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),$o))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)),product_case_prod(nat,nat,fun(product_prod(nat,nat),$o)),aTP_Lamp_pr(nat,fun(nat,fun(product_prod(nat,nat),$o)))),aa(int,product_prod(nat,nat),rep_Integ,X)),aa(int,product_prod(nat,nat),rep_Integ,Xa)) ) ).

% less_eq_int.rep_eq
tff(fact_3507_less__int_Orep__eq,axiom,
    ! [X: int,Xa: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),X),Xa)
    <=> aa(product_prod(nat,nat),$o,aa(product_prod(nat,nat),fun(product_prod(nat,nat),$o),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),$o))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)),product_case_prod(nat,nat,fun(product_prod(nat,nat),$o)),aTP_Lamp_pp(nat,fun(nat,fun(product_prod(nat,nat),$o)))),aa(int,product_prod(nat,nat),rep_Integ,X)),aa(int,product_prod(nat,nat),rep_Integ,Xa)) ) ).

% less_int.rep_eq
tff(fact_3508_arg__min__if__finite_I2_J,axiom,
    ! [B: $tType,A: $tType] :
      ( order(B)
     => ! [S: set(A),F: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),S)
         => ( ( S != bot_bot(set(A)) )
           => ~ ? [X3: A] :
                  ( aa(set(A),$o,member(A,X3),S)
                  & aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F,X3)),aa(A,B,F,lattic7623131987881927897min_on(A,B,F,S))) ) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_3509_image__split__eq__Sigma,axiom,
    ! [B: $tType,A: $tType,C: $tType,F: fun(C,A),G: fun(C,B),A4: set(C)] : aa(set(C),set(product_prod(A,B)),image2(C,product_prod(A,B),aa(fun(C,B),fun(C,product_prod(A,B)),aTP_Lamp_qv(fun(C,A),fun(fun(C,B),fun(C,product_prod(A,B))),F),G)),A4) = product_Sigma(A,B,aa(set(C),set(A),image2(C,A,F),A4),aa(set(C),fun(A,set(B)),aa(fun(C,B),fun(set(C),fun(A,set(B))),aTP_Lamp_qw(fun(C,A),fun(fun(C,B),fun(set(C),fun(A,set(B)))),F),G),A4)) ).

% image_split_eq_Sigma
tff(fact_3510_arg__min__least,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [S: set(A),Y: A,F: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),S)
         => ( ( S != bot_bot(set(A)) )
           => ( aa(set(A),$o,member(A,Y),S)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,lattic7623131987881927897min_on(A,B,F,S))),aa(A,B,F,Y)) ) ) ) ) ).

% arg_min_least
tff(fact_3511_SigmaI,axiom,
    ! [B: $tType,A: $tType,A3: A,A4: set(A),B2: B,B3: fun(A,set(B))] :
      ( aa(set(A),$o,member(A,A3),A4)
     => ( aa(set(B),$o,member(B,B2),aa(A,set(B),B3,A3))
       => aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),product_Sigma(A,B,A4,B3)) ) ) ).

% SigmaI
tff(fact_3512_mem__Sigma__iff,axiom,
    ! [B: $tType,A: $tType,A3: A,B2: B,A4: set(A),B3: fun(A,set(B))] :
      ( aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),product_Sigma(A,B,A4,B3))
    <=> ( aa(set(A),$o,member(A,A3),A4)
        & aa(set(B),$o,member(B,B2),aa(A,set(B),B3,A3)) ) ) ).

% mem_Sigma_iff
tff(fact_3513_Collect__case__prod,axiom,
    ! [A: $tType,B: $tType,P: fun(A,$o),Q: fun(B,$o)] : aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aa(fun(B,$o),fun(A,fun(B,$o)),aTP_Lamp_qx(fun(A,$o),fun(fun(B,$o),fun(A,fun(B,$o))),P),Q))) = product_Sigma(A,B,aa(fun(A,$o),set(A),collect(A),P),aTP_Lamp_qy(fun(B,$o),fun(A,set(B)),Q)) ).

% Collect_case_prod
tff(fact_3514_Sigma__empty1,axiom,
    ! [B: $tType,A: $tType,B3: fun(A,set(B))] : product_Sigma(A,B,bot_bot(set(A)),B3) = bot_bot(set(product_prod(A,B))) ).

% Sigma_empty1
tff(fact_3515_Times__empty,axiom,
    ! [A: $tType,B: $tType,A4: set(A),B3: set(B)] :
      ( ( product_Sigma(A,B,A4,aTP_Lamp_qz(set(B),fun(A,set(B)),B3)) = bot_bot(set(product_prod(A,B))) )
    <=> ( ( A4 = bot_bot(set(A)) )
        | ( B3 = bot_bot(set(B)) ) ) ) ).

% Times_empty
tff(fact_3516_Sigma__empty2,axiom,
    ! [B: $tType,A: $tType,A4: set(A)] : product_Sigma(A,B,A4,aTP_Lamp_ra(A,set(B))) = bot_bot(set(product_prod(A,B))) ).

% Sigma_empty2
tff(fact_3517_Sigma__UNIV__cancel,axiom,
    ! [B: $tType,A: $tType,A4: set(A),X5: 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))),minus_minus(set(product_prod(A,B))),product_Sigma(A,B,A4,aTP_Lamp_qz(set(B),fun(A,set(B)),X5))),product_Sigma(A,B,A4,aTP_Lamp_rb(A,set(B)))) = bot_bot(set(product_prod(A,B))) ).

% Sigma_UNIV_cancel
tff(fact_3518_Compl__Times__UNIV1,axiom,
    ! [A: $tType,B: $tType,A4: 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_qz(set(B),fun(A,set(B)),A4))) = product_Sigma(A,B,top_top(set(A)),aTP_Lamp_rc(set(B),fun(A,set(B)),A4)) ).

% Compl_Times_UNIV1
tff(fact_3519_Compl__Times__UNIV2,axiom,
    ! [B: $tType,A: $tType,A4: set(A)] : aa(set(product_prod(A,B)),set(product_prod(A,B)),uminus_uminus(set(product_prod(A,B))),product_Sigma(A,B,A4,aTP_Lamp_rb(A,set(B)))) = product_Sigma(A,B,aa(set(A),set(A),uminus_uminus(set(A)),A4),aTP_Lamp_rb(A,set(B))) ).

% Compl_Times_UNIV2
tff(fact_3520_finite__SigmaI,axiom,
    ! [B: $tType,A: $tType,A4: set(A),B3: fun(A,set(B))] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( ! [A6: A] :
            ( aa(set(A),$o,member(A,A6),A4)
           => aa(set(B),$o,finite_finite2(B),aa(A,set(B),B3,A6)) )
       => aa(set(product_prod(A,B)),$o,finite_finite2(product_prod(A,B)),product_Sigma(A,B,A4,B3)) ) ) ).

% finite_SigmaI
tff(fact_3521_UNIV__Times__UNIV,axiom,
    ! [B: $tType,A: $tType] : product_Sigma(A,B,top_top(set(A)),aTP_Lamp_rb(A,set(B))) = top_top(set(product_prod(A,B))) ).

% UNIV_Times_UNIV
tff(fact_3522_pairself__image__cart,axiom,
    ! [A: $tType,B: $tType,F: fun(B,A),A4: set(B),B3: set(B)] : aa(set(product_prod(B,B)),set(product_prod(A,A)),image2(product_prod(B,B),product_prod(A,A),pairself(B,A,F)),product_Sigma(B,B,A4,aTP_Lamp_rd(set(B),fun(B,set(B)),B3))) = product_Sigma(A,A,aa(set(B),set(A),image2(B,A,F),A4),aa(set(B),fun(A,set(A)),aTP_Lamp_re(fun(B,A),fun(set(B),fun(A,set(A))),F),B3)) ).

% pairself_image_cart
tff(fact_3523_insert__Times__insert,axiom,
    ! [A: $tType,B: $tType,A3: A,A4: set(A),B2: B,B3: set(B)] : product_Sigma(A,B,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),A4),aa(set(B),fun(A,set(B)),aTP_Lamp_rf(B,fun(set(B),fun(A,set(B))),B2),B3)) = aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(product_prod(A,B),fun(set(product_prod(A,B)),set(product_prod(A,B))),insert2(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),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,A4,aa(set(B),fun(A,set(B)),aTP_Lamp_rf(B,fun(set(B),fun(A,set(B))),B2),B3))),product_Sigma(A,B,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),A4),aTP_Lamp_qz(set(B),fun(A,set(B)),B3)))) ).

% insert_Times_insert
tff(fact_3524_card__SigmaI,axiom,
    ! [B: $tType,A: $tType,A4: set(A),B3: fun(A,set(B))] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( ! [X2: A] :
            ( aa(set(A),$o,member(A,X2),A4)
           => aa(set(B),$o,finite_finite2(B),aa(A,set(B),B3,X2)) )
       => ( aa(set(product_prod(A,B)),nat,finite_card(product_prod(A,B)),product_Sigma(A,B,A4,B3)) = aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aTP_Lamp_nd(fun(A,set(B)),fun(A,nat),B3)),A4) ) ) ) ).

% card_SigmaI
tff(fact_3525_Times__eq__cancel2,axiom,
    ! [A: $tType,B: $tType,X: A,C3: set(A),A4: set(B),B3: set(B)] :
      ( aa(set(A),$o,member(A,X),C3)
     => ( ( product_Sigma(B,A,A4,aTP_Lamp_ke(set(A),fun(B,set(A)),C3)) = product_Sigma(B,A,B3,aTP_Lamp_ke(set(A),fun(B,set(A)),C3)) )
      <=> ( A4 = B3 ) ) ) ).

% Times_eq_cancel2
tff(fact_3526_Sigma__cong,axiom,
    ! [B: $tType,A: $tType,A4: set(A),B3: set(A),C3: fun(A,set(B)),D4: fun(A,set(B))] :
      ( ( A4 = B3 )
     => ( ! [X2: A] :
            ( aa(set(A),$o,member(A,X2),B3)
           => ( aa(A,set(B),C3,X2) = aa(A,set(B),D4,X2) ) )
       => ( product_Sigma(A,B,A4,C3) = product_Sigma(A,B,B3,D4) ) ) ) ).

% Sigma_cong
tff(fact_3527_SigmaE,axiom,
    ! [A: $tType,B: $tType,C2: product_prod(A,B),A4: set(A),B3: fun(A,set(B))] :
      ( aa(set(product_prod(A,B)),$o,member(product_prod(A,B),C2),product_Sigma(A,B,A4,B3))
     => ~ ! [X2: A] :
            ( aa(set(A),$o,member(A,X2),A4)
           => ! [Y2: B] :
                ( aa(set(B),$o,member(B,Y2),aa(A,set(B),B3,X2))
               => ( C2 != aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Y2) ) ) ) ) ).

% SigmaE
tff(fact_3528_SigmaD1,axiom,
    ! [B: $tType,A: $tType,A3: A,B2: B,A4: set(A),B3: fun(A,set(B))] :
      ( aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),product_Sigma(A,B,A4,B3))
     => aa(set(A),$o,member(A,A3),A4) ) ).

% SigmaD1
tff(fact_3529_SigmaD2,axiom,
    ! [B: $tType,A: $tType,A3: A,B2: B,A4: set(A),B3: fun(A,set(B))] :
      ( aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),product_Sigma(A,B,A4,B3))
     => aa(set(B),$o,member(B,B2),aa(A,set(B),B3,A3)) ) ).

% SigmaD2
tff(fact_3530_SigmaE2,axiom,
    ! [B: $tType,A: $tType,A3: A,B2: B,A4: set(A),B3: fun(A,set(B))] :
      ( aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),product_Sigma(A,B,A4,B3))
     => ~ ( aa(set(A),$o,member(A,A3),A4)
         => ~ aa(set(B),$o,member(B,B2),aa(A,set(B),B3,A3)) ) ) ).

% SigmaE2
tff(fact_3531_times__eq__iff,axiom,
    ! [A: $tType,B: $tType,A4: set(A),B3: set(B),C3: set(A),D4: set(B)] :
      ( ( product_Sigma(A,B,A4,aTP_Lamp_qz(set(B),fun(A,set(B)),B3)) = product_Sigma(A,B,C3,aTP_Lamp_qz(set(B),fun(A,set(B)),D4)) )
    <=> ( ( ( A4 = C3 )
          & ( B3 = D4 ) )
        | ( ( ( A4 = bot_bot(set(A)) )
            | ( B3 = bot_bot(set(B)) ) )
          & ( ( C3 = bot_bot(set(A)) )
            | ( D4 = bot_bot(set(B)) ) ) ) ) ) ).

% times_eq_iff
tff(fact_3532_Times__Diff__distrib1,axiom,
    ! [A: $tType,B: $tType,A4: set(A),B3: set(A),C3: set(B)] : product_Sigma(A,B,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),B3),aTP_Lamp_qz(set(B),fun(A,set(B)),C3)) = 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,A4,aTP_Lamp_qz(set(B),fun(A,set(B)),C3))),product_Sigma(A,B,B3,aTP_Lamp_qz(set(B),fun(A,set(B)),C3))) ).

% Times_Diff_distrib1
tff(fact_3533_Sigma__Diff__distrib2,axiom,
    ! [B: $tType,A: $tType,I4: set(A),A4: fun(A,set(B)),B3: fun(A,set(B))] : product_Sigma(A,B,I4,aa(fun(A,set(B)),fun(A,set(B)),aTP_Lamp_rg(fun(A,set(B)),fun(fun(A,set(B)),fun(A,set(B))),A4),B3)) = 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,I4,A4)),product_Sigma(A,B,I4,B3)) ).

% Sigma_Diff_distrib2
tff(fact_3534_Collect__case__prod__Sigma,axiom,
    ! [B: $tType,A: $tType,P: fun(A,$o),Q: fun(A,fun(B,$o))] : aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aTP_Lamp_rh(fun(A,$o),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),P),Q))) = product_Sigma(A,B,aa(fun(A,$o),set(A),collect(A),P),aTP_Lamp_ri(fun(A,fun(B,$o)),fun(A,set(B)),Q)) ).

% Collect_case_prod_Sigma
tff(fact_3535_Pair__vimage__Sigma,axiom,
    ! [B: $tType,A: $tType,X: B,A4: set(B),F: fun(B,set(A))] :
      aa(set(product_prod(B,A)),set(A),aa(fun(A,product_prod(B,A)),fun(set(product_prod(B,A)),set(A)),vimage(A,product_prod(B,A)),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X)),product_Sigma(B,A,A4,F)) = $ite(aa(set(B),$o,member(B,X),A4),aa(B,set(A),F,X),bot_bot(set(A))) ).

% Pair_vimage_Sigma
tff(fact_3536_Sigma__Int__distrib1,axiom,
    ! [B: $tType,A: $tType,I4: set(A),J3: set(A),C3: 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)),I4),J3),C3) = 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,I4,C3)),product_Sigma(A,B,J3,C3)) ).

% Sigma_Int_distrib1
tff(fact_3537_Sigma__Un__distrib1,axiom,
    ! [B: $tType,A: $tType,I4: set(A),J3: set(A),C3: 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)),I4),J3),C3) = 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,I4,C3)),product_Sigma(A,B,J3,C3)) ).

% Sigma_Un_distrib1
tff(fact_3538_Times__subset__cancel2,axiom,
    ! [A: $tType,B: $tType,X: A,C3: set(A),A4: set(B),B3: set(B)] :
      ( aa(set(A),$o,member(A,X),C3)
     => ( aa(set(product_prod(B,A)),$o,aa(set(product_prod(B,A)),fun(set(product_prod(B,A)),$o),ord_less_eq(set(product_prod(B,A))),product_Sigma(B,A,A4,aTP_Lamp_ke(set(A),fun(B,set(A)),C3))),product_Sigma(B,A,B3,aTP_Lamp_ke(set(A),fun(B,set(A)),C3)))
      <=> aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A4),B3) ) ) ).

% Times_subset_cancel2
tff(fact_3539_Sigma__empty__iff,axiom,
    ! [A: $tType,B: $tType,I4: set(A),X5: fun(A,set(B))] :
      ( ( product_Sigma(A,B,I4,X5) = bot_bot(set(product_prod(A,B))) )
    <=> ! [X4: A] :
          ( aa(set(A),$o,member(A,X4),I4)
         => ( aa(A,set(B),X5,X4) = bot_bot(set(B)) ) ) ) ).

% Sigma_empty_iff
tff(fact_3540_Times__Int__Times,axiom,
    ! [A: $tType,B: $tType,A4: set(A),B3: set(B),C3: set(A),D4: 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,A4,aTP_Lamp_qz(set(B),fun(A,set(B)),B3))),product_Sigma(A,B,C3,aTP_Lamp_qz(set(B),fun(A,set(B)),D4))) = product_Sigma(A,B,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),C3),aa(set(B),fun(A,set(B)),aTP_Lamp_rj(set(B),fun(set(B),fun(A,set(B))),B3),D4)) ).

% Times_Int_Times
tff(fact_3541_Sigma__Int__distrib2,axiom,
    ! [B: $tType,A: $tType,I4: set(A),A4: fun(A,set(B)),B3: fun(A,set(B))] : product_Sigma(A,B,I4,aa(fun(A,set(B)),fun(A,set(B)),aTP_Lamp_rk(fun(A,set(B)),fun(fun(A,set(B)),fun(A,set(B))),A4),B3)) = 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,I4,A4)),product_Sigma(A,B,I4,B3)) ).

% Sigma_Int_distrib2
tff(fact_3542_Times__Int__distrib1,axiom,
    ! [A: $tType,B: $tType,A4: set(A),B3: set(A),C3: set(B)] : product_Sigma(A,B,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3),aTP_Lamp_qz(set(B),fun(A,set(B)),C3)) = 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,A4,aTP_Lamp_qz(set(B),fun(A,set(B)),C3))),product_Sigma(A,B,B3,aTP_Lamp_qz(set(B),fun(A,set(B)),C3))) ).

% Times_Int_distrib1
tff(fact_3543_finite__cartesian__product,axiom,
    ! [A: $tType,B: $tType,A4: set(A),B3: set(B)] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( aa(set(B),$o,finite_finite2(B),B3)
       => aa(set(product_prod(A,B)),$o,finite_finite2(product_prod(A,B)),product_Sigma(A,B,A4,aTP_Lamp_qz(set(B),fun(A,set(B)),B3))) ) ) ).

% finite_cartesian_product
tff(fact_3544_trancl__subset__Sigma,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A4: set(A)] :
      ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R2),product_Sigma(A,A,A4,aTP_Lamp_rl(set(A),fun(A,set(A)),A4)))
     => aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),transitive_trancl(A,R2)),product_Sigma(A,A,A4,aTP_Lamp_rl(set(A),fun(A,set(A)),A4))) ) ).

% trancl_subset_Sigma
tff(fact_3545_Sigma__Un__distrib2,axiom,
    ! [B: $tType,A: $tType,I4: set(A),A4: fun(A,set(B)),B3: fun(A,set(B))] : product_Sigma(A,B,I4,aa(fun(A,set(B)),fun(A,set(B)),aTP_Lamp_rm(fun(A,set(B)),fun(fun(A,set(B)),fun(A,set(B))),A4),B3)) = 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,I4,A4)),product_Sigma(A,B,I4,B3)) ).

% Sigma_Un_distrib2
tff(fact_3546_Times__Un__distrib1,axiom,
    ! [A: $tType,B: $tType,A4: set(A),B3: set(A),C3: set(B)] : product_Sigma(A,B,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3),aTP_Lamp_qz(set(B),fun(A,set(B)),C3)) = 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,A4,aTP_Lamp_qz(set(B),fun(A,set(B)),C3))),product_Sigma(A,B,B3,aTP_Lamp_qz(set(B),fun(A,set(B)),C3))) ).

% Times_Un_distrib1
tff(fact_3547_Sigma__Union,axiom,
    ! [B: $tType,A: $tType,X5: set(set(A)),B3: fun(A,set(B))] : product_Sigma(A,B,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),X5),B3) = aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),aa(set(set(A)),set(set(product_prod(A,B))),image2(set(A),set(product_prod(A,B)),aTP_Lamp_rn(fun(A,set(B)),fun(set(A),set(product_prod(A,B))),B3)),X5)) ).

% Sigma_Union
tff(fact_3548_Id__on__subset__Times,axiom,
    ! [A: $tType,A4: set(A)] : aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),id_on(A,A4)),product_Sigma(A,A,A4,aTP_Lamp_rl(set(A),fun(A,set(A)),A4))) ).

% Id_on_subset_Times
tff(fact_3549_rtrancl__last__touch,axiom,
    ! [A: $tType,Q3: A,Q4: A,R: set(product_prod(A,A)),S: set(A)] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q3),Q4)),transitive_rtrancl(A,R))
     => ( aa(set(A),$o,member(A,Q3),S)
       => ~ ! [Qt: A] :
              ( aa(set(A),$o,member(A,Qt),S)
             => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q3),Qt)),transitive_rtrancl(A,R))
               => ~ aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Qt),Q4)),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))),minus_minus(set(product_prod(A,A))),R),product_Sigma(A,A,top_top(set(A)),aTP_Lamp_rl(set(A),fun(A,set(A)),S))))) ) ) ) ) ).

% rtrancl_last_touch
tff(fact_3550_rtrancl__last__visit_H,axiom,
    ! [A: $tType,Q3: A,Q4: A,R: set(product_prod(A,A)),S: set(A)] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q3),Q4)),transitive_rtrancl(A,R))
     => ( ~ aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q3),Q4)),transitive_rtrancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),minus_minus(set(product_prod(A,A))),R),product_Sigma(A,A,top_top(set(A)),aTP_Lamp_rl(set(A),fun(A,set(A)),S)))))
       => ~ ! [Qt: A] :
              ( aa(set(A),$o,member(A,Qt),S)
             => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q3),Qt)),transitive_rtrancl(A,R))
               => ~ aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Qt),Q4)),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))),minus_minus(set(product_prod(A,A))),R),product_Sigma(A,A,top_top(set(A)),aTP_Lamp_rl(set(A),fun(A,set(A)),S))))) ) ) ) ) ).

% rtrancl_last_visit'
tff(fact_3551_times__subset__iff,axiom,
    ! [A: $tType,B: $tType,A4: set(A),C3: set(B),B3: set(A),D4: set(B)] :
      ( aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),product_Sigma(A,B,A4,aTP_Lamp_qz(set(B),fun(A,set(B)),C3))),product_Sigma(A,B,B3,aTP_Lamp_qz(set(B),fun(A,set(B)),D4)))
    <=> ( ( A4 = bot_bot(set(A)) )
        | ( C3 = bot_bot(set(B)) )
        | ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B3)
          & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),C3),D4) ) ) ) ).

% times_subset_iff
tff(fact_3552_trancl__subset__Sigma__aux,axiom,
    ! [A: $tType,A3: A,B2: A,R2: set(product_prod(A,A)),A4: set(A)] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_rtrancl(A,R2))
     => ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R2),product_Sigma(A,A,A4,aTP_Lamp_rl(set(A),fun(A,set(A)),A4)))
       => ( ( A3 = B2 )
          | aa(set(A),$o,member(A,A3),A4) ) ) ) ).

% trancl_subset_Sigma_aux
tff(fact_3553_finite__SigmaI2,axiom,
    ! [B: $tType,A: $tType,A4: set(A),B3: fun(A,set(B))] :
      ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,set(B)),fun(A,$o),aTP_Lamp_ro(set(A),fun(fun(A,set(B)),fun(A,$o)),A4),B3)))
     => ( ! [A6: A] :
            ( aa(set(A),$o,member(A,A6),A4)
           => aa(set(B),$o,finite_finite2(B),aa(A,set(B),B3,A6)) )
       => aa(set(product_prod(A,B)),$o,finite_finite2(product_prod(A,B)),product_Sigma(A,B,A4,B3)) ) ) ).

% finite_SigmaI2
tff(fact_3554_finite__cartesian__productD1,axiom,
    ! [B: $tType,A: $tType,A4: set(A),B3: set(B)] :
      ( aa(set(product_prod(A,B)),$o,finite_finite2(product_prod(A,B)),product_Sigma(A,B,A4,aTP_Lamp_qz(set(B),fun(A,set(B)),B3)))
     => ( ( B3 != bot_bot(set(B)) )
       => aa(set(A),$o,finite_finite2(A),A4) ) ) ).

% finite_cartesian_productD1
tff(fact_3555_finite__cartesian__productD2,axiom,
    ! [A: $tType,B: $tType,A4: set(A),B3: set(B)] :
      ( aa(set(product_prod(A,B)),$o,finite_finite2(product_prod(A,B)),product_Sigma(A,B,A4,aTP_Lamp_qz(set(B),fun(A,set(B)),B3)))
     => ( ( A4 != bot_bot(set(A)) )
       => aa(set(B),$o,finite_finite2(B),B3) ) ) ).

% finite_cartesian_productD2
tff(fact_3556_finite__cartesian__product__iff,axiom,
    ! [A: $tType,B: $tType,A4: set(A),B3: set(B)] :
      ( aa(set(product_prod(A,B)),$o,finite_finite2(product_prod(A,B)),product_Sigma(A,B,A4,aTP_Lamp_qz(set(B),fun(A,set(B)),B3)))
    <=> ( ( A4 = bot_bot(set(A)) )
        | ( B3 = bot_bot(set(B)) )
        | ( aa(set(A),$o,finite_finite2(A),A4)
          & aa(set(B),$o,finite_finite2(B),B3) ) ) ) ).

% finite_cartesian_product_iff
tff(fact_3557_Restr__rtrancl__mono,axiom,
    ! [A: $tType,V: A,W2: A,E3: set(product_prod(A,A)),U2: set(A)] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),V),W2)),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))),inf_inf(set(product_prod(A,A))),E3),product_Sigma(A,A,U2,aTP_Lamp_rl(set(A),fun(A,set(A)),U2)))))
     => aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),V),W2)),transitive_rtrancl(A,E3)) ) ).

% Restr_rtrancl_mono
tff(fact_3558_Restr__trancl__mono,axiom,
    ! [A: $tType,V: A,W2: A,E3: set(product_prod(A,A)),U2: set(A)] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),V),W2)),transitive_trancl(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))),E3),product_Sigma(A,A,U2,aTP_Lamp_rl(set(A),fun(A,set(A)),U2)))))
     => aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),V),W2)),transitive_trancl(A,E3)) ) ).

% Restr_trancl_mono
tff(fact_3559_UN__Times__distrib,axiom,
    ! [C: $tType,B: $tType,A: $tType,D: $tType,E3: fun(C,set(A)),F4: fun(D,set(B)),A4: set(C),B3: set(D)] : aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),aa(set(product_prod(C,D)),set(set(product_prod(A,B))),image2(product_prod(C,D),set(product_prod(A,B)),aa(fun(C,fun(D,set(product_prod(A,B)))),fun(product_prod(C,D),set(product_prod(A,B))),product_case_prod(C,D,set(product_prod(A,B))),aa(fun(D,set(B)),fun(C,fun(D,set(product_prod(A,B)))),aTP_Lamp_rq(fun(C,set(A)),fun(fun(D,set(B)),fun(C,fun(D,set(product_prod(A,B))))),E3),F4))),product_Sigma(C,D,A4,aTP_Lamp_rr(set(D),fun(C,set(D)),B3)))) = product_Sigma(A,B,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(C),set(set(A)),image2(C,set(A),E3),A4)),aa(set(D),fun(A,set(B)),aTP_Lamp_rs(fun(D,set(B)),fun(set(D),fun(A,set(B))),F4),B3)) ).

% UN_Times_distrib
tff(fact_3560_homo__rel__restrict__mono,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),B3: set(A),A4: set(A)] :
      ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R),product_Sigma(A,A,B3,aTP_Lamp_rl(set(A),fun(A,set(A)),B3)))
     => aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),rel_restrict(A,R,A4)),product_Sigma(A,A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B3),A4),aa(set(A),fun(A,set(A)),aTP_Lamp_rt(set(A),fun(set(A),fun(A,set(A))),B3),A4))) ) ).

% homo_rel_restrict_mono
tff(fact_3561_swap__product,axiom,
    ! [A: $tType,B: $tType,A4: set(B),B3: set(A)] : aa(set(product_prod(B,A)),set(product_prod(A,B)),image2(product_prod(B,A),product_prod(A,B),aa(fun(B,fun(A,product_prod(A,B))),fun(product_prod(B,A),product_prod(A,B)),product_case_prod(B,A,product_prod(A,B)),aTP_Lamp_ru(B,fun(A,product_prod(A,B))))),product_Sigma(B,A,A4,aTP_Lamp_ke(set(A),fun(B,set(A)),B3))) = product_Sigma(A,B,B3,aTP_Lamp_qz(set(B),fun(A,set(B)),A4)) ).

% swap_product
tff(fact_3562_rel__restrict__alt__def,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A4: set(A)] : rel_restrict(A,R,A4) = 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,aa(set(A),set(A),uminus_uminus(set(A)),A4),aTP_Lamp_rv(set(A),fun(A,set(A)),A4))) ).

% rel_restrict_alt_def
tff(fact_3563_card__cartesian__product,axiom,
    ! [A: $tType,B: $tType,A4: set(A),B3: set(B)] : aa(set(product_prod(A,B)),nat,finite_card(product_prod(A,B)),product_Sigma(A,B,A4,aTP_Lamp_qz(set(B),fun(A,set(B)),B3))) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(set(A),nat,finite_card(A),A4)),aa(set(B),nat,finite_card(B),B3)) ).

% card_cartesian_product
tff(fact_3564_sum_Ocartesian__product,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(B,fun(C,A)),B3: set(C),A4: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(set(C),fun(B,A),aTP_Lamp_cv(fun(B,fun(C,A)),fun(set(C),fun(B,A)),G),B3)),A4) = aa(set(product_prod(B,C)),A,aa(fun(product_prod(B,C),A),fun(set(product_prod(B,C)),A),groups7311177749621191930dd_sum(product_prod(B,C),A),aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),G)),product_Sigma(B,C,A4,aTP_Lamp_rw(set(C),fun(B,set(C)),B3))) ) ).

% sum.cartesian_product
tff(fact_3565_prod_Ocartesian__product,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(B,fun(C,A)),B3: set(C),A4: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(set(C),fun(B,A),aTP_Lamp_fe(fun(B,fun(C,A)),fun(set(C),fun(B,A)),G),B3)),A4) = aa(set(product_prod(B,C)),A,aa(fun(product_prod(B,C),A),fun(set(product_prod(B,C)),A),groups7121269368397514597t_prod(product_prod(B,C),A),aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),G)),product_Sigma(B,C,A4,aTP_Lamp_rw(set(C),fun(B,set(C)),B3))) ) ).

% prod.cartesian_product
tff(fact_3566_rtrancl__last__visit,axiom,
    ! [A: $tType,Q3: A,Q4: A,R: set(product_prod(A,A)),S: set(A)] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q3),Q4)),transitive_rtrancl(A,R))
     => ( ~ aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q3),Q4)),transitive_rtrancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),minus_minus(set(product_prod(A,A))),R),product_Sigma(A,A,top_top(set(A)),aTP_Lamp_rl(set(A),fun(A,set(A)),S)))))
       => ~ ! [Qt: A] :
              ( aa(set(A),$o,member(A,Qt),S)
             => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q3),Qt)),transitive_trancl(A,R))
               => ~ aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Qt),Q4)),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))),minus_minus(set(product_prod(A,A))),R),product_Sigma(A,A,top_top(set(A)),aTP_Lamp_rl(set(A),fun(A,set(A)),S))))) ) ) ) ) ).

% rtrancl_last_visit
tff(fact_3567_rtrancl__restrictI,axiom,
    ! [A: $tType,U: A,V: A,E3: set(product_prod(A,A)),R: set(A)] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),U),V)),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))),minus_minus(set(product_prod(A,A))),E3),product_Sigma(A,A,top_top(set(A)),aTP_Lamp_rl(set(A),fun(A,set(A)),R)))))
     => ( ~ aa(set(A),$o,member(A,U),R)
       => aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),U),V)),transitive_rtrancl(A,rel_restrict(A,E3,R))) ) ) ).

% rtrancl_restrictI
tff(fact_3568_image__paired__Times,axiom,
    ! [C: $tType,B: $tType,A: $tType,D: $tType,F: fun(C,A),G: fun(D,B),A4: set(C),B3: set(D)] : aa(set(product_prod(C,D)),set(product_prod(A,B)),image2(product_prod(C,D),product_prod(A,B),aa(fun(C,fun(D,product_prod(A,B))),fun(product_prod(C,D),product_prod(A,B)),product_case_prod(C,D,product_prod(A,B)),aa(fun(D,B),fun(C,fun(D,product_prod(A,B))),aTP_Lamp_rx(fun(C,A),fun(fun(D,B),fun(C,fun(D,product_prod(A,B)))),F),G))),product_Sigma(C,D,A4,aTP_Lamp_rr(set(D),fun(C,set(D)),B3))) = product_Sigma(A,B,aa(set(C),set(A),image2(C,A,F),A4),aa(set(D),fun(A,set(B)),aTP_Lamp_ry(fun(D,B),fun(set(D),fun(A,set(B))),G),B3)) ).

% image_paired_Times
tff(fact_3569_rel__restrict__Sigma__sub,axiom,
    ! [A: $tType,A4: set(A),R: set(A)] : aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),rel_restrict(A,transitive_trancl(A,product_Sigma(A,A,A4,aTP_Lamp_rl(set(A),fun(A,set(A)),A4))),R)),transitive_trancl(A,product_Sigma(A,A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),R),aa(set(A),fun(A,set(A)),aTP_Lamp_rt(set(A),fun(set(A),fun(A,set(A))),A4),R)))) ).

% rel_restrict_Sigma_sub
tff(fact_3570_card__cartesian__product__singleton,axiom,
    ! [A: $tType,B: $tType,X: A,A4: set(B)] : aa(set(product_prod(A,B)),nat,finite_card(product_prod(A,B)),product_Sigma(A,B,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))),aTP_Lamp_qz(set(B),fun(A,set(B)),A4))) = aa(set(B),nat,finite_card(B),A4) ).

% card_cartesian_product_singleton
tff(fact_3571_sum_OSigma,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( comm_monoid_add(C)
     => ! [A4: set(A),B3: fun(A,set(B)),G: fun(A,fun(B,C))] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ! [X2: A] :
                ( aa(set(A),$o,member(A,X2),A4)
               => aa(set(B),$o,finite_finite2(B),aa(A,set(B),B3,X2)) )
           => ( aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7311177749621191930dd_sum(A,C),aa(fun(A,fun(B,C)),fun(A,C),aTP_Lamp_rz(fun(A,set(B)),fun(fun(A,fun(B,C)),fun(A,C)),B3),G)),A4) = aa(set(product_prod(A,B)),C,aa(fun(product_prod(A,B),C),fun(set(product_prod(A,B)),C),groups7311177749621191930dd_sum(product_prod(A,B),C),aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),G)),product_Sigma(A,B,A4,B3)) ) ) ) ) ).

% sum.Sigma
tff(fact_3572_prod_OSigma,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( comm_monoid_mult(C)
     => ! [A4: set(A),B3: fun(A,set(B)),G: fun(A,fun(B,C))] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ! [X2: A] :
                ( aa(set(A),$o,member(A,X2),A4)
               => aa(set(B),$o,finite_finite2(B),aa(A,set(B),B3,X2)) )
           => ( aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7121269368397514597t_prod(A,C),aa(fun(A,fun(B,C)),fun(A,C),aTP_Lamp_sa(fun(A,set(B)),fun(fun(A,fun(B,C)),fun(A,C)),B3),G)),A4) = aa(set(product_prod(A,B)),C,aa(fun(product_prod(A,B),C),fun(set(product_prod(A,B)),C),groups7121269368397514597t_prod(product_prod(A,B),C),aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),G)),product_Sigma(A,B,A4,B3)) ) ) ) ) ).

% prod.Sigma
tff(fact_3573_arg__min__if__finite_I1_J,axiom,
    ! [B: $tType,A: $tType] :
      ( order(B)
     => ! [S: set(A),F: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),S)
         => ( ( S != bot_bot(set(A)) )
           => aa(set(A),$o,member(A,lattic7623131987881927897min_on(A,B,F,S)),S) ) ) ) ).

% arg_min_if_finite(1)
tff(fact_3574_Sigma__def,axiom,
    ! [B: $tType,A: $tType,A4: set(A),B3: fun(A,set(B))] : product_Sigma(A,B,A4,B3) = aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),aa(set(A),set(set(product_prod(A,B))),image2(A,set(product_prod(A,B)),aTP_Lamp_sc(fun(A,set(B)),fun(A,set(product_prod(A,B))),B3)),A4)) ).

% Sigma_def
tff(fact_3575_product__fold,axiom,
    ! [B: $tType,A: $tType,A4: set(A),B3: set(B)] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( aa(set(B),$o,finite_finite2(B),B3)
       => ( product_Sigma(A,B,A4,aTP_Lamp_qz(set(B),fun(A,set(B)),B3)) = finite_fold(A,set(product_prod(A,B)),aTP_Lamp_se(set(B),fun(A,fun(set(product_prod(A,B)),set(product_prod(A,B)))),B3),bot_bot(set(product_prod(A,B))),A4) ) ) ) ).

% product_fold
tff(fact_3576_trancl__multi__insert2,axiom,
    ! [A: $tType,A3: A,B2: A,R2: set(product_prod(A,A)),M: A,X5: set(A)] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_trancl(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))),R2),product_Sigma(A,A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),M),bot_bot(set(A))),aTP_Lamp_rl(set(A),fun(A,set(A)),X5)))))
     => ( ~ aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_trancl(A,R2))
       => ~ ! [X2: A] :
              ( aa(set(A),$o,member(A,X2),X5)
             => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),M)),transitive_rtrancl(A,R2))
               => ~ aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),B2)),transitive_rtrancl(A,R2)) ) ) ) ) ).

% trancl_multi_insert2
tff(fact_3577_trancl__multi__insert,axiom,
    ! [A: $tType,A3: A,B2: A,R2: set(product_prod(A,A)),X5: set(A),M: A] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_trancl(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))),R2),product_Sigma(A,A,X5,aTP_Lamp_sf(A,fun(A,set(A)),M)))))
     => ( ~ aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_trancl(A,R2))
       => ~ ! [X2: A] :
              ( aa(set(A),$o,member(A,X2),X5)
             => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),X2)),transitive_rtrancl(A,R2))
               => ~ aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),M),B2)),transitive_rtrancl(A,R2)) ) ) ) ) ).

% trancl_multi_insert
tff(fact_3578_of__int_Orep__eq,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [X: int] : aa(int,A,ring_1_of_int(A),X) = 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_pn(nat,fun(nat,A))),aa(int,product_prod(nat,nat),rep_Integ,X)) ) ).

% of_int.rep_eq
tff(fact_3579_rtrancl__last__visit__node,axiom,
    ! [A: $tType,S2: A,S4: A,R: set(product_prod(A,A)),Sh: A] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),S2),S4)),transitive_rtrancl(A,R))
     => ( ( ( S2 != Sh )
          & aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),S2),S4)),transitive_rtrancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R),product_Sigma(A,A,top_top(set(A)),aTP_Lamp_sg(A,fun(A,set(A)),Sh))))) )
        | ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),S2),Sh)),transitive_rtrancl(A,R))
          & aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Sh),S4)),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))),inf_inf(set(product_prod(A,A))),R),product_Sigma(A,A,top_top(set(A)),aTP_Lamp_sg(A,fun(A,set(A)),Sh))))) ) ) ) ).

% rtrancl_last_visit_node
tff(fact_3580_infinite__cartesian__product,axiom,
    ! [A: $tType,B: $tType,A4: set(A),B3: set(B)] :
      ( ~ aa(set(A),$o,finite_finite2(A),A4)
     => ( ~ aa(set(B),$o,finite_finite2(B),B3)
       => ~ aa(set(product_prod(A,B)),$o,finite_finite2(product_prod(A,B)),product_Sigma(A,B,A4,aTP_Lamp_qz(set(B),fun(A,set(B)),B3))) ) ) ).

% infinite_cartesian_product
tff(fact_3581_Restr__subset,axiom,
    ! [A: $tType,A4: set(A),B3: set(A),R2: set(product_prod(A,A))] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B3)
     => ( aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R2),product_Sigma(A,A,B3,aTP_Lamp_rl(set(A),fun(A,set(A)),B3)))),product_Sigma(A,A,A4,aTP_Lamp_rl(set(A),fun(A,set(A)),A4))) = aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R2),product_Sigma(A,A,A4,aTP_Lamp_rl(set(A),fun(A,set(A)),A4))) ) ) ).

% Restr_subset
tff(fact_3582_Gr__incl,axiom,
    ! [A: $tType,B: $tType,A4: set(A),F: fun(A,B),B3: set(B)] :
      ( aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),bNF_Gr(A,B,A4,F)),product_Sigma(A,B,A4,aTP_Lamp_qz(set(B),fun(A,set(B)),B3)))
    <=> aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F),A4)),B3) ) ).

% Gr_incl
tff(fact_3583_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_pm(nat,fun(nat,product_prod(nat,nat))))) ).

% uminus_int_def
tff(fact_3584_pred__nat__def,axiom,
    pred_nat = aa(fun(product_prod(nat,nat),$o),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aTP_Lamp_sh(nat,fun(nat,$o)))) ).

% pred_nat_def
tff(fact_3585_GrD2,axiom,
    ! [A: $tType,B: $tType,X: A,Fx: B,A4: set(A),F: fun(A,B)] :
      ( aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Fx)),bNF_Gr(A,B,A4,F))
     => ( aa(A,B,F,X) = Fx ) ) ).

% GrD2
tff(fact_3586_GrD1,axiom,
    ! [B: $tType,A: $tType,X: A,Fx: B,A4: set(A),F: fun(A,B)] :
      ( aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Fx)),bNF_Gr(A,B,A4,F))
     => aa(set(A),$o,member(A,X),A4) ) ).

% GrD1
tff(fact_3587_less__eq,axiom,
    ! [M: nat,N: nat] :
      ( aa(set(product_prod(nat,nat)),$o,member(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),M),N)),transitive_trancl(nat,pred_nat))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),N) ) ).

% less_eq
tff(fact_3588_pred__nat__trancl__eq__le,axiom,
    ! [M: nat,N: nat] :
      ( aa(set(product_prod(nat,nat)),$o,member(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),M),N)),transitive_rtrancl(nat,pred_nat))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),N) ) ).

% pred_nat_trancl_eq_le
tff(fact_3589_relChain__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [R2: set(product_prod(A,A)),As9: fun(A,B)] :
          ( bNF_Ca3754400796208372196lChain(A,B,R2,As9)
        <=> ! [I3: A,J4: A] :
              ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),I3),J4)),R2)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,As9,I3)),aa(A,B,As9,J4)) ) ) ) ).

% relChain_def
tff(fact_3590_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_pb(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))) ).

% times_int_def
tff(fact_3591_natLess__def,axiom,
    bNF_Ca8459412986667044542atLess = aa(fun(product_prod(nat,nat),$o),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),ord_less(nat))) ).

% natLess_def
tff(fact_3592_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_pv(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))) ).

% minus_int_def
tff(fact_3593_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_pt(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))) ).

% plus_int_def
tff(fact_3594_Gcd__remove0__nat,axiom,
    ! [M4: set(nat)] :
      ( aa(set(nat),$o,finite_finite2(nat),M4)
     => ( gcd_Gcd(nat,M4) = gcd_Gcd(nat,aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),M4),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),zero_zero(nat)),bot_bot(set(nat))))) ) ) ).

% Gcd_remove0_nat
tff(fact_3595_take__bit__numeral__minus__numeral__int,axiom,
    ! [M: num,N: num] : bit_se2584673776208193580ke_bit(int,aa(num,nat,numeral_numeral(nat),M),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = case_option(int,num,zero_zero(int),aTP_Lamp_si(num,fun(num,int),M),bit_take_bit_num(aa(num,nat,numeral_numeral(nat),M),N)) ).

% take_bit_numeral_minus_numeral_int
tff(fact_3596_rat__floor__lemma,axiom,
    ! [A3: int,B2: int] :
      ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),aa(int,rat,ring_1_of_int(rat),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B2))),aa(int,rat,aa(int,fun(int,rat),fract,A3),B2))
      & aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),aa(int,rat,aa(int,fun(int,rat),fract,A3),B2)),aa(int,rat,ring_1_of_int(rat),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B2)),one_one(int)))) ) ).

% rat_floor_lemma
tff(fact_3597_prod_Oinsert_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [I4: set(A),P3: fun(A,B),I: A] :
          ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_hl(set(A),fun(fun(A,B),fun(A,$o)),I4),P3)))
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1962203154675924110t_prod(A,B),P3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),I),I4)) = $ite(aa(set(A),$o,member(A,I),I4),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1962203154675924110t_prod(A,B),P3),I4),aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,P3,I)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1962203154675924110t_prod(A,B),P3),I4))) ) ) ) ).

% prod.insert'
tff(fact_3598_mult__ceiling__le__Ints,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(A)
        & linordered_idom(B) )
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
         => ( aa(set(A),$o,member(A,A3),ring_1_Ints(A))
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(int,B,ring_1_of_int(B),archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)))),aa(int,B,ring_1_of_int(B),aa(int,int,aa(int,fun(int,int),times_times(int),archimedean_ceiling(A,A3)),archimedean_ceiling(A,B2)))) ) ) ) ).

% mult_ceiling_le_Ints
tff(fact_3599_minus__rat__cancel,axiom,
    ! [A3: int,B2: int] : aa(int,rat,aa(int,fun(int,rat),fract,aa(int,int,uminus_uminus(int),A3)),aa(int,int,uminus_uminus(int),B2)) = aa(int,rat,aa(int,fun(int,rat),fract,A3),B2) ).

% minus_rat_cancel
tff(fact_3600_Gcd__empty,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ( gcd_Gcd(A,bot_bot(set(A))) = zero_zero(A) ) ) ).

% Gcd_empty
tff(fact_3601_Gcd__UNIV,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ( gcd_Gcd(A,top_top(set(A))) = one_one(A) ) ) ).

% Gcd_UNIV
tff(fact_3602_prod_Oempty_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [P3: fun(B,A)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),P3),bot_bot(set(B))) = one_one(A) ) ).

% prod.empty'
tff(fact_3603_minus__rat,axiom,
    ! [A3: int,B2: int] : aa(rat,rat,uminus_uminus(rat),aa(int,rat,aa(int,fun(int,rat),fract,A3),B2)) = aa(int,rat,aa(int,fun(int,rat),fract,aa(int,int,uminus_uminus(int),A3)),B2) ).

% minus_rat
tff(fact_3604_Gcd__0__iff,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A4: set(A)] :
          ( ( gcd_Gcd(A,A4) = zero_zero(A) )
        <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),zero_zero(A)),bot_bot(set(A)))) ) ) ).

% Gcd_0_iff
tff(fact_3605_take__bit__numeral__numeral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: num,N: num] : bit_se2584673776208193580ke_bit(A,aa(num,nat,numeral_numeral(nat),M),aa(num,A,numeral_numeral(A),N)) = case_option(A,num,zero_zero(A),numeral_numeral(A),bit_take_bit_num(aa(num,nat,numeral_numeral(nat),M),N)) ) ).

% take_bit_numeral_numeral
tff(fact_3606_Ints__1,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => aa(set(A),$o,member(A,one_one(A)),ring_1_Ints(A)) ) ).

% Ints_1
tff(fact_3607_option_Ocase__distrib,axiom,
    ! [B: $tType,A: $tType,C: $tType,H2: fun(B,A),F1: B,F22: fun(C,B),Option: option(C)] : aa(B,A,H2,case_option(B,C,F1,F22,Option)) = case_option(A,C,aa(B,A,H2,F1),aa(fun(C,B),fun(C,A),aTP_Lamp_ll(fun(B,A),fun(fun(C,B),fun(C,A)),H2),F22),Option) ).

% option.case_distrib
tff(fact_3608_Ints__mult,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A3: A,B2: A] :
          ( aa(set(A),$o,member(A,A3),ring_1_Ints(A))
         => ( aa(set(A),$o,member(A,B2),ring_1_Ints(A))
           => aa(set(A),$o,member(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),ring_1_Ints(A)) ) ) ) ).

% Ints_mult
tff(fact_3609_minus__in__Ints__iff,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [X: A] :
          ( aa(set(A),$o,member(A,aa(A,A,uminus_uminus(A),X)),ring_1_Ints(A))
        <=> aa(set(A),$o,member(A,X),ring_1_Ints(A)) ) ) ).

% minus_in_Ints_iff
tff(fact_3610_Ints__minus,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A3: A] :
          ( aa(set(A),$o,member(A,A3),ring_1_Ints(A))
         => aa(set(A),$o,member(A,aa(A,A,uminus_uminus(A),A3)),ring_1_Ints(A)) ) ) ).

% Ints_minus
tff(fact_3611_Gcd__nat__eq__one,axiom,
    ! [N4: set(nat)] :
      ( aa(set(nat),$o,member(nat,one_one(nat)),N4)
     => ( gcd_Gcd(nat,N4) = one_one(nat) ) ) ).

% Gcd_nat_eq_one
tff(fact_3612_Gcd__1,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A4: set(A)] :
          ( aa(set(A),$o,member(A,one_one(A)),A4)
         => ( gcd_Gcd(A,A4) = one_one(A) ) ) ) ).

% Gcd_1
tff(fact_3613_prod_Onon__neutral_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(B,A),I4: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),G),aa(fun(B,$o),set(B),collect(B),aa(set(B),fun(B,$o),aTP_Lamp_sj(fun(B,A),fun(set(B),fun(B,$o)),G),I4))) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),G),I4) ) ).

% prod.non_neutral'
tff(fact_3614_eq__rat_I2_J,axiom,
    ! [A3: int] : aa(int,rat,aa(int,fun(int,rat),fract,A3),zero_zero(int)) = aa(int,rat,aa(int,fun(int,rat),fract,zero_zero(int)),one_one(int)) ).

% eq_rat(2)
tff(fact_3615_Fract__of__nat__eq,axiom,
    ! [K: nat] : aa(int,rat,aa(int,fun(int,rat),fract,aa(nat,int,semiring_1_of_nat(int),K)),one_one(int)) = aa(nat,rat,semiring_1_of_nat(rat),K) ).

% Fract_of_nat_eq
tff(fact_3616_quotient__of__eq,axiom,
    ! [A3: int,B2: int,P3: int,Q3: int] :
      ( ( quotient_of(aa(int,rat,aa(int,fun(int,rat),fract,A3),B2)) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P3),Q3) )
     => ( aa(int,rat,aa(int,fun(int,rat),fract,P3),Q3) = aa(int,rat,aa(int,fun(int,rat),fract,A3),B2) ) ) ).

% quotient_of_eq
tff(fact_3617_One__rat__def,axiom,
    one_one(rat) = aa(int,rat,aa(int,fun(int,rat),fract,one_one(int)),one_one(int)) ).

% One_rat_def
tff(fact_3618_Fract__of__int__eq,axiom,
    ! [K: int] : aa(int,rat,aa(int,fun(int,rat),fract,K),one_one(int)) = aa(int,rat,ring_1_of_int(rat),K) ).

% Fract_of_int_eq
tff(fact_3619_normalize__eq,axiom,
    ! [A3: int,B2: int,P3: int,Q3: int] :
      ( ( normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),A3),B2)) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P3),Q3) )
     => ( aa(int,rat,aa(int,fun(int,rat),fract,P3),Q3) = aa(int,rat,aa(int,fun(int,rat),fract,A3),B2) ) ) ).

% normalize_eq
tff(fact_3620_Gcd__eq__1__I,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A3: A,A4: set(A)] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A3),one_one(A))
         => ( aa(set(A),$o,member(A,A3),A4)
           => ( gcd_Gcd(A,A4) = one_one(A) ) ) ) ) ).

% Gcd_eq_1_I
tff(fact_3621_finite__int__segment,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A3: A,B2: A] : aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(A,fun(A,$o),aTP_Lamp_sk(A,fun(A,fun(A,$o)),A3),B2))) ) ).

% finite_int_segment
tff(fact_3622_prod_Odistrib__triv_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [I4: set(A),G: fun(A,B),H2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),I4)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1962203154675924110t_prod(A,B),aa(fun(A,B),fun(A,B),aTP_Lamp_sl(fun(A,B),fun(fun(A,B),fun(A,B)),G),H2)),I4) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1962203154675924110t_prod(A,B),G),I4)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1962203154675924110t_prod(A,B),H2),I4)) ) ) ) ).

% prod.distrib_triv'
tff(fact_3623_Gcd__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_Gcd(B)
     => ! [A4: set(A),F: fun(A,B),G: fun(A,B)] :
          ( ! [X2: A] :
              ( aa(set(A),$o,member(A,X2),A4)
             => aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(A,B,F,X2)),aa(A,B,G,X2)) )
         => aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),gcd_Gcd(B,aa(set(A),set(B),image2(A,B,F),A4))),gcd_Gcd(B,aa(set(A),set(B),image2(A,B,G),A4))) ) ) ).

% Gcd_mono
tff(fact_3624_Zero__rat__def,axiom,
    zero_zero(rat) = aa(int,rat,aa(int,fun(int,rat),fract,zero_zero(int)),one_one(int)) ).

% Zero_rat_def
tff(fact_3625_Ints__odd__nonzero,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [A3: A] :
          ( aa(set(A),$o,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_3626_rat__number__expand_I3_J,axiom,
    ! [K: num] : aa(num,rat,numeral_numeral(rat),K) = aa(int,rat,aa(int,fun(int,rat),fract,aa(num,int,numeral_numeral(int),K)),one_one(int)) ).

% rat_number_expand(3)
tff(fact_3627_rat__number__collapse_I3_J,axiom,
    ! [W2: num] : aa(int,rat,aa(int,fun(int,rat),fract,aa(num,int,numeral_numeral(int),W2)),one_one(int)) = aa(num,rat,numeral_numeral(rat),W2) ).

% rat_number_collapse(3)
tff(fact_3628_quotient__of__Fract,axiom,
    ! [A3: int,B2: int] : quotient_of(aa(int,rat,aa(int,fun(int,rat),fract,A3),B2)) = normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),A3),B2)) ).

% quotient_of_Fract
tff(fact_3629_finite__abs__int__segment,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A3: A] : aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_sm(A,fun(A,$o),A3))) ) ).

% finite_abs_int_segment
tff(fact_3630_prod_Omono__neutral__left_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [S: set(A),T2: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),T2)
         => ( ! [X2: A] :
                ( aa(set(A),$o,member(A,X2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T2),S))
               => ( aa(A,B,G,X2) = one_one(B) ) )
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1962203154675924110t_prod(A,B),G),S) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1962203154675924110t_prod(A,B),G),T2) ) ) ) ) ).

% prod.mono_neutral_left'
tff(fact_3631_prod_Omono__neutral__right_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [S: set(A),T2: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),T2)
         => ( ! [X2: A] :
                ( aa(set(A),$o,member(A,X2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T2),S))
               => ( aa(A,B,G,X2) = one_one(B) ) )
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1962203154675924110t_prod(A,B),G),T2) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1962203154675924110t_prod(A,B),G),S) ) ) ) ) ).

% prod.mono_neutral_right'
tff(fact_3632_prod_Omono__neutral__cong__left_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [S: set(A),T2: set(A),H2: fun(A,B),G: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),T2)
         => ( ! [I2: A] :
                ( aa(set(A),$o,member(A,I2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T2),S))
               => ( aa(A,B,H2,I2) = one_one(B) ) )
           => ( ! [X2: A] :
                  ( aa(set(A),$o,member(A,X2),S)
                 => ( aa(A,B,G,X2) = aa(A,B,H2,X2) ) )
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1962203154675924110t_prod(A,B),G),S) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1962203154675924110t_prod(A,B),H2),T2) ) ) ) ) ) ).

% prod.mono_neutral_cong_left'
tff(fact_3633_prod_Omono__neutral__cong__right_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [S: set(A),T2: set(A),G: fun(A,B),H2: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),T2)
         => ( ! [X2: A] :
                ( aa(set(A),$o,member(A,X2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T2),S))
               => ( aa(A,B,G,X2) = one_one(B) ) )
           => ( ! [X2: A] :
                  ( aa(set(A),$o,member(A,X2),S)
                 => ( aa(A,B,G,X2) = aa(A,B,H2,X2) ) )
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1962203154675924110t_prod(A,B),G),T2) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1962203154675924110t_prod(A,B),H2),S) ) ) ) ) ) ).

% prod.mono_neutral_cong_right'
tff(fact_3634_Nats__altdef2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ( semiring_1_Nats(A) = aa(fun(A,$o),set(A),collect(A),aTP_Lamp_sn(A,$o)) ) ) ).

% Nats_altdef2
tff(fact_3635_Ints__def,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ( ring_1_Ints(A) = aa(set(int),set(A),image2(int,A,ring_1_of_int(A)),top_top(set(int))) ) ) ).

% Ints_def
tff(fact_3636_prod_Odistrib_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [I4: set(A),G: fun(A,B),H2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_hl(set(A),fun(fun(A,B),fun(A,$o)),I4),G)))
         => ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_hl(set(A),fun(fun(A,B),fun(A,$o)),I4),H2)))
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1962203154675924110t_prod(A,B),aa(fun(A,B),fun(A,B),aTP_Lamp_sl(fun(A,B),fun(fun(A,B),fun(A,B)),G),H2)),I4) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1962203154675924110t_prod(A,B),G),I4)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1962203154675924110t_prod(A,B),H2),I4)) ) ) ) ) ).

% prod.distrib'
tff(fact_3637_prod_OG__def,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [P3: fun(B,A),I4: set(B)] :
          aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),P3),I4) = $ite(aa(set(B),$o,finite_finite2(B),aa(fun(B,$o),set(B),collect(B),aa(set(B),fun(B,$o),aTP_Lamp_sj(fun(B,A),fun(set(B),fun(B,$o)),P3),I4))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),P3),aa(fun(B,$o),set(B),collect(B),aa(set(B),fun(B,$o),aTP_Lamp_sj(fun(B,A),fun(set(B),fun(B,$o)),P3),I4))),one_one(A)) ) ).

% prod.G_def
tff(fact_3638_Ints__odd__less__0,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( aa(set(A),$o,member(A,A3),ring_1_Ints(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),A3)),A3)),zero_zero(A))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A)) ) ) ) ).

% Ints_odd_less_0
tff(fact_3639_Ints__nonzero__abs__ge1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A] :
          ( aa(set(A),$o,member(A,X),ring_1_Ints(A))
         => ( ( X != zero_zero(A) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(A,A,abs_abs(A),X)) ) ) ) ).

% Ints_nonzero_abs_ge1
tff(fact_3640_Ints__nonzero__abs__less1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A] :
          ( aa(set(A),$o,member(A,X),ring_1_Ints(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),X)),one_one(A))
           => ( X = zero_zero(A) ) ) ) ) ).

% Ints_nonzero_abs_less1
tff(fact_3641_one__less__Fract__iff,axiom,
    ! [B2: int,A3: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
     => ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),one_one(rat)),aa(int,rat,aa(int,fun(int,rat),fract,A3),B2))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),B2),A3) ) ) ).

% one_less_Fract_iff
tff(fact_3642_Fract__less__one__iff,axiom,
    ! [B2: int,A3: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
     => ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),aa(int,rat,aa(int,fun(int,rat),fract,A3),B2)),one_one(rat))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),A3),B2) ) ) ).

% Fract_less_one_iff
tff(fact_3643_rat__number__collapse_I5_J,axiom,
    aa(int,rat,aa(int,fun(int,rat),fract,aa(int,int,uminus_uminus(int),one_one(int))),one_one(int)) = aa(rat,rat,uminus_uminus(rat),one_one(rat)) ).

% rat_number_collapse(5)
tff(fact_3644_Ints__eq__abs__less1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] :
          ( aa(set(A),$o,member(A,X),ring_1_Ints(A))
         => ( aa(set(A),$o,member(A,Y),ring_1_Ints(A))
           => ( ( X = Y )
            <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Y))),one_one(A)) ) ) ) ) ).

% Ints_eq_abs_less1
tff(fact_3645_Fract__add__one,axiom,
    ! [N: int,M: int] :
      ( ( N != zero_zero(int) )
     => ( aa(int,rat,aa(int,fun(int,rat),fract,aa(int,int,aa(int,fun(int,int),plus_plus(int),M),N)),N) = aa(rat,rat,aa(rat,fun(rat,rat),plus_plus(rat),aa(int,rat,aa(int,fun(int,rat),fract,M),N)),one_one(rat)) ) ) ).

% Fract_add_one
tff(fact_3646_one__le__Fract__iff,axiom,
    ! [B2: int,A3: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
     => ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),one_one(rat)),aa(int,rat,aa(int,fun(int,rat),fract,A3),B2))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),B2),A3) ) ) ).

% one_le_Fract_iff
tff(fact_3647_Fract__le__one__iff,axiom,
    ! [B2: int,A3: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
     => ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),aa(int,rat,aa(int,fun(int,rat),fract,A3),B2)),one_one(rat))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A3),B2) ) ) ).

% Fract_le_one_iff
tff(fact_3648_rat__number__collapse_I4_J,axiom,
    ! [W2: num] : aa(int,rat,aa(int,fun(int,rat),fract,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),W2))),one_one(int)) = aa(rat,rat,uminus_uminus(rat),aa(num,rat,numeral_numeral(rat),W2)) ).

% rat_number_collapse(4)
tff(fact_3649_rat__number__expand_I5_J,axiom,
    ! [K: num] : aa(rat,rat,uminus_uminus(rat),aa(num,rat,numeral_numeral(rat),K)) = aa(int,rat,aa(int,fun(int,rat),fract,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K))),one_one(int)) ).

% rat_number_expand(5)
tff(fact_3650_frac__neg,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          archimedean_frac(A,aa(A,A,uminus_uminus(A),X)) = $ite(aa(set(A),$o,member(A,X),ring_1_Ints(A)),zero_zero(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),archimedean_frac(A,X))) ) ).

% frac_neg
tff(fact_3651_frac__unique__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,A3: A] :
          ( ( archimedean_frac(A,X) = A3 )
        <=> ( aa(set(A),$o,member(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X),A3)),ring_1_Ints(A))
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),one_one(A)) ) ) ) ).

% frac_unique_iff
tff(fact_3652_le__mult__floor__Ints,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(A)
        & linordered_idom(B) )
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
         => ( aa(set(A),$o,member(A,A3),ring_1_Ints(A))
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(int,B,ring_1_of_int(B),aa(int,int,aa(int,fun(int,int),times_times(int),archim6421214686448440834_floor(A,A3)),archim6421214686448440834_floor(A,B2)))),aa(int,B,ring_1_of_int(B),archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)))) ) ) ) ).

% le_mult_floor_Ints
tff(fact_3653_and__minus__numerals_I3_J,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,N)))) = case_option(int,num,zero_zero(int),numeral_numeral(int),bit_and_not_num(M,bitM(N))) ).

% and_minus_numerals(3)
tff(fact_3654_and__minus__numerals_I7_J,axiom,
    ! [N: num,M: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,N)))),aa(num,int,numeral_numeral(int),M)) = case_option(int,num,zero_zero(int),numeral_numeral(int),bit_and_not_num(M,bitM(N))) ).

% and_minus_numerals(7)
tff(fact_3655_and__minus__numerals_I8_J,axiom,
    ! [N: num,M: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N)))),aa(num,int,numeral_numeral(int),M)) = case_option(int,num,zero_zero(int),numeral_numeral(int),bit_and_not_num(M,aa(num,num,bit0,N))) ).

% and_minus_numerals(8)
tff(fact_3656_and__minus__numerals_I4_J,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N)))) = case_option(int,num,zero_zero(int),numeral_numeral(int),bit_and_not_num(M,aa(num,num,bit0,N))) ).

% and_minus_numerals(4)
tff(fact_3657_Gcd__int__eq,axiom,
    ! [N4: set(nat)] : gcd_Gcd(int,aa(set(nat),set(int),image2(nat,int,semiring_1_of_nat(int)),N4)) = aa(nat,int,semiring_1_of_nat(int),gcd_Gcd(nat,N4)) ).

% Gcd_int_eq
tff(fact_3658_Gcd__abs__eq,axiom,
    ! [K5: set(int)] : gcd_Gcd(int,aa(set(int),set(int),image2(int,int,abs_abs(int)),K5)) = gcd_Gcd(int,K5) ).

% Gcd_abs_eq
tff(fact_3659_Gcd__nat__abs__eq,axiom,
    ! [K5: set(int)] : gcd_Gcd(nat,aa(set(int),set(nat),image2(int,nat,aTP_Lamp_so(int,nat)),K5)) = aa(int,nat,nat2,gcd_Gcd(int,K5)) ).

% Gcd_nat_abs_eq
tff(fact_3660_Gcd__int__def,axiom,
    ! [K5: set(int)] : gcd_Gcd(int,K5) = aa(nat,int,semiring_1_of_nat(int),gcd_Gcd(nat,aa(set(int),set(nat),image2(int,nat,aa(fun(int,int),fun(int,nat),comp(int,nat,int,nat2),abs_abs(int))),K5))) ).

% Gcd_int_def
tff(fact_3661_int__numeral__and__not__num,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),N))) = case_option(int,num,zero_zero(int),numeral_numeral(int),bit_and_not_num(M,N)) ).

% int_numeral_and_not_num
tff(fact_3662_int__numeral__not__and__num,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),M))),aa(num,int,numeral_numeral(int),N)) = case_option(int,num,zero_zero(int),numeral_numeral(int),bit_and_not_num(N,M)) ).

% int_numeral_not_and_num
tff(fact_3663_Gcd__eq__Max,axiom,
    ! [M4: set(nat)] :
      ( aa(set(nat),$o,finite_finite2(nat),M4)
     => ( ( M4 != bot_bot(set(nat)) )
       => ( ~ aa(set(nat),$o,member(nat,zero_zero(nat)),M4)
         => ( gcd_Gcd(nat,M4) = aa(set(nat),nat,lattic643756798349783984er_Max(nat),aa(set(set(nat)),set(nat),complete_Inf_Inf(set(nat)),aa(set(nat),set(set(nat)),image2(nat,set(nat),aTP_Lamp_sp(nat,set(nat))),M4))) ) ) ) ) ).

% Gcd_eq_Max
tff(fact_3664_semiring__char__def,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Uu: itself(A)] : semiri4206861660011772517g_char(A,Uu) = gcd_Gcd(nat,aa(fun(nat,$o),set(nat),collect(nat),aTP_Lamp_sq(nat,$o))) ) ).

% semiring_char_def
tff(fact_3665_take__bit__num__simps_I7_J,axiom,
    ! [R2: num,M: num] : bit_take_bit_num(aa(num,nat,numeral_numeral(nat),R2),aa(num,num,bit1,M)) = aa(num,option(num),some(num),case_option(num,num,one2,bit1,bit_take_bit_num(pred_numeral(R2),M))) ).

% take_bit_num_simps(7)
tff(fact_3666_set__decode__def,axiom,
    ! [X: nat] : nat_set_decode(X) = aa(fun(nat,$o),set(nat),collect(nat),aTP_Lamp_sr(nat,fun(nat,$o),X)) ).

% set_decode_def
tff(fact_3667_Max__singleton,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A] : aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) = X ) ).

% Max_singleton
tff(fact_3668_set__decode__zero,axiom,
    nat_set_decode(zero_zero(nat)) = bot_bot(set(nat)) ).

% set_decode_zero
tff(fact_3669_Max__divisors__self__nat,axiom,
    ! [N: nat] :
      ( ( N != zero_zero(nat) )
     => ( aa(set(nat),nat,lattic643756798349783984er_Max(nat),aa(fun(nat,$o),set(nat),collect(nat),aTP_Lamp_hx(nat,fun(nat,$o),N))) = N ) ) ).

% Max_divisors_self_nat
tff(fact_3670_Max_Obounded__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),X: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic643756798349783984er_Max(A),A4)),X)
            <=> ! [X4: A] :
                  ( aa(set(A),$o,member(A,X4),A4)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),X) ) ) ) ) ) ).

% Max.bounded_iff
tff(fact_3671_Max__less__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),X: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(A),A,lattic643756798349783984er_Max(A),A4)),X)
            <=> ! [X4: A] :
                  ( aa(set(A),$o,member(A,X4),A4)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),X) ) ) ) ) ) ).

% Max_less_iff
tff(fact_3672_Max__const,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(B)
     => ! [A4: set(A),C2: B] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( aa(set(B),B,lattic643756798349783984er_Max(B),aa(set(A),set(B),image2(A,B,aTP_Lamp_ss(B,fun(A,B),C2)),A4)) = C2 ) ) ) ) ).

% Max_const
tff(fact_3673_Max__insert,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),X: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A4)) = aa(A,A,aa(A,fun(A,A),ord_max(A),X),aa(set(A),A,lattic643756798349783984er_Max(A),A4)) ) ) ) ) ).

% Max_insert
tff(fact_3674_take__bit__num__simps_I4_J,axiom,
    ! [N: nat,M: num] : bit_take_bit_num(aa(nat,nat,suc,N),aa(num,num,bit1,M)) = aa(num,option(num),some(num),case_option(num,num,one2,bit1,bit_take_bit_num(N,M))) ).

% take_bit_num_simps(4)
tff(fact_3675_Max__in,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => aa(set(A),$o,member(A,aa(set(A),A,lattic643756798349783984er_Max(A),A4)),A4) ) ) ) ).

% Max_in
tff(fact_3676_Max_OboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),X: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( ! [A6: A] :
                  ( aa(set(A),$o,member(A,A6),A4)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A6),X) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic643756798349783984er_Max(A),A4)),X) ) ) ) ) ).

% Max.boundedI
tff(fact_3677_Max_OboundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),X: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic643756798349783984er_Max(A),A4)),X)
             => ! [A11: A] :
                  ( aa(set(A),$o,member(A,A11),A4)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A11),X) ) ) ) ) ) ).

% Max.boundedE
tff(fact_3678_eq__Max__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),M: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( ( M = aa(set(A),A,lattic643756798349783984er_Max(A),A4) )
            <=> ( aa(set(A),$o,member(A,M),A4)
                & ! [X4: A] :
                    ( aa(set(A),$o,member(A,X4),A4)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),M) ) ) ) ) ) ) ).

% eq_Max_iff
tff(fact_3679_Max__ge__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),X: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(set(A),A,lattic643756798349783984er_Max(A),A4))
            <=> ? [X4: A] :
                  ( aa(set(A),$o,member(A,X4),A4)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),X4) ) ) ) ) ) ).

% Max_ge_iff
tff(fact_3680_Max__eq__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),M: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( ( aa(set(A),A,lattic643756798349783984er_Max(A),A4) = M )
            <=> ( aa(set(A),$o,member(A,M),A4)
                & ! [X4: A] :
                    ( aa(set(A),$o,member(A,X4),A4)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),M) ) ) ) ) ) ) ).

% Max_eq_iff
tff(fact_3681_Max__gr__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),X: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),aa(set(A),A,lattic643756798349783984er_Max(A),A4))
            <=> ? [X4: A] :
                  ( aa(set(A),$o,member(A,X4),A4)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),X4) ) ) ) ) ) ).

% Max_gr_iff
tff(fact_3682_Max__Sup,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [A4: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( aa(set(A),A,lattic643756798349783984er_Max(A),A4) = aa(set(A),A,complete_Sup_Sup(A),A4) ) ) ) ) ).

% Max_Sup
tff(fact_3683_cSup__eq__Max,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X5: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),X5)
         => ( ( X5 != bot_bot(set(A)) )
           => ( aa(set(A),A,complete_Sup_Sup(A),X5) = aa(set(A),A,lattic643756798349783984er_Max(A),X5) ) ) ) ) ).

% cSup_eq_Max
tff(fact_3684_finite__range__Some,axiom,
    ! [A: $tType] :
      ( aa(set(option(A)),$o,finite_finite2(option(A)),aa(set(A),set(option(A)),image2(A,option(A),some(A)),top_top(set(A))))
    <=> aa(set(A),$o,finite_finite2(A),top_top(set(A))) ) ).

% finite_range_Some
tff(fact_3685_prod__decode__aux_Ocases,axiom,
    ! [X: product_prod(nat,nat)] :
      ~ ! [K2: nat,M3: nat] : X != aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),K2),M3) ).

% prod_decode_aux.cases
tff(fact_3686_Sup__nat__def,axiom,
    ! [X5: set(nat)] :
      aa(set(nat),nat,complete_Sup_Sup(nat),X5) = $ite(X5 = bot_bot(set(nat)),zero_zero(nat),aa(set(nat),nat,lattic643756798349783984er_Max(nat),X5)) ).

% Sup_nat_def
tff(fact_3687_Max_Osubset__imp,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),B3: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B3)
         => ( ( A4 != bot_bot(set(A)) )
           => ( aa(set(A),$o,finite_finite2(A),B3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic643756798349783984er_Max(A),A4)),aa(set(A),A,lattic643756798349783984er_Max(A),B3)) ) ) ) ) ).

% Max.subset_imp
tff(fact_3688_Max__mono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [M4: set(A),N4: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),M4),N4)
         => ( ( M4 != bot_bot(set(A)) )
           => ( aa(set(A),$o,finite_finite2(A),N4)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic643756798349783984er_Max(A),M4)),aa(set(A),A,lattic643756798349783984er_Max(A),N4)) ) ) ) ) ).

% Max_mono
tff(fact_3689_hom__Max__commute,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [H2: fun(A,A),N4: set(A)] :
          ( ! [X2: A,Y2: A] : aa(A,A,H2,aa(A,A,aa(A,fun(A,A),ord_max(A),X2),Y2)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,H2,X2)),aa(A,A,H2,Y2))
         => ( aa(set(A),$o,finite_finite2(A),N4)
           => ( ( N4 != bot_bot(set(A)) )
             => ( aa(A,A,H2,aa(set(A),A,lattic643756798349783984er_Max(A),N4)) = aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),image2(A,A,H2),N4)) ) ) ) ) ) ).

% hom_Max_commute
tff(fact_3690_Max_Osubset,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),B3: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( B3 != bot_bot(set(A)) )
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),A4)
             => ( aa(A,A,aa(A,fun(A,A),ord_max(A),aa(set(A),A,lattic643756798349783984er_Max(A),B3)),aa(set(A),A,lattic643756798349783984er_Max(A),A4)) = aa(set(A),A,lattic643756798349783984er_Max(A),A4) ) ) ) ) ) ).

% Max.subset
tff(fact_3691_Max_Oinsert__not__elem,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),X: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ~ aa(set(A),$o,member(A,X),A4)
           => ( ( A4 != bot_bot(set(A)) )
             => ( aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A4)) = aa(A,A,aa(A,fun(A,A),ord_max(A),X),aa(set(A),A,lattic643756798349783984er_Max(A),A4)) ) ) ) ) ) ).

% Max.insert_not_elem
tff(fact_3692_Max_Oclosed,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( ! [X2: A,Y2: A] : aa(set(A),$o,member(A,aa(A,A,aa(A,fun(A,A),ord_max(A),X2),Y2)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y2),bot_bot(set(A)))))
             => aa(set(A),$o,member(A,aa(set(A),A,lattic643756798349783984er_Max(A),A4)),A4) ) ) ) ) ).

% Max.closed
tff(fact_3693_Max_Ounion,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),B3: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( aa(set(A),$o,finite_finite2(A),B3)
             => ( ( B3 != bot_bot(set(A)) )
               => ( aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(set(A),A,lattic643756798349783984er_Max(A),A4)),aa(set(A),A,lattic643756798349783984er_Max(A),B3)) ) ) ) ) ) ) ).

% Max.union
tff(fact_3694_and__not__num_Osimps_I8_J,axiom,
    ! [M: num,N: num] : bit_and_not_num(aa(num,num,bit1,M),aa(num,num,bit0,N)) = case_option(option(num),num,aa(num,option(num),some(num),one2),aTP_Lamp_st(num,option(num)),bit_and_not_num(M,N)) ).

% and_not_num.simps(8)
tff(fact_3695_Max__add__commute,axiom,
    ! [A: $tType,B: $tType] :
      ( linord4140545234300271783up_add(B)
     => ! [S: set(A),F: fun(A,B),K: B] :
          ( aa(set(A),$o,finite_finite2(A),S)
         => ( ( S != bot_bot(set(A)) )
           => ( aa(set(B),B,lattic643756798349783984er_Max(B),aa(set(A),set(B),image2(A,B,aa(B,fun(A,B),aTP_Lamp_su(fun(A,B),fun(B,fun(A,B)),F),K)),S)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(B),B,lattic643756798349783984er_Max(B),aa(set(A),set(B),image2(A,B,F),S))),K) ) ) ) ) ).

% Max_add_commute
tff(fact_3696_divide__nat__def,axiom,
    ! [M: nat,N: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N) = $ite(N = zero_zero(nat),zero_zero(nat),aa(set(nat),nat,lattic643756798349783984er_Max(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_sv(nat,fun(nat,fun(nat,$o)),M),N)))) ).

% divide_nat_def
tff(fact_3697_Max_Oinsert__remove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),X: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A4)) = $ite(aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) = bot_bot(set(A)),X,aa(A,A,aa(A,fun(A,A),ord_max(A),X),aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))))))) ) ) ) ).

% Max.insert_remove
tff(fact_3698_Max_Oremove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),X: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),$o,member(A,X),A4)
           => ( aa(set(A),A,lattic643756798349783984er_Max(A),A4) = $ite(aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) = bot_bot(set(A)),X,aa(A,A,aa(A,fun(A,A),ord_max(A),X),aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))))))) ) ) ) ) ).

% Max.remove
tff(fact_3699_sup__Some,axiom,
    ! [A: $tType] :
      ( sup(A)
     => ! [X: A,Y: A] : aa(option(A),option(A),aa(option(A),fun(option(A),option(A)),sup_sup(option(A)),aa(A,option(A),some(A),X)),aa(A,option(A),some(A),Y)) = aa(A,option(A),some(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y)) ) ).

% sup_Some
tff(fact_3700_inf__Some,axiom,
    ! [A: $tType] :
      ( inf(A)
     => ! [X: A,Y: A] : aa(option(A),option(A),aa(option(A),fun(option(A),option(A)),inf_inf(option(A)),aa(A,option(A),some(A),X)),aa(A,option(A),some(A),Y)) = aa(A,option(A),some(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)) ) ).

% inf_Some
tff(fact_3701_Some__SUP,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A4: set(A),F: fun(A,B)] :
          ( ( A4 != bot_bot(set(A)) )
         => ( aa(B,option(B),some(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F),A4))) = aa(set(option(B)),option(B),complete_Sup_Sup(option(B)),aa(set(A),set(option(B)),image2(A,option(B),aTP_Lamp_sw(fun(A,B),fun(A,option(B)),F)),A4)) ) ) ) ).

% Some_SUP
tff(fact_3702_rel__of__def,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B)),P: fun(product_prod(A,B),$o)] : rel_of(A,B,M,P) = aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aa(fun(product_prod(A,B),$o),fun(A,fun(B,$o)),aTP_Lamp_sx(fun(A,option(B)),fun(fun(product_prod(A,B),$o),fun(A,fun(B,$o))),M),P))) ).

% rel_of_def
tff(fact_3703_Max__divisors__self__int,axiom,
    ! [N: int] :
      ( ( N != zero_zero(int) )
     => ( aa(set(int),int,lattic643756798349783984er_Max(int),aa(fun(int,$o),set(int),collect(int),aTP_Lamp_is(int,fun(int,$o),N))) = aa(int,int,abs_abs(int),N) ) ) ).

% Max_divisors_self_int
tff(fact_3704_less__eq__option__def,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: option(A),Y: option(A)] :
          ( aa(option(A),$o,aa(option(A),fun(option(A),$o),ord_less_eq(option(A)),X),Y)
        <=> case_option($o,A,$true,aTP_Lamp_sy(option(A),fun(A,$o),Y),X) ) ) ).

% less_eq_option_def
tff(fact_3705_less__option__def,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: option(A),Y: option(A)] :
          ( aa(option(A),$o,aa(option(A),fun(option(A),$o),ord_less(option(A)),X),Y)
        <=> case_option($o,A,$false,aTP_Lamp_ta(option(A),fun(A,$o),X),Y) ) ) ).

% less_option_def
tff(fact_3706_top__option__def,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ( top_top(option(A)) = aa(A,option(A),some(A),top_top(A)) ) ) ).

% top_option_def
tff(fact_3707_sup__option__def,axiom,
    ! [A: $tType] :
      ( sup(A)
     => ! [X: option(A),Y: option(A)] : aa(option(A),option(A),aa(option(A),fun(option(A),option(A)),sup_sup(option(A)),X),Y) = case_option(option(A),A,Y,aa(option(A),fun(A,option(A)),aTP_Lamp_tc(option(A),fun(option(A),fun(A,option(A))),X),Y),X) ) ).

% sup_option_def
tff(fact_3708_Some__Sup,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A4: set(A)] :
          ( ( A4 != bot_bot(set(A)) )
         => ( aa(A,option(A),some(A),aa(set(A),A,complete_Sup_Sup(A),A4)) = aa(set(option(A)),option(A),complete_Sup_Sup(option(A)),aa(set(A),set(option(A)),image2(A,option(A),some(A)),A4)) ) ) ) ).

% Some_Sup
tff(fact_3709_Some__INF,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F: fun(B,A),A4: set(B)] : aa(A,option(A),some(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F),A4))) = aa(set(option(A)),option(A),complete_Inf_Inf(option(A)),aa(set(B),set(option(A)),image2(B,option(A),aTP_Lamp_td(fun(B,A),fun(B,option(A)),F)),A4)) ) ).

% Some_INF
tff(fact_3710_finite__range__updI,axiom,
    ! [A: $tType,B: $tType,F: fun(B,option(A)),A3: B,B2: A] :
      ( aa(set(option(A)),$o,finite_finite2(option(A)),aa(set(B),set(option(A)),image2(B,option(A),F),top_top(set(B))))
     => aa(set(option(A)),$o,finite_finite2(option(A)),aa(set(B),set(option(A)),image2(B,option(A),fun_upd(B,option(A),F,A3,aa(A,option(A),some(A),B2))),top_top(set(B)))) ) ).

% finite_range_updI
tff(fact_3711_take__bit__num__simps_I6_J,axiom,
    ! [R2: num,M: num] : bit_take_bit_num(aa(num,nat,numeral_numeral(nat),R2),aa(num,num,bit0,M)) = case_option(option(num),num,none(num),aTP_Lamp_te(num,option(num)),bit_take_bit_num(pred_numeral(R2),M)) ).

% take_bit_num_simps(6)
tff(fact_3712_surj__int__encode,axiom,
    aa(set(int),set(nat),image2(int,nat,nat_int_encode),top_top(set(int))) = top_top(set(nat)) ).

% surj_int_encode
tff(fact_3713_prod__decode__aux_Oelims,axiom,
    ! [X: nat,Xa: nat,Y: product_prod(nat,nat)] :
      ( ( nat_prod_decode_aux(X,Xa) = Y )
     => ( Y = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xa),X),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),X),Xa)),nat_prod_decode_aux(aa(nat,nat,suc,X),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Xa),aa(nat,nat,suc,X)))) ) ) ).

% prod_decode_aux.elims
tff(fact_3714_empty__Sup,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ( aa(set(option(A)),option(A),complete_Sup_Sup(option(A)),bot_bot(set(option(A)))) = none(A) ) ) ).

% empty_Sup
tff(fact_3715_singleton__None__Sup,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ( aa(set(option(A)),option(A),complete_Sup_Sup(option(A)),aa(set(option(A)),set(option(A)),aa(option(A),fun(set(option(A)),set(option(A))),insert2(option(A)),none(A)),bot_bot(set(option(A))))) = none(A) ) ) ).

% singleton_None_Sup
tff(fact_3716_empty__upd__none,axiom,
    ! [A: $tType,B: $tType,X: A,X3: A] : aa(A,option(B),fun_upd(A,option(B),aTP_Lamp_tf(A,option(B)),X,none(B)),X3) = none(B) ).

% empty_upd_none
tff(fact_3717_inf__None__2,axiom,
    ! [A: $tType] :
      ( inf(A)
     => ! [X: option(A)] : aa(option(A),option(A),aa(option(A),fun(option(A),option(A)),inf_inf(option(A)),X),none(A)) = none(A) ) ).

% inf_None_2
tff(fact_3718_inf__None__1,axiom,
    ! [A: $tType] :
      ( inf(A)
     => ! [Y: option(A)] : aa(option(A),option(A),aa(option(A),fun(option(A),option(A)),inf_inf(option(A)),none(A)),Y) = none(A) ) ).

% inf_None_1
tff(fact_3719_sup__None__1,axiom,
    ! [A: $tType] :
      ( sup(A)
     => ! [Y: option(A)] : aa(option(A),option(A),aa(option(A),fun(option(A),option(A)),sup_sup(option(A)),none(A)),Y) = Y ) ).

% sup_None_1
tff(fact_3720_sup__None__2,axiom,
    ! [A: $tType] :
      ( sup(A)
     => ! [X: option(A)] : aa(option(A),option(A),aa(option(A),fun(option(A),option(A)),sup_sup(option(A)),X),none(A)) = X ) ).

% sup_None_2
tff(fact_3721_rel__of__empty,axiom,
    ! [B: $tType,A: $tType,P: fun(product_prod(A,B),$o)] : rel_of(A,B,aTP_Lamp_tf(A,option(B)),P) = bot_bot(set(product_prod(A,B))) ).

% rel_of_empty
tff(fact_3722_take__bit__num__simps_I3_J,axiom,
    ! [N: nat,M: num] : bit_take_bit_num(aa(nat,nat,suc,N),aa(num,num,bit0,M)) = case_option(option(num),num,none(num),aTP_Lamp_te(num,option(num)),bit_take_bit_num(N,M)) ).

% take_bit_num_simps(3)
tff(fact_3723_option_Odisc__eq__case_I2_J,axiom,
    ! [A: $tType,Option: option(A)] :
      ( ( Option != none(A) )
    <=> case_option($o,A,$false,aTP_Lamp_ar(A,$o),Option) ) ).

% option.disc_eq_case(2)
tff(fact_3724_option_Odisc__eq__case_I1_J,axiom,
    ! [A: $tType,Option: option(A)] :
      ( ( Option = none(A) )
    <=> case_option($o,A,$true,aTP_Lamp_ak(A,$o),Option) ) ).

% option.disc_eq_case(1)
tff(fact_3725_bot__option__def,axiom,
    ! [A: $tType] :
      ( order(A)
     => ( bot_bot(option(A)) = none(A) ) ) ).

% bot_option_def
tff(fact_3726_case__optionE,axiom,
    ! [A: $tType,P: $o,Q: fun(A,$o),X: option(A)] :
      ( case_option($o,A,(P),Q,X)
     => ( ( ( X = none(A) )
         => ~ (P) )
       => ~ ! [Y2: A] :
              ( ( X = aa(A,option(A),some(A),Y2) )
             => ~ aa(A,$o,Q,Y2) ) ) ) ).

% case_optionE
tff(fact_3727_notin__range__Some,axiom,
    ! [A: $tType,X: option(A)] :
      ( ~ aa(set(option(A)),$o,member(option(A),X),aa(set(A),set(option(A)),image2(A,option(A),some(A)),top_top(set(A))))
    <=> ( X = none(A) ) ) ).

% notin_range_Some
tff(fact_3728_UNIV__option__conv,axiom,
    ! [A: $tType] : top_top(set(option(A))) = aa(set(option(A)),set(option(A)),aa(option(A),fun(set(option(A)),set(option(A))),insert2(option(A)),none(A)),aa(set(A),set(option(A)),image2(A,option(A),some(A)),top_top(set(A)))) ).

% UNIV_option_conv
tff(fact_3729_inf__option__def,axiom,
    ! [A: $tType] :
      ( inf(A)
     => ! [X: option(A),Y: option(A)] : aa(option(A),option(A),aa(option(A),fun(option(A),option(A)),inf_inf(option(A)),X),Y) = case_option(option(A),A,none(A),aTP_Lamp_th(option(A),fun(A,option(A)),Y),X) ) ).

% inf_option_def
tff(fact_3730_prod__decode__aux_Osimps,axiom,
    ! [K: nat,M: nat] :
      nat_prod_decode_aux(K,M) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),K),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),M),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K),M)),nat_prod_decode_aux(aa(nat,nat,suc,K),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),aa(nat,nat,suc,K)))) ).

% prod_decode_aux.simps
tff(fact_3731_take__bit__num__code,axiom,
    ! [N: nat,M: num] : bit_take_bit_num(N,M) = aa(product_prod(nat,num),option(num),aa(fun(nat,fun(num,option(num))),fun(product_prod(nat,num),option(num)),product_case_prod(nat,num,option(num)),aTP_Lamp_tl(nat,fun(num,option(num)))),aa(num,product_prod(nat,num),aa(nat,fun(num,product_prod(nat,num)),product_Pair(nat,num),N),M)) ).

% take_bit_num_code
tff(fact_3732_disjE__realizer2,axiom,
    ! [B: $tType,A: $tType,P: $o,Q: fun(A,$o),X: option(A),R: fun(B,$o),F: B,G: fun(A,B)] :
      ( case_option($o,A,(P),Q,X)
     => ( ( (P)
         => aa(B,$o,R,F) )
       => ( ! [Q5: A] :
              ( aa(A,$o,Q,Q5)
             => aa(B,$o,R,aa(A,B,G,Q5)) )
         => aa(B,$o,R,case_option(B,A,F,G,X)) ) ) ) ).

% disjE_realizer2
tff(fact_3733_graph__map__upd,axiom,
    ! [A: $tType,B: $tType,M: fun(A,option(B)),K: A,V: B] : graph(A,B,fun_upd(A,option(B),M,K,aa(B,option(B),some(B),V))) = aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(product_prod(A,B),fun(set(product_prod(A,B)),set(product_prod(A,B))),insert2(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K),V)),graph(A,B,fun_upd(A,option(B),M,K,none(B)))) ).

% graph_map_upd
tff(fact_3734_dual__Min,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( lattices_Min(A,aTP_Lamp_tm(A,fun(A,$o))) = lattic643756798349783984er_Max(A) ) ) ).

% dual_Min
tff(fact_3735_not__Some__eq2,axiom,
    ! [B: $tType,A: $tType,V: option(product_prod(A,B))] :
      ( ! [X4: A,Y3: B] : V != aa(product_prod(A,B),option(product_prod(A,B)),some(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Y3))
    <=> ( V = none(product_prod(A,B)) ) ) ).

% not_Some_eq2
tff(fact_3736_graph__empty,axiom,
    ! [B: $tType,A: $tType] : graph(A,B,aTP_Lamp_tf(A,option(B))) = bot_bot(set(product_prod(A,B))) ).

% graph_empty
tff(fact_3737_num_Ocase__distrib,axiom,
    ! [B: $tType,A: $tType,H2: fun(B,A),F1: B,F22: fun(num,B),F32: fun(num,B),Num: num] : aa(B,A,H2,case_num(B,F1,F22,F32,Num)) = case_num(A,aa(B,A,H2,F1),aa(fun(num,B),fun(num,A),aTP_Lamp_tn(fun(B,A),fun(fun(num,B),fun(num,A)),H2),F22),aa(fun(num,B),fun(num,A),aTP_Lamp_tn(fun(B,A),fun(fun(num,B),fun(num,A)),H2),F32),Num) ).

% num.case_distrib
tff(fact_3738_in__graphI,axiom,
    ! [A: $tType,B: $tType,M: fun(B,option(A)),K: B,V: A] :
      ( ( aa(B,option(A),M,K) = aa(A,option(A),some(A),V) )
     => aa(set(product_prod(B,A)),$o,member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),K),V)),graph(B,A,M)) ) ).

% in_graphI
tff(fact_3739_in__graphD,axiom,
    ! [A: $tType,B: $tType,K: A,V: B,M: fun(A,option(B))] :
      ( aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K),V)),graph(A,B,M))
     => ( aa(A,option(B),M,K) = aa(B,option(B),some(B),V) ) ) ).

% in_graphD
tff(fact_3740_Sup__option__def,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A4: set(option(A))] :
          aa(set(option(A)),option(A),complete_Sup_Sup(option(A)),A4) = $ite(
            ( ( A4 = bot_bot(set(option(A))) )
            | ( A4 = aa(set(option(A)),set(option(A)),aa(option(A),fun(set(option(A)),set(option(A))),insert2(option(A)),none(A)),bot_bot(set(option(A)))) ) ),
            none(A),
            aa(A,option(A),some(A),aa(set(A),A,complete_Sup_Sup(A),these(A,A4))) ) ) ).

% Sup_option_def
tff(fact_3741_these__not__empty__eq,axiom,
    ! [A: $tType,B3: set(option(A))] :
      ( ( these(A,B3) != bot_bot(set(A)) )
    <=> ( ( B3 != bot_bot(set(option(A))) )
        & ( B3 != aa(set(option(A)),set(option(A)),aa(option(A),fun(set(option(A)),set(option(A))),insert2(option(A)),none(A)),bot_bot(set(option(A)))) ) ) ) ).

% these_not_empty_eq
tff(fact_3742_these__empty__eq,axiom,
    ! [A: $tType,B3: set(option(A))] :
      ( ( these(A,B3) = bot_bot(set(A)) )
    <=> ( ( B3 = bot_bot(set(option(A))) )
        | ( B3 = aa(set(option(A)),set(option(A)),aa(option(A),fun(set(option(A)),set(option(A))),insert2(option(A)),none(A)),bot_bot(set(option(A)))) ) ) ) ).

% these_empty_eq
tff(fact_3743_INT__greaterThan__UNIV,axiom,
    aa(set(set(nat)),set(nat),complete_Inf_Inf(set(nat)),aa(set(nat),set(set(nat)),image2(nat,set(nat),set_ord_greaterThan(nat)),top_top(set(nat)))) = bot_bot(set(nat)) ).

% INT_greaterThan_UNIV
tff(fact_3744_Compl__greaterThan,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [K: A] : aa(set(A),set(A),uminus_uminus(set(A)),aa(A,set(A),set_ord_greaterThan(A),K)) = aa(A,set(A),set_ord_atMost(A),K) ) ).

% Compl_greaterThan
tff(fact_3745_Compl__atMost,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [K: A] : aa(set(A),set(A),uminus_uminus(set(A)),aa(A,set(A),set_ord_atMost(A),K)) = aa(A,set(A),set_ord_greaterThan(A),K) ) ).

% Compl_atMost
tff(fact_3746_these__empty,axiom,
    ! [A: $tType] : these(A,bot_bot(set(option(A)))) = bot_bot(set(A)) ).

% these_empty
tff(fact_3747_Sup__greaterThanAtLeast,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),top_top(A))
         => ( aa(set(A),A,complete_Sup_Sup(A),aa(A,set(A),set_ord_greaterThan(A),X)) = top_top(A) ) ) ) ).

% Sup_greaterThanAtLeast
tff(fact_3748_image__uminus__greaterThan,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [X: A] : aa(set(A),set(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_3749_image__uminus__lessThan,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [X: A] : aa(set(A),set(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_3750_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_3751_greaterThan__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [L: A] : aa(A,set(A),set_ord_greaterThan(A),L) = aa(fun(A,$o),set(A),collect(A),aa(A,fun(A,$o),ord_less(A),L)) ) ).

% greaterThan_def
tff(fact_3752_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_3753_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_3754_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_3755_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_3756_greaterThan__0,axiom,
    aa(nat,set(nat),set_ord_greaterThan(nat),zero_zero(nat)) = aa(set(nat),set(nat),image2(nat,nat,suc),top_top(set(nat))) ).

% greaterThan_0
tff(fact_3757_greaterThan__Suc,axiom,
    ! [K: nat] : aa(nat,set(nat),set_ord_greaterThan(nat),aa(nat,nat,suc,K)) = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),aa(nat,set(nat),set_ord_greaterThan(nat),K)),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),aa(nat,nat,suc,K)),bot_bot(set(nat)))) ).

% greaterThan_Suc
tff(fact_3758_Some__image__these__eq,axiom,
    ! [A: $tType,A4: set(option(A))] : aa(set(A),set(option(A)),image2(A,option(A),some(A)),these(A,A4)) = aa(fun(option(A),$o),set(option(A)),collect(option(A)),aTP_Lamp_to(set(option(A)),fun(option(A),$o),A4)) ).

% Some_image_these_eq
tff(fact_3759_and__not__num_Oelims,axiom,
    ! [X: num,Xa: num,Y: option(num)] :
      ( ( bit_and_not_num(X,Xa) = Y )
     => ( ( ( X = one2 )
         => ( ( Xa = one2 )
           => ( Y != none(num) ) ) )
       => ( ( ( X = one2 )
           => ( ? [N3: num] : Xa = aa(num,num,bit0,N3)
             => ( Y != aa(num,option(num),some(num),one2) ) ) )
         => ( ( ( X = one2 )
             => ( ? [N3: num] : Xa = aa(num,num,bit1,N3)
               => ( Y != none(num) ) ) )
           => ( ! [M3: num] :
                  ( ( X = aa(num,num,bit0,M3) )
                 => ( ( Xa = one2 )
                   => ( Y != aa(num,option(num),some(num),aa(num,num,bit0,M3)) ) ) )
             => ( ! [M3: num] :
                    ( ( X = aa(num,num,bit0,M3) )
                   => ! [N3: num] :
                        ( ( Xa = aa(num,num,bit0,N3) )
                       => ( Y != aa(option(num),option(num),map_option(num,num,bit0),bit_and_not_num(M3,N3)) ) ) )
               => ( ! [M3: num] :
                      ( ( X = aa(num,num,bit0,M3) )
                     => ! [N3: num] :
                          ( ( Xa = aa(num,num,bit1,N3) )
                         => ( Y != aa(option(num),option(num),map_option(num,num,bit0),bit_and_not_num(M3,N3)) ) ) )
                 => ( ! [M3: num] :
                        ( ( X = aa(num,num,bit1,M3) )
                       => ( ( Xa = one2 )
                         => ( Y != aa(num,option(num),some(num),aa(num,num,bit0,M3)) ) ) )
                   => ( ! [M3: num] :
                          ( ( X = aa(num,num,bit1,M3) )
                         => ! [N3: num] :
                              ( ( Xa = aa(num,num,bit0,N3) )
                             => ( Y != case_option(option(num),num,aa(num,option(num),some(num),one2),aTP_Lamp_st(num,option(num)),bit_and_not_num(M3,N3)) ) ) )
                     => ~ ! [M3: num] :
                            ( ( X = aa(num,num,bit1,M3) )
                           => ! [N3: num] :
                                ( ( Xa = aa(num,num,bit1,N3) )
                               => ( Y != aa(option(num),option(num),map_option(num,num,bit0),bit_and_not_num(M3,N3)) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_not_num.elims
tff(fact_3760_surj__int__decode,axiom,
    aa(set(nat),set(int),image2(nat,int,nat_int_decode),top_top(set(nat))) = top_top(set(int)) ).

% surj_int_decode
tff(fact_3761_restrict__upd__same,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B)),X: A,Y: B] : restrict_map(A,B,fun_upd(A,option(B),M,X,aa(B,option(B),some(B),Y)),aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))))) = restrict_map(A,B,M,aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))))) ).

% restrict_upd_same
tff(fact_3762_finite__mono__remains__stable__implies__strict__prefix,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F: fun(nat,A)] :
          ( aa(set(A),$o,finite_finite2(A),aa(set(nat),set(A),image2(nat,A,F),top_top(set(nat))))
         => ( order_mono(nat,A,F)
           => ( ! [N3: nat] :
                  ( ( aa(nat,A,F,N3) = aa(nat,A,F,aa(nat,nat,suc,N3)) )
                 => ( aa(nat,A,F,aa(nat,nat,suc,N3)) = aa(nat,A,F,aa(nat,nat,suc,aa(nat,nat,suc,N3))) ) )
             => ? [N6: nat] :
                  ( ! [N7: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N7),N6)
                     => ! [M5: nat] :
                          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M5),N6)
                         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M5),N7)
                           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,F,M5)),aa(nat,A,F,N7)) ) ) )
                  & ! [N7: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N6),N7)
                     => ( aa(nat,A,F,N6) = aa(nat,A,F,N7) ) ) ) ) ) ) ) ).

% finite_mono_remains_stable_implies_strict_prefix
tff(fact_3763_restrict__map__UNIV,axiom,
    ! [B: $tType,A: $tType,F: fun(A,option(B))] : restrict_map(A,B,F,top_top(set(A))) = F ).

% restrict_map_UNIV
tff(fact_3764_restrict__restrict,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B)),A4: set(A),B3: set(A)] : restrict_map(A,B,restrict_map(A,B,M,A4),B3) = restrict_map(A,B,M,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3)) ).

% restrict_restrict
tff(fact_3765_map__option__o__empty,axiom,
    ! [A: $tType,C: $tType,B: $tType,F: fun(C,B),X3: A] : aa(A,option(B),aa(fun(A,option(C)),fun(A,option(B)),comp(option(C),option(B),A,map_option(C,B,F)),aTP_Lamp_tp(A,option(C))),X3) = none(B) ).

% map_option_o_empty
tff(fact_3766_restrict__map__empty,axiom,
    ! [A: $tType,B: $tType,D4: set(A),X3: A] : aa(A,option(B),restrict_map(A,B,aTP_Lamp_tf(A,option(B)),D4),X3) = none(B) ).

% restrict_map_empty
tff(fact_3767_restrict__map__to__empty,axiom,
    ! [A: $tType,B: $tType,M: fun(A,option(B)),X3: A] : aa(A,option(B),restrict_map(A,B,M,bot_bot(set(A))),X3) = none(B) ).

% restrict_map_to_empty
tff(fact_3768_restrict__fun__upd,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B)),X: A,Y: option(B),D4: set(A)] :
      restrict_map(A,B,fun_upd(A,option(B),M,X,Y),D4) = $ite(aa(set(A),$o,member(A,X),D4),fun_upd(A,option(B),restrict_map(A,B,M,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),D4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))))),X,Y),restrict_map(A,B,M,D4)) ).

% restrict_fun_upd
tff(fact_3769_fun__upd__restrict__conv,axiom,
    ! [A: $tType,B: $tType,X: A,D4: set(A),M: fun(A,option(B)),Y: option(B)] :
      ( aa(set(A),$o,member(A,X),D4)
     => ( fun_upd(A,option(B),restrict_map(A,B,M,D4),X,Y) = fun_upd(A,option(B),restrict_map(A,B,M,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),D4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))))),X,Y) ) ) ).

% fun_upd_restrict_conv
tff(fact_3770_fun__upd__None__restrict,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B)),D4: set(A),X: A] :
      fun_upd(A,option(B),restrict_map(A,B,M,D4),X,none(B)) = $ite(aa(set(A),$o,member(A,X),D4),restrict_map(A,B,M,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),D4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))))),restrict_map(A,B,M,D4)) ).

% fun_upd_None_restrict
tff(fact_3771_mono__strict__invE,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & order(B) )
     => ! [F: fun(A,B),X: A,Y: A] :
          ( order_mono(A,B,F)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F,X)),aa(A,B,F,Y))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y) ) ) ) ).

% mono_strict_invE
tff(fact_3772_mono__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F: fun(A,B)] :
          ( order_mono(A,B,F)
        <=> ! [X4: A,Y3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),Y3)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,X4)),aa(A,B,F,Y3)) ) ) ) ).

% mono_def
tff(fact_3773_monoI,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F: fun(A,B)] :
          ( ! [X2: A,Y2: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),Y2)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,X2)),aa(A,B,F,Y2)) )
         => order_mono(A,B,F) ) ) ).

% monoI
tff(fact_3774_monoE,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F: fun(A,B),X: A,Y: A] :
          ( order_mono(A,B,F)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,X)),aa(A,B,F,Y)) ) ) ) ).

% monoE
tff(fact_3775_monoD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F: fun(A,B),X: A,Y: A] :
          ( order_mono(A,B,F)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,X)),aa(A,B,F,Y)) ) ) ) ).

% monoD
tff(fact_3776_mono__add,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [A3: A] : order_mono(A,A,aa(A,fun(A,A),plus_plus(A),A3)) ) ).

% mono_add
tff(fact_3777_max__of__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & linorder(B) )
     => ! [F: fun(A,B),M: A,N: A] :
          ( order_mono(A,B,F)
         => ( aa(B,B,aa(B,fun(B,B),ord_max(B),aa(A,B,F,M)),aa(A,B,F,N)) = aa(A,B,F,aa(A,A,aa(A,fun(A,A),ord_max(A),M),N)) ) ) ) ).

% max_of_mono
tff(fact_3778_option_Omap__ident,axiom,
    ! [A: $tType,T3: option(A)] : aa(option(A),option(A),map_option(A,A,aTP_Lamp_au(A,A)),T3) = T3 ).

% option.map_ident
tff(fact_3779_mono__invE,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & order(B) )
     => ! [F: fun(A,B),X: A,Y: A] :
          ( order_mono(A,B,F)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F,X)),aa(A,B,F,Y))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y) ) ) ) ).

% mono_invE
tff(fact_3780_mono__inf,axiom,
    ! [B: $tType,A: $tType] :
      ( ( semilattice_inf(A)
        & semilattice_inf(B) )
     => ! [F: fun(A,B),A4: A,B3: A] :
          ( order_mono(A,B,F)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,aa(A,A,aa(A,fun(A,A),inf_inf(A),A4),B3))),aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(A,B,F,A4)),aa(A,B,F,B3))) ) ) ).

% mono_inf
tff(fact_3781_mono__sup,axiom,
    ! [B: $tType,A: $tType] :
      ( ( semilattice_sup(A)
        & semilattice_sup(B) )
     => ! [F: fun(A,B),A4: A,B3: A] :
          ( order_mono(A,B,F)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),sup_sup(B),aa(A,B,F,A4)),aa(A,B,F,B3))),aa(A,B,F,aa(A,A,aa(A,fun(A,A),sup_sup(A),A4),B3))) ) ) ).

% mono_sup
tff(fact_3782_graph__restrictD_I1_J,axiom,
    ! [B: $tType,A: $tType,K: A,V: B,M: fun(A,option(B)),A4: set(A)] :
      ( aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K),V)),graph(A,B,restrict_map(A,B,M,A4)))
     => aa(set(A),$o,member(A,K),A4) ) ).

% graph_restrictD(1)
tff(fact_3783_Rings_Omono__mult,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
         => order_mono(A,A,aa(A,fun(A,A),times_times(A),A3)) ) ) ).

% Rings.mono_mult
tff(fact_3784_mono__Sup,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comple6319245703460814977attice(A)
        & comple6319245703460814977attice(B) )
     => ! [F: fun(A,B),A4: set(A)] :
          ( order_mono(A,B,F)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F),A4))),aa(A,B,F,aa(set(A),A,complete_Sup_Sup(A),A4))) ) ) ).

% mono_Sup
tff(fact_3785_mono__SUP,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( comple6319245703460814977attice(A)
        & comple6319245703460814977attice(B) )
     => ! [F: fun(A,B),A4: fun(C,A),I4: set(C)] :
          ( order_mono(A,B,F)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(C),set(B),image2(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_tq(fun(A,B),fun(fun(C,A),fun(C,B)),F),A4)),I4))),aa(A,B,F,aa(set(A),A,complete_Sup_Sup(A),aa(set(C),set(A),image2(C,A,A4),I4)))) ) ) ).

% mono_SUP
tff(fact_3786_mono__INF,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( comple6319245703460814977attice(B)
        & comple6319245703460814977attice(A) )
     => ! [F: fun(A,B),A4: fun(C,A),I4: set(C)] :
          ( order_mono(A,B,F)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,aa(set(A),A,complete_Inf_Inf(A),aa(set(C),set(A),image2(C,A,A4),I4)))),aa(set(B),B,complete_Inf_Inf(B),aa(set(C),set(B),image2(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_tq(fun(A,B),fun(fun(C,A),fun(C,B)),F),A4)),I4))) ) ) ).

% mono_INF
tff(fact_3787_mono__Inf,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comple6319245703460814977attice(A)
        & comple6319245703460814977attice(B) )
     => ! [F: fun(A,B),A4: set(A)] :
          ( order_mono(A,B,F)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,aa(set(A),A,complete_Inf_Inf(A),A4))),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F),A4))) ) ) ).

% mono_Inf
tff(fact_3788_map__restrict__insert__none__simp,axiom,
    ! [A: $tType,B: $tType,M: fun(B,option(A)),X: B,S2: set(B)] :
      ( ( aa(B,option(A),M,X) = none(A) )
     => ( restrict_map(B,A,M,aa(set(B),set(B),uminus_uminus(set(B)),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),X),S2))) = restrict_map(B,A,M,aa(set(B),set(B),uminus_uminus(set(B)),S2)) ) ) ).

% map_restrict_insert_none_simp
tff(fact_3789_graph__restrictD_I2_J,axiom,
    ! [A: $tType,B: $tType,K: A,V: B,M: fun(A,option(B)),A4: set(A)] :
      ( aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K),V)),graph(A,B,restrict_map(A,B,M,A4)))
     => ( aa(A,option(B),M,K) = aa(B,option(B),some(B),V) ) ) ).

% graph_restrictD(2)
tff(fact_3790_map__option__case,axiom,
    ! [A: $tType,B: $tType,F: fun(B,A),Y: option(B)] : aa(option(B),option(A),map_option(B,A,F),Y) = case_option(option(A),B,none(A),aTP_Lamp_tr(fun(B,A),fun(B,option(A)),F),Y) ).

% map_option_case
tff(fact_3791_mono__Max__commute,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & linorder(B) )
     => ! [F: fun(A,B),A4: set(A)] :
          ( order_mono(A,B,F)
         => ( aa(set(A),$o,finite_finite2(A),A4)
           => ( ( A4 != bot_bot(set(A)) )
             => ( aa(A,B,F,aa(set(A),A,lattic643756798349783984er_Max(A),A4)) = aa(set(B),B,lattic643756798349783984er_Max(B),aa(set(A),set(B),image2(A,B,F),A4)) ) ) ) ) ) ).

% mono_Max_commute
tff(fact_3792_fun__upd__restrict,axiom,
    ! [A: $tType,B: $tType,M: fun(A,option(B)),D4: set(A),X: A,Y: option(B)] : fun_upd(A,option(B),restrict_map(A,B,M,D4),X,Y) = fun_upd(A,option(B),restrict_map(A,B,M,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),D4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))))),X,Y) ).

% fun_upd_restrict
tff(fact_3793_map__upd__eq__restrict,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B)),X: A] : fun_upd(A,option(B),M,X,none(B)) = restrict_map(A,B,M,aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))))) ).

% map_upd_eq_restrict
tff(fact_3794_restrict__complement__singleton__eq,axiom,
    ! [A: $tType,B: $tType,F: fun(A,option(B)),X: A] : restrict_map(A,B,F,aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))))) = fun_upd(A,option(B),F,X,none(B)) ).

% restrict_complement_singleton_eq
tff(fact_3795_mono__ge2__power__minus__self,axiom,
    ! [K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),K)
     => order_mono(nat,nat,aTP_Lamp_ts(nat,fun(nat,nat),K)) ) ).

% mono_ge2_power_minus_self
tff(fact_3796_and__num_Oelims,axiom,
    ! [X: num,Xa: num,Y: option(num)] :
      ( ( bit_un7362597486090784418nd_num(X,Xa) = Y )
     => ( ( ( X = one2 )
         => ( ( Xa = one2 )
           => ( Y != aa(num,option(num),some(num),one2) ) ) )
       => ( ( ( X = one2 )
           => ( ? [N3: num] : Xa = aa(num,num,bit0,N3)
             => ( Y != none(num) ) ) )
         => ( ( ( X = one2 )
             => ( ? [N3: num] : Xa = aa(num,num,bit1,N3)
               => ( Y != aa(num,option(num),some(num),one2) ) ) )
           => ( ( ? [M3: num] : X = aa(num,num,bit0,M3)
               => ( ( Xa = one2 )
                 => ( Y != none(num) ) ) )
             => ( ! [M3: num] :
                    ( ( X = aa(num,num,bit0,M3) )
                   => ! [N3: num] :
                        ( ( Xa = aa(num,num,bit0,N3) )
                       => ( Y != aa(option(num),option(num),map_option(num,num,bit0),bit_un7362597486090784418nd_num(M3,N3)) ) ) )
               => ( ! [M3: num] :
                      ( ( X = aa(num,num,bit0,M3) )
                     => ! [N3: num] :
                          ( ( Xa = aa(num,num,bit1,N3) )
                         => ( Y != aa(option(num),option(num),map_option(num,num,bit0),bit_un7362597486090784418nd_num(M3,N3)) ) ) )
                 => ( ( ? [M3: num] : X = aa(num,num,bit1,M3)
                     => ( ( Xa = one2 )
                       => ( Y != aa(num,option(num),some(num),one2) ) ) )
                   => ( ! [M3: num] :
                          ( ( X = aa(num,num,bit1,M3) )
                         => ! [N3: num] :
                              ( ( Xa = aa(num,num,bit0,N3) )
                             => ( Y != aa(option(num),option(num),map_option(num,num,bit0),bit_un7362597486090784418nd_num(M3,N3)) ) ) )
                     => ~ ! [M3: num] :
                            ( ( X = aa(num,num,bit1,M3) )
                           => ! [N3: num] :
                                ( ( Xa = aa(num,num,bit1,N3) )
                               => ( Y != case_option(option(num),num,aa(num,option(num),some(num),one2),aTP_Lamp_st(num,option(num)),bit_un7362597486090784418nd_num(M3,N3)) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_num.elims
tff(fact_3797_comp__fun__idem__on_Ocomp__comp__fun__idem__on,axiom,
    ! [B: $tType,A: $tType,C: $tType,S: set(A),F: fun(A,fun(B,B)),G: fun(C,A),R: set(C)] :
      ( finite673082921795544331dem_on(A,B,S,F)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(C),set(A),image2(C,A,G),top_top(set(C)))),S)
       => finite673082921795544331dem_on(C,B,R,aa(fun(C,A),fun(C,fun(B,B)),comp(A,fun(B,B),C,F),G)) ) ) ).

% comp_fun_idem_on.comp_comp_fun_idem_on
tff(fact_3798_ring__1__class_Oof__int__def,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ( ring_1_of_int(A) = aa(fun(product_prod(nat,nat),A),fun(int,A),map_fun(int,product_prod(nat,nat),A,A,rep_Integ,id(A)),aa(fun(nat,fun(nat,A)),fun(product_prod(nat,nat),A),product_case_prod(nat,nat,A),aTP_Lamp_pn(nat,fun(nat,A)))) ) ) ).

% ring_1_class.of_int_def
tff(fact_3799_INF__principal__finite,axiom,
    ! [B: $tType,A: $tType,X5: set(A),F: fun(A,set(B))] :
      ( aa(set(A),$o,finite_finite2(A),X5)
     => ( aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image2(A,filter(B),aTP_Lamp_tt(fun(A,set(B)),fun(A,filter(B)),F)),X5)) = principal(B,aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image2(A,set(B),F),X5))) ) ) ).

% INF_principal_finite
tff(fact_3800_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_3801_sup__principal,axiom,
    ! [A: $tType,A4: set(A),B3: set(A)] : aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),sup_sup(filter(A)),principal(A,A4)),principal(A,B3)) = principal(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3)) ).

% sup_principal
tff(fact_3802_group__add__class_Ominus__comp__minus,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ( aa(fun(A,A),fun(A,A),comp(A,A,A,uminus_uminus(A)),uminus_uminus(A)) = id(A) ) ) ).

% group_add_class.minus_comp_minus
tff(fact_3803_boolean__algebra__class_Ominus__comp__minus,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ( aa(fun(A,A),fun(A,A),comp(A,A,A,uminus_uminus(A)),uminus_uminus(A)) = id(A) ) ) ).

% boolean_algebra_class.minus_comp_minus
tff(fact_3804_inf__principal,axiom,
    ! [A: $tType,A4: set(A),B3: set(A)] : aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),inf_inf(filter(A)),principal(A,A4)),principal(A,B3)) = principal(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3)) ).

% inf_principal
tff(fact_3805_SUP__principal,axiom,
    ! [A: $tType,B: $tType,A4: fun(B,set(A)),I4: set(B)] : aa(set(filter(A)),filter(A),complete_Sup_Sup(filter(A)),aa(set(B),set(filter(A)),image2(B,filter(A),aTP_Lamp_tu(fun(B,set(A)),fun(B,filter(A)),A4)),I4)) = principal(A,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),A4),I4))) ).

% SUP_principal
tff(fact_3806_DEADID_Oin__rel,axiom,
    ! [A: $tType,A3: A,B2: A] :
      ( ( A3 = B2 )
    <=> ? [Z4: A] :
          ( aa(set(A),$o,member(A,Z4),top_top(set(A)))
          & ( aa(A,A,id(A),Z4) = A3 )
          & ( aa(A,A,id(A),Z4) = B2 ) ) ) ).

% DEADID.in_rel
tff(fact_3807_top__eq__principal__UNIV,axiom,
    ! [A: $tType] : top_top(filter(A)) = principal(A,top_top(set(A))) ).

% top_eq_principal_UNIV
tff(fact_3808_map__fun_Oidentity,axiom,
    ! [B: $tType,A: $tType] : map_fun(A,A,B,B,aTP_Lamp_au(A,A),aTP_Lamp_oc(B,B)) = id(fun(A,B)) ).

% map_fun.identity
tff(fact_3809_set_Oidentity,axiom,
    ! [A: $tType] : aa(fun(A,A),fun(set(A),set(A)),vimage(A,A),aTP_Lamp_au(A,A)) = id(set(A)) ).

% set.identity
tff(fact_3810_map__option_Oidentity,axiom,
    ! [A: $tType] : map_option(A,A,aTP_Lamp_au(A,A)) = id(option(A)) ).

% map_option.identity
tff(fact_3811_nat__def,axiom,
    nat2 = aa(fun(product_prod(nat,nat),nat),fun(int,nat),map_fun(int,product_prod(nat,nat),nat,nat,rep_Integ,id(nat)),aa(fun(nat,fun(nat,nat)),fun(product_prod(nat,nat),nat),product_case_prod(nat,nat,nat),minus_minus(nat))) ).

% nat_def
tff(fact_3812_mono__Int,axiom,
    ! [B: $tType,A: $tType,F: fun(set(A),set(B)),A4: set(A),B3: set(A)] :
      ( order_mono(set(A),set(B),F)
     => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),F,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3))),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(set(A),set(B),F,A4)),aa(set(A),set(B),F,B3))) ) ).

% mono_Int
tff(fact_3813_mono__Un,axiom,
    ! [B: $tType,A: $tType,F: fun(set(A),set(B)),A4: set(A),B3: set(A)] :
      ( order_mono(set(A),set(B),F)
     => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),aa(set(A),set(B),F,A4)),aa(set(A),set(B),F,B3))),aa(set(A),set(B),F,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3))) ) ).

% mono_Un
tff(fact_3814_surj__id,axiom,
    ! [A: $tType] : aa(set(A),set(A),image2(A,A,id(A)),top_top(set(A))) = top_top(set(A)) ).

% surj_id
tff(fact_3815_less__int__def,axiom,
    ord_less(int) = aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)),fun(int,fun(int,$o)),map_fun(int,product_prod(nat,nat),fun(product_prod(nat,nat),$o),fun(int,$o),rep_Integ,map_fun(int,product_prod(nat,nat),$o,$o,rep_Integ,id($o))),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),$o))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)),product_case_prod(nat,nat,fun(product_prod(nat,nat),$o)),aTP_Lamp_pp(nat,fun(nat,fun(product_prod(nat,nat),$o))))) ).

% less_int_def
tff(fact_3816_less__eq__int__def,axiom,
    ord_less_eq(int) = aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)),fun(int,fun(int,$o)),map_fun(int,product_prod(nat,nat),fun(product_prod(nat,nat),$o),fun(int,$o),rep_Integ,map_fun(int,product_prod(nat,nat),$o,$o,rep_Integ,id($o))),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),$o))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)),product_case_prod(nat,nat,fun(product_prod(nat,nat),$o)),aTP_Lamp_pr(nat,fun(nat,fun(product_prod(nat,nat),$o))))) ).

% less_eq_int_def
tff(fact_3817_bot__eq__principal__empty,axiom,
    ! [A: $tType] : bot_bot(filter(A)) = principal(A,bot_bot(set(A))) ).

% bot_eq_principal_empty
tff(fact_3818_principal__eq__bot__iff,axiom,
    ! [A: $tType,X5: set(A)] :
      ( ( principal(A,X5) = bot_bot(filter(A)) )
    <=> ( X5 = bot_bot(set(A)) ) ) ).

% principal_eq_bot_iff
tff(fact_3819_type__copy__map__id0,axiom,
    ! [B: $tType,A: $tType,Rep: fun(A,B),Abs: fun(B,A),M4: fun(B,B)] :
      ( type_definition(A,B,Rep,Abs,top_top(set(B)))
     => ( ( M4 = id(B) )
       => ( aa(fun(A,B),fun(A,A),comp(B,A,A,aa(fun(B,B),fun(B,A),comp(B,A,B,Abs),M4)),Rep) = id(A) ) ) ) ).

% type_copy_map_id0
tff(fact_3820_type__copy__Abs__o__Rep,axiom,
    ! [B: $tType,A: $tType,Rep: fun(A,B),Abs: fun(B,A)] :
      ( type_definition(A,B,Rep,Abs,top_top(set(B)))
     => ( aa(fun(A,B),fun(A,A),comp(B,A,A,Abs),Rep) = id(A) ) ) ).

% type_copy_Abs_o_Rep
tff(fact_3821_type__copy__Rep__o__Abs,axiom,
    ! [A: $tType,B: $tType,Rep: fun(A,B),Abs: fun(B,A)] :
      ( type_definition(A,B,Rep,Abs,top_top(set(B)))
     => ( aa(fun(B,A),fun(B,B),comp(A,B,B,Rep),Abs) = id(B) ) ) ).

% type_copy_Rep_o_Abs
tff(fact_3822_numeral__and__num,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M: num,N: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),N)) = case_option(A,num,zero_zero(A),numeral_numeral(A),bit_un7362597486090784418nd_num(M,N)) ) ).

% numeral_and_num
tff(fact_3823_and__num_Osimps_I9_J,axiom,
    ! [M: num,N: num] : bit_un7362597486090784418nd_num(aa(num,num,bit1,M),aa(num,num,bit1,N)) = case_option(option(num),num,aa(num,option(num),some(num),one2),aTP_Lamp_st(num,option(num)),bit_un7362597486090784418nd_num(M,N)) ).

% and_num.simps(9)
tff(fact_3824_at__bot__def,axiom,
    ! [A: $tType] :
      ( order(A)
     => ( at_bot(A) = aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(A),set(filter(A)),image2(A,filter(A),aTP_Lamp_tv(A,filter(A))),top_top(set(A)))) ) ) ).

% at_bot_def
tff(fact_3825_at__bot__sub,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A] : at_bot(A) = aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(A),set(filter(A)),image2(A,filter(A),aTP_Lamp_tw(A,filter(A))),aa(A,set(A),set_ord_atMost(A),C2))) ) ).

% at_bot_sub
tff(fact_3826_finite__subsets__at__top__finite,axiom,
    ! [A: $tType,A4: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( finite5375528669736107172at_top(A,A4) = principal(set(A),aa(set(set(A)),set(set(A)),aa(set(A),fun(set(set(A)),set(set(A))),insert2(set(A)),A4),bot_bot(set(set(A))))) ) ) ).

% finite_subsets_at_top_finite
tff(fact_3827_dual__min,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( min(A,aTP_Lamp_tm(A,fun(A,$o))) = ord_max(A) ) ) ).

% dual_min
tff(fact_3828_ord_Omin__def,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,$o)),A3: A,B2: A] :
      aa(A,A,aa(A,fun(A,A),min(A,Less_eq),A3),B2) = $ite(aa(A,$o,aa(A,fun(A,$o),Less_eq,A3),B2),A3,B2) ).

% ord.min_def
tff(fact_3829_ord_Omin_Ocong,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,$o))] : min(A,Less_eq) = min(A,Less_eq) ).

% ord.min.cong
tff(fact_3830_finite__subsets__at__top__neq__bot,axiom,
    ! [A: $tType,A4: set(A)] : finite5375528669736107172at_top(A,A4) != bot_bot(filter(set(A))) ).

% finite_subsets_at_top_neq_bot
tff(fact_3831_trivial__limit__at__bot__linorder,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( at_bot(A) != bot_bot(filter(A)) ) ) ).

% trivial_limit_at_bot_linorder
tff(fact_3832_uminus__integer__def,axiom,
    uminus_uminus(code_integer) = aa(fun(int,int),fun(code_integer,code_integer),map_fun(code_integer,int,int,code_integer,code_int_of_integer,code_integer_of_int),uminus_uminus(int)) ).

% uminus_integer_def
tff(fact_3833_finite__subsets__at__top__def,axiom,
    ! [A: $tType,A4: set(A)] : finite5375528669736107172at_top(A,A4) = aa(set(filter(set(A))),filter(set(A)),complete_Inf_Inf(filter(set(A))),aa(set(set(A)),set(filter(set(A))),image2(set(A),filter(set(A)),aTP_Lamp_ty(set(A),fun(set(A),filter(set(A))),A4)),aa(fun(set(A),$o),set(set(A)),collect(set(A)),aTP_Lamp_tz(set(A),fun(set(A),$o),A4)))) ).

% finite_subsets_at_top_def
tff(fact_3834_coinduct3__mono__lemma,axiom,
    ! [A: $tType,B: $tType] :
      ( order(A)
     => ! [F: fun(A,set(B)),X5: set(B),B3: set(B)] :
          ( order_mono(A,set(B),F)
         => order_mono(A,set(B),aa(set(B),fun(A,set(B)),aa(set(B),fun(set(B),fun(A,set(B))),aTP_Lamp_ua(fun(A,set(B)),fun(set(B),fun(set(B),fun(A,set(B)))),F),X5),B3)) ) ) ).

% coinduct3_mono_lemma
tff(fact_3835_mask__mod__exp,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [N: nat,M: nat] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),M)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),M),N))),one_one(A)) ) ).

% mask_mod_exp
tff(fact_3836_image__o__collect,axiom,
    ! [B: $tType,C: $tType,A: $tType,G: fun(C,B),F4: set(fun(A,set(C)))] : bNF_collect(A,B,aa(set(fun(A,set(C))),set(fun(A,set(B))),image2(fun(A,set(C)),fun(A,set(B)),comp(set(C),set(B),A,image2(C,B,G))),F4)) = aa(fun(A,set(C)),fun(A,set(B)),comp(set(C),set(B),A,image2(C,B,G)),bNF_collect(A,C,F4)) ).

% image_o_collect
tff(fact_3837_Max_Oeq__fold_H,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A)] : aa(set(A),A,lattic643756798349783984er_Max(A),A4) = aa(option(A),A,the2(A),finite_fold(A,option(A),aTP_Lamp_ub(A,fun(option(A),option(A))),none(A),A4)) ) ).

% Max.eq_fold'
tff(fact_3838_min_Oright__idem,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2)),B2) = aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2) ) ).

% min.right_idem
tff(fact_3839_min_Oleft__idem,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),A3),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2) ) ).

% min.left_idem
tff(fact_3840_min_Oidem,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),A3),A3) = A3 ) ).

% min.idem
tff(fact_3841_min_Obounded__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),C2))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),C2) ) ) ) ).

% min.bounded_iff
tff(fact_3842_min_Oabsorb2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3)
         => ( aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2) = B2 ) ) ) ).

% min.absorb2
tff(fact_3843_min_Oabsorb1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
         => ( aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2) = A3 ) ) ) ).

% min.absorb1
tff(fact_3844_min__less__iff__conj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Z2: A,X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z2),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z2),X)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z2),Y) ) ) ) ).

% min_less_iff_conj
tff(fact_3845_min_Oabsorb4,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3)
         => ( aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2) = B2 ) ) ) ).

% min.absorb4
tff(fact_3846_min_Oabsorb3,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
         => ( aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2) = A3 ) ) ) ).

% min.absorb3
tff(fact_3847_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_3848_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_3849_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_3850_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_3851_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_3852_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_3853_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_3854_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_3855_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_3856_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_3857_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_3858_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_3859_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_3860_min__number__of_I2_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V: num] :
          aa(A,A,aa(A,fun(A,A),ord_min(A),aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),aa(num,A,numeral_numeral(A),U),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))) ) ).

% min_number_of(2)
tff(fact_3861_min__number__of_I3_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V: num] :
          aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(num,A,numeral_numeral(A),V)) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(num,A,numeral_numeral(A),V)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V)) ) ).

% min_number_of(3)
tff(fact_3862_min__number__of_I4_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V: num] :
          aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))) ) ).

% min_number_of(4)
tff(fact_3863_Int__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A,C2: A,D3: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or1337092689740270186AtMost(A,A3,B2)),set_or1337092689740270186AtMost(A,C2,D3)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),ord_max(A),A3),C2),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),D3)) ) ).

% Int_atLeastAtMost
tff(fact_3864_Int__atLeastLessThan,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A,C2: A,D3: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or7035219750837199246ssThan(A,A3,B2)),set_or7035219750837199246ssThan(A,C2,D3)) = set_or7035219750837199246ssThan(A,aa(A,A,aa(A,fun(A,A),ord_max(A),A3),C2),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),D3)) ) ).

% Int_atLeastLessThan
tff(fact_3865_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_3866_Int__atLeastAtMostR1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,C2: 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,C2,D3)) = set_or1337092689740270186AtMost(A,C2,aa(A,A,aa(A,fun(A,A),ord_min(A),B2),D3)) ) ).

% Int_atLeastAtMostR1
tff(fact_3867_Int__greaterThanLessThan,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A,C2: A,D3: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or5935395276787703475ssThan(A,A3,B2)),set_or5935395276787703475ssThan(A,C2,D3)) = set_or5935395276787703475ssThan(A,aa(A,A,aa(A,fun(A,A),ord_max(A),A3),C2),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),D3)) ) ).

% Int_greaterThanLessThan
tff(fact_3868_min__of__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & linorder(B) )
     => ! [F: fun(A,B),M: A,N: A] :
          ( order_mono(A,B,F)
         => ( aa(B,B,aa(B,fun(B,B),ord_min(B),aa(A,B,F,M)),aa(A,B,F,N)) = aa(A,B,F,aa(A,A,aa(A,fun(A,A),ord_min(A),M),N)) ) ) ) ).

% min_of_mono
tff(fact_3869_min__def__raw,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X3: A,Xa3: A] :
          aa(A,A,aa(A,fun(A,A),ord_min(A),X3),Xa3) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Xa3),X3,Xa3) ) ).

% min_def_raw
tff(fact_3870_inf__nat__def,axiom,
    inf_inf(nat) = ord_min(nat) ).

% inf_nat_def
tff(fact_3871_inf__min,axiom,
    ! [A: $tType] :
      ( ( semilattice_inf(A)
        & linorder(A) )
     => ( inf_inf(A) = ord_min(A) ) ) ).

% inf_min
tff(fact_3872_complete__linorder__inf__min,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ( inf_inf(A) = ord_min(A) ) ) ).

% complete_linorder_inf_min
tff(fact_3873_max__min__distrib1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,C2: A,A3: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),C2)),A3) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),A3)),aa(A,A,aa(A,fun(A,A),ord_max(A),C2),A3)) ) ).

% max_min_distrib1
tff(fact_3874_max__min__distrib2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),A3),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2)),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),C2)) ) ).

% max_min_distrib2
tff(fact_3875_min__max__distrib1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,C2: A,A3: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),C2)),A3) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),A3)),aa(A,A,aa(A,fun(A,A),ord_min(A),C2),A3)) ) ).

% min_max_distrib1
tff(fact_3876_min__max__distrib2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),A3),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),ord_max(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),A3),C2)) ) ).

% min_max_distrib2
tff(fact_3877_min_Oleft__commute,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A3: A,C2: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),B2),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),C2)) = aa(A,A,aa(A,fun(A,A),ord_min(A),A3),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),C2)) ) ).

% min.left_commute
tff(fact_3878_min_Ocommute,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: 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),B2),A3) ) ).

% min.commute
tff(fact_3879_min_Oassoc,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),ord_min(A),A3),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),C2)) ) ).

% min.assoc
tff(fact_3880_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_3881_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_3882_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_3883_min__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A3: A,B2: A] :
          aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2),A3,B2) ) ).

% min_def
tff(fact_3884_min__absorb1,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
         => ( aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y) = X ) ) ) ).

% min_absorb1
tff(fact_3885_min__absorb2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Y: A,X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X)
         => ( aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y) = Y ) ) ) ).

% min_absorb2
tff(fact_3886_min_Omono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,C2: A,B2: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2)),aa(A,A,aa(A,fun(A,A),ord_min(A),C2),D3)) ) ) ) ).

% min.mono
tff(fact_3887_min_OorderE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
         => ( A3 = aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2) ) ) ) ).

% min.orderE
tff(fact_3888_min_OorderI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] :
          ( ( A3 = aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2) ) ) ).

% min.orderI
tff(fact_3889_min_OboundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),C2))
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),C2) ) ) ) ).

% min.boundedE
tff(fact_3890_min_OboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),C2)) ) ) ) ).

% min.boundedI
tff(fact_3891_min_Oorder__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),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_3892_min_Ocobounded1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2)),A3) ) ).

% min.cobounded1
tff(fact_3893_min_Ocobounded2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2)),B2) ) ).

% min.cobounded2
tff(fact_3894_min_Oabsorb__iff1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),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_3895_min_Oabsorb__iff2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),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_3896_min_OcoboundedI1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),C2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2)),C2) ) ) ).

% min.coboundedI1
tff(fact_3897_min_OcoboundedI2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,C2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),C2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2)),C2) ) ) ).

% min.coboundedI2
tff(fact_3898_min__le__iff__disj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A,Z2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)),Z2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Z2)
            | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z2) ) ) ) ).

% min_le_iff_disj
tff(fact_3899_min__less__iff__disj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A,Z2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)),Z2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Z2)
            | aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Z2) ) ) ) ).

% min_less_iff_disj
tff(fact_3900_min_Ostrict__boundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),C2))
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),C2) ) ) ) ).

% min.strict_boundedE
tff(fact_3901_min_Ostrict__order__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
        <=> ( ( A3 = aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2) )
            & ( A3 != B2 ) ) ) ) ).

% min.strict_order_iff
tff(fact_3902_min_Ostrict__coboundedI1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),C2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2)),C2) ) ) ).

% min.strict_coboundedI1
tff(fact_3903_min_Ostrict__coboundedI2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,C2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),C2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2)),C2) ) ) ).

% min.strict_coboundedI2
tff(fact_3904_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_3905_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_3906_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_3907_Option_Othese__def,axiom,
    ! [A: $tType,A4: set(option(A))] : these(A,A4) = aa(set(option(A)),set(A),image2(option(A),A,the2(A)),aa(fun(option(A),$o),set(option(A)),collect(option(A)),aTP_Lamp_to(set(option(A)),fun(option(A),$o),A4))) ).

% Option.these_def
tff(fact_3908_max__mult__distrib__left,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [P3: A,X: A,Y: A] :
          aa(A,A,aa(A,fun(A,A),times_times(A),P3),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y)) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),P3),aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),times_times(A),P3),X)),aa(A,A,aa(A,fun(A,A),times_times(A),P3),Y)),aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),times_times(A),P3),X)),aa(A,A,aa(A,fun(A,A),times_times(A),P3),Y))) ) ).

% max_mult_distrib_left
tff(fact_3909_min__mult__distrib__left,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [P3: A,X: A,Y: A] :
          aa(A,A,aa(A,fun(A,A),times_times(A),P3),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),P3),aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),times_times(A),P3),X)),aa(A,A,aa(A,fun(A,A),times_times(A),P3),Y)),aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),times_times(A),P3),X)),aa(A,A,aa(A,fun(A,A),times_times(A),P3),Y))) ) ).

% min_mult_distrib_left
tff(fact_3910_max__mult__distrib__right,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A,P3: A] :
          aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y)),P3) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),P3),aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),P3)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),P3)),aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),P3)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),P3))) ) ).

% max_mult_distrib_right
tff(fact_3911_min__mult__distrib__right,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A,P3: A] :
          aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)),P3) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),P3),aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),P3)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),P3)),aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),P3)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),P3))) ) ).

% min_mult_distrib_right
tff(fact_3912_Inf__insert__finite,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [S: set(A),X: A] :
          ( aa(set(A),$o,finite_finite2(A),S)
         => ( aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),S)) = $ite(S = bot_bot(set(A)),X,aa(A,A,aa(A,fun(A,A),ord_min(A),X),aa(set(A),A,complete_Inf_Inf(A),S))) ) ) ) ).

% Inf_insert_finite
tff(fact_3913_min__Suc2,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),M),aa(nat,nat,suc,N)) = aa(nat,nat,aa(fun(nat,nat),fun(nat,nat),aa(nat,fun(fun(nat,nat),fun(nat,nat)),case_nat(nat),zero_zero(nat)),aTP_Lamp_uc(nat,fun(nat,nat),N)),M) ).

% min_Suc2
tff(fact_3914_min__Suc1,axiom,
    ! [N: nat,M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(nat,nat,suc,N)),M) = aa(nat,nat,aa(fun(nat,nat),fun(nat,nat),aa(nat,fun(fun(nat,nat),fun(nat,nat)),case_nat(nat),zero_zero(nat)),aTP_Lamp_ud(nat,fun(nat,nat),N)),M) ).

% min_Suc1
tff(fact_3915_collect__comp,axiom,
    ! [A: $tType,B: $tType,C: $tType,F4: set(fun(C,set(B))),G: fun(A,C)] : aa(fun(A,C),fun(A,set(B)),comp(C,set(B),A,bNF_collect(C,B,F4)),G) = bNF_collect(A,B,aa(set(fun(C,set(B))),set(fun(A,set(B))),image2(fun(C,set(B)),fun(A,set(B)),aTP_Lamp_ue(fun(A,C),fun(fun(C,set(B)),fun(A,set(B))),G)),F4)) ).

% collect_comp
tff(fact_3916_collect__def,axiom,
    ! [A: $tType,B: $tType,F4: set(fun(B,set(A))),X: B] : aa(B,set(A),bNF_collect(B,A,F4),X) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(fun(B,set(A))),set(set(A)),image2(fun(B,set(A)),set(A),aTP_Lamp_uf(B,fun(fun(B,set(A)),set(A)),X)),F4)) ).

% collect_def
tff(fact_3917_Code__Numeral_Odup__def,axiom,
    code_dup = aa(fun(int,int),fun(code_integer,code_integer),map_fun(code_integer,int,int,code_integer,code_int_of_integer,code_integer_of_int),aTP_Lamp_ug(int,int)) ).

% Code_Numeral.dup_def
tff(fact_3918_Sup__fin_Oeq__fold_H,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A4: set(A)] : aa(set(A),A,lattic5882676163264333800up_fin(A),A4) = aa(option(A),A,the2(A),finite_fold(A,option(A),aTP_Lamp_uh(A,fun(option(A),option(A))),none(A),A4)) ) ).

% Sup_fin.eq_fold'
tff(fact_3919_Sup__fin_Osingleton,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A] : aa(set(A),A,lattic5882676163264333800up_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) = X ) ).

% Sup_fin.singleton
tff(fact_3920_inf__Sup__absorb,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [A4: set(A),A3: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),$o,member(A,A3),A4)
           => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),aa(set(A),A,lattic5882676163264333800up_fin(A),A4)) = A3 ) ) ) ) ).

% inf_Sup_absorb
tff(fact_3921_Sup__fin_Oinsert,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A4: set(A),X: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( aa(set(A),A,lattic5882676163264333800up_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A4)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(set(A),A,lattic5882676163264333800up_fin(A),A4)) ) ) ) ) ).

% Sup_fin.insert
tff(fact_3922_Sup__fin_Oin__idem,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A4: set(A),X: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),$o,member(A,X),A4)
           => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(set(A),A,lattic5882676163264333800up_fin(A),A4)) = aa(set(A),A,lattic5882676163264333800up_fin(A),A4) ) ) ) ) ).

% Sup_fin.in_idem
tff(fact_3923_Sup__fin_OboundedE,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A4: set(A),X: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic5882676163264333800up_fin(A),A4)),X)
             => ! [A11: A] :
                  ( aa(set(A),$o,member(A,A11),A4)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A11),X) ) ) ) ) ) ).

% Sup_fin.boundedE
tff(fact_3924_Sup__fin_OboundedI,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A4: set(A),X: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( ! [A6: A] :
                  ( aa(set(A),$o,member(A,A6),A4)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A6),X) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic5882676163264333800up_fin(A),A4)),X) ) ) ) ) ).

% Sup_fin.boundedI
tff(fact_3925_Sup__fin_Obounded__iff,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A4: set(A),X: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic5882676163264333800up_fin(A),A4)),X)
            <=> ! [X4: A] :
                  ( aa(set(A),$o,member(A,X4),A4)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),X) ) ) ) ) ) ).

% Sup_fin.bounded_iff
tff(fact_3926_cSup__eq__Sup__fin,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X5: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),X5)
         => ( ( X5 != bot_bot(set(A)) )
           => ( aa(set(A),A,complete_Sup_Sup(A),X5) = aa(set(A),A,lattic5882676163264333800up_fin(A),X5) ) ) ) ) ).

% cSup_eq_Sup_fin
tff(fact_3927_Sup__fin__Sup,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A4: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( aa(set(A),A,lattic5882676163264333800up_fin(A),A4) = aa(set(A),A,complete_Sup_Sup(A),A4) ) ) ) ) ).

% Sup_fin_Sup
tff(fact_3928_Sup__fin_Osubset__imp,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A4: set(A),B3: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B3)
         => ( ( A4 != bot_bot(set(A)) )
           => ( aa(set(A),$o,finite_finite2(A),B3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic5882676163264333800up_fin(A),A4)),aa(set(A),A,lattic5882676163264333800up_fin(A),B3)) ) ) ) ) ).

% Sup_fin.subset_imp
tff(fact_3929_Sup__fin_Ohom__commute,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [H2: fun(A,A),N4: set(A)] :
          ( ! [X2: A,Y2: A] : aa(A,A,H2,aa(A,A,aa(A,fun(A,A),sup_sup(A),X2),Y2)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,H2,X2)),aa(A,A,H2,Y2))
         => ( aa(set(A),$o,finite_finite2(A),N4)
           => ( ( N4 != bot_bot(set(A)) )
             => ( aa(A,A,H2,aa(set(A),A,lattic5882676163264333800up_fin(A),N4)) = aa(set(A),A,lattic5882676163264333800up_fin(A),aa(set(A),set(A),image2(A,A,H2),N4)) ) ) ) ) ) ).

% Sup_fin.hom_commute
tff(fact_3930_Sup__fin_Osubset,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A4: set(A),B3: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( B3 != bot_bot(set(A)) )
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),A4)
             => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,lattic5882676163264333800up_fin(A),B3)),aa(set(A),A,lattic5882676163264333800up_fin(A),A4)) = aa(set(A),A,lattic5882676163264333800up_fin(A),A4) ) ) ) ) ) ).

% Sup_fin.subset
tff(fact_3931_Sup__fin_Oinsert__not__elem,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A4: set(A),X: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ~ aa(set(A),$o,member(A,X),A4)
           => ( ( A4 != bot_bot(set(A)) )
             => ( aa(set(A),A,lattic5882676163264333800up_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A4)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(set(A),A,lattic5882676163264333800up_fin(A),A4)) ) ) ) ) ) ).

% Sup_fin.insert_not_elem
tff(fact_3932_Sup__fin_Oclosed,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A4: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( ! [X2: A,Y2: A] : aa(set(A),$o,member(A,aa(A,A,aa(A,fun(A,A),sup_sup(A),X2),Y2)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y2),bot_bot(set(A)))))
             => aa(set(A),$o,member(A,aa(set(A),A,lattic5882676163264333800up_fin(A),A4)),A4) ) ) ) ) ).

% Sup_fin.closed
tff(fact_3933_Sup__fin_Ounion,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A4: set(A),B3: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( aa(set(A),$o,finite_finite2(A),B3)
             => ( ( B3 != bot_bot(set(A)) )
               => ( aa(set(A),A,lattic5882676163264333800up_fin(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,lattic5882676163264333800up_fin(A),A4)),aa(set(A),A,lattic5882676163264333800up_fin(A),B3)) ) ) ) ) ) ) ).

% Sup_fin.union
tff(fact_3934_Sup__fin_Oeq__fold,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A4: set(A),X: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),A,lattic5882676163264333800up_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A4)) = finite_fold(A,A,sup_sup(A),X,A4) ) ) ) ).

% Sup_fin.eq_fold
tff(fact_3935_Sup__fin_Oinsert__remove,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A4: set(A),X: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),A,lattic5882676163264333800up_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A4)) = $ite(aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) = bot_bot(set(A)),X,aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(set(A),A,lattic5882676163264333800up_fin(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))))))) ) ) ) ).

% Sup_fin.insert_remove
tff(fact_3936_Sup__fin_Oremove,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A4: set(A),X: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),$o,member(A,X),A4)
           => ( aa(set(A),A,lattic5882676163264333800up_fin(A),A4) = $ite(aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) = bot_bot(set(A)),X,aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(set(A),A,lattic5882676163264333800up_fin(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))))))) ) ) ) ) ).

% Sup_fin.remove
tff(fact_3937_Inf__fin_Oeq__fold_H,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A4: set(A)] : aa(set(A),A,lattic7752659483105999362nf_fin(A),A4) = aa(option(A),A,the2(A),finite_fold(A,option(A),aTP_Lamp_ui(A,fun(option(A),option(A))),none(A),A4)) ) ).

% Inf_fin.eq_fold'
tff(fact_3938_Code__Numeral_Osub__code_I9_J,axiom,
    ! [M: num,N: num] : aa(num,code_integer,aa(num,fun(num,code_integer),code_sub,aa(num,num,bit0,M)),aa(num,num,bit1,N)) = aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),aa(code_integer,code_integer,code_dup,aa(num,code_integer,aa(num,fun(num,code_integer),code_sub,M),N))),one_one(code_integer)) ).

% Code_Numeral.sub_code(9)
tff(fact_3939_Code__Numeral_Osub__code_I8_J,axiom,
    ! [M: num,N: num] : aa(num,code_integer,aa(num,fun(num,code_integer),code_sub,aa(num,num,bit1,M)),aa(num,num,bit0,N)) = aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,code_dup,aa(num,code_integer,aa(num,fun(num,code_integer),code_sub,M),N))),one_one(code_integer)) ).

% Code_Numeral.sub_code(8)
tff(fact_3940_Min_Oeq__fold_H,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A)] : aa(set(A),A,lattic643756798350308766er_Min(A),A4) = aa(option(A),A,the2(A),finite_fold(A,option(A),aTP_Lamp_uj(A,fun(option(A),option(A))),none(A),A4)) ) ).

% Min.eq_fold'
tff(fact_3941_Min__singleton,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A] : aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) = X ) ).

% Min_singleton
tff(fact_3942_Inf__fin_Osingleton,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [X: A] : aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) = X ) ).

% Inf_fin.singleton
tff(fact_3943_sup__Inf__absorb,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [A4: set(A),A3: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),$o,member(A,A3),A4)
           => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,lattic7752659483105999362nf_fin(A),A4)),A3) = A3 ) ) ) ) ).

% sup_Inf_absorb
tff(fact_3944_Min_Obounded__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),X: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(set(A),A,lattic643756798350308766er_Min(A),A4))
            <=> ! [X4: A] :
                  ( aa(set(A),$o,member(A,X4),A4)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),X4) ) ) ) ) ) ).

% Min.bounded_iff
tff(fact_3945_Min__gr__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),X: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),aa(set(A),A,lattic643756798350308766er_Min(A),A4))
            <=> ! [X4: A] :
                  ( aa(set(A),$o,member(A,X4),A4)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),X4) ) ) ) ) ) ).

% Min_gr_iff
tff(fact_3946_Min__const,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(B)
     => ! [A4: set(A),C2: B] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( aa(set(B),B,lattic643756798350308766er_Min(B),aa(set(A),set(B),image2(A,B,aTP_Lamp_ss(B,fun(A,B),C2)),A4)) = C2 ) ) ) ) ).

% Min_const
tff(fact_3947_Inf__fin_Oinsert,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A4: set(A),X: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A4)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(set(A),A,lattic7752659483105999362nf_fin(A),A4)) ) ) ) ) ).

% Inf_fin.insert
tff(fact_3948_Min__insert,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),X: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A4)) = aa(A,A,aa(A,fun(A,A),ord_min(A),X),aa(set(A),A,lattic643756798350308766er_Min(A),A4)) ) ) ) ) ).

% Min_insert
tff(fact_3949_minus__Min__eq__Max,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [S: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),S)
         => ( ( S != bot_bot(set(A)) )
           => ( aa(A,A,uminus_uminus(A),aa(set(A),A,lattic643756798350308766er_Min(A),S)) = aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),image2(A,A,uminus_uminus(A)),S)) ) ) ) ) ).

% minus_Min_eq_Max
tff(fact_3950_minus__Max__eq__Min,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [S: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),S)
         => ( ( S != bot_bot(set(A)) )
           => ( aa(A,A,uminus_uminus(A),aa(set(A),A,lattic643756798349783984er_Max(A),S)) = aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(A),set(A),image2(A,A,uminus_uminus(A)),S)) ) ) ) ) ).

% minus_Max_eq_Min
tff(fact_3951_Min__in,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => aa(set(A),$o,member(A,aa(set(A),A,lattic643756798350308766er_Min(A),A4)),A4) ) ) ) ).

% Min_in
tff(fact_3952_Inf__fin_Oin__idem,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A4: set(A),X: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),$o,member(A,X),A4)
           => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(set(A),A,lattic7752659483105999362nf_fin(A),A4)) = aa(set(A),A,lattic7752659483105999362nf_fin(A),A4) ) ) ) ) ).

% Inf_fin.in_idem
tff(fact_3953_Min__eq__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),M: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( ( aa(set(A),A,lattic643756798350308766er_Min(A),A4) = M )
            <=> ( aa(set(A),$o,member(A,M),A4)
                & ! [X4: A] :
                    ( aa(set(A),$o,member(A,X4),A4)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),M),X4) ) ) ) ) ) ) ).

% Min_eq_iff
tff(fact_3954_Min__le__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),X: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic643756798350308766er_Min(A),A4)),X)
            <=> ? [X4: A] :
                  ( aa(set(A),$o,member(A,X4),A4)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),X) ) ) ) ) ) ).

% Min_le_iff
tff(fact_3955_eq__Min__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),M: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( ( M = aa(set(A),A,lattic643756798350308766er_Min(A),A4) )
            <=> ( aa(set(A),$o,member(A,M),A4)
                & ! [X4: A] :
                    ( aa(set(A),$o,member(A,X4),A4)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),M),X4) ) ) ) ) ) ) ).

% eq_Min_iff
tff(fact_3956_Min_OboundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),X: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(set(A),A,lattic643756798350308766er_Min(A),A4))
             => ! [A11: A] :
                  ( aa(set(A),$o,member(A,A11),A4)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),A11) ) ) ) ) ) ).

% Min.boundedE
tff(fact_3957_Min_OboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),X: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( ! [A6: A] :
                  ( aa(set(A),$o,member(A,A6),A4)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),A6) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(set(A),A,lattic643756798350308766er_Min(A),A4)) ) ) ) ) ).

% Min.boundedI
tff(fact_3958_Min__less__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),X: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(A),A,lattic643756798350308766er_Min(A),A4)),X)
            <=> ? [X4: A] :
                  ( aa(set(A),$o,member(A,X4),A4)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),X) ) ) ) ) ) ).

% Min_less_iff
tff(fact_3959_cInf__eq__Min,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X5: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),X5)
         => ( ( X5 != bot_bot(set(A)) )
           => ( aa(set(A),A,complete_Inf_Inf(A),X5) = aa(set(A),A,lattic643756798350308766er_Min(A),X5) ) ) ) ) ).

% cInf_eq_Min
tff(fact_3960_Min__Inf,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [A4: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( aa(set(A),A,lattic643756798350308766er_Min(A),A4) = aa(set(A),A,complete_Inf_Inf(A),A4) ) ) ) ) ).

% Min_Inf
tff(fact_3961_Inf__fin_Obounded__iff,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A4: set(A),X: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(set(A),A,lattic7752659483105999362nf_fin(A),A4))
            <=> ! [X4: A] :
                  ( aa(set(A),$o,member(A,X4),A4)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),X4) ) ) ) ) ) ).

% Inf_fin.bounded_iff
tff(fact_3962_Inf__fin_OboundedI,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A4: set(A),X: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( ! [A6: A] :
                  ( aa(set(A),$o,member(A,A6),A4)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),A6) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(set(A),A,lattic7752659483105999362nf_fin(A),A4)) ) ) ) ) ).

% Inf_fin.boundedI
tff(fact_3963_Inf__fin_OboundedE,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A4: set(A),X: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(set(A),A,lattic7752659483105999362nf_fin(A),A4))
             => ! [A11: A] :
                  ( aa(set(A),$o,member(A,A11),A4)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),A11) ) ) ) ) ) ).

% Inf_fin.boundedE
tff(fact_3964_cInf__eq__Inf__fin,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X5: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),X5)
         => ( ( X5 != bot_bot(set(A)) )
           => ( aa(set(A),A,complete_Inf_Inf(A),X5) = aa(set(A),A,lattic7752659483105999362nf_fin(A),X5) ) ) ) ) ).

% cInf_eq_Inf_fin
tff(fact_3965_Inf__fin__Inf,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A4: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( aa(set(A),A,lattic7752659483105999362nf_fin(A),A4) = aa(set(A),A,complete_Inf_Inf(A),A4) ) ) ) ) ).

% Inf_fin_Inf
tff(fact_3966_Min_Osubset__imp,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),B3: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B3)
         => ( ( A4 != bot_bot(set(A)) )
           => ( aa(set(A),$o,finite_finite2(A),B3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic643756798350308766er_Min(A),B3)),aa(set(A),A,lattic643756798350308766er_Min(A),A4)) ) ) ) ) ).

% Min.subset_imp
tff(fact_3967_Min__antimono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [M4: set(A),N4: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),M4),N4)
         => ( ( M4 != bot_bot(set(A)) )
           => ( aa(set(A),$o,finite_finite2(A),N4)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic643756798350308766er_Min(A),N4)),aa(set(A),A,lattic643756798350308766er_Min(A),M4)) ) ) ) ) ).

% Min_antimono
tff(fact_3968_hom__Min__commute,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [H2: fun(A,A),N4: set(A)] :
          ( ! [X2: A,Y2: A] : aa(A,A,H2,aa(A,A,aa(A,fun(A,A),ord_min(A),X2),Y2)) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,H2,X2)),aa(A,A,H2,Y2))
         => ( aa(set(A),$o,finite_finite2(A),N4)
           => ( ( N4 != bot_bot(set(A)) )
             => ( aa(A,A,H2,aa(set(A),A,lattic643756798350308766er_Min(A),N4)) = aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(A),set(A),image2(A,A,H2),N4)) ) ) ) ) ) ).

% hom_Min_commute
tff(fact_3969_Min_Osubset,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),B3: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( B3 != bot_bot(set(A)) )
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),A4)
             => ( aa(A,A,aa(A,fun(A,A),ord_min(A),aa(set(A),A,lattic643756798350308766er_Min(A),B3)),aa(set(A),A,lattic643756798350308766er_Min(A),A4)) = aa(set(A),A,lattic643756798350308766er_Min(A),A4) ) ) ) ) ) ).

% Min.subset
tff(fact_3970_Min_Oinsert__not__elem,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),X: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ~ aa(set(A),$o,member(A,X),A4)
           => ( ( A4 != bot_bot(set(A)) )
             => ( aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A4)) = aa(A,A,aa(A,fun(A,A),ord_min(A),X),aa(set(A),A,lattic643756798350308766er_Min(A),A4)) ) ) ) ) ) ).

% Min.insert_not_elem
tff(fact_3971_Min_Oclosed,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( ! [X2: A,Y2: A] : aa(set(A),$o,member(A,aa(A,A,aa(A,fun(A,A),ord_min(A),X2),Y2)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y2),bot_bot(set(A)))))
             => aa(set(A),$o,member(A,aa(set(A),A,lattic643756798350308766er_Min(A),A4)),A4) ) ) ) ) ).

% Min.closed
tff(fact_3972_mono__Min__commute,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & linorder(B) )
     => ! [F: fun(A,B),A4: set(A)] :
          ( order_mono(A,B,F)
         => ( aa(set(A),$o,finite_finite2(A),A4)
           => ( ( A4 != bot_bot(set(A)) )
             => ( aa(A,B,F,aa(set(A),A,lattic643756798350308766er_Min(A),A4)) = aa(set(B),B,lattic643756798350308766er_Min(B),aa(set(A),set(B),image2(A,B,F),A4)) ) ) ) ) ) ).

% mono_Min_commute
tff(fact_3973_Min_Ounion,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),B3: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( aa(set(A),$o,finite_finite2(A),B3)
             => ( ( B3 != bot_bot(set(A)) )
               => ( aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3)) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(set(A),A,lattic643756798350308766er_Min(A),A4)),aa(set(A),A,lattic643756798350308766er_Min(A),B3)) ) ) ) ) ) ) ).

% Min.union
tff(fact_3974_Min__add__commute,axiom,
    ! [A: $tType,B: $tType] :
      ( linord4140545234300271783up_add(B)
     => ! [S: set(A),F: fun(A,B),K: B] :
          ( aa(set(A),$o,finite_finite2(A),S)
         => ( ( S != bot_bot(set(A)) )
           => ( aa(set(B),B,lattic643756798350308766er_Min(B),aa(set(A),set(B),image2(A,B,aa(B,fun(A,B),aTP_Lamp_su(fun(A,B),fun(B,fun(A,B)),F),K)),S)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(B),B,lattic643756798350308766er_Min(B),aa(set(A),set(B),image2(A,B,F),S))),K) ) ) ) ) ).

% Min_add_commute
tff(fact_3975_Inf__fin_Osubset__imp,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A4: set(A),B3: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B3)
         => ( ( A4 != bot_bot(set(A)) )
           => ( aa(set(A),$o,finite_finite2(A),B3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic7752659483105999362nf_fin(A),B3)),aa(set(A),A,lattic7752659483105999362nf_fin(A),A4)) ) ) ) ) ).

% Inf_fin.subset_imp
tff(fact_3976_Inf__fin_Ohom__commute,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [H2: fun(A,A),N4: set(A)] :
          ( ! [X2: A,Y2: A] : aa(A,A,H2,aa(A,A,aa(A,fun(A,A),inf_inf(A),X2),Y2)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,H2,X2)),aa(A,A,H2,Y2))
         => ( aa(set(A),$o,finite_finite2(A),N4)
           => ( ( N4 != bot_bot(set(A)) )
             => ( aa(A,A,H2,aa(set(A),A,lattic7752659483105999362nf_fin(A),N4)) = aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(set(A),set(A),image2(A,A,H2),N4)) ) ) ) ) ) ).

% Inf_fin.hom_commute
tff(fact_3977_Inf__fin_Osubset,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A4: set(A),B3: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( B3 != bot_bot(set(A)) )
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),A4)
             => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,lattic7752659483105999362nf_fin(A),B3)),aa(set(A),A,lattic7752659483105999362nf_fin(A),A4)) = aa(set(A),A,lattic7752659483105999362nf_fin(A),A4) ) ) ) ) ) ).

% Inf_fin.subset
tff(fact_3978_Inf__fin_Oclosed,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A4: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( ! [X2: A,Y2: A] : aa(set(A),$o,member(A,aa(A,A,aa(A,fun(A,A),inf_inf(A),X2),Y2)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y2),bot_bot(set(A)))))
             => aa(set(A),$o,member(A,aa(set(A),A,lattic7752659483105999362nf_fin(A),A4)),A4) ) ) ) ) ).

% Inf_fin.closed
tff(fact_3979_Inf__fin_Oinsert__not__elem,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A4: set(A),X: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ~ aa(set(A),$o,member(A,X),A4)
           => ( ( A4 != bot_bot(set(A)) )
             => ( aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A4)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(set(A),A,lattic7752659483105999362nf_fin(A),A4)) ) ) ) ) ) ).

% Inf_fin.insert_not_elem
tff(fact_3980_Inf__fin_Ounion,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A4: set(A),B3: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( aa(set(A),$o,finite_finite2(A),B3)
             => ( ( B3 != bot_bot(set(A)) )
               => ( aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,lattic7752659483105999362nf_fin(A),A4)),aa(set(A),A,lattic7752659483105999362nf_fin(A),B3)) ) ) ) ) ) ) ).

% Inf_fin.union
tff(fact_3981_Inf__fin_Oeq__fold,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A4: set(A),X: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A4)) = finite_fold(A,A,inf_inf(A),X,A4) ) ) ) ).

% Inf_fin.eq_fold
tff(fact_3982_Inf__fin__le__Sup__fin,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [A4: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic7752659483105999362nf_fin(A),A4)),aa(set(A),A,lattic5882676163264333800up_fin(A),A4)) ) ) ) ).

% Inf_fin_le_Sup_fin
tff(fact_3983_Code__Numeral_Osub__def,axiom,
    code_sub = aa(fun(num,fun(num,int)),fun(num,fun(num,code_integer)),map_fun(num,num,fun(num,int),fun(num,code_integer),id(num),map_fun(num,num,int,code_integer,id(num),code_integer_of_int)),aTP_Lamp_uk(num,fun(num,int))) ).

% Code_Numeral.sub_def
tff(fact_3984_Min_Oinsert__remove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),X: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A4)) = $ite(aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) = bot_bot(set(A)),X,aa(A,A,aa(A,fun(A,A),ord_min(A),X),aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))))))) ) ) ) ).

% Min.insert_remove
tff(fact_3985_Min_Oremove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),X: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),$o,member(A,X),A4)
           => ( aa(set(A),A,lattic643756798350308766er_Min(A),A4) = $ite(aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) = bot_bot(set(A)),X,aa(A,A,aa(A,fun(A,A),ord_min(A),X),aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))))))) ) ) ) ) ).

% Min.remove
tff(fact_3986_Inf__fin_Oremove,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A4: set(A),X: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),$o,member(A,X),A4)
           => ( aa(set(A),A,lattic7752659483105999362nf_fin(A),A4) = $ite(aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) = bot_bot(set(A)),X,aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))))))) ) ) ) ) ).

% Inf_fin.remove
tff(fact_3987_Inf__fin_Oinsert__remove,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A4: set(A),X: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A4)) = $ite(aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) = bot_bot(set(A)),X,aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))))))) ) ) ) ).

% Inf_fin.insert_remove
tff(fact_3988_dual__Max,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( lattices_Max(A,aTP_Lamp_tm(A,fun(A,$o))) = lattic643756798350308766er_Min(A) ) ) ).

% dual_Max
tff(fact_3989_dual__max,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( max(A,aTP_Lamp_tm(A,fun(A,$o))) = ord_min(A) ) ) ).

% dual_max
tff(fact_3990_total__on__singleton,axiom,
    ! [A: $tType,X: A] : total_on(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(product_prod(A,A),fun(set(product_prod(A,A)),set(product_prod(A,A))),insert2(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),X)),bot_bot(set(product_prod(A,A))))) ).

% total_on_singleton
tff(fact_3991_comp__fun__commute__product__fold,axiom,
    ! [B: $tType,A: $tType,B3: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),B3)
     => finite6289374366891150609ommute(B,set(product_prod(B,A)),aTP_Lamp_ul(set(A),fun(B,fun(set(product_prod(B,A)),set(product_prod(B,A)))),B3)) ) ).

% comp_fun_commute_product_fold
tff(fact_3992_total__on__empty,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : total_on(A,bot_bot(set(A)),R2) ).

% total_on_empty
tff(fact_3993_ord_Omax__def,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,$o)),A3: A,B2: A] :
      aa(A,A,aa(A,fun(A,A),max(A,Less_eq),A3),B2) = $ite(aa(A,$o,aa(A,fun(A,$o),Less_eq,A3),B2),B2,A3) ).

% ord.max_def
tff(fact_3994_ord_Omax_Ocong,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,$o))] : max(A,Less_eq) = max(A,Less_eq) ).

% ord.max.cong
tff(fact_3995_comp__fun__commute__const,axiom,
    ! [A: $tType,B: $tType,F: fun(B,B)] : finite6289374366891150609ommute(A,B,aTP_Lamp_um(fun(B,B),fun(A,fun(B,B)),F)) ).

% comp_fun_commute_const
tff(fact_3996_total__onI,axiom,
    ! [A: $tType,A4: set(A),R2: set(product_prod(A,A))] :
      ( ! [X2: A,Y2: A] :
          ( aa(set(A),$o,member(A,X2),A4)
         => ( aa(set(A),$o,member(A,Y2),A4)
           => ( ( X2 != Y2 )
             => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y2)),R2)
                | aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),X2)),R2) ) ) ) )
     => total_on(A,A4,R2) ) ).

% total_onI
tff(fact_3997_total__on__def,axiom,
    ! [A: $tType,A4: set(A),R2: set(product_prod(A,A))] :
      ( total_on(A,A4,R2)
    <=> ! [X4: A] :
          ( aa(set(A),$o,member(A,X4),A4)
         => ! [Xa2: A] :
              ( aa(set(A),$o,member(A,Xa2),A4)
             => ( ( X4 != Xa2 )
               => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Xa2)),R2)
                  | aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa2),X4)),R2) ) ) ) ) ) ).

% total_on_def
tff(fact_3998_total__on__lex__prod,axiom,
    ! [A: $tType,B: $tType,A4: set(A),R2: set(product_prod(A,A)),B3: set(B),S2: set(product_prod(B,B))] :
      ( total_on(A,A4,R2)
     => ( total_on(B,B3,S2)
       => total_on(product_prod(A,B),product_Sigma(A,B,A4,aTP_Lamp_qz(set(B),fun(A,set(B)),B3)),lex_prod(A,B,R2,S2)) ) ) ).

% total_on_lex_prod
tff(fact_3999_comp__fun__commute__filter__fold,axiom,
    ! [A: $tType,P: fun(A,$o)] : finite6289374366891150609ommute(A,set(A),aTP_Lamp_om(fun(A,$o),fun(A,fun(set(A),set(A))),P)) ).

% comp_fun_commute_filter_fold
tff(fact_4000_total__lex__prod,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(A,A)),S2: set(product_prod(B,B))] :
      ( total_on(A,top_top(set(A)),R2)
     => ( total_on(B,top_top(set(B)),S2)
       => total_on(product_prod(A,B),top_top(set(product_prod(A,B))),lex_prod(A,B,R2,S2)) ) ) ).

% total_lex_prod
tff(fact_4001_comp__fun__commute__def_H,axiom,
    ! [B: $tType,A: $tType,F: fun(A,fun(B,B))] :
      ( finite6289374366891150609ommute(A,B,F)
    <=> finite4664212375090638736ute_on(A,B,top_top(set(A)),F) ) ).

% comp_fun_commute_def'
tff(fact_4002_comp__fun__commute__relcomp__fold,axiom,
    ! [C: $tType,B: $tType,A: $tType,S: set(product_prod(A,B))] :
      ( aa(set(product_prod(A,B)),$o,finite_finite2(product_prod(A,B)),S)
     => 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_uo(set(product_prod(A,B)),fun(C,fun(A,fun(set(product_prod(C,B)),set(product_prod(C,B))))),S))) ) ).

% comp_fun_commute_relcomp_fold
tff(fact_4003_and__not__num_Opelims,axiom,
    ! [X: num,Xa: num,Y: option(num)] :
      ( ( bit_and_not_num(X,Xa) = Y )
     => ( accp(product_prod(num,num),bit_and_not_num_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),X),Xa))
       => ( ( ( X = one2 )
           => ( ( Xa = one2 )
             => ( ( Y = none(num) )
               => ~ accp(product_prod(num,num),bit_and_not_num_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),one2)) ) ) )
         => ( ( ( X = one2 )
             => ! [N3: num] :
                  ( ( Xa = aa(num,num,bit0,N3) )
                 => ( ( Y = aa(num,option(num),some(num),one2) )
                   => ~ accp(product_prod(num,num),bit_and_not_num_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),aa(num,num,bit0,N3))) ) ) )
           => ( ( ( X = one2 )
               => ! [N3: num] :
                    ( ( Xa = aa(num,num,bit1,N3) )
                   => ( ( Y = none(num) )
                     => ~ accp(product_prod(num,num),bit_and_not_num_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),aa(num,num,bit1,N3))) ) ) )
             => ( ! [M3: num] :
                    ( ( X = aa(num,num,bit0,M3) )
                   => ( ( Xa = one2 )
                     => ( ( Y = aa(num,option(num),some(num),aa(num,num,bit0,M3)) )
                       => ~ accp(product_prod(num,num),bit_and_not_num_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit0,M3)),one2)) ) ) )
               => ( ! [M3: num] :
                      ( ( X = aa(num,num,bit0,M3) )
                     => ! [N3: num] :
                          ( ( Xa = aa(num,num,bit0,N3) )
                         => ( ( Y = aa(option(num),option(num),map_option(num,num,bit0),bit_and_not_num(M3,N3)) )
                           => ~ accp(product_prod(num,num),bit_and_not_num_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit0,M3)),aa(num,num,bit0,N3))) ) ) )
                 => ( ! [M3: num] :
                        ( ( X = aa(num,num,bit0,M3) )
                       => ! [N3: num] :
                            ( ( Xa = aa(num,num,bit1,N3) )
                           => ( ( Y = aa(option(num),option(num),map_option(num,num,bit0),bit_and_not_num(M3,N3)) )
                             => ~ accp(product_prod(num,num),bit_and_not_num_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit0,M3)),aa(num,num,bit1,N3))) ) ) )
                   => ( ! [M3: num] :
                          ( ( X = aa(num,num,bit1,M3) )
                         => ( ( Xa = one2 )
                           => ( ( Y = aa(num,option(num),some(num),aa(num,num,bit0,M3)) )
                             => ~ accp(product_prod(num,num),bit_and_not_num_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M3)),one2)) ) ) )
                     => ( ! [M3: num] :
                            ( ( X = aa(num,num,bit1,M3) )
                           => ! [N3: num] :
                                ( ( Xa = aa(num,num,bit0,N3) )
                               => ( ( Y = case_option(option(num),num,aa(num,option(num),some(num),one2),aTP_Lamp_st(num,option(num)),bit_and_not_num(M3,N3)) )
                                 => ~ accp(product_prod(num,num),bit_and_not_num_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M3)),aa(num,num,bit0,N3))) ) ) )
                       => ~ ! [M3: num] :
                              ( ( X = aa(num,num,bit1,M3) )
                             => ! [N3: num] :
                                  ( ( Xa = aa(num,num,bit1,N3) )
                                 => ( ( Y = aa(option(num),option(num),map_option(num,num,bit0),bit_and_not_num(M3,N3)) )
                                   => ~ accp(product_prod(num,num),bit_and_not_num_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M3)),aa(num,num,bit1,N3))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_not_num.pelims
tff(fact_4004_and__num_Opelims,axiom,
    ! [X: num,Xa: num,Y: option(num)] :
      ( ( bit_un7362597486090784418nd_num(X,Xa) = Y )
     => ( accp(product_prod(num,num),bit_un4731106466462545111um_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),X),Xa))
       => ( ( ( X = one2 )
           => ( ( Xa = one2 )
             => ( ( Y = aa(num,option(num),some(num),one2) )
               => ~ accp(product_prod(num,num),bit_un4731106466462545111um_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),one2)) ) ) )
         => ( ( ( X = one2 )
             => ! [N3: num] :
                  ( ( Xa = aa(num,num,bit0,N3) )
                 => ( ( Y = none(num) )
                   => ~ accp(product_prod(num,num),bit_un4731106466462545111um_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),aa(num,num,bit0,N3))) ) ) )
           => ( ( ( X = one2 )
               => ! [N3: num] :
                    ( ( Xa = aa(num,num,bit1,N3) )
                   => ( ( Y = aa(num,option(num),some(num),one2) )
                     => ~ accp(product_prod(num,num),bit_un4731106466462545111um_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),aa(num,num,bit1,N3))) ) ) )
             => ( ! [M3: num] :
                    ( ( X = aa(num,num,bit0,M3) )
                   => ( ( Xa = one2 )
                     => ( ( Y = none(num) )
                       => ~ accp(product_prod(num,num),bit_un4731106466462545111um_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit0,M3)),one2)) ) ) )
               => ( ! [M3: num] :
                      ( ( X = aa(num,num,bit0,M3) )
                     => ! [N3: num] :
                          ( ( Xa = aa(num,num,bit0,N3) )
                         => ( ( Y = aa(option(num),option(num),map_option(num,num,bit0),bit_un7362597486090784418nd_num(M3,N3)) )
                           => ~ accp(product_prod(num,num),bit_un4731106466462545111um_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit0,M3)),aa(num,num,bit0,N3))) ) ) )
                 => ( ! [M3: num] :
                        ( ( X = aa(num,num,bit0,M3) )
                       => ! [N3: num] :
                            ( ( Xa = aa(num,num,bit1,N3) )
                           => ( ( Y = aa(option(num),option(num),map_option(num,num,bit0),bit_un7362597486090784418nd_num(M3,N3)) )
                             => ~ accp(product_prod(num,num),bit_un4731106466462545111um_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit0,M3)),aa(num,num,bit1,N3))) ) ) )
                   => ( ! [M3: num] :
                          ( ( X = aa(num,num,bit1,M3) )
                         => ( ( Xa = one2 )
                           => ( ( Y = aa(num,option(num),some(num),one2) )
                             => ~ accp(product_prod(num,num),bit_un4731106466462545111um_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M3)),one2)) ) ) )
                     => ( ! [M3: num] :
                            ( ( X = aa(num,num,bit1,M3) )
                           => ! [N3: num] :
                                ( ( Xa = aa(num,num,bit0,N3) )
                               => ( ( Y = aa(option(num),option(num),map_option(num,num,bit0),bit_un7362597486090784418nd_num(M3,N3)) )
                                 => ~ accp(product_prod(num,num),bit_un4731106466462545111um_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M3)),aa(num,num,bit0,N3))) ) ) )
                       => ~ ! [M3: num] :
                              ( ( X = aa(num,num,bit1,M3) )
                             => ! [N3: num] :
                                  ( ( Xa = aa(num,num,bit1,N3) )
                                 => ( ( Y = case_option(option(num),num,aa(num,option(num),some(num),one2),aTP_Lamp_st(num,option(num)),bit_un7362597486090784418nd_num(M3,N3)) )
                                   => ~ accp(product_prod(num,num),bit_un4731106466462545111um_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M3)),aa(num,num,bit1,N3))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_num.pelims
tff(fact_4005_power_Opower__eq__if,axiom,
    ! [A: $tType,One: A,Times: fun(A,fun(A,A)),P3: A,M: nat] :
      power2(A,One,Times,P3,M) = $ite(M = zero_zero(nat),One,aa(A,A,aa(A,fun(A,A),Times,P3),power2(A,One,Times,P3,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),one_one(nat))))) ).

% power.power_eq_if
tff(fact_4006_comp__fun__commute__on_Ofold__graph__insertE__aux,axiom,
    ! [A: $tType,B: $tType,S: set(A),F: fun(A,fun(B,B)),A4: set(A),Z2: B,Y: B,A3: A] :
      ( finite4664212375090638736ute_on(A,B,S,F)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),S)
       => ( aa(B,$o,finite_fold_graph(A,B,F,Z2,A4),Y)
         => ( aa(set(A),$o,member(A,A3),A4)
           => ? [Y7: B] :
                ( ( Y = aa(B,B,aa(A,fun(B,B),F,A3),Y7) )
                & aa(B,$o,finite_fold_graph(A,B,F,Z2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A))))),Y7) ) ) ) ) ) ).

% comp_fun_commute_on.fold_graph_insertE_aux
tff(fact_4007_empty__fold__graphE,axiom,
    ! [A: $tType,B: $tType,F: fun(A,fun(B,B)),Z2: B,X: B] :
      ( aa(B,$o,finite_fold_graph(A,B,F,Z2,bot_bot(set(A))),X)
     => ( X = Z2 ) ) ).

% empty_fold_graphE
tff(fact_4008_fold__graph_OemptyI,axiom,
    ! [A: $tType,B: $tType,F: fun(A,fun(B,B)),Z2: B] : aa(B,$o,finite_fold_graph(A,B,F,Z2,bot_bot(set(A))),Z2) ).

% fold_graph.emptyI
tff(fact_4009_comp__fun__commute__Image__fold,axiom,
    ! [B: $tType,A: $tType,S: set(A)] : finite6289374366891150609ommute(product_prod(A,B),set(B),aa(fun(A,fun(B,fun(set(B),set(B)))),fun(product_prod(A,B),fun(set(B),set(B))),product_case_prod(A,B,fun(set(B),set(B))),aTP_Lamp_up(set(A),fun(A,fun(B,fun(set(B),set(B)))),S))) ).

% comp_fun_commute_Image_fold
tff(fact_4010_fold__graph_Ocases,axiom,
    ! [A: $tType,B: $tType,F: fun(A,fun(B,B)),Z2: B,A1: set(A),A22: B] :
      ( aa(B,$o,finite_fold_graph(A,B,F,Z2,A1),A22)
     => ( ( ( A1 = bot_bot(set(A)) )
         => ( A22 != Z2 ) )
       => ~ ! [X2: A,A8: set(A)] :
              ( ( A1 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X2),A8) )
             => ! [Y2: B] :
                  ( ( A22 = aa(B,B,aa(A,fun(B,B),F,X2),Y2) )
                 => ( ~ aa(set(A),$o,member(A,X2),A8)
                   => ~ aa(B,$o,finite_fold_graph(A,B,F,Z2,A8),Y2) ) ) ) ) ) ).

% fold_graph.cases
tff(fact_4011_fold__graph_Osimps,axiom,
    ! [A: $tType,B: $tType,F: fun(A,fun(B,B)),Z2: B,A1: set(A),A22: B] :
      ( aa(B,$o,finite_fold_graph(A,B,F,Z2,A1),A22)
    <=> ( ( ( A1 = bot_bot(set(A)) )
          & ( A22 = Z2 ) )
        | ? [X4: A,A9: set(A),Y3: B] :
            ( ( A1 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X4),A9) )
            & ( A22 = aa(B,B,aa(A,fun(B,B),F,X4),Y3) )
            & ~ aa(set(A),$o,member(A,X4),A9)
            & aa(B,$o,finite_fold_graph(A,B,F,Z2,A9),Y3) ) ) ) ).

% fold_graph.simps
tff(fact_4012_Finite__Set_Ofold__def,axiom,
    ! [B: $tType,A: $tType,F: fun(B,fun(A,A)),Z2: A,A4: set(B)] :
      finite_fold(B,A,F,Z2,A4) = $ite(aa(set(B),$o,finite_finite2(B),A4),the(A,finite_fold_graph(B,A,F,Z2,A4)),Z2) ).

% Finite_Set.fold_def
tff(fact_4013_comp__fun__commute__Pow__fold,axiom,
    ! [A: $tType] : finite6289374366891150609ommute(A,set(set(A)),aTP_Lamp_op(A,fun(set(set(A)),set(set(A))))) ).

% comp_fun_commute_Pow_fold
tff(fact_4014_or__not__num__neg_Opelims,axiom,
    ! [X: num,Xa: num,Y: num] :
      ( ( bit_or_not_num_neg(X,Xa) = Y )
     => ( accp(product_prod(num,num),bit_or3848514188828904588eg_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),X),Xa))
       => ( ( ( X = one2 )
           => ( ( Xa = one2 )
             => ( ( Y = one2 )
               => ~ accp(product_prod(num,num),bit_or3848514188828904588eg_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),one2)) ) ) )
         => ( ( ( X = one2 )
             => ! [M3: num] :
                  ( ( Xa = aa(num,num,bit0,M3) )
                 => ( ( Y = aa(num,num,bit1,M3) )
                   => ~ accp(product_prod(num,num),bit_or3848514188828904588eg_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),aa(num,num,bit0,M3))) ) ) )
           => ( ( ( X = one2 )
               => ! [M3: num] :
                    ( ( Xa = aa(num,num,bit1,M3) )
                   => ( ( Y = aa(num,num,bit1,M3) )
                     => ~ accp(product_prod(num,num),bit_or3848514188828904588eg_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),aa(num,num,bit1,M3))) ) ) )
             => ( ! [N3: num] :
                    ( ( X = aa(num,num,bit0,N3) )
                   => ( ( Xa = one2 )
                     => ( ( Y = aa(num,num,bit0,one2) )
                       => ~ accp(product_prod(num,num),bit_or3848514188828904588eg_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit0,N3)),one2)) ) ) )
               => ( ! [N3: num] :
                      ( ( X = aa(num,num,bit0,N3) )
                     => ! [M3: num] :
                          ( ( Xa = aa(num,num,bit0,M3) )
                         => ( ( Y = bitM(bit_or_not_num_neg(N3,M3)) )
                           => ~ accp(product_prod(num,num),bit_or3848514188828904588eg_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit0,N3)),aa(num,num,bit0,M3))) ) ) )
                 => ( ! [N3: num] :
                        ( ( X = aa(num,num,bit0,N3) )
                       => ! [M3: num] :
                            ( ( Xa = aa(num,num,bit1,M3) )
                           => ( ( Y = aa(num,num,bit0,bit_or_not_num_neg(N3,M3)) )
                             => ~ accp(product_prod(num,num),bit_or3848514188828904588eg_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit0,N3)),aa(num,num,bit1,M3))) ) ) )
                   => ( ! [N3: num] :
                          ( ( X = aa(num,num,bit1,N3) )
                         => ( ( Xa = one2 )
                           => ( ( Y = one2 )
                             => ~ accp(product_prod(num,num),bit_or3848514188828904588eg_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,N3)),one2)) ) ) )
                     => ( ! [N3: num] :
                            ( ( X = aa(num,num,bit1,N3) )
                           => ! [M3: num] :
                                ( ( Xa = aa(num,num,bit0,M3) )
                               => ( ( Y = bitM(bit_or_not_num_neg(N3,M3)) )
                                 => ~ accp(product_prod(num,num),bit_or3848514188828904588eg_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,N3)),aa(num,num,bit0,M3))) ) ) )
                       => ~ ! [N3: num] :
                              ( ( X = aa(num,num,bit1,N3) )
                             => ! [M3: num] :
                                  ( ( Xa = aa(num,num,bit1,M3) )
                                 => ( ( Y = bitM(bit_or_not_num_neg(N3,M3)) )
                                   => ~ accp(product_prod(num,num),bit_or3848514188828904588eg_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,N3)),aa(num,num,bit1,M3))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% or_not_num_neg.pelims
tff(fact_4015_relcomp__fold,axiom,
    ! [C: $tType,B: $tType,A: $tType,R: set(product_prod(A,B)),S: set(product_prod(B,C))] :
      ( aa(set(product_prod(A,B)),$o,finite_finite2(product_prod(A,B)),R)
     => ( aa(set(product_prod(B,C)),$o,finite_finite2(product_prod(B,C)),S)
       => ( relcomp(A,B,C,R,S) = 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_ur(set(product_prod(B,C)),fun(A,fun(B,fun(set(product_prod(A,C)),set(product_prod(A,C))))),S)),bot_bot(set(product_prod(A,C))),R) ) ) ) ).

% relcomp_fold
tff(fact_4016_xor__num_Opelims,axiom,
    ! [X: num,Xa: num,Y: option(num)] :
      ( ( bit_un2480387367778600638or_num(X,Xa) = Y )
     => ( accp(product_prod(num,num),bit_un2901131394128224187um_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),X),Xa))
       => ( ( ( X = one2 )
           => ( ( Xa = one2 )
             => ( ( Y = none(num) )
               => ~ accp(product_prod(num,num),bit_un2901131394128224187um_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),one2)) ) ) )
         => ( ( ( X = one2 )
             => ! [N3: num] :
                  ( ( Xa = aa(num,num,bit0,N3) )
                 => ( ( Y = aa(num,option(num),some(num),aa(num,num,bit1,N3)) )
                   => ~ accp(product_prod(num,num),bit_un2901131394128224187um_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),aa(num,num,bit0,N3))) ) ) )
           => ( ( ( X = one2 )
               => ! [N3: num] :
                    ( ( Xa = aa(num,num,bit1,N3) )
                   => ( ( Y = aa(num,option(num),some(num),aa(num,num,bit0,N3)) )
                     => ~ accp(product_prod(num,num),bit_un2901131394128224187um_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),aa(num,num,bit1,N3))) ) ) )
             => ( ! [M3: num] :
                    ( ( X = aa(num,num,bit0,M3) )
                   => ( ( Xa = one2 )
                     => ( ( Y = aa(num,option(num),some(num),aa(num,num,bit1,M3)) )
                       => ~ accp(product_prod(num,num),bit_un2901131394128224187um_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit0,M3)),one2)) ) ) )
               => ( ! [M3: num] :
                      ( ( X = aa(num,num,bit0,M3) )
                     => ! [N3: num] :
                          ( ( Xa = aa(num,num,bit0,N3) )
                         => ( ( Y = aa(option(num),option(num),map_option(num,num,bit0),bit_un2480387367778600638or_num(M3,N3)) )
                           => ~ accp(product_prod(num,num),bit_un2901131394128224187um_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit0,M3)),aa(num,num,bit0,N3))) ) ) )
                 => ( ! [M3: num] :
                        ( ( X = aa(num,num,bit0,M3) )
                       => ! [N3: num] :
                            ( ( Xa = aa(num,num,bit1,N3) )
                           => ( ( Y = aa(num,option(num),some(num),case_option(num,num,one2,bit1,bit_un2480387367778600638or_num(M3,N3))) )
                             => ~ accp(product_prod(num,num),bit_un2901131394128224187um_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit0,M3)),aa(num,num,bit1,N3))) ) ) )
                   => ( ! [M3: num] :
                          ( ( X = aa(num,num,bit1,M3) )
                         => ( ( Xa = one2 )
                           => ( ( Y = aa(num,option(num),some(num),aa(num,num,bit0,M3)) )
                             => ~ accp(product_prod(num,num),bit_un2901131394128224187um_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M3)),one2)) ) ) )
                     => ( ! [M3: num] :
                            ( ( X = aa(num,num,bit1,M3) )
                           => ! [N3: num] :
                                ( ( Xa = aa(num,num,bit0,N3) )
                               => ( ( Y = aa(num,option(num),some(num),case_option(num,num,one2,bit1,bit_un2480387367778600638or_num(M3,N3))) )
                                 => ~ accp(product_prod(num,num),bit_un2901131394128224187um_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M3)),aa(num,num,bit0,N3))) ) ) )
                       => ~ ! [M3: num] :
                              ( ( X = aa(num,num,bit1,M3) )
                             => ! [N3: num] :
                                  ( ( Xa = aa(num,num,bit1,N3) )
                                 => ( ( Y = aa(option(num),option(num),map_option(num,num,bit0),bit_un2480387367778600638or_num(M3,N3)) )
                                   => ~ accp(product_prod(num,num),bit_un2901131394128224187um_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M3)),aa(num,num,bit1,N3))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% xor_num.pelims
tff(fact_4017_Field__insert,axiom,
    ! [A: $tType,A3: A,B2: A,R2: set(product_prod(A,A))] : field2(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(product_prod(A,A),fun(set(product_prod(A,A)),set(product_prod(A,A))),insert2(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R2)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A))))),field2(A,R2)) ).

% Field_insert
tff(fact_4018_relcomp__empty2,axiom,
    ! [C: $tType,B: $tType,A: $tType,R: set(product_prod(A,C))] : relcomp(A,C,B,R,bot_bot(set(product_prod(C,B)))) = bot_bot(set(product_prod(A,B))) ).

% relcomp_empty2
tff(fact_4019_relcomp__empty1,axiom,
    ! [C: $tType,B: $tType,A: $tType,R: set(product_prod(C,B))] : relcomp(A,C,B,bot_bot(set(product_prod(A,C))),R) = bot_bot(set(product_prod(A,B))) ).

% relcomp_empty1
tff(fact_4020_relcomp__distrib2,axiom,
    ! [A: $tType,B: $tType,C: $tType,S: set(product_prod(A,C)),T2: set(product_prod(A,C)),R: set(product_prod(C,B))] : relcomp(A,C,B,aa(set(product_prod(A,C)),set(product_prod(A,C)),aa(set(product_prod(A,C)),fun(set(product_prod(A,C)),set(product_prod(A,C))),sup_sup(set(product_prod(A,C))),S),T2),R) = 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))),relcomp(A,C,B,S,R)),relcomp(A,C,B,T2,R)) ).

% relcomp_distrib2
tff(fact_4021_relcomp__distrib,axiom,
    ! [A: $tType,B: $tType,C: $tType,R: set(product_prod(A,C)),S: set(product_prod(C,B)),T2: set(product_prod(C,B))] : relcomp(A,C,B,R,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))),S),T2)) = 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))),relcomp(A,C,B,R,S)),relcomp(A,C,B,R,T2)) ).

% relcomp_distrib
tff(fact_4022_Field__square,axiom,
    ! [A: $tType,X: set(A)] : field2(A,product_Sigma(A,A,X,aTP_Lamp_rl(set(A),fun(A,set(A)),X))) = X ).

% Field_square
tff(fact_4023_Field__empty,axiom,
    ! [A: $tType] : field2(A,bot_bot(set(product_prod(A,A)))) = bot_bot(set(A)) ).

% Field_empty
tff(fact_4024_Field__Un,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),S2: set(product_prod(A,A))] : field2(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),R2),S2)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),field2(A,R2)),field2(A,S2)) ).

% Field_Un
tff(fact_4025_FieldI2,axiom,
    ! [A: $tType,I: A,J: A,R: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),I),J)),R)
     => aa(set(A),$o,member(A,J),field2(A,R)) ) ).

% FieldI2
tff(fact_4026_FieldI1,axiom,
    ! [A: $tType,I: A,J: A,R: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),I),J)),R)
     => aa(set(A),$o,member(A,I),field2(A,R)) ) ).

% FieldI1
tff(fact_4027_relcompEpair,axiom,
    ! [A: $tType,B: $tType,C: $tType,A3: A,C2: B,R2: set(product_prod(A,C)),S2: set(product_prod(C,B))] :
      ( aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),C2)),relcomp(A,C,B,R2,S2))
     => ~ ! [B5: C] :
            ( aa(set(product_prod(A,C)),$o,member(product_prod(A,C),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),A3),B5)),R2)
           => ~ aa(set(product_prod(C,B)),$o,member(product_prod(C,B),aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),B5),C2)),S2) ) ) ).

% relcompEpair
tff(fact_4028_relcompE,axiom,
    ! [A: $tType,B: $tType,C: $tType,Xz: product_prod(A,B),R2: set(product_prod(A,C)),S2: set(product_prod(C,B))] :
      ( aa(set(product_prod(A,B)),$o,member(product_prod(A,B),Xz),relcomp(A,C,B,R2,S2))
     => ~ ! [X2: A,Y2: C,Z3: B] :
            ( ( Xz = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Z3) )
           => ( aa(set(product_prod(A,C)),$o,member(product_prod(A,C),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),X2),Y2)),R2)
             => ~ aa(set(product_prod(C,B)),$o,member(product_prod(C,B),aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),Y2),Z3)),S2) ) ) ) ).

% relcompE
tff(fact_4029_relcomp_OrelcompI,axiom,
    ! [A: $tType,C: $tType,B: $tType,A3: A,B2: B,R2: set(product_prod(A,B)),C2: C,S2: set(product_prod(B,C))] :
      ( aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),R2)
     => ( aa(set(product_prod(B,C)),$o,member(product_prod(B,C),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),B2),C2)),S2)
       => aa(set(product_prod(A,C)),$o,member(product_prod(A,C),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),A3),C2)),relcomp(A,B,C,R2,S2)) ) ) ).

% relcomp.relcompI
tff(fact_4030_relcomp_Osimps,axiom,
    ! [A: $tType,B: $tType,C: $tType,A1: A,A22: B,R2: set(product_prod(A,C)),S2: set(product_prod(C,B))] :
      ( aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A1),A22)),relcomp(A,C,B,R2,S2))
    <=> ? [A10: A,B6: C,C5: B] :
          ( ( A1 = A10 )
          & ( A22 = C5 )
          & aa(set(product_prod(A,C)),$o,member(product_prod(A,C),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),A10),B6)),R2)
          & aa(set(product_prod(C,B)),$o,member(product_prod(C,B),aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),B6),C5)),S2) ) ) ).

% relcomp.simps
tff(fact_4031_relcomp_Ocases,axiom,
    ! [A: $tType,B: $tType,C: $tType,A1: A,A22: B,R2: set(product_prod(A,C)),S2: set(product_prod(C,B))] :
      ( aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A1),A22)),relcomp(A,C,B,R2,S2))
     => ~ ! [B5: C] :
            ( aa(set(product_prod(A,C)),$o,member(product_prod(A,C),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),A1),B5)),R2)
           => ~ aa(set(product_prod(C,B)),$o,member(product_prod(C,B),aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),B5),A22)),S2) ) ) ).

% relcomp.cases
tff(fact_4032_trancl__unfold,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : transitive_trancl(A,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))),R2),relcomp(A,A,A,transitive_trancl(A,R2),R2)) ).

% trancl_unfold
tff(fact_4033_union__comp__emptyR,axiom,
    ! [A: $tType,A4: set(product_prod(A,A)),B3: set(product_prod(A,A)),C3: set(product_prod(A,A))] :
      ( ( relcomp(A,A,A,A4,B3) = bot_bot(set(product_prod(A,A))) )
     => ( ( relcomp(A,A,A,A4,C3) = bot_bot(set(product_prod(A,A))) )
       => ( relcomp(A,A,A,A4,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),B3),C3)) = bot_bot(set(product_prod(A,A))) ) ) ) ).

% union_comp_emptyR
tff(fact_4034_union__comp__emptyL,axiom,
    ! [A: $tType,A4: set(product_prod(A,A)),C3: set(product_prod(A,A)),B3: set(product_prod(A,A))] :
      ( ( relcomp(A,A,A,A4,C3) = bot_bot(set(product_prod(A,A))) )
     => ( ( relcomp(A,A,A,B3,C3) = bot_bot(set(product_prod(A,A))) )
       => ( relcomp(A,A,A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),A4),B3),C3) = bot_bot(set(product_prod(A,A))) ) ) ) ).

% union_comp_emptyL
tff(fact_4035_relcomp__subset__Sigma,axiom,
    ! [B: $tType,A: $tType,C: $tType,R2: set(product_prod(A,B)),A4: set(A),B3: set(B),S2: set(product_prod(B,C)),C3: set(C)] :
      ( aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),R2),product_Sigma(A,B,A4,aTP_Lamp_qz(set(B),fun(A,set(B)),B3)))
     => ( aa(set(product_prod(B,C)),$o,aa(set(product_prod(B,C)),fun(set(product_prod(B,C)),$o),ord_less_eq(set(product_prod(B,C))),S2),product_Sigma(B,C,B3,aTP_Lamp_rw(set(C),fun(B,set(C)),C3)))
       => aa(set(product_prod(A,C)),$o,aa(set(product_prod(A,C)),fun(set(product_prod(A,C)),$o),ord_less_eq(set(product_prod(A,C))),relcomp(A,B,C,R2,S2)),product_Sigma(A,C,A4,aTP_Lamp_us(set(C),fun(A,set(C)),C3))) ) ) ).

% relcomp_subset_Sigma
tff(fact_4036_R__subset__Field,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] : aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R),product_Sigma(A,A,field2(A,R),aTP_Lamp_ut(set(product_prod(A,A)),fun(A,set(A)),R))) ).

% R_subset_Field
tff(fact_4037_Restr__Field,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R2),product_Sigma(A,A,field2(A,R2),aTP_Lamp_ut(set(product_prod(A,A)),fun(A,set(A)),R2))) = R2 ).

% Restr_Field
tff(fact_4038_relcomp__UNION__distrib,axiom,
    ! [C: $tType,B: $tType,A: $tType,D: $tType,S2: set(product_prod(A,C)),R2: fun(D,set(product_prod(C,B))),I4: set(D)] : relcomp(A,C,B,S2,aa(set(set(product_prod(C,B))),set(product_prod(C,B)),complete_Sup_Sup(set(product_prod(C,B))),aa(set(D),set(set(product_prod(C,B))),image2(D,set(product_prod(C,B)),R2),I4))) = aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),aa(set(D),set(set(product_prod(A,B))),image2(D,set(product_prod(A,B)),aa(fun(D,set(product_prod(C,B))),fun(D,set(product_prod(A,B))),aTP_Lamp_uu(set(product_prod(A,C)),fun(fun(D,set(product_prod(C,B))),fun(D,set(product_prod(A,B)))),S2),R2)),I4)) ).

% relcomp_UNION_distrib
tff(fact_4039_relcomp__UNION__distrib2,axiom,
    ! [C: $tType,B: $tType,A: $tType,D: $tType,R2: fun(D,set(product_prod(A,C))),I4: set(D),S2: set(product_prod(C,B))] : relcomp(A,C,B,aa(set(set(product_prod(A,C))),set(product_prod(A,C)),complete_Sup_Sup(set(product_prod(A,C))),aa(set(D),set(set(product_prod(A,C))),image2(D,set(product_prod(A,C)),R2),I4)),S2) = aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),aa(set(D),set(set(product_prod(A,B))),image2(D,set(product_prod(A,B)),aa(set(product_prod(C,B)),fun(D,set(product_prod(A,B))),aTP_Lamp_uv(fun(D,set(product_prod(A,C))),fun(set(product_prod(C,B)),fun(D,set(product_prod(A,B)))),R2),S2)),I4)) ).

% relcomp_UNION_distrib2
tff(fact_4040_trancl__Int__subset,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),S2: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R2),S2)
     => ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),relcomp(A,A,A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),transitive_trancl(A,R2)),S2),R2)),S2)
       => aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),transitive_trancl(A,R2)),S2) ) ) ).

% trancl_Int_subset
tff(fact_4041_rel__restrict__Int__empty,axiom,
    ! [A: $tType,A4: set(A),R: set(product_prod(A,A))] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),field2(A,R)) = bot_bot(set(A)) )
     => ( rel_restrict(A,R,A4) = R ) ) ).

% rel_restrict_Int_empty
tff(fact_4042_Field__Restr__subset,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A4: set(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),field2(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R2),product_Sigma(A,A,A4,aTP_Lamp_rl(set(A),fun(A,set(A)),A4))))),A4) ).

% Field_Restr_subset
tff(fact_4043_trancl__subset__Field2,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),transitive_trancl(A,R2)),product_Sigma(A,A,field2(A,R2),aTP_Lamp_ut(set(product_prod(A,A)),fun(A,set(A)),R2))) ).

% trancl_subset_Field2
tff(fact_4044_Field__natLeq__on,axiom,
    ! [N: nat] : field2(nat,aa(fun(product_prod(nat,nat),$o),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aTP_Lamp_uw(nat,fun(nat,fun(nat,$o)),N)))) = aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_bk(nat,fun(nat,$o)),N)) ).

% Field_natLeq_on
tff(fact_4045_Total__Restr,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A4: set(A)] :
      ( total_on(A,field2(A,R2),R2)
     => total_on(A,field2(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R2),product_Sigma(A,A,A4,aTP_Lamp_rl(set(A),fun(A,set(A)),A4)))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R2),product_Sigma(A,A,A4,aTP_Lamp_rl(set(A),fun(A,set(A)),A4)))) ) ).

% Total_Restr
tff(fact_4046_total__on__imp__Total__Restr,axiom,
    ! [A: $tType,A4: set(A),R2: set(product_prod(A,A))] :
      ( total_on(A,A4,R2)
     => total_on(A,field2(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R2),product_Sigma(A,A,A4,aTP_Lamp_rl(set(A),fun(A,set(A)),A4)))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R2),product_Sigma(A,A,A4,aTP_Lamp_rl(set(A),fun(A,set(A)),A4)))) ) ).

% total_on_imp_Total_Restr
tff(fact_4047_numeral__xor__num,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M: num,N: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),N)) = case_option(A,num,zero_zero(A),numeral_numeral(A),bit_un2480387367778600638or_num(M,N)) ) ).

% numeral_xor_num
tff(fact_4048_min__ext__compat,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),S: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),relcomp(A,A,A,R,S)),R)
     => aa(set(product_prod(set(A),set(A))),$o,aa(set(product_prod(set(A),set(A))),fun(set(product_prod(set(A),set(A))),$o),ord_less_eq(set(product_prod(set(A),set(A)))),relcomp(set(A),set(A),set(A),min_ext(A,R),aa(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A))),aa(set(product_prod(set(A),set(A))),fun(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A)))),sup_sup(set(product_prod(set(A),set(A)))),min_ext(A,S)),aa(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A))),aa(product_prod(set(A),set(A)),fun(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A)))),insert2(product_prod(set(A),set(A))),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),bot_bot(set(A))),bot_bot(set(A)))),bot_bot(set(product_prod(set(A),set(A)))))))),min_ext(A,R)) ) ).

% min_ext_compat
tff(fact_4049_cofinal__def,axiom,
    ! [A: $tType,A4: set(A),R2: set(product_prod(A,A))] :
      ( bNF_Ca7293521722713021262ofinal(A,A4,R2)
    <=> ! [X4: A] :
          ( aa(set(A),$o,member(A,X4),field2(A,R2))
         => ? [Xa2: A] :
              ( aa(set(A),$o,member(A,Xa2),A4)
              & ( X4 != Xa2 )
              & aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Xa2)),R2) ) ) ) ).

% cofinal_def
tff(fact_4050_max__ext__compat,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),S: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),relcomp(A,A,A,R,S)),R)
     => aa(set(product_prod(set(A),set(A))),$o,aa(set(product_prod(set(A),set(A))),fun(set(product_prod(set(A),set(A))),$o),ord_less_eq(set(product_prod(set(A),set(A)))),relcomp(set(A),set(A),set(A),max_ext(A,R),aa(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A))),aa(set(product_prod(set(A),set(A))),fun(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A)))),sup_sup(set(product_prod(set(A),set(A)))),max_ext(A,S)),aa(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A))),aa(product_prod(set(A),set(A)),fun(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A)))),insert2(product_prod(set(A),set(A))),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),bot_bot(set(A))),bot_bot(set(A)))),bot_bot(set(product_prod(set(A),set(A)))))))),max_ext(A,R)) ) ).

% max_ext_compat
tff(fact_4051_ntrancl__Suc,axiom,
    ! [A: $tType,N: nat,R: set(product_prod(A,A))] : transitive_ntrancl(A,aa(nat,nat,suc,N),R) = relcomp(A,A,A,transitive_ntrancl(A,N,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))),id2(A)),R)) ).

% ntrancl_Suc
tff(fact_4052_IdI,axiom,
    ! [A: $tType,A3: A] : aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),A3)),id2(A)) ).

% IdI
tff(fact_4053_pair__in__Id__conv,axiom,
    ! [A: $tType,A3: A,B2: A] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),id2(A))
    <=> ( A3 = B2 ) ) ).

% pair_in_Id_conv
tff(fact_4054_rtrancl__empty,axiom,
    ! [A: $tType] : transitive_rtrancl(A,bot_bot(set(product_prod(A,A)))) = id2(A) ).

% rtrancl_empty
tff(fact_4055_rtrancl__reflcl__absorb,axiom,
    ! [A: $tType,R: 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))),sup_sup(set(product_prod(A,A))),transitive_rtrancl(A,R)),id2(A)) = transitive_rtrancl(A,R) ).

% rtrancl_reflcl_absorb
tff(fact_4056_rtrancl__reflcl,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] : 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))),R),id2(A))) = transitive_rtrancl(A,R) ).

% rtrancl_reflcl
tff(fact_4057_trancl__reflcl,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : transitive_trancl(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))),R2),id2(A))) = transitive_rtrancl(A,R2) ).

% trancl_reflcl
tff(fact_4058_IdE,axiom,
    ! [A: $tType,P3: product_prod(A,A)] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),P3),id2(A))
     => ~ ! [X2: A] : P3 != aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),X2) ) ).

% IdE
tff(fact_4059_BNF__Greatest__Fixpoint_OIdD,axiom,
    ! [A: $tType,A3: A,B2: A] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),id2(A))
     => ( A3 = B2 ) ) ).

% BNF_Greatest_Fixpoint.IdD
tff(fact_4060_max__ext__additive,axiom,
    ! [A: $tType,A4: set(A),B3: set(A),R: set(product_prod(A,A)),C3: set(A),D4: set(A)] :
      ( aa(set(product_prod(set(A),set(A))),$o,member(product_prod(set(A),set(A)),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),A4),B3)),max_ext(A,R))
     => ( aa(set(product_prod(set(A),set(A))),$o,member(product_prod(set(A),set(A)),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),C3),D4)),max_ext(A,R))
       => aa(set(product_prod(set(A),set(A))),$o,member(product_prod(set(A),set(A)),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),C3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B3),D4))),max_ext(A,R)) ) ) ).

% max_ext_additive
tff(fact_4061_rtrancl__trancl__reflcl,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : transitive_rtrancl(A,R2) = aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),transitive_trancl(A,R2)),id2(A)) ).

% rtrancl_trancl_reflcl
tff(fact_4062_rtrancl__unfold,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : transitive_rtrancl(A,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)),relcomp(A,A,A,transitive_rtrancl(A,R2),R2)) ).

% rtrancl_unfold
tff(fact_4063_reflcl__set__eq,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),X3: A,Xa3: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(fun(A,fun(A,$o)),fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),sup_sup(fun(A,fun(A,$o))),aTP_Lamp_ux(set(product_prod(A,A)),fun(A,fun(A,$o)),R2)),fequal(A)),X3),Xa3)
    <=> aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Xa3)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),R2),id2(A))) ) ).

% reflcl_set_eq
tff(fact_4064_rtrancl__Int__subset,axiom,
    ! [A: $tType,S2: set(product_prod(A,A)),R2: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),id2(A)),S2)
     => ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),relcomp(A,A,A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),transitive_rtrancl(A,R2)),S2),R2)),S2)
       => aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),transitive_rtrancl(A,R2)),S2) ) ) ).

% rtrancl_Int_subset
tff(fact_4065_max__ext_Omax__extI,axiom,
    ! [A: $tType,X5: set(A),Y4: set(A),R: set(product_prod(A,A))] :
      ( aa(set(A),$o,finite_finite2(A),X5)
     => ( aa(set(A),$o,finite_finite2(A),Y4)
       => ( ( Y4 != bot_bot(set(A)) )
         => ( ! [X2: A] :
                ( aa(set(A),$o,member(A,X2),X5)
               => ? [Xa3: A] :
                    ( aa(set(A),$o,member(A,Xa3),Y4)
                    & aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Xa3)),R) ) )
           => aa(set(product_prod(set(A),set(A))),$o,member(product_prod(set(A),set(A)),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),X5),Y4)),max_ext(A,R)) ) ) ) ) ).

% max_ext.max_extI
tff(fact_4066_max__ext_Osimps,axiom,
    ! [A: $tType,A1: set(A),A22: set(A),R: set(product_prod(A,A))] :
      ( aa(set(product_prod(set(A),set(A))),$o,member(product_prod(set(A),set(A)),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),A1),A22)),max_ext(A,R))
    <=> ( aa(set(A),$o,finite_finite2(A),A1)
        & aa(set(A),$o,finite_finite2(A),A22)
        & ( A22 != bot_bot(set(A)) )
        & ! [X4: A] :
            ( aa(set(A),$o,member(A,X4),A1)
           => ? [Xa2: A] :
                ( aa(set(A),$o,member(A,Xa2),A22)
                & aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Xa2)),R) ) ) ) ) ).

% max_ext.simps
tff(fact_4067_max__ext_Ocases,axiom,
    ! [A: $tType,A1: set(A),A22: set(A),R: set(product_prod(A,A))] :
      ( aa(set(product_prod(set(A),set(A))),$o,member(product_prod(set(A),set(A)),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),A1),A22)),max_ext(A,R))
     => ~ ( aa(set(A),$o,finite_finite2(A),A1)
         => ( aa(set(A),$o,finite_finite2(A),A22)
           => ( ( A22 != bot_bot(set(A)) )
             => ~ ! [X3: A] :
                    ( aa(set(A),$o,member(A,X3),A1)
                   => ? [Xa4: A] :
                        ( aa(set(A),$o,member(A,Xa4),A22)
                        & aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Xa4)),R) ) ) ) ) ) ) ).

% max_ext.cases
tff(fact_4068_Total__subset__Id,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( total_on(A,field2(A,R2),R2)
     => ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R2),id2(A))
       => ( ( R2 = bot_bot(set(product_prod(A,A))) )
          | ? [A6: A] : R2 = aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(product_prod(A,A),fun(set(product_prod(A,A)),set(product_prod(A,A))),insert2(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A6),A6)),bot_bot(set(product_prod(A,A)))) ) ) ) ).

% Total_subset_Id
tff(fact_4069_bsqr__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : bNF_Wellorder_bsqr(A,R2) = aa(fun(product_prod(product_prod(A,A),product_prod(A,A)),$o),set(product_prod(product_prod(A,A),product_prod(A,A))),collect(product_prod(product_prod(A,A),product_prod(A,A))),aa(fun(product_prod(A,A),fun(product_prod(A,A),$o)),fun(product_prod(product_prod(A,A),product_prod(A,A)),$o),product_case_prod(product_prod(A,A),product_prod(A,A),$o),aa(fun(A,fun(A,fun(product_prod(A,A),$o))),fun(product_prod(A,A),fun(product_prod(A,A),$o)),product_case_prod(A,A,fun(product_prod(A,A),$o)),aTP_Lamp_uz(set(product_prod(A,A)),fun(A,fun(A,fun(product_prod(A,A),$o))),R2)))) ).

% bsqr_def
tff(fact_4070_max__ext__def,axiom,
    ! [A: $tType,X3: set(product_prod(A,A))] : max_ext(A,X3) = aa(fun(product_prod(set(A),set(A)),$o),set(product_prod(set(A),set(A))),collect(product_prod(set(A),set(A))),aa(fun(set(A),fun(set(A),$o)),fun(product_prod(set(A),set(A)),$o),product_case_prod(set(A),set(A),$o),max_extp(A,aTP_Lamp_ux(set(product_prod(A,A)),fun(A,fun(A,$o)),X3)))) ).

% max_ext_def
tff(fact_4071_max__extp__eq,axiom,
    ! [A: $tType,R2: fun(A,fun(A,$o)),X: set(A),Y: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),max_extp(A,R2),X),Y)
    <=> aa(set(product_prod(set(A),set(A))),$o,member(product_prod(set(A),set(A)),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),X),Y)),max_ext(A,aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),R2)))) ) ).

% max_extp_eq
tff(fact_4072_Field__bsqr,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : field2(product_prod(A,A),bNF_Wellorder_bsqr(A,R2)) = product_Sigma(A,A,field2(A,R2),aTP_Lamp_ut(set(product_prod(A,A)),fun(A,set(A)),R2)) ).

% Field_bsqr
tff(fact_4073_max__extp__max__ext__eq,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),X3: set(A),Xa3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),max_extp(A,aTP_Lamp_ux(set(product_prod(A,A)),fun(A,fun(A,$o)),R)),X3),Xa3)
    <=> aa(set(product_prod(set(A),set(A))),$o,member(product_prod(set(A),set(A)),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),X3),Xa3)),max_ext(A,R)) ) ).

% max_extp_max_ext_eq
tff(fact_4074_linear__order__on__singleton,axiom,
    ! [A: $tType,X: A] : order_679001287576687338der_on(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(product_prod(A,A),fun(set(product_prod(A,A)),set(product_prod(A,A))),insert2(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),X)),bot_bot(set(product_prod(A,A))))) ).

% linear_order_on_singleton
tff(fact_4075_Preorder__Restr,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A4: set(A)] :
      ( order_preorder_on(A,field2(A,R2),R2)
     => order_preorder_on(A,field2(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R2),product_Sigma(A,A,A4,aTP_Lamp_rl(set(A),fun(A,set(A)),A4)))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R2),product_Sigma(A,A,A4,aTP_Lamp_rl(set(A),fun(A,set(A)),A4)))) ) ).

% Preorder_Restr
tff(fact_4076_finite__enumerate__initial__segment,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S: set(A),N: nat,S2: A] :
          ( aa(set(A),$o,finite_finite2(A),S)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S),aa(A,set(A),set_ord_lessThan(A),S2))))
           => ( aa(nat,A,infini527867602293511546merate(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S),aa(A,set(A),set_ord_lessThan(A),S2))),N) = aa(nat,A,infini527867602293511546merate(A,S),N) ) ) ) ) ).

% finite_enumerate_initial_segment
tff(fact_4077_remove__def,axiom,
    ! [A: $tType,X: A,A4: set(A)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),remove(A),X),A4) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) ).

% remove_def
tff(fact_4078_lnear__order__on__empty,axiom,
    ! [A: $tType] : order_679001287576687338der_on(A,bot_bot(set(A)),bot_bot(set(product_prod(A,A)))) ).

% lnear_order_on_empty
tff(fact_4079_preorder__on__empty,axiom,
    ! [A: $tType] : order_preorder_on(A,bot_bot(set(A)),bot_bot(set(product_prod(A,A)))) ).

% preorder_on_empty
tff(fact_4080_range__enumerate,axiom,
    ! [S: set(nat)] :
      ( ~ aa(set(nat),$o,finite_finite2(nat),S)
     => ( aa(set(nat),set(nat),image2(nat,nat,infini527867602293511546merate(nat,S)),top_top(set(nat))) = S ) ) ).

% range_enumerate
tff(fact_4081_Linear__order__Restr,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A4: set(A)] :
      ( order_679001287576687338der_on(A,field2(A,R2),R2)
     => order_679001287576687338der_on(A,field2(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R2),product_Sigma(A,A,A4,aTP_Lamp_rl(set(A),fun(A,set(A)),A4)))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R2),product_Sigma(A,A,A4,aTP_Lamp_rl(set(A),fun(A,set(A)),A4)))) ) ).

% Linear_order_Restr
tff(fact_4082_Linear__order__in__diff__Id,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A3: A,B2: A] :
      ( order_679001287576687338der_on(A,field2(A,R2),R2)
     => ( aa(set(A),$o,member(A,A3),field2(A,R2))
       => ( aa(set(A),$o,member(A,B2),field2(A,R2))
         => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R2)
          <=> ~ aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),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))),R2),id2(A))) ) ) ) ) ).

% Linear_order_in_diff_Id
tff(fact_4083_enumerate__Suc_H,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S: set(A),N: nat] : aa(nat,A,infini527867602293511546merate(A,S),aa(nat,nat,suc,N)) = aa(nat,A,infini527867602293511546merate(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),aa(nat,A,infini527867602293511546merate(A,S),zero_zero(nat))),bot_bot(set(A))))),N) ) ).

% enumerate_Suc'
tff(fact_4084_linear__order__on__Restr,axiom,
    ! [A: $tType,A4: set(A),R2: set(product_prod(A,A)),X: A] :
      ( order_679001287576687338der_on(A,A4,R2)
     => order_679001287576687338der_on(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),order_above(A,R2,X)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R2),product_Sigma(A,A,order_above(A,R2,X),aa(A,fun(A,set(A)),aTP_Lamp_va(set(product_prod(A,A)),fun(A,fun(A,set(A))),R2),X)))) ) ).

% linear_order_on_Restr
tff(fact_4085_finite__enumerate__Suc_H_H,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S: set(A),N: nat] :
          ( aa(set(A),$o,finite_finite2(A),S)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,N)),aa(set(A),nat,finite_card(A),S))
           => ( aa(nat,A,infini527867602293511546merate(A,S),aa(nat,nat,suc,N)) = ord_Least(A,aa(nat,fun(A,$o),aTP_Lamp_vb(set(A),fun(nat,fun(A,$o)),S),N)) ) ) ) ) ).

% finite_enumerate_Suc''
tff(fact_4086_enumerate__Suc,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S: set(A),N: nat] : aa(nat,A,infini527867602293511546merate(A,S),aa(nat,nat,suc,N)) = aa(nat,A,infini527867602293511546merate(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),ord_Least(A,aTP_Lamp_vc(set(A),fun(A,$o),S))),bot_bot(set(A))))),N) ) ).

% enumerate_Suc
tff(fact_4087_Linear__order__wf__diff__Id,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( order_679001287576687338der_on(A,field2(A,R2),R2)
     => ( wf(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),minus_minus(set(product_prod(A,A))),R2),id2(A)))
      <=> ! [A9: set(A)] :
            ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A9),field2(A,R2))
           => ( ( A9 != bot_bot(set(A)) )
             => ? [X4: A] :
                  ( aa(set(A),$o,member(A,X4),A9)
                  & ! [Xa2: A] :
                      ( aa(set(A),$o,member(A,Xa2),A9)
                     => aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Xa2)),R2) ) ) ) ) ) ) ).

% Linear_order_wf_diff_Id
tff(fact_4088_wf__empty,axiom,
    ! [A: $tType] : wf(A,bot_bot(set(product_prod(A,A)))) ).

% wf_empty
tff(fact_4089_wf__insert,axiom,
    ! [A: $tType,Y: A,X: A,R2: set(product_prod(A,A))] :
      ( wf(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(product_prod(A,A),fun(set(product_prod(A,A)),set(product_prod(A,A))),insert2(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),X)),R2))
    <=> ( wf(A,R2)
        & ~ aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),transitive_rtrancl(A,R2)) ) ) ).

% wf_insert
tff(fact_4090_LeastI2,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P: fun(A,$o),A3: A,Q: fun(A,$o)] :
          ( aa(A,$o,P,A3)
         => ( ! [X2: A] :
                ( aa(A,$o,P,X2)
               => aa(A,$o,Q,X2) )
           => aa(A,$o,Q,ord_Least(A,P)) ) ) ) ).

% LeastI2
tff(fact_4091_LeastI__ex,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P: fun(A,$o)] :
          ( ? [X_13: A] : aa(A,$o,P,X_13)
         => aa(A,$o,P,ord_Least(A,P)) ) ) ).

% LeastI_ex
tff(fact_4092_LeastI2__ex,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P: fun(A,$o),Q: fun(A,$o)] :
          ( ? [X_13: A] : aa(A,$o,P,X_13)
         => ( ! [X2: A] :
                ( aa(A,$o,P,X2)
               => aa(A,$o,Q,X2) )
           => aa(A,$o,Q,ord_Least(A,P)) ) ) ) ).

% LeastI2_ex
tff(fact_4093_LeastI,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P: fun(A,$o),K: A] :
          ( aa(A,$o,P,K)
         => aa(A,$o,P,ord_Least(A,P)) ) ) ).

% LeastI
tff(fact_4094_wf__induct__rule,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),P: fun(A,$o),A3: A] :
      ( wf(A,R2)
     => ( ! [X2: A] :
            ( ! [Y5: A] :
                ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y5),X2)),R2)
               => aa(A,$o,P,Y5) )
           => aa(A,$o,P,X2) )
       => aa(A,$o,P,A3) ) ) ).

% wf_induct_rule
tff(fact_4095_wf__eq__minimal,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( wf(A,R2)
    <=> ! [Q7: set(A)] :
          ( ? [X4: A] : aa(set(A),$o,member(A,X4),Q7)
         => ? [X4: A] :
              ( aa(set(A),$o,member(A,X4),Q7)
              & ! [Y3: A] :
                  ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),X4)),R2)
                 => ~ aa(set(A),$o,member(A,Y3),Q7) ) ) ) ) ).

% wf_eq_minimal
tff(fact_4096_wf__not__refl,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A3: A] :
      ( wf(A,R2)
     => ~ aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),A3)),R2) ) ).

% wf_not_refl
tff(fact_4097_wf__not__sym,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A3: A,X: A] :
      ( wf(A,R2)
     => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),X)),R2)
       => ~ aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),A3)),R2) ) ) ).

% wf_not_sym
tff(fact_4098_wf__irrefl,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A3: A] :
      ( wf(A,R2)
     => ~ aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),A3)),R2) ) ).

% wf_irrefl
tff(fact_4099_wf__induct,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),P: fun(A,$o),A3: A] :
      ( wf(A,R2)
     => ( ! [X2: A] :
            ( ! [Y5: A] :
                ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y5),X2)),R2)
               => aa(A,$o,P,Y5) )
           => aa(A,$o,P,X2) )
       => aa(A,$o,P,A3) ) ) ).

% wf_induct
tff(fact_4100_wf__asym,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A3: A,X: A] :
      ( wf(A,R2)
     => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),X)),R2)
       => ~ aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),A3)),R2) ) ) ).

% wf_asym
tff(fact_4101_wfUNIVI,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( ! [P2: fun(A,$o),X2: A] :
          ( ! [Xa3: A] :
              ( ! [Y2: A] :
                  ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),Xa3)),R2)
                 => aa(A,$o,P2,Y2) )
             => aa(A,$o,P2,Xa3) )
         => aa(A,$o,P2,X2) )
     => wf(A,R2) ) ).

% wfUNIVI
tff(fact_4102_wfI__min,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( ! [X2: A,Q2: set(A)] :
          ( aa(set(A),$o,member(A,X2),Q2)
         => ? [Xa3: A] :
              ( aa(set(A),$o,member(A,Xa3),Q2)
              & ! [Y2: A] :
                  ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),Xa3)),R)
                 => ~ aa(set(A),$o,member(A,Y2),Q2) ) ) )
     => wf(A,R) ) ).

% wfI_min
tff(fact_4103_wfE__min,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),X: A,Q: set(A)] :
      ( wf(A,R)
     => ( aa(set(A),$o,member(A,X),Q)
       => ~ ! [Z3: A] :
              ( aa(set(A),$o,member(A,Z3),Q)
             => ~ ! [Y5: A] :
                    ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y5),Z3)),R)
                   => ~ aa(set(A),$o,member(A,Y5),Q) ) ) ) ) ).

% wfE_min
tff(fact_4104_wf__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( wf(A,R2)
    <=> ! [P5: fun(A,$o)] :
          ( ! [X4: A] :
              ( ! [Y3: A] :
                  ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),X4)),R2)
                 => aa(A,$o,P5,Y3) )
             => aa(A,$o,P5,X4) )
         => ! [X_12: A] : aa(A,$o,P5,X_12) ) ) ).

% wf_def
tff(fact_4105_wf__Int2,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),R3: set(product_prod(A,A))] :
      ( wf(A,R2)
     => wf(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R3),R2)) ) ).

% wf_Int2
tff(fact_4106_wf__Int1,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),R3: set(product_prod(A,A))] :
      ( wf(A,R2)
     => wf(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R2),R3)) ) ).

% wf_Int1
tff(fact_4107_LeastI2__wellorder__ex,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P: fun(A,$o),Q: fun(A,$o)] :
          ( ? [X_13: A] : aa(A,$o,P,X_13)
         => ( ! [A6: A] :
                ( aa(A,$o,P,A6)
               => ( ! [B12: A] :
                      ( aa(A,$o,P,B12)
                     => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A6),B12) )
                 => aa(A,$o,Q,A6) ) )
           => aa(A,$o,Q,ord_Least(A,P)) ) ) ) ).

% LeastI2_wellorder_ex
tff(fact_4108_LeastI2__wellorder,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P: fun(A,$o),A3: A,Q: fun(A,$o)] :
          ( aa(A,$o,P,A3)
         => ( ! [A6: A] :
                ( aa(A,$o,P,A6)
               => ( ! [B12: A] :
                      ( aa(A,$o,P,B12)
                     => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A6),B12) )
                 => aa(A,$o,Q,A6) ) )
           => aa(A,$o,Q,ord_Least(A,P)) ) ) ) ).

% LeastI2_wellorder
tff(fact_4109_Least__equality,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P: fun(A,$o),X: A] :
          ( aa(A,$o,P,X)
         => ( ! [Y2: A] :
                ( aa(A,$o,P,Y2)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y2) )
           => ( ord_Least(A,P) = X ) ) ) ) ).

% Least_equality
tff(fact_4110_LeastI2__order,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P: fun(A,$o),X: A,Q: fun(A,$o)] :
          ( aa(A,$o,P,X)
         => ( ! [Y2: A] :
                ( aa(A,$o,P,Y2)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y2) )
           => ( ! [X2: A] :
                  ( aa(A,$o,P,X2)
                 => ( ! [Y5: A] :
                        ( aa(A,$o,P,Y5)
                       => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),Y5) )
                   => aa(A,$o,Q,X2) ) )
             => aa(A,$o,Q,ord_Least(A,P)) ) ) ) ) ).

% LeastI2_order
tff(fact_4111_Least1__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P: fun(A,$o),Z2: A] :
          ( ? [X3: A] :
              ( aa(A,$o,P,X3)
              & ! [Y2: A] :
                  ( aa(A,$o,P,Y2)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Y2) )
              & ! [Y2: A] :
                  ( ( aa(A,$o,P,Y2)
                    & ! [Ya2: A] :
                        ( aa(A,$o,P,Ya2)
                       => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y2),Ya2) ) )
                 => ( Y2 = X3 ) ) )
         => ( aa(A,$o,P,Z2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),ord_Least(A,P)),Z2) ) ) ) ).

% Least1_le
tff(fact_4112_Least1I,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P: fun(A,$o)] :
          ( ? [X3: A] :
              ( aa(A,$o,P,X3)
              & ! [Y2: A] :
                  ( aa(A,$o,P,Y2)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Y2) )
              & ! [Y2: A] :
                  ( ( aa(A,$o,P,Y2)
                    & ! [Ya2: A] :
                        ( aa(A,$o,P,Ya2)
                       => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y2),Ya2) ) )
                 => ( Y2 = X3 ) ) )
         => aa(A,$o,P,ord_Least(A,P)) ) ) ).

% Least1I
tff(fact_4113_wfE__min_H,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),Q: set(A)] :
      ( wf(A,R)
     => ( ( Q != bot_bot(set(A)) )
       => ~ ! [Z3: A] :
              ( aa(set(A),$o,member(A,Z3),Q)
             => ~ ! [Y5: A] :
                    ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y5),Z3)),R)
                   => ~ aa(set(A),$o,member(A,Y5),Q) ) ) ) ) ).

% wfE_min'
tff(fact_4114_wf__no__infinite__down__chainE,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),F: fun(nat,A)] :
      ( wf(A,R2)
     => ~ ! [K2: nat] : aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(nat,A,F,aa(nat,nat,suc,K2))),aa(nat,A,F,K2))),R2) ) ).

% wf_no_infinite_down_chainE
tff(fact_4115_wf__iff__no__infinite__down__chain,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( wf(A,R2)
    <=> ~ ? [F6: fun(nat,A)] :
          ! [I3: nat] : aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(nat,A,F6,aa(nat,nat,suc,I3))),aa(nat,A,F6,I3))),R2) ) ).

% wf_iff_no_infinite_down_chain
tff(fact_4116_Least__le,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P: fun(A,$o),K: A] :
          ( aa(A,$o,P,K)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),ord_Least(A,P)),K) ) ) ).

% Least_le
tff(fact_4117_not__less__Least,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [K: A,P: fun(A,$o)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),K),ord_Least(A,P))
         => ~ aa(A,$o,P,K) ) ) ).

% not_less_Least
tff(fact_4118_wf__no__loop,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( ( relcomp(A,A,A,R,R) = bot_bot(set(product_prod(A,A))) )
     => wf(A,R) ) ).

% wf_no_loop
tff(fact_4119_wf__union__merge,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),S: set(product_prod(A,A))] :
      ( wf(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),R),S))
    <=> wf(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),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))),relcomp(A,A,A,R,R)),relcomp(A,A,A,S,R))),S)) ) ).

% wf_union_merge
tff(fact_4120_Inf__nat__def,axiom,
    ! [X5: set(nat)] : aa(set(nat),nat,complete_Inf_Inf(nat),X5) = ord_Least(nat,aTP_Lamp_vd(set(nat),fun(nat,$o),X5)) ).

% Inf_nat_def
tff(fact_4121_wf,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => wf(A,aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),ord_less(A)))) ) ).

% wf
tff(fact_4122_wf__if__measure,axiom,
    ! [A: $tType,P: fun(A,$o),F: fun(A,nat),G: fun(A,A)] :
      ( ! [X2: A] :
          ( aa(A,$o,P,X2)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,F,aa(A,A,G,X2))),aa(A,nat,F,X2)) )
     => wf(A,aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),aa(fun(A,A),fun(A,fun(A,$o)),aTP_Lamp_ve(fun(A,$o),fun(fun(A,A),fun(A,fun(A,$o))),P),G)))) ) ).

% wf_if_measure
tff(fact_4123_wf__less,axiom,
    wf(nat,aa(fun(product_prod(nat,nat),$o),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),ord_less(nat)))) ).

% wf_less
tff(fact_4124_wf__bounded__measure,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),Ub: fun(A,nat),F: fun(A,nat)] :
      ( ! [A6: A,B5: A] :
          ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B5),A6)),R2)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,Ub,B5)),aa(A,nat,Ub,A6))
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,F,B5)),aa(A,nat,Ub,A6))
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,F,A6)),aa(A,nat,F,B5)) ) )
     => wf(A,R2) ) ).

% wf_bounded_measure
tff(fact_4125_Least__Suc,axiom,
    ! [P: fun(nat,$o),N: nat] :
      ( aa(nat,$o,P,N)
     => ( ~ aa(nat,$o,P,zero_zero(nat))
       => ( ord_Least(nat,P) = aa(nat,nat,suc,ord_Least(nat,aTP_Lamp_vf(fun(nat,$o),fun(nat,$o),P))) ) ) ) ).

% Least_Suc
tff(fact_4126_wf__linord__ex__has__least,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,A)),P: fun(B,$o),K: B,M: fun(B,A)] :
      ( wf(A,R2)
     => ( ! [X2: A,Y2: A] :
            ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y2)),transitive_trancl(A,R2))
          <=> ~ aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),X2)),transitive_rtrancl(A,R2)) )
       => ( aa(B,$o,P,K)
         => ? [X2: B] :
              ( aa(B,$o,P,X2)
              & ! [Y5: B] :
                  ( aa(B,$o,P,Y5)
                 => aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(B,A,M,X2)),aa(B,A,M,Y5))),transitive_rtrancl(A,R2)) ) ) ) ) ) ).

% wf_linord_ex_has_least
tff(fact_4127_wf__union__compatible,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),S: set(product_prod(A,A))] :
      ( wf(A,R)
     => ( wf(A,S)
       => ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),relcomp(A,A,A,R,S)),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))),sup_sup(set(product_prod(A,A))),R),S)) ) ) ) ).

% wf_union_compatible
tff(fact_4128_wfI,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A4: set(A),B3: set(A)] :
      ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R2),product_Sigma(A,A,A4,aTP_Lamp_rl(set(A),fun(A,set(A)),B3)))
     => ( ! [X2: A,P2: fun(A,$o)] :
            ( ! [Xa3: A] :
                ( ! [Y2: A] :
                    ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),Xa3)),R2)
                   => aa(A,$o,P2,Y2) )
               => aa(A,$o,P2,Xa3) )
           => ( aa(set(A),$o,member(A,X2),A4)
             => ( aa(set(A),$o,member(A,X2),B3)
               => aa(A,$o,P2,X2) ) ) )
       => wf(A,R2) ) ) ).

% wfI
tff(fact_4129_Least__Min,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,$o)] :
          ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),P))
         => ( ? [X_13: A] : aa(A,$o,P,X_13)
           => ( ord_Least(A,P) = aa(set(A),A,lattic643756798350308766er_Min(A),aa(fun(A,$o),set(A),collect(A),P)) ) ) ) ) ).

% Least_Min
tff(fact_4130_enumerate__0,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S: set(A)] : aa(nat,A,infini527867602293511546merate(A,S),zero_zero(nat)) = ord_Least(A,aTP_Lamp_vc(set(A),fun(A,$o),S)) ) ).

% enumerate_0
tff(fact_4131_wf__eq__minimal2,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( wf(A,R2)
    <=> ! [A9: set(A)] :
          ( ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A9),field2(A,R2))
            & ( A9 != bot_bot(set(A)) ) )
         => ? [X4: A] :
              ( aa(set(A),$o,member(A,X4),A9)
              & ! [Xa2: A] :
                  ( aa(set(A),$o,member(A,Xa2),A9)
                 => ~ aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa2),X4)),R2) ) ) ) ) ).

% wf_eq_minimal2
tff(fact_4132_wf__bounded__set,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,A)),Ub: fun(A,set(B)),F: fun(A,set(B))] :
      ( ! [A6: A,B5: A] :
          ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B5),A6)),R2)
         => ( aa(set(B),$o,finite_finite2(B),aa(A,set(B),Ub,A6))
            & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(A,set(B),Ub,B5)),aa(A,set(B),Ub,A6))
            & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(A,set(B),F,B5)),aa(A,set(B),Ub,A6))
            & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less(set(B)),aa(A,set(B),F,A6)),aa(A,set(B),F,B5)) ) )
     => wf(A,R2) ) ).

% wf_bounded_set
tff(fact_4133_qc__wf__relto__iff,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),S: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),relcomp(A,A,A,R,S)),relcomp(A,A,A,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))),R),S)),R))
     => ( wf(A,relcomp(A,A,A,transitive_rtrancl(A,S),relcomp(A,A,A,R,transitive_rtrancl(A,S))))
      <=> wf(A,R) ) ) ).

% qc_wf_relto_iff
tff(fact_4134_above__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A3: A] : order_above(A,R2,A3) = aa(fun(A,$o),set(A),collect(A),aa(A,fun(A,$o),aTP_Lamp_ux(set(product_prod(A,A)),fun(A,fun(A,$o)),R2),A3)) ).

% above_def
tff(fact_4135_wf__bounded__supset,axiom,
    ! [A: $tType,S: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),S)
     => wf(set(A),aa(fun(product_prod(set(A),set(A)),$o),set(product_prod(set(A),set(A))),collect(product_prod(set(A),set(A))),aa(fun(set(A),fun(set(A),$o)),fun(product_prod(set(A),set(A)),$o),product_case_prod(set(A),set(A),$o),aTP_Lamp_vg(set(A),fun(set(A),fun(set(A),$o)),S)))) ) ).

% wf_bounded_supset
tff(fact_4136_finite__subset__wf,axiom,
    ! [A: $tType,A4: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => wf(set(A),aa(fun(product_prod(set(A),set(A)),$o),set(product_prod(set(A),set(A))),collect(product_prod(set(A),set(A))),aa(fun(set(A),fun(set(A),$o)),fun(product_prod(set(A),set(A)),$o),product_case_prod(set(A),set(A),$o),aTP_Lamp_vh(set(A),fun(set(A),fun(set(A),$o)),A4)))) ) ).

% finite_subset_wf
tff(fact_4137_Least__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F: fun(A,B),S: set(A)] :
          ( order_mono(A,B,F)
         => ( ? [X3: A] :
                ( aa(set(A),$o,member(A,X3),S)
                & ! [Xa4: A] :
                    ( aa(set(A),$o,member(A,Xa4),S)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Xa4) ) )
           => ( ord_Least(B,aa(set(A),fun(B,$o),aTP_Lamp_vi(fun(A,B),fun(set(A),fun(B,$o)),F),S)) = aa(A,B,F,ord_Least(A,aTP_Lamp_vj(set(A),fun(A,$o),S))) ) ) ) ) ).

% Least_mono
tff(fact_4138_enumerate__Suc_H_H,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S: set(A),N: nat] :
          ( ~ aa(set(A),$o,finite_finite2(A),S)
         => ( aa(nat,A,infini527867602293511546merate(A,S),aa(nat,nat,suc,N)) = ord_Least(A,aa(nat,fun(A,$o),aTP_Lamp_vb(set(A),fun(nat,fun(A,$o)),S),N)) ) ) ) ).

% enumerate_Suc''
tff(fact_4139_reduction__pairI,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),S: set(product_prod(A,A))] :
      ( wf(A,R)
     => ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),relcomp(A,A,A,R,S)),R)
       => fun_reduction_pair(A,aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R),S)) ) ) ).

% reduction_pairI
tff(fact_4140_dependent__wf__choice,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,A)),P: fun(fun(A,B),fun(A,fun(B,$o)))] :
      ( wf(A,R)
     => ( ! [F2: fun(A,B),G3: fun(A,B),X2: A,R4: B] :
            ( ! [Z6: A] :
                ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z6),X2)),R)
               => ( aa(A,B,F2,Z6) = aa(A,B,G3,Z6) ) )
           => ( aa(B,$o,aa(A,fun(B,$o),aa(fun(A,B),fun(A,fun(B,$o)),P,F2),X2),R4)
            <=> aa(B,$o,aa(A,fun(B,$o),aa(fun(A,B),fun(A,fun(B,$o)),P,G3),X2),R4) ) )
       => ( ! [X2: A,F2: fun(A,B)] :
              ( ! [Y5: A] :
                  ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y5),X2)),R)
                 => aa(B,$o,aa(A,fun(B,$o),aa(fun(A,B),fun(A,fun(B,$o)),P,F2),Y5),aa(A,B,F2,Y5)) )
             => ? [X_13: B] : aa(B,$o,aa(A,fun(B,$o),aa(fun(A,B),fun(A,fun(B,$o)),P,F2),X2),X_13) )
         => ? [F2: fun(A,B)] :
            ! [X3: A] : aa(B,$o,aa(A,fun(B,$o),aa(fun(A,B),fun(A,fun(B,$o)),P,F2),X3),aa(A,B,F2,X3)) ) ) ) ).

% dependent_wf_choice
tff(fact_4141_abort__Bleast__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [S: set(A),P: fun(A,$o)] : abort_Bleast(A,S,P) = ord_Least(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_vk(set(A),fun(fun(A,$o),fun(A,$o)),S),P)) ) ).

% abort_Bleast_def
tff(fact_4142_Bleast__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [S: set(A),P: fun(A,$o)] : bleast(A,S,P) = ord_Least(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_vk(set(A),fun(fun(A,$o),fun(A,$o)),S),P)) ) ).

% Bleast_def
tff(fact_4143_chains__extend,axiom,
    ! [A: $tType,C2: set(set(A)),S: set(set(A)),Z2: set(A)] :
      ( aa(set(set(set(A))),$o,member(set(set(A)),C2),chains2(A,S))
     => ( aa(set(set(A)),$o,member(set(A),Z2),S)
       => ( ! [X2: set(A)] :
              ( aa(set(set(A)),$o,member(set(A),X2),C2)
             => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X2),Z2) )
         => aa(set(set(set(A))),$o,member(set(set(A)),aa(set(set(A)),set(set(A)),aa(set(set(A)),fun(set(set(A)),set(set(A))),sup_sup(set(set(A))),aa(set(set(A)),set(set(A)),aa(set(A),fun(set(set(A)),set(set(A))),insert2(set(A)),Z2),bot_bot(set(set(A))))),C2)),chains2(A,S)) ) ) ) ).

% chains_extend
tff(fact_4144_rat__sgn__code,axiom,
    ! [P3: rat] : quotient_of(aa(rat,rat,sgn_sgn(rat),P3)) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,sgn_sgn(int),aa(product_prod(int,int),int,product_fst(int,int),quotient_of(P3)))),one_one(int)) ).

% rat_sgn_code
tff(fact_4145_abstract__boolean__algebra__sym__diff__def,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xor: fun(A,fun(A,A))] :
      ( boolea3799213064322606851m_diff(A,Conj,Disj,Compl,Zero,One,Xor)
    <=> ( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
        & boolea5476839437570043046axioms(A,Conj,Disj,Compl,Xor) ) ) ).

% abstract_boolean_algebra_sym_diff_def
tff(fact_4146_abstract__boolean__algebra__sym__diff_Ointro,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xor: fun(A,fun(A,A))] :
      ( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
     => ( boolea5476839437570043046axioms(A,Conj,Disj,Compl,Xor)
       => boolea3799213064322606851m_diff(A,Conj,Disj,Compl,Zero,One,Xor) ) ) ).

% abstract_boolean_algebra_sym_diff.intro
tff(fact_4147_img__fst,axiom,
    ! [B: $tType,A: $tType,A3: A,B2: B,S: set(product_prod(A,B))] :
      ( aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),S)
     => aa(set(A),$o,member(A,A3),aa(set(product_prod(A,B)),set(A),image2(product_prod(A,B),A,product_fst(A,B)),S)) ) ).

% img_fst
tff(fact_4148_range__fst,axiom,
    ! [B: $tType,A: $tType] : aa(set(product_prod(A,B)),set(A),image2(product_prod(A,B),A,product_fst(A,B)),top_top(set(product_prod(A,B)))) = top_top(set(A)) ).

% range_fst
tff(fact_4149_fst__image__times,axiom,
    ! [B: $tType,A: $tType,A4: set(A),B3: set(B)] :
      aa(set(product_prod(A,B)),set(A),image2(product_prod(A,B),A,product_fst(A,B)),product_Sigma(A,B,A4,aTP_Lamp_qz(set(B),fun(A,set(B)),B3))) = $ite(B3 = bot_bot(set(B)),bot_bot(set(A)),A4) ).

% fst_image_times
tff(fact_4150_one__div__numeral,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [N: num] : aa(A,A,aa(A,fun(A,A),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_4151_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_4152_fstE,axiom,
    ! [B: $tType,A: $tType,X: product_prod(A,B),A3: A,B2: B,P: fun(A,$o)] :
      ( ( X = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2) )
     => ( aa(A,$o,P,aa(product_prod(A,B),A,product_fst(A,B),X))
       => aa(A,$o,P,A3) ) ) ).

% fstE
tff(fact_4153_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_4154_fst__conv,axiom,
    ! [B: $tType,A: $tType,X1: A,X22: B] : aa(product_prod(A,B),A,product_fst(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X1),X22)) = X1 ).

% fst_conv
tff(fact_4155_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_vl(A,product_prod(A,A))),product_fst(A,B))) = product_fst(A,B) ).

% fst_diag_fst
tff(fact_4156_abstract__boolean__algebra__sym__diff__axioms_Ointro,axiom,
    ! [A: $tType,Xor: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Conj: fun(A,fun(A,A)),Compl: fun(A,A)] :
      ( ! [X2: A,Y2: A] : aa(A,A,aa(A,fun(A,A),Xor,X2),Y2) = aa(A,A,aa(A,fun(A,A),Disj,aa(A,A,aa(A,fun(A,A),Conj,X2),aa(A,A,Compl,Y2))),aa(A,A,aa(A,fun(A,A),Conj,aa(A,A,Compl,X2)),Y2))
     => boolea5476839437570043046axioms(A,Conj,Disj,Compl,Xor) ) ).

% abstract_boolean_algebra_sym_diff_axioms.intro
tff(fact_4157_abstract__boolean__algebra__sym__diff__axioms__def,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Xor: fun(A,fun(A,A))] :
      ( boolea5476839437570043046axioms(A,Conj,Disj,Compl,Xor)
    <=> ! [X4: A,Y3: A] : aa(A,A,aa(A,fun(A,A),Xor,X4),Y3) = aa(A,A,aa(A,fun(A,A),Disj,aa(A,A,aa(A,fun(A,A),Conj,X4),aa(A,A,Compl,Y3))),aa(A,A,aa(A,fun(A,A),Conj,aa(A,A,Compl,X4)),Y3)) ) ).

% abstract_boolean_algebra_sym_diff_axioms_def
tff(fact_4158_fn__fst__conv,axiom,
    ! [B: $tType,C: $tType,A: $tType,F: fun(A,C)] : aTP_Lamp_vm(fun(A,C),fun(product_prod(A,B),C),F) = aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),aTP_Lamp_vn(fun(A,C),fun(A,fun(B,C)),F)) ).

% fn_fst_conv
tff(fact_4159_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_kf(A,fun(B,A))),Prod) ).

% fst_def
tff(fact_4160_in__fst__imageE,axiom,
    ! [B: $tType,A: $tType,X: A,S: set(product_prod(A,B))] :
      ( aa(set(A),$o,member(A,X),aa(set(product_prod(A,B)),set(A),image2(product_prod(A,B),A,product_fst(A,B)),S))
     => ~ ! [Y2: B] : ~ aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y2)),S) ) ).

% in_fst_imageE
tff(fact_4161_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_vl(A,product_prod(A,A))),Z2) = aa(A,A,id(A),Z2) ).

% fst_diag_id
tff(fact_4162_vimage__fst,axiom,
    ! [B: $tType,A: $tType,A4: set(A)] : aa(set(A),set(product_prod(A,B)),aa(fun(product_prod(A,B),A),fun(set(A),set(product_prod(A,B))),vimage(product_prod(A,B),A),product_fst(A,B)),A4) = product_Sigma(A,B,A4,aTP_Lamp_rb(A,set(B))) ).

% vimage_fst
tff(fact_4163_fst__image__mp,axiom,
    ! [B: $tType,A: $tType,A4: set(product_prod(A,B)),B3: set(A),X: A,Y: B] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(product_prod(A,B)),set(A),image2(product_prod(A,B),A,product_fst(A,B)),A4)),B3)
     => ( aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y)),A4)
       => aa(set(A),$o,member(A,X),B3) ) ) ).

% fst_image_mp
tff(fact_4164_fst__image__Sigma,axiom,
    ! [B: $tType,A: $tType,A4: set(A),B3: fun(A,set(B))] : aa(set(product_prod(A,B)),set(A),image2(product_prod(A,B),A,product_fst(A,B)),product_Sigma(A,B,A4,B3)) = aa(fun(A,$o),set(A),collect(A),aa(fun(A,set(B)),fun(A,$o),aTP_Lamp_ro(set(A),fun(fun(A,set(B)),fun(A,$o)),A4),B3)) ).

% fst_image_Sigma
tff(fact_4165_abstract__boolean__algebra__sym__diff_Oaxioms_I2_J,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xor: fun(A,fun(A,A))] :
      ( boolea3799213064322606851m_diff(A,Conj,Disj,Compl,Zero,One,Xor)
     => boolea5476839437570043046axioms(A,Conj,Disj,Compl,Xor) ) ).

% abstract_boolean_algebra_sym_diff.axioms(2)
tff(fact_4166_graph__fun__upd__None,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B)),K: A] : graph(A,B,fun_upd(A,option(B),M,K,none(B))) = aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(A,fun(product_prod(A,B),$o),aTP_Lamp_vo(fun(A,option(B)),fun(A,fun(product_prod(A,B),$o)),M),K)) ).

% graph_fun_upd_None
tff(fact_4167_bezw_Oelims,axiom,
    ! [X: nat,Xa: nat,Y: product_prod(int,int)] :
      ( ( bezw(X,Xa) = Y )
     => ( Y = $ite(Xa = zero_zero(nat),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),zero_zero(int)),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Xa,modulo_modulo(nat,X,Xa)))),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(product_prod(int,int),int,product_fst(int,int),bezw(Xa,modulo_modulo(nat,X,Xa)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Xa,modulo_modulo(nat,X,Xa)))),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),X),Xa)))))) ) ) ).

% bezw.elims
tff(fact_4168_bezw_Osimps,axiom,
    ! [X: nat,Y: nat] :
      bezw(X,Y) = $ite(Y = zero_zero(nat),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),zero_zero(int)),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Y,modulo_modulo(nat,X,Y)))),aa(int,int,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),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),X),Y)))))) ).

% bezw.simps
tff(fact_4169_insert__relcomp__union__fold,axiom,
    ! [C: $tType,B: $tType,A: $tType,S: set(product_prod(A,B)),X: product_prod(C,A),X5: set(product_prod(C,B))] :
      ( aa(set(product_prod(A,B)),$o,finite_finite2(product_prod(A,B)),S)
     => ( aa(set(product_prod(C,B)),set(product_prod(C,B)),aa(set(product_prod(C,B)),fun(set(product_prod(C,B)),set(product_prod(C,B))),sup_sup(set(product_prod(C,B))),relcomp(C,A,B,aa(set(product_prod(C,A)),set(product_prod(C,A)),aa(product_prod(C,A),fun(set(product_prod(C,A)),set(product_prod(C,A))),insert2(product_prod(C,A)),X),bot_bot(set(product_prod(C,A)))),S)),X5) = finite_fold(product_prod(A,B),set(product_prod(C,B)),aa(fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B))))),fun(product_prod(A,B),fun(set(product_prod(C,B)),set(product_prod(C,B)))),product_case_prod(A,B,fun(set(product_prod(C,B)),set(product_prod(C,B)))),aTP_Lamp_vp(product_prod(C,A),fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B))))),X)),X5,S) ) ) ).

% insert_relcomp_union_fold
tff(fact_4170_bezw__non__0,axiom,
    ! [Y: nat,X: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Y)
     => ( bezw(X,Y) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Y,modulo_modulo(nat,X,Y)))),aa(int,int,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),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),X),Y))))) ) ) ).

% bezw_non_0
tff(fact_4171_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_4172_img__snd,axiom,
    ! [B: $tType,A: $tType,A3: A,B2: B,S: set(product_prod(A,B))] :
      ( aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),S)
     => aa(set(B),$o,member(B,B2),aa(set(product_prod(A,B)),set(B),image2(product_prod(A,B),B,product_snd(A,B)),S)) ) ).

% img_snd
tff(fact_4173_range__snd,axiom,
    ! [B: $tType,A: $tType] : aa(set(product_prod(B,A)),set(A),image2(product_prod(B,A),A,product_snd(B,A)),top_top(set(product_prod(B,A)))) = top_top(set(A)) ).

% range_snd
tff(fact_4174_snd__image__times,axiom,
    ! [B: $tType,A: $tType,A4: set(B),B3: set(A)] :
      aa(set(product_prod(B,A)),set(A),image2(product_prod(B,A),A,product_snd(B,A)),product_Sigma(B,A,A4,aTP_Lamp_ke(set(A),fun(B,set(A)),B3))) = $ite(A4 = bot_bot(set(B)),bot_bot(set(A)),B3) ).

% snd_image_times
tff(fact_4175_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_4176_snd__def,axiom,
    ! [A: $tType,B: $tType,Prod: product_prod(B,A)] : aa(product_prod(B,A),A,product_snd(B,A),Prod) = aa(product_prod(B,A),A,aa(fun(B,fun(A,A)),fun(product_prod(B,A),A),product_case_prod(B,A,A),aTP_Lamp_vq(B,fun(A,A))),Prod) ).

% snd_def
tff(fact_4177_fn__snd__conv,axiom,
    ! [A: $tType,C: $tType,B: $tType,F: fun(B,C)] : aTP_Lamp_vr(fun(B,C),fun(product_prod(A,B),C),F) = aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),aTP_Lamp_vs(fun(B,C),fun(A,fun(B,C)),F)) ).

% fn_snd_conv
tff(fact_4178_snd__conv,axiom,
    ! [B: $tType,A: $tType,X1: B,X22: A] : aa(product_prod(B,A),A,product_snd(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X1),X22)) = X22 ).

% snd_conv
tff(fact_4179_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_4180_sndE,axiom,
    ! [A: $tType,B: $tType,X: product_prod(A,B),A3: A,B2: B,P: fun(B,$o)] :
      ( ( X = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2) )
     => ( aa(B,$o,P,aa(product_prod(A,B),B,product_snd(A,B),X))
       => aa(B,$o,P,B2) ) ) ).

% sndE
tff(fact_4181_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_vt(B,product_prod(B,B))),product_snd(A,B))) = product_snd(A,B) ).

% snd_diag_snd
tff(fact_4182_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_4183_surjective__pairing,axiom,
    ! [B: $tType,A: $tType,T3: product_prod(A,B)] : T3 = 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),T3)),aa(product_prod(A,B),B,product_snd(A,B),T3)) ).

% surjective_pairing
tff(fact_4184_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_4185_exI__realizer,axiom,
    ! [B: $tType,A: $tType,P: fun(A,fun(B,$o)),Y: A,X: B] :
      ( aa(B,$o,aa(A,fun(B,$o),P,Y),X)
     => aa(B,$o,aa(A,fun(B,$o),P,aa(product_prod(B,A),A,product_snd(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X),Y))),aa(product_prod(B,A),B,product_fst(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X),Y))) ) ).

% exI_realizer
tff(fact_4186_conjI__realizer,axiom,
    ! [A: $tType,B: $tType,P: fun(A,$o),P3: A,Q: fun(B,$o),Q3: B] :
      ( aa(A,$o,P,P3)
     => ( aa(B,$o,Q,Q3)
       => ( aa(A,$o,P,aa(product_prod(A,B),A,product_fst(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),P3),Q3)))
          & aa(B,$o,Q,aa(product_prod(A,B),B,product_snd(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),P3),Q3))) ) ) ) ).

% conjI_realizer
tff(fact_4187_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [B: $tType,A: $tType,P: fun(A,fun(B,$o)),X: A,Y: B,A3: product_prod(A,B)] :
      ( aa(B,$o,aa(A,fun(B,$o),P,X),Y)
     => ( ( A3 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y) )
       => aa(B,$o,aa(A,fun(B,$o),P,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_4188_in__snd__imageE,axiom,
    ! [A: $tType,B: $tType,Y: A,S: set(product_prod(B,A))] :
      ( aa(set(A),$o,member(A,Y),aa(set(product_prod(B,A)),set(A),image2(product_prod(B,A),A,product_snd(B,A)),S))
     => ~ ! [X2: B] : ~ aa(set(product_prod(B,A)),$o,member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X2),Y)),S) ) ).

% in_snd_imageE
tff(fact_4189_reduction__pair__lemma,axiom,
    ! [A: $tType,P: product_prod(set(product_prod(A,A)),set(product_prod(A,A))),R: set(product_prod(A,A)),S: set(product_prod(A,A))] :
      ( fun_reduction_pair(A,P)
     => ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R),aa(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),set(product_prod(A,A)),product_fst(set(product_prod(A,A)),set(product_prod(A,A))),P))
       => ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),S),aa(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),set(product_prod(A,A)),product_snd(set(product_prod(A,A)),set(product_prod(A,A))),P))
         => ( wf(A,S)
           => wf(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),R),S)) ) ) ) ) ).

% reduction_pair_lemma
tff(fact_4190_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_vl(A,product_prod(A,A))),product_fst(A,B))) = product_fst(A,B) ).

% snd_diag_fst
tff(fact_4191_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_vt(B,product_prod(B,B))),product_snd(A,B))) = product_snd(A,B) ).

% fst_diag_snd
tff(fact_4192_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_vl(A,product_prod(A,A))),Z2) = aa(A,A,id(A),Z2) ).

% snd_diag_id
tff(fact_4193_vimage__snd,axiom,
    ! [A: $tType,B: $tType,A4: set(B)] : aa(set(B),set(product_prod(A,B)),aa(fun(product_prod(A,B),B),fun(set(B),set(product_prod(A,B))),vimage(product_prod(A,B),B),product_snd(A,B)),A4) = product_Sigma(A,B,top_top(set(A)),aTP_Lamp_qz(set(B),fun(A,set(B)),A4)) ).

% vimage_snd
tff(fact_4194_exE__realizer,axiom,
    ! [C: $tType,A: $tType,B: $tType,P: fun(A,fun(B,$o)),P3: product_prod(B,A),Q: fun(C,$o),F: fun(B,fun(A,C))] :
      ( aa(B,$o,aa(A,fun(B,$o),P,aa(product_prod(B,A),A,product_snd(B,A),P3)),aa(product_prod(B,A),B,product_fst(B,A),P3))
     => ( ! [X2: B,Y2: A] :
            ( aa(B,$o,aa(A,fun(B,$o),P,Y2),X2)
           => aa(C,$o,Q,aa(A,C,aa(B,fun(A,C),F,X2),Y2)) )
       => aa(C,$o,Q,aa(product_prod(B,A),C,aa(fun(B,fun(A,C)),fun(product_prod(B,A),C),product_case_prod(B,A,C),F),P3)) ) ) ).

% exE_realizer
tff(fact_4195_case__prod__unfold,axiom,
    ! [C: $tType,B: $tType,A: $tType,X3: fun(A,fun(B,C)),Xa3: product_prod(A,B)] : aa(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),X3),Xa3) = aa(B,C,aa(A,fun(B,C),X3,aa(product_prod(A,B),A,product_fst(A,B),Xa3)),aa(product_prod(A,B),B,product_snd(A,B),Xa3)) ).

% case_prod_unfold
tff(fact_4196_case__prod__beta_H,axiom,
    ! [C: $tType,B: $tType,A: $tType,F: fun(A,fun(B,C)),X3: product_prod(A,B)] : aa(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F),X3) = aa(B,C,aa(A,fun(B,C),F,aa(product_prod(A,B),A,product_fst(A,B),X3)),aa(product_prod(A,B),B,product_snd(A,B),X3)) ).

% case_prod_beta'
tff(fact_4197_split__comp__eq,axiom,
    ! [B: $tType,C: $tType,D: $tType,A: $tType,F: fun(D,fun(B,C)),G: fun(A,D)] : aa(fun(A,D),fun(product_prod(A,B),C),aTP_Lamp_vu(fun(D,fun(B,C)),fun(fun(A,D),fun(product_prod(A,B),C)),F),G) = aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),aa(fun(A,D),fun(A,fun(B,C)),aTP_Lamp_vv(fun(D,fun(B,C)),fun(fun(A,D),fun(A,fun(B,C))),F),G)) ).

% split_comp_eq
tff(fact_4198_mem__Times__iff,axiom,
    ! [A: $tType,B: $tType,X: product_prod(A,B),A4: set(A),B3: set(B)] :
      ( aa(set(product_prod(A,B)),$o,member(product_prod(A,B),X),product_Sigma(A,B,A4,aTP_Lamp_qz(set(B),fun(A,set(B)),B3)))
    <=> ( aa(set(A),$o,member(A,aa(product_prod(A,B),A,product_fst(A,B),X)),A4)
        & aa(set(B),$o,member(B,aa(product_prod(A,B),B,product_snd(A,B),X)),B3) ) ) ).

% mem_Times_iff
tff(fact_4199_in__prod__fst__sndI,axiom,
    ! [A: $tType,B: $tType,X: product_prod(A,B),A4: set(A),B3: set(B)] :
      ( aa(set(A),$o,member(A,aa(product_prod(A,B),A,product_fst(A,B),X)),A4)
     => ( aa(set(B),$o,member(B,aa(product_prod(A,B),B,product_snd(A,B),X)),B3)
       => aa(set(product_prod(A,B)),$o,member(product_prod(A,B),X),product_Sigma(A,B,A4,aTP_Lamp_qz(set(B),fun(A,set(B)),B3))) ) ) ).

% in_prod_fst_sndI
tff(fact_4200_Id__fstsnd__eq,axiom,
    ! [A: $tType] : id2(A) = aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aTP_Lamp_vw(product_prod(A,A),$o)) ).

% Id_fstsnd_eq
tff(fact_4201_prod_Osplit__sel__asm,axiom,
    ! [A: $tType,C: $tType,B: $tType,P: fun(A,$o),F: fun(B,fun(C,A)),Prod: product_prod(B,C)] :
      ( aa(A,$o,P,aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),F),Prod))
    <=> ~ ( ( Prod = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),aa(product_prod(B,C),B,product_fst(B,C),Prod)),aa(product_prod(B,C),C,product_snd(B,C),Prod)) )
          & ~ aa(A,$o,P,aa(C,A,aa(B,fun(C,A),F,aa(product_prod(B,C),B,product_fst(B,C),Prod)),aa(product_prod(B,C),C,product_snd(B,C),Prod))) ) ) ).

% prod.split_sel_asm
tff(fact_4202_prod_Osplit__sel,axiom,
    ! [A: $tType,C: $tType,B: $tType,P: fun(A,$o),F: fun(B,fun(C,A)),Prod: product_prod(B,C)] :
      ( aa(A,$o,P,aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),F),Prod))
    <=> ( ( Prod = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),aa(product_prod(B,C),B,product_fst(B,C),Prod)),aa(product_prod(B,C),C,product_snd(B,C),Prod)) )
       => aa(A,$o,P,aa(C,A,aa(B,fun(C,A),F,aa(product_prod(B,C),B,product_fst(B,C),Prod)),aa(product_prod(B,C),C,product_snd(B,C),Prod))) ) ) ).

% prod.split_sel
tff(fact_4203_snd__image__mp,axiom,
    ! [B: $tType,A: $tType,A4: set(product_prod(B,A)),B3: set(A),X: B,Y: A] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(product_prod(B,A)),set(A),image2(product_prod(B,A),A,product_snd(B,A)),A4)),B3)
     => ( aa(set(product_prod(B,A)),$o,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)),A4)
       => aa(set(A),$o,member(A,Y),B3) ) ) ).

% snd_image_mp
tff(fact_4204_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_ru(B,fun(A,product_prod(A,B))))),Xy) ).

% snd_fst_flip
tff(fact_4205_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_vx(A,fun(B,product_prod(B,A))))),Xy) ).

% fst_snd_flip
tff(fact_4206_case__prod__comp,axiom,
    ! [D: $tType,A: $tType,C: $tType,B: $tType,F: fun(D,fun(C,A)),G: 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,F),G)),X) = aa(C,A,aa(D,fun(C,A),F,aa(B,D,G,aa(product_prod(B,C),B,product_fst(B,C),X))),aa(product_prod(B,C),C,product_snd(B,C),X)) ).

% case_prod_comp
tff(fact_4207_The__case__prod,axiom,
    ! [B: $tType,A: $tType,P: fun(A,fun(B,$o))] : the(product_prod(A,B),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),P)) = the(product_prod(A,B),aTP_Lamp_vy(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),P)) ).

% The_case_prod
tff(fact_4208_snd__image__Sigma,axiom,
    ! [A: $tType,B: $tType,A4: set(B),B3: fun(B,set(A))] : aa(set(product_prod(B,A)),set(A),image2(product_prod(B,A),A,product_snd(B,A)),product_Sigma(B,A,A4,B3)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B3),A4)) ).

% snd_image_Sigma
tff(fact_4209_subset__fst__snd,axiom,
    ! [B: $tType,A: $tType,A4: set(product_prod(A,B))] : aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),A4),product_Sigma(A,B,aa(set(product_prod(A,B)),set(A),image2(product_prod(A,B),A,product_fst(A,B)),A4),aTP_Lamp_vz(set(product_prod(A,B)),fun(A,set(B)),A4))) ).

% subset_fst_snd
tff(fact_4210_vimage__Times,axiom,
    ! [A: $tType,B: $tType,C: $tType,F: fun(A,product_prod(B,C)),A4: set(B),B3: set(C)] : aa(set(product_prod(B,C)),set(A),aa(fun(A,product_prod(B,C)),fun(set(product_prod(B,C)),set(A)),vimage(A,product_prod(B,C)),F),product_Sigma(B,C,A4,aTP_Lamp_rw(set(C),fun(B,set(C)),B3))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(B),set(A),aa(fun(A,B),fun(set(B),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)),F)),A4)),aa(set(C),set(A),aa(fun(A,C),fun(set(C),set(A)),vimage(A,C),aa(fun(A,product_prod(B,C)),fun(A,C),comp(product_prod(B,C),C,A,product_snd(B,C)),F)),B3)) ).

% vimage_Times
tff(fact_4211_finite__range__prod,axiom,
    ! [A: $tType,C: $tType,B: $tType,F: fun(B,product_prod(A,C))] :
      ( aa(set(A),$o,finite_finite2(A),aa(set(B),set(A),image2(B,A,aa(fun(B,product_prod(A,C)),fun(B,A),comp(product_prod(A,C),A,B,product_fst(A,C)),F)),top_top(set(B))))
     => ( aa(set(C),$o,finite_finite2(C),aa(set(B),set(C),image2(B,C,aa(fun(B,product_prod(A,C)),fun(B,C),comp(product_prod(A,C),C,B,product_snd(A,C)),F)),top_top(set(B))))
       => aa(set(product_prod(A,C)),$o,finite_finite2(product_prod(A,C)),aa(set(B),set(product_prod(A,C)),image2(B,product_prod(A,C),F),top_top(set(B)))) ) ) ).

% finite_range_prod
tff(fact_4212_range__prod,axiom,
    ! [B: $tType,A: $tType,C: $tType,F: fun(C,product_prod(A,B))] : aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),aa(set(C),set(product_prod(A,B)),image2(C,product_prod(A,B),F),top_top(set(C)))),product_Sigma(A,B,aa(set(C),set(A),image2(C,A,aa(fun(C,product_prod(A,B)),fun(C,A),comp(product_prod(A,B),A,C,product_fst(A,B)),F)),top_top(set(C))),aTP_Lamp_wa(fun(C,product_prod(A,B)),fun(A,set(B)),F))) ).

% range_prod
tff(fact_4213_ID_Oin__rel,axiom,
    ! [A: $tType,B: $tType,R: fun(A,fun(B,$o)),A3: A,B2: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),bNF_id_bnf(fun(A,fun(B,$o))),R),A3),B2)
    <=> ? [Z4: product_prod(A,B)] :
          ( aa(set(product_prod(A,B)),$o,member(product_prod(A,B),Z4),aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aTP_Lamp_wb(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),R)))
          & ( aa(product_prod(A,B),A,aa(fun(product_prod(A,B),A),fun(product_prod(A,B),A),bNF_id_bnf(fun(product_prod(A,B),A)),product_fst(A,B)),Z4) = A3 )
          & ( aa(product_prod(A,B),B,aa(fun(product_prod(A,B),B),fun(product_prod(A,B),B),bNF_id_bnf(fun(product_prod(A,B),B)),product_snd(A,B)),Z4) = B2 ) ) ) ).

% ID.in_rel
tff(fact_4214_insert__relcomp__fold,axiom,
    ! [C: $tType,B: $tType,A: $tType,S: set(product_prod(A,B)),X: product_prod(C,A),R: set(product_prod(C,A))] :
      ( aa(set(product_prod(A,B)),$o,finite_finite2(product_prod(A,B)),S)
     => ( relcomp(C,A,B,aa(set(product_prod(C,A)),set(product_prod(C,A)),aa(product_prod(C,A),fun(set(product_prod(C,A)),set(product_prod(C,A))),insert2(product_prod(C,A)),X),R),S) = 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_vp(product_prod(C,A),fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B))))),X)),relcomp(C,A,B,R,S),S) ) ) ).

% insert_relcomp_fold
tff(fact_4215_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_4216_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_4217_minus__numeral__mod__numeral,axiom,
    ! [M: num,N: num] : modulo_modulo(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M)),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,M,N))) ).

% minus_numeral_mod_numeral
tff(fact_4218_numeral__mod__minus__numeral,axiom,
    ! [M: num,N: num] : modulo_modulo(int,aa(num,int,numeral_numeral(int),M),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,M,N)))) ).

% numeral_mod_minus_numeral
tff(fact_4219_bezw_Opelims,axiom,
    ! [X: nat,Xa: nat,Y: product_prod(int,int)] :
      ( ( bezw(X,Xa) = 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),Xa))
       => ~ ( ( Y = $ite(Xa = zero_zero(nat),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),zero_zero(int)),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Xa,modulo_modulo(nat,X,Xa)))),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(product_prod(int,int),int,product_fst(int,int),bezw(Xa,modulo_modulo(nat,X,Xa)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Xa,modulo_modulo(nat,X,Xa)))),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),X),Xa)))))) )
           => ~ accp(product_prod(nat,nat),bezw_rel,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X),Xa)) ) ) ) ).

% bezw.pelims
tff(fact_4220_normalize__def,axiom,
    ! [P3: product_prod(int,int)] :
      normalize(P3) = $ite(
        aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),aa(product_prod(int,int),int,product_snd(int,int),P3)),
        $let(
          a2: int,
          a2:= aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(product_prod(int,int),int,product_fst(int,int),P3)),aa(product_prod(int,int),int,product_snd(int,int),P3)),
          aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(product_prod(int,int),int,product_fst(int,int),P3)),a2)),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(product_prod(int,int),int,product_snd(int,int),P3)),a2)) ),
        $ite(
          aa(product_prod(int,int),int,product_snd(int,int),P3) = zero_zero(int),
          aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int)),
          $let(
            a2: int,
            a2:= aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(product_prod(int,int),int,product_fst(int,int),P3)),aa(product_prod(int,int),int,product_snd(int,int),P3))),
            aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(product_prod(int,int),int,product_fst(int,int),P3)),a2)),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(product_prod(int,int),int,product_snd(int,int),P3)),a2)) ) ) ) ).

% normalize_def
tff(fact_4221_chains__def,axiom,
    ! [A: $tType,A4: set(set(A))] : chains2(A,A4) = aa(fun(set(set(A)),$o),set(set(set(A))),collect(set(set(A))),aTP_Lamp_wc(set(set(A)),fun(set(set(A)),$o),A4)) ).

% chains_def
tff(fact_4222_fun_Oin__rel,axiom,
    ! [B: $tType,C: $tType,A: $tType,R: fun(B,fun(C,$o)),A3: fun(A,B),B2: fun(A,C)] :
      ( aa(fun(A,C),$o,aa(fun(A,B),fun(fun(A,C),$o),bNF_rel_fun(A,A,B,C,fequal(A),R),A3),B2)
    <=> ? [Z4: fun(A,product_prod(B,C))] :
          ( aa(set(fun(A,product_prod(B,C))),$o,member(fun(A,product_prod(B,C)),Z4),aa(fun(fun(A,product_prod(B,C)),$o),set(fun(A,product_prod(B,C))),collect(fun(A,product_prod(B,C))),aTP_Lamp_wd(fun(B,fun(C,$o)),fun(fun(A,product_prod(B,C)),$o),R)))
          & ( aa(fun(A,product_prod(B,C)),fun(A,B),comp(product_prod(B,C),B,A,product_fst(B,C)),Z4) = A3 )
          & ( aa(fun(A,product_prod(B,C)),fun(A,C),comp(product_prod(B,C),C,A,product_snd(B,C)),Z4) = B2 ) ) ) ).

% fun.in_rel
tff(fact_4223_gcd_Obottom__right__bottom,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),one_one(A)) = one_one(A) ) ).

% gcd.bottom_right_bottom
tff(fact_4224_gcd_Obottom__left__bottom,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),gcd_gcd(A),one_one(A)),A3) = one_one(A) ) ).

% gcd.bottom_left_bottom
tff(fact_4225_gcd__neg2,axiom,
    ! [A: $tType] :
      ( ring_gcd(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),aa(A,A,uminus_uminus(A),B2)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2) ) ).

% gcd_neg2
tff(fact_4226_gcd__neg1,axiom,
    ! [A: $tType] :
      ( ring_gcd(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(A,A,uminus_uminus(A),A3)),B2) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2) ) ).

% gcd_neg1
tff(fact_4227_gcd__1__int,axiom,
    ! [M: int] : aa(int,int,aa(int,fun(int,int),gcd_gcd(int),M),one_one(int)) = one_one(int) ).

% gcd_1_int
tff(fact_4228_gcd__neg1__int,axiom,
    ! [X: int,Y: int] : aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(int,int,uminus_uminus(int),X)),Y) = aa(int,int,aa(int,fun(int,int),gcd_gcd(int),X),Y) ).

% gcd_neg1_int
tff(fact_4229_gcd__neg2__int,axiom,
    ! [X: int,Y: int] : aa(int,int,aa(int,fun(int,int),gcd_gcd(int),X),aa(int,int,uminus_uminus(int),Y)) = aa(int,int,aa(int,fun(int,int),gcd_gcd(int),X),Y) ).

% gcd_neg2_int
tff(fact_4230_gcd__neg__numeral__2,axiom,
    ! [A: $tType] :
      ( ring_gcd(A)
     => ! [A3: A,N: num] : aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),aa(num,A,numeral_numeral(A),N)) ) ).

% gcd_neg_numeral_2
tff(fact_4231_gcd__neg__numeral__1,axiom,
    ! [A: $tType] :
      ( ring_gcd(A)
     => ! [N: num,A3: A] : aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))),A3) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(num,A,numeral_numeral(A),N)),A3) ) ).

% gcd_neg_numeral_1
tff(fact_4232_is__unit__gcd__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2)),one_one(A))
        <=> ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2) = one_one(A) ) ) ) ).

% is_unit_gcd_iff
tff(fact_4233_gcd__neg__numeral__2__int,axiom,
    ! [X: int,N: num] : aa(int,int,aa(int,fun(int,int),gcd_gcd(int),X),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = aa(int,int,aa(int,fun(int,int),gcd_gcd(int),X),aa(num,int,numeral_numeral(int),N)) ).

% gcd_neg_numeral_2_int
tff(fact_4234_gcd__neg__numeral__1__int,axiom,
    ! [N: num,X: int] : aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))),X) = aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(num,int,numeral_numeral(int),N)),X) ).

% gcd_neg_numeral_1_int
tff(fact_4235_Gcd__2,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A3: A,B2: A] : gcd_Gcd(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A))))) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2) ) ).

% Gcd_2
tff(fact_4236_dup_Orsp,axiom,
    aa(fun(int,int),$o,aa(fun(int,int),fun(fun(int,int),$o),bNF_rel_fun(int,int,int,int,fequal(int),fequal(int)),aTP_Lamp_ug(int,int)),aTP_Lamp_ug(int,int)) ).

% dup.rsp
tff(fact_4237_transfer__rule__of__int,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ring_1(B)
        & ring_1(A) )
     => ! [R: fun(A,fun(B,$o))] :
          ( aa(B,$o,aa(A,fun(B,$o),R,zero_zero(A)),zero_zero(B))
         => ( aa(B,$o,aa(A,fun(B,$o),R,one_one(A)),one_one(B))
           => ( aa(fun(B,fun(B,B)),$o,aa(fun(A,fun(A,A)),fun(fun(B,fun(B,B)),$o),bNF_rel_fun(A,B,fun(A,A),fun(B,B),R,bNF_rel_fun(A,B,A,B,R,R)),plus_plus(A)),plus_plus(B))
             => ( aa(fun(B,B),$o,aa(fun(A,A),fun(fun(B,B),$o),bNF_rel_fun(A,B,A,B,R,R),uminus_uminus(A)),uminus_uminus(B))
               => aa(fun(int,B),$o,aa(fun(int,A),fun(fun(int,B),$o),bNF_rel_fun(int,int,A,B,fequal(int),R),ring_1_of_int(A)),ring_1_of_int(B)) ) ) ) ) ) ).

% transfer_rule_of_int
tff(fact_4238_power__transfer,axiom,
    ! [A: $tType,B: $tType] :
      ( ( power(B)
        & power(A) )
     => ! [R: fun(A,fun(B,$o))] :
          ( aa(B,$o,aa(A,fun(B,$o),R,one_one(A)),one_one(B))
         => ( aa(fun(B,fun(B,B)),$o,aa(fun(A,fun(A,A)),fun(fun(B,fun(B,B)),$o),bNF_rel_fun(A,B,fun(A,A),fun(B,B),R,bNF_rel_fun(A,B,A,B,R,R)),times_times(A)),times_times(B))
           => aa(fun(B,fun(nat,B)),$o,aa(fun(A,fun(nat,A)),fun(fun(B,fun(nat,B)),$o),bNF_rel_fun(A,B,fun(nat,A),fun(nat,B),R,bNF_rel_fun(nat,nat,A,B,fequal(nat),R)),power_power(A)),power_power(B)) ) ) ) ).

% power_transfer
tff(fact_4239_transfer__rule__numeral,axiom,
    ! [A: $tType,B: $tType] :
      ( ( monoid_add(B)
        & semiring_numeral(B)
        & monoid_add(A)
        & semiring_numeral(A) )
     => ! [R: fun(A,fun(B,$o))] :
          ( aa(B,$o,aa(A,fun(B,$o),R,zero_zero(A)),zero_zero(B))
         => ( aa(B,$o,aa(A,fun(B,$o),R,one_one(A)),one_one(B))
           => ( aa(fun(B,fun(B,B)),$o,aa(fun(A,fun(A,A)),fun(fun(B,fun(B,B)),$o),bNF_rel_fun(A,B,fun(A,A),fun(B,B),R,bNF_rel_fun(A,B,A,B,R,R)),plus_plus(A)),plus_plus(B))
             => aa(fun(num,B),$o,aa(fun(num,A),fun(fun(num,B),$o),bNF_rel_fun(num,num,A,B,fequal(num),R),numeral_numeral(A)),numeral_numeral(B)) ) ) ) ) ).

% transfer_rule_numeral
tff(fact_4240_sub_Orsp,axiom,
    aa(fun(num,fun(num,int)),$o,aa(fun(num,fun(num,int)),fun(fun(num,fun(num,int)),$o),bNF_rel_fun(num,num,fun(num,int),fun(num,int),fequal(num),bNF_rel_fun(num,num,int,int,fequal(num),fequal(int))),aTP_Lamp_uk(num,fun(num,int))),aTP_Lamp_uk(num,fun(num,int))) ).

% sub.rsp
tff(fact_4241_uminus__integer_Orsp,axiom,
    aa(fun(int,int),$o,aa(fun(int,int),fun(fun(int,int),$o),bNF_rel_fun(int,int,int,int,fequal(int),fequal(int)),uminus_uminus(int)),uminus_uminus(int)) ).

% uminus_integer.rsp
tff(fact_4242_fun_Orel__map_I2_J,axiom,
    ! [B: $tType,C: $tType,D: $tType,A: $tType,Sa: fun(B,fun(C,$o)),X: fun(A,B),G: fun(D,C),Y: fun(A,D)] :
      ( aa(fun(A,C),$o,aa(fun(A,B),fun(fun(A,C),$o),bNF_rel_fun(A,A,B,C,fequal(A),Sa),X),aa(fun(A,D),fun(A,C),comp(D,C,A,G),Y))
    <=> aa(fun(A,D),$o,aa(fun(A,B),fun(fun(A,D),$o),bNF_rel_fun(A,A,B,D,fequal(A),aa(fun(D,C),fun(B,fun(D,$o)),aTP_Lamp_we(fun(B,fun(C,$o)),fun(fun(D,C),fun(B,fun(D,$o))),Sa),G)),X),Y) ) ).

% fun.rel_map(2)
tff(fact_4243_fun_Orel__map_I1_J,axiom,
    ! [D: $tType,B: $tType,C: $tType,A: $tType,Sb: fun(B,fun(C,$o)),I: fun(D,B),X: fun(A,D),Y: fun(A,C)] :
      ( aa(fun(A,C),$o,aa(fun(A,B),fun(fun(A,C),$o),bNF_rel_fun(A,A,B,C,fequal(A),Sb),aa(fun(A,D),fun(A,B),comp(D,B,A,I),X)),Y)
    <=> aa(fun(A,C),$o,aa(fun(A,D),fun(fun(A,C),$o),bNF_rel_fun(A,A,D,C,fequal(A),aa(fun(D,B),fun(D,fun(C,$o)),aTP_Lamp_wf(fun(B,fun(C,$o)),fun(fun(D,B),fun(D,fun(C,$o))),Sb),I)),X),Y) ) ).

% fun.rel_map(1)
tff(fact_4244_fun_Orel__cong,axiom,
    ! [B: $tType,C: $tType,A: $tType,X: fun(A,B),Ya: fun(A,B),Y: fun(A,C),Xa: fun(A,C),R: fun(B,fun(C,$o)),Ra: fun(B,fun(C,$o))] :
      ( ( X = Ya )
     => ( ( Y = Xa )
       => ( ! [Z3: B,Yb: C] :
              ( aa(set(B),$o,member(B,Z3),aa(set(A),set(B),image2(A,B,Ya),top_top(set(A))))
             => ( aa(set(C),$o,member(C,Yb),aa(set(A),set(C),image2(A,C,Xa),top_top(set(A))))
               => ( aa(C,$o,aa(B,fun(C,$o),R,Z3),Yb)
                <=> aa(C,$o,aa(B,fun(C,$o),Ra,Z3),Yb) ) ) )
         => ( aa(fun(A,C),$o,aa(fun(A,B),fun(fun(A,C),$o),bNF_rel_fun(A,A,B,C,fequal(A),R),X),Y)
          <=> aa(fun(A,C),$o,aa(fun(A,B),fun(fun(A,C),$o),bNF_rel_fun(A,A,B,C,fequal(A),Ra),Ya),Xa) ) ) ) ) ).

% fun.rel_cong
tff(fact_4245_fun_Orel__mono__strong,axiom,
    ! [B: $tType,C: $tType,A: $tType,R: fun(B,fun(C,$o)),X: fun(A,B),Y: fun(A,C),Ra: fun(B,fun(C,$o))] :
      ( aa(fun(A,C),$o,aa(fun(A,B),fun(fun(A,C),$o),bNF_rel_fun(A,A,B,C,fequal(A),R),X),Y)
     => ( ! [Z3: B,Yb: C] :
            ( aa(set(B),$o,member(B,Z3),aa(set(A),set(B),image2(A,B,X),top_top(set(A))))
           => ( aa(set(C),$o,member(C,Yb),aa(set(A),set(C),image2(A,C,Y),top_top(set(A))))
             => ( aa(C,$o,aa(B,fun(C,$o),R,Z3),Yb)
               => aa(C,$o,aa(B,fun(C,$o),Ra,Z3),Yb) ) ) )
       => aa(fun(A,C),$o,aa(fun(A,B),fun(fun(A,C),$o),bNF_rel_fun(A,A,B,C,fequal(A),Ra),X),Y) ) ) ).

% fun.rel_mono_strong
tff(fact_4246_fun_Orel__refl__strong,axiom,
    ! [A: $tType,B: $tType,X: fun(B,A),Ra: fun(A,fun(A,$o))] :
      ( ! [Z3: A] :
          ( aa(set(A),$o,member(A,Z3),aa(set(B),set(A),image2(B,A,X),top_top(set(B))))
         => aa(A,$o,aa(A,fun(A,$o),Ra,Z3),Z3) )
     => aa(fun(B,A),$o,aa(fun(B,A),fun(fun(B,A),$o),bNF_rel_fun(B,B,A,A,fequal(B),Ra),X),X) ) ).

% fun.rel_refl_strong
tff(fact_4247_gcd__add__mult,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [M: A,K: A,N: A] : aa(A,A,aa(A,fun(A,A),gcd_gcd(A),M),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),K),M)),N)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),M),N) ) ).

% gcd_add_mult
tff(fact_4248_gcd__dvd__prod,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A,K: A] : aa(A,$o,aa(A,fun(A,$o),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),K),B2)) ) ).

% gcd_dvd_prod
tff(fact_4249_predicate2__transferD,axiom,
    ! [A: $tType,B: $tType,D: $tType,C: $tType,R1: fun(A,fun(B,$o)),R22: fun(C,fun(D,$o)),P: fun(A,fun(C,$o)),Q: fun(B,fun(D,$o)),A3: product_prod(A,B),A4: set(product_prod(A,B)),B2: product_prod(C,D),B3: set(product_prod(C,D))] :
      ( aa(fun(B,fun(D,$o)),$o,aa(fun(A,fun(C,$o)),fun(fun(B,fun(D,$o)),$o),bNF_rel_fun(A,B,fun(C,$o),fun(D,$o),R1,bNF_rel_fun(C,D,$o,$o,R22,fequal($o))),P),Q)
     => ( aa(set(product_prod(A,B)),$o,member(product_prod(A,B),A3),A4)
       => ( aa(set(product_prod(C,D)),$o,member(product_prod(C,D),B2),B3)
         => ( aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),A4),aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),R1)))
           => ( aa(set(product_prod(C,D)),$o,aa(set(product_prod(C,D)),fun(set(product_prod(C,D)),$o),ord_less_eq(set(product_prod(C,D))),B3),aa(fun(product_prod(C,D),$o),set(product_prod(C,D)),collect(product_prod(C,D)),aa(fun(C,fun(D,$o)),fun(product_prod(C,D),$o),product_case_prod(C,D,$o),R22)))
             => ( aa(C,$o,aa(A,fun(C,$o),P,aa(product_prod(A,B),A,product_fst(A,B),A3)),aa(product_prod(C,D),C,product_fst(C,D),B2))
              <=> aa(D,$o,aa(B,fun(D,$o),Q,aa(product_prod(A,B),B,product_snd(A,B),A3)),aa(product_prod(C,D),D,product_snd(C,D),B2)) ) ) ) ) ) ) ).

% predicate2_transferD
tff(fact_4250_gcd__mult__unit2,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),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),C2),A3)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),B2),C2) ) ) ) ).

% gcd_mult_unit2
tff(fact_4251_gcd__mult__unit1,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),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)),C2) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),B2),C2) ) ) ) ).

% gcd_mult_unit1
tff(fact_4252_gcd__div__unit2,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),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),divide_divide(A),C2),A3)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),B2),C2) ) ) ) ).

% gcd_div_unit2
tff(fact_4253_gcd__div__unit1,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),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),divide_divide(A),B2),A3)),C2) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),B2),C2) ) ) ) ).

% gcd_div_unit1
tff(fact_4254_gcd__cases__int,axiom,
    ! [X: int,Y: int,P: fun(int,$o)] :
      ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),X)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Y)
         => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),gcd_gcd(int),X),Y)) ) )
     => ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),X)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Y),zero_zero(int))
           => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),gcd_gcd(int),X),aa(int,int,uminus_uminus(int),Y))) ) )
       => ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),X),zero_zero(int))
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Y)
             => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(int,int,uminus_uminus(int),X)),Y)) ) )
         => ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),X),zero_zero(int))
             => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Y),zero_zero(int))
               => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(int,int,uminus_uminus(int),X)),aa(int,int,uminus_uminus(int),Y))) ) )
           => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),gcd_gcd(int),X),Y)) ) ) ) ) ).

% gcd_cases_int
tff(fact_4255_gcd__is__Max__divisors__int,axiom,
    ! [N: int,M: int] :
      ( ( N != zero_zero(int) )
     => ( aa(int,int,aa(int,fun(int,int),gcd_gcd(int),M),N) = aa(set(int),int,lattic643756798349783984er_Max(int),aa(fun(int,$o),set(int),collect(int),aa(int,fun(int,$o),aTP_Lamp_wg(int,fun(int,fun(int,$o)),N),M))) ) ) ).

% gcd_is_Max_divisors_int
tff(fact_4256_prod__decode__aux_Opelims,axiom,
    ! [X: nat,Xa: nat,Y: product_prod(nat,nat)] :
      ( ( nat_prod_decode_aux(X,Xa) = 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),Xa))
       => ~ ( ( Y = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xa),X),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),X),Xa)),nat_prod_decode_aux(aa(nat,nat,suc,X),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Xa),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),Xa)) ) ) ) ).

% prod_decode_aux.pelims
tff(fact_4257_transfer__rule__of__nat,axiom,
    ! [A: $tType,B: $tType] :
      ( ( semiring_1(B)
        & semiring_1(A) )
     => ! [R: fun(A,fun(B,$o))] :
          ( aa(B,$o,aa(A,fun(B,$o),R,zero_zero(A)),zero_zero(B))
         => ( aa(B,$o,aa(A,fun(B,$o),R,one_one(A)),one_one(B))
           => ( aa(fun(B,fun(B,B)),$o,aa(fun(A,fun(A,A)),fun(fun(B,fun(B,B)),$o),bNF_rel_fun(A,B,fun(A,A),fun(B,B),R,bNF_rel_fun(A,B,A,B,R,R)),plus_plus(A)),plus_plus(B))
             => aa(fun(nat,B),$o,aa(fun(nat,A),fun(fun(nat,B),$o),bNF_rel_fun(nat,nat,A,B,fequal(nat),R),semiring_1_of_nat(A)),semiring_1_of_nat(B)) ) ) ) ) ).

% transfer_rule_of_nat
tff(fact_4258_transfer__rule__of__bool,axiom,
    ! [A: $tType,B: $tType] :
      ( ( zero_neq_one(B)
        & zero_neq_one(A) )
     => ! [R: fun(A,fun(B,$o))] :
          ( aa(B,$o,aa(A,fun(B,$o),R,zero_zero(A)),zero_zero(B))
         => ( aa(B,$o,aa(A,fun(B,$o),R,one_one(A)),one_one(B))
           => aa(fun($o,B),$o,aa(fun($o,A),fun(fun($o,B),$o),bNF_rel_fun($o,$o,A,B,fequal($o),R),zero_neq_one_of_bool(A)),zero_neq_one_of_bool(B)) ) ) ) ).

% transfer_rule_of_bool
tff(fact_4259_plus__rat_Otransfer,axiom,
    aa(fun(rat,fun(rat,rat)),$o,aa(fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),fun(fun(rat,fun(rat,rat)),$o),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_wh(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))),plus_plus(rat)) ).

% plus_rat.transfer
tff(fact_4260_gcd__1__nat,axiom,
    ! [M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),M),one_one(nat)) = one_one(nat) ).

% gcd_1_nat
tff(fact_4261_Gcd__in,axiom,
    ! [A4: set(nat)] :
      ( ! [A6: nat,B5: nat] :
          ( aa(set(nat),$o,member(nat,A6),A4)
         => ( aa(set(nat),$o,member(nat,B5),A4)
           => aa(set(nat),$o,member(nat,aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A6),B5)),A4) ) )
     => ( ( A4 != bot_bot(set(nat)) )
       => aa(set(nat),$o,member(nat,gcd_Gcd(nat,A4)),A4) ) ) ).

% Gcd_in
tff(fact_4262_gcd__nat_Osemilattice__neutr__axioms,axiom,
    semilattice_neutr(nat,gcd_gcd(nat),zero_zero(nat)) ).

% gcd_nat.semilattice_neutr_axioms
tff(fact_4263_gcd__nat_Ocomm__monoid__axioms,axiom,
    comm_monoid(nat,gcd_gcd(nat),zero_zero(nat)) ).

% gcd_nat.comm_monoid_axioms
tff(fact_4264_gcd__nat_Omonoid__axioms,axiom,
    monoid(nat,gcd_gcd(nat),zero_zero(nat)) ).

% gcd_nat.monoid_axioms
tff(fact_4265_gcd__nat_Osemilattice__neutr__order__axioms,axiom,
    semila1105856199041335345_order(nat,gcd_gcd(nat),zero_zero(nat),dvd_dvd(nat),aTP_Lamp_cc(nat,fun(nat,$o))) ).

% gcd_nat.semilattice_neutr_order_axioms
tff(fact_4266_one__rat_Otransfer,axiom,
    aa(rat,$o,aa(product_prod(int,int),fun(rat,$o),pcr_rat,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),one_one(int))),one_one(rat)) ).

% one_rat.transfer
tff(fact_4267_gcd__is__Max__divisors__nat,axiom,
    ! [N: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),M),N) = aa(set(nat),nat,lattic643756798349783984er_Max(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_wi(nat,fun(nat,fun(nat,$o)),N),M))) ) ) ).

% gcd_is_Max_divisors_nat
tff(fact_4268_zero__rat_Otransfer,axiom,
    aa(rat,$o,aa(product_prod(int,int),fun(rat,$o),pcr_rat,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int))),zero_zero(rat)) ).

% zero_rat.transfer
tff(fact_4269_Fract_Otransfer,axiom,
    aa(fun(int,fun(int,rat)),$o,aa(fun(int,fun(int,product_prod(int,int))),fun(fun(int,fun(int,rat)),$o),bNF_rel_fun(int,int,fun(int,product_prod(int,int)),fun(int,rat),fequal(int),bNF_rel_fun(int,int,product_prod(int,int),rat,fequal(int),pcr_rat)),aTP_Lamp_wj(int,fun(int,product_prod(int,int)))),fract) ).

% Fract.transfer
tff(fact_4270_uminus__rat_Otransfer,axiom,
    aa(fun(rat,rat),$o,aa(fun(product_prod(int,int),product_prod(int,int)),fun(fun(rat,rat),$o),bNF_rel_fun(product_prod(int,int),rat,product_prod(int,int),rat,pcr_rat,pcr_rat),aTP_Lamp_wk(product_prod(int,int),product_prod(int,int))),uminus_uminus(rat)) ).

% uminus_rat.transfer
tff(fact_4271_times__rat_Otransfer,axiom,
    aa(fun(rat,fun(rat,rat)),$o,aa(fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),fun(fun(rat,fun(rat,rat)),$o),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_wl(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))),times_times(rat)) ).

% times_rat.transfer
tff(fact_4272_inverse__rat_Otransfer,axiom,
    aa(fun(rat,rat),$o,aa(fun(product_prod(int,int),product_prod(int,int)),fun(fun(rat,rat),$o),bNF_rel_fun(product_prod(int,int),rat,product_prod(int,int),rat,pcr_rat,pcr_rat),aTP_Lamp_wm(product_prod(int,int),product_prod(int,int))),inverse_inverse(rat)) ).

% inverse_rat.transfer
tff(fact_4273_gcd__nat_Opelims,axiom,
    ! [X: nat,Xa: nat,Y: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),X),Xa) = 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),Xa))
       => ~ ( ( Y = $ite(Xa = zero_zero(nat),X,aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Xa),modulo_modulo(nat,X,Xa))) )
           => ~ accp(product_prod(nat,nat),gcd_nat_rel,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X),Xa)) ) ) ) ).

% gcd_nat.pelims
tff(fact_4274_positive_Otransfer,axiom,
    aa(fun(rat,$o),$o,aa(fun(product_prod(int,int),$o),fun(fun(rat,$o),$o),bNF_rel_fun(product_prod(int,int),rat,$o,$o,pcr_rat,fequal($o)),aTP_Lamp_wn(product_prod(int,int),$o)),positive) ).

% positive.transfer
tff(fact_4275_times__int_Otransfer,axiom,
    aa(fun(int,fun(int,int)),$o,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),fun(fun(int,fun(int,int)),$o),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_pb(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))),times_times(int)) ).

% times_int.transfer
tff(fact_4276_typedef__rep__transfer,axiom,
    ! [B: $tType,A: $tType,Rep: fun(A,B),Abs: fun(B,A),A4: set(B),T2: fun(B,fun(A,$o))] :
      ( type_definition(A,B,Rep,Abs,A4)
     => ( ! [X2: B,Xa4: A] :
            ( aa(A,$o,aa(B,fun(A,$o),T2,X2),Xa4)
          <=> ( X2 = aa(A,B,Rep,Xa4) ) )
       => aa(fun(A,B),$o,aa(fun(B,B),fun(fun(A,B),$o),bNF_rel_fun(B,A,B,B,T2,fequal(B)),aTP_Lamp_oc(B,B)),Rep) ) ) ).

% typedef_rep_transfer
tff(fact_4277_zero__int_Otransfer,axiom,
    aa(int,$o,aa(product_prod(nat,nat),fun(int,$o),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_4278_positive__minus,axiom,
    ! [X: rat] :
      ( ~ aa(rat,$o,positive,X)
     => ( ( X != zero_zero(rat) )
       => aa(rat,$o,positive,aa(rat,rat,uminus_uminus(rat),X)) ) ) ).

% positive_minus
tff(fact_4279_int__transfer,axiom,
    aa(fun(nat,int),$o,aa(fun(nat,product_prod(nat,nat)),fun(fun(nat,int),$o),bNF_rel_fun(nat,nat,product_prod(nat,nat),int,fequal(nat),pcr_int),aTP_Lamp_wo(nat,product_prod(nat,nat))),semiring_1_of_nat(int)) ).

% int_transfer
tff(fact_4280_uminus__int_Otransfer,axiom,
    aa(fun(int,int),$o,aa(fun(product_prod(nat,nat),product_prod(nat,nat)),fun(fun(int,int),$o),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_pm(nat,fun(nat,product_prod(nat,nat))))),uminus_uminus(int)) ).

% uminus_int.transfer
tff(fact_4281_nat_Otransfer,axiom,
    aa(fun(int,nat),$o,aa(fun(product_prod(nat,nat),nat),fun(fun(int,nat),$o),bNF_rel_fun(product_prod(nat,nat),int,nat,nat,pcr_int,fequal(nat)),aa(fun(nat,fun(nat,nat)),fun(product_prod(nat,nat),nat),product_case_prod(nat,nat,nat),minus_minus(nat))),nat2) ).

% nat.transfer
tff(fact_4282_one__int_Otransfer,axiom,
    aa(int,$o,aa(product_prod(nat,nat),fun(int,$o),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_4283_of__int_Otransfer,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => aa(fun(int,A),$o,aa(fun(product_prod(nat,nat),A),fun(fun(int,A),$o),bNF_rel_fun(product_prod(nat,nat),int,A,A,pcr_int,fequal(A)),aa(fun(nat,fun(nat,A)),fun(product_prod(nat,nat),A),product_case_prod(nat,nat,A),aTP_Lamp_pn(nat,fun(nat,A)))),ring_1_of_int(A)) ) ).

% of_int.transfer
tff(fact_4284_less__int_Otransfer,axiom,
    aa(fun(int,fun(int,$o)),$o,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)),fun(fun(int,fun(int,$o)),$o),bNF_rel_fun(product_prod(nat,nat),int,fun(product_prod(nat,nat),$o),fun(int,$o),pcr_int,bNF_rel_fun(product_prod(nat,nat),int,$o,$o,pcr_int,fequal($o))),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),$o))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)),product_case_prod(nat,nat,fun(product_prod(nat,nat),$o)),aTP_Lamp_pp(nat,fun(nat,fun(product_prod(nat,nat),$o))))),ord_less(int)) ).

% less_int.transfer
tff(fact_4285_less__eq__int_Otransfer,axiom,
    aa(fun(int,fun(int,$o)),$o,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)),fun(fun(int,fun(int,$o)),$o),bNF_rel_fun(product_prod(nat,nat),int,fun(product_prod(nat,nat),$o),fun(int,$o),pcr_int,bNF_rel_fun(product_prod(nat,nat),int,$o,$o,pcr_int,fequal($o))),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),$o))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)),product_case_prod(nat,nat,fun(product_prod(nat,nat),$o)),aTP_Lamp_pr(nat,fun(nat,fun(product_prod(nat,nat),$o))))),ord_less_eq(int)) ).

% less_eq_int.transfer
tff(fact_4286_plus__int_Otransfer,axiom,
    aa(fun(int,fun(int,int)),$o,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),fun(fun(int,fun(int,int)),$o),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_pt(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))),plus_plus(int)) ).

% plus_int.transfer
tff(fact_4287_minus__int_Otransfer,axiom,
    aa(fun(int,fun(int,int)),$o,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),fun(fun(int,fun(int,int)),$o),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_pv(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))),minus_minus(int)) ).

% minus_int.transfer
tff(fact_4288_positive__def,axiom,
    positive = aa(fun(product_prod(int,int),$o),fun(rat,$o),map_fun(rat,product_prod(int,int),$o,$o,rep_Rat,id($o)),aTP_Lamp_wn(product_prod(int,int),$o)) ).

% positive_def
tff(fact_4289_of__rat_Otransfer,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => aa(fun(rat,A),$o,aa(fun(product_prod(int,int),A),fun(fun(rat,A),$o),bNF_rel_fun(product_prod(int,int),rat,A,A,pcr_rat,fequal(A)),aTP_Lamp_wp(product_prod(int,int),A)),field_char_0_of_rat(A)) ) ).

% of_rat.transfer
tff(fact_4290_apfst__apsnd,axiom,
    ! [A: $tType,B: $tType,D: $tType,C: $tType,F: fun(C,A),G: fun(D,B),X: product_prod(C,D)] : aa(product_prod(C,B),product_prod(A,B),product_apfst(C,A,B,F),aa(product_prod(C,D),product_prod(C,B),aa(fun(D,B),fun(product_prod(C,D),product_prod(C,B)),product_apsnd(D,B,C),G),X)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,F,aa(product_prod(C,D),C,product_fst(C,D),X))),aa(D,B,G,aa(product_prod(C,D),D,product_snd(C,D),X))) ).

% apfst_apsnd
tff(fact_4291_apsnd__apfst,axiom,
    ! [A: $tType,B: $tType,C: $tType,D: $tType,F: fun(C,B),G: fun(D,A),X: product_prod(D,C)] : aa(product_prod(A,C),product_prod(A,B),aa(fun(C,B),fun(product_prod(A,C),product_prod(A,B)),product_apsnd(C,B,A),F),aa(product_prod(D,C),product_prod(A,C),product_apfst(D,A,C,G),X)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(D,A,G,aa(product_prod(D,C),D,product_fst(D,C),X))),aa(C,B,F,aa(product_prod(D,C),C,product_snd(D,C),X))) ).

% apsnd_apfst
tff(fact_4292_apfst__conv,axiom,
    ! [C: $tType,A: $tType,B: $tType,F: fun(C,A),X: C,Y: B] : aa(product_prod(C,B),product_prod(A,B),product_apfst(C,A,B,F),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,F,X)),Y) ).

% apfst_conv
tff(fact_4293_one__eq__of__rat__iff,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: rat] :
          ( ( one_one(A) = aa(rat,A,field_char_0_of_rat(A),A3) )
        <=> ( one_one(rat) = A3 ) ) ) ).

% one_eq_of_rat_iff
tff(fact_4294_of__rat__eq__1__iff,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: rat] :
          ( ( aa(rat,A,field_char_0_of_rat(A),A3) = one_one(A) )
        <=> ( A3 = one_one(rat) ) ) ) ).

% of_rat_eq_1_iff
tff(fact_4295_of__rat__1,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ( aa(rat,A,field_char_0_of_rat(A),one_one(rat)) = one_one(A) ) ) ).

% of_rat_1
tff(fact_4296_of__rat__neg__one,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ( aa(rat,A,field_char_0_of_rat(A),aa(rat,rat,uminus_uminus(rat),one_one(rat))) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ).

% of_rat_neg_one
tff(fact_4297_one__less__of__rat__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R2: rat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(rat,A,field_char_0_of_rat(A),R2))
        <=> aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),one_one(rat)),R2) ) ) ).

% one_less_of_rat_iff
tff(fact_4298_of__rat__less__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R2: rat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(rat,A,field_char_0_of_rat(A),R2)),one_one(A))
        <=> aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),R2),one_one(rat)) ) ) ).

% of_rat_less_1_iff
tff(fact_4299_one__le__of__rat__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R2: rat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(rat,A,field_char_0_of_rat(A),R2))
        <=> aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),one_one(rat)),R2) ) ) ).

% one_le_of_rat_iff
tff(fact_4300_of__rat__le__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R2: rat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(rat,A,field_char_0_of_rat(A),R2)),one_one(A))
        <=> aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),R2),one_one(rat)) ) ) ).

% of_rat_le_1_iff
tff(fact_4301_of__rat__neg__numeral__eq,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [W2: num] : aa(rat,A,field_char_0_of_rat(A),aa(rat,rat,uminus_uminus(rat),aa(num,rat,numeral_numeral(rat),W2))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)) ) ).

% of_rat_neg_numeral_eq
tff(fact_4302_of__rat__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( field_char_0(A)
     => ! [F: fun(B,rat),A4: set(B)] : aa(rat,A,field_char_0_of_rat(A),aa(set(B),rat,aa(fun(B,rat),fun(set(B),rat),groups7311177749621191930dd_sum(B,rat),F),A4)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aTP_Lamp_wq(fun(B,rat),fun(B,A),F)),A4) ) ).

% of_rat_sum
tff(fact_4303_of__rat__prod,axiom,
    ! [A: $tType,B: $tType] :
      ( field_char_0(A)
     => ! [F: fun(B,rat),A4: set(B)] : aa(rat,A,field_char_0_of_rat(A),aa(set(B),rat,aa(fun(B,rat),fun(set(B),rat),groups7121269368397514597t_prod(B,rat),F),A4)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aTP_Lamp_wq(fun(B,rat),fun(B,A),F)),A4) ) ).

% of_rat_prod
tff(fact_4304_of__rat__minus,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: rat] : aa(rat,A,field_char_0_of_rat(A),aa(rat,rat,uminus_uminus(rat),A3)) = aa(A,A,uminus_uminus(A),aa(rat,A,field_char_0_of_rat(A),A3)) ) ).

% of_rat_minus
tff(fact_4305_of__rat__mult,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: rat,B2: rat] : aa(rat,A,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),aa(rat,A,field_char_0_of_rat(A),A3)),aa(rat,A,field_char_0_of_rat(A),B2)) ) ).

% of_rat_mult
tff(fact_4306_of__rat__def,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ( field_char_0_of_rat(A) = aa(fun(product_prod(int,int),A),fun(rat,A),map_fun(rat,product_prod(int,int),A,A,rep_Rat,id(A)),aTP_Lamp_wp(product_prod(int,int),A)) ) ) ).

% of_rat_def
tff(fact_4307_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_wh(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))) ).

% plus_rat_def
tff(fact_4308_inverse__rat__def,axiom,
    inverse_inverse(rat) = aa(fun(product_prod(int,int),product_prod(int,int)),fun(rat,rat),map_fun(rat,product_prod(int,int),product_prod(int,int),rat,rep_Rat,abs_Rat),aTP_Lamp_wm(product_prod(int,int),product_prod(int,int))) ).

% inverse_rat_def
tff(fact_4309_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_wl(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))) ).

% times_rat_def
tff(fact_4310_uminus__rat__def,axiom,
    uminus_uminus(rat) = aa(fun(product_prod(int,int),product_prod(int,int)),fun(rat,rat),map_fun(rat,product_prod(int,int),product_prod(int,int),rat,rep_Rat,abs_Rat),aTP_Lamp_wk(product_prod(int,int),product_prod(int,int))) ).

% uminus_rat_def
tff(fact_4311_one__rat__def,axiom,
    one_one(rat) = aa(product_prod(int,int),rat,abs_Rat,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),one_one(int))) ).

% one_rat_def
tff(fact_4312_Fract_Oabs__eq,axiom,
    ! [Xa: int,X: int] :
      aa(int,rat,aa(int,fun(int,rat),fract,Xa),X) = aa(product_prod(int,int),rat,abs_Rat,
        $ite(X = zero_zero(int),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int)),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Xa),X))) ).

% Fract.abs_eq
tff(fact_4313_zero__rat__def,axiom,
    zero_zero(rat) = aa(product_prod(int,int),rat,abs_Rat,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int))) ).

% zero_rat_def
tff(fact_4314_Fract__def,axiom,
    fract = aa(fun(int,fun(int,product_prod(int,int))),fun(int,fun(int,rat)),map_fun(int,int,fun(int,product_prod(int,int)),fun(int,rat),id(int),map_fun(int,int,product_prod(int,int),rat,id(int),abs_Rat)),aTP_Lamp_wj(int,fun(int,product_prod(int,int)))) ).

% Fract_def
tff(fact_4315_plus__rat_Oabs__eq,axiom,
    ! [Xa: product_prod(int,int),X: product_prod(int,int)] :
      ( aa(product_prod(int,int),$o,aa(product_prod(int,int),fun(product_prod(int,int),$o),ratrel,Xa),Xa)
     => ( aa(product_prod(int,int),$o,aa(product_prod(int,int),fun(product_prod(int,int),$o),ratrel,X),X)
       => ( aa(rat,rat,aa(rat,fun(rat,rat),plus_plus(rat),aa(product_prod(int,int),rat,abs_Rat,Xa)),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),Xa)),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),Xa)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),Xa)),aa(product_prod(int,int),int,product_snd(int,int),X)))) ) ) ) ).

% plus_rat.abs_eq
tff(fact_4316_inverse__rat_Oabs__eq,axiom,
    ! [X: product_prod(int,int)] :
      ( aa(product_prod(int,int),$o,aa(product_prod(int,int),fun(product_prod(int,int),$o),ratrel,X),X)
     => ( aa(rat,rat,inverse_inverse(rat),aa(product_prod(int,int),rat,abs_Rat,X)) = aa(product_prod(int,int),rat,abs_Rat,
            $ite(aa(product_prod(int,int),int,product_fst(int,int),X) = zero_zero(int),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int)),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(product_prod(int,int),int,product_snd(int,int),X)),aa(product_prod(int,int),int,product_fst(int,int),X)))) ) ) ).

% inverse_rat.abs_eq
tff(fact_4317_apfst__convE,axiom,
    ! [C: $tType,A: $tType,B: $tType,Q3: product_prod(A,B),F: fun(C,A),P3: product_prod(C,B)] :
      ( ( Q3 = aa(product_prod(C,B),product_prod(A,B),product_apfst(C,A,B,F),P3) )
     => ~ ! [X2: C,Y2: B] :
            ( ( P3 = aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),X2),Y2) )
           => ( Q3 != aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,F,X2)),Y2) ) ) ) ).

% apfst_convE
tff(fact_4318_times__rat_Oabs__eq,axiom,
    ! [Xa: product_prod(int,int),X: product_prod(int,int)] :
      ( aa(product_prod(int,int),$o,aa(product_prod(int,int),fun(product_prod(int,int),$o),ratrel,Xa),Xa)
     => ( aa(product_prod(int,int),$o,aa(product_prod(int,int),fun(product_prod(int,int),$o),ratrel,X),X)
       => ( aa(rat,rat,aa(rat,fun(rat,rat),times_times(rat),aa(product_prod(int,int),rat,abs_Rat,Xa)),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),Xa)),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),Xa)),aa(product_prod(int,int),int,product_snd(int,int),X)))) ) ) ) ).

% times_rat.abs_eq
tff(fact_4319_one__rat_Orsp,axiom,
    aa(product_prod(int,int),$o,aa(product_prod(int,int),fun(product_prod(int,int),$o),ratrel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),one_one(int))),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),one_one(int))) ).

% one_rat.rsp
tff(fact_4320_zero__rat_Orsp,axiom,
    aa(product_prod(int,int),$o,aa(product_prod(int,int),fun(product_prod(int,int),$o),ratrel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int))),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int))) ).

% zero_rat.rsp
tff(fact_4321_Fract_Orsp,axiom,
    aa(fun(int,fun(int,product_prod(int,int))),$o,aa(fun(int,fun(int,product_prod(int,int))),fun(fun(int,fun(int,product_prod(int,int))),$o),bNF_rel_fun(int,int,fun(int,product_prod(int,int)),fun(int,product_prod(int,int)),fequal(int),bNF_rel_fun(int,int,product_prod(int,int),product_prod(int,int),fequal(int),ratrel)),aTP_Lamp_wj(int,fun(int,product_prod(int,int)))),aTP_Lamp_wj(int,fun(int,product_prod(int,int)))) ).

% Fract.rsp
tff(fact_4322_of__rat_Orsp,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => aa(fun(product_prod(int,int),A),$o,aa(fun(product_prod(int,int),A),fun(fun(product_prod(int,int),A),$o),bNF_rel_fun(product_prod(int,int),product_prod(int,int),A,A,ratrel,fequal(A)),aTP_Lamp_wp(product_prod(int,int),A)),aTP_Lamp_wp(product_prod(int,int),A)) ) ).

% of_rat.rsp
tff(fact_4323_ratrel__def,axiom,
    ! [X3: product_prod(int,int),Xa3: product_prod(int,int)] :
      ( aa(product_prod(int,int),$o,aa(product_prod(int,int),fun(product_prod(int,int),$o),ratrel,X3),Xa3)
    <=> ( ( aa(product_prod(int,int),int,product_snd(int,int),X3) != zero_zero(int) )
        & ( aa(product_prod(int,int),int,product_snd(int,int),Xa3) != zero_zero(int) )
        & ( aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),X3)),aa(product_prod(int,int),int,product_snd(int,int),Xa3)) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),Xa3)),aa(product_prod(int,int),int,product_snd(int,int),X3)) ) ) ) ).

% ratrel_def
tff(fact_4324_times__rat_Orsp,axiom,
    aa(fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),$o,aa(fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),fun(fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),$o),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_wl(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))),aTP_Lamp_wl(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))) ).

% times_rat.rsp
tff(fact_4325_uminus__rat_Orsp,axiom,
    aa(fun(product_prod(int,int),product_prod(int,int)),$o,aa(fun(product_prod(int,int),product_prod(int,int)),fun(fun(product_prod(int,int),product_prod(int,int)),$o),bNF_rel_fun(product_prod(int,int),product_prod(int,int),product_prod(int,int),product_prod(int,int),ratrel,ratrel),aTP_Lamp_wk(product_prod(int,int),product_prod(int,int))),aTP_Lamp_wk(product_prod(int,int),product_prod(int,int))) ).

% uminus_rat.rsp
tff(fact_4326_positive_Orsp,axiom,
    aa(fun(product_prod(int,int),$o),$o,aa(fun(product_prod(int,int),$o),fun(fun(product_prod(int,int),$o),$o),bNF_rel_fun(product_prod(int,int),product_prod(int,int),$o,$o,ratrel,fequal($o)),aTP_Lamp_wn(product_prod(int,int),$o)),aTP_Lamp_wn(product_prod(int,int),$o)) ).

% positive.rsp
tff(fact_4327_inverse__rat_Orsp,axiom,
    aa(fun(product_prod(int,int),product_prod(int,int)),$o,aa(fun(product_prod(int,int),product_prod(int,int)),fun(fun(product_prod(int,int),product_prod(int,int)),$o),bNF_rel_fun(product_prod(int,int),product_prod(int,int),product_prod(int,int),product_prod(int,int),ratrel,ratrel),aTP_Lamp_wm(product_prod(int,int),product_prod(int,int))),aTP_Lamp_wm(product_prod(int,int),product_prod(int,int))) ).

% inverse_rat.rsp
tff(fact_4328_plus__rat_Orsp,axiom,
    aa(fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),$o,aa(fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),fun(fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),$o),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_wh(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))),aTP_Lamp_wh(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))) ).

% plus_rat.rsp
tff(fact_4329_uminus__rat_Oabs__eq,axiom,
    ! [X: product_prod(int,int)] :
      ( aa(product_prod(int,int),$o,aa(product_prod(int,int),fun(product_prod(int,int),$o),ratrel,X),X)
     => ( aa(rat,rat,uminus_uminus(rat),aa(product_prod(int,int),rat,abs_Rat,X)) = aa(product_prod(int,int),rat,abs_Rat,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,uminus_uminus(int),aa(product_prod(int,int),int,product_fst(int,int),X))),aa(product_prod(int,int),int,product_snd(int,int),X))) ) ) ).

% uminus_rat.abs_eq
tff(fact_4330_cr__rat__def,axiom,
    ! [X3: product_prod(int,int),Xa3: rat] :
      ( cr_rat(X3,Xa3)
    <=> ( aa(product_prod(int,int),$o,aa(product_prod(int,int),fun(product_prod(int,int),$o),ratrel,X3),X3)
        & ( aa(product_prod(int,int),rat,abs_Rat,X3) = Xa3 ) ) ) ).

% cr_rat_def
tff(fact_4331_eq__snd__iff,axiom,
    ! [B: $tType,A: $tType,B2: A,P3: product_prod(B,A)] :
      ( ( B2 = aa(product_prod(B,A),A,product_snd(B,A),P3) )
    <=> ? [A10: B] : P3 = aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),A10),B2) ) ).

% eq_snd_iff
tff(fact_4332_eq__fst__iff,axiom,
    ! [A: $tType,B: $tType,A3: A,P3: product_prod(A,B)] :
      ( ( A3 = aa(product_prod(A,B),A,product_fst(A,B),P3) )
    <=> ? [B6: B] : P3 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B6) ) ).

% eq_fst_iff
tff(fact_4333_times__int_Orsp,axiom,
    aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),$o,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),fun(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),$o),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_pb(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_pb(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))) ).

% times_int.rsp
tff(fact_4334_intrel__iff,axiom,
    ! [X: nat,Y: nat,U: nat,V: nat] :
      ( aa(product_prod(nat,nat),$o,aa(product_prod(nat,nat),fun(product_prod(nat,nat),$o),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),V))
    <=> ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),X),V) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),U),Y) ) ) ).

% intrel_iff
tff(fact_4335_zero__int_Orsp,axiom,
    aa(product_prod(nat,nat),$o,aa(product_prod(nat,nat),fun(product_prod(nat,nat),$o),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_4336_uminus__int_Orsp,axiom,
    aa(fun(product_prod(nat,nat),product_prod(nat,nat)),$o,aa(fun(product_prod(nat,nat),product_prod(nat,nat)),fun(fun(product_prod(nat,nat),product_prod(nat,nat)),$o),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_pm(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_pm(nat,fun(nat,product_prod(nat,nat))))) ).

% uminus_int.rsp
tff(fact_4337_nat_Orsp,axiom,
    aa(fun(product_prod(nat,nat),nat),$o,aa(fun(product_prod(nat,nat),nat),fun(fun(product_prod(nat,nat),nat),$o),bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),nat,nat,intrel,fequal(nat)),aa(fun(nat,fun(nat,nat)),fun(product_prod(nat,nat),nat),product_case_prod(nat,nat,nat),minus_minus(nat))),aa(fun(nat,fun(nat,nat)),fun(product_prod(nat,nat),nat),product_case_prod(nat,nat,nat),minus_minus(nat))) ).

% nat.rsp
tff(fact_4338_one__int_Orsp,axiom,
    aa(product_prod(nat,nat),$o,aa(product_prod(nat,nat),fun(product_prod(nat,nat),$o),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_4339_of__int_Orsp,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => aa(fun(product_prod(nat,nat),A),$o,aa(fun(product_prod(nat,nat),A),fun(fun(product_prod(nat,nat),A),$o),bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),A,A,intrel,fequal(A)),aa(fun(nat,fun(nat,A)),fun(product_prod(nat,nat),A),product_case_prod(nat,nat,A),aTP_Lamp_pn(nat,fun(nat,A)))),aa(fun(nat,fun(nat,A)),fun(product_prod(nat,nat),A),product_case_prod(nat,nat,A),aTP_Lamp_pn(nat,fun(nat,A)))) ) ).

% of_int.rsp
tff(fact_4340_intrel__def,axiom,
    intrel = aa(fun(nat,fun(nat,fun(product_prod(nat,nat),$o))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)),product_case_prod(nat,nat,fun(product_prod(nat,nat),$o)),aTP_Lamp_ws(nat,fun(nat,fun(product_prod(nat,nat),$o)))) ).

% intrel_def
tff(fact_4341_less__int_Orsp,axiom,
    aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)),$o,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)),fun(fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)),$o),bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),fun(product_prod(nat,nat),$o),fun(product_prod(nat,nat),$o),intrel,bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),$o,$o,intrel,fequal($o))),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),$o))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)),product_case_prod(nat,nat,fun(product_prod(nat,nat),$o)),aTP_Lamp_pp(nat,fun(nat,fun(product_prod(nat,nat),$o))))),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),$o))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)),product_case_prod(nat,nat,fun(product_prod(nat,nat),$o)),aTP_Lamp_pp(nat,fun(nat,fun(product_prod(nat,nat),$o))))) ).

% less_int.rsp
tff(fact_4342_less__eq__int_Orsp,axiom,
    aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)),$o,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)),fun(fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)),$o),bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),fun(product_prod(nat,nat),$o),fun(product_prod(nat,nat),$o),intrel,bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),$o,$o,intrel,fequal($o))),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),$o))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)),product_case_prod(nat,nat,fun(product_prod(nat,nat),$o)),aTP_Lamp_pr(nat,fun(nat,fun(product_prod(nat,nat),$o))))),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),$o))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)),product_case_prod(nat,nat,fun(product_prod(nat,nat),$o)),aTP_Lamp_pr(nat,fun(nat,fun(product_prod(nat,nat),$o))))) ).

% less_eq_int.rsp
tff(fact_4343_plus__int_Orsp,axiom,
    aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),$o,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),fun(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),$o),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_pt(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_pt(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))) ).

% plus_int.rsp
tff(fact_4344_minus__int_Orsp,axiom,
    aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),$o,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),fun(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),$o),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_pv(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_pv(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))) ).

% minus_int.rsp
tff(fact_4345_quotient__of__def,axiom,
    ! [X: rat] : quotient_of(X) = the(product_prod(int,int),aTP_Lamp_wt(rat,fun(product_prod(int,int),$o),X)) ).

% quotient_of_def
tff(fact_4346_Image__fold,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,B)),S: set(A)] :
      ( aa(set(product_prod(A,B)),$o,finite_finite2(product_prod(A,B)),R)
     => ( aa(set(A),set(B),image(A,B,R),S) = finite_fold(product_prod(A,B),set(B),aa(fun(A,fun(B,fun(set(B),set(B)))),fun(product_prod(A,B),fun(set(B),set(B))),product_case_prod(A,B,fun(set(B),set(B))),aTP_Lamp_up(set(A),fun(A,fun(B,fun(set(B),set(B)))),S)),bot_bot(set(B)),R) ) ) ).

% Image_fold
tff(fact_4347_signed__take__bit__eq__concat__bit,axiom,
    ! [N: nat,K: int] : bit_ri4674362597316999326ke_bit(int,N,K) = bit_concat_bit(N,K,aa(int,int,uminus_uminus(int),aa($o,int,zero_neq_one_of_bool(int),aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),N)))) ).

% signed_take_bit_eq_concat_bit
tff(fact_4348_trancl__set__ntrancl,axiom,
    ! [A: $tType,Xs: list(product_prod(A,A))] : transitive_trancl(A,aa(list(product_prod(A,A)),set(product_prod(A,A)),set2(product_prod(A,A)),Xs)) = transitive_ntrancl(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(product_prod(A,A)),nat,finite_card(product_prod(A,A)),aa(list(product_prod(A,A)),set(product_prod(A,A)),set2(product_prod(A,A)),Xs))),one_one(nat)),aa(list(product_prod(A,A)),set(product_prod(A,A)),set2(product_prod(A,A)),Xs)) ).

% trancl_set_ntrancl
tff(fact_4349_ImageI,axiom,
    ! [B: $tType,A: $tType,A3: A,B2: B,R2: set(product_prod(A,B)),A4: set(A)] :
      ( aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),R2)
     => ( aa(set(A),$o,member(A,A3),A4)
       => aa(set(B),$o,member(B,B2),aa(set(A),set(B),image(A,B,R2),A4)) ) ) ).

% ImageI
tff(fact_4350_Image__empty2,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(B,A))] : aa(set(B),set(A),image(B,A,R),bot_bot(set(B))) = bot_bot(set(A)) ).

% Image_empty2
tff(fact_4351_coprime__mult__right__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [C2: A,A3: A,B2: A] :
          ( algebr8660921524188924756oprime(A,C2,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2))
        <=> ( algebr8660921524188924756oprime(A,C2,A3)
            & algebr8660921524188924756oprime(A,C2,B2) ) ) ) ).

% coprime_mult_right_iff
tff(fact_4352_coprime__mult__left__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A,C2: A] :
          ( algebr8660921524188924756oprime(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2),C2)
        <=> ( algebr8660921524188924756oprime(A,A3,C2)
            & algebr8660921524188924756oprime(A,B2,C2) ) ) ) ).

% coprime_mult_left_iff
tff(fact_4353_coprime__minus__left__iff,axiom,
    ! [A: $tType] :
      ( ring_gcd(A)
     => ! [A3: A,B2: A] :
          ( algebr8660921524188924756oprime(A,aa(A,A,uminus_uminus(A),A3),B2)
        <=> algebr8660921524188924756oprime(A,A3,B2) ) ) ).

% coprime_minus_left_iff
tff(fact_4354_coprime__minus__right__iff,axiom,
    ! [A: $tType] :
      ( ring_gcd(A)
     => ! [A3: A,B2: A] :
          ( algebr8660921524188924756oprime(A,A3,aa(A,A,uminus_uminus(A),B2))
        <=> algebr8660921524188924756oprime(A,A3,B2) ) ) ).

% coprime_minus_right_iff
tff(fact_4355_coprime__self,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A] :
          ( algebr8660921524188924756oprime(A,A3,A3)
        <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A3),one_one(A)) ) ) ).

% coprime_self
tff(fact_4356_coprime__imp__gcd__eq__1,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A] :
          ( algebr8660921524188924756oprime(A,A3,B2)
         => ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2) = one_one(A) ) ) ) ).

% coprime_imp_gcd_eq_1
tff(fact_4357_Image__empty1,axiom,
    ! [B: $tType,A: $tType,X5: set(B)] : aa(set(B),set(A),image(B,A,bot_bot(set(product_prod(B,A)))),X5) = bot_bot(set(A)) ).

% Image_empty1
tff(fact_4358_Image__Id__on,axiom,
    ! [A: $tType,A4: set(A),B3: set(A)] : aa(set(A),set(A),image(A,A,id_on(A,A4)),B3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3) ).

% Image_Id_on
tff(fact_4359_coprime__0__left__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A] :
          ( algebr8660921524188924756oprime(A,zero_zero(A),A3)
        <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A3),one_one(A)) ) ) ).

% coprime_0_left_iff
tff(fact_4360_coprime__0__right__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A] :
          ( algebr8660921524188924756oprime(A,A3,zero_zero(A))
        <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A3),one_one(A)) ) ) ).

% coprime_0_right_iff
tff(fact_4361_Image__singleton__iff,axiom,
    ! [A: $tType,B: $tType,B2: A,R2: set(product_prod(B,A)),A3: B] :
      ( aa(set(A),$o,member(A,B2),aa(set(B),set(A),image(B,A,R2),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),A3),bot_bot(set(B)))))
    <=> aa(set(product_prod(B,A)),$o,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)),R2) ) ).

% Image_singleton_iff
tff(fact_4362_coprime__mult__self__right__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,C2: A,B2: A] :
          ( algebr8660921524188924756oprime(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))
        <=> ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C2),one_one(A))
            & algebr8660921524188924756oprime(A,A3,B2) ) ) ) ).

% coprime_mult_self_right_iff
tff(fact_4363_coprime__mult__self__left__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,A3: A,B2: A] :
          ( algebr8660921524188924756oprime(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
        <=> ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C2),one_one(A))
            & algebr8660921524188924756oprime(A,A3,B2) ) ) ) ).

% coprime_mult_self_left_iff
tff(fact_4364_is__unit__gcd,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2)),one_one(A))
        <=> algebr8660921524188924756oprime(A,A3,B2) ) ) ).

% is_unit_gcd
tff(fact_4365_normalize__stable,axiom,
    ! [Q3: int,P3: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Q3)
     => ( algebr8660921524188924756oprime(int,P3,Q3)
       => ( normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P3),Q3)) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P3),Q3) ) ) ) ).

% normalize_stable
tff(fact_4366_pair__vimage__is__Image,axiom,
    ! [A: $tType,B: $tType,U: B,E3: set(product_prod(B,A))] : aa(set(product_prod(B,A)),set(A),aa(fun(A,product_prod(B,A)),fun(set(product_prod(B,A)),set(A)),vimage(A,product_prod(B,A)),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),U)),E3) = aa(set(B),set(A),image(B,A,E3),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),U),bot_bot(set(B)))) ).

% pair_vimage_is_Image
tff(fact_4367_prod__coprime__left,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_gcd(B)
     => ! [A4: set(A),F: fun(A,B),A3: B] :
          ( ! [I2: A] :
              ( aa(set(A),$o,member(A,I2),A4)
             => algebr8660921524188924756oprime(B,aa(A,B,F,I2),A3) )
         => algebr8660921524188924756oprime(B,aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F),A4),A3) ) ) ).

% prod_coprime_left
tff(fact_4368_prod__coprime__right,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_gcd(B)
     => ! [A4: set(A),A3: B,F: fun(A,B)] :
          ( ! [I2: A] :
              ( aa(set(A),$o,member(A,I2),A4)
             => algebr8660921524188924756oprime(B,A3,aa(A,B,F,I2)) )
         => algebr8660921524188924756oprime(B,A3,aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F),A4)) ) ) ).

% prod_coprime_right
tff(fact_4369_ImageE,axiom,
    ! [A: $tType,B: $tType,B2: A,R2: set(product_prod(B,A)),A4: set(B)] :
      ( aa(set(A),$o,member(A,B2),aa(set(B),set(A),image(B,A,R2),A4))
     => ~ ! [X2: B] :
            ( aa(set(product_prod(B,A)),$o,member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X2),B2)),R2)
           => ~ aa(set(B),$o,member(B,X2),A4) ) ) ).

% ImageE
tff(fact_4370_Image__iff,axiom,
    ! [A: $tType,B: $tType,B2: A,R2: set(product_prod(B,A)),A4: set(B)] :
      ( aa(set(A),$o,member(A,B2),aa(set(B),set(A),image(B,A,R2),A4))
    <=> ? [X4: B] :
          ( aa(set(B),$o,member(B,X4),A4)
          & aa(set(product_prod(B,A)),$o,member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X4),B2)),R2) ) ) ).

% Image_iff
tff(fact_4371_rev__ImageI,axiom,
    ! [B: $tType,A: $tType,A3: A,A4: set(A),B2: B,R2: set(product_prod(A,B))] :
      ( aa(set(A),$o,member(A,A3),A4)
     => ( aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),R2)
       => aa(set(B),$o,member(B,B2),aa(set(A),set(B),image(A,B,R2),A4)) ) ) ).

% rev_ImageI
tff(fact_4372_Image__Un,axiom,
    ! [A: $tType,B: $tType,R: set(product_prod(B,A)),A4: set(B),B3: set(B)] : aa(set(B),set(A),image(B,A,R),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A4),B3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(B),set(A),image(B,A,R),A4)),aa(set(B),set(A),image(B,A,R),B3)) ).

% Image_Un
tff(fact_4373_coprime__1__right,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A] : algebr8660921524188924756oprime(A,A3,one_one(A)) ) ).

% coprime_1_right
tff(fact_4374_coprime__1__left,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A] : algebr8660921524188924756oprime(A,one_one(A),A3) ) ).

% coprime_1_left
tff(fact_4375_coprime__add__one__right,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A] : algebr8660921524188924756oprime(A,A3,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A))) ) ).

% coprime_add_one_right
tff(fact_4376_coprime__add__one__left,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A] : algebr8660921524188924756oprime(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A)),A3) ) ).

% coprime_add_one_left
tff(fact_4377_coprime__doff__one__right,axiom,
    ! [A: $tType] :
      ( ring_gcd(A)
     => ! [A3: A] : algebr8660921524188924756oprime(A,A3,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),one_one(A))) ) ).

% coprime_doff_one_right
tff(fact_4378_coprime__diff__one__left,axiom,
    ! [A: $tType] :
      ( ring_gcd(A)
     => ! [A3: A] : algebr8660921524188924756oprime(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),one_one(A)),A3) ) ).

% coprime_diff_one_left
tff(fact_4379_divides__mult,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A3),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),C2)
           => ( algebr8660921524188924756oprime(A,A3,B2)
             => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),C2) ) ) ) ) ).

% divides_mult
tff(fact_4380_coprime__dvd__mult__left__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,C2: A,B2: A] :
          ( algebr8660921524188924756oprime(A,A3,C2)
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))
          <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A3),B2) ) ) ) ).

% coprime_dvd_mult_left_iff
tff(fact_4381_coprime__dvd__mult__right__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,C2: A,B2: A] :
          ( algebr8660921524188924756oprime(A,A3,C2)
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
          <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A3),B2) ) ) ) ).

% coprime_dvd_mult_right_iff
tff(fact_4382_coprimeI,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A] :
          ( ! [C4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C4),A3)
             => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C4),B2)
               => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C4),one_one(A)) ) )
         => algebr8660921524188924756oprime(A,A3,B2) ) ) ).

% coprimeI
tff(fact_4383_coprime__def,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A] :
          ( algebr8660921524188924756oprime(A,A3,B2)
        <=> ! [C5: A] :
              ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C5),A3)
             => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C5),B2)
               => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C5),one_one(A)) ) ) ) ) ).

% coprime_def
tff(fact_4384_not__coprimeE,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A] :
          ( ~ algebr8660921524188924756oprime(A,A3,B2)
         => ~ ! [C4: A] :
                ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C4),A3)
               => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C4),B2)
                 => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C4),one_one(A)) ) ) ) ) ).

% not_coprimeE
tff(fact_4385_not__coprimeI,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C2),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C2),B2)
           => ( ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C2),one_one(A))
             => ~ algebr8660921524188924756oprime(A,A3,B2) ) ) ) ) ).

% not_coprimeI
tff(fact_4386_coprime__absorb__left,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),X),Y)
         => ( algebr8660921524188924756oprime(A,X,Y)
          <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),X),one_one(A)) ) ) ) ).

% coprime_absorb_left
tff(fact_4387_coprime__imp__coprime,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,D3: A,A3: A,B2: A] :
          ( algebr8660921524188924756oprime(A,C2,D3)
         => ( ! [E2: A] :
                ( ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),E2),one_one(A))
               => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),E2),A3)
                 => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),E2),B2)
                   => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),E2),C2) ) ) )
           => ( ! [E2: A] :
                  ( ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),E2),one_one(A))
                 => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),E2),A3)
                   => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),E2),B2)
                     => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),E2),D3) ) ) )
             => algebr8660921524188924756oprime(A,A3,B2) ) ) ) ) ).

% coprime_imp_coprime
tff(fact_4388_coprime__absorb__right,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [Y: A,X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),Y),X)
         => ( algebr8660921524188924756oprime(A,X,Y)
          <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),Y),one_one(A)) ) ) ) ).

% coprime_absorb_right
tff(fact_4389_coprime__common__divisor,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A,C2: A] :
          ( algebr8660921524188924756oprime(A,A3,B2)
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C2),A3)
           => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C2),B2)
             => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C2),one_one(A)) ) ) ) ) ).

% coprime_common_divisor
tff(fact_4390_is__unit__left__imp__coprime,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A3),one_one(A))
         => algebr8660921524188924756oprime(A,A3,B2) ) ) ).

% is_unit_left_imp_coprime
tff(fact_4391_is__unit__right__imp__coprime,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),one_one(A))
         => algebr8660921524188924756oprime(A,A3,B2) ) ) ).

% is_unit_right_imp_coprime
tff(fact_4392_Image__Int__subset,axiom,
    ! [A: $tType,B: $tType,R: set(product_prod(B,A)),A4: set(B),B3: set(B)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(B),set(A),image(B,A,R),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),B3))),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(B),set(A),image(B,A,R),A4)),aa(set(B),set(A),image(B,A,R),B3))) ).

% Image_Int_subset
tff(fact_4393_rtrancl__image__advance__rtrancl,axiom,
    ! [A: $tType,Q3: A,R: set(product_prod(A,A)),Q0: set(A),X: A] :
      ( aa(set(A),$o,member(A,Q3),aa(set(A),set(A),image(A,A,transitive_rtrancl(A,R)),Q0))
     => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q3),X)),transitive_rtrancl(A,R))
       => aa(set(A),$o,member(A,X),aa(set(A),set(A),image(A,A,transitive_rtrancl(A,R)),Q0)) ) ) ).

% rtrancl_image_advance_rtrancl
tff(fact_4394_rtrancl__image__advance,axiom,
    ! [A: $tType,Q3: A,R: set(product_prod(A,A)),Q0: set(A),X: A] :
      ( aa(set(A),$o,member(A,Q3),aa(set(A),set(A),image(A,A,transitive_rtrancl(A,R)),Q0))
     => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q3),X)),R)
       => aa(set(A),$o,member(A,X),aa(set(A),set(A),image(A,A,transitive_rtrancl(A,R)),Q0)) ) ) ).

% rtrancl_image_advance
tff(fact_4395_mult__mod__cancel__right,axiom,
    ! [A: $tType] :
      ( ( euclid8851590272496341667cancel(A)
        & semiring_gcd(A) )
     => ! [A3: A,N: A,M: A,B2: A] :
          ( ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),N),M) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),B2),N),M) )
         => ( algebr8660921524188924756oprime(A,M,N)
           => ( modulo_modulo(A,A3,M) = modulo_modulo(A,B2,M) ) ) ) ) ).

% mult_mod_cancel_right
tff(fact_4396_mult__mod__cancel__left,axiom,
    ! [A: $tType] :
      ( ( euclid8851590272496341667cancel(A)
        & semiring_gcd(A) )
     => ! [N: A,A3: A,M: A,B2: A] :
          ( ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),N),A3),M) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),N),B2),M) )
         => ( algebr8660921524188924756oprime(A,M,N)
           => ( modulo_modulo(A,A3,M) = modulo_modulo(A,B2,M) ) ) ) ) ).

% mult_mod_cancel_left
tff(fact_4397_gcd__mult__right__right__cancel,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,C2: A,B2: A] :
          ( algebr8660921524188924756oprime(A,A3,C2)
         => ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2) ) ) ) ).

% gcd_mult_right_right_cancel
tff(fact_4398_gcd__mult__right__left__cancel,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,C2: A,B2: A] :
          ( algebr8660921524188924756oprime(A,A3,C2)
         => ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2) ) ) ) ).

% gcd_mult_right_left_cancel
tff(fact_4399_gcd__mult__left__right__cancel,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [B2: A,C2: A,A3: A] :
          ( algebr8660921524188924756oprime(A,B2,C2)
         => ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),B2) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2) ) ) ) ).

% gcd_mult_left_right_cancel
tff(fact_4400_gcd__mult__left__left__cancel,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [B2: A,C2: A,A3: A] :
          ( algebr8660921524188924756oprime(A,B2,C2)
         => ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),B2) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2) ) ) ) ).

% gcd_mult_left_left_cancel
tff(fact_4401_coprime__iff__gcd__eq__1,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A] :
          ( algebr8660921524188924756oprime(A,A3,B2)
        <=> ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2) = one_one(A) ) ) ) ).

% coprime_iff_gcd_eq_1
tff(fact_4402_gcd__eq__1__imp__coprime,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2) = one_one(A) )
         => algebr8660921524188924756oprime(A,A3,B2) ) ) ).

% gcd_eq_1_imp_coprime
tff(fact_4403_Un__Image,axiom,
    ! [A: $tType,B: $tType,R: set(product_prod(B,A)),S: set(product_prod(B,A)),A4: set(B)] : aa(set(B),set(A),image(B,A,aa(set(product_prod(B,A)),set(product_prod(B,A)),aa(set(product_prod(B,A)),fun(set(product_prod(B,A)),set(product_prod(B,A))),sup_sup(set(product_prod(B,A))),R),S)),A4) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(B),set(A),image(B,A,R),A4)),aa(set(B),set(A),image(B,A,S),A4)) ).

% Un_Image
tff(fact_4404_quotient__of__coprime,axiom,
    ! [R2: rat,P3: int,Q3: int] :
      ( ( quotient_of(R2) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P3),Q3) )
     => algebr8660921524188924756oprime(int,P3,Q3) ) ).

% quotient_of_coprime
tff(fact_4405_normalize__coprime,axiom,
    ! [R2: product_prod(int,int),P3: int,Q3: int] :
      ( ( normalize(R2) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P3),Q3) )
     => algebr8660921524188924756oprime(int,P3,Q3) ) ).

% normalize_coprime
tff(fact_4406_Image__UN,axiom,
    ! [B: $tType,A: $tType,C: $tType,R2: set(product_prod(B,A)),B3: fun(C,set(B)),A4: set(C)] : aa(set(B),set(A),image(B,A,R2),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),image2(C,set(B),B3),A4))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(C),set(set(A)),image2(C,set(A),aa(fun(C,set(B)),fun(C,set(A)),aTP_Lamp_wu(set(product_prod(B,A)),fun(fun(C,set(B)),fun(C,set(A))),R2),B3)),A4)) ).

% Image_UN
tff(fact_4407_invertible__coprime,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A3: A,B2: A,C2: A] :
          ( ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2),C2) = one_one(A) )
         => algebr8660921524188924756oprime(A,A3,C2) ) ) ).

% invertible_coprime
tff(fact_4408_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) )
         => ? [A13: A,B8: A] :
              ( ( A3 = aa(A,A,aa(A,fun(A,A),times_times(A),A13),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),B8),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2)) )
              & algebr8660921524188924756oprime(A,A13,B8) ) ) ) ).

% gcd_coprime_exists
tff(fact_4409_gcd__coprime,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A,A5: A,B4: 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),A5),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),B4),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2)) )
             => algebr8660921524188924756oprime(A,A5,B4) ) ) ) ) ).

% gcd_coprime
tff(fact_4410_Image__empty__rtrancl__Image__id,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),V: A] :
      ( ( aa(set(A),set(A),image(A,A,R),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),V),bot_bot(set(A)))) = bot_bot(set(A)) )
     => ( aa(set(A),set(A),image(A,A,transitive_rtrancl(A,R)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),V),bot_bot(set(A)))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),V),bot_bot(set(A))) ) ) ).

% Image_empty_rtrancl_Image_id
tff(fact_4411_wfI__pf,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( ! [A8: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A8),aa(set(A),set(A),image(A,A,R),A8))
         => ( A8 = bot_bot(set(A)) ) )
     => wf(A,R) ) ).

% wfI_pf
tff(fact_4412_wfE__pf,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A4: set(A)] :
      ( wf(A,R)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),aa(set(A),set(A),image(A,A,R),A4))
       => ( A4 = bot_bot(set(A)) ) ) ) ).

% wfE_pf
tff(fact_4413_Image__empty__trancl__Image__empty,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),V: A] :
      ( ( aa(set(A),set(A),image(A,A,R),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),V),bot_bot(set(A)))) = bot_bot(set(A)) )
     => ( aa(set(A),set(A),image(A,A,transitive_trancl(A,R)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),V),bot_bot(set(A)))) = bot_bot(set(A)) ) ) ).

% Image_empty_trancl_Image_empty
tff(fact_4414_coprime__common__divisor__int,axiom,
    ! [A3: int,B2: int,X: int] :
      ( algebr8660921524188924756oprime(int,A3,B2)
     => ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),X),A3)
       => ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),X),B2)
         => ( aa(int,int,abs_abs(int),X) = one_one(int) ) ) ) ) ).

% coprime_common_divisor_int
tff(fact_4415_trancl__image__by__rtrancl,axiom,
    ! [A: $tType,E3: set(product_prod(A,A)),Vi: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),image(A,A,transitive_trancl(A,E3)),Vi)),Vi) = aa(set(A),set(A),image(A,A,transitive_rtrancl(A,E3)),Vi) ).

% trancl_image_by_rtrancl
tff(fact_4416_Image__singleton,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(B,A)),A3: B] : aa(set(B),set(A),image(B,A,R2),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),A3),bot_bot(set(B)))) = aa(fun(A,$o),set(A),collect(A),aa(B,fun(A,$o),aTP_Lamp_wv(set(product_prod(B,A)),fun(B,fun(A,$o)),R2),A3)) ).

% Image_singleton
tff(fact_4417_Image__subset,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(A,B)),A4: set(A),B3: set(B),C3: set(A)] :
      ( aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),R2),product_Sigma(A,B,A4,aTP_Lamp_qz(set(B),fun(A,set(B)),B3)))
     => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,R2),C3)),B3) ) ).

% Image_subset
tff(fact_4418_Image__INT__subset,axiom,
    ! [B: $tType,A: $tType,C: $tType,R2: set(product_prod(B,A)),B3: fun(C,set(B)),A4: set(C)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(B),set(A),image(B,A,R2),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(C),set(set(B)),image2(C,set(B),B3),A4)))),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(C),set(set(A)),image2(C,set(A),aa(fun(C,set(B)),fun(C,set(A)),aTP_Lamp_wu(set(product_prod(B,A)),fun(fun(C,set(B)),fun(C,set(A))),R2),B3)),A4))) ).

% Image_INT_subset
tff(fact_4419_rtrancl__apply__insert,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),X: A,S: set(A)] : aa(set(A),set(A),image(A,A,transitive_rtrancl(A,R)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),S)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),aa(set(A),set(A),image(A,A,transitive_rtrancl(A,R)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),S),aa(set(A),set(A),image(A,A,R),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))))))) ).

% rtrancl_apply_insert
tff(fact_4420_rtrancl__Image__in__Field,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),V3: set(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),image(A,A,transitive_rtrancl(A,R)),V3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),field2(A,R)),V3)) ).

% rtrancl_Image_in_Field
tff(fact_4421_E__closed__restr__reach__cases,axiom,
    ! [A: $tType,U: A,V: A,E3: set(product_prod(A,A)),R: set(A)] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),U),V)),transitive_rtrancl(A,E3))
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),image(A,A,E3),R)),R)
       => ( ~ aa(set(A),$o,member(A,V),R)
         => ~ ( ~ aa(set(A),$o,member(A,U),R)
             => ~ aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),U),V)),transitive_rtrancl(A,rel_restrict(A,E3,R))) ) ) ) ) ).

% E_closed_restr_reach_cases
tff(fact_4422_rel__restrict__tranclI,axiom,
    ! [A: $tType,X: A,Y: A,E3: set(product_prod(A,A)),R: set(A)] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),transitive_trancl(A,E3))
     => ( ~ aa(set(A),$o,member(A,X),R)
       => ( ~ aa(set(A),$o,member(A,Y),R)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),image(A,A,E3),R)),R)
           => aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),transitive_trancl(A,rel_restrict(A,E3,R))) ) ) ) ) ).

% rel_restrict_tranclI
tff(fact_4423_subset__Image__Image__iff,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A4: set(A),B3: set(A)] :
      ( order_preorder_on(A,field2(A,R2),R2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),field2(A,R2))
       => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),field2(A,R2))
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),image(A,A,R2),A4)),aa(set(A),set(A),image(A,A,R2),B3))
          <=> ! [X4: A] :
                ( aa(set(A),$o,member(A,X4),A4)
               => ? [Xa2: A] :
                    ( aa(set(A),$o,member(A,Xa2),B3)
                    & aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa2),X4)),R2) ) ) ) ) ) ) ).

% subset_Image_Image_iff
tff(fact_4424_Image__eq__UN,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(B,A)),B3: set(B)] : aa(set(B),set(A),image(B,A,R2),B3) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aTP_Lamp_ww(set(product_prod(B,A)),fun(B,set(A)),R2)),B3)) ).

% Image_eq_UN
tff(fact_4425_Sigma__Image,axiom,
    ! [A: $tType,B: $tType,A4: set(B),B3: fun(B,set(A)),X5: set(B)] : aa(set(B),set(A),image(B,A,product_Sigma(B,A,A4,B3)),X5) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),X5),A4))) ).

% Sigma_Image
tff(fact_4426_UN__Image,axiom,
    ! [B: $tType,A: $tType,C: $tType,X5: fun(C,set(product_prod(B,A))),I4: set(C),S: set(B)] : aa(set(B),set(A),image(B,A,aa(set(set(product_prod(B,A))),set(product_prod(B,A)),complete_Sup_Sup(set(product_prod(B,A))),aa(set(C),set(set(product_prod(B,A))),image2(C,set(product_prod(B,A)),X5),I4))),S) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(C),set(set(A)),image2(C,set(A),aa(set(B),fun(C,set(A)),aTP_Lamp_wx(fun(C,set(product_prod(B,A))),fun(set(B),fun(C,set(A))),X5),S)),I4)) ).

% UN_Image
tff(fact_4427_finite__reachable__advance,axiom,
    ! [A: $tType,E3: set(product_prod(A,A)),V0: A,V: A] :
      ( aa(set(A),$o,finite_finite2(A),aa(set(A),set(A),image(A,A,transitive_rtrancl(A,E3)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),V0),bot_bot(set(A)))))
     => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),V0),V)),transitive_rtrancl(A,E3))
       => aa(set(A),$o,finite_finite2(A),aa(set(A),set(A),image(A,A,transitive_rtrancl(A,E3)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),V),bot_bot(set(A))))) ) ) ).

% finite_reachable_advance
tff(fact_4428_rtrancl__Image__advance__ss,axiom,
    ! [A: $tType,U: A,V: A,E3: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),U),V)),E3)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),image(A,A,transitive_rtrancl(A,E3)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),V),bot_bot(set(A))))),aa(set(A),set(A),image(A,A,transitive_rtrancl(A,E3)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),U),bot_bot(set(A))))) ) ).

% rtrancl_Image_advance_ss
tff(fact_4429_trancl__Image__advance__ss,axiom,
    ! [A: $tType,U: A,V: A,E3: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),U),V)),E3)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),image(A,A,transitive_trancl(A,E3)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),V),bot_bot(set(A))))),aa(set(A),set(A),image(A,A,transitive_trancl(A,E3)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),U),bot_bot(set(A))))) ) ).

% trancl_Image_advance_ss
tff(fact_4430_trancl__restrict__reachable,axiom,
    ! [A: $tType,U: A,V: A,E3: set(product_prod(A,A)),S: set(A)] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),U),V)),transitive_trancl(A,E3))
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),image(A,A,E3),S)),S)
       => ( aa(set(A),$o,member(A,U),S)
         => aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),U),V)),transitive_trancl(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))),E3),product_Sigma(A,A,S,aTP_Lamp_rl(set(A),fun(A,set(A)),S))))) ) ) ) ).

% trancl_restrict_reachable
tff(fact_4431_subset__Image1__Image1__iff,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A3: A,B2: A] :
      ( order_preorder_on(A,field2(A,R2),R2)
     => ( aa(set(A),$o,member(A,A3),field2(A,R2))
       => ( aa(set(A),$o,member(A,B2),field2(A,R2))
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),image(A,A,R2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A))))),aa(set(A),set(A),image(A,A,R2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A)))))
          <=> aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),A3)),R2) ) ) ) ) ).

% subset_Image1_Image1_iff
tff(fact_4432_set__union,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] : aa(list(A),set(A),set2(A),union(A,Xs,Ys)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys)) ).

% set_union
tff(fact_4433_quotient__def,axiom,
    ! [A: $tType,A4: set(A),R2: set(product_prod(A,A))] : equiv_quotient(A,A4,R2) = aa(set(set(set(A))),set(set(A)),complete_Sup_Sup(set(set(A))),aa(set(A),set(set(set(A))),image2(A,set(set(A)),aTP_Lamp_wy(set(product_prod(A,A)),fun(A,set(set(A))),R2)),A4)) ).

% quotient_def
tff(fact_4434_set__product,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B)] : aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),product(A,B,Xs,Ys)) = product_Sigma(A,B,aa(list(A),set(A),set2(A),Xs),aTP_Lamp_wz(list(B),fun(A,set(B)),Ys)) ).

% set_product
tff(fact_4435_card__lists__length__le,axiom,
    ! [A: $tType,A4: set(A),N: nat] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( aa(set(list(A)),nat,finite_card(list(A)),aa(fun(list(A),$o),set(list(A)),collect(list(A)),aa(nat,fun(list(A),$o),aTP_Lamp_xa(set(A),fun(nat,fun(list(A),$o)),A4),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),A4))),aa(nat,set(nat),set_ord_atMost(nat),N)) ) ) ).

% card_lists_length_le
tff(fact_4436_quotient__empty,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : equiv_quotient(A,bot_bot(set(A)),R2) = bot_bot(set(set(A))) ).

% quotient_empty
tff(fact_4437_quotient__is__empty,axiom,
    ! [A: $tType,A4: set(A),R2: set(product_prod(A,A))] :
      ( ( equiv_quotient(A,A4,R2) = bot_bot(set(set(A))) )
    <=> ( A4 = bot_bot(set(A)) ) ) ).

% quotient_is_empty
tff(fact_4438_quotient__is__empty2,axiom,
    ! [A: $tType,A4: set(A),R2: set(product_prod(A,A))] :
      ( ( bot_bot(set(set(A))) = equiv_quotient(A,A4,R2) )
    <=> ( A4 = bot_bot(set(A)) ) ) ).

% quotient_is_empty2
tff(fact_4439_coprime__common__divisor__nat,axiom,
    ! [A3: nat,B2: nat,X: nat] :
      ( algebr8660921524188924756oprime(nat,A3,B2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),X),A3)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),X),B2)
         => ( X = one_one(nat) ) ) ) ) ).

% coprime_common_divisor_nat
tff(fact_4440_quotientE,axiom,
    ! [A: $tType,X5: set(A),A4: set(A),R2: set(product_prod(A,A))] :
      ( aa(set(set(A)),$o,member(set(A),X5),equiv_quotient(A,A4,R2))
     => ~ ! [X2: A] :
            ( ( X5 = aa(set(A),set(A),image(A,A,R2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X2),bot_bot(set(A)))) )
           => ~ aa(set(A),$o,member(A,X2),A4) ) ) ).

% quotientE
tff(fact_4441_quotientI,axiom,
    ! [A: $tType,X: A,A4: set(A),R2: set(product_prod(A,A))] :
      ( aa(set(A),$o,member(A,X),A4)
     => aa(set(set(A)),$o,member(set(A),aa(set(A),set(A),image(A,A,R2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))))),equiv_quotient(A,A4,R2)) ) ).

% quotientI
tff(fact_4442_finite__lists__length__eq,axiom,
    ! [A: $tType,A4: set(A),N: nat] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => aa(set(list(A)),$o,finite_finite2(list(A)),aa(fun(list(A),$o),set(list(A)),collect(list(A)),aa(nat,fun(list(A),$o),aTP_Lamp_xb(set(A),fun(nat,fun(list(A),$o)),A4),N))) ) ).

% finite_lists_length_eq
tff(fact_4443_coprime__diff__one__left__nat,axiom,
    ! [N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),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_4444_coprime__diff__one__right__nat,axiom,
    ! [N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),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_4445_finite__equiv__class,axiom,
    ! [A: $tType,A4: set(A),R2: set(product_prod(A,A)),X5: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R2),product_Sigma(A,A,A4,aTP_Lamp_rl(set(A),fun(A,set(A)),A4)))
       => ( aa(set(set(A)),$o,member(set(A),X5),equiv_quotient(A,A4,R2))
         => aa(set(A),$o,finite_finite2(A),X5) ) ) ) ).

% finite_equiv_class
tff(fact_4446_finite__lists__length__le,axiom,
    ! [A: $tType,A4: set(A),N: nat] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => aa(set(list(A)),$o,finite_finite2(list(A)),aa(fun(list(A),$o),set(list(A)),collect(list(A)),aa(nat,fun(list(A),$o),aTP_Lamp_xa(set(A),fun(nat,fun(list(A),$o)),A4),N))) ) ).

% finite_lists_length_le
tff(fact_4447_finite__quotient,axiom,
    ! [A: $tType,A4: set(A),R2: set(product_prod(A,A))] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R2),product_Sigma(A,A,A4,aTP_Lamp_rl(set(A),fun(A,set(A)),A4)))
       => aa(set(set(A)),$o,finite_finite2(set(A)),equiv_quotient(A,A4,R2)) ) ) ).

% finite_quotient
tff(fact_4448_card__lists__length__eq,axiom,
    ! [A: $tType,A4: set(A),N: nat] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( aa(set(list(A)),nat,finite_card(list(A)),aa(fun(list(A),$o),set(list(A)),collect(list(A)),aa(nat,fun(list(A),$o),aTP_Lamp_xb(set(A),fun(nat,fun(list(A),$o)),A4),N))) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(set(A),nat,finite_card(A),A4)),N) ) ) ).

% card_lists_length_eq
tff(fact_4449_singleton__quotient,axiom,
    ! [A: $tType,X: A,R2: set(product_prod(A,A))] : equiv_quotient(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))),R2) = aa(set(set(A)),set(set(A)),aa(set(A),fun(set(set(A)),set(set(A))),insert2(set(A)),aa(set(A),set(A),image(A,A,R2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))))),bot_bot(set(set(A)))) ).

% singleton_quotient
tff(fact_4450_card__lists__distinct__length__eq,axiom,
    ! [A: $tType,A4: set(A),K: nat] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),aa(set(A),nat,finite_card(A),A4))
       => ( aa(set(list(A)),nat,finite_card(list(A)),aa(fun(list(A),$o),set(list(A)),collect(list(A)),aa(nat,fun(list(A),$o),aTP_Lamp_xc(set(A),fun(nat,fun(list(A),$o)),A4),K))) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),aTP_Lamp_dj(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),A4)),K)),one_one(nat)),aa(set(A),nat,finite_card(A),A4))) ) ) ) ).

% card_lists_distinct_length_eq
tff(fact_4451_card__lists__distinct__length__eq_H,axiom,
    ! [A: $tType,K: nat,A4: set(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),aa(set(A),nat,finite_card(A),A4))
     => ( aa(set(list(A)),nat,finite_card(list(A)),aa(fun(list(A),$o),set(list(A)),collect(list(A)),aa(set(A),fun(list(A),$o),aTP_Lamp_xd(nat,fun(set(A),fun(list(A),$o)),K),A4))) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),aTP_Lamp_dj(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),A4)),K)),one_one(nat)),aa(set(A),nat,finite_card(A),A4))) ) ) ).

% card_lists_distinct_length_eq'
tff(fact_4452_mergesort__by__rel__split__length,axiom,
    ! [A: $tType,Xs1: list(A),Xs2: list(A),Xs: list(A)] :
      ( ( aa(list(A),nat,size_size(list(A)),aa(product_prod(list(A),list(A)),list(A),product_fst(list(A),list(A)),merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs1),Xs2),Xs))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),Xs1)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),modulo_modulo(nat,aa(list(A),nat,size_size(list(A)),Xs),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) )
      & ( aa(list(A),nat,size_size(list(A)),aa(product_prod(list(A),list(A)),list(A),product_snd(list(A),list(A)),merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs1),Xs2),Xs))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),Xs2)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ) ) ).

% mergesort_by_rel_split_length
tff(fact_4453_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_4454_finite__lists__distinct__length__eq,axiom,
    ! [A: $tType,A4: set(A),N: nat] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => aa(set(list(A)),$o,finite_finite2(list(A)),aa(fun(list(A),$o),set(list(A)),collect(list(A)),aa(nat,fun(list(A),$o),aTP_Lamp_xc(set(A),fun(nat,fun(list(A),$o)),A4),N))) ) ).

% finite_lists_distinct_length_eq
tff(fact_4455_distinct__disjoint__shuffles,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),Zs: list(A)] :
      ( distinct(A,Xs)
     => ( distinct(A,Ys)
       => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys)) = bot_bot(set(A)) )
         => ( aa(set(list(A)),$o,member(list(A),Zs),shuffles(A,Xs,Ys))
           => distinct(A,Zs) ) ) ) ) ).

% distinct_disjoint_shuffles
tff(fact_4456_distinct__finite__set,axiom,
    ! [A: $tType,X: set(A)] : aa(set(list(A)),$o,finite_finite2(list(A)),aa(fun(list(A),$o),set(list(A)),collect(list(A)),aTP_Lamp_xe(set(A),fun(list(A),$o),X))) ).

% distinct_finite_set
tff(fact_4457_infinite__UNIV__listI,axiom,
    ! [A: $tType] : ~ aa(set(list(A)),$o,finite_finite2(list(A)),top_top(set(list(A)))) ).

% infinite_UNIV_listI
tff(fact_4458_set__shuffles,axiom,
    ! [A: $tType,Zs: list(A),Xs: list(A),Ys: list(A)] :
      ( aa(set(list(A)),$o,member(list(A),Zs),shuffles(A,Xs,Ys))
     => ( aa(list(A),set(A),set2(A),Zs) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys)) ) ) ).

% set_shuffles
tff(fact_4459_distinct__finite__subset,axiom,
    ! [A: $tType,X: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),X)
     => aa(set(list(A)),$o,finite_finite2(list(A)),aa(fun(list(A),$o),set(list(A)),collect(list(A)),aTP_Lamp_xf(set(A),fun(list(A),$o),X))) ) ).

% distinct_finite_subset
tff(fact_4460_set__remove1__eq,axiom,
    ! [A: $tType,Xs: list(A),X: A] :
      ( distinct(A,Xs)
     => ( aa(list(A),set(A),set2(A),remove1(A,X,Xs)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) ) ) ).

% set_remove1_eq
tff(fact_4461_set__removeAll,axiom,
    ! [A: $tType,X: A,Xs: list(A)] : aa(list(A),set(A),set2(A),aa(list(A),list(A),removeAll(A,X),Xs)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) ).

% set_removeAll
tff(fact_4462_set__list__bind,axiom,
    ! [A: $tType,B: $tType,Xs: list(B),F: fun(B,list(A))] : aa(list(A),set(A),set2(A),bind(B,A,Xs,F)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aTP_Lamp_xg(fun(B,list(A)),fun(B,set(A)),F)),aa(list(B),set(B),set2(B),Xs))) ).

% set_list_bind
tff(fact_4463_set__empty,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( aa(list(A),set(A),set2(A),Xs) = bot_bot(set(A)) )
    <=> ( Xs = nil(A) ) ) ).

% set_empty
tff(fact_4464_set__empty2,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( bot_bot(set(A)) = aa(list(A),set(A),set2(A),Xs) )
    <=> ( Xs = nil(A) ) ) ).

% set_empty2
tff(fact_4465_shuffles_Osimps_I2_J,axiom,
    ! [A: $tType,Xs: list(A)] : shuffles(A,Xs,nil(A)) = aa(set(list(A)),set(list(A)),aa(list(A),fun(set(list(A)),set(list(A))),insert2(list(A)),Xs),bot_bot(set(list(A)))) ).

% shuffles.simps(2)
tff(fact_4466_shuffles_Osimps_I1_J,axiom,
    ! [A: $tType,Ys: list(A)] : shuffles(A,nil(A),Ys) = aa(set(list(A)),set(list(A)),aa(list(A),fun(set(list(A)),set(list(A))),insert2(list(A)),Ys),bot_bot(set(list(A)))) ).

% shuffles.simps(1)
tff(fact_4467_empty__set,axiom,
    ! [A: $tType] : bot_bot(set(A)) = aa(list(A),set(A),set2(A),nil(A)) ).

% empty_set
tff(fact_4468_mergesort__by__rel__split_Osimps_I1_J,axiom,
    ! [A: $tType,Xs1: list(A),Xs2: list(A)] : merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs1),Xs2),nil(A)) = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs1),Xs2) ).

% mergesort_by_rel_split.simps(1)
tff(fact_4469_length__remove1,axiom,
    ! [A: $tType,X: A,Xs: list(A)] :
      aa(list(A),nat,size_size(list(A)),remove1(A,X,Xs)) = $ite(aa(set(A),$o,member(A,X),aa(list(A),set(A),set2(A),Xs)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat)),aa(list(A),nat,size_size(list(A)),Xs)) ).

% length_remove1
tff(fact_4470_Pow__set_I1_J,axiom,
    ! [A: $tType] : pow2(A,aa(list(A),set(A),set2(A),nil(A))) = aa(set(set(A)),set(set(A)),aa(set(A),fun(set(set(A)),set(set(A))),insert2(set(A)),bot_bot(set(A))),bot_bot(set(set(A)))) ).

% Pow_set(1)
tff(fact_4471_mergesort__by__rel_Opinduct,axiom,
    ! [A: $tType,A0: fun(A,fun(A,$o)),A1: list(A),P: fun(fun(A,fun(A,$o)),fun(list(A),$o))] :
      ( accp(product_prod(fun(A,fun(A,$o)),list(A)),mergesort_by_rel_rel(A),aa(list(A),product_prod(fun(A,fun(A,$o)),list(A)),aa(fun(A,fun(A,$o)),fun(list(A),product_prod(fun(A,fun(A,$o)),list(A))),product_Pair(fun(A,fun(A,$o)),list(A)),A0),A1))
     => ( ! [R6: fun(A,fun(A,$o)),Xs3: list(A)] :
            ( accp(product_prod(fun(A,fun(A,$o)),list(A)),mergesort_by_rel_rel(A),aa(list(A),product_prod(fun(A,fun(A,$o)),list(A)),aa(fun(A,fun(A,$o)),fun(list(A),product_prod(fun(A,fun(A,$o)),list(A))),product_Pair(fun(A,fun(A,$o)),list(A)),R6),Xs3))
           => ( ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),Xs3)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))
               => aa(list(A),$o,aa(fun(A,fun(A,$o)),fun(list(A),$o),P,R6),aa(product_prod(list(A),list(A)),list(A),product_fst(list(A),list(A)),merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),Xs3))) )
             => ( ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),Xs3)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))
                 => aa(list(A),$o,aa(fun(A,fun(A,$o)),fun(list(A),$o),P,R6),aa(product_prod(list(A),list(A)),list(A),product_snd(list(A),list(A)),merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),Xs3))) )
               => aa(list(A),$o,aa(fun(A,fun(A,$o)),fun(list(A),$o),P,R6),Xs3) ) ) )
       => aa(list(A),$o,aa(fun(A,fun(A,$o)),fun(list(A),$o),P,A0),A1) ) ) ).

% mergesort_by_rel.pinduct
tff(fact_4472_mergesort__by__rel_Osimps,axiom,
    ! [A: $tType,R: fun(A,fun(A,$o)),Xs: list(A)] :
      aa(list(A),list(A),mergesort_by_rel(A,R),Xs) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Xs,merges9089515139780605204_merge(A,R,aa(list(A),list(A),mergesort_by_rel(A,R),aa(product_prod(list(A),list(A)),list(A),product_fst(list(A),list(A)),merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),Xs))),aa(list(A),list(A),mergesort_by_rel(A,R),aa(product_prod(list(A),list(A)),list(A),product_snd(list(A),list(A)),merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),Xs))))) ).

% mergesort_by_rel.simps
tff(fact_4473_mergesort__by__rel_Oelims,axiom,
    ! [A: $tType,X: fun(A,fun(A,$o)),Xa: list(A),Y: list(A)] :
      ( ( aa(list(A),list(A),mergesort_by_rel(A,X),Xa) = Y )
     => ( Y = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),Xa)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Xa,merges9089515139780605204_merge(A,X,aa(list(A),list(A),mergesort_by_rel(A,X),aa(product_prod(list(A),list(A)),list(A),product_fst(list(A),list(A)),merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),Xa))),aa(list(A),list(A),mergesort_by_rel(A,X),aa(product_prod(list(A),list(A)),list(A),product_snd(list(A),list(A)),merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),Xa))))) ) ) ).

% mergesort_by_rel.elims
tff(fact_4474_distinct__foldl__invar,axiom,
    ! [B: $tType,A: $tType,S: list(A),I4: fun(set(A),fun(B,$o)),Sigma_0: B,F: fun(B,fun(A,B))] :
      ( distinct(A,S)
     => ( aa(B,$o,aa(set(A),fun(B,$o),I4,aa(list(A),set(A),set2(A),S)),Sigma_0)
       => ( ! [X2: A,It: set(A),Sigma: B] :
              ( aa(set(A),$o,member(A,X2),It)
             => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),It),aa(list(A),set(A),set2(A),S))
               => ( aa(B,$o,aa(set(A),fun(B,$o),I4,It),Sigma)
                 => aa(B,$o,aa(set(A),fun(B,$o),I4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),It),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X2),bot_bot(set(A))))),aa(A,B,aa(B,fun(A,B),F,Sigma),X2)) ) ) )
         => aa(B,$o,aa(set(A),fun(B,$o),I4,bot_bot(set(A))),foldl(B,A,F,Sigma_0,S)) ) ) ) ).

% distinct_foldl_invar
tff(fact_4475_set__mergesort__by__rel__merge,axiom,
    ! [A: $tType,R: fun(A,fun(A,$o)),Xs: list(A),Ys: list(A)] : aa(list(A),set(A),set2(A),merges9089515139780605204_merge(A,R,Xs,Ys)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys)) ).

% set_mergesort_by_rel_merge
tff(fact_4476_foldl__length,axiom,
    ! [A: $tType,L: list(A)] : foldl(nat,A,aTP_Lamp_xh(nat,fun(A,nat)),zero_zero(nat),L) = aa(list(A),nat,size_size(list(A)),L) ).

% foldl_length
tff(fact_4477_mergesort__by__rel_Ocases,axiom,
    ! [A: $tType,X: product_prod(fun(A,fun(A,$o)),list(A))] :
      ~ ! [R6: fun(A,fun(A,$o)),Xs3: list(A)] : X != aa(list(A),product_prod(fun(A,fun(A,$o)),list(A)),aa(fun(A,fun(A,$o)),fun(list(A),product_prod(fun(A,fun(A,$o)),list(A))),product_Pair(fun(A,fun(A,$o)),list(A)),R6),Xs3) ).

% mergesort_by_rel.cases
tff(fact_4478_comp__fun__commute_Ofoldl__f__commute,axiom,
    ! [B: $tType,A: $tType,F: fun(A,fun(B,B)),A3: A,B2: B,Xs: list(A)] :
      ( finite6289374366891150609ommute(A,B,F)
     => ( aa(B,B,aa(A,fun(B,B),F,A3),foldl(B,A,aTP_Lamp_xi(fun(A,fun(B,B)),fun(B,fun(A,B)),F),B2,Xs)) = foldl(B,A,aTP_Lamp_xi(fun(A,fun(B,B)),fun(B,fun(A,B)),F),aa(B,B,aa(A,fun(B,B),F,A3),B2),Xs) ) ) ).

% comp_fun_commute.foldl_f_commute
tff(fact_4479_fst__foldl,axiom,
    ! [B: $tType,A: $tType,C: $tType,F: fun(A,fun(C,A)),G: fun(A,fun(B,fun(C,B))),A3: A,B2: B,Xs: list(C)] : aa(product_prod(A,B),A,product_fst(A,B),foldl(product_prod(A,B),C,aa(fun(A,fun(B,fun(C,product_prod(A,B)))),fun(product_prod(A,B),fun(C,product_prod(A,B))),product_case_prod(A,B,fun(C,product_prod(A,B))),aa(fun(A,fun(B,fun(C,B))),fun(A,fun(B,fun(C,product_prod(A,B)))),aTP_Lamp_xj(fun(A,fun(C,A)),fun(fun(A,fun(B,fun(C,B))),fun(A,fun(B,fun(C,product_prod(A,B))))),F),G)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2),Xs)) = foldl(A,C,F,A3,Xs) ).

% fst_foldl
tff(fact_4480_foldl__absorb1,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [X: A,Zs: list(A)] : aa(A,A,aa(A,fun(A,A),times_times(A),X),foldl(A,A,times_times(A),one_one(A),Zs)) = foldl(A,A,times_times(A),X,Zs) ) ).

% foldl_absorb1
tff(fact_4481_foldl__un__empty__eq,axiom,
    ! [A: $tType,I: set(A),Ww: list(set(A))] : foldl(set(A),set(A),sup_sup(set(A)),I,Ww) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),I),foldl(set(A),set(A),sup_sup(set(A)),bot_bot(set(A)),Ww)) ).

% foldl_un_empty_eq
tff(fact_4482_mergesort__by__rel_Opsimps,axiom,
    ! [A: $tType,R: fun(A,fun(A,$o)),Xs: list(A)] :
      ( accp(product_prod(fun(A,fun(A,$o)),list(A)),mergesort_by_rel_rel(A),aa(list(A),product_prod(fun(A,fun(A,$o)),list(A)),aa(fun(A,fun(A,$o)),fun(list(A),product_prod(fun(A,fun(A,$o)),list(A))),product_Pair(fun(A,fun(A,$o)),list(A)),R),Xs))
     => ( aa(list(A),list(A),mergesort_by_rel(A,R),Xs) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Xs,merges9089515139780605204_merge(A,R,aa(list(A),list(A),mergesort_by_rel(A,R),aa(product_prod(list(A),list(A)),list(A),product_fst(list(A),list(A)),merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),Xs))),aa(list(A),list(A),mergesort_by_rel(A,R),aa(product_prod(list(A),list(A)),list(A),product_snd(list(A),list(A)),merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),Xs))))) ) ) ).

% mergesort_by_rel.psimps
tff(fact_4483_mergesort__by__rel_Opelims,axiom,
    ! [A: $tType,X: fun(A,fun(A,$o)),Xa: list(A),Y: list(A)] :
      ( ( aa(list(A),list(A),mergesort_by_rel(A,X),Xa) = Y )
     => ( accp(product_prod(fun(A,fun(A,$o)),list(A)),mergesort_by_rel_rel(A),aa(list(A),product_prod(fun(A,fun(A,$o)),list(A)),aa(fun(A,fun(A,$o)),fun(list(A),product_prod(fun(A,fun(A,$o)),list(A))),product_Pair(fun(A,fun(A,$o)),list(A)),X),Xa))
       => ~ ( ( Y = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),Xa)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Xa,merges9089515139780605204_merge(A,X,aa(list(A),list(A),mergesort_by_rel(A,X),aa(product_prod(list(A),list(A)),list(A),product_fst(list(A),list(A)),merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),Xa))),aa(list(A),list(A),mergesort_by_rel(A,X),aa(product_prod(list(A),list(A)),list(A),product_snd(list(A),list(A)),merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),Xa))))) )
           => ~ accp(product_prod(fun(A,fun(A,$o)),list(A)),mergesort_by_rel_rel(A),aa(list(A),product_prod(fun(A,fun(A,$o)),list(A)),aa(fun(A,fun(A,$o)),fun(list(A),product_prod(fun(A,fun(A,$o)),list(A))),product_Pair(fun(A,fun(A,$o)),list(A)),X),Xa)) ) ) ) ).

% mergesort_by_rel.pelims
tff(fact_4484_foldl__length__aux,axiom,
    ! [A: $tType,A3: nat,L: list(A)] : foldl(nat,A,aTP_Lamp_xh(nat,fun(A,nat)),A3,L) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A3),aa(list(A),nat,size_size(list(A)),L)) ).

% foldl_length_aux
tff(fact_4485_foldl__set,axiom,
    ! [A: $tType,L: list(set(A))] : foldl(set(A),set(A),sup_sup(set(A)),bot_bot(set(A)),L) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(fun(set(A),$o),set(set(A)),collect(set(A)),aTP_Lamp_xk(list(set(A)),fun(set(A),$o),L))) ).

% foldl_set
tff(fact_4486_listset_Osimps_I1_J,axiom,
    ! [A: $tType] : listset(A,nil(set(A))) = aa(set(list(A)),set(list(A)),aa(list(A),fun(set(list(A)),set(list(A))),insert2(list(A)),nil(A)),bot_bot(set(list(A)))) ).

% listset.simps(1)
tff(fact_4487_mergesort__by__rel__simps_I3_J,axiom,
    ! [A: $tType,R: fun(A,fun(A,$o)),X1: A,X22: A,Xs: list(A)] : aa(list(A),list(A),mergesort_by_rel(A,R),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X22),Xs))) = aa(product_prod(list(A),list(A)),list(A),aa(fun(list(A),fun(list(A),list(A))),fun(product_prod(list(A),list(A)),list(A)),product_case_prod(list(A),list(A),list(A)),aTP_Lamp_xl(fun(A,fun(A,$o)),fun(list(A),fun(list(A),list(A))),R)),merges295452479951948502_split(A,aa(list(A),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),X1),nil(A))),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X22),nil(A))),Xs)) ).

% mergesort_by_rel_simps(3)
tff(fact_4488_set__n__lists,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : aa(list(list(A)),set(list(A)),set2(list(A)),n_lists(A,N,Xs)) = aa(fun(list(A),$o),set(list(A)),collect(list(A)),aa(list(A),fun(list(A),$o),aTP_Lamp_xm(nat,fun(list(A),fun(list(A),$o)),N),Xs)) ).

% set_n_lists
tff(fact_4489_nth__step__trancl,axiom,
    ! [A: $tType,Xs: list(A),R: set(product_prod(A,A)),N: nat,M: nat] :
      ( ! [N3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N3),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat)))
         => aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(nat,A,nth(A,Xs),aa(nat,nat,suc,N3))),aa(nat,A,nth(A,Xs),N3))),R) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs))
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),N)
         => aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(nat,A,nth(A,Xs),N)),aa(nat,A,nth(A,Xs),M))),transitive_trancl(A,R)) ) ) ) ).

% nth_step_trancl
tff(fact_4490_nth__Cons__numeral,axiom,
    ! [A: $tType,X: A,Xs: list(A),V: num] : aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),aa(num,nat,numeral_numeral(nat),V)) = aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(num,nat,numeral_numeral(nat),V)),one_one(nat))) ).

% nth_Cons_numeral
tff(fact_4491_nth__Cons__pos,axiom,
    ! [A: $tType,N: nat,X: A,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),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_4492_shuffles_Osimps_I3_J,axiom,
    ! [A: $tType,X: A,Xs: list(A),Y: A,Ys: list(A)] : shuffles(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)) = aa(set(list(A)),set(list(A)),aa(set(list(A)),fun(set(list(A)),set(list(A))),sup_sup(set(list(A))),aa(set(list(A)),set(list(A)),image2(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X)),shuffles(A,Xs,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)))),aa(set(list(A)),set(list(A)),image2(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y)),shuffles(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs),Ys))) ).

% shuffles.simps(3)
tff(fact_4493_nth__Cons,axiom,
    ! [A: $tType,X: A,Xs: list(A),N: nat] : aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),N) = aa(nat,A,aa(fun(nat,A),fun(nat,A),aa(A,fun(fun(nat,A),fun(nat,A)),case_nat(A),X),nth(A,Xs)),N) ).

% nth_Cons
tff(fact_4494_nth__Cons_H,axiom,
    ! [A: $tType,X: A,Xs: list(A),N: nat] :
      aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),N) = $ite(N = zero_zero(nat),X,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_4495_nth__non__equal__first__eq,axiom,
    ! [A: $tType,X: A,Y: A,Xs: list(A),N: nat] :
      ( ( X != Y )
     => ( ( aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),N) = Y )
      <=> ( ( aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))) = Y )
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N) ) ) ) ).

% nth_non_equal_first_eq
tff(fact_4496_mergesort__by__rel__split_Ocases,axiom,
    ! [A: $tType,X: product_prod(product_prod(list(A),list(A)),list(A))] :
      ( ! [Xs12: list(A),Xs22: list(A)] : X != aa(list(A),product_prod(product_prod(list(A),list(A)),list(A)),aa(product_prod(list(A),list(A)),fun(list(A),product_prod(product_prod(list(A),list(A)),list(A))),product_Pair(product_prod(list(A),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)),Xs12),Xs22)),nil(A))
     => ( ! [Xs12: list(A),Xs22: list(A),X2: A] : X != aa(list(A),product_prod(product_prod(list(A),list(A)),list(A)),aa(product_prod(list(A),list(A)),fun(list(A),product_prod(product_prod(list(A),list(A)),list(A))),product_Pair(product_prod(list(A),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)),Xs12),Xs22)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),nil(A)))
       => ~ ! [Xs12: list(A),Xs22: list(A),X12: A,X23: A,Xs3: list(A)] : X != aa(list(A),product_prod(product_prod(list(A),list(A)),list(A)),aa(product_prod(list(A),list(A)),fun(list(A),product_prod(product_prod(list(A),list(A)),list(A))),product_Pair(product_prod(list(A),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)),Xs12),Xs22)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X12),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X23),Xs3))) ) ) ).

% mergesort_by_rel_split.cases
tff(fact_4497_mergesort__by__rel__merge_Ocases,axiom,
    ! [A: $tType,X: product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A)))] :
      ( ! [R6: fun(A,fun(A,$o)),X2: A,Xs3: list(A),Y2: A,Ys2: list(A)] : X != aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),aa(fun(A,fun(A,$o)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,$o)),product_prod(list(A),list(A))),R6),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs3)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys2)))
     => ( ! [R6: fun(A,fun(A,$o)),Xs3: list(A)] : X != aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),aa(fun(A,fun(A,$o)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,$o)),product_prod(list(A),list(A))),R6),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs3),nil(A)))
       => ~ ! [R6: fun(A,fun(A,$o)),V2: A,Va: list(A)] : X != aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),aa(fun(A,fun(A,$o)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,$o)),product_prod(list(A),list(A))),R6),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V2),Va))) ) ) ).

% mergesort_by_rel_merge.cases
tff(fact_4498_quicksort__by__rel_Ocases,axiom,
    ! [A: $tType,X: product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A)))] :
      ( ! [R6: fun(A,fun(A,$o)),Sl: list(A)] : X != aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),aa(fun(A,fun(A,$o)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,$o)),product_prod(list(A),list(A))),R6),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Sl),nil(A)))
     => ~ ! [R6: fun(A,fun(A,$o)),Sl: list(A),X2: A,Xs3: list(A)] : X != aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),aa(fun(A,fun(A,$o)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,$o)),product_prod(list(A),list(A))),R6),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Sl),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs3))) ) ).

% quicksort_by_rel.cases
tff(fact_4499_map__tailrec__rev_Ocases,axiom,
    ! [A: $tType,B: $tType,X: product_prod(fun(A,B),product_prod(list(A),list(B)))] :
      ( ! [F2: fun(A,B),Bs2: list(B)] : X != aa(product_prod(list(A),list(B)),product_prod(fun(A,B),product_prod(list(A),list(B))),aa(fun(A,B),fun(product_prod(list(A),list(B)),product_prod(fun(A,B),product_prod(list(A),list(B)))),product_Pair(fun(A,B),product_prod(list(A),list(B))),F2),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),nil(A)),Bs2))
     => ~ ! [F2: fun(A,B),A6: A,As4: list(A),Bs2: list(B)] : X != aa(product_prod(list(A),list(B)),product_prod(fun(A,B),product_prod(list(A),list(B))),aa(fun(A,B),fun(product_prod(list(A),list(B)),product_prod(fun(A,B),product_prod(list(A),list(B)))),product_Pair(fun(A,B),product_prod(list(A),list(B))),F2),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A6),As4)),Bs2)) ) ).

% map_tailrec_rev.cases
tff(fact_4500_partition__rev_Ocases,axiom,
    ! [A: $tType,X: product_prod(fun(A,$o),product_prod(product_prod(list(A),list(A)),list(A)))] :
      ( ! [P2: fun(A,$o),Yes: list(A),No: list(A)] : X != aa(product_prod(product_prod(list(A),list(A)),list(A)),product_prod(fun(A,$o),product_prod(product_prod(list(A),list(A)),list(A))),aa(fun(A,$o),fun(product_prod(product_prod(list(A),list(A)),list(A)),product_prod(fun(A,$o),product_prod(product_prod(list(A),list(A)),list(A)))),product_Pair(fun(A,$o),product_prod(product_prod(list(A),list(A)),list(A))),P2),aa(list(A),product_prod(product_prod(list(A),list(A)),list(A)),aa(product_prod(list(A),list(A)),fun(list(A),product_prod(product_prod(list(A),list(A)),list(A))),product_Pair(product_prod(list(A),list(A)),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes),No)),nil(A)))
     => ~ ! [P2: fun(A,$o),Yes: list(A),No: list(A),X2: A,Xs3: list(A)] : X != aa(product_prod(product_prod(list(A),list(A)),list(A)),product_prod(fun(A,$o),product_prod(product_prod(list(A),list(A)),list(A))),aa(fun(A,$o),fun(product_prod(product_prod(list(A),list(A)),list(A)),product_prod(fun(A,$o),product_prod(product_prod(list(A),list(A)),list(A)))),product_Pair(fun(A,$o),product_prod(product_prod(list(A),list(A)),list(A))),P2),aa(list(A),product_prod(product_prod(list(A),list(A)),list(A)),aa(product_prod(list(A),list(A)),fun(list(A),product_prod(product_prod(list(A),list(A)),list(A))),product_Pair(product_prod(list(A),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)),Yes),No)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs3))) ) ).

% partition_rev.cases
tff(fact_4501_list__all__zip_Ocases,axiom,
    ! [A: $tType,B: $tType,X: product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B)))] :
      ( ! [P2: fun(A,fun(B,$o))] : X != aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),aa(fun(A,fun(B,$o)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,$o)),product_prod(list(A),list(B))),P2),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),nil(A)),nil(B)))
     => ( ! [P2: fun(A,fun(B,$o)),A6: A,As4: list(A),B5: B,Bs2: list(B)] : X != aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),aa(fun(A,fun(B,$o)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,$o)),product_prod(list(A),list(B))),P2),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),A6),As4)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B5),Bs2)))
       => ( ! [P2: fun(A,fun(B,$o)),V2: A,Va: list(A)] : X != aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),aa(fun(A,fun(B,$o)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,$o)),product_prod(list(A),list(B))),P2),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V2),Va)),nil(B)))
         => ~ ! [P2: fun(A,fun(B,$o)),V2: B,Va: list(B)] : X != aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),aa(fun(A,fun(B,$o)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,$o)),product_prod(list(A),list(B))),P2),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),nil(A)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),V2),Va))) ) ) ) ).

% list_all_zip.cases
tff(fact_4502_shuffles_Ocases,axiom,
    ! [A: $tType,X: product_prod(list(A),list(A))] :
      ( ! [Ys2: list(A)] : X != aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),Ys2)
     => ( ! [Xs3: list(A)] : X != aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs3),nil(A))
       => ~ ! [X2: A,Xs3: list(A),Y2: A,Ys2: list(A)] : X != aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs3)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys2)) ) ) ).

% shuffles.cases
tff(fact_4503_merge_Ocases,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: product_prod(list(A),list(A))] :
          ( ! [L22: list(A)] : X != aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),L22)
         => ( ! [V2: A,Va: list(A)] : X != aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V2),Va)),nil(A))
           => ~ ! [X12: A,L1: list(A),X23: A,L22: list(A)] : X != aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X12),L1)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X23),L22)) ) ) ) ).

% merge.cases
tff(fact_4504_zipf_Ocases,axiom,
    ! [C: $tType,A: $tType,B: $tType,X: product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B)))] :
      ( ! [F2: fun(A,fun(B,C))] : X != aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B))),aa(fun(A,fun(B,C)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,C)),product_prod(list(A),list(B))),F2),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),nil(A)),nil(B)))
     => ( ! [F2: fun(A,fun(B,C)),A6: A,As4: list(A),B5: B,Bs2: list(B)] : X != aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B))),aa(fun(A,fun(B,C)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,C)),product_prod(list(A),list(B))),F2),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A6),As4)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B5),Bs2)))
       => ( ! [A6: fun(A,fun(B,C)),V2: A,Va: list(A)] : X != aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B))),aa(fun(A,fun(B,C)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,C)),product_prod(list(A),list(B))),A6),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),V2),Va)),nil(B)))
         => ~ ! [A6: fun(A,fun(B,C)),V2: B,Va: list(B)] : X != aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B))),aa(fun(A,fun(B,C)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,C)),product_prod(list(A),list(B))),A6),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),nil(A)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),V2),Va))) ) ) ) ).

% zipf.cases
tff(fact_4505_subset__eq__mset__impl_Ocases,axiom,
    ! [A: $tType,X: product_prod(list(A),list(A))] :
      ( ! [Ys2: list(A)] : X != aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),Ys2)
     => ~ ! [X2: A,Xs3: list(A),Ys2: list(A)] : X != aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs3)),Ys2) ) ).

% subset_eq_mset_impl.cases
tff(fact_4506_arg__min__list_Ocases,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [X: product_prod(fun(A,B),list(A))] :
          ( ! [F2: fun(A,B),X2: A] : X != aa(list(A),product_prod(fun(A,B),list(A)),aa(fun(A,B),fun(list(A),product_prod(fun(A,B),list(A))),product_Pair(fun(A,B),list(A)),F2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),nil(A)))
         => ( ! [F2: fun(A,B),X2: A,Y2: A,Zs2: list(A)] : X != aa(list(A),product_prod(fun(A,B),list(A)),aa(fun(A,B),fun(list(A),product_prod(fun(A,B),list(A))),product_Pair(fun(A,B),list(A)),F2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Zs2)))
           => ~ ! [A6: fun(A,B)] : X != aa(list(A),product_prod(fun(A,B),list(A)),aa(fun(A,B),fun(list(A),product_prod(fun(A,B),list(A))),product_Pair(fun(A,B),list(A)),A6),nil(A)) ) ) ) ).

% arg_min_list.cases
tff(fact_4507_sorted__wrt_Ocases,axiom,
    ! [A: $tType,X: product_prod(fun(A,fun(A,$o)),list(A))] :
      ( ! [P2: fun(A,fun(A,$o))] : X != aa(list(A),product_prod(fun(A,fun(A,$o)),list(A)),aa(fun(A,fun(A,$o)),fun(list(A),product_prod(fun(A,fun(A,$o)),list(A))),product_Pair(fun(A,fun(A,$o)),list(A)),P2),nil(A))
     => ~ ! [P2: fun(A,fun(A,$o)),X2: A,Ys2: list(A)] : X != aa(list(A),product_prod(fun(A,fun(A,$o)),list(A)),aa(fun(A,fun(A,$o)),fun(list(A),product_prod(fun(A,fun(A,$o)),list(A))),product_Pair(fun(A,fun(A,$o)),list(A)),P2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Ys2)) ) ).

% sorted_wrt.cases
tff(fact_4508_successively_Ocases,axiom,
    ! [A: $tType,X: product_prod(fun(A,fun(A,$o)),list(A))] :
      ( ! [P2: fun(A,fun(A,$o))] : X != aa(list(A),product_prod(fun(A,fun(A,$o)),list(A)),aa(fun(A,fun(A,$o)),fun(list(A),product_prod(fun(A,fun(A,$o)),list(A))),product_Pair(fun(A,fun(A,$o)),list(A)),P2),nil(A))
     => ( ! [P2: fun(A,fun(A,$o)),X2: A] : X != aa(list(A),product_prod(fun(A,fun(A,$o)),list(A)),aa(fun(A,fun(A,$o)),fun(list(A),product_prod(fun(A,fun(A,$o)),list(A))),product_Pair(fun(A,fun(A,$o)),list(A)),P2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),nil(A)))
       => ~ ! [P2: fun(A,fun(A,$o)),X2: A,Y2: A,Xs3: list(A)] : X != aa(list(A),product_prod(fun(A,fun(A,$o)),list(A)),aa(fun(A,fun(A,$o)),fun(list(A),product_prod(fun(A,fun(A,$o)),list(A))),product_Pair(fun(A,fun(A,$o)),list(A)),P2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Xs3))) ) ) ).

% successively.cases
tff(fact_4509_mergesort__by__rel__split_Osimps_I3_J,axiom,
    ! [A: $tType,Xs1: list(A),Xs2: list(A),X1: A,X22: A,Xs: list(A)] : merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs1),Xs2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X22),Xs))) = merges295452479951948502_split(A,aa(list(A),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),X1),Xs1)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X22),Xs2)),Xs) ).

% mergesort_by_rel_split.simps(3)
tff(fact_4510_list__decomp__1,axiom,
    ! [A: $tType,L: list(A)] :
      ( ( aa(list(A),nat,size_size(list(A)),L) = one_one(nat) )
     => ? [A6: A] : L = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A6),nil(A)) ) ).

% list_decomp_1
tff(fact_4511_mergesort__by__rel__split_Oelims,axiom,
    ! [A: $tType,X: product_prod(list(A),list(A)),Xa: list(A),Y: product_prod(list(A),list(A))] :
      ( ( merges295452479951948502_split(A,X,Xa) = Y )
     => ( ! [Xs12: list(A),Xs22: list(A)] :
            ( ( X = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs12),Xs22) )
           => ( ( Xa = nil(A) )
             => ( Y != aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs12),Xs22) ) ) )
       => ( ! [Xs12: list(A),Xs22: list(A)] :
              ( ( X = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs12),Xs22) )
             => ! [X2: A] :
                  ( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),nil(A)) )
                 => ( Y != aa(list(A),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),X2),Xs12)),Xs22) ) ) )
         => ~ ! [Xs12: list(A),Xs22: list(A)] :
                ( ( X = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs12),Xs22) )
               => ! [X12: A,X23: A,Xs3: list(A)] :
                    ( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X12),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X23),Xs3)) )
                   => ( Y != merges295452479951948502_split(A,aa(list(A),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),X12),Xs12)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X23),Xs22)),Xs3) ) ) ) ) ) ) ).

% mergesort_by_rel_split.elims
tff(fact_4512_mergesort__by__rel__split_Osimps_I2_J,axiom,
    ! [A: $tType,Xs1: list(A),Xs2: list(A),X: A] : merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs1),Xs2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A))) = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs1)),Xs2) ).

% mergesort_by_rel_split.simps(2)
tff(fact_4513_product__nth,axiom,
    ! [A: $tType,B: $tType,N: nat,Xs: list(A),Ys: list(B)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),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),aa(nat,nat,aa(nat,fun(nat,nat),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_4514_lists__length__Suc__eq,axiom,
    ! [A: $tType,A4: set(A),N: nat] : aa(fun(list(A),$o),set(list(A)),collect(list(A)),aa(nat,fun(list(A),$o),aTP_Lamp_xn(set(A),fun(nat,fun(list(A),$o)),A4),N)) = aa(set(product_prod(list(A),A)),set(list(A)),image2(product_prod(list(A),A),list(A),aa(fun(list(A),fun(A,list(A))),fun(product_prod(list(A),A),list(A)),product_case_prod(list(A),A,list(A)),aTP_Lamp_xo(list(A),fun(A,list(A))))),product_Sigma(list(A),A,aa(fun(list(A),$o),set(list(A)),collect(list(A)),aa(nat,fun(list(A),$o),aTP_Lamp_xb(set(A),fun(nat,fun(list(A),$o)),A4),N)),aTP_Lamp_xp(set(A),fun(list(A),set(A)),A4))) ).

% lists_length_Suc_eq
tff(fact_4515_shuffles_Oelims,axiom,
    ! [A: $tType,X: list(A),Xa: list(A),Y: set(list(A))] :
      ( ( shuffles(A,X,Xa) = Y )
     => ( ( ( X = nil(A) )
         => ( Y != aa(set(list(A)),set(list(A)),aa(list(A),fun(set(list(A)),set(list(A))),insert2(list(A)),Xa),bot_bot(set(list(A)))) ) )
       => ( ( ( Xa = nil(A) )
           => ( Y != aa(set(list(A)),set(list(A)),aa(list(A),fun(set(list(A)),set(list(A))),insert2(list(A)),X),bot_bot(set(list(A)))) ) )
         => ~ ! [X2: A,Xs3: list(A)] :
                ( ( X = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs3) )
               => ! [Y2: A,Ys2: list(A)] :
                    ( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys2) )
                   => ( Y != aa(set(list(A)),set(list(A)),aa(set(list(A)),fun(set(list(A)),set(list(A))),sup_sup(set(list(A))),aa(set(list(A)),set(list(A)),image2(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2)),shuffles(A,Xs3,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys2)))),aa(set(list(A)),set(list(A)),image2(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2)),shuffles(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs3),Ys2))) ) ) ) ) ) ) ).

% shuffles.elims
tff(fact_4516_Pow__set_I2_J,axiom,
    ! [A: $tType,X: A,Xs: list(A)] :
      pow2(A,aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs))) = $let(
        a3: set(set(A)),
        a3:= pow2(A,aa(list(A),set(A),set2(A),Xs)),
        aa(set(set(A)),set(set(A)),aa(set(set(A)),fun(set(set(A)),set(set(A))),sup_sup(set(set(A))),a3),aa(set(set(A)),set(set(A)),image2(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X)),a3)) ) ).

% Pow_set(2)
tff(fact_4517_set__Cons__sing__Nil,axiom,
    ! [A: $tType,A4: set(A)] : set_Cons(A,A4,aa(set(list(A)),set(list(A)),aa(list(A),fun(set(list(A)),set(list(A))),insert2(list(A)),nil(A)),bot_bot(set(list(A))))) = aa(set(A),set(list(A)),image2(A,list(A),aTP_Lamp_xq(A,list(A))),A4) ).

% set_Cons_sing_Nil
tff(fact_4518_mergesort__by__rel__merge_Opelims,axiom,
    ! [A: $tType,X: fun(A,fun(A,$o)),Xa: list(A),Xb: list(A),Y: list(A)] :
      ( ( merges9089515139780605204_merge(A,X,Xa,Xb) = Y )
     => ( accp(product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),merges2244889521215249637ge_rel(A),aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),aa(fun(A,fun(A,$o)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,$o)),product_prod(list(A),list(A))),X),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xa),Xb)))
       => ( ! [X2: A,Xs3: list(A)] :
              ( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs3) )
             => ! [Y2: A,Ys2: list(A)] :
                  ( ( Xb = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys2) )
                 => ( ( Y = $ite(aa(A,$o,aa(A,fun(A,$o),X,X2),Y2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),merges9089515139780605204_merge(A,X,Xs3,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys2))),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),merges9089515139780605204_merge(A,X,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs3),Ys2))) )
                   => ~ accp(product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),merges2244889521215249637ge_rel(A),aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),aa(fun(A,fun(A,$o)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,$o)),product_prod(list(A),list(A))),X),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs3)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys2)))) ) ) )
         => ( ( ( Xb = nil(A) )
             => ( ( Y = Xa )
               => ~ accp(product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),merges2244889521215249637ge_rel(A),aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),aa(fun(A,fun(A,$o)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,$o)),product_prod(list(A),list(A))),X),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xa),nil(A)))) ) )
           => ~ ( ( Xa = nil(A) )
               => ! [V2: A,Va: list(A)] :
                    ( ( Xb = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V2),Va) )
                   => ( ( Y = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V2),Va) )
                     => ~ accp(product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),merges2244889521215249637ge_rel(A),aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),aa(fun(A,fun(A,$o)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,$o)),product_prod(list(A),list(A))),X),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V2),Va)))) ) ) ) ) ) ) ) ).

% mergesort_by_rel_merge.pelims
tff(fact_4519_remdups__adj__altdef,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( remdups_adj(A,Xs) = Ys )
    <=> ? [F6: fun(nat,nat)] :
          ( order_mono(nat,nat,F6)
          & ( aa(set(nat),set(nat),image2(nat,nat,F6),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xs))) = set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(A),nat,size_size(list(A)),Ys)) )
          & ! [I3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),Xs))
             => ( aa(nat,A,nth(A,Xs),I3) = aa(nat,A,nth(A,Ys),aa(nat,nat,F6,I3)) ) )
          & ! [I3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I3),one_one(nat))),aa(list(A),nat,size_size(list(A)),Xs))
             => ( ( aa(nat,A,nth(A,Xs),I3) = aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I3),one_one(nat))) )
              <=> ( aa(nat,nat,F6,I3) = aa(nat,nat,F6,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I3),one_one(nat))) ) ) ) ) ) ).

% remdups_adj_altdef
tff(fact_4520_ran__nth__set__encoding__conv,axiom,
    ! [A: $tType,L: list(A)] : ran(nat,A,aTP_Lamp_xr(list(A),fun(nat,option(A)),L)) = aa(list(A),set(A),set2(A),L) ).

% ran_nth_set_encoding_conv
tff(fact_4521_ran__empty,axiom,
    ! [B: $tType,A: $tType] : ran(B,A,aTP_Lamp_xs(B,option(A))) = bot_bot(set(A)) ).

% ran_empty
tff(fact_4522_map__update__eta__repair_I2_J,axiom,
    ! [A: $tType,B: $tType,M: fun(B,option(A)),K: B,V: A] :
      ( ( aa(B,option(A),M,K) = none(A) )
     => ( ran(B,A,aa(A,fun(B,option(A)),aa(B,fun(A,fun(B,option(A))),aTP_Lamp_xt(fun(B,option(A)),fun(B,fun(A,fun(B,option(A)))),M),K),V)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),V),ran(B,A,M)) ) ) ).

% map_update_eta_repair(2)
tff(fact_4523_ran__map__option,axiom,
    ! [A: $tType,C: $tType,B: $tType,F: fun(C,A),M: fun(B,option(C))] : ran(B,A,aa(fun(B,option(C)),fun(B,option(A)),aTP_Lamp_xu(fun(C,A),fun(fun(B,option(C)),fun(B,option(A))),F),M)) = aa(set(C),set(A),image2(C,A,F),ran(B,C,M)) ).

% ran_map_option
tff(fact_4524_listrel__Cons,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(B,A)),X: B,Xs: list(B)] : aa(set(list(B)),set(list(A)),image(list(B),list(A),listrel(B,A,R2)),aa(set(list(B)),set(list(B)),aa(list(B),fun(set(list(B)),set(list(B))),insert2(list(B)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Xs)),bot_bot(set(list(B))))) = set_Cons(A,aa(set(B),set(A),image(B,A,R2),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),X),bot_bot(set(B)))),aa(set(list(B)),set(list(A)),image(list(B),list(A),listrel(B,A,R2)),aa(set(list(B)),set(list(B)),aa(list(B),fun(set(list(B)),set(list(B))),insert2(list(B)),Xs),bot_bot(set(list(B)))))) ).

% listrel_Cons
tff(fact_4525_upto__aux__rec,axiom,
    ! [I: int,J: int,Js: list(int)] :
      upto_aux(I,J,Js) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less(int),J),I),Js,upto_aux(I,aa(int,int,aa(int,fun(int,int),minus_minus(int),J),one_one(int)),aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),J),Js))) ).

% upto_aux_rec
tff(fact_4526_slice__Cons,axiom,
    ! [A: $tType,Begin: nat,End: nat,X: A,Xs: list(A)] :
      slice(A,Begin,End,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = $ite(
        ( ( Begin = zero_zero(nat) )
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),End) ),
        aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),slice(A,Begin,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),End),one_one(nat)),Xs)),
        slice(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Begin),one_one(nat)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),End),one_one(nat)),Xs) ) ).

% slice_Cons
tff(fact_4527_horner__sum__eq__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(A)
     => ! [F: 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),F),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_xv(fun(B,A),fun(A,fun(list(B),fun(nat,A))),F),A3),Xs)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(B),nat,size_size(list(B)),Xs))) ) ).

% horner_sum_eq_sum
tff(fact_4528_listrel__rtrancl__refl,axiom,
    ! [A: $tType,Xs: list(A),R2: set(product_prod(A,A))] : aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Xs)),listrel(A,A,transitive_rtrancl(A,R2))) ).

% listrel_rtrancl_refl
tff(fact_4529_horner__sum__simps_I2_J,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_0(A)
     => ! [F: 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),F),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,F,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),F),A3),Xs))) ) ).

% horner_sum_simps(2)
tff(fact_4530_listrel__Nil,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(B,A))] : aa(set(list(B)),set(list(A)),image(list(B),list(A),listrel(B,A,R2)),aa(set(list(B)),set(list(B)),aa(list(B),fun(set(list(B)),set(list(B))),insert2(list(B)),nil(B)),bot_bot(set(list(B))))) = aa(set(list(A)),set(list(A)),aa(list(A),fun(set(list(A)),set(list(A))),insert2(list(A)),nil(A)),bot_bot(set(list(A)))) ).

% listrel_Nil
tff(fact_4531_listrel__Nil2,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),R2: set(product_prod(A,B))] :
      ( aa(set(product_prod(list(A),list(B))),$o,member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xs),nil(B))),listrel(A,B,R2))
     => ( Xs = nil(A) ) ) ).

% listrel_Nil2
tff(fact_4532_listrel__Nil1,axiom,
    ! [A: $tType,B: $tType,Xs: list(B),R2: set(product_prod(A,B))] :
      ( aa(set(product_prod(list(A),list(B))),$o,member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),nil(A)),Xs)),listrel(A,B,R2))
     => ( Xs = nil(B) ) ) ).

% listrel_Nil1
tff(fact_4533_listrel_ONil,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,B))] : aa(set(product_prod(list(A),list(B))),$o,member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),nil(A)),nil(B))),listrel(A,B,R2)) ).

% listrel.Nil
tff(fact_4534_listrel__eq__len,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),R2: set(product_prod(A,B))] :
      ( aa(set(product_prod(list(A),list(B))),$o,member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xs),Ys)),listrel(A,B,R2))
     => ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) ) ) ).

% listrel_eq_len
tff(fact_4535_listrel__rtrancl__trans,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),R2: set(product_prod(A,A)),Zs: list(A)] :
      ( aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),listrel(A,A,transitive_rtrancl(A,R2)))
     => ( aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Ys),Zs)),listrel(A,A,transitive_rtrancl(A,R2)))
       => aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Zs)),listrel(A,A,transitive_rtrancl(A,R2))) ) ) ).

% listrel_rtrancl_trans
tff(fact_4536_listrel_OCons,axiom,
    ! [B: $tType,A: $tType,X: A,Y: B,R2: set(product_prod(A,B)),Xs: list(A),Ys: list(B)] :
      ( aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y)),R2)
     => ( aa(set(product_prod(list(A),list(B))),$o,member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xs),Ys)),listrel(A,B,R2))
       => aa(set(product_prod(list(A),list(B))),$o,member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),Ys))),listrel(A,B,R2)) ) ) ).

% listrel.Cons
tff(fact_4537_listrel__Cons1,axiom,
    ! [B: $tType,A: $tType,Y: A,Ys: list(A),Xs: list(B),R2: set(product_prod(A,B))] :
      ( aa(set(product_prod(list(A),list(B))),$o,member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)),Xs)),listrel(A,B,R2))
     => ~ ! [Y2: B,Ys2: list(B)] :
            ( ( Xs = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y2),Ys2) )
           => ( aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Y),Y2)),R2)
             => ~ aa(set(product_prod(list(A),list(B))),$o,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),Ys2)),listrel(A,B,R2)) ) ) ) ).

% listrel_Cons1
tff(fact_4538_listrel__Cons2,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Y: B,Ys: list(B),R2: set(product_prod(A,B))] :
      ( aa(set(product_prod(list(A),list(B))),$o,member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xs),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),Ys))),listrel(A,B,R2))
     => ~ ! [X2: A,Xs3: list(A)] :
            ( ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs3) )
           => ( aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Y)),R2)
             => ~ aa(set(product_prod(list(A),list(B))),$o,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),Ys)),listrel(A,B,R2)) ) ) ) ).

% listrel_Cons2
tff(fact_4539_listrel_Osimps,axiom,
    ! [B: $tType,A: $tType,A1: list(A),A22: list(B),R2: set(product_prod(A,B))] :
      ( aa(set(product_prod(list(A),list(B))),$o,member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),A1),A22)),listrel(A,B,R2))
    <=> ( ( ( A1 = nil(A) )
          & ( A22 = nil(B) ) )
        | ? [X4: A,Y3: B,Xs4: list(A),Ys3: list(B)] :
            ( ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs4) )
            & ( A22 = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y3),Ys3) )
            & aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Y3)),R2)
            & aa(set(product_prod(list(A),list(B))),$o,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)),Xs4),Ys3)),listrel(A,B,R2)) ) ) ) ).

% listrel.simps
tff(fact_4540_listrel_Ocases,axiom,
    ! [B: $tType,A: $tType,A1: list(A),A22: list(B),R2: set(product_prod(A,B))] :
      ( aa(set(product_prod(list(A),list(B))),$o,member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),A1),A22)),listrel(A,B,R2))
     => ( ( ( A1 = nil(A) )
         => ( A22 != nil(B) ) )
       => ~ ! [X2: A,Y2: B,Xs3: list(A)] :
              ( ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs3) )
             => ! [Ys2: list(B)] :
                  ( ( A22 = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y2),Ys2) )
                 => ( aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Y2)),R2)
                   => ~ aa(set(product_prod(list(A),list(B))),$o,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),Ys2)),listrel(A,B,R2)) ) ) ) ) ) ).

% listrel.cases
tff(fact_4541_listrel__iff__nth,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys: list(B),R2: set(product_prod(A,B))] :
      ( aa(set(product_prod(list(A),list(B))),$o,member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xs),Ys)),listrel(A,B,R2))
    <=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
        & ! [N2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N2),aa(list(A),nat,size_size(list(A)),Xs))
           => aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(nat,A,nth(A,Xs),N2)),aa(nat,B,nth(B,Ys),N2))),R2) ) ) ) ).

% listrel_iff_nth
tff(fact_4542_upto_Opsimps,axiom,
    ! [I: int,J: int] :
      ( accp(product_prod(int,int),upto_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),I),J))
     => ( upto(I,J) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I),J),aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),I),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I),one_one(int)),J)),nil(int)) ) ) ).

% upto.psimps
tff(fact_4543_upto_Opelims,axiom,
    ! [X: int,Xa: int,Y: list(int)] :
      ( ( upto(X,Xa) = Y )
     => ( accp(product_prod(int,int),upto_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),X),Xa))
       => ~ ( ( Y = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),X),Xa),aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),X),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),X),one_one(int)),Xa)),nil(int)) )
           => ~ accp(product_prod(int,int),upto_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),X),Xa)) ) ) ) ).

% upto.pelims
tff(fact_4544_upto__rec__numeral_I4_J,axiom,
    ! [M: num,N: num] :
      upto(aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))),aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),one_one(int)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N)))),nil(int)) ).

% upto_rec_numeral(4)
tff(fact_4545_upto__rec__numeral_I3_J,axiom,
    ! [M: num,N: num] :
      upto(aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M)),aa(num,int,numeral_numeral(int),N)) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),aa(num,int,numeral_numeral(int),N)),aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),one_one(int)),aa(num,int,numeral_numeral(int),N))),nil(int)) ).

% upto_rec_numeral(3)
tff(fact_4546_length__upto,axiom,
    ! [I: int,J: int] : aa(list(int),nat,size_size(list(int)),upto(I,J)) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),J),I)),one_one(int))) ).

% length_upto
tff(fact_4547_upto__rec__numeral_I1_J,axiom,
    ! [M: num,N: num] :
      upto(aa(num,int,numeral_numeral(int),M),aa(num,int,numeral_numeral(int),N)) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(num,int,numeral_numeral(int),M)),aa(num,int,numeral_numeral(int),N)),aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),aa(num,int,numeral_numeral(int),M)),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,numeral_numeral(int),M)),one_one(int)),aa(num,int,numeral_numeral(int),N))),nil(int)) ).

% upto_rec_numeral(1)
tff(fact_4548_upto__rec__numeral_I2_J,axiom,
    ! [M: num,N: num] :
      upto(aa(num,int,numeral_numeral(int),M),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))),aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),aa(num,int,numeral_numeral(int),M)),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,numeral_numeral(int),M)),one_one(int)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N)))),nil(int)) ).

% upto_rec_numeral(2)
tff(fact_4549_atLeastLessThan__upto,axiom,
    ! [I: int,J: int] : set_or7035219750837199246ssThan(int,I,J) = aa(list(int),set(int),set2(int),upto(I,aa(int,int,aa(int,fun(int,int),minus_minus(int),J),one_one(int)))) ).

% atLeastLessThan_upto
tff(fact_4550_upto_Oelims,axiom,
    ! [X: int,Xa: int,Y: list(int)] :
      ( ( upto(X,Xa) = Y )
     => ( Y = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),X),Xa),aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),X),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),X),one_one(int)),Xa)),nil(int)) ) ) ).

% upto.elims
tff(fact_4551_upto_Osimps,axiom,
    ! [I: int,J: int] :
      upto(I,J) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I),J),aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),I),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I),one_one(int)),J)),nil(int)) ).

% upto.simps
tff(fact_4552_upto__rec1,axiom,
    ! [I: int,J: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I),J)
     => ( upto(I,J) = aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),I),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I),one_one(int)),J)) ) ) ).

% upto_rec1
tff(fact_4553_greaterThanLessThan__upto,axiom,
    ! [I: int,J: int] : set_or5935395276787703475ssThan(int,I,J) = aa(list(int),set(int),set2(int),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I),one_one(int)),aa(int,int,aa(int,fun(int,int),minus_minus(int),J),one_one(int)))) ).

% greaterThanLessThan_upto
tff(fact_4554_sorted__in__between,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [I: nat,J: nat,L: list(A),X: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),zero_zero(nat)),I)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),L))
             => ( sorted_wrt(A,ord_less_eq(A),L)
               => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,nth(A,L),I)),X)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),aa(nat,A,nth(A,L),J))
                   => ~ ! [K2: nat] :
                          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),K2)
                         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K2),J)
                           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,nth(A,L),K2)),X)
                             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),aa(nat,A,nth(A,L),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K2),one_one(nat)))) ) ) ) ) ) ) ) ) ) ) ).

% sorted_in_between
tff(fact_4555_part__code_I2_J,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F: fun(A,B),Pivot: B,X: A,Xs: list(A)] : linorder_part(A,B,F,Pivot,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(product_prod(list(A),product_prod(list(A),list(A))),product_prod(list(A),product_prod(list(A),list(A))),aa(fun(list(A),fun(product_prod(list(A),list(A)),product_prod(list(A),product_prod(list(A),list(A))))),fun(product_prod(list(A),product_prod(list(A),list(A))),product_prod(list(A),product_prod(list(A),list(A)))),product_case_prod(list(A),product_prod(list(A),list(A)),product_prod(list(A),product_prod(list(A),list(A)))),aa(A,fun(list(A),fun(product_prod(list(A),list(A)),product_prod(list(A),product_prod(list(A),list(A))))),aa(B,fun(A,fun(list(A),fun(product_prod(list(A),list(A)),product_prod(list(A),product_prod(list(A),list(A)))))),aTP_Lamp_xx(fun(A,B),fun(B,fun(A,fun(list(A),fun(product_prod(list(A),list(A)),product_prod(list(A),product_prod(list(A),list(A))))))),F),Pivot),X)),linorder_part(A,B,F,Pivot,Xs)) ) ).

% part_code(2)
tff(fact_4556_sorted__list__of__set__nonempty,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( aa(set(A),list(A),linord4507533701916653071of_set(A),A4) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(set(A),A,lattic643756798350308766er_Min(A),A4)),aa(set(A),list(A),linord4507533701916653071of_set(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),aa(set(A),A,lattic643756798350308766er_Min(A),A4)),bot_bot(set(A)))))) ) ) ) ) ).

% sorted_list_of_set_nonempty
tff(fact_4557_Un__set__drop__extend,axiom,
    ! [A: $tType,J: nat,L: list(set(A))] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(set(A)),nat,size_size(list(set(A))),L))
       => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(nat,set(A),nth(set(A),L),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),aa(nat,nat,suc,zero_zero(nat))))),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(list(set(A)),set(set(A)),set2(set(A)),drop(set(A),J,L)))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(list(set(A)),set(set(A)),set2(set(A)),drop(set(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),aa(nat,nat,suc,zero_zero(nat))),L))) ) ) ) ).

% Un_set_drop_extend
tff(fact_4558_sorted__list__of__set_Osorted__key__list__of__set__empty,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( aa(set(A),list(A),linord4507533701916653071of_set(A),bot_bot(set(A))) = nil(A) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_empty
tff(fact_4559_sorted__list__of__set_Osorted__key__list__of__set__eq__Nil__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( aa(set(A),list(A),linord4507533701916653071of_set(A),A4) = nil(A) )
          <=> ( A4 = bot_bot(set(A)) ) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_eq_Nil_iff
tff(fact_4560_drop__Cons__numeral,axiom,
    ! [A: $tType,V: num,X: A,Xs: list(A)] : drop(A,aa(num,nat,numeral_numeral(nat),V),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = drop(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(num,nat,numeral_numeral(nat),V)),one_one(nat)),Xs) ).

% drop_Cons_numeral
tff(fact_4561_sorted__wrt__true,axiom,
    ! [A: $tType,Xs: list(A)] : sorted_wrt(A,aTP_Lamp_xy(A,fun(A,$o)),Xs) ).

% sorted_wrt_true
tff(fact_4562_drop__Cons,axiom,
    ! [A: $tType,N: nat,X: A,Xs: list(A)] : drop(A,N,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(nat,list(A),aa(fun(nat,list(A)),fun(nat,list(A)),aa(list(A),fun(fun(nat,list(A)),fun(nat,list(A))),case_nat(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),aTP_Lamp_xz(list(A),fun(nat,list(A)),Xs)),N) ).

% drop_Cons
tff(fact_4563_sorted__wrt01,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,fun(A,$o))] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat))
     => sorted_wrt(A,P,Xs) ) ).

% sorted_wrt01
tff(fact_4564_sorted01,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat))
         => sorted_wrt(A,ord_less_eq(A),Xs) ) ) ).

% sorted01
tff(fact_4565_drop__Cons_H,axiom,
    ! [A: $tType,N: nat,X: A,Xs: list(A)] :
      drop(A,N,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = $ite(N = zero_zero(nat),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_4566_part__code_I1_J,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F: fun(A,B),Pivot: B] : linorder_part(A,B,F,Pivot,nil(A)) = aa(product_prod(list(A),list(A)),product_prod(list(A),product_prod(list(A),list(A))),aa(list(A),fun(product_prod(list(A),list(A)),product_prod(list(A),product_prod(list(A),list(A)))),product_Pair(list(A),product_prod(list(A),list(A))),nil(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A))) ) ).

% part_code(1)
tff(fact_4567_sorted__list__of__set_Osorted__key__list__of__set__remove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),X: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),list(A),linord4507533701916653071of_set(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))))) = remove1(A,X,aa(set(A),list(A),linord4507533701916653071of_set(A),A4)) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_remove
tff(fact_4568_sorted__find__Min,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),P: fun(A,$o)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => ( ? [X3: A] :
                ( aa(set(A),$o,member(A,X3),aa(list(A),set(A),set2(A),Xs))
                & aa(A,$o,P,X3) )
           => ( find(A,P,Xs) = aa(A,option(A),some(A),aa(set(A),A,lattic643756798350308766er_Min(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_ya(list(A),fun(fun(A,$o),fun(A,$o)),Xs),P)))) ) ) ) ) ).

% sorted_find_Min
tff(fact_4569_extract__Cons__code,axiom,
    ! [A: $tType,P: fun(A,$o),X: A,Xs: list(A)] :
      extract(A,P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = $ite(aa(A,$o,P,X),aa(product_prod(list(A),product_prod(A,list(A))),option(product_prod(list(A),product_prod(A,list(A)))),some(product_prod(list(A),product_prod(A,list(A)))),aa(product_prod(A,list(A)),product_prod(list(A),product_prod(A,list(A))),aa(list(A),fun(product_prod(A,list(A)),product_prod(list(A),product_prod(A,list(A)))),product_Pair(list(A),product_prod(A,list(A))),nil(A)),aa(list(A),product_prod(A,list(A)),aa(A,fun(list(A),product_prod(A,list(A))),product_Pair(A,list(A)),X),Xs))),case_option(option(product_prod(list(A),product_prod(A,list(A)))),product_prod(list(A),product_prod(A,list(A))),none(product_prod(list(A),product_prod(A,list(A)))),aa(fun(list(A),fun(product_prod(A,list(A)),option(product_prod(list(A),product_prod(A,list(A)))))),fun(product_prod(list(A),product_prod(A,list(A))),option(product_prod(list(A),product_prod(A,list(A))))),product_case_prod(list(A),product_prod(A,list(A)),option(product_prod(list(A),product_prod(A,list(A))))),aTP_Lamp_yc(A,fun(list(A),fun(product_prod(A,list(A)),option(product_prod(list(A),product_prod(A,list(A)))))),X)),extract(A,P,Xs))) ).

% extract_Cons_code
tff(fact_4570_set__update__distinct,axiom,
    ! [A: $tType,Xs: list(A),N: nat,X: A] :
      ( distinct(A,Xs)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),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),aa(A,fun(set(A),set(A)),insert2(A),X),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),aa(nat,A,nth(A,Xs),N)),bot_bot(set(A))))) ) ) ) ).

% set_update_distinct
tff(fact_4571_nth__enumerate__eq,axiom,
    ! [A: $tType,M: nat,Xs: list(A),N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(nat,product_prod(nat,A),nth(product_prod(nat,A),enumerate(A,N,Xs)),M) = aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M)),aa(nat,A,nth(A,Xs),M)) ) ) ).

% nth_enumerate_eq
tff(fact_4572_enumerate__simps_I2_J,axiom,
    ! [A: $tType,N: nat,X: A,Xs: list(A)] : enumerate(A,N,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(list(product_prod(nat,A)),list(product_prod(nat,A)),aa(product_prod(nat,A),fun(list(product_prod(nat,A)),list(product_prod(nat,A))),cons(product_prod(nat,A)),aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),N),X)),enumerate(A,aa(nat,nat,suc,N),Xs)) ).

% enumerate_simps(2)
tff(fact_4573_list__update_Osimps_I2_J,axiom,
    ! [A: $tType,X: A,Xs: list(A),I: nat,V: A] : list_update(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs),I,V) = aa(nat,list(A),aa(fun(nat,list(A)),fun(nat,list(A)),aa(list(A),fun(fun(nat,list(A)),fun(nat,list(A))),case_nat(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V),Xs)),aa(A,fun(nat,list(A)),aa(list(A),fun(A,fun(nat,list(A))),aTP_Lamp_yd(A,fun(list(A),fun(A,fun(nat,list(A)))),X),Xs),V)),I) ).

% list_update.simps(2)
tff(fact_4574_distinct__list__update,axiom,
    ! [A: $tType,Xs: list(A),A3: A,I: nat] :
      ( distinct(A,Xs)
     => ( ~ aa(set(A),$o,member(A,A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),aa(nat,A,nth(A,Xs),I)),bot_bot(set(A)))))
       => distinct(A,list_update(A,Xs,I,A3)) ) ) ).

% distinct_list_update
tff(fact_4575_horner__sum__eq__sum__funpow,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_0(A)
     => ! [F: 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),F),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_ye(fun(B,A),fun(A,fun(list(B),fun(nat,A))),F),A3),Xs)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(B),nat,size_size(list(B)),Xs))) ) ).

% horner_sum_eq_sum_funpow
tff(fact_4576_extract__SomeE,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A),Ys: list(A),Y: A,Zs: list(A)] :
      ( ( extract(A,P,Xs) = aa(product_prod(list(A),product_prod(A,list(A))),option(product_prod(list(A),product_prod(A,list(A)))),some(product_prod(list(A),product_prod(A,list(A)))),aa(product_prod(A,list(A)),product_prod(list(A),product_prod(A,list(A))),aa(list(A),fun(product_prod(A,list(A)),product_prod(list(A),product_prod(A,list(A)))),product_Pair(list(A),product_prod(A,list(A))),Ys),aa(list(A),product_prod(A,list(A)),aa(A,fun(list(A),product_prod(A,list(A))),product_Pair(A,list(A)),Y),Zs))) )
     => ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Zs)) )
        & aa(A,$o,P,Y)
        & ~ ? [X3: A] :
              ( aa(set(A),$o,member(A,X3),aa(list(A),set(A),set2(A),Ys))
              & aa(A,$o,P,X3) ) ) ) ).

% extract_SomeE
tff(fact_4577_extract__Some__iff,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A),Ys: list(A),Y: A,Zs: list(A)] :
      ( ( extract(A,P,Xs) = aa(product_prod(list(A),product_prod(A,list(A))),option(product_prod(list(A),product_prod(A,list(A)))),some(product_prod(list(A),product_prod(A,list(A)))),aa(product_prod(A,list(A)),product_prod(list(A),product_prod(A,list(A))),aa(list(A),fun(product_prod(A,list(A)),product_prod(list(A),product_prod(A,list(A)))),product_Pair(list(A),product_prod(A,list(A))),Ys),aa(list(A),product_prod(A,list(A)),aa(A,fun(list(A),product_prod(A,list(A))),product_Pair(A,list(A)),Y),Zs))) )
    <=> ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Zs)) )
        & aa(A,$o,P,Y)
        & ~ ? [X4: A] :
              ( aa(set(A),$o,member(A,X4),aa(list(A),set(A),set2(A),Ys))
              & aa(A,$o,P,X4) ) ) ) ).

% extract_Some_iff
tff(fact_4578_relpowp__1,axiom,
    ! [A: $tType,P: fun(A,fun(A,$o))] : compow(fun(A,fun(A,$o)),one_one(nat),P) = P ).

% relpowp_1
tff(fact_4579_surj__fn,axiom,
    ! [A: $tType,F: fun(A,A),N: nat] :
      ( ( aa(set(A),set(A),image2(A,A,F),top_top(set(A))) = top_top(set(A)) )
     => ( aa(set(A),set(A),image2(A,A,compow(fun(A,A),N,F)),top_top(set(A))) = top_top(set(A)) ) ) ).

% surj_fn
tff(fact_4580_distinct__append,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( distinct(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys))
    <=> ( distinct(A,Xs)
        & distinct(A,Ys)
        & ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys)) = bot_bot(set(A)) ) ) ) ).

% distinct_append
tff(fact_4581_append_Omonoid__axioms,axiom,
    ! [A: $tType] : monoid(list(A),append(A),nil(A)) ).

% append.monoid_axioms
tff(fact_4582_comp__fun__commute_Ocomp__fun__commute__funpow,axiom,
    ! [B: $tType,A: $tType,F: fun(A,fun(B,B)),G: fun(A,nat)] :
      ( finite6289374366891150609ommute(A,B,F)
     => finite6289374366891150609ommute(A,B,aa(fun(A,nat),fun(A,fun(B,B)),aTP_Lamp_yf(fun(A,fun(B,B)),fun(fun(A,nat),fun(A,fun(B,B))),F),G)) ) ).

% comp_fun_commute.comp_fun_commute_funpow
tff(fact_4583_comp__fun__commute__on_Ocomp__fun__commute__on__funpow,axiom,
    ! [B: $tType,A: $tType,S: set(A),F: fun(A,fun(B,B)),G: fun(A,nat)] :
      ( finite4664212375090638736ute_on(A,B,S,F)
     => finite4664212375090638736ute_on(A,B,S,aa(fun(A,nat),fun(A,fun(B,B)),aTP_Lamp_yf(fun(A,fun(B,B)),fun(fun(A,nat),fun(A,fun(B,B))),F),G)) ) ).

% comp_fun_commute_on.comp_fun_commute_on_funpow
tff(fact_4584_funpow__times__power,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [F: fun(A,nat),X: A] : compow(fun(A,A),aa(A,nat,F,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,F,X))) ) ).

% funpow_times_power
tff(fact_4585_set__union__code,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys)) = aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) ).

% set_union_code
tff(fact_4586_relpowp__relpow__eq,axiom,
    ! [A: $tType,N: nat,R: set(product_prod(A,A)),X3: A,Xa3: A] :
      ( aa(A,$o,aa(A,fun(A,$o),compow(fun(A,fun(A,$o)),N,aTP_Lamp_ux(set(product_prod(A,A)),fun(A,fun(A,$o)),R)),X3),Xa3)
    <=> aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Xa3)),compow(set(product_prod(A,A)),N,R)) ) ).

% relpowp_relpow_eq
tff(fact_4587_Kleene__iter__lpfp,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [F: fun(A,A),P3: A,K: nat] :
          ( order_mono(A,A,F)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,F,P3)),P3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,compow(fun(A,A),K,F),bot_bot(A))),P3) ) ) ) ).

% Kleene_iter_lpfp
tff(fact_4588_Kleene__iter__gpfp,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [F: fun(A,A),P3: A,K: nat] :
          ( order_mono(A,A,F)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),P3),aa(A,A,F,P3))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),P3),aa(A,A,compow(fun(A,A),K,F),top_top(A))) ) ) ) ).

% Kleene_iter_gpfp
tff(fact_4589_relpowp__bot,axiom,
    ! [A: $tType,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
     => ( compow(fun(A,fun(A,$o)),N,bot_bot(fun(A,fun(A,$o)))) = bot_bot(fun(A,fun(A,$o))) ) ) ).

% relpowp_bot
tff(fact_4590_of__nat__def,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [N: nat] : aa(nat,A,semiring_1_of_nat(A),N) = aa(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_4591_numeral__add__unfold__funpow,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [K: num,A3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),K)),A3) = aa(A,A,compow(fun(A,A),aa(num,nat,numeral_numeral(nat),K),aa(A,fun(A,A),plus_plus(A),one_one(A))),A3) ) ).

% numeral_add_unfold_funpow
tff(fact_4592_mono__funpow,axiom,
    ! [A: $tType] :
      ( ( lattice(A)
        & order_bot(A) )
     => ! [Q: fun(A,A)] :
          ( order_mono(A,A,Q)
         => order_mono(nat,A,aTP_Lamp_yg(fun(A,A),fun(nat,A),Q)) ) ) ).

% mono_funpow
tff(fact_4593_funpow__decreasing,axiom,
    ! [A: $tType] :
      ( ( lattice(A)
        & order_bot(A) )
     => ! [M: nat,N: nat,F: fun(A,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),N)
         => ( order_mono(A,A,F)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,compow(fun(A,A),M,F),bot_bot(A))),aa(A,A,compow(fun(A,A),N,F),bot_bot(A))) ) ) ) ).

% funpow_decreasing
tff(fact_4594_funpow__increasing,axiom,
    ! [A: $tType] :
      ( ( lattice(A)
        & order_top(A) )
     => ! [M: nat,N: nat,F: fun(A,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),N)
         => ( order_mono(A,A,F)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,compow(fun(A,A),N,F),top_top(A))),aa(A,A,compow(fun(A,A),M,F),top_top(A))) ) ) ) ).

% funpow_increasing
tff(fact_4595_upto__split2,axiom,
    ! [I: int,J: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I),J)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),J),K)
       => ( upto(I,K) = aa(list(int),list(int),aa(list(int),fun(list(int),list(int)),append(int),upto(I,J)),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),J),one_one(int)),K)) ) ) ) ).

% upto_split2
tff(fact_4596_upto__split1,axiom,
    ! [I: int,J: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I),J)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),J),K)
       => ( upto(I,K) = aa(list(int),list(int),aa(list(int),fun(list(int),list(int)),append(int),upto(I,aa(int,int,aa(int,fun(int,int),minus_minus(int),J),one_one(int)))),upto(J,K)) ) ) ) ).

% upto_split1
tff(fact_4597_numeral__unfold__funpow,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [K: num] : aa(num,A,numeral_numeral(A),K) = aa(A,A,compow(fun(A,A),aa(num,nat,numeral_numeral(nat),K),aa(A,fun(A,A),plus_plus(A),one_one(A))),zero_zero(A)) ) ).

% numeral_unfold_funpow
tff(fact_4598_slice__prepend,axiom,
    ! [A: $tType,I: nat,K: nat,Xs: list(A),Ys: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),K)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),aa(list(A),nat,size_size(list(A)),Xs))
       => $let(
            p: nat,
            p:= aa(list(A),nat,size_size(list(A)),Ys),
            slice(A,I,K,Xs) = slice(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),p),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),p),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Xs)) ) ) ) ).

% slice_prepend
tff(fact_4599_horner__sum__append,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(A)
     => ! [F: 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),F),A3),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Xs),Ys)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(list(B),A,aa(A,fun(list(B),A),aa(fun(B,A),fun(A,fun(list(B),A)),groups4207007520872428315er_sum(B,A),F),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),F),A3),Ys))) ) ).

% horner_sum_append
tff(fact_4600_upto__split3,axiom,
    ! [I: int,J: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I),J)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),J),K)
       => ( upto(I,K) = aa(list(int),list(int),aa(list(int),fun(list(int),list(int)),append(int),upto(I,aa(int,int,aa(int,fun(int,int),minus_minus(int),J),one_one(int)))),aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),J),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),J),one_one(int)),K))) ) ) ) ).

% upto_split3
tff(fact_4601_upto__rec2,axiom,
    ! [I: int,J: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I),J)
     => ( upto(I,J) = aa(list(int),list(int),aa(list(int),fun(list(int),list(int)),append(int),upto(I,aa(int,int,aa(int,fun(int,int),minus_minus(int),J),one_one(int)))),aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),J),nil(int))) ) ) ).

% upto_rec2
tff(fact_4602_subset__eq__mset__impl_Oelims,axiom,
    ! [A: $tType,X: list(A),Xa: list(A),Y: option($o)] :
      ( ( subset_eq_mset_impl(A,X,Xa) = Y )
     => ( ( ( X = nil(A) )
         => ( Y != aa($o,option($o),some($o),Xa != nil(A)) ) )
       => ~ ! [X2: A,Xs3: list(A)] :
              ( ( X = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs3) )
             => ( Y != case_option(option($o),product_prod(list(A),product_prod(A,list(A))),none($o),aa(fun(list(A),fun(product_prod(A,list(A)),option($o))),fun(product_prod(list(A),product_prod(A,list(A))),option($o)),product_case_prod(list(A),product_prod(A,list(A)),option($o)),aTP_Lamp_yi(list(A),fun(list(A),fun(product_prod(A,list(A)),option($o))),Xs3)),extract(A,aa(A,fun(A,$o),fequal(A),X2),Xa)) ) ) ) ) ).

% subset_eq_mset_impl.elims
tff(fact_4603_Succ__def,axiom,
    ! [A: $tType,Kl: set(list(A)),Kl2: list(A)] : bNF_Greatest_Succ(A,Kl,Kl2) = aa(fun(A,$o),set(A),collect(A),aa(list(A),fun(A,$o),aTP_Lamp_yj(set(list(A)),fun(list(A),fun(A,$o)),Kl),Kl2)) ).

% Succ_def
tff(fact_4604_sort__key__by__quicksort__code,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F: fun(A,B),Xs: list(A)] : aa(list(A),list(A),linorder_sort_key(A,B,F),Xs) = aa(list(A),list(A),case_list(list(A),A,nil(A),aa(list(A),fun(A,fun(list(A),list(A))),aTP_Lamp_yo(fun(A,B),fun(list(A),fun(A,fun(list(A),list(A)))),F),Xs)),Xs) ) ).

% sort_key_by_quicksort_code
tff(fact_4605_sort__upto,axiom,
    ! [I: int,J: int] : aa(list(int),list(int),linorder_sort_key(int,int,aTP_Lamp_dl(int,int)),upto(I,J)) = upto(I,J) ).

% sort_upto
tff(fact_4606_list_Ocase__distrib,axiom,
    ! [B: $tType,A: $tType,C: $tType,H2: fun(B,A),F1: B,F22: fun(C,fun(list(C),B)),List: list(C)] : aa(B,A,H2,aa(list(C),B,case_list(B,C,F1,F22),List)) = aa(list(C),A,case_list(A,C,aa(B,A,H2,F1),aa(fun(C,fun(list(C),B)),fun(C,fun(list(C),A)),aTP_Lamp_yp(fun(B,A),fun(fun(C,fun(list(C),B)),fun(C,fun(list(C),A))),H2),F22)),List) ).

% list.case_distrib
tff(fact_4607_sort__key__const,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [C2: B,Xs: list(A)] : aa(list(A),list(A),linorder_sort_key(A,B,aTP_Lamp_ss(B,fun(A,B),C2)),Xs) = Xs ) ).

% sort_key_const
tff(fact_4608_sorted__sort,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] : sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),linorder_sort_key(A,A,aTP_Lamp_yq(A,A)),Xs)) ) ).

% sorted_sort
tff(fact_4609_sorted__sort__id,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => ( aa(list(A),list(A),linorder_sort_key(A,A,aTP_Lamp_yq(A,A)),Xs) = Xs ) ) ) ).

% sorted_sort_id
tff(fact_4610_sort__mergesort__by__rel,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( linorder_sort_key(A,A,aTP_Lamp_yq(A,A)) = mergesort_by_rel(A,ord_less_eq(A)) ) ) ).

% sort_mergesort_by_rel
tff(fact_4611_remdups__adj__Cons,axiom,
    ! [A: $tType,X: A,Xs: list(A)] : remdups_adj(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(list(A),list(A),case_list(list(A),A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)),aTP_Lamp_yr(A,fun(A,fun(list(A),list(A))),X)),remdups_adj(A,Xs)) ).

% remdups_adj_Cons
tff(fact_4612_subset__eq__mset__impl_Osimps_I2_J,axiom,
    ! [A: $tType,X: A,Xs: list(A),Ys: list(A)] : subset_eq_mset_impl(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs),Ys) = case_option(option($o),product_prod(list(A),product_prod(A,list(A))),none($o),aa(fun(list(A),fun(product_prod(A,list(A)),option($o))),fun(product_prod(list(A),product_prod(A,list(A))),option($o)),product_case_prod(list(A),product_prod(A,list(A)),option($o)),aTP_Lamp_yi(list(A),fun(list(A),fun(product_prod(A,list(A)),option($o))),Xs)),extract(A,aa(A,fun(A,$o),fequal(A),X),Ys)) ).

% subset_eq_mset_impl.simps(2)
tff(fact_4613_subset__eq__mset__impl_Opelims,axiom,
    ! [A: $tType,X: list(A),Xa: list(A),Y: option($o)] :
      ( ( subset_eq_mset_impl(A,X,Xa) = Y )
     => ( accp(product_prod(list(A),list(A)),subset751672762298770561pl_rel(A),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X),Xa))
       => ( ( ( X = nil(A) )
           => ( ( Y = aa($o,option($o),some($o),Xa != nil(A)) )
             => ~ accp(product_prod(list(A),list(A)),subset751672762298770561pl_rel(A),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),Xa)) ) )
         => ~ ! [X2: A,Xs3: list(A)] :
                ( ( X = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs3) )
               => ( ( Y = case_option(option($o),product_prod(list(A),product_prod(A,list(A))),none($o),aa(fun(list(A),fun(product_prod(A,list(A)),option($o))),fun(product_prod(list(A),product_prod(A,list(A))),option($o)),product_case_prod(list(A),product_prod(A,list(A)),option($o)),aTP_Lamp_yi(list(A),fun(list(A),fun(product_prod(A,list(A)),option($o))),Xs3)),extract(A,aa(A,fun(A,$o),fequal(A),X2),Xa)) )
                 => ~ accp(product_prod(list(A),list(A)),subset751672762298770561pl_rel(A),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs3)),Xa)) ) ) ) ) ) ).

% subset_eq_mset_impl.pelims
tff(fact_4614_sort__by__quicksort,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] : aa(list(A),list(A),linorder_sort_key(A,A,aTP_Lamp_yq(A,A)),Xs) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),linorder_sort_key(A,A,aTP_Lamp_yq(A,A)),aa(list(A),list(A),filter2(A,aTP_Lamp_ys(list(A),fun(A,$o),Xs)),Xs))),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),filter2(A,aTP_Lamp_yt(list(A),fun(A,$o),Xs)),Xs)),aa(list(A),list(A),linorder_sort_key(A,A,aTP_Lamp_yq(A,A)),aa(list(A),list(A),filter2(A,aa(A,fun(A,$o),ord_less(A),aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),Xs)))) ) ).

% sort_by_quicksort
tff(fact_4615_sort__key__by__quicksort,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F: fun(A,B),Xs: list(A)] : aa(list(A),list(A),linorder_sort_key(A,B,F),Xs) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),linorder_sort_key(A,B,F),aa(list(A),list(A),filter2(A,aa(list(A),fun(A,$o),aTP_Lamp_yu(fun(A,B),fun(list(A),fun(A,$o)),F),Xs)),Xs))),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),filter2(A,aa(list(A),fun(A,$o),aTP_Lamp_yv(fun(A,B),fun(list(A),fun(A,$o)),F),Xs)),Xs)),aa(list(A),list(A),linorder_sort_key(A,B,F),aa(list(A),list(A),filter2(A,aa(list(A),fun(A,$o),aTP_Lamp_yw(fun(A,B),fun(list(A),fun(A,$o)),F),Xs)),Xs)))) ) ).

% sort_key_by_quicksort
tff(fact_4616_filter__filter,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o),Xs: list(A)] : aa(list(A),list(A),filter2(A,P),aa(list(A),list(A),filter2(A,Q),Xs)) = aa(list(A),list(A),filter2(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_yx(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q)),Xs) ).

% filter_filter
tff(fact_4617_set__filter,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] : aa(list(A),set(A),set2(A),aa(list(A),list(A),filter2(A,P),Xs)) = aa(fun(A,$o),set(A),collect(A),aa(list(A),fun(A,$o),aTP_Lamp_yy(fun(A,$o),fun(list(A),fun(A,$o)),P),Xs)) ).

% set_filter
tff(fact_4618_sort__key__stable,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F: fun(A,B),K: B,Xs: list(A)] : aa(list(A),list(A),filter2(A,aa(B,fun(A,$o),aTP_Lamp_yz(fun(A,B),fun(B,fun(A,$o)),F),K)),aa(list(A),list(A),linorder_sort_key(A,B,F),Xs)) = aa(list(A),list(A),filter2(A,aa(B,fun(A,$o),aTP_Lamp_yz(fun(A,B),fun(B,fun(A,$o)),F),K)),Xs) ) ).

% sort_key_stable
tff(fact_4619_partition__in__shuffles,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o)] : aa(set(list(A)),$o,member(list(A),Xs),shuffles(A,aa(list(A),list(A),filter2(A,P),Xs),aa(list(A),list(A),filter2(A,aTP_Lamp_aj(fun(A,$o),fun(A,$o),P)),Xs))) ).

% partition_in_shuffles
tff(fact_4620_removeAll__filter__not__eq,axiom,
    ! [A: $tType,X: A] : removeAll(A,X) = filter2(A,aa(A,fun(A,$o),aTP_Lamp_za(A,fun(A,$o)),X)) ).

% removeAll_filter_not_eq
tff(fact_4621_list_Odisc__eq__case_I2_J,axiom,
    ! [A: $tType,List: list(A)] :
      ( ( List != nil(A) )
    <=> aa(list(A),$o,case_list($o,A,$false,aTP_Lamp_zb(A,fun(list(A),$o))),List) ) ).

% list.disc_eq_case(2)
tff(fact_4622_list_Odisc__eq__case_I1_J,axiom,
    ! [A: $tType,List: list(A)] :
      ( ( List = nil(A) )
    <=> aa(list(A),$o,case_list($o,A,$true,aTP_Lamp_zc(A,fun(list(A),$o))),List) ) ).

% list.disc_eq_case(1)
tff(fact_4623_sum__length__filter__compl,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),filter2(A,P),Xs))),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),filter2(A,aTP_Lamp_aj(fun(A,$o),fun(A,$o),P)),Xs))) = aa(list(A),nat,size_size(list(A)),Xs) ).

% sum_length_filter_compl
tff(fact_4624_inter__set__filter,axiom,
    ! [A: $tType,A4: set(A),Xs: list(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),aa(list(A),set(A),set2(A),Xs)) = aa(list(A),set(A),set2(A),aa(list(A),list(A),filter2(A,aa(set(A),fun(A,$o),aTP_Lamp_ag(set(A),fun(A,$o)),A4)),Xs)) ).

% inter_set_filter
tff(fact_4625_sorted__same,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [G: fun(list(A),A),Xs: list(A)] : sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),filter2(A,aa(list(A),fun(A,$o),aTP_Lamp_zd(fun(list(A),A),fun(list(A),fun(A,$o)),G),Xs)),Xs)) ) ).

% sorted_same
tff(fact_4626_set__minus__filter__out,axiom,
    ! [A: $tType,Xs: list(A),Y: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y),bot_bot(set(A)))) = aa(list(A),set(A),set2(A),aa(list(A),list(A),filter2(A,aa(A,fun(A,$o),aTP_Lamp_ze(A,fun(A,$o)),Y)),Xs)) ).

% set_minus_filter_out
tff(fact_4627_filter__shuffles__disjoint2_I1_J,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),Zs: list(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys)) = bot_bot(set(A)) )
     => ( aa(set(list(A)),$o,member(list(A),Zs),shuffles(A,Xs,Ys))
       => ( aa(list(A),list(A),filter2(A,aTP_Lamp_zf(list(A),fun(A,$o),Ys)),Zs) = Ys ) ) ) ).

% filter_shuffles_disjoint2(1)
tff(fact_4628_filter__shuffles__disjoint2_I2_J,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),Zs: list(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys)) = bot_bot(set(A)) )
     => ( aa(set(list(A)),$o,member(list(A),Zs),shuffles(A,Xs,Ys))
       => ( aa(list(A),list(A),filter2(A,aTP_Lamp_zg(list(A),fun(A,$o),Ys)),Zs) = Xs ) ) ) ).

% filter_shuffles_disjoint2(2)
tff(fact_4629_filter__shuffles__disjoint1_I1_J,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),Zs: list(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys)) = bot_bot(set(A)) )
     => ( aa(set(list(A)),$o,member(list(A),Zs),shuffles(A,Xs,Ys))
       => ( aa(list(A),list(A),filter2(A,aTP_Lamp_zf(list(A),fun(A,$o),Xs)),Zs) = Xs ) ) ) ).

% filter_shuffles_disjoint1(1)
tff(fact_4630_filter__shuffles__disjoint1_I2_J,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),Zs: list(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys)) = bot_bot(set(A)) )
     => ( aa(set(list(A)),$o,member(list(A),Zs),shuffles(A,Xs,Ys))
       => ( aa(list(A),list(A),filter2(A,aTP_Lamp_zg(list(A),fun(A,$o),Xs)),Zs) = Ys ) ) ) ).

% filter_shuffles_disjoint1(2)
tff(fact_4631_length__filter__conv__card,axiom,
    ! [A: $tType,P3: fun(A,$o),Xs: list(A)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),filter2(A,P3),Xs)) = aa(set(nat),nat,finite_card(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(list(A),fun(nat,$o),aTP_Lamp_zh(fun(A,$o),fun(list(A),fun(nat,$o)),P3),Xs))) ).

% length_filter_conv_card
tff(fact_4632_distinct__length__filter,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o)] :
      ( distinct(A,Xs)
     => ( aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),filter2(A,P),Xs)) = aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(fun(A,$o),set(A),collect(A),P)),aa(list(A),set(A),set2(A),Xs))) ) ) ).

% distinct_length_filter
tff(fact_4633_part__def,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F: fun(A,B),Pivot: B,Xs: list(A)] : linorder_part(A,B,F,Pivot,Xs) = aa(product_prod(list(A),list(A)),product_prod(list(A),product_prod(list(A),list(A))),aa(list(A),fun(product_prod(list(A),list(A)),product_prod(list(A),product_prod(list(A),list(A)))),product_Pair(list(A),product_prod(list(A),list(A))),aa(list(A),list(A),filter2(A,aa(B,fun(A,$o),aTP_Lamp_zi(fun(A,B),fun(B,fun(A,$o)),F),Pivot)),Xs)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),filter2(A,aa(B,fun(A,$o),aTP_Lamp_yz(fun(A,B),fun(B,fun(A,$o)),F),Pivot)),Xs)),aa(list(A),list(A),filter2(A,aa(B,fun(A,$o),aTP_Lamp_zj(fun(A,B),fun(B,fun(A,$o)),F),Pivot)),Xs))) ) ).

% part_def
tff(fact_4634_Bleast__code,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),P: fun(A,$o)] : bleast(A,aa(list(A),set(A),set2(A),Xs),P) = aa(list(A),A,case_list(A,A,abort_Bleast(A,aa(list(A),set(A),set2(A),Xs),P),aTP_Lamp_zk(A,fun(list(A),A))),aa(list(A),list(A),filter2(A,P),aa(list(A),list(A),linorder_sort_key(A,A,aTP_Lamp_yq(A,A)),Xs))) ) ).

% Bleast_code
tff(fact_4635_shuffles_Opelims,axiom,
    ! [A: $tType,X: list(A),Xa: list(A),Y: set(list(A))] :
      ( ( shuffles(A,X,Xa) = Y )
     => ( accp(product_prod(list(A),list(A)),shuffles_rel(A),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X),Xa))
       => ( ( ( X = nil(A) )
           => ( ( Y = aa(set(list(A)),set(list(A)),aa(list(A),fun(set(list(A)),set(list(A))),insert2(list(A)),Xa),bot_bot(set(list(A)))) )
             => ~ accp(product_prod(list(A),list(A)),shuffles_rel(A),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),Xa)) ) )
         => ( ( ( Xa = nil(A) )
             => ( ( Y = aa(set(list(A)),set(list(A)),aa(list(A),fun(set(list(A)),set(list(A))),insert2(list(A)),X),bot_bot(set(list(A)))) )
               => ~ accp(product_prod(list(A),list(A)),shuffles_rel(A),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X),nil(A))) ) )
           => ~ ! [X2: A,Xs3: list(A)] :
                  ( ( X = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs3) )
                 => ! [Y2: A,Ys2: list(A)] :
                      ( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys2) )
                     => ( ( Y = aa(set(list(A)),set(list(A)),aa(set(list(A)),fun(set(list(A)),set(list(A))),sup_sup(set(list(A))),aa(set(list(A)),set(list(A)),image2(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2)),shuffles(A,Xs3,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys2)))),aa(set(list(A)),set(list(A)),image2(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2)),shuffles(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs3),Ys2))) )
                       => ~ accp(product_prod(list(A),list(A)),shuffles_rel(A),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs3)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys2))) ) ) ) ) ) ) ) ).

% shuffles.pelims
tff(fact_4636_shuffles_Opsimps_I3_J,axiom,
    ! [A: $tType,X: A,Xs: list(A),Y: A,Ys: list(A)] :
      ( accp(product_prod(list(A),list(A)),shuffles_rel(A),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)))
     => ( shuffles(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)) = aa(set(list(A)),set(list(A)),aa(set(list(A)),fun(set(list(A)),set(list(A))),sup_sup(set(list(A))),aa(set(list(A)),set(list(A)),image2(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X)),shuffles(A,Xs,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)))),aa(set(list(A)),set(list(A)),image2(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y)),shuffles(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs),Ys))) ) ) ).

% shuffles.psimps(3)
tff(fact_4637_shuffles_Opinduct,axiom,
    ! [A: $tType,A0: list(A),A1: list(A),P: fun(list(A),fun(list(A),$o))] :
      ( accp(product_prod(list(A),list(A)),shuffles_rel(A),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),A0),A1))
     => ( ! [Ys2: list(A)] :
            ( accp(product_prod(list(A),list(A)),shuffles_rel(A),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),Ys2))
           => aa(list(A),$o,aa(list(A),fun(list(A),$o),P,nil(A)),Ys2) )
       => ( ! [Xs3: list(A)] :
              ( accp(product_prod(list(A),list(A)),shuffles_rel(A),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs3),nil(A)))
             => aa(list(A),$o,aa(list(A),fun(list(A),$o),P,Xs3),nil(A)) )
         => ( ! [X2: A,Xs3: list(A),Y2: A,Ys2: list(A)] :
                ( accp(product_prod(list(A),list(A)),shuffles_rel(A),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs3)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys2)))
               => ( aa(list(A),$o,aa(list(A),fun(list(A),$o),P,Xs3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys2))
                 => ( aa(list(A),$o,aa(list(A),fun(list(A),$o),P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs3)),Ys2)
                   => aa(list(A),$o,aa(list(A),fun(list(A),$o),P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs3)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys2)) ) ) )
           => aa(list(A),$o,aa(list(A),fun(list(A),$o),P,A0),A1) ) ) ) ) ).

% shuffles.pinduct
tff(fact_4638_shuffles_Opsimps_I1_J,axiom,
    ! [A: $tType,Ys: list(A)] :
      ( accp(product_prod(list(A),list(A)),shuffles_rel(A),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),Ys))
     => ( shuffles(A,nil(A),Ys) = aa(set(list(A)),set(list(A)),aa(list(A),fun(set(list(A)),set(list(A))),insert2(list(A)),Ys),bot_bot(set(list(A)))) ) ) ).

% shuffles.psimps(1)
tff(fact_4639_shuffles_Opsimps_I2_J,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( accp(product_prod(list(A),list(A)),shuffles_rel(A),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),nil(A)))
     => ( shuffles(A,Xs,nil(A)) = aa(set(list(A)),set(list(A)),aa(list(A),fun(set(list(A)),set(list(A))),insert2(list(A)),Xs),bot_bot(set(list(A)))) ) ) ).

% shuffles.psimps(2)
tff(fact_4640_quicksort_Oelims,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: list(A),Y: list(A)] :
          ( ( aa(list(A),list(A),linorder_quicksort(A),X) = Y )
         => ( ( ( X = nil(A) )
             => ( Y != nil(A) ) )
           => ~ ! [X2: A,Xs3: list(A)] :
                  ( ( X = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs3) )
                 => ( Y != aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),linorder_quicksort(A),aa(list(A),list(A),filter2(A,aTP_Lamp_zl(A,fun(A,$o),X2)),Xs3))),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),nil(A))),aa(list(A),list(A),linorder_quicksort(A),aa(list(A),list(A),filter2(A,aa(A,fun(A,$o),ord_less_eq(A),X2)),Xs3)))) ) ) ) ) ) ).

% quicksort.elims
tff(fact_4641_quicksort_Osimps_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Xs: list(A)] : aa(list(A),list(A),linorder_quicksort(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),linorder_quicksort(A),aa(list(A),list(A),filter2(A,aTP_Lamp_zl(A,fun(A,$o),X)),Xs))),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A))),aa(list(A),list(A),linorder_quicksort(A),aa(list(A),list(A),filter2(A,aa(A,fun(A,$o),ord_less_eq(A),X)),Xs)))) ) ).

% quicksort.simps(2)
tff(fact_4642_splice_Opinduct,axiom,
    ! [A: $tType,A0: list(A),A1: list(A),P: fun(list(A),fun(list(A),$o))] :
      ( accp(product_prod(list(A),list(A)),splice_rel(A),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),A0),A1))
     => ( ! [Ys2: list(A)] :
            ( accp(product_prod(list(A),list(A)),splice_rel(A),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),Ys2))
           => aa(list(A),$o,aa(list(A),fun(list(A),$o),P,nil(A)),Ys2) )
       => ( ! [X2: A,Xs3: list(A),Ys2: list(A)] :
              ( accp(product_prod(list(A),list(A)),splice_rel(A),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs3)),Ys2))
             => ( aa(list(A),$o,aa(list(A),fun(list(A),$o),P,Ys2),Xs3)
               => aa(list(A),$o,aa(list(A),fun(list(A),$o),P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs3)),Ys2) ) )
         => aa(list(A),$o,aa(list(A),fun(list(A),$o),P,A0),A1) ) ) ) ).

% splice.pinduct
tff(fact_4643_sort__quicksort,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( linorder_sort_key(A,A,aTP_Lamp_yq(A,A)) = linorder_quicksort(A) ) ) ).

% sort_quicksort
tff(fact_4644_splice_Opelims,axiom,
    ! [A: $tType,X: list(A),Xa: list(A),Y: list(A)] :
      ( ( splice(A,X,Xa) = Y )
     => ( accp(product_prod(list(A),list(A)),splice_rel(A),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X),Xa))
       => ( ( ( X = nil(A) )
           => ( ( Y = Xa )
             => ~ accp(product_prod(list(A),list(A)),splice_rel(A),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),Xa)) ) )
         => ~ ! [X2: A,Xs3: list(A)] :
                ( ( X = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs3) )
               => ( ( Y = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),splice(A,Xa,Xs3)) )
                 => ~ accp(product_prod(list(A),list(A)),splice_rel(A),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs3)),Xa)) ) ) ) ) ) ).

% splice.pelims
tff(fact_4645_min__list_Osimps,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X: A,Xs: list(A)] : min_list(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(list(A),A,case_list(A,A,X,aa(list(A),fun(A,fun(list(A),A)),aTP_Lamp_zm(A,fun(list(A),fun(A,fun(list(A),A))),X),Xs)),Xs) ) ).

% min_list.simps
tff(fact_4646_transpose__aux__max,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Xss: list(list(B))] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,suc,aa(list(A),nat,size_size(list(A)),Xs))),aa(nat,nat,foldr(list(B),nat,aTP_Lamp_zn(list(B),fun(nat,nat)),Xss),zero_zero(nat))) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(nat,nat,foldr(list(B),nat,aTP_Lamp_zo(list(B),fun(nat,nat)),aa(list(list(B)),list(list(B)),filter2(list(B),aTP_Lamp_zp(list(B),$o)),Xss)),zero_zero(nat)))) ).

% transpose_aux_max
tff(fact_4647_foldr__length,axiom,
    ! [A: $tType,L: list(A)] : aa(nat,nat,foldr(A,nat,aTP_Lamp_oo(A,fun(nat,nat)),L),zero_zero(nat)) = aa(list(A),nat,size_size(list(A)),L) ).

% foldr_length
tff(fact_4648_foldr__filter,axiom,
    ! [A: $tType,B: $tType,F: fun(B,fun(A,A)),P: fun(B,$o),Xs: list(B)] : foldr(B,A,F,aa(list(B),list(B),filter2(B,P),Xs)) = foldr(B,A,aa(fun(B,$o),fun(B,fun(A,A)),aTP_Lamp_zq(fun(B,fun(A,A)),fun(fun(B,$o),fun(B,fun(A,A))),F),P),Xs) ).

% foldr_filter
tff(fact_4649_comp__fun__commute_Ofoldr__conv__foldl,axiom,
    ! [B: $tType,A: $tType,F: fun(A,fun(B,B)),Xs: list(A),A3: B] :
      ( finite6289374366891150609ommute(A,B,F)
     => ( aa(B,B,foldr(A,B,F,Xs),A3) = foldl(B,A,aTP_Lamp_xi(fun(A,fun(B,B)),fun(B,fun(A,B)),F),A3,Xs) ) ) ).

% comp_fun_commute.foldr_conv_foldl
tff(fact_4650_foldr__length__aux,axiom,
    ! [A: $tType,L: list(A),A3: nat] : aa(nat,nat,foldr(A,nat,aTP_Lamp_oo(A,fun(nat,nat)),L),A3) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A3),aa(list(A),nat,size_size(list(A)),L)) ).

% foldr_length_aux
tff(fact_4651_horner__sum__foldr,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_semiring_0(A)
     => ! [F: 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),F),A3),Xs) = aa(A,A,foldr(B,A,aa(A,fun(B,fun(A,A)),aTP_Lamp_zr(fun(B,A),fun(A,fun(B,fun(A,A))),F),A3),Xs),zero_zero(A)) ) ).

% horner_sum_foldr
tff(fact_4652_splice_Opsimps_I2_J,axiom,
    ! [A: $tType,X: A,Xs: list(A),Ys: list(A)] :
      ( accp(product_prod(list(A),list(A)),splice_rel(A),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),Ys))
     => ( splice(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs),Ys) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),splice(A,Ys,Xs)) ) ) ).

% splice.psimps(2)
tff(fact_4653_splice_Opsimps_I1_J,axiom,
    ! [A: $tType,Ys: list(A)] :
      ( accp(product_prod(list(A),list(A)),splice_rel(A),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),Ys))
     => ( splice(A,nil(A),Ys) = Ys ) ) ).

% splice.psimps(1)
tff(fact_4654_transpose__max__length,axiom,
    ! [A: $tType,Xs: list(list(A))] : aa(nat,nat,foldr(list(A),nat,aTP_Lamp_zs(list(A),fun(nat,nat)),transpose(A,Xs)),zero_zero(nat)) = aa(list(list(A)),nat,size_size(list(list(A))),aa(list(list(A)),list(list(A)),filter2(list(A),aTP_Lamp_zt(list(A),$o)),Xs)) ).

% transpose_max_length
tff(fact_4655_distinct__concat_H,axiom,
    ! [A: $tType,Xs: list(list(A))] :
      ( distinct(list(A),aa(list(list(A)),list(list(A)),filter2(list(A),aTP_Lamp_zt(list(A),$o)),Xs))
     => ( ! [Ys2: list(A)] :
            ( aa(set(list(A)),$o,member(list(A),Ys2),aa(list(list(A)),set(list(A)),set2(list(A)),Xs))
           => distinct(A,Ys2) )
       => ( ! [Ys2: list(A),Zs2: list(A)] :
              ( aa(set(list(A)),$o,member(list(A),Ys2),aa(list(list(A)),set(list(A)),set2(list(A)),Xs))
             => ( aa(set(list(A)),$o,member(list(A),Zs2),aa(list(list(A)),set(list(A)),set2(list(A)),Xs))
               => ( ( Ys2 != Zs2 )
                 => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Ys2)),aa(list(A),set(A),set2(A),Zs2)) = bot_bot(set(A)) ) ) ) )
         => distinct(A,concat(A,Xs)) ) ) ) ).

% distinct_concat'
tff(fact_4656_Cons__lenlex__iff,axiom,
    ! [A: $tType,M: A,Ms: list(A),N: A,Ns: list(A),R2: set(product_prod(A,A))] :
      ( aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),M),Ms)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),N),Ns))),lenlex(A,R2))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),Ms)),aa(list(A),nat,size_size(list(A)),Ns))
        | ( ( aa(list(A),nat,size_size(list(A)),Ms) = aa(list(A),nat,size_size(list(A)),Ns) )
          & aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),M),N)),R2) )
        | ( ( M = N )
          & aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Ms),Ns)),lenlex(A,R2)) ) ) ) ).

% Cons_lenlex_iff
tff(fact_4657_Nil__lenlex__iff1,axiom,
    ! [A: $tType,Ns: list(A),R2: set(product_prod(A,A))] :
      ( aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),Ns)),lenlex(A,R2))
    <=> ( Ns != nil(A) ) ) ).

% Nil_lenlex_iff1
tff(fact_4658_set__concat,axiom,
    ! [A: $tType,Xs: list(list(A))] : aa(list(A),set(A),set2(A),concat(A,Xs)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(list(A)),set(set(A)),image2(list(A),set(A),set2(A)),aa(list(list(A)),set(list(A)),set2(list(A)),Xs))) ).

% set_concat
tff(fact_4659_concat__filter__neq__Nil,axiom,
    ! [A: $tType,Xs: list(list(A))] : concat(A,aa(list(list(A)),list(list(A)),filter2(list(A),aTP_Lamp_zt(list(A),$o)),Xs)) = concat(A,Xs) ).

% concat_filter_neq_Nil
tff(fact_4660_lenlex__irreflexive,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),Xs: list(A)] :
      ( ! [X2: A] : ~ aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),X2)),R2)
     => ~ aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Xs)),lenlex(A,R2)) ) ).

% lenlex_irreflexive
tff(fact_4661_Nil__lenlex__iff2,axiom,
    ! [A: $tType,Ns: list(A),R2: set(product_prod(A,A))] : ~ aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Ns),nil(A))),lenlex(A,R2)) ).

% Nil_lenlex_iff2
tff(fact_4662_filter__conv__foldr,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] : aa(list(A),list(A),filter2(A,P),Xs) = aa(list(A),list(A),foldr(A,list(A),aTP_Lamp_zu(fun(A,$o),fun(A,fun(list(A),list(A))),P),Xs),nil(A)) ).

% filter_conv_foldr
tff(fact_4663_lenlex__length,axiom,
    ! [A: $tType,Ms: list(A),Ns: list(A),R2: set(product_prod(A,A))] :
      ( aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Ms),Ns)),lenlex(A,R2))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Ms)),aa(list(A),nat,size_size(list(A)),Ns)) ) ).

% lenlex_length
tff(fact_4664_lenlex__append1,axiom,
    ! [A: $tType,Us: list(A),Xs: list(A),R: set(product_prod(A,A)),Vs: list(A),Ys: list(A)] :
      ( aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Us),Xs)),lenlex(A,R))
     => ( ( aa(list(A),nat,size_size(list(A)),Vs) = aa(list(A),nat,size_size(list(A)),Ys) )
       => aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us),Vs)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys))),lenlex(A,R)) ) ) ).

% lenlex_append1
tff(fact_4665_total__lenlex,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( total_on(A,top_top(set(A)),R2)
     => total_on(list(A),top_top(set(list(A))),lenlex(A,R2)) ) ).

% total_lenlex
tff(fact_4666_distinct__concat,axiom,
    ! [A: $tType,Xs: list(list(A))] :
      ( distinct(list(A),Xs)
     => ( ! [Ys2: list(A)] :
            ( aa(set(list(A)),$o,member(list(A),Ys2),aa(list(list(A)),set(list(A)),set2(list(A)),Xs))
           => distinct(A,Ys2) )
       => ( ! [Ys2: list(A),Zs2: list(A)] :
              ( aa(set(list(A)),$o,member(list(A),Ys2),aa(list(list(A)),set(list(A)),set2(list(A)),Xs))
             => ( aa(set(list(A)),$o,member(list(A),Zs2),aa(list(list(A)),set(list(A)),set2(list(A)),Xs))
               => ( ( Ys2 != Zs2 )
                 => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Ys2)),aa(list(A),set(A),set2(A),Zs2)) = bot_bot(set(A)) ) ) ) )
         => distinct(A,concat(A,Xs)) ) ) ) ).

% distinct_concat
tff(fact_4667_length__transpose,axiom,
    ! [A: $tType,Xs: list(list(A))] : aa(list(list(A)),nat,size_size(list(list(A))),transpose(A,Xs)) = aa(nat,nat,foldr(list(A),nat,aTP_Lamp_zs(list(A),fun(nat,nat)),Xs),zero_zero(nat)) ).

% length_transpose
tff(fact_4668_distinct__concat__iff,axiom,
    ! [A: $tType,Xs: list(list(A))] :
      ( distinct(A,concat(A,Xs))
    <=> ( distinct(list(A),aa(list(list(A)),list(list(A)),removeAll(list(A),nil(A)),Xs))
        & ! [Ys3: list(A)] :
            ( aa(set(list(A)),$o,member(list(A),Ys3),aa(list(list(A)),set(list(A)),set2(list(A)),Xs))
           => distinct(A,Ys3) )
        & ! [Ys3: list(A),Zs3: list(A)] :
            ( ( aa(set(list(A)),$o,member(list(A),Ys3),aa(list(list(A)),set(list(A)),set2(list(A)),Xs))
              & aa(set(list(A)),$o,member(list(A),Zs3),aa(list(list(A)),set(list(A)),set2(list(A)),Xs))
              & ( Ys3 != Zs3 ) )
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Ys3)),aa(list(A),set(A),set2(A),Zs3)) = bot_bot(set(A)) ) ) ) ) ).

% distinct_concat_iff
tff(fact_4669_length__remdups__concat,axiom,
    ! [A: $tType,Xss: list(list(A))] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),remdups(A),concat(A,Xss))) = aa(set(A),nat,finite_card(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(list(A)),set(set(A)),image2(list(A),set(A),set2(A)),aa(list(list(A)),set(list(A)),set2(list(A)),Xss)))) ).

% length_remdups_concat
tff(fact_4670_sort__mergesort,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( linorder_sort_key(A,A,aTP_Lamp_yq(A,A)) = mergesort(A) ) ) ).

% sort_mergesort
tff(fact_4671_min__list_Oelims,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X: list(A),Y: A] :
          ( ( min_list(A,X) = Y )
         => ( ! [X2: A,Xs3: list(A)] :
                ( ( X = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs3) )
               => ( Y != aa(list(A),A,case_list(A,A,X2,aa(list(A),fun(A,fun(list(A),A)),aTP_Lamp_zm(A,fun(list(A),fun(A,fun(list(A),A))),X2),Xs3)),Xs3) ) )
           => ~ ( ( X = nil(A) )
               => ( Y != undefined(A) ) ) ) ) ) ).

% min_list.elims
tff(fact_4672_mod__h__bot__normalize,axiom,
    ! [A: $tType,H2: heap_ext(product_unit),P: assn] :
      ( syntax7388354845996824322omatch(A,heap_ext(product_unit),undefined(A),H2)
     => ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,P),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),bot_bot(set(nat))))
      <=> aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,P),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),undefined(heap_ext(product_unit))),bot_bot(set(nat)))) ) ) ).

% mod_h_bot_normalize
tff(fact_4673_option_Othe__def,axiom,
    ! [A: $tType,Option: option(A)] : aa(option(A),A,the2(A),Option) = case_option(A,A,undefined(A),aTP_Lamp_au(A,A),Option) ).

% option.the_def
tff(fact_4674_sorted__list__of__set__sort__remdups,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] : aa(set(A),list(A),linord4507533701916653071of_set(A),aa(list(A),set(A),set2(A),Xs)) = aa(list(A),list(A),linorder_sort_key(A,A,aTP_Lamp_yq(A,A)),aa(list(A),list(A),remdups(A),Xs)) ) ).

% sorted_list_of_set_sort_remdups
tff(fact_4675_arg__min__list_Oelims,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [X: fun(A,B),Xa: list(A),Y: A] :
          ( ( arg_min_list(A,B,X,Xa) = Y )
         => ( ! [X2: A] :
                ( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),nil(A)) )
               => ( Y != X2 ) )
           => ( ! [X2: A,Y2: A,Zs2: list(A)] :
                  ( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Zs2)) )
                 => ( Y != $let(
                        m: A,
                        m:= arg_min_list(A,B,X,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Zs2)),
                        $ite(aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,X,X2)),aa(A,B,X,m)),X2,m) ) ) )
             => ~ ( ( Xa = nil(A) )
                 => ( Y != undefined(A) ) ) ) ) ) ) ).

% arg_min_list.elims
tff(fact_4676_arg__min__list_Opelims,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [X: fun(A,B),Xa: list(A),Y: A] :
          ( ( arg_min_list(A,B,X,Xa) = 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),Xa))
           => ( ! [X2: A] :
                  ( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),nil(A)) )
                 => ( ( Y = X2 )
                   => ~ 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),X2),nil(A)))) ) )
             => ( ! [X2: A,Y2: A,Zs2: list(A)] :
                    ( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Zs2)) )
                   => ( ( Y = $let(
                            m: A,
                            m:= arg_min_list(A,B,X,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Zs2)),
                            $ite(aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,X,X2)),aa(A,B,X,m)),X2,m) ) )
                     => ~ 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),X2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Zs2)))) ) )
               => ~ ( ( Xa = 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_4677_Cons__in__lex,axiom,
    ! [A: $tType,X: A,Xs: list(A),Y: A,Ys: list(A),R2: set(product_prod(A,A))] :
      ( aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys))),lex(A,R2))
    <=> ( ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R2)
          & ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys) ) )
        | ( ( X = Y )
          & aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),lex(A,R2)) ) ) ) ).

% Cons_in_lex
tff(fact_4678_Nil2__notin__lex,axiom,
    ! [A: $tType,Xs: list(A),R2: set(product_prod(A,A))] : ~ aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),nil(A))),lex(A,R2)) ).

% Nil2_notin_lex
tff(fact_4679_Nil__notin__lex,axiom,
    ! [A: $tType,Ys: list(A),R2: set(product_prod(A,A))] : ~ aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),Ys)),lex(A,R2)) ).

% Nil_notin_lex
tff(fact_4680_lex__append__leftI,axiom,
    ! [A: $tType,Ys: list(A),Zs: list(A),R2: set(product_prod(A,A)),Xs: list(A)] :
      ( aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Ys),Zs)),lex(A,R2))
     => aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Zs))),lex(A,R2)) ) ).

% lex_append_leftI
tff(fact_4681_lex__append__left__iff,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),Xs: list(A),Ys: list(A),Zs: list(A)] :
      ( ! [X2: A] : ~ aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),X2)),R2)
     => ( aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Zs))),lex(A,R2))
      <=> aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Ys),Zs)),lex(A,R2)) ) ) ).

% lex_append_left_iff
tff(fact_4682_lex__append__leftD,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),Xs: list(A),Ys: list(A),Zs: list(A)] :
      ( ! [X2: A] : ~ aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),X2)),R2)
     => ( aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Zs))),lex(A,R2))
       => aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Ys),Zs)),lex(A,R2)) ) ) ).

% lex_append_leftD
tff(fact_4683_lex__append__rightI,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),R2: set(product_prod(A,A)),Vs: list(A),Us: list(A)] :
      ( aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),lex(A,R2))
     => ( ( aa(list(A),nat,size_size(list(A)),Vs) = aa(list(A),nat,size_size(list(A)),Us) )
       => aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Us)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Vs))),lex(A,R2)) ) ) ).

% lex_append_rightI
tff(fact_4684_arg__min__list_Osimps_I2_J,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F: fun(A,B),X: A,Y: A,Zs: list(A)] :
          arg_min_list(A,B,F,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))) = $let(
            m: A,
            m:= arg_min_list(A,B,F,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Zs)),
            $ite(aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,X)),aa(A,B,F,m)),X,m) ) ) ).

% arg_min_list.simps(2)
tff(fact_4685_lenlex__conv,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : lenlex(A,R2) = aa(fun(product_prod(list(A),list(A)),$o),set(product_prod(list(A),list(A))),collect(product_prod(list(A),list(A))),aa(fun(list(A),fun(list(A),$o)),fun(product_prod(list(A),list(A)),$o),product_case_prod(list(A),list(A),$o),aTP_Lamp_zv(set(product_prod(A,A)),fun(list(A),fun(list(A),$o)),R2))) ).

% lenlex_conv
tff(fact_4686_sorted__list__of__set_Osorted__key__list__of__set__insert__remove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),X: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),list(A),linord4507533701916653071of_set(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A4)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_yq(A,A)),X),aa(set(A),list(A),linord4507533701916653071of_set(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))))) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_insert_remove
tff(fact_4687_remdup__sort__mergesort__remdups,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( aa(fun(list(A),list(A)),fun(list(A),list(A)),comp(list(A),list(A),list(A),remdups(A)),linorder_sort_key(A,A,aTP_Lamp_yq(A,A))) = mergesort_remdups(A) ) ) ).

% remdup_sort_mergesort_remdups
tff(fact_4688_merge__correct,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L12: list(A),L23: list(A)] :
          ( ( distinct(A,L12)
            & sorted_wrt(A,ord_less_eq(A),L12) )
         => ( ( distinct(A,L23)
              & sorted_wrt(A,ord_less_eq(A),L23) )
           => ( distinct(A,merge(A,L12,L23))
              & sorted_wrt(A,ord_less_eq(A),merge(A,L12,L23))
              & ( aa(list(A),set(A),set2(A),merge(A,L12,L23)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(list(A),set(A),set2(A),L12)),aa(list(A),set(A),set2(A),L23)) ) ) ) ) ) ).

% merge_correct
tff(fact_4689_sorted__list__of__set_Osorted__key__list__of__set__insert,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),X: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ~ aa(set(A),$o,member(A,X),A4)
           => ( aa(set(A),list(A),linord4507533701916653071of_set(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A4)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_yq(A,A)),X),aa(set(A),list(A),linord4507533701916653071of_set(A),A4)) ) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_insert
tff(fact_4690_sorted__list__of__set_Ofold__insort__key_Ocomp__fun__commute__on,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Y: A,X: A] : aa(fun(list(A),list(A)),fun(list(A),list(A)),comp(list(A),list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_yq(A,A)),Y)),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_yq(A,A)),X)) = aa(fun(list(A),list(A)),fun(list(A),list(A)),comp(list(A),list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_yq(A,A)),X)),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_yq(A,A)),Y)) ) ).

% sorted_list_of_set.fold_insort_key.comp_fun_commute_on
tff(fact_4691_insort__left__comm,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A,Xs: list(A)] : aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_yq(A,A)),X),aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_yq(A,A)),Y),Xs)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_yq(A,A)),Y),aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_yq(A,A)),X),Xs)) ) ).

% insort_left_comm
tff(fact_4692_comp__fun__commute__insort,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => finite6289374366891150609ommute(A,list(A),linorder_insort_key(A,A,aTP_Lamp_yq(A,A))) ) ).

% comp_fun_commute_insort
tff(fact_4693_sorted__insort,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_yq(A,A)),X),Xs))
        <=> sorted_wrt(A,ord_less_eq(A),Xs) ) ) ).

% sorted_insort
tff(fact_4694_sorted__list__of__set_Ofold__insort__key_Oeq__fold,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A)] : aa(set(A),list(A),linord4507533701916653071of_set(A),A4) = finite_fold(A,list(A),linorder_insort_key(A,A,aTP_Lamp_yq(A,A)),nil(A),A4) ) ).

% sorted_list_of_set.fold_insort_key.eq_fold
tff(fact_4695_insort__remove1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,Xs: list(A)] :
          ( aa(set(A),$o,member(A,A3),aa(list(A),set(A),set2(A),Xs))
         => ( sorted_wrt(A,ord_less_eq(A),Xs)
           => ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_yq(A,A)),A3),remove1(A,A3,Xs)) = Xs ) ) ) ) ).

% insort_remove1
tff(fact_4696_sorted__list__of__set_Ofold__insort__key_Oremove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),X: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),$o,member(A,X),A4)
           => ( aa(set(A),list(A),linord4507533701916653071of_set(A),A4) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_yq(A,A)),X),aa(set(A),list(A),linord4507533701916653071of_set(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))))) ) ) ) ) ).

% sorted_list_of_set.fold_insort_key.remove
tff(fact_4697_sorted__insort__is__snoc,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),A3: A] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => ( ! [X2: A] :
                ( aa(set(A),$o,member(A,X2),aa(list(A),set(A),set2(A),Xs))
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),A3) )
           => ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_yq(A,A)),A3),Xs) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),nil(A))) ) ) ) ) ).

% sorted_insort_is_snoc
tff(fact_4698_merge_Opelims,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: list(A),Xa: list(A),Y: list(A)] :
          ( ( merge(A,X,Xa) = Y )
         => ( accp(product_prod(list(A),list(A)),merge_rel(A),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X),Xa))
           => ( ( ( X = nil(A) )
               => ( ( Y = Xa )
                 => ~ accp(product_prod(list(A),list(A)),merge_rel(A),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),Xa)) ) )
             => ( ! [V2: A,Va: list(A)] :
                    ( ( X = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V2),Va) )
                   => ( ( Xa = nil(A) )
                     => ( ( Y = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V2),Va) )
                       => ~ accp(product_prod(list(A),list(A)),merge_rel(A),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V2),Va)),nil(A))) ) ) )
               => ~ ! [X12: A,L1: list(A)] :
                      ( ( X = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X12),L1) )
                     => ! [X23: A,L22: list(A)] :
                          ( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X23),L22) )
                         => ( ( Y = $ite(
                                  aa(A,$o,aa(A,fun(A,$o),ord_less(A),X12),X23),
                                  aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X12),merge(A,L1,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X23),L22))),
                                  $ite(X12 = X23,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X12),merge(A,L1,L22)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X23),merge(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X12),L1),L22))) ) )
                           => ~ accp(product_prod(list(A),list(A)),merge_rel(A),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X12),L1)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X23),L22))) ) ) ) ) ) ) ) ) ).

% merge.pelims
tff(fact_4699_listrel1__iff__update,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),R2: set(product_prod(A,A))] :
      ( aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),listrel1(A,R2))
    <=> ? [Y3: A,N2: nat] :
          ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(nat,A,nth(A,Xs),N2)),Y3)),R2)
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N2),aa(list(A),nat,size_size(list(A)),Xs))
          & ( Ys = list_update(A,Xs,N2,Y3) ) ) ) ).

% listrel1_iff_update
tff(fact_4700_sorted__list__of__set__def,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( linord4507533701916653071of_set(A) = linord144544945434240204of_set(A,A,aTP_Lamp_yq(A,A)) ) ) ).

% sorted_list_of_set_def
tff(fact_4701_Cons__listrel1__Cons,axiom,
    ! [A: $tType,X: A,Xs: list(A),Y: A,Ys: list(A),R2: set(product_prod(A,A))] :
      ( aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys))),listrel1(A,R2))
    <=> ( ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R2)
          & ( Xs = Ys ) )
        | ( ( X = Y )
          & aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),listrel1(A,R2)) ) ) ) ).

% Cons_listrel1_Cons
tff(fact_4702_rtrancl__listrel1__ConsI1,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),R2: set(product_prod(A,A)),X: A] :
      ( aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),transitive_rtrancl(list(A),listrel1(A,R2)))
     => aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Ys))),transitive_rtrancl(list(A),listrel1(A,R2))) ) ).

% rtrancl_listrel1_ConsI1
tff(fact_4703_listrel1I2,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),R2: set(product_prod(A,A)),X: A] :
      ( aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),listrel1(A,R2))
     => aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Ys))),listrel1(A,R2)) ) ).

% listrel1I2
tff(fact_4704_not__listrel1__Nil,axiom,
    ! [A: $tType,Xs: list(A),R2: set(product_prod(A,A))] : ~ aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),nil(A))),listrel1(A,R2)) ).

% not_listrel1_Nil
tff(fact_4705_not__Nil__listrel1,axiom,
    ! [A: $tType,Xs: list(A),R2: set(product_prod(A,A))] : ~ aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),Xs)),listrel1(A,R2)) ).

% not_Nil_listrel1
tff(fact_4706_listrel1__eq__len,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),R2: set(product_prod(A,A))] :
      ( aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),listrel1(A,R2))
     => ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys) ) ) ).

% listrel1_eq_len
tff(fact_4707_rtrancl__listrel1__eq__len,axiom,
    ! [A: $tType,X: list(A),Y: list(A),R2: set(product_prod(A,A))] :
      ( aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X),Y)),transitive_rtrancl(list(A),listrel1(A,R2)))
     => ( aa(list(A),nat,size_size(list(A)),X) = aa(list(A),nat,size_size(list(A)),Y) ) ) ).

% rtrancl_listrel1_eq_len
tff(fact_4708_append__listrel1I,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),R2: set(product_prod(A,A)),Us: list(A),Vs: list(A)] :
      ( ( ( aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),listrel1(A,R2))
          & ( Us = Vs ) )
        | ( ( Xs = Ys )
          & aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Us),Vs)),listrel1(A,R2)) ) )
     => aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Us)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Vs))),listrel1(A,R2)) ) ).

% append_listrel1I
tff(fact_4709_rtrancl__listrel1__if__listrel,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),R2: set(product_prod(A,A))] :
      ( aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),listrel(A,A,R2))
     => aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),transitive_rtrancl(list(A),listrel1(A,R2))) ) ).

% rtrancl_listrel1_if_listrel
tff(fact_4710_listrel1I1,axiom,
    ! [A: $tType,X: A,Y: A,R2: set(product_prod(A,A)),Xs: list(A)] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R2)
     => aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Xs))),listrel1(A,R2)) ) ).

% listrel1I1
tff(fact_4711_Cons__listrel1E1,axiom,
    ! [A: $tType,X: A,Xs: list(A),Ys: list(A),R2: set(product_prod(A,A))] :
      ( aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),Ys)),listrel1(A,R2))
     => ( ! [Y2: A] :
            ( ( Ys = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Xs) )
           => ~ aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y2)),R2) )
       => ~ ! [Zs2: list(A)] :
              ( ( Ys = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Zs2) )
             => ~ aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Zs2)),listrel1(A,R2)) ) ) ) ).

% Cons_listrel1E1
tff(fact_4712_Cons__listrel1E2,axiom,
    ! [A: $tType,Xs: list(A),Y: A,Ys: list(A),R2: set(product_prod(A,A))] :
      ( aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys))),listrel1(A,R2))
     => ( ! [X2: A] :
            ( ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Ys) )
           => ~ aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y)),R2) )
       => ~ ! [Zs2: list(A)] :
              ( ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Zs2) )
             => ~ aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Zs2),Ys)),listrel1(A,R2)) ) ) ) ).

% Cons_listrel1E2
tff(fact_4713_listrel__reflcl__if__listrel1,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),R2: set(product_prod(A,A))] :
      ( aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),listrel1(A,R2))
     => aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),listrel(A,A,transitive_rtrancl(A,R2))) ) ).

% listrel_reflcl_if_listrel1
tff(fact_4714_listrel1I,axiom,
    ! [A: $tType,X: A,Y: A,R2: set(product_prod(A,A)),Xs: list(A),Us: list(A),Vs: list(A),Ys: list(A)] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R2)
     => ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Vs)) )
       => ( ( Ys = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Vs)) )
         => aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),listrel1(A,R2)) ) ) ) ).

% listrel1I
tff(fact_4715_listrel1E,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),R2: set(product_prod(A,A))] :
      ( aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),listrel1(A,R2))
     => ~ ! [X2: A,Y2: A] :
            ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y2)),R2)
           => ! [Us2: list(A),Vs2: list(A)] :
                ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Vs2)) )
               => ( Ys != aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Vs2)) ) ) ) ) ).

% listrel1E
tff(fact_4716_rtrancl__listrel1__ConsI2,axiom,
    ! [A: $tType,X: A,Y: A,R2: set(product_prod(A,A)),Xs: list(A),Ys: list(A)] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),transitive_rtrancl(A,R2))
     => ( aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),transitive_rtrancl(list(A),listrel1(A,R2)))
       => aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys))),transitive_rtrancl(list(A),listrel1(A,R2))) ) ) ).

% rtrancl_listrel1_ConsI2
tff(fact_4717_snoc__listrel1__snoc__iff,axiom,
    ! [A: $tType,Xs: list(A),X: A,Ys: list(A),Y: A,R2: set(product_prod(A,A))] :
      ( aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)))),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),nil(A))))),listrel1(A,R2))
    <=> ( ( aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),listrel1(A,R2))
          & ( X = Y ) )
        | ( ( Xs = Ys )
          & aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R2) ) ) ) ).

% snoc_listrel1_snoc_iff
tff(fact_4718_listrel1p__def,axiom,
    ! [A: $tType,R2: fun(A,fun(A,$o)),Xs: list(A),Ys: list(A)] :
      ( listrel1p(A,R2,Xs,Ys)
    <=> aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),listrel1(A,aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),R2)))) ) ).

% listrel1p_def
tff(fact_4719_nth__transpose,axiom,
    ! [A: $tType,I: nat,Xs: list(list(A))] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(list(A)),nat,size_size(list(list(A))),transpose(A,Xs)))
     => ( aa(nat,list(A),nth(list(A),transpose(A,Xs)),I) = aa(list(list(A)),list(A),map(list(A),A,aTP_Lamp_zw(nat,fun(list(A),A),I)),aa(list(list(A)),list(list(A)),filter2(list(A),aTP_Lamp_zx(nat,fun(list(A),$o),I)),Xs)) ) ) ).

% nth_transpose
tff(fact_4720_listrel__def,axiom,
    ! [B: $tType,A: $tType,X3: set(product_prod(A,B))] : listrel(A,B,X3) = aa(fun(product_prod(list(A),list(B)),$o),set(product_prod(list(A),list(B))),collect(product_prod(list(A),list(B))),aa(fun(list(A),fun(list(B),$o)),fun(product_prod(list(A),list(B)),$o),product_case_prod(list(A),list(B),$o),listrelp(A,B,aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_as(set(product_prod(A,B)),fun(A,fun(B,$o))),X3)))) ).

% listrel_def
tff(fact_4721_map__ident,axiom,
    ! [A: $tType,X3: list(A)] : aa(list(A),list(A),map(A,A,aTP_Lamp_au(A,A)),X3) = X3 ).

% map_ident
tff(fact_4722_concat__map__singleton,axiom,
    ! [A: $tType,B: $tType,F: fun(B,A),Xs: list(B)] : concat(A,aa(list(B),list(list(A)),map(B,list(A),aTP_Lamp_zy(fun(B,A),fun(B,list(A)),F)),Xs)) = aa(list(B),list(A),map(B,A,F),Xs) ).

% concat_map_singleton
tff(fact_4723_sorted__wrt__map,axiom,
    ! [A: $tType,B: $tType,R: fun(A,fun(A,$o)),F: fun(B,A),Xs: list(B)] :
      ( sorted_wrt(A,R,aa(list(B),list(A),map(B,A,F),Xs))
    <=> sorted_wrt(B,aa(fun(B,A),fun(B,fun(B,$o)),aTP_Lamp_zz(fun(A,fun(A,$o)),fun(fun(B,A),fun(B,fun(B,$o))),R),F),Xs) ) ).

% sorted_wrt_map
tff(fact_4724_foldl__map,axiom,
    ! [A: $tType,B: $tType,C: $tType,G: fun(A,fun(B,A)),A3: A,F: fun(C,B),Xs: list(C)] : foldl(A,B,G,A3,aa(list(C),list(B),map(C,B,F),Xs)) = foldl(A,C,aa(fun(C,B),fun(A,fun(C,A)),aTP_Lamp_aaa(fun(A,fun(B,A)),fun(fun(C,B),fun(A,fun(C,A))),G),F),A3,Xs) ).

% foldl_map
tff(fact_4725_list_Omap__ident,axiom,
    ! [A: $tType,T3: list(A)] : aa(list(A),list(A),map(A,A,aTP_Lamp_au(A,A)),T3) = T3 ).

% list.map_ident
tff(fact_4726_List_Omap_Oidentity,axiom,
    ! [A: $tType] : map(A,A,aTP_Lamp_au(A,A)) = id(list(A)) ).

% List.map.identity
tff(fact_4727_sorted__map,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F: fun(B,A),Xs: list(B)] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),map(B,A,F),Xs))
        <=> sorted_wrt(B,aTP_Lamp_aab(fun(B,A),fun(B,fun(B,$o)),F),Xs) ) ) ).

% sorted_map
tff(fact_4728_transpose_Oelims,axiom,
    ! [A: $tType,X: list(list(A)),Y: list(list(A))] :
      ( ( transpose(A,X) = Y )
     => ( ( ( X = nil(list(A)) )
         => ( Y != nil(list(A)) ) )
       => ( ! [Xss2: list(list(A))] :
              ( ( X = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),nil(A)),Xss2) )
             => ( Y != transpose(A,Xss2) ) )
         => ~ ! [X2: A,Xs3: list(A),Xss2: list(list(A))] :
                ( ( X = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs3)),Xss2) )
               => ( Y != aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),concat(A,aa(list(list(A)),list(list(A)),map(list(A),list(A),case_list(list(A),A,nil(A),aTP_Lamp_aac(A,fun(list(A),list(A))))),Xss2)))),transpose(A,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),Xs3),concat(list(A),aa(list(list(A)),list(list(list(A))),map(list(A),list(list(A)),case_list(list(list(A)),A,nil(list(A)),aTP_Lamp_aad(A,fun(list(A),list(list(A)))))),Xss2))))) ) ) ) ) ) ).

% transpose.elims
tff(fact_4729_transpose_Osimps_I3_J,axiom,
    ! [A: $tType,X: A,Xs: list(A),Xss: list(list(A))] : transpose(A,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),Xss)) = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),concat(A,aa(list(list(A)),list(list(A)),map(list(A),list(A),case_list(list(A),A,nil(A),aTP_Lamp_aac(A,fun(list(A),list(A))))),Xss)))),transpose(A,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),Xs),concat(list(A),aa(list(list(A)),list(list(list(A))),map(list(A),list(list(A)),case_list(list(list(A)),A,nil(list(A)),aTP_Lamp_aad(A,fun(list(A),list(list(A)))))),Xss))))) ).

% transpose.simps(3)
tff(fact_4730_map__by__foldl,axiom,
    ! [A: $tType,B: $tType,F: fun(B,A),L: list(B)] : foldl(list(A),B,aTP_Lamp_aae(fun(B,A),fun(list(A),fun(B,list(A))),F),nil(A),L) = aa(list(B),list(A),map(B,A,F),L) ).

% map_by_foldl
tff(fact_4731_sorted__map__same,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F: fun(B,A),G: fun(list(B),A),Xs: list(B)] : sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),map(B,A,F),aa(list(B),list(B),filter2(B,aa(list(B),fun(B,$o),aa(fun(list(B),A),fun(list(B),fun(B,$o)),aTP_Lamp_aaf(fun(B,A),fun(fun(list(B),A),fun(list(B),fun(B,$o))),F),G),Xs)),Xs))) ) ).

% sorted_map_same
tff(fact_4732_listrelp__listrel__eq,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,B)),X3: list(A),Xa3: list(B)] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),listrelp(A,B,aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_as(set(product_prod(A,B)),fun(A,fun(B,$o))),R2)),X3),Xa3)
    <=> aa(set(product_prod(list(A),list(B))),$o,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)),X3),Xa3)),listrel(A,B,R2)) ) ).

% listrelp_listrel_eq
tff(fact_4733_map__filter__def,axiom,
    ! [A: $tType,B: $tType,F: fun(B,option(A)),Xs: list(B)] : map_filter(B,A,F,Xs) = aa(list(B),list(A),map(B,A,aa(fun(B,option(A)),fun(B,A),comp(option(A),A,B,the2(A)),F)),aa(list(B),list(B),filter2(B,aTP_Lamp_aag(fun(B,option(A)),fun(B,$o),F)),Xs)) ).

% map_filter_def
tff(fact_4734_nth__nth__transpose__sorted,axiom,
    ! [A: $tType,Xs: list(list(A)),I: nat,J: nat] :
      ( sorted_wrt(nat,ord_less_eq(nat),rev(nat,aa(list(list(A)),list(nat),map(list(A),nat,size_size(list(A))),Xs)))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(list(A)),nat,size_size(list(list(A))),transpose(A,Xs)))
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(list(A)),nat,size_size(list(list(A))),aa(list(list(A)),list(list(A)),filter2(list(A),aTP_Lamp_zx(nat,fun(list(A),$o),I)),Xs)))
         => ( aa(nat,A,nth(A,aa(nat,list(A),nth(list(A),transpose(A,Xs)),I)),J) = aa(nat,A,nth(A,aa(nat,list(A),nth(list(A),Xs),J)),I) ) ) ) ) ).

% nth_nth_transpose_sorted
tff(fact_4735_transpose__column,axiom,
    ! [A: $tType,Xs: list(list(A)),I: nat] :
      ( sorted_wrt(nat,ord_less_eq(nat),rev(nat,aa(list(list(A)),list(nat),map(list(A),nat,size_size(list(A))),Xs)))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(list(A)),nat,size_size(list(list(A))),Xs))
       => ( aa(list(list(A)),list(A),map(list(A),A,aTP_Lamp_zw(nat,fun(list(A),A),I)),aa(list(list(A)),list(list(A)),filter2(list(A),aTP_Lamp_zx(nat,fun(list(A),$o),I)),transpose(A,Xs))) = aa(nat,list(A),nth(list(A),Xs),I) ) ) ) ).

% transpose_column
tff(fact_4736_map__fst__mk__snd,axiom,
    ! [B: $tType,A: $tType,K: B,L: list(A)] : aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),aa(list(A),list(product_prod(A,B)),map(A,product_prod(A,B),aa(B,fun(A,product_prod(A,B)),aTP_Lamp_ru(B,fun(A,product_prod(A,B))),K)),L)) = L ).

% map_fst_mk_snd
tff(fact_4737_map__snd__mk__fst,axiom,
    ! [B: $tType,A: $tType,K: B,L: list(A)] : aa(list(product_prod(B,A)),list(A),map(product_prod(B,A),A,product_snd(B,A)),aa(list(A),list(product_prod(B,A)),map(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),K)),L)) = L ).

% map_snd_mk_fst
tff(fact_4738_sorted__wrt__map__linord,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(A)
     => ! [L: list(product_prod(A,B))] :
          ( sorted_wrt(product_prod(A,B),aTP_Lamp_aah(product_prod(A,B),fun(product_prod(A,B),$o)),L)
        <=> sorted_wrt(A,ord_less_eq(A),aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),L)) ) ) ).

% sorted_wrt_map_linord
tff(fact_4739_sorted__wrt__rev__linord,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: list(A)] :
          ( sorted_wrt(A,aTP_Lamp_tm(A,fun(A,$o)),L)
        <=> sorted_wrt(A,ord_less_eq(A),rev(A,L)) ) ) ).

% sorted_wrt_rev_linord
tff(fact_4740_sorted__wrt__map__rev__linord,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(A)
     => ! [L: list(product_prod(A,B))] :
          ( sorted_wrt(product_prod(A,B),aTP_Lamp_aai(product_prod(A,B),fun(product_prod(A,B),$o)),L)
        <=> sorted_wrt(A,ord_less_eq(A),rev(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),L))) ) ) ).

% sorted_wrt_map_rev_linord
tff(fact_4741_product__concat__map,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys: list(B)] : product(A,B,Xs,Ys) = concat(product_prod(A,B),aa(list(A),list(list(product_prod(A,B))),map(A,list(product_prod(A,B)),aTP_Lamp_aaj(list(B),fun(A,list(product_prod(A,B))),Ys)),Xs)) ).

% product_concat_map
tff(fact_4742_sorted__wrt__rev,axiom,
    ! [A: $tType,P: fun(A,fun(A,$o)),Xs: list(A)] :
      ( sorted_wrt(A,P,rev(A,Xs))
    <=> sorted_wrt(A,aTP_Lamp_aak(fun(A,fun(A,$o)),fun(A,fun(A,$o)),P),Xs) ) ).

% sorted_wrt_rev
tff(fact_4743_merge__list_Ocases,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: product_prod(list(list(A)),list(list(A)))] :
          ( ( X != aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),nil(list(A))),nil(list(A))) )
         => ( ! [L2: list(A)] : X != aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),nil(list(A))),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L2),nil(list(A))))
           => ( ! [La: list(A),Acc22: list(list(A))] : X != aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc22)),nil(list(A)))
             => ( ! [La: list(A),Acc22: list(list(A)),L2: list(A)] : X != aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc22)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L2),nil(list(A))))
               => ~ ! [Acc22: list(list(A)),L1: list(A),L22: list(A),Ls: list(list(A))] : X != aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),Acc22),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L1),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L22),Ls))) ) ) ) ) ) ).

% merge_list.cases
tff(fact_4744_foldl__conv__foldr,axiom,
    ! [B: $tType,A: $tType,F: fun(A,fun(B,A)),A3: A,Xs: list(B)] : foldl(A,B,F,A3,Xs) = aa(A,A,foldr(B,A,aTP_Lamp_aal(fun(A,fun(B,A)),fun(B,fun(A,A)),F),rev(B,Xs)),A3) ).

% foldl_conv_foldr
tff(fact_4745_foldr__conv__foldl,axiom,
    ! [A: $tType,B: $tType,F: fun(B,fun(A,A)),Xs: list(B),A3: A] : aa(A,A,foldr(B,A,F,Xs),A3) = foldl(A,B,aTP_Lamp_aam(fun(B,fun(A,A)),fun(A,fun(B,A)),F),A3,rev(B,Xs)) ).

% foldr_conv_foldl
tff(fact_4746_map__filter__simps_I1_J,axiom,
    ! [A: $tType,B: $tType,F: fun(B,option(A)),X: B,Xs: list(B)] : map_filter(B,A,F,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Xs)) = case_option(list(A),A,map_filter(B,A,F,Xs),aa(list(B),fun(A,list(A)),aTP_Lamp_aan(fun(B,option(A)),fun(list(B),fun(A,list(A))),F),Xs),aa(B,option(A),F,X)) ).

% map_filter_simps(1)
tff(fact_4747_product_Osimps_I2_J,axiom,
    ! [A: $tType,B: $tType,X: A,Xs: list(A),Ys: list(B)] : product(A,B,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs),Ys) = aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(list(product_prod(A,B)),fun(list(product_prod(A,B)),list(product_prod(A,B))),append(product_prod(A,B)),aa(list(B),list(product_prod(A,B)),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_4748_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),aa(list(list(A)),list(list(list(A))),map(list(A),list(list(A)),aTP_Lamp_aao(list(A),fun(list(A),list(list(A))),Xs)),n_lists(A,N,Xs))) ).

% n_lists.simps(2)
tff(fact_4749_eq__key__imp__eq__value,axiom,
    ! [A: $tType,B: $tType,Xs: list(product_prod(A,B)),K: A,V1: B,V22: B] :
      ( distinct(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xs))
     => ( aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K),V1)),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Xs))
       => ( aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K),V22)),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Xs))
         => ( V1 = V22 ) ) ) ) ).

% eq_key_imp_eq_value
tff(fact_4750_distinct__map__fstD,axiom,
    ! [A: $tType,B: $tType,Xs: list(product_prod(A,B)),X: A,Y: B,Z2: B] :
      ( distinct(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xs))
     => ( aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y)),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Xs))
       => ( aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Z2)),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Xs))
         => ( Y = Z2 ) ) ) ) ).

% distinct_map_fstD
tff(fact_4751_Id__on__set,axiom,
    ! [A: $tType,Xs: list(A)] : id_on(A,aa(list(A),set(A),set2(A),Xs)) = aa(list(product_prod(A,A)),set(product_prod(A,A)),set2(product_prod(A,A)),aa(list(A),list(product_prod(A,A)),map(A,product_prod(A,A),aTP_Lamp_vl(A,product_prod(A,A))),Xs)) ).

% Id_on_set
tff(fact_4752_rev__update,axiom,
    ! [A: $tType,K: nat,Xs: list(A),Y: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),aa(list(A),nat,size_size(list(A)),Xs))
     => ( rev(A,list_update(A,Xs,K,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)),K)),one_one(nat)),Y) ) ) ).

% rev_update
tff(fact_4753_set__relcomp,axiom,
    ! [B: $tType,C: $tType,A: $tType,Xys: list(product_prod(A,C)),Yzs: list(product_prod(C,B))] : relcomp(A,C,B,aa(list(product_prod(A,C)),set(product_prod(A,C)),set2(product_prod(A,C)),Xys),aa(list(product_prod(C,B)),set(product_prod(C,B)),set2(product_prod(C,B)),Yzs)) = aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),concat(product_prod(A,B),aa(list(product_prod(A,C)),list(list(product_prod(A,B))),map(product_prod(A,C),list(product_prod(A,B)),aTP_Lamp_aaq(list(product_prod(C,B)),fun(product_prod(A,C),list(product_prod(A,B))),Yzs)),Xys))) ).

% set_relcomp
tff(fact_4754_transpose__column__length,axiom,
    ! [A: $tType,Xs: list(list(A)),I: nat] :
      ( sorted_wrt(nat,ord_less_eq(nat),rev(nat,aa(list(list(A)),list(nat),map(list(A),nat,size_size(list(A))),Xs)))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(list(A)),nat,size_size(list(list(A))),Xs))
       => ( aa(list(list(A)),nat,size_size(list(list(A))),aa(list(list(A)),list(list(A)),filter2(list(A),aTP_Lamp_zx(nat,fun(list(A),$o),I)),transpose(A,Xs))) = aa(list(A),nat,size_size(list(A)),aa(nat,list(A),nth(list(A),Xs),I)) ) ) ) ).

% transpose_column_length
tff(fact_4755_map__filter__map__filter,axiom,
    ! [A: $tType,B: $tType,F: fun(B,A),P: fun(B,$o),Xs: list(B)] : aa(list(B),list(A),map(B,A,F),aa(list(B),list(B),filter2(B,P),Xs)) = map_filter(B,A,aa(fun(B,$o),fun(B,option(A)),aTP_Lamp_aar(fun(B,A),fun(fun(B,$o),fun(B,option(A))),F),P),Xs) ).

% map_filter_map_filter
tff(fact_4756_product__lists_Osimps_I2_J,axiom,
    ! [A: $tType,Xs: list(A),Xss: list(list(A))] : product_lists(A,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),Xs),Xss)) = concat(list(A),aa(list(A),list(list(list(A))),map(A,list(list(A)),aTP_Lamp_aas(list(list(A)),fun(A,list(list(A))),Xss)),Xs)) ).

% product_lists.simps(2)
tff(fact_4757_product__code,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys: list(B)] : product_product(A,B,aa(list(A),set(A),set2(A),Xs),aa(list(B),set(B),set2(B),Ys)) = aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),concat(product_prod(A,B),aa(list(A),list(list(product_prod(A,B))),map(A,list(product_prod(A,B)),aTP_Lamp_aaj(list(B),fun(A,list(product_prod(A,B))),Ys)),Xs))) ).

% product_code
tff(fact_4758_length__product__lists,axiom,
    ! [A: $tType,Xss: list(list(A))] : aa(list(list(A)),nat,size_size(list(list(A))),product_lists(A,Xss)) = aa(nat,nat,foldr(nat,nat,times_times(nat),aa(list(list(A)),list(nat),map(list(A),nat,size_size(list(A))),Xss)),one_one(nat)) ).

% length_product_lists
tff(fact_4759_member__product,axiom,
    ! [A: $tType,B: $tType,X: product_prod(A,B),A4: set(A),B3: set(B)] :
      ( aa(set(product_prod(A,B)),$o,member(product_prod(A,B),X),product_product(A,B,A4,B3))
    <=> aa(set(product_prod(A,B)),$o,member(product_prod(A,B),X),product_Sigma(A,B,A4,aTP_Lamp_qz(set(B),fun(A,set(B)),B3))) ) ).

% member_product
tff(fact_4760_Product__Type_Oproduct__def,axiom,
    ! [A: $tType,B: $tType,A4: set(A),B3: set(B)] : product_product(A,B,A4,B3) = product_Sigma(A,B,A4,aTP_Lamp_qz(set(B),fun(A,set(B)),B3)) ).

% Product_Type.product_def
tff(fact_4761_remove__rev__alt__def,axiom,
    ! [A: $tType,X: A,Xs: list(A)] : remove_rev(A,X,Xs) = aa(list(A),list(A),filter2(A,aa(A,fun(A,$o),aTP_Lamp_ze(A,fun(A,$o)),X)),rev(A,Xs)) ).

% remove_rev_alt_def
tff(fact_4762_transpose__aux__filter__tail,axiom,
    ! [A: $tType,Xss: list(list(A))] : concat(list(A),aa(list(list(A)),list(list(list(A))),map(list(A),list(list(A)),case_list(list(list(A)),A,nil(list(A)),aTP_Lamp_aad(A,fun(list(A),list(list(A)))))),Xss)) = aa(list(list(A)),list(list(A)),map(list(A),list(A),tl(A)),aa(list(list(A)),list(list(A)),filter2(list(A),aTP_Lamp_zt(list(A),$o)),Xss)) ).

% transpose_aux_filter_tail
tff(fact_4763_transpose__transpose,axiom,
    ! [A: $tType,Xs: list(list(A))] :
      ( sorted_wrt(nat,ord_less_eq(nat),rev(nat,aa(list(list(A)),list(nat),map(list(A),nat,size_size(list(A))),Xs)))
     => ( transpose(A,transpose(A,Xs)) = takeWhile(list(A),aTP_Lamp_zt(list(A),$o),Xs) ) ) ).

% transpose_transpose
tff(fact_4764_length__tl,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),tl(A),Xs)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat)) ).

% length_tl
tff(fact_4765_tl__def,axiom,
    ! [A: $tType,List: list(A)] : aa(list(A),list(A),tl(A),List) = aa(list(A),list(A),case_list(list(A),A,nil(A),aTP_Lamp_aat(A,fun(list(A),list(A)))),List) ).

% tl_def
tff(fact_4766_tl__append,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] : aa(list(A),list(A),tl(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(list(A),list(A),case_list(list(A),A,aa(list(A),list(A),tl(A),Ys),aTP_Lamp_aau(list(A),fun(A,fun(list(A),list(A))),Ys)),Xs) ).

% tl_append
tff(fact_4767_filter__equals__takeWhile__sorted__rev,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F: fun(B,A),Xs: list(B),T3: A] :
          ( sorted_wrt(A,ord_less_eq(A),rev(A,aa(list(B),list(A),map(B,A,F),Xs)))
         => ( aa(list(B),list(B),filter2(B,aa(A,fun(B,$o),aTP_Lamp_aav(fun(B,A),fun(A,fun(B,$o)),F),T3)),Xs) = takeWhile(B,aa(A,fun(B,$o),aTP_Lamp_aav(fun(B,A),fun(A,fun(B,$o)),F),T3),Xs) ) ) ) ).

% filter_equals_takeWhile_sorted_rev
tff(fact_4768_extract__def,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] : extract(A,P,Xs) = aa(list(A),option(product_prod(list(A),product_prod(A,list(A)))),case_list(option(product_prod(list(A),product_prod(A,list(A)))),A,none(product_prod(list(A),product_prod(A,list(A)))),aa(list(A),fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A)))))),aTP_Lamp_aaw(fun(A,$o),fun(list(A),fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A))))))),P),Xs)),dropWhile(A,aa(fun(A,$o),fun(A,$o),comp($o,$o,A,fNot),P),Xs)) ).

% extract_def
tff(fact_4769_transpose_Opsimps_I3_J,axiom,
    ! [A: $tType,X: A,Xs: list(A),Xss: list(list(A))] :
      ( accp(list(list(A)),transpose_rel(A),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),Xss))
     => ( transpose(A,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),Xss)) = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),concat(A,aa(list(list(A)),list(list(A)),map(list(A),list(A),case_list(list(A),A,nil(A),aTP_Lamp_aac(A,fun(list(A),list(A))))),Xss)))),transpose(A,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),Xs),concat(list(A),aa(list(list(A)),list(list(list(A))),map(list(A),list(list(A)),case_list(list(list(A)),A,nil(list(A)),aTP_Lamp_aad(A,fun(list(A),list(list(A)))))),Xss))))) ) ) ).

% transpose.psimps(3)
tff(fact_4770_transpose_Opelims,axiom,
    ! [A: $tType,X: list(list(A)),Y: list(list(A))] :
      ( ( transpose(A,X) = Y )
     => ( accp(list(list(A)),transpose_rel(A),X)
       => ( ( ( X = nil(list(A)) )
           => ( ( Y = nil(list(A)) )
             => ~ accp(list(list(A)),transpose_rel(A),nil(list(A))) ) )
         => ( ! [Xss2: list(list(A))] :
                ( ( X = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),nil(A)),Xss2) )
               => ( ( Y = transpose(A,Xss2) )
                 => ~ accp(list(list(A)),transpose_rel(A),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),nil(A)),Xss2)) ) )
           => ~ ! [X2: A,Xs3: list(A),Xss2: list(list(A))] :
                  ( ( X = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs3)),Xss2) )
                 => ( ( Y = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),concat(A,aa(list(list(A)),list(list(A)),map(list(A),list(A),case_list(list(A),A,nil(A),aTP_Lamp_aac(A,fun(list(A),list(A))))),Xss2)))),transpose(A,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),Xs3),concat(list(A),aa(list(list(A)),list(list(list(A))),map(list(A),list(list(A)),case_list(list(list(A)),A,nil(list(A)),aTP_Lamp_aad(A,fun(list(A),list(list(A)))))),Xss2))))) )
                   => ~ accp(list(list(A)),transpose_rel(A),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs3)),Xss2)) ) ) ) ) ) ) ).

% transpose.pelims
tff(fact_4771_conj__comp__iff,axiom,
    ! [B: $tType,A: $tType,P: fun(B,$o),Q: fun(B,$o),G: fun(A,B),X3: A] :
      ( aa(A,$o,aa(fun(A,B),fun(A,$o),comp(B,$o,A,aa(fun(B,$o),fun(B,$o),aTP_Lamp_aax(fun(B,$o),fun(fun(B,$o),fun(B,$o)),P),Q)),G),X3)
    <=> ( aa(A,$o,aa(fun(A,B),fun(A,$o),comp(B,$o,A,P),G),X3)
        & aa(A,$o,aa(fun(A,B),fun(A,$o),comp(B,$o,A,Q),G),X3) ) ) ).

% conj_comp_iff
tff(fact_4772_remdups__adj__Cons_H,axiom,
    ! [A: $tType,X: A,Xs: list(A)] : remdups_adj(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),remdups_adj(A,dropWhile(A,aTP_Lamp_cf(A,fun(A,$o),X),Xs))) ).

% remdups_adj_Cons'
tff(fact_4773_find__dropWhile,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] : find(A,P,Xs) = aa(list(A),option(A),case_list(option(A),A,none(A),aTP_Lamp_aay(A,fun(list(A),option(A)))),dropWhile(A,aa(fun(A,$o),fun(A,$o),comp($o,$o,A,fNot),P),Xs)) ).

% find_dropWhile
tff(fact_4774_remdups__adj__append__dropWhile,axiom,
    ! [A: $tType,Xs: list(A),Y: A,Ys: list(A)] : remdups_adj(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys))) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),remdups_adj(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),nil(A))))),remdups_adj(A,dropWhile(A,aTP_Lamp_cf(A,fun(A,$o),Y),Ys))) ).

% remdups_adj_append_dropWhile
tff(fact_4775_transpose_Opinduct,axiom,
    ! [A: $tType,A0: list(list(A)),P: fun(list(list(A)),$o)] :
      ( accp(list(list(A)),transpose_rel(A),A0)
     => ( ( accp(list(list(A)),transpose_rel(A),nil(list(A)))
         => aa(list(list(A)),$o,P,nil(list(A))) )
       => ( ! [Xss2: list(list(A))] :
              ( accp(list(list(A)),transpose_rel(A),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),nil(A)),Xss2))
             => ( aa(list(list(A)),$o,P,Xss2)
               => aa(list(list(A)),$o,P,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),nil(A)),Xss2)) ) )
         => ( ! [X2: A,Xs3: list(A),Xss2: list(list(A))] :
                ( accp(list(list(A)),transpose_rel(A),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs3)),Xss2))
               => ( aa(list(list(A)),$o,P,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),Xs3),concat(list(A),aa(list(list(A)),list(list(list(A))),map(list(A),list(list(A)),case_list(list(list(A)),A,nil(list(A)),aTP_Lamp_aad(A,fun(list(A),list(list(A)))))),Xss2))))
                 => aa(list(list(A)),$o,P,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs3)),Xss2)) ) )
           => aa(list(list(A)),$o,P,A0) ) ) ) ) ).

% transpose.pinduct
tff(fact_4776_dropWhile__neq__rev,axiom,
    ! [A: $tType,Xs: list(A),X: A] :
      ( distinct(A,Xs)
     => ( aa(set(A),$o,member(A,X),aa(list(A),set(A),set2(A),Xs))
       => ( dropWhile(A,aa(A,fun(A,$o),aTP_Lamp_ze(A,fun(A,$o)),X),rev(A,Xs)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),rev(A,takeWhile(A,aa(A,fun(A,$o),aTP_Lamp_ze(A,fun(A,$o)),X),Xs))) ) ) ) ).

% dropWhile_neq_rev
tff(fact_4777_takeWhile__neq__rev,axiom,
    ! [A: $tType,Xs: list(A),X: A] :
      ( distinct(A,Xs)
     => ( aa(set(A),$o,member(A,X),aa(list(A),set(A),set2(A),Xs))
       => ( takeWhile(A,aa(A,fun(A,$o),aTP_Lamp_ze(A,fun(A,$o)),X),rev(A,Xs)) = rev(A,aa(list(A),list(A),tl(A),dropWhile(A,aa(A,fun(A,$o),aTP_Lamp_ze(A,fun(A,$o)),X),Xs))) ) ) ) ).

% takeWhile_neq_rev
tff(fact_4778_min__list_Opelims,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X: list(A),Y: A] :
          ( ( min_list(A,X) = Y )
         => ( accp(list(A),min_list_rel(A),X)
           => ( ! [X2: A,Xs3: list(A)] :
                  ( ( X = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs3) )
                 => ( ( Y = aa(list(A),A,case_list(A,A,X2,aa(list(A),fun(A,fun(list(A),A)),aTP_Lamp_zm(A,fun(list(A),fun(A,fun(list(A),A))),X2),Xs3)),Xs3) )
                   => ~ accp(list(A),min_list_rel(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs3)) ) )
             => ~ ( ( X = nil(A) )
                 => ( ( Y = undefined(A) )
                   => ~ accp(list(A),min_list_rel(A),nil(A)) ) ) ) ) ) ) ).

% min_list.pelims
tff(fact_4779_quicksort_Opelims,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: list(A),Y: list(A)] :
          ( ( aa(list(A),list(A),linorder_quicksort(A),X) = Y )
         => ( accp(list(A),linord6200660962353139674rt_rel(A),X)
           => ( ( ( X = nil(A) )
               => ( ( Y = nil(A) )
                 => ~ accp(list(A),linord6200660962353139674rt_rel(A),nil(A)) ) )
             => ~ ! [X2: A,Xs3: list(A)] :
                    ( ( X = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs3) )
                   => ( ( Y = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),linorder_quicksort(A),aa(list(A),list(A),filter2(A,aTP_Lamp_zl(A,fun(A,$o),X2)),Xs3))),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),nil(A))),aa(list(A),list(A),linorder_quicksort(A),aa(list(A),list(A),filter2(A,aa(A,fun(A,$o),ord_less_eq(A),X2)),Xs3)))) )
                     => ~ accp(list(A),linord6200660962353139674rt_rel(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs3)) ) ) ) ) ) ) ).

% quicksort.pelims
tff(fact_4780_transpose__rectangle,axiom,
    ! [A: $tType,Xs: list(list(A)),N: nat] :
      ( ( ( Xs = nil(list(A)) )
       => ( N = zero_zero(nat) ) )
     => ( ! [I2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(list(A)),nat,size_size(list(list(A))),Xs))
           => ( aa(list(A),nat,size_size(list(A)),aa(nat,list(A),nth(list(A),Xs),I2)) = N ) )
       => ( transpose(A,Xs) = aa(list(nat),list(list(A)),map(nat,list(A),aTP_Lamp_aba(list(list(A)),fun(nat,list(A)),Xs)),upt(zero_zero(nat),N)) ) ) ) ).

% transpose_rectangle
tff(fact_4781_sort__upt,axiom,
    ! [M: nat,N: nat] : aa(list(nat),list(nat),linorder_sort_key(nat,nat,aTP_Lamp_dj(nat,nat)),upt(M,N)) = upt(M,N) ).

% sort_upt
tff(fact_4782_map__add__upt_H,axiom,
    ! [Ofs: nat,A3: nat,B2: nat] : aa(list(nat),list(nat),map(nat,nat,aTP_Lamp_abb(nat,fun(nat,nat),Ofs)),upt(A3,B2)) = upt(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A3),Ofs),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),B2),Ofs)) ).

% map_add_upt'
tff(fact_4783_map__add__upt,axiom,
    ! [N: nat,M: nat] : aa(list(nat),list(nat),map(nat,nat,aTP_Lamp_abb(nat,fun(nat,nat),N)),upt(zero_zero(nat),M)) = upt(N,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)) ).

% map_add_upt
tff(fact_4784_enumerate__map__upt,axiom,
    ! [A: $tType,N: nat,F: fun(nat,A),M: nat] : enumerate(A,N,aa(list(nat),list(A),map(nat,A,F),upt(N,M))) = aa(list(nat),list(product_prod(nat,A)),map(nat,product_prod(nat,A),aTP_Lamp_abc(fun(nat,A),fun(nat,product_prod(nat,A)),F)),upt(N,M)) ).

% enumerate_map_upt
tff(fact_4785_upt__eq__Cons__conv,axiom,
    ! [I: nat,J: nat,X: nat,Xs: list(nat)] :
      ( ( upt(I,J) = aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),X),Xs) )
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
        & ( I = X )
        & ( upt(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),one_one(nat)),J) = Xs ) ) ) ).

% upt_eq_Cons_conv
tff(fact_4786_map__upt__Suc,axiom,
    ! [A: $tType,F: fun(nat,A),N: nat] : aa(list(nat),list(A),map(nat,A,F),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,F,zero_zero(nat))),aa(list(nat),list(A),map(nat,A,aTP_Lamp_abd(fun(nat,A),fun(nat,A),F)),upt(zero_zero(nat),N))) ).

% map_upt_Suc
tff(fact_4787_upt__filter__extend,axiom,
    ! [U: nat,U3: nat,P: fun(nat,$o)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),U),U3)
     => ( ! [I2: nat] :
            ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),U),I2)
              & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),U3) )
           => ~ aa(nat,$o,P,I2) )
       => ( aa(list(nat),list(nat),filter2(nat,P),upt(zero_zero(nat),U)) = aa(list(nat),list(nat),filter2(nat,P),upt(zero_zero(nat),U3)) ) ) ) ).

% upt_filter_extend
tff(fact_4788_map__decr__upt,axiom,
    ! [M: nat,N: nat] : aa(list(nat),list(nat),map(nat,nat,aTP_Lamp_jn(nat,nat)),upt(aa(nat,nat,suc,M),aa(nat,nat,suc,N))) = upt(M,N) ).

% map_decr_upt
tff(fact_4789_map__nth,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(nat),list(A),map(nat,A,nth(A,Xs)),upt(zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xs))) = Xs ).

% map_nth
tff(fact_4790_filter__upt__last,axiom,
    ! [A: $tType,P: fun(A,$o),L: list(A),Js: list(nat),J: nat,I: nat] :
      ( ( aa(list(nat),list(nat),filter2(nat,aa(list(A),fun(nat,$o),aTP_Lamp_abe(fun(A,$o),fun(list(A),fun(nat,$o)),P),L)),upt(zero_zero(nat),aa(list(A),nat,size_size(list(A)),L))) = aa(list(nat),list(nat),aa(list(nat),fun(list(nat),list(nat)),append(nat),Js),aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),J),nil(nat))) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),I)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),L))
         => ~ aa(A,$o,P,aa(nat,A,nth(A,L),I)) ) ) ) ).

% filter_upt_last
tff(fact_4791_sorted__wrt__less__sum__mono__lowerbound,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [F: fun(nat,A),Ns: list(nat)] :
          ( ! [X2: nat,Y2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X2),Y2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F,X2)),aa(nat,A,F,Y2)) )
         => ( sorted_wrt(nat,ord_less(nat),Ns)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(nat),nat,size_size(list(nat)),Ns)))),groups8242544230860333062m_list(A,aa(list(nat),list(A),map(nat,A,F),Ns))) ) ) ) ).

% sorted_wrt_less_sum_mono_lowerbound
tff(fact_4792_map__of__distinct__upd4,axiom,
    ! [A: $tType,B: $tType,X: A,Xs: list(product_prod(A,B)),Ys: list(product_prod(A,B)),Y: B] :
      ( ~ aa(set(A),$o,member(A,X),aa(list(A),set(A),set2(A),aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xs)))
     => ( ~ aa(set(A),$o,member(A,X),aa(list(A),set(A),set2(A),aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Ys)))
       => ( map_of(A,B,aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(list(product_prod(A,B)),fun(list(product_prod(A,B)),list(product_prod(A,B))),append(product_prod(A,B)),Xs),Ys)) = fun_upd(A,option(B),map_of(A,B,aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(list(product_prod(A,B)),fun(list(product_prod(A,B)),list(product_prod(A,B))),append(product_prod(A,B)),Xs),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)),Ys))),X,none(B)) ) ) ) ).

% map_of_distinct_upd4
tff(fact_4793_map__of__distinct__upd3,axiom,
    ! [A: $tType,B: $tType,X: A,Xs: list(product_prod(A,B)),Ys: list(product_prod(A,B)),Y: B,Y6: B] :
      ( ~ aa(set(A),$o,member(A,X),aa(list(A),set(A),set2(A),aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xs)))
     => ( ~ aa(set(A),$o,member(A,X),aa(list(A),set(A),set2(A),aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Ys)))
       => ( map_of(A,B,aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(list(product_prod(A,B)),fun(list(product_prod(A,B)),list(product_prod(A,B))),append(product_prod(A,B)),Xs),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)),Ys))) = fun_upd(A,option(B),map_of(A,B,aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(list(product_prod(A,B)),fun(list(product_prod(A,B)),list(product_prod(A,B))),append(product_prod(A,B)),Xs),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),Y6)),Ys))),X,aa(B,option(B),some(B),Y)) ) ) ) ).

% map_of_distinct_upd3
tff(fact_4794_sum__list__0,axiom,
    ! [B: $tType,A: $tType] :
      ( monoid_add(A)
     => ! [Xs: list(B)] : groups8242544230860333062m_list(A,aa(list(B),list(A),map(B,A,aTP_Lamp_abf(B,A)),Xs)) = zero_zero(A) ) ).

% sum_list_0
tff(fact_4795_empty__eq__map__of__iff,axiom,
    ! [B: $tType,A: $tType,Xys: list(product_prod(A,B))] :
      ( ( aTP_Lamp_tf(A,option(B)) = map_of(A,B,Xys) )
    <=> ( Xys = nil(product_prod(A,B)) ) ) ).

% empty_eq_map_of_iff
tff(fact_4796_sum__list__upt,axiom,
    ! [M: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),N)
     => ( groups8242544230860333062m_list(nat,upt(M,N)) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_dj(nat,nat)),set_or7035219750837199246ssThan(nat,M,N)) ) ) ).

% sum_list_upt
tff(fact_4797_map__of__is__SomeI,axiom,
    ! [A: $tType,B: $tType,Xys: list(product_prod(A,B)),X: A,Y: B] :
      ( distinct(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xys))
     => ( aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),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_4798_Some__eq__map__of__iff,axiom,
    ! [B: $tType,A: $tType,Xys: list(product_prod(A,B)),Y: B,X: A] :
      ( distinct(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xys))
     => ( ( aa(B,option(B),some(B),Y) = aa(A,option(B),map_of(A,B,Xys),X) )
      <=> aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),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_4799_map__of__eq__Some__iff,axiom,
    ! [B: $tType,A: $tType,Xys: list(product_prod(A,B)),X: A,Y: B] :
      ( distinct(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xys))
     => ( ( aa(A,option(B),map_of(A,B,Xys),X) = aa(B,option(B),some(B),Y) )
      <=> aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),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_4800_sum__list__addf,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [F: fun(B,A),G: fun(B,A),Xs: list(B)] : groups8242544230860333062m_list(A,aa(list(B),list(A),map(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_cy(fun(B,A),fun(fun(B,A),fun(B,A)),F),G)),Xs)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),groups8242544230860333062m_list(A,aa(list(B),list(A),map(B,A,F),Xs))),groups8242544230860333062m_list(A,aa(list(B),list(A),map(B,A,G),Xs))) ) ).

% sum_list_addf
tff(fact_4801_sum__list__const__mult,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_0(A)
     => ! [C2: A,F: fun(B,A),Xs: list(B)] : groups8242544230860333062m_list(A,aa(list(B),list(A),map(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_dc(A,fun(fun(B,A),fun(B,A)),C2),F)),Xs)) = aa(A,A,aa(A,fun(A,A),times_times(A),C2),groups8242544230860333062m_list(A,aa(list(B),list(A),map(B,A,F),Xs))) ) ).

% sum_list_const_mult
tff(fact_4802_sum__list__mult__const,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_0(A)
     => ! [F: fun(B,A),C2: A,Xs: list(B)] : groups8242544230860333062m_list(A,aa(list(B),list(A),map(B,A,aa(A,fun(B,A),aTP_Lamp_db(fun(B,A),fun(A,fun(B,A)),F),C2)),Xs)) = aa(A,A,aa(A,fun(A,A),times_times(A),groups8242544230860333062m_list(A,aa(list(B),list(A),map(B,A,F),Xs))),C2) ) ).

% sum_list_mult_const
tff(fact_4803_sum__list__subtractf,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_add(A)
     => ! [F: fun(B,A),G: fun(B,A),Xs: list(B)] : groups8242544230860333062m_list(A,aa(list(B),list(A),map(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_dd(fun(B,A),fun(fun(B,A),fun(B,A)),F),G)),Xs)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),groups8242544230860333062m_list(A,aa(list(B),list(A),map(B,A,F),Xs))),groups8242544230860333062m_list(A,aa(list(B),list(A),map(B,A,G),Xs))) ) ).

% sum_list_subtractf
tff(fact_4804_weak__map__of__SomeI,axiom,
    ! [A: $tType,B: $tType,K: A,X: B,L: list(product_prod(A,B))] :
      ( aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K),X)),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),L))
     => ? [X2: B] : aa(A,option(B),map_of(A,B,L),K) = aa(B,option(B),some(B),X2) ) ).

% weak_map_of_SomeI
tff(fact_4805_map__of__SomeD,axiom,
    ! [A: $tType,B: $tType,Xs: list(product_prod(B,A)),K: B,Y: A] :
      ( ( aa(B,option(A),map_of(B,A,Xs),K) = aa(A,option(A),some(A),Y) )
     => aa(set(product_prod(B,A)),$o,member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),K),Y)),aa(list(product_prod(B,A)),set(product_prod(B,A)),set2(product_prod(B,A)),Xs)) ) ).

% map_of_SomeD
tff(fact_4806_uminus__sum__list__map,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_add(A)
     => ! [F: fun(B,A),Xs: list(B)] : aa(A,A,uminus_uminus(A),groups8242544230860333062m_list(A,aa(list(B),list(A),map(B,A,F),Xs))) = groups8242544230860333062m_list(A,aa(list(B),list(A),map(B,A,aa(fun(B,A),fun(B,A),comp(A,A,B,uminus_uminus(A)),F)),Xs)) ) ).

% uminus_sum_list_map
tff(fact_4807_map__of__Cons__code_I2_J,axiom,
    ! [A: $tType,B: $tType,L: B,V: A,Ps: list(product_prod(B,A)),K: B] :
      aa(B,option(A),map_of(B,A,aa(list(product_prod(B,A)),list(product_prod(B,A)),aa(product_prod(B,A),fun(list(product_prod(B,A)),list(product_prod(B,A))),cons(product_prod(B,A)),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),L),V)),Ps)),K) = $ite(L = K,aa(A,option(A),some(A),V),aa(B,option(A),map_of(B,A,Ps),K)) ).

% map_of_Cons_code(2)
tff(fact_4808_sum__list__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( monoid_add(B)
        & ordere6658533253407199908up_add(B) )
     => ! [Xs: list(A),F: fun(A,B),G: fun(A,B)] :
          ( ! [X2: A] :
              ( aa(set(A),$o,member(A,X2),aa(list(A),set(A),set2(A),Xs))
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,X2)),aa(A,B,G,X2)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),groups8242544230860333062m_list(B,aa(list(A),list(B),map(A,B,F),Xs))),groups8242544230860333062m_list(B,aa(list(A),list(B),map(A,B,G),Xs))) ) ) ).

% sum_list_mono
tff(fact_4809_sum__list__map__filter_H,axiom,
    ! [A: $tType,B: $tType] :
      ( monoid_add(A)
     => ! [F: fun(B,A),P: fun(B,$o),Xs: list(B)] : groups8242544230860333062m_list(A,aa(list(B),list(A),map(B,A,F),aa(list(B),list(B),filter2(B,P),Xs))) = groups8242544230860333062m_list(A,aa(list(B),list(A),map(B,A,aa(fun(B,$o),fun(B,A),aTP_Lamp_abg(fun(B,A),fun(fun(B,$o),fun(B,A)),F),P)),Xs)) ) ).

% sum_list_map_filter'
tff(fact_4810_distinct__sum__list__conv__Sum,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Xs: list(A)] :
          ( distinct(A,Xs)
         => ( groups8242544230860333062m_list(A,Xs) = aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_abh(A,A)),aa(list(A),set(A),set2(A),Xs)) ) ) ) ).

% distinct_sum_list_conv_Sum
tff(fact_4811_finite__range__map__of,axiom,
    ! [A: $tType,B: $tType,Xys: list(product_prod(B,A))] : aa(set(option(A)),$o,finite_finite2(option(A)),aa(set(B),set(option(A)),image2(B,option(A),map_of(B,A,Xys)),top_top(set(B)))) ).

% finite_range_map_of
tff(fact_4812_card__length__sum__list__rec,axiom,
    ! [M: nat,N4: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),M)
     => ( aa(set(list(nat)),nat,finite_card(list(nat)),aa(fun(list(nat),$o),set(list(nat)),collect(list(nat)),aa(nat,fun(list(nat),$o),aTP_Lamp_abi(nat,fun(nat,fun(list(nat),$o)),M),N4))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(list(nat)),nat,finite_card(list(nat)),aa(fun(list(nat),$o),set(list(nat)),collect(list(nat)),aa(nat,fun(list(nat),$o),aTP_Lamp_abj(nat,fun(nat,fun(list(nat),$o)),M),N4)))),aa(set(list(nat)),nat,finite_card(list(nat)),aa(fun(list(nat),$o),set(list(nat)),collect(list(nat)),aa(nat,fun(list(nat),$o),aTP_Lamp_abk(nat,fun(nat,fun(list(nat),$o)),M),N4)))) ) ) ).

% card_length_sum_list_rec
tff(fact_4813_card__length__sum__list,axiom,
    ! [M: nat,N4: nat] : aa(set(list(nat)),nat,finite_card(list(nat)),aa(fun(list(nat),$o),set(list(nat)),collect(list(nat)),aa(nat,fun(list(nat),$o),aTP_Lamp_abi(nat,fun(nat,fun(list(nat),$o)),M),N4))) = 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),N4),M)),one_one(nat))),N4) ).

% card_length_sum_list
tff(fact_4814_sum__list__triv,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(A)
     => ! [R2: A,Xs: list(B)] : groups8242544230860333062m_list(A,aa(list(B),list(A),map(B,A,aTP_Lamp_ja(A,fun(B,A),R2)),Xs)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(list(B),nat,size_size(list(B)),Xs))),R2) ) ).

% sum_list_triv
tff(fact_4815_sum__list__Suc,axiom,
    ! [A: $tType,F: fun(A,nat),Xs: list(A)] : groups8242544230860333062m_list(nat,aa(list(A),list(nat),map(A,nat,aTP_Lamp_jj(fun(A,nat),fun(A,nat),F)),Xs)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),groups8242544230860333062m_list(nat,aa(list(A),list(nat),map(A,nat,F),Xs))),aa(list(A),nat,size_size(list(A)),Xs)) ).

% sum_list_Suc
tff(fact_4816_map__of__map,axiom,
    ! [B: $tType,C: $tType,A: $tType,F: fun(C,B),Xs: list(product_prod(A,C))] : map_of(A,B,aa(list(product_prod(A,C)),list(product_prod(A,B)),map(product_prod(A,C),product_prod(A,B),aa(fun(A,fun(C,product_prod(A,B))),fun(product_prod(A,C),product_prod(A,B)),product_case_prod(A,C,product_prod(A,B)),aTP_Lamp_abl(fun(C,B),fun(A,fun(C,product_prod(A,B))),F))),Xs)) = aa(fun(A,option(C)),fun(A,option(B)),comp(option(C),option(B),A,map_option(C,B,F)),map_of(A,C,Xs)) ).

% map_of_map
tff(fact_4817_map__of__Some__split,axiom,
    ! [B: $tType,A: $tType,Xs: list(product_prod(B,A)),K: B,V: A] :
      ( ( aa(B,option(A),map_of(B,A,Xs),K) = aa(A,option(A),some(A),V) )
     => ? [Ys2: list(product_prod(B,A)),Zs2: list(product_prod(B,A))] :
          ( ( Xs = aa(list(product_prod(B,A)),list(product_prod(B,A)),aa(list(product_prod(B,A)),fun(list(product_prod(B,A)),list(product_prod(B,A))),append(product_prod(B,A)),Ys2),aa(list(product_prod(B,A)),list(product_prod(B,A)),aa(product_prod(B,A),fun(list(product_prod(B,A)),list(product_prod(B,A))),cons(product_prod(B,A)),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),K),V)),Zs2)) )
          & ( aa(B,option(A),map_of(B,A,Ys2),K) = none(A) ) ) ) ).

% map_of_Some_split
tff(fact_4818_sum__list__sum__nth,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Xs: list(A)] : groups8242544230860333062m_list(A,Xs) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),nth(A,Xs)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xs))) ) ).

% sum_list_sum_nth
tff(fact_4819_map__of__Some__filter__not__in,axiom,
    ! [B: $tType,A: $tType,Xs: list(product_prod(B,A)),K: B,V: A,P: fun(product_prod(B,A),$o)] :
      ( ( aa(B,option(A),map_of(B,A,Xs),K) = aa(A,option(A),some(A),V) )
     => ( ~ aa(product_prod(B,A),$o,P,aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),K),V))
       => ( distinct(B,aa(list(product_prod(B,A)),list(B),map(product_prod(B,A),B,product_fst(B,A)),Xs))
         => ( aa(B,option(A),map_of(B,A,aa(list(product_prod(B,A)),list(product_prod(B,A)),filter2(product_prod(B,A),P),Xs)),K) = none(A) ) ) ) ) ).

% map_of_Some_filter_not_in
tff(fact_4820_set__map__of__compr,axiom,
    ! [B: $tType,A: $tType,Xs: list(product_prod(A,B))] :
      ( distinct(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xs))
     => ( aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Xs) = aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aTP_Lamp_abm(list(product_prod(A,B)),fun(A,fun(B,$o)),Xs))) ) ) ).

% set_map_of_compr
tff(fact_4821_map__of__map__restrict,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),Ks: list(A)] : map_of(A,B,aa(list(A),list(product_prod(A,B)),map(A,product_prod(A,B),aTP_Lamp_abn(fun(A,B),fun(A,product_prod(A,B)),F)),Ks)) = restrict_map(A,B,aa(fun(A,B),fun(A,option(B)),comp(B,option(B),A,some(B)),F),aa(list(A),set(A),set2(A),Ks)) ).

% map_of_map_restrict
tff(fact_4822_map__of__distinct__lookup,axiom,
    ! [A: $tType,B: $tType,X: A,Xs: list(product_prod(A,B)),Ys: list(product_prod(A,B)),Y: B] :
      ( ~ aa(set(A),$o,member(A,X),aa(list(A),set(A),set2(A),aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xs)))
     => ( ~ aa(set(A),$o,member(A,X),aa(list(A),set(A),set2(A),aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Ys)))
       => ( aa(A,option(B),map_of(A,B,aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(list(product_prod(A,B)),fun(list(product_prod(A,B)),list(product_prod(A,B))),append(product_prod(A,B)),Xs),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)),Ys))),X) = aa(B,option(B),some(B),Y) ) ) ) ).

% map_of_distinct_lookup
tff(fact_4823_map__of__distinct__upd2,axiom,
    ! [A: $tType,B: $tType,X: A,Xs: list(product_prod(A,B)),Ys: list(product_prod(A,B)),Y: B] :
      ( ~ aa(set(A),$o,member(A,X),aa(list(A),set(A),set2(A),aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xs)))
     => ( ~ aa(set(A),$o,member(A,X),aa(list(A),set(A),set2(A),aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Ys)))
       => ( map_of(A,B,aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(list(product_prod(A,B)),fun(list(product_prod(A,B)),list(product_prod(A,B))),append(product_prod(A,B)),Xs),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)),Ys))) = fun_upd(A,option(B),map_of(A,B,aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(list(product_prod(A,B)),fun(list(product_prod(A,B)),list(product_prod(A,B))),append(product_prod(A,B)),Xs),Ys)),X,aa(B,option(B),some(B),Y)) ) ) ) ).

% map_of_distinct_upd2
tff(fact_4824_sum__list__map__eq__sum__count2,axiom,
    ! [A: $tType,Xs: list(A),X5: set(A),F: fun(A,nat)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),X5)
     => ( aa(set(A),$o,finite_finite2(A),X5)
       => ( groups8242544230860333062m_list(nat,aa(list(A),list(nat),map(A,nat,F),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_abo(list(A),fun(fun(A,nat),fun(A,nat)),Xs),F)),X5) ) ) ) ).

% sum_list_map_eq_sum_count2
tff(fact_4825_sum__list__map__eq__sum__count,axiom,
    ! [A: $tType,F: fun(A,nat),Xs: list(A)] : groups8242544230860333062m_list(nat,aa(list(A),list(nat),map(A,nat,F),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_abp(fun(A,nat),fun(list(A),fun(A,nat)),F),Xs)),aa(list(A),set(A),set2(A),Xs)) ).

% sum_list_map_eq_sum_count
tff(fact_4826_transpose__aux__filter__head,axiom,
    ! [A: $tType,Xss: list(list(A))] : concat(A,aa(list(list(A)),list(list(A)),map(list(A),list(A),case_list(list(A),A,nil(A),aTP_Lamp_aac(A,fun(list(A),list(A))))),Xss)) = aa(list(list(A)),list(A),map(list(A),A,hd(A)),aa(list(list(A)),list(list(A)),filter2(list(A),aTP_Lamp_zt(list(A),$o)),Xss)) ).

% transpose_aux_filter_head
tff(fact_4827_hd__def,axiom,
    ! [A: $tType,List: list(A)] : aa(list(A),A,hd(A),List) = aa(list(A),A,case_list(A,A,undefined(A),aTP_Lamp_abq(A,fun(list(A),A))),List) ).

% hd_def
tff(fact_4828_tl__remdups__adj,axiom,
    ! [A: $tType,Ys: list(A)] :
      ( ( Ys != nil(A) )
     => ( aa(list(A),list(A),tl(A),remdups_adj(A,Ys)) = remdups_adj(A,dropWhile(A,aTP_Lamp_abr(list(A),fun(A,$o),Ys),aa(list(A),list(A),tl(A),Ys))) ) ) ).

% tl_remdups_adj
tff(fact_4829_count__list_Osimps_I2_J,axiom,
    ! [A: $tType,X: A,Xs: list(A),Y: A] :
      count_list(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs),Y) = $ite(X = Y,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),count_list(A,Xs,Y)),one_one(nat)),count_list(A,Xs,Y)) ).

% count_list.simps(2)
tff(fact_4830_insort__key__remove1,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [A3: A,Xs: list(A),F: fun(A,B)] :
          ( aa(set(A),$o,member(A,A3),aa(list(A),set(A),set2(A),Xs))
         => ( sorted_wrt(B,ord_less_eq(B),aa(list(A),list(B),map(A,B,F),Xs))
           => ( ( aa(list(A),A,hd(A),aa(list(A),list(A),filter2(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_abs(A,fun(fun(A,B),fun(A,$o)),A3),F)),Xs)) = A3 )
             => ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,B,F),A3),remove1(A,A3,Xs)) = Xs ) ) ) ) ) ).

% insort_key_remove1
tff(fact_4831_partition__filter__conv,axiom,
    ! [A: $tType,F: fun(A,$o),Xs: list(A)] : aa(list(A),product_prod(list(A),list(A)),partition(A,F),Xs) = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),filter2(A,F),Xs)),aa(list(A),list(A),filter2(A,aa(fun(A,$o),fun(A,$o),comp($o,$o,A,fNot),F)),Xs)) ).

% partition_filter_conv
tff(fact_4832_partition__rev__filter__conv,axiom,
    ! [A: $tType,P: fun(A,$o),Yes2: list(A),No2: list(A),Xs: list(A)] : partition_rev(A,P,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes2),No2),Xs) = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),rev(A,aa(list(A),list(A),filter2(A,P),Xs))),Yes2)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),rev(A,aa(list(A),list(A),filter2(A,aa(fun(A,$o),fun(A,$o),comp($o,$o,A,fNot),P)),Xs))),No2)) ).

% partition_rev_filter_conv
tff(fact_4833_mergesort__remdups__def,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] : aa(list(A),list(A),mergesort_remdups(A),Xs) = merge_list(A,nil(list(A)),aa(list(A),list(list(A)),map(A,list(A),aTP_Lamp_abt(A,list(A))),Xs)) ) ).

% mergesort_remdups_def
tff(fact_4834_partition__rev_Osimps_I2_J,axiom,
    ! [A: $tType,P: fun(A,$o),Yes2: list(A),No2: list(A),X: A,Xs: list(A)] :
      partition_rev(A,P,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes2),No2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = partition_rev(A,P,
        $ite(aa(A,$o,P,X),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Yes2)),No2),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),No2))),
        Xs) ).

% partition_rev.simps(2)
tff(fact_4835_partition__rev_Osimps_I1_J,axiom,
    ! [A: $tType,P: fun(A,$o),Yes2: list(A),No2: list(A)] : partition_rev(A,P,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes2),No2),nil(A)) = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes2),No2) ).

% partition_rev.simps(1)
tff(fact_4836_partition_Osimps_I1_J,axiom,
    ! [A: $tType,P: fun(A,$o)] : aa(list(A),product_prod(list(A),list(A)),partition(A,P),nil(A)) = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)) ).

% partition.simps(1)
tff(fact_4837_partition__P,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A),Yes2: list(A),No2: list(A)] :
      ( ( aa(list(A),product_prod(list(A),list(A)),partition(A,P),Xs) = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes2),No2) )
     => ( ! [X3: A] :
            ( aa(set(A),$o,member(A,X3),aa(list(A),set(A),set2(A),Yes2))
           => aa(A,$o,P,X3) )
        & ! [X3: A] :
            ( aa(set(A),$o,member(A,X3),aa(list(A),set(A),set2(A),No2))
           => ~ aa(A,$o,P,X3) ) ) ) ).

% partition_P
tff(fact_4838_partition_Osimps_I2_J,axiom,
    ! [A: $tType,P: fun(A,$o),X: A,Xs: list(A)] : aa(list(A),product_prod(list(A),list(A)),partition(A,P),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(product_prod(list(A),list(A)),product_prod(list(A),list(A)),aa(fun(list(A),fun(list(A),product_prod(list(A),list(A)))),fun(product_prod(list(A),list(A)),product_prod(list(A),list(A))),product_case_prod(list(A),list(A),product_prod(list(A),list(A))),aa(A,fun(list(A),fun(list(A),product_prod(list(A),list(A)))),aTP_Lamp_abu(fun(A,$o),fun(A,fun(list(A),fun(list(A),product_prod(list(A),list(A))))),P),X)),aa(list(A),product_prod(list(A),list(A)),partition(A,P),Xs)) ).

% partition.simps(2)
tff(fact_4839_partition__rev_Oelims,axiom,
    ! [A: $tType,X: fun(A,$o),Xa: product_prod(list(A),list(A)),Xb: list(A),Y: product_prod(list(A),list(A))] :
      ( ( partition_rev(A,X,Xa,Xb) = Y )
     => ( ! [Yes: list(A),No: list(A)] :
            ( ( Xa = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes),No) )
           => ( ( Xb = nil(A) )
             => ( Y != aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes),No) ) ) )
       => ~ ! [Yes: list(A),No: list(A)] :
              ( ( Xa = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes),No) )
             => ! [X2: A,Xs3: list(A)] :
                  ( ( Xb = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs3) )
                 => ( Y != partition_rev(A,X,
                        $ite(aa(A,$o,X,X2),aa(list(A),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),X2),Yes)),No),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),No))),
                        Xs3) ) ) ) ) ) ).

% partition_rev.elims
tff(fact_4840_partition__set,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A),Yes2: list(A),No2: list(A)] :
      ( ( aa(list(A),product_prod(list(A),list(A)),partition(A,P),Xs) = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes2),No2) )
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(list(A),set(A),set2(A),Yes2)),aa(list(A),set(A),set2(A),No2)) = aa(list(A),set(A),set2(A),Xs) ) ) ).

% partition_set
tff(fact_4841_inv__image__partition,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o),Ys: list(A)] :
      ( ! [X2: A] :
          ( aa(set(A),$o,member(A,X2),aa(list(A),set(A),set2(A),Xs))
         => aa(A,$o,P,X2) )
     => ( ! [Y2: A] :
            ( aa(set(A),$o,member(A,Y2),aa(list(A),set(A),set2(A),Ys))
           => ~ aa(A,$o,P,Y2) )
       => ( aa(set(product_prod(list(A),list(A))),set(list(A)),aa(fun(list(A),product_prod(list(A),list(A))),fun(set(product_prod(list(A),list(A))),set(list(A))),vimage(list(A),product_prod(list(A),list(A))),partition(A,P)),aa(set(product_prod(list(A),list(A))),set(product_prod(list(A),list(A))),aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),set(product_prod(list(A),list(A)))),insert2(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),bot_bot(set(product_prod(list(A),list(A)))))) = shuffles(A,Xs,Ys) ) ) ) ).

% inv_image_partition
tff(fact_4842_partition__rev_Opelims,axiom,
    ! [A: $tType,X: fun(A,$o),Xa: product_prod(list(A),list(A)),Xb: list(A),Y: product_prod(list(A),list(A))] :
      ( ( partition_rev(A,X,Xa,Xb) = Y )
     => ( accp(product_prod(fun(A,$o),product_prod(product_prod(list(A),list(A)),list(A))),partition_rev_rel(A),aa(product_prod(product_prod(list(A),list(A)),list(A)),product_prod(fun(A,$o),product_prod(product_prod(list(A),list(A)),list(A))),aa(fun(A,$o),fun(product_prod(product_prod(list(A),list(A)),list(A)),product_prod(fun(A,$o),product_prod(product_prod(list(A),list(A)),list(A)))),product_Pair(fun(A,$o),product_prod(product_prod(list(A),list(A)),list(A))),X),aa(list(A),product_prod(product_prod(list(A),list(A)),list(A)),aa(product_prod(list(A),list(A)),fun(list(A),product_prod(product_prod(list(A),list(A)),list(A))),product_Pair(product_prod(list(A),list(A)),list(A)),Xa),Xb)))
       => ( ! [Yes: list(A),No: list(A)] :
              ( ( Xa = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes),No) )
             => ( ( Xb = nil(A) )
               => ( ( Y = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes),No) )
                 => ~ accp(product_prod(fun(A,$o),product_prod(product_prod(list(A),list(A)),list(A))),partition_rev_rel(A),aa(product_prod(product_prod(list(A),list(A)),list(A)),product_prod(fun(A,$o),product_prod(product_prod(list(A),list(A)),list(A))),aa(fun(A,$o),fun(product_prod(product_prod(list(A),list(A)),list(A)),product_prod(fun(A,$o),product_prod(product_prod(list(A),list(A)),list(A)))),product_Pair(fun(A,$o),product_prod(product_prod(list(A),list(A)),list(A))),X),aa(list(A),product_prod(product_prod(list(A),list(A)),list(A)),aa(product_prod(list(A),list(A)),fun(list(A),product_prod(product_prod(list(A),list(A)),list(A))),product_Pair(product_prod(list(A),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)),Yes),No)),nil(A)))) ) ) )
         => ~ ! [Yes: list(A),No: list(A)] :
                ( ( Xa = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes),No) )
               => ! [X2: A,Xs3: list(A)] :
                    ( ( Xb = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs3) )
                   => ( ( Y = partition_rev(A,X,
                            $ite(aa(A,$o,X,X2),aa(list(A),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),X2),Yes)),No),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),No))),
                            Xs3) )
                     => ~ accp(product_prod(fun(A,$o),product_prod(product_prod(list(A),list(A)),list(A))),partition_rev_rel(A),aa(product_prod(product_prod(list(A),list(A)),list(A)),product_prod(fun(A,$o),product_prod(product_prod(list(A),list(A)),list(A))),aa(fun(A,$o),fun(product_prod(product_prod(list(A),list(A)),list(A)),product_prod(fun(A,$o),product_prod(product_prod(list(A),list(A)),list(A)))),product_Pair(fun(A,$o),product_prod(product_prod(list(A),list(A)),list(A))),X),aa(list(A),product_prod(product_prod(list(A),list(A)),list(A)),aa(product_prod(list(A),list(A)),fun(list(A),product_prod(product_prod(list(A),list(A)),list(A))),product_Pair(product_prod(list(A),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)),Yes),No)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs3)))) ) ) ) ) ) ) ).

% partition_rev.pelims
tff(fact_4843_merge__list_Opelims,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: list(list(A)),Xa: list(list(A)),Y: list(A)] :
          ( ( merge_list(A,X,Xa) = Y )
         => ( accp(product_prod(list(list(A)),list(list(A))),merge_list_rel(A),aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),X),Xa))
           => ( ( ( X = nil(list(A)) )
               => ( ( Xa = nil(list(A)) )
                 => ( ( Y = nil(A) )
                   => ~ accp(product_prod(list(list(A)),list(list(A))),merge_list_rel(A),aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),nil(list(A))),nil(list(A)))) ) ) )
             => ( ( ( X = nil(list(A)) )
                 => ! [L2: list(A)] :
                      ( ( Xa = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L2),nil(list(A))) )
                     => ( ( Y = L2 )
                       => ~ accp(product_prod(list(list(A)),list(list(A))),merge_list_rel(A),aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),nil(list(A))),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L2),nil(list(A))))) ) ) )
               => ( ! [La: list(A),Acc22: list(list(A))] :
                      ( ( X = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc22) )
                     => ( ( Xa = nil(list(A)) )
                       => ( ( Y = merge_list(A,nil(list(A)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc22)) )
                         => ~ accp(product_prod(list(list(A)),list(list(A))),merge_list_rel(A),aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc22)),nil(list(A)))) ) ) )
                 => ( ! [La: list(A),Acc22: list(list(A))] :
                        ( ( X = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc22) )
                       => ! [L2: list(A)] :
                            ( ( Xa = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L2),nil(list(A))) )
                           => ( ( Y = merge_list(A,nil(list(A)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L2),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc22))) )
                             => ~ accp(product_prod(list(list(A)),list(list(A))),merge_list_rel(A),aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc22)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L2),nil(list(A))))) ) ) )
                   => ~ ! [L1: list(A),L22: list(A),Ls: list(list(A))] :
                          ( ( Xa = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L1),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L22),Ls)) )
                         => ( ( Y = merge_list(A,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),merge(A,L1,L22)),X),Ls) )
                           => ~ accp(product_prod(list(list(A)),list(list(A))),merge_list_rel(A),aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),X),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L1),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L22),Ls)))) ) ) ) ) ) ) ) ) ) ).

% merge_list.pelims
tff(fact_4844_map__of__distinct__upd,axiom,
    ! [A: $tType,B: $tType,X: A,Xs: list(product_prod(A,B)),Y: B] :
      ( ~ aa(set(A),$o,member(A,X),aa(list(A),set(A),set2(A),aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xs)))
     => ( map_add(A,B,fun_upd(A,option(B),aTP_Lamp_tf(A,option(B)),X,aa(B,option(B),some(B),Y)),map_of(A,B,Xs)) = fun_upd(A,option(B),map_of(A,B,Xs),X,aa(B,option(B),some(B),Y)) ) ) ).

% map_of_distinct_upd
tff(fact_4845_empty__eq__map__add__iff,axiom,
    ! [A: $tType,B: $tType,F: fun(A,option(B)),G: fun(A,option(B))] :
      ( ( aTP_Lamp_tf(A,option(B)) = map_add(A,B,F,G) )
    <=> ( ! [X4: A] : aa(A,option(B),F,X4) = none(B)
        & ! [X4: A] : aa(A,option(B),G,X4) = none(B) ) ) ).

% empty_eq_map_add_iff
tff(fact_4846_map__add__empty,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B))] : map_add(A,B,M,aTP_Lamp_tf(A,option(B))) = M ).

% map_add_empty
tff(fact_4847_empty__map__add,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B))] : map_add(A,B,aTP_Lamp_tf(A,option(B)),M) = M ).

% empty_map_add
tff(fact_4848_map__add__def,axiom,
    ! [B: $tType,A: $tType,M12: fun(A,option(B)),M23: fun(A,option(B)),X3: A] : aa(A,option(B),map_add(A,B,M12,M23),X3) = case_option(option(B),B,aa(A,option(B),M12,X3),some(B),aa(A,option(B),M23,X3)) ).

% map_add_def
tff(fact_4849_map__add__map__of__foldr,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B)),Ps: list(product_prod(A,B))] : map_add(A,B,M,map_of(A,B,Ps)) = aa(fun(A,option(B)),fun(A,option(B)),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_abv(A,fun(B,fun(fun(A,option(B)),fun(A,option(B)))))),Ps),M) ).

% map_add_map_of_foldr
tff(fact_4850_map__of__concat,axiom,
    ! [B: $tType,A: $tType,Xss: list(list(product_prod(A,B)))] : map_of(A,B,concat(product_prod(A,B),Xss)) = aa(fun(A,option(B)),fun(A,option(B)),foldr(list(product_prod(A,B)),fun(A,option(B)),aTP_Lamp_abw(list(product_prod(A,B)),fun(fun(A,option(B)),fun(A,option(B)))),Xss),aTP_Lamp_tf(A,option(B))) ).

% map_of_concat
tff(fact_4851_finite__range__map__of__map__add,axiom,
    ! [A: $tType,B: $tType,F: fun(B,option(A)),L: list(product_prod(B,A))] :
      ( aa(set(option(A)),$o,finite_finite2(option(A)),aa(set(B),set(option(A)),image2(B,option(A),F),top_top(set(B))))
     => aa(set(option(A)),$o,finite_finite2(option(A)),aa(set(B),set(option(A)),image2(B,option(A),map_add(B,A,F,map_of(B,A,L))),top_top(set(B)))) ) ).

% finite_range_map_of_map_add
tff(fact_4852_merge__list_Opsimps_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( accp(product_prod(list(list(A)),list(list(A))),merge_list_rel(A),aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),nil(list(A))),nil(list(A))))
       => ( merge_list(A,nil(list(A)),nil(list(A))) = nil(A) ) ) ) ).

% merge_list.psimps(1)
tff(fact_4853_merge__list_Opsimps_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: list(A)] :
          ( accp(product_prod(list(list(A)),list(list(A))),merge_list_rel(A),aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),nil(list(A))),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L),nil(list(A)))))
         => ( merge_list(A,nil(list(A)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L),nil(list(A)))) = L ) ) ) ).

% merge_list.psimps(2)
tff(fact_4854_merge__list_Opsimps_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [La2: list(A),Acc23: list(list(A))] :
          ( accp(product_prod(list(list(A)),list(list(A))),merge_list_rel(A),aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La2),Acc23)),nil(list(A))))
         => ( merge_list(A,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La2),Acc23),nil(list(A))) = merge_list(A,nil(list(A)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La2),Acc23)) ) ) ) ).

% merge_list.psimps(3)
tff(fact_4855_merge__list_Opsimps_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [La2: list(A),Acc23: list(list(A)),L: list(A)] :
          ( accp(product_prod(list(list(A)),list(list(A))),merge_list_rel(A),aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La2),Acc23)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L),nil(list(A)))))
         => ( merge_list(A,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La2),Acc23),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L),nil(list(A)))) = merge_list(A,nil(list(A)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La2),Acc23))) ) ) ) ).

% merge_list.psimps(4)
tff(fact_4856_merge__list_Opinduct,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A0: list(list(A)),A1: list(list(A)),P: fun(list(list(A)),fun(list(list(A)),$o))] :
          ( accp(product_prod(list(list(A)),list(list(A))),merge_list_rel(A),aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),A0),A1))
         => ( ( accp(product_prod(list(list(A)),list(list(A))),merge_list_rel(A),aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),nil(list(A))),nil(list(A))))
             => aa(list(list(A)),$o,aa(list(list(A)),fun(list(list(A)),$o),P,nil(list(A))),nil(list(A))) )
           => ( ! [L2: list(A)] :
                  ( accp(product_prod(list(list(A)),list(list(A))),merge_list_rel(A),aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),nil(list(A))),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L2),nil(list(A)))))
                 => aa(list(list(A)),$o,aa(list(list(A)),fun(list(list(A)),$o),P,nil(list(A))),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L2),nil(list(A)))) )
             => ( ! [La: list(A),Acc22: list(list(A))] :
                    ( accp(product_prod(list(list(A)),list(list(A))),merge_list_rel(A),aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc22)),nil(list(A))))
                   => ( aa(list(list(A)),$o,aa(list(list(A)),fun(list(list(A)),$o),P,nil(list(A))),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc22))
                     => aa(list(list(A)),$o,aa(list(list(A)),fun(list(list(A)),$o),P,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc22)),nil(list(A))) ) )
               => ( ! [La: list(A),Acc22: list(list(A)),L2: list(A)] :
                      ( accp(product_prod(list(list(A)),list(list(A))),merge_list_rel(A),aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc22)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L2),nil(list(A)))))
                     => ( aa(list(list(A)),$o,aa(list(list(A)),fun(list(list(A)),$o),P,nil(list(A))),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L2),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc22)))
                       => aa(list(list(A)),$o,aa(list(list(A)),fun(list(list(A)),$o),P,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc22)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L2),nil(list(A)))) ) )
                 => ( ! [Acc22: list(list(A)),L1: list(A),L22: list(A),Ls: list(list(A))] :
                        ( accp(product_prod(list(list(A)),list(list(A))),merge_list_rel(A),aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),Acc22),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L1),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L22),Ls))))
                       => ( aa(list(list(A)),$o,aa(list(list(A)),fun(list(list(A)),$o),P,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),merge(A,L1,L22)),Acc22)),Ls)
                         => aa(list(list(A)),$o,aa(list(list(A)),fun(list(list(A)),$o),P,Acc22),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L1),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L22),Ls))) ) )
                   => aa(list(list(A)),$o,aa(list(list(A)),fun(list(list(A)),$o),P,A0),A1) ) ) ) ) ) ) ) ).

% merge_list.pinduct
tff(fact_4857_merge__list_Opsimps_I5_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Acc23: list(list(A)),L12: list(A),L23: list(A),Ls2: list(list(A))] :
          ( accp(product_prod(list(list(A)),list(list(A))),merge_list_rel(A),aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),Acc23),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L12),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L23),Ls2))))
         => ( merge_list(A,Acc23,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L12),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L23),Ls2))) = merge_list(A,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),merge(A,L12,L23)),Acc23),Ls2) ) ) ) ).

% merge_list.psimps(5)
tff(fact_4858_quicksort__by__rel_Osimps_I2_J,axiom,
    ! [A: $tType,R: fun(A,fun(A,$o)),Sl2: list(A),X: A,Xs: list(A)] : aa(list(A),list(A),quicksort_by_rel(A,R,Sl2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(product_prod(list(A),list(A)),list(A),aa(fun(list(A),fun(list(A),list(A))),fun(product_prod(list(A),list(A)),list(A)),product_case_prod(list(A),list(A),list(A)),aa(A,fun(list(A),fun(list(A),list(A))),aa(list(A),fun(A,fun(list(A),fun(list(A),list(A)))),aTP_Lamp_abx(fun(A,fun(A,$o)),fun(list(A),fun(A,fun(list(A),fun(list(A),list(A))))),R),Sl2),X)),partition_rev(A,aa(A,fun(A,$o),aTP_Lamp_aak(fun(A,fun(A,$o)),fun(A,fun(A,$o)),R),X),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),Xs)) ).

% quicksort_by_rel.simps(2)
tff(fact_4859_quicksort__by__rel_Oelims,axiom,
    ! [A: $tType,X: fun(A,fun(A,$o)),Xa: list(A),Xb: list(A),Y: list(A)] :
      ( ( aa(list(A),list(A),quicksort_by_rel(A,X,Xa),Xb) = Y )
     => ( ( ( Xb = nil(A) )
         => ( Y != Xa ) )
       => ~ ! [X2: A,Xs3: list(A)] :
              ( ( Xb = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs3) )
             => ( Y != aa(product_prod(list(A),list(A)),list(A),aa(fun(list(A),fun(list(A),list(A))),fun(product_prod(list(A),list(A)),list(A)),product_case_prod(list(A),list(A),list(A)),aa(A,fun(list(A),fun(list(A),list(A))),aa(list(A),fun(A,fun(list(A),fun(list(A),list(A)))),aTP_Lamp_abx(fun(A,fun(A,$o)),fun(list(A),fun(A,fun(list(A),fun(list(A),list(A))))),X),Xa),X2)),partition_rev(A,aa(A,fun(A,$o),aTP_Lamp_aak(fun(A,fun(A,$o)),fun(A,fun(A,$o)),X),X2),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),Xs3)) ) ) ) ) ).

% quicksort_by_rel.elims
tff(fact_4860_enumerate__replicate__eq,axiom,
    ! [A: $tType,N: nat,M: nat,A3: A] : enumerate(A,N,replicate(A,M,A3)) = aa(list(nat),list(product_prod(nat,A)),map(nat,product_prod(nat,A),aTP_Lamp_aby(A,fun(nat,product_prod(nat,A)),A3)),upt(N,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M))) ).

% enumerate_replicate_eq
tff(fact_4861_tl__replicate,axiom,
    ! [A: $tType,N: nat,X: A] : aa(list(A),list(A),tl(A),replicate(A,N,X)) = replicate(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)),X) ).

% tl_replicate
tff(fact_4862_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),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))) ) ) ).

% set_replicate
tff(fact_4863_map__fst__mk__fst,axiom,
    ! [B: $tType,A: $tType,K: A,L: list(B)] : aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),aa(list(B),list(product_prod(A,B)),map(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K)),L)) = replicate(A,aa(list(B),nat,size_size(list(B)),L),K) ).

% map_fst_mk_fst
tff(fact_4864_map__snd__mk__snd,axiom,
    ! [B: $tType,A: $tType,K: A,L: list(B)] : aa(list(product_prod(B,A)),list(A),map(product_prod(B,A),A,product_snd(B,A)),aa(list(B),list(product_prod(B,A)),map(B,product_prod(B,A),aa(A,fun(B,product_prod(B,A)),aTP_Lamp_vx(A,fun(B,product_prod(B,A))),K)),L)) = replicate(A,aa(list(B),nat,size_size(list(B)),L),K) ).

% map_snd_mk_snd
tff(fact_4865_map__replicate__const,axiom,
    ! [B: $tType,A: $tType,K: A,Lst: list(B)] : aa(list(B),list(A),map(B,A,aa(A,fun(B,A),aTP_Lamp_kf(A,fun(B,A)),K)),Lst) = replicate(A,aa(list(B),nat,size_size(list(B)),Lst),K) ).

% map_replicate_const
tff(fact_4866_replicate__length__filter,axiom,
    ! [A: $tType,X: A,Xs: list(A)] : replicate(A,aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),filter2(A,aa(A,fun(A,$o),fequal(A),X)),Xs)),X) = aa(list(A),list(A),filter2(A,aa(A,fun(A,$o),fequal(A),X)),Xs) ).

% replicate_length_filter
tff(fact_4867_sum__list__replicate,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [N: nat,C2: A] : groups8242544230860333062m_list(A,replicate(A,N,C2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),C2) ) ).

% sum_list_replicate
tff(fact_4868_map__replicate__trivial,axiom,
    ! [A: $tType,X: A,I: nat] : aa(list(nat),list(A),map(nat,A,aTP_Lamp_abz(A,fun(nat,A),X)),upt(zero_zero(nat),I)) = replicate(A,I,X) ).

% map_replicate_trivial
tff(fact_4869_set__replicate__conv__if,axiom,
    ! [A: $tType,N: nat,X: A] :
      aa(list(A),set(A),set2(A),replicate(A,N,X)) = $ite(N = zero_zero(nat),bot_bot(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) ).

% set_replicate_conv_if
tff(fact_4870_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),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))) ).

% set_replicate_Suc
tff(fact_4871_sort__quicksort__by__rel,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( linorder_sort_key(A,A,aTP_Lamp_yq(A,A)) = quicksort_by_rel(A,ord_less_eq(A),nil(A)) ) ) ).

% sort_quicksort_by_rel
tff(fact_4872_Cons__replicate__eq,axiom,
    ! [A: $tType,X: A,Xs: list(A),N: nat,Y: A] :
      ( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) = replicate(A,N,Y) )
    <=> ( ( X = Y )
        & aa(nat,$o,aa(nat,fun(nat,$o),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_4873_comm__append__is__replicate,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( Xs != nil(A) )
     => ( ( Ys != nil(A) )
       => ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Xs) )
         => ? [N3: nat,Zs2: list(A)] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),one_one(nat)),N3)
              & ( concat(A,replicate(list(A),N3,Zs2)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys) ) ) ) ) ) ).

% comm_append_is_replicate
tff(fact_4874_quicksort__by__rel_Opsimps_I2_J,axiom,
    ! [A: $tType,R: fun(A,fun(A,$o)),Sl2: list(A),X: A,Xs: list(A)] :
      ( accp(product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),quicksort_by_rel_rel(A),aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),aa(fun(A,fun(A,$o)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,$o)),product_prod(list(A),list(A))),R),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Sl2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs))))
     => ( aa(list(A),list(A),quicksort_by_rel(A,R,Sl2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(product_prod(list(A),list(A)),list(A),aa(fun(list(A),fun(list(A),list(A))),fun(product_prod(list(A),list(A)),list(A)),product_case_prod(list(A),list(A),list(A)),aa(A,fun(list(A),fun(list(A),list(A))),aa(list(A),fun(A,fun(list(A),fun(list(A),list(A)))),aTP_Lamp_abx(fun(A,fun(A,$o)),fun(list(A),fun(A,fun(list(A),fun(list(A),list(A))))),R),Sl2),X)),partition_rev(A,aa(A,fun(A,$o),aTP_Lamp_aak(fun(A,fun(A,$o)),fun(A,fun(A,$o)),R),X),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),Xs)) ) ) ).

% quicksort_by_rel.psimps(2)
tff(fact_4875_quicksort__by__rel_Opelims,axiom,
    ! [A: $tType,X: fun(A,fun(A,$o)),Xa: list(A),Xb: list(A),Y: list(A)] :
      ( ( aa(list(A),list(A),quicksort_by_rel(A,X,Xa),Xb) = Y )
     => ( accp(product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),quicksort_by_rel_rel(A),aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),aa(fun(A,fun(A,$o)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,$o)),product_prod(list(A),list(A))),X),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xa),Xb)))
       => ( ( ( Xb = nil(A) )
           => ( ( Y = Xa )
             => ~ accp(product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),quicksort_by_rel_rel(A),aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),aa(fun(A,fun(A,$o)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,$o)),product_prod(list(A),list(A))),X),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xa),nil(A)))) ) )
         => ~ ! [X2: A,Xs3: list(A)] :
                ( ( Xb = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs3) )
               => ( ( Y = aa(product_prod(list(A),list(A)),list(A),aa(fun(list(A),fun(list(A),list(A))),fun(product_prod(list(A),list(A)),list(A)),product_case_prod(list(A),list(A),list(A)),aa(A,fun(list(A),fun(list(A),list(A))),aa(list(A),fun(A,fun(list(A),fun(list(A),list(A)))),aTP_Lamp_abx(fun(A,fun(A,$o)),fun(list(A),fun(A,fun(list(A),fun(list(A),list(A))))),X),Xa),X2)),partition_rev(A,aa(A,fun(A,$o),aTP_Lamp_aak(fun(A,fun(A,$o)),fun(A,fun(A,$o)),X),X2),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),Xs3)) )
                 => ~ accp(product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),quicksort_by_rel_rel(A),aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),aa(fun(A,fun(A,$o)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,$o)),product_prod(list(A),list(A))),X),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xa),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs3)))) ) ) ) ) ) ).

% quicksort_by_rel.pelims
tff(fact_4876_quicksort__by__rel_Opinduct,axiom,
    ! [A: $tType,A0: fun(A,fun(A,$o)),A1: list(A),A22: list(A),P: fun(fun(A,fun(A,$o)),fun(list(A),fun(list(A),$o)))] :
      ( accp(product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),quicksort_by_rel_rel(A),aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),aa(fun(A,fun(A,$o)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,$o)),product_prod(list(A),list(A))),A0),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),A1),A22)))
     => ( ! [R6: fun(A,fun(A,$o)),Sl: list(A)] :
            ( accp(product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),quicksort_by_rel_rel(A),aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),aa(fun(A,fun(A,$o)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,$o)),product_prod(list(A),list(A))),R6),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Sl),nil(A))))
           => aa(list(A),$o,aa(list(A),fun(list(A),$o),aa(fun(A,fun(A,$o)),fun(list(A),fun(list(A),$o)),P,R6),Sl),nil(A)) )
       => ( ! [R6: fun(A,fun(A,$o)),Sl: list(A),X2: A,Xs3: list(A)] :
              ( accp(product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),quicksort_by_rel_rel(A),aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),aa(fun(A,fun(A,$o)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,$o)),product_prod(list(A),list(A))),R6),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Sl),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs3))))
             => ( ! [Xa3: product_prod(list(A),list(A)),Xb2: list(A),Y5: list(A)] :
                    ( ( Xa3 = partition_rev(A,aa(A,fun(A,$o),aTP_Lamp_aak(fun(A,fun(A,$o)),fun(A,fun(A,$o)),R6),X2),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),Xs3) )
                   => ( ( aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xb2),Y5) = Xa3 )
                     => aa(list(A),$o,aa(list(A),fun(list(A),$o),aa(fun(A,fun(A,$o)),fun(list(A),fun(list(A),$o)),P,R6),Sl),Y5) ) )
               => ( ! [Xa3: product_prod(list(A),list(A)),Xb2: list(A),Y5: list(A)] :
                      ( ( Xa3 = partition_rev(A,aa(A,fun(A,$o),aTP_Lamp_aak(fun(A,fun(A,$o)),fun(A,fun(A,$o)),R6),X2),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),Xs3) )
                     => ( ( aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xb2),Y5) = Xa3 )
                       => aa(list(A),$o,aa(list(A),fun(list(A),$o),aa(fun(A,fun(A,$o)),fun(list(A),fun(list(A),$o)),P,R6),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),aa(list(A),list(A),quicksort_by_rel(A,R6,Sl),Y5))),Xb2) ) )
                 => aa(list(A),$o,aa(list(A),fun(list(A),$o),aa(fun(A,fun(A,$o)),fun(list(A),fun(list(A),$o)),P,R6),Sl),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs3)) ) ) )
         => aa(list(A),$o,aa(list(A),fun(list(A),$o),aa(fun(A,fun(A,$o)),fun(list(A),fun(list(A),$o)),P,A0),A1),A22) ) ) ) ).

% quicksort_by_rel.pinduct
tff(fact_4877_quicksort__by__rel_Opsimps_I1_J,axiom,
    ! [A: $tType,R: fun(A,fun(A,$o)),Sl2: list(A)] :
      ( accp(product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),quicksort_by_rel_rel(A),aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),aa(fun(A,fun(A,$o)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,$o)),product_prod(list(A),list(A))),R),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Sl2),nil(A))))
     => ( aa(list(A),list(A),quicksort_by_rel(A,R,Sl2),nil(A)) = Sl2 ) ) ).

% quicksort_by_rel.psimps(1)
tff(fact_4878_total__lexord,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( total_on(A,top_top(set(A)),R2)
     => total_on(list(A),top_top(set(list(A))),lexord(A,R2)) ) ).

% total_lexord
tff(fact_4879_nths__Cons,axiom,
    ! [A: $tType,X: A,L: list(A),A4: set(nat)] :
      nths(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),L),A4) = aa(list(A),list(A),
        aa(list(A),fun(list(A),list(A)),append(A),
          $ite(aa(set(nat),$o,member(nat,zero_zero(nat)),A4),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,$o),set(nat),collect(nat),aTP_Lamp_aca(set(nat),fun(nat,$o),A4)))) ).

% nths_Cons
tff(fact_4880_map__of__mapk__SomeI,axiom,
    ! [A: $tType,B: $tType,C: $tType,F: fun(A,B),T3: list(product_prod(A,C)),K: A,X: C] :
      ( inj_on(A,B,F,top_top(set(A)))
     => ( ( aa(A,option(C),map_of(A,C,T3),K) = aa(C,option(C),some(C),X) )
       => ( aa(B,option(C),map_of(B,C,aa(list(product_prod(A,C)),list(product_prod(B,C)),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_acb(fun(A,B),fun(A,fun(C,product_prod(B,C))),F))),T3)),aa(A,B,F,K)) = aa(C,option(C),some(C),X) ) ) ) ).

% map_of_mapk_SomeI
tff(fact_4881_inj__on__empty,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B)] : inj_on(A,B,F,bot_bot(set(A))) ).

% inj_on_empty
tff(fact_4882_inj__uminus,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A4: set(A)] : inj_on(A,A,uminus_uminus(A),A4) ) ).

% inj_uminus
tff(fact_4883_inj__map__eq__map,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),Xs: list(A),Ys: list(A)] :
      ( inj_on(A,B,F,top_top(set(A)))
     => ( ( aa(list(A),list(B),map(A,B,F),Xs) = aa(list(A),list(B),map(A,B,F),Ys) )
      <=> ( Xs = Ys ) ) ) ).

% inj_map_eq_map
tff(fact_4884_nths__empty,axiom,
    ! [A: $tType,Xs: list(A)] : nths(A,Xs,bot_bot(set(nat))) = nil(A) ).

% nths_empty
tff(fact_4885_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_4886_lexord__cons__cons,axiom,
    ! [A: $tType,A3: A,X: list(A),B2: A,Y: list(A),R2: set(product_prod(A,A))] :
      ( aa(set(product_prod(list(A),list(A))),$o,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,R2))
    <=> ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R2)
        | ( ( A3 = B2 )
          & aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X),Y)),lexord(A,R2)) ) ) ) ).

% lexord_cons_cons
tff(fact_4887_lexord__Nil__left,axiom,
    ! [A: $tType,Y: list(A),R2: set(product_prod(A,A))] :
      ( aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),Y)),lexord(A,R2))
    <=> ? [A10: A,X4: list(A)] : Y = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A10),X4) ) ).

% lexord_Nil_left
tff(fact_4888_inj__divide__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A] :
          ( inj_on(A,A,aTP_Lamp_acc(A,fun(A,A),A3),top_top(set(A)))
        <=> ( A3 != zero_zero(A) ) ) ) ).

% inj_divide_right
tff(fact_4889_inj__mapI,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B)] :
      ( inj_on(A,B,F,top_top(set(A)))
     => inj_on(list(A),list(B),map(A,B,F),top_top(set(list(A)))) ) ).

% inj_mapI
tff(fact_4890_inj__map,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B)] :
      ( inj_on(list(A),list(B),map(A,B,F),top_top(set(list(A))))
    <=> inj_on(A,B,F,top_top(set(A))) ) ).

% inj_map
tff(fact_4891_inj__apfst,axiom,
    ! [B: $tType,C: $tType,A: $tType,F: fun(A,C)] :
      ( inj_on(product_prod(A,B),product_prod(C,B),product_apfst(A,C,B,F),top_top(set(product_prod(A,B))))
    <=> inj_on(A,C,F,top_top(set(A))) ) ).

% inj_apfst
tff(fact_4892_inj__apsnd,axiom,
    ! [A: $tType,C: $tType,B: $tType,F: 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),F),top_top(set(product_prod(A,B))))
    <=> inj_on(B,C,F,top_top(set(B))) ) ).

% inj_apsnd
tff(fact_4893_inj__on__apfst,axiom,
    ! [B: $tType,C: $tType,A: $tType,F: fun(A,C),A4: set(A)] :
      ( inj_on(product_prod(A,B),product_prod(C,B),product_apfst(A,C,B,F),product_Sigma(A,B,A4,aTP_Lamp_rb(A,set(B))))
    <=> inj_on(A,C,F,A4) ) ).

% inj_on_apfst
tff(fact_4894_inj__on__apsnd,axiom,
    ! [A: $tType,C: $tType,B: $tType,F: fun(B,C),A4: 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),F),product_Sigma(A,B,top_top(set(A)),aTP_Lamp_qz(set(B),fun(A,set(B)),A4)))
    <=> inj_on(B,C,F,A4) ) ).

% inj_on_apsnd
tff(fact_4895_inj__on__insert,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),A3: A,A4: set(A)] :
      ( inj_on(A,B,F,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),A4))
    <=> ( inj_on(A,B,F,A4)
        & ~ aa(set(B),$o,member(B,aa(A,B,F,A3)),aa(set(A),set(B),image2(A,B,F),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A)))))) ) ) ).

% inj_on_insert
tff(fact_4896_inj__mapD,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B)] :
      ( inj_on(list(A),list(B),map(A,B,F),top_top(set(list(A))))
     => inj_on(A,B,F,top_top(set(A))) ) ).

% inj_mapD
tff(fact_4897_fun_Oinj__map,axiom,
    ! [B: $tType,A: $tType,C: $tType,F: fun(A,B)] :
      ( inj_on(A,B,F,top_top(set(A)))
     => inj_on(fun(C,A),fun(C,B),comp(A,B,C,F),top_top(set(fun(C,A)))) ) ).

% fun.inj_map
tff(fact_4898_option_Oinj__map,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B)] :
      ( inj_on(A,B,F,top_top(set(A)))
     => inj_on(option(A),option(B),map_option(A,B,F),top_top(set(option(A)))) ) ).

% option.inj_map
tff(fact_4899_inj__fun,axiom,
    ! [B: $tType,C: $tType,A: $tType,F: fun(A,B)] :
      ( inj_on(A,B,F,top_top(set(A)))
     => inj_on(A,fun(C,B),aTP_Lamp_acd(fun(A,B),fun(A,fun(C,B)),F),top_top(set(A))) ) ).

% inj_fun
tff(fact_4900_injD,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),X: A,Y: A] :
      ( inj_on(A,B,F,top_top(set(A)))
     => ( ( aa(A,B,F,X) = aa(A,B,F,Y) )
       => ( X = Y ) ) ) ).

% injD
tff(fact_4901_injI,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B)] :
      ( ! [X2: A,Y2: A] :
          ( ( aa(A,B,F,X2) = aa(A,B,F,Y2) )
         => ( X2 = Y2 ) )
     => inj_on(A,B,F,top_top(set(A))) ) ).

% injI
tff(fact_4902_inj__eq,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),X: A,Y: A] :
      ( inj_on(A,B,F,top_top(set(A)))
     => ( ( aa(A,B,F,X) = aa(A,B,F,Y) )
      <=> ( X = Y ) ) ) ).

% inj_eq
tff(fact_4903_inj__def,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B)] :
      ( inj_on(A,B,F,top_top(set(A)))
    <=> ! [X4: A,Y3: A] :
          ( ( aa(A,B,F,X4) = aa(A,B,F,Y3) )
         => ( X4 = Y3 ) ) ) ).

% inj_def
tff(fact_4904_sorted__list__of__set_Oinj__on,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => inj_on(A,A,aTP_Lamp_yq(A,A),top_top(set(A))) ) ).

% sorted_list_of_set.inj_on
tff(fact_4905_inj__on__id2,axiom,
    ! [A: $tType,A4: set(A)] : inj_on(A,A,aTP_Lamp_au(A,A),A4) ).

% inj_on_id2
tff(fact_4906_inj__on__add_H,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [A3: A,A4: set(A)] : inj_on(A,A,aTP_Lamp_ace(A,fun(A,A),A3),A4) ) ).

% inj_on_add'
tff(fact_4907_inj__on__Int,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),A4: set(A),B3: set(A)] :
      ( ( inj_on(A,B,F,A4)
        | inj_on(A,B,F,B3) )
     => inj_on(A,B,F,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3)) ) ).

% inj_on_Int
tff(fact_4908_finite__inverse__image__gen,axiom,
    ! [A: $tType,B: $tType,A4: set(A),F: fun(B,A),D4: set(B)] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( inj_on(B,A,F,D4)
       => aa(set(B),$o,finite_finite2(B),aa(fun(B,$o),set(B),collect(B),aa(set(B),fun(B,$o),aa(fun(B,A),fun(set(B),fun(B,$o)),aTP_Lamp_acf(set(A),fun(fun(B,A),fun(set(B),fun(B,$o))),A4),F),D4))) ) ) ).

% finite_inverse_image_gen
tff(fact_4909_inj__on__convol__ident,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),X5: set(A)] : inj_on(A,product_prod(A,B),aTP_Lamp_abn(fun(A,B),fun(A,product_prod(A,B)),F),X5) ).

% inj_on_convol_ident
tff(fact_4910_inj__Pair_I1_J,axiom,
    ! [B: $tType,A: $tType,C2: fun(A,B),S: set(A)] : inj_on(A,product_prod(A,B),aTP_Lamp_abn(fun(A,B),fun(A,product_prod(A,B)),C2),S) ).

% inj_Pair(1)
tff(fact_4911_inj__Pair_I2_J,axiom,
    ! [B: $tType,A: $tType,C2: fun(A,B),S: set(A)] : inj_on(A,product_prod(B,A),aTP_Lamp_acg(fun(A,B),fun(A,product_prod(B,A)),C2),S) ).

% inj_Pair(2)
tff(fact_4912_inj__on__mult,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A3: A,A4: set(A)] :
          ( ( A3 != zero_zero(A) )
         => inj_on(A,A,aa(A,fun(A,A),times_times(A),A3),A4) ) ) ).

% inj_on_mult
tff(fact_4913_linorder__injI,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(A)
     => ! [F: fun(A,B)] :
          ( ! [X2: A,Y2: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X2),Y2)
             => ( aa(A,B,F,X2) != aa(A,B,F,Y2) ) )
         => inj_on(A,B,F,top_top(set(A))) ) ) ).

% linorder_injI
tff(fact_4914_inj__add__left,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [A3: A] : inj_on(A,A,aa(A,fun(A,A),plus_plus(A),A3),top_top(set(A))) ) ).

% inj_add_left
tff(fact_4915_inj__image__mem__iff,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),A3: A,A4: set(A)] :
      ( inj_on(A,B,F,top_top(set(A)))
     => ( aa(set(B),$o,member(B,aa(A,B,F,A3)),aa(set(A),set(B),image2(A,B,F),A4))
      <=> aa(set(A),$o,member(A,A3),A4) ) ) ).

% inj_image_mem_iff
tff(fact_4916_inj__image__eq__iff,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),A4: set(A),B3: set(A)] :
      ( inj_on(A,B,F,top_top(set(A)))
     => ( ( aa(set(A),set(B),image2(A,B,F),A4) = aa(set(A),set(B),image2(A,B,F),B3) )
      <=> ( A4 = B3 ) ) ) ).

% inj_image_eq_iff
tff(fact_4917_range__ex1__eq,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),B2: B] :
      ( inj_on(A,B,F,top_top(set(A)))
     => ( aa(set(B),$o,member(B,B2),aa(set(A),set(B),image2(A,B,F),top_top(set(A))))
      <=> ? [X4: A] :
            ( ( B2 = aa(A,B,F,X4) )
            & ! [Y3: A] :
                ( ( B2 = aa(A,B,F,Y3) )
               => ( Y3 = X4 ) ) ) ) ) ).

% range_ex1_eq
tff(fact_4918_inj__on__Un__image__eq__iff,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),A4: set(A),B3: set(A)] :
      ( inj_on(A,B,F,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3))
     => ( ( aa(set(A),set(B),image2(A,B,F),A4) = aa(set(A),set(B),image2(A,B,F),B3) )
      <=> ( A4 = B3 ) ) ) ).

% inj_on_Un_image_eq_iff
tff(fact_4919_inj__compose,axiom,
    ! [A: $tType,B: $tType,C: $tType,F: fun(A,B),G: fun(C,A)] :
      ( inj_on(A,B,F,top_top(set(A)))
     => ( inj_on(C,A,G,top_top(set(C)))
       => inj_on(C,B,aa(fun(C,A),fun(C,B),comp(A,B,C,F),G),top_top(set(C))) ) ) ).

% inj_compose
tff(fact_4920_map__injective,axiom,
    ! [A: $tType,B: $tType,F: fun(B,A),Xs: list(B),Ys: list(B)] :
      ( ( aa(list(B),list(A),map(B,A,F),Xs) = aa(list(B),list(A),map(B,A,F),Ys) )
     => ( inj_on(B,A,F,top_top(set(B)))
       => ( Xs = Ys ) ) ) ).

% map_injective
tff(fact_4921_inj__fn,axiom,
    ! [A: $tType,F: fun(A,A),N: nat] :
      ( inj_on(A,A,F,top_top(set(A)))
     => inj_on(A,A,compow(fun(A,A),N,F),top_top(set(A))) ) ).

% inj_fn
tff(fact_4922_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_4923_inj__on__Inter,axiom,
    ! [B: $tType,A: $tType,S: set(set(A)),F: fun(A,B)] :
      ( ( S != bot_bot(set(set(A))) )
     => ( ! [A8: set(A)] :
            ( aa(set(set(A)),$o,member(set(A),A8),S)
           => inj_on(A,B,F,A8) )
       => inj_on(A,B,F,aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),S)) ) ) ).

% inj_on_Inter
tff(fact_4924_inj__diff__right,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A] : inj_on(A,A,aTP_Lamp_nl(A,fun(A,A),A3),top_top(set(A))) ) ).

% inj_diff_right
tff(fact_4925_inj__singleton,axiom,
    ! [A: $tType,A4: set(A)] : inj_on(A,set(A),aTP_Lamp_kg(A,set(A)),A4) ).

% inj_singleton
tff(fact_4926_finite__Collect,axiom,
    ! [A: $tType,B: $tType,S: set(A),F: fun(B,A)] :
      ( aa(set(A),$o,finite_finite2(A),S)
     => ( inj_on(B,A,F,top_top(set(B)))
       => aa(set(B),$o,finite_finite2(B),aa(fun(B,$o),set(B),collect(B),aa(fun(B,A),fun(B,$o),aTP_Lamp_ach(set(A),fun(fun(B,A),fun(B,$o)),S),F))) ) ) ).

% finite_Collect
tff(fact_4927_finite__inverse__image,axiom,
    ! [A: $tType,B: $tType,A4: set(A),F: fun(B,A)] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( inj_on(B,A,F,top_top(set(B)))
       => aa(set(B),$o,finite_finite2(B),aa(fun(B,$o),set(B),collect(B),aa(fun(B,A),fun(B,$o),aTP_Lamp_ach(set(A),fun(fun(B,A),fun(B,$o)),A4),F))) ) ) ).

% finite_inverse_image
tff(fact_4928_sum_Oimage__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [G: fun(A,B),A4: set(A)] :
          ( inj_on(A,B,G,A4)
         => ( aa(set(B),B,aa(fun(B,B),fun(set(B),B),groups7311177749621191930dd_sum(B,B),aTP_Lamp_aci(B,B)),aa(set(A),set(B),image2(A,B,G),A4)) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),A4) ) ) ) ).

% sum.image_eq
tff(fact_4929_prod_Oimage__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [G: fun(A,B),A4: set(A)] :
          ( inj_on(A,B,G,A4)
         => ( aa(set(B),B,aa(fun(B,B),fun(set(B),B),groups7121269368397514597t_prod(B,B),aTP_Lamp_acj(B,B)),aa(set(A),set(B),image2(A,B,G),A4)) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A4) ) ) ) ).

% prod.image_eq
tff(fact_4930_inj__on__diff__nat,axiom,
    ! [N4: set(nat),K: nat] :
      ( ! [N3: nat] :
          ( aa(set(nat),$o,member(nat,N3),N4)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),N3) )
     => inj_on(nat,nat,aTP_Lamp_ob(nat,fun(nat,nat),K),N4) ) ).

% inj_on_diff_nat
tff(fact_4931_mult__inj__if__coprime__nat,axiom,
    ! [A: $tType,B: $tType,F: fun(A,nat),A4: set(A),G: fun(B,nat),B3: set(B)] :
      ( inj_on(A,nat,F,A4)
     => ( inj_on(B,nat,G,B3)
       => ( ! [A6: A,B5: B] :
              ( aa(set(A),$o,member(A,A6),A4)
             => ( aa(set(B),$o,member(B,B5),B3)
               => algebr8660921524188924756oprime(nat,aa(A,nat,F,A6),aa(B,nat,G,B5)) ) )
         => 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_ack(fun(A,nat),fun(fun(B,nat),fun(A,fun(B,nat))),F),G)),product_Sigma(A,B,A4,aTP_Lamp_qz(set(B),fun(A,set(B)),B3))) ) ) ) ).

% mult_inj_if_coprime_nat
tff(fact_4932_swap__inj__on,axiom,
    ! [B: $tType,A: $tType,A4: 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_vx(A,fun(B,product_prod(B,A)))),A4) ).

% swap_inj_on
tff(fact_4933_lexord__irreflexive,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),Xs: list(A)] :
      ( ! [X2: A] : ~ aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),X2)),R2)
     => ~ aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Xs)),lexord(A,R2)) ) ).

% lexord_irreflexive
tff(fact_4934_lexord__linear,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),X: list(A),Y: list(A)] :
      ( ! [A6: A,B5: A] :
          ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A6),B5)),R2)
          | ( A6 = B5 )
          | aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B5),A6)),R2) )
     => ( aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X),Y)),lexord(A,R2))
        | ( X = Y )
        | aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Y),X)),lexord(A,R2)) ) ) ).

% lexord_linear
tff(fact_4935_lexord__Nil__right,axiom,
    ! [A: $tType,X: list(A),R2: set(product_prod(A,A))] : ~ aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X),nil(A))),lexord(A,R2)) ).

% lexord_Nil_right
tff(fact_4936_lexord__append__leftI,axiom,
    ! [A: $tType,U: list(A),V: list(A),R2: set(product_prod(A,A)),X: list(A)] :
      ( aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),U),V)),lexord(A,R2))
     => aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),X),U)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),X),V))),lexord(A,R2)) ) ).

% lexord_append_leftI
tff(fact_4937_inj__split__Cons,axiom,
    ! [A: $tType,X5: set(product_prod(list(A),A))] : inj_on(product_prod(list(A),A),list(A),aa(fun(list(A),fun(A,list(A))),fun(product_prod(list(A),A),list(A)),product_case_prod(list(A),A,list(A)),aTP_Lamp_xo(list(A),fun(A,list(A)))),X5) ).

% inj_split_Cons
tff(fact_4938_inj__of__char,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => inj_on(char,A,comm_s6883823935334413003f_char(A),top_top(set(char))) ) ).

% inj_of_char
tff(fact_4939_inj__on__iff__surj,axiom,
    ! [B: $tType,A: $tType,A4: set(A),A14: set(B)] :
      ( ( A4 != bot_bot(set(A)) )
     => ( ? [F6: fun(A,B)] :
            ( inj_on(A,B,F6,A4)
            & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F6),A4)),A14) )
      <=> ? [G5: fun(B,A)] : aa(set(B),set(A),image2(B,A,G5),A14) = A4 ) ) ).

% inj_on_iff_surj
tff(fact_4940_finite__UNIV__inj__surj,axiom,
    ! [A: $tType,F: fun(A,A)] :
      ( aa(set(A),$o,finite_finite2(A),top_top(set(A)))
     => ( inj_on(A,A,F,top_top(set(A)))
       => ( aa(set(A),set(A),image2(A,A,F),top_top(set(A))) = top_top(set(A)) ) ) ) ).

% finite_UNIV_inj_surj
tff(fact_4941_finite__UNIV__surj__inj,axiom,
    ! [A: $tType,F: fun(A,A)] :
      ( aa(set(A),$o,finite_finite2(A),top_top(set(A)))
     => ( ( aa(set(A),set(A),image2(A,A,F),top_top(set(A))) = top_top(set(A)) )
       => inj_on(A,A,F,top_top(set(A))) ) ) ).

% finite_UNIV_surj_inj
tff(fact_4942_inj__image__subset__iff,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),A4: set(A),B3: set(A)] :
      ( inj_on(A,B,F,top_top(set(A)))
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F),A4)),aa(set(A),set(B),image2(A,B,F),B3))
      <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B3) ) ) ).

% inj_image_subset_iff
tff(fact_4943_inj__on__image__Int,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),C3: set(A),A4: set(A),B3: set(A)] :
      ( inj_on(A,B,F,C3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),C3)
       => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),C3)
         => ( aa(set(A),set(B),image2(A,B,F),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3)) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(set(A),set(B),image2(A,B,F),A4)),aa(set(A),set(B),image2(A,B,F),B3)) ) ) ) ) ).

% inj_on_image_Int
tff(fact_4944_image__Int,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),A4: set(A),B3: set(A)] :
      ( inj_on(A,B,F,top_top(set(A)))
     => ( aa(set(A),set(B),image2(A,B,F),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3)) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(set(A),set(B),image2(A,B,F),A4)),aa(set(A),set(B),image2(A,B,F),B3)) ) ) ).

% image_Int
tff(fact_4945_image__set__diff,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),A4: set(A),B3: set(A)] :
      ( inj_on(A,B,F,top_top(set(A)))
     => ( aa(set(A),set(B),image2(A,B,F),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),B3)) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),aa(set(A),set(B),image2(A,B,F),A4)),aa(set(A),set(B),image2(A,B,F),B3)) ) ) ).

% image_set_diff
tff(fact_4946_map__inj__on,axiom,
    ! [A: $tType,B: $tType,F: fun(B,A),Xs: list(B),Ys: list(B)] :
      ( ( aa(list(B),list(A),map(B,A,F),Xs) = aa(list(B),list(A),map(B,A,F),Ys) )
     => ( inj_on(B,A,F,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),aa(list(B),set(B),set2(B),Xs)),aa(list(B),set(B),set2(B),Ys)))
       => ( Xs = Ys ) ) ) ).

% map_inj_on
tff(fact_4947_inj__on__map__eq__map,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),Xs: list(A),Ys: list(A)] :
      ( inj_on(A,B,F,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys)))
     => ( ( aa(list(A),list(B),map(A,B,F),Xs) = aa(list(A),list(B),map(A,B,F),Ys) )
      <=> ( Xs = Ys ) ) ) ).

% inj_on_map_eq_map
tff(fact_4948_inj__vimage__image__eq,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),A4: set(A)] :
      ( inj_on(A,B,F,top_top(set(A)))
     => ( aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F),aa(set(A),set(B),image2(A,B,F),A4)) = A4 ) ) ).

% inj_vimage_image_eq
tff(fact_4949_finite__vimageI,axiom,
    ! [B: $tType,A: $tType,F4: set(A),H2: fun(B,A)] :
      ( aa(set(A),$o,finite_finite2(A),F4)
     => ( inj_on(B,A,H2,top_top(set(B)))
       => aa(set(B),$o,finite_finite2(B),aa(set(A),set(B),aa(fun(B,A),fun(set(A),set(B)),vimage(B,A),H2),F4)) ) ) ).

% finite_vimageI
tff(fact_4950_finite__vimage__IntI,axiom,
    ! [A: $tType,B: $tType,F4: set(A),H2: fun(B,A),A4: set(B)] :
      ( aa(set(A),$o,finite_finite2(A),F4)
     => ( inj_on(B,A,H2,A4)
       => aa(set(B),$o,finite_finite2(B),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(set(A),set(B),aa(fun(B,A),fun(set(A),set(B)),vimage(B,A),H2),F4)),A4)) ) ) ).

% finite_vimage_IntI
tff(fact_4951_remdups__adj__map__injective,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),Xs: list(A)] :
      ( inj_on(A,B,F,top_top(set(A)))
     => ( remdups_adj(B,aa(list(A),list(B),map(A,B,F),Xs)) = aa(list(A),list(B),map(A,B,F),remdups_adj(A,Xs)) ) ) ).

% remdups_adj_map_injective
tff(fact_4952_inj__graph,axiom,
    ! [B: $tType,A: $tType] : inj_on(fun(A,B),set(product_prod(A,B)),aTP_Lamp_acm(fun(A,B),set(product_prod(A,B))),top_top(set(fun(A,B)))) ).

% inj_graph
tff(fact_4953_range__inj__infinite,axiom,
    ! [A: $tType,F: fun(nat,A)] :
      ( inj_on(nat,A,F,top_top(set(nat)))
     => ~ aa(set(A),$o,finite_finite2(A),aa(set(nat),set(A),image2(nat,A,F),top_top(set(nat)))) ) ).

% range_inj_infinite
tff(fact_4954_map__removeAll__inj,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),X: A,Xs: list(A)] :
      ( inj_on(A,B,F,top_top(set(A)))
     => ( aa(list(A),list(B),map(A,B,F),aa(list(A),list(A),removeAll(A,X),Xs)) = aa(list(B),list(B),removeAll(B,aa(A,B,F,X)),aa(list(A),list(B),map(A,B,F),Xs)) ) ) ).

% map_removeAll_inj
tff(fact_4955_inj__enumerate,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S: set(A)] :
          ( ~ aa(set(A),$o,finite_finite2(A),S)
         => inj_on(nat,A,infini527867602293511546merate(A,S),top_top(set(nat))) ) ) ).

% inj_enumerate
tff(fact_4956_drop__eq__nths,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : drop(A,N,Xs) = nths(A,Xs,aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),ord_less_eq(nat),N))) ).

% drop_eq_nths
tff(fact_4957_inj__on__UNION__chain,axiom,
    ! [C: $tType,B: $tType,A: $tType,I4: set(A),A4: fun(A,set(B)),F: fun(B,C)] :
      ( ! [I2: A,J2: A] :
          ( aa(set(A),$o,member(A,I2),I4)
         => ( aa(set(A),$o,member(A,J2),I4)
           => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(A,set(B),A4,I2)),aa(A,set(B),A4,J2))
              | aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(A,set(B),A4,J2)),aa(A,set(B),A4,I2)) ) ) )
     => ( ! [I2: A] :
            ( aa(set(A),$o,member(A,I2),I4)
           => inj_on(B,C,F,aa(A,set(B),A4,I2)) )
       => inj_on(B,C,F,aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),A4),I4))) ) ) ).

% inj_on_UNION_chain
tff(fact_4958_inj__on__filter__key__eq,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),Y: A,Xs: list(A)] :
      ( inj_on(A,B,F,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y),aa(list(A),set(A),set2(A),Xs)))
     => ( aa(list(A),list(A),filter2(A,aa(A,fun(A,$o),aTP_Lamp_acn(fun(A,B),fun(A,fun(A,$o)),F),Y)),Xs) = aa(list(A),list(A),filter2(A,aa(A,fun(A,$o),fequal(A),Y)),Xs) ) ) ).

% inj_on_filter_key_eq
tff(fact_4959_inj__on__INTER,axiom,
    ! [C: $tType,B: $tType,A: $tType,I4: set(A),F: fun(B,C),A4: fun(A,set(B))] :
      ( ( I4 != bot_bot(set(A)) )
     => ( ! [I2: A] :
            ( aa(set(A),$o,member(A,I2),I4)
           => inj_on(B,C,F,aa(A,set(B),A4,I2)) )
       => inj_on(B,C,F,aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image2(A,set(B),A4),I4))) ) ) ).

% inj_on_INTER
tff(fact_4960_finite__imp__nat__seg__image__inj__on,axiom,
    ! [A: $tType,A4: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ? [N3: nat,F2: fun(nat,A)] :
          ( ( A4 = aa(set(nat),set(A),image2(nat,A,F2),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_bk(nat,fun(nat,$o)),N3))) )
          & inj_on(nat,A,F2,aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_bk(nat,fun(nat,$o)),N3))) ) ) ).

% finite_imp_nat_seg_image_inj_on
tff(fact_4961_finite__imp__inj__to__nat__seg_H,axiom,
    ! [A: $tType,A4: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ~ ! [F2: fun(A,nat)] :
            ( ? [N3: nat] : aa(set(A),set(nat),image2(A,nat,F2),A4) = aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_bk(nat,fun(nat,$o)),N3))
           => ~ inj_on(A,nat,F2,A4) ) ) ).

% finite_imp_inj_to_nat_seg'
tff(fact_4962_finite__imp__inj__to__nat__seg,axiom,
    ! [A: $tType,A4: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ? [F2: fun(A,nat),N3: nat] :
          ( ( aa(set(A),set(nat),image2(A,nat,F2),A4) = aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_bk(nat,fun(nat,$o)),N3)) )
          & inj_on(A,nat,F2,A4) ) ) ).

% finite_imp_inj_to_nat_seg
tff(fact_4963_lexord__partial__trans,axiom,
    ! [A: $tType,Xs: list(A),R2: set(product_prod(A,A)),Ys: list(A),Zs: list(A)] :
      ( ! [X2: A,Y2: A,Z3: A] :
          ( aa(set(A),$o,member(A,X2),aa(list(A),set(A),set2(A),Xs))
         => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y2)),R2)
           => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),Z3)),R2)
             => aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Z3)),R2) ) ) )
     => ( aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),lexord(A,R2))
       => ( aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Ys),Zs)),lexord(A,R2))
         => aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Zs)),lexord(A,R2)) ) ) ) ).

% lexord_partial_trans
tff(fact_4964_lexord__append__leftD,axiom,
    ! [A: $tType,X: list(A),U: list(A),V: list(A),R2: set(product_prod(A,A))] :
      ( aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),X),U)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),X),V))),lexord(A,R2))
     => ( ! [A6: A] : ~ aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A6),A6)),R2)
       => aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),U),V)),lexord(A,R2)) ) ) ).

% lexord_append_leftD
tff(fact_4965_lexord__append__rightI,axiom,
    ! [A: $tType,Y: list(A),X: list(A),R2: set(product_prod(A,A))] :
      ( ? [B12: A,Z6: list(A)] : Y = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),B12),Z6)
     => aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),X),Y))),lexord(A,R2)) ) ).

% lexord_append_rightI
tff(fact_4966_lexord__sufE,axiom,
    ! [A: $tType,Xs: list(A),Zs: list(A),Ys: list(A),Qs: list(A),R2: set(product_prod(A,A))] :
      ( aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Zs)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Qs))),lexord(A,R2))
     => ( ( Xs != Ys )
       => ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys) )
         => ( ( aa(list(A),nat,size_size(list(A)),Zs) = aa(list(A),nat,size_size(list(A)),Qs) )
           => aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),lexord(A,R2)) ) ) ) ) ).

% lexord_sufE
tff(fact_4967_vimage__subsetI,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),B3: set(B),A4: set(A)] :
      ( inj_on(A,B,F,top_top(set(A)))
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),B3),aa(set(A),set(B),image2(A,B,F),A4))
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F),B3)),A4) ) ) ).

% vimage_subsetI
tff(fact_4968_inj__image__Compl__subset,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),A4: set(A)] :
      ( inj_on(A,B,F,top_top(set(A)))
     => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F),aa(set(A),set(A),uminus_uminus(set(A)),A4))),aa(set(B),set(B),uminus_uminus(set(B)),aa(set(A),set(B),image2(A,B,F),A4))) ) ).

% inj_image_Compl_subset
tff(fact_4969_lexord__lex,axiom,
    ! [A: $tType,X: list(A),Y: list(A),R2: set(product_prod(A,A))] :
      ( aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X),Y)),lex(A,R2))
    <=> ( aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X),Y)),lexord(A,R2))
        & ( aa(list(A),nat,size_size(list(A)),X) = aa(list(A),nat,size_size(list(A)),Y) ) ) ) ).

% lexord_lex
tff(fact_4970_infinite__countable__subset,axiom,
    ! [A: $tType,S: set(A)] :
      ( ~ aa(set(A),$o,finite_finite2(A),S)
     => ? [F2: fun(nat,A)] :
          ( inj_on(nat,A,F2,top_top(set(nat)))
          & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(nat),set(A),image2(nat,A,F2),top_top(set(nat)))),S) ) ) ).

% infinite_countable_subset
tff(fact_4971_infinite__iff__countable__subset,axiom,
    ! [A: $tType,S: set(A)] :
      ( ~ aa(set(A),$o,finite_finite2(A),S)
    <=> ? [F6: fun(nat,A)] :
          ( inj_on(nat,A,F6,top_top(set(nat)))
          & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(nat),set(A),image2(nat,A,F6),top_top(set(nat)))),S) ) ) ).

% infinite_iff_countable_subset
tff(fact_4972_inj__on__disjoint__Un,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),A4: set(A),G: fun(A,B),B3: set(A)] :
      ( inj_on(A,B,F,A4)
     => ( inj_on(A,B,G,B3)
       => ( ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(set(A),set(B),image2(A,B,F),A4)),aa(set(A),set(B),image2(A,B,G),B3)) = bot_bot(set(B)) )
         => inj_on(A,B,aa(fun(A,B),fun(A,B),aa(set(A),fun(fun(A,B),fun(A,B)),aTP_Lamp_aco(fun(A,B),fun(set(A),fun(fun(A,B),fun(A,B))),F),A4),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3)) ) ) ) ).

% inj_on_disjoint_Un
tff(fact_4973_image__INT,axiom,
    ! [A: $tType,B: $tType,C: $tType,F: fun(A,B),C3: set(A),A4: set(C),B3: fun(C,set(A)),J: C] :
      ( inj_on(A,B,F,C3)
     => ( ! [X2: C] :
            ( aa(set(C),$o,member(C,X2),A4)
           => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(C,set(A),B3,X2)),C3) )
       => ( aa(set(C),$o,member(C,J),A4)
         => ( aa(set(A),set(B),image2(A,B,F),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(C),set(set(A)),image2(C,set(A),B3),A4))) = aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(C),set(set(B)),image2(C,set(B),aa(fun(C,set(A)),fun(C,set(B)),aTP_Lamp_acp(fun(A,B),fun(fun(C,set(A)),fun(C,set(B))),F),B3)),A4)) ) ) ) ) ).

% image_INT
tff(fact_4974_inj__on__funpow__least,axiom,
    ! [A: $tType,N: nat,F: fun(A,A),S2: A] :
      ( ( aa(A,A,compow(fun(A,A),N,F),S2) = S2 )
     => ( ! [M3: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M3)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M3),N)
             => ( aa(A,A,compow(fun(A,A),M3,F),S2) != S2 ) ) )
       => inj_on(nat,A,aa(A,fun(nat,A),aTP_Lamp_acq(fun(A,A),fun(A,fun(nat,A)),F),S2),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)) ) ) ).

% inj_on_funpow_least
tff(fact_4975_quotient__diff1,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A4: set(A),A3: A] :
      ( inj_on(A,set(set(A)),aTP_Lamp_acr(set(product_prod(A,A)),fun(A,set(set(A))),R2),A4)
     => ( aa(set(A),$o,member(A,A3),A4)
       => ( equiv_quotient(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A)))),R2) = aa(set(set(A)),set(set(A)),aa(set(set(A)),fun(set(set(A)),set(set(A))),minus_minus(set(set(A))),equiv_quotient(A,A4,R2)),equiv_quotient(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A))),R2)) ) ) ) ).

% quotient_diff1
tff(fact_4976_nths__append,axiom,
    ! [A: $tType,L: list(A),L3: list(A),A4: set(nat)] : nths(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L),L3),A4) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),nths(A,L,A4)),nths(A,L3,aa(fun(nat,$o),set(nat),collect(nat),aa(set(nat),fun(nat,$o),aTP_Lamp_acs(list(A),fun(set(nat),fun(nat,$o)),L),A4)))) ).

% nths_append
tff(fact_4977_filter__in__nths,axiom,
    ! [A: $tType,Xs: list(A),S2: set(nat)] :
      ( distinct(A,Xs)
     => ( aa(list(A),list(A),filter2(A,aa(set(nat),fun(A,$o),aTP_Lamp_act(list(A),fun(set(nat),fun(A,$o)),Xs),S2)),Xs) = nths(A,Xs,S2) ) ) ).

% filter_in_nths
tff(fact_4978_length__nths,axiom,
    ! [A: $tType,Xs: list(A),I4: set(nat)] : aa(list(A),nat,size_size(list(A)),nths(A,Xs,I4)) = aa(set(nat),nat,finite_card(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(set(nat),fun(nat,$o),aTP_Lamp_acu(list(A),fun(set(nat),fun(nat,$o)),Xs),I4))) ).

% length_nths
tff(fact_4979_inj__on__Un,axiom,
    ! [A: $tType,B: $tType,F: fun(A,B),A4: set(A),B3: set(A)] :
      ( inj_on(A,B,F,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3))
    <=> ( inj_on(A,B,F,A4)
        & inj_on(A,B,F,B3)
        & ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(set(A),set(B),image2(A,B,F),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),B3))),aa(set(A),set(B),image2(A,B,F),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B3),A4))) = bot_bot(set(B)) ) ) ) ).

% inj_on_Un
tff(fact_4980_lexord__append__left__rightI,axiom,
    ! [A: $tType,A3: A,B2: A,R2: set(product_prod(A,A)),U: list(A),X: list(A),Y: list(A)] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R2)
     => aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),U),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),X))),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),U),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),B2),Y)))),lexord(A,R2)) ) ).

% lexord_append_left_rightI
tff(fact_4981_card__vimage__inj,axiom,
    ! [A: $tType,B: $tType,F: fun(A,B),A4: set(B)] :
      ( inj_on(A,B,F,top_top(set(A)))
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A4),aa(set(A),set(B),image2(A,B,F),top_top(set(A))))
       => ( aa(set(A),nat,finite_card(A),aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F),A4)) = aa(set(B),nat,finite_card(B),A4) ) ) ) ).

% card_vimage_inj
tff(fact_4982_lexord__same__pref__iff,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),Zs: list(A),R2: set(product_prod(A,A))] :
      ( aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Zs))),lexord(A,R2))
    <=> ( ? [X4: A] :
            ( aa(set(A),$o,member(A,X4),aa(list(A),set(A),set2(A),Xs))
            & aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),X4)),R2) )
        | aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Ys),Zs)),lexord(A,R2)) ) ) ).

% lexord_same_pref_iff
tff(fact_4983_card__vimage__inj__on__le,axiom,
    ! [A: $tType,B: $tType,F: fun(A,B),D4: set(A),A4: set(B)] :
      ( inj_on(A,B,F,D4)
     => ( aa(set(B),$o,finite_finite2(B),A4)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F),A4)),D4))),aa(set(B),nat,finite_card(B),A4)) ) ) ).

% card_vimage_inj_on_le
tff(fact_4984_lexord__sufI,axiom,
    ! [A: $tType,U: list(A),W2: list(A),R2: set(product_prod(A,A)),V: list(A),Z2: list(A)] :
      ( aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),U),W2)),lexord(A,R2))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),W2)),aa(list(A),nat,size_size(list(A)),U))
       => aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),U),V)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),W2),Z2))),lexord(A,R2)) ) ) ).

% lexord_sufI
tff(fact_4985_Ex__inj__on__UNION__Sigma,axiom,
    ! [A: $tType,B: $tType,A4: fun(B,set(A)),I4: set(B)] :
    ? [F2: fun(A,product_prod(B,A))] :
      ( inj_on(A,product_prod(B,A),F2,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),A4),I4)))
      & aa(set(product_prod(B,A)),$o,aa(set(product_prod(B,A)),fun(set(product_prod(B,A)),$o),ord_less_eq(set(product_prod(B,A))),aa(set(A),set(product_prod(B,A)),image2(A,product_prod(B,A),F2),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),A4),I4)))),product_Sigma(B,A,I4,A4)) ) ).

% Ex_inj_on_UNION_Sigma
tff(fact_4986_filter__eq__nths,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] : aa(list(A),list(A),filter2(A,P),Xs) = nths(A,Xs,aa(fun(nat,$o),set(nat),collect(nat),aa(list(A),fun(nat,$o),aTP_Lamp_zh(fun(A,$o),fun(list(A),fun(nat,$o)),P),Xs))) ).

% filter_eq_nths
tff(fact_4987_map__sorted__distinct__set__unique,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F: fun(A,B),Xs: list(A),Ys: list(A)] :
          ( inj_on(A,B,F,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys)))
         => ( sorted_wrt(B,ord_less_eq(B),aa(list(A),list(B),map(A,B,F),Xs))
           => ( distinct(B,aa(list(A),list(B),map(A,B,F),Xs))
             => ( sorted_wrt(B,ord_less_eq(B),aa(list(A),list(B),map(A,B,F),Ys))
               => ( distinct(B,aa(list(A),list(B),map(A,B,F),Ys))
                 => ( ( aa(list(A),set(A),set2(A),Xs) = aa(list(A),set(A),set2(A),Ys) )
                   => ( Xs = Ys ) ) ) ) ) ) ) ) ).

% map_sorted_distinct_set_unique
tff(fact_4988_inj__vimage__singleton,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),A3: B] :
      ( inj_on(A,B,F,top_top(set(A)))
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),A3),bot_bot(set(B))))),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),the(A,aa(B,fun(A,$o),aTP_Lamp_acv(fun(A,B),fun(B,fun(A,$o)),F),A3))),bot_bot(set(A)))) ) ).

% inj_vimage_singleton
tff(fact_4989_inj__on__vimage__singleton,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),A4: set(A),A3: B] :
      ( inj_on(A,B,F,A4)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),A3),bot_bot(set(B))))),A4)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),the(A,aa(B,fun(A,$o),aa(set(A),fun(B,fun(A,$o)),aTP_Lamp_acw(fun(A,B),fun(set(A),fun(B,fun(A,$o))),F),A4),A3))),bot_bot(set(A)))) ) ).

% inj_on_vimage_singleton
tff(fact_4990_card__quotient__disjoint,axiom,
    ! [A: $tType,A4: set(A),R2: set(product_prod(A,A))] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( inj_on(A,set(set(A)),aTP_Lamp_acr(set(product_prod(A,A)),fun(A,set(set(A))),R2),A4)
       => ( aa(set(set(A)),nat,finite_card(set(A)),equiv_quotient(A,A4,R2)) = aa(set(A),nat,finite_card(A),A4) ) ) ) ).

% card_quotient_disjoint
tff(fact_4991_List_Olexordp__def,axiom,
    ! [A: $tType,R2: fun(A,fun(A,$o)),Xs: list(A),Ys: list(A)] :
      ( lexordp(A,R2,Xs,Ys)
    <=> aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),lexord(A,aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),R2)))) ) ).

% List.lexordp_def
tff(fact_4992_lexord__take__index__conv,axiom,
    ! [A: $tType,X: list(A),Y: list(A),R2: set(product_prod(A,A))] :
      ( aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X),Y)),lexord(A,R2))
    <=> ( ( aa(nat,$o,aa(nat,fun(nat,$o),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 ) )
        | ? [I3: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),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,I3,X) = take(A,I3,Y) )
            & aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(nat,A,nth(A,X),I3)),aa(nat,A,nth(A,Y),I3))),R2) ) ) ) ).

% lexord_take_index_conv
tff(fact_4993_If__the__inv__into__f__f,axiom,
    ! [B: $tType,A: $tType,I: A,C3: set(A),G: fun(A,B),X: A] :
      ( aa(set(A),$o,member(A,I),C3)
     => ( inj_on(A,B,G,C3)
       => ( aa(A,A,aa(fun(A,B),fun(A,A),comp(B,A,A,aa(A,fun(B,A),aa(fun(A,B),fun(A,fun(B,A)),aTP_Lamp_acx(set(A),fun(fun(A,B),fun(A,fun(B,A))),C3),G),X)),G),I) = aa(A,A,id(A),I) ) ) ) ).

% If_the_inv_into_f_f
tff(fact_4994_take__Cons__numeral,axiom,
    ! [A: $tType,V: num,X: A,Xs: list(A)] : take(A,aa(num,nat,numeral_numeral(nat),V),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),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),aa(num,nat,numeral_numeral(nat),V)),one_one(nat)),Xs)) ).

% take_Cons_numeral
tff(fact_4995_the__inv__into__def,axiom,
    ! [B: $tType,A: $tType,A4: set(B),F: fun(B,A),X3: A] : aa(A,B,the_inv_into(B,A,A4,F),X3) = the(B,aa(A,fun(B,$o),aa(fun(B,A),fun(A,fun(B,$o)),aTP_Lamp_acy(set(B),fun(fun(B,A),fun(A,fun(B,$o))),A4),F),X3)) ).

% the_inv_into_def
tff(fact_4996_tl__take,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : aa(list(A),list(A),tl(A),take(A,N,Xs)) = take(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)),aa(list(A),list(A),tl(A),Xs)) ).

% tl_take
tff(fact_4997_the__inv__f__f,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),X: A] :
      ( inj_on(A,B,F,top_top(set(A)))
     => ( aa(B,A,the_inv_into(A,B,top_top(set(A)),F),aa(A,B,F,X)) = X ) ) ).

% the_inv_f_f
tff(fact_4998_filter__upt__take__conv,axiom,
    ! [A: $tType,P: fun(A,$o),M: nat,L: list(A),N: nat] : aa(list(nat),list(nat),filter2(nat,aa(list(A),fun(nat,$o),aa(nat,fun(list(A),fun(nat,$o)),aTP_Lamp_acz(fun(A,$o),fun(nat,fun(list(A),fun(nat,$o))),P),M),L)),upt(N,M)) = aa(list(nat),list(nat),filter2(nat,aa(list(A),fun(nat,$o),aTP_Lamp_abe(fun(A,$o),fun(list(A),fun(nat,$o)),P),L)),upt(N,M)) ).

% filter_upt_take_conv
tff(fact_4999_take__Cons,axiom,
    ! [A: $tType,N: nat,X: A,Xs: list(A)] : take(A,N,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(nat,list(A),aa(fun(nat,list(A)),fun(nat,list(A)),aa(list(A),fun(fun(nat,list(A)),fun(nat,list(A))),case_nat(list(A)),nil(A)),aa(list(A),fun(nat,list(A)),aTP_Lamp_ada(A,fun(list(A),fun(nat,list(A))),X),Xs)),N) ).

% take_Cons
tff(fact_5000_take__Cons_H,axiom,
    ! [A: $tType,N: nat,X: A,Xs: list(A)] :
      take(A,N,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = $ite(N = zero_zero(nat),nil(A),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_5001_Union__take__drop__id,axiom,
    ! [A: $tType,N: nat,L: list(set(A))] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(list(set(A)),set(set(A)),set2(set(A)),drop(set(A),N,L)))),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(list(set(A)),set(set(A)),set2(set(A)),take(set(A),N,L)))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(list(set(A)),set(set(A)),set2(set(A)),L)) ).

% Union_take_drop_id
tff(fact_5002_set__take__disj__set__drop__if__distinct,axiom,
    ! [A: $tType,Vs: list(A),I: nat,J: nat] :
      ( distinct(A,Vs)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
       => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),take(A,I,Vs))),aa(list(A),set(A),set2(A),drop(A,J,Vs))) = bot_bot(set(A)) ) ) ) ).

% set_take_disj_set_drop_if_distinct
tff(fact_5003_lex__take__index,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),R2: set(product_prod(A,A))] :
      ( aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),lex(A,R2))
     => ~ ! [I2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs))
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Ys))
             => ( ( take(A,I2,Xs) = take(A,I2,Ys) )
               => ~ aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(nat,A,nth(A,Xs),I2)),aa(nat,A,nth(A,Ys),I2))),R2) ) ) ) ) ).

% lex_take_index
tff(fact_5004_the__inv__f__o__f__id,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),Z2: A] :
      ( inj_on(A,B,F,top_top(set(A)))
     => ( aa(A,A,aa(fun(A,B),fun(A,A),comp(B,A,A,the_inv_into(A,B,top_top(set(A)),F)),F),Z2) = aa(A,A,id(A),Z2) ) ) ).

% the_inv_f_o_f_id
tff(fact_5005_If__the__inv__into__in__Func,axiom,
    ! [B: $tType,A: $tType,G: fun(A,B),C3: set(A),B3: set(A),X: A] :
      ( inj_on(A,B,G,C3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))))
       => aa(set(fun(B,A)),$o,member(fun(B,A),aa(A,fun(B,A),aa(set(A),fun(A,fun(B,A)),aTP_Lamp_adb(fun(A,B),fun(set(A),fun(A,fun(B,A))),G),C3),X)),bNF_Wellorder_Func(B,A,top_top(set(B)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))))) ) ) ).

% If_the_inv_into_in_Func
tff(fact_5006_ex__inj,axiom,
    ! [A: $tType] :
      ( countable(A)
     => ? [To_nat: fun(A,nat)] : inj_on(A,nat,To_nat,top_top(set(A))) ) ).

% ex_inj
tff(fact_5007_ran__map__upd__Some,axiom,
    ! [B: $tType,A: $tType,M: fun(B,option(A)),X: B,Y: A,Z2: A] :
      ( ( aa(B,option(A),M,X) = aa(A,option(A),some(A),Y) )
     => ( inj_on(B,option(A),M,dom(B,A,M))
       => ( ~ aa(set(A),$o,member(A,Z2),ran(B,A,M))
         => ( ran(B,A,fun_upd(B,option(A),M,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,M)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y),bot_bot(set(A))))),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Z2),bot_bot(set(A)))) ) ) ) ) ).

% ran_map_upd_Some
tff(fact_5008_dom__eq__empty__conv,axiom,
    ! [A: $tType,B: $tType,F: fun(A,option(B))] :
      ( ( dom(A,B,F) = bot_bot(set(A)) )
    <=> ! [X4: A] : aa(A,option(B),F,X4) = none(B) ) ).

% dom_eq_empty_conv
tff(fact_5009_dom__map__add,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B)),N: fun(A,option(B))] : dom(A,B,map_add(A,B,M,N)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),dom(A,B,N)),dom(A,B,M)) ).

% dom_map_add
tff(fact_5010_dom__restrict,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B)),A4: set(A)] : dom(A,B,restrict_map(A,B,M,A4)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),dom(A,B,M)),A4) ).

% dom_restrict
tff(fact_5011_dom__empty,axiom,
    ! [B: $tType,A: $tType] : dom(A,B,aTP_Lamp_tf(A,option(B))) = bot_bot(set(A)) ).

% dom_empty
tff(fact_5012_map__update__eta__repair_I1_J,axiom,
    ! [B: $tType,A: $tType,K: A,V: B,M: fun(A,option(B))] : dom(A,B,aa(fun(A,option(B)),fun(A,option(B)),aa(B,fun(fun(A,option(B)),fun(A,option(B))),aTP_Lamp_adc(A,fun(B,fun(fun(A,option(B)),fun(A,option(B)))),K),V),M)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),K),dom(A,B,M)) ).

% map_update_eta_repair(1)
tff(fact_5013_dom__const_H,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B)] : dom(A,B,aTP_Lamp_add(fun(A,B),fun(A,option(B)),F)) = top_top(set(A)) ).

% dom_const'
tff(fact_5014_restrict__map__inv,axiom,
    ! [A: $tType,B: $tType,F: fun(A,option(B)),X3: A] : aa(A,option(B),restrict_map(A,B,F,aa(set(A),set(A),uminus_uminus(set(A)),dom(A,B,F))),X3) = none(B) ).

% restrict_map_inv
tff(fact_5015_dom__fun__upd,axiom,
    ! [B: $tType,A: $tType,F: fun(A,option(B)),X: A,Y: option(B)] :
      dom(A,B,fun_upd(A,option(B),F,X,Y)) = $ite(Y = none(B),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),dom(A,B,F)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),dom(A,B,F))) ).

% dom_fun_upd
tff(fact_5016_ran__map__add,axiom,
    ! [B: $tType,A: $tType,M12: fun(A,option(B)),M23: fun(A,option(B))] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),dom(A,B,M12)),dom(A,B,M23)) = bot_bot(set(A)) )
     => ( ran(A,B,map_add(A,B,M12,M23)) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),ran(A,B,M12)),ran(A,B,M23)) ) ) ).

% ran_map_add
tff(fact_5017_ran__add,axiom,
    ! [B: $tType,A: $tType,F: fun(A,option(B)),G: fun(A,option(B))] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),dom(A,B,F)),dom(A,B,G)) = bot_bot(set(A)) )
     => ( ran(A,B,map_add(A,B,F,G)) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),ran(A,B,F)),ran(A,B,G)) ) ) ).

% ran_add
tff(fact_5018_Func__is__emp,axiom,
    ! [A: $tType,B: $tType,A4: set(A),B3: set(B)] :
      ( ( bNF_Wellorder_Func(A,B,A4,B3) = bot_bot(set(fun(A,B))) )
    <=> ( ( A4 != bot_bot(set(A)) )
        & ( B3 = bot_bot(set(B)) ) ) ) ).

% Func_is_emp
tff(fact_5019_Func__non__emp,axiom,
    ! [A: $tType,B: $tType,B3: set(A),A4: set(B)] :
      ( ( B3 != bot_bot(set(A)) )
     => ( bNF_Wellorder_Func(B,A,A4,B3) != bot_bot(set(fun(B,A))) ) ) ).

% Func_non_emp
tff(fact_5020_dom__map__option,axiom,
    ! [B: $tType,C: $tType,A: $tType,F: fun(A,fun(C,B)),M: fun(A,option(C))] : dom(A,B,aa(fun(A,option(C)),fun(A,option(B)),aTP_Lamp_ade(fun(A,fun(C,B)),fun(fun(A,option(C)),fun(A,option(B))),F),M)) = dom(A,C,M) ).

% dom_map_option
tff(fact_5021_dom__def,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B))] : dom(A,B,M) = aa(fun(A,$o),set(A),collect(A),aTP_Lamp_adf(fun(A,option(B)),fun(A,$o),M)) ).

% dom_def
tff(fact_5022_dom__if,axiom,
    ! [B: $tType,A: $tType,P: fun(A,$o),F: fun(A,option(B)),G: fun(A,option(B))] : dom(A,B,aa(fun(A,option(B)),fun(A,option(B)),aa(fun(A,option(B)),fun(fun(A,option(B)),fun(A,option(B))),aTP_Lamp_adg(fun(A,$o),fun(fun(A,option(B)),fun(fun(A,option(B)),fun(A,option(B)))),P),F),G)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),dom(A,B,F)),aa(fun(A,$o),set(A),collect(A),P))),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),dom(A,B,G)),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_aj(fun(A,$o),fun(A,$o),P)))) ).

% dom_if
tff(fact_5023_finite__map__freshness,axiom,
    ! [A: $tType,B: $tType,F: fun(A,option(B))] :
      ( aa(set(A),$o,finite_finite2(A),dom(A,B,F))
     => ( ~ aa(set(A),$o,finite_finite2(A),top_top(set(A)))
       => ? [X2: A] : aa(A,option(B),F,X2) = none(B) ) ) ).

% finite_map_freshness
tff(fact_5024_map__add__left__comm,axiom,
    ! [B: $tType,A: $tType,A4: fun(A,option(B)),B3: fun(A,option(B)),C3: fun(A,option(B))] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),dom(A,B,A4)),dom(A,B,B3)) = bot_bot(set(A)) )
     => ( map_add(A,B,A4,map_add(A,B,B3,C3)) = map_add(A,B,B3,map_add(A,B,A4,C3)) ) ) ).

% map_add_left_comm
tff(fact_5025_map__add__comm,axiom,
    ! [B: $tType,A: $tType,M12: fun(A,option(B)),M23: fun(A,option(B))] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),dom(A,B,M12)),dom(A,B,M23)) = bot_bot(set(A)) )
     => ( map_add(A,B,M12,M23) = map_add(A,B,M23,M12) ) ) ).

% map_add_comm
tff(fact_5026_map__add__distinct__le,axiom,
    ! [B: $tType,A: $tType] :
      ( preorder(B)
     => ! [M: fun(A,option(B)),M6: fun(A,option(B)),N: fun(A,option(B)),N8: fun(A,option(B))] :
          ( aa(fun(A,option(B)),$o,aa(fun(A,option(B)),fun(fun(A,option(B)),$o),ord_less_eq(fun(A,option(B))),M),M6)
         => ( aa(fun(A,option(B)),$o,aa(fun(A,option(B)),fun(fun(A,option(B)),$o),ord_less_eq(fun(A,option(B))),N),N8)
           => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),dom(A,B,M6)),dom(A,B,N8)) = bot_bot(set(A)) )
             => aa(fun(A,option(B)),$o,aa(fun(A,option(B)),fun(fun(A,option(B)),$o),ord_less_eq(fun(A,option(B))),map_add(A,B,M,N)),map_add(A,B,M6,N8)) ) ) ) ) ).

% map_add_distinct_le
tff(fact_5027_restrict__map__eq_I1_J,axiom,
    ! [A: $tType,B: $tType,M: fun(B,option(A)),A4: set(B),K: B] :
      ( ( aa(B,option(A),restrict_map(B,A,M,A4),K) = none(A) )
    <=> ~ aa(set(B),$o,member(B,K),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),dom(B,A,M)),A4)) ) ).

% restrict_map_eq(1)
tff(fact_5028_finite__set__of__finite__maps,axiom,
    ! [A: $tType,B: $tType,A4: set(A),B3: set(B)] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( aa(set(B),$o,finite_finite2(B),B3)
       => aa(set(fun(A,option(B))),$o,finite_finite2(fun(A,option(B))),aa(fun(fun(A,option(B)),$o),set(fun(A,option(B))),collect(fun(A,option(B))),aa(set(B),fun(fun(A,option(B)),$o),aTP_Lamp_adh(set(A),fun(set(B),fun(fun(A,option(B)),$o)),A4),B3))) ) ) ).

% finite_set_of_finite_maps
tff(fact_5029_rat__denum,axiom,
    ? [F2: fun(nat,rat)] : aa(set(nat),set(rat),image2(nat,rat,F2),top_top(set(nat))) = top_top(set(rat)) ).

% rat_denum
tff(fact_5030_Func__empty,axiom,
    ! [B: $tType,A: $tType,B3: set(B)] : bNF_Wellorder_Func(A,B,bot_bot(set(A)),B3) = aa(set(fun(A,B)),set(fun(A,B)),aa(fun(A,B),fun(set(fun(A,B)),set(fun(A,B))),insert2(fun(A,B)),aTP_Lamp_adi(A,B)),bot_bot(set(fun(A,B)))) ).

% Func_empty
tff(fact_5031_finite__Map__induct,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B)),P: fun(fun(A,option(B)),$o)] :
      ( aa(set(A),$o,finite_finite2(A),dom(A,B,M))
     => ( aa(fun(A,option(B)),$o,P,aTP_Lamp_tf(A,option(B)))
       => ( ! [K2: A,V2: B,M3: fun(A,option(B))] :
              ( aa(set(A),$o,finite_finite2(A),dom(A,B,M3))
             => ( ~ aa(set(A),$o,member(A,K2),dom(A,B,M3))
               => ( aa(fun(A,option(B)),$o,P,M3)
                 => aa(fun(A,option(B)),$o,P,fun_upd(A,option(B),M3,K2,aa(B,option(B),some(B),V2))) ) ) )
         => aa(fun(A,option(B)),$o,P,M) ) ) ) ).

% finite_Map_induct
tff(fact_5032_graph__eq__to__snd__dom,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B))] : graph(A,B,M) = aa(set(A),set(product_prod(A,B)),image2(A,product_prod(A,B),aTP_Lamp_adj(fun(A,option(B)),fun(A,product_prod(A,B)),M)),dom(A,B,M)) ).

% graph_eq_to_snd_dom
tff(fact_5033_graph__map__add,axiom,
    ! [B: $tType,A: $tType,M12: fun(A,option(B)),M23: fun(A,option(B))] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),dom(A,B,M12)),dom(A,B,M23)) = bot_bot(set(A)) )
     => ( graph(A,B,map_add(A,B,M12,M23)) = aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),sup_sup(set(product_prod(A,B))),graph(A,B,M12)),graph(A,B,M23)) ) ) ).

% graph_map_add
tff(fact_5034_dom__eq__singleton__conv,axiom,
    ! [A: $tType,B: $tType,F: fun(A,option(B)),X: A] :
      ( ( dom(A,B,F) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))) )
    <=> ? [V4: B] : F = fun_upd(A,option(B),aTP_Lamp_tf(A,option(B)),X,aa(B,option(B),some(B),V4)) ) ).

% dom_eq_singleton_conv
tff(fact_5035_map__of__map__keys,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),M: fun(A,option(B))] :
      ( ( aa(list(A),set(A),set2(A),Xs) = dom(A,B,M) )
     => ( map_of(A,B,aa(list(A),list(product_prod(A,B)),map(A,product_prod(A,B),aTP_Lamp_adj(fun(A,option(B)),fun(A,product_prod(A,B)),M)),Xs)) = M ) ) ).

% map_of_map_keys
tff(fact_5036_Func__map__surj,axiom,
    ! [C: $tType,A: $tType,D: $tType,B: $tType,F1: fun(B,A),A15: set(B),B1: set(A),F22: fun(C,D),B22: set(C),A24: set(D)] :
      ( ( aa(set(B),set(A),image2(B,A,F1),A15) = B1 )
     => ( inj_on(C,D,F22,B22)
       => ( aa(set(D),$o,aa(set(D),fun(set(D),$o),ord_less_eq(set(D)),aa(set(C),set(D),image2(C,D,F22),B22)),A24)
         => ( ( ( B22 = bot_bot(set(C)) )
             => ( A24 = bot_bot(set(D)) ) )
           => ( bNF_Wellorder_Func(C,A,B22,B1) = aa(set(fun(D,B)),set(fun(C,A)),image2(fun(D,B),fun(C,A),bNF_We4925052301507509544nc_map(C,B,A,D,B22,F1,F22)),bNF_Wellorder_Func(D,B,A24,A15)) ) ) ) ) ) ).

% Func_map_surj
tff(fact_5037_dom__override__on,axiom,
    ! [B: $tType,A: $tType,F: fun(A,option(B)),G: fun(A,option(B)),A4: set(A)] : dom(A,B,override_on(A,option(B),F,G,A4)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),dom(A,B,F)),aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_adk(fun(A,option(B)),fun(set(A),fun(A,$o)),G),A4)))),aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_adl(fun(A,option(B)),fun(set(A),fun(A,$o)),G),A4))) ).

% dom_override_on
tff(fact_5038_surj__from__nat,axiom,
    ! [A: $tType] :
      ( countable(A)
     => ( aa(set(nat),set(A),image2(nat,A,from_nat(A)),top_top(set(nat))) = top_top(set(A)) ) ) ).

% surj_from_nat
tff(fact_5039_override__on__emptyset,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),G: fun(A,B)] : override_on(A,B,F,G,bot_bot(set(A))) = F ).

% override_on_emptyset
tff(fact_5040_surj__nat__to__rat__surj,axiom,
    aa(set(nat),set(rat),image2(nat,rat,nat_to_rat_surj),top_top(set(nat))) = top_top(set(rat)) ).

% surj_nat_to_rat_surj
tff(fact_5041_dom__map__upds,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B)),Xs: list(A),Ys: list(B)] : dom(A,B,map_upds(A,B,M,Xs,Ys)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(list(A),set(A),set2(A),take(A,aa(list(B),nat,size_size(list(B)),Ys),Xs))),dom(A,B,M)) ).

% dom_map_upds
tff(fact_5042_set__to__map__simp,axiom,
    ! [B: $tType,A: $tType,S: set(product_prod(A,B)),K: A,V: B] :
      ( inj_on(product_prod(A,B),A,product_fst(A,B),S)
     => ( ( aa(A,option(B),set_to_map(A,B,S),K) = aa(B,option(B),some(B),V) )
      <=> aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K),V)),S) ) ) ).

% set_to_map_simp
tff(fact_5043_set__to__map__empty,axiom,
    ! [A: $tType,B: $tType,X3: A] : aa(A,option(B),set_to_map(A,B,bot_bot(set(product_prod(A,B)))),X3) = none(B) ).

% set_to_map_empty
tff(fact_5044_set__to__map__empty__iff_I1_J,axiom,
    ! [B: $tType,A: $tType,S: set(product_prod(A,B))] :
      ( ! [X4: A] : aa(A,option(B),set_to_map(A,B,S),X4) = none(B)
    <=> ( S = bot_bot(set(product_prod(A,B))) ) ) ).

% set_to_map_empty_iff(1)
tff(fact_5045_set__to__map__empty__iff_I2_J,axiom,
    ! [B: $tType,A: $tType,S: set(product_prod(A,B))] :
      ( ( aTP_Lamp_tf(A,option(B)) = set_to_map(A,B,S) )
    <=> ( S = bot_bot(set(product_prod(A,B))) ) ) ).

% set_to_map_empty_iff(2)
tff(fact_5046_last__take__nth__conv,axiom,
    ! [A: $tType,N: nat,L: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),aa(list(A),nat,size_size(list(A)),L))
     => ( ( N != zero_zero(nat) )
       => ( last(A,take(A,N,L)) = aa(nat,A,nth(A,L),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))) ) ) ) ).

% last_take_nth_conv
tff(fact_5047_nth__zip,axiom,
    ! [A: $tType,B: $tType,I: nat,Xs: list(A),Ys: list(B)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(B),nat,size_size(list(B)),Ys))
       => ( aa(nat,product_prod(A,B),nth(product_prod(A,B),zip(A,B,Xs,Ys)),I) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(nat,A,nth(A,Xs),I)),aa(nat,B,nth(B,Ys),I)) ) ) ) ).

% nth_zip
tff(fact_5048_lexn_Osimps_I1_J,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : aa(nat,set(product_prod(list(A),list(A))),lexn(A,R2),zero_zero(nat)) = bot_bot(set(product_prod(list(A),list(A)))) ).

% lexn.simps(1)
tff(fact_5049_last__upt,axiom,
    ! [I: nat,J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
     => ( last(nat,upt(I,J)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),one_one(nat)) ) ) ).

% last_upt
tff(fact_5050_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_5051_zip__replicate,axiom,
    ! [A: $tType,B: $tType,I: nat,X: A,J: nat,Y: B] : zip(A,B,replicate(A,I,X),replicate(B,J,Y)) = replicate(product_prod(A,B),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),I),J),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y)) ).

% zip_replicate
tff(fact_5052_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_5053_zip__assoc,axiom,
    ! [B: $tType,A: $tType,C: $tType,Xs: list(A),Ys: list(B),Zs: list(C)] : zip(A,product_prod(B,C),Xs,zip(B,C,Ys,Zs)) = aa(list(product_prod(product_prod(A,B),C)),list(product_prod(A,product_prod(B,C))),map(product_prod(product_prod(A,B),C),product_prod(A,product_prod(B,C)),aa(fun(product_prod(A,B),fun(C,product_prod(A,product_prod(B,C)))),fun(product_prod(product_prod(A,B),C),product_prod(A,product_prod(B,C))),product_case_prod(product_prod(A,B),C,product_prod(A,product_prod(B,C))),aa(fun(A,fun(B,fun(C,product_prod(A,product_prod(B,C))))),fun(product_prod(A,B),fun(C,product_prod(A,product_prod(B,C)))),product_case_prod(A,B,fun(C,product_prod(A,product_prod(B,C)))),aTP_Lamp_adm(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_5054_zip__update,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),I: nat,X: A,Ys: list(B),Y: B] : zip(A,B,list_update(A,Xs,I,X),list_update(B,Ys,I,Y)) = list_update(product_prod(A,B),zip(A,B,Xs,Ys),I,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y)) ).

% zip_update
tff(fact_5055_zip__left__commute,axiom,
    ! [B: $tType,A: $tType,C: $tType,Xs: list(A),Ys: list(B),Zs: list(C)] : zip(A,product_prod(B,C),Xs,zip(B,C,Ys,Zs)) = aa(list(product_prod(B,product_prod(A,C))),list(product_prod(A,product_prod(B,C))),map(product_prod(B,product_prod(A,C)),product_prod(A,product_prod(B,C)),aa(fun(B,fun(product_prod(A,C),product_prod(A,product_prod(B,C)))),fun(product_prod(B,product_prod(A,C)),product_prod(A,product_prod(B,C))),product_case_prod(B,product_prod(A,C),product_prod(A,product_prod(B,C))),aTP_Lamp_ado(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_5056_zip__same__conv__map,axiom,
    ! [A: $tType,Xs: list(A)] : zip(A,A,Xs,Xs) = aa(list(A),list(product_prod(A,A)),map(A,product_prod(A,A),aTP_Lamp_vl(A,product_prod(A,A))),Xs) ).

% zip_same_conv_map
tff(fact_5057_nths__shift__lemma__Suc,axiom,
    ! [A: $tType,P: fun(nat,$o),Xs: list(A),Is: list(nat)] : aa(list(product_prod(A,nat)),list(A),map(product_prod(A,nat),A,product_fst(A,nat)),aa(list(product_prod(A,nat)),list(product_prod(A,nat)),filter2(product_prod(A,nat),aTP_Lamp_adp(fun(nat,$o),fun(product_prod(A,nat),$o),P)),zip(A,nat,Xs,Is))) = aa(list(product_prod(A,nat)),list(A),map(product_prod(A,nat),A,product_fst(A,nat)),aa(list(product_prod(A,nat)),list(product_prod(A,nat)),filter2(product_prod(A,nat),aTP_Lamp_adq(fun(nat,$o),fun(product_prod(A,nat),$o),P)),zip(A,nat,Xs,aa(list(nat),list(nat),map(nat,nat,suc),Is)))) ).

% nths_shift_lemma_Suc
tff(fact_5058_hd__zip,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B)] :
      ( ( Xs != nil(A) )
     => ( ( Ys != nil(B) )
       => ( aa(list(product_prod(A,B)),product_prod(A,B),hd(product_prod(A,B)),zip(A,B,Xs,Ys)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(list(A),A,hd(A),Xs)),aa(list(B),B,hd(B),Ys)) ) ) ) ).

% hd_zip
tff(fact_5059_zip__same,axiom,
    ! [A: $tType,A3: A,B2: A,Xs: list(A)] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),aa(list(product_prod(A,A)),set(product_prod(A,A)),set2(product_prod(A,A)),zip(A,A,Xs,Xs)))
    <=> ( aa(set(A),$o,member(A,A3),aa(list(A),set(A),set2(A),Xs))
        & ( A3 = B2 ) ) ) ).

% zip_same
tff(fact_5060_in__set__zipE,axiom,
    ! [A: $tType,B: $tType,X: A,Y: B,Xs: list(A),Ys: list(B)] :
      ( aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y)),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys)))
     => ~ ( aa(set(A),$o,member(A,X),aa(list(A),set(A),set2(A),Xs))
         => ~ aa(set(B),$o,member(B,Y),aa(list(B),set(B),set2(B),Ys)) ) ) ).

% in_set_zipE
tff(fact_5061_set__zip__leftD,axiom,
    ! [B: $tType,A: $tType,X: A,Y: B,Xs: list(A),Ys: list(B)] :
      ( aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y)),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys)))
     => aa(set(A),$o,member(A,X),aa(list(A),set(A),set2(A),Xs)) ) ).

% set_zip_leftD
tff(fact_5062_set__zip__rightD,axiom,
    ! [A: $tType,B: $tType,X: A,Y: B,Xs: list(A),Ys: list(B)] :
      ( aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y)),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys)))
     => aa(set(B),$o,member(B,Y),aa(list(B),set(B),set2(B),Ys)) ) ).

% set_zip_rightD
tff(fact_5063_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) )
     => ~ ! [X2: A,Xs5: list(A)] :
            ( ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs5) )
           => ! [Y2: B,Ys4: list(B)] :
                ( ( Ys = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y2),Ys4) )
               => ( ( Xy = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Y2) )
                 => ( Xys != zip(A,B,Xs5,Ys4) ) ) ) ) ) ).

% zip_eq_ConsE
tff(fact_5064_map2__map__map,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType,H2: fun(B,fun(C,A)),F: fun(D,B),Xs: list(D),G: fun(D,C)] : aa(list(product_prod(B,C)),list(A),map(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),H2)),zip(B,C,aa(list(D),list(B),map(D,B,F),Xs),aa(list(D),list(C),map(D,C,G),Xs))) = aa(list(D),list(A),map(D,A,aa(fun(D,C),fun(D,A),aa(fun(D,B),fun(fun(D,C),fun(D,A)),aTP_Lamp_adr(fun(B,fun(C,A)),fun(fun(D,B),fun(fun(D,C),fun(D,A))),H2),F),G)),Xs) ).

% map2_map_map
tff(fact_5065_set__zip__cart,axiom,
    ! [A: $tType,B: $tType,X: product_prod(A,B),L: list(A),L3: list(B)] :
      ( aa(set(product_prod(A,B)),$o,member(product_prod(A,B),X),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,L,L3)))
     => aa(set(product_prod(A,B)),$o,member(product_prod(A,B),X),product_Sigma(A,B,aa(list(A),set(A),set2(A),L),aTP_Lamp_wz(list(B),fun(A,set(B)),L3))) ) ).

% set_zip_cart
tff(fact_5066_zip__commute,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys: list(B)] : zip(A,B,Xs,Ys) = aa(list(product_prod(B,A)),list(product_prod(A,B)),map(product_prod(B,A),product_prod(A,B),aa(fun(B,fun(A,product_prod(A,B))),fun(product_prod(B,A),product_prod(A,B)),product_case_prod(B,A,product_prod(A,B)),aTP_Lamp_ru(B,fun(A,product_prod(A,B))))),zip(B,A,Ys,Xs)) ).

% zip_commute
tff(fact_5067_in__set__impl__in__set__zip2,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),Y: B] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( aa(set(B),$o,member(B,Y),aa(list(B),set(B),set2(B),Ys))
       => ~ ! [X2: A] : ~ aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),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_5068_in__set__impl__in__set__zip1,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),X: A] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( aa(set(A),$o,member(A,X),aa(list(A),set(A),set2(A),Xs))
       => ~ ! [Y2: B] : ~ aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y2)),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip1
tff(fact_5069_map__upds__fold__map__upd,axiom,
    ! [A: $tType,B: $tType,M: fun(A,option(B)),Ks: list(A),Vs: list(B)] : map_upds(A,B,M,Ks,Vs) = foldl(fun(A,option(B)),product_prod(A,B),aTP_Lamp_adt(fun(A,option(B)),fun(product_prod(A,B),fun(A,option(B)))),M,zip(A,B,Ks,Vs)) ).

% map_upds_fold_map_upd
tff(fact_5070_zip__replicate1,axiom,
    ! [A: $tType,B: $tType,N: nat,X: A,Ys: list(B)] : zip(A,B,replicate(A,N,X),Ys) = aa(list(B),list(product_prod(A,B)),map(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X)),take(B,N,Ys)) ).

% zip_replicate1
tff(fact_5071_nths__shift__lemma,axiom,
    ! [A: $tType,A4: set(nat),Xs: list(A),I: nat] : aa(list(product_prod(A,nat)),list(A),map(product_prod(A,nat),A,product_fst(A,nat)),aa(list(product_prod(A,nat)),list(product_prod(A,nat)),filter2(product_prod(A,nat),aTP_Lamp_adu(set(nat),fun(product_prod(A,nat),$o),A4)),zip(A,nat,Xs,upt(I,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),aa(list(A),nat,size_size(list(A)),Xs)))))) = aa(list(product_prod(A,nat)),list(A),map(product_prod(A,nat),A,product_fst(A,nat)),aa(list(product_prod(A,nat)),list(product_prod(A,nat)),filter2(product_prod(A,nat),aa(nat,fun(product_prod(A,nat),$o),aTP_Lamp_adv(set(nat),fun(nat,fun(product_prod(A,nat),$o)),A4),I)),zip(A,nat,Xs,upt(zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xs))))) ).

% nths_shift_lemma
tff(fact_5072_map__zip__map2,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType,F: fun(product_prod(B,C),A),Xs: list(B),G: fun(D,C),Ys: list(D)] : aa(list(product_prod(B,C)),list(A),map(product_prod(B,C),A,F),zip(B,C,Xs,aa(list(D),list(C),map(D,C,G),Ys))) = aa(list(product_prod(B,D)),list(A),map(product_prod(B,D),A,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_adw(fun(product_prod(B,C),A),fun(fun(D,C),fun(B,fun(D,A))),F),G))),zip(B,D,Xs,Ys)) ).

% map_zip_map2
tff(fact_5073_map__zip__map,axiom,
    ! [B: $tType,A: $tType,D: $tType,C: $tType,F: fun(product_prod(B,C),A),G: fun(D,B),Xs: list(D),Ys: list(C)] : aa(list(product_prod(B,C)),list(A),map(product_prod(B,C),A,F),zip(B,C,aa(list(D),list(B),map(D,B,G),Xs),Ys)) = aa(list(product_prod(D,C)),list(A),map(product_prod(D,C),A,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_adx(fun(product_prod(B,C),A),fun(fun(D,B),fun(D,fun(C,A))),F),G))),zip(D,C,Xs,Ys)) ).

% map_zip_map
tff(fact_5074_foldl__snd__zip,axiom,
    ! [B: $tType,C: $tType,A: $tType,Ys: list(A),Xs: list(B),F: fun(C,fun(A,C)),B2: C] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Ys)),aa(list(B),nat,size_size(list(B)),Xs))
     => ( foldl(C,product_prod(B,A),aTP_Lamp_adz(fun(C,fun(A,C)),fun(C,fun(product_prod(B,A),C)),F),B2,zip(B,A,Xs,Ys)) = foldl(C,A,F,B2,Ys) ) ) ).

% foldl_snd_zip
tff(fact_5075_lexn__length,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),R2: set(product_prod(A,A)),N: nat] :
      ( aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),aa(nat,set(product_prod(list(A),list(A))),lexn(A,R2),N))
     => ( ( aa(list(A),nat,size_size(list(A)),Xs) = N )
        & ( aa(list(A),nat,size_size(list(A)),Ys) = N ) ) ) ).

% lexn_length
tff(fact_5076_map__zip1,axiom,
    ! [A: $tType,B: $tType,K: B,L: list(A)] : aa(list(A),list(product_prod(A,B)),map(A,product_prod(A,B),aa(B,fun(A,product_prod(A,B)),aTP_Lamp_ru(B,fun(A,product_prod(A,B))),K)),L) = zip(A,B,L,replicate(B,aa(list(A),nat,size_size(list(A)),L),K)) ).

% map_zip1
tff(fact_5077_map__zip2,axiom,
    ! [A: $tType,B: $tType,K: A,L: list(B)] : aa(list(B),list(product_prod(A,B)),map(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K)),L) = zip(A,B,replicate(A,aa(list(B),nat,size_size(list(B)),L),K),L) ).

% map_zip2
tff(fact_5078_nths__def,axiom,
    ! [A: $tType,Xs: list(A),A4: set(nat)] : nths(A,Xs,A4) = aa(list(product_prod(A,nat)),list(A),map(product_prod(A,nat),A,product_fst(A,nat)),aa(list(product_prod(A,nat)),list(product_prod(A,nat)),filter2(product_prod(A,nat),aTP_Lamp_adu(set(nat),fun(product_prod(A,nat),$o),A4)),zip(A,nat,Xs,upt(zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xs))))) ).

% nths_def
tff(fact_5079_zip__replicate2,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),N: nat,Y: B] : zip(A,B,Xs,replicate(B,N,Y)) = aa(list(A),list(product_prod(A,B)),map(A,product_prod(A,B),aa(B,fun(A,product_prod(A,B)),aTP_Lamp_ru(B,fun(A,product_prod(A,B))),Y)),take(A,N,Xs)) ).

% zip_replicate2
tff(fact_5080_map__prod__fun__zip,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType,F: fun(C,A),G: fun(D,B),Xs: list(C),Ys: list(D)] : aa(list(product_prod(C,D)),list(product_prod(A,B)),map(product_prod(C,D),product_prod(A,B),aa(fun(C,fun(D,product_prod(A,B))),fun(product_prod(C,D),product_prod(A,B)),product_case_prod(C,D,product_prod(A,B)),aa(fun(D,B),fun(C,fun(D,product_prod(A,B))),aTP_Lamp_rx(fun(C,A),fun(fun(D,B),fun(C,fun(D,product_prod(A,B)))),F),G))),zip(C,D,Xs,Ys)) = zip(A,B,aa(list(C),list(A),map(C,A,F),Xs),aa(list(D),list(B),map(D,B,G),Ys)) ).

% map_prod_fun_zip
tff(fact_5081_zip__map2,axiom,
    ! [B: $tType,A: $tType,C: $tType,Xs: list(A),F: fun(C,B),Ys: list(C)] : zip(A,B,Xs,aa(list(C),list(B),map(C,B,F),Ys)) = aa(list(product_prod(A,C)),list(product_prod(A,B)),map(product_prod(A,C),product_prod(A,B),aa(fun(A,fun(C,product_prod(A,B))),fun(product_prod(A,C),product_prod(A,B)),product_case_prod(A,C,product_prod(A,B)),aTP_Lamp_abl(fun(C,B),fun(A,fun(C,product_prod(A,B))),F))),zip(A,C,Xs,Ys)) ).

% zip_map2
tff(fact_5082_zip__map1,axiom,
    ! [A: $tType,C: $tType,B: $tType,F: fun(C,A),Xs: list(C),Ys: list(B)] : zip(A,B,aa(list(C),list(A),map(C,A,F),Xs),Ys) = aa(list(product_prod(C,B)),list(product_prod(A,B)),map(product_prod(C,B),product_prod(A,B),aa(fun(C,fun(B,product_prod(A,B))),fun(product_prod(C,B),product_prod(A,B)),product_case_prod(C,B,product_prod(A,B)),aTP_Lamp_aea(fun(C,A),fun(C,fun(B,product_prod(A,B))),F))),zip(C,B,Xs,Ys)) ).

% zip_map1
tff(fact_5083_zip__Cons,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Y: B,Ys: list(B)] : zip(A,B,Xs,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),Ys)) = aa(list(A),list(product_prod(A,B)),case_list(list(product_prod(A,B)),A,nil(product_prod(A,B)),aa(list(B),fun(A,fun(list(A),list(product_prod(A,B)))),aTP_Lamp_aeb(B,fun(list(B),fun(A,fun(list(A),list(product_prod(A,B))))),Y),Ys)),Xs) ).

% zip_Cons
tff(fact_5084_zip__Cons1,axiom,
    ! [A: $tType,B: $tType,X: A,Xs: list(A),Ys: list(B)] : zip(A,B,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs),Ys) = aa(list(B),list(product_prod(A,B)),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_aec(A,fun(list(A),fun(B,fun(list(B),list(product_prod(A,B))))),X),Xs)),Ys) ).

% zip_Cons1
tff(fact_5085_remdups__adj__append_H_H,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( Xs != nil(A) )
     => ( remdups_adj(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),remdups_adj(A,Xs)),remdups_adj(A,dropWhile(A,aTP_Lamp_aed(list(A),fun(A,$o),Xs),Ys))) ) ) ).

% remdups_adj_append''
tff(fact_5086_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_5087_last__list__update,axiom,
    ! [A: $tType,Xs: list(A),K: nat,X: A] :
      ( ( Xs != nil(A) )
     => ( last(A,list_update(A,Xs,K,X)) = $ite(K = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat)),X,last(A,Xs)) ) ) ).

% last_list_update
tff(fact_5088_lex__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : lex(A,R2) = aa(set(set(product_prod(list(A),list(A)))),set(product_prod(list(A),list(A))),complete_Sup_Sup(set(product_prod(list(A),list(A)))),aa(set(nat),set(set(product_prod(list(A),list(A)))),image2(nat,set(product_prod(list(A),list(A))),lexn(A,R2)),top_top(set(nat)))) ).

% lex_def
tff(fact_5089_foldr__snd__zip,axiom,
    ! [B: $tType,A: $tType,C: $tType,Ys: list(A),Xs: list(B),F: fun(A,fun(C,C)),B2: C] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Ys)),aa(list(B),nat,size_size(list(B)),Xs))
     => ( aa(C,C,foldr(product_prod(B,A),C,aa(fun(B,fun(A,fun(C,C))),fun(product_prod(B,A),fun(C,C)),product_case_prod(B,A,fun(C,C)),aTP_Lamp_aee(fun(A,fun(C,C)),fun(B,fun(A,fun(C,C))),F)),zip(B,A,Xs,Ys)),B2) = aa(C,C,foldr(A,C,F,Ys),B2) ) ) ).

% foldr_snd_zip
tff(fact_5090_lexn_Osimps_I2_J,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),N: nat] : aa(nat,set(product_prod(list(A),list(A))),lexn(A,R2),aa(nat,nat,suc,N)) = aa(set(product_prod(list(A),list(A))),set(product_prod(list(A),list(A))),aa(set(product_prod(list(A),list(A))),fun(set(product_prod(list(A),list(A))),set(product_prod(list(A),list(A)))),inf_inf(set(product_prod(list(A),list(A)))),aa(set(product_prod(product_prod(A,list(A)),product_prod(A,list(A)))),set(product_prod(list(A),list(A))),image2(product_prod(product_prod(A,list(A)),product_prod(A,list(A))),product_prod(list(A),list(A)),product_map_prod(product_prod(A,list(A)),list(A),product_prod(A,list(A)),list(A),aa(fun(A,fun(list(A),list(A))),fun(product_prod(A,list(A)),list(A)),product_case_prod(A,list(A),list(A)),cons(A)),aa(fun(A,fun(list(A),list(A))),fun(product_prod(A,list(A)),list(A)),product_case_prod(A,list(A),list(A)),cons(A)))),lex_prod(A,list(A),R2,aa(nat,set(product_prod(list(A),list(A))),lexn(A,R2),N)))),aa(fun(product_prod(list(A),list(A)),$o),set(product_prod(list(A),list(A))),collect(product_prod(list(A),list(A))),aa(fun(list(A),fun(list(A),$o)),fun(product_prod(list(A),list(A)),$o),product_case_prod(list(A),list(A),$o),aTP_Lamp_aef(nat,fun(list(A),fun(list(A),$o)),N)))) ).

% lexn.simps(2)
tff(fact_5091_set__zip,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B)] : aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys)) = aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(list(B),fun(product_prod(A,B),$o),aTP_Lamp_aeg(list(A),fun(list(B),fun(product_prod(A,B),$o)),Xs),Ys)) ).

% set_zip
tff(fact_5092_listrel__iff__zip,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys: list(B),R2: set(product_prod(A,B))] :
      ( aa(set(product_prod(list(A),list(B))),$o,member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xs),Ys)),listrel(A,B,R2))
    <=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
        & ! [X4: product_prod(A,B)] :
            ( aa(set(product_prod(A,B)),$o,member(product_prod(A,B),X4),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys)))
           => aa(product_prod(A,B),$o,aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_as(set(product_prod(A,B)),fun(A,fun(B,$o))),R2)),X4) ) ) ) ).

% listrel_iff_zip
tff(fact_5093_map__prod__ident,axiom,
    ! [B: $tType,A: $tType,X3: product_prod(A,B)] : aa(product_prod(A,B),product_prod(A,B),product_map_prod(A,A,B,B,aTP_Lamp_au(A,A),aTP_Lamp_oc(B,B)),X3) = X3 ).

% map_prod_ident
tff(fact_5094_ball__empty,axiom,
    ! [A: $tType,P: fun(A,$o),X3: A] :
      ( aa(set(A),$o,member(A,X3),bot_bot(set(A)))
     => aa(A,$o,P,X3) ) ).

% ball_empty
tff(fact_5095_map__prod__simp,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType,F: fun(C,A),G: fun(D,B),A3: C,B2: D] : aa(product_prod(C,D),product_prod(A,B),product_map_prod(C,A,D,B,F,G),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,F,A3)),aa(D,B,G,B2)) ).

% map_prod_simp
tff(fact_5096_finite__Collect__bounded__ex,axiom,
    ! [B: $tType,A: $tType,P: fun(A,$o),Q: fun(B,fun(A,$o))] :
      ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),P))
     => ( aa(set(B),$o,finite_finite2(B),aa(fun(B,$o),set(B),collect(B),aa(fun(B,fun(A,$o)),fun(B,$o),aTP_Lamp_aeh(fun(A,$o),fun(fun(B,fun(A,$o)),fun(B,$o)),P),Q)))
      <=> ! [Y3: A] :
            ( aa(A,$o,P,Y3)
           => aa(set(B),$o,finite_finite2(B),aa(fun(B,$o),set(B),collect(B),aa(A,fun(B,$o),aTP_Lamp_aei(fun(B,fun(A,$o)),fun(A,fun(B,$o)),Q),Y3))) ) ) ) ).

% finite_Collect_bounded_ex
tff(fact_5097_eq__or__mem__image__simp,axiom,
    ! [A: $tType,B: $tType,F: fun(B,A),A3: B,B3: set(B)] : aa(fun(A,$o),set(A),collect(A),aa(set(B),fun(A,$o),aa(B,fun(set(B),fun(A,$o)),aTP_Lamp_aej(fun(B,A),fun(B,fun(set(B),fun(A,$o))),F),A3),B3)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),aa(B,A,F,A3)),aa(fun(A,$o),set(A),collect(A),aa(set(B),fun(A,$o),aTP_Lamp_aek(fun(B,A),fun(set(B),fun(A,$o)),F),B3))) ).

% eq_or_mem_image_simp
tff(fact_5098_map__prod__imageI,axiom,
    ! [D: $tType,C: $tType,B: $tType,A: $tType,A3: A,B2: B,R: set(product_prod(A,B)),F: fun(A,C),G: fun(B,D)] :
      ( aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),R)
     => aa(set(product_prod(C,D)),$o,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,F,A3)),aa(B,D,G,B2))),aa(set(product_prod(A,B)),set(product_prod(C,D)),image2(product_prod(A,B),product_prod(C,D),product_map_prod(A,C,B,D,F,G)),R)) ) ).

% map_prod_imageI
tff(fact_5099_INF__bool__eq,axiom,
    ! [A: $tType] : aTP_Lamp_ael(set(A),fun(fun(A,$o),$o)) = ball(A) ).

% INF_bool_eq
tff(fact_5100_pairself__image__eq,axiom,
    ! [A: $tType,B: $tType,F: fun(B,A),P: fun(B,fun(B,$o))] : aa(set(product_prod(B,B)),set(product_prod(A,A)),image2(product_prod(B,B),product_prod(A,A),pairself(B,A,F)),aa(fun(product_prod(B,B),$o),set(product_prod(B,B)),collect(product_prod(B,B)),aa(fun(B,fun(B,$o)),fun(product_prod(B,B),$o),product_case_prod(B,B,$o),P))) = aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(B,fun(B,$o)),fun(product_prod(A,A),$o),aTP_Lamp_aem(fun(B,A),fun(fun(B,fun(B,$o)),fun(product_prod(A,A),$o)),F),P)) ).

% pairself_image_eq
tff(fact_5101_finite__image__set2,axiom,
    ! [C: $tType,B: $tType,A: $tType,P: fun(A,$o),Q: fun(B,$o),F: fun(A,fun(B,C))] :
      ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),P))
     => ( aa(set(B),$o,finite_finite2(B),aa(fun(B,$o),set(B),collect(B),Q))
       => aa(set(C),$o,finite_finite2(C),aa(fun(C,$o),set(C),collect(C),aa(fun(A,fun(B,C)),fun(C,$o),aa(fun(B,$o),fun(fun(A,fun(B,C)),fun(C,$o)),aTP_Lamp_aen(fun(A,$o),fun(fun(B,$o),fun(fun(A,fun(B,C)),fun(C,$o))),P),Q),F))) ) ) ).

% finite_image_set2
tff(fact_5102_finite__image__set,axiom,
    ! [B: $tType,A: $tType,P: fun(A,$o),F: fun(A,B)] :
      ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),P))
     => aa(set(B),$o,finite_finite2(B),aa(fun(B,$o),set(B),collect(B),aa(fun(A,B),fun(B,$o),aTP_Lamp_aeo(fun(A,$o),fun(fun(A,B),fun(B,$o)),P),F))) ) ).

% finite_image_set
tff(fact_5103_Ball__fold,axiom,
    ! [A: $tType,A4: set(A),P: fun(A,$o)] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( ! [X4: A] :
            ( aa(set(A),$o,member(A,X4),A4)
           => aa(A,$o,P,X4) )
      <=> finite_fold(A,$o,aTP_Lamp_aep(fun(A,$o),fun(A,fun($o,$o)),P),$true,A4) ) ) ).

% Ball_fold
tff(fact_5104_finite_Omono,axiom,
    ! [A: $tType] : order_mono(fun(set(A),$o),fun(set(A),$o),aTP_Lamp_aeq(fun(set(A),$o),fun(set(A),$o))) ).

% finite.mono
tff(fact_5105_fs__contract,axiom,
    ! [A: $tType,B: $tType,C: $tType,F: fun(A,fun(B,C)),S: set(C)] : aa(set(product_prod(A,B)),set(A),image2(product_prod(A,B),A,product_fst(A,B)),aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(set(C),fun(product_prod(A,B),$o),aTP_Lamp_aer(fun(A,fun(B,C)),fun(set(C),fun(product_prod(A,B),$o)),F),S))) = aa(fun(A,$o),set(A),collect(A),aa(set(C),fun(A,$o),aTP_Lamp_aes(fun(A,fun(B,C)),fun(set(C),fun(A,$o)),F),S)) ).

% fs_contract
tff(fact_5106_setcompr__eq__image,axiom,
    ! [A: $tType,B: $tType,F: fun(B,A),P: fun(B,$o)] : aa(fun(A,$o),set(A),collect(A),aa(fun(B,$o),fun(A,$o),aTP_Lamp_aet(fun(B,A),fun(fun(B,$o),fun(A,$o)),F),P)) = aa(set(B),set(A),image2(B,A,F),aa(fun(B,$o),set(B),collect(B),P)) ).

% setcompr_eq_image
tff(fact_5107_Setcompr__eq__image,axiom,
    ! [A: $tType,B: $tType,F: fun(B,A),A4: set(B)] : aa(fun(A,$o),set(A),collect(A),aa(set(B),fun(A,$o),aTP_Lamp_aek(fun(B,A),fun(set(B),fun(A,$o)),F),A4)) = aa(set(B),set(A),image2(B,A,F),A4) ).

% Setcompr_eq_image
tff(fact_5108_case__prod__map__prod,axiom,
    ! [C: $tType,A: $tType,B: $tType,E: $tType,D: $tType,H2: fun(B,fun(C,A)),F: fun(D,B),G: fun(E,C),X: product_prod(D,E)] : aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),H2),aa(product_prod(D,E),product_prod(B,C),product_map_prod(D,B,E,C,F,G),X)) = aa(product_prod(D,E),A,aa(fun(D,fun(E,A)),fun(product_prod(D,E),A),product_case_prod(D,E,A),aa(fun(E,C),fun(D,fun(E,A)),aa(fun(D,B),fun(fun(E,C),fun(D,fun(E,A))),aTP_Lamp_aeu(fun(B,fun(C,A)),fun(fun(D,B),fun(fun(E,C),fun(D,fun(E,A)))),H2),F),G)),X) ).

% case_prod_map_prod
tff(fact_5109_Union__SetCompr__eq,axiom,
    ! [A: $tType,B: $tType,F: fun(B,set(A)),P: fun(B,$o)] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(fun(set(A),$o),set(set(A)),collect(set(A)),aa(fun(B,$o),fun(set(A),$o),aTP_Lamp_aev(fun(B,set(A)),fun(fun(B,$o),fun(set(A),$o)),F),P))) = aa(fun(A,$o),set(A),collect(A),aa(fun(B,$o),fun(A,$o),aTP_Lamp_aew(fun(B,set(A)),fun(fun(B,$o),fun(A,$o)),F),P)) ).

% Union_SetCompr_eq
tff(fact_5110_rtranclp_Omono,axiom,
    ! [A: $tType,R2: fun(A,fun(A,$o))] : order_mono(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aTP_Lamp_aex(fun(A,fun(A,$o)),fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),R2)) ).

% rtranclp.mono
tff(fact_5111_tranclp_Omono,axiom,
    ! [A: $tType,R2: fun(A,fun(A,$o))] : order_mono(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aTP_Lamp_aey(fun(A,fun(A,$o)),fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),R2)) ).

% tranclp.mono
tff(fact_5112_prod_Omap__ident,axiom,
    ! [B: $tType,A: $tType,T3: product_prod(A,B)] : aa(product_prod(A,B),product_prod(A,B),product_map_prod(A,A,B,B,aTP_Lamp_au(A,A),aTP_Lamp_oc(B,B)),T3) = T3 ).

% prod.map_ident
tff(fact_5113_lexordp_Omono,axiom,
    ! [A: $tType] :
      ( ord(A)
     => order_mono(fun(list(A),fun(list(A),$o)),fun(list(A),fun(list(A),$o)),aTP_Lamp_aez(fun(list(A),fun(list(A),$o)),fun(list(A),fun(list(A),$o)))) ) ).

% lexordp.mono
tff(fact_5114_set__Cons__def,axiom,
    ! [A: $tType,A4: set(A),XS: set(list(A))] : set_Cons(A,A4,XS) = aa(fun(list(A),$o),set(list(A)),collect(list(A)),aa(set(list(A)),fun(list(A),$o),aTP_Lamp_afa(set(A),fun(set(list(A)),fun(list(A),$o)),A4),XS)) ).

% set_Cons_def
tff(fact_5115_ord_Olexordp_Omono,axiom,
    ! [A: $tType,Less: fun(A,fun(A,$o))] : order_mono(fun(list(A),fun(list(A),$o)),fun(list(A),fun(list(A),$o)),aTP_Lamp_afb(fun(A,fun(A,$o)),fun(fun(list(A),fun(list(A),$o)),fun(list(A),fun(list(A),$o))),Less)) ).

% ord.lexordp.mono
tff(fact_5116_map__prod_Oidentity,axiom,
    ! [B: $tType,A: $tType] : product_map_prod(A,A,B,B,aTP_Lamp_au(A,A),aTP_Lamp_oc(B,B)) = id(product_prod(A,B)) ).

% map_prod.identity
tff(fact_5117_full__SetCompr__eq,axiom,
    ! [A: $tType,B: $tType,F: fun(B,A)] : aa(fun(A,$o),set(A),collect(A),aTP_Lamp_afc(fun(B,A),fun(A,$o),F)) = aa(set(B),set(A),image2(B,A,F),top_top(set(B))) ).

% full_SetCompr_eq
tff(fact_5118_finite__inf__Sup,axiom,
    ! [A: $tType] :
      ( finite8700451911770168679attice(A)
     => ! [A3: A,A4: set(A)] : aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),aa(set(A),A,complete_Sup_Sup(A),A4)) = aa(set(A),A,complete_Sup_Sup(A),aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_afd(A,fun(set(A),fun(A,$o)),A3),A4))) ) ).

% finite_inf_Sup
tff(fact_5119_prod__fun__imageE,axiom,
    ! [B: $tType,A: $tType,D: $tType,C: $tType,C2: product_prod(A,B),F: fun(C,A),G: fun(D,B),R: set(product_prod(C,D))] :
      ( aa(set(product_prod(A,B)),$o,member(product_prod(A,B),C2),aa(set(product_prod(C,D)),set(product_prod(A,B)),image2(product_prod(C,D),product_prod(A,B),product_map_prod(C,A,D,B,F,G)),R))
     => ~ ! [X2: C,Y2: D] :
            ( ( C2 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,F,X2)),aa(D,B,G,Y2)) )
           => ~ aa(set(product_prod(C,D)),$o,member(product_prod(C,D),aa(D,product_prod(C,D),aa(C,fun(D,product_prod(C,D)),product_Pair(C,D),X2),Y2)),R) ) ) ).

% prod_fun_imageE
tff(fact_5120_Field__not__elem,axiom,
    ! [A: $tType,V: A,R: set(product_prod(A,A))] :
      ( ~ aa(set(A),$o,member(A,V),field2(A,R))
     => ! [X3: product_prod(A,A)] :
          ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),X3),R)
         => aa(product_prod(A,A),$o,aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),aTP_Lamp_afe(A,fun(A,fun(A,$o)),V)),X3) ) ) ).

% Field_not_elem
tff(fact_5121_INTER__eq,axiom,
    ! [A: $tType,B: $tType,B3: fun(B,set(A)),A4: set(B)] : aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B3),A4)) = aa(fun(A,$o),set(A),collect(A),aa(set(B),fun(A,$o),aTP_Lamp_aff(fun(B,set(A)),fun(set(B),fun(A,$o)),B3),A4)) ).

% INTER_eq
tff(fact_5122_Collect__ball__eq,axiom,
    ! [A: $tType,B: $tType,A4: set(B),P: fun(A,fun(B,$o))] : aa(fun(A,$o),set(A),collect(A),aa(fun(A,fun(B,$o)),fun(A,$o),aTP_Lamp_afg(set(B),fun(fun(A,fun(B,$o)),fun(A,$o)),A4),P)) = aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aTP_Lamp_afi(fun(A,fun(B,$o)),fun(B,set(A)),P)),A4)) ).

% Collect_ball_eq
tff(fact_5123_Id__def,axiom,
    ! [A: $tType] : id2(A) = aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aTP_Lamp_afj(product_prod(A,A),$o)) ).

% Id_def
tff(fact_5124_map__prod__def,axiom,
    ! [A: $tType,C: $tType,D: $tType,B: $tType,F: fun(A,C),G: fun(B,D)] : product_map_prod(A,C,B,D,F,G) = aa(fun(A,fun(B,product_prod(C,D))),fun(product_prod(A,B),product_prod(C,D)),product_case_prod(A,B,product_prod(C,D)),aa(fun(B,D),fun(A,fun(B,product_prod(C,D))),aTP_Lamp_afk(fun(A,C),fun(fun(B,D),fun(A,fun(B,product_prod(C,D)))),F),G)) ).

% map_prod_def
tff(fact_5125_ran__def,axiom,
    ! [A: $tType,B: $tType,M: fun(B,option(A))] : ran(B,A,M) = aa(fun(A,$o),set(A),collect(A),aTP_Lamp_afl(fun(B,option(A)),fun(A,$o),M)) ).

% ran_def
tff(fact_5126_Gr__def,axiom,
    ! [B: $tType,A: $tType,A4: set(A),F: fun(A,B)] : bNF_Gr(A,B,A4,F) = aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,B),fun(product_prod(A,B),$o),aTP_Lamp_afm(set(A),fun(fun(A,B),fun(product_prod(A,B),$o)),A4),F)) ).

% Gr_def
tff(fact_5127_Ball__comp__iff,axiom,
    ! [A: $tType,B: $tType,C: $tType,A4: fun(B,set(C)),F: fun(C,$o),G: fun(A,B),X3: A] :
      ( aa(A,$o,aa(fun(A,B),fun(A,$o),comp(B,$o,A,aa(fun(C,$o),fun(B,$o),aTP_Lamp_afn(fun(B,set(C)),fun(fun(C,$o),fun(B,$o)),A4),F)),G),X3)
    <=> ! [Xa2: C] :
          ( aa(set(C),$o,member(C,Xa2),aa(A,set(C),aa(fun(A,B),fun(A,set(C)),comp(B,set(C),A,A4),G),X3))
         => aa(C,$o,F,Xa2) ) ) ).

% Ball_comp_iff
tff(fact_5128_listrel1__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : listrel1(A,R2) = aa(fun(product_prod(list(A),list(A)),$o),set(product_prod(list(A),list(A))),collect(product_prod(list(A),list(A))),aa(fun(list(A),fun(list(A),$o)),fun(product_prod(list(A),list(A)),$o),product_case_prod(list(A),list(A),$o),aTP_Lamp_afo(set(product_prod(A,A)),fun(list(A),fun(list(A),$o)),R2))) ).

% listrel1_def
tff(fact_5129_lexord__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : lexord(A,R2) = aa(fun(product_prod(list(A),list(A)),$o),set(product_prod(list(A),list(A))),collect(product_prod(list(A),list(A))),aa(fun(list(A),fun(list(A),$o)),fun(product_prod(list(A),list(A)),$o),product_case_prod(list(A),list(A),$o),aTP_Lamp_afp(set(product_prod(A,A)),fun(list(A),fun(list(A),$o)),R2))) ).

% lexord_def
tff(fact_5130_case__prod__o__map__prod,axiom,
    ! [A: $tType,D: $tType,C: $tType,E: $tType,B: $tType,F: fun(D,fun(E,C)),G1: fun(A,D),G22: fun(B,E)] : aa(fun(product_prod(A,B),product_prod(D,E)),fun(product_prod(A,B),C),comp(product_prod(D,E),C,product_prod(A,B),aa(fun(D,fun(E,C)),fun(product_prod(D,E),C),product_case_prod(D,E,C),F)),product_map_prod(A,D,B,E,G1,G22)) = aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),aa(fun(B,E),fun(A,fun(B,C)),aa(fun(A,D),fun(fun(B,E),fun(A,fun(B,C))),aTP_Lamp_afq(fun(D,fun(E,C)),fun(fun(A,D),fun(fun(B,E),fun(A,fun(B,C)))),F),G1),G22)) ).

% case_prod_o_map_prod
tff(fact_5131_ID_Opred__set,axiom,
    ! [A: $tType,P: fun(A,$o),X3: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),bNF_id_bnf(fun(A,$o)),P),X3)
    <=> ! [Xa2: A] :
          ( aa(set(A),$o,member(A,Xa2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X3),bot_bot(set(A))))
         => aa(A,$o,P,Xa2) ) ) ).

% ID.pred_set
tff(fact_5132_Un__interval,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(A)
     => ! [B13: A,B23: A,B32: A,F: fun(A,B)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B13),B23)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B23),B32)
           => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),aa(fun(B,$o),set(B),collect(B),aa(fun(A,B),fun(B,$o),aa(A,fun(fun(A,B),fun(B,$o)),aTP_Lamp_afr(A,fun(A,fun(fun(A,B),fun(B,$o))),B13),B23),F))),aa(fun(B,$o),set(B),collect(B),aa(fun(A,B),fun(B,$o),aa(A,fun(fun(A,B),fun(B,$o)),aTP_Lamp_afr(A,fun(A,fun(fun(A,B),fun(B,$o))),B23),B32),F))) = aa(fun(B,$o),set(B),collect(B),aa(fun(A,B),fun(B,$o),aa(A,fun(fun(A,B),fun(B,$o)),aTP_Lamp_afr(A,fun(A,fun(fun(A,B),fun(B,$o))),B13),B32),F)) ) ) ) ) ).

% Un_interval
tff(fact_5133_Sup__eq__Inf,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A4: set(A)] : aa(set(A),A,complete_Sup_Sup(A),A4) = aa(set(A),A,complete_Inf_Inf(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_afs(set(A),fun(A,$o),A4))) ) ).

% Sup_eq_Inf
tff(fact_5134_Inf__eq__Sup,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A4: set(A)] : aa(set(A),A,complete_Inf_Inf(A),A4) = aa(set(A),A,complete_Sup_Sup(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_aft(set(A),fun(A,$o),A4))) ) ).

% Inf_eq_Sup
tff(fact_5135_lex__conv,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : lex(A,R2) = aa(fun(product_prod(list(A),list(A)),$o),set(product_prod(list(A),list(A))),collect(product_prod(list(A),list(A))),aa(fun(list(A),fun(list(A),$o)),fun(product_prod(list(A),list(A)),$o),product_case_prod(list(A),list(A),$o),aTP_Lamp_afu(set(product_prod(A,A)),fun(list(A),fun(list(A),$o)),R2))) ).

% lex_conv
tff(fact_5136_Collect__ex__eq,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,$o))] : aa(fun(A,$o),set(A),collect(A),aTP_Lamp_afv(fun(A,fun(B,$o)),fun(A,$o),P)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aTP_Lamp_afi(fun(A,fun(B,$o)),fun(B,set(A)),P)),top_top(set(B)))) ).

% Collect_ex_eq
tff(fact_5137_lexn__conv,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),N: nat] : aa(nat,set(product_prod(list(A),list(A))),lexn(A,R2),N) = aa(fun(product_prod(list(A),list(A)),$o),set(product_prod(list(A),list(A))),collect(product_prod(list(A),list(A))),aa(fun(list(A),fun(list(A),$o)),fun(product_prod(list(A),list(A)),$o),product_case_prod(list(A),list(A),$o),aa(nat,fun(list(A),fun(list(A),$o)),aTP_Lamp_afw(set(product_prod(A,A)),fun(nat,fun(list(A),fun(list(A),$o))),R2),N))) ).

% lexn_conv
tff(fact_5138_relcomp__unfold,axiom,
    ! [A: $tType,B: $tType,C: $tType,R2: set(product_prod(A,C)),S2: set(product_prod(C,B))] : relcomp(A,C,B,R2,S2) = aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aa(set(product_prod(C,B)),fun(A,fun(B,$o)),aTP_Lamp_afx(set(product_prod(A,C)),fun(set(product_prod(C,B)),fun(A,fun(B,$o))),R2),S2))) ).

% relcomp_unfold
tff(fact_5139_takeWhile__append,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A),Ys: list(A)] :
      takeWhile(A,P,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = $ite(
        ! [X4: A] :
          ( aa(set(A),$o,member(A,X4),aa(list(A),set(A),set2(A),Xs))
         => aa(A,$o,P,X4) ),
        aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),takeWhile(A,P,Ys)),
        takeWhile(A,P,Xs) ) ).

% takeWhile_append
tff(fact_5140_dropWhile__append,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A),Ys: list(A)] :
      dropWhile(A,P,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = $ite(
        ! [X4: A] :
          ( aa(set(A),$o,member(A,X4),aa(list(A),set(A),set2(A),Xs))
         => aa(A,$o,P,X4) ),
        dropWhile(A,P,Ys),
        aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),dropWhile(A,P,Xs)),Ys) ) ).

% dropWhile_append
tff(fact_5141_graph__def,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B))] : graph(A,B,M) = aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aTP_Lamp_afy(fun(A,option(B)),fun(product_prod(A,B),$o),M)) ).

% graph_def
tff(fact_5142_map__prod__surj__on,axiom,
    ! [D: $tType,B: $tType,A: $tType,C: $tType,F: fun(B,A),A4: set(B),A14: set(A),G: fun(D,C),B3: set(D),B14: set(C)] :
      ( ( aa(set(B),set(A),image2(B,A,F),A4) = A14 )
     => ( ( aa(set(D),set(C),image2(D,C,G),B3) = B14 )
       => ( aa(set(product_prod(B,D)),set(product_prod(A,C)),image2(product_prod(B,D),product_prod(A,C),product_map_prod(B,A,D,C,F,G)),product_Sigma(B,D,A4,aTP_Lamp_afz(set(D),fun(B,set(D)),B3))) = product_Sigma(A,C,A14,aTP_Lamp_us(set(C),fun(A,set(C)),B14)) ) ) ) ).

% map_prod_surj_on
tff(fact_5143_set__map__filter,axiom,
    ! [A: $tType,B: $tType,G: fun(B,option(A)),Xs: list(B)] : aa(list(A),set(A),set2(A),map_filter(B,A,G,Xs)) = aa(fun(A,$o),set(A),collect(A),aa(list(B),fun(A,$o),aTP_Lamp_aga(fun(B,option(A)),fun(list(B),fun(A,$o)),G),Xs)) ).

% set_map_filter
tff(fact_5144_map__prod__inj__on,axiom,
    ! [D: $tType,B: $tType,A: $tType,C: $tType,F: fun(A,B),A4: set(A),G: fun(C,D),B3: set(C)] :
      ( inj_on(A,B,F,A4)
     => ( inj_on(C,D,G,B3)
       => inj_on(product_prod(A,C),product_prod(B,D),product_map_prod(A,B,C,D,F,G),product_Sigma(A,C,A4,aTP_Lamp_us(set(C),fun(A,set(C)),B3))) ) ) ).

% map_prod_inj_on
tff(fact_5145_set__conv__nth,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(A),set(A),set2(A),Xs) = aa(fun(A,$o),set(A),collect(A),aTP_Lamp_agb(list(A),fun(A,$o),Xs)) ).

% set_conv_nth
tff(fact_5146_inf__Sup2__distrib,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [A4: set(A),B3: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( aa(set(A),$o,finite_finite2(A),B3)
             => ( ( B3 != bot_bot(set(A)) )
               => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,lattic5882676163264333800up_fin(A),A4)),aa(set(A),A,lattic5882676163264333800up_fin(A),B3)) = aa(set(A),A,lattic5882676163264333800up_fin(A),aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_agc(set(A),fun(set(A),fun(A,$o)),A4),B3))) ) ) ) ) ) ) ).

% inf_Sup2_distrib
tff(fact_5147_inf__Sup1__distrib,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [A4: set(A),X: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(set(A),A,lattic5882676163264333800up_fin(A),A4)) = aa(set(A),A,lattic5882676163264333800up_fin(A),aa(fun(A,$o),set(A),collect(A),aa(A,fun(A,$o),aTP_Lamp_agd(set(A),fun(A,fun(A,$o)),A4),X))) ) ) ) ) ).

% inf_Sup1_distrib
tff(fact_5148_sup__Inf2__distrib,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [A4: set(A),B3: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( aa(set(A),$o,finite_finite2(A),B3)
             => ( ( B3 != bot_bot(set(A)) )
               => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,lattic7752659483105999362nf_fin(A),A4)),aa(set(A),A,lattic7752659483105999362nf_fin(A),B3)) = aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_age(set(A),fun(set(A),fun(A,$o)),A4),B3))) ) ) ) ) ) ) ).

% sup_Inf2_distrib
tff(fact_5149_sup__Inf1__distrib,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [A4: set(A),X: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(set(A),A,lattic7752659483105999362nf_fin(A),A4)) = aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(fun(A,$o),set(A),collect(A),aa(A,fun(A,$o),aTP_Lamp_agf(set(A),fun(A,fun(A,$o)),A4),X))) ) ) ) ) ).

% sup_Inf1_distrib
tff(fact_5150_wf__map__prod__image,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,A)),F: fun(A,B)] :
      ( wf(A,R2)
     => ( inj_on(A,B,F,top_top(set(A)))
       => wf(B,aa(set(product_prod(A,A)),set(product_prod(B,B)),image2(product_prod(A,A),product_prod(B,B),product_map_prod(A,B,A,B,F,F)),R2)) ) ) ).

% wf_map_prod_image
tff(fact_5151_list__eq__iff__zip__eq,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( Xs = Ys )
    <=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys) )
        & ! [X4: product_prod(A,A)] :
            ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),X4),aa(list(product_prod(A,A)),set(product_prod(A,A)),set2(product_prod(A,A)),zip(A,A,Xs,Ys)))
           => aa(product_prod(A,A),$o,aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),fequal(A)),X4) ) ) ) ).

% list_eq_iff_zip_eq
tff(fact_5152_map__prod__surj,axiom,
    ! [B: $tType,D: $tType,C: $tType,A: $tType,F: fun(B,A),G: fun(D,C)] :
      ( ( aa(set(B),set(A),image2(B,A,F),top_top(set(B))) = top_top(set(A)) )
     => ( ( aa(set(D),set(C),image2(D,C,G),top_top(set(D))) = top_top(set(C)) )
       => ( aa(set(product_prod(B,D)),set(product_prod(A,C)),image2(product_prod(B,D),product_prod(A,C),product_map_prod(B,A,D,C,F,G)),top_top(set(product_prod(B,D)))) = top_top(set(product_prod(A,C))) ) ) ) ).

% map_prod_surj
tff(fact_5153_prod_Oinj__map,axiom,
    ! [D: $tType,B: $tType,C: $tType,A: $tType,F1: fun(A,B),F22: fun(C,D)] :
      ( inj_on(A,B,F1,top_top(set(A)))
     => ( inj_on(C,D,F22,top_top(set(C)))
       => inj_on(product_prod(A,C),product_prod(B,D),product_map_prod(A,B,C,D,F1,F22),top_top(set(product_prod(A,C)))) ) ) ).

% prod.inj_map
tff(fact_5154_concat__eq__concat__iff,axiom,
    ! [A: $tType,Xs: list(list(A)),Ys: list(list(A))] :
      ( ! [X2: product_prod(list(A),list(A))] :
          ( aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),X2),aa(list(product_prod(list(A),list(A))),set(product_prod(list(A),list(A))),set2(product_prod(list(A),list(A))),zip(list(A),list(A),Xs,Ys)))
         => aa(product_prod(list(A),list(A)),$o,aa(fun(list(A),fun(list(A),$o)),fun(product_prod(list(A),list(A)),$o),product_case_prod(list(A),list(A),$o),aTP_Lamp_agg(list(A),fun(list(A),$o))),X2) )
     => ( ( 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_5155_concat__injective,axiom,
    ! [A: $tType,Xs: list(list(A)),Ys: list(list(A))] :
      ( ( concat(A,Xs) = concat(A,Ys) )
     => ( ( aa(list(list(A)),nat,size_size(list(list(A))),Xs) = aa(list(list(A)),nat,size_size(list(list(A))),Ys) )
       => ( ! [X2: product_prod(list(A),list(A))] :
              ( aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),X2),aa(list(product_prod(list(A),list(A))),set(product_prod(list(A),list(A))),set2(product_prod(list(A),list(A))),zip(list(A),list(A),Xs,Ys)))
             => aa(product_prod(list(A),list(A)),$o,aa(fun(list(A),fun(list(A),$o)),fun(product_prod(list(A),list(A)),$o),product_case_prod(list(A),list(A),$o),aTP_Lamp_agg(list(A),fun(list(A),$o))),X2) )
         => ( Xs = Ys ) ) ) ) ).

% concat_injective
tff(fact_5156_set__nths,axiom,
    ! [A: $tType,Xs: list(A),I4: set(nat)] : aa(list(A),set(A),set2(A),nths(A,Xs,I4)) = aa(fun(A,$o),set(A),collect(A),aa(set(nat),fun(A,$o),aTP_Lamp_agh(list(A),fun(set(nat),fun(A,$o)),Xs),I4)) ).

% set_nths
tff(fact_5157_funpow__inj__finite,axiom,
    ! [A: $tType,P3: fun(A,A),X: A] :
      ( inj_on(A,A,P3,top_top(set(A)))
     => ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(A,fun(A,$o),aTP_Lamp_agi(fun(A,A),fun(A,fun(A,$o)),P3),X)))
       => ~ ! [N3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N3)
             => ( aa(A,A,compow(fun(A,A),N3,P3),X) != X ) ) ) ) ).

% funpow_inj_finite
tff(fact_5158_set__drop__conv,axiom,
    ! [A: $tType,N: nat,L: list(A)] : aa(list(A),set(A),set2(A),drop(A,N,L)) = aa(fun(A,$o),set(A),collect(A),aa(list(A),fun(A,$o),aTP_Lamp_agj(nat,fun(list(A),fun(A,$o)),N),L)) ).

% set_drop_conv
tff(fact_5159_sum__mult__sum__if__inj,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( semiring_0(C)
     => ! [F: fun(A,C),G: fun(B,C),A4: set(A),B3: set(B)] :
          ( inj_on(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),aa(fun(B,C),fun(A,fun(B,C)),aTP_Lamp_agk(fun(A,C),fun(fun(B,C),fun(A,fun(B,C))),F),G)),product_Sigma(A,B,A4,aTP_Lamp_qz(set(B),fun(A,set(B)),B3)))
         => ( aa(C,C,aa(C,fun(C,C),times_times(C),aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7311177749621191930dd_sum(A,C),F),A4)),aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),G),B3)) = aa(set(C),C,aa(fun(C,C),fun(set(C),C),groups7311177749621191930dd_sum(C,C),id(C)),aa(fun(C,$o),set(C),collect(C),aa(set(B),fun(C,$o),aa(set(A),fun(set(B),fun(C,$o)),aa(fun(B,C),fun(set(A),fun(set(B),fun(C,$o))),aTP_Lamp_agl(fun(A,C),fun(fun(B,C),fun(set(A),fun(set(B),fun(C,$o)))),F),G),A4),B3))) ) ) ) ).

% sum_mult_sum_if_inj
tff(fact_5160_UnderS__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A4: set(A)] : order_UnderS(A,R2,A4) = aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_agm(set(product_prod(A,A)),fun(set(A),fun(A,$o)),R2),A4)) ).

% UnderS_def
tff(fact_5161_Under__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A4: set(A)] : order_Under(A,R2,A4) = aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_agn(set(product_prod(A,A)),fun(set(A),fun(A,$o)),R2),A4)) ).

% Under_def
tff(fact_5162_sorted__wrt_Opelims_I2_J,axiom,
    ! [A: $tType,X: fun(A,fun(A,$o)),Xa: list(A)] :
      ( sorted_wrt(A,X,Xa)
     => ( accp(product_prod(fun(A,fun(A,$o)),list(A)),sorted_wrt_rel(A),aa(list(A),product_prod(fun(A,fun(A,$o)),list(A)),aa(fun(A,fun(A,$o)),fun(list(A),product_prod(fun(A,fun(A,$o)),list(A))),product_Pair(fun(A,fun(A,$o)),list(A)),X),Xa))
       => ( ( ( Xa = nil(A) )
           => ~ accp(product_prod(fun(A,fun(A,$o)),list(A)),sorted_wrt_rel(A),aa(list(A),product_prod(fun(A,fun(A,$o)),list(A)),aa(fun(A,fun(A,$o)),fun(list(A),product_prod(fun(A,fun(A,$o)),list(A))),product_Pair(fun(A,fun(A,$o)),list(A)),X),nil(A))) )
         => ~ ! [X2: A,Ys2: list(A)] :
                ( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Ys2) )
               => ( accp(product_prod(fun(A,fun(A,$o)),list(A)),sorted_wrt_rel(A),aa(list(A),product_prod(fun(A,fun(A,$o)),list(A)),aa(fun(A,fun(A,$o)),fun(list(A),product_prod(fun(A,fun(A,$o)),list(A))),product_Pair(fun(A,fun(A,$o)),list(A)),X),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Ys2)))
                 => ~ ( ! [Xa3: A] :
                          ( aa(set(A),$o,member(A,Xa3),aa(list(A),set(A),set2(A),Ys2))
                         => aa(A,$o,aa(A,fun(A,$o),X,X2),Xa3) )
                      & sorted_wrt(A,X,Ys2) ) ) ) ) ) ) ).

% sorted_wrt.pelims(2)
tff(fact_5163_Inf__Sup,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [A4: set(set(A))] : aa(set(A),A,complete_Inf_Inf(A),aa(set(set(A)),set(A),image2(set(A),A,complete_Sup_Sup(A)),A4)) = aa(set(A),A,complete_Sup_Sup(A),aa(set(set(A)),set(A),image2(set(A),A,complete_Inf_Inf(A)),aa(fun(set(A),$o),set(set(A)),collect(set(A)),aTP_Lamp_ago(set(set(A)),fun(set(A),$o),A4)))) ) ).

% Inf_Sup
tff(fact_5164_Sup__Inf,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [A4: set(set(A))] : aa(set(A),A,complete_Sup_Sup(A),aa(set(set(A)),set(A),image2(set(A),A,complete_Inf_Inf(A)),A4)) = aa(set(A),A,complete_Inf_Inf(A),aa(set(set(A)),set(A),image2(set(A),A,complete_Sup_Sup(A)),aa(fun(set(A),$o),set(set(A)),collect(set(A)),aTP_Lamp_ago(set(set(A)),fun(set(A),$o),A4)))) ) ).

% Sup_Inf
tff(fact_5165_Inter__eq,axiom,
    ! [A: $tType,A4: set(set(A))] : aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),A4) = aa(fun(A,$o),set(A),collect(A),aTP_Lamp_agp(set(set(A)),fun(A,$o),A4)) ).

% Inter_eq
tff(fact_5166_mono__compose,axiom,
    ! [A: $tType,C: $tType,B: $tType,D: $tType] :
      ( ( order(C)
        & order(A) )
     => ! [Q: fun(A,fun(B,C)),F: fun(D,B)] :
          ( order_mono(A,fun(B,C),Q)
         => order_mono(A,fun(D,C),aa(fun(D,B),fun(A,fun(D,C)),aTP_Lamp_agq(fun(A,fun(B,C)),fun(fun(D,B),fun(A,fun(D,C))),Q),F)) ) ) ).

% mono_compose
tff(fact_5167_Inf__Sup__le,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [A4: set(set(A))] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(set(A)),set(A),image2(set(A),A,complete_Sup_Sup(A)),A4))),aa(set(A),A,complete_Sup_Sup(A),aa(set(set(A)),set(A),image2(set(A),A,complete_Inf_Inf(A)),aa(fun(set(A),$o),set(set(A)),collect(set(A)),aTP_Lamp_ago(set(set(A)),fun(set(A),$o),A4))))) ) ).

% Inf_Sup_le
tff(fact_5168_Sup__Inf__le,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A4: set(set(A))] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(set(A)),set(A),image2(set(A),A,complete_Inf_Inf(A)),aa(fun(set(A),$o),set(set(A)),collect(set(A)),aTP_Lamp_agr(set(set(A)),fun(set(A),$o),A4))))),aa(set(A),A,complete_Inf_Inf(A),aa(set(set(A)),set(A),image2(set(A),A,complete_Sup_Sup(A)),A4))) ) ).

% Sup_Inf_le
tff(fact_5169_finite__Inf__Sup,axiom,
    ! [A: $tType] :
      ( finite8700451911770168679attice(A)
     => ! [A4: set(set(A))] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(set(A)),set(A),image2(set(A),A,complete_Sup_Sup(A)),A4))),aa(set(A),A,complete_Sup_Sup(A),aa(set(set(A)),set(A),image2(set(A),A,complete_Inf_Inf(A)),aa(fun(set(A),$o),set(set(A)),collect(set(A)),aTP_Lamp_ags(set(set(A)),fun(set(A),$o),A4))))) ) ).

% finite_Inf_Sup
tff(fact_5170_SUP__INF__set,axiom,
    ! [A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [G: fun(B,A),A4: set(set(B))] : aa(set(A),A,complete_Sup_Sup(A),aa(set(set(B)),set(A),image2(set(B),A,aTP_Lamp_agt(fun(B,A),fun(set(B),A),G)),A4)) = aa(set(A),A,complete_Inf_Inf(A),aa(set(set(B)),set(A),image2(set(B),A,aTP_Lamp_agu(fun(B,A),fun(set(B),A),G)),aa(fun(set(B),$o),set(set(B)),collect(set(B)),aTP_Lamp_agv(set(set(B)),fun(set(B),$o),A4)))) ) ).

% SUP_INF_set
tff(fact_5171_INF__SUP__set,axiom,
    ! [A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [G: fun(B,A),A4: set(set(B))] : aa(set(A),A,complete_Inf_Inf(A),aa(set(set(B)),set(A),image2(set(B),A,aTP_Lamp_agu(fun(B,A),fun(set(B),A),G)),A4)) = aa(set(A),A,complete_Sup_Sup(A),aa(set(set(B)),set(A),image2(set(B),A,aTP_Lamp_agt(fun(B,A),fun(set(B),A),G)),aa(fun(set(B),$o),set(set(B)),collect(set(B)),aTP_Lamp_agv(set(set(B)),fun(set(B),$o),A4)))) ) ).

% INF_SUP_set
tff(fact_5172_option__Inf__Sup,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [A4: set(set(option(A)))] : aa(option(A),$o,aa(option(A),fun(option(A),$o),ord_less_eq(option(A)),aa(set(option(A)),option(A),complete_Inf_Inf(option(A)),aa(set(set(option(A))),set(option(A)),image2(set(option(A)),option(A),complete_Sup_Sup(option(A))),A4))),aa(set(option(A)),option(A),complete_Sup_Sup(option(A)),aa(set(set(option(A))),set(option(A)),image2(set(option(A)),option(A),complete_Inf_Inf(option(A))),aa(fun(set(option(A)),$o),set(set(option(A))),collect(set(option(A))),aTP_Lamp_agw(set(set(option(A))),fun(set(option(A)),$o),A4))))) ) ).

% option_Inf_Sup
tff(fact_5173_Pow__Compl,axiom,
    ! [A: $tType,A4: set(A)] : pow2(A,aa(set(A),set(A),uminus_uminus(set(A)),A4)) = aa(fun(set(A),$o),set(set(A)),collect(set(A)),aTP_Lamp_agx(set(A),fun(set(A),$o),A4)) ).

% Pow_Compl
tff(fact_5174_Union__maximal__sets,axiom,
    ! [A: $tType,F7: set(set(A))] :
      ( aa(set(set(A)),$o,finite_finite2(set(A)),F7)
     => ( aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(fun(set(A),$o),set(set(A)),collect(set(A)),aTP_Lamp_agy(set(set(A)),fun(set(A),$o),F7))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),F7) ) ) ).

% Union_maximal_sets
tff(fact_5175_Inf__filter__def,axiom,
    ! [A: $tType,S: set(filter(A))] : aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),S) = aa(set(filter(A)),filter(A),complete_Sup_Sup(filter(A)),aa(fun(filter(A),$o),set(filter(A)),collect(filter(A)),aTP_Lamp_agz(set(filter(A)),fun(filter(A),$o),S))) ).

% Inf_filter_def
tff(fact_5176_Nats__altdef1,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ( semiring_1_Nats(A) = aa(fun(A,$o),set(A),collect(A),aTP_Lamp_aha(A,$o)) ) ) ).

% Nats_altdef1
tff(fact_5177_Sup__int__def,axiom,
    ! [X5: set(int)] : aa(set(int),int,complete_Sup_Sup(int),X5) = the(int,aTP_Lamp_ahb(set(int),fun(int,$o),X5)) ).

% Sup_int_def
tff(fact_5178_sorted__wrt_Opelims_I3_J,axiom,
    ! [A: $tType,X: fun(A,fun(A,$o)),Xa: list(A)] :
      ( ~ sorted_wrt(A,X,Xa)
     => ( accp(product_prod(fun(A,fun(A,$o)),list(A)),sorted_wrt_rel(A),aa(list(A),product_prod(fun(A,fun(A,$o)),list(A)),aa(fun(A,fun(A,$o)),fun(list(A),product_prod(fun(A,fun(A,$o)),list(A))),product_Pair(fun(A,fun(A,$o)),list(A)),X),Xa))
       => ~ ! [X2: A,Ys2: list(A)] :
              ( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Ys2) )
             => ( accp(product_prod(fun(A,fun(A,$o)),list(A)),sorted_wrt_rel(A),aa(list(A),product_prod(fun(A,fun(A,$o)),list(A)),aa(fun(A,fun(A,$o)),fun(list(A),product_prod(fun(A,fun(A,$o)),list(A))),product_Pair(fun(A,fun(A,$o)),list(A)),X),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Ys2)))
               => ( ! [Xa4: A] :
                      ( aa(set(A),$o,member(A,Xa4),aa(list(A),set(A),set2(A),Ys2))
                     => aa(A,$o,aa(A,fun(A,$o),X,X2),Xa4) )
                  & sorted_wrt(A,X,Ys2) ) ) ) ) ) ).

% sorted_wrt.pelims(3)
tff(fact_5179_sorted__wrt_Opelims_I1_J,axiom,
    ! [A: $tType,X: fun(A,fun(A,$o)),Xa: list(A),Y: $o] :
      ( ( sorted_wrt(A,X,Xa)
      <=> (Y) )
     => ( accp(product_prod(fun(A,fun(A,$o)),list(A)),sorted_wrt_rel(A),aa(list(A),product_prod(fun(A,fun(A,$o)),list(A)),aa(fun(A,fun(A,$o)),fun(list(A),product_prod(fun(A,fun(A,$o)),list(A))),product_Pair(fun(A,fun(A,$o)),list(A)),X),Xa))
       => ( ( ( Xa = nil(A) )
           => ( (Y)
             => ~ accp(product_prod(fun(A,fun(A,$o)),list(A)),sorted_wrt_rel(A),aa(list(A),product_prod(fun(A,fun(A,$o)),list(A)),aa(fun(A,fun(A,$o)),fun(list(A),product_prod(fun(A,fun(A,$o)),list(A))),product_Pair(fun(A,fun(A,$o)),list(A)),X),nil(A))) ) )
         => ~ ! [X2: A,Ys2: list(A)] :
                ( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Ys2) )
               => ( ( (Y)
                  <=> ( ! [Xa2: A] :
                          ( aa(set(A),$o,member(A,Xa2),aa(list(A),set(A),set2(A),Ys2))
                         => aa(A,$o,aa(A,fun(A,$o),X,X2),Xa2) )
                      & sorted_wrt(A,X,Ys2) ) )
                 => ~ accp(product_prod(fun(A,fun(A,$o)),list(A)),sorted_wrt_rel(A),aa(list(A),product_prod(fun(A,fun(A,$o)),list(A)),aa(fun(A,fun(A,$o)),fun(list(A),product_prod(fun(A,fun(A,$o)),list(A))),product_Pair(fun(A,fun(A,$o)),list(A)),X),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Ys2))) ) ) ) ) ) ).

% sorted_wrt.pelims(1)
tff(fact_5180_Above__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A4: set(A)] : order_Above(A,R2,A4) = aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_ahc(set(product_prod(A,A)),fun(set(A),fun(A,$o)),R2),A4)) ).

% Above_def
tff(fact_5181_map__to__set__upd,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B)),K: A,V: B] : map_to_set(A,B,fun_upd(A,option(B),M,K,aa(B,option(B),some(B),V))) = aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(product_prod(A,B),fun(set(product_prod(A,B)),set(product_prod(A,B))),insert2(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K),V)),aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),minus_minus(set(product_prod(A,B))),map_to_set(A,B,M)),aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aTP_Lamp_ahd(A,fun(product_prod(A,B),$o),K)))) ).

% map_to_set_upd
tff(fact_5182_brk__rel__def,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,B))] : brk_rel(A,B,R) = aa(set(product_prod(product_prod($o,A),product_prod($o,B))),set(product_prod(product_prod($o,A),product_prod($o,B))),aa(set(product_prod(product_prod($o,A),product_prod($o,B))),fun(set(product_prod(product_prod($o,A),product_prod($o,B))),set(product_prod(product_prod($o,A),product_prod($o,B)))),sup_sup(set(product_prod(product_prod($o,A),product_prod($o,B)))),aa(fun(product_prod(product_prod($o,A),product_prod($o,B)),$o),set(product_prod(product_prod($o,A),product_prod($o,B))),collect(product_prod(product_prod($o,A),product_prod($o,B))),aTP_Lamp_ahe(set(product_prod(A,B)),fun(product_prod(product_prod($o,A),product_prod($o,B)),$o),R))),aa(fun(product_prod(product_prod($o,A),product_prod($o,B)),$o),set(product_prod(product_prod($o,A),product_prod($o,B))),collect(product_prod(product_prod($o,A),product_prod($o,B))),aTP_Lamp_ahf(product_prod(product_prod($o,A),product_prod($o,B)),$o))) ).

% brk_rel_def
tff(fact_5183_map__to__set__empty,axiom,
    ! [B: $tType,A: $tType] : map_to_set(A,B,aTP_Lamp_tf(A,option(B))) = bot_bot(set(product_prod(A,B))) ).

% map_to_set_empty
tff(fact_5184_map__to__set__empty__iff_I2_J,axiom,
    ! [A: $tType,B: $tType,M: fun(A,option(B))] :
      ( ( bot_bot(set(product_prod(A,B))) = map_to_set(A,B,M) )
    <=> ! [X4: A] : aa(A,option(B),M,X4) = none(B) ) ).

% map_to_set_empty_iff(2)
tff(fact_5185_map__to__set__empty__iff_I1_J,axiom,
    ! [A: $tType,B: $tType,M: fun(A,option(B))] :
      ( ( map_to_set(A,B,M) = bot_bot(set(product_prod(A,B))) )
    <=> ! [X4: A] : aa(A,option(B),M,X4) = none(B) ) ).

% map_to_set_empty_iff(1)
tff(fact_5186_map__to__set__def,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B))] : map_to_set(A,B,M) = aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aTP_Lamp_ahg(fun(A,option(B)),fun(A,fun(B,$o)),M))) ).

% map_to_set_def
tff(fact_5187_rel__pred__comp__def,axiom,
    ! [A: $tType,B: $tType,R: fun(A,fun(B,$o)),P: fun(B,$o),X3: A] :
      ( rel_pred_comp(A,B,R,P,X3)
    <=> ? [Y3: B] :
          ( aa(B,$o,aa(A,fun(B,$o),R,X3),Y3)
          & aa(B,$o,P,Y3) ) ) ).

% rel_pred_comp_def
tff(fact_5188_iteratesp_Omono,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [F: fun(A,A)] : order_mono(fun(A,$o),fun(A,$o),aTP_Lamp_ahh(fun(A,A),fun(fun(A,$o),fun(A,$o)),F)) ) ).

% iteratesp.mono
tff(fact_5189_list__collect__set__alt,axiom,
    ! [A: $tType,B: $tType,F: fun(B,set(A)),L: list(B)] : list_collect_set(B,A,F,L) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(fun(set(A),$o),set(set(A)),collect(set(A)),aa(list(B),fun(set(A),$o),aTP_Lamp_ahi(fun(B,set(A)),fun(list(B),fun(set(A),$o)),F),L))) ).

% list_collect_set_alt
tff(fact_5190_list__collect__set__simps_I1_J,axiom,
    ! [B: $tType,A: $tType,F: fun(B,set(A))] : list_collect_set(B,A,F,nil(B)) = bot_bot(set(A)) ).

% list_collect_set_simps(1)
tff(fact_5191_list__collect__set__simps_I3_J,axiom,
    ! [A: $tType,B: $tType,F: fun(B,set(A)),A3: B,L: list(B)] : list_collect_set(B,A,F,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),A3),L)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(B,set(A),F,A3)),list_collect_set(B,A,F,L)) ).

% list_collect_set_simps(3)
tff(fact_5192_list__collect__set__simps_I4_J,axiom,
    ! [A: $tType,B: $tType,F: fun(B,set(A)),L: list(B),L3: list(B)] : list_collect_set(B,A,F,aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),L),L3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),list_collect_set(B,A,F,L)),list_collect_set(B,A,F,L3)) ).

% list_collect_set_simps(4)
tff(fact_5193_list__collect__set__map__simps_I1_J,axiom,
    ! [C: $tType,B: $tType,A: $tType,F: fun(B,set(A)),X: fun(C,B)] : list_collect_set(B,A,F,aa(list(C),list(B),map(C,B,X),nil(C))) = bot_bot(set(A)) ).

% list_collect_set_map_simps(1)
tff(fact_5194_list__collect__set__map__simps_I3_J,axiom,
    ! [A: $tType,B: $tType,C: $tType,F: fun(B,set(A)),X: fun(C,B),A3: C,L: list(C)] : list_collect_set(B,A,F,aa(list(C),list(B),map(C,B,X),aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),A3),L))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(B,set(A),F,aa(C,B,X,A3))),list_collect_set(B,A,F,aa(list(C),list(B),map(C,B,X),L))) ).

% list_collect_set_map_simps(3)
tff(fact_5195_list__collect__set__map__simps_I4_J,axiom,
    ! [A: $tType,B: $tType,C: $tType,F: fun(B,set(A)),X: fun(C,B),L: list(C),L3: list(C)] : list_collect_set(B,A,F,aa(list(C),list(B),map(C,B,X),aa(list(C),list(C),aa(list(C),fun(list(C),list(C)),append(C),L),L3))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),list_collect_set(B,A,F,aa(list(C),list(B),map(C,B,X),L))),list_collect_set(B,A,F,aa(list(C),list(B),map(C,B,X),L3))) ).

% list_collect_set_map_simps(4)
tff(fact_5196_chain__compr,axiom,
    ! [A: $tType,Ord: fun(A,fun(A,$o)),A4: set(A),P: fun(A,$o)] :
      ( comple1602240252501008431_chain(A,Ord,A4)
     => comple1602240252501008431_chain(A,Ord,aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_bi(set(A),fun(fun(A,$o),fun(A,$o)),A4),P))) ) ).

% chain_compr
tff(fact_5197_chain__empty,axiom,
    ! [A: $tType,Ord: fun(A,fun(A,$o))] : comple1602240252501008431_chain(A,Ord,bot_bot(set(A))) ).

% chain_empty
tff(fact_5198_chain__singleton,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [X: A] : comple1602240252501008431_chain(A,ord_less_eq(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) ) ).

% chain_singleton
tff(fact_5199_list__collect__set__def,axiom,
    ! [A: $tType,B: $tType,F: fun(B,set(A)),L: list(B)] : list_collect_set(B,A,F,L) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(fun(set(A),$o),set(set(A)),collect(set(A)),aa(list(B),fun(set(A),$o),aTP_Lamp_ahj(fun(B,set(A)),fun(list(B),fun(set(A),$o)),F),L))) ).

% list_collect_set_def
tff(fact_5200_in__chain__finite,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [A4: set(A)] :
          ( comple1602240252501008431_chain(A,ord_less_eq(A),A4)
         => ( aa(set(A),$o,finite_finite2(A),A4)
           => ( ( A4 != bot_bot(set(A)) )
             => aa(set(A),$o,member(A,aa(set(A),A,complete_Sup_Sup(A),A4)),A4) ) ) ) ) ).

% in_chain_finite
tff(fact_5201_image2__def,axiom,
    ! [A: $tType,B: $tType,C: $tType,A4: set(C),F: fun(C,A),G: fun(C,B)] : bNF_Greatest_image2(C,A,B,A4,F,G) = aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(C,B),fun(product_prod(A,B),$o),aa(fun(C,A),fun(fun(C,B),fun(product_prod(A,B),$o)),aTP_Lamp_ahk(set(C),fun(fun(C,A),fun(fun(C,B),fun(product_prod(A,B),$o))),A4),F),G)) ).

% image2_def
tff(fact_5202_finite__def,axiom,
    ! [A: $tType] : finite_finite2(A) = complete_lattice_lfp(fun(set(A),$o),aTP_Lamp_aeq(fun(set(A),$o),fun(set(A),$o))) ).

% finite_def
tff(fact_5203_lfp__lfp,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F: fun(A,fun(A,A))] :
          ( ! [X2: A,Y2: A,W: A,Z3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),Y2)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),W),Z3)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),F,X2),W)),aa(A,A,aa(A,fun(A,A),F,Y2),Z3)) ) )
         => ( complete_lattice_lfp(A,aTP_Lamp_ahl(fun(A,fun(A,A)),fun(A,A),F)) = complete_lattice_lfp(A,aTP_Lamp_ahm(fun(A,fun(A,A)),fun(A,A),F)) ) ) ) ).

% lfp_lfp
tff(fact_5204_lfp__const,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [T3: A] : complete_lattice_lfp(A,aTP_Lamp_ahn(A,fun(A,A),T3)) = T3 ) ).

% lfp_const
tff(fact_5205_lfp__rolling,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comple6319245703460814977attice(B)
        & comple6319245703460814977attice(A) )
     => ! [G: fun(A,B),F: fun(B,A)] :
          ( order_mono(A,B,G)
         => ( order_mono(B,A,F)
           => ( aa(A,B,G,complete_lattice_lfp(A,aa(fun(B,A),fun(A,A),aTP_Lamp_aho(fun(A,B),fun(fun(B,A),fun(A,A)),G),F))) = complete_lattice_lfp(B,aa(fun(B,A),fun(B,B),aTP_Lamp_ahp(fun(A,B),fun(fun(B,A),fun(B,B)),G),F)) ) ) ) ) ).

% lfp_rolling
tff(fact_5206_lfp__def,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F: fun(A,A)] : complete_lattice_lfp(A,F) = aa(set(A),A,complete_Inf_Inf(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_ahq(fun(A,A),fun(A,$o),F))) ) ).

% lfp_def
tff(fact_5207_image2__eqI,axiom,
    ! [A: $tType,C: $tType,B: $tType,B2: A,F: fun(B,A),X: B,C2: C,G: fun(B,C),A4: set(B)] :
      ( ( B2 = aa(B,A,F,X) )
     => ( ( C2 = aa(B,C,G,X) )
       => ( aa(set(B),$o,member(B,X),A4)
         => aa(set(product_prod(A,C)),$o,member(product_prod(A,C),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),B2),C2)),bNF_Greatest_image2(B,A,C,A4,F,G)) ) ) ) ).

% image2_eqI
tff(fact_5208_def__lfp__induct,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A4: A,F: fun(A,A),P: A] :
          ( ( A4 = complete_lattice_lfp(A,F) )
         => ( order_mono(A,A,F)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,F,aa(A,A,aa(A,fun(A,A),inf_inf(A),A4),P))),P)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),P) ) ) ) ) ).

% def_lfp_induct
tff(fact_5209_lfp__induct,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F: fun(A,A),P: A] :
          ( order_mono(A,A,F)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,F,aa(A,A,aa(A,fun(A,A),inf_inf(A),complete_lattice_lfp(A,F)),P))),P)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),complete_lattice_lfp(A,F)),P) ) ) ) ).

% lfp_induct
tff(fact_5210_lfp__Kleene__iter,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F: fun(A,A),K: nat] :
          ( order_mono(A,A,F)
         => ( ( aa(A,A,compow(fun(A,A),aa(nat,nat,suc,K),F),bot_bot(A)) = aa(A,A,compow(fun(A,A),K,F),bot_bot(A)) )
           => ( complete_lattice_lfp(A,F) = aa(A,A,compow(fun(A,A),K,F),bot_bot(A)) ) ) ) ) ).

% lfp_Kleene_iter
tff(fact_5211_iteratesp__def,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [X3: fun(A,A)] : comple7512665784863727008ratesp(A,X3) = complete_lattice_lfp(fun(A,$o),aTP_Lamp_ahh(fun(A,A),fun(fun(A,$o),fun(A,$o)),X3)) ) ).

% iteratesp_def
tff(fact_5212_flat__lub__def,axiom,
    ! [A: $tType,B2: A,A4: set(A)] :
      partial_flat_lub(A,B2,A4) = $ite(aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A)))),B2,the(A,aa(set(A),fun(A,$o),aTP_Lamp_ahr(A,fun(set(A),fun(A,$o)),B2),A4))) ).

% flat_lub_def
tff(fact_5213_Rats__eq__range__of__rat__o__nat__to__rat__surj,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ( field_char_0_Rats(A) = aa(set(nat),set(A),image2(nat,A,aa(fun(nat,rat),fun(nat,A),comp(rat,A,nat,field_char_0_of_rat(A)),nat_to_rat_surj)),top_top(set(nat))) ) ) ).

% Rats_eq_range_of_rat_o_nat_to_rat_surj
tff(fact_5214_Rats__minus__iff,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A] :
          ( aa(set(A),$o,member(A,aa(A,A,uminus_uminus(A),A3)),field_char_0_Rats(A))
        <=> aa(set(A),$o,member(A,A3),field_char_0_Rats(A)) ) ) ).

% Rats_minus_iff
tff(fact_5215_Rats__1,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => aa(set(A),$o,member(A,one_one(A)),field_char_0_Rats(A)) ) ).

% Rats_1
tff(fact_5216_Rats__mult,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,B2: A] :
          ( aa(set(A),$o,member(A,A3),field_char_0_Rats(A))
         => ( aa(set(A),$o,member(A,B2),field_char_0_Rats(A))
           => aa(set(A),$o,member(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),field_char_0_Rats(A)) ) ) ) ).

% Rats_mult
tff(fact_5217_def__lfp__induct__set,axiom,
    ! [A: $tType,A4: set(A),F: fun(set(A),set(A)),A3: A,P: fun(A,$o)] :
      ( ( A4 = complete_lattice_lfp(set(A),F) )
     => ( order_mono(set(A),set(A),F)
       => ( aa(set(A),$o,member(A,A3),A4)
         => ( ! [X2: A] :
                ( aa(set(A),$o,member(A,X2),aa(set(A),set(A),F,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),aa(fun(A,$o),set(A),collect(A),P))))
               => aa(A,$o,P,X2) )
           => aa(A,$o,P,A3) ) ) ) ) ).

% def_lfp_induct_set
tff(fact_5218_lfp__induct__set,axiom,
    ! [A: $tType,A3: A,F: fun(set(A),set(A)),P: fun(A,$o)] :
      ( aa(set(A),$o,member(A,A3),complete_lattice_lfp(set(A),F))
     => ( order_mono(set(A),set(A),F)
       => ( ! [X2: A] :
              ( aa(set(A),$o,member(A,X2),aa(set(A),set(A),F,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),complete_lattice_lfp(set(A),F)),aa(fun(A,$o),set(A),collect(A),P))))
             => aa(A,$o,P,X2) )
         => aa(A,$o,P,A3) ) ) ) ).

% lfp_induct_set
tff(fact_5219_Rats__def,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ( field_char_0_Rats(A) = aa(set(rat),set(A),image2(rat,A,field_char_0_of_rat(A)),top_top(set(rat))) ) ) ).

% Rats_def
tff(fact_5220_lfp__induct2,axiom,
    ! [A: $tType,B: $tType,A3: A,B2: B,F: fun(set(product_prod(A,B)),set(product_prod(A,B))),P: fun(A,fun(B,$o))] :
      ( aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),complete_lattice_lfp(set(product_prod(A,B)),F))
     => ( order_mono(set(product_prod(A,B)),set(product_prod(A,B)),F)
       => ( ! [A6: A,B5: B] :
              ( aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A6),B5)),aa(set(product_prod(A,B)),set(product_prod(A,B)),F,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)),F)),aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),P)))))
             => aa(B,$o,aa(A,fun(B,$o),P,A6),B5) )
         => aa(B,$o,aa(A,fun(B,$o),P,A3),B2) ) ) ) ).

% lfp_induct2
tff(fact_5221_Rats__eq__range__nat__to__rat__surj,axiom,
    field_char_0_Rats(rat) = aa(set(nat),set(rat),image2(nat,rat,nat_to_rat_surj),top_top(set(nat))) ).

% Rats_eq_range_nat_to_rat_surj
tff(fact_5222_ord__class_Olexordp__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ( ord_lexordp(A) = complete_lattice_lfp(fun(list(A),fun(list(A),$o)),aTP_Lamp_aez(fun(list(A),fun(list(A),$o)),fun(list(A),fun(list(A),$o)))) ) ) ).

% ord_class.lexordp_def
tff(fact_5223_eq__f__restr__ss__eq,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F: fun(fun(A,option(B)),fun(A,option(B))),A4: fun(A,option(B))] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S2),dom(A,B,aa(fun(A,option(B)),fun(A,option(B)),F,A4)))
     => ( ( A4 = restrict_map(A,B,aa(fun(A,option(B)),fun(A,option(B)),F,A4),aa(set(A),set(A),uminus_uminus(set(A)),S2)) )
      <=> ( map_le(A,B,A4,aa(fun(A,option(B)),fun(A,option(B)),F,A4))
          & ( S2 = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),dom(A,B,aa(fun(A,option(B)),fun(A,option(B)),F,A4))),dom(A,B,A4)) ) ) ) ) ).

% eq_f_restr_ss_eq
tff(fact_5224_eq__f__restr__conv,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F: fun(fun(A,option(B)),fun(A,option(B))),A4: fun(A,option(B))] :
      ( ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S2),dom(A,B,aa(fun(A,option(B)),fun(A,option(B)),F,A4)))
        & ( A4 = restrict_map(A,B,aa(fun(A,option(B)),fun(A,option(B)),F,A4),aa(set(A),set(A),uminus_uminus(set(A)),S2)) ) )
    <=> ( map_le(A,B,A4,aa(fun(A,option(B)),fun(A,option(B)),F,A4))
        & ( S2 = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),dom(A,B,aa(fun(A,option(B)),fun(A,option(B)),F,A4))),dom(A,B,A4)) ) ) ) ).

% eq_f_restr_conv
tff(fact_5225_map__le__empty,axiom,
    ! [B: $tType,A: $tType,G: fun(A,option(B))] : map_le(A,B,aTP_Lamp_tf(A,option(B)),G) ).

% map_le_empty
tff(fact_5226_lexordp__conv__lexord,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Ys: list(A)] :
          ( aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),Xs),Ys)
        <=> aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),lexord(A,aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),ord_less(A))))) ) ) ).

% lexordp_conv_lexord
tff(fact_5227_ord_Olexordp__def,axiom,
    ! [A: $tType,Less: fun(A,fun(A,$o))] : lexordp2(A,Less) = complete_lattice_lfp(fun(list(A),fun(list(A),$o)),aTP_Lamp_afb(fun(A,fun(A,$o)),fun(fun(list(A),fun(list(A),$o)),fun(list(A),fun(list(A),$o))),Less)) ).

% ord.lexordp_def
tff(fact_5228_min__ext__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : min_ext(A,R2) = aa(fun(product_prod(set(A),set(A)),$o),set(product_prod(set(A),set(A))),collect(product_prod(set(A),set(A))),aTP_Lamp_ahs(set(product_prod(A,A)),fun(product_prod(set(A),set(A)),$o),R2)) ).

% min_ext_def
tff(fact_5229_lists__empty,axiom,
    ! [A: $tType] : lists(A,bot_bot(set(A))) = aa(set(list(A)),set(list(A)),aa(list(A),fun(set(list(A)),set(list(A))),insert2(list(A)),nil(A)),bot_bot(set(list(A)))) ).

% lists_empty
tff(fact_5230_bex__empty,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ~ ? [X3: A] :
          ( aa(set(A),$o,member(A,X3),bot_bot(set(A)))
          & aa(A,$o,P,X3) ) ).

% bex_empty
tff(fact_5231_lists__Int__eq,axiom,
    ! [A: $tType,A4: set(A),B3: set(A)] : lists(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3)) = aa(set(list(A)),set(list(A)),aa(set(list(A)),fun(set(list(A)),set(list(A))),inf_inf(set(list(A))),lists(A,A4)),lists(A,B3)) ).

% lists_Int_eq
tff(fact_5232_finite__Collect__bex,axiom,
    ! [B: $tType,A: $tType,A4: set(A),Q: fun(B,fun(A,$o))] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( aa(set(B),$o,finite_finite2(B),aa(fun(B,$o),set(B),collect(B),aa(fun(B,fun(A,$o)),fun(B,$o),aTP_Lamp_aht(set(A),fun(fun(B,fun(A,$o)),fun(B,$o)),A4),Q)))
      <=> ! [X4: A] :
            ( aa(set(A),$o,member(A,X4),A4)
           => aa(set(B),$o,finite_finite2(B),aa(fun(B,$o),set(B),collect(B),aa(A,fun(B,$o),aTP_Lamp_aei(fun(B,fun(A,$o)),fun(A,fun(B,$o)),Q),X4))) ) ) ) ).

% finite_Collect_bex
tff(fact_5233_bex__UNIV,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( ? [X4: A] :
          ( aa(set(A),$o,member(A,X4),top_top(set(A)))
          & aa(A,$o,P,X4) )
    <=> ? [X_12: A] : aa(A,$o,P,X_12) ) ).

% bex_UNIV
tff(fact_5234_lists__UNIV,axiom,
    ! [A: $tType] : lists(A,top_top(set(A))) = top_top(set(list(A))) ).

% lists_UNIV
tff(fact_5235_Image__Collect__case__prod,axiom,
    ! [A: $tType,B: $tType,P: fun(B,fun(A,$o)),A4: set(B)] : aa(set(B),set(A),image(B,A,aa(fun(product_prod(B,A),$o),set(product_prod(B,A)),collect(product_prod(B,A)),aa(fun(B,fun(A,$o)),fun(product_prod(B,A),$o),product_case_prod(B,A,$o),P))),A4) = aa(fun(A,$o),set(A),collect(A),aa(set(B),fun(A,$o),aTP_Lamp_ahu(fun(B,fun(A,$o)),fun(set(B),fun(A,$o)),P),A4)) ).

% Image_Collect_case_prod
tff(fact_5236_SUP__bool__eq,axiom,
    ! [A: $tType] : aTP_Lamp_ahv(set(A),fun(fun(A,$o),$o)) = bex(A) ).

% SUP_bool_eq
tff(fact_5237_image__def,axiom,
    ! [A: $tType,B: $tType,F: fun(B,A),A4: set(B)] : aa(set(B),set(A),image2(B,A,F),A4) = aa(fun(A,$o),set(A),collect(A),aa(set(B),fun(A,$o),aTP_Lamp_ahw(fun(B,A),fun(set(B),fun(A,$o)),F),A4)) ).

% image_def
tff(fact_5238_Union__eq,axiom,
    ! [A: $tType,A4: set(set(A))] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A4) = aa(fun(A,$o),set(A),collect(A),aTP_Lamp_ahx(set(set(A)),fun(A,$o),A4)) ).

% Union_eq
tff(fact_5239_lists__IntI,axiom,
    ! [A: $tType,L: list(A),A4: set(A),B3: set(A)] :
      ( aa(set(list(A)),$o,member(list(A),L),lists(A,A4))
     => ( aa(set(list(A)),$o,member(list(A),L),lists(A,B3))
       => aa(set(list(A)),$o,member(list(A),L),lists(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3))) ) ) ).

% lists_IntI
tff(fact_5240_Image__def,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(B,A)),S2: set(B)] : aa(set(B),set(A),image(B,A,R2),S2) = aa(fun(A,$o),set(A),collect(A),aa(set(B),fun(A,$o),aTP_Lamp_ahy(set(product_prod(B,A)),fun(set(B),fun(A,$o)),R2),S2)) ).

% Image_def
tff(fact_5241_UNION__eq,axiom,
    ! [A: $tType,B: $tType,B3: fun(B,set(A)),A4: set(B)] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B3),A4)) = aa(fun(A,$o),set(A),collect(A),aa(set(B),fun(A,$o),aTP_Lamp_ahz(fun(B,set(A)),fun(set(B),fun(A,$o)),B3),A4)) ).

% UNION_eq
tff(fact_5242_Collect__bex__eq,axiom,
    ! [A: $tType,B: $tType,A4: set(B),P: fun(A,fun(B,$o))] : aa(fun(A,$o),set(A),collect(A),aa(fun(A,fun(B,$o)),fun(A,$o),aTP_Lamp_aia(set(B),fun(fun(A,fun(B,$o)),fun(A,$o)),A4),P)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aTP_Lamp_afi(fun(A,fun(B,$o)),fun(B,set(A)),P)),A4)) ).

% Collect_bex_eq
tff(fact_5243_vimage__image__eq,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),A4: set(A)] : aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F),aa(set(A),set(B),image2(A,B,F),A4)) = aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_aib(fun(A,B),fun(set(A),fun(A,$o)),F),A4)) ).

% vimage_image_eq
tff(fact_5244_lists__eq__set,axiom,
    ! [A: $tType,A4: set(A)] : lists(A,A4) = aa(fun(list(A),$o),set(list(A)),collect(list(A)),aTP_Lamp_aic(set(A),fun(list(A),$o),A4)) ).

% lists_eq_set
tff(fact_5245_Bex__fold,axiom,
    ! [A: $tType,A4: set(A),P: fun(A,$o)] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( ? [X4: A] :
            ( aa(set(A),$o,member(A,X4),A4)
            & aa(A,$o,P,X4) )
      <=> finite_fold(A,$o,aTP_Lamp_aid(fun(A,$o),fun(A,fun($o,$o)),P),$false,A4) ) ) ).

% Bex_fold
tff(fact_5246_nths__nths,axiom,
    ! [A: $tType,Xs: list(A),A4: set(nat),B3: set(nat)] : nths(A,nths(A,Xs,A4),B3) = nths(A,Xs,aa(fun(nat,$o),set(nat),collect(nat),aa(set(nat),fun(nat,$o),aTP_Lamp_aif(set(nat),fun(set(nat),fun(nat,$o)),A4),B3))) ).

% nths_nths
tff(fact_5247_max__extp_Omax__extI,axiom,
    ! [A: $tType,X5: set(A),Y4: set(A),R: fun(A,fun(A,$o))] :
      ( aa(set(A),$o,finite_finite2(A),X5)
     => ( aa(set(A),$o,finite_finite2(A),Y4)
       => ( ( Y4 != aa(fun(A,$o),set(A),collect(A),bot_bot(fun(A,$o))) )
         => ( ! [X2: A] :
                ( aa(set(A),$o,member(A,X2),X5)
               => ? [Xa3: A] :
                    ( aa(set(A),$o,member(A,Xa3),Y4)
                    & aa(A,$o,aa(A,fun(A,$o),R,X2),Xa3) ) )
           => aa(set(A),$o,aa(set(A),fun(set(A),$o),max_extp(A,R),X5),Y4) ) ) ) ) ).

% max_extp.max_extI
tff(fact_5248_max__extp_Osimps,axiom,
    ! [A: $tType,R: fun(A,fun(A,$o)),A1: set(A),A22: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),max_extp(A,R),A1),A22)
    <=> ( aa(set(A),$o,finite_finite2(A),A1)
        & aa(set(A),$o,finite_finite2(A),A22)
        & ( A22 != aa(fun(A,$o),set(A),collect(A),bot_bot(fun(A,$o))) )
        & ! [X4: A] :
            ( aa(set(A),$o,member(A,X4),A1)
           => ? [Xa2: A] :
                ( aa(set(A),$o,member(A,Xa2),A22)
                & aa(A,$o,aa(A,fun(A,$o),R,X4),Xa2) ) ) ) ) ).

% max_extp.simps
tff(fact_5249_max__extp_Ocases,axiom,
    ! [A: $tType,R: fun(A,fun(A,$o)),A1: set(A),A22: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),max_extp(A,R),A1),A22)
     => ~ ( aa(set(A),$o,finite_finite2(A),A1)
         => ( aa(set(A),$o,finite_finite2(A),A22)
           => ( ( A22 != aa(fun(A,$o),set(A),collect(A),bot_bot(fun(A,$o))) )
             => ~ ! [X3: A] :
                    ( aa(set(A),$o,member(A,X3),A1)
                   => ? [Xa4: A] :
                        ( aa(set(A),$o,member(A,Xa4),A22)
                        & aa(A,$o,aa(A,fun(A,$o),R,X3),Xa4) ) ) ) ) ) ) ).

% max_extp.cases
tff(fact_5250_listrel__subset,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A4: set(A)] :
      ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R2),product_Sigma(A,A,A4,aTP_Lamp_rl(set(A),fun(A,set(A)),A4)))
     => aa(set(product_prod(list(A),list(A))),$o,aa(set(product_prod(list(A),list(A))),fun(set(product_prod(list(A),list(A))),$o),ord_less_eq(set(product_prod(list(A),list(A)))),listrel(A,A,R2)),product_Sigma(list(A),list(A),lists(A,A4),aTP_Lamp_aig(set(A),fun(list(A),set(list(A))),A4))) ) ).

% listrel_subset
tff(fact_5251_lists__of__len__fin2,axiom,
    ! [A: $tType,P: set(A),N: nat] :
      ( aa(set(A),$o,finite_finite2(A),P)
     => aa(set(list(A)),$o,finite_finite2(list(A)),aa(set(list(A)),set(list(A)),aa(set(list(A)),fun(set(list(A)),set(list(A))),inf_inf(set(list(A))),lists(A,P)),aa(fun(list(A),$o),set(list(A)),collect(list(A)),aTP_Lamp_aih(nat,fun(list(A),$o),N)))) ) ).

% lists_of_len_fin2
tff(fact_5252_lists__of__len__fin1,axiom,
    ! [A: $tType,P: set(A),N: nat] :
      ( aa(set(A),$o,finite_finite2(A),P)
     => aa(set(list(A)),$o,finite_finite2(list(A)),aa(set(list(A)),set(list(A)),aa(set(list(A)),fun(set(list(A)),set(list(A))),inf_inf(set(list(A))),lists(A,P)),aa(fun(list(A),$o),set(list(A)),collect(list(A)),aTP_Lamp_aii(nat,fun(list(A),$o),N)))) ) ).

% lists_of_len_fin1
tff(fact_5253_map__project__def,axiom,
    ! [A: $tType,B: $tType,F: fun(B,option(A)),A4: set(B)] : map_project(B,A,F,A4) = aa(fun(A,$o),set(A),collect(A),aa(set(B),fun(A,$o),aTP_Lamp_aij(fun(B,option(A)),fun(set(B),fun(A,$o)),F),A4)) ).

% map_project_def
tff(fact_5254_max__ext__eq,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] : max_ext(A,R) = aa(fun(product_prod(set(A),set(A)),$o),set(product_prod(set(A),set(A))),collect(product_prod(set(A),set(A))),aa(fun(set(A),fun(set(A),$o)),fun(product_prod(set(A),set(A)),$o),product_case_prod(set(A),set(A),$o),aTP_Lamp_aik(set(product_prod(A,A)),fun(set(A),fun(set(A),$o)),R))) ).

% max_ext_eq
tff(fact_5255_antimono__funpow,axiom,
    ! [A: $tType] :
      ( ( lattice(A)
        & order_top(A) )
     => ! [Q: fun(A,A)] :
          ( order_mono(A,A,Q)
         => order_antimono(nat,A,aTP_Lamp_ail(fun(A,A),fun(nat,A),Q)) ) ) ).

% antimono_funpow
tff(fact_5256_ball__UNIV,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( ! [X4: A] :
          ( aa(set(A),$o,member(A,X4),top_top(set(A)))
         => aa(A,$o,P,X4) )
    <=> ! [X_12: A] : aa(A,$o,P,X_12) ) ).

% ball_UNIV
tff(fact_5257_rel__fun__def,axiom,
    ! [A: $tType,B: $tType,D: $tType,C: $tType,A4: fun(A,fun(C,$o)),B3: fun(B,fun(D,$o)),X3: fun(A,B),Xa3: fun(C,D)] :
      ( aa(fun(C,D),$o,aa(fun(A,B),fun(fun(C,D),$o),bNF_rel_fun(A,C,B,D,A4,B3),X3),Xa3)
    <=> ! [Xb4: A,Y3: C] :
          ( aa(C,$o,aa(A,fun(C,$o),A4,Xb4),Y3)
         => aa(D,$o,aa(B,fun(D,$o),B3,aa(A,B,X3,Xb4)),aa(C,D,Xa3,Y3)) ) ) ).

% rel_fun_def
tff(fact_5258_rel__fun__eq__rel,axiom,
    ! [B: $tType,C: $tType,A: $tType,R: fun(B,fun(C,$o)),X3: fun(A,B),Xa3: fun(A,C)] :
      ( aa(fun(A,C),$o,aa(fun(A,B),fun(fun(A,C),$o),bNF_rel_fun(A,A,B,C,fequal(A),R),X3),Xa3)
    <=> ! [Xb4: A] : aa(C,$o,aa(B,fun(C,$o),R,aa(A,B,X3,Xb4)),aa(A,C,Xa3,Xb4)) ) ).

% rel_fun_eq_rel
tff(fact_5259_antimonoD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F: fun(A,B),X: A,Y: A] :
          ( order_antimono(A,B,F)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,Y)),aa(A,B,F,X)) ) ) ) ).

% antimonoD
tff(fact_5260_antimonoE,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F: fun(A,B),X: A,Y: A] :
          ( order_antimono(A,B,F)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,Y)),aa(A,B,F,X)) ) ) ) ).

% antimonoE
tff(fact_5261_antimonoI,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F: fun(A,B)] :
          ( ! [X2: A,Y2: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),Y2)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,Y2)),aa(A,B,F,X2)) )
         => order_antimono(A,B,F) ) ) ).

% antimonoI
tff(fact_5262_antimono__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F: fun(A,B)] :
          ( order_antimono(A,B,F)
        <=> ! [X4: A,Y3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),Y3)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,Y3)),aa(A,B,F,X4)) ) ) ) ).

% antimono_def
tff(fact_5263_finite__set__of__finite__funs,axiom,
    ! [A: $tType,B: $tType,A4: set(A),B3: set(B),D3: B] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( aa(set(B),$o,finite_finite2(B),B3)
       => aa(set(fun(A,B)),$o,finite_finite2(fun(A,B)),aa(fun(fun(A,B),$o),set(fun(A,B)),collect(fun(A,B)),aa(B,fun(fun(A,B),$o),aa(set(B),fun(B,fun(fun(A,B),$o)),aTP_Lamp_aim(set(A),fun(set(B),fun(B,fun(fun(A,B),$o))),A4),B3),D3))) ) ) ).

% finite_set_of_finite_funs
tff(fact_5264_Collect__all__eq,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,$o))] : aa(fun(A,$o),set(A),collect(A),aTP_Lamp_ain(fun(A,fun(B,$o)),fun(A,$o),P)) = aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aTP_Lamp_afi(fun(A,fun(B,$o)),fun(B,set(A)),P)),top_top(set(B)))) ).

% Collect_all_eq
tff(fact_5265_Least__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [P: fun(A,$o)] : ord_Least(A,P) = the(A,aTP_Lamp_aio(fun(A,$o),fun(A,$o),P)) ) ).

% Least_def
tff(fact_5266_Func__def,axiom,
    ! [A: $tType,B: $tType,A4: set(A),B3: set(B)] : bNF_Wellorder_Func(A,B,A4,B3) = aa(fun(fun(A,B),$o),set(fun(A,B)),collect(fun(A,B)),aa(set(B),fun(fun(A,B),$o),aTP_Lamp_aip(set(A),fun(set(B),fun(fun(A,B),$o)),A4),B3)) ).

% Func_def
tff(fact_5267_min__of__antimono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & linorder(B) )
     => ! [F: fun(A,B),X: A,Y: A] :
          ( order_antimono(A,B,F)
         => ( aa(B,B,aa(B,fun(B,B),ord_min(B),aa(A,B,F,X)),aa(A,B,F,Y)) = aa(A,B,F,aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y)) ) ) ) ).

% min_of_antimono
tff(fact_5268_max__of__antimono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & linorder(B) )
     => ! [F: fun(A,B),X: A,Y: A] :
          ( order_antimono(A,B,F)
         => ( aa(B,B,aa(B,fun(B,B),ord_max(B),aa(A,B,F,X)),aa(A,B,F,Y)) = aa(A,B,F,aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)) ) ) ) ).

% max_of_antimono
tff(fact_5269_Greatest__def,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P: fun(A,$o)] : order_Greatest(A,P) = the(A,aTP_Lamp_aiq(fun(A,$o),fun(A,$o),P)) ) ).

% Greatest_def
tff(fact_5270_ord_OLeast__def,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,$o)),P: fun(A,$o)] : aa(fun(A,$o),A,least(A,Less_eq),P) = the(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_air(fun(A,fun(A,$o)),fun(fun(A,$o),fun(A,$o)),Less_eq),P)) ).

% ord.Least_def
tff(fact_5271_transfer__bforall__def,axiom,
    ! [A: $tType,X3: fun(A,$o),Xa3: fun(A,$o)] :
      ( transfer_bforall(A,X3,Xa3)
    <=> ! [Xb4: A] :
          ( aa(A,$o,X3,Xb4)
         => aa(A,$o,Xa3,Xb4) ) ) ).

% transfer_bforall_def
tff(fact_5272_ord_OLeast_Ocong,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,$o))] : least(A,Less_eq) = least(A,Less_eq) ).

% ord.Least.cong
tff(fact_5273_Greatest__equality,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P: fun(A,$o),X: A] :
          ( aa(A,$o,P,X)
         => ( ! [Y2: A] :
                ( aa(A,$o,P,Y2)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y2),X) )
           => ( order_Greatest(A,P) = X ) ) ) ) ).

% Greatest_equality
tff(fact_5274_GreatestI2__order,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P: fun(A,$o),X: A,Q: fun(A,$o)] :
          ( aa(A,$o,P,X)
         => ( ! [Y2: A] :
                ( aa(A,$o,P,Y2)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y2),X) )
           => ( ! [X2: A] :
                  ( aa(A,$o,P,X2)
                 => ( ! [Y5: A] :
                        ( aa(A,$o,P,Y5)
                       => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y5),X2) )
                   => aa(A,$o,Q,X2) ) )
             => aa(A,$o,Q,order_Greatest(A,P)) ) ) ) ) ).

% GreatestI2_order
tff(fact_5275_listrel1__subset__listrel,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),R3: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R2),R3)
     => ( refl_on(A,top_top(set(A)),R3)
       => aa(set(product_prod(list(A),list(A))),$o,aa(set(product_prod(list(A),list(A))),fun(set(product_prod(list(A),list(A))),$o),ord_less_eq(set(product_prod(list(A),list(A)))),listrel1(A,R2)),listrel(A,A,R3)) ) ) ).

% listrel1_subset_listrel
tff(fact_5276_list_Osize__gen_I2_J,axiom,
    ! [A: $tType,X: fun(A,nat),X21: A,X222: list(A)] : size_list(A,X,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X21),X222)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,X,X21)),size_list(A,X,X222))),aa(nat,nat,suc,zero_zero(nat))) ).

% list.size_gen(2)
tff(fact_5277_INF__set__fold,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F: fun(B,A),Xs: list(B)] : aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F),aa(list(B),set(B),set2(B),Xs))) = aa(A,A,fold(B,A,aa(fun(B,A),fun(B,fun(A,A)),comp(A,fun(A,A),B,inf_inf(A)),F),Xs),top_top(A)) ) ).

% INF_set_fold
tff(fact_5278_foldl__conv__fold,axiom,
    ! [B: $tType,A: $tType,F: fun(A,fun(B,A)),S2: A,Xs: list(B)] : foldl(A,B,F,S2,Xs) = aa(A,A,fold(B,A,aTP_Lamp_aal(fun(A,fun(B,A)),fun(B,fun(A,A)),F),Xs),S2) ).

% foldl_conv_fold
tff(fact_5279_refl__onD2,axiom,
    ! [A: $tType,A4: set(A),R2: set(product_prod(A,A)),X: A,Y: A] :
      ( refl_on(A,A4,R2)
     => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R2)
       => aa(set(A),$o,member(A,Y),A4) ) ) ).

% refl_onD2
tff(fact_5280_refl__onD1,axiom,
    ! [A: $tType,A4: set(A),R2: set(product_prod(A,A)),X: A,Y: A] :
      ( refl_on(A,A4,R2)
     => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R2)
       => aa(set(A),$o,member(A,X),A4) ) ) ).

% refl_onD1
tff(fact_5281_refl__onD,axiom,
    ! [A: $tType,A4: set(A),R2: set(product_prod(A,A)),A3: A] :
      ( refl_on(A,A4,R2)
     => ( aa(set(A),$o,member(A,A3),A4)
       => aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),A3)),R2) ) ) ).

% refl_onD
tff(fact_5282_refl__on__domain,axiom,
    ! [A: $tType,A4: set(A),R2: set(product_prod(A,A)),A3: A,B2: A] :
      ( refl_on(A,A4,R2)
     => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R2)
       => ( aa(set(A),$o,member(A,A3),A4)
          & aa(set(A),$o,member(A,B2),A4) ) ) ) ).

% refl_on_domain
tff(fact_5283_refl__rtrancl,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : refl_on(A,top_top(set(A)),transitive_rtrancl(A,R2)) ).

% refl_rtrancl
tff(fact_5284_refl__on__empty,axiom,
    ! [A: $tType] : refl_on(A,bot_bot(set(A)),bot_bot(set(product_prod(A,A)))) ).

% refl_on_empty
tff(fact_5285_refl__on__Int,axiom,
    ! [A: $tType,A4: set(A),R2: set(product_prod(A,A)),B3: set(A),S2: set(product_prod(A,A))] :
      ( refl_on(A,A4,R2)
     => ( refl_on(A,B3,S2)
       => refl_on(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R2),S2)) ) ) ).

% refl_on_Int
tff(fact_5286_refl__Id,axiom,
    ! [A: $tType] : refl_on(A,top_top(set(A)),id2(A)) ).

% refl_Id
tff(fact_5287_refl__on__Un,axiom,
    ! [A: $tType,A4: set(A),R2: set(product_prod(A,A)),B3: set(A),S2: set(product_prod(A,A))] :
      ( refl_on(A,A4,R2)
     => ( refl_on(A,B3,S2)
       => refl_on(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),R2),S2)) ) ) ).

% refl_on_Un
tff(fact_5288_fold__filter,axiom,
    ! [A: $tType,B: $tType,F: fun(B,fun(A,A)),P: fun(B,$o),Xs: list(B)] : fold(B,A,F,aa(list(B),list(B),filter2(B,P),Xs)) = fold(B,A,aa(fun(B,$o),fun(B,fun(A,A)),aTP_Lamp_zq(fun(B,fun(A,A)),fun(fun(B,$o),fun(B,fun(A,A))),F),P),Xs) ).

% fold_filter
tff(fact_5289_union__set__fold,axiom,
    ! [A: $tType,Xs: list(A),A4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(list(A),set(A),set2(A),Xs)),A4) = aa(set(A),set(A),fold(A,set(A),insert2(A),Xs),A4) ).

% union_set_fold
tff(fact_5290_refl__reflcl,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : refl_on(A,top_top(set(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))),R2),id2(A))) ).

% refl_reflcl
tff(fact_5291_sort__conv__fold,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] : aa(list(A),list(A),linorder_sort_key(A,A,aTP_Lamp_yq(A,A)),Xs) = aa(list(A),list(A),fold(A,list(A),linorder_insort_key(A,A,aTP_Lamp_yq(A,A)),Xs),nil(A)) ) ).

% sort_conv_fold
tff(fact_5292_refl__on__def,axiom,
    ! [A: $tType,A4: set(A),R2: set(product_prod(A,A))] :
      ( refl_on(A,A4,R2)
    <=> ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R2),product_Sigma(A,A,A4,aTP_Lamp_rl(set(A),fun(A,set(A)),A4)))
        & ! [X4: A] :
            ( aa(set(A),$o,member(A,X4),A4)
           => aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),X4)),R2) ) ) ) ).

% refl_on_def
tff(fact_5293_refl__onI,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A4: set(A)] :
      ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R2),product_Sigma(A,A,A4,aTP_Lamp_rl(set(A),fun(A,set(A)),A4)))
     => ( ! [X2: A] :
            ( aa(set(A),$o,member(A,X2),A4)
           => aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),X2)),R2) )
       => refl_on(A,A4,R2) ) ) ).

% refl_onI
tff(fact_5294_Refl__Restr,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A4: set(A)] :
      ( refl_on(A,field2(A,R2),R2)
     => refl_on(A,field2(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R2),product_Sigma(A,A,A4,aTP_Lamp_rl(set(A),fun(A,set(A)),A4)))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R2),product_Sigma(A,A,A4,aTP_Lamp_rl(set(A),fun(A,set(A)),A4)))) ) ).

% Refl_Restr
tff(fact_5295_Sup__set__fold,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Xs: list(A)] : aa(set(A),A,complete_Sup_Sup(A),aa(list(A),set(A),set2(A),Xs)) = aa(A,A,fold(A,A,sup_sup(A),Xs),bot_bot(A)) ) ).

% Sup_set_fold
tff(fact_5296_Inf__set__fold,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Xs: list(A)] : aa(set(A),A,complete_Inf_Inf(A),aa(list(A),set(A),set2(A),Xs)) = aa(A,A,fold(A,A,inf_inf(A),Xs),top_top(A)) ) ).

% Inf_set_fold
tff(fact_5297_refl__on__def_H,axiom,
    ! [A: $tType,A4: set(A),R2: set(product_prod(A,A))] :
      ( refl_on(A,A4,R2)
    <=> ( ! [X4: product_prod(A,A)] :
            ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),X4),R2)
           => aa(product_prod(A,A),$o,aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),aTP_Lamp_ais(set(A),fun(A,fun(A,$o)),A4)),X4) )
        & ! [X4: A] :
            ( aa(set(A),$o,member(A,X4),A4)
           => aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),X4)),R2) ) ) ) ).

% refl_on_def'
tff(fact_5298_Inf__fin_Oset__eq__fold,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [X: A,Xs: list(A)] : aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs))) = aa(A,A,fold(A,A,inf_inf(A),Xs),X) ) ).

% Inf_fin.set_eq_fold
tff(fact_5299_Sup__fin_Oset__eq__fold,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,Xs: list(A)] : aa(set(A),A,lattic5882676163264333800up_fin(A),aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs))) = aa(A,A,fold(A,A,sup_sup(A),Xs),X) ) ).

% Sup_fin.set_eq_fold
tff(fact_5300_refl__on__reflcl__Image,axiom,
    ! [A: $tType,B3: set(A),A4: set(product_prod(A,A)),C3: set(A)] :
      ( refl_on(A,B3,A4)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C3),B3)
       => ( aa(set(A),set(A),image(A,A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),A4),id2(A))),C3) = aa(set(A),set(A),image(A,A,A4),C3) ) ) ) ).

% refl_on_reflcl_Image
tff(fact_5301_Refl__Field__Restr2,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A4: set(A)] :
      ( refl_on(A,field2(A,R2),R2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),field2(A,R2))
       => ( field2(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R2),product_Sigma(A,A,A4,aTP_Lamp_rl(set(A),fun(A,set(A)),A4)))) = A4 ) ) ) ).

% Refl_Field_Restr2
tff(fact_5302_Refl__Field__Restr,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A4: set(A)] :
      ( refl_on(A,field2(A,R2),R2)
     => ( field2(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R2),product_Sigma(A,A,A4,aTP_Lamp_rl(set(A),fun(A,set(A)),A4)))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),field2(A,R2)),A4) ) ) ).

% Refl_Field_Restr
tff(fact_5303_refl__on__singleton,axiom,
    ! [A: $tType,X: A] : refl_on(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(product_prod(A,A),fun(set(product_prod(A,A)),set(product_prod(A,A))),insert2(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),X)),bot_bot(set(product_prod(A,A))))) ).

% refl_on_singleton
tff(fact_5304_SUP__set__fold,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F: fun(B,A),Xs: list(B)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F),aa(list(B),set(B),set2(B),Xs))) = aa(A,A,fold(B,A,aa(fun(B,A),fun(B,fun(A,A)),comp(A,fun(A,A),B,sup_sup(A)),F),Xs),bot_bot(A)) ) ).

% SUP_set_fold
tff(fact_5305_Refl__antisym__eq__Image1__Image1__iff,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A3: A,B2: A] :
      ( refl_on(A,field2(A,R2),R2)
     => ( antisym(A,R2)
       => ( aa(set(A),$o,member(A,A3),field2(A,R2))
         => ( aa(set(A),$o,member(A,B2),field2(A,R2))
           => ( ( aa(set(A),set(A),image(A,A,R2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A)))) = aa(set(A),set(A),image(A,A,R2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A)))) )
            <=> ( A3 = B2 ) ) ) ) ) ) ).

% Refl_antisym_eq_Image1_Image1_iff
tff(fact_5306_rotate1__length01,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat))
     => ( rotate1(A,Xs) = Xs ) ) ).

% rotate1_length01
tff(fact_5307_list_Oin__rel,axiom,
    ! [A: $tType,B: $tType,R: fun(A,fun(B,$o)),A3: list(A),B2: list(B)] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),list_all2(A,B,R),A3),B2)
    <=> ? [Z4: list(product_prod(A,B))] :
          ( aa(set(list(product_prod(A,B))),$o,member(list(product_prod(A,B)),Z4),aa(fun(list(product_prod(A,B)),$o),set(list(product_prod(A,B))),collect(list(product_prod(A,B))),aTP_Lamp_ait(fun(A,fun(B,$o)),fun(list(product_prod(A,B)),$o),R)))
          & ( aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Z4) = A3 )
          & ( aa(list(product_prod(A,B)),list(B),map(product_prod(A,B),B,product_snd(A,B)),Z4) = B2 ) ) ) ).

% list.in_rel
tff(fact_5308_antisym__reflcl,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( antisym(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))),R2),id2(A)))
    <=> antisym(A,R2) ) ).

% antisym_reflcl
tff(fact_5309_antisym__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( antisym(A,R2)
    <=> ! [X4: A,Y3: A] :
          ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Y3)),R2)
         => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),X4)),R2)
           => ( X4 = Y3 ) ) ) ) ).

% antisym_def
tff(fact_5310_antisymI,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( ! [X2: A,Y2: A] :
          ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y2)),R2)
         => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),X2)),R2)
           => ( X2 = Y2 ) ) )
     => antisym(A,R2) ) ).

% antisymI
tff(fact_5311_antisymD,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A3: A,B2: A] :
      ( antisym(A,R2)
     => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R2)
       => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),A3)),R2)
         => ( A3 = B2 ) ) ) ) ).

% antisymD
tff(fact_5312_list__all2__map2,axiom,
    ! [A: $tType,B: $tType,C: $tType,P: fun(A,fun(B,$o)),As: list(A),F: fun(C,B),Bs: list(C)] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),list_all2(A,B,P),As),aa(list(C),list(B),map(C,B,F),Bs))
    <=> aa(list(C),$o,aa(list(A),fun(list(C),$o),list_all2(A,C,aa(fun(C,B),fun(A,fun(C,$o)),aTP_Lamp_aiu(fun(A,fun(B,$o)),fun(fun(C,B),fun(A,fun(C,$o))),P),F)),As),Bs) ) ).

% list_all2_map2
tff(fact_5313_list__all2__map1,axiom,
    ! [C: $tType,A: $tType,B: $tType,P: fun(A,fun(B,$o)),F: fun(C,A),As: list(C),Bs: list(B)] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),list_all2(A,B,P),aa(list(C),list(A),map(C,A,F),As)),Bs)
    <=> aa(list(B),$o,aa(list(C),fun(list(B),$o),list_all2(C,B,aa(fun(C,A),fun(C,fun(B,$o)),aTP_Lamp_aiv(fun(A,fun(B,$o)),fun(fun(C,A),fun(C,fun(B,$o))),P),F)),As),Bs) ) ).

% list_all2_map1
tff(fact_5314_list_Orel__map_I1_J,axiom,
    ! [C: $tType,A: $tType,B: $tType,Sb: fun(A,fun(B,$o)),I: fun(C,A),X: list(C),Y: list(B)] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),list_all2(A,B,Sb),aa(list(C),list(A),map(C,A,I),X)),Y)
    <=> aa(list(B),$o,aa(list(C),fun(list(B),$o),list_all2(C,B,aa(fun(C,A),fun(C,fun(B,$o)),aTP_Lamp_aiv(fun(A,fun(B,$o)),fun(fun(C,A),fun(C,fun(B,$o))),Sb),I)),X),Y) ) ).

% list.rel_map(1)
tff(fact_5315_list_Orel__map_I2_J,axiom,
    ! [A: $tType,B: $tType,C: $tType,Sa: fun(A,fun(B,$o)),X: list(A),G: fun(C,B),Y: list(C)] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),list_all2(A,B,Sa),X),aa(list(C),list(B),map(C,B,G),Y))
    <=> aa(list(C),$o,aa(list(A),fun(list(C),$o),list_all2(A,C,aa(fun(C,B),fun(A,fun(C,$o)),aTP_Lamp_aiu(fun(A,fun(B,$o)),fun(fun(C,B),fun(A,fun(C,$o))),Sa),G)),X),Y) ) ).

% list.rel_map(2)
tff(fact_5316_list_Odisc__transfer_I2_J,axiom,
    ! [A: $tType,B: $tType,R: fun(A,fun(B,$o))] : aa(fun(list(B),$o),$o,aa(fun(list(A),$o),fun(fun(list(B),$o),$o),bNF_rel_fun(list(A),list(B),$o,$o,list_all2(A,B,R),fequal($o)),aTP_Lamp_zt(list(A),$o)),aTP_Lamp_zp(list(B),$o)) ).

% list.disc_transfer(2)
tff(fact_5317_list_Odisc__transfer_I1_J,axiom,
    ! [A: $tType,B: $tType,R: fun(A,fun(B,$o))] : aa(fun(list(B),$o),$o,aa(fun(list(A),$o),fun(fun(list(B),$o),$o),bNF_rel_fun(list(A),list(B),$o,$o,list_all2(A,B,R),fequal($o)),aTP_Lamp_aiw(list(A),$o)),aTP_Lamp_aix(list(B),$o)) ).

% list.disc_transfer(1)
tff(fact_5318_antisym__empty,axiom,
    ! [A: $tType] : antisym(A,bot_bot(set(product_prod(A,A)))) ).

% antisym_empty
tff(fact_5319_antisym__singleton,axiom,
    ! [A: $tType,X: product_prod(A,A)] : antisym(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(product_prod(A,A),fun(set(product_prod(A,A)),set(product_prod(A,A))),insert2(product_prod(A,A)),X),bot_bot(set(product_prod(A,A))))) ).

% antisym_singleton
tff(fact_5320_antisym__Restr,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A4: set(A)] :
      ( antisym(A,R2)
     => antisym(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R2),product_Sigma(A,A,A4,aTP_Lamp_rl(set(A),fun(A,set(A)),A4)))) ) ).

% antisym_Restr
tff(fact_5321_product__lists__set,axiom,
    ! [A: $tType,Xss: list(list(A))] : aa(list(list(A)),set(list(A)),set2(list(A)),product_lists(A,Xss)) = aa(fun(list(A),$o),set(list(A)),collect(list(A)),aTP_Lamp_aiz(list(list(A)),fun(list(A),$o),Xss)) ).

% product_lists_set
tff(fact_5322_list__all2__iff,axiom,
    ! [B: $tType,A: $tType,P: fun(A,fun(B,$o)),Xs: list(A),Ys: list(B)] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),list_all2(A,B,P),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)] :
            ( aa(set(product_prod(A,B)),$o,member(product_prod(A,B),X4),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys)))
           => aa(product_prod(A,B),$o,aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),P),X4) ) ) ) ).

% list_all2_iff
tff(fact_5323_horner__sum__transfer,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType] :
      ( ( comm_semiring_0(B)
        & comm_semiring_0(A) )
     => ! [A4: fun(A,fun(B,$o)),B3: fun(C,fun(D,$o))] :
          ( aa(B,$o,aa(A,fun(B,$o),A4,zero_zero(A)),zero_zero(B))
         => ( aa(fun(B,fun(B,B)),$o,aa(fun(A,fun(A,A)),fun(fun(B,fun(B,B)),$o),bNF_rel_fun(A,B,fun(A,A),fun(B,B),A4,bNF_rel_fun(A,B,A,B,A4,A4)),plus_plus(A)),plus_plus(B))
           => ( aa(fun(B,fun(B,B)),$o,aa(fun(A,fun(A,A)),fun(fun(B,fun(B,B)),$o),bNF_rel_fun(A,B,fun(A,A),fun(B,B),A4,bNF_rel_fun(A,B,A,B,A4,A4)),times_times(A)),times_times(B))
             => aa(fun(fun(D,B),fun(B,fun(list(D),B))),$o,aa(fun(fun(C,A),fun(A,fun(list(C),A))),fun(fun(fun(D,B),fun(B,fun(list(D),B))),$o),bNF_rel_fun(fun(C,A),fun(D,B),fun(A,fun(list(C),A)),fun(B,fun(list(D),B)),bNF_rel_fun(C,D,A,B,B3,A4),bNF_rel_fun(A,B,fun(list(C),A),fun(list(D),B),A4,bNF_rel_fun(list(C),list(D),A,B,list_all2(C,D,B3),A4))),groups4207007520872428315er_sum(C,A)),groups4207007520872428315er_sum(D,B)) ) ) ) ) ).

% horner_sum_transfer
tff(fact_5324_prod__list__transfer,axiom,
    ! [A: $tType,B: $tType] :
      ( ( monoid_mult(B)
        & monoid_mult(A) )
     => ! [A4: fun(A,fun(B,$o))] :
          ( aa(B,$o,aa(A,fun(B,$o),A4,one_one(A)),one_one(B))
         => ( aa(fun(B,fun(B,B)),$o,aa(fun(A,fun(A,A)),fun(fun(B,fun(B,B)),$o),bNF_rel_fun(A,B,fun(A,A),fun(B,B),A4,bNF_rel_fun(A,B,A,B,A4,A4)),times_times(A)),times_times(B))
           => aa(fun(list(B),B),$o,aa(fun(list(A),A),fun(fun(list(B),B),$o),bNF_rel_fun(list(A),list(B),A,B,list_all2(A,B,A4),A4),groups5270119922927024881d_list(A)),groups5270119922927024881d_list(B)) ) ) ) ).

% prod_list_transfer
tff(fact_5325_inter__coset__fold,axiom,
    ! [A: $tType,A4: set(A),Xs: list(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),coset(A,Xs)) = aa(set(A),set(A),fold(A,set(A),remove(A),Xs),A4) ).

% inter_coset_fold
tff(fact_5326_congruent__def,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,A)),F: fun(A,B)] :
      ( equiv_congruent(A,B,R2,F)
    <=> ! [X4: product_prod(A,A)] :
          ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),X4),R2)
         => aa(product_prod(A,A),$o,aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),aTP_Lamp_acn(fun(A,B),fun(A,fun(A,$o)),F)),X4) ) ) ).

% congruent_def
tff(fact_5327_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_5328_prod__list_ONil,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ( aa(list(A),A,groups5270119922927024881d_list(A),nil(A)) = one_one(A) ) ) ).

% prod_list.Nil
tff(fact_5329_prod__list_Oappend,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Xs: list(A),Ys: list(A)] : aa(list(A),A,groups5270119922927024881d_list(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(list(A),A,groups5270119922927024881d_list(A),Xs)),aa(list(A),A,groups5270119922927024881d_list(A),Ys)) ) ).

% prod_list.append
tff(fact_5330_congruentD,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,A)),F: fun(A,B),Y: A,Z2: A] :
      ( equiv_congruent(A,B,R2,F)
     => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z2)),R2)
       => ( aa(A,B,F,Y) = aa(A,B,F,Z2) ) ) ) ).

% congruentD
tff(fact_5331_congruentI,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,A)),F: fun(A,B)] :
      ( ! [Y2: A,Z3: A] :
          ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),Z3)),R2)
         => ( aa(A,B,F,Y2) = aa(A,B,F,Z3) ) )
     => equiv_congruent(A,B,R2,F) ) ).

% congruentI
tff(fact_5332_UNIV__coset,axiom,
    ! [A: $tType] : top_top(set(A)) = coset(A,nil(A)) ).

% UNIV_coset
tff(fact_5333_coset__def,axiom,
    ! [A: $tType,Xs: list(A)] : coset(A,Xs) = aa(set(A),set(A),uminus_uminus(set(A)),aa(list(A),set(A),set2(A),Xs)) ).

% coset_def
tff(fact_5334_compl__coset,axiom,
    ! [A: $tType,Xs: list(A)] : aa(set(A),set(A),uminus_uminus(set(A)),coset(A,Xs)) = aa(list(A),set(A),set2(A),Xs) ).

% compl_coset
tff(fact_5335_union__coset__filter,axiom,
    ! [A: $tType,Xs: list(A),A4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),coset(A,Xs)),A4) = coset(A,aa(list(A),list(A),filter2(A,aTP_Lamp_ai(set(A),fun(A,$o),A4)),Xs)) ).

% union_coset_filter
tff(fact_5336_prod__list_Oeq__foldr,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Xs: list(A)] : aa(list(A),A,groups5270119922927024881d_list(A),Xs) = aa(A,A,foldr(A,A,times_times(A),Xs),one_one(A)) ) ).

% prod_list.eq_foldr
tff(fact_5337_minus__coset__filter,axiom,
    ! [A: $tType,A4: set(A),Xs: list(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),coset(A,Xs)) = aa(list(A),set(A),set2(A),aa(list(A),list(A),filter2(A,aa(set(A),fun(A,$o),aTP_Lamp_ag(set(A),fun(A,$o)),A4)),Xs)) ).

% minus_coset_filter
tff(fact_5338_congruent2__def,axiom,
    ! [B: $tType,C: $tType,A: $tType,R12: set(product_prod(A,A)),R23: set(product_prod(B,B)),F: fun(A,fun(B,C))] :
      ( equiv_congruent2(A,B,C,R12,R23,F)
    <=> ! [X4: product_prod(A,A)] :
          ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),X4),R12)
         => aa(product_prod(A,A),$o,aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),aa(fun(A,fun(B,C)),fun(A,fun(A,$o)),aTP_Lamp_ajb(set(product_prod(B,B)),fun(fun(A,fun(B,C)),fun(A,fun(A,$o))),R23),F)),X4) ) ) ).

% congruent2_def
tff(fact_5339_properties__for__sort__key,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Ys: list(A),Xs: list(A),F: fun(A,B)] :
          ( ( aa(list(A),multiset(A),mset(A),Ys) = aa(list(A),multiset(A),mset(A),Xs) )
         => ( ! [K2: A] :
                ( aa(set(A),$o,member(A,K2),aa(list(A),set(A),set2(A),Ys))
               => ( aa(list(A),list(A),filter2(A,aa(A,fun(A,$o),aTP_Lamp_ajc(fun(A,B),fun(A,fun(A,$o)),F),K2)),Ys) = aa(list(A),list(A),filter2(A,aa(A,fun(A,$o),aTP_Lamp_ajc(fun(A,B),fun(A,fun(A,$o)),F),K2)),Xs) ) )
           => ( sorted_wrt(B,ord_less_eq(B),aa(list(A),list(B),map(A,B,F),Ys))
             => ( aa(list(A),list(A),linorder_sort_key(A,B,F),Xs) = Ys ) ) ) ) ) ).

% properties_for_sort_key
tff(fact_5340_Gcd__fin_Oeq__fold,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A4: set(A)] :
          aa(set(A),A,semiring_gcd_Gcd_fin(A),A4) = $ite(aa(set(A),$o,finite_finite2(A),A4),finite_fold(A,A,gcd_gcd(A),zero_zero(A),A4),one_one(A)) ) ).

% Gcd_fin.eq_fold
tff(fact_5341_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_5342_Gcd__fin_Oinfinite,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A4: set(A)] :
          ( ~ aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),A,semiring_gcd_Gcd_fin(A),A4) = one_one(A) ) ) ) ).

% Gcd_fin.infinite
tff(fact_5343_is__unit__Gcd__fin__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A4: set(A)] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(set(A),A,semiring_gcd_Gcd_fin(A),A4)),one_one(A))
        <=> ( aa(set(A),A,semiring_gcd_Gcd_fin(A),A4) = one_one(A) ) ) ) ).

% is_unit_Gcd_fin_iff
tff(fact_5344_mset__mergesort__by__rel__split,axiom,
    ! [A: $tType,Xs1: list(A),Xs2: list(A),Xs: list(A)] : aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(list(A),multiset(A),mset(A),aa(product_prod(list(A),list(A)),list(A),product_fst(list(A),list(A)),merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs1),Xs2),Xs)))),aa(list(A),multiset(A),mset(A),aa(product_prod(list(A),list(A)),list(A),product_snd(list(A),list(A)),merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs1),Xs2),Xs)))) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(list(A),multiset(A),mset(A),Xs)),aa(list(A),multiset(A),mset(A),Xs1))),aa(list(A),multiset(A),mset(A),Xs2)) ).

% mset_mergesort_by_rel_split
tff(fact_5345_surj__mset,axiom,
    ! [A: $tType] : aa(set(list(A)),set(multiset(A)),image2(list(A),multiset(A),mset(A)),top_top(set(list(A)))) = top_top(set(multiset(A))) ).

% surj_mset
tff(fact_5346_mset__eq__finite,axiom,
    ! [A: $tType,Xs: list(A)] : aa(set(list(A)),$o,finite_finite2(list(A)),aa(fun(list(A),$o),set(list(A)),collect(list(A)),aTP_Lamp_ajd(list(A),fun(list(A),$o),Xs))) ).

% mset_eq_finite
tff(fact_5347_mset__eq__length__filter,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),Z2: A] :
      ( ( aa(list(A),multiset(A),mset(A),Xs) = aa(list(A),multiset(A),mset(A),Ys) )
     => ( aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),filter2(A,aa(A,fun(A,$o),fequal(A),Z2)),Xs)) = aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),filter2(A,aa(A,fun(A,$o),fequal(A),Z2)),Ys)) ) ) ).

% mset_eq_length_filter
tff(fact_5348_congruent2I_H,axiom,
    ! [C: $tType,B: $tType,A: $tType,R12: set(product_prod(A,A)),R23: set(product_prod(B,B)),F: fun(A,fun(B,C))] :
      ( ! [Y12: A,Z1: A,Y23: B,Z22: B] :
          ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y12),Z1)),R12)
         => ( aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Y23),Z22)),R23)
           => ( aa(B,C,aa(A,fun(B,C),F,Y12),Y23) = aa(B,C,aa(A,fun(B,C),F,Z1),Z22) ) ) )
     => equiv_congruent2(A,B,C,R12,R23,F) ) ).

% congruent2I'
tff(fact_5349_congruent2D,axiom,
    ! [A: $tType,C: $tType,B: $tType,R12: set(product_prod(A,A)),R23: set(product_prod(B,B)),F: fun(A,fun(B,C)),Y1: A,Z12: A,Y22: B,Z23: B] :
      ( equiv_congruent2(A,B,C,R12,R23,F)
     => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y1),Z12)),R12)
       => ( aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Y22),Z23)),R23)
         => ( aa(B,C,aa(A,fun(B,C),F,Y1),Y22) = aa(B,C,aa(A,fun(B,C),F,Z12),Z23) ) ) ) ) ).

% congruent2D
tff(fact_5350_Gcd__fin_Ounion,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A4: set(A),B3: set(A)] : aa(set(A),A,semiring_gcd_Gcd_fin(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(set(A),A,semiring_gcd_Gcd_fin(A),A4)),aa(set(A),A,semiring_gcd_Gcd_fin(A),B3)) ) ).

% Gcd_fin.union
tff(fact_5351_properties__for__sort,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ys: list(A),Xs: list(A)] :
          ( ( aa(list(A),multiset(A),mset(A),Ys) = aa(list(A),multiset(A),mset(A),Xs) )
         => ( sorted_wrt(A,ord_less_eq(A),Ys)
           => ( aa(list(A),list(A),linorder_sort_key(A,A,aTP_Lamp_yq(A,A)),Xs) = Ys ) ) ) ) ).

% properties_for_sort
tff(fact_5352_Gcd__fin_Oinsert__remove,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,A4: set(A)] : aa(set(A),A,semiring_gcd_Gcd_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),A4)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),aa(set(A),A,semiring_gcd_Gcd_fin(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A)))))) ) ).

% Gcd_fin.insert_remove
tff(fact_5353_Gcd__fin_Oremove,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,A4: set(A)] :
          ( aa(set(A),$o,member(A,A3),A4)
         => ( aa(set(A),A,semiring_gcd_Gcd_fin(A),A4) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),aa(set(A),A,semiring_gcd_Gcd_fin(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A)))))) ) ) ) ).

% Gcd_fin.remove
tff(fact_5354_Gcd__fin__0__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A4: set(A)] :
          ( ( aa(set(A),A,semiring_gcd_Gcd_fin(A),A4) = zero_zero(A) )
        <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),zero_zero(A)),bot_bot(set(A))))
            & aa(set(A),$o,finite_finite2(A),A4) ) ) ) ).

% Gcd_fin_0_iff
tff(fact_5355_mset__zip__take__Cons__drop__twice,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),J: nat,X: A,Y: B] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),aa(list(A),nat,size_size(list(A)),Xs))
       => ( aa(list(product_prod(A,B)),multiset(product_prod(A,B)),mset(product_prod(A,B)),zip(A,B,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),take(A,J,Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),drop(A,J,Xs))),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),take(B,J,Ys)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),drop(B,J,Ys))))) = aa(multiset(product_prod(A,B)),multiset(product_prod(A,B)),aa(product_prod(A,B),fun(multiset(product_prod(A,B)),multiset(product_prod(A,B))),add_mset(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)),multiset(product_prod(A,B)),mset(product_prod(A,B)),zip(A,B,Xs,Ys))) ) ) ) ).

% mset_zip_take_Cons_drop_twice
tff(fact_5356_sorted__list__of__multiset__mset,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] : linord6283353356039996273ltiset(A,aa(list(A),multiset(A),mset(A),Xs)) = aa(list(A),list(A),linorder_sort_key(A,A,aTP_Lamp_yq(A,A)),Xs) ) ).

% sorted_list_of_multiset_mset
tff(fact_5357_image__mset__map__of,axiom,
    ! [B: $tType,A: $tType,Xs: list(product_prod(A,B))] :
      ( distinct(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xs))
     => ( aa(multiset(A),multiset(B),image_mset(A,B,aTP_Lamp_aje(list(product_prod(A,B)),fun(A,B),Xs)),aa(list(A),multiset(A),mset(A),aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xs))) = aa(list(B),multiset(B),mset(B),aa(list(product_prod(A,B)),list(B),map(product_prod(A,B),B,product_snd(A,B)),Xs)) ) ) ).

% image_mset_map_of
tff(fact_5358_image__mset_Oidentity,axiom,
    ! [A: $tType] : image_mset(A,A,aTP_Lamp_au(A,A)) = id(multiset(A)) ).

% image_mset.identity
tff(fact_5359_Multiset_Omset__insort,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Xs: list(A)] : aa(list(A),multiset(A),mset(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_yq(A,A)),X),Xs)) = aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),X),aa(list(A),multiset(A),mset(A),Xs)) ) ).

% Multiset.mset_insort
tff(fact_5360_sorted__list__of__multiset__insert,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,M4: multiset(A)] : linord6283353356039996273ltiset(A,aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),X),M4)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_yq(A,A)),X),linord6283353356039996273ltiset(A,M4)) ) ).

% sorted_list_of_multiset_insert
tff(fact_5361_multiset_Oinj__map,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B)] :
      ( inj_on(A,B,F,top_top(set(A)))
     => inj_on(multiset(A),multiset(B),image_mset(A,B,F),top_top(set(multiset(A)))) ) ).

% multiset.inj_map
tff(fact_5362_multiset_Omap__ident,axiom,
    ! [A: $tType,T3: multiset(A)] : aa(multiset(A),multiset(A),image_mset(A,A,aTP_Lamp_au(A,A)),T3) = T3 ).

% multiset.map_ident
tff(fact_5363_nat__to__rat__surj__def,axiom,
    ! [N: nat] : aa(nat,rat,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_ajf(nat,fun(nat,rat))),aa(nat,product_prod(nat,nat),nat_prod_decode,N)) ).

% nat_to_rat_surj_def
tff(fact_5364_lenlex__append2,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),Us: list(A),Xs: list(A),Ys: list(A)] :
      ( irrefl(A,R)
     => ( aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us),Xs)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us),Ys))),lenlex(A,R))
      <=> aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),lenlex(A,R)) ) ) ).

% lenlex_append2
tff(fact_5365_relation__of__def,axiom,
    ! [A: $tType,P: fun(A,fun(A,$o)),A4: set(A)] : order_relation_of(A,P,A4) = aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),aa(set(A),fun(A,fun(A,$o)),aTP_Lamp_ajg(fun(A,fun(A,$o)),fun(set(A),fun(A,fun(A,$o))),P),A4))) ).

% relation_of_def
tff(fact_5366_lexord__same__pref__if__irrefl,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),Xs: list(A),Ys: list(A),Zs: list(A)] :
      ( irrefl(A,R2)
     => ( aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Zs))),lexord(A,R2))
      <=> aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Ys),Zs)),lexord(A,R2)) ) ) ).

% lexord_same_pref_if_irrefl
tff(fact_5367_surj__prod__decode,axiom,
    aa(set(nat),set(product_prod(nat,nat)),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_5368_irreflI,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( ! [A6: A] : ~ aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A6),A6)),R)
     => irrefl(A,R) ) ).

% irreflI
tff(fact_5369_irrefl__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( irrefl(A,R2)
    <=> ! [A10: A] : ~ aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A10),A10)),R2) ) ).

% irrefl_def
tff(fact_5370_lexl__not__refl,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),X: list(A)] :
      ( irrefl(A,R2)
     => ~ aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X),X)),lex(A,R2)) ) ).

% lexl_not_refl
tff(fact_5371_irrefl__distinct,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( irrefl(A,R2)
    <=> ! [X4: product_prod(A,A)] :
          ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),X4),R2)
         => aa(product_prod(A,A),$o,aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),aTP_Lamp_za(A,fun(A,$o))),X4) ) ) ).

% irrefl_distinct
tff(fact_5372_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_ajh(nat,fun(nat,list(nat)))),aa(nat,product_prod(nat,nat),nat_prod_decode,N3)) ) ) ) ) ).

% list_decode.elims
tff(fact_5373_stable__sort__key__def,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Sk: fun(fun(A,B),fun(list(A),list(A)))] :
          ( linord3483353639454293061rt_key(A,B,Sk)
        <=> ! [F6: fun(A,B),Xs4: list(A),K4: B] : aa(list(A),list(A),filter2(A,aa(B,fun(A,$o),aTP_Lamp_yz(fun(A,B),fun(B,fun(A,$o)),F6),K4)),aa(list(A),list(A),aa(fun(A,B),fun(list(A),list(A)),Sk,F6),Xs4)) = aa(list(A),list(A),filter2(A,aa(B,fun(A,$o),aTP_Lamp_yz(fun(A,B),fun(B,fun(A,$o)),F6),K4)),Xs4) ) ) ).

% stable_sort_key_def
tff(fact_5374_filterlim__base__iff,axiom,
    ! [A: $tType,B: $tType,C: $tType,D: $tType,I4: set(A),F4: fun(A,set(B)),F: fun(B,C),G4: fun(D,set(C)),J3: set(D)] :
      ( ( I4 != bot_bot(set(A)) )
     => ( ! [I2: A] :
            ( aa(set(A),$o,member(A,I2),I4)
           => ! [J2: A] :
                ( aa(set(A),$o,member(A,J2),I4)
               => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(A,set(B),F4,I2)),aa(A,set(B),F4,J2))
                  | aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(A,set(B),F4,J2)),aa(A,set(B),F4,I2)) ) ) )
       => ( filterlim(B,C,F,aa(set(filter(C)),filter(C),complete_Inf_Inf(filter(C)),aa(set(D),set(filter(C)),image2(D,filter(C),aTP_Lamp_aji(fun(D,set(C)),fun(D,filter(C)),G4)),J3)),aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image2(A,filter(B),aTP_Lamp_tt(fun(A,set(B)),fun(A,filter(B)),F4)),I4)))
        <=> ! [X4: D] :
              ( aa(set(D),$o,member(D,X4),J3)
             => ? [Xa2: A] :
                  ( aa(set(A),$o,member(A,Xa2),I4)
                  & ! [Xb4: B] :
                      ( aa(set(B),$o,member(B,Xb4),aa(A,set(B),F4,Xa2))
                     => aa(set(C),$o,member(C,aa(B,C,F,Xb4)),aa(D,set(C),G4,X4)) ) ) ) ) ) ) ).

% filterlim_base_iff
tff(fact_5375_filterlim__compose,axiom,
    ! [B: $tType,A: $tType,C: $tType,G: fun(A,B),F33: filter(B),F23: filter(A),F: fun(C,A),F12: filter(C)] :
      ( filterlim(A,B,G,F33,F23)
     => ( filterlim(C,A,F,F23,F12)
       => filterlim(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_ajj(fun(A,B),fun(fun(C,A),fun(C,B)),G),F),F33,F12) ) ) ).

% filterlim_compose
tff(fact_5376_filterlim__ident,axiom,
    ! [A: $tType,F4: filter(A)] : filterlim(A,A,aTP_Lamp_au(A,A),F4,F4) ).

% filterlim_ident
tff(fact_5377_filterlim__sup,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),F4: filter(B),F12: filter(A),F23: filter(A)] :
      ( filterlim(A,B,F,F4,F12)
     => ( filterlim(A,B,F,F4,F23)
       => filterlim(A,B,F,F4,aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),sup_sup(filter(A)),F12),F23)) ) ) ).

% filterlim_sup
tff(fact_5378_filterlim__inf,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),F23: filter(B),F33: filter(B),F12: filter(A)] :
      ( filterlim(A,B,F,aa(filter(B),filter(B),aa(filter(B),fun(filter(B),filter(B)),inf_inf(filter(B)),F23),F33),F12)
    <=> ( filterlim(A,B,F,F23,F12)
        & filterlim(A,B,F,F33,F12) ) ) ).

% filterlim_inf
tff(fact_5379_surj__list__decode,axiom,
    aa(set(nat),set(list(nat)),image2(nat,list(nat),nat_list_decode),top_top(set(nat))) = top_top(set(list(nat))) ).

% surj_list_decode
tff(fact_5380_filterlim__top,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),F4: filter(A)] : filterlim(A,B,F,top_top(filter(B)),F4) ).

% filterlim_top
tff(fact_5381_filterlim__INF,axiom,
    ! [C: $tType,B: $tType,A: $tType,F: fun(A,B),G4: fun(C,filter(B)),B3: set(C),F4: filter(A)] :
      ( filterlim(A,B,F,aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(C),set(filter(B)),image2(C,filter(B),G4),B3)),F4)
    <=> ! [X4: C] :
          ( aa(set(C),$o,member(C,X4),B3)
         => filterlim(A,B,F,aa(C,filter(B),G4,X4),F4) ) ) ).

% filterlim_INF
tff(fact_5382_filterlim__INF_H,axiom,
    ! [C: $tType,B: $tType,A: $tType,X: A,A4: set(A),F: fun(B,C),F4: filter(C),G4: fun(A,filter(B))] :
      ( aa(set(A),$o,member(A,X),A4)
     => ( filterlim(B,C,F,F4,aa(A,filter(B),G4,X))
       => filterlim(B,C,F,F4,aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image2(A,filter(B),G4),A4))) ) ) ).

% filterlim_INF'
tff(fact_5383_filterlim__If,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),G4: filter(B),F4: filter(A),P: fun(A,$o),G: fun(A,B)] :
      ( filterlim(A,B,F,G4,aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),inf_inf(filter(A)),F4),principal(A,aa(fun(A,$o),set(A),collect(A),P))))
     => ( filterlim(A,B,G,G4,aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),inf_inf(filter(A)),F4),principal(A,aa(fun(A,$o),set(A),collect(A),aTP_Lamp_aj(fun(A,$o),fun(A,$o),P)))))
       => filterlim(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,$o),fun(fun(A,B),fun(A,B)),aTP_Lamp_ajk(fun(A,B),fun(fun(A,$o),fun(fun(A,B),fun(A,B))),F),P),G),G4,F4) ) ) ).

% filterlim_If
tff(fact_5384_filterlim__base,axiom,
    ! [B: $tType,A: $tType,E: $tType,D: $tType,C: $tType,J3: set(A),I: fun(A,C),I4: set(C),F4: fun(C,set(D)),F: fun(D,E),G4: fun(A,set(E))] :
      ( ! [M3: A,X2: B] :
          ( aa(set(A),$o,member(A,M3),J3)
         => aa(set(C),$o,member(C,aa(A,C,I,M3)),I4) )
     => ( ! [M3: A,X2: D] :
            ( aa(set(A),$o,member(A,M3),J3)
           => ( aa(set(D),$o,member(D,X2),aa(C,set(D),F4,aa(A,C,I,M3)))
             => aa(set(E),$o,member(E,aa(D,E,F,X2)),aa(A,set(E),G4,M3)) ) )
       => filterlim(D,E,F,aa(set(filter(E)),filter(E),complete_Inf_Inf(filter(E)),aa(set(A),set(filter(E)),image2(A,filter(E),aTP_Lamp_ajl(fun(A,set(E)),fun(A,filter(E)),G4)),J3)),aa(set(filter(D)),filter(D),complete_Inf_Inf(filter(D)),aa(set(C),set(filter(D)),image2(C,filter(D),aTP_Lamp_ajm(fun(C,set(D)),fun(C,filter(D)),F4)),I4))) ) ) ).

% filterlim_base
tff(fact_5385_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_ajh(nat,fun(nat,list(nat)))),aa(nat,product_prod(nat,nat),nat_prod_decode,N)) ).

% list_decode.simps(2)
tff(fact_5386_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_ajh(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_5387_map__rec,axiom,
    ! [A: $tType,B: $tType,F: fun(B,A),Xs: list(B)] : aa(list(B),list(A),map(B,A,F),Xs) = aa(list(B),list(A),rec_list(list(A),B,nil(A),aTP_Lamp_ajn(fun(B,A),fun(B,fun(list(B),fun(list(A),list(A)))),F)),Xs) ).

% map_rec
tff(fact_5388_zipf__zip,axiom,
    ! [A: $tType,B: $tType,L12: list(A),L23: list(B)] :
      ( ( aa(list(A),nat,size_size(list(A)),L12) = aa(list(B),nat,size_size(list(B)),L23) )
     => ( zipf(A,B,product_prod(A,B),product_Pair(A,B),L12,L23) = zip(A,B,L12,L23) ) ) ).

% zipf_zip
tff(fact_5389_rec__list__Cons__imp,axiom,
    ! [B: $tType,A: $tType,F: fun(list(A),B),F1: B,F22: fun(A,fun(list(A),fun(B,B))),X: A,Xs: list(A)] :
      ( ( F = rec_list(B,A,F1,F22) )
     => ( aa(list(A),B,F,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(B,B,aa(list(A),fun(B,B),aa(A,fun(list(A),fun(B,B)),F22,X),Xs),aa(list(A),B,F,Xs)) ) ) ).

% rec_list_Cons_imp
tff(fact_5390_rec__list__Nil__imp,axiom,
    ! [A: $tType,B: $tType,F: fun(list(A),B),F1: B,F22: fun(A,fun(list(A),fun(B,B)))] :
      ( ( F = rec_list(B,A,F1,F22) )
     => ( aa(list(A),B,F,nil(A)) = F1 ) ) ).

% rec_list_Nil_imp
tff(fact_5391_list__decode_Opinduct,axiom,
    ! [A0: nat,P: fun(nat,$o)] :
      ( accp(nat,nat_list_decode_rel,A0)
     => ( ( accp(nat,nat_list_decode_rel,zero_zero(nat))
         => aa(nat,$o,P,zero_zero(nat)) )
       => ( ! [N3: nat] :
              ( accp(nat,nat_list_decode_rel,aa(nat,nat,suc,N3))
             => ( ! [X3: nat,Y5: nat] :
                    ( ( aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X3),Y5) = aa(nat,product_prod(nat,nat),nat_prod_decode,N3) )
                   => aa(nat,$o,P,Y5) )
               => aa(nat,$o,P,aa(nat,nat,suc,N3)) ) )
         => aa(nat,$o,P,A0) ) ) ) ).

% list_decode.pinduct
tff(fact_5392_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_ajh(nat,fun(nat,list(nat)))),aa(nat,product_prod(nat,nat),nat_prod_decode,N)) ) ) ).

% list_decode.psimps(2)
tff(fact_5393_list_Orec__o__map,axiom,
    ! [B: $tType,C: $tType,A: $tType,G: B,Ga: fun(C,fun(list(C),fun(B,B))),F: fun(A,C)] : aa(fun(list(A),list(C)),fun(list(A),B),comp(list(C),B,list(A),rec_list(B,C,G,Ga)),map(A,C,F)) = rec_list(B,A,G,aa(fun(A,C),fun(A,fun(list(A),fun(B,B))),aTP_Lamp_ajo(fun(C,fun(list(C),fun(B,B))),fun(fun(A,C),fun(A,fun(list(A),fun(B,B)))),Ga),F)) ).

% list.rec_o_map
tff(fact_5394_zipf_Opelims,axiom,
    ! [A: $tType,B: $tType,C: $tType,X: fun(B,fun(C,A)),Xa: list(B),Xb: list(C),Y: list(A)] :
      ( ( zipf(B,C,A,X,Xa,Xb) = Y )
     => ( accp(product_prod(fun(B,fun(C,A)),product_prod(list(B),list(C))),zipf_rel(B,C,A),aa(product_prod(list(B),list(C)),product_prod(fun(B,fun(C,A)),product_prod(list(B),list(C))),aa(fun(B,fun(C,A)),fun(product_prod(list(B),list(C)),product_prod(fun(B,fun(C,A)),product_prod(list(B),list(C)))),product_Pair(fun(B,fun(C,A)),product_prod(list(B),list(C))),X),aa(list(C),product_prod(list(B),list(C)),aa(list(B),fun(list(C),product_prod(list(B),list(C))),product_Pair(list(B),list(C)),Xa),Xb)))
       => ( ( ( Xa = nil(B) )
           => ( ( Xb = nil(C) )
             => ( ( Y = nil(A) )
               => ~ accp(product_prod(fun(B,fun(C,A)),product_prod(list(B),list(C))),zipf_rel(B,C,A),aa(product_prod(list(B),list(C)),product_prod(fun(B,fun(C,A)),product_prod(list(B),list(C))),aa(fun(B,fun(C,A)),fun(product_prod(list(B),list(C)),product_prod(fun(B,fun(C,A)),product_prod(list(B),list(C)))),product_Pair(fun(B,fun(C,A)),product_prod(list(B),list(C))),X),aa(list(C),product_prod(list(B),list(C)),aa(list(B),fun(list(C),product_prod(list(B),list(C))),product_Pair(list(B),list(C)),nil(B)),nil(C)))) ) ) )
         => ( ! [A6: B,As4: list(B)] :
                ( ( Xa = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),A6),As4) )
               => ! [B5: C,Bs2: list(C)] :
                    ( ( Xb = aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),B5),Bs2) )
                   => ( ( Y = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(C,A,aa(B,fun(C,A),X,A6),B5)),zipf(B,C,A,X,As4,Bs2)) )
                     => ~ accp(product_prod(fun(B,fun(C,A)),product_prod(list(B),list(C))),zipf_rel(B,C,A),aa(product_prod(list(B),list(C)),product_prod(fun(B,fun(C,A)),product_prod(list(B),list(C))),aa(fun(B,fun(C,A)),fun(product_prod(list(B),list(C)),product_prod(fun(B,fun(C,A)),product_prod(list(B),list(C)))),product_Pair(fun(B,fun(C,A)),product_prod(list(B),list(C))),X),aa(list(C),product_prod(list(B),list(C)),aa(list(B),fun(list(C),product_prod(list(B),list(C))),product_Pair(list(B),list(C)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),A6),As4)),aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),B5),Bs2)))) ) ) )
           => ( ! [V2: B,Va: list(B)] :
                  ( ( Xa = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),V2),Va) )
                 => ( ( Xb = nil(C) )
                   => ( ( Y = undefined(list(A)) )
                     => ~ accp(product_prod(fun(B,fun(C,A)),product_prod(list(B),list(C))),zipf_rel(B,C,A),aa(product_prod(list(B),list(C)),product_prod(fun(B,fun(C,A)),product_prod(list(B),list(C))),aa(fun(B,fun(C,A)),fun(product_prod(list(B),list(C)),product_prod(fun(B,fun(C,A)),product_prod(list(B),list(C)))),product_Pair(fun(B,fun(C,A)),product_prod(list(B),list(C))),X),aa(list(C),product_prod(list(B),list(C)),aa(list(B),fun(list(C),product_prod(list(B),list(C))),product_Pair(list(B),list(C)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),V2),Va)),nil(C)))) ) ) )
             => ~ ( ( Xa = nil(B) )
                 => ! [V2: C,Va: list(C)] :
                      ( ( Xb = aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),V2),Va) )
                     => ( ( Y = undefined(list(A)) )
                       => ~ accp(product_prod(fun(B,fun(C,A)),product_prod(list(B),list(C))),zipf_rel(B,C,A),aa(product_prod(list(B),list(C)),product_prod(fun(B,fun(C,A)),product_prod(list(B),list(C))),aa(fun(B,fun(C,A)),fun(product_prod(list(B),list(C)),product_prod(fun(B,fun(C,A)),product_prod(list(B),list(C)))),product_Pair(fun(B,fun(C,A)),product_prod(list(B),list(C))),X),aa(list(C),product_prod(list(B),list(C)),aa(list(B),fun(list(C),product_prod(list(B),list(C))),product_Pair(list(B),list(C)),nil(B)),aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),V2),Va)))) ) ) ) ) ) ) ) ) ).

% zipf.pelims
tff(fact_5395_set__rec,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(A),set(A),set2(A),Xs) = aa(list(A),set(A),rec_list(set(A),A,bot_bot(set(A)),aTP_Lamp_ajp(A,fun(list(A),fun(set(A),set(A))))),Xs) ).

% set_rec
tff(fact_5396_map__tailrec__rev_Opelims,axiom,
    ! [B: $tType,A: $tType,X: fun(B,A),Xa: list(B),Xb: list(A),Y: list(A)] :
      ( ( map_tailrec_rev(B,A,X,Xa,Xb) = Y )
     => ( accp(product_prod(fun(B,A),product_prod(list(B),list(A))),map_tailrec_rev_rel(B,A),aa(product_prod(list(B),list(A)),product_prod(fun(B,A),product_prod(list(B),list(A))),aa(fun(B,A),fun(product_prod(list(B),list(A)),product_prod(fun(B,A),product_prod(list(B),list(A)))),product_Pair(fun(B,A),product_prod(list(B),list(A))),X),aa(list(A),product_prod(list(B),list(A)),aa(list(B),fun(list(A),product_prod(list(B),list(A))),product_Pair(list(B),list(A)),Xa),Xb)))
       => ( ( ( Xa = nil(B) )
           => ( ( Y = Xb )
             => ~ accp(product_prod(fun(B,A),product_prod(list(B),list(A))),map_tailrec_rev_rel(B,A),aa(product_prod(list(B),list(A)),product_prod(fun(B,A),product_prod(list(B),list(A))),aa(fun(B,A),fun(product_prod(list(B),list(A)),product_prod(fun(B,A),product_prod(list(B),list(A)))),product_Pair(fun(B,A),product_prod(list(B),list(A))),X),aa(list(A),product_prod(list(B),list(A)),aa(list(B),fun(list(A),product_prod(list(B),list(A))),product_Pair(list(B),list(A)),nil(B)),Xb))) ) )
         => ~ ! [A6: B,As4: list(B)] :
                ( ( Xa = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),A6),As4) )
               => ( ( Y = map_tailrec_rev(B,A,X,As4,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(B,A,X,A6)),Xb)) )
                 => ~ accp(product_prod(fun(B,A),product_prod(list(B),list(A))),map_tailrec_rev_rel(B,A),aa(product_prod(list(B),list(A)),product_prod(fun(B,A),product_prod(list(B),list(A))),aa(fun(B,A),fun(product_prod(list(B),list(A)),product_prod(fun(B,A),product_prod(list(B),list(A)))),product_Pair(fun(B,A),product_prod(list(B),list(A))),X),aa(list(A),product_prod(list(B),list(A)),aa(list(B),fun(list(A),product_prod(list(B),list(A))),product_Pair(list(B),list(A)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),A6),As4)),Xb))) ) ) ) ) ) ).

% map_tailrec_rev.pelims
tff(fact_5397_the__dflt__None__empty,axiom,
    ! [A: $tType] : dflt_None_set(A,bot_bot(set(A))) = none(set(A)) ).

% the_dflt_None_empty
tff(fact_5398_the__dflt__None__nonempty,axiom,
    ! [A: $tType,S: set(A)] :
      ( ( S != bot_bot(set(A)) )
     => ( dflt_None_set(A,S) = aa(set(A),option(set(A)),some(set(A)),S) ) ) ).

% the_dflt_None_nonempty
tff(fact_5399_curr__surj,axiom,
    ! [C: $tType,B: $tType,A: $tType,G: fun(A,fun(B,C)),A4: set(A),B3: set(B),C3: set(C)] :
      ( aa(set(fun(A,fun(B,C))),$o,member(fun(A,fun(B,C)),G),bNF_Wellorder_Func(A,fun(B,C),A4,bNF_Wellorder_Func(B,C,B3,C3)))
     => ? [X2: fun(product_prod(A,B),C)] :
          ( aa(set(fun(product_prod(A,B),C)),$o,member(fun(product_prod(A,B),C),X2),bNF_Wellorder_Func(product_prod(A,B),C,product_Sigma(A,B,A4,aTP_Lamp_qz(set(B),fun(A,set(B)),B3)),C3))
          & ( aa(fun(product_prod(A,B),C),fun(A,fun(B,C)),bNF_Wellorder_curr(A,B,C,A4),X2) = G ) ) ) ).

% curr_surj
tff(fact_5400_curr__def,axiom,
    ! [A: $tType,C: $tType,B: $tType,A4: set(A),F: fun(product_prod(A,B),C),X3: A] :
      aa(A,fun(B,C),aa(fun(product_prod(A,B),C),fun(A,fun(B,C)),bNF_Wellorder_curr(A,B,C,A4),F),X3) = $ite(aa(set(A),$o,member(A,X3),A4),aa(A,fun(B,C),aTP_Lamp_bq(fun(product_prod(A,B),C),fun(A,fun(B,C)),F),X3),undefined(fun(B,C))) ).

% curr_def
tff(fact_5401_dflt__None__set__def,axiom,
    ! [A: $tType,S: set(A)] :
      dflt_None_set(A,S) = $ite(S = bot_bot(set(A)),none(set(A)),aa(set(A),option(set(A)),some(set(A)),S)) ).

% dflt_None_set_def
tff(fact_5402_curr__inj,axiom,
    ! [C: $tType,B: $tType,A: $tType,F1: fun(product_prod(A,B),C),A4: set(A),B3: set(B),C3: set(C),F22: fun(product_prod(A,B),C)] :
      ( aa(set(fun(product_prod(A,B),C)),$o,member(fun(product_prod(A,B),C),F1),bNF_Wellorder_Func(product_prod(A,B),C,product_Sigma(A,B,A4,aTP_Lamp_qz(set(B),fun(A,set(B)),B3)),C3))
     => ( aa(set(fun(product_prod(A,B),C)),$o,member(fun(product_prod(A,B),C),F22),bNF_Wellorder_Func(product_prod(A,B),C,product_Sigma(A,B,A4,aTP_Lamp_qz(set(B),fun(A,set(B)),B3)),C3))
       => ( ( aa(fun(product_prod(A,B),C),fun(A,fun(B,C)),bNF_Wellorder_curr(A,B,C,A4),F1) = aa(fun(product_prod(A,B),C),fun(A,fun(B,C)),bNF_Wellorder_curr(A,B,C,A4),F22) )
        <=> ( F1 = F22 ) ) ) ) ).

% curr_inj
tff(fact_5403_curr__in,axiom,
    ! [A: $tType,B: $tType,C: $tType,F: fun(product_prod(A,B),C),A4: set(A),B3: set(B),C3: set(C)] :
      ( aa(set(fun(product_prod(A,B),C)),$o,member(fun(product_prod(A,B),C),F),bNF_Wellorder_Func(product_prod(A,B),C,product_Sigma(A,B,A4,aTP_Lamp_qz(set(B),fun(A,set(B)),B3)),C3))
     => aa(set(fun(A,fun(B,C))),$o,member(fun(A,fun(B,C)),aa(fun(product_prod(A,B),C),fun(A,fun(B,C)),bNF_Wellorder_curr(A,B,C,A4),F)),bNF_Wellorder_Func(A,fun(B,C),A4,bNF_Wellorder_Func(B,C,B3,C3))) ) ).

% curr_in
tff(fact_5404_the__dflt__None__set,axiom,
    ! [A: $tType,X: set(A)] : the_default(set(A),bot_bot(set(A)),dflt_None_set(A,X)) = X ).

% the_dflt_None_set
tff(fact_5405_bij__betw__curr,axiom,
    ! [A: $tType,B: $tType,C: $tType,A4: set(A),B3: set(B),C3: set(C)] : bij_betw(fun(product_prod(A,B),C),fun(A,fun(B,C)),bNF_Wellorder_curr(A,B,C,A4),bNF_Wellorder_Func(product_prod(A,B),C,product_Sigma(A,B,A4,aTP_Lamp_qz(set(B),fun(A,set(B)),B3)),C3),bNF_Wellorder_Func(A,fun(B,C),A4,bNF_Wellorder_Func(B,C,B3,C3))) ).

% bij_betw_curr
tff(fact_5406_init__seg__of__def,axiom,
    ! [A: $tType] : init_seg_of(A) = aa(fun(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),$o),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)),$o)),fun(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),$o),product_case_prod(set(product_prod(A,A)),set(product_prod(A,A)),$o),aTP_Lamp_ajq(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o)))) ).

% init_seg_of_def
tff(fact_5407_bij__is__inj,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B)] :
      ( bij_betw(A,B,F,top_top(set(A)),top_top(set(B)))
     => inj_on(A,B,F,top_top(set(A))) ) ).

% bij_is_inj
tff(fact_5408_bij__id,axiom,
    ! [A: $tType] : bij_betw(A,A,id(A),top_top(set(A)),top_top(set(A))) ).

% bij_id
tff(fact_5409_bij__fn,axiom,
    ! [A: $tType,F: fun(A,A),N: nat] :
      ( bij_betw(A,A,F,top_top(set(A)),top_top(set(A)))
     => bij_betw(A,A,compow(fun(A,A),N,F),top_top(set(A)),top_top(set(A))) ) ).

% bij_fn
tff(fact_5410_bij__betw__imp__surj,axiom,
    ! [A: $tType,B: $tType,F: fun(A,B),A4: set(A)] :
      ( bij_betw(A,B,F,A4,top_top(set(B)))
     => ( aa(set(A),set(B),image2(A,B,F),top_top(set(A))) = top_top(set(B)) ) ) ).

% bij_betw_imp_surj
tff(fact_5411_bij__is__surj,axiom,
    ! [A: $tType,B: $tType,F: fun(A,B)] :
      ( bij_betw(A,B,F,top_top(set(A)),top_top(set(B)))
     => ( aa(set(A),set(B),image2(A,B,F),top_top(set(A))) = top_top(set(B)) ) ) ).

% bij_is_surj
tff(fact_5412_bij__uminus,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => bij_betw(A,A,uminus_uminus(A),top_top(set(A)),top_top(set(A))) ) ).

% bij_uminus
tff(fact_5413_bijI_H,axiom,
    ! [A: $tType,B: $tType,F: fun(A,B)] :
      ( ! [X2: A,Y2: A] :
          ( ( aa(A,B,F,X2) = aa(A,B,F,Y2) )
        <=> ( X2 = Y2 ) )
     => ( ! [Y2: B] :
          ? [X3: A] : Y2 = aa(A,B,F,X3)
       => bij_betw(A,B,F,top_top(set(A)),top_top(set(B))) ) ) ).

% bijI'
tff(fact_5414_bij__iff,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B)] :
      ( bij_betw(A,B,F,top_top(set(A)),top_top(set(B)))
    <=> ! [X4: B] :
        ? [Xa2: A] :
          ( ( aa(A,B,F,Xa2) = X4 )
          & ! [Y3: A] :
              ( ( aa(A,B,F,Y3) = X4 )
             => ( Y3 = Xa2 ) ) ) ) ).

% bij_iff
tff(fact_5415_bij__pointE,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),Y: B] :
      ( bij_betw(A,B,F,top_top(set(A)),top_top(set(B)))
     => ~ ! [X2: A] :
            ( ( Y = aa(A,B,F,X2) )
           => ~ ! [X7: A] :
                  ( ( Y = aa(A,B,F,X7) )
                 => ( X7 = X2 ) ) ) ) ).

% bij_pointE
tff(fact_5416_involuntory__imp__bij,axiom,
    ! [A: $tType,F: fun(A,A)] :
      ( ! [X2: A] : aa(A,A,F,aa(A,A,F,X2)) = X2
     => bij_betw(A,A,F,top_top(set(A)),top_top(set(A))) ) ).

% involuntory_imp_bij
tff(fact_5417_bij__betw__empty2,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),A4: set(A)] :
      ( bij_betw(A,B,F,A4,bot_bot(set(B)))
     => ( A4 = bot_bot(set(A)) ) ) ).

% bij_betw_empty2
tff(fact_5418_bij__betw__empty1,axiom,
    ! [A: $tType,B: $tType,F: fun(A,B),A4: set(B)] :
      ( bij_betw(A,B,F,bot_bot(set(A)),A4)
     => ( A4 = bot_bot(set(B)) ) ) ).

% bij_betw_empty1
tff(fact_5419_sum_Oreindex__bij__betw,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [H2: fun(A,B),S: set(A),T2: set(B),G: fun(B,C)] :
          ( bij_betw(A,B,H2,S,T2)
         => ( 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_ajr(fun(A,B),fun(fun(B,C),fun(A,C)),H2),G)),S) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),G),T2) ) ) ) ).

% sum.reindex_bij_betw
tff(fact_5420_prod_Oreindex__bij__betw,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [H2: fun(A,B),S: set(A),T2: set(B),G: fun(B,C)] :
          ( bij_betw(A,B,H2,S,T2)
         => ( aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7121269368397514597t_prod(A,C),aa(fun(B,C),fun(A,C),aTP_Lamp_ajs(fun(A,B),fun(fun(B,C),fun(A,C)),H2),G)),S) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),G),T2) ) ) ) ).

% prod.reindex_bij_betw
tff(fact_5421_bij__comp,axiom,
    ! [A: $tType,B: $tType,C: $tType,F: fun(A,B),G: fun(B,C)] :
      ( bij_betw(A,B,F,top_top(set(A)),top_top(set(B)))
     => ( bij_betw(B,C,G,top_top(set(B)),top_top(set(C)))
       => bij_betw(A,C,aa(fun(A,B),fun(A,C),comp(B,C,A,G),F),top_top(set(A)),top_top(set(C))) ) ) ).

% bij_comp
tff(fact_5422_trans__init__seg__of,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),S2: set(product_prod(A,A)),T3: set(product_prod(A,A))] :
      ( aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),$o,member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R2),S2)),init_seg_of(A))
     => ( aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),$o,member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),S2),T3)),init_seg_of(A))
       => aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),$o,member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R2),T3)),init_seg_of(A)) ) ) ).

% trans_init_seg_of
tff(fact_5423_antisym__init__seg__of,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),S2: set(product_prod(A,A))] :
      ( aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),$o,member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R2),S2)),init_seg_of(A))
     => ( aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),$o,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))),S2),R2)),init_seg_of(A))
       => ( R2 = S2 ) ) ) ).

% antisym_init_seg_of
tff(fact_5424_refl__on__init__seg__of,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),$o,member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R2),R2)),init_seg_of(A)) ).

% refl_on_init_seg_of
tff(fact_5425_notIn__Un__bij__betw,axiom,
    ! [A: $tType,B: $tType,B2: A,A4: set(A),F: fun(A,B),A14: set(B)] :
      ( ~ aa(set(A),$o,member(A,B2),A4)
     => ( ~ aa(set(B),$o,member(B,aa(A,B,F,B2)),A14)
       => ( bij_betw(A,B,F,A4,A14)
         => bij_betw(A,B,F,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A)))),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A14),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),aa(A,B,F,B2)),bot_bot(set(B))))) ) ) ) ).

% notIn_Un_bij_betw
tff(fact_5426_notIn__Un__bij__betw3,axiom,
    ! [A: $tType,B: $tType,B2: A,A4: set(A),F: fun(A,B),A14: set(B)] :
      ( ~ aa(set(A),$o,member(A,B2),A4)
     => ( ~ aa(set(B),$o,member(B,aa(A,B,F,B2)),A14)
       => ( bij_betw(A,B,F,A4,A14)
        <=> bij_betw(A,B,F,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A)))),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A14),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),aa(A,B,F,B2)),bot_bot(set(B))))) ) ) ) ).

% notIn_Un_bij_betw3
tff(fact_5427_bijI,axiom,
    ! [A: $tType,B: $tType,F: fun(A,B)] :
      ( inj_on(A,B,F,top_top(set(A)))
     => ( ( aa(set(A),set(B),image2(A,B,F),top_top(set(A))) = top_top(set(B)) )
       => bij_betw(A,B,F,top_top(set(A)),top_top(set(B))) ) ) ).

% bijI
tff(fact_5428_bij__def,axiom,
    ! [A: $tType,B: $tType,F: fun(A,B)] :
      ( bij_betw(A,B,F,top_top(set(A)),top_top(set(B)))
    <=> ( inj_on(A,B,F,top_top(set(A)))
        & ( aa(set(A),set(B),image2(A,B,F),top_top(set(A))) = top_top(set(B)) ) ) ) ).

% bij_def
tff(fact_5429_bij__betw__combine,axiom,
    ! [A: $tType,B: $tType,F: fun(A,B),A4: set(A),B3: set(B),C3: set(A),D4: set(B)] :
      ( bij_betw(A,B,F,A4,B3)
     => ( bij_betw(A,B,F,C3,D4)
       => ( ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),B3),D4) = bot_bot(set(B)) )
         => bij_betw(A,B,F,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),C3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),B3),D4)) ) ) ) ).

% bij_betw_combine
tff(fact_5430_bij__betw__partition,axiom,
    ! [A: $tType,B: $tType,F: fun(A,B),A4: set(A),C3: set(A),B3: set(B),D4: set(B)] :
      ( bij_betw(A,B,F,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),C3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),B3),D4))
     => ( bij_betw(A,B,F,C3,D4)
       => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),C3) = bot_bot(set(A)) )
         => ( ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),B3),D4) = bot_bot(set(B)) )
           => bij_betw(A,B,F,A4,B3) ) ) ) ) ).

% bij_betw_partition
tff(fact_5431_o__bij,axiom,
    ! [A: $tType,B: $tType,G: fun(B,A),F: fun(A,B)] :
      ( ( aa(fun(A,B),fun(A,A),comp(B,A,A,G),F) = id(A) )
     => ( ( aa(fun(B,A),fun(B,B),comp(A,B,B,F),G) = id(B) )
       => bij_betw(A,B,F,top_top(set(A)),top_top(set(B))) ) ) ).

% o_bij
tff(fact_5432_finite__vimage__iff,axiom,
    ! [A: $tType,B: $tType,H2: fun(A,B),F4: set(B)] :
      ( bij_betw(A,B,H2,top_top(set(A)),top_top(set(B)))
     => ( aa(set(A),$o,finite_finite2(A),aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),H2),F4))
      <=> aa(set(B),$o,finite_finite2(B),F4) ) ) ).

% finite_vimage_iff
tff(fact_5433_bij__image__Compl__eq,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),A4: set(A)] :
      ( bij_betw(A,B,F,top_top(set(A)),top_top(set(B)))
     => ( aa(set(A),set(B),image2(A,B,F),aa(set(A),set(A),uminus_uminus(set(A)),A4)) = aa(set(B),set(B),uminus_uminus(set(B)),aa(set(A),set(B),image2(A,B,F),A4)) ) ) ).

% bij_image_Compl_eq
tff(fact_5434_bij__betw__disjoint__Un,axiom,
    ! [A: $tType,B: $tType,F: fun(A,B),A4: set(A),C3: set(B),G: fun(A,B),B3: set(A),D4: set(B)] :
      ( bij_betw(A,B,F,A4,C3)
     => ( bij_betw(A,B,G,B3,D4)
       => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3) = bot_bot(set(A)) )
         => ( ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),C3),D4) = bot_bot(set(B)) )
           => bij_betw(A,B,aa(fun(A,B),fun(A,B),aa(set(A),fun(fun(A,B),fun(A,B)),aTP_Lamp_aco(fun(A,B),fun(set(A),fun(fun(A,B),fun(A,B))),F),A4),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),C3),D4)) ) ) ) ) ).

% bij_betw_disjoint_Un
tff(fact_5435_bij__betw__UNION__chain,axiom,
    ! [B: $tType,C: $tType,A: $tType,I4: set(A),A4: fun(A,set(B)),F: fun(B,C),A14: fun(A,set(C))] :
      ( ! [I2: A,J2: A] :
          ( aa(set(A),$o,member(A,I2),I4)
         => ( aa(set(A),$o,member(A,J2),I4)
           => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(A,set(B),A4,I2)),aa(A,set(B),A4,J2))
              | aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(A,set(B),A4,J2)),aa(A,set(B),A4,I2)) ) ) )
     => ( ! [I2: A] :
            ( aa(set(A),$o,member(A,I2),I4)
           => bij_betw(B,C,F,aa(A,set(B),A4,I2),aa(A,set(C),A14,I2)) )
       => bij_betw(B,C,F,aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),A4),I4)),aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(A),set(set(C)),image2(A,set(C),A14),I4))) ) ) ).

% bij_betw_UNION_chain
tff(fact_5436_infinite__imp__bij__betw2,axiom,
    ! [A: $tType,A4: set(A),A3: A] :
      ( ~ aa(set(A),$o,finite_finite2(A),A4)
     => ? [H: fun(A,A)] : bij_betw(A,A,H,A4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A))))) ) ).

% infinite_imp_bij_betw2
tff(fact_5437_infinite__imp__bij__betw,axiom,
    ! [A: $tType,A4: set(A),A3: A] :
      ( ~ aa(set(A),$o,finite_finite2(A),A4)
     => ? [H: fun(A,A)] : bij_betw(A,A,H,A4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A))))) ) ).

% infinite_imp_bij_betw
tff(fact_5438_vimage__subset__eq,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),B3: set(B),A4: set(A)] :
      ( bij_betw(A,B,F,top_top(set(A)),top_top(set(B)))
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F),B3)),A4)
      <=> aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),B3),aa(set(A),set(B),image2(A,B,F),A4)) ) ) ).

% vimage_subset_eq
tff(fact_5439_mono__bij__Inf,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comple5582772986160207858norder(A)
        & comple5582772986160207858norder(B) )
     => ! [F: fun(A,B),A4: set(A)] :
          ( order_mono(A,B,F)
         => ( bij_betw(A,B,F,top_top(set(A)),top_top(set(B)))
           => ( aa(A,B,F,aa(set(A),A,complete_Inf_Inf(A),A4)) = aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F),A4)) ) ) ) ) ).

% mono_bij_Inf
tff(fact_5440_sum_Oreindex__bij__betw__not__neutral,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [S3: set(A),T4: set(B),H2: fun(A,B),S: set(A),T2: set(B),G: fun(B,C)] :
          ( aa(set(A),$o,finite_finite2(A),S3)
         => ( aa(set(B),$o,finite_finite2(B),T4)
           => ( bij_betw(A,B,H2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S),S3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T2),T4))
             => ( ! [A6: A] :
                    ( aa(set(A),$o,member(A,A6),S3)
                   => ( aa(B,C,G,aa(A,B,H2,A6)) = zero_zero(C) ) )
               => ( ! [B5: B] :
                      ( aa(set(B),$o,member(B,B5),T4)
                     => ( aa(B,C,G,B5) = zero_zero(C) ) )
                 => ( aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7311177749621191930dd_sum(A,C),aa(fun(B,C),fun(A,C),aTP_Lamp_ajr(fun(A,B),fun(fun(B,C),fun(A,C)),H2),G)),S) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),G),T2) ) ) ) ) ) ) ) ).

% sum.reindex_bij_betw_not_neutral
tff(fact_5441_initial__segment__of__Diff,axiom,
    ! [A: $tType,P3: set(product_prod(A,A)),Q3: set(product_prod(A,A)),S2: set(product_prod(A,A))] :
      ( aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),$o,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))),P3),Q3)),init_seg_of(A))
     => aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),$o,member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),minus_minus(set(product_prod(A,A))),P3),S2)),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))),Q3),S2))),init_seg_of(A)) ) ).

% initial_segment_of_Diff
tff(fact_5442_prod_Oreindex__bij__betw__not__neutral,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [S3: set(A),T4: set(B),H2: fun(A,B),S: set(A),T2: set(B),G: fun(B,C)] :
          ( aa(set(A),$o,finite_finite2(A),S3)
         => ( aa(set(B),$o,finite_finite2(B),T4)
           => ( bij_betw(A,B,H2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S),S3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T2),T4))
             => ( ! [A6: A] :
                    ( aa(set(A),$o,member(A,A6),S3)
                   => ( aa(B,C,G,aa(A,B,H2,A6)) = one_one(C) ) )
               => ( ! [B5: B] :
                      ( aa(set(B),$o,member(B,B5),T4)
                     => ( aa(B,C,G,B5) = one_one(C) ) )
                 => ( aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7121269368397514597t_prod(A,C),aa(fun(B,C),fun(A,C),aTP_Lamp_ajs(fun(A,B),fun(fun(B,C),fun(A,C)),H2),G)),S) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),G),T2) ) ) ) ) ) ) ) ).

% prod.reindex_bij_betw_not_neutral
tff(fact_5443_bij__image__INT,axiom,
    ! [A: $tType,B: $tType,C: $tType,F: fun(A,B),B3: fun(C,set(A)),A4: set(C)] :
      ( bij_betw(A,B,F,top_top(set(A)),top_top(set(B)))
     => ( aa(set(A),set(B),image2(A,B,F),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(C),set(set(A)),image2(C,set(A),B3),A4))) = aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(C),set(set(B)),image2(C,set(B),aa(fun(C,set(A)),fun(C,set(B)),aTP_Lamp_acp(fun(A,B),fun(fun(C,set(A)),fun(C,set(B))),F),B3)),A4)) ) ) ).

% bij_image_INT
tff(fact_5444_Chains__init__seg__of__Union,axiom,
    ! [A: $tType,R: set(set(product_prod(A,A))),R2: set(product_prod(A,A))] :
      ( aa(set(set(set(product_prod(A,A)))),$o,member(set(set(product_prod(A,A))),R),chains(set(product_prod(A,A)),init_seg_of(A)))
     => ( aa(set(set(product_prod(A,A))),$o,member(set(product_prod(A,A)),R2),R)
       => aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),$o,member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R2),aa(set(set(product_prod(A,A))),set(product_prod(A,A)),complete_Sup_Sup(set(product_prod(A,A))),R))),init_seg_of(A)) ) ) ).

% Chains_init_seg_of_Union
tff(fact_5445_arg__min__inj__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( order(B)
     => ! [F: fun(A,B),P: fun(A,$o),A3: A] :
          ( inj_on(A,B,F,aa(fun(A,$o),set(A),collect(A),P))
         => ( aa(A,$o,P,A3)
           => ( ! [Y2: A] :
                  ( aa(A,$o,P,Y2)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,A3)),aa(A,B,F,Y2)) )
             => ( lattices_ord_arg_min(A,B,F,P) = A3 ) ) ) ) ) ).

% arg_min_inj_eq
tff(fact_5446_sorted__list__of__multiset__def,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [M4: multiset(A)] : linord6283353356039996273ltiset(A,M4) = fold_mset(A,list(A),linorder_insort_key(A,A,aTP_Lamp_yq(A,A)),nil(A),M4) ) ).

% sorted_list_of_multiset_def
tff(fact_5447_bij__int__decode,axiom,
    bij_betw(nat,int,nat_int_decode,top_top(set(nat)),top_top(set(int))) ).

% bij_int_decode
tff(fact_5448_bij__enumerate,axiom,
    ! [S: set(nat)] :
      ( ~ aa(set(nat),$o,finite_finite2(nat),S)
     => bij_betw(nat,nat,infini527867602293511546merate(nat,S),top_top(set(nat)),S) ) ).

% bij_enumerate
tff(fact_5449_bij__int__encode,axiom,
    bij_betw(int,nat,nat_int_encode,top_top(set(int)),top_top(set(nat))) ).

% bij_int_encode
tff(fact_5450_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_5451_ex__bij__betw__nat__finite__1,axiom,
    ! [A: $tType,M4: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),M4)
     => ? [H: fun(nat,A)] : bij_betw(nat,A,H,set_or1337092689740270186AtMost(nat,one_one(nat),aa(set(A),nat,finite_card(A),M4)),M4) ) ).

% ex_bij_betw_nat_finite_1
tff(fact_5452_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_5453_Chains__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : chains(A,R2) = aa(fun(set(A),$o),set(set(A)),collect(set(A)),aTP_Lamp_ajt(set(product_prod(A,A)),fun(set(A),$o),R2)) ).

% Chains_def
tff(fact_5454_Chains__inits__DiffI,axiom,
    ! [A: $tType,R: set(set(product_prod(A,A))),S2: set(product_prod(A,A))] :
      ( aa(set(set(set(product_prod(A,A)))),$o,member(set(set(product_prod(A,A))),R),chains(set(product_prod(A,A)),init_seg_of(A)))
     => aa(set(set(set(product_prod(A,A)))),$o,member(set(set(product_prod(A,A))),aa(fun(set(product_prod(A,A)),$o),set(set(product_prod(A,A))),collect(set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),aTP_Lamp_aju(set(set(product_prod(A,A))),fun(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o)),R),S2))),chains(set(product_prod(A,A)),init_seg_of(A))) ) ).

% Chains_inits_DiffI
tff(fact_5455_arg__min__on__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F: fun(A,B),S: set(A)] : lattic7623131987881927897min_on(A,B,F,S) = lattices_ord_arg_min(A,B,F,aa(set(A),fun(A,$o),aTP_Lamp_ag(set(A),fun(A,$o)),S)) ) ).

% arg_min_on_def
tff(fact_5456_Chains__subset_H,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( refl_on(A,top_top(set(A)),R2)
     => aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),aa(fun(set(A),$o),set(set(A)),collect(set(A)),pred_chain(A,top_top(set(A)),aTP_Lamp_ux(set(product_prod(A,A)),fun(A,fun(A,$o)),R2)))),chains(A,R2)) ) ).

% Chains_subset'
tff(fact_5457_Chains__subset,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),chains(A,R2)),aa(fun(set(A),$o),set(set(A)),collect(set(A)),pred_chain(A,top_top(set(A)),aTP_Lamp_ux(set(product_prod(A,A)),fun(A,fun(A,$o)),R2)))) ).

% Chains_subset
tff(fact_5458_Chains__alt__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( refl_on(A,top_top(set(A)),R2)
     => ( chains(A,R2) = aa(fun(set(A),$o),set(set(A)),collect(set(A)),pred_chain(A,top_top(set(A)),aTP_Lamp_ux(set(product_prod(A,A)),fun(A,fun(A,$o)),R2))) ) ) ).

% Chains_alt_def
tff(fact_5459_pred__on_Ochain__empty,axiom,
    ! [A: $tType,A4: set(A),P: fun(A,fun(A,$o))] : aa(set(A),$o,pred_chain(A,A4,P),bot_bot(set(A))) ).

% pred_on.chain_empty
tff(fact_5460_subset_Ochain__total,axiom,
    ! [A: $tType,A4: set(set(A)),C3: set(set(A)),X: set(A),Y: set(A)] :
      ( aa(set(set(A)),$o,pred_chain(set(A),A4,ord_less(set(A))),C3)
     => ( aa(set(set(A)),$o,member(set(A),X),C3)
       => ( aa(set(set(A)),$o,member(set(A),Y),C3)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),aa(fun(set(A),fun(set(A),$o)),fun(set(A),fun(set(A),$o)),aa(fun(set(A),fun(set(A),$o)),fun(fun(set(A),fun(set(A),$o)),fun(set(A),fun(set(A),$o))),sup_sup(fun(set(A),fun(set(A),$o))),ord_less(set(A))),fequal(set(A))),X),Y)
            | aa(set(A),$o,aa(set(A),fun(set(A),$o),aa(fun(set(A),fun(set(A),$o)),fun(set(A),fun(set(A),$o)),aa(fun(set(A),fun(set(A),$o)),fun(fun(set(A),fun(set(A),$o)),fun(set(A),fun(set(A),$o))),sup_sup(fun(set(A),fun(set(A),$o))),ord_less(set(A))),fequal(set(A))),Y),X) ) ) ) ) ).

% subset.chain_total
tff(fact_5461_pred__on_Ochain__total,axiom,
    ! [A: $tType,A4: set(A),P: fun(A,fun(A,$o)),C3: set(A),X: A,Y: A] :
      ( aa(set(A),$o,pred_chain(A,A4,P),C3)
     => ( aa(set(A),$o,member(A,X),C3)
       => ( aa(set(A),$o,member(A,Y),C3)
         => ( aa(A,$o,aa(A,fun(A,$o),aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(fun(A,fun(A,$o)),fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),sup_sup(fun(A,fun(A,$o))),P),fequal(A)),X),Y)
            | aa(A,$o,aa(A,fun(A,$o),aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(fun(A,fun(A,$o)),fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),sup_sup(fun(A,fun(A,$o))),P),fequal(A)),Y),X) ) ) ) ) ).

% pred_on.chain_total
tff(fact_5462_subset_Ochain__def,axiom,
    ! [A: $tType,A4: set(set(A)),C3: set(set(A))] :
      ( aa(set(set(A)),$o,pred_chain(set(A),A4,ord_less(set(A))),C3)
    <=> ( aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),C3),A4)
        & ! [X4: set(A)] :
            ( aa(set(set(A)),$o,member(set(A),X4),C3)
           => ! [Xa2: set(A)] :
                ( aa(set(set(A)),$o,member(set(A),Xa2),C3)
               => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),aa(fun(set(A),fun(set(A),$o)),fun(set(A),fun(set(A),$o)),aa(fun(set(A),fun(set(A),$o)),fun(fun(set(A),fun(set(A),$o)),fun(set(A),fun(set(A),$o))),sup_sup(fun(set(A),fun(set(A),$o))),ord_less(set(A))),fequal(set(A))),X4),Xa2)
                  | aa(set(A),$o,aa(set(A),fun(set(A),$o),aa(fun(set(A),fun(set(A),$o)),fun(set(A),fun(set(A),$o)),aa(fun(set(A),fun(set(A),$o)),fun(fun(set(A),fun(set(A),$o)),fun(set(A),fun(set(A),$o))),sup_sup(fun(set(A),fun(set(A),$o))),ord_less(set(A))),fequal(set(A))),Xa2),X4) ) ) ) ) ) ).

% subset.chain_def
tff(fact_5463_subset_OchainI,axiom,
    ! [A: $tType,C3: set(set(A)),A4: set(set(A))] :
      ( aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),C3),A4)
     => ( ! [X2: set(A),Y2: set(A)] :
            ( aa(set(set(A)),$o,member(set(A),X2),C3)
           => ( aa(set(set(A)),$o,member(set(A),Y2),C3)
             => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),aa(fun(set(A),fun(set(A),$o)),fun(set(A),fun(set(A),$o)),aa(fun(set(A),fun(set(A),$o)),fun(fun(set(A),fun(set(A),$o)),fun(set(A),fun(set(A),$o))),sup_sup(fun(set(A),fun(set(A),$o))),ord_less(set(A))),fequal(set(A))),X2),Y2)
                | aa(set(A),$o,aa(set(A),fun(set(A),$o),aa(fun(set(A),fun(set(A),$o)),fun(set(A),fun(set(A),$o)),aa(fun(set(A),fun(set(A),$o)),fun(fun(set(A),fun(set(A),$o)),fun(set(A),fun(set(A),$o))),sup_sup(fun(set(A),fun(set(A),$o))),ord_less(set(A))),fequal(set(A))),Y2),X2) ) ) )
       => aa(set(set(A)),$o,pred_chain(set(A),A4,ord_less(set(A))),C3) ) ) ).

% subset.chainI
tff(fact_5464_subset_Ochain__empty,axiom,
    ! [A: $tType,A4: set(set(A))] : aa(set(set(A)),$o,pred_chain(set(A),A4,ord_less(set(A))),bot_bot(set(set(A)))) ).

% subset.chain_empty
tff(fact_5465_pred__on_OchainI,axiom,
    ! [A: $tType,C3: set(A),A4: set(A),P: fun(A,fun(A,$o))] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C3),A4)
     => ( ! [X2: A,Y2: A] :
            ( aa(set(A),$o,member(A,X2),C3)
           => ( aa(set(A),$o,member(A,Y2),C3)
             => ( aa(A,$o,aa(A,fun(A,$o),aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(fun(A,fun(A,$o)),fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),sup_sup(fun(A,fun(A,$o))),P),fequal(A)),X2),Y2)
                | aa(A,$o,aa(A,fun(A,$o),aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(fun(A,fun(A,$o)),fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),sup_sup(fun(A,fun(A,$o))),P),fequal(A)),Y2),X2) ) ) )
       => aa(set(A),$o,pred_chain(A,A4,P),C3) ) ) ).

% pred_on.chainI
tff(fact_5466_pred__on_Ochain__def,axiom,
    ! [A: $tType,A4: set(A),P: fun(A,fun(A,$o)),C3: set(A)] :
      ( aa(set(A),$o,pred_chain(A,A4,P),C3)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C3),A4)
        & ! [X4: A] :
            ( aa(set(A),$o,member(A,X4),C3)
           => ! [Xa2: A] :
                ( aa(set(A),$o,member(A,Xa2),C3)
               => ( aa(A,$o,aa(A,fun(A,$o),aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(fun(A,fun(A,$o)),fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),sup_sup(fun(A,fun(A,$o))),P),fequal(A)),X4),Xa2)
                  | aa(A,$o,aa(A,fun(A,$o),aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(fun(A,fun(A,$o)),fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),sup_sup(fun(A,fun(A,$o))),P),fequal(A)),Xa2),X4) ) ) ) ) ) ).

% pred_on.chain_def
tff(fact_5467_chains__alt__def,axiom,
    ! [A: $tType,A4: set(set(A))] : chains2(A,A4) = aa(fun(set(set(A)),$o),set(set(set(A))),collect(set(set(A))),pred_chain(set(A),A4,ord_less(set(A)))) ).

% chains_alt_def
tff(fact_5468_chain__subset__alt__def,axiom,
    ! [A: $tType,C3: set(set(A))] :
      ( chain_subset(A,C3)
    <=> aa(set(set(A)),$o,pred_chain(set(A),top_top(set(set(A))),ord_less(set(A))),C3) ) ).

% chain_subset_alt_def
tff(fact_5469_subset__Zorn__nonempty,axiom,
    ! [A: $tType,A16: set(set(A))] :
      ( ( A16 != bot_bot(set(set(A))) )
     => ( ! [C8: set(set(A))] :
            ( ( C8 != bot_bot(set(set(A))) )
           => ( aa(set(set(A)),$o,pred_chain(set(A),A16,ord_less(set(A))),C8)
             => aa(set(set(A)),$o,member(set(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C8)),A16) ) )
       => ? [X2: set(A)] :
            ( aa(set(set(A)),$o,member(set(A),X2),A16)
            & ! [Xa3: set(A)] :
                ( aa(set(set(A)),$o,member(set(A),Xa3),A16)
               => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X2),Xa3)
                 => ( Xa3 = X2 ) ) ) ) ) ) ).

% subset_Zorn_nonempty
tff(fact_5470_Union__in__chain,axiom,
    ! [A: $tType,B11: set(set(A)),A16: set(set(A))] :
      ( aa(set(set(A)),$o,finite_finite2(set(A)),B11)
     => ( ( B11 != bot_bot(set(set(A))) )
       => ( aa(set(set(A)),$o,pred_chain(set(A),A16,ord_less(set(A))),B11)
         => aa(set(set(A)),$o,member(set(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),B11)),B11) ) ) ) ).

% Union_in_chain
tff(fact_5471_Inter__in__chain,axiom,
    ! [A: $tType,B11: set(set(A)),A16: set(set(A))] :
      ( aa(set(set(A)),$o,finite_finite2(set(A)),B11)
     => ( ( B11 != bot_bot(set(set(A))) )
       => ( aa(set(set(A)),$o,pred_chain(set(A),A16,ord_less(set(A))),B11)
         => aa(set(set(A)),$o,member(set(A),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),B11)),B11) ) ) ) ).

% Inter_in_chain
tff(fact_5472_subset_Ochain__extend,axiom,
    ! [A: $tType,A4: set(set(A)),C3: set(set(A)),Z2: set(A)] :
      ( aa(set(set(A)),$o,pred_chain(set(A),A4,ord_less(set(A))),C3)
     => ( aa(set(set(A)),$o,member(set(A),Z2),A4)
       => ( ! [X2: set(A)] :
              ( aa(set(set(A)),$o,member(set(A),X2),C3)
             => aa(set(A),$o,aa(set(A),fun(set(A),$o),aa(fun(set(A),fun(set(A),$o)),fun(set(A),fun(set(A),$o)),aa(fun(set(A),fun(set(A),$o)),fun(fun(set(A),fun(set(A),$o)),fun(set(A),fun(set(A),$o))),sup_sup(fun(set(A),fun(set(A),$o))),ord_less(set(A))),fequal(set(A))),X2),Z2) )
         => aa(set(set(A)),$o,pred_chain(set(A),A4,ord_less(set(A))),aa(set(set(A)),set(set(A)),aa(set(set(A)),fun(set(set(A)),set(set(A))),sup_sup(set(set(A))),aa(set(set(A)),set(set(A)),aa(set(A),fun(set(set(A)),set(set(A))),insert2(set(A)),Z2),bot_bot(set(set(A))))),C3)) ) ) ) ).

% subset.chain_extend
tff(fact_5473_pred__on_Ochain__extend,axiom,
    ! [A: $tType,A4: set(A),P: fun(A,fun(A,$o)),C3: set(A),Z2: A] :
      ( aa(set(A),$o,pred_chain(A,A4,P),C3)
     => ( aa(set(A),$o,member(A,Z2),A4)
       => ( ! [X2: A] :
              ( aa(set(A),$o,member(A,X2),C3)
             => aa(A,$o,aa(A,fun(A,$o),aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(fun(A,fun(A,$o)),fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),sup_sup(fun(A,fun(A,$o))),P),fequal(A)),X2),Z2) )
         => aa(set(A),$o,pred_chain(A,A4,P),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Z2),bot_bot(set(A)))),C3)) ) ) ) ).

% pred_on.chain_extend
tff(fact_5474_finite__subset__Union__chain,axiom,
    ! [A: $tType,A4: set(A),B11: set(set(A)),A16: set(set(A))] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),B11))
       => ( ( B11 != bot_bot(set(set(A))) )
         => ( aa(set(set(A)),$o,pred_chain(set(A),A16,ord_less(set(A))),B11)
           => ~ ! [B10: set(A)] :
                  ( aa(set(set(A)),$o,member(set(A),B10),B11)
                 => ~ aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B10) ) ) ) ) ) ).

% finite_subset_Union_chain
tff(fact_5475_dom__mmupd,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B)),K5: set(A),V: B] : dom(A,B,map_mmupd(A,B,M,K5,V)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),dom(A,B,M)),K5) ).

% dom_mmupd
tff(fact_5476_in__measures_I2_J,axiom,
    ! [A: $tType,X: A,Y: A,F: fun(A,nat),Fs: list(fun(A,nat))] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),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)),F),Fs)))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,F,X)),aa(A,nat,F,Y))
        | ( ( aa(A,nat,F,X) = aa(A,nat,F,Y) )
          & aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),measures(A,Fs)) ) ) ) ).

% in_measures(2)
tff(fact_5477_Rep__unit,axiom,
    ! [X: product_unit] : aa(set($o),$o,member($o,aa(product_unit,$o,product_Rep_unit,X)),aa(set($o),set($o),aa($o,fun(set($o),set($o)),insert2($o),$true),bot_bot(set($o)))) ).

% Rep_unit
tff(fact_5478_map__mmupd__empty,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B)),V: B] : map_mmupd(A,B,M,bot_bot(set(A)),V) = M ).

% map_mmupd_empty
tff(fact_5479_in__measures_I1_J,axiom,
    ! [A: $tType,X: A,Y: A] : ~ aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),measures(A,nil(fun(A,nat)))) ).

% in_measures(1)
tff(fact_5480_measures__less,axiom,
    ! [A: $tType,F: fun(A,nat),X: A,Y: A,Fs: list(fun(A,nat))] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,F,X)),aa(A,nat,F,Y))
     => aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),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)),F),Fs))) ) ).

% measures_less
tff(fact_5481_measures__lesseq,axiom,
    ! [A: $tType,F: fun(A,nat),X: A,Y: A,Fs: list(fun(A,nat))] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,F,X)),aa(A,nat,F,Y))
     => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),measures(A,Fs))
       => aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),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)),F),Fs))) ) ) ).

% measures_lesseq
tff(fact_5482_Rep__unit__induct,axiom,
    ! [Y: $o,P: fun($o,$o)] :
      ( aa(set($o),$o,member($o,(Y)),aa(set($o),set($o),aa($o,fun(set($o),set($o)),insert2($o),$true),bot_bot(set($o))))
     => ( ! [X2: product_unit] : aa($o,$o,P,aa(product_unit,$o,product_Rep_unit,X2))
       => aa($o,$o,P,(Y)) ) ) ).

% Rep_unit_induct
tff(fact_5483_Rep__unit__cases,axiom,
    ! [Y: $o] :
      ( aa(set($o),$o,member($o,(Y)),aa(set($o),set($o),aa($o,fun(set($o),set($o)),insert2($o),$true),bot_bot(set($o))))
     => ~ ! [X2: product_unit] :
            ( (Y)
          <=> ~ aa(product_unit,$o,product_Rep_unit,X2) ) ) ).

% Rep_unit_cases
tff(fact_5484_type__definition__unit,axiom,
    type_definition(product_unit,$o,product_Rep_unit,product_Abs_unit,aa(set($o),set($o),aa($o,fun(set($o),set($o)),insert2($o),$true),bot_bot(set($o)))) ).

% type_definition_unit
tff(fact_5485_Abs__unit__inverse,axiom,
    ! [Y: $o] :
      ( aa(set($o),$o,member($o,(Y)),aa(set($o),set($o),aa($o,fun(set($o),set($o)),insert2($o),$true),bot_bot(set($o))))
     => ( aa(product_unit,$o,product_Rep_unit,aa($o,product_unit,product_Abs_unit,(Y)))
      <=> (Y) ) ) ).

% Abs_unit_inverse
tff(fact_5486_Abs__unit__inject,axiom,
    ! [X: $o,Y: $o] :
      ( aa(set($o),$o,member($o,(X)),aa(set($o),set($o),aa($o,fun(set($o),set($o)),insert2($o),$true),bot_bot(set($o))))
     => ( aa(set($o),$o,member($o,(Y)),aa(set($o),set($o),aa($o,fun(set($o),set($o)),insert2($o),$true),bot_bot(set($o))))
       => ( ( aa($o,product_unit,product_Abs_unit,(X)) = aa($o,product_unit,product_Abs_unit,(Y)) )
        <=> ( (X)
          <=> (Y) ) ) ) ) ).

% Abs_unit_inject
tff(fact_5487_Abs__unit__cases,axiom,
    ! [X: product_unit] :
      ~ ! [Y2: $o] :
          ( ( X = aa($o,product_unit,product_Abs_unit,(Y2)) )
         => ~ aa(set($o),$o,member($o,(Y2)),aa(set($o),set($o),aa($o,fun(set($o),set($o)),insert2($o),$true),bot_bot(set($o)))) ) ).

% Abs_unit_cases
tff(fact_5488_Abs__unit__induct,axiom,
    ! [P: fun(product_unit,$o),X: product_unit] :
      ( ! [Y2: $o] :
          ( aa(set($o),$o,member($o,(Y2)),aa(set($o),set($o),aa($o,fun(set($o),set($o)),insert2($o),$true),bot_bot(set($o))))
         => aa(product_unit,$o,P,aa($o,product_unit,product_Abs_unit,(Y2))) )
     => aa(product_unit,$o,P,X) ) ).

% Abs_unit_induct
tff(fact_5489_power__int__def,axiom,
    ! [A: $tType] :
      ( ( inverse(A)
        & power(A) )
     => ! [X: A,N: int] :
          power_int(A,X,N) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),N),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(int,nat,nat2,N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,inverse_inverse(A),X)),aa(int,nat,nat2,aa(int,int,uminus_uminus(int),N)))) ) ).

% power_int_def
tff(fact_5490_mset__set__Union,axiom,
    ! [A: $tType,A4: set(A),B3: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( aa(set(A),$o,finite_finite2(A),B3)
       => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3) = bot_bot(set(A)) )
         => ( mset_set(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),mset_set(A,A4)),mset_set(A,B3)) ) ) ) ) ).

% mset_set_Union
tff(fact_5491_Partial__order__eq__Image1__Image1__iff,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A3: A,B2: A] :
      ( order_7125193373082350890der_on(A,field2(A,R2),R2)
     => ( aa(set(A),$o,member(A,A3),field2(A,R2))
       => ( aa(set(A),$o,member(A,B2),field2(A,R2))
         => ( ( aa(set(A),set(A),image(A,A,R2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A)))) = aa(set(A),set(A),image(A,A,R2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A)))) )
          <=> ( A3 = B2 ) ) ) ) ) ).

% Partial_order_eq_Image1_Image1_iff
tff(fact_5492_power__int__1__left,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [N: int] : power_int(A,one_one(A),N) = one_one(A) ) ).

% power_int_1_left
tff(fact_5493_power__int__1__right,axiom,
    ! [A: $tType] :
      ( ( inverse(A)
        & monoid_mult(A) )
     => ! [Y: A] : power_int(A,Y,one_one(int)) = Y ) ).

% power_int_1_right
tff(fact_5494_power__int__mult__distrib__numeral1,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [W2: num,Y: A,M: int] : power_int(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),Y),M) = aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,aa(num,A,numeral_numeral(A),W2),M)),power_int(A,Y,M)) ) ).

% power_int_mult_distrib_numeral1
tff(fact_5495_power__int__mult__distrib__numeral2,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [X: A,W2: num,M: int] : power_int(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),aa(num,A,numeral_numeral(A),W2)),M) = aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,X,M)),power_int(A,aa(num,A,numeral_numeral(A),W2),M)) ) ).

% power_int_mult_distrib_numeral2
tff(fact_5496_power__int__0__right,axiom,
    ! [A: $tType] :
      ( ( inverse(A)
        & power(A) )
     => ! [X: A] : power_int(A,X,zero_zero(int)) = one_one(A) ) ).

% power_int_0_right
tff(fact_5497_abs__power__int__minus,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,N: int] : aa(A,A,abs_abs(A),power_int(A,aa(A,A,uminus_uminus(A),A3),N)) = aa(A,A,abs_abs(A),power_int(A,A3,N)) ) ).

% abs_power_int_minus
tff(fact_5498_mset__set_Oempty,axiom,
    ! [A: $tType] : mset_set(A,bot_bot(set(A))) = zero_zero(multiset(A)) ).

% mset_set.empty
tff(fact_5499_sum__multiset__singleton,axiom,
    ! [A: $tType,A4: set(A)] : aa(set(A),multiset(A),aa(fun(A,multiset(A)),fun(set(A),multiset(A)),groups7311177749621191930dd_sum(A,multiset(A)),aTP_Lamp_ajv(A,multiset(A))),A4) = mset_set(A,A4) ).

% sum_multiset_singleton
tff(fact_5500_power__int__minus__one__mult__self,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [M: int] : aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,aa(A,A,uminus_uminus(A),one_one(A)),M)),power_int(A,aa(A,A,uminus_uminus(A),one_one(A)),M)) = one_one(A) ) ).

% power_int_minus_one_mult_self
tff(fact_5501_power__int__minus__one__mult__self_H,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [M: 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)),M)),aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,aa(A,A,uminus_uminus(A),one_one(A)),M)),B2)) = B2 ) ).

% power_int_minus_one_mult_self'
tff(fact_5502_power__int__add__numeral,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [X: A,M: num,N: num] : aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,X,aa(num,int,numeral_numeral(int),M))),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),M),N))) ) ).

% power_int_add_numeral
tff(fact_5503_power__int__add__numeral2,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [X: A,M: 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),M))),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),M),N)))),B2) ) ).

% power_int_add_numeral2
tff(fact_5504_power__int__minus1__right,axiom,
    ! [A: $tType] :
      ( ( inverse(A)
        & monoid_mult(A) )
     => ! [Y: A] : power_int(A,Y,aa(int,int,uminus_uminus(int),one_one(int))) = aa(A,A,inverse_inverse(A),Y) ) ).

% power_int_minus1_right
tff(fact_5505_power__int__minus__left__odd,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [N: int,A3: A] :
          ( ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N)
         => ( power_int(A,aa(A,A,uminus_uminus(A),A3),N) = aa(A,A,uminus_uminus(A),power_int(A,A3,N)) ) ) ) ).

% power_int_minus_left_odd
tff(fact_5506_power__int__minus__left__even,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [N: int,A3: A] :
          ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N)
         => ( power_int(A,aa(A,A,uminus_uminus(A),A3),N) = power_int(A,A3,N) ) ) ) ).

% power_int_minus_left_even
tff(fact_5507_power__int__mult__distrib,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [X: A,Y: A,M: int] : power_int(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),Y),M) = aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,X,M)),power_int(A,Y,M)) ) ).

% power_int_mult_distrib
tff(fact_5508_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_5509_power__int__one__over,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [X: A,N: int] : power_int(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),X),N) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),power_int(A,X,N)) ) ).

% power_int_one_over
tff(fact_5510_power__int__minus,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [X: A,N: int] : power_int(A,X,aa(int,int,uminus_uminus(int),N)) = aa(A,A,inverse_inverse(A),power_int(A,X,N)) ) ).

% power_int_minus
tff(fact_5511_power__int__0__left__If,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [M: int] :
          power_int(A,zero_zero(A),M) = $ite(M = zero_zero(int),one_one(A),zero_zero(A)) ) ).

% power_int_0_left_If
tff(fact_5512_power__int__increasing,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [N: int,N4: int,A3: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),N),N4)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),A3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),power_int(A,A3,N)),power_int(A,A3,N4)) ) ) ) ).

% power_int_increasing
tff(fact_5513_power__int__strict__increasing,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [N: int,N4: int,A3: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),N),N4)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),power_int(A,A3,N)),power_int(A,A3,N4)) ) ) ) ).

% power_int_strict_increasing
tff(fact_5514_power__int__minus__one__minus,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [N: int] : power_int(A,aa(A,A,uminus_uminus(A),one_one(A)),aa(int,int,uminus_uminus(int),N)) = power_int(A,aa(A,A,uminus_uminus(A),one_one(A)),N) ) ).

% power_int_minus_one_minus
tff(fact_5515_power__int__minus__one__diff__commute,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: int,B2: int] : power_int(A,aa(A,A,uminus_uminus(A),one_one(A)),aa(int,int,aa(int,fun(int,int),minus_minus(int),A3),B2)) = power_int(A,aa(A,A,uminus_uminus(A),one_one(A)),aa(int,int,aa(int,fun(int,int),minus_minus(int),B2),A3)) ) ).

% power_int_minus_one_diff_commute
tff(fact_5516_partial__order__on__empty,axiom,
    ! [A: $tType] : order_7125193373082350890der_on(A,bot_bot(set(A)),bot_bot(set(product_prod(A,A)))) ).

% partial_order_on_empty
tff(fact_5517_mset__set__empty__iff,axiom,
    ! [A: $tType,A4: set(A)] :
      ( ( mset_set(A,A4) = zero_zero(multiset(A)) )
    <=> ( ( A4 = bot_bot(set(A)) )
        | ~ aa(set(A),$o,finite_finite2(A),A4) ) ) ).

% mset_set_empty_iff
tff(fact_5518_power__int__strict__decreasing,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [N: int,N4: int,A3: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),N),N4)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),one_one(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),power_int(A,A3,N4)),power_int(A,A3,N)) ) ) ) ) ).

% power_int_strict_decreasing
tff(fact_5519_one__le__power__int,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,N: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),X)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),N)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),power_int(A,X,N)) ) ) ) ).

% one_le_power_int
tff(fact_5520_one__less__power__int,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,N: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A3)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),N)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),power_int(A,A3,N)) ) ) ) ).

% one_less_power_int
tff(fact_5521_power__int__add,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [X: A,M: int,N: int] :
          ( ( ( X != zero_zero(A) )
            | ( aa(int,int,aa(int,fun(int,int),plus_plus(int),M),N) != zero_zero(int) ) )
         => ( power_int(A,X,aa(int,int,aa(int,fun(int,int),plus_plus(int),M),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,X,M)),power_int(A,X,N)) ) ) ) ).

% power_int_add
tff(fact_5522_power__int__minus__left__distrib,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( division_ring(C)
        & one(A)
        & uminus(A) )
     => ! [X: B,A3: C,N: int] :
          ( nO_MATCH(A,B,aa(A,A,uminus_uminus(A),one_one(A)),X)
         => ( power_int(C,aa(C,C,uminus_uminus(C),A3),N) = aa(C,C,aa(C,fun(C,C),times_times(C),power_int(C,aa(C,C,uminus_uminus(C),one_one(C)),N)),power_int(C,A3,N)) ) ) ) ).

% power_int_minus_left_distrib
tff(fact_5523_power__int__decreasing,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [N: int,N4: int,A3: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),N),N4)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),one_one(A))
             => ( ( ( A3 != zero_zero(A) )
                  | ( N4 != zero_zero(int) )
                  | ( N = zero_zero(int) ) )
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),power_int(A,A3,N4)),power_int(A,A3,N)) ) ) ) ) ) ).

% power_int_decreasing
tff(fact_5524_power__int__le__one,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,N: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),X)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),N)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),one_one(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),power_int(A,X,N)),one_one(A)) ) ) ) ) ).

% power_int_le_one
tff(fact_5525_power__int__le__imp__le__exp,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,M: int,N: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),X)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),power_int(A,X,M)),power_int(A,X,N))
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),N)
             => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),M),N) ) ) ) ) ).

% power_int_le_imp_le_exp
tff(fact_5526_power__int__le__imp__less__exp,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,M: int,N: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),X)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),power_int(A,X,M)),power_int(A,X,N))
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),N)
             => aa(int,$o,aa(int,fun(int,$o),ord_less(int),M),N) ) ) ) ) ).

% power_int_le_imp_less_exp
tff(fact_5527_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_5528_power__int__add__1_H,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [X: A,M: int] :
          ( ( ( X != zero_zero(A) )
            | ( M != aa(int,int,uminus_uminus(int),one_one(int)) ) )
         => ( power_int(A,X,aa(int,int,aa(int,fun(int,int),plus_plus(int),M),one_one(int))) = aa(A,A,aa(A,fun(A,A),times_times(A),X),power_int(A,X,M)) ) ) ) ).

% power_int_add_1'
tff(fact_5529_power__int__add__1,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [X: A,M: int] :
          ( ( ( X != zero_zero(A) )
            | ( M != aa(int,int,uminus_uminus(int),one_one(int)) ) )
         => ( power_int(A,X,aa(int,int,aa(int,fun(int,int),plus_plus(int),M),one_one(int))) = aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,X,M)),X) ) ) ) ).

% power_int_add_1
tff(fact_5530_power__int__minus__left,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,N: int] :
          power_int(A,aa(A,A,uminus_uminus(A),A3),N) = $ite(aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N),power_int(A,A3,N),aa(A,A,uminus_uminus(A),power_int(A,A3,N))) ) ).

% power_int_minus_left
tff(fact_5531_Zorns__po__lemma,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( order_7125193373082350890der_on(A,field2(A,R2),R2)
     => ( ! [C7: set(A)] :
            ( aa(set(set(A)),$o,member(set(A),C7),chains(A,R2))
           => ? [X3: A] :
                ( aa(set(A),$o,member(A,X3),field2(A,R2))
                & ! [Xa4: A] :
                    ( aa(set(A),$o,member(A,Xa4),C7)
                   => aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa4),X3)),R2) ) ) )
       => ? [X2: A] :
            ( aa(set(A),$o,member(A,X2),field2(A,R2))
            & ! [Xa3: A] :
                ( aa(set(A),$o,member(A,Xa3),field2(A,R2))
               => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Xa3)),R2)
                 => ( Xa3 = X2 ) ) ) ) ) ) ).

% Zorns_po_lemma
tff(fact_5532_Partial__order__Restr,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A4: set(A)] :
      ( order_7125193373082350890der_on(A,field2(A,R2),R2)
     => order_7125193373082350890der_on(A,field2(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R2),product_Sigma(A,A,A4,aTP_Lamp_rl(set(A),fun(A,set(A)),A4)))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R2),product_Sigma(A,A,A4,aTP_Lamp_rl(set(A),fun(A,set(A)),A4)))) ) ).

% Partial_order_Restr
tff(fact_5533_mset__set_Oinsert__remove,axiom,
    ! [A: $tType,A4: set(A),X: A] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( mset_set(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A4)) = aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),X),mset_set(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))))) ) ) ).

% mset_set.insert_remove
tff(fact_5534_mset__set_Oremove,axiom,
    ! [A: $tType,A4: set(A),X: A] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( aa(set(A),$o,member(A,X),A4)
       => ( mset_set(A,A4) = aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),X),mset_set(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))))) ) ) ) ).

% mset_set.remove
tff(fact_5535_power__int__numeral__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [M: num,N: num] : power_int(A,aa(num,A,numeral_numeral(A),M),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = aa(A,A,inverse_inverse(A),aa(num,A,numeral_numeral(A),pow(M,N))) ) ).

% power_int_numeral_neg_numeral
tff(fact_5536_UN__equiv__class2,axiom,
    ! [A: $tType,C: $tType,B: $tType,A15: set(A),R12: set(product_prod(A,A)),A24: set(B),R23: set(product_prod(B,B)),F: fun(A,fun(B,set(C))),A1: A,A22: B] :
      ( equiv_equiv(A,A15,R12)
     => ( equiv_equiv(B,A24,R23)
       => ( equiv_congruent2(A,B,set(C),R12,R23,F)
         => ( aa(set(A),$o,member(A,A1),A15)
           => ( aa(set(B),$o,member(B,A22),A24)
             => ( aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(A),set(set(C)),image2(A,set(C),aa(B,fun(A,set(C)),aa(fun(A,fun(B,set(C))),fun(B,fun(A,set(C))),aTP_Lamp_ajw(set(product_prod(B,B)),fun(fun(A,fun(B,set(C))),fun(B,fun(A,set(C)))),R23),F),A22)),aa(set(A),set(A),image(A,A,R12),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A1),bot_bot(set(A)))))) = aa(B,set(C),aa(A,fun(B,set(C)),F,A1),A22) ) ) ) ) ) ) ).

% UN_equiv_class2
tff(fact_5537_congruent2__implies__congruent__UN,axiom,
    ! [A: $tType,C: $tType,B: $tType,A15: set(A),R12: set(product_prod(A,A)),A24: set(B),R23: set(product_prod(B,B)),F: fun(A,fun(B,set(C))),A3: B] :
      ( equiv_equiv(A,A15,R12)
     => ( equiv_equiv(B,A24,R23)
       => ( equiv_congruent2(A,B,set(C),R12,R23,F)
         => ( aa(set(B),$o,member(B,A3),A24)
           => equiv_congruent(A,set(C),R12,aa(B,fun(A,set(C)),aa(fun(A,fun(B,set(C))),fun(B,fun(A,set(C))),aTP_Lamp_ajw(set(product_prod(B,B)),fun(fun(A,fun(B,set(C))),fun(B,fun(A,set(C)))),R23),F),A3)) ) ) ) ) ).

% congruent2_implies_congruent_UN
tff(fact_5538_quotient__eqI,axiom,
    ! [A: $tType,A4: set(A),R2: set(product_prod(A,A)),X5: set(A),Y4: set(A),X: A,Y: A] :
      ( equiv_equiv(A,A4,R2)
     => ( aa(set(set(A)),$o,member(set(A),X5),equiv_quotient(A,A4,R2))
       => ( aa(set(set(A)),$o,member(set(A),Y4),equiv_quotient(A,A4,R2))
         => ( aa(set(A),$o,member(A,X),X5)
           => ( aa(set(A),$o,member(A,Y),Y4)
             => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R2)
               => ( X5 = Y4 ) ) ) ) ) ) ) ).

% quotient_eqI
tff(fact_5539_quotient__eq__iff,axiom,
    ! [A: $tType,A4: set(A),R2: set(product_prod(A,A)),X5: set(A),Y4: set(A),X: A,Y: A] :
      ( equiv_equiv(A,A4,R2)
     => ( aa(set(set(A)),$o,member(set(A),X5),equiv_quotient(A,A4,R2))
       => ( aa(set(set(A)),$o,member(set(A),Y4),equiv_quotient(A,A4,R2))
         => ( aa(set(A),$o,member(A,X),X5)
           => ( aa(set(A),$o,member(A,Y),Y4)
             => ( ( X5 = Y4 )
              <=> aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R2) ) ) ) ) ) ) ).

% quotient_eq_iff
tff(fact_5540_in__quotient__imp__closed,axiom,
    ! [A: $tType,A4: set(A),R2: set(product_prod(A,A)),X5: set(A),X: A,Y: A] :
      ( equiv_equiv(A,A4,R2)
     => ( aa(set(set(A)),$o,member(set(A),X5),equiv_quotient(A,A4,R2))
       => ( aa(set(A),$o,member(A,X),X5)
         => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R2)
           => aa(set(A),$o,member(A,Y),X5) ) ) ) ) ).

% in_quotient_imp_closed
tff(fact_5541_in__quotient__imp__non__empty,axiom,
    ! [A: $tType,A4: set(A),R2: set(product_prod(A,A)),X5: set(A)] :
      ( equiv_equiv(A,A4,R2)
     => ( aa(set(set(A)),$o,member(set(A),X5),equiv_quotient(A,A4,R2))
       => ( X5 != bot_bot(set(A)) ) ) ) ).

% in_quotient_imp_non_empty
tff(fact_5542_congruent2__commuteI,axiom,
    ! [B: $tType,A: $tType,A4: set(A),R2: set(product_prod(A,A)),F: fun(A,fun(A,B))] :
      ( equiv_equiv(A,A4,R2)
     => ( ! [Y2: A,Z3: A] :
            ( aa(set(A),$o,member(A,Y2),A4)
           => ( aa(set(A),$o,member(A,Z3),A4)
             => ( aa(A,B,aa(A,fun(A,B),F,Y2),Z3) = aa(A,B,aa(A,fun(A,B),F,Z3),Y2) ) ) )
       => ( ! [Y2: A,Z3: A,W: A] :
              ( aa(set(A),$o,member(A,W),A4)
             => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),Z3)),R2)
               => ( aa(A,B,aa(A,fun(A,B),F,W),Y2) = aa(A,B,aa(A,fun(A,B),F,W),Z3) ) ) )
         => equiv_congruent2(A,A,B,R2,R2,F) ) ) ) ).

% congruent2_commuteI
tff(fact_5543_congruent2I,axiom,
    ! [C: $tType,B: $tType,A: $tType,A15: set(A),R12: set(product_prod(A,A)),A24: set(B),R23: set(product_prod(B,B)),F: fun(A,fun(B,C))] :
      ( equiv_equiv(A,A15,R12)
     => ( equiv_equiv(B,A24,R23)
       => ( ! [Y2: A,Z3: A,W: B] :
              ( aa(set(B),$o,member(B,W),A24)
             => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),Z3)),R12)
               => ( aa(B,C,aa(A,fun(B,C),F,Y2),W) = aa(B,C,aa(A,fun(B,C),F,Z3),W) ) ) )
         => ( ! [Y2: B,Z3: B,W: A] :
                ( aa(set(A),$o,member(A,W),A15)
               => ( aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Y2),Z3)),R23)
                 => ( aa(B,C,aa(A,fun(B,C),F,W),Y2) = aa(B,C,aa(A,fun(B,C),F,W),Z3) ) ) )
           => equiv_congruent2(A,B,C,R12,R23,F) ) ) ) ) ).

% congruent2I
tff(fact_5544_equiv__type,axiom,
    ! [A: $tType,A4: set(A),R2: set(product_prod(A,A))] :
      ( equiv_equiv(A,A4,R2)
     => aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R2),product_Sigma(A,A,A4,aTP_Lamp_rl(set(A),fun(A,set(A)),A4))) ) ).

% equiv_type
tff(fact_5545_equiv__class__self,axiom,
    ! [A: $tType,A4: set(A),R2: set(product_prod(A,A)),A3: A] :
      ( equiv_equiv(A,A4,R2)
     => ( aa(set(A),$o,member(A,A3),A4)
       => aa(set(A),$o,member(A,A3),aa(set(A),set(A),image(A,A,R2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A))))) ) ) ).

% equiv_class_self
tff(fact_5546_quotient__disj,axiom,
    ! [A: $tType,A4: set(A),R2: set(product_prod(A,A)),X5: set(A),Y4: set(A)] :
      ( equiv_equiv(A,A4,R2)
     => ( aa(set(set(A)),$o,member(set(A),X5),equiv_quotient(A,A4,R2))
       => ( aa(set(set(A)),$o,member(set(A),Y4),equiv_quotient(A,A4,R2))
         => ( ( X5 = Y4 )
            | ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),X5),Y4) = bot_bot(set(A)) ) ) ) ) ) ).

% quotient_disj
tff(fact_5547_UN__equiv__class__type,axiom,
    ! [A: $tType,B: $tType,A4: set(A),R2: set(product_prod(A,A)),F: fun(A,set(B)),X5: set(A),B3: set(set(B))] :
      ( equiv_equiv(A,A4,R2)
     => ( equiv_congruent(A,set(B),R2,F)
       => ( aa(set(set(A)),$o,member(set(A),X5),equiv_quotient(A,A4,R2))
         => ( ! [X2: A] :
                ( aa(set(A),$o,member(A,X2),A4)
               => aa(set(set(B)),$o,member(set(B),aa(A,set(B),F,X2)),B3) )
           => aa(set(set(B)),$o,member(set(B),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),F),X5))),B3) ) ) ) ) ).

% UN_equiv_class_type
tff(fact_5548_UN__equiv__class__type2,axiom,
    ! [A: $tType,B: $tType,C: $tType,A15: set(A),R12: set(product_prod(A,A)),A24: set(B),R23: set(product_prod(B,B)),F: fun(A,fun(B,set(C))),X13: set(A),X24: set(B),B3: set(set(C))] :
      ( equiv_equiv(A,A15,R12)
     => ( equiv_equiv(B,A24,R23)
       => ( equiv_congruent2(A,B,set(C),R12,R23,F)
         => ( aa(set(set(A)),$o,member(set(A),X13),equiv_quotient(A,A15,R12))
           => ( aa(set(set(B)),$o,member(set(B),X24),equiv_quotient(B,A24,R23))
             => ( ! [X12: A,X23: B] :
                    ( aa(set(A),$o,member(A,X12),A15)
                   => ( aa(set(B),$o,member(B,X23),A24)
                     => aa(set(set(C)),$o,member(set(C),aa(B,set(C),aa(A,fun(B,set(C)),F,X12),X23)),B3) ) )
               => aa(set(set(C)),$o,member(set(C),aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(A),set(set(C)),image2(A,set(C),aa(set(B),fun(A,set(C)),aTP_Lamp_ajx(fun(A,fun(B,set(C))),fun(set(B),fun(A,set(C))),F),X24)),X13))),B3) ) ) ) ) ) ) ).

% UN_equiv_class_type2
tff(fact_5549_equiv__class__eq__iff,axiom,
    ! [A: $tType,A4: set(A),R2: set(product_prod(A,A)),X: A,Y: A] :
      ( equiv_equiv(A,A4,R2)
     => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R2)
      <=> ( ( aa(set(A),set(A),image(A,A,R2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) = aa(set(A),set(A),image(A,A,R2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y),bot_bot(set(A)))) )
          & aa(set(A),$o,member(A,X),A4)
          & aa(set(A),$o,member(A,Y),A4) ) ) ) ).

% equiv_class_eq_iff
tff(fact_5550_eq__equiv__class__iff,axiom,
    ! [A: $tType,A4: set(A),R2: set(product_prod(A,A)),X: A,Y: A] :
      ( equiv_equiv(A,A4,R2)
     => ( aa(set(A),$o,member(A,X),A4)
       => ( aa(set(A),$o,member(A,Y),A4)
         => ( ( aa(set(A),set(A),image(A,A,R2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) = aa(set(A),set(A),image(A,A,R2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y),bot_bot(set(A)))) )
          <=> aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R2) ) ) ) ) ).

% eq_equiv_class_iff
tff(fact_5551_equiv__class__eq,axiom,
    ! [A: $tType,A4: set(A),R2: set(product_prod(A,A)),A3: A,B2: A] :
      ( equiv_equiv(A,A4,R2)
     => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R2)
       => ( aa(set(A),set(A),image(A,A,R2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A)))) = aa(set(A),set(A),image(A,A,R2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A)))) ) ) ) ).

% equiv_class_eq
tff(fact_5552_eq__equiv__class,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A3: A,B2: A,A4: set(A)] :
      ( ( aa(set(A),set(A),image(A,A,R2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A)))) = aa(set(A),set(A),image(A,A,R2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A)))) )
     => ( equiv_equiv(A,A4,R2)
       => ( aa(set(A),$o,member(A,B2),A4)
         => aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R2) ) ) ) ).

% eq_equiv_class
tff(fact_5553_eq__equiv__class__iff2,axiom,
    ! [A: $tType,A4: set(A),R2: set(product_prod(A,A)),X: A,Y: A] :
      ( equiv_equiv(A,A4,R2)
     => ( aa(set(A),$o,member(A,X),A4)
       => ( aa(set(A),$o,member(A,Y),A4)
         => ( ( equiv_quotient(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))),R2) = equiv_quotient(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y),bot_bot(set(A))),R2) )
          <=> aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R2) ) ) ) ) ).

% eq_equiv_class_iff2
tff(fact_5554_refines__equiv__class__eq2,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),S: set(product_prod(A,A)),A4: set(A),A3: A] :
      ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R),S)
     => ( equiv_equiv(A,A4,R)
       => ( equiv_equiv(A,A4,S)
         => ( aa(set(A),set(A),image(A,A,S),aa(set(A),set(A),image(A,A,R),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A))))) = aa(set(A),set(A),image(A,A,S),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A)))) ) ) ) ) ).

% refines_equiv_class_eq2
tff(fact_5555_refines__equiv__class__eq,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),S: set(product_prod(A,A)),A4: set(A),A3: A] :
      ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R),S)
     => ( equiv_equiv(A,A4,R)
       => ( equiv_equiv(A,A4,S)
         => ( aa(set(A),set(A),image(A,A,R),aa(set(A),set(A),image(A,A,S),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A))))) = aa(set(A),set(A),image(A,A,S),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A)))) ) ) ) ) ).

% refines_equiv_class_eq
tff(fact_5556_UN__equiv__class__inject,axiom,
    ! [B: $tType,A: $tType,A4: set(A),R2: set(product_prod(A,A)),F: fun(A,set(B)),X5: set(A),Y4: set(A)] :
      ( equiv_equiv(A,A4,R2)
     => ( equiv_congruent(A,set(B),R2,F)
       => ( ( aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),F),X5)) = aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),F),Y4)) )
         => ( aa(set(set(A)),$o,member(set(A),X5),equiv_quotient(A,A4,R2))
           => ( aa(set(set(A)),$o,member(set(A),Y4),equiv_quotient(A,A4,R2))
             => ( ! [X2: A,Y2: A] :
                    ( aa(set(A),$o,member(A,X2),A4)
                   => ( aa(set(A),$o,member(A,Y2),A4)
                     => ( ( aa(A,set(B),F,X2) = aa(A,set(B),F,Y2) )
                       => aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y2)),R2) ) ) )
               => ( X5 = Y4 ) ) ) ) ) ) ) ).

% UN_equiv_class_inject
tff(fact_5557_refines__equiv__image__eq,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),S: set(product_prod(A,A)),A4: set(A)] :
      ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R),S)
     => ( equiv_equiv(A,A4,R)
       => ( equiv_equiv(A,A4,S)
         => ( aa(set(set(A)),set(set(A)),image2(set(A),set(A),image(A,A,S)),equiv_quotient(A,A4,R)) = equiv_quotient(A,A4,S) ) ) ) ) ).

% refines_equiv_image_eq
tff(fact_5558_subset__equiv__class,axiom,
    ! [A: $tType,A4: set(A),R2: set(product_prod(A,A)),B2: A,A3: A] :
      ( equiv_equiv(A,A4,R2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),image(A,A,R2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A))))),aa(set(A),set(A),image(A,A,R2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A)))))
       => ( aa(set(A),$o,member(A,B2),A4)
         => aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R2) ) ) ) ).

% subset_equiv_class
tff(fact_5559_equiv__class__subset,axiom,
    ! [A: $tType,A4: set(A),R2: set(product_prod(A,A)),A3: A,B2: A] :
      ( equiv_equiv(A,A4,R2)
     => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R2)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),image(A,A,R2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A))))),aa(set(A),set(A),image(A,A,R2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A))))) ) ) ).

% equiv_class_subset
tff(fact_5560_equiv__class__nondisjoint,axiom,
    ! [A: $tType,A4: set(A),R2: set(product_prod(A,A)),X: A,A3: A,B2: A] :
      ( equiv_equiv(A,A4,R2)
     => ( aa(set(A),$o,member(A,X),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),image(A,A,R2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A))))),aa(set(A),set(A),image(A,A,R2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A))))))
       => aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R2) ) ) ).

% equiv_class_nondisjoint
tff(fact_5561_in__quotient__imp__in__rel,axiom,
    ! [A: $tType,A4: set(A),R2: set(product_prod(A,A)),X5: set(A),X: A,Y: A] :
      ( equiv_equiv(A,A4,R2)
     => ( aa(set(set(A)),$o,member(set(A),X5),equiv_quotient(A,A4,R2))
       => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y),bot_bot(set(A))))),X5)
         => aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R2) ) ) ) ).

% in_quotient_imp_in_rel
tff(fact_5562_UN__equiv__class,axiom,
    ! [B: $tType,A: $tType,A4: set(A),R2: set(product_prod(A,A)),F: fun(A,set(B)),A3: A] :
      ( equiv_equiv(A,A4,R2)
     => ( equiv_congruent(A,set(B),R2,F)
       => ( aa(set(A),$o,member(A,A3),A4)
         => ( aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),F),aa(set(A),set(A),image(A,A,R2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A)))))) = aa(A,set(B),F,A3) ) ) ) ) ).

% UN_equiv_class
tff(fact_5563_proj__iff,axiom,
    ! [A: $tType,A4: set(A),R2: set(product_prod(A,A)),X: A,Y: A] :
      ( equiv_equiv(A,A4,R2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y),bot_bot(set(A))))),A4)
       => ( ( equiv_proj(A,A,R2,X) = equiv_proj(A,A,R2,Y) )
        <=> aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R2) ) ) ) ).

% proj_iff
tff(fact_5564_disjnt__equiv__class,axiom,
    ! [A: $tType,A4: set(A),R2: set(product_prod(A,A)),A3: A,B2: A] :
      ( equiv_equiv(A,A4,R2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),disjnt(A),aa(set(A),set(A),image(A,A,R2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A))))),aa(set(A),set(A),image(A,A,R2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A)))))
      <=> ~ aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R2) ) ) ).

% disjnt_equiv_class
tff(fact_5565_wo__rel_Ocases__Total3,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A3: A,B2: A,Phi: fun(A,fun(A,$o))] :
      ( bNF_Wellorder_wo_rel(A,R2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A))))),field2(A,R2))
       => ( ( ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),minus_minus(set(product_prod(A,A))),R2),id2(A)))
              | aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),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))),R2),id2(A))) )
           => aa(A,$o,aa(A,fun(A,$o),Phi,A3),B2) )
         => ( ( ( A3 = B2 )
             => aa(A,$o,aa(A,fun(A,$o),Phi,A3),B2) )
           => aa(A,$o,aa(A,fun(A,$o),Phi,A3),B2) ) ) ) ) ).

% wo_rel.cases_Total3
tff(fact_5566_disjnt__self__iff__empty,axiom,
    ! [A: $tType,S: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),disjnt(A),S),S)
    <=> ( S = bot_bot(set(A)) ) ) ).

% disjnt_self_iff_empty
tff(fact_5567_disjnt__Un1,axiom,
    ! [A: $tType,A4: set(A),B3: set(A),C3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),disjnt(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3)),C3)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),disjnt(A),A4),C3)
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),disjnt(A),B3),C3) ) ) ).

% disjnt_Un1
tff(fact_5568_disjnt__Un2,axiom,
    ! [A: $tType,C3: set(A),A4: set(A),B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),disjnt(A),C3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3))
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),disjnt(A),C3),A4)
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),disjnt(A),C3),B3) ) ) ).

% disjnt_Un2
tff(fact_5569_disjnt__Times2__iff,axiom,
    ! [B: $tType,A: $tType,A4: set(A),C3: set(B),B3: set(A)] :
      ( aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),disjnt(product_prod(A,B)),product_Sigma(A,B,A4,aTP_Lamp_qz(set(B),fun(A,set(B)),C3))),product_Sigma(A,B,B3,aTP_Lamp_qz(set(B),fun(A,set(B)),C3)))
    <=> ( ( C3 = bot_bot(set(B)) )
        | aa(set(A),$o,aa(set(A),fun(set(A),$o),disjnt(A),A4),B3) ) ) ).

% disjnt_Times2_iff
tff(fact_5570_disjnt__Times1__iff,axiom,
    ! [A: $tType,B: $tType,C3: set(A),A4: set(B),B3: set(B)] :
      ( aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),disjnt(product_prod(A,B)),product_Sigma(A,B,C3,aTP_Lamp_qz(set(B),fun(A,set(B)),A4))),product_Sigma(A,B,C3,aTP_Lamp_qz(set(B),fun(A,set(B)),B3)))
    <=> ( ( C3 = bot_bot(set(A)) )
        | aa(set(B),$o,aa(set(B),fun(set(B),$o),disjnt(B),A4),B3) ) ) ).

% disjnt_Times1_iff
tff(fact_5571_disjnt__Sigma__iff,axiom,
    ! [B: $tType,A: $tType,A4: set(A),C3: fun(A,set(B)),B3: set(A)] :
      ( aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),disjnt(product_prod(A,B)),product_Sigma(A,B,A4,C3)),product_Sigma(A,B,B3,C3))
    <=> ( ! [X4: A] :
            ( aa(set(A),$o,member(A,X4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3))
           => ( aa(A,set(B),C3,X4) = bot_bot(set(B)) ) )
        | aa(set(A),$o,aa(set(A),fun(set(A),$o),disjnt(A),A4),B3) ) ) ).

% disjnt_Sigma_iff
tff(fact_5572_wo__rel_Owell__order__induct,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),P: fun(A,$o),A3: A] :
      ( bNF_Wellorder_wo_rel(A,R2)
     => ( ! [X2: A] :
            ( ! [Y5: A] :
                ( ( ( Y5 != X2 )
                  & aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y5),X2)),R2) )
               => aa(A,$o,P,Y5) )
           => aa(A,$o,P,X2) )
       => aa(A,$o,P,A3) ) ) ).

% wo_rel.well_order_induct
tff(fact_5573_wo__rel_OTOTALS,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( bNF_Wellorder_wo_rel(A,R2)
     => ! [X3: A] :
          ( aa(set(A),$o,member(A,X3),field2(A,R2))
         => ! [Xa3: A] :
              ( aa(set(A),$o,member(A,Xa3),field2(A,R2))
             => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Xa3)),R2)
                | aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa3),X3)),R2) ) ) ) ) ).

% wo_rel.TOTALS
tff(fact_5574_wo__rel_Omax2__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A3: A,B2: A] :
      ( bNF_Wellorder_wo_rel(A,R2)
     => ( bNF_We1388413361240627857o_max2(A,R2,A3,B2) = $ite(aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R2),B2,A3) ) ) ).

% wo_rel.max2_def
tff(fact_5575_disjnt__empty1,axiom,
    ! [A: $tType,A4: set(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),disjnt(A),bot_bot(set(A))),A4) ).

% disjnt_empty1
tff(fact_5576_disjnt__empty2,axiom,
    ! [A: $tType,A4: set(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),disjnt(A),A4),bot_bot(set(A))) ).

% disjnt_empty2
tff(fact_5577_well__order__induct__imp,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),P: fun(A,$o),A3: A] :
      ( bNF_Wellorder_wo_rel(A,R2)
     => ( ! [X2: A] :
            ( ! [Y5: A] :
                ( ( ( Y5 != X2 )
                  & aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y5),X2)),R2) )
               => ( aa(set(A),$o,member(A,Y5),field2(A,R2))
                 => aa(A,$o,P,Y5) ) )
           => ( aa(set(A),$o,member(A,X2),field2(A,R2))
             => aa(A,$o,P,X2) ) )
       => ( aa(set(A),$o,member(A,A3),field2(A,R2))
         => aa(A,$o,P,A3) ) ) ) ).

% well_order_induct_imp
tff(fact_5578_disjnt__def,axiom,
    ! [A: $tType,A4: set(A),B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),disjnt(A),A4),B3)
    <=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3) = bot_bot(set(A)) ) ) ).

% disjnt_def
tff(fact_5579_wo__rel_Omax2__greater,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A3: A,B2: A] :
      ( bNF_Wellorder_wo_rel(A,R2)
     => ( aa(set(A),$o,member(A,A3),field2(A,R2))
       => ( aa(set(A),$o,member(A,B2),field2(A,R2))
         => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),bNF_We1388413361240627857o_max2(A,R2,A3,B2))),R2)
            & aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),bNF_We1388413361240627857o_max2(A,R2,A3,B2))),R2) ) ) ) ) ).

% wo_rel.max2_greater
tff(fact_5580_wo__rel_Omax2__equals2,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A3: A,B2: A] :
      ( bNF_Wellorder_wo_rel(A,R2)
     => ( aa(set(A),$o,member(A,A3),field2(A,R2))
       => ( aa(set(A),$o,member(A,B2),field2(A,R2))
         => ( ( bNF_We1388413361240627857o_max2(A,R2,A3,B2) = B2 )
          <=> aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R2) ) ) ) ) ).

% wo_rel.max2_equals2
tff(fact_5581_wo__rel_Omax2__equals1,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A3: A,B2: A] :
      ( bNF_Wellorder_wo_rel(A,R2)
     => ( aa(set(A),$o,member(A,A3),field2(A,R2))
       => ( aa(set(A),$o,member(A,B2),field2(A,R2))
         => ( ( bNF_We1388413361240627857o_max2(A,R2,A3,B2) = A3 )
          <=> aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),A3)),R2) ) ) ) ) ).

% wo_rel.max2_equals1
tff(fact_5582_disjoint__UN__iff,axiom,
    ! [A: $tType,B: $tType,A4: set(A),B3: fun(B,set(A)),I4: set(B)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),disjnt(A),A4),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B3),I4)))
    <=> ! [X4: B] :
          ( aa(set(B),$o,member(B,X4),I4)
         => aa(set(A),$o,aa(set(A),fun(set(A),$o),disjnt(A),A4),aa(B,set(A),B3,X4)) ) ) ).

% disjoint_UN_iff
tff(fact_5583_wo__rel_Omax2__among,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A3: A,B2: A] :
      ( bNF_Wellorder_wo_rel(A,R2)
     => ( aa(set(A),$o,member(A,A3),field2(A,R2))
       => ( aa(set(A),$o,member(A,B2),field2(A,R2))
         => aa(set(A),$o,member(A,bNF_We1388413361240627857o_max2(A,R2,A3,B2)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A))))) ) ) ) ).

% wo_rel.max2_among
tff(fact_5584_wo__rel_Ocases__Total,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A3: A,B2: A,Phi: fun(A,fun(A,$o))] :
      ( bNF_Wellorder_wo_rel(A,R2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A))))),field2(A,R2))
       => ( ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R2)
           => aa(A,$o,aa(A,fun(A,$o),Phi,A3),B2) )
         => ( ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),A3)),R2)
             => aa(A,$o,aa(A,fun(A,$o),Phi,A3),B2) )
           => aa(A,$o,aa(A,fun(A,$o),Phi,A3),B2) ) ) ) ) ).

% wo_rel.cases_Total
tff(fact_5585_natLeq__on__wo__rel,axiom,
    ! [N: nat] : bNF_Wellorder_wo_rel(nat,aa(fun(product_prod(nat,nat),$o),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aTP_Lamp_uw(nat,fun(nat,fun(nat,$o)),N)))) ).

% natLeq_on_wo_rel
tff(fact_5586_wo__rel_Omax2__greater__among,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A3: A,B2: A] :
      ( bNF_Wellorder_wo_rel(A,R2)
     => ( aa(set(A),$o,member(A,A3),field2(A,R2))
       => ( aa(set(A),$o,member(A,B2),field2(A,R2))
         => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),bNF_We1388413361240627857o_max2(A,R2,A3,B2))),R2)
            & aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),bNF_We1388413361240627857o_max2(A,R2,A3,B2))),R2)
            & aa(set(A),$o,member(A,bNF_We1388413361240627857o_max2(A,R2,A3,B2)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A))))) ) ) ) ) ).

% wo_rel.max2_greater_among
tff(fact_5587_card__Un__disjnt,axiom,
    ! [A: $tType,A4: set(A),B3: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( aa(set(A),$o,finite_finite2(A),B3)
       => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),disjnt(A),A4),B3)
         => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(A),nat,finite_card(A),A4)),aa(set(A),nat,finite_card(A),B3)) ) ) ) ) ).

% card_Un_disjnt
tff(fact_5588_proj__def,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(B,A)),X: B] : equiv_proj(B,A,R2,X) = aa(set(B),set(A),image(B,A,R2),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),X),bot_bot(set(B)))) ).

% proj_def
tff(fact_5589_wo__rel_OWell__order__isMinim__exists,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),B3: set(A)] :
      ( bNF_Wellorder_wo_rel(A,R2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),field2(A,R2))
       => ( ( B3 != bot_bot(set(A)) )
         => ? [X_1: A] : aa(A,$o,bNF_We4791949203932849705sMinim(A,R2,B3),X_1) ) ) ) ).

% wo_rel.Well_order_isMinim_exists
tff(fact_5590_sum__card__image,axiom,
    ! [B: $tType,A: $tType,A4: set(A),F: fun(A,set(B))] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( pairwise(A,aTP_Lamp_ajy(fun(A,set(B)),fun(A,fun(A,$o)),F),A4)
       => ( aa(set(set(B)),nat,aa(fun(set(B),nat),fun(set(set(B)),nat),groups7311177749621191930dd_sum(set(B),nat),finite_card(B)),aa(set(A),set(set(B)),image2(A,set(B),F),A4)) = aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aTP_Lamp_nd(fun(A,set(B)),fun(A,nat),F)),A4) ) ) ) ).

% sum_card_image
tff(fact_5591_wo__rel_Ominim__in,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),B3: set(A)] :
      ( bNF_Wellorder_wo_rel(A,R2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),field2(A,R2))
       => ( ( B3 != bot_bot(set(A)) )
         => aa(set(A),$o,member(A,bNF_We6954850376910717587_minim(A,R2,B3)),B3) ) ) ) ).

% wo_rel.minim_in
tff(fact_5592_pairwise__image,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(A,$o)),F: fun(B,A),S2: set(B)] :
      ( pairwise(A,R2,aa(set(B),set(A),image2(B,A,F),S2))
    <=> pairwise(B,aa(fun(B,A),fun(B,fun(B,$o)),aTP_Lamp_ajz(fun(A,fun(A,$o)),fun(fun(B,A),fun(B,fun(B,$o))),R2),F),S2) ) ).

% pairwise_image
tff(fact_5593_pairwise__empty,axiom,
    ! [A: $tType,P: fun(A,fun(A,$o))] : pairwise(A,P,bot_bot(set(A))) ).

% pairwise_empty
tff(fact_5594_pairwise__trivial,axiom,
    ! [A: $tType,I4: set(A)] : pairwise(A,aTP_Lamp_ze(A,fun(A,$o)),I4) ).

% pairwise_trivial
tff(fact_5595_wo__rel_Ominim__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A4: set(A)] :
      ( bNF_Wellorder_wo_rel(A,R2)
     => ( bNF_We6954850376910717587_minim(A,R2,A4) = the(A,bNF_We4791949203932849705sMinim(A,R2,A4)) ) ) ).

% wo_rel.minim_def
tff(fact_5596_pairwise__singleton,axiom,
    ! [A: $tType,P: fun(A,fun(A,$o)),A4: A] : pairwise(A,P,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),bot_bot(set(A)))) ).

% pairwise_singleton
tff(fact_5597_wo__rel_Ominim__isMinim,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),B3: set(A)] :
      ( bNF_Wellorder_wo_rel(A,R2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),field2(A,R2))
       => ( ( B3 != bot_bot(set(A)) )
         => aa(A,$o,bNF_We4791949203932849705sMinim(A,R2,B3),bNF_We6954850376910717587_minim(A,R2,B3)) ) ) ) ).

% wo_rel.minim_isMinim
tff(fact_5598_pairwise__alt,axiom,
    ! [A: $tType,R: fun(A,fun(A,$o)),S: set(A)] :
      ( pairwise(A,R,S)
    <=> ! [X4: A] :
          ( aa(set(A),$o,member(A,X4),S)
         => ! [Xa2: A] :
              ( aa(set(A),$o,member(A,Xa2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X4),bot_bot(set(A)))))
             => aa(A,$o,aa(A,fun(A,$o),R,X4),Xa2) ) ) ) ).

% pairwise_alt
tff(fact_5599_wo__rel_OisMinim__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A4: set(A),B2: A] :
      ( bNF_Wellorder_wo_rel(A,R2)
     => ( aa(A,$o,bNF_We4791949203932849705sMinim(A,R2,A4),B2)
      <=> ( aa(set(A),$o,member(A,B2),A4)
          & ! [X4: A] :
              ( aa(set(A),$o,member(A,X4),A4)
             => aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),X4)),R2) ) ) ) ) ).

% wo_rel.isMinim_def
tff(fact_5600_wo__rel_Oequals__minim,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),B3: set(A),A3: A] :
      ( bNF_Wellorder_wo_rel(A,R2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),field2(A,R2))
       => ( aa(set(A),$o,member(A,A3),B3)
         => ( ! [B5: A] :
                ( aa(set(A),$o,member(A,B5),B3)
               => aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B5)),R2) )
           => ( A3 = bNF_We6954850376910717587_minim(A,R2,B3) ) ) ) ) ) ).

% wo_rel.equals_minim
tff(fact_5601_wo__rel_Ominim__least,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),B3: set(A),B2: A] :
      ( bNF_Wellorder_wo_rel(A,R2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),field2(A,R2))
       => ( aa(set(A),$o,member(A,B2),B3)
         => aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),bNF_We6954850376910717587_minim(A,R2,B3)),B2)),R2) ) ) ) ).

% wo_rel.minim_least
tff(fact_5602_wo__rel_Ominim__inField,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),B3: set(A)] :
      ( bNF_Wellorder_wo_rel(A,R2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),field2(A,R2))
       => ( ( B3 != bot_bot(set(A)) )
         => aa(set(A),$o,member(A,bNF_We6954850376910717587_minim(A,R2,B3)),field2(A,R2)) ) ) ) ).

% wo_rel.minim_inField
tff(fact_5603_butlast__take,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),aa(list(A),nat,size_size(list(A)),Xs))
     => ( 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_5604_eventually__INF__base,axiom,
    ! [B: $tType,A: $tType,B3: set(A),F4: fun(A,filter(B)),P: fun(B,$o)] :
      ( ( B3 != bot_bot(set(A)) )
     => ( ! [A6: A] :
            ( aa(set(A),$o,member(A,A6),B3)
           => ! [B5: A] :
                ( aa(set(A),$o,member(A,B5),B3)
               => ? [X3: A] :
                    ( aa(set(A),$o,member(A,X3),B3)
                    & aa(filter(B),$o,aa(filter(B),fun(filter(B),$o),ord_less_eq(filter(B)),aa(A,filter(B),F4,X3)),aa(filter(B),filter(B),aa(filter(B),fun(filter(B),filter(B)),inf_inf(filter(B)),aa(A,filter(B),F4,A6)),aa(A,filter(B),F4,B5))) ) ) )
       => ( eventually(B,P,aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image2(A,filter(B),F4),B3)))
        <=> ? [X4: A] :
              ( aa(set(A),$o,member(A,X4),B3)
              & eventually(B,P,aa(A,filter(B),F4,X4)) ) ) ) ) ).

% eventually_INF_base
tff(fact_5605_pair__lessI2,axiom,
    ! [A3: nat,B2: nat,S2: nat,T3: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A3),B2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),S2),T3)
       => aa(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),$o,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),T3))),fun_pair_less) ) ) ).

% pair_lessI2
tff(fact_5606_eventually__top,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( eventually(A,P,top_top(filter(A)))
    <=> ! [X_12: A] : aa(A,$o,P,X_12) ) ).

% eventually_top
tff(fact_5607_eventually__const,axiom,
    ! [A: $tType,F4: filter(A),P: $o] :
      ( ( F4 != bot_bot(filter(A)) )
     => ( eventually(A,aTP_Lamp_ah($o,fun(A,$o),(P)),F4)
      <=> (P) ) ) ).

% eventually_const
tff(fact_5608_length__butlast,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(A),nat,size_size(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_5609_pair__less__iff1,axiom,
    ! [X: nat,Y: nat,Z2: nat] :
      ( aa(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),$o,member(product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X),Y)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X),Z2))),fun_pair_less)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y),Z2) ) ).

% pair_less_iff1
tff(fact_5610_eventually__ex,axiom,
    ! [B: $tType,A: $tType,P: fun(A,fun(B,$o)),F4: filter(A)] :
      ( eventually(A,aTP_Lamp_afv(fun(A,fun(B,$o)),fun(A,$o),P),F4)
    <=> ? [Y8: fun(A,B)] : eventually(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_aka(fun(A,fun(B,$o)),fun(fun(A,B),fun(A,$o)),P),Y8),F4) ) ).

% eventually_ex
tff(fact_5611_eventually__compose__filterlim,axiom,
    ! [A: $tType,B: $tType,P: fun(A,$o),F4: filter(A),F: fun(B,A),G4: filter(B)] :
      ( eventually(A,P,F4)
     => ( filterlim(B,A,F,F4,G4)
       => eventually(B,aa(fun(B,A),fun(B,$o),aTP_Lamp_akb(fun(A,$o),fun(fun(B,A),fun(B,$o)),P),F),G4) ) ) ).

% eventually_compose_filterlim
tff(fact_5612_filterlim__cong,axiom,
    ! [A: $tType,B: $tType,F12: filter(A),F13: filter(A),F23: filter(B),F24: filter(B),F: fun(B,A),G: fun(B,A)] :
      ( ( F12 = F13 )
     => ( ( F23 = F24 )
       => ( eventually(B,aa(fun(B,A),fun(B,$o),aTP_Lamp_akc(fun(B,A),fun(fun(B,A),fun(B,$o)),F),G),F23)
         => ( filterlim(B,A,F,F12,F23)
          <=> filterlim(B,A,G,F13,F24) ) ) ) ) ).

% filterlim_cong
tff(fact_5613_filterlim__iff,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),F23: filter(B),F12: filter(A)] :
      ( filterlim(A,B,F,F23,F12)
    <=> ! [P5: fun(B,$o)] :
          ( eventually(B,P5,F23)
         => eventually(A,aa(fun(B,$o),fun(A,$o),aTP_Lamp_av(fun(A,B),fun(fun(B,$o),fun(A,$o)),F),P5),F12) ) ) ).

% filterlim_iff
tff(fact_5614_eventually__all__finite,axiom,
    ! [A: $tType,B: $tType] :
      ( finite_finite(A)
     => ! [P: fun(B,fun(A,$o)),Net: filter(B)] :
          ( ! [Y2: A] : eventually(B,aa(A,fun(B,$o),aTP_Lamp_akd(fun(B,fun(A,$o)),fun(A,fun(B,$o)),P),Y2),Net)
         => eventually(B,aTP_Lamp_ake(fun(B,fun(A,$o)),fun(B,$o),P),Net) ) ) ).

% eventually_all_finite
tff(fact_5615_eventually__inf,axiom,
    ! [A: $tType,P: fun(A,$o),F4: filter(A),F8: filter(A)] :
      ( eventually(A,P,aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),inf_inf(filter(A)),F4),F8))
    <=> ? [Q7: fun(A,$o),R7: fun(A,$o)] :
          ( eventually(A,Q7,F4)
          & eventually(A,R7,F8)
          & ! [X4: A] :
              ( ( aa(A,$o,Q7,X4)
                & aa(A,$o,R7,X4) )
             => aa(A,$o,P,X4) ) ) ) ).

% eventually_inf
tff(fact_5616_eventually__at__bot__not__equal,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_bot(A) )
     => ! [C2: A] : eventually(A,aTP_Lamp_akf(A,fun(A,$o),C2),at_bot(A)) ) ).

% eventually_at_bot_not_equal
tff(fact_5617_eventually__frequently__const__simps_I6_J,axiom,
    ! [A: $tType,C3: $o,P: fun(A,$o),F4: filter(A)] :
      ( eventually(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_akg($o,fun(fun(A,$o),fun(A,$o)),(C3)),P),F4)
    <=> ( (C3)
       => eventually(A,P,F4) ) ) ).

% eventually_frequently_const_simps(6)
tff(fact_5618_eventually__frequently__const__simps_I4_J,axiom,
    ! [A: $tType,C3: $o,P: fun(A,$o),F4: filter(A)] :
      ( eventually(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_akh($o,fun(fun(A,$o),fun(A,$o)),(C3)),P),F4)
    <=> ( (C3)
        | eventually(A,P,F4) ) ) ).

% eventually_frequently_const_simps(4)
tff(fact_5619_eventually__frequently__const__simps_I3_J,axiom,
    ! [A: $tType,P: fun(A,$o),C3: $o,F4: filter(A)] :
      ( eventually(A,aa($o,fun(A,$o),aTP_Lamp_aki(fun(A,$o),fun($o,fun(A,$o)),P),(C3)),F4)
    <=> ( eventually(A,P,F4)
        | (C3) ) ) ).

% eventually_frequently_const_simps(3)
tff(fact_5620_eventually__mp,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o),F4: filter(A)] :
      ( eventually(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ao(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q),F4)
     => ( eventually(A,P,F4)
       => eventually(A,Q,F4) ) ) ).

% eventually_mp
tff(fact_5621_eventually__True,axiom,
    ! [A: $tType,F4: filter(A)] : eventually(A,aTP_Lamp_ar(A,$o),F4) ).

% eventually_True
tff(fact_5622_eventually__conj,axiom,
    ! [A: $tType,P: fun(A,$o),F4: filter(A),Q: fun(A,$o)] :
      ( eventually(A,P,F4)
     => ( eventually(A,Q,F4)
       => eventually(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_am(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q),F4) ) ) ).

% eventually_conj
tff(fact_5623_eventually__elim2,axiom,
    ! [A: $tType,P: fun(A,$o),F4: filter(A),Q: fun(A,$o),R: fun(A,$o)] :
      ( eventually(A,P,F4)
     => ( eventually(A,Q,F4)
       => ( ! [I2: A] :
              ( aa(A,$o,P,I2)
             => ( aa(A,$o,Q,I2)
               => aa(A,$o,R,I2) ) )
         => eventually(A,R,F4) ) ) ) ).

% eventually_elim2
tff(fact_5624_eventually__subst,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o),F4: filter(A)] :
      ( eventually(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_akj(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q),F4)
     => ( eventually(A,P,F4)
      <=> eventually(A,Q,F4) ) ) ).

% eventually_subst
tff(fact_5625_eventually__rev__mp,axiom,
    ! [A: $tType,P: fun(A,$o),F4: filter(A),Q: fun(A,$o)] :
      ( eventually(A,P,F4)
     => ( eventually(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ao(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q),F4)
       => eventually(A,Q,F4) ) ) ).

% eventually_rev_mp
tff(fact_5626_eventually__conj__iff,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o),F4: filter(A)] :
      ( eventually(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_am(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q),F4)
    <=> ( eventually(A,P,F4)
        & eventually(A,Q,F4) ) ) ).

% eventually_conj_iff
tff(fact_5627_not__eventually__impI,axiom,
    ! [A: $tType,P: fun(A,$o),F4: filter(A),Q: fun(A,$o)] :
      ( eventually(A,P,F4)
     => ( ~ eventually(A,Q,F4)
       => ~ eventually(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ao(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q),F4) ) ) ).

% not_eventually_impI
tff(fact_5628_eventually__happens_H,axiom,
    ! [A: $tType,F4: filter(A),P: fun(A,$o)] :
      ( ( F4 != bot_bot(filter(A)) )
     => ( eventually(A,P,F4)
       => ? [X_1: A] : aa(A,$o,P,X_1) ) ) ).

% eventually_happens'
tff(fact_5629_eventually__happens,axiom,
    ! [A: $tType,P: fun(A,$o),Net: filter(A)] :
      ( eventually(A,P,Net)
     => ( ( Net = bot_bot(filter(A)) )
        | ? [X_1: A] : aa(A,$o,P,X_1) ) ) ).

% eventually_happens
tff(fact_5630_eventually__bot,axiom,
    ! [A: $tType,P: fun(A,$o)] : eventually(A,P,bot_bot(filter(A))) ).

% eventually_bot
tff(fact_5631_trivial__limit__def,axiom,
    ! [A: $tType,F4: filter(A)] :
      ( ( F4 = bot_bot(filter(A)) )
    <=> eventually(A,aTP_Lamp_ak(A,$o),F4) ) ).

% trivial_limit_def
tff(fact_5632_eventually__const__iff,axiom,
    ! [A: $tType,P: $o,F4: filter(A)] :
      ( eventually(A,aTP_Lamp_ah($o,fun(A,$o),(P)),F4)
    <=> ( (P)
        | ( F4 = bot_bot(filter(A)) ) ) ) ).

% eventually_const_iff
tff(fact_5633_False__imp__not__eventually,axiom,
    ! [A: $tType,P: fun(A,$o),Net: filter(A)] :
      ( ! [X2: A] : ~ aa(A,$o,P,X2)
     => ( ( Net != bot_bot(filter(A)) )
       => ~ eventually(A,P,Net) ) ) ).

% False_imp_not_eventually
tff(fact_5634_eventually__sup,axiom,
    ! [A: $tType,P: fun(A,$o),F4: filter(A),F8: filter(A)] :
      ( eventually(A,P,aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),sup_sup(filter(A)),F4),F8))
    <=> ( eventually(A,P,F4)
        & eventually(A,P,F8) ) ) ).

% eventually_sup
tff(fact_5635_eventually__ball__finite,axiom,
    ! [A: $tType,B: $tType,A4: set(A),P: fun(B,fun(A,$o)),Net: filter(B)] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( ! [X2: A] :
            ( aa(set(A),$o,member(A,X2),A4)
           => eventually(B,aa(A,fun(B,$o),aTP_Lamp_aei(fun(B,fun(A,$o)),fun(A,fun(B,$o)),P),X2),Net) )
       => eventually(B,aa(fun(B,fun(A,$o)),fun(B,$o),aTP_Lamp_akk(set(A),fun(fun(B,fun(A,$o)),fun(B,$o)),A4),P),Net) ) ) ).

% eventually_ball_finite
tff(fact_5636_eventually__ball__finite__distrib,axiom,
    ! [A: $tType,B: $tType,A4: set(A),P: fun(B,fun(A,$o)),Net: filter(B)] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( eventually(B,aa(fun(B,fun(A,$o)),fun(B,$o),aTP_Lamp_akk(set(A),fun(fun(B,fun(A,$o)),fun(B,$o)),A4),P),Net)
      <=> ! [X4: A] :
            ( aa(set(A),$o,member(A,X4),A4)
           => eventually(B,aa(A,fun(B,$o),aTP_Lamp_aei(fun(B,fun(A,$o)),fun(A,fun(B,$o)),P),X4),Net) ) ) ) ).

% eventually_ball_finite_distrib
tff(fact_5637_eventually__le__at__bot,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A] : eventually(A,aa(A,fun(A,$o),aTP_Lamp_tm(A,fun(A,$o)),C2),at_bot(A)) ) ).

% eventually_le_at_bot
tff(fact_5638_eventually__gt__at__bot,axiom,
    ! [A: $tType] :
      ( unboun7993243217541854897norder(A)
     => ! [C2: A] : eventually(A,aTP_Lamp_akl(A,fun(A,$o),C2),at_bot(A)) ) ).

% eventually_gt_at_bot
tff(fact_5639_filterlim__mono__eventually,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),F4: filter(B),G4: filter(A),F8: filter(B),G6: filter(A),F9: fun(A,B)] :
      ( filterlim(A,B,F,F4,G4)
     => ( aa(filter(B),$o,aa(filter(B),fun(filter(B),$o),ord_less_eq(filter(B)),F4),F8)
       => ( aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),G6),G4)
         => ( eventually(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_ck(fun(A,B),fun(fun(A,B),fun(A,$o)),F),F9),G6)
           => filterlim(A,B,F9,F8,G6) ) ) ) ) ).

% filterlim_mono_eventually
tff(fact_5640_filterlim__principal,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),S: set(B),F4: filter(A)] :
      ( filterlim(A,B,F,principal(B,S),F4)
    <=> eventually(A,aa(set(B),fun(A,$o),aTP_Lamp_ay(fun(A,B),fun(set(B),fun(A,$o)),F),S),F4) ) ).

% filterlim_principal
tff(fact_5641_le__principal,axiom,
    ! [A: $tType,F4: filter(A),A4: set(A)] :
      ( aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),F4),principal(A,A4))
    <=> eventually(A,aa(set(A),fun(A,$o),aTP_Lamp_ag(set(A),fun(A,$o)),A4),F4) ) ).

% le_principal
tff(fact_5642_eventually__INF1,axiom,
    ! [B: $tType,A: $tType,I: A,I4: set(A),P: fun(B,$o),F4: fun(A,filter(B))] :
      ( aa(set(A),$o,member(A,I),I4)
     => ( eventually(B,P,aa(A,filter(B),F4,I))
       => eventually(B,P,aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image2(A,filter(B),F4),I4))) ) ) ).

% eventually_INF1
tff(fact_5643_eventually__inf__principal,axiom,
    ! [A: $tType,P: fun(A,$o),F4: filter(A),S2: set(A)] :
      ( eventually(A,P,aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),inf_inf(filter(A)),F4),principal(A,S2)))
    <=> eventually(A,aa(set(A),fun(A,$o),aTP_Lamp_akm(fun(A,$o),fun(set(A),fun(A,$o)),P),S2),F4) ) ).

% eventually_inf_principal
tff(fact_5644_eventually__Inf__base,axiom,
    ! [A: $tType,B3: set(filter(A)),P: fun(A,$o)] :
      ( ( B3 != bot_bot(set(filter(A))) )
     => ( ! [F5: filter(A)] :
            ( aa(set(filter(A)),$o,member(filter(A),F5),B3)
           => ! [G7: filter(A)] :
                ( aa(set(filter(A)),$o,member(filter(A),G7),B3)
               => ? [X3: filter(A)] :
                    ( aa(set(filter(A)),$o,member(filter(A),X3),B3)
                    & aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),X3),aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),inf_inf(filter(A)),F5),G7)) ) ) )
       => ( eventually(A,P,aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),B3))
        <=> ? [X4: filter(A)] :
              ( aa(set(filter(A)),$o,member(filter(A),X4),B3)
              & eventually(A,P,X4) ) ) ) ) ).

% eventually_Inf_base
tff(fact_5645_eventually__INF__finite,axiom,
    ! [A: $tType,B: $tType,A4: set(A),P: fun(B,$o),F4: fun(A,filter(B))] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( eventually(B,P,aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image2(A,filter(B),F4),A4)))
      <=> ? [Q7: fun(A,fun(B,$o))] :
            ( ! [X4: A] :
                ( aa(set(A),$o,member(A,X4),A4)
               => eventually(B,aa(A,fun(B,$o),Q7,X4),aa(A,filter(B),F4,X4)) )
            & ! [Y3: B] :
                ( ! [X4: A] :
                    ( aa(set(A),$o,member(A,X4),A4)
                   => aa(B,$o,aa(A,fun(B,$o),Q7,X4),Y3) )
               => aa(B,$o,P,Y3) ) ) ) ) ).

% eventually_INF_finite
tff(fact_5646_filterlim__finite__subsets__at__top,axiom,
    ! [B: $tType,A: $tType,F: fun(A,set(B)),A4: set(B),F4: filter(A)] :
      ( filterlim(A,set(B),F,finite5375528669736107172at_top(B,A4),F4)
    <=> ! [X8: set(B)] :
          ( ( aa(set(B),$o,finite_finite2(B),X8)
            & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),X8),A4) )
         => eventually(A,aa(set(B),fun(A,$o),aa(set(B),fun(set(B),fun(A,$o)),aTP_Lamp_akn(fun(A,set(B)),fun(set(B),fun(set(B),fun(A,$o))),F),A4),X8),F4) ) ) ).

% filterlim_finite_subsets_at_top
tff(fact_5647_filterlim__at__bot,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F: fun(A,B),F4: filter(A)] :
          ( filterlim(A,B,F,at_bot(B),F4)
        <=> ! [Z8: B] : eventually(A,aa(B,fun(A,$o),aTP_Lamp_ako(fun(A,B),fun(B,fun(A,$o)),F),Z8),F4) ) ) ).

% filterlim_at_bot
tff(fact_5648_filterlim__at__bot__le,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F: fun(A,B),F4: filter(A),C2: B] :
          ( filterlim(A,B,F,at_bot(B),F4)
        <=> ! [Z8: B] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Z8),C2)
             => eventually(A,aa(B,fun(A,$o),aTP_Lamp_ako(fun(A,B),fun(B,fun(A,$o)),F),Z8),F4) ) ) ) ).

% filterlim_at_bot_le
tff(fact_5649_butlast__upt,axiom,
    ! [M: nat,N: nat] : butlast(nat,upt(M,N)) = upt(M,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))) ).

% butlast_upt
tff(fact_5650_filterlim__at__bot__dense,axiom,
    ! [B: $tType,A: $tType] :
      ( ( dense_linorder(B)
        & no_bot(B) )
     => ! [F: fun(A,B),F4: filter(A)] :
          ( filterlim(A,B,F,at_bot(B),F4)
        <=> ! [Z8: B] : eventually(A,aa(B,fun(A,$o),aTP_Lamp_akp(fun(A,B),fun(B,fun(A,$o)),F),Z8),F4) ) ) ).

% filterlim_at_bot_dense
tff(fact_5651_pair__lessI1,axiom,
    ! [A3: nat,B2: nat,S2: nat,T3: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A3),B2)
     => aa(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),$o,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),T3))),fun_pair_less) ) ).

% pair_lessI1
tff(fact_5652_takeWhile__not__last,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( distinct(A,Xs)
     => ( takeWhile(A,aTP_Lamp_akq(list(A),fun(A,$o),Xs),Xs) = butlast(A,Xs) ) ) ).

% takeWhile_not_last
tff(fact_5653_eventually__INF,axiom,
    ! [A: $tType,B: $tType,P: fun(A,$o),F4: fun(B,filter(A)),B3: set(B)] :
      ( eventually(A,P,aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(B),set(filter(A)),image2(B,filter(A),F4),B3)))
    <=> ? [X8: set(B)] :
          ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),X8),B3)
          & aa(set(B),$o,finite_finite2(B),X8)
          & eventually(A,P,aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(B),set(filter(A)),image2(B,filter(A),F4),X8))) ) ) ).

% eventually_INF
tff(fact_5654_filterlim__at__bot__lt,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [F: fun(A,B),F4: filter(A),C2: B] :
          ( filterlim(A,B,F,at_bot(B),F4)
        <=> ! [Z8: B] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),Z8),C2)
             => eventually(A,aa(B,fun(A,$o),aTP_Lamp_akr(fun(A,B),fun(B,fun(A,$o)),F),Z8),F4) ) ) ) ).

% filterlim_at_bot_lt
tff(fact_5655_butlast__conv__take,axiom,
    ! [A: $tType,Xs: 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_5656_hd__butlast,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),one_one(nat)),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(list(A),A,hd(A),butlast(A,Xs)) = aa(list(A),A,hd(A),Xs) ) ) ).

% hd_butlast
tff(fact_5657_butlast__list__update,axiom,
    ! [A: $tType,Xs: list(A),K: nat,X: A] :
      butlast(A,list_update(A,Xs,K,X)) = $ite(K = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat)),butlast(A,Xs),list_update(A,butlast(A,Xs),K,X)) ).

% butlast_list_update
tff(fact_5658_map__filter__on__comp,axiom,
    ! [A: $tType,C: $tType,B: $tType,G: fun(B,A),Y4: set(B),X5: set(A),F4: filter(B),F: fun(A,C)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(B),set(A),image2(B,A,G),Y4)),X5)
     => ( eventually(B,aTP_Lamp_aks(set(B),fun(B,$o),Y4),F4)
       => ( aa(filter(A),filter(C),aa(fun(A,C),fun(filter(A),filter(C)),map_filter_on(A,C,X5),F),aa(filter(B),filter(A),aa(fun(B,A),fun(filter(B),filter(A)),map_filter_on(B,A,Y4),G),F4)) = aa(filter(B),filter(C),aa(fun(B,C),fun(filter(B),filter(C)),map_filter_on(B,C,Y4),aa(fun(B,A),fun(B,C),comp(A,C,B,F),G)),F4) ) ) ) ).

% map_filter_on_comp
tff(fact_5659_smin__insertI,axiom,
    ! [X: product_prod(nat,nat),XS: set(product_prod(nat,nat)),Y: product_prod(nat,nat),YS: set(product_prod(nat,nat))] :
      ( aa(set(product_prod(nat,nat)),$o,member(product_prod(nat,nat),X),XS)
     => ( aa(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),$o,member(product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),X),Y)),fun_pair_less)
       => ( aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),$o,member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),XS),YS)),fun_min_strict)
         => aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),$o,member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),XS),aa(set(product_prod(nat,nat)),set(product_prod(nat,nat)),aa(product_prod(nat,nat),fun(set(product_prod(nat,nat)),set(product_prod(nat,nat))),insert2(product_prod(nat,nat)),Y),YS))),fun_min_strict) ) ) ) ).

% smin_insertI
tff(fact_5660_Sup__filter__def,axiom,
    ! [A: $tType,S: set(filter(A))] : aa(set(filter(A)),filter(A),complete_Sup_Sup(filter(A)),S) = abs_filter(A,aTP_Lamp_akt(set(filter(A)),fun(fun(A,$o),$o),S)) ).

% Sup_filter_def
tff(fact_5661_smin__emptyI,axiom,
    ! [X5: set(product_prod(nat,nat))] :
      ( ( X5 != bot_bot(set(product_prod(nat,nat))) )
     => aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),$o,member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),X5),bot_bot(set(product_prod(nat,nat))))),fun_min_strict) ) ).

% smin_emptyI
tff(fact_5662_map__filter__on__def,axiom,
    ! [A: $tType,B: $tType,X5: set(B),F: fun(B,A),F4: filter(B)] : aa(filter(B),filter(A),aa(fun(B,A),fun(filter(B),filter(A)),map_filter_on(B,A,X5),F),F4) = abs_filter(A,aa(filter(B),fun(fun(A,$o),$o),aa(fun(B,A),fun(filter(B),fun(fun(A,$o),$o)),aTP_Lamp_akv(set(B),fun(fun(B,A),fun(filter(B),fun(fun(A,$o),$o))),X5),F),F4)) ).

% map_filter_on_def
tff(fact_5663_bot__filter__def,axiom,
    ! [A: $tType] : bot_bot(filter(A)) = abs_filter(A,aTP_Lamp_akw(fun(A,$o),$o)) ).

% bot_filter_def
tff(fact_5664_eventually__map__filter__on,axiom,
    ! [B: $tType,A: $tType,X5: set(A),F4: filter(A),P: fun(B,$o),F: fun(A,B)] :
      ( eventually(A,aa(set(A),fun(A,$o),aTP_Lamp_ag(set(A),fun(A,$o)),X5),F4)
     => ( eventually(B,P,aa(filter(A),filter(B),aa(fun(A,B),fun(filter(A),filter(B)),map_filter_on(A,B,X5),F),F4))
      <=> eventually(A,aa(fun(A,B),fun(A,$o),aa(fun(B,$o),fun(fun(A,B),fun(A,$o)),aTP_Lamp_akx(set(A),fun(fun(B,$o),fun(fun(A,B),fun(A,$o))),X5),P),F),F4) ) ) ).

% eventually_map_filter_on
tff(fact_5665_sup__filter__def,axiom,
    ! [A: $tType,F4: filter(A),F8: filter(A)] : aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),sup_sup(filter(A)),F4),F8) = abs_filter(A,aa(filter(A),fun(fun(A,$o),$o),aTP_Lamp_aky(filter(A),fun(filter(A),fun(fun(A,$o),$o)),F4),F8)) ).

% sup_filter_def
tff(fact_5666_principal__def,axiom,
    ! [A: $tType,S: set(A)] : principal(A,S) = abs_filter(A,aa(set(A),fun(fun(A,$o),$o),ball(A),S)) ).

% principal_def
tff(fact_5667_top__filter__def,axiom,
    ! [A: $tType] : top_top(filter(A)) = abs_filter(A,fAll(A)) ).

% top_filter_def
tff(fact_5668_inf__filter__def,axiom,
    ! [A: $tType,F4: filter(A),F8: filter(A)] : aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),inf_inf(filter(A)),F4),F8) = abs_filter(A,aa(filter(A),fun(fun(A,$o),$o),aTP_Lamp_akz(filter(A),fun(filter(A),fun(fun(A,$o),$o)),F4),F8)) ).

% inf_filter_def
tff(fact_5669_smax__insertI,axiom,
    ! [Y: product_prod(nat,nat),Y4: set(product_prod(nat,nat)),X: product_prod(nat,nat),X5: set(product_prod(nat,nat))] :
      ( aa(set(product_prod(nat,nat)),$o,member(product_prod(nat,nat),Y),Y4)
     => ( aa(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),$o,member(product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),X),Y)),fun_pair_less)
       => ( aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),$o,member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),X5),Y4)),fun_max_strict)
         => aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),$o,member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),set(product_prod(nat,nat)),aa(product_prod(nat,nat),fun(set(product_prod(nat,nat)),set(product_prod(nat,nat))),insert2(product_prod(nat,nat)),X),X5)),Y4)),fun_max_strict) ) ) ) ).

% smax_insertI
tff(fact_5670_filterlim__at__top__gt,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [F: fun(A,B),F4: filter(A),C2: B] :
          ( filterlim(A,B,F,at_top(B),F4)
        <=> ! [Z8: B] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),C2),Z8)
             => eventually(A,aa(B,fun(A,$o),aTP_Lamp_ala(fun(A,B),fun(B,fun(A,$o)),F),Z8),F4) ) ) ) ).

% filterlim_at_top_gt
tff(fact_5671_pair__leqI2,axiom,
    ! [A3: nat,B2: nat,S2: nat,T3: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A3),B2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),S2),T3)
       => aa(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),$o,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),T3))),fun_pair_leq) ) ) ).

% pair_leqI2
tff(fact_5672_eventually__sequentially__Suc,axiom,
    ! [P: fun(nat,$o)] :
      ( eventually(nat,aTP_Lamp_vf(fun(nat,$o),fun(nat,$o),P),at_top(nat))
    <=> eventually(nat,P,at_top(nat)) ) ).

% eventually_sequentially_Suc
tff(fact_5673_eventually__sequentially__seg,axiom,
    ! [P: fun(nat,$o),K: nat] :
      ( eventually(nat,aa(nat,fun(nat,$o),aTP_Lamp_alb(fun(nat,$o),fun(nat,fun(nat,$o)),P),K),at_top(nat))
    <=> eventually(nat,P,at_top(nat)) ) ).

% eventually_sequentially_seg
tff(fact_5674_trivial__limit__at__top__linorder,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( at_top(A) != bot_bot(filter(A)) ) ) ).

% trivial_limit_at_top_linorder
tff(fact_5675_filterlim__sequentially__Suc,axiom,
    ! [A: $tType,F: fun(nat,A),F4: filter(A)] :
      ( filterlim(nat,A,aTP_Lamp_abd(fun(nat,A),fun(nat,A),F),F4,at_top(nat))
    <=> filterlim(nat,A,F,F4,at_top(nat)) ) ).

% filterlim_sequentially_Suc
tff(fact_5676_le__sequentially,axiom,
    ! [F4: filter(nat)] :
      ( aa(filter(nat),$o,aa(filter(nat),fun(filter(nat),$o),ord_less_eq(filter(nat)),F4),at_top(nat))
    <=> ! [N9: nat] : eventually(nat,aa(nat,fun(nat,$o),ord_less_eq(nat),N9),F4) ) ).

% le_sequentially
tff(fact_5677_eventually__at__top__not__equal,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_top(A) )
     => ! [C2: A] : eventually(A,aTP_Lamp_alc(A,fun(A,$o),C2),at_top(A)) ) ).

% eventually_at_top_not_equal
tff(fact_5678_pair__leq__def,axiom,
    fun_pair_leq = aa(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),aa(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),fun(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),set(product_prod(product_prod(nat,nat),product_prod(nat,nat)))),sup_sup(set(product_prod(product_prod(nat,nat),product_prod(nat,nat)))),fun_pair_less),id2(product_prod(nat,nat))) ).

% pair_leq_def
tff(fact_5679_smax__emptyI,axiom,
    ! [Y4: set(product_prod(nat,nat))] :
      ( aa(set(product_prod(nat,nat)),$o,finite_finite2(product_prod(nat,nat)),Y4)
     => ( ( Y4 != bot_bot(set(product_prod(nat,nat))) )
       => aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),$o,member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),bot_bot(set(product_prod(nat,nat)))),Y4)),fun_max_strict) ) ) ).

% smax_emptyI
tff(fact_5680_filterlim__atMost__at__top,axiom,
    filterlim(nat,set(nat),set_ord_atMost(nat),finite5375528669736107172at_top(nat,top_top(set(nat))),at_top(nat)) ).

% filterlim_atMost_at_top
tff(fact_5681_filterlim__lessThan__at__top,axiom,
    filterlim(nat,set(nat),set_ord_lessThan(nat),finite5375528669736107172at_top(nat,top_top(set(nat))),at_top(nat)) ).

% filterlim_lessThan_at_top
tff(fact_5682_eventually__ge__at__top,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A] : eventually(A,aa(A,fun(A,$o),ord_less_eq(A),C2),at_top(A)) ) ).

% eventually_ge_at_top
tff(fact_5683_eventually__gt__at__top,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_top(A) )
     => ! [C2: A] : eventually(A,aa(A,fun(A,$o),ord_less(A),C2),at_top(A)) ) ).

% eventually_gt_at_top
tff(fact_5684_eventually__all__ge__at__top,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,$o)] :
          ( eventually(A,P,at_top(A))
         => eventually(A,aTP_Lamp_ald(fun(A,$o),fun(A,$o),P),at_top(A)) ) ) ).

% eventually_all_ge_at_top
tff(fact_5685_filterlim__at__top,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F: fun(A,B),F4: filter(A)] :
          ( filterlim(A,B,F,at_top(B),F4)
        <=> ! [Z8: B] : eventually(A,aa(B,fun(A,$o),aTP_Lamp_ale(fun(A,B),fun(B,fun(A,$o)),F),Z8),F4) ) ) ).

% filterlim_at_top
tff(fact_5686_filterlim__at__top__ge,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F: fun(A,B),F4: filter(A),C2: B] :
          ( filterlim(A,B,F,at_top(B),F4)
        <=> ! [Z8: B] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),C2),Z8)
             => eventually(A,aa(B,fun(A,$o),aTP_Lamp_ale(fun(A,B),fun(B,fun(A,$o)),F),Z8),F4) ) ) ) ).

% filterlim_at_top_ge
tff(fact_5687_filterlim__at__top__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F: fun(A,B),F4: filter(A),G: fun(A,B)] :
          ( filterlim(A,B,F,at_top(B),F4)
         => ( eventually(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_alf(fun(A,B),fun(fun(A,B),fun(A,$o)),F),G),F4)
           => filterlim(A,B,G,at_top(B),F4) ) ) ) ).

% filterlim_at_top_mono
tff(fact_5688_filterlim__at__top__dense,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [F: fun(A,B),F4: filter(A)] :
          ( filterlim(A,B,F,at_top(B),F4)
        <=> ! [Z8: B] : eventually(A,aa(B,fun(A,$o),aTP_Lamp_alg(fun(A,B),fun(B,fun(A,$o)),F),Z8),F4) ) ) ).

% filterlim_at_top_dense
tff(fact_5689_pair__leqI1,axiom,
    ! [A3: nat,B2: nat,S2: nat,T3: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A3),B2)
     => aa(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),$o,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),T3))),fun_pair_leq) ) ).

% pair_leqI1
tff(fact_5690_wmin__insertI,axiom,
    ! [X: product_prod(nat,nat),XS: set(product_prod(nat,nat)),Y: product_prod(nat,nat),YS: set(product_prod(nat,nat))] :
      ( aa(set(product_prod(nat,nat)),$o,member(product_prod(nat,nat),X),XS)
     => ( aa(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),$o,member(product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),X),Y)),fun_pair_leq)
       => ( aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),$o,member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),XS),YS)),fun_min_weak)
         => aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),$o,member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),XS),aa(set(product_prod(nat,nat)),set(product_prod(nat,nat)),aa(product_prod(nat,nat),fun(set(product_prod(nat,nat)),set(product_prod(nat,nat))),insert2(product_prod(nat,nat)),Y),YS))),fun_min_weak) ) ) ) ).

% wmin_insertI
tff(fact_5691_wmax__insertI,axiom,
    ! [Y: product_prod(nat,nat),YS: set(product_prod(nat,nat)),X: product_prod(nat,nat),XS: set(product_prod(nat,nat))] :
      ( aa(set(product_prod(nat,nat)),$o,member(product_prod(nat,nat),Y),YS)
     => ( aa(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),$o,member(product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),X),Y)),fun_pair_leq)
       => ( aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),$o,member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),XS),YS)),fun_max_weak)
         => aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),$o,member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),set(product_prod(nat,nat)),aa(product_prod(nat,nat),fun(set(product_prod(nat,nat)),set(product_prod(nat,nat))),insert2(product_prod(nat,nat)),X),XS)),YS)),fun_max_weak) ) ) ) ).

% wmax_insertI
tff(fact_5692_max__weak__def,axiom,
    fun_max_weak = aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),fun(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))))),sup_sup(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))))),max_ext(product_prod(nat,nat),fun_pair_leq)),aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),aa(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),fun(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))))),insert2(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),bot_bot(set(product_prod(nat,nat)))),bot_bot(set(product_prod(nat,nat))))),bot_bot(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))))))) ).

% max_weak_def
tff(fact_5693_trivial__limit__sequentially,axiom,
    at_top(nat) != bot_bot(filter(nat)) ).

% trivial_limit_sequentially
tff(fact_5694_eventually__False__sequentially,axiom,
    ~ eventually(nat,aTP_Lamp_ej(nat,$o),at_top(nat)) ).

% eventually_False_sequentially
tff(fact_5695_max__rpair__set,axiom,
    fun_reduction_pair(set(product_prod(nat,nat)),aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_prod(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))))),aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),fun(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_prod(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))))),product_Pair(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))))),fun_max_strict),fun_max_weak)) ).

% max_rpair_set
tff(fact_5696_min__rpair__set,axiom,
    fun_reduction_pair(set(product_prod(nat,nat)),aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_prod(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))))),aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),fun(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_prod(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))))),product_Pair(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))))),fun_min_strict),fun_min_weak)) ).

% min_rpair_set
tff(fact_5697_wmax__emptyI,axiom,
    ! [X5: set(product_prod(nat,nat))] :
      ( aa(set(product_prod(nat,nat)),$o,finite_finite2(product_prod(nat,nat)),X5)
     => aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),$o,member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),bot_bot(set(product_prod(nat,nat)))),X5)),fun_max_weak) ) ).

% wmax_emptyI
tff(fact_5698_wmin__emptyI,axiom,
    ! [X5: set(product_prod(nat,nat))] : aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),$o,member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),X5),bot_bot(set(product_prod(nat,nat))))),fun_min_weak) ).

% wmin_emptyI
tff(fact_5699_min__weak__def,axiom,
    fun_min_weak = aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),fun(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))))),sup_sup(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))))),min_ext(product_prod(nat,nat),fun_pair_leq)),aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),aa(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),fun(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))))),insert2(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),bot_bot(set(product_prod(nat,nat)))),bot_bot(set(product_prod(nat,nat))))),bot_bot(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))))))) ).

% min_weak_def
tff(fact_5700_at__top__def,axiom,
    ! [A: $tType] :
      ( order(A)
     => ( at_top(A) = aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(A),set(filter(A)),image2(A,filter(A),aTP_Lamp_alh(A,filter(A))),top_top(set(A)))) ) ) ).

% at_top_def
tff(fact_5701_filtercomap__def,axiom,
    ! [A: $tType,B: $tType,F: fun(A,B),F4: filter(B)] : filtercomap(A,B,F,F4) = abs_filter(A,aa(filter(B),fun(fun(A,$o),$o),aTP_Lamp_ali(fun(A,B),fun(filter(B),fun(fun(A,$o),$o)),F),F4)) ).

% filtercomap_def
tff(fact_5702_at__top__sub,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A] : at_top(A) = aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(A),set(filter(A)),image2(A,filter(A),aTP_Lamp_alj(A,filter(A))),aa(A,set(A),set_ord_atLeast(A),C2))) ) ).

% at_top_sub
tff(fact_5703_atLeast__0,axiom,
    aa(nat,set(nat),set_ord_atLeast(nat),zero_zero(nat)) = top_top(set(nat)) ).

% atLeast_0
tff(fact_5704_filtercomap__bot,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B)] : filtercomap(A,B,F,bot_bot(filter(B))) = bot_bot(filter(A)) ).

% filtercomap_bot
tff(fact_5705_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_5706_eventually__filtercomapI,axiom,
    ! [B: $tType,A: $tType,P: fun(A,$o),F4: filter(A),F: fun(B,A)] :
      ( eventually(A,P,F4)
     => eventually(B,aa(fun(B,A),fun(B,$o),aTP_Lamp_akb(fun(A,$o),fun(fun(B,A),fun(B,$o)),P),F),filtercomap(B,A,F,F4)) ) ).

% eventually_filtercomapI
tff(fact_5707_filtercomap__top,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B)] : filtercomap(A,B,F,top_top(filter(B))) = top_top(filter(A)) ).

% filtercomap_top
tff(fact_5708_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_5709_Compl__lessThan,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [K: A] : aa(set(A),set(A),uminus_uminus(set(A)),aa(A,set(A),set_ord_lessThan(A),K)) = aa(A,set(A),set_ord_atLeast(A),K) ) ).

% Compl_lessThan
tff(fact_5710_Compl__atLeast,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [K: A] : aa(set(A),set(A),uminus_uminus(set(A)),aa(A,set(A),set_ord_atLeast(A),K)) = aa(A,set(A),set_ord_lessThan(A),K) ) ).

% Compl_atLeast
tff(fact_5711_image__uminus__atLeast,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [X: A] : aa(set(A),set(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_5712_image__uminus__atMost,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [X: A] : aa(set(A),set(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_5713_Int__atLeastAtMostL2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A,C2: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or1337092689740270186AtMost(A,A3,B2)),aa(A,set(A),set_ord_atLeast(A),C2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),ord_max(A),A3),C2),B2) ) ).

% Int_atLeastAtMostL2
tff(fact_5714_Int__atLeastAtMostR2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,C2: 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,C2,D3)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),ord_max(A),A3),C2),D3) ) ).

% Int_atLeastAtMostR2
tff(fact_5715_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_5716_atLeast__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [L: A] : aa(A,set(A),set_ord_atLeast(A),L) = aa(fun(A,$o),set(A),collect(A),aa(A,fun(A,$o),ord_less_eq(A),L)) ) ).

% atLeast_def
tff(fact_5717_filtercomap__inf,axiom,
    ! [A: $tType,B: $tType,F: fun(A,B),F12: filter(B),F23: filter(B)] : filtercomap(A,B,F,aa(filter(B),filter(B),aa(filter(B),fun(filter(B),filter(B)),inf_inf(filter(B)),F12),F23)) = aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),inf_inf(filter(A)),filtercomap(A,B,F,F12)),filtercomap(A,B,F,F23)) ).

% filtercomap_inf
tff(fact_5718_filtercomap__ident,axiom,
    ! [A: $tType,F4: filter(A)] : filtercomap(A,A,aTP_Lamp_au(A,A),F4) = F4 ).

% filtercomap_ident
tff(fact_5719_filtercomap__filtercomap,axiom,
    ! [B: $tType,A: $tType,C: $tType,F: fun(A,B),G: fun(B,C),F4: filter(C)] : filtercomap(A,B,F,filtercomap(B,C,G,F4)) = filtercomap(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_alk(fun(A,B),fun(fun(B,C),fun(A,C)),F),G),F4) ).

% filtercomap_filtercomap
tff(fact_5720_not__UNIV__eq__Ici,axiom,
    ! [A: $tType] :
      ( no_bot(A)
     => ! [L3: A] : top_top(set(A)) != aa(A,set(A),set_ord_atLeast(A),L3) ) ).

% not_UNIV_eq_Ici
tff(fact_5721_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_5722_not__UNIV__le__Ici,axiom,
    ! [A: $tType] :
      ( no_bot(A)
     => ! [L: A] : ~ aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),top_top(set(A))),aa(A,set(A),set_ord_atLeast(A),L)) ) ).

% not_UNIV_le_Ici
tff(fact_5723_filtercomap__neq__bot,axiom,
    ! [A: $tType,B: $tType,F4: filter(A),F: fun(B,A)] :
      ( ! [P2: fun(A,$o)] :
          ( eventually(A,P2,F4)
         => ? [X3: B] : aa(A,$o,P2,aa(B,A,F,X3)) )
     => ( filtercomap(B,A,F,F4) != bot_bot(filter(B)) ) ) ).

% filtercomap_neq_bot
tff(fact_5724_filtercomap__sup,axiom,
    ! [A: $tType,B: $tType,F: fun(A,B),F12: filter(B),F23: filter(B)] : aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),sup_sup(filter(A)),filtercomap(A,B,F,F12)),filtercomap(A,B,F,F23))),filtercomap(A,B,F,aa(filter(B),filter(B),aa(filter(B),fun(filter(B),filter(B)),sup_sup(filter(B)),F12),F23))) ).

% filtercomap_sup
tff(fact_5725_filtercomap__INF,axiom,
    ! [B: $tType,A: $tType,C: $tType,F: fun(A,B),F4: fun(C,filter(B)),B3: set(C)] : filtercomap(A,B,F,aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(C),set(filter(B)),image2(C,filter(B),F4),B3))) = aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(C),set(filter(A)),image2(C,filter(A),aa(fun(C,filter(B)),fun(C,filter(A)),aTP_Lamp_all(fun(A,B),fun(fun(C,filter(B)),fun(C,filter(A))),F),F4)),B3)) ).

% filtercomap_INF
tff(fact_5726_ivl__disj__un__one_I8_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),U)
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or7035219750837199246ssThan(A,L,U)),aa(A,set(A),set_ord_atLeast(A),U)) = aa(A,set(A),set_ord_atLeast(A),L) ) ) ) ).

% ivl_disj_un_one(8)
tff(fact_5727_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_5728_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_5729_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_5730_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_5731_filtercomap__neq__bot__surj,axiom,
    ! [A: $tType,B: $tType,F4: filter(A),F: fun(B,A)] :
      ( ( F4 != bot_bot(filter(A)) )
     => ( ( aa(set(B),set(A),image2(B,A,F),top_top(set(B))) = top_top(set(A)) )
       => ( filtercomap(B,A,F,F4) != bot_bot(filter(B)) ) ) ) ).

% filtercomap_neq_bot_surj
tff(fact_5732_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),aa(A,fun(set(A),set(A)),insert2(A),N),bot_bot(set(A))) ) ).

% atMost_Int_atLeast
tff(fact_5733_ivl__disj__un__singleton_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),L),bot_bot(set(A)))),aa(A,set(A),set_ord_greaterThan(A),L)) = aa(A,set(A),set_ord_atLeast(A),L) ) ).

% ivl_disj_un_singleton(1)
tff(fact_5734_ivl__disj__un__one_I7_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),U)
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or1337092689740270186AtMost(A,L,U)),aa(A,set(A),set_ord_greaterThan(A),U)) = aa(A,set(A),set_ord_atLeast(A),L) ) ) ) ).

% ivl_disj_un_one(7)
tff(fact_5735_filtercomap__SUP,axiom,
    ! [A: $tType,C: $tType,B: $tType,F: fun(A,C),F4: fun(B,filter(C)),B3: set(B)] : aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),aa(set(filter(A)),filter(A),complete_Sup_Sup(filter(A)),aa(set(B),set(filter(A)),image2(B,filter(A),aa(fun(B,filter(C)),fun(B,filter(A)),aTP_Lamp_alm(fun(A,C),fun(fun(B,filter(C)),fun(B,filter(A))),F),F4)),B3))),filtercomap(A,C,F,aa(set(filter(C)),filter(C),complete_Sup_Sup(filter(C)),aa(set(B),set(filter(C)),image2(B,filter(C),F4),B3)))) ).

% filtercomap_SUP
tff(fact_5736_ivl__disj__un__one_I6_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),L),U)
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or5935395276787703475ssThan(A,L,U)),aa(A,set(A),set_ord_atLeast(A),U)) = aa(A,set(A),set_ord_greaterThan(A),L) ) ) ) ).

% ivl_disj_un_one(6)
tff(fact_5737_atLeast__Suc,axiom,
    ! [K: nat] : aa(nat,set(nat),set_ord_atLeast(nat),aa(nat,nat,suc,K)) = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),aa(nat,set(nat),set_ord_atLeast(nat),K)),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),K),bot_bot(set(nat)))) ).

% atLeast_Suc
tff(fact_5738_UN__atLeast__UNIV,axiom,
    aa(set(set(nat)),set(nat),complete_Sup_Sup(set(nat)),aa(set(nat),set(set(nat)),image2(nat,set(nat),set_ord_atLeast(nat)),top_top(set(nat)))) = top_top(set(nat)) ).

% UN_atLeast_UNIV
tff(fact_5739_interval__cases,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [S: set(A)] :
          ( ! [A6: A,B5: A,X2: A] :
              ( aa(set(A),$o,member(A,A6),S)
             => ( aa(set(A),$o,member(A,B5),S)
               => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A6),X2)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),B5)
                   => aa(set(A),$o,member(A,X2),S) ) ) ) )
         => ? [A6: A,B5: A] :
              ( ( S = bot_bot(set(A)) )
              | ( S = top_top(set(A)) )
              | ( S = aa(A,set(A),set_ord_lessThan(A),B5) )
              | ( S = aa(A,set(A),set_ord_atMost(A),B5) )
              | ( S = aa(A,set(A),set_ord_greaterThan(A),A6) )
              | ( S = aa(A,set(A),set_ord_atLeast(A),A6) )
              | ( S = set_or5935395276787703475ssThan(A,A6,B5) )
              | ( S = set_or3652927894154168847AtMost(A,A6,B5) )
              | ( S = set_or7035219750837199246ssThan(A,A6,B5) )
              | ( S = set_or1337092689740270186AtMost(A,A6,B5) ) ) ) ) ).

% interval_cases
tff(fact_5740_euclidean__size__times__nonunit,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [A3: A,B2: A] :
          ( ( A3 != zero_zero(A) )
         => ( ( B2 != zero_zero(A) )
           => ( ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A3),one_one(A))
             => aa(nat,$o,aa(nat,fun(nat,$o),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_5741_coinduct3,axiom,
    ! [A: $tType,F: fun(set(A),set(A)),A3: A,X5: set(A)] :
      ( order_mono(set(A),set(A),F)
     => ( aa(set(A),$o,member(A,A3),X5)
       => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X5),aa(set(A),set(A),F,complete_lattice_lfp(set(A),aa(set(A),fun(set(A),set(A)),aTP_Lamp_aln(fun(set(A),set(A)),fun(set(A),fun(set(A),set(A))),F),X5))))
         => aa(set(A),$o,member(A,A3),complete_lattice_gfp(set(A),F)) ) ) ) ).

% coinduct3
tff(fact_5742_greaterThanAtMost__empty,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,K: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),K)
         => ( set_or3652927894154168847AtMost(A,K,L) = bot_bot(set(A)) ) ) ) ).

% greaterThanAtMost_empty
tff(fact_5743_greaterThanAtMost__empty__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [K: A,L: A] :
          ( ( set_or3652927894154168847AtMost(A,K,L) = bot_bot(set(A)) )
        <=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),K),L) ) ) ).

% greaterThanAtMost_empty_iff
tff(fact_5744_greaterThanAtMost__empty__iff2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [K: A,L: A] :
          ( ( bot_bot(set(A)) = set_or3652927894154168847AtMost(A,K,L) )
        <=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),K),L) ) ) ).

% greaterThanAtMost_empty_iff2
tff(fact_5745_euclidean__size__1,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ( euclid6346220572633701492n_size(A,one_one(A)) = one_one(nat) ) ) ).

% euclidean_size_1
tff(fact_5746_image__uminus__atLeastLessThan,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [X: A,Y: A] : aa(set(A),set(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_5747_image__uminus__greaterThanAtMost,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [X: A,Y: A] : aa(set(A),set(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_5748_Int__greaterThanAtMost,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A,C2: A,D3: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or3652927894154168847AtMost(A,A3,B2)),set_or3652927894154168847AtMost(A,C2,D3)) = set_or3652927894154168847AtMost(A,aa(A,A,aa(A,fun(A,A),ord_max(A),A3),C2),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),D3)) ) ).

% Int_greaterThanAtMost
tff(fact_5749_gfp__rolling,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comple6319245703460814977attice(B)
        & comple6319245703460814977attice(A) )
     => ! [G: fun(A,B),F: fun(B,A)] :
          ( order_mono(A,B,G)
         => ( order_mono(B,A,F)
           => ( aa(A,B,G,complete_lattice_gfp(A,aa(fun(B,A),fun(A,A),aTP_Lamp_aho(fun(A,B),fun(fun(B,A),fun(A,A)),G),F))) = complete_lattice_gfp(B,aa(fun(B,A),fun(B,B),aTP_Lamp_ahp(fun(A,B),fun(fun(B,A),fun(B,B)),G),F)) ) ) ) ) ).

% gfp_rolling
tff(fact_5750_gfp__const,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [T3: A] : complete_lattice_gfp(A,aTP_Lamp_ahn(A,fun(A,A),T3)) = T3 ) ).

% gfp_const
tff(fact_5751_gfp__gfp,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F: fun(A,fun(A,A))] :
          ( ! [X2: A,Y2: A,W: A,Z3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),Y2)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),W),Z3)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),F,X2),W)),aa(A,A,aa(A,fun(A,A),F,Y2),Z3)) ) )
         => ( complete_lattice_gfp(A,aTP_Lamp_alo(fun(A,fun(A,A)),fun(A,A),F)) = complete_lattice_gfp(A,aTP_Lamp_ahm(fun(A,fun(A,A)),fun(A,A),F)) ) ) ) ).

% gfp_gfp
tff(fact_5752_ivl__disj__un__two_I6_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),M)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),M),U)
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or3652927894154168847AtMost(A,L,M)),set_or3652927894154168847AtMost(A,M,U)) = set_or3652927894154168847AtMost(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(6)
tff(fact_5753_ivl__disj__int__two_I6_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,M: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or3652927894154168847AtMost(A,L,M)),set_or3652927894154168847AtMost(A,M,U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_two(6)
tff(fact_5754_euclidean__size__unit,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A3),one_one(A))
         => ( euclid6346220572633701492n_size(A,A3) = euclid6346220572633701492n_size(A,one_one(A)) ) ) ) ).

% euclidean_size_unit
tff(fact_5755_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_5756_gfp__fun__UnI2,axiom,
    ! [A: $tType,F: fun(set(A),set(A)),A3: A,X5: set(A)] :
      ( order_mono(set(A),set(A),F)
     => ( aa(set(A),$o,member(A,A3),complete_lattice_gfp(set(A),F))
       => aa(set(A),$o,member(A,A3),aa(set(A),set(A),F,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),X5),complete_lattice_gfp(set(A),F)))) ) ) ).

% gfp_fun_UnI2
tff(fact_5757_gfp__def,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F: fun(A,A)] : complete_lattice_gfp(A,F) = aa(set(A),A,complete_Sup_Sup(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_alp(fun(A,A),fun(A,$o),F))) ) ).

% gfp_def
tff(fact_5758_def__Collect__coinduct,axiom,
    ! [A: $tType,A4: set(A),P: fun(set(A),fun(A,$o)),A3: A,X5: set(A)] :
      ( ( A4 = complete_lattice_gfp(set(A),aTP_Lamp_alq(fun(set(A),fun(A,$o)),fun(set(A),set(A)),P)) )
     => ( order_mono(set(A),set(A),aTP_Lamp_alq(fun(set(A),fun(A,$o)),fun(set(A),set(A)),P))
       => ( aa(set(A),$o,member(A,A3),X5)
         => ( ! [Z3: A] :
                ( aa(set(A),$o,member(A,Z3),X5)
               => aa(A,$o,aa(set(A),fun(A,$o),P,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),X5),A4)),Z3) )
           => aa(set(A),$o,member(A,A3),A4) ) ) ) ) ).

% def_Collect_coinduct
tff(fact_5759_Ioc__disjoint,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A,C2: A,D3: A] :
          ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or3652927894154168847AtMost(A,A3,B2)),set_or3652927894154168847AtMost(A,C2,D3)) = bot_bot(set(A)) )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3)
            | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),D3),C2)
            | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),C2)
            | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),D3),A3) ) ) ) ).

% Ioc_disjoint
tff(fact_5760_ivl__disj__un__two_I8_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),M)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),M),U)
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or1337092689740270186AtMost(A,L,M)),set_or3652927894154168847AtMost(A,M,U)) = set_or1337092689740270186AtMost(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(8)
tff(fact_5761_ivl__disj__int__two_I8_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,M: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or1337092689740270186AtMost(A,L,M)),set_or3652927894154168847AtMost(A,M,U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_two(8)
tff(fact_5762_ivl__disj__un__one_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),U)
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(A,set(A),set_ord_atMost(A),L)),set_or3652927894154168847AtMost(A,L,U)) = aa(A,set(A),set_ord_atMost(A),U) ) ) ) ).

% ivl_disj_un_one(3)
tff(fact_5763_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_5764_unit__iff__euclidean__size,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),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_5765_size__mult__mono_H,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [B2: A,A3: A] :
          ( ( B2 != zero_zero(A) )
         => aa(nat,$o,aa(nat,fun(nat,$o),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_5766_size__mult__mono,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [B2: A,A3: A] :
          ( ( B2 != zero_zero(A) )
         => aa(nat,$o,aa(nat,fun(nat,$o),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_5767_euclidean__size__times__unit,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),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_5768_ivl__disj__int__two_I2_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,M: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or3652927894154168847AtMost(A,L,M)),set_or5935395276787703475ssThan(A,M,U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_two(2)
tff(fact_5769_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_5770_ivl__disj__un__one_I5_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),U)
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or3652927894154168847AtMost(A,L,U)),aa(A,set(A),set_ord_greaterThan(A),U)) = aa(A,set(A),set_ord_greaterThan(A),L) ) ) ) ).

% ivl_disj_un_one(5)
tff(fact_5771_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_5772_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_5773_coinduct__set,axiom,
    ! [A: $tType,F: fun(set(A),set(A)),A3: A,X5: set(A)] :
      ( order_mono(set(A),set(A),F)
     => ( aa(set(A),$o,member(A,A3),X5)
       => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X5),aa(set(A),set(A),F,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),X5),complete_lattice_gfp(set(A),F))))
         => aa(set(A),$o,member(A,A3),complete_lattice_gfp(set(A),F)) ) ) ) ).

% coinduct_set
tff(fact_5774_def__coinduct__set,axiom,
    ! [A: $tType,A4: set(A),F: fun(set(A),set(A)),A3: A,X5: set(A)] :
      ( ( A4 = complete_lattice_gfp(set(A),F) )
     => ( order_mono(set(A),set(A),F)
       => ( aa(set(A),$o,member(A,A3),X5)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X5),aa(set(A),set(A),F,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),X5),A4)))
           => aa(set(A),$o,member(A,A3),A4) ) ) ) ) ).

% def_coinduct_set
tff(fact_5775_coinduct__lemma,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [X5: A,F: fun(A,A)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X5),aa(A,A,F,aa(A,A,aa(A,fun(A,A),sup_sup(A),X5),complete_lattice_gfp(A,F))))
         => ( order_mono(A,A,F)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),X5),complete_lattice_gfp(A,F))),aa(A,A,F,aa(A,A,aa(A,fun(A,A),sup_sup(A),X5),complete_lattice_gfp(A,F)))) ) ) ) ).

% coinduct_lemma
tff(fact_5776_def__coinduct,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A4: A,F: fun(A,A),X5: A] :
          ( ( A4 = complete_lattice_gfp(A,F) )
         => ( order_mono(A,A,F)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X5),aa(A,A,F,aa(A,A,aa(A,fun(A,A),sup_sup(A),X5),A4)))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X5),A4) ) ) ) ) ).

% def_coinduct
tff(fact_5777_coinduct,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F: fun(A,A),X5: A] :
          ( order_mono(A,A,F)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X5),aa(A,A,F,aa(A,A,aa(A,fun(A,A),sup_sup(A),X5),complete_lattice_gfp(A,F))))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X5),complete_lattice_gfp(A,F)) ) ) ) ).

% coinduct
tff(fact_5778_prod_Ohead,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,N: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),N)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,M)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or3652927894154168847AtMost(nat,M,N))) ) ) ) ).

% prod.head
tff(fact_5779_ivl__disj__un__two__touch_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),L),M)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),M),U)
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or3652927894154168847AtMost(A,L,M)),set_or1337092689740270186AtMost(A,M,U)) = set_or3652927894154168847AtMost(A,L,U) ) ) ) ) ).

% ivl_disj_un_two_touch(3)
tff(fact_5780_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),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A)))) ) ).

% greaterThanAtMost_eq_atLeastAtMost_diff
tff(fact_5781_ivl__disj__un__two_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),M)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),M),U)
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or3652927894154168847AtMost(A,L,M)),set_or5935395276787703475ssThan(A,M,U)) = set_or5935395276787703475ssThan(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(2)
tff(fact_5782_ivl__disj__un__two__touch_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),L),M)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),M),U)
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or3652927894154168847AtMost(A,L,M)),set_or7035219750837199246ssThan(A,M,U)) = set_or5935395276787703475ssThan(A,L,U) ) ) ) ) ).

% ivl_disj_un_two_touch(1)
tff(fact_5783_atLeastPlusOneAtMost__greaterThanAtMost__integer,axiom,
    ! [L: code_integer,U: code_integer] : set_or1337092689740270186AtMost(code_integer,aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),L),one_one(code_integer)),U) = set_or3652927894154168847AtMost(code_integer,L,U) ).

% atLeastPlusOneAtMost_greaterThanAtMost_integer
tff(fact_5784_greaterThanAtMost__upto,axiom,
    ! [I: int,J: int] : set_or3652927894154168847AtMost(int,I,J) = aa(list(int),set(int),set2(int),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I),one_one(int)),J)) ).

% greaterThanAtMost_upto
tff(fact_5785_gfp__Kleene__iter,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F: fun(A,A),K: nat] :
          ( order_mono(A,A,F)
         => ( ( aa(A,A,compow(fun(A,A),aa(nat,nat,suc,K),F),top_top(A)) = aa(A,A,compow(fun(A,A),K,F),top_top(A)) )
           => ( complete_lattice_gfp(A,F) = aa(A,A,compow(fun(A,A),K,F),top_top(A)) ) ) ) ) ).

% gfp_Kleene_iter
tff(fact_5786_ivl__disj__un__singleton_I5_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),U)
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),L),bot_bot(set(A)))),set_or3652927894154168847AtMost(A,L,U)) = set_or1337092689740270186AtMost(A,L,U) ) ) ) ).

% ivl_disj_un_singleton(5)
tff(fact_5787_ivl__disj__un__singleton_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),L),U)
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or5935395276787703475ssThan(A,L,U)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),U),bot_bot(set(A)))) = set_or3652927894154168847AtMost(A,L,U) ) ) ) ).

% ivl_disj_un_singleton(4)
tff(fact_5788_ivl__disj__un__two_I5_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),L),M)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),M),U)
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or5935395276787703475ssThan(A,L,M)),set_or1337092689740270186AtMost(A,M,U)) = set_or3652927894154168847AtMost(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(5)
tff(fact_5789_coinduct3__lemma,axiom,
    ! [A: $tType,X5: set(A),F: fun(set(A),set(A))] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X5),aa(set(A),set(A),F,complete_lattice_lfp(set(A),aa(fun(set(A),set(A)),fun(set(A),set(A)),aTP_Lamp_alr(set(A),fun(fun(set(A),set(A)),fun(set(A),set(A))),X5),F))))
     => ( order_mono(set(A),set(A),F)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),complete_lattice_lfp(set(A),aa(fun(set(A),set(A)),fun(set(A),set(A)),aTP_Lamp_alr(set(A),fun(fun(set(A),set(A)),fun(set(A),set(A))),X5),F))),aa(set(A),set(A),F,complete_lattice_lfp(set(A),aa(fun(set(A),set(A)),fun(set(A),set(A)),aTP_Lamp_alr(set(A),fun(fun(set(A),set(A)),fun(set(A),set(A))),X5),F)))) ) ) ).

% coinduct3_lemma
tff(fact_5790_def__coinduct3,axiom,
    ! [A: $tType,A4: set(A),F: fun(set(A),set(A)),A3: A,X5: set(A)] :
      ( ( A4 = complete_lattice_gfp(set(A),F) )
     => ( order_mono(set(A),set(A),F)
       => ( aa(set(A),$o,member(A,A3),X5)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X5),aa(set(A),set(A),F,complete_lattice_lfp(set(A),aa(set(A),fun(set(A),set(A)),aa(fun(set(A),set(A)),fun(set(A),fun(set(A),set(A))),aTP_Lamp_als(set(A),fun(fun(set(A),set(A)),fun(set(A),fun(set(A),set(A)))),A4),F),X5))))
           => aa(set(A),$o,member(A,A3),A4) ) ) ) ) ).

% def_coinduct3
tff(fact_5791_divmod__cases,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [B2: A,A3: A] :
          ( ( ( B2 != zero_zero(A) )
           => ( ( modulo_modulo(A,A3,B2) = zero_zero(A) )
             => ( A3 != aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),B2) ) ) )
         => ( ( ( B2 != zero_zero(A) )
             => ! [Q5: A,R4: A] :
                  ( ( euclid7384307370059645450egment(A,R4) = euclid7384307370059645450egment(A,B2) )
                 => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),euclid6346220572633701492n_size(A,R4)),euclid6346220572633701492n_size(A,B2))
                   => ( ( R4 != zero_zero(A) )
                     => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) = Q5 )
                       => ( ( modulo_modulo(A,A3,B2) = R4 )
                         => ( A3 != aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Q5),B2)),R4) ) ) ) ) ) ) )
           => ( B2 = zero_zero(A) ) ) ) ) ).

% divmod_cases
tff(fact_5792_mod__eqI,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [B2: A,R2: A,Q3: A,A3: A] :
          ( ( B2 != zero_zero(A) )
         => ( ( euclid7384307370059645450egment(A,R2) = euclid7384307370059645450egment(A,B2) )
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),euclid6346220572633701492n_size(A,R2)),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),Q3),B2)),R2) = A3 )
               => ( modulo_modulo(A,A3,B2) = R2 ) ) ) ) ) ) ).

% mod_eqI
tff(fact_5793_div__eqI,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [B2: A,R2: A,Q3: A,A3: A] :
          ( ( B2 != zero_zero(A) )
         => ( ( euclid7384307370059645450egment(A,R2) = euclid7384307370059645450egment(A,B2) )
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),euclid6346220572633701492n_size(A,R2)),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),Q3),B2)),R2) = A3 )
               => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) = Q3 ) ) ) ) ) ) ).

% div_eqI
tff(fact_5794_abs__division__segment,axiom,
    ! [K: int] : aa(int,int,abs_abs(int),euclid7384307370059645450egment(int,K)) = one_one(int) ).

% abs_division_segment
tff(fact_5795_division__segment__1,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ( euclid7384307370059645450egment(A,one_one(A)) = one_one(A) ) ) ).

% division_segment_1
tff(fact_5796_division__segment__numeral,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [K: num] : euclid7384307370059645450egment(A,aa(num,A,numeral_numeral(A),K)) = one_one(A) ) ).

% division_segment_numeral
tff(fact_5797_division__segment__of__nat,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [N: nat] : euclid7384307370059645450egment(A,aa(nat,A,semiring_1_of_nat(A),N)) = one_one(A) ) ).

% division_segment_of_nat
tff(fact_5798_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_5799_division__segment__nat__def,axiom,
    ! [N: nat] : euclid7384307370059645450egment(nat,N) = one_one(nat) ).

% division_segment_nat_def
tff(fact_5800_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_5801_is__unit__division__segment,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [A3: A] : aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),euclid7384307370059645450egment(A,A3)),one_one(A)) ) ).

% is_unit_division_segment
tff(fact_5802_division__segment__int__def,axiom,
    ! [K: int] :
      euclid7384307370059645450egment(int,K) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K),one_one(int),aa(int,int,uminus_uminus(int),one_one(int))) ).

% division_segment_int_def
tff(fact_5803_div__bounded,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [B2: A,R2: A,Q3: A] :
          ( ( B2 != zero_zero(A) )
         => ( ( euclid7384307370059645450egment(A,R2) = euclid7384307370059645450egment(A,B2) )
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),euclid6346220572633701492n_size(A,R2)),euclid6346220572633701492n_size(A,B2))
             => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Q3),B2)),R2)),B2) = Q3 ) ) ) ) ) ).

% div_bounded
tff(fact_5804_surj__swap,axiom,
    ! [B: $tType,A: $tType] : aa(set(product_prod(B,A)),set(product_prod(A,B)),image2(product_prod(B,A),product_prod(A,B),product_swap(B,A)),top_top(set(product_prod(B,A)))) = top_top(set(product_prod(A,B))) ).

% surj_swap
tff(fact_5805_AboveS__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A4: set(A)] : order_AboveS(A,R2,A4) = aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_alt(set(product_prod(A,A)),fun(set(A),fun(A,$o)),R2),A4)) ).

% AboveS_def
tff(fact_5806_total__inv__image,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),R2: set(product_prod(B,B))] :
      ( inj_on(A,B,F,top_top(set(A)))
     => ( total_on(B,top_top(set(B)),R2)
       => total_on(A,top_top(set(A)),inv_image(B,A,R2,F)) ) ) ).

% total_inv_image
tff(fact_5807_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_5808_in__inv__image,axiom,
    ! [A: $tType,B: $tType,X: A,Y: A,R2: set(product_prod(B,B)),F: fun(A,B)] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),inv_image(B,A,R2,F))
    <=> aa(set(product_prod(B,B)),$o,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,F,X)),aa(A,B,F,Y))),R2) ) ).

% in_inv_image
tff(fact_5809_case__swap,axiom,
    ! [A: $tType,B: $tType,C: $tType,F: fun(C,fun(B,A)),P3: 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_alu(fun(C,fun(B,A)),fun(B,fun(C,A)),F)),aa(product_prod(C,B),product_prod(B,C),product_swap(C,B),P3)) = aa(product_prod(C,B),A,aa(fun(C,fun(B,A)),fun(product_prod(C,B),A),product_case_prod(C,B,A),F),P3) ).

% case_swap
tff(fact_5810_pair__in__swap__image,axiom,
    ! [A: $tType,B: $tType,Y: A,X: B,A4: set(product_prod(B,A))] :
      ( aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Y),X)),aa(set(product_prod(B,A)),set(product_prod(A,B)),image2(product_prod(B,A),product_prod(A,B),product_swap(B,A)),A4))
    <=> aa(set(product_prod(B,A)),$o,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)),A4) ) ).

% pair_in_swap_image
tff(fact_5811_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_5812_AboveS__disjoint,axiom,
    ! [A: $tType,A4: set(A),R2: set(product_prod(A,A))] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),order_AboveS(A,R2,A4)) = bot_bot(set(A)) ).

% AboveS_disjoint
tff(fact_5813_inv__image__def,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(B,B)),F: fun(A,B)] : inv_image(B,A,R2,F) = aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),aa(fun(A,B),fun(A,fun(A,$o)),aTP_Lamp_alv(set(product_prod(B,B)),fun(fun(A,B),fun(A,fun(A,$o))),R2),F))) ).

% inv_image_def
tff(fact_5814_product__swap,axiom,
    ! [A: $tType,B: $tType,A4: set(B),B3: set(A)] : aa(set(product_prod(B,A)),set(product_prod(A,B)),image2(product_prod(B,A),product_prod(A,B),product_swap(B,A)),product_Sigma(B,A,A4,aTP_Lamp_ke(set(A),fun(B,set(A)),B3))) = product_Sigma(A,B,B3,aTP_Lamp_qz(set(B),fun(A,set(B)),A4)) ).

% product_swap
tff(fact_5815_prod_Oswap__def,axiom,
    ! [A: $tType,B: $tType,P3: product_prod(B,A)] : aa(product_prod(B,A),product_prod(A,B),product_swap(B,A),P3) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(product_prod(B,A),A,product_snd(B,A),P3)),aa(product_prod(B,A),B,product_fst(B,A),P3)) ).

% prod.swap_def
tff(fact_5816_rp__inv__image__def,axiom,
    ! [B: $tType,A: $tType] : fun_rp_inv_image(A,B) = aa(fun(set(product_prod(A,A)),fun(set(product_prod(A,A)),fun(fun(B,A),product_prod(set(product_prod(B,B)),set(product_prod(B,B)))))),fun(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),fun(fun(B,A),product_prod(set(product_prod(B,B)),set(product_prod(B,B))))),product_case_prod(set(product_prod(A,A)),set(product_prod(A,A)),fun(fun(B,A),product_prod(set(product_prod(B,B)),set(product_prod(B,B))))),aTP_Lamp_alw(set(product_prod(A,A)),fun(set(product_prod(A,A)),fun(fun(B,A),product_prod(set(product_prod(B,B)),set(product_prod(B,B))))))) ).

% rp_inv_image_def
tff(fact_5817_wo__rel_Osuc__greater,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),B3: set(A),B2: A] :
      ( bNF_Wellorder_wo_rel(A,R2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),field2(A,R2))
       => ( ( order_AboveS(A,R2,B3) != bot_bot(set(A)) )
         => ( aa(set(A),$o,member(A,B2),B3)
           => ( ( bNF_Wellorder_wo_suc(A,R2,B3) != B2 )
              & aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),bNF_Wellorder_wo_suc(A,R2,B3))),R2) ) ) ) ) ) ).

% wo_rel.suc_greater
tff(fact_5818_wo__rel_Osuc__inField,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),B3: set(A)] :
      ( bNF_Wellorder_wo_rel(A,R2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),field2(A,R2))
       => ( ( order_AboveS(A,R2,B3) != bot_bot(set(A)) )
         => aa(set(A),$o,member(A,bNF_Wellorder_wo_suc(A,R2,B3)),field2(A,R2)) ) ) ) ).

% wo_rel.suc_inField
tff(fact_5819_wo__rel_Osuc__least__AboveS,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A3: A,B3: set(A)] :
      ( bNF_Wellorder_wo_rel(A,R2)
     => ( aa(set(A),$o,member(A,A3),order_AboveS(A,R2,B3))
       => aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),bNF_Wellorder_wo_suc(A,R2,B3)),A3)),R2) ) ) ).

% wo_rel.suc_least_AboveS
tff(fact_5820_wo__rel_Oequals__suc__AboveS,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),B3: set(A),A3: A] :
      ( bNF_Wellorder_wo_rel(A,R2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),field2(A,R2))
       => ( aa(set(A),$o,member(A,A3),order_AboveS(A,R2,B3))
         => ( ! [A13: A] :
                ( aa(set(A),$o,member(A,A13),order_AboveS(A,R2,B3))
               => aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),A13)),R2) )
           => ( A3 = bNF_Wellorder_wo_suc(A,R2,B3) ) ) ) ) ) ).

% wo_rel.equals_suc_AboveS
tff(fact_5821_wo__rel_Osuc__AboveS,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),B3: set(A)] :
      ( bNF_Wellorder_wo_rel(A,R2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),field2(A,R2))
       => ( ( order_AboveS(A,R2,B3) != bot_bot(set(A)) )
         => aa(set(A),$o,member(A,bNF_Wellorder_wo_suc(A,R2,B3)),order_AboveS(A,R2,B3)) ) ) ) ).

% wo_rel.suc_AboveS
tff(fact_5822_lenlex__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : lenlex(A,R2) = inv_image(product_prod(nat,list(A)),list(A),lex_prod(nat,list(A),less_than,lex(A,R2)),aTP_Lamp_alx(list(A),product_prod(nat,list(A)))) ).

% lenlex_def
tff(fact_5823_wo__rel_Osuc__ofilter__in,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A4: set(A),B2: A] :
      ( bNF_Wellorder_wo_rel(A,R2)
     => ( order_ofilter(A,R2,A4)
       => ( ( order_AboveS(A,R2,A4) != bot_bot(set(A)) )
         => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),bNF_Wellorder_wo_suc(A,R2,A4))),R2)
           => ( ( B2 != bNF_Wellorder_wo_suc(A,R2,A4) )
             => aa(set(A),$o,member(A,B2),A4) ) ) ) ) ) ).

% wo_rel.suc_ofilter_in
tff(fact_5824_cSUP__UNION,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( condit1219197933456340205attice(C)
     => ! [A4: set(A),B3: fun(A,set(B)),F: fun(B,C)] :
          ( ( A4 != bot_bot(set(A)) )
         => ( ! [X2: A] :
                ( aa(set(A),$o,member(A,X2),A4)
               => ( aa(A,set(B),B3,X2) != bot_bot(set(B)) ) )
           => ( condit941137186595557371_above(C,aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(A),set(set(C)),image2(A,set(C),aa(fun(B,C),fun(A,set(C)),aTP_Lamp_aly(fun(A,set(B)),fun(fun(B,C),fun(A,set(C))),B3),F)),A4)))
             => ( aa(set(C),C,complete_Sup_Sup(C),aa(set(B),set(C),image2(B,C,F),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),B3),A4)))) = aa(set(C),C,complete_Sup_Sup(C),aa(set(A),set(C),image2(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_alz(fun(A,set(B)),fun(fun(B,C),fun(A,C)),B3),F)),A4)) ) ) ) ) ) ).

% cSUP_UNION
tff(fact_5825_bdd__above__empty,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => condit941137186595557371_above(A,bot_bot(set(A))) ) ).

% bdd_above_empty
tff(fact_5826_bdd__above__Un,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [A4: set(A),B3: set(A)] :
          ( condit941137186595557371_above(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3))
        <=> ( condit941137186595557371_above(A,A4)
            & condit941137186595557371_above(A,B3) ) ) ) ).

% bdd_above_Un
tff(fact_5827_less__than__iff,axiom,
    ! [X: nat,Y: nat] :
      ( aa(set(product_prod(nat,nat)),$o,member(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X),Y)),less_than)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),Y) ) ).

% less_than_iff
tff(fact_5828_bdd__above__image__sup,axiom,
    ! [A: $tType,B: $tType] :
      ( lattice(A)
     => ! [F: fun(B,A),G: fun(B,A),A4: set(B)] :
          ( condit941137186595557371_above(A,aa(set(B),set(A),image2(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_ama(fun(B,A),fun(fun(B,A),fun(B,A)),F),G)),A4))
        <=> ( condit941137186595557371_above(A,aa(set(B),set(A),image2(B,A,F),A4))
            & condit941137186595557371_above(A,aa(set(B),set(A),image2(B,A,G),A4)) ) ) ) ).

% bdd_above_image_sup
tff(fact_5829_bdd__above__UN,axiom,
    ! [B: $tType,A: $tType] :
      ( lattice(B)
     => ! [I4: set(A),A4: fun(A,set(B))] :
          ( aa(set(A),$o,finite_finite2(A),I4)
         => ( condit941137186595557371_above(B,aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),A4),I4)))
          <=> ! [X4: A] :
                ( aa(set(A),$o,member(A,X4),I4)
               => condit941137186595557371_above(B,aa(A,set(B),A4,X4)) ) ) ) ) ).

% bdd_above_UN
tff(fact_5830_bdd__above__Int2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B3: set(A),A4: set(A)] :
          ( condit941137186595557371_above(A,B3)
         => condit941137186595557371_above(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3)) ) ) ).

% bdd_above_Int2
tff(fact_5831_bdd__above__Int1,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A4: set(A),B3: set(A)] :
          ( condit941137186595557371_above(A,A4)
         => condit941137186595557371_above(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3)) ) ) ).

% bdd_above_Int1
tff(fact_5832_total__less__than,axiom,
    total_on(nat,top_top(set(nat)),less_than) ).

% total_less_than
tff(fact_5833_cSup__mono,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [B3: set(A),A4: set(A)] :
          ( ( B3 != bot_bot(set(A)) )
         => ( condit941137186595557371_above(A,A4)
           => ( ! [B5: A] :
                  ( aa(set(A),$o,member(A,B5),B3)
                 => ? [X3: A] :
                      ( aa(set(A),$o,member(A,X3),A4)
                      & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B5),X3) ) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),B3)),aa(set(A),A,complete_Sup_Sup(A),A4)) ) ) ) ) ).

% cSup_mono
tff(fact_5834_cSup__le__iff,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [S: set(A),A3: A] :
          ( ( S != bot_bot(set(A)) )
         => ( condit941137186595557371_above(A,S)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),S)),A3)
            <=> ! [X4: A] :
                  ( aa(set(A),$o,member(A,X4),S)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),A3) ) ) ) ) ) ).

% cSup_le_iff
tff(fact_5835_less__cSup__iff,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X5: set(A),Y: A] :
          ( ( X5 != bot_bot(set(A)) )
         => ( condit941137186595557371_above(A,X5)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),aa(set(A),A,complete_Sup_Sup(A),X5))
            <=> ? [X4: A] :
                  ( aa(set(A),$o,member(A,X4),X5)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),X4) ) ) ) ) ) ).

% less_cSup_iff
tff(fact_5836_cSUP__lessD,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [F: fun(B,A),A4: set(B),Y: A,I: B] :
          ( condit941137186595557371_above(A,aa(set(B),set(A),image2(B,A,F),A4))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F),A4))),Y)
           => ( aa(set(B),$o,member(B,I),A4)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,F,I)),Y) ) ) ) ) ).

% cSUP_lessD
tff(fact_5837_measures__def,axiom,
    ! [A: $tType,Fs: list(fun(A,nat))] : measures(A,Fs) = inv_image(list(nat),A,lex(nat,less_than),aTP_Lamp_amc(list(fun(A,nat)),fun(A,list(nat)),Fs)) ).

% measures_def
tff(fact_5838_cSUP__le__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [A4: set(A),F: fun(A,B),U: B] :
          ( ( A4 != bot_bot(set(A)) )
         => ( condit941137186595557371_above(B,aa(set(A),set(B),image2(A,B,F),A4))
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F),A4))),U)
            <=> ! [X4: A] :
                  ( aa(set(A),$o,member(A,X4),A4)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,X4)),U) ) ) ) ) ) ).

% cSUP_le_iff
tff(fact_5839_cSUP__mono,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [A4: set(A),G: fun(C,B),B3: set(C),F: fun(A,B)] :
          ( ( A4 != bot_bot(set(A)) )
         => ( condit941137186595557371_above(B,aa(set(C),set(B),image2(C,B,G),B3))
           => ( ! [N3: A] :
                  ( aa(set(A),$o,member(A,N3),A4)
                 => ? [X3: C] :
                      ( aa(set(C),$o,member(C,X3),B3)
                      & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,N3)),aa(C,B,G,X3)) ) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F),A4))),aa(set(B),B,complete_Sup_Sup(B),aa(set(C),set(B),image2(C,B,G),B3))) ) ) ) ) ).

% cSUP_mono
tff(fact_5840_cSup__subset__mono,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [A4: set(A),B3: set(A)] :
          ( ( A4 != bot_bot(set(A)) )
         => ( condit941137186595557371_above(A,B3)
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A4)),aa(set(A),A,complete_Sup_Sup(A),B3)) ) ) ) ) ).

% cSup_subset_mono
tff(fact_5841_cSup__insert__If,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X5: set(A),A3: A] :
          ( condit941137186595557371_above(A,X5)
         => ( aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),X5)) = $ite(X5 = bot_bot(set(A)),A3,aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),aa(set(A),A,complete_Sup_Sup(A),X5))) ) ) ) ).

% cSup_insert_If
tff(fact_5842_cSup__insert,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X5: set(A),A3: A] :
          ( ( X5 != bot_bot(set(A)) )
         => ( condit941137186595557371_above(A,X5)
           => ( aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),X5)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),aa(set(A),A,complete_Sup_Sup(A),X5)) ) ) ) ) ).

% cSup_insert
tff(fact_5843_cSup__union__distrib,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [A4: set(A),B3: set(A)] :
          ( ( A4 != bot_bot(set(A)) )
         => ( condit941137186595557371_above(A,A4)
           => ( ( B3 != bot_bot(set(A)) )
             => ( condit941137186595557371_above(A,B3)
               => ( aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Sup_Sup(A),A4)),aa(set(A),A,complete_Sup_Sup(A),B3)) ) ) ) ) ) ) ).

% cSup_union_distrib
tff(fact_5844_wo__rel_Oofilter__UNION,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(A,A)),I4: set(B),A4: fun(B,set(A))] :
      ( bNF_Wellorder_wo_rel(A,R2)
     => ( ! [I2: B] :
            ( aa(set(B),$o,member(B,I2),I4)
           => order_ofilter(A,R2,aa(B,set(A),A4,I2)) )
       => order_ofilter(A,R2,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),A4),I4))) ) ) ).

% wo_rel.ofilter_UNION
tff(fact_5845_less__cSUP__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( condit6923001295902523014norder(B)
     => ! [A4: set(A),F: fun(A,B),A3: B] :
          ( ( A4 != bot_bot(set(A)) )
         => ( condit941137186595557371_above(B,aa(set(A),set(B),image2(A,B,F),A4))
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A3),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F),A4)))
            <=> ? [X4: A] :
                  ( aa(set(A),$o,member(A,X4),A4)
                  & aa(B,$o,aa(B,fun(B,$o),ord_less(B),A3),aa(A,B,F,X4)) ) ) ) ) ) ).

% less_cSUP_iff
tff(fact_5846_conditionally__complete__lattice__class_OSUP__sup__distrib,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [A4: set(A),F: fun(A,B),G: fun(A,B)] :
          ( ( A4 != bot_bot(set(A)) )
         => ( condit941137186595557371_above(B,aa(set(A),set(B),image2(A,B,F),A4))
           => ( condit941137186595557371_above(B,aa(set(A),set(B),image2(A,B,G),A4))
             => ( aa(B,B,aa(B,fun(B,B),sup_sup(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F),A4))),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,G),A4))) = aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_amd(fun(A,B),fun(fun(A,B),fun(A,B)),F),G)),A4)) ) ) ) ) ) ).

% conditionally_complete_lattice_class.SUP_sup_distrib
tff(fact_5847_cSUP__subset__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [A4: set(A),G: fun(A,B),B3: set(A),F: fun(A,B)] :
          ( ( A4 != bot_bot(set(A)) )
         => ( condit941137186595557371_above(B,aa(set(A),set(B),image2(A,B,G),B3))
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B3)
             => ( ! [X2: A] :
                    ( aa(set(A),$o,member(A,X2),A4)
                   => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,X2)),aa(A,B,G,X2)) )
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F),A4))),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,G),B3))) ) ) ) ) ) ).

% cSUP_subset_mono
tff(fact_5848_cSup__inter__less__eq,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [A4: set(A),B3: set(A)] :
          ( condit941137186595557371_above(A,A4)
         => ( condit941137186595557371_above(A,B3)
           => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3) != bot_bot(set(A)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3))),aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Sup_Sup(A),A4)),aa(set(A),A,complete_Sup_Sup(A),B3))) ) ) ) ) ).

% cSup_inter_less_eq
tff(fact_5849_cSUP__insert,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [A4: set(A),F: fun(A,B),A3: A] :
          ( ( A4 != bot_bot(set(A)) )
         => ( condit941137186595557371_above(B,aa(set(A),set(B),image2(A,B,F),A4))
           => ( aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),A4))) = aa(B,B,aa(B,fun(B,B),sup_sup(B),aa(A,B,F,A3)),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F),A4))) ) ) ) ) ).

% cSUP_insert
tff(fact_5850_cSUP__union,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [A4: set(A),F: fun(A,B),B3: set(A)] :
          ( ( A4 != bot_bot(set(A)) )
         => ( condit941137186595557371_above(B,aa(set(A),set(B),image2(A,B,F),A4))
           => ( ( B3 != bot_bot(set(A)) )
             => ( condit941137186595557371_above(B,aa(set(A),set(B),image2(A,B,F),B3))
               => ( aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3))) = aa(B,B,aa(B,fun(B,B),sup_sup(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F),A4))),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F),B3))) ) ) ) ) ) ) ).

% cSUP_union
tff(fact_5851_wo__rel_Oofilter__AboveS__Field,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A4: set(A)] :
      ( bNF_Wellorder_wo_rel(A,R2)
     => ( order_ofilter(A,R2,A4)
       => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),order_AboveS(A,R2,A4)) = field2(A,R2) ) ) ) ).

% wo_rel.ofilter_AboveS_Field
tff(fact_5852_mlex__prod__def,axiom,
    ! [A: $tType,F: fun(A,nat),R: set(product_prod(A,A))] : mlex_prod(A,F,R) = inv_image(product_prod(nat,A),A,lex_prod(nat,A,less_than,R),aTP_Lamp_ame(fun(A,nat),fun(A,product_prod(nat,A)),F)) ).

% mlex_prod_def
tff(fact_5853_cSup__cInf,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [S: set(A)] :
          ( ( S != bot_bot(set(A)) )
         => ( condit941137186595557371_above(A,S)
           => ( aa(set(A),A,complete_Sup_Sup(A),S) = aa(set(A),A,complete_Inf_Inf(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_amf(set(A),fun(A,$o),S))) ) ) ) ) ).

% cSup_cInf
tff(fact_5854_mono__cSUP,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( condit1219197933456340205attice(A)
        & condit1219197933456340205attice(B) )
     => ! [F: fun(A,B),A4: fun(C,A),I4: set(C)] :
          ( order_mono(A,B,F)
         => ( condit941137186595557371_above(A,aa(set(C),set(A),image2(C,A,A4),I4))
           => ( ( I4 != bot_bot(set(C)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(C),set(B),image2(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_amg(fun(A,B),fun(fun(C,A),fun(C,B)),F),A4)),I4))),aa(A,B,F,aa(set(A),A,complete_Sup_Sup(A),aa(set(C),set(A),image2(C,A,A4),I4)))) ) ) ) ) ).

% mono_cSUP
tff(fact_5855_mono__cSup,axiom,
    ! [B: $tType,A: $tType] :
      ( ( condit1219197933456340205attice(A)
        & condit1219197933456340205attice(B) )
     => ! [F: fun(A,B),A4: set(A)] :
          ( order_mono(A,B,F)
         => ( condit941137186595557371_above(A,A4)
           => ( ( A4 != bot_bot(set(A)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F),A4))),aa(A,B,F,aa(set(A),A,complete_Sup_Sup(A),A4))) ) ) ) ) ).

% mono_cSup
tff(fact_5856_ofilterIncl__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : bNF_We413866401316099525erIncl(A,R2) = aa(fun(product_prod(set(A),set(A)),$o),set(product_prod(set(A),set(A))),collect(product_prod(set(A),set(A))),aa(fun(set(A),fun(set(A),$o)),fun(product_prod(set(A),set(A)),$o),product_case_prod(set(A),set(A),$o),aTP_Lamp_amh(set(product_prod(A,A)),fun(set(A),fun(set(A),$o)),R2))) ).

% ofilterIncl_def
tff(fact_5857_wo__rel_Oofilter__under__UNION,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A4: set(A)] :
      ( bNF_Wellorder_wo_rel(A,R2)
     => ( order_ofilter(A,R2,A4)
       => ( A4 = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(A),set(set(A)),image2(A,set(A),order_under(A,R2)),A4)) ) ) ) ).

% wo_rel.ofilter_under_UNION
tff(fact_5858_cINF__UNION,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( condit1219197933456340205attice(C)
     => ! [A4: set(A),B3: fun(A,set(B)),F: fun(B,C)] :
          ( ( A4 != bot_bot(set(A)) )
         => ( ! [X2: A] :
                ( aa(set(A),$o,member(A,X2),A4)
               => ( aa(A,set(B),B3,X2) != bot_bot(set(B)) ) )
           => ( aa(set(C),$o,condit1013018076250108175_below(C),aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(A),set(set(C)),image2(A,set(C),aa(fun(B,C),fun(A,set(C)),aTP_Lamp_aly(fun(A,set(B)),fun(fun(B,C),fun(A,set(C))),B3),F)),A4)))
             => ( aa(set(C),C,complete_Inf_Inf(C),aa(set(B),set(C),image2(B,C,F),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),B3),A4)))) = aa(set(C),C,complete_Inf_Inf(C),aa(set(A),set(C),image2(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_ami(fun(A,set(B)),fun(fun(B,C),fun(A,C)),B3),F)),A4)) ) ) ) ) ) ).

% cINF_UNION
tff(fact_5859_bdd__below__empty,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => aa(set(A),$o,condit1013018076250108175_below(A),bot_bot(set(A))) ) ).

% bdd_below_empty
tff(fact_5860_bdd__below__Un,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [A4: set(A),B3: set(A)] :
          ( aa(set(A),$o,condit1013018076250108175_below(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3))
        <=> ( aa(set(A),$o,condit1013018076250108175_below(A),A4)
            & aa(set(A),$o,condit1013018076250108175_below(A),B3) ) ) ) ).

% bdd_below_Un
tff(fact_5861_bdd__below__image__inf,axiom,
    ! [A: $tType,B: $tType] :
      ( lattice(A)
     => ! [F: fun(B,A),G: fun(B,A),A4: set(B)] :
          ( aa(set(A),$o,condit1013018076250108175_below(A),aa(set(B),set(A),image2(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_amj(fun(B,A),fun(fun(B,A),fun(B,A)),F),G)),A4))
        <=> ( aa(set(A),$o,condit1013018076250108175_below(A),aa(set(B),set(A),image2(B,A,F),A4))
            & aa(set(A),$o,condit1013018076250108175_below(A),aa(set(B),set(A),image2(B,A,G),A4)) ) ) ) ).

% bdd_below_image_inf
tff(fact_5862_bdd__below__uminus,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [X5: set(A)] :
          ( aa(set(A),$o,condit1013018076250108175_below(A),aa(set(A),set(A),image2(A,A,uminus_uminus(A)),X5))
        <=> condit941137186595557371_above(A,X5) ) ) ).

% bdd_below_uminus
tff(fact_5863_bdd__above__uminus,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [X5: set(A)] :
          ( condit941137186595557371_above(A,aa(set(A),set(A),image2(A,A,uminus_uminus(A)),X5))
        <=> aa(set(A),$o,condit1013018076250108175_below(A),X5) ) ) ).

% bdd_above_uminus
tff(fact_5864_bdd__below__UN,axiom,
    ! [B: $tType,A: $tType] :
      ( lattice(B)
     => ! [I4: set(A),A4: fun(A,set(B))] :
          ( aa(set(A),$o,finite_finite2(A),I4)
         => ( aa(set(B),$o,condit1013018076250108175_below(B),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),A4),I4)))
          <=> ! [X4: A] :
                ( aa(set(A),$o,member(A,X4),I4)
               => aa(set(B),$o,condit1013018076250108175_below(B),aa(A,set(B),A4,X4)) ) ) ) ) ).

% bdd_below_UN
tff(fact_5865_under__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A3: A] : aa(A,set(A),order_under(A,R2),A3) = aa(fun(A,$o),set(A),collect(A),aa(A,fun(A,$o),aTP_Lamp_amk(set(product_prod(A,A)),fun(A,fun(A,$o)),R2),A3)) ).

% under_def
tff(fact_5866_bdd__below__Int1,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A4: set(A),B3: set(A)] :
          ( aa(set(A),$o,condit1013018076250108175_below(A),A4)
         => aa(set(A),$o,condit1013018076250108175_below(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3)) ) ) ).

% bdd_below_Int1
tff(fact_5867_bdd__below__Int2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B3: set(A),A4: set(A)] :
          ( aa(set(A),$o,condit1013018076250108175_below(A),B3)
         => aa(set(A),$o,condit1013018076250108175_below(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3)) ) ) ).

% bdd_below_Int2
tff(fact_5868_le__cInf__iff,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [S: set(A),A3: A] :
          ( ( S != bot_bot(set(A)) )
         => ( aa(set(A),$o,condit1013018076250108175_below(A),S)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(set(A),A,complete_Inf_Inf(A),S))
            <=> ! [X4: A] :
                  ( aa(set(A),$o,member(A,X4),S)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),X4) ) ) ) ) ) ).

% le_cInf_iff
tff(fact_5869_cInf__mono,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [B3: set(A),A4: set(A)] :
          ( ( B3 != bot_bot(set(A)) )
         => ( aa(set(A),$o,condit1013018076250108175_below(A),A4)
           => ( ! [B5: A] :
                  ( aa(set(A),$o,member(A,B5),B3)
                 => ? [X3: A] :
                      ( aa(set(A),$o,member(A,X3),A4)
                      & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),B5) ) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A4)),aa(set(A),A,complete_Inf_Inf(A),B3)) ) ) ) ) ).

% cInf_mono
tff(fact_5870_cInf__less__iff,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X5: set(A),Y: A] :
          ( ( X5 != bot_bot(set(A)) )
         => ( aa(set(A),$o,condit1013018076250108175_below(A),X5)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(A),A,complete_Inf_Inf(A),X5)),Y)
            <=> ? [X4: A] :
                  ( aa(set(A),$o,member(A,X4),X5)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Y) ) ) ) ) ) ).

% cInf_less_iff
tff(fact_5871_less__cINF__D,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [F: fun(B,A),A4: set(B),Y: A,I: B] :
          ( aa(set(A),$o,condit1013018076250108175_below(A),aa(set(B),set(A),image2(B,A,F),A4))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F),A4)))
           => ( aa(set(B),$o,member(B,I),A4)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),aa(B,A,F,I)) ) ) ) ) ).

% less_cINF_D
tff(fact_5872_le__cINF__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [A4: set(A),F: fun(A,B),U: B] :
          ( ( A4 != bot_bot(set(A)) )
         => ( aa(set(B),$o,condit1013018076250108175_below(B),aa(set(A),set(B),image2(A,B,F),A4))
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),U),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F),A4)))
            <=> ! [X4: A] :
                  ( aa(set(A),$o,member(A,X4),A4)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),U),aa(A,B,F,X4)) ) ) ) ) ) ).

% le_cINF_iff
tff(fact_5873_cINF__mono,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [B3: set(A),F: fun(C,B),A4: set(C),G: fun(A,B)] :
          ( ( B3 != bot_bot(set(A)) )
         => ( aa(set(B),$o,condit1013018076250108175_below(B),aa(set(C),set(B),image2(C,B,F),A4))
           => ( ! [M3: A] :
                  ( aa(set(A),$o,member(A,M3),B3)
                 => ? [X3: C] :
                      ( aa(set(C),$o,member(C,X3),A4)
                      & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(C,B,F,X3)),aa(A,B,G,M3)) ) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(C),set(B),image2(C,B,F),A4))),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,G),B3))) ) ) ) ) ).

% cINF_mono
tff(fact_5874_cInf__superset__mono,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [A4: set(A),B3: set(A)] :
          ( ( A4 != bot_bot(set(A)) )
         => ( aa(set(A),$o,condit1013018076250108175_below(A),B3)
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),B3)),aa(set(A),A,complete_Inf_Inf(A),A4)) ) ) ) ) ).

% cInf_superset_mono
tff(fact_5875_cInf__insert,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X5: set(A),A3: A] :
          ( ( X5 != bot_bot(set(A)) )
         => ( aa(set(A),$o,condit1013018076250108175_below(A),X5)
           => ( aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),X5)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),aa(set(A),A,complete_Inf_Inf(A),X5)) ) ) ) ) ).

% cInf_insert
tff(fact_5876_cInf__insert__If,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X5: set(A),A3: A] :
          ( aa(set(A),$o,condit1013018076250108175_below(A),X5)
         => ( aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),X5)) = $ite(X5 = bot_bot(set(A)),A3,aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),aa(set(A),A,complete_Inf_Inf(A),X5))) ) ) ) ).

% cInf_insert_If
tff(fact_5877_cInf__union__distrib,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [A4: set(A),B3: set(A)] :
          ( ( A4 != bot_bot(set(A)) )
         => ( aa(set(A),$o,condit1013018076250108175_below(A),A4)
           => ( ( B3 != bot_bot(set(A)) )
             => ( aa(set(A),$o,condit1013018076250108175_below(A),B3)
               => ( aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Inf_Inf(A),A4)),aa(set(A),A,complete_Inf_Inf(A),B3)) ) ) ) ) ) ) ).

% cInf_union_distrib
tff(fact_5878_cINF__less__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( condit6923001295902523014norder(B)
     => ! [A4: set(A),F: fun(A,B),A3: B] :
          ( ( A4 != bot_bot(set(A)) )
         => ( aa(set(B),$o,condit1013018076250108175_below(B),aa(set(A),set(B),image2(A,B,F),A4))
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F),A4))),A3)
            <=> ? [X4: A] :
                  ( aa(set(A),$o,member(A,X4),A4)
                  & aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F,X4)),A3) ) ) ) ) ) ).

% cINF_less_iff
tff(fact_5879_cINF__inf__distrib,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [A4: set(A),F: fun(A,B),G: fun(A,B)] :
          ( ( A4 != bot_bot(set(A)) )
         => ( aa(set(B),$o,condit1013018076250108175_below(B),aa(set(A),set(B),image2(A,B,F),A4))
           => ( aa(set(B),$o,condit1013018076250108175_below(B),aa(set(A),set(B),image2(A,B,G),A4))
             => ( aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F),A4))),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,G),A4))) = aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_aml(fun(A,B),fun(fun(A,B),fun(A,B)),F),G)),A4)) ) ) ) ) ) ).

% cINF_inf_distrib
tff(fact_5880_cSUP__eq__cINF__D,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [F: fun(B,A),A4: set(B),A3: B] :
          ( ( aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F),A4)) = aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F),A4)) )
         => ( condit941137186595557371_above(A,aa(set(B),set(A),image2(B,A,F),A4))
           => ( aa(set(A),$o,condit1013018076250108175_below(A),aa(set(B),set(A),image2(B,A,F),A4))
             => ( aa(set(B),$o,member(B,A3),A4)
               => ( aa(B,A,F,A3) = aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F),A4)) ) ) ) ) ) ) ).

% cSUP_eq_cINF_D
tff(fact_5881_cINF__superset__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [A4: set(A),G: fun(A,B),B3: set(A),F: fun(A,B)] :
          ( ( A4 != bot_bot(set(A)) )
         => ( aa(set(B),$o,condit1013018076250108175_below(B),aa(set(A),set(B),image2(A,B,G),B3))
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B3)
             => ( ! [X2: A] :
                    ( aa(set(A),$o,member(A,X2),B3)
                   => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,G,X2)),aa(A,B,F,X2)) )
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,G),B3))),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F),A4))) ) ) ) ) ) ).

% cINF_superset_mono
tff(fact_5882_less__eq__cInf__inter,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [A4: set(A),B3: set(A)] :
          ( aa(set(A),$o,condit1013018076250108175_below(A),A4)
         => ( aa(set(A),$o,condit1013018076250108175_below(A),B3)
           => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3) != bot_bot(set(A)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Inf_Inf(A),A4)),aa(set(A),A,complete_Inf_Inf(A),B3))),aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3))) ) ) ) ) ).

% less_eq_cInf_inter
tff(fact_5883_cINF__insert,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [A4: set(A),F: fun(A,B),A3: A] :
          ( ( A4 != bot_bot(set(A)) )
         => ( aa(set(B),$o,condit1013018076250108175_below(B),aa(set(A),set(B),image2(A,B,F),A4))
           => ( aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),A4))) = aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(A,B,F,A3)),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F),A4))) ) ) ) ) ).

% cINF_insert
tff(fact_5884_cINF__union,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [A4: set(A),F: fun(A,B),B3: set(A)] :
          ( ( A4 != bot_bot(set(A)) )
         => ( aa(set(B),$o,condit1013018076250108175_below(B),aa(set(A),set(B),image2(A,B,F),A4))
           => ( ( B3 != bot_bot(set(A)) )
             => ( aa(set(B),$o,condit1013018076250108175_below(B),aa(set(A),set(B),image2(A,B,F),B3))
               => ( aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3))) = aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F),A4))),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F),B3))) ) ) ) ) ) ) ).

% cINF_union
tff(fact_5885_cInf__le__cSup,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [A4: set(A)] :
          ( ( A4 != bot_bot(set(A)) )
         => ( condit941137186595557371_above(A,A4)
           => ( aa(set(A),$o,condit1013018076250108175_below(A),A4)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A4)),aa(set(A),A,complete_Sup_Sup(A),A4)) ) ) ) ) ).

% cInf_le_cSup
tff(fact_5886_cInf__cSup,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [S: set(A)] :
          ( ( S != bot_bot(set(A)) )
         => ( aa(set(A),$o,condit1013018076250108175_below(A),S)
           => ( aa(set(A),A,complete_Inf_Inf(A),S) = aa(set(A),A,complete_Sup_Sup(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_amm(set(A),fun(A,$o),S))) ) ) ) ) ).

% cInf_cSup
tff(fact_5887_mono__cInf,axiom,
    ! [B: $tType,A: $tType] :
      ( ( condit1219197933456340205attice(A)
        & condit1219197933456340205attice(B) )
     => ! [F: fun(A,B),A4: set(A)] :
          ( order_mono(A,B,F)
         => ( aa(set(A),$o,condit1013018076250108175_below(A),A4)
           => ( ( A4 != bot_bot(set(A)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,aa(set(A),A,complete_Inf_Inf(A),A4))),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F),A4))) ) ) ) ) ).

% mono_cInf
tff(fact_5888_mono__cINF,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( condit1219197933456340205attice(B)
        & condit1219197933456340205attice(A) )
     => ! [F: fun(A,B),A4: fun(C,A),I4: set(C)] :
          ( order_mono(A,B,F)
         => ( aa(set(A),$o,condit1013018076250108175_below(A),aa(set(C),set(A),image2(C,A,A4),I4))
           => ( ( I4 != bot_bot(set(C)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,aa(set(A),A,complete_Inf_Inf(A),aa(set(C),set(A),image2(C,A,A4),I4)))),aa(set(B),B,complete_Inf_Inf(B),aa(set(C),set(B),image2(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_amg(fun(A,B),fun(fun(C,A),fun(C,B)),F),A4)),I4))) ) ) ) ) ).

% mono_cINF
tff(fact_5889_bsqr__ofilter,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),D4: set(product_prod(A,A))] :
      ( order_well_order_on(A,field2(A,R2),R2)
     => ( order_ofilter(product_prod(A,A),bNF_Wellorder_bsqr(A,R2),D4)
       => ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less(set(product_prod(A,A))),D4),product_Sigma(A,A,field2(A,R2),aTP_Lamp_ut(set(product_prod(A,A)),fun(A,set(A)),R2)))
         => ( ~ ? [A6: A] : field2(A,R2) = aa(A,set(A),order_under(A,R2),A6)
           => ? [A8: set(A)] :
                ( order_ofilter(A,R2,A8)
                & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A8),field2(A,R2))
                & aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),D4),product_Sigma(A,A,A8,aTP_Lamp_rl(set(A),fun(A,set(A)),A8))) ) ) ) ) ) ).

% bsqr_ofilter
tff(fact_5890_Refl__under__underS,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A3: A] :
      ( refl_on(A,field2(A,R2),R2)
     => ( aa(set(A),$o,member(A,A3),field2(A,R2))
       => ( aa(A,set(A),order_under(A,R2),A3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),order_underS(A,R2,A3)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A)))) ) ) ) ).

% Refl_under_underS
tff(fact_5891_ofilter__Restr__under,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A4: set(A),A3: A] :
      ( order_well_order_on(A,field2(A,R2),R2)
     => ( order_ofilter(A,R2,A4)
       => ( aa(set(A),$o,member(A,A3),A4)
         => ( aa(A,set(A),order_under(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R2),product_Sigma(A,A,A4,aTP_Lamp_rl(set(A),fun(A,set(A)),A4)))),A3) = aa(A,set(A),order_under(A,R2),A3) ) ) ) ) ).

% ofilter_Restr_under
tff(fact_5892_well__order__on__empty,axiom,
    ! [A: $tType] : order_well_order_on(A,bot_bot(set(A)),bot_bot(set(product_prod(A,A)))) ).

% well_order_on_empty
tff(fact_5893_well__ordering,axiom,
    ! [A: $tType] :
    ? [R4: set(product_prod(A,A))] :
      ( order_well_order_on(A,field2(A,R4),R4)
      & ( field2(A,R4) = top_top(set(A)) ) ) ).

% well_ordering
tff(fact_5894_underS__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A3: A] : order_underS(A,R2,A3) = aa(fun(A,$o),set(A),collect(A),aa(A,fun(A,$o),aTP_Lamp_amn(set(product_prod(A,A)),fun(A,fun(A,$o)),R2),A3)) ).

% underS_def
tff(fact_5895_underS__E,axiom,
    ! [A: $tType,I: A,R: set(product_prod(A,A)),J: A] :
      ( aa(set(A),$o,member(A,I),order_underS(A,R,J))
     => ( ( I != J )
        & aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),I),J)),R) ) ) ).

% underS_E
tff(fact_5896_underS__I,axiom,
    ! [A: $tType,I: A,J: A,R: set(product_prod(A,A))] :
      ( ( I != J )
     => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),I),J)),R)
       => aa(set(A),$o,member(A,I),order_underS(A,R,J)) ) ) ).

% underS_I
tff(fact_5897_well__order__on__domain,axiom,
    ! [A: $tType,A4: set(A),R2: set(product_prod(A,A)),A3: A,B2: A] :
      ( order_well_order_on(A,A4,R2)
     => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R2)
       => ( aa(set(A),$o,member(A,A3),A4)
          & aa(set(A),$o,member(A,B2),A4) ) ) ) ).

% well_order_on_domain
tff(fact_5898_underS__empty,axiom,
    ! [A: $tType,A3: A,R2: set(product_prod(A,A))] :
      ( ~ aa(set(A),$o,member(A,A3),field2(A,R2))
     => ( order_underS(A,R2,A3) = bot_bot(set(A)) ) ) ).

% underS_empty
tff(fact_5899_Well__order__Restr,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A4: set(A)] :
      ( order_well_order_on(A,field2(A,R2),R2)
     => order_well_order_on(A,field2(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R2),product_Sigma(A,A,A4,aTP_Lamp_rl(set(A),fun(A,set(A)),A4)))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R2),product_Sigma(A,A,A4,aTP_Lamp_rl(set(A),fun(A,set(A)),A4)))) ) ).

% Well_order_Restr
tff(fact_5900_natLeq__on__well__order__on,axiom,
    ! [N: nat] : order_well_order_on(nat,aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_bk(nat,fun(nat,$o)),N)),aa(fun(product_prod(nat,nat),$o),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aTP_Lamp_uw(nat,fun(nat,fun(nat,$o)),N)))) ).

% natLeq_on_well_order_on
tff(fact_5901_underS__Field3,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A3: A] :
      ( ( field2(A,R2) != bot_bot(set(A)) )
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),order_underS(A,R2,A3)),field2(A,R2)) ) ).

% underS_Field3
tff(fact_5902_well__order__on__Restr,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A4: set(A)] :
      ( order_well_order_on(A,field2(A,R2),R2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),field2(A,R2))
       => order_well_order_on(A,A4,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R2),product_Sigma(A,A,A4,aTP_Lamp_rl(set(A),fun(A,set(A)),A4)))) ) ) ).

% well_order_on_Restr
tff(fact_5903_Field__Restr__ofilter,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A4: set(A)] :
      ( order_well_order_on(A,field2(A,R2),R2)
     => ( order_ofilter(A,R2,A4)
       => ( field2(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R2),product_Sigma(A,A,A4,aTP_Lamp_rl(set(A),fun(A,set(A)),A4)))) = A4 ) ) ) ).

% Field_Restr_ofilter
tff(fact_5904_natLeq__on__Well__order,axiom,
    ! [N: nat] : order_well_order_on(nat,field2(nat,aa(fun(product_prod(nat,nat),$o),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aTP_Lamp_uw(nat,fun(nat,fun(nat,$o)),N)))),aa(fun(product_prod(nat,nat),$o),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aTP_Lamp_uw(nat,fun(nat,fun(nat,$o)),N)))) ).

% natLeq_on_Well_order
tff(fact_5905_underS__incl__iff,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A3: A,B2: A] :
      ( order_679001287576687338der_on(A,field2(A,R2),R2)
     => ( aa(set(A),$o,member(A,A3),field2(A,R2))
       => ( aa(set(A),$o,member(A,B2),field2(A,R2))
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),order_underS(A,R2,A3)),order_underS(A,R2,B2))
          <=> aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R2) ) ) ) ) ).

% underS_incl_iff
tff(fact_5906_Linear__order__Well__order__iff,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( order_679001287576687338der_on(A,field2(A,R2),R2)
     => ( order_well_order_on(A,field2(A,R2),R2)
      <=> ! [A9: set(A)] :
            ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A9),field2(A,R2))
           => ( ( A9 != bot_bot(set(A)) )
             => ? [X4: A] :
                  ( aa(set(A),$o,member(A,X4),A9)
                  & ! [Xa2: A] :
                      ( aa(set(A),$o,member(A,Xa2),A9)
                     => aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Xa2)),R2) ) ) ) ) ) ) ).

% Linear_order_Well_order_iff
tff(fact_5907_ofilter__Restr__subset,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A4: set(A),B3: set(A)] :
      ( order_well_order_on(A,field2(A,R2),R2)
     => ( order_ofilter(A,R2,A4)
       => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B3)
         => order_ofilter(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R2),product_Sigma(A,A,B3,aTP_Lamp_rl(set(A),fun(A,set(A)),B3))),A4) ) ) ) ).

% ofilter_Restr_subset
tff(fact_5908_ofilter__Restr__Int,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A4: set(A),B3: set(A)] :
      ( order_well_order_on(A,field2(A,R2),R2)
     => ( order_ofilter(A,R2,A4)
       => order_ofilter(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R2),product_Sigma(A,A,B3,aTP_Lamp_rl(set(A),fun(A,set(A)),B3))),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3)) ) ) ).

% ofilter_Restr_Int
tff(fact_5909_bsqr__max2,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A1: A,A22: A,B13: A,B23: A] :
      ( order_well_order_on(A,field2(A,R2),R2)
     => ( aa(set(product_prod(product_prod(A,A),product_prod(A,A))),$o,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),A1),A22)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B13),B23))),bNF_Wellorder_bsqr(A,R2))
       => aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),bNF_We1388413361240627857o_max2(A,R2,A1,A22)),bNF_We1388413361240627857o_max2(A,R2,B13,B23))),R2) ) ) ).

% bsqr_max2
tff(fact_5910_UNION__inj__on__ofilter,axiom,
    ! [C: $tType,A: $tType,B: $tType,R2: set(product_prod(A,A)),I4: set(B),A4: fun(B,set(A)),F: fun(A,C)] :
      ( order_well_order_on(A,field2(A,R2),R2)
     => ( ! [I2: B] :
            ( aa(set(B),$o,member(B,I2),I4)
           => order_ofilter(A,R2,aa(B,set(A),A4,I2)) )
       => ( ! [I2: B] :
              ( aa(set(B),$o,member(B,I2),I4)
             => inj_on(A,C,F,aa(B,set(A),A4,I2)) )
         => inj_on(A,C,F,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),A4),I4))) ) ) ) ).

% UNION_inj_on_ofilter
tff(fact_5911_ofilter__subset__embedS,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A4: set(A),B3: set(A)] :
      ( order_well_order_on(A,field2(A,R2),R2)
     => ( order_ofilter(A,R2,A4)
       => ( order_ofilter(A,R2,B3)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A4),B3)
          <=> bNF_Wellorder_embedS(A,A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R2),product_Sigma(A,A,A4,aTP_Lamp_rl(set(A),fun(A,set(A)),A4))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R2),product_Sigma(A,A,B3,aTP_Lamp_rl(set(A),fun(A,set(A)),B3))),id(A)) ) ) ) ) ).

% ofilter_subset_embedS
tff(fact_5912_ofilter__subset__ordLess,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A4: set(A),B3: set(A)] :
      ( order_well_order_on(A,field2(A,R2),R2)
     => ( order_ofilter(A,R2,A4)
       => ( order_ofilter(A,R2,B3)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A4),B3)
          <=> aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),$o,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))),R2),product_Sigma(A,A,A4,aTP_Lamp_rl(set(A),fun(A,set(A)),A4)))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R2),product_Sigma(A,A,B3,aTP_Lamp_rl(set(A),fun(A,set(A)),B3))))),bNF_We4044943003108391690rdLess(A,A)) ) ) ) ) ).

% ofilter_subset_ordLess
tff(fact_5913_ordLess__transitive,axiom,
    ! [B: $tType,C: $tType,A: $tType,R2: set(product_prod(A,A)),R3: set(product_prod(B,B)),R8: set(product_prod(C,C))] :
      ( aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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))),R2),R3)),bNF_We4044943003108391690rdLess(A,B))
     => ( aa(set(product_prod(set(product_prod(B,B)),set(product_prod(C,C)))),$o,member(product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(B,B)),fun(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C)))),product_Pair(set(product_prod(B,B)),set(product_prod(C,C))),R3),R8)),bNF_We4044943003108391690rdLess(B,C))
       => aa(set(product_prod(set(product_prod(A,A)),set(product_prod(C,C)))),$o,member(product_prod(set(product_prod(A,A)),set(product_prod(C,C))),aa(set(product_prod(C,C)),product_prod(set(product_prod(A,A)),set(product_prod(C,C))),aa(set(product_prod(A,A)),fun(set(product_prod(C,C)),product_prod(set(product_prod(A,A)),set(product_prod(C,C)))),product_Pair(set(product_prod(A,A)),set(product_prod(C,C))),R2),R8)),bNF_We4044943003108391690rdLess(A,C)) ) ) ).

% ordLess_transitive
tff(fact_5914_ordLess__irreflexive,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : ~ aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),$o,member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R2),R2)),bNF_We4044943003108391690rdLess(A,A)) ).

% ordLess_irreflexive
tff(fact_5915_ordLess__def,axiom,
    ! [B: $tType,A: $tType] : bNF_We4044943003108391690rdLess(A,B) = aa(fun(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),$o),set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),collect(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),aa(fun(set(product_prod(A,A)),fun(set(product_prod(B,B)),$o)),fun(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),$o),product_case_prod(set(product_prod(A,A)),set(product_prod(B,B)),$o),aTP_Lamp_amo(set(product_prod(A,A)),fun(set(product_prod(B,B)),$o)))) ).

% ordLess_def
tff(fact_5916_finite__ordLess__infinite,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,A)),R3: set(product_prod(B,B))] :
      ( order_well_order_on(A,field2(A,R2),R2)
     => ( order_well_order_on(B,field2(B,R3),R3)
       => ( aa(set(A),$o,finite_finite2(A),field2(A,R2))
         => ( ~ aa(set(B),$o,finite_finite2(B),field2(B,R3))
           => aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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))),R2),R3)),bNF_We4044943003108391690rdLess(A,B)) ) ) ) ) ).

% finite_ordLess_infinite
tff(fact_5917_underS__Restr__ordLess,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A3: A] :
      ( order_well_order_on(A,field2(A,R2),R2)
     => ( ( field2(A,R2) != bot_bot(set(A)) )
       => aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),$o,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))),R2),product_Sigma(A,A,order_underS(A,R2,A3),aa(A,fun(A,set(A)),aTP_Lamp_amp(set(product_prod(A,A)),fun(A,fun(A,set(A))),R2),A3)))),R2)),bNF_We4044943003108391690rdLess(A,A)) ) ) ).

% underS_Restr_ordLess
tff(fact_5918_ofilter__ordLess,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A4: set(A)] :
      ( order_well_order_on(A,field2(A,R2),R2)
     => ( order_ofilter(A,R2,A4)
       => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A4),field2(A,R2))
        <=> aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),$o,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))),R2),product_Sigma(A,A,A4,aTP_Lamp_rl(set(A),fun(A,set(A)),A4)))),R2)),bNF_We4044943003108391690rdLess(A,A)) ) ) ) ).

% ofilter_ordLess
tff(fact_5919_ofilter__subset__embedS__iso,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A4: set(A),B3: set(A)] :
      ( order_well_order_on(A,field2(A,R2),R2)
     => ( order_ofilter(A,R2,A4)
       => ( order_ofilter(A,R2,B3)
         => ( ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A4),B3)
            <=> bNF_Wellorder_embedS(A,A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R2),product_Sigma(A,A,A4,aTP_Lamp_rl(set(A),fun(A,set(A)),A4))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R2),product_Sigma(A,A,B3,aTP_Lamp_rl(set(A),fun(A,set(A)),B3))),id(A)) )
            & ( ( A4 = B3 )
            <=> bNF_Wellorder_iso(A,A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R2),product_Sigma(A,A,A4,aTP_Lamp_rl(set(A),fun(A,set(A)),A4))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R2),product_Sigma(A,A,B3,aTP_Lamp_rl(set(A),fun(A,set(A)),B3))),id(A)) ) ) ) ) ) ).

% ofilter_subset_embedS_iso
tff(fact_5920_ordLess__iff__ordIso__Restr,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(A,A)),R3: set(product_prod(B,B))] :
      ( order_well_order_on(A,field2(A,R2),R2)
     => ( order_well_order_on(B,field2(B,R3),R3)
       => ( aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),$o,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))),R3),R2)),bNF_We4044943003108391690rdLess(B,A))
        <=> ? [X4: A] :
              ( aa(set(A),$o,member(A,X4),field2(A,R2))
              & aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),$o,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))),R3),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R2),product_Sigma(A,A,order_underS(A,R2,X4),aa(A,fun(A,set(A)),aTP_Lamp_amp(set(product_prod(A,A)),fun(A,fun(A,set(A))),R2),X4))))),bNF_Wellorder_ordIso(B,A)) ) ) ) ) ).

% ordLess_iff_ordIso_Restr
tff(fact_5921_ordLeq__iff__ordLess__Restr,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,A)),R3: set(product_prod(B,B))] :
      ( order_well_order_on(A,field2(A,R2),R2)
     => ( order_well_order_on(B,field2(B,R3),R3)
       => ( aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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))),R2),R3)),bNF_Wellorder_ordLeq(A,B))
        <=> ! [X4: A] :
              ( aa(set(A),$o,member(A,X4),field2(A,R2))
             => aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R2),product_Sigma(A,A,order_underS(A,R2,X4),aa(A,fun(A,set(A)),aTP_Lamp_amp(set(product_prod(A,A)),fun(A,fun(A,set(A))),R2),X4)))),R3)),bNF_We4044943003108391690rdLess(A,B)) ) ) ) ) ).

% ordLeq_iff_ordLess_Restr
tff(fact_5922_not__ordLess__ordIso,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,A)),R3: set(product_prod(B,B))] :
      ( aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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))),R2),R3)),bNF_We4044943003108391690rdLess(A,B))
     => ~ aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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))),R2),R3)),bNF_Wellorder_ordIso(A,B)) ) ).

% not_ordLess_ordIso
tff(fact_5923_not__ordLess__ordLeq,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(A,A)),R3: set(product_prod(B,B))] :
      ( aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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))),R2),R3)),bNF_We4044943003108391690rdLess(A,B))
     => ~ aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),$o,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))),R3),R2)),bNF_Wellorder_ordLeq(B,A)) ) ).

% not_ordLess_ordLeq
tff(fact_5924_ordLess__imp__ordLeq,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,A)),R3: set(product_prod(B,B))] :
      ( aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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))),R2),R3)),bNF_We4044943003108391690rdLess(A,B))
     => aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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))),R2),R3)),bNF_Wellorder_ordLeq(A,B)) ) ).

% ordLess_imp_ordLeq
tff(fact_5925_ordIso__ordLess__trans,axiom,
    ! [B: $tType,C: $tType,A: $tType,R2: set(product_prod(A,A)),R3: set(product_prod(B,B)),R8: set(product_prod(C,C))] :
      ( aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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))),R2),R3)),bNF_Wellorder_ordIso(A,B))
     => ( aa(set(product_prod(set(product_prod(B,B)),set(product_prod(C,C)))),$o,member(product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(B,B)),fun(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C)))),product_Pair(set(product_prod(B,B)),set(product_prod(C,C))),R3),R8)),bNF_We4044943003108391690rdLess(B,C))
       => aa(set(product_prod(set(product_prod(A,A)),set(product_prod(C,C)))),$o,member(product_prod(set(product_prod(A,A)),set(product_prod(C,C))),aa(set(product_prod(C,C)),product_prod(set(product_prod(A,A)),set(product_prod(C,C))),aa(set(product_prod(A,A)),fun(set(product_prod(C,C)),product_prod(set(product_prod(A,A)),set(product_prod(C,C)))),product_Pair(set(product_prod(A,A)),set(product_prod(C,C))),R2),R8)),bNF_We4044943003108391690rdLess(A,C)) ) ) ).

% ordIso_ordLess_trans
tff(fact_5926_ordLeq__ordLess__trans,axiom,
    ! [B: $tType,C: $tType,A: $tType,R2: set(product_prod(A,A)),R3: set(product_prod(B,B)),R8: set(product_prod(C,C))] :
      ( aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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))),R2),R3)),bNF_Wellorder_ordLeq(A,B))
     => ( aa(set(product_prod(set(product_prod(B,B)),set(product_prod(C,C)))),$o,member(product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(B,B)),fun(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C)))),product_Pair(set(product_prod(B,B)),set(product_prod(C,C))),R3),R8)),bNF_We4044943003108391690rdLess(B,C))
       => aa(set(product_prod(set(product_prod(A,A)),set(product_prod(C,C)))),$o,member(product_prod(set(product_prod(A,A)),set(product_prod(C,C))),aa(set(product_prod(C,C)),product_prod(set(product_prod(A,A)),set(product_prod(C,C))),aa(set(product_prod(A,A)),fun(set(product_prod(C,C)),product_prod(set(product_prod(A,A)),set(product_prod(C,C)))),product_Pair(set(product_prod(A,A)),set(product_prod(C,C))),R2),R8)),bNF_We4044943003108391690rdLess(A,C)) ) ) ).

% ordLeq_ordLess_trans
tff(fact_5927_ordLess__ordIso__trans,axiom,
    ! [B: $tType,C: $tType,A: $tType,R2: set(product_prod(A,A)),R3: set(product_prod(B,B)),R8: set(product_prod(C,C))] :
      ( aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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))),R2),R3)),bNF_We4044943003108391690rdLess(A,B))
     => ( aa(set(product_prod(set(product_prod(B,B)),set(product_prod(C,C)))),$o,member(product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(B,B)),fun(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C)))),product_Pair(set(product_prod(B,B)),set(product_prod(C,C))),R3),R8)),bNF_Wellorder_ordIso(B,C))
       => aa(set(product_prod(set(product_prod(A,A)),set(product_prod(C,C)))),$o,member(product_prod(set(product_prod(A,A)),set(product_prod(C,C))),aa(set(product_prod(C,C)),product_prod(set(product_prod(A,A)),set(product_prod(C,C))),aa(set(product_prod(A,A)),fun(set(product_prod(C,C)),product_prod(set(product_prod(A,A)),set(product_prod(C,C)))),product_Pair(set(product_prod(A,A)),set(product_prod(C,C))),R2),R8)),bNF_We4044943003108391690rdLess(A,C)) ) ) ).

% ordLess_ordIso_trans
tff(fact_5928_ordLess__ordLeq__trans,axiom,
    ! [B: $tType,C: $tType,A: $tType,R2: set(product_prod(A,A)),R3: set(product_prod(B,B)),R8: set(product_prod(C,C))] :
      ( aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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))),R2),R3)),bNF_We4044943003108391690rdLess(A,B))
     => ( aa(set(product_prod(set(product_prod(B,B)),set(product_prod(C,C)))),$o,member(product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(B,B)),fun(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C)))),product_Pair(set(product_prod(B,B)),set(product_prod(C,C))),R3),R8)),bNF_Wellorder_ordLeq(B,C))
       => aa(set(product_prod(set(product_prod(A,A)),set(product_prod(C,C)))),$o,member(product_prod(set(product_prod(A,A)),set(product_prod(C,C))),aa(set(product_prod(C,C)),product_prod(set(product_prod(A,A)),set(product_prod(C,C))),aa(set(product_prod(A,A)),fun(set(product_prod(C,C)),product_prod(set(product_prod(A,A)),set(product_prod(C,C)))),product_Pair(set(product_prod(A,A)),set(product_prod(C,C))),R2),R8)),bNF_We4044943003108391690rdLess(A,C)) ) ) ).

% ordLess_ordLeq_trans
tff(fact_5929_ordLeq__iff__ordLess__or__ordIso,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,A)),R3: set(product_prod(B,B))] :
      ( aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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))),R2),R3)),bNF_Wellorder_ordLeq(A,B))
    <=> ( aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R2),R3)),bNF_We4044943003108391690rdLess(A,B))
        | aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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))),R2),R3)),bNF_Wellorder_ordIso(A,B)) ) ) ).

% ordLeq_iff_ordLess_or_ordIso
tff(fact_5930_iso__forward,axiom,
    ! [A: $tType,B: $tType,X: A,Y: A,R2: set(product_prod(A,A)),R3: set(product_prod(B,B)),F: fun(A,B)] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R2)
     => ( bNF_Wellorder_iso(A,B,R2,R3,F)
       => aa(set(product_prod(B,B)),$o,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,F,X)),aa(A,B,F,Y))),R3) ) ) ).

% iso_forward
tff(fact_5931_ordLeq__ordIso__trans,axiom,
    ! [B: $tType,C: $tType,A: $tType,R2: set(product_prod(A,A)),R3: set(product_prod(B,B)),R8: set(product_prod(C,C))] :
      ( aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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))),R2),R3)),bNF_Wellorder_ordLeq(A,B))
     => ( aa(set(product_prod(set(product_prod(B,B)),set(product_prod(C,C)))),$o,member(product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(B,B)),fun(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C)))),product_Pair(set(product_prod(B,B)),set(product_prod(C,C))),R3),R8)),bNF_Wellorder_ordIso(B,C))
       => aa(set(product_prod(set(product_prod(A,A)),set(product_prod(C,C)))),$o,member(product_prod(set(product_prod(A,A)),set(product_prod(C,C))),aa(set(product_prod(C,C)),product_prod(set(product_prod(A,A)),set(product_prod(C,C))),aa(set(product_prod(A,A)),fun(set(product_prod(C,C)),product_prod(set(product_prod(A,A)),set(product_prod(C,C)))),product_Pair(set(product_prod(A,A)),set(product_prod(C,C))),R2),R8)),bNF_Wellorder_ordLeq(A,C)) ) ) ).

% ordLeq_ordIso_trans
tff(fact_5932_ordIso__ordLeq__trans,axiom,
    ! [B: $tType,C: $tType,A: $tType,R2: set(product_prod(A,A)),R3: set(product_prod(B,B)),R8: set(product_prod(C,C))] :
      ( aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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))),R2),R3)),bNF_Wellorder_ordIso(A,B))
     => ( aa(set(product_prod(set(product_prod(B,B)),set(product_prod(C,C)))),$o,member(product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(B,B)),fun(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C)))),product_Pair(set(product_prod(B,B)),set(product_prod(C,C))),R3),R8)),bNF_Wellorder_ordLeq(B,C))
       => aa(set(product_prod(set(product_prod(A,A)),set(product_prod(C,C)))),$o,member(product_prod(set(product_prod(A,A)),set(product_prod(C,C))),aa(set(product_prod(C,C)),product_prod(set(product_prod(A,A)),set(product_prod(C,C))),aa(set(product_prod(A,A)),fun(set(product_prod(C,C)),product_prod(set(product_prod(A,A)),set(product_prod(C,C)))),product_Pair(set(product_prod(A,A)),set(product_prod(C,C))),R2),R8)),bNF_Wellorder_ordLeq(A,C)) ) ) ).

% ordIso_ordLeq_trans
tff(fact_5933_ordLeq__transitive,axiom,
    ! [B: $tType,C: $tType,A: $tType,R2: set(product_prod(A,A)),R3: set(product_prod(B,B)),R8: set(product_prod(C,C))] :
      ( aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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))),R2),R3)),bNF_Wellorder_ordLeq(A,B))
     => ( aa(set(product_prod(set(product_prod(B,B)),set(product_prod(C,C)))),$o,member(product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(B,B)),fun(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C)))),product_Pair(set(product_prod(B,B)),set(product_prod(C,C))),R3),R8)),bNF_Wellorder_ordLeq(B,C))
       => aa(set(product_prod(set(product_prod(A,A)),set(product_prod(C,C)))),$o,member(product_prod(set(product_prod(A,A)),set(product_prod(C,C))),aa(set(product_prod(C,C)),product_prod(set(product_prod(A,A)),set(product_prod(C,C))),aa(set(product_prod(A,A)),fun(set(product_prod(C,C)),product_prod(set(product_prod(A,A)),set(product_prod(C,C)))),product_Pair(set(product_prod(A,A)),set(product_prod(C,C))),R2),R8)),bNF_Wellorder_ordLeq(A,C)) ) ) ).

% ordLeq_transitive
tff(fact_5934_ordIso__transitive,axiom,
    ! [B: $tType,C: $tType,A: $tType,R2: set(product_prod(A,A)),R3: set(product_prod(B,B)),R8: set(product_prod(C,C))] :
      ( aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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))),R2),R3)),bNF_Wellorder_ordIso(A,B))
     => ( aa(set(product_prod(set(product_prod(B,B)),set(product_prod(C,C)))),$o,member(product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(B,B)),fun(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C)))),product_Pair(set(product_prod(B,B)),set(product_prod(C,C))),R3),R8)),bNF_Wellorder_ordIso(B,C))
       => aa(set(product_prod(set(product_prod(A,A)),set(product_prod(C,C)))),$o,member(product_prod(set(product_prod(A,A)),set(product_prod(C,C))),aa(set(product_prod(C,C)),product_prod(set(product_prod(A,A)),set(product_prod(C,C))),aa(set(product_prod(A,A)),fun(set(product_prod(C,C)),product_prod(set(product_prod(A,A)),set(product_prod(C,C)))),product_Pair(set(product_prod(A,A)),set(product_prod(C,C))),R2),R8)),bNF_Wellorder_ordIso(A,C)) ) ) ).

% ordIso_transitive
tff(fact_5935_ordIso__imp__ordLeq,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,A)),R3: set(product_prod(B,B))] :
      ( aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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))),R2),R3)),bNF_Wellorder_ordIso(A,B))
     => aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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))),R2),R3)),bNF_Wellorder_ordLeq(A,B)) ) ).

% ordIso_imp_ordLeq
tff(fact_5936_ordIso__iff__ordLeq,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(A,A)),R3: set(product_prod(B,B))] :
      ( aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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))),R2),R3)),bNF_Wellorder_ordIso(A,B))
    <=> ( aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R2),R3)),bNF_Wellorder_ordLeq(A,B))
        & aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),$o,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))),R3),R2)),bNF_Wellorder_ordLeq(B,A)) ) ) ).

% ordIso_iff_ordLeq
tff(fact_5937_ordIso__symmetric,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(A,A)),R3: set(product_prod(B,B))] :
      ( aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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))),R2),R3)),bNF_Wellorder_ordIso(A,B))
     => aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),$o,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))),R3),R2)),bNF_Wellorder_ordIso(B,A)) ) ).

% ordIso_symmetric
tff(fact_5938_internalize__ordLeq,axiom,
    ! [A: $tType,B: $tType,R3: set(product_prod(A,A)),R2: set(product_prod(B,B))] :
      ( aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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))),R3),R2)),bNF_Wellorder_ordLeq(A,B))
    <=> ? [P6: set(product_prod(B,B))] :
          ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),field2(B,P6)),field2(B,R2))
          & aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R3),P6)),bNF_Wellorder_ordIso(A,B))
          & aa(set(product_prod(set(product_prod(B,B)),set(product_prod(B,B)))),$o,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))),P6),R2)),bNF_Wellorder_ordLeq(B,B)) ) ) ).

% internalize_ordLeq
tff(fact_5939_ordIso__def,axiom,
    ! [B: $tType,A: $tType] : bNF_Wellorder_ordIso(A,B) = aa(fun(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),$o),set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),collect(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),aa(fun(set(product_prod(A,A)),fun(set(product_prod(B,B)),$o)),fun(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),$o),product_case_prod(set(product_prod(A,A)),set(product_prod(B,B)),$o),aTP_Lamp_amq(set(product_prod(A,A)),fun(set(product_prod(B,B)),$o)))) ).

% ordIso_def
tff(fact_5940_finite__well__order__on__ordIso,axiom,
    ! [A: $tType,A4: set(A),R2: set(product_prod(A,A)),R3: set(product_prod(A,A))] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( order_well_order_on(A,A4,R2)
       => ( order_well_order_on(A,A4,R3)
         => aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),$o,member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R2),R3)),bNF_Wellorder_ordIso(A,A)) ) ) ) ).

% finite_well_order_on_ordIso
tff(fact_5941_ordLeq__total,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(A,A)),R3: set(product_prod(B,B))] :
      ( order_well_order_on(A,field2(A,R2),R2)
     => ( order_well_order_on(B,field2(B,R3),R3)
       => ( aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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))),R2),R3)),bNF_Wellorder_ordLeq(A,B))
          | aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),$o,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))),R3),R2)),bNF_Wellorder_ordLeq(B,A)) ) ) ) ).

% ordLeq_total
tff(fact_5942_ordLeq__reflexive,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( order_well_order_on(A,field2(A,R2),R2)
     => aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),$o,member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R2),R2)),bNF_Wellorder_ordLeq(A,A)) ) ).

% ordLeq_reflexive
tff(fact_5943_ordLeq__Well__order__simp,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(A,A)),R3: set(product_prod(B,B))] :
      ( aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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))),R2),R3)),bNF_Wellorder_ordLeq(A,B))
     => ( order_well_order_on(A,field2(A,R2),R2)
        & order_well_order_on(B,field2(B,R3),R3) ) ) ).

% ordLeq_Well_order_simp
tff(fact_5944_exists__minim__Well__order,axiom,
    ! [A: $tType,R: set(set(product_prod(A,A)))] :
      ( ( R != bot_bot(set(set(product_prod(A,A)))) )
     => ( ! [X2: set(product_prod(A,A))] :
            ( aa(set(set(product_prod(A,A))),$o,member(set(product_prod(A,A)),X2),R)
           => order_well_order_on(A,field2(A,X2),X2) )
       => ? [X2: set(product_prod(A,A))] :
            ( aa(set(set(product_prod(A,A))),$o,member(set(product_prod(A,A)),X2),R)
            & ! [Xa3: set(product_prod(A,A))] :
                ( aa(set(set(product_prod(A,A))),$o,member(set(product_prod(A,A)),Xa3),R)
               => aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),$o,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))),X2),Xa3)),bNF_Wellorder_ordLeq(A,A)) ) ) ) ) ).

% exists_minim_Well_order
tff(fact_5945_ordIso__reflexive,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( order_well_order_on(A,field2(A,R2),R2)
     => aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),$o,member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R2),R2)),bNF_Wellorder_ordIso(A,A)) ) ).

% ordIso_reflexive
tff(fact_5946_Well__order__iso__copy,axiom,
    ! [B: $tType,A: $tType,A4: set(A),R2: set(product_prod(A,A)),F: fun(A,B),A14: set(B)] :
      ( order_well_order_on(A,A4,R2)
     => ( bij_betw(A,B,F,A4,A14)
       => ? [R9: set(product_prod(B,B))] :
            ( order_well_order_on(B,A14,R9)
            & aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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))),R2),R9)),bNF_Wellorder_ordIso(A,B)) ) ) ) ).

% Well_order_iso_copy
tff(fact_5947_iso__iff2,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(A,A)),R3: set(product_prod(B,B)),F: fun(A,B)] :
      ( bNF_Wellorder_iso(A,B,R2,R3,F)
    <=> ( bij_betw(A,B,F,field2(A,R2),field2(B,R3))
        & ! [X4: A] :
            ( aa(set(A),$o,member(A,X4),field2(A,R2))
           => ! [Xa2: A] :
                ( aa(set(A),$o,member(A,Xa2),field2(A,R2))
               => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Xa2)),R2)
                <=> aa(set(product_prod(B,B)),$o,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,F,X4)),aa(A,B,F,Xa2))),R3) ) ) ) ) ) ).

% iso_iff2
tff(fact_5948_internalize__ordLess,axiom,
    ! [A: $tType,B: $tType,R3: set(product_prod(A,A)),R2: set(product_prod(B,B))] :
      ( aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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))),R3),R2)),bNF_We4044943003108391690rdLess(A,B))
    <=> ? [P6: set(product_prod(B,B))] :
          ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less(set(B)),field2(B,P6)),field2(B,R2))
          & aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R3),P6)),bNF_Wellorder_ordIso(A,B))
          & aa(set(product_prod(set(product_prod(B,B)),set(product_prod(B,B)))),$o,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))),P6),R2)),bNF_We4044943003108391690rdLess(B,B)) ) ) ).

% internalize_ordLess
tff(fact_5949_not__ordLess__iff__ordLeq,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,A)),R3: set(product_prod(B,B))] :
      ( order_well_order_on(A,field2(A,R2),R2)
     => ( order_well_order_on(B,field2(B,R3),R3)
       => ( ~ aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),$o,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))),R3),R2)),bNF_We4044943003108391690rdLess(B,A))
        <=> aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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))),R2),R3)),bNF_Wellorder_ordLeq(A,B)) ) ) ) ).

% not_ordLess_iff_ordLeq
tff(fact_5950_not__ordLeq__iff__ordLess,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,A)),R3: set(product_prod(B,B))] :
      ( order_well_order_on(A,field2(A,R2),R2)
     => ( order_well_order_on(B,field2(B,R3),R3)
       => ( ~ aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),$o,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))),R3),R2)),bNF_Wellorder_ordLeq(B,A))
        <=> aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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))),R2),R3)),bNF_We4044943003108391690rdLess(A,B)) ) ) ) ).

% not_ordLeq_iff_ordLess
tff(fact_5951_ordLess__or__ordLeq,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(A,A)),R3: set(product_prod(B,B))] :
      ( order_well_order_on(A,field2(A,R2),R2)
     => ( order_well_order_on(B,field2(B,R3),R3)
       => ( aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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))),R2),R3)),bNF_We4044943003108391690rdLess(A,B))
          | aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),$o,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))),R3),R2)),bNF_Wellorder_ordLeq(B,A)) ) ) ) ).

% ordLess_or_ordLeq
tff(fact_5952_ofilter__subset__ordLeq,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A4: set(A),B3: set(A)] :
      ( order_well_order_on(A,field2(A,R2),R2)
     => ( order_ofilter(A,R2,A4)
       => ( order_ofilter(A,R2,B3)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B3)
          <=> aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),$o,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))),R2),product_Sigma(A,A,A4,aTP_Lamp_rl(set(A),fun(A,set(A)),A4)))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R2),product_Sigma(A,A,B3,aTP_Lamp_rl(set(A),fun(A,set(A)),B3))))),bNF_Wellorder_ordLeq(A,A)) ) ) ) ) ).

% ofilter_subset_ordLeq
tff(fact_5953_ofilter__embed,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A4: set(A)] :
      ( order_well_order_on(A,field2(A,R2),R2)
     => ( order_ofilter(A,R2,A4)
      <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),field2(A,R2))
          & aa(fun(A,A),$o,bNF_Wellorder_embed(A,A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R2),product_Sigma(A,A,A4,aTP_Lamp_rl(set(A),fun(A,set(A)),A4))),R2),id(A)) ) ) ) ).

% ofilter_embed
tff(fact_5954_ofilter__subset__embed,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A4: set(A),B3: set(A)] :
      ( order_well_order_on(A,field2(A,R2),R2)
     => ( order_ofilter(A,R2,A4)
       => ( order_ofilter(A,R2,B3)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B3)
          <=> aa(fun(A,A),$o,bNF_Wellorder_embed(A,A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R2),product_Sigma(A,A,A4,aTP_Lamp_rl(set(A),fun(A,set(A)),A4))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R2),product_Sigma(A,A,B3,aTP_Lamp_rl(set(A),fun(A,set(A)),B3)))),id(A)) ) ) ) ) ).

% ofilter_subset_embed
tff(fact_5955_embed__ordLess__ofilterIncl,axiom,
    ! [B: $tType,A: $tType,C: $tType,R12: set(product_prod(A,A)),R23: set(product_prod(B,B)),R32: set(product_prod(C,C)),F132: fun(A,C),F232: fun(B,C)] :
      ( aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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))),R12),R23)),bNF_We4044943003108391690rdLess(A,B))
     => ( aa(set(product_prod(set(product_prod(B,B)),set(product_prod(C,C)))),$o,member(product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(B,B)),fun(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C)))),product_Pair(set(product_prod(B,B)),set(product_prod(C,C))),R23),R32)),bNF_We4044943003108391690rdLess(B,C))
       => ( aa(fun(A,C),$o,bNF_Wellorder_embed(A,C,R12,R32),F132)
         => ( aa(fun(B,C),$o,bNF_Wellorder_embed(B,C,R23,R32),F232)
           => aa(set(product_prod(set(C),set(C))),$o,member(product_prod(set(C),set(C)),aa(set(C),product_prod(set(C),set(C)),aa(set(C),fun(set(C),product_prod(set(C),set(C))),product_Pair(set(C),set(C)),aa(set(A),set(C),image2(A,C,F132),field2(A,R12))),aa(set(B),set(C),image2(B,C,F232),field2(B,R23)))),bNF_We413866401316099525erIncl(C,R32)) ) ) ) ) ).

% embed_ordLess_ofilterIncl
tff(fact_5956_ordLess__not__embed,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(A,A)),R3: set(product_prod(B,B))] :
      ( aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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))),R2),R3)),bNF_We4044943003108391690rdLess(A,B))
     => ~ ? [X_13: fun(B,A)] : aa(fun(B,A),$o,bNF_Wellorder_embed(B,A,R3,R2),X_13) ) ).

% ordLess_not_embed
tff(fact_5957_iso__defs_I2_J,axiom,
    ! [A: $tType,B: $tType,X3: set(product_prod(A,A)),Xa3: set(product_prod(B,B)),Xb2: fun(A,B)] :
      ( bNF_Wellorder_iso(A,B,X3,Xa3,Xb2)
    <=> ( aa(fun(A,B),$o,bNF_Wellorder_embed(A,B,X3,Xa3),Xb2)
        & bij_betw(A,B,Xb2,field2(A,X3),field2(B,Xa3)) ) ) ).

% iso_defs(2)
tff(fact_5958_BNF__Wellorder__Constructions_OordLess__Field,axiom,
    ! [A: $tType,B: $tType,R12: set(product_prod(A,A)),R23: set(product_prod(B,B)),F: fun(A,B)] :
      ( aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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))),R12),R23)),bNF_We4044943003108391690rdLess(A,B))
     => ( aa(fun(A,B),$o,bNF_Wellorder_embed(A,B,R12,R23),F)
       => ( aa(set(A),set(B),image2(A,B,F),field2(A,R12)) != field2(B,R23) ) ) ) ).

% BNF_Wellorder_Constructions.ordLess_Field
tff(fact_5959_embedS__defs_I2_J,axiom,
    ! [A: $tType,B: $tType,X3: set(product_prod(A,A)),Xa3: set(product_prod(B,B)),Xb2: fun(A,B)] :
      ( bNF_Wellorder_embedS(A,B,X3,Xa3,Xb2)
    <=> ( aa(fun(A,B),$o,bNF_Wellorder_embed(A,B,X3,Xa3),Xb2)
        & ~ bij_betw(A,B,Xb2,field2(A,X3),field2(B,Xa3)) ) ) ).

% embedS_defs(2)
tff(fact_5960_ordLess__iff,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(A,A)),R3: set(product_prod(B,B))] :
      ( aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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))),R2),R3)),bNF_We4044943003108391690rdLess(A,B))
    <=> ( order_well_order_on(A,field2(A,R2),R2)
        & order_well_order_on(B,field2(B,R3),R3)
        & ~ ? [X_12: fun(B,A)] : aa(fun(B,A),$o,bNF_Wellorder_embed(B,A,R3,R2),X_12) ) ) ).

% ordLess_iff
tff(fact_5961_embed__defs_I2_J,axiom,
    ! [B: $tType,A: $tType,X3: set(product_prod(A,A)),Xa3: set(product_prod(B,B)),Xb2: fun(A,B)] :
      ( aa(fun(A,B),$o,bNF_Wellorder_embed(A,B,X3,Xa3),Xb2)
    <=> ! [Xc2: A] :
          ( aa(set(A),$o,member(A,Xc2),field2(A,X3))
         => bij_betw(A,B,Xb2,aa(A,set(A),order_under(A,X3),Xc2),aa(B,set(B),order_under(B,Xa3),aa(A,B,Xb2,Xc2))) ) ) ).

% embed_defs(2)
tff(fact_5962_ordLeq__def,axiom,
    ! [B: $tType,A: $tType] : bNF_Wellorder_ordLeq(A,B) = aa(fun(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),$o),set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),collect(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),aa(fun(set(product_prod(A,A)),fun(set(product_prod(B,B)),$o)),fun(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),$o),product_case_prod(set(product_prod(A,A)),set(product_prod(B,B)),$o),aTP_Lamp_amr(set(product_prod(A,A)),fun(set(product_prod(B,B)),$o)))) ).

% ordLeq_def
tff(fact_5963_embed__implies__iso__Restr,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(A,A)),R3: set(product_prod(B,B)),F: fun(B,A)] :
      ( order_well_order_on(A,field2(A,R2),R2)
     => ( order_well_order_on(B,field2(B,R3),R3)
       => ( aa(fun(B,A),$o,bNF_Wellorder_embed(B,A,R3,R2),F)
         => bNF_Wellorder_iso(B,A,R3,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R2),product_Sigma(A,A,aa(set(B),set(A),image2(B,A,F),field2(B,R3)),aa(fun(B,A),fun(A,set(A)),aTP_Lamp_ams(set(product_prod(B,B)),fun(fun(B,A),fun(A,set(A))),R3),F))),F) ) ) ) ).

% embed_implies_iso_Restr
tff(fact_5964_comp__set__bd__Union__o__collect,axiom,
    ! [C: $tType,B: $tType,A: $tType,X: C,X5: set(fun(C,set(set(A)))),Hbd: set(product_prod(B,B))] :
      ( aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(set(set(A))),set(set(A)),complete_Sup_Sup(set(set(A))),aa(set(fun(C,set(set(A)))),set(set(set(A))),image2(fun(C,set(set(A))),set(set(A)),aTP_Lamp_amt(C,fun(fun(C,set(set(A))),set(set(A))),X)),X5))))),Hbd)),bNF_Wellorder_ordLeq(A,B))
     => aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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,aa(C,set(A),aa(fun(C,set(set(A))),fun(C,set(A)),comp(set(set(A)),set(A),C,complete_Sup_Sup(set(A))),bNF_collect(C,set(A),X5)),X))),Hbd)),bNF_Wellorder_ordLeq(A,B)) ) ).

% comp_set_bd_Union_o_collect
tff(fact_5965_dir__image__ordIso,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,A)),F: fun(A,B)] :
      ( order_well_order_on(A,field2(A,R2),R2)
     => ( inj_on(A,B,F,field2(A,R2))
       => aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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))),R2),bNF_We2720479622203943262_image(A,B,R2,F))),bNF_Wellorder_ordIso(A,B)) ) ) ).

% dir_image_ordIso
tff(fact_5966_card__of__bool,axiom,
    ! [A: $tType,A1: A,A22: A] :
      ( ( A1 != A22 )
     => aa(set(product_prod(set(product_prod($o,$o)),set(product_prod(A,A)))),$o,member(product_prod(set(product_prod($o,$o)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod($o,$o)),set(product_prod(A,A))),aa(set(product_prod($o,$o)),fun(set(product_prod(A,A)),product_prod(set(product_prod($o,$o)),set(product_prod(A,A)))),product_Pair(set(product_prod($o,$o)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of($o,top_top(set($o)))),bNF_Ca6860139660246222851ard_of(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A1),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A22),bot_bot(set(A))))))),bNF_Wellorder_ordIso($o,A)) ) ).

% card_of_bool
tff(fact_5967_card__of__Times1,axiom,
    ! [A: $tType,B: $tType,A4: set(A),B3: set(B)] :
      ( ( A4 != bot_bot(set(A)) )
     => aa(set(product_prod(set(product_prod(B,B)),set(product_prod(product_prod(B,A),product_prod(B,A))))),$o,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,B3)),bNF_Ca6860139660246222851ard_of(product_prod(B,A),product_Sigma(B,A,B3,aTP_Lamp_ke(set(A),fun(B,set(A)),A4))))),bNF_Wellorder_ordLeq(B,product_prod(B,A))) ) ).

% card_of_Times1
tff(fact_5968_card__of__Times2,axiom,
    ! [A: $tType,B: $tType,A4: set(A),B3: set(B)] :
      ( ( A4 != bot_bot(set(A)) )
     => aa(set(product_prod(set(product_prod(B,B)),set(product_prod(product_prod(A,B),product_prod(A,B))))),$o,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,B3)),bNF_Ca6860139660246222851ard_of(product_prod(A,B),product_Sigma(A,B,A4,aTP_Lamp_qz(set(B),fun(A,set(B)),B3))))),bNF_Wellorder_ordLeq(B,product_prod(A,B))) ) ).

% card_of_Times2
tff(fact_5969_card__of__Pow,axiom,
    ! [A: $tType,A4: set(A)] : aa(set(product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A))))),$o,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,A4)),bNF_Ca6860139660246222851ard_of(set(A),pow2(A,A4)))),bNF_We4044943003108391690rdLess(A,set(A))) ).

% card_of_Pow
tff(fact_5970_infinite__iff__card__of__nat,axiom,
    ! [A: $tType,A4: set(A)] :
      ( ~ aa(set(A),$o,finite_finite2(A),A4)
    <=> aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(A,A)))),$o,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,A4))),bNF_Wellorder_ordLeq(nat,A)) ) ).

% infinite_iff_card_of_nat
tff(fact_5971_card__of__Sigma__ordLeq__infinite,axiom,
    ! [A: $tType,C: $tType,B: $tType,B3: set(A),I4: set(B),A4: fun(B,set(C))] :
      ( ~ aa(set(A),$o,finite_finite2(A),B3)
     => ( aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),$o,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,I4)),bNF_Ca6860139660246222851ard_of(A,B3))),bNF_Wellorder_ordLeq(B,A))
       => ( ! [X2: B] :
              ( aa(set(B),$o,member(B,X2),I4)
             => aa(set(product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),$o,member(product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(C,C)),fun(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),product_Pair(set(product_prod(C,C)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(C,aa(B,set(C),A4,X2))),bNF_Ca6860139660246222851ard_of(A,B3))),bNF_Wellorder_ordLeq(C,A)) )
         => aa(set(product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A)))),$o,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,I4,A4))),bNF_Ca6860139660246222851ard_of(A,B3))),bNF_Wellorder_ordLeq(product_prod(B,C),A)) ) ) ) ).

% card_of_Sigma_ordLeq_infinite
tff(fact_5972_card__of__Times__same__infinite,axiom,
    ! [A: $tType,A4: set(A)] :
      ( ~ aa(set(A),$o,finite_finite2(A),A4)
     => aa(set(product_prod(set(product_prod(product_prod(A,A),product_prod(A,A))),set(product_prod(A,A)))),$o,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,A4,aTP_Lamp_rl(set(A),fun(A,set(A)),A4)))),bNF_Ca6860139660246222851ard_of(A,A4))),bNF_Wellorder_ordIso(product_prod(A,A),A)) ) ).

% card_of_Times_same_infinite
tff(fact_5973_card__of__Times__commute,axiom,
    ! [B: $tType,A: $tType,A4: set(A),B3: set(B)] : aa(set(product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(product_prod(B,A),product_prod(B,A))))),$o,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,A4,aTP_Lamp_qz(set(B),fun(A,set(B)),B3)))),bNF_Ca6860139660246222851ard_of(product_prod(B,A),product_Sigma(B,A,B3,aTP_Lamp_ke(set(A),fun(B,set(A)),A4))))),bNF_Wellorder_ordIso(product_prod(A,B),product_prod(B,A))) ).

% card_of_Times_commute
tff(fact_5974_card__of__Times__mono2,axiom,
    ! [B: $tType,A: $tType,C: $tType,A4: set(A),B3: set(B),C3: set(C)] :
      ( aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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,A4)),bNF_Ca6860139660246222851ard_of(B,B3))),bNF_Wellorder_ordLeq(A,B))
     => 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))))),$o,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,C3,aTP_Lamp_amu(set(A),fun(C,set(A)),A4)))),bNF_Ca6860139660246222851ard_of(product_prod(C,B),product_Sigma(C,B,C3,aTP_Lamp_amv(set(B),fun(C,set(B)),B3))))),bNF_Wellorder_ordLeq(product_prod(C,A),product_prod(C,B))) ) ).

% card_of_Times_mono2
tff(fact_5975_card__of__Times__mono1,axiom,
    ! [B: $tType,C: $tType,A: $tType,A4: set(A),B3: set(B),C3: set(C)] :
      ( aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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,A4)),bNF_Ca6860139660246222851ard_of(B,B3))),bNF_Wellorder_ordLeq(A,B))
     => 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))))),$o,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,A4,aTP_Lamp_us(set(C),fun(A,set(C)),C3)))),bNF_Ca6860139660246222851ard_of(product_prod(B,C),product_Sigma(B,C,B3,aTP_Lamp_rw(set(C),fun(B,set(C)),C3))))),bNF_Wellorder_ordLeq(product_prod(A,C),product_prod(B,C))) ) ).

% card_of_Times_mono1
tff(fact_5976_card__of__Sigma__mono1,axiom,
    ! [C: $tType,B: $tType,A: $tType,I4: set(A),A4: fun(A,set(B)),B3: fun(A,set(C))] :
      ( ! [X2: A] :
          ( aa(set(A),$o,member(A,X2),I4)
         => aa(set(product_prod(set(product_prod(B,B)),set(product_prod(C,C)))),$o,member(product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(B,B)),fun(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C)))),product_Pair(set(product_prod(B,B)),set(product_prod(C,C))),bNF_Ca6860139660246222851ard_of(B,aa(A,set(B),A4,X2))),bNF_Ca6860139660246222851ard_of(C,aa(A,set(C),B3,X2)))),bNF_Wellorder_ordLeq(B,C)) )
     => aa(set(product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(product_prod(A,C),product_prod(A,C))))),$o,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,I4,A4))),bNF_Ca6860139660246222851ard_of(product_prod(A,C),product_Sigma(A,C,I4,B3)))),bNF_Wellorder_ordLeq(product_prod(A,B),product_prod(A,C))) ) ).

% card_of_Sigma_mono1
tff(fact_5977_card__of__Times3,axiom,
    ! [A: $tType,A4: set(A)] : aa(set(product_prod(set(product_prod(A,A)),set(product_prod(product_prod(A,A),product_prod(A,A))))),$o,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,A4)),bNF_Ca6860139660246222851ard_of(product_prod(A,A),product_Sigma(A,A,A4,aTP_Lamp_rl(set(A),fun(A,set(A)),A4))))),bNF_Wellorder_ordLeq(A,product_prod(A,A))) ).

% card_of_Times3
tff(fact_5978_card__of__refl,axiom,
    ! [A: $tType,A4: set(A)] : aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),$o,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,A4)),bNF_Ca6860139660246222851ard_of(A,A4))),bNF_Wellorder_ordIso(A,A)) ).

% card_of_refl
tff(fact_5979_card__of__UNION__Sigma,axiom,
    ! [B: $tType,A: $tType,A4: fun(B,set(A)),I4: set(B)] : aa(set(product_prod(set(product_prod(A,A)),set(product_prod(product_prod(B,A),product_prod(B,A))))),$o,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)),aa(set(B),set(set(A)),image2(B,set(A),A4),I4)))),bNF_Ca6860139660246222851ard_of(product_prod(B,A),product_Sigma(B,A,I4,A4)))),bNF_Wellorder_ordLeq(A,product_prod(B,A))) ).

% card_of_UNION_Sigma
tff(fact_5980_card__of__Pow__Func,axiom,
    ! [A: $tType,A4: set(A)] : aa(set(product_prod(set(product_prod(set(A),set(A))),set(product_prod(fun(A,$o),fun(A,$o))))),$o,member(product_prod(set(product_prod(set(A),set(A))),set(product_prod(fun(A,$o),fun(A,$o)))),aa(set(product_prod(fun(A,$o),fun(A,$o))),product_prod(set(product_prod(set(A),set(A))),set(product_prod(fun(A,$o),fun(A,$o)))),aa(set(product_prod(set(A),set(A))),fun(set(product_prod(fun(A,$o),fun(A,$o))),product_prod(set(product_prod(set(A),set(A))),set(product_prod(fun(A,$o),fun(A,$o))))),product_Pair(set(product_prod(set(A),set(A))),set(product_prod(fun(A,$o),fun(A,$o)))),bNF_Ca6860139660246222851ard_of(set(A),pow2(A,A4))),bNF_Ca6860139660246222851ard_of(fun(A,$o),bNF_Wellorder_Func(A,$o,A4,top_top(set($o)))))),bNF_Wellorder_ordIso(set(A),fun(A,$o))) ).

% card_of_Pow_Func
tff(fact_5981_Func__Times__Range,axiom,
    ! [C: $tType,B: $tType,A: $tType,A4: set(A),B3: set(B),C3: set(C)] : aa(set(product_prod(set(product_prod(fun(A,product_prod(B,C)),fun(A,product_prod(B,C)))),set(product_prod(product_prod(fun(A,B),fun(A,C)),product_prod(fun(A,B),fun(A,C)))))),$o,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),A4,product_Sigma(B,C,B3,aTP_Lamp_rw(set(C),fun(B,set(C)),C3))))),bNF_Ca6860139660246222851ard_of(product_prod(fun(A,B),fun(A,C)),product_Sigma(fun(A,B),fun(A,C),bNF_Wellorder_Func(A,B,A4,B3),aa(set(C),fun(fun(A,B),set(fun(A,C))),aTP_Lamp_amw(set(A),fun(set(C),fun(fun(A,B),set(fun(A,C)))),A4),C3))))),bNF_Wellorder_ordIso(fun(A,product_prod(B,C)),product_prod(fun(A,B),fun(A,C)))) ).

% Func_Times_Range
tff(fact_5982_card__of__Times__infinite,axiom,
    ! [A: $tType,B: $tType,A4: set(A),B3: set(B)] :
      ( ~ aa(set(A),$o,finite_finite2(A),A4)
     => ( ( B3 != bot_bot(set(B)) )
       => ( aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),$o,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,B3)),bNF_Ca6860139660246222851ard_of(A,A4))),bNF_Wellorder_ordLeq(B,A))
         => ( aa(set(product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A)))),$o,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,A4,aTP_Lamp_qz(set(B),fun(A,set(B)),B3)))),bNF_Ca6860139660246222851ard_of(A,A4))),bNF_Wellorder_ordIso(product_prod(A,B),A))
            & aa(set(product_prod(set(product_prod(product_prod(B,A),product_prod(B,A))),set(product_prod(A,A)))),$o,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,B3,aTP_Lamp_ke(set(A),fun(B,set(A)),A4)))),bNF_Ca6860139660246222851ard_of(A,A4))),bNF_Wellorder_ordIso(product_prod(B,A),A)) ) ) ) ) ).

% card_of_Times_infinite
tff(fact_5983_card__of__Times__infinite__simps_I1_J,axiom,
    ! [B: $tType,A: $tType,A4: set(A),B3: set(B)] :
      ( ~ aa(set(A),$o,finite_finite2(A),A4)
     => ( ( B3 != bot_bot(set(B)) )
       => ( aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),$o,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,B3)),bNF_Ca6860139660246222851ard_of(A,A4))),bNF_Wellorder_ordLeq(B,A))
         => aa(set(product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A)))),$o,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,A4,aTP_Lamp_qz(set(B),fun(A,set(B)),B3)))),bNF_Ca6860139660246222851ard_of(A,A4))),bNF_Wellorder_ordIso(product_prod(A,B),A)) ) ) ) ).

% card_of_Times_infinite_simps(1)
tff(fact_5984_card__of__Times__infinite__simps_I2_J,axiom,
    ! [B: $tType,A: $tType,A4: set(A),B3: set(B)] :
      ( ~ aa(set(A),$o,finite_finite2(A),A4)
     => ( ( B3 != bot_bot(set(B)) )
       => ( aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),$o,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,B3)),bNF_Ca6860139660246222851ard_of(A,A4))),bNF_Wellorder_ordLeq(B,A))
         => aa(set(product_prod(set(product_prod(A,A)),set(product_prod(product_prod(A,B),product_prod(A,B))))),$o,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,A4)),bNF_Ca6860139660246222851ard_of(product_prod(A,B),product_Sigma(A,B,A4,aTP_Lamp_qz(set(B),fun(A,set(B)),B3))))),bNF_Wellorder_ordIso(A,product_prod(A,B))) ) ) ) ).

% card_of_Times_infinite_simps(2)
tff(fact_5985_card__of__Times__infinite__simps_I3_J,axiom,
    ! [A: $tType,B: $tType,A4: set(A),B3: set(B)] :
      ( ~ aa(set(A),$o,finite_finite2(A),A4)
     => ( ( B3 != bot_bot(set(B)) )
       => ( aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),$o,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,B3)),bNF_Ca6860139660246222851ard_of(A,A4))),bNF_Wellorder_ordLeq(B,A))
         => aa(set(product_prod(set(product_prod(product_prod(B,A),product_prod(B,A))),set(product_prod(A,A)))),$o,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,B3,aTP_Lamp_ke(set(A),fun(B,set(A)),A4)))),bNF_Ca6860139660246222851ard_of(A,A4))),bNF_Wellorder_ordIso(product_prod(B,A),A)) ) ) ) ).

% card_of_Times_infinite_simps(3)
tff(fact_5986_card__of__Times__infinite__simps_I4_J,axiom,
    ! [B: $tType,A: $tType,A4: set(A),B3: set(B)] :
      ( ~ aa(set(A),$o,finite_finite2(A),A4)
     => ( ( B3 != bot_bot(set(B)) )
       => ( aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),$o,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,B3)),bNF_Ca6860139660246222851ard_of(A,A4))),bNF_Wellorder_ordLeq(B,A))
         => aa(set(product_prod(set(product_prod(A,A)),set(product_prod(product_prod(B,A),product_prod(B,A))))),$o,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,A4)),bNF_Ca6860139660246222851ard_of(product_prod(B,A),product_Sigma(B,A,B3,aTP_Lamp_ke(set(A),fun(B,set(A)),A4))))),bNF_Wellorder_ordIso(A,product_prod(B,A))) ) ) ) ).

% card_of_Times_infinite_simps(4)
tff(fact_5987_card__of__Func__Times,axiom,
    ! [C: $tType,B: $tType,A: $tType,A4: set(A),B3: set(B),C3: set(C)] : aa(set(product_prod(set(product_prod(fun(product_prod(A,B),C),fun(product_prod(A,B),C))),set(product_prod(fun(A,fun(B,C)),fun(A,fun(B,C)))))),$o,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,A4,aTP_Lamp_qz(set(B),fun(A,set(B)),B3)),C3))),bNF_Ca6860139660246222851ard_of(fun(A,fun(B,C)),bNF_Wellorder_Func(A,fun(B,C),A4,bNF_Wellorder_Func(B,C,B3,C3))))),bNF_Wellorder_ordIso(fun(product_prod(A,B),C),fun(A,fun(B,C)))) ).

% card_of_Func_Times
tff(fact_5988_card__of__image,axiom,
    ! [B: $tType,A: $tType,F: fun(B,A),A4: set(B)] : aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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,aa(set(B),set(A),image2(B,A,F),A4))),bNF_Ca6860139660246222851ard_of(B,A4))),bNF_Wellorder_ordLeq(A,B)) ).

% card_of_image
tff(fact_5989_card__of__empty3,axiom,
    ! [B: $tType,A: $tType,A4: set(A)] :
      ( aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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,A4)),bNF_Ca6860139660246222851ard_of(B,bot_bot(set(B))))),bNF_Wellorder_ordLeq(A,B))
     => ( A4 = bot_bot(set(A)) ) ) ).

% card_of_empty3
tff(fact_5990_card__of__empty,axiom,
    ! [B: $tType,A: $tType,A4: set(B)] : aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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,A4))),bNF_Wellorder_ordLeq(A,B)) ).

% card_of_empty
tff(fact_5991_card__of__ordLeq__infinite,axiom,
    ! [A: $tType,B: $tType,A4: set(A),B3: set(B)] :
      ( aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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,A4)),bNF_Ca6860139660246222851ard_of(B,B3))),bNF_Wellorder_ordLeq(A,B))
     => ( ~ aa(set(A),$o,finite_finite2(A),A4)
       => ~ aa(set(B),$o,finite_finite2(B),B3) ) ) ).

% card_of_ordLeq_infinite
tff(fact_5992_card__of__ordLeq__finite,axiom,
    ! [B: $tType,A: $tType,A4: set(A),B3: set(B)] :
      ( aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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,A4)),bNF_Ca6860139660246222851ard_of(B,B3))),bNF_Wellorder_ordLeq(A,B))
     => ( aa(set(B),$o,finite_finite2(B),B3)
       => aa(set(A),$o,finite_finite2(A),A4) ) ) ).

% card_of_ordLeq_finite
tff(fact_5993_card__of__mono1,axiom,
    ! [A: $tType,A4: set(A),B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B3)
     => aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),$o,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,A4)),bNF_Ca6860139660246222851ard_of(A,B3))),bNF_Wellorder_ordLeq(A,A)) ) ).

% card_of_mono1
tff(fact_5994_card__of__empty__ordIso,axiom,
    ! [B: $tType,A: $tType] : aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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_5995_card__of__empty2,axiom,
    ! [B: $tType,A: $tType,A4: set(A)] :
      ( aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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,A4)),bNF_Ca6860139660246222851ard_of(B,bot_bot(set(B))))),bNF_Wellorder_ordIso(A,B))
     => ( A4 = bot_bot(set(A)) ) ) ).

% card_of_empty2
tff(fact_5996_card__of__ordIso__finite,axiom,
    ! [A: $tType,B: $tType,A4: set(A),B3: set(B)] :
      ( aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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,A4)),bNF_Ca6860139660246222851ard_of(B,B3))),bNF_Wellorder_ordIso(A,B))
     => ( aa(set(A),$o,finite_finite2(A),A4)
      <=> aa(set(B),$o,finite_finite2(B),B3) ) ) ).

% card_of_ordIso_finite
tff(fact_5997_card__of__mono2,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,A)),R3: set(product_prod(B,B))] :
      ( aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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))),R2),R3)),bNF_Wellorder_ordLeq(A,B))
     => aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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,R2))),bNF_Ca6860139660246222851ard_of(B,field2(B,R3)))),bNF_Wellorder_ordLeq(A,B)) ) ).

% card_of_mono2
tff(fact_5998_card__of__cong,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,A)),R3: set(product_prod(B,B))] :
      ( aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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))),R2),R3)),bNF_Wellorder_ordIso(A,B))
     => aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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,R2))),bNF_Ca6860139660246222851ard_of(B,field2(B,R3)))),bNF_Wellorder_ordIso(A,B)) ) ).

% card_of_cong
tff(fact_5999_card__of__ordLeqI,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),A4: set(A),B3: set(B)] :
      ( inj_on(A,B,F,A4)
     => ( ! [A6: A] :
            ( aa(set(A),$o,member(A,A6),A4)
           => aa(set(B),$o,member(B,aa(A,B,F,A6)),B3) )
       => aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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,A4)),bNF_Ca6860139660246222851ard_of(B,B3))),bNF_Wellorder_ordLeq(A,B)) ) ) ).

% card_of_ordLeqI
tff(fact_6000_dir__image__def,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(B,B)),F: fun(B,A)] : bNF_We2720479622203943262_image(B,A,R2,F) = aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(B,A),fun(product_prod(A,A),$o),aTP_Lamp_amx(set(product_prod(B,B)),fun(fun(B,A),fun(product_prod(A,A),$o)),R2),F)) ).

% dir_image_def
tff(fact_6001_ex__bij__betw,axiom,
    ! [B: $tType,A: $tType,A4: set(A),R2: set(product_prod(B,B))] :
      ( aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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,A4)),R2)),bNF_Wellorder_ordLeq(A,B))
     => ? [F2: fun(B,A),B10: set(B)] : bij_betw(B,A,F2,B10,A4) ) ).

% ex_bij_betw
tff(fact_6002_card__of__least,axiom,
    ! [A: $tType,A4: set(A),R2: set(product_prod(A,A))] :
      ( order_well_order_on(A,A4,R2)
     => aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),$o,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,A4)),R2)),bNF_Wellorder_ordLeq(A,A)) ) ).

% card_of_least
tff(fact_6003_BNF__Cardinal__Order__Relation_OordLess__Field,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,A)),R3: set(product_prod(B,B))] :
      ( aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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))),R2),R3)),bNF_We4044943003108391690rdLess(A,B))
     => aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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,R2))),R3)),bNF_We4044943003108391690rdLess(A,B)) ) ).

% BNF_Cardinal_Order_Relation.ordLess_Field
tff(fact_6004_card__of__ordIsoI,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),A4: set(A),B3: set(B)] :
      ( bij_betw(A,B,F,A4,B3)
     => aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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,A4)),bNF_Ca6860139660246222851ard_of(B,B3))),bNF_Wellorder_ordIso(A,B)) ) ).

% card_of_ordIsoI
tff(fact_6005_card__of__ordIso,axiom,
    ! [B: $tType,A: $tType,A4: set(A),B3: set(B)] :
      ( ? [F6: fun(A,B)] : bij_betw(A,B,F6,A4,B3)
    <=> aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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,A4)),bNF_Ca6860139660246222851ard_of(B,B3))),bNF_Wellorder_ordIso(A,B)) ) ).

% card_of_ordIso
tff(fact_6006_type__copy__set__bd,axiom,
    ! [A: $tType,D: $tType,C: $tType,B: $tType,S: fun(A,set(B)),Bd: set(product_prod(C,C)),Rep: fun(D,A),X: D] :
      ( ! [Y2: A] : aa(set(product_prod(set(product_prod(B,B)),set(product_prod(C,C)))),$o,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),S,Y2))),Bd)),bNF_Wellorder_ordLeq(B,C))
     => aa(set(product_prod(set(product_prod(B,B)),set(product_prod(C,C)))),$o,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(D,set(B),aa(fun(D,A),fun(D,set(B)),comp(A,set(B),D,S),Rep),X))),Bd)),bNF_Wellorder_ordLeq(B,C)) ) ).

% type_copy_set_bd
tff(fact_6007_card__of__ordLeq2,axiom,
    ! [B: $tType,A: $tType,A4: set(A),B3: set(B)] :
      ( ( A4 != bot_bot(set(A)) )
     => ( ? [G5: fun(B,A)] : aa(set(B),set(A),image2(B,A,G5),B3) = A4
      <=> aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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,A4)),bNF_Ca6860139660246222851ard_of(B,B3))),bNF_Wellorder_ordLeq(A,B)) ) ) ).

% card_of_ordLeq2
tff(fact_6008_surj__imp__ordLeq,axiom,
    ! [B: $tType,A: $tType,B3: set(A),F: fun(B,A),A4: set(B)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),aa(set(B),set(A),image2(B,A,F),A4))
     => aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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,B3)),bNF_Ca6860139660246222851ard_of(B,A4))),bNF_Wellorder_ordLeq(A,B)) ) ).

% surj_imp_ordLeq
tff(fact_6009_card__of__singl__ordLeq,axiom,
    ! [A: $tType,B: $tType,A4: set(A),B2: B] :
      ( ( A4 != bot_bot(set(A)) )
     => aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),$o,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(B,fun(set(B),set(B)),insert2(B),B2),bot_bot(set(B))))),bNF_Ca6860139660246222851ard_of(A,A4))),bNF_Wellorder_ordLeq(B,A)) ) ).

% card_of_singl_ordLeq
tff(fact_6010_card__of__ordLess2,axiom,
    ! [A: $tType,B: $tType,B3: set(A),A4: set(B)] :
      ( ( B3 != bot_bot(set(A)) )
     => ( ~ ? [F6: fun(B,A)] : aa(set(B),set(A),image2(B,A,F6),A4) = B3
      <=> aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),$o,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,A4)),bNF_Ca6860139660246222851ard_of(A,B3))),bNF_We4044943003108391690rdLess(B,A)) ) ) ).

% card_of_ordLess2
tff(fact_6011_internalize__card__of__ordLeq2,axiom,
    ! [A: $tType,B: $tType,A4: set(A),C3: set(B)] :
      ( aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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,A4)),bNF_Ca6860139660246222851ard_of(B,C3))),bNF_Wellorder_ordLeq(A,B))
    <=> ? [B9: set(B)] :
          ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),B9),C3)
          & aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Ca6860139660246222851ard_of(B,B9))),bNF_Wellorder_ordIso(A,B))
          & aa(set(product_prod(set(product_prod(B,B)),set(product_prod(B,B)))),$o,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,B9)),bNF_Ca6860139660246222851ard_of(B,C3))),bNF_Wellorder_ordLeq(B,B)) ) ) ).

% internalize_card_of_ordLeq2
tff(fact_6012_card__of__Field__ordLess,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( order_well_order_on(A,field2(A,R2),R2)
     => aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),$o,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,R2))),R2)),bNF_Wellorder_ordLeq(A,A)) ) ).

% card_of_Field_ordLess
tff(fact_6013_card__of__Func__UNIV,axiom,
    ! [B: $tType,A: $tType,B3: set(B)] : aa(set(product_prod(set(product_prod(fun(A,B),fun(A,B))),set(product_prod(fun(A,B),fun(A,B))))),$o,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)),B3))),bNF_Ca6860139660246222851ard_of(fun(A,B),aa(fun(fun(A,B),$o),set(fun(A,B)),collect(fun(A,B)),aTP_Lamp_amy(set(B),fun(fun(A,B),$o),B3))))),bNF_Wellorder_ordIso(fun(A,B),fun(A,B))) ).

% card_of_Func_UNIV
tff(fact_6014_ordLeq__Times__mono2,axiom,
    ! [B: $tType,A: $tType,C: $tType,R2: set(product_prod(A,A)),R3: set(product_prod(B,B)),A4: set(C)] :
      ( aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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))),R2),R3)),bNF_Wellorder_ordLeq(A,B))
     => 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))))),$o,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,A4,aTP_Lamp_amz(set(product_prod(A,A)),fun(C,set(A)),R2)))),bNF_Ca6860139660246222851ard_of(product_prod(C,B),product_Sigma(C,B,A4,aTP_Lamp_ana(set(product_prod(B,B)),fun(C,set(B)),R3))))),bNF_Wellorder_ordLeq(product_prod(C,A),product_prod(C,B))) ) ).

% ordLeq_Times_mono2
tff(fact_6015_ordLeq__Times__mono1,axiom,
    ! [B: $tType,C: $tType,A: $tType,R2: set(product_prod(A,A)),R3: set(product_prod(B,B)),C3: set(C)] :
      ( aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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))),R2),R3)),bNF_Wellorder_ordLeq(A,B))
     => 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))))),$o,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,R2),aTP_Lamp_us(set(C),fun(A,set(C)),C3)))),bNF_Ca6860139660246222851ard_of(product_prod(B,C),product_Sigma(B,C,field2(B,R3),aTP_Lamp_rw(set(C),fun(B,set(C)),C3))))),bNF_Wellorder_ordLeq(product_prod(A,C),product_prod(B,C))) ) ).

% ordLeq_Times_mono1
tff(fact_6016_card__of__ordLeq,axiom,
    ! [B: $tType,A: $tType,A4: set(A),B3: set(B)] :
      ( ? [F6: fun(A,B)] :
          ( inj_on(A,B,F6,A4)
          & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F6),A4)),B3) )
    <=> aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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,A4)),bNF_Ca6860139660246222851ard_of(B,B3))),bNF_Wellorder_ordLeq(A,B)) ) ).

% card_of_ordLeq
tff(fact_6017_internalize__card__of__ordLeq,axiom,
    ! [A: $tType,B: $tType,A4: set(A),R2: set(product_prod(B,B))] :
      ( aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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,A4)),R2)),bNF_Wellorder_ordLeq(A,B))
    <=> ? [B9: set(B)] :
          ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),B9),field2(B,R2))
          & aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Ca6860139660246222851ard_of(B,B9))),bNF_Wellorder_ordIso(A,B))
          & aa(set(product_prod(set(product_prod(B,B)),set(product_prod(B,B)))),$o,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,B9)),R2)),bNF_Wellorder_ordLeq(B,B)) ) ) ).

% internalize_card_of_ordLeq
tff(fact_6018_card__of__ordLess,axiom,
    ! [A: $tType,B: $tType,A4: set(A),B3: set(B)] :
      ( ~ ? [F6: fun(A,B)] :
            ( inj_on(A,B,F6,A4)
            & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F6),A4)),B3) )
    <=> aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),$o,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,B3)),bNF_Ca6860139660246222851ard_of(A,A4))),bNF_We4044943003108391690rdLess(B,A)) ) ).

% card_of_ordLess
tff(fact_6019_card__of__UNION__ordLeq__infinite,axiom,
    ! [B: $tType,A: $tType,C: $tType,B3: set(A),I4: set(B),A4: fun(B,set(C))] :
      ( ~ aa(set(A),$o,finite_finite2(A),B3)
     => ( aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),$o,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,I4)),bNF_Ca6860139660246222851ard_of(A,B3))),bNF_Wellorder_ordLeq(B,A))
       => ( ! [X2: B] :
              ( aa(set(B),$o,member(B,X2),I4)
             => aa(set(product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),$o,member(product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(C,C)),fun(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),product_Pair(set(product_prod(C,C)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(C,aa(B,set(C),A4,X2))),bNF_Ca6860139660246222851ard_of(A,B3))),bNF_Wellorder_ordLeq(C,A)) )
         => aa(set(product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),$o,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)),aa(set(B),set(set(C)),image2(B,set(C),A4),I4)))),bNF_Ca6860139660246222851ard_of(A,B3))),bNF_Wellorder_ordLeq(C,A)) ) ) ) ).

% card_of_UNION_ordLeq_infinite
tff(fact_6020_regularCard__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( bNF_Ca7133664381575040944arCard(A,R2)
    <=> ! [K6: set(A)] :
          ( ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),K6),field2(A,R2))
            & bNF_Ca7293521722713021262ofinal(A,K6,R2) )
         => aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),$o,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)),R2)),bNF_Wellorder_ordIso(A,A)) ) ) ).

% regularCard_def
tff(fact_6021_card__of__ordIso__subst,axiom,
    ! [A: $tType,A4: set(A),B3: set(A)] :
      ( ( A4 = B3 )
     => aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),$o,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,A4)),bNF_Ca6860139660246222851ard_of(A,B3))),bNF_Wellorder_ordIso(A,A)) ) ).

% card_of_ordIso_subst
tff(fact_6022_SIGMA__CSUM,axiom,
    ! [B: $tType,A: $tType,I4: set(A),As9: fun(A,set(B))] : bNF_Ca6860139660246222851ard_of(product_prod(A,B),product_Sigma(A,B,I4,As9)) = bNF_Cardinal_Csum(A,B,bNF_Ca6860139660246222851ard_of(A,I4),aTP_Lamp_anb(fun(A,set(B)),fun(A,set(product_prod(B,B))),As9)) ).

% SIGMA_CSUM
tff(fact_6023_card__of__Times__ordLeq__infinite__Field,axiom,
    ! [A: $tType,C: $tType,B: $tType,R2: set(product_prod(A,A)),A4: set(B),B3: set(C)] :
      ( ~ aa(set(A),$o,finite_finite2(A),field2(A,R2))
     => ( aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),$o,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,A4)),R2)),bNF_Wellorder_ordLeq(B,A))
       => ( aa(set(product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),$o,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,B3)),R2)),bNF_Wellorder_ordLeq(C,A))
         => ( aa(set(product_prod(A,A)),$o,bNF_Ca8970107618336181345der_on(A,field2(A,R2)),R2)
           => aa(set(product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A)))),$o,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,A4,aTP_Lamp_rw(set(C),fun(B,set(C)),B3)))),R2)),bNF_Wellorder_ordLeq(product_prod(B,C),A)) ) ) ) ) ).

% card_of_Times_ordLeq_infinite_Field
tff(fact_6024_card__order__on__ordIso,axiom,
    ! [A: $tType,A4: set(A),R2: set(product_prod(A,A)),R3: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,bNF_Ca8970107618336181345der_on(A,A4),R2)
     => ( aa(set(product_prod(A,A)),$o,bNF_Ca8970107618336181345der_on(A,A4),R3)
       => aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),$o,member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R2),R3)),bNF_Wellorder_ordIso(A,A)) ) ) ).

% card_order_on_ordIso
tff(fact_6025_Card__order__trans,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),X: A,Y: A,Z2: A] :
      ( aa(set(product_prod(A,A)),$o,bNF_Ca8970107618336181345der_on(A,field2(A,R2)),R2)
     => ( ( X != Y )
       => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R2)
         => ( ( Y != Z2 )
           => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z2)),R2)
             => ( ( X != Z2 )
                & aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z2)),R2) ) ) ) ) ) ) ).

% Card_order_trans
tff(fact_6026_Field__card__order,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,bNF_Ca8970107618336181345der_on(A,top_top(set(A))),R2)
     => ( field2(A,R2) = top_top(set(A)) ) ) ).

% Field_card_order
tff(fact_6027_infinite__Card__order__limit,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A3: A] :
      ( aa(set(product_prod(A,A)),$o,bNF_Ca8970107618336181345der_on(A,field2(A,R2)),R2)
     => ( ~ aa(set(A),$o,finite_finite2(A),field2(A,R2))
       => ( aa(set(A),$o,member(A,A3),field2(A,R2))
         => ? [X2: A] :
              ( aa(set(A),$o,member(A,X2),field2(A,R2))
              & ( A3 != X2 )
              & aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),X2)),R2) ) ) ) ) ).

% infinite_Card_order_limit
tff(fact_6028_ordLeq__refl,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,bNF_Ca8970107618336181345der_on(A,field2(A,R2)),R2)
     => aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),$o,member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R2),R2)),bNF_Wellorder_ordLeq(A,A)) ) ).

% ordLeq_refl
tff(fact_6029_Card__order__ordIso,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(A,A)),R3: set(product_prod(B,B))] :
      ( aa(set(product_prod(A,A)),$o,bNF_Ca8970107618336181345der_on(A,field2(A,R2)),R2)
     => ( aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),$o,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),R3),R2)),bNF_Wellorder_ordIso(B,A))
       => aa(set(product_prod(B,B)),$o,bNF_Ca8970107618336181345der_on(B,field2(B,R3)),R3) ) ) ).

% Card_order_ordIso
tff(fact_6030_Card__order__ordIso2,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(A,A)),R3: set(product_prod(B,B))] :
      ( aa(set(product_prod(A,A)),$o,bNF_Ca8970107618336181345der_on(A,field2(A,R2)),R2)
     => ( aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R2),R3)),bNF_Wellorder_ordIso(A,B))
       => aa(set(product_prod(B,B)),$o,bNF_Ca8970107618336181345der_on(B,field2(B,R3)),R3) ) ) ).

% Card_order_ordIso2
tff(fact_6031_ordIso__refl,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,bNF_Ca8970107618336181345der_on(A,field2(A,R2)),R2)
     => aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),$o,member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R2),R2)),bNF_Wellorder_ordIso(A,A)) ) ).

% ordIso_refl
tff(fact_6032_card__of__unique,axiom,
    ! [A: $tType,A4: set(A),R2: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,bNF_Ca8970107618336181345der_on(A,A4),R2)
     => aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),$o,member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R2),bNF_Ca6860139660246222851ard_of(A,A4))),bNF_Wellorder_ordIso(A,A)) ) ).

% card_of_unique
tff(fact_6033_card__order__on__def,axiom,
    ! [A: $tType,A4: set(A),R2: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,bNF_Ca8970107618336181345der_on(A,A4),R2)
    <=> ( order_well_order_on(A,A4,R2)
        & ! [R10: set(product_prod(A,A))] :
            ( order_well_order_on(A,A4,R10)
           => aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),$o,member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R2),R10)),bNF_Wellorder_ordLeq(A,A)) ) ) ) ).

% card_order_on_def
tff(fact_6034_card__order__dir__image,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),R2: set(product_prod(A,A))] :
      ( bij_betw(A,B,F,top_top(set(A)),top_top(set(B)))
     => ( aa(set(product_prod(A,A)),$o,bNF_Ca8970107618336181345der_on(A,top_top(set(A))),R2)
       => aa(set(product_prod(B,B)),$o,bNF_Ca8970107618336181345der_on(B,top_top(set(B))),bNF_We2720479622203943262_image(A,B,R2,F)) ) ) ).

% card_order_dir_image
tff(fact_6035_Card__order__iff__ordLeq__card__of,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,bNF_Ca8970107618336181345der_on(A,field2(A,R2)),R2)
    <=> aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),$o,member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R2),bNF_Ca6860139660246222851ard_of(A,field2(A,R2)))),bNF_Wellorder_ordLeq(A,A)) ) ).

% Card_order_iff_ordLeq_card_of
tff(fact_6036_ordIso__card__of__imp__Card__order,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,A)),A4: set(B)] :
      ( aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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))),R2),bNF_Ca6860139660246222851ard_of(B,A4))),bNF_Wellorder_ordIso(A,B))
     => aa(set(product_prod(A,A)),$o,bNF_Ca8970107618336181345der_on(A,field2(A,R2)),R2) ) ).

% ordIso_card_of_imp_Card_order
tff(fact_6037_Card__order__iff__ordIso__card__of,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,bNF_Ca8970107618336181345der_on(A,field2(A,R2)),R2)
    <=> aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),$o,member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R2),bNF_Ca6860139660246222851ard_of(A,field2(A,R2)))),bNF_Wellorder_ordIso(A,A)) ) ).

% Card_order_iff_ordIso_card_of
tff(fact_6038_card__of__Field__ordIso,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,bNF_Ca8970107618336181345der_on(A,field2(A,R2)),R2)
     => aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),$o,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,R2))),R2)),bNF_Wellorder_ordIso(A,A)) ) ).

% card_of_Field_ordIso
tff(fact_6039_dir__image,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),R2: set(product_prod(A,A))] :
      ( ! [X2: A,Y2: A] :
          ( ( aa(A,B,F,X2) = aa(A,B,F,Y2) )
        <=> ( X2 = Y2 ) )
     => ( aa(set(product_prod(A,A)),$o,bNF_Ca8970107618336181345der_on(A,field2(A,R2)),R2)
       => aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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))),R2),bNF_We2720479622203943262_image(A,B,R2,F))),bNF_Wellorder_ordIso(A,B)) ) ) ).

% dir_image
tff(fact_6040_exists__minim__Card__order,axiom,
    ! [A: $tType,R: set(set(product_prod(A,A)))] :
      ( ( R != bot_bot(set(set(product_prod(A,A)))) )
     => ( ! [X2: set(product_prod(A,A))] :
            ( aa(set(set(product_prod(A,A))),$o,member(set(product_prod(A,A)),X2),R)
           => aa(set(product_prod(A,A)),$o,bNF_Ca8970107618336181345der_on(A,field2(A,X2)),X2) )
       => ? [X2: set(product_prod(A,A))] :
            ( aa(set(set(product_prod(A,A))),$o,member(set(product_prod(A,A)),X2),R)
            & ! [Xa3: set(product_prod(A,A))] :
                ( aa(set(set(product_prod(A,A))),$o,member(set(product_prod(A,A)),Xa3),R)
               => aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),$o,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))),X2),Xa3)),bNF_Wellorder_ordLeq(A,A)) ) ) ) ) ).

% exists_minim_Card_order
tff(fact_6041_Card__order__empty,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,bNF_Ca8970107618336181345der_on(A,field2(A,R2)),R2)
     => aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),$o,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)))),R2)),bNF_Wellorder_ordLeq(B,A)) ) ).

% Card_order_empty
tff(fact_6042_card__of__ordIso__finite__Field,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(A,A)),A4: set(B)] :
      ( aa(set(product_prod(A,A)),$o,bNF_Ca8970107618336181345der_on(A,field2(A,R2)),R2)
     => ( aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R2),bNF_Ca6860139660246222851ard_of(B,A4))),bNF_Wellorder_ordIso(A,B))
       => ( aa(set(A),$o,finite_finite2(A),field2(A,R2))
        <=> aa(set(B),$o,finite_finite2(B),A4) ) ) ) ).

% card_of_ordIso_finite_Field
tff(fact_6043_card__of__underS,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A3: A] :
      ( aa(set(product_prod(A,A)),$o,bNF_Ca8970107618336181345der_on(A,field2(A,R2)),R2)
     => ( aa(set(A),$o,member(A,A3),field2(A,R2))
       => aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),$o,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,R2,A3))),R2)),bNF_We4044943003108391690rdLess(A,A)) ) ) ).

% card_of_underS
tff(fact_6044_Card__order__singl__ordLeq,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(A,A)),B2: B] :
      ( aa(set(product_prod(A,A)),$o,bNF_Ca8970107618336181345der_on(A,field2(A,R2)),R2)
     => ( ( field2(A,R2) != bot_bot(set(A)) )
       => aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),$o,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(B,fun(set(B),set(B)),insert2(B),B2),bot_bot(set(B))))),R2)),bNF_Wellorder_ordLeq(B,A)) ) ) ).

% Card_order_singl_ordLeq
tff(fact_6045_card__of__Un__ordLeq__infinite__Field,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(A,A)),A4: set(B),B3: set(B)] :
      ( ~ aa(set(A),$o,finite_finite2(A),field2(A,R2))
     => ( aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),$o,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,A4)),R2)),bNF_Wellorder_ordLeq(B,A))
       => ( aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),$o,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,B3)),R2)),bNF_Wellorder_ordLeq(B,A))
         => ( aa(set(product_prod(A,A)),$o,bNF_Ca8970107618336181345der_on(A,field2(A,R2)),R2)
           => aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),$o,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)),A4),B3))),R2)),bNF_Wellorder_ordLeq(B,A)) ) ) ) ) ).

% card_of_Un_ordLeq_infinite_Field
tff(fact_6046_card__of__empty1,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(A,A))] :
      ( ( order_well_order_on(A,field2(A,R2),R2)
        | aa(set(product_prod(A,A)),$o,bNF_Ca8970107618336181345der_on(A,field2(A,R2)),R2) )
     => aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),$o,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)))),R2)),bNF_Wellorder_ordLeq(B,A)) ) ).

% card_of_empty1
tff(fact_6047_Card__order__Pow,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,bNF_Ca8970107618336181345der_on(A,field2(A,R2)),R2)
     => aa(set(product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A))))),$o,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)))),R2),bNF_Ca6860139660246222851ard_of(set(A),pow2(A,field2(A,R2))))),bNF_We4044943003108391690rdLess(A,set(A))) ) ).

% Card_order_Pow
tff(fact_6048_Card__order__Times1,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,A)),B3: set(B)] :
      ( aa(set(product_prod(A,A)),$o,bNF_Ca8970107618336181345der_on(A,field2(A,R2)),R2)
     => ( ( B3 != bot_bot(set(B)) )
       => aa(set(product_prod(set(product_prod(A,A)),set(product_prod(product_prod(A,B),product_prod(A,B))))),$o,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)))),R2),bNF_Ca6860139660246222851ard_of(product_prod(A,B),product_Sigma(A,B,field2(A,R2),aTP_Lamp_qz(set(B),fun(A,set(B)),B3))))),bNF_Wellorder_ordLeq(A,product_prod(A,B))) ) ) ).

% Card_order_Times1
tff(fact_6049_Card__order__Times2,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,A)),A4: set(B)] :
      ( aa(set(product_prod(A,A)),$o,bNF_Ca8970107618336181345der_on(A,field2(A,R2)),R2)
     => ( ( A4 != bot_bot(set(B)) )
       => aa(set(product_prod(set(product_prod(A,A)),set(product_prod(product_prod(B,A),product_prod(B,A))))),$o,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)))),R2),bNF_Ca6860139660246222851ard_of(product_prod(B,A),product_Sigma(B,A,A4,aTP_Lamp_anc(set(product_prod(A,A)),fun(B,set(A)),R2))))),bNF_Wellorder_ordLeq(A,product_prod(B,A))) ) ) ).

% Card_order_Times2
tff(fact_6050_Card__order__Times__same__infinite,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,bNF_Ca8970107618336181345der_on(A,field2(A,R2)),R2)
     => ( ~ aa(set(A),$o,finite_finite2(A),field2(A,R2))
       => aa(set(product_prod(set(product_prod(product_prod(A,A),product_prod(A,A))),set(product_prod(A,A)))),$o,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,R2),aTP_Lamp_ut(set(product_prod(A,A)),fun(A,set(A)),R2)))),R2)),bNF_Wellorder_ordLeq(product_prod(A,A),A)) ) ) ).

% Card_order_Times_same_infinite
tff(fact_6051_card__of__UNION__ordLeq__infinite__Field,axiom,
    ! [B: $tType,A: $tType,C: $tType,R2: set(product_prod(A,A)),I4: set(B),A4: fun(B,set(C))] :
      ( ~ aa(set(A),$o,finite_finite2(A),field2(A,R2))
     => ( aa(set(product_prod(A,A)),$o,bNF_Ca8970107618336181345der_on(A,field2(A,R2)),R2)
       => ( aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),$o,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,I4)),R2)),bNF_Wellorder_ordLeq(B,A))
         => ( ! [X2: B] :
                ( aa(set(B),$o,member(B,X2),I4)
               => aa(set(product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),$o,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),A4,X2))),R2)),bNF_Wellorder_ordLeq(C,A)) )
           => aa(set(product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),$o,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)),aa(set(B),set(set(C)),image2(B,set(C),A4),I4)))),R2)),bNF_Wellorder_ordLeq(C,A)) ) ) ) ) ).

% card_of_UNION_ordLeq_infinite_Field
tff(fact_6052_Card__order__iff__Restr__underS,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( order_well_order_on(A,field2(A,R2),R2)
     => ( aa(set(product_prod(A,A)),$o,bNF_Ca8970107618336181345der_on(A,field2(A,R2)),R2)
      <=> ! [X4: A] :
            ( aa(set(A),$o,member(A,X4),field2(A,R2))
           => aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),$o,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))),R2),product_Sigma(A,A,order_underS(A,R2,X4),aa(A,fun(A,set(A)),aTP_Lamp_amp(set(product_prod(A,A)),fun(A,fun(A,set(A))),R2),X4)))),bNF_Ca6860139660246222851ard_of(A,field2(A,R2)))),bNF_We4044943003108391690rdLess(A,A)) ) ) ) ).

% Card_order_iff_Restr_underS
tff(fact_6053_regularCard__UNION,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,A)),As9: fun(A,set(B)),B3: set(B)] :
      ( aa(set(product_prod(A,A)),$o,bNF_Ca8970107618336181345der_on(A,field2(A,R2)),R2)
     => ( bNF_Ca7133664381575040944arCard(A,R2)
       => ( bNF_Ca3754400796208372196lChain(A,set(B),R2,As9)
         => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),B3),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),As9),field2(A,R2))))
           => ( aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),$o,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,B3)),R2)),bNF_We4044943003108391690rdLess(B,A))
             => ? [X2: A] :
                  ( aa(set(A),$o,member(A,X2),field2(A,R2))
                  & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),B3),aa(A,set(B),As9,X2)) ) ) ) ) ) ) ).

% regularCard_UNION
tff(fact_6054_Card__order__Times__infinite,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(A,A)),P3: set(product_prod(B,B))] :
      ( ~ aa(set(A),$o,finite_finite2(A),field2(A,R2))
     => ( aa(set(product_prod(A,A)),$o,bNF_Ca8970107618336181345der_on(A,field2(A,R2)),R2)
       => ( ( field2(B,P3) != bot_bot(set(B)) )
         => ( aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),$o,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))),P3),R2)),bNF_Wellorder_ordLeq(B,A))
           => ( aa(set(product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A)))),$o,member(product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A))),aa(set(product_prod(product_prod(A,B),product_prod(A,B))),fun(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A)))),product_Pair(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(product_prod(A,B),product_Sigma(A,B,field2(A,R2),aTP_Lamp_and(set(product_prod(B,B)),fun(A,set(B)),P3)))),R2)),bNF_Wellorder_ordIso(product_prod(A,B),A))
              & aa(set(product_prod(set(product_prod(product_prod(B,A),product_prod(B,A))),set(product_prod(A,A)))),$o,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,P3),aTP_Lamp_anc(set(product_prod(A,A)),fun(B,set(A)),R2)))),R2)),bNF_Wellorder_ordIso(product_prod(B,A),A)) ) ) ) ) ) ).

% Card_order_Times_infinite
tff(fact_6055_Csum__def,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,A)),Rs: fun(A,set(product_prod(B,B)))] : bNF_Cardinal_Csum(A,B,R2,Rs) = bNF_Ca6860139660246222851ard_of(product_prod(A,B),product_Sigma(A,B,field2(A,R2),aTP_Lamp_ane(fun(A,set(product_prod(B,B))),fun(A,set(B)),Rs))) ).

% Csum_def
tff(fact_6056_card__of__Sigma__ordLeq__infinite__Field,axiom,
    ! [A: $tType,C: $tType,B: $tType,R2: set(product_prod(A,A)),I4: set(B),A4: fun(B,set(C))] :
      ( ~ aa(set(A),$o,finite_finite2(A),field2(A,R2))
     => ( aa(set(product_prod(A,A)),$o,bNF_Ca8970107618336181345der_on(A,field2(A,R2)),R2)
       => ( aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),$o,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,I4)),R2)),bNF_Wellorder_ordLeq(B,A))
         => ( ! [X2: B] :
                ( aa(set(B),$o,member(B,X2),I4)
               => aa(set(product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),$o,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),A4,X2))),R2)),bNF_Wellorder_ordLeq(C,A)) )
           => aa(set(product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A)))),$o,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,I4,A4))),R2)),bNF_Wellorder_ordLeq(product_prod(B,C),A)) ) ) ) ) ).

% card_of_Sigma_ordLeq_infinite_Field
tff(fact_6057_ex__toCard__pred,axiom,
    ! [B: $tType,A: $tType,A4: set(A),R2: set(product_prod(B,B))] :
      ( aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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,A4)),R2)),bNF_Wellorder_ordLeq(A,B))
     => ( aa(set(product_prod(B,B)),$o,bNF_Ca8970107618336181345der_on(B,field2(B,R2)),R2)
       => ? [X_1: fun(A,B)] : aa(fun(A,B),$o,bNF_Gr1419584066657907630d_pred(A,B,A4,R2),X_1) ) ) ).

% ex_toCard_pred
tff(fact_6058_cardSuc__UNION,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,A)),As9: fun(set(A),set(B)),B3: set(B)] :
      ( aa(set(product_prod(A,A)),$o,bNF_Ca8970107618336181345der_on(A,field2(A,R2)),R2)
     => ( ~ aa(set(A),$o,finite_finite2(A),field2(A,R2))
       => ( bNF_Ca3754400796208372196lChain(set(A),set(B),bNF_Ca8387033319878233205ardSuc(A,R2),As9)
         => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),B3),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(set(A)),set(set(B)),image2(set(A),set(B),As9),field2(set(A),bNF_Ca8387033319878233205ardSuc(A,R2)))))
           => ( aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),$o,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,B3)),R2)),bNF_Wellorder_ordLeq(B,A))
             => ? [X2: set(A)] :
                  ( aa(set(set(A)),$o,member(set(A),X2),field2(set(A),bNF_Ca8387033319878233205ardSuc(A,R2)))
                  & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),B3),aa(set(A),set(B),As9,X2)) ) ) ) ) ) ) ).

% cardSuc_UNION
tff(fact_6059_cardSuc__ordLeq,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,bNF_Ca8970107618336181345der_on(A,field2(A,R2)),R2)
     => aa(set(product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A))))),$o,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)))),R2),bNF_Ca8387033319878233205ardSuc(A,R2))),bNF_Wellorder_ordLeq(A,set(A))) ) ).

% cardSuc_ordLeq
tff(fact_6060_cardSuc__greater,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,bNF_Ca8970107618336181345der_on(A,field2(A,R2)),R2)
     => aa(set(product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A))))),$o,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)))),R2),bNF_Ca8387033319878233205ardSuc(A,R2))),bNF_We4044943003108391690rdLess(A,set(A))) ) ).

% cardSuc_greater
tff(fact_6061_cardSuc__least,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,A)),R3: set(product_prod(B,B))] :
      ( aa(set(product_prod(A,A)),$o,bNF_Ca8970107618336181345der_on(A,field2(A,R2)),R2)
     => ( aa(set(product_prod(B,B)),$o,bNF_Ca8970107618336181345der_on(B,field2(B,R3)),R3)
       => ( aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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))),R2),R3)),bNF_We4044943003108391690rdLess(A,B))
         => aa(set(product_prod(set(product_prod(set(A),set(A))),set(product_prod(B,B)))),$o,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,R2)),R3)),bNF_Wellorder_ordLeq(set(A),B)) ) ) ) ).

% cardSuc_least
tff(fact_6062_cardSuc__ordLess__ordLeq,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,A)),R3: set(product_prod(B,B))] :
      ( aa(set(product_prod(A,A)),$o,bNF_Ca8970107618336181345der_on(A,field2(A,R2)),R2)
     => ( aa(set(product_prod(B,B)),$o,bNF_Ca8970107618336181345der_on(B,field2(B,R3)),R3)
       => ( aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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))),R2),R3)),bNF_We4044943003108391690rdLess(A,B))
        <=> aa(set(product_prod(set(product_prod(set(A),set(A))),set(product_prod(B,B)))),$o,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,R2)),R3)),bNF_Wellorder_ordLeq(set(A),B)) ) ) ) ).

% cardSuc_ordLess_ordLeq
tff(fact_6063_cardSuc__mono__ordLeq,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,A)),R3: set(product_prod(B,B))] :
      ( aa(set(product_prod(A,A)),$o,bNF_Ca8970107618336181345der_on(A,field2(A,R2)),R2)
     => ( aa(set(product_prod(B,B)),$o,bNF_Ca8970107618336181345der_on(B,field2(B,R3)),R3)
       => ( aa(set(product_prod(set(product_prod(set(A),set(A))),set(product_prod(set(B),set(B))))),$o,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,R2)),bNF_Ca8387033319878233205ardSuc(B,R3))),bNF_Wellorder_ordLeq(set(A),set(B)))
        <=> aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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))),R2),R3)),bNF_Wellorder_ordLeq(A,B)) ) ) ) ).

% cardSuc_mono_ordLeq
tff(fact_6064_cardSuc__invar__ordIso,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,A)),R3: set(product_prod(B,B))] :
      ( aa(set(product_prod(A,A)),$o,bNF_Ca8970107618336181345der_on(A,field2(A,R2)),R2)
     => ( aa(set(product_prod(B,B)),$o,bNF_Ca8970107618336181345der_on(B,field2(B,R3)),R3)
       => ( aa(set(product_prod(set(product_prod(set(A),set(A))),set(product_prod(set(B),set(B))))),$o,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,R2)),bNF_Ca8387033319878233205ardSuc(B,R3))),bNF_Wellorder_ordIso(set(A),set(B)))
        <=> aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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))),R2),R3)),bNF_Wellorder_ordIso(A,B)) ) ) ) ).

% cardSuc_invar_ordIso
tff(fact_6065_cardSuc__least__aux,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),R3: set(product_prod(set(A),set(A)))] :
      ( aa(set(product_prod(A,A)),$o,bNF_Ca8970107618336181345der_on(A,field2(A,R2)),R2)
     => ( aa(set(product_prod(set(A),set(A))),$o,bNF_Ca8970107618336181345der_on(set(A),field2(set(A),R3)),R3)
       => ( aa(set(product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A))))),$o,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)))),R2),R3)),bNF_We4044943003108391690rdLess(A,set(A)))
         => aa(set(product_prod(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A))))),$o,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,R2)),R3)),bNF_Wellorder_ordLeq(set(A),set(A))) ) ) ) ).

% cardSuc_least_aux
tff(fact_6066_cardSuc__ordLeq__ordLess,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(A,A)),R3: set(product_prod(B,B))] :
      ( aa(set(product_prod(A,A)),$o,bNF_Ca8970107618336181345der_on(A,field2(A,R2)),R2)
     => ( aa(set(product_prod(B,B)),$o,bNF_Ca8970107618336181345der_on(B,field2(B,R3)),R3)
       => ( aa(set(product_prod(set(product_prod(B,B)),set(product_prod(set(A),set(A))))),$o,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)))),R3),bNF_Ca8387033319878233205ardSuc(A,R2))),bNF_We4044943003108391690rdLess(B,set(A)))
        <=> aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),$o,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))),R3),R2)),bNF_Wellorder_ordLeq(B,A)) ) ) ) ).

% cardSuc_ordLeq_ordLess
tff(fact_6067_isCardSuc__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),R3: set(product_prod(set(A),set(A)))] :
      ( aa(set(product_prod(set(A),set(A))),$o,bNF_Ca6246979054910435723ardSuc(A,R2),R3)
    <=> ( aa(set(product_prod(set(A),set(A))),$o,bNF_Ca8970107618336181345der_on(set(A),field2(set(A),R3)),R3)
        & aa(set(product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A))))),$o,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)))),R2),R3)),bNF_We4044943003108391690rdLess(A,set(A)))
        & ! [R11: set(product_prod(set(A),set(A)))] :
            ( ( aa(set(product_prod(set(A),set(A))),$o,bNF_Ca8970107618336181345der_on(set(A),field2(set(A),R11)),R11)
              & aa(set(product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A))))),$o,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)))),R2),R11)),bNF_We4044943003108391690rdLess(A,set(A))) )
           => aa(set(product_prod(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A))))),$o,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)))),R3),R11)),bNF_Wellorder_ordLeq(set(A),set(A))) ) ) ) ).

% isCardSuc_def
tff(fact_6068_toCard__pred__toCard,axiom,
    ! [A: $tType,B: $tType,A4: set(A),R2: set(product_prod(B,B))] :
      ( aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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,A4)),R2)),bNF_Wellorder_ordLeq(A,B))
     => ( aa(set(product_prod(B,B)),$o,bNF_Ca8970107618336181345der_on(B,field2(B,R2)),R2)
       => aa(fun(A,B),$o,bNF_Gr1419584066657907630d_pred(A,B,A4,R2),bNF_Greatest_toCard(A,B,A4,R2)) ) ) ).

% toCard_pred_toCard
tff(fact_6069_toCard__inj,axiom,
    ! [B: $tType,A: $tType,A4: set(A),R2: set(product_prod(B,B)),X: A,Y: A] :
      ( aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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,A4)),R2)),bNF_Wellorder_ordLeq(A,B))
     => ( aa(set(product_prod(B,B)),$o,bNF_Ca8970107618336181345der_on(B,field2(B,R2)),R2)
       => ( aa(set(A),$o,member(A,X),A4)
         => ( aa(set(A),$o,member(A,Y),A4)
           => ( ( aa(A,B,bNF_Greatest_toCard(A,B,A4,R2),X) = aa(A,B,bNF_Greatest_toCard(A,B,A4,R2),Y) )
            <=> ( X = Y ) ) ) ) ) ) ).

% toCard_inj
tff(fact_6070_fromCard__toCard,axiom,
    ! [B: $tType,A: $tType,A4: set(A),R2: set(product_prod(B,B)),B2: A] :
      ( aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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,A4)),R2)),bNF_Wellorder_ordLeq(A,B))
     => ( aa(set(product_prod(B,B)),$o,bNF_Ca8970107618336181345der_on(B,field2(B,R2)),R2)
       => ( aa(set(A),$o,member(A,B2),A4)
         => ( bNF_Gr5436034075474128252omCard(A,B,A4,R2,aa(A,B,bNF_Greatest_toCard(A,B,A4,R2),B2)) = B2 ) ) ) ) ).

% fromCard_toCard
tff(fact_6071_cardSuc__UNION__Cinfinite,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,A)),As9: fun(set(A),set(B)),B3: set(B)] :
      ( ( bNF_Ca4139267488887388095finite(A,R2)
        & aa(set(product_prod(A,A)),$o,bNF_Ca8970107618336181345der_on(A,field2(A,R2)),R2) )
     => ( bNF_Ca3754400796208372196lChain(set(A),set(B),bNF_Ca8387033319878233205ardSuc(A,R2),As9)
       => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),B3),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(set(A)),set(set(B)),image2(set(A),set(B),As9),field2(set(A),bNF_Ca8387033319878233205ardSuc(A,R2)))))
         => ( aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),$o,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,B3)),R2)),bNF_Wellorder_ordLeq(B,A))
           => ? [X2: set(A)] :
                ( aa(set(set(A)),$o,member(set(A),X2),field2(set(A),bNF_Ca8387033319878233205ardSuc(A,R2)))
                & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),B3),aa(set(A),set(B),As9,X2)) ) ) ) ) ) ).

% cardSuc_UNION_Cinfinite
tff(fact_6072_cinfinite__mono,axiom,
    ! [A: $tType,B: $tType,R12: set(product_prod(A,A)),R23: set(product_prod(B,B))] :
      ( aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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))),R12),R23)),bNF_Wellorder_ordLeq(A,B))
     => ( bNF_Ca4139267488887388095finite(A,R12)
       => bNF_Ca4139267488887388095finite(B,R23) ) ) ).

% cinfinite_mono
tff(fact_6073_Cinfinite__limit,axiom,
    ! [A: $tType,X: A,R2: set(product_prod(A,A))] :
      ( aa(set(A),$o,member(A,X),field2(A,R2))
     => ( ( bNF_Ca4139267488887388095finite(A,R2)
          & aa(set(product_prod(A,A)),$o,bNF_Ca8970107618336181345der_on(A,field2(A,R2)),R2) )
       => ? [X2: A] :
            ( aa(set(A),$o,member(A,X2),field2(A,R2))
            & ( X != X2 )
            & aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),X2)),R2) ) ) ) ).

% Cinfinite_limit
tff(fact_6074_Cinfinite__limit2,axiom,
    ! [A: $tType,X1: A,R2: set(product_prod(A,A)),X22: A] :
      ( aa(set(A),$o,member(A,X1),field2(A,R2))
     => ( aa(set(A),$o,member(A,X22),field2(A,R2))
       => ( ( bNF_Ca4139267488887388095finite(A,R2)
            & aa(set(product_prod(A,A)),$o,bNF_Ca8970107618336181345der_on(A,field2(A,R2)),R2) )
         => ? [X2: A] :
              ( aa(set(A),$o,member(A,X2),field2(A,R2))
              & ( X1 != X2 )
              & aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X1),X2)),R2)
              & ( X22 != X2 )
              & aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X22),X2)),R2) ) ) ) ) ).

% Cinfinite_limit2
tff(fact_6075_Cinfinite__cong,axiom,
    ! [A: $tType,B: $tType,R12: set(product_prod(A,A)),R23: set(product_prod(B,B))] :
      ( aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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))),R12),R23)),bNF_Wellorder_ordIso(A,B))
     => ( ( bNF_Ca4139267488887388095finite(A,R12)
          & aa(set(product_prod(A,A)),$o,bNF_Ca8970107618336181345der_on(A,field2(A,R12)),R12) )
       => ( bNF_Ca4139267488887388095finite(B,R23)
          & aa(set(product_prod(B,B)),$o,bNF_Ca8970107618336181345der_on(B,field2(B,R23)),R23) ) ) ) ).

% Cinfinite_cong
tff(fact_6076_Cinfinite__limit__finite,axiom,
    ! [A: $tType,X5: set(A),R2: set(product_prod(A,A))] :
      ( aa(set(A),$o,finite_finite2(A),X5)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X5),field2(A,R2))
       => ( ( bNF_Ca4139267488887388095finite(A,R2)
            & aa(set(product_prod(A,A)),$o,bNF_Ca8970107618336181345der_on(A,field2(A,R2)),R2) )
         => ? [X2: A] :
              ( aa(set(A),$o,member(A,X2),field2(A,R2))
              & ! [Xa3: A] :
                  ( aa(set(A),$o,member(A,Xa3),X5)
                 => ( ( Xa3 != X2 )
                    & aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa3),X2)),R2) ) ) ) ) ) ) ).

% Cinfinite_limit_finite
tff(fact_6077_Un__Cinfinite__bound,axiom,
    ! [B: $tType,A: $tType,A4: set(A),R2: set(product_prod(B,B)),B3: set(A)] :
      ( aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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,A4)),R2)),bNF_Wellorder_ordLeq(A,B))
     => ( aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,B3)),R2)),bNF_Wellorder_ordLeq(A,B))
       => ( ( bNF_Ca4139267488887388095finite(B,R2)
            & aa(set(product_prod(B,B)),$o,bNF_Ca8970107618336181345der_on(B,field2(B,R2)),R2) )
         => aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3))),R2)),bNF_Wellorder_ordLeq(A,B)) ) ) ) ).

% Un_Cinfinite_bound
tff(fact_6078_UNION__Cinfinite__bound,axiom,
    ! [A: $tType,B: $tType,C: $tType,I4: set(A),R2: set(product_prod(B,B)),A4: fun(A,set(C))] :
      ( aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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,I4)),R2)),bNF_Wellorder_ordLeq(A,B))
     => ( ! [X2: A] :
            ( aa(set(A),$o,member(A,X2),I4)
           => aa(set(product_prod(set(product_prod(C,C)),set(product_prod(B,B)))),$o,member(product_prod(set(product_prod(C,C)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(C,C)),set(product_prod(B,B))),aa(set(product_prod(C,C)),fun(set(product_prod(B,B)),product_prod(set(product_prod(C,C)),set(product_prod(B,B)))),product_Pair(set(product_prod(C,C)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(C,aa(A,set(C),A4,X2))),R2)),bNF_Wellorder_ordLeq(C,B)) )
       => ( ( bNF_Ca4139267488887388095finite(B,R2)
            & aa(set(product_prod(B,B)),$o,bNF_Ca8970107618336181345der_on(B,field2(B,R2)),R2) )
         => aa(set(product_prod(set(product_prod(C,C)),set(product_prod(B,B)))),$o,member(product_prod(set(product_prod(C,C)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(C,C)),set(product_prod(B,B))),aa(set(product_prod(C,C)),fun(set(product_prod(B,B)),product_prod(set(product_prod(C,C)),set(product_prod(B,B)))),product_Pair(set(product_prod(C,C)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(C,aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(A),set(set(C)),image2(A,set(C),A4),I4)))),R2)),bNF_Wellorder_ordLeq(C,B)) ) ) ) ).

% UNION_Cinfinite_bound
tff(fact_6079_card__of__Csum__Times_H,axiom,
    ! [A: $tType,C: $tType,B: $tType,R2: set(product_prod(A,A)),I4: set(B),A4: fun(B,set(C))] :
      ( aa(set(product_prod(A,A)),$o,bNF_Ca8970107618336181345der_on(A,field2(A,R2)),R2)
     => ( ! [X2: B] :
            ( aa(set(B),$o,member(B,X2),I4)
           => aa(set(product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),$o,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),A4,X2))),R2)),bNF_Wellorder_ordLeq(C,A)) )
       => aa(set(product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(product_prod(B,A),product_prod(B,A))))),$o,member(product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),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(B,C),product_prod(B,C))),set(product_prod(product_prod(B,A),product_prod(B,A)))),aa(set(product_prod(product_prod(B,C),product_prod(B,C))),fun(set(product_prod(product_prod(B,A),product_prod(B,A))),product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(product_prod(B,A),product_prod(B,A))))),product_Pair(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(product_prod(B,A),product_prod(B,A)))),bNF_Cardinal_Csum(B,C,bNF_Ca6860139660246222851ard_of(B,I4),aTP_Lamp_anf(fun(B,set(C)),fun(B,set(product_prod(C,C))),A4))),bNF_Cardinal_cprod(B,A,bNF_Ca6860139660246222851ard_of(B,I4),R2))),bNF_Wellorder_ordLeq(product_prod(B,C),product_prod(B,A))) ) ) ).

% card_of_Csum_Times'
tff(fact_6080_Cfinite__ordLess__Cinfinite,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,A)),S2: set(product_prod(B,B))] :
      ( ( bNF_Cardinal_cfinite(A,R2)
        & aa(set(product_prod(A,A)),$o,bNF_Ca8970107618336181345der_on(A,field2(A,R2)),R2) )
     => ( ( bNF_Ca4139267488887388095finite(B,S2)
          & aa(set(product_prod(B,B)),$o,bNF_Ca8970107618336181345der_on(B,field2(B,S2)),S2) )
       => aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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))),R2),S2)),bNF_We4044943003108391690rdLess(A,B)) ) ) ).

% Cfinite_ordLess_Cinfinite
tff(fact_6081_cprod__com,axiom,
    ! [B: $tType,A: $tType,P1: set(product_prod(A,A)),P22: set(product_prod(B,B))] : aa(set(product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(product_prod(B,A),product_prod(B,A))))),$o,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_Cardinal_cprod(A,B,P1,P22)),bNF_Cardinal_cprod(B,A,P22,P1))),bNF_Wellorder_ordIso(product_prod(A,B),product_prod(B,A))) ).

% cprod_com
tff(fact_6082_card__order__cprod,axiom,
    ! [A: $tType,B: $tType,R12: set(product_prod(A,A)),R23: set(product_prod(B,B))] :
      ( aa(set(product_prod(A,A)),$o,bNF_Ca8970107618336181345der_on(A,top_top(set(A))),R12)
     => ( aa(set(product_prod(B,B)),$o,bNF_Ca8970107618336181345der_on(B,top_top(set(B))),R23)
       => aa(set(product_prod(product_prod(A,B),product_prod(A,B))),$o,bNF_Ca8970107618336181345der_on(product_prod(A,B),top_top(set(product_prod(A,B)))),bNF_Cardinal_cprod(A,B,R12,R23)) ) ) ).

% card_order_cprod
tff(fact_6083_Cfinite__cprod__Cinfinite,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,A)),S2: set(product_prod(B,B))] :
      ( ( bNF_Cardinal_cfinite(A,R2)
        & aa(set(product_prod(A,A)),$o,bNF_Ca8970107618336181345der_on(A,field2(A,R2)),R2) )
     => ( ( bNF_Ca4139267488887388095finite(B,S2)
          & aa(set(product_prod(B,B)),$o,bNF_Ca8970107618336181345der_on(B,field2(B,S2)),S2) )
       => aa(set(product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(B,B)))),$o,member(product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(B,B))),aa(set(product_prod(product_prod(A,B),product_prod(A,B))),fun(set(product_prod(B,B)),product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(B,B)))),product_Pair(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(B,B))),bNF_Cardinal_cprod(A,B,R2,S2)),S2)),bNF_Wellorder_ordLeq(product_prod(A,B),B)) ) ) ).

% Cfinite_cprod_Cinfinite
tff(fact_6084_cprod__def,axiom,
    ! [A: $tType,B: $tType,R12: set(product_prod(A,A)),R23: set(product_prod(B,B))] : bNF_Cardinal_cprod(A,B,R12,R23) = bNF_Ca6860139660246222851ard_of(product_prod(A,B),product_Sigma(A,B,field2(A,R12),aTP_Lamp_and(set(product_prod(B,B)),fun(A,set(B)),R23))) ).

% cprod_def
tff(fact_6085_cprod__mono2,axiom,
    ! [B: $tType,A: $tType,C: $tType,P22: set(product_prod(A,A)),R23: set(product_prod(B,B)),Q3: set(product_prod(C,C))] :
      ( aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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))),P22),R23)),bNF_Wellorder_ordLeq(A,B))
     => 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))))),$o,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_Cardinal_cprod(C,A,Q3,P22)),bNF_Cardinal_cprod(C,B,Q3,R23))),bNF_Wellorder_ordLeq(product_prod(C,A),product_prod(C,B))) ) ).

% cprod_mono2
tff(fact_6086_cprod__mono1,axiom,
    ! [B: $tType,C: $tType,A: $tType,P1: set(product_prod(A,A)),R12: set(product_prod(B,B)),Q3: set(product_prod(C,C))] :
      ( aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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))),P1),R12)),bNF_Wellorder_ordLeq(A,B))
     => 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))))),$o,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_Cardinal_cprod(A,C,P1,Q3)),bNF_Cardinal_cprod(B,C,R12,Q3))),bNF_Wellorder_ordLeq(product_prod(A,C),product_prod(B,C))) ) ).

% cprod_mono1
tff(fact_6087_cprod__mono,axiom,
    ! [D: $tType,B: $tType,C: $tType,A: $tType,P1: set(product_prod(A,A)),R12: set(product_prod(B,B)),P22: set(product_prod(C,C)),R23: set(product_prod(D,D))] :
      ( aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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))),P1),R12)),bNF_Wellorder_ordLeq(A,B))
     => ( aa(set(product_prod(set(product_prod(C,C)),set(product_prod(D,D)))),$o,member(product_prod(set(product_prod(C,C)),set(product_prod(D,D))),aa(set(product_prod(D,D)),product_prod(set(product_prod(C,C)),set(product_prod(D,D))),aa(set(product_prod(C,C)),fun(set(product_prod(D,D)),product_prod(set(product_prod(C,C)),set(product_prod(D,D)))),product_Pair(set(product_prod(C,C)),set(product_prod(D,D))),P22),R23)),bNF_Wellorder_ordLeq(C,D))
       => aa(set(product_prod(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,D),product_prod(B,D))))),$o,member(product_prod(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,D),product_prod(B,D)))),aa(set(product_prod(product_prod(B,D),product_prod(B,D))),product_prod(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,D),product_prod(B,D)))),aa(set(product_prod(product_prod(A,C),product_prod(A,C))),fun(set(product_prod(product_prod(B,D),product_prod(B,D))),product_prod(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,D),product_prod(B,D))))),product_Pair(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,D),product_prod(B,D)))),bNF_Cardinal_cprod(A,C,P1,P22)),bNF_Cardinal_cprod(B,D,R12,R23))),bNF_Wellorder_ordLeq(product_prod(A,C),product_prod(B,D))) ) ) ).

% cprod_mono
tff(fact_6088_cprod__cong2,axiom,
    ! [B: $tType,A: $tType,C: $tType,P22: set(product_prod(A,A)),R23: set(product_prod(B,B)),Q3: set(product_prod(C,C))] :
      ( aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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))),P22),R23)),bNF_Wellorder_ordIso(A,B))
     => 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))))),$o,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_Cardinal_cprod(C,A,Q3,P22)),bNF_Cardinal_cprod(C,B,Q3,R23))),bNF_Wellorder_ordIso(product_prod(C,A),product_prod(C,B))) ) ).

% cprod_cong2
tff(fact_6089_cprod__cong1,axiom,
    ! [B: $tType,C: $tType,A: $tType,P1: set(product_prod(A,A)),R12: set(product_prod(B,B)),P22: set(product_prod(C,C))] :
      ( aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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))),P1),R12)),bNF_Wellorder_ordIso(A,B))
     => 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))))),$o,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_Cardinal_cprod(A,C,P1,P22)),bNF_Cardinal_cprod(B,C,R12,P22))),bNF_Wellorder_ordIso(product_prod(A,C),product_prod(B,C))) ) ).

% cprod_cong1
tff(fact_6090_cprod__cong,axiom,
    ! [D: $tType,B: $tType,C: $tType,A: $tType,P1: set(product_prod(A,A)),R12: set(product_prod(B,B)),P22: set(product_prod(C,C)),R23: set(product_prod(D,D))] :
      ( aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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))),P1),R12)),bNF_Wellorder_ordIso(A,B))
     => ( aa(set(product_prod(set(product_prod(C,C)),set(product_prod(D,D)))),$o,member(product_prod(set(product_prod(C,C)),set(product_prod(D,D))),aa(set(product_prod(D,D)),product_prod(set(product_prod(C,C)),set(product_prod(D,D))),aa(set(product_prod(C,C)),fun(set(product_prod(D,D)),product_prod(set(product_prod(C,C)),set(product_prod(D,D)))),product_Pair(set(product_prod(C,C)),set(product_prod(D,D))),P22),R23)),bNF_Wellorder_ordIso(C,D))
       => aa(set(product_prod(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,D),product_prod(B,D))))),$o,member(product_prod(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,D),product_prod(B,D)))),aa(set(product_prod(product_prod(B,D),product_prod(B,D))),product_prod(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,D),product_prod(B,D)))),aa(set(product_prod(product_prod(A,C),product_prod(A,C))),fun(set(product_prod(product_prod(B,D),product_prod(B,D))),product_prod(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,D),product_prod(B,D))))),product_Pair(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,D),product_prod(B,D)))),bNF_Cardinal_cprod(A,C,P1,P22)),bNF_Cardinal_cprod(B,D,R12,R23))),bNF_Wellorder_ordIso(product_prod(A,C),product_prod(B,D))) ) ) ).

% cprod_cong
tff(fact_6091_cprod__infinite,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( ( bNF_Ca4139267488887388095finite(A,R2)
        & aa(set(product_prod(A,A)),$o,bNF_Ca8970107618336181345der_on(A,field2(A,R2)),R2) )
     => aa(set(product_prod(set(product_prod(product_prod(A,A),product_prod(A,A))),set(product_prod(A,A)))),$o,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_Cardinal_cprod(A,A,R2,R2)),R2)),bNF_Wellorder_ordIso(product_prod(A,A),A)) ) ).

% cprod_infinite
tff(fact_6092_cprod__cinfinite__bound,axiom,
    ! [B: $tType,C: $tType,A: $tType,P3: set(product_prod(A,A)),R2: set(product_prod(B,B)),Q3: set(product_prod(C,C))] :
      ( aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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))),P3),R2)),bNF_Wellorder_ordLeq(A,B))
     => ( aa(set(product_prod(set(product_prod(C,C)),set(product_prod(B,B)))),$o,member(product_prod(set(product_prod(C,C)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(C,C)),set(product_prod(B,B))),aa(set(product_prod(C,C)),fun(set(product_prod(B,B)),product_prod(set(product_prod(C,C)),set(product_prod(B,B)))),product_Pair(set(product_prod(C,C)),set(product_prod(B,B))),Q3),R2)),bNF_Wellorder_ordLeq(C,B))
       => ( aa(set(product_prod(A,A)),$o,bNF_Ca8970107618336181345der_on(A,field2(A,P3)),P3)
         => ( aa(set(product_prod(C,C)),$o,bNF_Ca8970107618336181345der_on(C,field2(C,Q3)),Q3)
           => ( ( bNF_Ca4139267488887388095finite(B,R2)
                & aa(set(product_prod(B,B)),$o,bNF_Ca8970107618336181345der_on(B,field2(B,R2)),R2) )
             => aa(set(product_prod(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(B,B)))),$o,member(product_prod(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(B,B))),aa(set(product_prod(product_prod(A,C),product_prod(A,C))),fun(set(product_prod(B,B)),product_prod(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(B,B)))),product_Pair(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(B,B))),bNF_Cardinal_cprod(A,C,P3,Q3)),R2)),bNF_Wellorder_ordLeq(product_prod(A,C),B)) ) ) ) ) ) ).

% cprod_cinfinite_bound
tff(fact_6093_card__of__Csum__Times,axiom,
    ! [C: $tType,B: $tType,A: $tType,I4: set(A),A4: fun(A,set(B)),B3: set(C)] :
      ( ! [X2: A] :
          ( aa(set(A),$o,member(A,X2),I4)
         => aa(set(product_prod(set(product_prod(B,B)),set(product_prod(C,C)))),$o,member(product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(B,B)),fun(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C)))),product_Pair(set(product_prod(B,B)),set(product_prod(C,C))),bNF_Ca6860139660246222851ard_of(B,aa(A,set(B),A4,X2))),bNF_Ca6860139660246222851ard_of(C,B3))),bNF_Wellorder_ordLeq(B,C)) )
     => aa(set(product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(product_prod(A,C),product_prod(A,C))))),$o,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_Cardinal_Csum(A,B,bNF_Ca6860139660246222851ard_of(A,I4),aTP_Lamp_anb(fun(A,set(B)),fun(A,set(product_prod(B,B))),A4))),bNF_Cardinal_cprod(A,C,bNF_Ca6860139660246222851ard_of(A,I4),bNF_Ca6860139660246222851ard_of(C,B3)))),bNF_Wellorder_ordLeq(product_prod(A,B),product_prod(A,C))) ) ).

% card_of_Csum_Times
tff(fact_6094_cprod__dup,axiom,
    ! [A: $tType,C: $tType,B: $tType,R2: set(product_prod(A,A)),P3: set(product_prod(B,B)),P7: set(product_prod(C,C))] :
      ( bNF_Ca4139267488887388095finite(A,R2)
     => ( aa(set(product_prod(A,A)),$o,bNF_Ca8970107618336181345der_on(A,field2(A,R2)),R2)
       => ( aa(set(product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(product_prod(A,A),product_prod(A,A))))),$o,member(product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(product_prod(A,A),product_prod(A,A)))),aa(set(product_prod(product_prod(A,A),product_prod(A,A))),product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(product_prod(A,A),product_prod(A,A)))),aa(set(product_prod(product_prod(B,C),product_prod(B,C))),fun(set(product_prod(product_prod(A,A),product_prod(A,A))),product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(product_prod(A,A),product_prod(A,A))))),product_Pair(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(product_prod(A,A),product_prod(A,A)))),bNF_Cardinal_cprod(B,C,P3,P7)),bNF_Cardinal_cprod(A,A,R2,R2))),bNF_Wellorder_ordIso(product_prod(B,C),product_prod(A,A)))
         => aa(set(product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A)))),$o,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_Cardinal_cprod(B,C,P3,P7)),R2)),bNF_Wellorder_ordIso(product_prod(B,C),A)) ) ) ) ).

% cprod_dup
tff(fact_6095_comp__single__set__bd,axiom,
    ! [B: $tType,D: $tType,A: $tType,E: $tType,C: $tType,Fbd: set(product_prod(A,A)),Fset: fun(B,set(C)),Gset: fun(D,set(B)),Gbd: set(product_prod(E,E)),X: D] :
      ( aa(set(product_prod(A,A)),$o,bNF_Ca8970107618336181345der_on(A,field2(A,Fbd)),Fbd)
     => ( ! [X2: B] : aa(set(product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),$o,member(product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(C,C)),fun(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),product_Pair(set(product_prod(C,C)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(C,aa(B,set(C),Fset,X2))),Fbd)),bNF_Wellorder_ordLeq(C,A))
       => ( ! [X2: D] : aa(set(product_prod(set(product_prod(B,B)),set(product_prod(E,E)))),$o,member(product_prod(set(product_prod(B,B)),set(product_prod(E,E))),aa(set(product_prod(E,E)),product_prod(set(product_prod(B,B)),set(product_prod(E,E))),aa(set(product_prod(B,B)),fun(set(product_prod(E,E)),product_prod(set(product_prod(B,B)),set(product_prod(E,E)))),product_Pair(set(product_prod(B,B)),set(product_prod(E,E))),bNF_Ca6860139660246222851ard_of(B,aa(D,set(B),Gset,X2))),Gbd)),bNF_Wellorder_ordLeq(B,E))
         => aa(set(product_prod(set(product_prod(C,C)),set(product_prod(product_prod(E,A),product_prod(E,A))))),$o,member(product_prod(set(product_prod(C,C)),set(product_prod(product_prod(E,A),product_prod(E,A)))),aa(set(product_prod(product_prod(E,A),product_prod(E,A))),product_prod(set(product_prod(C,C)),set(product_prod(product_prod(E,A),product_prod(E,A)))),aa(set(product_prod(C,C)),fun(set(product_prod(product_prod(E,A),product_prod(E,A))),product_prod(set(product_prod(C,C)),set(product_prod(product_prod(E,A),product_prod(E,A))))),product_Pair(set(product_prod(C,C)),set(product_prod(product_prod(E,A),product_prod(E,A)))),bNF_Ca6860139660246222851ard_of(C,aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(B),set(set(C)),image2(B,set(C),Fset),aa(D,set(B),Gset,X))))),bNF_Cardinal_cprod(E,A,Gbd,Fbd))),bNF_Wellorder_ordLeq(C,product_prod(E,A))) ) ) ) ).

% comp_single_set_bd
tff(fact_6096_cprod__infinite1_H,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,A)),P3: set(product_prod(B,B))] :
      ( ( bNF_Ca4139267488887388095finite(A,R2)
        & aa(set(product_prod(A,A)),$o,bNF_Ca8970107618336181345der_on(A,field2(A,R2)),R2) )
     => ( ( ~ aa(set(product_prod(set(product_prod(B,B)),set(product_prod(B,B)))),$o,member(product_prod(set(product_prod(B,B)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(B,B)),set(product_prod(B,B))),aa(set(product_prod(B,B)),fun(set(product_prod(B,B)),product_prod(set(product_prod(B,B)),set(product_prod(B,B)))),product_Pair(set(product_prod(B,B)),set(product_prod(B,B))),P3),bNF_Cardinal_czero(B))),bNF_Wellorder_ordIso(B,B))
          & aa(set(product_prod(B,B)),$o,bNF_Ca8970107618336181345der_on(B,field2(B,P3)),P3) )
       => ( aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),$o,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),P3),R2)),bNF_Wellorder_ordLeq(B,A))
         => aa(set(product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A)))),$o,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_Cardinal_cprod(A,B,R2,P3)),R2)),bNF_Wellorder_ordIso(product_prod(A,B),A)) ) ) ) ).

% cprod_infinite1'
tff(fact_6097_Cinfinite__cprod2,axiom,
    ! [A: $tType,B: $tType,R12: set(product_prod(A,A)),R23: set(product_prod(B,B))] :
      ( ( ~ aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),$o,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))),R12),bNF_Cardinal_czero(A))),bNF_Wellorder_ordIso(A,A))
        & aa(set(product_prod(A,A)),$o,bNF_Ca8970107618336181345der_on(A,field2(A,R12)),R12) )
     => ( ( bNF_Ca4139267488887388095finite(B,R23)
          & aa(set(product_prod(B,B)),$o,bNF_Ca8970107618336181345der_on(B,field2(B,R23)),R23) )
       => ( bNF_Ca4139267488887388095finite(product_prod(A,B),bNF_Cardinal_cprod(A,B,R12,R23))
          & aa(set(product_prod(product_prod(A,B),product_prod(A,B))),$o,bNF_Ca8970107618336181345der_on(product_prod(A,B),field2(product_prod(A,B),bNF_Cardinal_cprod(A,B,R12,R23))),bNF_Cardinal_cprod(A,B,R12,R23)) ) ) ) ).

% Cinfinite_cprod2
tff(fact_6098_czero__def,axiom,
    ! [A: $tType] : bNF_Cardinal_czero(A) = bNF_Ca6860139660246222851ard_of(A,bot_bot(set(A))) ).

% czero_def
tff(fact_6099_czero__ordIso,axiom,
    ! [B: $tType,A: $tType] : aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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_Cardinal_czero(A)),bNF_Cardinal_czero(B))),bNF_Wellorder_ordIso(A,B)) ).

% czero_ordIso
tff(fact_6100_cinfinite__not__czero,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,A))] :
      ( bNF_Ca4139267488887388095finite(A,R2)
     => ~ aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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))),R2),bNF_Cardinal_czero(B))),bNF_Wellorder_ordIso(A,B)) ) ).

% cinfinite_not_czero
tff(fact_6101_czeroE,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,A))] :
      ( aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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))),R2),bNF_Cardinal_czero(B))),bNF_Wellorder_ordIso(A,B))
     => ( field2(A,R2) = bot_bot(set(A)) ) ) ).

% czeroE
tff(fact_6102_card__of__ordIso__czero__iff__empty,axiom,
    ! [B: $tType,A: $tType,A4: set(A)] :
      ( aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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,A4)),bNF_Cardinal_czero(B))),bNF_Wellorder_ordIso(A,B))
    <=> ( A4 = bot_bot(set(A)) ) ) ).

% card_of_ordIso_czero_iff_empty
tff(fact_6103_czeroI,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,bNF_Ca8970107618336181345der_on(A,field2(A,R2)),R2)
     => ( ( field2(A,R2) = bot_bot(set(A)) )
       => aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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))),R2),bNF_Cardinal_czero(B))),bNF_Wellorder_ordIso(A,B)) ) ) ).

% czeroI
tff(fact_6104_Cnotzero__imp__not__empty,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( ( ~ aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),$o,member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R2),bNF_Cardinal_czero(A))),bNF_Wellorder_ordIso(A,A))
        & aa(set(product_prod(A,A)),$o,bNF_Ca8970107618336181345der_on(A,field2(A,R2)),R2) )
     => ( field2(A,R2) != bot_bot(set(A)) ) ) ).

% Cnotzero_imp_not_empty
tff(fact_6105_Cnotzero__mono,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(A,A)),Q3: set(product_prod(B,B))] :
      ( ( ~ aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),$o,member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R2),bNF_Cardinal_czero(A))),bNF_Wellorder_ordIso(A,A))
        & aa(set(product_prod(A,A)),$o,bNF_Ca8970107618336181345der_on(A,field2(A,R2)),R2) )
     => ( aa(set(product_prod(B,B)),$o,bNF_Ca8970107618336181345der_on(B,field2(B,Q3)),Q3)
       => ( aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R2),Q3)),bNF_Wellorder_ordLeq(A,B))
         => ( ~ aa(set(product_prod(set(product_prod(B,B)),set(product_prod(B,B)))),$o,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))),Q3),bNF_Cardinal_czero(B))),bNF_Wellorder_ordIso(B,B))
            & aa(set(product_prod(B,B)),$o,bNF_Ca8970107618336181345der_on(B,field2(B,Q3)),Q3) ) ) ) ) ).

% Cnotzero_mono
tff(fact_6106_Cinfinite__Cnotzero,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( ( bNF_Ca4139267488887388095finite(A,R2)
        & aa(set(product_prod(A,A)),$o,bNF_Ca8970107618336181345der_on(A,field2(A,R2)),R2) )
     => ( ~ aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),$o,member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R2),bNF_Cardinal_czero(A))),bNF_Wellorder_ordIso(A,A))
        & aa(set(product_prod(A,A)),$o,bNF_Ca8970107618336181345der_on(A,field2(A,R2)),R2) ) ) ).

% Cinfinite_Cnotzero
tff(fact_6107_Cnotzero__UNIV,axiom,
    ! [A: $tType] :
      ( ~ aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),$o,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,top_top(set(A)))),bNF_Cardinal_czero(A))),bNF_Wellorder_ordIso(A,A))
      & aa(set(product_prod(A,A)),$o,bNF_Ca8970107618336181345der_on(A,field2(A,bNF_Ca6860139660246222851ard_of(A,top_top(set(A))))),bNF_Ca6860139660246222851ard_of(A,top_top(set(A)))) ) ).

% Cnotzero_UNIV
tff(fact_6108_cinfinite__cprod2,axiom,
    ! [A: $tType,B: $tType,R12: set(product_prod(A,A)),R23: set(product_prod(B,B))] :
      ( ( ~ aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),$o,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))),R12),bNF_Cardinal_czero(A))),bNF_Wellorder_ordIso(A,A))
        & aa(set(product_prod(A,A)),$o,bNF_Ca8970107618336181345der_on(A,field2(A,R12)),R12) )
     => ( ( bNF_Ca4139267488887388095finite(B,R23)
          & aa(set(product_prod(B,B)),$o,bNF_Ca8970107618336181345der_on(B,field2(B,R23)),R23) )
       => bNF_Ca4139267488887388095finite(product_prod(A,B),bNF_Cardinal_cprod(A,B,R12,R23)) ) ) ).

% cinfinite_cprod2
tff(fact_6109_ordLeq__cprod2,axiom,
    ! [A: $tType,B: $tType,P1: set(product_prod(A,A)),P22: set(product_prod(B,B))] :
      ( ( ~ aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),$o,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))),P1),bNF_Cardinal_czero(A))),bNF_Wellorder_ordIso(A,A))
        & aa(set(product_prod(A,A)),$o,bNF_Ca8970107618336181345der_on(A,field2(A,P1)),P1) )
     => ( aa(set(product_prod(B,B)),$o,bNF_Ca8970107618336181345der_on(B,field2(B,P22)),P22)
       => aa(set(product_prod(set(product_prod(B,B)),set(product_prod(product_prod(A,B),product_prod(A,B))))),$o,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)))),P22),bNF_Cardinal_cprod(A,B,P1,P22))),bNF_Wellorder_ordLeq(B,product_prod(A,B))) ) ) ).

% ordLeq_cprod2
tff(fact_6110_cone__ordLeq__Cnotzero,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( ( ~ aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),$o,member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R2),bNF_Cardinal_czero(A))),bNF_Wellorder_ordIso(A,A))
        & aa(set(product_prod(A,A)),$o,bNF_Ca8970107618336181345der_on(A,field2(A,R2)),R2) )
     => aa(set(product_prod(set(product_prod(product_unit,product_unit)),set(product_prod(A,A)))),$o,member(product_prod(set(product_prod(product_unit,product_unit)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(product_unit,product_unit)),set(product_prod(A,A))),aa(set(product_prod(product_unit,product_unit)),fun(set(product_prod(A,A)),product_prod(set(product_prod(product_unit,product_unit)),set(product_prod(A,A)))),product_Pair(set(product_prod(product_unit,product_unit)),set(product_prod(A,A))),bNF_Cardinal_cone),R2)),bNF_Wellorder_ordLeq(product_unit,A)) ) ).

% cone_ordLeq_Cnotzero
tff(fact_6111_cexp__mono,axiom,
    ! [E: $tType,F3: $tType,B: $tType,D: $tType,A: $tType,C: $tType,P1: set(product_prod(A,A)),R12: set(product_prod(B,B)),P22: set(product_prod(C,C)),R23: set(product_prod(D,D))] :
      ( aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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))),P1),R12)),bNF_Wellorder_ordLeq(A,B))
     => ( aa(set(product_prod(set(product_prod(C,C)),set(product_prod(D,D)))),$o,member(product_prod(set(product_prod(C,C)),set(product_prod(D,D))),aa(set(product_prod(D,D)),product_prod(set(product_prod(C,C)),set(product_prod(D,D))),aa(set(product_prod(C,C)),fun(set(product_prod(D,D)),product_prod(set(product_prod(C,C)),set(product_prod(D,D)))),product_Pair(set(product_prod(C,C)),set(product_prod(D,D))),P22),R23)),bNF_Wellorder_ordLeq(C,D))
       => ( ( aa(set(product_prod(set(product_prod(C,C)),set(product_prod(E,E)))),$o,member(product_prod(set(product_prod(C,C)),set(product_prod(E,E))),aa(set(product_prod(E,E)),product_prod(set(product_prod(C,C)),set(product_prod(E,E))),aa(set(product_prod(C,C)),fun(set(product_prod(E,E)),product_prod(set(product_prod(C,C)),set(product_prod(E,E)))),product_Pair(set(product_prod(C,C)),set(product_prod(E,E))),P22),bNF_Cardinal_czero(E))),bNF_Wellorder_ordIso(C,E))
           => aa(set(product_prod(set(product_prod(D,D)),set(product_prod(F3,F3)))),$o,member(product_prod(set(product_prod(D,D)),set(product_prod(F3,F3))),aa(set(product_prod(F3,F3)),product_prod(set(product_prod(D,D)),set(product_prod(F3,F3))),aa(set(product_prod(D,D)),fun(set(product_prod(F3,F3)),product_prod(set(product_prod(D,D)),set(product_prod(F3,F3)))),product_Pair(set(product_prod(D,D)),set(product_prod(F3,F3))),R23),bNF_Cardinal_czero(F3))),bNF_Wellorder_ordIso(D,F3)) )
         => ( aa(set(product_prod(C,C)),$o,bNF_Ca8970107618336181345der_on(C,field2(C,P22)),P22)
           => aa(set(product_prod(set(product_prod(fun(C,A),fun(C,A))),set(product_prod(fun(D,B),fun(D,B))))),$o,member(product_prod(set(product_prod(fun(C,A),fun(C,A))),set(product_prod(fun(D,B),fun(D,B)))),aa(set(product_prod(fun(D,B),fun(D,B))),product_prod(set(product_prod(fun(C,A),fun(C,A))),set(product_prod(fun(D,B),fun(D,B)))),aa(set(product_prod(fun(C,A),fun(C,A))),fun(set(product_prod(fun(D,B),fun(D,B))),product_prod(set(product_prod(fun(C,A),fun(C,A))),set(product_prod(fun(D,B),fun(D,B))))),product_Pair(set(product_prod(fun(C,A),fun(C,A))),set(product_prod(fun(D,B),fun(D,B)))),bNF_Cardinal_cexp(A,C,P1,P22)),bNF_Cardinal_cexp(B,D,R12,R23))),bNF_Wellorder_ordLeq(fun(C,A),fun(D,B))) ) ) ) ) ).

% cexp_mono
tff(fact_6112_cexp__cprod,axiom,
    ! [A: $tType,C: $tType,B: $tType,R12: set(product_prod(A,A)),R23: set(product_prod(C,C)),R32: set(product_prod(B,B))] :
      ( aa(set(product_prod(A,A)),$o,bNF_Ca8970107618336181345der_on(A,field2(A,R12)),R12)
     => aa(set(product_prod(set(product_prod(fun(B,fun(C,A)),fun(B,fun(C,A)))),set(product_prod(fun(product_prod(C,B),A),fun(product_prod(C,B),A))))),$o,member(product_prod(set(product_prod(fun(B,fun(C,A)),fun(B,fun(C,A)))),set(product_prod(fun(product_prod(C,B),A),fun(product_prod(C,B),A)))),aa(set(product_prod(fun(product_prod(C,B),A),fun(product_prod(C,B),A))),product_prod(set(product_prod(fun(B,fun(C,A)),fun(B,fun(C,A)))),set(product_prod(fun(product_prod(C,B),A),fun(product_prod(C,B),A)))),aa(set(product_prod(fun(B,fun(C,A)),fun(B,fun(C,A)))),fun(set(product_prod(fun(product_prod(C,B),A),fun(product_prod(C,B),A))),product_prod(set(product_prod(fun(B,fun(C,A)),fun(B,fun(C,A)))),set(product_prod(fun(product_prod(C,B),A),fun(product_prod(C,B),A))))),product_Pair(set(product_prod(fun(B,fun(C,A)),fun(B,fun(C,A)))),set(product_prod(fun(product_prod(C,B),A),fun(product_prod(C,B),A)))),bNF_Cardinal_cexp(fun(C,A),B,bNF_Cardinal_cexp(A,C,R12,R23),R32)),bNF_Cardinal_cexp(A,product_prod(C,B),R12,bNF_Cardinal_cprod(C,B,R23,R32)))),bNF_Wellorder_ordIso(fun(B,fun(C,A)),fun(product_prod(C,B),A))) ) ).

% cexp_cprod
tff(fact_6113_cexp__cone,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,bNF_Ca8970107618336181345der_on(A,field2(A,R2)),R2)
     => aa(set(product_prod(set(product_prod(fun(product_unit,A),fun(product_unit,A))),set(product_prod(A,A)))),$o,member(product_prod(set(product_prod(fun(product_unit,A),fun(product_unit,A))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(fun(product_unit,A),fun(product_unit,A))),set(product_prod(A,A))),aa(set(product_prod(fun(product_unit,A),fun(product_unit,A))),fun(set(product_prod(A,A)),product_prod(set(product_prod(fun(product_unit,A),fun(product_unit,A))),set(product_prod(A,A)))),product_Pair(set(product_prod(fun(product_unit,A),fun(product_unit,A))),set(product_prod(A,A))),bNF_Cardinal_cexp(A,product_unit,R2,bNF_Cardinal_cone)),R2)),bNF_Wellorder_ordIso(fun(product_unit,A),A)) ) ).

% cexp_cone
tff(fact_6114_cone__not__czero,axiom,
    ! [A: $tType] : ~ aa(set(product_prod(set(product_prod(product_unit,product_unit)),set(product_prod(A,A)))),$o,member(product_prod(set(product_prod(product_unit,product_unit)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(product_unit,product_unit)),set(product_prod(A,A))),aa(set(product_prod(product_unit,product_unit)),fun(set(product_prod(A,A)),product_prod(set(product_prod(product_unit,product_unit)),set(product_prod(A,A)))),product_Pair(set(product_prod(product_unit,product_unit)),set(product_prod(A,A))),bNF_Cardinal_cone),bNF_Cardinal_czero(A))),bNF_Wellorder_ordIso(product_unit,A)) ).

% cone_not_czero
tff(fact_6115_cprod__cexp,axiom,
    ! [C: $tType,B: $tType,A: $tType,R2: set(product_prod(B,B)),S2: set(product_prod(C,C)),T3: set(product_prod(A,A))] : aa(set(product_prod(set(product_prod(fun(A,product_prod(B,C)),fun(A,product_prod(B,C)))),set(product_prod(product_prod(fun(A,B),fun(A,C)),product_prod(fun(A,B),fun(A,C)))))),$o,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_Cardinal_cexp(product_prod(B,C),A,bNF_Cardinal_cprod(B,C,R2,S2),T3)),bNF_Cardinal_cprod(fun(A,B),fun(A,C),bNF_Cardinal_cexp(B,A,R2,T3),bNF_Cardinal_cexp(C,A,S2,T3)))),bNF_Wellorder_ordIso(fun(A,product_prod(B,C)),product_prod(fun(A,B),fun(A,C)))) ).

% cprod_cexp
tff(fact_6116_cexp__cprod__ordLeq,axiom,
    ! [A: $tType,B: $tType,C: $tType,R12: set(product_prod(A,A)),R23: set(product_prod(B,B)),R32: set(product_prod(C,C))] :
      ( aa(set(product_prod(A,A)),$o,bNF_Ca8970107618336181345der_on(A,field2(A,R12)),R12)
     => ( ( bNF_Ca4139267488887388095finite(B,R23)
          & aa(set(product_prod(B,B)),$o,bNF_Ca8970107618336181345der_on(B,field2(B,R23)),R23) )
       => ( ( ~ aa(set(product_prod(set(product_prod(C,C)),set(product_prod(C,C)))),$o,member(product_prod(set(product_prod(C,C)),set(product_prod(C,C))),aa(set(product_prod(C,C)),product_prod(set(product_prod(C,C)),set(product_prod(C,C))),aa(set(product_prod(C,C)),fun(set(product_prod(C,C)),product_prod(set(product_prod(C,C)),set(product_prod(C,C)))),product_Pair(set(product_prod(C,C)),set(product_prod(C,C))),R32),bNF_Cardinal_czero(C))),bNF_Wellorder_ordIso(C,C))
            & aa(set(product_prod(C,C)),$o,bNF_Ca8970107618336181345der_on(C,field2(C,R32)),R32) )
         => ( aa(set(product_prod(set(product_prod(C,C)),set(product_prod(B,B)))),$o,member(product_prod(set(product_prod(C,C)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(C,C)),set(product_prod(B,B))),aa(set(product_prod(C,C)),fun(set(product_prod(B,B)),product_prod(set(product_prod(C,C)),set(product_prod(B,B)))),product_Pair(set(product_prod(C,C)),set(product_prod(B,B))),R32),R23)),bNF_Wellorder_ordLeq(C,B))
           => aa(set(product_prod(set(product_prod(fun(C,fun(B,A)),fun(C,fun(B,A)))),set(product_prod(fun(B,A),fun(B,A))))),$o,member(product_prod(set(product_prod(fun(C,fun(B,A)),fun(C,fun(B,A)))),set(product_prod(fun(B,A),fun(B,A)))),aa(set(product_prod(fun(B,A),fun(B,A))),product_prod(set(product_prod(fun(C,fun(B,A)),fun(C,fun(B,A)))),set(product_prod(fun(B,A),fun(B,A)))),aa(set(product_prod(fun(C,fun(B,A)),fun(C,fun(B,A)))),fun(set(product_prod(fun(B,A),fun(B,A))),product_prod(set(product_prod(fun(C,fun(B,A)),fun(C,fun(B,A)))),set(product_prod(fun(B,A),fun(B,A))))),product_Pair(set(product_prod(fun(C,fun(B,A)),fun(C,fun(B,A)))),set(product_prod(fun(B,A),fun(B,A)))),bNF_Cardinal_cexp(fun(B,A),C,bNF_Cardinal_cexp(A,B,R12,R23),R32)),bNF_Cardinal_cexp(A,B,R12,R23))),bNF_Wellorder_ordIso(fun(C,fun(B,A)),fun(B,A))) ) ) ) ) ).

% cexp_cprod_ordLeq
tff(fact_6117_cexp__mono_H,axiom,
    ! [B: $tType,D: $tType,A: $tType,C: $tType,P1: set(product_prod(A,A)),R12: set(product_prod(B,B)),P22: set(product_prod(C,C)),R23: set(product_prod(D,D))] :
      ( aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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))),P1),R12)),bNF_Wellorder_ordLeq(A,B))
     => ( aa(set(product_prod(set(product_prod(C,C)),set(product_prod(D,D)))),$o,member(product_prod(set(product_prod(C,C)),set(product_prod(D,D))),aa(set(product_prod(D,D)),product_prod(set(product_prod(C,C)),set(product_prod(D,D))),aa(set(product_prod(C,C)),fun(set(product_prod(D,D)),product_prod(set(product_prod(C,C)),set(product_prod(D,D)))),product_Pair(set(product_prod(C,C)),set(product_prod(D,D))),P22),R23)),bNF_Wellorder_ordLeq(C,D))
       => ( ( ( field2(C,P22) = bot_bot(set(C)) )
           => ( field2(D,R23) = bot_bot(set(D)) ) )
         => aa(set(product_prod(set(product_prod(fun(C,A),fun(C,A))),set(product_prod(fun(D,B),fun(D,B))))),$o,member(product_prod(set(product_prod(fun(C,A),fun(C,A))),set(product_prod(fun(D,B),fun(D,B)))),aa(set(product_prod(fun(D,B),fun(D,B))),product_prod(set(product_prod(fun(C,A),fun(C,A))),set(product_prod(fun(D,B),fun(D,B)))),aa(set(product_prod(fun(C,A),fun(C,A))),fun(set(product_prod(fun(D,B),fun(D,B))),product_prod(set(product_prod(fun(C,A),fun(C,A))),set(product_prod(fun(D,B),fun(D,B))))),product_Pair(set(product_prod(fun(C,A),fun(C,A))),set(product_prod(fun(D,B),fun(D,B)))),bNF_Cardinal_cexp(A,C,P1,P22)),bNF_Cardinal_cexp(B,D,R12,R23))),bNF_Wellorder_ordLeq(fun(C,A),fun(D,B))) ) ) ) ).

% cexp_mono'
tff(fact_6118_cexp__mono1,axiom,
    ! [B: $tType,A: $tType,C: $tType,P1: set(product_prod(A,A)),R12: set(product_prod(B,B)),Q3: set(product_prod(C,C))] :
      ( aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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))),P1),R12)),bNF_Wellorder_ordLeq(A,B))
     => ( aa(set(product_prod(C,C)),$o,bNF_Ca8970107618336181345der_on(C,field2(C,Q3)),Q3)
       => aa(set(product_prod(set(product_prod(fun(C,A),fun(C,A))),set(product_prod(fun(C,B),fun(C,B))))),$o,member(product_prod(set(product_prod(fun(C,A),fun(C,A))),set(product_prod(fun(C,B),fun(C,B)))),aa(set(product_prod(fun(C,B),fun(C,B))),product_prod(set(product_prod(fun(C,A),fun(C,A))),set(product_prod(fun(C,B),fun(C,B)))),aa(set(product_prod(fun(C,A),fun(C,A))),fun(set(product_prod(fun(C,B),fun(C,B))),product_prod(set(product_prod(fun(C,A),fun(C,A))),set(product_prod(fun(C,B),fun(C,B))))),product_Pair(set(product_prod(fun(C,A),fun(C,A))),set(product_prod(fun(C,B),fun(C,B)))),bNF_Cardinal_cexp(A,C,P1,Q3)),bNF_Cardinal_cexp(B,C,R12,Q3))),bNF_Wellorder_ordLeq(fun(C,A),fun(C,B))) ) ) ).

% cexp_mono1
tff(fact_6119_cexp__mono2_H,axiom,
    ! [B: $tType,C: $tType,A: $tType,P22: set(product_prod(A,A)),R23: set(product_prod(B,B)),Q3: set(product_prod(C,C))] :
      ( aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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))),P22),R23)),bNF_Wellorder_ordLeq(A,B))
     => ( aa(set(product_prod(C,C)),$o,bNF_Ca8970107618336181345der_on(C,field2(C,Q3)),Q3)
       => ( ( ( field2(A,P22) = bot_bot(set(A)) )
           => ( field2(B,R23) = bot_bot(set(B)) ) )
         => aa(set(product_prod(set(product_prod(fun(A,C),fun(A,C))),set(product_prod(fun(B,C),fun(B,C))))),$o,member(product_prod(set(product_prod(fun(A,C),fun(A,C))),set(product_prod(fun(B,C),fun(B,C)))),aa(set(product_prod(fun(B,C),fun(B,C))),product_prod(set(product_prod(fun(A,C),fun(A,C))),set(product_prod(fun(B,C),fun(B,C)))),aa(set(product_prod(fun(A,C),fun(A,C))),fun(set(product_prod(fun(B,C),fun(B,C))),product_prod(set(product_prod(fun(A,C),fun(A,C))),set(product_prod(fun(B,C),fun(B,C))))),product_Pair(set(product_prod(fun(A,C),fun(A,C))),set(product_prod(fun(B,C),fun(B,C)))),bNF_Cardinal_cexp(C,A,Q3,P22)),bNF_Cardinal_cexp(C,B,Q3,R23))),bNF_Wellorder_ordLeq(fun(A,C),fun(B,C))) ) ) ) ).

% cexp_mono2'
tff(fact_6120_ordLeq__cexp1,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(A,A)),Q3: set(product_prod(B,B))] :
      ( ( ~ aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),$o,member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R2),bNF_Cardinal_czero(A))),bNF_Wellorder_ordIso(A,A))
        & aa(set(product_prod(A,A)),$o,bNF_Ca8970107618336181345der_on(A,field2(A,R2)),R2) )
     => ( aa(set(product_prod(B,B)),$o,bNF_Ca8970107618336181345der_on(B,field2(B,Q3)),Q3)
       => aa(set(product_prod(set(product_prod(B,B)),set(product_prod(fun(A,B),fun(A,B))))),$o,member(product_prod(set(product_prod(B,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(B,B)),set(product_prod(fun(A,B),fun(A,B)))),aa(set(product_prod(B,B)),fun(set(product_prod(fun(A,B),fun(A,B))),product_prod(set(product_prod(B,B)),set(product_prod(fun(A,B),fun(A,B))))),product_Pair(set(product_prod(B,B)),set(product_prod(fun(A,B),fun(A,B)))),Q3),bNF_Cardinal_cexp(B,A,Q3,R2))),bNF_Wellorder_ordLeq(B,fun(A,B))) ) ) ).

% ordLeq_cexp1
tff(fact_6121_cexp__cong,axiom,
    ! [B: $tType,D: $tType,A: $tType,C: $tType,P1: set(product_prod(A,A)),R12: set(product_prod(B,B)),P22: set(product_prod(C,C)),R23: set(product_prod(D,D))] :
      ( aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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))),P1),R12)),bNF_Wellorder_ordIso(A,B))
     => ( aa(set(product_prod(set(product_prod(C,C)),set(product_prod(D,D)))),$o,member(product_prod(set(product_prod(C,C)),set(product_prod(D,D))),aa(set(product_prod(D,D)),product_prod(set(product_prod(C,C)),set(product_prod(D,D))),aa(set(product_prod(C,C)),fun(set(product_prod(D,D)),product_prod(set(product_prod(C,C)),set(product_prod(D,D)))),product_Pair(set(product_prod(C,C)),set(product_prod(D,D))),P22),R23)),bNF_Wellorder_ordIso(C,D))
       => ( aa(set(product_prod(D,D)),$o,bNF_Ca8970107618336181345der_on(D,field2(D,R23)),R23)
         => ( aa(set(product_prod(C,C)),$o,bNF_Ca8970107618336181345der_on(C,field2(C,P22)),P22)
           => aa(set(product_prod(set(product_prod(fun(C,A),fun(C,A))),set(product_prod(fun(D,B),fun(D,B))))),$o,member(product_prod(set(product_prod(fun(C,A),fun(C,A))),set(product_prod(fun(D,B),fun(D,B)))),aa(set(product_prod(fun(D,B),fun(D,B))),product_prod(set(product_prod(fun(C,A),fun(C,A))),set(product_prod(fun(D,B),fun(D,B)))),aa(set(product_prod(fun(C,A),fun(C,A))),fun(set(product_prod(fun(D,B),fun(D,B))),product_prod(set(product_prod(fun(C,A),fun(C,A))),set(product_prod(fun(D,B),fun(D,B))))),product_Pair(set(product_prod(fun(C,A),fun(C,A))),set(product_prod(fun(D,B),fun(D,B)))),bNF_Cardinal_cexp(A,C,P1,P22)),bNF_Cardinal_cexp(B,D,R12,R23))),bNF_Wellorder_ordIso(fun(C,A),fun(D,B))) ) ) ) ) ).

% cexp_cong
tff(fact_6122_cexp__cong1,axiom,
    ! [B: $tType,A: $tType,C: $tType,P1: set(product_prod(A,A)),R12: set(product_prod(B,B)),Q3: set(product_prod(C,C))] :
      ( aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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))),P1),R12)),bNF_Wellorder_ordIso(A,B))
     => ( aa(set(product_prod(C,C)),$o,bNF_Ca8970107618336181345der_on(C,field2(C,Q3)),Q3)
       => aa(set(product_prod(set(product_prod(fun(C,A),fun(C,A))),set(product_prod(fun(C,B),fun(C,B))))),$o,member(product_prod(set(product_prod(fun(C,A),fun(C,A))),set(product_prod(fun(C,B),fun(C,B)))),aa(set(product_prod(fun(C,B),fun(C,B))),product_prod(set(product_prod(fun(C,A),fun(C,A))),set(product_prod(fun(C,B),fun(C,B)))),aa(set(product_prod(fun(C,A),fun(C,A))),fun(set(product_prod(fun(C,B),fun(C,B))),product_prod(set(product_prod(fun(C,A),fun(C,A))),set(product_prod(fun(C,B),fun(C,B))))),product_Pair(set(product_prod(fun(C,A),fun(C,A))),set(product_prod(fun(C,B),fun(C,B)))),bNF_Cardinal_cexp(A,C,P1,Q3)),bNF_Cardinal_cexp(B,C,R12,Q3))),bNF_Wellorder_ordIso(fun(C,A),fun(C,B))) ) ) ).

% cexp_cong1
tff(fact_6123_cexp__cong2,axiom,
    ! [B: $tType,C: $tType,A: $tType,P22: set(product_prod(A,A)),R23: set(product_prod(B,B)),Q3: set(product_prod(C,C))] :
      ( aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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))),P22),R23)),bNF_Wellorder_ordIso(A,B))
     => ( aa(set(product_prod(C,C)),$o,bNF_Ca8970107618336181345der_on(C,field2(C,Q3)),Q3)
       => ( aa(set(product_prod(A,A)),$o,bNF_Ca8970107618336181345der_on(A,field2(A,P22)),P22)
         => aa(set(product_prod(set(product_prod(fun(A,C),fun(A,C))),set(product_prod(fun(B,C),fun(B,C))))),$o,member(product_prod(set(product_prod(fun(A,C),fun(A,C))),set(product_prod(fun(B,C),fun(B,C)))),aa(set(product_prod(fun(B,C),fun(B,C))),product_prod(set(product_prod(fun(A,C),fun(A,C))),set(product_prod(fun(B,C),fun(B,C)))),aa(set(product_prod(fun(A,C),fun(A,C))),fun(set(product_prod(fun(B,C),fun(B,C))),product_prod(set(product_prod(fun(A,C),fun(A,C))),set(product_prod(fun(B,C),fun(B,C))))),product_Pair(set(product_prod(fun(A,C),fun(A,C))),set(product_prod(fun(B,C),fun(B,C)))),bNF_Cardinal_cexp(C,A,Q3,P22)),bNF_Cardinal_cexp(C,B,Q3,R23))),bNF_Wellorder_ordIso(fun(A,C),fun(B,C))) ) ) ) ).

% cexp_cong2
tff(fact_6124_cexp__mono2__Cnotzero,axiom,
    ! [B: $tType,C: $tType,A: $tType,P22: set(product_prod(A,A)),R23: set(product_prod(B,B)),Q3: set(product_prod(C,C))] :
      ( aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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))),P22),R23)),bNF_Wellorder_ordLeq(A,B))
     => ( aa(set(product_prod(C,C)),$o,bNF_Ca8970107618336181345der_on(C,field2(C,Q3)),Q3)
       => ( ( ~ aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),$o,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))),P22),bNF_Cardinal_czero(A))),bNF_Wellorder_ordIso(A,A))
            & aa(set(product_prod(A,A)),$o,bNF_Ca8970107618336181345der_on(A,field2(A,P22)),P22) )
         => aa(set(product_prod(set(product_prod(fun(A,C),fun(A,C))),set(product_prod(fun(B,C),fun(B,C))))),$o,member(product_prod(set(product_prod(fun(A,C),fun(A,C))),set(product_prod(fun(B,C),fun(B,C)))),aa(set(product_prod(fun(B,C),fun(B,C))),product_prod(set(product_prod(fun(A,C),fun(A,C))),set(product_prod(fun(B,C),fun(B,C)))),aa(set(product_prod(fun(A,C),fun(A,C))),fun(set(product_prod(fun(B,C),fun(B,C))),product_prod(set(product_prod(fun(A,C),fun(A,C))),set(product_prod(fun(B,C),fun(B,C))))),product_Pair(set(product_prod(fun(A,C),fun(A,C))),set(product_prod(fun(B,C),fun(B,C)))),bNF_Cardinal_cexp(C,A,Q3,P22)),bNF_Cardinal_cexp(C,B,Q3,R23))),bNF_Wellorder_ordLeq(fun(A,C),fun(B,C))) ) ) ) ).

% cexp_mono2_Cnotzero
tff(fact_6125_cexp__mono2,axiom,
    ! [D: $tType,E: $tType,B: $tType,C: $tType,A: $tType,P22: set(product_prod(A,A)),R23: set(product_prod(B,B)),Q3: set(product_prod(C,C))] :
      ( aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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))),P22),R23)),bNF_Wellorder_ordLeq(A,B))
     => ( aa(set(product_prod(C,C)),$o,bNF_Ca8970107618336181345der_on(C,field2(C,Q3)),Q3)
       => ( ( aa(set(product_prod(set(product_prod(A,A)),set(product_prod(D,D)))),$o,member(product_prod(set(product_prod(A,A)),set(product_prod(D,D))),aa(set(product_prod(D,D)),product_prod(set(product_prod(A,A)),set(product_prod(D,D))),aa(set(product_prod(A,A)),fun(set(product_prod(D,D)),product_prod(set(product_prod(A,A)),set(product_prod(D,D)))),product_Pair(set(product_prod(A,A)),set(product_prod(D,D))),P22),bNF_Cardinal_czero(D))),bNF_Wellorder_ordIso(A,D))
           => aa(set(product_prod(set(product_prod(B,B)),set(product_prod(E,E)))),$o,member(product_prod(set(product_prod(B,B)),set(product_prod(E,E))),aa(set(product_prod(E,E)),product_prod(set(product_prod(B,B)),set(product_prod(E,E))),aa(set(product_prod(B,B)),fun(set(product_prod(E,E)),product_prod(set(product_prod(B,B)),set(product_prod(E,E)))),product_Pair(set(product_prod(B,B)),set(product_prod(E,E))),R23),bNF_Cardinal_czero(E))),bNF_Wellorder_ordIso(B,E)) )
         => ( aa(set(product_prod(A,A)),$o,bNF_Ca8970107618336181345der_on(A,field2(A,P22)),P22)
           => aa(set(product_prod(set(product_prod(fun(A,C),fun(A,C))),set(product_prod(fun(B,C),fun(B,C))))),$o,member(product_prod(set(product_prod(fun(A,C),fun(A,C))),set(product_prod(fun(B,C),fun(B,C)))),aa(set(product_prod(fun(B,C),fun(B,C))),product_prod(set(product_prod(fun(A,C),fun(A,C))),set(product_prod(fun(B,C),fun(B,C)))),aa(set(product_prod(fun(A,C),fun(A,C))),fun(set(product_prod(fun(B,C),fun(B,C))),product_prod(set(product_prod(fun(A,C),fun(A,C))),set(product_prod(fun(B,C),fun(B,C))))),product_Pair(set(product_prod(fun(A,C),fun(A,C))),set(product_prod(fun(B,C),fun(B,C)))),bNF_Cardinal_cexp(C,A,Q3,P22)),bNF_Cardinal_cexp(C,B,Q3,R23))),bNF_Wellorder_ordLeq(fun(A,C),fun(B,C))) ) ) ) ) ).

% cexp_mono2
tff(fact_6126_ordLeq__cexp2,axiom,
    ! [A: $tType,B: $tType,Q3: set(product_prod(A,A)),R2: set(product_prod(B,B))] :
      ( aa(set(product_prod(set(product_prod($o,$o)),set(product_prod(A,A)))),$o,member(product_prod(set(product_prod($o,$o)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod($o,$o)),set(product_prod(A,A))),aa(set(product_prod($o,$o)),fun(set(product_prod(A,A)),product_prod(set(product_prod($o,$o)),set(product_prod(A,A)))),product_Pair(set(product_prod($o,$o)),set(product_prod(A,A))),bNF_Cardinal_ctwo),Q3)),bNF_Wellorder_ordLeq($o,A))
     => ( aa(set(product_prod(B,B)),$o,bNF_Ca8970107618336181345der_on(B,field2(B,R2)),R2)
       => aa(set(product_prod(set(product_prod(B,B)),set(product_prod(fun(B,A),fun(B,A))))),$o,member(product_prod(set(product_prod(B,B)),set(product_prod(fun(B,A),fun(B,A)))),aa(set(product_prod(fun(B,A),fun(B,A))),product_prod(set(product_prod(B,B)),set(product_prod(fun(B,A),fun(B,A)))),aa(set(product_prod(B,B)),fun(set(product_prod(fun(B,A),fun(B,A))),product_prod(set(product_prod(B,B)),set(product_prod(fun(B,A),fun(B,A))))),product_Pair(set(product_prod(B,B)),set(product_prod(fun(B,A),fun(B,A)))),R2),bNF_Cardinal_cexp(A,B,Q3,R2))),bNF_Wellorder_ordLeq(B,fun(B,A))) ) ) ).

% ordLeq_cexp2
tff(fact_6127_Cfinite__cexp__Cinfinite,axiom,
    ! [A: $tType,B: $tType,S2: set(product_prod(A,A)),T3: set(product_prod(B,B))] :
      ( ( bNF_Cardinal_cfinite(A,S2)
        & aa(set(product_prod(A,A)),$o,bNF_Ca8970107618336181345der_on(A,field2(A,S2)),S2) )
     => ( ( bNF_Ca4139267488887388095finite(B,T3)
          & aa(set(product_prod(B,B)),$o,bNF_Ca8970107618336181345der_on(B,field2(B,T3)),T3) )
       => aa(set(product_prod(set(product_prod(fun(B,A),fun(B,A))),set(product_prod(fun(B,$o),fun(B,$o))))),$o,member(product_prod(set(product_prod(fun(B,A),fun(B,A))),set(product_prod(fun(B,$o),fun(B,$o)))),aa(set(product_prod(fun(B,$o),fun(B,$o))),product_prod(set(product_prod(fun(B,A),fun(B,A))),set(product_prod(fun(B,$o),fun(B,$o)))),aa(set(product_prod(fun(B,A),fun(B,A))),fun(set(product_prod(fun(B,$o),fun(B,$o))),product_prod(set(product_prod(fun(B,A),fun(B,A))),set(product_prod(fun(B,$o),fun(B,$o))))),product_Pair(set(product_prod(fun(B,A),fun(B,A))),set(product_prod(fun(B,$o),fun(B,$o)))),bNF_Cardinal_cexp(A,B,S2,T3)),bNF_Cardinal_cexp($o,B,bNF_Cardinal_ctwo,T3))),bNF_Wellorder_ordLeq(fun(B,A),fun(B,$o))) ) ) ).

% Cfinite_cexp_Cinfinite
tff(fact_6128_ctwo__Cnotzero,axiom,
    ( ~ aa(set(product_prod(set(product_prod($o,$o)),set(product_prod($o,$o)))),$o,member(product_prod(set(product_prod($o,$o)),set(product_prod($o,$o))),aa(set(product_prod($o,$o)),product_prod(set(product_prod($o,$o)),set(product_prod($o,$o))),aa(set(product_prod($o,$o)),fun(set(product_prod($o,$o)),product_prod(set(product_prod($o,$o)),set(product_prod($o,$o)))),product_Pair(set(product_prod($o,$o)),set(product_prod($o,$o))),bNF_Cardinal_ctwo),bNF_Cardinal_czero($o))),bNF_Wellorder_ordIso($o,$o))
    & aa(set(product_prod($o,$o)),$o,bNF_Ca8970107618336181345der_on($o,field2($o,bNF_Cardinal_ctwo)),bNF_Cardinal_ctwo) ) ).

% ctwo_Cnotzero
tff(fact_6129_ctwo__def,axiom,
    bNF_Cardinal_ctwo = bNF_Ca6860139660246222851ard_of($o,top_top(set($o))) ).

% ctwo_def
tff(fact_6130_ordLess__ctwo__cexp,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,bNF_Ca8970107618336181345der_on(A,field2(A,R2)),R2)
     => aa(set(product_prod(set(product_prod(A,A)),set(product_prod(fun(A,$o),fun(A,$o))))),$o,member(product_prod(set(product_prod(A,A)),set(product_prod(fun(A,$o),fun(A,$o)))),aa(set(product_prod(fun(A,$o),fun(A,$o))),product_prod(set(product_prod(A,A)),set(product_prod(fun(A,$o),fun(A,$o)))),aa(set(product_prod(A,A)),fun(set(product_prod(fun(A,$o),fun(A,$o))),product_prod(set(product_prod(A,A)),set(product_prod(fun(A,$o),fun(A,$o))))),product_Pair(set(product_prod(A,A)),set(product_prod(fun(A,$o),fun(A,$o)))),R2),bNF_Cardinal_cexp($o,A,bNF_Cardinal_ctwo,R2))),bNF_We4044943003108391690rdLess(A,fun(A,$o))) ) ).

% ordLess_ctwo_cexp
tff(fact_6131_ctwo__not__czero,axiom,
    ! [A: $tType] : ~ aa(set(product_prod(set(product_prod($o,$o)),set(product_prod(A,A)))),$o,member(product_prod(set(product_prod($o,$o)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod($o,$o)),set(product_prod(A,A))),aa(set(product_prod($o,$o)),fun(set(product_prod(A,A)),product_prod(set(product_prod($o,$o)),set(product_prod(A,A)))),product_Pair(set(product_prod($o,$o)),set(product_prod(A,A))),bNF_Cardinal_ctwo),bNF_Cardinal_czero(A))),bNF_Wellorder_ordIso($o,A)) ).

% ctwo_not_czero
tff(fact_6132_ctwo__ordLeq__Cinfinite,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( ( bNF_Ca4139267488887388095finite(A,R2)
        & aa(set(product_prod(A,A)),$o,bNF_Ca8970107618336181345der_on(A,field2(A,R2)),R2) )
     => aa(set(product_prod(set(product_prod($o,$o)),set(product_prod(A,A)))),$o,member(product_prod(set(product_prod($o,$o)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod($o,$o)),set(product_prod(A,A))),aa(set(product_prod($o,$o)),fun(set(product_prod(A,A)),product_prod(set(product_prod($o,$o)),set(product_prod(A,A)))),product_Pair(set(product_prod($o,$o)),set(product_prod(A,A))),bNF_Cardinal_ctwo),R2)),bNF_Wellorder_ordLeq($o,A)) ) ).

% ctwo_ordLeq_Cinfinite
tff(fact_6133_ctwo__ordLess__Cinfinite,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( ( bNF_Ca4139267488887388095finite(A,R2)
        & aa(set(product_prod(A,A)),$o,bNF_Ca8970107618336181345der_on(A,field2(A,R2)),R2) )
     => aa(set(product_prod(set(product_prod($o,$o)),set(product_prod(A,A)))),$o,member(product_prod(set(product_prod($o,$o)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod($o,$o)),set(product_prod(A,A))),aa(set(product_prod($o,$o)),fun(set(product_prod(A,A)),product_prod(set(product_prod($o,$o)),set(product_prod(A,A)))),product_Pair(set(product_prod($o,$o)),set(product_prod(A,A))),bNF_Cardinal_ctwo),R2)),bNF_We4044943003108391690rdLess($o,A)) ) ).

% ctwo_ordLess_Cinfinite
tff(fact_6134_cinfinite__cexp,axiom,
    ! [A: $tType,B: $tType,Q3: set(product_prod(A,A)),R2: set(product_prod(B,B))] :
      ( aa(set(product_prod(set(product_prod($o,$o)),set(product_prod(A,A)))),$o,member(product_prod(set(product_prod($o,$o)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod($o,$o)),set(product_prod(A,A))),aa(set(product_prod($o,$o)),fun(set(product_prod(A,A)),product_prod(set(product_prod($o,$o)),set(product_prod(A,A)))),product_Pair(set(product_prod($o,$o)),set(product_prod(A,A))),bNF_Cardinal_ctwo),Q3)),bNF_Wellorder_ordLeq($o,A))
     => ( ( bNF_Ca4139267488887388095finite(B,R2)
          & aa(set(product_prod(B,B)),$o,bNF_Ca8970107618336181345der_on(B,field2(B,R2)),R2) )
       => bNF_Ca4139267488887388095finite(fun(B,A),bNF_Cardinal_cexp(A,B,Q3,R2)) ) ) ).

% cinfinite_cexp
tff(fact_6135_Cinfinite__cexp,axiom,
    ! [A: $tType,B: $tType,Q3: set(product_prod(A,A)),R2: set(product_prod(B,B))] :
      ( aa(set(product_prod(set(product_prod($o,$o)),set(product_prod(A,A)))),$o,member(product_prod(set(product_prod($o,$o)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod($o,$o)),set(product_prod(A,A))),aa(set(product_prod($o,$o)),fun(set(product_prod(A,A)),product_prod(set(product_prod($o,$o)),set(product_prod(A,A)))),product_Pair(set(product_prod($o,$o)),set(product_prod(A,A))),bNF_Cardinal_ctwo),Q3)),bNF_Wellorder_ordLeq($o,A))
     => ( ( bNF_Ca4139267488887388095finite(B,R2)
          & aa(set(product_prod(B,B)),$o,bNF_Ca8970107618336181345der_on(B,field2(B,R2)),R2) )
       => ( bNF_Ca4139267488887388095finite(fun(B,A),bNF_Cardinal_cexp(A,B,Q3,R2))
          & aa(set(product_prod(fun(B,A),fun(B,A))),$o,bNF_Ca8970107618336181345der_on(fun(B,A),field2(fun(B,A),bNF_Cardinal_cexp(A,B,Q3,R2))),bNF_Cardinal_cexp(A,B,Q3,R2)) ) ) ) ).

% Cinfinite_cexp
tff(fact_6136_card__of__Plus__Times__aux,axiom,
    ! [B: $tType,A: $tType,A1: A,A22: A,A4: set(A),B3: set(B)] :
      ( ( ( A1 != A22 )
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A1),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A22),bot_bot(set(A))))),A4) )
     => ( aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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,A4)),bNF_Ca6860139660246222851ard_of(B,B3))),bNF_Wellorder_ordLeq(A,B))
       => 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))))),$o,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,A4,B3))),bNF_Ca6860139660246222851ard_of(product_prod(A,B),product_Sigma(A,B,A4,aTP_Lamp_qz(set(B),fun(A,set(B)),B3))))),bNF_Wellorder_ordLeq(sum_sum(A,B),product_prod(A,B))) ) ) ).

% card_of_Plus_Times_aux
tff(fact_6137_natLeq__ordLeq__cinfinite,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( ( bNF_Ca4139267488887388095finite(A,R2)
        & aa(set(product_prod(A,A)),$o,bNF_Ca8970107618336181345der_on(A,field2(A,R2)),R2) )
     => aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(A,A)))),$o,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),R2)),bNF_Wellorder_ordLeq(nat,A)) ) ).

% natLeq_ordLeq_cinfinite
tff(fact_6138_Card__order__Plus2,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,A)),A4: set(B)] :
      ( aa(set(product_prod(A,A)),$o,bNF_Ca8970107618336181345der_on(A,field2(A,R2)),R2)
     => aa(set(product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(B,A),sum_sum(B,A))))),$o,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)))),R2),bNF_Ca6860139660246222851ard_of(sum_sum(B,A),sum_Plus(B,A,A4,field2(A,R2))))),bNF_Wellorder_ordLeq(A,sum_sum(B,A))) ) ).

% Card_order_Plus2
tff(fact_6139_Card__order__Plus1,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,A)),B3: set(B)] :
      ( aa(set(product_prod(A,A)),$o,bNF_Ca8970107618336181345der_on(A,field2(A,R2)),R2)
     => aa(set(product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,B),sum_sum(A,B))))),$o,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)))),R2),bNF_Ca6860139660246222851ard_of(sum_sum(A,B),sum_Plus(A,B,field2(A,R2),B3)))),bNF_Wellorder_ordLeq(A,sum_sum(A,B))) ) ).

% Card_order_Plus1
tff(fact_6140_ctwo__ordLess__natLeq,axiom,
    aa(set(product_prod(set(product_prod($o,$o)),set(product_prod(nat,nat)))),$o,member(product_prod(set(product_prod($o,$o)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod($o,$o)),set(product_prod(nat,nat))),aa(set(product_prod($o,$o)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod($o,$o)),set(product_prod(nat,nat)))),product_Pair(set(product_prod($o,$o)),set(product_prod(nat,nat))),bNF_Cardinal_ctwo),bNF_Ca8665028551170535155natLeq)),bNF_We4044943003108391690rdLess($o,nat)) ).

% ctwo_ordLess_natLeq
tff(fact_6141_card__of__Plus__Times__bool,axiom,
    ! [A: $tType,A4: set(A)] : aa(set(product_prod(set(product_prod(sum_sum(A,A),sum_sum(A,A))),set(product_prod(product_prod(A,$o),product_prod(A,$o))))),$o,member(product_prod(set(product_prod(sum_sum(A,A),sum_sum(A,A))),set(product_prod(product_prod(A,$o),product_prod(A,$o)))),aa(set(product_prod(product_prod(A,$o),product_prod(A,$o))),product_prod(set(product_prod(sum_sum(A,A),sum_sum(A,A))),set(product_prod(product_prod(A,$o),product_prod(A,$o)))),aa(set(product_prod(sum_sum(A,A),sum_sum(A,A))),fun(set(product_prod(product_prod(A,$o),product_prod(A,$o))),product_prod(set(product_prod(sum_sum(A,A),sum_sum(A,A))),set(product_prod(product_prod(A,$o),product_prod(A,$o))))),product_Pair(set(product_prod(sum_sum(A,A),sum_sum(A,A))),set(product_prod(product_prod(A,$o),product_prod(A,$o)))),bNF_Ca6860139660246222851ard_of(sum_sum(A,A),sum_Plus(A,A,A4,A4))),bNF_Ca6860139660246222851ard_of(product_prod(A,$o),product_Sigma(A,$o,A4,aTP_Lamp_ang(A,set($o)))))),bNF_Wellorder_ordIso(sum_sum(A,A),product_prod(A,$o))) ).

% card_of_Plus_Times_bool
tff(fact_6142_card__of__Plus__commute,axiom,
    ! [B: $tType,A: $tType,A4: set(A),B3: set(B)] : aa(set(product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(sum_sum(B,A),sum_sum(B,A))))),$o,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,A4,B3))),bNF_Ca6860139660246222851ard_of(sum_sum(B,A),sum_Plus(B,A,B3,A4)))),bNF_Wellorder_ordIso(sum_sum(A,B),sum_sum(B,A))) ).

% card_of_Plus_commute
tff(fact_6143_card__of__Plus__assoc,axiom,
    ! [C: $tType,B: $tType,A: $tType,A4: set(A),B3: set(B),C3: set(C)] : aa(set(product_prod(set(product_prod(sum_sum(sum_sum(A,B),C),sum_sum(sum_sum(A,B),C))),set(product_prod(sum_sum(A,sum_sum(B,C)),sum_sum(A,sum_sum(B,C)))))),$o,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,A4,B3),C3))),bNF_Ca6860139660246222851ard_of(sum_sum(A,sum_sum(B,C)),sum_Plus(A,sum_sum(B,C),A4,sum_Plus(B,C,B3,C3))))),bNF_Wellorder_ordIso(sum_sum(sum_sum(A,B),C),sum_sum(A,sum_sum(B,C)))) ).

% card_of_Plus_assoc
tff(fact_6144_card__of__Plus2,axiom,
    ! [B: $tType,A: $tType,B3: set(A),A4: set(B)] : aa(set(product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(B,A),sum_sum(B,A))))),$o,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,B3)),bNF_Ca6860139660246222851ard_of(sum_sum(B,A),sum_Plus(B,A,A4,B3)))),bNF_Wellorder_ordLeq(A,sum_sum(B,A))) ).

% card_of_Plus2
tff(fact_6145_card__of__Plus1,axiom,
    ! [B: $tType,A: $tType,A4: set(A),B3: set(B)] : aa(set(product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,B),sum_sum(A,B))))),$o,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,A4)),bNF_Ca6860139660246222851ard_of(sum_sum(A,B),sum_Plus(A,B,A4,B3)))),bNF_Wellorder_ordLeq(A,sum_sum(A,B))) ).

% card_of_Plus1
tff(fact_6146_card__of__Times__Plus__distrib,axiom,
    ! [C: $tType,B: $tType,A: $tType,A4: set(A),B3: set(B),C3: set(C)] : aa(set(product_prod(set(product_prod(product_prod(A,sum_sum(B,C)),product_prod(A,sum_sum(B,C)))),set(product_prod(sum_sum(product_prod(A,B),product_prod(A,C)),sum_sum(product_prod(A,B),product_prod(A,C)))))),$o,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),A4,aa(set(C),fun(A,set(sum_sum(B,C))),aTP_Lamp_anh(set(B),fun(set(C),fun(A,set(sum_sum(B,C)))),B3),C3)))),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,A4,aTP_Lamp_qz(set(B),fun(A,set(B)),B3)),product_Sigma(A,C,A4,aTP_Lamp_us(set(C),fun(A,set(C)),C3)))))),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_6147_card__of__Un__Plus__ordLeq,axiom,
    ! [A: $tType,A4: set(A),B3: set(A)] : aa(set(product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,A),sum_sum(A,A))))),$o,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)),A4),B3))),bNF_Ca6860139660246222851ard_of(sum_sum(A,A),sum_Plus(A,A,A4,B3)))),bNF_Wellorder_ordLeq(A,sum_sum(A,A))) ).

% card_of_Un_Plus_ordLeq
tff(fact_6148_card__of__Plus__empty1,axiom,
    ! [B: $tType,A: $tType,A4: set(A)] : aa(set(product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,B),sum_sum(A,B))))),$o,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,A4)),bNF_Ca6860139660246222851ard_of(sum_sum(A,B),sum_Plus(A,B,A4,bot_bot(set(B)))))),bNF_Wellorder_ordIso(A,sum_sum(A,B))) ).

% card_of_Plus_empty1
tff(fact_6149_card__of__Plus__empty2,axiom,
    ! [B: $tType,A: $tType,A4: set(A)] : aa(set(product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(B,A),sum_sum(B,A))))),$o,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,A4)),bNF_Ca6860139660246222851ard_of(sum_sum(B,A),sum_Plus(B,A,bot_bot(set(B)),A4)))),bNF_Wellorder_ordIso(A,sum_sum(B,A))) ).

% card_of_Plus_empty2
tff(fact_6150_natLeq__card__order,axiom,
    aa(set(product_prod(nat,nat)),$o,bNF_Ca8970107618336181345der_on(nat,top_top(set(nat))),bNF_Ca8665028551170535155natLeq) ).

% natLeq_card_order
tff(fact_6151_natLeq__underS__less,axiom,
    ! [N: nat] : order_underS(nat,bNF_Ca8665028551170535155natLeq,N) = aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_bk(nat,fun(nat,$o)),N)) ).

% natLeq_underS_less
tff(fact_6152_ordLeq__Plus__mono2,axiom,
    ! [B: $tType,A: $tType,C: $tType,R2: set(product_prod(A,A)),R3: set(product_prod(B,B)),A4: set(C)] :
      ( aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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))),R2),R3)),bNF_Wellorder_ordLeq(A,B))
     => 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))))),$o,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,A4,field2(A,R2)))),bNF_Ca6860139660246222851ard_of(sum_sum(C,B),sum_Plus(C,B,A4,field2(B,R3))))),bNF_Wellorder_ordLeq(sum_sum(C,A),sum_sum(C,B))) ) ).

% ordLeq_Plus_mono2
tff(fact_6153_ordLeq__Plus__mono1,axiom,
    ! [B: $tType,C: $tType,A: $tType,R2: set(product_prod(A,A)),R3: set(product_prod(B,B)),C3: set(C)] :
      ( aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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))),R2),R3)),bNF_Wellorder_ordLeq(A,B))
     => 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))))),$o,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,R2),C3))),bNF_Ca6860139660246222851ard_of(sum_sum(B,C),sum_Plus(B,C,field2(B,R3),C3)))),bNF_Wellorder_ordLeq(sum_sum(A,C),sum_sum(B,C))) ) ).

% ordLeq_Plus_mono1
tff(fact_6154_ordLeq__Plus__mono,axiom,
    ! [D: $tType,B: $tType,C: $tType,A: $tType,R2: set(product_prod(A,A)),R3: set(product_prod(B,B)),P3: set(product_prod(C,C)),P7: set(product_prod(D,D))] :
      ( aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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))),R2),R3)),bNF_Wellorder_ordLeq(A,B))
     => ( aa(set(product_prod(set(product_prod(C,C)),set(product_prod(D,D)))),$o,member(product_prod(set(product_prod(C,C)),set(product_prod(D,D))),aa(set(product_prod(D,D)),product_prod(set(product_prod(C,C)),set(product_prod(D,D))),aa(set(product_prod(C,C)),fun(set(product_prod(D,D)),product_prod(set(product_prod(C,C)),set(product_prod(D,D)))),product_Pair(set(product_prod(C,C)),set(product_prod(D,D))),P3),P7)),bNF_Wellorder_ordLeq(C,D))
       => aa(set(product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D))))),$o,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,R2),field2(C,P3)))),bNF_Ca6860139660246222851ard_of(sum_sum(B,D),sum_Plus(B,D,field2(B,R3),field2(D,P7))))),bNF_Wellorder_ordLeq(sum_sum(A,C),sum_sum(B,D))) ) ) ).

% ordLeq_Plus_mono
tff(fact_6155_ordIso__Plus__cong2,axiom,
    ! [B: $tType,A: $tType,C: $tType,R2: set(product_prod(A,A)),R3: set(product_prod(B,B)),A4: set(C)] :
      ( aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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))),R2),R3)),bNF_Wellorder_ordIso(A,B))
     => 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))))),$o,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,A4,field2(A,R2)))),bNF_Ca6860139660246222851ard_of(sum_sum(C,B),sum_Plus(C,B,A4,field2(B,R3))))),bNF_Wellorder_ordIso(sum_sum(C,A),sum_sum(C,B))) ) ).

% ordIso_Plus_cong2
tff(fact_6156_ordIso__Plus__cong1,axiom,
    ! [B: $tType,C: $tType,A: $tType,R2: set(product_prod(A,A)),R3: set(product_prod(B,B)),C3: set(C)] :
      ( aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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))),R2),R3)),bNF_Wellorder_ordIso(A,B))
     => 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))))),$o,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,R2),C3))),bNF_Ca6860139660246222851ard_of(sum_sum(B,C),sum_Plus(B,C,field2(B,R3),C3)))),bNF_Wellorder_ordIso(sum_sum(A,C),sum_sum(B,C))) ) ).

% ordIso_Plus_cong1
tff(fact_6157_ordIso__Plus__cong,axiom,
    ! [D: $tType,B: $tType,C: $tType,A: $tType,R2: set(product_prod(A,A)),R3: set(product_prod(B,B)),P3: set(product_prod(C,C)),P7: set(product_prod(D,D))] :
      ( aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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))),R2),R3)),bNF_Wellorder_ordIso(A,B))
     => ( aa(set(product_prod(set(product_prod(C,C)),set(product_prod(D,D)))),$o,member(product_prod(set(product_prod(C,C)),set(product_prod(D,D))),aa(set(product_prod(D,D)),product_prod(set(product_prod(C,C)),set(product_prod(D,D))),aa(set(product_prod(C,C)),fun(set(product_prod(D,D)),product_prod(set(product_prod(C,C)),set(product_prod(D,D)))),product_Pair(set(product_prod(C,C)),set(product_prod(D,D))),P3),P7)),bNF_Wellorder_ordIso(C,D))
       => aa(set(product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D))))),$o,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,R2),field2(C,P3)))),bNF_Ca6860139660246222851ard_of(sum_sum(B,D),sum_Plus(B,D,field2(B,R3),field2(D,P7))))),bNF_Wellorder_ordIso(sum_sum(A,C),sum_sum(B,D))) ) ) ).

% ordIso_Plus_cong
tff(fact_6158_card__of__Plus__mono,axiom,
    ! [D: $tType,B: $tType,C: $tType,A: $tType,A4: set(A),B3: set(B),C3: set(C),D4: set(D)] :
      ( aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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,A4)),bNF_Ca6860139660246222851ard_of(B,B3))),bNF_Wellorder_ordLeq(A,B))
     => ( aa(set(product_prod(set(product_prod(C,C)),set(product_prod(D,D)))),$o,member(product_prod(set(product_prod(C,C)),set(product_prod(D,D))),aa(set(product_prod(D,D)),product_prod(set(product_prod(C,C)),set(product_prod(D,D))),aa(set(product_prod(C,C)),fun(set(product_prod(D,D)),product_prod(set(product_prod(C,C)),set(product_prod(D,D)))),product_Pair(set(product_prod(C,C)),set(product_prod(D,D))),bNF_Ca6860139660246222851ard_of(C,C3)),bNF_Ca6860139660246222851ard_of(D,D4))),bNF_Wellorder_ordLeq(C,D))
       => aa(set(product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D))))),$o,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,A4,C3))),bNF_Ca6860139660246222851ard_of(sum_sum(B,D),sum_Plus(B,D,B3,D4)))),bNF_Wellorder_ordLeq(sum_sum(A,C),sum_sum(B,D))) ) ) ).

% card_of_Plus_mono
tff(fact_6159_card__of__Plus__mono1,axiom,
    ! [B: $tType,C: $tType,A: $tType,A4: set(A),B3: set(B),C3: set(C)] :
      ( aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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,A4)),bNF_Ca6860139660246222851ard_of(B,B3))),bNF_Wellorder_ordLeq(A,B))
     => 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))))),$o,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,A4,C3))),bNF_Ca6860139660246222851ard_of(sum_sum(B,C),sum_Plus(B,C,B3,C3)))),bNF_Wellorder_ordLeq(sum_sum(A,C),sum_sum(B,C))) ) ).

% card_of_Plus_mono1
tff(fact_6160_card__of__Plus__mono2,axiom,
    ! [B: $tType,A: $tType,C: $tType,A4: set(A),B3: set(B),C3: set(C)] :
      ( aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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,A4)),bNF_Ca6860139660246222851ard_of(B,B3))),bNF_Wellorder_ordLeq(A,B))
     => 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))))),$o,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,C3,A4))),bNF_Ca6860139660246222851ard_of(sum_sum(C,B),sum_Plus(C,B,C3,B3)))),bNF_Wellorder_ordLeq(sum_sum(C,A),sum_sum(C,B))) ) ).

% card_of_Plus_mono2
tff(fact_6161_card__of__Plus__cong,axiom,
    ! [D: $tType,B: $tType,C: $tType,A: $tType,A4: set(A),B3: set(B),C3: set(C),D4: set(D)] :
      ( aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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,A4)),bNF_Ca6860139660246222851ard_of(B,B3))),bNF_Wellorder_ordIso(A,B))
     => ( aa(set(product_prod(set(product_prod(C,C)),set(product_prod(D,D)))),$o,member(product_prod(set(product_prod(C,C)),set(product_prod(D,D))),aa(set(product_prod(D,D)),product_prod(set(product_prod(C,C)),set(product_prod(D,D))),aa(set(product_prod(C,C)),fun(set(product_prod(D,D)),product_prod(set(product_prod(C,C)),set(product_prod(D,D)))),product_Pair(set(product_prod(C,C)),set(product_prod(D,D))),bNF_Ca6860139660246222851ard_of(C,C3)),bNF_Ca6860139660246222851ard_of(D,D4))),bNF_Wellorder_ordIso(C,D))
       => aa(set(product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D))))),$o,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,A4,C3))),bNF_Ca6860139660246222851ard_of(sum_sum(B,D),sum_Plus(B,D,B3,D4)))),bNF_Wellorder_ordIso(sum_sum(A,C),sum_sum(B,D))) ) ) ).

% card_of_Plus_cong
tff(fact_6162_card__of__Plus__cong1,axiom,
    ! [B: $tType,C: $tType,A: $tType,A4: set(A),B3: set(B),C3: set(C)] :
      ( aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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,A4)),bNF_Ca6860139660246222851ard_of(B,B3))),bNF_Wellorder_ordIso(A,B))
     => 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))))),$o,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,A4,C3))),bNF_Ca6860139660246222851ard_of(sum_sum(B,C),sum_Plus(B,C,B3,C3)))),bNF_Wellorder_ordIso(sum_sum(A,C),sum_sum(B,C))) ) ).

% card_of_Plus_cong1
tff(fact_6163_card__of__Plus__cong2,axiom,
    ! [B: $tType,A: $tType,C: $tType,A4: set(A),B3: set(B),C3: set(C)] :
      ( aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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,A4)),bNF_Ca6860139660246222851ard_of(B,B3))),bNF_Wellorder_ordIso(A,B))
     => 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))))),$o,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,C3,A4))),bNF_Ca6860139660246222851ard_of(sum_sum(C,B),sum_Plus(C,B,C3,B3)))),bNF_Wellorder_ordIso(sum_sum(C,A),sum_sum(C,B))) ) ).

% card_of_Plus_cong2
tff(fact_6164_Field__natLeq,axiom,
    field2(nat,bNF_Ca8665028551170535155natLeq) = top_top(set(nat)) ).

% Field_natLeq
tff(fact_6165_natLeq__def,axiom,
    bNF_Ca8665028551170535155natLeq = aa(fun(product_prod(nat,nat),$o),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),ord_less_eq(nat))) ).

% natLeq_def
tff(fact_6166_card__of__Plus__infinite,axiom,
    ! [A: $tType,B: $tType,A4: set(A),B3: set(B)] :
      ( ~ aa(set(A),$o,finite_finite2(A),A4)
     => ( aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),$o,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,B3)),bNF_Ca6860139660246222851ard_of(A,A4))),bNF_Wellorder_ordLeq(B,A))
       => ( aa(set(product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(A,A)))),$o,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,A4,B3))),bNF_Ca6860139660246222851ard_of(A,A4))),bNF_Wellorder_ordIso(sum_sum(A,B),A))
          & aa(set(product_prod(set(product_prod(sum_sum(B,A),sum_sum(B,A))),set(product_prod(A,A)))),$o,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,B3,A4))),bNF_Ca6860139660246222851ard_of(A,A4))),bNF_Wellorder_ordIso(sum_sum(B,A),A)) ) ) ) ).

% card_of_Plus_infinite
tff(fact_6167_card__of__Plus__infinite1,axiom,
    ! [B: $tType,A: $tType,A4: set(A),B3: set(B)] :
      ( ~ aa(set(A),$o,finite_finite2(A),A4)
     => ( aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),$o,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,B3)),bNF_Ca6860139660246222851ard_of(A,A4))),bNF_Wellorder_ordLeq(B,A))
       => aa(set(product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(A,A)))),$o,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,A4,B3))),bNF_Ca6860139660246222851ard_of(A,A4))),bNF_Wellorder_ordIso(sum_sum(A,B),A)) ) ) ).

% card_of_Plus_infinite1
tff(fact_6168_card__of__Plus__infinite2,axiom,
    ! [A: $tType,B: $tType,A4: set(A),B3: set(B)] :
      ( ~ aa(set(A),$o,finite_finite2(A),A4)
     => ( aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),$o,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,B3)),bNF_Ca6860139660246222851ard_of(A,A4))),bNF_Wellorder_ordLeq(B,A))
       => aa(set(product_prod(set(product_prod(sum_sum(B,A),sum_sum(B,A))),set(product_prod(A,A)))),$o,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,B3,A4))),bNF_Ca6860139660246222851ard_of(A,A4))),bNF_Wellorder_ordIso(sum_sum(B,A),A)) ) ) ).

% card_of_Plus_infinite2
tff(fact_6169_card__of__Plus__ordLess__infinite,axiom,
    ! [A: $tType,C: $tType,B: $tType,C3: set(A),A4: set(B),B3: set(C)] :
      ( ~ aa(set(A),$o,finite_finite2(A),C3)
     => ( aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),$o,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,A4)),bNF_Ca6860139660246222851ard_of(A,C3))),bNF_We4044943003108391690rdLess(B,A))
       => ( aa(set(product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),$o,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,B3)),bNF_Ca6860139660246222851ard_of(A,C3))),bNF_We4044943003108391690rdLess(C,A))
         => aa(set(product_prod(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(A,A)))),$o,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,A4,B3))),bNF_Ca6860139660246222851ard_of(A,C3))),bNF_We4044943003108391690rdLess(sum_sum(B,C),A)) ) ) ) ).

% card_of_Plus_ordLess_infinite
tff(fact_6170_finite__iff__ordLess__natLeq,axiom,
    ! [A: $tType,A4: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A4)
    <=> aa(set(product_prod(set(product_prod(A,A)),set(product_prod(nat,nat)))),$o,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,A4)),bNF_Ca8665028551170535155natLeq)),bNF_We4044943003108391690rdLess(A,nat)) ) ).

% finite_iff_ordLess_natLeq
tff(fact_6171_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_ani(nat,fun(nat,set(nat)),N))) = aa(fun(product_prod(nat,nat),$o),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aTP_Lamp_uw(nat,fun(nat,fun(nat,$o)),N))) ).

% Restr_natLeq2
tff(fact_6172_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,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_bk(nat,fun(nat,$o)),N)),aTP_Lamp_anj(nat,fun(nat,set(nat)),N))) = aa(fun(product_prod(nat,nat),$o),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aTP_Lamp_uw(nat,fun(nat,fun(nat,$o)),N))) ).

% Restr_natLeq
tff(fact_6173_Card__order__Plus__infinite,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(A,A)),P3: set(product_prod(B,B))] :
      ( ~ aa(set(A),$o,finite_finite2(A),field2(A,R2))
     => ( aa(set(product_prod(A,A)),$o,bNF_Ca8970107618336181345der_on(A,field2(A,R2)),R2)
       => ( aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),$o,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))),P3),R2)),bNF_Wellorder_ordLeq(B,A))
         => ( aa(set(product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(A,A)))),$o,member(product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(A,A))),aa(set(product_prod(sum_sum(A,B),sum_sum(A,B))),fun(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(A,A)))),product_Pair(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(sum_sum(A,B),sum_Plus(A,B,field2(A,R2),field2(B,P3)))),R2)),bNF_Wellorder_ordIso(sum_sum(A,B),A))
            & aa(set(product_prod(set(product_prod(sum_sum(B,A),sum_sum(B,A))),set(product_prod(A,A)))),$o,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,P3),field2(A,R2)))),R2)),bNF_Wellorder_ordIso(sum_sum(B,A),A)) ) ) ) ) ).

% Card_order_Plus_infinite
tff(fact_6174_card__of__nat,axiom,
    aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),$o,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_6175_card__of__Plus__Times,axiom,
    ! [B: $tType,A: $tType,A1: A,A22: A,A4: set(A),B13: B,B23: B,B3: set(B)] :
      ( ( ( A1 != A22 )
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A1),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A22),bot_bot(set(A))))),A4) )
     => ( ( ( B13 != B23 )
          & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),B13),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),B23),bot_bot(set(B))))),B3) )
       => aa(set(product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(product_prod(A,B),product_prod(A,B))))),$o,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,A4,B3))),bNF_Ca6860139660246222851ard_of(product_prod(A,B),product_Sigma(A,B,A4,aTP_Lamp_qz(set(B),fun(A,set(B)),B3))))),bNF_Wellorder_ordLeq(sum_sum(A,B),product_prod(A,B))) ) ) ).

% card_of_Plus_Times
tff(fact_6176_card__of__Field__natLeq,axiom,
    aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),$o,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_6177_card__of__Plus__ordLeq__infinite__Field,axiom,
    ! [A: $tType,C: $tType,B: $tType,R2: set(product_prod(A,A)),A4: set(B),B3: set(C)] :
      ( ~ aa(set(A),$o,finite_finite2(A),field2(A,R2))
     => ( aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),$o,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,A4)),R2)),bNF_Wellorder_ordLeq(B,A))
       => ( aa(set(product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),$o,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,B3)),R2)),bNF_Wellorder_ordLeq(C,A))
         => ( aa(set(product_prod(A,A)),$o,bNF_Ca8970107618336181345der_on(A,field2(A,R2)),R2)
           => aa(set(product_prod(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(A,A)))),$o,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,A4,B3))),R2)),bNF_Wellorder_ordLeq(sum_sum(B,C),A)) ) ) ) ) ).

% card_of_Plus_ordLeq_infinite_Field
tff(fact_6178_card__of__Plus__ordLess__infinite__Field,axiom,
    ! [A: $tType,C: $tType,B: $tType,R2: set(product_prod(A,A)),A4: set(B),B3: set(C)] :
      ( ~ aa(set(A),$o,finite_finite2(A),field2(A,R2))
     => ( aa(set(product_prod(A,A)),$o,bNF_Ca8970107618336181345der_on(A,field2(A,R2)),R2)
       => ( aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),$o,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,A4)),R2)),bNF_We4044943003108391690rdLess(B,A))
         => ( aa(set(product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),$o,member(product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(C,C)),fun(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),product_Pair(set(product_prod(C,C)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(C,B3)),R2)),bNF_We4044943003108391690rdLess(C,A))
           => aa(set(product_prod(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(A,A)))),$o,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,A4,B3))),R2)),bNF_We4044943003108391690rdLess(sum_sum(B,C),A)) ) ) ) ) ).

% card_of_Plus_ordLess_infinite_Field
tff(fact_6179_infinite__iff__natLeq__ordLeq,axiom,
    ! [A: $tType,A4: set(A)] :
      ~ ( aa(set(A),$o,finite_finite2(A),A4)
      <=> aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(A,A)))),$o,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,A4))),bNF_Wellorder_ordLeq(nat,A)) ) ).

% infinite_iff_natLeq_ordLeq
tff(fact_6180_UNIV__Plus__UNIV,axiom,
    ! [B: $tType,A: $tType] : sum_Plus(A,B,top_top(set(A)),top_top(set(B))) = top_top(set(sum_sum(A,B))) ).

% UNIV_Plus_UNIV
tff(fact_6181_Plus__eq__empty__conv,axiom,
    ! [A: $tType,B: $tType,A4: set(A),B3: set(B)] :
      ( ( sum_Plus(A,B,A4,B3) = bot_bot(set(sum_sum(A,B))) )
    <=> ( ( A4 = bot_bot(set(A)) )
        & ( B3 = bot_bot(set(B)) ) ) ) ).

% Plus_eq_empty_conv
tff(fact_6182_csum__dup,axiom,
    ! [A: $tType,C: $tType,B: $tType,R2: set(product_prod(A,A)),P3: set(product_prod(B,B)),P7: set(product_prod(C,C))] :
      ( bNF_Ca4139267488887388095finite(A,R2)
     => ( aa(set(product_prod(A,A)),$o,bNF_Ca8970107618336181345der_on(A,field2(A,R2)),R2)
       => ( aa(set(product_prod(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(sum_sum(A,A),sum_sum(A,A))))),$o,member(product_prod(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(sum_sum(A,A),sum_sum(A,A)))),aa(set(product_prod(sum_sum(A,A),sum_sum(A,A))),product_prod(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(sum_sum(A,A),sum_sum(A,A)))),aa(set(product_prod(sum_sum(B,C),sum_sum(B,C))),fun(set(product_prod(sum_sum(A,A),sum_sum(A,A))),product_prod(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(sum_sum(A,A),sum_sum(A,A))))),product_Pair(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(sum_sum(A,A),sum_sum(A,A)))),bNF_Cardinal_csum(B,C,P3,P7)),bNF_Cardinal_csum(A,A,R2,R2))),bNF_Wellorder_ordIso(sum_sum(B,C),sum_sum(A,A)))
         => aa(set(product_prod(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(A,A)))),$o,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_Cardinal_csum(B,C,P3,P7)),R2)),bNF_Wellorder_ordIso(sum_sum(B,C),A)) ) ) ) ).

% csum_dup
tff(fact_6183_csum__Cnotzero1,axiom,
    ! [A: $tType,B: $tType,R12: set(product_prod(A,A)),R23: set(product_prod(B,B))] :
      ( ( ~ aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),$o,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))),R12),bNF_Cardinal_czero(A))),bNF_Wellorder_ordIso(A,A))
        & aa(set(product_prod(A,A)),$o,bNF_Ca8970107618336181345der_on(A,field2(A,R12)),R12) )
     => ( ~ aa(set(product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(sum_sum(A,B),sum_sum(A,B))))),$o,member(product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),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(sum_sum(A,B),sum_sum(A,B))),set(product_prod(sum_sum(A,B),sum_sum(A,B)))),aa(set(product_prod(sum_sum(A,B),sum_sum(A,B))),fun(set(product_prod(sum_sum(A,B),sum_sum(A,B))),product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(sum_sum(A,B),sum_sum(A,B))))),product_Pair(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(sum_sum(A,B),sum_sum(A,B)))),bNF_Cardinal_csum(A,B,R12,R23)),bNF_Cardinal_czero(sum_sum(A,B)))),bNF_Wellorder_ordIso(sum_sum(A,B),sum_sum(A,B)))
        & aa(set(product_prod(sum_sum(A,B),sum_sum(A,B))),$o,bNF_Ca8970107618336181345der_on(sum_sum(A,B),field2(sum_sum(A,B),bNF_Cardinal_csum(A,B,R12,R23))),bNF_Cardinal_csum(A,B,R12,R23)) ) ) ).

% csum_Cnotzero1
tff(fact_6184_csum__assoc,axiom,
    ! [C: $tType,B: $tType,A: $tType,P1: set(product_prod(A,A)),P22: set(product_prod(B,B)),P32: set(product_prod(C,C))] : aa(set(product_prod(set(product_prod(sum_sum(sum_sum(A,B),C),sum_sum(sum_sum(A,B),C))),set(product_prod(sum_sum(A,sum_sum(B,C)),sum_sum(A,sum_sum(B,C)))))),$o,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_Cardinal_csum(sum_sum(A,B),C,bNF_Cardinal_csum(A,B,P1,P22),P32)),bNF_Cardinal_csum(A,sum_sum(B,C),P1,bNF_Cardinal_csum(B,C,P22,P32)))),bNF_Wellorder_ordIso(sum_sum(sum_sum(A,B),C),sum_sum(A,sum_sum(B,C)))) ).

% csum_assoc
tff(fact_6185_csum__csum,axiom,
    ! [D: $tType,C: $tType,B: $tType,A: $tType,R12: set(product_prod(A,A)),R23: set(product_prod(B,B)),R32: set(product_prod(C,C)),R42: set(product_prod(D,D))] : aa(set(product_prod(set(product_prod(sum_sum(sum_sum(A,B),sum_sum(C,D)),sum_sum(sum_sum(A,B),sum_sum(C,D)))),set(product_prod(sum_sum(sum_sum(A,C),sum_sum(B,D)),sum_sum(sum_sum(A,C),sum_sum(B,D)))))),$o,member(product_prod(set(product_prod(sum_sum(sum_sum(A,B),sum_sum(C,D)),sum_sum(sum_sum(A,B),sum_sum(C,D)))),set(product_prod(sum_sum(sum_sum(A,C),sum_sum(B,D)),sum_sum(sum_sum(A,C),sum_sum(B,D))))),aa(set(product_prod(sum_sum(sum_sum(A,C),sum_sum(B,D)),sum_sum(sum_sum(A,C),sum_sum(B,D)))),product_prod(set(product_prod(sum_sum(sum_sum(A,B),sum_sum(C,D)),sum_sum(sum_sum(A,B),sum_sum(C,D)))),set(product_prod(sum_sum(sum_sum(A,C),sum_sum(B,D)),sum_sum(sum_sum(A,C),sum_sum(B,D))))),aa(set(product_prod(sum_sum(sum_sum(A,B),sum_sum(C,D)),sum_sum(sum_sum(A,B),sum_sum(C,D)))),fun(set(product_prod(sum_sum(sum_sum(A,C),sum_sum(B,D)),sum_sum(sum_sum(A,C),sum_sum(B,D)))),product_prod(set(product_prod(sum_sum(sum_sum(A,B),sum_sum(C,D)),sum_sum(sum_sum(A,B),sum_sum(C,D)))),set(product_prod(sum_sum(sum_sum(A,C),sum_sum(B,D)),sum_sum(sum_sum(A,C),sum_sum(B,D)))))),product_Pair(set(product_prod(sum_sum(sum_sum(A,B),sum_sum(C,D)),sum_sum(sum_sum(A,B),sum_sum(C,D)))),set(product_prod(sum_sum(sum_sum(A,C),sum_sum(B,D)),sum_sum(sum_sum(A,C),sum_sum(B,D))))),bNF_Cardinal_csum(sum_sum(A,B),sum_sum(C,D),bNF_Cardinal_csum(A,B,R12,R23),bNF_Cardinal_csum(C,D,R32,R42))),bNF_Cardinal_csum(sum_sum(A,C),sum_sum(B,D),bNF_Cardinal_csum(A,C,R12,R32),bNF_Cardinal_csum(B,D,R23,R42)))),bNF_Wellorder_ordIso(sum_sum(sum_sum(A,B),sum_sum(C,D)),sum_sum(sum_sum(A,C),sum_sum(B,D)))) ).

% csum_csum
tff(fact_6186_cprod__csum__distrib1,axiom,
    ! [C: $tType,B: $tType,A: $tType,R12: set(product_prod(A,A)),R23: set(product_prod(B,B)),R32: set(product_prod(C,C))] : aa(set(product_prod(set(product_prod(sum_sum(product_prod(A,B),product_prod(A,C)),sum_sum(product_prod(A,B),product_prod(A,C)))),set(product_prod(product_prod(A,sum_sum(B,C)),product_prod(A,sum_sum(B,C)))))),$o,member(product_prod(set(product_prod(sum_sum(product_prod(A,B),product_prod(A,C)),sum_sum(product_prod(A,B),product_prod(A,C)))),set(product_prod(product_prod(A,sum_sum(B,C)),product_prod(A,sum_sum(B,C))))),aa(set(product_prod(product_prod(A,sum_sum(B,C)),product_prod(A,sum_sum(B,C)))),product_prod(set(product_prod(sum_sum(product_prod(A,B),product_prod(A,C)),sum_sum(product_prod(A,B),product_prod(A,C)))),set(product_prod(product_prod(A,sum_sum(B,C)),product_prod(A,sum_sum(B,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)))),fun(set(product_prod(product_prod(A,sum_sum(B,C)),product_prod(A,sum_sum(B,C)))),product_prod(set(product_prod(sum_sum(product_prod(A,B),product_prod(A,C)),sum_sum(product_prod(A,B),product_prod(A,C)))),set(product_prod(product_prod(A,sum_sum(B,C)),product_prod(A,sum_sum(B,C)))))),product_Pair(set(product_prod(sum_sum(product_prod(A,B),product_prod(A,C)),sum_sum(product_prod(A,B),product_prod(A,C)))),set(product_prod(product_prod(A,sum_sum(B,C)),product_prod(A,sum_sum(B,C))))),bNF_Cardinal_csum(product_prod(A,B),product_prod(A,C),bNF_Cardinal_cprod(A,B,R12,R23),bNF_Cardinal_cprod(A,C,R12,R32))),bNF_Cardinal_cprod(A,sum_sum(B,C),R12,bNF_Cardinal_csum(B,C,R23,R32)))),bNF_Wellorder_ordIso(sum_sum(product_prod(A,B),product_prod(A,C)),product_prod(A,sum_sum(B,C)))) ).

% cprod_csum_distrib1
tff(fact_6187_cprod__csum__cexp,axiom,
    ! [B: $tType,A: $tType,R12: set(product_prod(A,A)),R23: set(product_prod(B,B))] : aa(set(product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(fun($o,sum_sum(A,B)),fun($o,sum_sum(A,B)))))),$o,member(product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(fun($o,sum_sum(A,B)),fun($o,sum_sum(A,B))))),aa(set(product_prod(fun($o,sum_sum(A,B)),fun($o,sum_sum(A,B)))),product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(fun($o,sum_sum(A,B)),fun($o,sum_sum(A,B))))),aa(set(product_prod(product_prod(A,B),product_prod(A,B))),fun(set(product_prod(fun($o,sum_sum(A,B)),fun($o,sum_sum(A,B)))),product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(fun($o,sum_sum(A,B)),fun($o,sum_sum(A,B)))))),product_Pair(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(fun($o,sum_sum(A,B)),fun($o,sum_sum(A,B))))),bNF_Cardinal_cprod(A,B,R12,R23)),bNF_Cardinal_cexp(sum_sum(A,B),$o,bNF_Cardinal_csum(A,B,R12,R23),bNF_Cardinal_ctwo))),bNF_Wellorder_ordLeq(product_prod(A,B),fun($o,sum_sum(A,B)))) ).

% cprod_csum_cexp
tff(fact_6188_csum__com,axiom,
    ! [B: $tType,A: $tType,P1: set(product_prod(A,A)),P22: set(product_prod(B,B))] : aa(set(product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(sum_sum(B,A),sum_sum(B,A))))),$o,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_Cardinal_csum(A,B,P1,P22)),bNF_Cardinal_csum(B,A,P22,P1))),bNF_Wellorder_ordIso(sum_sum(A,B),sum_sum(B,A))) ).

% csum_com
tff(fact_6189_csum__Cfinite__cexp__Cinfinite,axiom,
    ! [B: $tType,A: $tType,C: $tType,R2: set(product_prod(A,A)),S2: set(product_prod(B,B)),T3: set(product_prod(C,C))] :
      ( aa(set(product_prod(A,A)),$o,bNF_Ca8970107618336181345der_on(A,field2(A,R2)),R2)
     => ( ( bNF_Cardinal_cfinite(B,S2)
          & aa(set(product_prod(B,B)),$o,bNF_Ca8970107618336181345der_on(B,field2(B,S2)),S2) )
       => ( ( bNF_Ca4139267488887388095finite(C,T3)
            & aa(set(product_prod(C,C)),$o,bNF_Ca8970107618336181345der_on(C,field2(C,T3)),T3) )
         => aa(set(product_prod(set(product_prod(fun(C,sum_sum(A,B)),fun(C,sum_sum(A,B)))),set(product_prod(fun(C,sum_sum(A,$o)),fun(C,sum_sum(A,$o)))))),$o,member(product_prod(set(product_prod(fun(C,sum_sum(A,B)),fun(C,sum_sum(A,B)))),set(product_prod(fun(C,sum_sum(A,$o)),fun(C,sum_sum(A,$o))))),aa(set(product_prod(fun(C,sum_sum(A,$o)),fun(C,sum_sum(A,$o)))),product_prod(set(product_prod(fun(C,sum_sum(A,B)),fun(C,sum_sum(A,B)))),set(product_prod(fun(C,sum_sum(A,$o)),fun(C,sum_sum(A,$o))))),aa(set(product_prod(fun(C,sum_sum(A,B)),fun(C,sum_sum(A,B)))),fun(set(product_prod(fun(C,sum_sum(A,$o)),fun(C,sum_sum(A,$o)))),product_prod(set(product_prod(fun(C,sum_sum(A,B)),fun(C,sum_sum(A,B)))),set(product_prod(fun(C,sum_sum(A,$o)),fun(C,sum_sum(A,$o)))))),product_Pair(set(product_prod(fun(C,sum_sum(A,B)),fun(C,sum_sum(A,B)))),set(product_prod(fun(C,sum_sum(A,$o)),fun(C,sum_sum(A,$o))))),bNF_Cardinal_cexp(sum_sum(A,B),C,bNF_Cardinal_csum(A,B,R2,S2),T3)),bNF_Cardinal_cexp(sum_sum(A,$o),C,bNF_Cardinal_csum(A,$o,R2,bNF_Cardinal_ctwo),T3))),bNF_Wellorder_ordLeq(fun(C,sum_sum(A,B)),fun(C,sum_sum(A,$o)))) ) ) ) ).

% csum_Cfinite_cexp_Cinfinite
tff(fact_6190_card__order__csum,axiom,
    ! [A: $tType,B: $tType,R12: set(product_prod(A,A)),R23: set(product_prod(B,B))] :
      ( aa(set(product_prod(A,A)),$o,bNF_Ca8970107618336181345der_on(A,top_top(set(A))),R12)
     => ( aa(set(product_prod(B,B)),$o,bNF_Ca8970107618336181345der_on(B,top_top(set(B))),R23)
       => aa(set(product_prod(sum_sum(A,B),sum_sum(A,B))),$o,bNF_Ca8970107618336181345der_on(sum_sum(A,B),top_top(set(sum_sum(A,B)))),bNF_Cardinal_csum(A,B,R12,R23)) ) ) ).

% card_order_csum
tff(fact_6191_Un__csum,axiom,
    ! [A: $tType,A4: set(A),B3: set(A)] : aa(set(product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,A),sum_sum(A,A))))),$o,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)),A4),B3))),bNF_Cardinal_csum(A,A,bNF_Ca6860139660246222851ard_of(A,A4),bNF_Ca6860139660246222851ard_of(A,B3)))),bNF_Wellorder_ordLeq(A,sum_sum(A,A))) ).

% Un_csum
tff(fact_6192_ordLeq__csum1,axiom,
    ! [B: $tType,A: $tType,P1: set(product_prod(A,A)),P22: set(product_prod(B,B))] :
      ( aa(set(product_prod(A,A)),$o,bNF_Ca8970107618336181345der_on(A,field2(A,P1)),P1)
     => aa(set(product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,B),sum_sum(A,B))))),$o,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)))),P1),bNF_Cardinal_csum(A,B,P1,P22))),bNF_Wellorder_ordLeq(A,sum_sum(A,B))) ) ).

% ordLeq_csum1
tff(fact_6193_ordLeq__csum2,axiom,
    ! [B: $tType,A: $tType,P22: set(product_prod(A,A)),P1: set(product_prod(B,B))] :
      ( aa(set(product_prod(A,A)),$o,bNF_Ca8970107618336181345der_on(A,field2(A,P22)),P22)
     => aa(set(product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(B,A),sum_sum(B,A))))),$o,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)))),P22),bNF_Cardinal_csum(B,A,P1,P22))),bNF_Wellorder_ordLeq(A,sum_sum(B,A))) ) ).

% ordLeq_csum2
tff(fact_6194_csum__cong,axiom,
    ! [D: $tType,B: $tType,C: $tType,A: $tType,P1: set(product_prod(A,A)),R12: set(product_prod(B,B)),P22: set(product_prod(C,C)),R23: set(product_prod(D,D))] :
      ( aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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))),P1),R12)),bNF_Wellorder_ordIso(A,B))
     => ( aa(set(product_prod(set(product_prod(C,C)),set(product_prod(D,D)))),$o,member(product_prod(set(product_prod(C,C)),set(product_prod(D,D))),aa(set(product_prod(D,D)),product_prod(set(product_prod(C,C)),set(product_prod(D,D))),aa(set(product_prod(C,C)),fun(set(product_prod(D,D)),product_prod(set(product_prod(C,C)),set(product_prod(D,D)))),product_Pair(set(product_prod(C,C)),set(product_prod(D,D))),P22),R23)),bNF_Wellorder_ordIso(C,D))
       => aa(set(product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D))))),$o,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_Cardinal_csum(A,C,P1,P22)),bNF_Cardinal_csum(B,D,R12,R23))),bNF_Wellorder_ordIso(sum_sum(A,C),sum_sum(B,D))) ) ) ).

% csum_cong
tff(fact_6195_csum__cong1,axiom,
    ! [B: $tType,C: $tType,A: $tType,P1: set(product_prod(A,A)),R12: set(product_prod(B,B)),Q3: set(product_prod(C,C))] :
      ( aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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))),P1),R12)),bNF_Wellorder_ordIso(A,B))
     => 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))))),$o,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_Cardinal_csum(A,C,P1,Q3)),bNF_Cardinal_csum(B,C,R12,Q3))),bNF_Wellorder_ordIso(sum_sum(A,C),sum_sum(B,C))) ) ).

% csum_cong1
tff(fact_6196_csum__cong2,axiom,
    ! [B: $tType,A: $tType,C: $tType,P22: set(product_prod(A,A)),R23: set(product_prod(B,B)),Q3: set(product_prod(C,C))] :
      ( aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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))),P22),R23)),bNF_Wellorder_ordIso(A,B))
     => 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))))),$o,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_Cardinal_csum(C,A,Q3,P22)),bNF_Cardinal_csum(C,B,Q3,R23))),bNF_Wellorder_ordIso(sum_sum(C,A),sum_sum(C,B))) ) ).

% csum_cong2
tff(fact_6197_csum__mono,axiom,
    ! [D: $tType,B: $tType,C: $tType,A: $tType,P1: set(product_prod(A,A)),R12: set(product_prod(B,B)),P22: set(product_prod(C,C)),R23: set(product_prod(D,D))] :
      ( aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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))),P1),R12)),bNF_Wellorder_ordLeq(A,B))
     => ( aa(set(product_prod(set(product_prod(C,C)),set(product_prod(D,D)))),$o,member(product_prod(set(product_prod(C,C)),set(product_prod(D,D))),aa(set(product_prod(D,D)),product_prod(set(product_prod(C,C)),set(product_prod(D,D))),aa(set(product_prod(C,C)),fun(set(product_prod(D,D)),product_prod(set(product_prod(C,C)),set(product_prod(D,D)))),product_Pair(set(product_prod(C,C)),set(product_prod(D,D))),P22),R23)),bNF_Wellorder_ordLeq(C,D))
       => aa(set(product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D))))),$o,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_Cardinal_csum(A,C,P1,P22)),bNF_Cardinal_csum(B,D,R12,R23))),bNF_Wellorder_ordLeq(sum_sum(A,C),sum_sum(B,D))) ) ) ).

% csum_mono
tff(fact_6198_csum__mono1,axiom,
    ! [B: $tType,C: $tType,A: $tType,P1: set(product_prod(A,A)),R12: set(product_prod(B,B)),Q3: set(product_prod(C,C))] :
      ( aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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))),P1),R12)),bNF_Wellorder_ordLeq(A,B))
     => 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))))),$o,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_Cardinal_csum(A,C,P1,Q3)),bNF_Cardinal_csum(B,C,R12,Q3))),bNF_Wellorder_ordLeq(sum_sum(A,C),sum_sum(B,C))) ) ).

% csum_mono1
tff(fact_6199_csum__mono2,axiom,
    ! [B: $tType,A: $tType,C: $tType,P22: set(product_prod(A,A)),R23: set(product_prod(B,B)),Q3: set(product_prod(C,C))] :
      ( aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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))),P22),R23)),bNF_Wellorder_ordLeq(A,B))
     => 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))))),$o,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_Cardinal_csum(C,A,Q3,P22)),bNF_Cardinal_csum(C,B,Q3,R23))),bNF_Wellorder_ordLeq(sum_sum(C,A),sum_sum(C,B))) ) ).

% csum_mono2
tff(fact_6200_Plus__csum,axiom,
    ! [B: $tType,A: $tType,A4: set(A),B3: set(B)] : aa(set(product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(sum_sum(A,B),sum_sum(A,B))))),$o,member(product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),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(sum_sum(A,B),sum_sum(A,B))),set(product_prod(sum_sum(A,B),sum_sum(A,B)))),aa(set(product_prod(sum_sum(A,B),sum_sum(A,B))),fun(set(product_prod(sum_sum(A,B),sum_sum(A,B))),product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(sum_sum(A,B),sum_sum(A,B))))),product_Pair(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(sum_sum(A,B),sum_sum(A,B)))),bNF_Ca6860139660246222851ard_of(sum_sum(A,B),sum_Plus(A,B,A4,B3))),bNF_Cardinal_csum(A,B,bNF_Ca6860139660246222851ard_of(A,A4),bNF_Ca6860139660246222851ard_of(B,B3)))),bNF_Wellorder_ordIso(sum_sum(A,B),sum_sum(A,B))) ).

% Plus_csum
tff(fact_6201_cprod__cexp__csum__cexp__Cinfinite,axiom,
    ! [C: $tType,B: $tType,A: $tType,T3: set(product_prod(A,A)),R2: set(product_prod(B,B)),S2: set(product_prod(C,C))] :
      ( ( bNF_Ca4139267488887388095finite(A,T3)
        & aa(set(product_prod(A,A)),$o,bNF_Ca8970107618336181345der_on(A,field2(A,T3)),T3) )
     => aa(set(product_prod(set(product_prod(fun(A,product_prod(B,C)),fun(A,product_prod(B,C)))),set(product_prod(fun(A,sum_sum(B,C)),fun(A,sum_sum(B,C)))))),$o,member(product_prod(set(product_prod(fun(A,product_prod(B,C)),fun(A,product_prod(B,C)))),set(product_prod(fun(A,sum_sum(B,C)),fun(A,sum_sum(B,C))))),aa(set(product_prod(fun(A,sum_sum(B,C)),fun(A,sum_sum(B,C)))),product_prod(set(product_prod(fun(A,product_prod(B,C)),fun(A,product_prod(B,C)))),set(product_prod(fun(A,sum_sum(B,C)),fun(A,sum_sum(B,C))))),aa(set(product_prod(fun(A,product_prod(B,C)),fun(A,product_prod(B,C)))),fun(set(product_prod(fun(A,sum_sum(B,C)),fun(A,sum_sum(B,C)))),product_prod(set(product_prod(fun(A,product_prod(B,C)),fun(A,product_prod(B,C)))),set(product_prod(fun(A,sum_sum(B,C)),fun(A,sum_sum(B,C)))))),product_Pair(set(product_prod(fun(A,product_prod(B,C)),fun(A,product_prod(B,C)))),set(product_prod(fun(A,sum_sum(B,C)),fun(A,sum_sum(B,C))))),bNF_Cardinal_cexp(product_prod(B,C),A,bNF_Cardinal_cprod(B,C,R2,S2),T3)),bNF_Cardinal_cexp(sum_sum(B,C),A,bNF_Cardinal_csum(B,C,R2,S2),T3))),bNF_Wellorder_ordLeq(fun(A,product_prod(B,C)),fun(A,sum_sum(B,C)))) ) ).

% cprod_cexp_csum_cexp_Cinfinite
tff(fact_6202_csum__absorb1,axiom,
    ! [B: $tType,A: $tType,R23: set(product_prod(A,A)),R12: set(product_prod(B,B))] :
      ( ( bNF_Ca4139267488887388095finite(A,R23)
        & aa(set(product_prod(A,A)),$o,bNF_Ca8970107618336181345der_on(A,field2(A,R23)),R23) )
     => ( aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),$o,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),R12),R23)),bNF_Wellorder_ordLeq(B,A))
       => aa(set(product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(A,A)))),$o,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_Cardinal_csum(A,B,R23,R12)),R23)),bNF_Wellorder_ordIso(sum_sum(A,B),A)) ) ) ).

% csum_absorb1
tff(fact_6203_csum__absorb1_H,axiom,
    ! [B: $tType,A: $tType,R23: set(product_prod(A,A)),R12: set(product_prod(B,B))] :
      ( aa(set(product_prod(A,A)),$o,bNF_Ca8970107618336181345der_on(A,field2(A,R23)),R23)
     => ( aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),$o,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),R12),R23)),bNF_Wellorder_ordLeq(B,A))
       => ( ( bNF_Ca4139267488887388095finite(B,R12)
            | bNF_Ca4139267488887388095finite(A,R23) )
         => aa(set(product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(A,A)))),$o,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_Cardinal_csum(A,B,R23,R12)),R23)),bNF_Wellorder_ordIso(sum_sum(A,B),A)) ) ) ) ).

% csum_absorb1'
tff(fact_6204_csum__absorb2_H,axiom,
    ! [A: $tType,B: $tType,R23: set(product_prod(A,A)),R12: set(product_prod(B,B))] :
      ( aa(set(product_prod(A,A)),$o,bNF_Ca8970107618336181345der_on(A,field2(A,R23)),R23)
     => ( aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),$o,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),R12),R23)),bNF_Wellorder_ordLeq(B,A))
       => ( ( bNF_Ca4139267488887388095finite(B,R12)
            | bNF_Ca4139267488887388095finite(A,R23) )
         => aa(set(product_prod(set(product_prod(sum_sum(B,A),sum_sum(B,A))),set(product_prod(A,A)))),$o,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_Cardinal_csum(B,A,R12,R23)),R23)),bNF_Wellorder_ordIso(sum_sum(B,A),A)) ) ) ) ).

% csum_absorb2'
tff(fact_6205_csum__cinfinite__bound,axiom,
    ! [B: $tType,C: $tType,A: $tType,P3: set(product_prod(A,A)),R2: set(product_prod(B,B)),Q3: set(product_prod(C,C))] :
      ( aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),$o,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))),P3),R2)),bNF_Wellorder_ordLeq(A,B))
     => ( aa(set(product_prod(set(product_prod(C,C)),set(product_prod(B,B)))),$o,member(product_prod(set(product_prod(C,C)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(C,C)),set(product_prod(B,B))),aa(set(product_prod(C,C)),fun(set(product_prod(B,B)),product_prod(set(product_prod(C,C)),set(product_prod(B,B)))),product_Pair(set(product_prod(C,C)),set(product_prod(B,B))),Q3),R2)),bNF_Wellorder_ordLeq(C,B))
       => ( aa(set(product_prod(A,A)),$o,bNF_Ca8970107618336181345der_on(A,field2(A,P3)),P3)
         => ( aa(set(product_prod(C,C)),$o,bNF_Ca8970107618336181345der_on(C,field2(C,Q3)),Q3)
           => ( ( bNF_Ca4139267488887388095finite(B,R2)
                & aa(set(product_prod(B,B)),$o,bNF_Ca8970107618336181345der_on(B,field2(B,R2)),R2) )
             => aa(set(product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(B,B)))),$o,member(product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(B,B))),aa(set(product_prod(sum_sum(A,C),sum_sum(A,C))),fun(set(product_prod(B,B)),product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(B,B)))),product_Pair(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(B,B))),bNF_Cardinal_csum(A,C,P3,Q3)),R2)),bNF_Wellorder_ordLeq(sum_sum(A,C),B)) ) ) ) ) ) ).

% csum_cinfinite_bound
tff(fact_6206_numeral__num__of__nat__unfold,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [N: nat] :
          aa(num,A,numeral_numeral(A),num_of_nat(N)) = $ite(N = zero_zero(nat),one_one(A),aa(nat,A,semiring_1_of_nat(A),N)) ) ).

% numeral_num_of_nat_unfold
tff(fact_6207_filterlim__INF__INF,axiom,
    ! [A: $tType,C: $tType,D: $tType,B: $tType,J3: set(A),I4: set(B),F: fun(D,C),F4: fun(B,filter(D)),G4: fun(A,filter(C))] :
      ( ! [M3: A] :
          ( aa(set(A),$o,member(A,M3),J3)
         => ? [X3: B] :
              ( aa(set(B),$o,member(B,X3),I4)
              & aa(filter(C),$o,aa(filter(C),fun(filter(C),$o),ord_less_eq(filter(C)),aa(filter(D),filter(C),aa(fun(D,C),fun(filter(D),filter(C)),filtermap(D,C),F),aa(B,filter(D),F4,X3))),aa(A,filter(C),G4,M3)) ) )
     => filterlim(D,C,F,aa(set(filter(C)),filter(C),complete_Inf_Inf(filter(C)),aa(set(A),set(filter(C)),image2(A,filter(C),G4),J3)),aa(set(filter(D)),filter(D),complete_Inf_Inf(filter(D)),aa(set(B),set(filter(D)),image2(B,filter(D),F4),I4))) ) ).

% filterlim_INF_INF
tff(fact_6208_filtermap__id_H,axiom,
    ! [A: $tType,X3: filter(A)] : aa(filter(A),filter(A),aa(fun(A,A),fun(filter(A),filter(A)),filtermap(A,A),aTP_Lamp_au(A,A)),X3) = X3 ).

% filtermap_id'
tff(fact_6209_filtermap__bot,axiom,
    ! [B: $tType,A: $tType,F: fun(B,A)] : aa(filter(B),filter(A),aa(fun(B,A),fun(filter(B),filter(A)),filtermap(B,A),F),bot_bot(filter(B))) = bot_bot(filter(A)) ).

% filtermap_bot
tff(fact_6210_eventually__filtermap,axiom,
    ! [A: $tType,B: $tType,P: fun(A,$o),F: fun(B,A),F4: filter(B)] :
      ( eventually(A,P,aa(filter(B),filter(A),aa(fun(B,A),fun(filter(B),filter(A)),filtermap(B,A),F),F4))
    <=> eventually(B,aa(fun(B,A),fun(B,$o),aTP_Lamp_akb(fun(A,$o),fun(fun(B,A),fun(B,$o)),P),F),F4) ) ).

% eventually_filtermap
tff(fact_6211_filtermap__sup,axiom,
    ! [A: $tType,B: $tType,F: fun(B,A),F12: filter(B),F23: filter(B)] : aa(filter(B),filter(A),aa(fun(B,A),fun(filter(B),filter(A)),filtermap(B,A),F),aa(filter(B),filter(B),aa(filter(B),fun(filter(B),filter(B)),sup_sup(filter(B)),F12),F23)) = aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),sup_sup(filter(A)),aa(filter(B),filter(A),aa(fun(B,A),fun(filter(B),filter(A)),filtermap(B,A),F),F12)),aa(filter(B),filter(A),aa(fun(B,A),fun(filter(B),filter(A)),filtermap(B,A),F),F23)) ).

% filtermap_sup
tff(fact_6212_filterlim__filtermap,axiom,
    ! [B: $tType,A: $tType,C: $tType,F: fun(A,B),F12: filter(B),G: fun(C,A),F23: filter(C)] :
      ( filterlim(A,B,F,F12,aa(filter(C),filter(A),aa(fun(C,A),fun(filter(C),filter(A)),filtermap(C,A),G),F23))
    <=> filterlim(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_ajj(fun(A,B),fun(fun(C,A),fun(C,B)),F),G),F12,F23) ) ).

% filterlim_filtermap
tff(fact_6213_filtermap__ident,axiom,
    ! [A: $tType,F4: filter(A)] : aa(filter(A),filter(A),aa(fun(A,A),fun(filter(A),filter(A)),filtermap(A,A),aTP_Lamp_au(A,A)),F4) = F4 ).

% filtermap_ident
tff(fact_6214_filtermap__filtermap,axiom,
    ! [B: $tType,A: $tType,C: $tType,F: fun(B,A),G: fun(C,B),F4: filter(C)] : aa(filter(B),filter(A),aa(fun(B,A),fun(filter(B),filter(A)),filtermap(B,A),F),aa(filter(C),filter(B),aa(fun(C,B),fun(filter(C),filter(B)),filtermap(C,B),G),F4)) = aa(filter(C),filter(A),aa(fun(C,A),fun(filter(C),filter(A)),filtermap(C,A),aa(fun(C,B),fun(C,A),aTP_Lamp_ll(fun(B,A),fun(fun(C,B),fun(C,A)),F),G)),F4) ).

% filtermap_filtermap
tff(fact_6215_filtermap__eq__strong,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),F4: filter(A),G4: filter(A)] :
      ( inj_on(A,B,F,top_top(set(A)))
     => ( ( aa(filter(A),filter(B),aa(fun(A,B),fun(filter(A),filter(B)),filtermap(A,B),F),F4) = aa(filter(A),filter(B),aa(fun(A,B),fun(filter(A),filter(B)),filtermap(A,B),F),G4) )
      <=> ( F4 = G4 ) ) ) ).

% filtermap_eq_strong
tff(fact_6216_filtermap__bot__iff,axiom,
    ! [A: $tType,B: $tType,F: fun(B,A),F4: filter(B)] :
      ( ( aa(filter(B),filter(A),aa(fun(B,A),fun(filter(B),filter(A)),filtermap(B,A),F),F4) = bot_bot(filter(A)) )
    <=> ( F4 = bot_bot(filter(B)) ) ) ).

% filtermap_bot_iff
tff(fact_6217_map__filter__on__UNIV,axiom,
    ! [B: $tType,A: $tType] : map_filter_on(A,B,top_top(set(A))) = filtermap(A,B) ).

% map_filter_on_UNIV
tff(fact_6218_filtermap__sequentually__ne__bot,axiom,
    ! [A: $tType,F: fun(nat,A)] : aa(filter(nat),filter(A),aa(fun(nat,A),fun(filter(nat),filter(A)),filtermap(nat,A),F),at_top(nat)) != bot_bot(filter(A)) ).

% filtermap_sequentually_ne_bot
tff(fact_6219_filtermap__inf,axiom,
    ! [A: $tType,B: $tType,F: fun(B,A),F12: filter(B),F23: filter(B)] : aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),aa(filter(B),filter(A),aa(fun(B,A),fun(filter(B),filter(A)),filtermap(B,A),F),aa(filter(B),filter(B),aa(filter(B),fun(filter(B),filter(B)),inf_inf(filter(B)),F12),F23))),aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),inf_inf(filter(A)),aa(filter(B),filter(A),aa(fun(B,A),fun(filter(B),filter(A)),filtermap(B,A),F),F12)),aa(filter(B),filter(A),aa(fun(B,A),fun(filter(B),filter(A)),filtermap(B,A),F),F23))) ).

% filtermap_inf
tff(fact_6220_filtermap__fun__inverse,axiom,
    ! [B: $tType,A: $tType,G: fun(A,B),F4: filter(B),G4: filter(A),F: fun(B,A)] :
      ( filterlim(A,B,G,F4,G4)
     => ( filterlim(B,A,F,G4,F4)
       => ( eventually(A,aa(fun(B,A),fun(A,$o),aTP_Lamp_ank(fun(A,B),fun(fun(B,A),fun(A,$o)),G),F),G4)
         => ( aa(filter(B),filter(A),aa(fun(B,A),fun(filter(B),filter(A)),filtermap(B,A),F),F4) = G4 ) ) ) ) ).

% filtermap_fun_inverse
tff(fact_6221_filtermap__SUP,axiom,
    ! [B: $tType,A: $tType,C: $tType,F: fun(B,A),F4: fun(C,filter(B)),B3: set(C)] : aa(filter(B),filter(A),aa(fun(B,A),fun(filter(B),filter(A)),filtermap(B,A),F),aa(set(filter(B)),filter(B),complete_Sup_Sup(filter(B)),aa(set(C),set(filter(B)),image2(C,filter(B),F4),B3))) = aa(set(filter(A)),filter(A),complete_Sup_Sup(filter(A)),aa(set(C),set(filter(A)),image2(C,filter(A),aa(fun(C,filter(B)),fun(C,filter(A)),aTP_Lamp_anl(fun(B,A),fun(fun(C,filter(B)),fun(C,filter(A))),F),F4)),B3)) ).

% filtermap_SUP
tff(fact_6222_filtermap__def,axiom,
    ! [A: $tType,B: $tType,F: fun(B,A),F4: filter(B)] : aa(filter(B),filter(A),aa(fun(B,A),fun(filter(B),filter(A)),filtermap(B,A),F),F4) = abs_filter(A,aa(filter(B),fun(fun(A,$o),$o),aTP_Lamp_ann(fun(B,A),fun(filter(B),fun(fun(A,$o),$o)),F),F4)) ).

% filtermap_def
tff(fact_6223_filtermap__mono__strong,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),F4: filter(A),G4: filter(A)] :
      ( inj_on(A,B,F,top_top(set(A)))
     => ( aa(filter(B),$o,aa(filter(B),fun(filter(B),$o),ord_less_eq(filter(B)),aa(filter(A),filter(B),aa(fun(A,B),fun(filter(A),filter(B)),filtermap(A,B),F),F4)),aa(filter(A),filter(B),aa(fun(A,B),fun(filter(A),filter(B)),filtermap(A,B),F),G4))
      <=> aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),F4),G4) ) ) ).

% filtermap_mono_strong
tff(fact_6224_num__of__nat__One,axiom,
    ! [N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),one_one(nat))
     => ( num_of_nat(N) = one2 ) ) ).

% num_of_nat_One
tff(fact_6225_filtermap__INF,axiom,
    ! [B: $tType,A: $tType,C: $tType,F: fun(B,A),F4: fun(C,filter(B)),B3: set(C)] : aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),aa(filter(B),filter(A),aa(fun(B,A),fun(filter(B),filter(A)),filtermap(B,A),F),aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(C),set(filter(B)),image2(C,filter(B),F4),B3)))),aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(C),set(filter(A)),image2(C,filter(A),aa(fun(C,filter(B)),fun(C,filter(A)),aTP_Lamp_anl(fun(B,A),fun(fun(C,filter(B)),fun(C,filter(A))),F),F4)),B3))) ).

% filtermap_INF
tff(fact_6226_prod__filter__principal__singleton2,axiom,
    ! [B: $tType,A: $tType,F4: filter(A),X: B] : prod_filter(A,B,F4,principal(B,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),X),bot_bot(set(B))))) = aa(filter(A),filter(product_prod(A,B)),aa(fun(A,product_prod(A,B)),fun(filter(A),filter(product_prod(A,B))),filtermap(A,product_prod(A,B)),aa(B,fun(A,product_prod(A,B)),aTP_Lamp_ru(B,fun(A,product_prod(A,B))),X)),F4) ).

% prod_filter_principal_singleton2
tff(fact_6227_Max_Osemilattice__order__set__axioms,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => lattic4895041142388067077er_set(A,ord_max(A),aTP_Lamp_tm(A,fun(A,$o)),aTP_Lamp_ano(A,fun(A,$o))) ) ).

% Max.semilattice_order_set_axioms
tff(fact_6228_prod__filter__assoc,axiom,
    ! [A: $tType,B: $tType,C: $tType,F4: filter(A),G4: filter(B),H8: filter(C)] : prod_filter(product_prod(A,B),C,prod_filter(A,B,F4,G4),H8) = aa(filter(product_prod(A,product_prod(B,C))),filter(product_prod(product_prod(A,B),C)),aa(fun(product_prod(A,product_prod(B,C)),product_prod(product_prod(A,B),C)),fun(filter(product_prod(A,product_prod(B,C))),filter(product_prod(product_prod(A,B),C))),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_anq(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,G4,H8))) ).

% prod_filter_assoc
tff(fact_6229_prod__filter__eq__bot,axiom,
    ! [A: $tType,B: $tType,A4: filter(A),B3: filter(B)] :
      ( ( prod_filter(A,B,A4,B3) = bot_bot(filter(product_prod(A,B))) )
    <=> ( ( A4 = bot_bot(filter(A)) )
        | ( B3 = bot_bot(filter(B)) ) ) ) ).

% prod_filter_eq_bot
tff(fact_6230_eventually__prod__same,axiom,
    ! [A: $tType,P: fun(product_prod(A,A),$o),F4: filter(A)] :
      ( eventually(product_prod(A,A),P,prod_filter(A,A,F4,F4))
    <=> ? [Q7: fun(A,$o)] :
          ( eventually(A,Q7,F4)
          & ! [X4: A,Y3: A] :
              ( aa(A,$o,Q7,X4)
             => ( aa(A,$o,Q7,Y3)
               => aa(product_prod(A,A),$o,P,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Y3)) ) ) ) ) ).

% eventually_prod_same
tff(fact_6231_eventually__prod__filter,axiom,
    ! [A: $tType,B: $tType,P: fun(product_prod(A,B),$o),F4: filter(A),G4: filter(B)] :
      ( eventually(product_prod(A,B),P,prod_filter(A,B,F4,G4))
    <=> ? [Pf: fun(A,$o),Pg: fun(B,$o)] :
          ( eventually(A,Pf,F4)
          & eventually(B,Pg,G4)
          & ! [X4: A,Y3: B] :
              ( aa(A,$o,Pf,X4)
             => ( aa(B,$o,Pg,Y3)
               => aa(product_prod(A,B),$o,P,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Y3)) ) ) ) ) ).

% eventually_prod_filter
tff(fact_6232_prod__filter__mono__iff,axiom,
    ! [A: $tType,B: $tType,A4: filter(A),B3: filter(B),C3: filter(A),D4: filter(B)] :
      ( ( A4 != bot_bot(filter(A)) )
     => ( ( B3 != bot_bot(filter(B)) )
       => ( aa(filter(product_prod(A,B)),$o,aa(filter(product_prod(A,B)),fun(filter(product_prod(A,B)),$o),ord_less_eq(filter(product_prod(A,B))),prod_filter(A,B,A4,B3)),prod_filter(A,B,C3,D4))
        <=> ( aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),A4),C3)
            & aa(filter(B),$o,aa(filter(B),fun(filter(B),$o),ord_less_eq(filter(B)),B3),D4) ) ) ) ) ).

% prod_filter_mono_iff
tff(fact_6233_filterlim__Pair,axiom,
    ! [C: $tType,B: $tType,A: $tType,F: fun(A,B),G4: filter(B),F4: filter(A),G: fun(A,C),H8: filter(C)] :
      ( filterlim(A,B,F,G4,F4)
     => ( filterlim(A,C,G,H8,F4)
       => filterlim(A,product_prod(B,C),aa(fun(A,C),fun(A,product_prod(B,C)),aTP_Lamp_anr(fun(A,B),fun(fun(A,C),fun(A,product_prod(B,C))),F),G),prod_filter(B,C,G4,H8),F4) ) ) ).

% filterlim_Pair
tff(fact_6234_filtermap__Pair,axiom,
    ! [A: $tType,B: $tType,C: $tType,F: fun(C,A),G: fun(C,B),F4: filter(C)] : aa(filter(product_prod(A,B)),$o,aa(filter(product_prod(A,B)),fun(filter(product_prod(A,B)),$o),ord_less_eq(filter(product_prod(A,B))),aa(filter(C),filter(product_prod(A,B)),aa(fun(C,product_prod(A,B)),fun(filter(C),filter(product_prod(A,B))),filtermap(C,product_prod(A,B)),aa(fun(C,B),fun(C,product_prod(A,B)),aTP_Lamp_qv(fun(C,A),fun(fun(C,B),fun(C,product_prod(A,B))),F),G)),F4)),prod_filter(A,B,aa(filter(C),filter(A),aa(fun(C,A),fun(filter(C),filter(A)),filtermap(C,A),F),F4),aa(filter(C),filter(B),aa(fun(C,B),fun(filter(C),filter(B)),filtermap(C,B),G),F4))) ).

% filtermap_Pair
tff(fact_6235_principal__prod__principal,axiom,
    ! [A: $tType,B: $tType,A4: set(A),B3: set(B)] : prod_filter(A,B,principal(A,A4),principal(B,B3)) = principal(product_prod(A,B),product_Sigma(A,B,A4,aTP_Lamp_qz(set(B),fun(A,set(B)),B3))) ).

% principal_prod_principal
tff(fact_6236_prod__filter__def,axiom,
    ! [A: $tType,B: $tType,F4: filter(A),G4: filter(B)] : prod_filter(A,B,F4,G4) = aa(set(filter(product_prod(A,B))),filter(product_prod(A,B)),complete_Inf_Inf(filter(product_prod(A,B))),aa(set(product_prod(fun(A,$o),fun(B,$o))),set(filter(product_prod(A,B))),image2(product_prod(fun(A,$o),fun(B,$o)),filter(product_prod(A,B)),aa(fun(fun(A,$o),fun(fun(B,$o),filter(product_prod(A,B)))),fun(product_prod(fun(A,$o),fun(B,$o)),filter(product_prod(A,B))),product_case_prod(fun(A,$o),fun(B,$o),filter(product_prod(A,B))),aTP_Lamp_ans(fun(A,$o),fun(fun(B,$o),filter(product_prod(A,B)))))),aa(fun(product_prod(fun(A,$o),fun(B,$o)),$o),set(product_prod(fun(A,$o),fun(B,$o))),collect(product_prod(fun(A,$o),fun(B,$o))),aa(fun(fun(A,$o),fun(fun(B,$o),$o)),fun(product_prod(fun(A,$o),fun(B,$o)),$o),product_case_prod(fun(A,$o),fun(B,$o),$o),aa(filter(B),fun(fun(A,$o),fun(fun(B,$o),$o)),aTP_Lamp_ant(filter(A),fun(filter(B),fun(fun(A,$o),fun(fun(B,$o),$o))),F4),G4))))) ).

% prod_filter_def
tff(fact_6237_eventually__prod__sequentially,axiom,
    ! [P: fun(product_prod(nat,nat),$o)] :
      ( eventually(product_prod(nat,nat),P,prod_filter(nat,nat,at_top(nat),at_top(nat)))
    <=> ? [N9: nat] :
        ! [M2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N9),M2)
         => ! [N2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N9),N2)
             => aa(product_prod(nat,nat),$o,P,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),N2),M2)) ) ) ) ).

% eventually_prod_sequentially
tff(fact_6238_eventually__prodI,axiom,
    ! [A: $tType,B: $tType,P: fun(A,$o),F4: filter(A),Q: fun(B,$o),G4: filter(B)] :
      ( eventually(A,P,F4)
     => ( eventually(B,Q,G4)
       => eventually(product_prod(A,B),aa(fun(B,$o),fun(product_prod(A,B),$o),aTP_Lamp_anu(fun(A,$o),fun(fun(B,$o),fun(product_prod(A,B),$o)),P),Q),prod_filter(A,B,F4,G4)) ) ) ).

% eventually_prodI
tff(fact_6239_eventually__prod1,axiom,
    ! [A: $tType,B: $tType,B3: filter(A),P: fun(B,$o),A4: filter(B)] :
      ( ( B3 != bot_bot(filter(A)) )
     => ( eventually(product_prod(B,A),aa(fun(B,fun(A,$o)),fun(product_prod(B,A),$o),product_case_prod(B,A,$o),aTP_Lamp_anv(fun(B,$o),fun(B,fun(A,$o)),P)),prod_filter(B,A,A4,B3))
      <=> eventually(B,P,A4) ) ) ).

% eventually_prod1
tff(fact_6240_eventually__prod2,axiom,
    ! [A: $tType,B: $tType,A4: filter(A),P: fun(B,$o),B3: filter(B)] :
      ( ( A4 != bot_bot(filter(A)) )
     => ( eventually(product_prod(A,B),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aTP_Lamp_anw(fun(B,$o),fun(A,fun(B,$o)),P)),prod_filter(A,B,A4,B3))
      <=> eventually(B,P,B3) ) ) ).

% eventually_prod2
tff(fact_6241_prod__filter__INF2,axiom,
    ! [C: $tType,B: $tType,A: $tType,J3: set(A),A4: filter(B),B3: fun(A,filter(C))] :
      ( ( J3 != bot_bot(set(A)) )
     => ( prod_filter(B,C,A4,aa(set(filter(C)),filter(C),complete_Inf_Inf(filter(C)),aa(set(A),set(filter(C)),image2(A,filter(C),B3),J3))) = aa(set(filter(product_prod(B,C))),filter(product_prod(B,C)),complete_Inf_Inf(filter(product_prod(B,C))),aa(set(A),set(filter(product_prod(B,C))),image2(A,filter(product_prod(B,C)),aa(fun(A,filter(C)),fun(A,filter(product_prod(B,C))),aTP_Lamp_anx(filter(B),fun(fun(A,filter(C)),fun(A,filter(product_prod(B,C)))),A4),B3)),J3)) ) ) ).

% prod_filter_INF2
tff(fact_6242_prod__filter__INF1,axiom,
    ! [C: $tType,B: $tType,A: $tType,I4: set(A),A4: fun(A,filter(B)),B3: filter(C)] :
      ( ( I4 != bot_bot(set(A)) )
     => ( prod_filter(B,C,aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image2(A,filter(B),A4),I4)),B3) = aa(set(filter(product_prod(B,C))),filter(product_prod(B,C)),complete_Inf_Inf(filter(product_prod(B,C))),aa(set(A),set(filter(product_prod(B,C))),image2(A,filter(product_prod(B,C)),aa(filter(C),fun(A,filter(product_prod(B,C))),aTP_Lamp_any(fun(A,filter(B)),fun(filter(C),fun(A,filter(product_prod(B,C)))),A4),B3)),I4)) ) ) ).

% prod_filter_INF1
tff(fact_6243_prod__filter__INF,axiom,
    ! [B: $tType,D: $tType,C: $tType,A: $tType,I4: set(A),J3: set(B),A4: fun(A,filter(C)),B3: fun(B,filter(D))] :
      ( ( I4 != bot_bot(set(A)) )
     => ( ( J3 != bot_bot(set(B)) )
       => ( prod_filter(C,D,aa(set(filter(C)),filter(C),complete_Inf_Inf(filter(C)),aa(set(A),set(filter(C)),image2(A,filter(C),A4),I4)),aa(set(filter(D)),filter(D),complete_Inf_Inf(filter(D)),aa(set(B),set(filter(D)),image2(B,filter(D),B3),J3))) = aa(set(filter(product_prod(C,D))),filter(product_prod(C,D)),complete_Inf_Inf(filter(product_prod(C,D))),aa(set(A),set(filter(product_prod(C,D))),image2(A,filter(product_prod(C,D)),aa(fun(B,filter(D)),fun(A,filter(product_prod(C,D))),aa(fun(A,filter(C)),fun(fun(B,filter(D)),fun(A,filter(product_prod(C,D)))),aTP_Lamp_aoa(set(B),fun(fun(A,filter(C)),fun(fun(B,filter(D)),fun(A,filter(product_prod(C,D))))),J3),A4),B3)),I4)) ) ) ) ).

% prod_filter_INF
tff(fact_6244_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_6245_prod__filter__principal__singleton,axiom,
    ! [A: $tType,B: $tType,X: A,F4: filter(B)] : prod_filter(A,B,principal(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))),F4) = aa(filter(B),filter(product_prod(A,B)),aa(fun(B,product_prod(A,B)),fun(filter(B),filter(product_prod(A,B))),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_6246_Sup__fin_Osemilattice__order__set__axioms,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => lattic4895041142388067077er_set(A,sup_sup(A),aTP_Lamp_aob(A,fun(A,$o)),aTP_Lamp_aoc(A,fun(A,$o))) ) ).

% Sup_fin.semilattice_order_set_axioms
tff(fact_6247_set__to__map__def,axiom,
    ! [A: $tType,B: $tType,S: set(product_prod(B,A)),K: B] : aa(B,option(A),set_to_map(B,A,S),K) = eps_Opt(A,aa(B,fun(A,$o),aTP_Lamp_wv(set(product_prod(B,A)),fun(B,fun(A,$o)),S),K)) ).

% set_to_map_def
tff(fact_6248_subset__singleton__iff__Uniq,axiom,
    ! [A: $tType,A4: set(A)] :
      ( ? [A10: A] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A10),bot_bot(set(A))))
    <=> uniq(A,aa(set(A),fun(A,$o),aTP_Lamp_ag(set(A),fun(A,$o)),A4)) ) ).

% subset_singleton_iff_Uniq
tff(fact_6249_some__opt__eq__trivial,axiom,
    ! [A: $tType,X: A] : eps_Opt(A,aTP_Lamp_cf(A,fun(A,$o),X)) = aa(A,option(A),some(A),X) ).

% some_opt_eq_trivial
tff(fact_6250_some__opt__false__trivial,axiom,
    ! [A: $tType] : eps_Opt(A,aTP_Lamp_ak(A,$o)) = none(A) ).

% some_opt_false_trivial
tff(fact_6251_alt__ex1E_H,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( ? [X3: A] :
          ( aa(A,$o,P,X3)
          & ! [Y2: A] :
              ( aa(A,$o,P,Y2)
             => ( Y2 = X3 ) ) )
     => ~ ( ? [X_1: A] : aa(A,$o,P,X_1)
         => ~ uniq(A,P) ) ) ).

% alt_ex1E'
tff(fact_6252_ex1__iff__ex__Uniq,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( ? [X4: A] :
          ( aa(A,$o,P,X4)
          & ! [Y3: A] :
              ( aa(A,$o,P,Y3)
             => ( Y3 = X4 ) ) )
    <=> ( ? [X_12: A] : aa(A,$o,P,X_12)
        & uniq(A,P) ) ) ).

% ex1_iff_ex_Uniq
tff(fact_6253_inj__on__iff__Uniq,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),A4: set(A)] :
      ( inj_on(A,B,F,A4)
    <=> ! [X4: A] :
          ( aa(set(A),$o,member(A,X4),A4)
         => uniq(A,aa(A,fun(A,$o),aa(set(A),fun(A,fun(A,$o)),aTP_Lamp_aod(fun(A,B),fun(set(A),fun(A,fun(A,$o))),F),A4),X4)) ) ) ).

% inj_on_iff_Uniq
tff(fact_6254_pairwise__disjnt__iff,axiom,
    ! [A: $tType,A16: set(set(A))] :
      ( pairwise(set(A),disjnt(A),A16)
    <=> ! [X4: A] : uniq(set(A),aa(A,fun(set(A),$o),aTP_Lamp_aoe(set(set(A)),fun(A,fun(set(A),$o)),A16),X4)) ) ).

% pairwise_disjnt_iff
tff(fact_6255_the1__equality_H,axiom,
    ! [A: $tType,P: fun(A,$o),A3: A] :
      ( uniq(A,P)
     => ( aa(A,$o,P,A3)
       => ( the(A,P) = A3 ) ) ) ).

% the1_equality'
tff(fact_6256_strict__sorted__equal__Uniq,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A)] : uniq(list(A),aTP_Lamp_aof(set(A),fun(list(A),$o),A4)) ) ).

% strict_sorted_equal_Uniq
tff(fact_6257_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_6258_ex__is__arg__min__if__finite,axiom,
    ! [B: $tType,A: $tType] :
      ( order(B)
     => ! [S: set(A),F: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),S)
         => ( ( S != bot_bot(set(A)) )
           => ? [X_1: A] : aa(A,$o,lattic501386751177426532rg_min(A,B,F,aa(set(A),fun(A,$o),aTP_Lamp_ag(set(A),fun(A,$o)),S)),X_1) ) ) ) ).

% ex_is_arg_min_if_finite
tff(fact_6259_bounded__quasi__semilattice__set_Oremove,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Top: A,Bot: A,Normalize: fun(A,A),A3: A,A4: set(A)] :
      ( bounde6485984586167503788ce_set(A,F,Top,Bot,Normalize)
     => ( aa(set(A),$o,member(A,A3),A4)
       => ( aa(set(A),A,bounde2362111253966948842tice_F(A,F,Top,Bot),A4) = aa(A,A,aa(A,fun(A,A),F,A3),aa(set(A),A,bounde2362111253966948842tice_F(A,F,Top,Bot),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A)))))) ) ) ) ).

% bounded_quasi_semilattice_set.remove
tff(fact_6260_bounded__quasi__semilattice__set_Oinsert__remove,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Top: A,Bot: A,Normalize: fun(A,A),A3: A,A4: set(A)] :
      ( bounde6485984586167503788ce_set(A,F,Top,Bot,Normalize)
     => ( aa(set(A),A,bounde2362111253966948842tice_F(A,F,Top,Bot),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),A4)) = aa(A,A,aa(A,fun(A,A),F,A3),aa(set(A),A,bounde2362111253966948842tice_F(A,F,Top,Bot),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A)))))) ) ) ).

% bounded_quasi_semilattice_set.insert_remove
tff(fact_6261_bounded__quasi__semilattice__set_Oempty,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Top: A,Bot: A,Normalize: fun(A,A)] :
      ( bounde6485984586167503788ce_set(A,F,Top,Bot,Normalize)
     => ( aa(set(A),A,bounde2362111253966948842tice_F(A,F,Top,Bot),bot_bot(set(A))) = Top ) ) ).

% bounded_quasi_semilattice_set.empty
tff(fact_6262_bounded__quasi__semilattice__set_Ounion,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Top: A,Bot: A,Normalize: fun(A,A),A4: set(A),B3: set(A)] :
      ( bounde6485984586167503788ce_set(A,F,Top,Bot,Normalize)
     => ( aa(set(A),A,bounde2362111253966948842tice_F(A,F,Top,Bot),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3)) = aa(A,A,aa(A,fun(A,A),F,aa(set(A),A,bounde2362111253966948842tice_F(A,F,Top,Bot),A4)),aa(set(A),A,bounde2362111253966948842tice_F(A,F,Top,Bot),B3)) ) ) ).

% bounded_quasi_semilattice_set.union
tff(fact_6263_relImage__def,axiom,
    ! [A: $tType,B: $tType,R: set(product_prod(B,B)),F: fun(B,A)] : bNF_Gr4221423524335903396lImage(B,A,R,F) = aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(B,A),fun(product_prod(A,A),$o),aTP_Lamp_amx(set(product_prod(B,B)),fun(fun(B,A),fun(product_prod(A,A),$o)),R),F)) ).

% relImage_def
tff(fact_6264_card__def,axiom,
    ! [A: $tType] : finite_card(A) = finite_folding_F(A,nat,aTP_Lamp_oo(A,fun(nat,nat)),zero_zero(nat)) ).

% card_def
tff(fact_6265_relImage__relInvImage,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,A)),F: fun(B,A),A4: set(B)] :
      ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R),product_Sigma(A,A,aa(set(B),set(A),image2(B,A,F),A4),aa(set(B),fun(A,set(A)),aTP_Lamp_re(fun(B,A),fun(set(B),fun(A,set(A))),F),A4)))
     => ( bNF_Gr4221423524335903396lImage(B,A,bNF_Gr7122648621184425601vImage(B,A,A4,R,F),F) = R ) ) ).

% relImage_relInvImage
tff(fact_6266_folding__on_Oinsert__remove,axiom,
    ! [B: $tType,A: $tType,S: set(A),F: fun(A,fun(B,B)),X: A,A4: set(A),Z2: B] :
      ( finite_folding_on(A,B,S,F)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A4)),S)
       => ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),B,finite_folding_F(A,B,F,Z2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A4)) = aa(B,B,aa(A,fun(B,B),F,X),aa(set(A),B,finite_folding_F(A,B,F,Z2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))))) ) ) ) ) ).

% folding_on.insert_remove
tff(fact_6267_mset__set_Ofolding__on__axioms,axiom,
    ! [A: $tType] : finite_folding_on(A,multiset(A),top_top(set(A)),add_mset(A)) ).

% mset_set.folding_on_axioms
tff(fact_6268_folding__on_Oempty,axiom,
    ! [A: $tType,B: $tType,S: set(A),F: fun(A,fun(B,B)),Z2: B] :
      ( finite_folding_on(A,B,S,F)
     => ( aa(set(A),B,finite_folding_F(A,B,F,Z2),bot_bot(set(A))) = Z2 ) ) ).

% folding_on.empty
tff(fact_6269_card_Ofolding__on__axioms,axiom,
    ! [A: $tType] : finite_folding_on(A,nat,top_top(set(A)),aTP_Lamp_oo(A,fun(nat,nat))) ).

% card.folding_on_axioms
tff(fact_6270_sorted__list__of__set_Ofold__insort__key_Ofolding__on__axioms,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => finite_folding_on(A,list(A),top_top(set(A)),linorder_insort_key(A,A,aTP_Lamp_yq(A,A))) ) ).

% sorted_list_of_set.fold_insort_key.folding_on_axioms
tff(fact_6271_relInvImage__def,axiom,
    ! [B: $tType,A: $tType,A4: set(A),R: set(product_prod(B,B)),F: fun(A,B)] : bNF_Gr7122648621184425601vImage(A,B,A4,R,F) = aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,B),fun(product_prod(A,A),$o),aa(set(product_prod(B,B)),fun(fun(A,B),fun(product_prod(A,A),$o)),aTP_Lamp_aog(set(A),fun(set(product_prod(B,B)),fun(fun(A,B),fun(product_prod(A,A),$o))),A4),R),F)) ).

% relInvImage_def
tff(fact_6272_relInvImage__UNIV__relImage,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,A)),F: fun(A,B)] : aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R),bNF_Gr7122648621184425601vImage(A,B,top_top(set(A)),bNF_Gr4221423524335903396lImage(A,B,R,F),F)) ).

% relInvImage_UNIV_relImage
tff(fact_6273_folding__on_Oremove,axiom,
    ! [B: $tType,A: $tType,S: set(A),F: fun(A,fun(B,B)),A4: set(A),X: A,Z2: B] :
      ( finite_folding_on(A,B,S,F)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),S)
       => ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),$o,member(A,X),A4)
           => ( aa(set(A),B,finite_folding_F(A,B,F,Z2),A4) = aa(B,B,aa(A,fun(B,B),F,X),aa(set(A),B,finite_folding_F(A,B,F,Z2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))))) ) ) ) ) ) ).

% folding_on.remove
tff(fact_6274_relInvImage__Gr,axiom,
    ! [A: $tType,B: $tType,R: set(product_prod(A,A)),B3: set(A),A4: set(B),F: fun(B,A)] :
      ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R),product_Sigma(A,A,B3,aTP_Lamp_rl(set(A),fun(A,set(A)),B3)))
     => ( bNF_Gr7122648621184425601vImage(B,A,A4,R,F) = relcomp(B,A,B,bNF_Gr(B,A,A4,F),relcomp(A,A,B,R,converse(B,A,bNF_Gr(B,A,A4,F)))) ) ) ).

% relInvImage_Gr
tff(fact_6275_relImage__Gr,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,A)),A4: set(A),F: fun(A,B)] :
      ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R),product_Sigma(A,A,A4,aTP_Lamp_rl(set(A),fun(A,set(A)),A4)))
     => ( bNF_Gr4221423524335903396lImage(A,B,R,F) = relcomp(B,A,B,converse(A,B,bNF_Gr(A,B,A4,F)),relcomp(A,A,B,R,bNF_Gr(A,B,A4,F))) ) ) ).

% relImage_Gr
tff(fact_6276_converse__iff,axiom,
    ! [A: $tType,B: $tType,A3: A,B2: B,R2: set(product_prod(B,A))] :
      ( aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),converse(B,A,R2))
    <=> aa(set(product_prod(B,A)),$o,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)),R2) ) ).

% converse_iff
tff(fact_6277_converse__empty,axiom,
    ! [B: $tType,A: $tType] : converse(B,A,bot_bot(set(product_prod(B,A)))) = bot_bot(set(product_prod(A,B))) ).

% converse_empty
tff(fact_6278_converse__UNIV,axiom,
    ! [B: $tType,A: $tType] : converse(B,A,top_top(set(product_prod(B,A)))) = top_top(set(product_prod(A,B))) ).

% converse_UNIV
tff(fact_6279_pair__set__inverse,axiom,
    ! [A: $tType,B: $tType,P: fun(B,fun(A,$o))] : converse(B,A,aa(fun(product_prod(B,A),$o),set(product_prod(B,A)),collect(product_prod(B,A)),aa(fun(B,fun(A,$o)),fun(product_prod(B,A),$o),product_case_prod(B,A,$o),P))) = aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aTP_Lamp_aei(fun(B,fun(A,$o)),fun(A,fun(B,$o)),P))) ).

% pair_set_inverse
tff(fact_6280_rtrancl__converseD,axiom,
    ! [A: $tType,X: A,Y: A,R2: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),transitive_rtrancl(A,converse(A,A,R2)))
     => aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),X)),transitive_rtrancl(A,R2)) ) ).

% rtrancl_converseD
tff(fact_6281_rtrancl__converseI,axiom,
    ! [A: $tType,Y: A,X: A,R2: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),X)),transitive_rtrancl(A,R2))
     => aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),transitive_rtrancl(A,converse(A,A,R2))) ) ).

% rtrancl_converseI
tff(fact_6282_in__listrel1__converse,axiom,
    ! [A: $tType,X: list(A),Y: list(A),R2: set(product_prod(A,A))] :
      ( aa(set(product_prod(list(A),list(A))),$o,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)),listrel1(A,converse(A,A,R2)))
    <=> aa(set(product_prod(list(A),list(A))),$o,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)),converse(list(A),list(A),listrel1(A,R2))) ) ).

% in_listrel1_converse
tff(fact_6283_converseI,axiom,
    ! [B: $tType,A: $tType,A3: A,B2: B,R2: set(product_prod(A,B))] :
      ( aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),R2)
     => aa(set(product_prod(B,A)),$o,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,R2)) ) ).

% converseI
tff(fact_6284_converseE,axiom,
    ! [A: $tType,B: $tType,Yx: product_prod(A,B),R2: set(product_prod(B,A))] :
      ( aa(set(product_prod(A,B)),$o,member(product_prod(A,B),Yx),converse(B,A,R2))
     => ~ ! [X2: B,Y2: A] :
            ( ( Yx = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Y2),X2) )
           => ~ aa(set(product_prod(B,A)),$o,member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X2),Y2)),R2) ) ) ).

% converseE
tff(fact_6285_converseD,axiom,
    ! [A: $tType,B: $tType,A3: A,B2: B,R2: set(product_prod(B,A))] :
      ( aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),converse(B,A,R2))
     => aa(set(product_prod(B,A)),$o,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)),R2) ) ).

% converseD
tff(fact_6286_converse_Osimps,axiom,
    ! [A: $tType,B: $tType,A1: A,A22: B,R2: set(product_prod(B,A))] :
      ( aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A1),A22)),converse(B,A,R2))
    <=> ? [A10: B,B6: A] :
          ( ( A1 = B6 )
          & ( A22 = A10 )
          & aa(set(product_prod(B,A)),$o,member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),A10),B6)),R2) ) ) ).

% converse.simps
tff(fact_6287_converse_Ocases,axiom,
    ! [A: $tType,B: $tType,A1: A,A22: B,R2: set(product_prod(B,A))] :
      ( aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A1),A22)),converse(B,A,R2))
     => aa(set(product_prod(B,A)),$o,member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),A22),A1)),R2) ) ).

% converse.cases
tff(fact_6288_trancl__converseD,axiom,
    ! [A: $tType,X: A,Y: A,R2: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),transitive_trancl(A,converse(A,A,R2)))
     => aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),converse(A,A,transitive_trancl(A,R2))) ) ).

% trancl_converseD
tff(fact_6289_trancl__converseI,axiom,
    ! [A: $tType,X: A,Y: A,R2: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),converse(A,A,transitive_trancl(A,R2)))
     => aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),transitive_trancl(A,converse(A,A,R2))) ) ).

% trancl_converseI
tff(fact_6290_converse__unfold,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(B,A))] : converse(B,A,R2) = aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aTP_Lamp_aoh(set(product_prod(B,A)),fun(A,fun(B,$o)),R2))) ).

% converse_unfold
tff(fact_6291_converse__Int,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(B,A)),S2: set(product_prod(B,A))] : converse(B,A,aa(set(product_prod(B,A)),set(product_prod(B,A)),aa(set(product_prod(B,A)),fun(set(product_prod(B,A)),set(product_prod(B,A))),inf_inf(set(product_prod(B,A))),R2),S2)) = 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))),converse(B,A,R2)),converse(B,A,S2)) ).

% converse_Int
tff(fact_6292_converse__Times,axiom,
    ! [A: $tType,B: $tType,A4: set(B),B3: set(A)] : converse(B,A,product_Sigma(B,A,A4,aTP_Lamp_ke(set(A),fun(B,set(A)),B3))) = product_Sigma(A,B,B3,aTP_Lamp_qz(set(B),fun(A,set(B)),A4)) ).

% converse_Times
tff(fact_6293_converse__Un,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(B,A)),S2: set(product_prod(B,A))] : converse(B,A,aa(set(product_prod(B,A)),set(product_prod(B,A)),aa(set(product_prod(B,A)),fun(set(product_prod(B,A)),set(product_prod(B,A))),sup_sup(set(product_prod(B,A))),R2),S2)) = 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))),converse(B,A,R2)),converse(B,A,S2)) ).

% converse_Un
tff(fact_6294_converse__UNION,axiom,
    ! [B: $tType,A: $tType,C: $tType,R2: fun(C,set(product_prod(B,A))),S: set(C)] : converse(B,A,aa(set(set(product_prod(B,A))),set(product_prod(B,A)),complete_Sup_Sup(set(product_prod(B,A))),aa(set(C),set(set(product_prod(B,A))),image2(C,set(product_prod(B,A)),R2),S))) = aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),aa(set(C),set(set(product_prod(A,B))),image2(C,set(product_prod(A,B)),aTP_Lamp_aoi(fun(C,set(product_prod(B,A))),fun(C,set(product_prod(A,B))),R2)),S)) ).

% converse_UNION
tff(fact_6295_converse__INTER,axiom,
    ! [B: $tType,A: $tType,C: $tType,R2: fun(C,set(product_prod(B,A))),S: set(C)] : converse(B,A,aa(set(set(product_prod(B,A))),set(product_prod(B,A)),complete_Inf_Inf(set(product_prod(B,A))),aa(set(C),set(set(product_prod(B,A))),image2(C,set(product_prod(B,A)),R2),S))) = aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Inf_Inf(set(product_prod(A,B))),aa(set(C),set(set(product_prod(A,B))),image2(C,set(product_prod(A,B)),aTP_Lamp_aoi(fun(C,set(product_prod(B,A))),fun(C,set(product_prod(A,B))),R2)),S)) ).

% converse_INTER
tff(fact_6296_Image__subset__eq,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(B,A)),A4: set(B),B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(B),set(A),image(B,A,R2),A4)),B3)
    <=> aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A4),aa(set(B),set(B),uminus_uminus(set(B)),aa(set(A),set(B),image(A,B,converse(B,A,R2)),aa(set(A),set(A),uminus_uminus(set(A)),B3)))) ) ).

% Image_subset_eq
tff(fact_6297_irrefl__tranclI,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),X: A] :
      ( ( aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),converse(A,A,R2)),transitive_rtrancl(A,R2)) = bot_bot(set(product_prod(A,A))) )
     => ~ aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),X)),transitive_trancl(A,R2)) ) ).

% irrefl_tranclI
tff(fact_6298_trans__wf__iff,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( trans(A,R2)
     => ( wf(A,R2)
      <=> ! [A10: A] : 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))),inf_inf(set(product_prod(A,A))),R2),product_Sigma(A,A,aa(set(A),set(A),image(A,A,converse(A,A,R2)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A10),bot_bot(set(A)))),aa(A,fun(A,set(A)),aTP_Lamp_aoj(set(product_prod(A,A)),fun(A,fun(A,set(A))),R2),A10)))) ) ) ).

% trans_wf_iff
tff(fact_6299_Image__INT__eq,axiom,
    ! [B: $tType,A: $tType,C: $tType,R2: set(product_prod(B,A)),A4: set(C),B3: fun(C,set(B))] :
      ( single_valued(A,B,converse(B,A,R2))
     => ( ( A4 != bot_bot(set(C)) )
       => ( aa(set(B),set(A),image(B,A,R2),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(C),set(set(B)),image2(C,set(B),B3),A4))) = aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(C),set(set(A)),image2(C,set(A),aa(fun(C,set(B)),fun(C,set(A)),aTP_Lamp_wu(set(product_prod(B,A)),fun(fun(C,set(B)),fun(C,set(A))),R2),B3)),A4)) ) ) ) ).

% Image_INT_eq
tff(fact_6300_trans__reflclI,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( trans(A,R2)
     => trans(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))),R2),id2(A))) ) ).

% trans_reflclI
tff(fact_6301_trans__Int,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),S2: set(product_prod(A,A))] :
      ( trans(A,R2)
     => ( trans(A,S2)
       => trans(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R2),S2)) ) ) ).

% trans_Int
tff(fact_6302_single__valued__inter1,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,B)),S: set(product_prod(A,B))] :
      ( single_valued(A,B,R)
     => single_valued(A,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))),R),S)) ) ).

% single_valued_inter1
tff(fact_6303_single__valued__inter2,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,B)),S: set(product_prod(A,B))] :
      ( single_valued(A,B,R)
     => single_valued(A,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))),S),R)) ) ).

% single_valued_inter2
tff(fact_6304_lexord__trans,axiom,
    ! [A: $tType,X: list(A),Y: list(A),R2: set(product_prod(A,A)),Z2: list(A)] :
      ( aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X),Y)),lexord(A,R2))
     => ( aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Y),Z2)),lexord(A,R2))
       => ( trans(A,R2)
         => aa(set(product_prod(list(A),list(A))),$o,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),Z2)),lexord(A,R2)) ) ) ) ).

% lexord_trans
tff(fact_6305_transD,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),X: A,Y: A,Z2: A] :
      ( trans(A,R2)
     => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R2)
       => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z2)),R2)
         => aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z2)),R2) ) ) ) ).

% transD
tff(fact_6306_transE,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),X: A,Y: A,Z2: A] :
      ( trans(A,R2)
     => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R2)
       => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z2)),R2)
         => aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z2)),R2) ) ) ) ).

% transE
tff(fact_6307_transI,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( ! [X2: A,Y2: A,Z3: A] :
          ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y2)),R2)
         => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),Z3)),R2)
           => aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Z3)),R2) ) )
     => trans(A,R2) ) ).

% transI
tff(fact_6308_trans__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( trans(A,R2)
    <=> ! [X4: A,Y3: A,Z4: A] :
          ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Y3)),R2)
         => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),Z4)),R2)
           => aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Z4)),R2) ) ) ) ).

% trans_def
tff(fact_6309_single__valuedD,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(A,B)),X: A,Y: B,Z2: B] :
      ( single_valued(A,B,R2)
     => ( aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y)),R2)
       => ( aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Z2)),R2)
         => ( Y = Z2 ) ) ) ) ).

% single_valuedD
tff(fact_6310_single__valuedI,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,B))] :
      ( ! [X2: A,Y2: B,Z3: B] :
          ( aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Y2)),R2)
         => ( aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Z3)),R2)
           => ( Y2 = Z3 ) ) )
     => single_valued(A,B,R2) ) ).

% single_valuedI
tff(fact_6311_single__valued__def,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(A,B))] :
      ( single_valued(A,B,R2)
    <=> ! [X4: A,Y3: B] :
          ( aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Y3)),R2)
         => ! [Z4: B] :
              ( aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Z4)),R2)
             => ( Y3 = Z4 ) ) ) ) ).

% single_valued_def
tff(fact_6312_single__valued__empty,axiom,
    ! [B: $tType,A: $tType] : single_valued(A,B,bot_bot(set(product_prod(A,B)))) ).

% single_valued_empty
tff(fact_6313_trans__empty,axiom,
    ! [A: $tType] : trans(A,bot_bot(set(product_prod(A,A)))) ).

% trans_empty
tff(fact_6314_lenlex__trans,axiom,
    ! [A: $tType,X: list(A),Y: list(A),R2: set(product_prod(A,A)),Z2: list(A)] :
      ( aa(set(product_prod(list(A),list(A))),$o,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)),lenlex(A,R2))
     => ( aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Y),Z2)),lenlex(A,R2))
       => ( trans(A,R2)
         => aa(set(product_prod(list(A),list(A))),$o,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),Z2)),lenlex(A,R2)) ) ) ) ).

% lenlex_trans
tff(fact_6315_trans__Restr,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A4: set(A)] :
      ( trans(A,R2)
     => trans(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R2),product_Sigma(A,A,A4,aTP_Lamp_rl(set(A),fun(A,set(A)),A4)))) ) ).

% trans_Restr
tff(fact_6316_single__valued__confluent,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),X: A,Y: A,Z2: A] :
      ( single_valued(A,A,R2)
     => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),transitive_rtrancl(A,R2))
       => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z2)),transitive_rtrancl(A,R2))
         => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z2)),transitive_rtrancl(A,R2))
            | aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z2),Y)),transitive_rtrancl(A,R2)) ) ) ) ) ).

% single_valued_confluent
tff(fact_6317_under__incr,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A3: A,B2: A] :
      ( trans(A,R2)
     => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R2)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(A,set(A),order_under(A,R2),A3)),aa(A,set(A),order_under(A,R2),B2)) ) ) ).

% under_incr
tff(fact_6318_trans__singleton,axiom,
    ! [A: $tType,A3: A] : trans(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(product_prod(A,A),fun(set(product_prod(A,A)),set(product_prod(A,A))),insert2(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),A3)),bot_bot(set(product_prod(A,A))))) ).

% trans_singleton
tff(fact_6319_trans__rtrancl__eq__reflcl,axiom,
    ! [A: $tType,A4: set(product_prod(A,A))] :
      ( trans(A,A4)
     => ( transitive_rtrancl(A,A4) = 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))),A4),id2(A)) ) ) ).

% trans_rtrancl_eq_reflcl
tff(fact_6320_trans__join,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( trans(A,R2)
    <=> ! [X4: product_prod(A,A)] :
          ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),X4),R2)
         => aa(product_prod(A,A),$o,aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),aTP_Lamp_aol(set(product_prod(A,A)),fun(A,fun(A,$o)),R2)),X4) ) ) ).

% trans_join
tff(fact_6321_underS__incr,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A3: A,B2: A] :
      ( trans(A,R2)
     => ( antisym(A,R2)
       => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R2)
         => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),order_underS(A,R2,A3)),order_underS(A,R2,B2)) ) ) ) ).

% underS_incr
tff(fact_6322_Image__Int__eq,axiom,
    ! [A: $tType,B: $tType,R: set(product_prod(B,A)),A4: set(B),B3: set(B)] :
      ( single_valued(A,B,converse(B,A,R))
     => ( aa(set(B),set(A),image(B,A,R),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),B3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(B),set(A),image(B,A,R),A4)),aa(set(B),set(A),image(B,A,R),B3)) ) ) ).

% Image_Int_eq
tff(fact_6323_wf__finite__segments,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( irrefl(A,R2)
     => ( trans(A,R2)
       => ( ! [X2: A] : aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(A,fun(A,$o),aTP_Lamp_amk(set(product_prod(A,A)),fun(A,fun(A,$o)),R2),X2)))
         => wf(A,R2) ) ) ) ).

% wf_finite_segments
tff(fact_6324_ord__to__filter__compat,axiom,
    ! [A: $tType,R0: set(product_prod(A,A))] : bNF_Wellorder_compat(set(product_prod(A,A)),set(A),aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),aa(set(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)))),set(product_prod(set(product_prod(A,A)),set(product_prod(A,A))))),inf_inf(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A))))),bNF_We4044943003108391690rdLess(A,A)),product_Sigma(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(set(product_prod(A,A))),set(set(product_prod(A,A))),image(set(product_prod(A,A)),set(product_prod(A,A)),converse(set(product_prod(A,A)),set(product_prod(A,A)),bNF_We4044943003108391690rdLess(A,A))),aa(set(set(product_prod(A,A))),set(set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(set(product_prod(A,A))),set(set(product_prod(A,A)))),insert2(set(product_prod(A,A))),R0),bot_bot(set(set(product_prod(A,A)))))),aTP_Lamp_aom(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(set(product_prod(A,A)))),R0))),bNF_We413866401316099525erIncl(A,R0),bNF_We8469521843155493636filter(A,R0)) ).

% ord_to_filter_compat
tff(fact_6325_rel__filter_Ocases,axiom,
    ! [A: $tType,B: $tType,R: fun(A,fun(B,$o)),F4: filter(A),G4: filter(B)] :
      ( aa(filter(B),$o,aa(filter(A),fun(filter(B),$o),rel_filter(A,B,R),F4),G4)
     => ~ ! [Z9: filter(product_prod(A,B))] :
            ( eventually(product_prod(A,B),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),R),Z9)
           => ( ( aa(filter(product_prod(A,B)),filter(A),aa(fun(product_prod(A,B),A),fun(filter(product_prod(A,B)),filter(A)),map_filter_on(product_prod(A,B),A,aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),R))),product_fst(A,B)),Z9) = F4 )
             => ( aa(filter(product_prod(A,B)),filter(B),aa(fun(product_prod(A,B),B),fun(filter(product_prod(A,B)),filter(B)),map_filter_on(product_prod(A,B),B,aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),R))),product_snd(A,B)),Z9) != G4 ) ) ) ) ).

% rel_filter.cases
tff(fact_6326_bot__filter__parametric,axiom,
    ! [A: $tType,B: $tType,A4: fun(A,fun(B,$o))] : aa(filter(B),$o,aa(filter(A),fun(filter(B),$o),rel_filter(A,B,A4),bot_bot(filter(A))),bot_bot(filter(B))) ).

% bot_filter_parametric
tff(fact_6327_sup__filter__parametric,axiom,
    ! [A: $tType,B: $tType,A4: fun(A,fun(B,$o))] : aa(fun(filter(B),fun(filter(B),filter(B))),$o,aa(fun(filter(A),fun(filter(A),filter(A))),fun(fun(filter(B),fun(filter(B),filter(B))),$o),bNF_rel_fun(filter(A),filter(B),fun(filter(A),filter(A)),fun(filter(B),filter(B)),rel_filter(A,B,A4),bNF_rel_fun(filter(A),filter(B),filter(A),filter(B),rel_filter(A,B,A4),rel_filter(A,B,A4))),sup_sup(filter(A))),sup_sup(filter(B))) ).

% sup_filter_parametric
tff(fact_6328_compat__def,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(A,A)),R3: set(product_prod(B,B)),F: fun(A,B)] :
      ( bNF_Wellorder_compat(A,B,R2,R3,F)
    <=> ! [A10: A,B6: A] :
          ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A10),B6)),R2)
         => aa(set(product_prod(B,B)),$o,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,F,A10)),aa(A,B,F,B6))),R3) ) ) ).

% compat_def
tff(fact_6329_rel__filter_Ointros,axiom,
    ! [A: $tType,B: $tType,R: fun(A,fun(B,$o)),Z5: filter(product_prod(A,B)),F4: filter(A),G4: filter(B)] :
      ( eventually(product_prod(A,B),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),R),Z5)
     => ( ( aa(filter(product_prod(A,B)),filter(A),aa(fun(product_prod(A,B),A),fun(filter(product_prod(A,B)),filter(A)),map_filter_on(product_prod(A,B),A,aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),R))),product_fst(A,B)),Z5) = F4 )
       => ( ( aa(filter(product_prod(A,B)),filter(B),aa(fun(product_prod(A,B),B),fun(filter(product_prod(A,B)),filter(B)),map_filter_on(product_prod(A,B),B,aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),R))),product_snd(A,B)),Z5) = G4 )
         => aa(filter(B),$o,aa(filter(A),fun(filter(B),$o),rel_filter(A,B,R),F4),G4) ) ) ) ).

% rel_filter.intros
tff(fact_6330_rel__filter_Osimps,axiom,
    ! [A: $tType,B: $tType,R: fun(A,fun(B,$o)),F4: filter(A),G4: filter(B)] :
      ( aa(filter(B),$o,aa(filter(A),fun(filter(B),$o),rel_filter(A,B,R),F4),G4)
    <=> ? [Z8: filter(product_prod(A,B))] :
          ( eventually(product_prod(A,B),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),R),Z8)
          & ( aa(filter(product_prod(A,B)),filter(A),aa(fun(product_prod(A,B),A),fun(filter(product_prod(A,B)),filter(A)),map_filter_on(product_prod(A,B),A,aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),R))),product_fst(A,B)),Z8) = F4 )
          & ( aa(filter(product_prod(A,B)),filter(B),aa(fun(product_prod(A,B),B),fun(filter(product_prod(A,B)),filter(B)),map_filter_on(product_prod(A,B),B,aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),R))),product_snd(A,B)),Z8) = G4 ) ) ) ).

% rel_filter.simps
tff(fact_6331_ord__to__filter__def,axiom,
    ! [A: $tType,R0: set(product_prod(A,A)),R2: set(product_prod(A,A))] : aa(set(product_prod(A,A)),set(A),bNF_We8469521843155493636filter(A,R0),R2) = aa(set(A),set(A),image2(A,A,fChoice(fun(A,A),bNF_Wellorder_embed(A,A,R2,R0))),field2(A,R2)) ).

% ord_to_filter_def
tff(fact_6332_arg__min__SOME__Min,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [S: set(A),F: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),S)
         => ( lattic7623131987881927897min_on(A,B,F,S) = fChoice(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_aon(set(A),fun(fun(A,B),fun(A,$o)),S),F)) ) ) ) ).

% arg_min_SOME_Min
tff(fact_6333_some__equality,axiom,
    ! [A: $tType,P: fun(A,$o),A3: A] :
      ( aa(A,$o,P,A3)
     => ( ! [X2: A] :
            ( aa(A,$o,P,X2)
           => ( X2 = A3 ) )
       => ( fChoice(A,P) = A3 ) ) ) ).

% some_equality
tff(fact_6334_some__eq__trivial,axiom,
    ! [A: $tType,X: A] : fChoice(A,aTP_Lamp_cf(A,fun(A,$o),X)) = X ).

% some_eq_trivial
tff(fact_6335_some__sym__eq__trivial,axiom,
    ! [A: $tType,X: A] : fChoice(A,aa(A,fun(A,$o),fequal(A),X)) = X ).

% some_sym_eq_trivial
tff(fact_6336_Eps__case__prod__eq,axiom,
    ! [A: $tType,B: $tType,X: A,Y: B] : fChoice(product_prod(A,B),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aa(B,fun(A,fun(B,$o)),aTP_Lamp_qt(A,fun(B,fun(A,fun(B,$o))),X),Y))) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y) ).

% Eps_case_prod_eq
tff(fact_6337_some__insert__self,axiom,
    ! [A: $tType,S: set(A)] :
      ( ( S != bot_bot(set(A)) )
     => ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),fChoice(A,aa(set(A),fun(A,$o),aTP_Lamp_ag(set(A),fun(A,$o)),S))),S) = S ) ) ).

% some_insert_self
tff(fact_6338_verit__sko__ex_H,axiom,
    ! [A: $tType,P: fun(A,$o),A4: $o] :
      ( ( aa(A,$o,P,fChoice(A,P))
      <=> (A4) )
     => ( ? [X_12: A] : aa(A,$o,P,X_12)
      <=> (A4) ) ) ).

% verit_sko_ex'
tff(fact_6339_verit__sko__forall,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( ! [X_12: A] : aa(A,$o,P,X_12)
    <=> aa(A,$o,P,fChoice(A,aTP_Lamp_aj(fun(A,$o),fun(A,$o),P))) ) ).

% verit_sko_forall
tff(fact_6340_verit__sko__forall_H,axiom,
    ! [A: $tType,P: fun(A,$o),A4: $o] :
      ( ( aa(A,$o,P,fChoice(A,aTP_Lamp_aj(fun(A,$o),fun(A,$o),P)))
      <=> (A4) )
     => ( ! [X_12: A] : aa(A,$o,P,X_12)
      <=> (A4) ) ) ).

% verit_sko_forall'
tff(fact_6341_verit__sko__forall_H_H,axiom,
    ! [A: $tType,B3: A,A4: A,P: fun(A,$o)] :
      ( ( B3 = A4 )
     => ( ( fChoice(A,P) = A4 )
      <=> ( fChoice(A,P) = B3 ) ) ) ).

% verit_sko_forall''
tff(fact_6342_verit__sko__ex__indirect,axiom,
    ! [A: $tType,X: A,P: fun(A,$o)] :
      ( ( X = fChoice(A,P) )
     => ( ? [X_12: A] : aa(A,$o,P,X_12)
      <=> aa(A,$o,P,X) ) ) ).

% verit_sko_ex_indirect
tff(fact_6343_verit__sko__ex__indirect2,axiom,
    ! [A: $tType,X: A,P: fun(A,$o),P4: fun(A,$o)] :
      ( ( X = fChoice(A,P) )
     => ( ! [X2: A] :
            ( aa(A,$o,P,X2)
          <=> aa(A,$o,P4,X2) )
       => ( ? [X_12: A] : aa(A,$o,P4,X_12)
        <=> aa(A,$o,P,X) ) ) ) ).

% verit_sko_ex_indirect2
tff(fact_6344_verit__sko__forall__indirect,axiom,
    ! [A: $tType,X: A,P: fun(A,$o)] :
      ( ( X = fChoice(A,aTP_Lamp_aj(fun(A,$o),fun(A,$o),P)) )
     => ( ! [X_12: A] : aa(A,$o,P,X_12)
      <=> aa(A,$o,P,X) ) ) ).

% verit_sko_forall_indirect
tff(fact_6345_verit__sko__forall__indirect2,axiom,
    ! [A: $tType,X: A,P: fun(A,$o),P4: fun(A,$o)] :
      ( ( X = fChoice(A,aTP_Lamp_aj(fun(A,$o),fun(A,$o),P)) )
     => ( ! [X2: A] :
            ( aa(A,$o,P,X2)
          <=> aa(A,$o,P4,X2) )
       => ( ! [X_12: A] : aa(A,$o,P4,X_12)
        <=> aa(A,$o,P,X) ) ) ) ).

% verit_sko_forall_indirect2
tff(fact_6346_someI2,axiom,
    ! [A: $tType,P: fun(A,$o),A3: A,Q: fun(A,$o)] :
      ( aa(A,$o,P,A3)
     => ( ! [X2: A] :
            ( aa(A,$o,P,X2)
           => aa(A,$o,Q,X2) )
       => aa(A,$o,Q,fChoice(A,P)) ) ) ).

% someI2
tff(fact_6347_someI__ex,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( ? [X_13: A] : aa(A,$o,P,X_13)
     => aa(A,$o,P,fChoice(A,P)) ) ).

% someI_ex
tff(fact_6348_someI2__ex,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] :
      ( ? [X_13: A] : aa(A,$o,P,X_13)
     => ( ! [X2: A] :
            ( aa(A,$o,P,X2)
           => aa(A,$o,Q,X2) )
       => aa(A,$o,Q,fChoice(A,P)) ) ) ).

% someI2_ex
tff(fact_6349_someI2__bex,axiom,
    ! [A: $tType,A4: set(A),P: fun(A,$o),Q: fun(A,$o)] :
      ( ? [X3: A] :
          ( aa(set(A),$o,member(A,X3),A4)
          & aa(A,$o,P,X3) )
     => ( ! [X2: A] :
            ( ( aa(set(A),$o,member(A,X2),A4)
              & aa(A,$o,P,X2) )
           => aa(A,$o,Q,X2) )
       => aa(A,$o,Q,fChoice(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_bi(set(A),fun(fun(A,$o),fun(A,$o)),A4),P))) ) ) ).

% someI2_bex
tff(fact_6350_some__eq__ex,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( aa(A,$o,P,fChoice(A,P))
    <=> ? [X_12: A] : aa(A,$o,P,X_12) ) ).

% some_eq_ex
tff(fact_6351_some1__equality,axiom,
    ! [A: $tType,P: fun(A,$o),A3: A] :
      ( ? [X3: A] :
          ( aa(A,$o,P,X3)
          & ! [Y2: A] :
              ( aa(A,$o,P,Y2)
             => ( Y2 = X3 ) ) )
     => ( aa(A,$o,P,A3)
       => ( fChoice(A,P) = A3 ) ) ) ).

% some1_equality
tff(fact_6352_some__in__eq,axiom,
    ! [A: $tType,A4: set(A)] :
      ( aa(set(A),$o,member(A,fChoice(A,aa(set(A),fun(A,$o),aTP_Lamp_ag(set(A),fun(A,$o)),A4))),A4)
    <=> ( A4 != bot_bot(set(A)) ) ) ).

% some_in_eq
tff(fact_6353_some__elem,axiom,
    ! [A: $tType,S: set(A)] :
      ( ( S != bot_bot(set(A)) )
     => aa(set(A),$o,member(A,fChoice(A,aa(set(A),fun(A,$o),aTP_Lamp_ag(set(A),fun(A,$o)),S))),S) ) ).

% some_elem
tff(fact_6354_split__paired__Eps,axiom,
    ! [B: $tType,A: $tType,P: fun(product_prod(A,B),$o)] : fChoice(product_prod(A,B),P) = fChoice(product_prod(A,B),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aTP_Lamp_aoo(fun(product_prod(A,B),$o),fun(A,fun(B,$o)),P))) ).

% split_paired_Eps
tff(fact_6355_card__of__def,axiom,
    ! [A: $tType,A4: set(A)] : bNF_Ca6860139660246222851ard_of(A,A4) = fChoice(set(product_prod(A,A)),bNF_Ca8970107618336181345der_on(A,A4)) ).

% card_of_def
tff(fact_6356_some__theI,axiom,
    ! [B: $tType,A: $tType,P: fun(A,fun(B,$o))] :
      ( ? [A11: A,X_13: B] : aa(B,$o,aa(A,fun(B,$o),P,A11),X_13)
     => ( ! [B15: B,B24: B] :
            ( ? [A6: A] : aa(B,$o,aa(A,fun(B,$o),P,A6),B15)
           => ( ? [A6: A] : aa(B,$o,aa(A,fun(B,$o),P,A6),B24)
             => ( B15 = B24 ) ) )
       => aa(B,$o,aa(A,fun(B,$o),P,fChoice(A,aTP_Lamp_afv(fun(A,fun(B,$o)),fun(A,$o),P))),the(B,aTP_Lamp_aop(fun(A,fun(B,$o)),fun(B,$o),P))) ) ) ).

% some_theI
tff(fact_6357_equiv__Eps__preserves,axiom,
    ! [A: $tType,A4: set(A),R2: set(product_prod(A,A)),X5: set(A)] :
      ( equiv_equiv(A,A4,R2)
     => ( aa(set(set(A)),$o,member(set(A),X5),equiv_quotient(A,A4,R2))
       => aa(set(A),$o,member(A,fChoice(A,aa(set(A),fun(A,$o),aTP_Lamp_ag(set(A),fun(A,$o)),X5))),A4) ) ) ).

% equiv_Eps_preserves
tff(fact_6358_equiv__Eps__in,axiom,
    ! [A: $tType,A4: set(A),R2: set(product_prod(A,A)),X5: set(A)] :
      ( equiv_equiv(A,A4,R2)
     => ( aa(set(set(A)),$o,member(set(A),X5),equiv_quotient(A,A4,R2))
       => aa(set(A),$o,member(A,fChoice(A,aa(set(A),fun(A,$o),aTP_Lamp_ag(set(A),fun(A,$o)),X5))),X5) ) ) ).

% equiv_Eps_in
tff(fact_6359_cardSuc__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : bNF_Ca8387033319878233205ardSuc(A,R2) = fChoice(set(product_prod(set(A),set(A))),bNF_Ca6246979054910435723ardSuc(A,R2)) ).

% cardSuc_def
tff(fact_6360_arg__min__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F: fun(A,B),P: fun(A,$o)] : lattices_ord_arg_min(A,B,F,P) = fChoice(A,lattic501386751177426532rg_min(A,B,F,P)) ) ).

% arg_min_def
tff(fact_6361_Eps__case__prod,axiom,
    ! [B: $tType,A: $tType,P: fun(A,fun(B,$o))] : fChoice(product_prod(A,B),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),P)) = fChoice(product_prod(A,B),aTP_Lamp_vy(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),P)) ).

% Eps_case_prod
tff(fact_6362_toCard__def,axiom,
    ! [A: $tType,B: $tType,A4: set(A),R2: set(product_prod(B,B))] : bNF_Greatest_toCard(A,B,A4,R2) = fChoice(fun(A,B),bNF_Gr1419584066657907630d_pred(A,B,A4,R2)) ).

% toCard_def
tff(fact_6363_proj__Eps,axiom,
    ! [A: $tType,A4: set(A),R2: set(product_prod(A,A)),X5: set(A)] :
      ( equiv_equiv(A,A4,R2)
     => ( aa(set(set(A)),$o,member(set(A),X5),equiv_quotient(A,A4,R2))
       => ( equiv_proj(A,A,R2,fChoice(A,aa(set(A),fun(A,$o),aTP_Lamp_ag(set(A),fun(A,$o)),X5))) = X5 ) ) ) ).

% proj_Eps
tff(fact_6364_fromCard__def,axiom,
    ! [A: $tType,B: $tType,A4: set(A),R2: set(product_prod(B,B)),K: B] : bNF_Gr5436034075474128252omCard(A,B,A4,R2,K) = fChoice(A,aa(B,fun(A,$o),aa(set(product_prod(B,B)),fun(B,fun(A,$o)),aTP_Lamp_aoq(set(A),fun(set(product_prod(B,B)),fun(B,fun(A,$o))),A4),R2),K)) ).

% fromCard_def
tff(fact_6365_Eps__Opt__def,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      eps_Opt(A,P) = $ite(
        ? [X8: A] : aa(A,$o,P,X8),
        aa(A,option(A),some(A),fChoice(A,P)),
        none(A) ) ).

% Eps_Opt_def
tff(fact_6366_fun__of__rel__def,axiom,
    ! [A: $tType,B: $tType,R: set(product_prod(B,A)),X: B] : fun_of_rel(B,A,R,X) = fChoice(A,aa(B,fun(A,$o),aTP_Lamp_wv(set(product_prod(B,A)),fun(B,fun(A,$o)),R),X)) ).

% fun_of_rel_def
tff(fact_6367_to__nat__def,axiom,
    ! [A: $tType] :
      ( countable(A)
     => ( to_nat(A) = fChoice(fun(A,nat),aTP_Lamp_aor(fun(A,nat),$o)) ) ) ).

% to_nat_def
tff(fact_6368_inj__to__nat,axiom,
    ! [A: $tType] :
      ( countable(A)
     => inj_on(A,nat,to_nat(A),top_top(set(A))) ) ).

% inj_to_nat
tff(fact_6369_pred__on_Onot__maxchain__Some,axiom,
    ! [A: $tType,A4: set(A),P: fun(A,fun(A,$o)),C3: set(A)] :
      ( aa(set(A),$o,pred_chain(A,A4,P),C3)
     => ( ~ pred_maxchain(A,A4,P,C3)
       => ( aa(set(A),$o,pred_chain(A,A4,P),fChoice(set(A),aa(set(A),fun(set(A),$o),aa(fun(A,fun(A,$o)),fun(set(A),fun(set(A),$o)),aTP_Lamp_aos(set(A),fun(fun(A,fun(A,$o)),fun(set(A),fun(set(A),$o))),A4),P),C3)))
          & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),C3),fChoice(set(A),aa(set(A),fun(set(A),$o),aa(fun(A,fun(A,$o)),fun(set(A),fun(set(A),$o)),aTP_Lamp_aos(set(A),fun(fun(A,fun(A,$o)),fun(set(A),fun(set(A),$o))),A4),P),C3))) ) ) ) ).

% pred_on.not_maxchain_Some
tff(fact_6370_map__comp__def,axiom,
    ! [B: $tType,C: $tType,A: $tType,F: fun(C,option(B)),G: fun(A,option(C)),X3: A] : map_comp(C,B,A,F,G,X3) = case_option(option(B),C,none(B),F,aa(A,option(C),G,X3)) ).

% map_comp_def
tff(fact_6371_map__comp__empty_I1_J,axiom,
    ! [A: $tType,C: $tType,B: $tType,M: fun(C,option(B)),X3: A] : map_comp(C,B,A,M,aTP_Lamp_tp(A,option(C)),X3) = none(B) ).

% map_comp_empty(1)
tff(fact_6372_map__comp__empty_I2_J,axiom,
    ! [A: $tType,C: $tType,B: $tType,M: fun(A,option(C)),X3: A] : map_comp(C,B,A,aTP_Lamp_aot(C,option(B)),M,X3) = none(B) ).

% map_comp_empty(2)
tff(fact_6373_subset_Onot__maxchain__Some,axiom,
    ! [A: $tType,A4: set(set(A)),C3: set(set(A))] :
      ( aa(set(set(A)),$o,pred_chain(set(A),A4,ord_less(set(A))),C3)
     => ( ~ pred_maxchain(set(A),A4,ord_less(set(A)),C3)
       => ( aa(set(set(A)),$o,pred_chain(set(A),A4,ord_less(set(A))),fChoice(set(set(A)),aa(set(set(A)),fun(set(set(A)),$o),aTP_Lamp_aou(set(set(A)),fun(set(set(A)),fun(set(set(A)),$o)),A4),C3)))
          & aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less(set(set(A))),C3),fChoice(set(set(A)),aa(set(set(A)),fun(set(set(A)),$o),aTP_Lamp_aou(set(set(A)),fun(set(set(A)),fun(set(set(A)),$o)),A4),C3))) ) ) ) ).

% subset.not_maxchain_Some
tff(fact_6374_pred__on_Osuc__def,axiom,
    ! [A: $tType,A4: set(A),P: fun(A,fun(A,$o)),C3: set(A)] :
      pred_suc(A,A4,P,C3) = $ite(
        ( ~ aa(set(A),$o,pred_chain(A,A4,P),C3)
        | pred_maxchain(A,A4,P,C3) ),
        C3,
        fChoice(set(A),aa(set(A),fun(set(A),$o),aa(fun(A,fun(A,$o)),fun(set(A),fun(set(A),$o)),aTP_Lamp_aos(set(A),fun(fun(A,fun(A,$o)),fun(set(A),fun(set(A),$o))),A4),P),C3)) ) ).

% pred_on.suc_def
tff(fact_6375_univ__def,axiom,
    ! [A: $tType,B: $tType,F: fun(B,A),X5: set(B)] : bNF_Greatest_univ(B,A,F,X5) = aa(B,A,F,fChoice(B,aTP_Lamp_aks(set(B),fun(B,$o),X5))) ).

% univ_def
tff(fact_6376_subset_Osuc__def,axiom,
    ! [A: $tType,A4: set(set(A)),C3: set(set(A))] :
      pred_suc(set(A),A4,ord_less(set(A)),C3) = $ite(
        ( ~ aa(set(set(A)),$o,pred_chain(set(A),A4,ord_less(set(A))),C3)
        | pred_maxchain(set(A),A4,ord_less(set(A)),C3) ),
        C3,
        fChoice(set(set(A)),aa(set(set(A)),fun(set(set(A)),$o),aTP_Lamp_aou(set(set(A)),fun(set(set(A)),fun(set(set(A)),$o)),A4),C3)) ) ).

% subset.suc_def
tff(fact_6377_flip__pred,axiom,
    ! [A: $tType,B: $tType,A4: set(product_prod(A,B)),R: fun(B,fun(A,$o))] :
      ( aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),A4),aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),conversep(B,A,R))))
     => aa(set(product_prod(B,A)),$o,aa(set(product_prod(B,A)),fun(set(product_prod(B,A)),$o),ord_less_eq(set(product_prod(B,A))),aa(set(product_prod(A,B)),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_vx(A,fun(B,product_prod(B,A))))),A4)),aa(fun(product_prod(B,A),$o),set(product_prod(B,A)),collect(product_prod(B,A)),aa(fun(B,fun(A,$o)),fun(product_prod(B,A),$o),product_case_prod(B,A,$o),R))) ) ).

% flip_pred
tff(fact_6378_prod__set__simps_I2_J,axiom,
    ! [B: $tType,A: $tType,X: B,Y: A] : basic_snds(B,A,aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X),Y)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y),bot_bot(set(A))) ).

% prod_set_simps(2)
tff(fact_6379_conversep__noteq,axiom,
    ! [A: $tType,X3: A,Xa3: A] :
      ( aa(A,$o,aa(A,fun(A,$o),conversep(A,A,aTP_Lamp_za(A,fun(A,$o))),X3),Xa3)
    <=> ( X3 != Xa3 ) ) ).

% conversep_noteq
tff(fact_6380_conversep__converse__eq,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(B,A)),X3: A,Xa3: B] :
      ( aa(B,$o,aa(A,fun(B,$o),conversep(B,A,aTP_Lamp_wv(set(product_prod(B,A)),fun(B,fun(A,$o)),R2)),X3),Xa3)
    <=> aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Xa3)),converse(B,A,R2)) ) ).

% conversep_converse_eq
tff(fact_6381_converse__meet,axiom,
    ! [A: $tType,B: $tType,R2: fun(B,fun(A,$o)),S2: fun(B,fun(A,$o))] : conversep(B,A,aa(fun(B,fun(A,$o)),fun(B,fun(A,$o)),aa(fun(B,fun(A,$o)),fun(fun(B,fun(A,$o)),fun(B,fun(A,$o))),inf_inf(fun(B,fun(A,$o))),R2),S2)) = aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),inf_inf(fun(A,fun(B,$o))),conversep(B,A,R2)),conversep(B,A,S2)) ).

% converse_meet
tff(fact_6382_converse__join,axiom,
    ! [A: $tType,B: $tType,R2: fun(B,fun(A,$o)),S2: fun(B,fun(A,$o))] : conversep(B,A,aa(fun(B,fun(A,$o)),fun(B,fun(A,$o)),aa(fun(B,fun(A,$o)),fun(fun(B,fun(A,$o)),fun(B,fun(A,$o))),sup_sup(fun(B,fun(A,$o))),R2),S2)) = aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),sup_sup(fun(A,fun(B,$o))),conversep(B,A,R2)),conversep(B,A,S2)) ).

% converse_join
tff(fact_6383_converse__def,axiom,
    ! [B: $tType,A: $tType,X3: set(product_prod(A,B))] : converse(A,B,X3) = aa(fun(product_prod(B,A),$o),set(product_prod(B,A)),collect(product_prod(B,A)),aa(fun(B,fun(A,$o)),fun(product_prod(B,A),$o),product_case_prod(B,A,$o),conversep(A,B,aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_as(set(product_prod(A,B)),fun(A,fun(B,$o))),X3)))) ).

% converse_def
tff(fact_6384_prod__set__defs_I2_J,axiom,
    ! [A: $tType,B: $tType,X3: product_prod(A,B)] : basic_snds(A,B,X3) = aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),aa(product_prod(A,B),B,product_snd(A,B),X3)),bot_bot(set(B))) ).

% prod_set_defs(2)
tff(fact_6385_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),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))) ).

% prod_set_simps(1)
tff(fact_6386_set__encode__empty,axiom,
    nat_set_encode(bot_bot(set(nat))) = zero_zero(nat) ).

% set_encode_empty
tff(fact_6387_prod__set__defs_I1_J,axiom,
    ! [B: $tType,A: $tType,X3: product_prod(A,B)] : basic_fsts(A,B,X3) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),aa(product_prod(A,B),A,product_fst(A,B),X3)),bot_bot(set(A))) ).

% prod_set_defs(1)
tff(fact_6388_prod_Oin__rel,axiom,
    ! [A: $tType,C: $tType,D: $tType,B: $tType,R1: fun(A,fun(B,$o)),R22: fun(C,fun(D,$o)),A3: product_prod(A,C),B2: product_prod(B,D)] :
      ( aa(product_prod(B,D),$o,aa(product_prod(A,C),fun(product_prod(B,D),$o),basic_rel_prod(A,B,C,D,R1,R22),A3),B2)
    <=> ? [Z4: product_prod(product_prod(A,B),product_prod(C,D))] :
          ( aa(set(product_prod(product_prod(A,B),product_prod(C,D))),$o,member(product_prod(product_prod(A,B),product_prod(C,D)),Z4),aa(fun(product_prod(product_prod(A,B),product_prod(C,D)),$o),set(product_prod(product_prod(A,B),product_prod(C,D))),collect(product_prod(product_prod(A,B),product_prod(C,D))),aa(fun(C,fun(D,$o)),fun(product_prod(product_prod(A,B),product_prod(C,D)),$o),aTP_Lamp_aov(fun(A,fun(B,$o)),fun(fun(C,fun(D,$o)),fun(product_prod(product_prod(A,B),product_prod(C,D)),$o)),R1),R22)))
          & ( aa(product_prod(product_prod(A,B),product_prod(C,D)),product_prod(A,C),product_map_prod(product_prod(A,B),A,product_prod(C,D),C,product_fst(A,B),product_fst(C,D)),Z4) = A3 )
          & ( aa(product_prod(product_prod(A,B),product_prod(C,D)),product_prod(B,D),product_map_prod(product_prod(A,B),B,product_prod(C,D),D,product_snd(A,B),product_snd(C,D)),Z4) = B2 ) ) ) ).

% prod.in_rel
tff(fact_6389_single__valuedp__single__valued__eq,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,B))] :
      ( single_valuedp(A,B,aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_as(set(product_prod(A,B)),fun(A,fun(B,$o))),R2))
    <=> single_valued(A,B,R2) ) ).

% single_valuedp_single_valued_eq
tff(fact_6390_rel__prod__inject,axiom,
    ! [B: $tType,A: $tType,C: $tType,D: $tType,R1: fun(A,fun(B,$o)),R22: fun(C,fun(D,$o)),A3: A,B2: C,C2: B,D3: D] :
      ( aa(product_prod(B,D),$o,aa(product_prod(A,C),fun(product_prod(B,D),$o),basic_rel_prod(A,B,C,D,R1,R22),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),A3),B2)),aa(D,product_prod(B,D),aa(B,fun(D,product_prod(B,D)),product_Pair(B,D),C2),D3))
    <=> ( aa(B,$o,aa(A,fun(B,$o),R1,A3),C2)
        & aa(D,$o,aa(C,fun(D,$o),R22,B2),D3) ) ) ).

% rel_prod_inject
tff(fact_6391_prod_Orel__map_I2_J,axiom,
    ! [A: $tType,C: $tType,B: $tType,D: $tType,F3: $tType,E: $tType,S1a: fun(A,fun(B,$o)),S2a: fun(C,fun(D,$o)),X: product_prod(A,C),G1: fun(E,B),G22: fun(F3,D),Y: product_prod(E,F3)] :
      ( aa(product_prod(B,D),$o,aa(product_prod(A,C),fun(product_prod(B,D),$o),basic_rel_prod(A,B,C,D,S1a,S2a),X),aa(product_prod(E,F3),product_prod(B,D),product_map_prod(E,B,F3,D,G1,G22),Y))
    <=> aa(product_prod(E,F3),$o,aa(product_prod(A,C),fun(product_prod(E,F3),$o),basic_rel_prod(A,E,C,F3,aa(fun(E,B),fun(A,fun(E,$o)),aTP_Lamp_aow(fun(A,fun(B,$o)),fun(fun(E,B),fun(A,fun(E,$o))),S1a),G1),aa(fun(F3,D),fun(C,fun(F3,$o)),aTP_Lamp_aox(fun(C,fun(D,$o)),fun(fun(F3,D),fun(C,fun(F3,$o))),S2a),G22)),X),Y) ) ).

% prod.rel_map(2)
tff(fact_6392_prod_Orel__map_I1_J,axiom,
    ! [E: $tType,F3: $tType,A: $tType,C: $tType,D: $tType,B: $tType,S1b: fun(A,fun(B,$o)),S2b: fun(C,fun(D,$o)),I1: fun(E,A),I22: fun(F3,C),X: product_prod(E,F3),Y: product_prod(B,D)] :
      ( aa(product_prod(B,D),$o,aa(product_prod(A,C),fun(product_prod(B,D),$o),basic_rel_prod(A,B,C,D,S1b,S2b),aa(product_prod(E,F3),product_prod(A,C),product_map_prod(E,A,F3,C,I1,I22),X)),Y)
    <=> aa(product_prod(B,D),$o,aa(product_prod(E,F3),fun(product_prod(B,D),$o),basic_rel_prod(E,B,F3,D,aa(fun(E,A),fun(E,fun(B,$o)),aTP_Lamp_aoy(fun(A,fun(B,$o)),fun(fun(E,A),fun(E,fun(B,$o))),S1b),I1),aa(fun(F3,C),fun(F3,fun(D,$o)),aTP_Lamp_aoz(fun(C,fun(D,$o)),fun(fun(F3,C),fun(F3,fun(D,$o))),S2b),I22)),X),Y) ) ).

% prod.rel_map(1)
tff(fact_6393_rel__prod_Ocases,axiom,
    ! [B: $tType,A: $tType,C: $tType,D: $tType,R1: fun(A,fun(B,$o)),R22: fun(C,fun(D,$o)),A1: product_prod(A,C),A22: product_prod(B,D)] :
      ( aa(product_prod(B,D),$o,aa(product_prod(A,C),fun(product_prod(B,D),$o),basic_rel_prod(A,B,C,D,R1,R22),A1),A22)
     => ~ ! [A6: A,B5: B,C4: C] :
            ( ( A1 = aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),A6),C4) )
           => ! [D2: D] :
                ( ( A22 = aa(D,product_prod(B,D),aa(B,fun(D,product_prod(B,D)),product_Pair(B,D),B5),D2) )
               => ( aa(B,$o,aa(A,fun(B,$o),R1,A6),B5)
                 => ~ aa(D,$o,aa(C,fun(D,$o),R22,C4),D2) ) ) ) ) ).

% rel_prod.cases
tff(fact_6394_rel__prod_Osimps,axiom,
    ! [B: $tType,A: $tType,C: $tType,D: $tType,R1: fun(A,fun(B,$o)),R22: fun(C,fun(D,$o)),A1: product_prod(A,C),A22: product_prod(B,D)] :
      ( aa(product_prod(B,D),$o,aa(product_prod(A,C),fun(product_prod(B,D),$o),basic_rel_prod(A,B,C,D,R1,R22),A1),A22)
    <=> ? [A10: A,B6: B,C5: C,D5: D] :
          ( ( A1 = aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),A10),C5) )
          & ( A22 = aa(D,product_prod(B,D),aa(B,fun(D,product_prod(B,D)),product_Pair(B,D),B6),D5) )
          & aa(B,$o,aa(A,fun(B,$o),R1,A10),B6)
          & aa(D,$o,aa(C,fun(D,$o),R22,C5),D5) ) ) ).

% rel_prod.simps
tff(fact_6395_rel__prod_Ointros,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType,R1: fun(A,fun(B,$o)),A3: A,B2: B,R22: fun(C,fun(D,$o)),C2: C,D3: D] :
      ( aa(B,$o,aa(A,fun(B,$o),R1,A3),B2)
     => ( aa(D,$o,aa(C,fun(D,$o),R22,C2),D3)
       => aa(product_prod(B,D),$o,aa(product_prod(A,C),fun(product_prod(B,D),$o),basic_rel_prod(A,B,C,D,R1,R22),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),A3),C2)),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_6396_rel__prod__conv,axiom,
    ! [A: $tType,C: $tType,D: $tType,B: $tType,R1: fun(A,fun(C,$o)),R22: fun(B,fun(D,$o))] : basic_rel_prod(A,C,B,D,R1,R22) = aa(fun(A,fun(B,fun(product_prod(C,D),$o))),fun(product_prod(A,B),fun(product_prod(C,D),$o)),product_case_prod(A,B,fun(product_prod(C,D),$o)),aa(fun(B,fun(D,$o)),fun(A,fun(B,fun(product_prod(C,D),$o))),aTP_Lamp_apb(fun(A,fun(C,$o)),fun(fun(B,fun(D,$o)),fun(A,fun(B,fun(product_prod(C,D),$o)))),R1),R22)) ).

% rel_prod_conv
tff(fact_6397_int_Oid__abs__transfer,axiom,
    aa(fun(product_prod(nat,nat),int),$o,aa(fun(product_prod(nat,nat),product_prod(nat,nat)),fun(fun(product_prod(nat,nat),int),$o),bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),product_prod(nat,nat),int,basic_rel_prod(nat,nat,nat,nat,fequal(nat),fequal(nat)),pcr_int),aTP_Lamp_apc(product_prod(nat,nat),product_prod(nat,nat))),abs_Integ) ).

% int.id_abs_transfer
tff(fact_6398_single__valuedp__bot,axiom,
    ! [B: $tType,A: $tType] : single_valuedp(A,B,bot_bot(fun(A,fun(B,$o)))) ).

% single_valuedp_bot
tff(fact_6399_single__valuedp__iff__Uniq,axiom,
    ! [B: $tType,A: $tType,R2: fun(A,fun(B,$o))] :
      ( single_valuedp(A,B,R2)
    <=> ! [X4: A] : uniq(B,aa(A,fun(B,$o),R2,X4)) ) ).

% single_valuedp_iff_Uniq
tff(fact_6400_Pair__transfer,axiom,
    ! [A: $tType,C: $tType,D: $tType,B: $tType,A4: fun(A,fun(B,$o)),B3: fun(C,fun(D,$o))] : aa(fun(B,fun(D,product_prod(B,D))),$o,aa(fun(A,fun(C,product_prod(A,C))),fun(fun(B,fun(D,product_prod(B,D))),$o),bNF_rel_fun(A,B,fun(C,product_prod(A,C)),fun(D,product_prod(B,D)),A4,bNF_rel_fun(C,D,product_prod(A,C),product_prod(B,D),B3,basic_rel_prod(A,B,C,D,A4,B3))),product_Pair(A,C)),product_Pair(B,D)) ).

% Pair_transfer
tff(fact_6401_prod_Orel__compp__Grp,axiom,
    ! [A: $tType,C: $tType,D: $tType,B: $tType,R1: fun(A,fun(C,$o)),R22: fun(B,fun(D,$o))] : basic_rel_prod(A,C,B,D,R1,R22) = aa(fun(product_prod(product_prod(A,C),product_prod(B,D)),fun(product_prod(C,D),$o)),fun(product_prod(A,B),fun(product_prod(C,D),$o)),aa(fun(product_prod(A,B),fun(product_prod(product_prod(A,C),product_prod(B,D)),$o)),fun(fun(product_prod(product_prod(A,C),product_prod(B,D)),fun(product_prod(C,D),$o)),fun(product_prod(A,B),fun(product_prod(C,D),$o))),relcompp(product_prod(A,B),product_prod(product_prod(A,C),product_prod(B,D)),product_prod(C,D)),conversep(product_prod(product_prod(A,C),product_prod(B,D)),product_prod(A,B),bNF_Grp(product_prod(product_prod(A,C),product_prod(B,D)),product_prod(A,B),aa(fun(product_prod(product_prod(A,C),product_prod(B,D)),$o),set(product_prod(product_prod(A,C),product_prod(B,D))),collect(product_prod(product_prod(A,C),product_prod(B,D))),aa(fun(B,fun(D,$o)),fun(product_prod(product_prod(A,C),product_prod(B,D)),$o),aTP_Lamp_apd(fun(A,fun(C,$o)),fun(fun(B,fun(D,$o)),fun(product_prod(product_prod(A,C),product_prod(B,D)),$o)),R1),R22)),product_map_prod(product_prod(A,C),A,product_prod(B,D),B,product_fst(A,C),product_fst(B,D))))),bNF_Grp(product_prod(product_prod(A,C),product_prod(B,D)),product_prod(C,D),aa(fun(product_prod(product_prod(A,C),product_prod(B,D)),$o),set(product_prod(product_prod(A,C),product_prod(B,D))),collect(product_prod(product_prod(A,C),product_prod(B,D))),aa(fun(B,fun(D,$o)),fun(product_prod(product_prod(A,C),product_prod(B,D)),$o),aTP_Lamp_apd(fun(A,fun(C,$o)),fun(fun(B,fun(D,$o)),fun(product_prod(product_prod(A,C),product_prod(B,D)),$o)),R1),R22)),product_map_prod(product_prod(A,C),C,product_prod(B,D),D,product_snd(A,C),product_snd(B,D)))) ).

% prod.rel_compp_Grp
tff(fact_6402_revg_Opelims,axiom,
    ! [A: $tType,X: list(A),Xa: list(A),Y: list(A)] :
      ( ( revg(A,X,Xa) = Y )
     => ( accp(product_prod(list(A),list(A)),revg_rel(A),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X),Xa))
       => ( ( ( X = nil(A) )
           => ( ( Y = Xa )
             => ~ accp(product_prod(list(A),list(A)),revg_rel(A),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),Xa)) ) )
         => ~ ! [A6: A,As4: list(A)] :
                ( ( X = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A6),As4) )
               => ( ( Y = revg(A,As4,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A6),Xa)) )
                 => ~ accp(product_prod(list(A),list(A)),revg_rel(A),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A6),As4)),Xa)) ) ) ) ) ) ).

% revg.pelims
tff(fact_6403_relcompp__distrib,axiom,
    ! [A: $tType,B: $tType,C: $tType,R: fun(A,fun(C,$o)),S: fun(C,fun(B,$o)),T2: fun(C,fun(B,$o))] : aa(fun(C,fun(B,$o)),fun(A,fun(B,$o)),aa(fun(A,fun(C,$o)),fun(fun(C,fun(B,$o)),fun(A,fun(B,$o))),relcompp(A,C,B),R),aa(fun(C,fun(B,$o)),fun(C,fun(B,$o)),aa(fun(C,fun(B,$o)),fun(fun(C,fun(B,$o)),fun(C,fun(B,$o))),sup_sup(fun(C,fun(B,$o))),S),T2)) = aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),sup_sup(fun(A,fun(B,$o))),aa(fun(C,fun(B,$o)),fun(A,fun(B,$o)),aa(fun(A,fun(C,$o)),fun(fun(C,fun(B,$o)),fun(A,fun(B,$o))),relcompp(A,C,B),R),S)),aa(fun(C,fun(B,$o)),fun(A,fun(B,$o)),aa(fun(A,fun(C,$o)),fun(fun(C,fun(B,$o)),fun(A,fun(B,$o))),relcompp(A,C,B),R),T2)) ).

% relcompp_distrib
tff(fact_6404_relcompp__distrib2,axiom,
    ! [A: $tType,B: $tType,C: $tType,S: fun(A,fun(C,$o)),T2: fun(A,fun(C,$o)),R: fun(C,fun(B,$o))] : aa(fun(C,fun(B,$o)),fun(A,fun(B,$o)),aa(fun(A,fun(C,$o)),fun(fun(C,fun(B,$o)),fun(A,fun(B,$o))),relcompp(A,C,B),aa(fun(A,fun(C,$o)),fun(A,fun(C,$o)),aa(fun(A,fun(C,$o)),fun(fun(A,fun(C,$o)),fun(A,fun(C,$o))),sup_sup(fun(A,fun(C,$o))),S),T2)),R) = aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),sup_sup(fun(A,fun(B,$o))),aa(fun(C,fun(B,$o)),fun(A,fun(B,$o)),aa(fun(A,fun(C,$o)),fun(fun(C,fun(B,$o)),fun(A,fun(B,$o))),relcompp(A,C,B),S),R)),aa(fun(C,fun(B,$o)),fun(A,fun(B,$o)),aa(fun(A,fun(C,$o)),fun(fun(C,fun(B,$o)),fun(A,fun(B,$o))),relcompp(A,C,B),T2),R)) ).

% relcompp_distrib2
tff(fact_6405_relcompp__bot2,axiom,
    ! [C: $tType,B: $tType,A: $tType,R: fun(A,fun(C,$o))] : aa(fun(C,fun(B,$o)),fun(A,fun(B,$o)),aa(fun(A,fun(C,$o)),fun(fun(C,fun(B,$o)),fun(A,fun(B,$o))),relcompp(A,C,B),R),bot_bot(fun(C,fun(B,$o)))) = bot_bot(fun(A,fun(B,$o))) ).

% relcompp_bot2
tff(fact_6406_relcompp__bot1,axiom,
    ! [C: $tType,B: $tType,A: $tType,R: fun(C,fun(B,$o))] : aa(fun(C,fun(B,$o)),fun(A,fun(B,$o)),aa(fun(A,fun(C,$o)),fun(fun(C,fun(B,$o)),fun(A,fun(B,$o))),relcompp(A,C,B),bot_bot(fun(A,fun(C,$o)))),R) = bot_bot(fun(A,fun(B,$o))) ).

% relcompp_bot1
tff(fact_6407_OO__Grp__alt,axiom,
    ! [A: $tType,C: $tType,B: $tType,A4: set(C),F: fun(C,A),G: fun(C,B),X3: A,Xa3: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(fun(C,fun(B,$o)),fun(A,fun(B,$o)),aa(fun(A,fun(C,$o)),fun(fun(C,fun(B,$o)),fun(A,fun(B,$o))),relcompp(A,C,B),conversep(C,A,bNF_Grp(C,A,A4,F))),bNF_Grp(C,B,A4,G)),X3),Xa3)
    <=> ? [Z4: C] :
          ( aa(set(C),$o,member(C,Z4),A4)
          & ( aa(C,A,F,Z4) = X3 )
          & ( aa(C,B,G,Z4) = Xa3 ) ) ) ).

% OO_Grp_alt
tff(fact_6408_Grp__UNIV__id,axiom,
    ! [A: $tType,F: fun(A,A)] :
      ( ( F = id(A) )
     => ( aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(fun(A,fun(A,$o)),fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),relcompp(A,A,A),conversep(A,A,bNF_Grp(A,A,top_top(set(A)),F))),bNF_Grp(A,A,top_top(set(A)),F)) = bNF_Grp(A,A,top_top(set(A)),F) ) ) ).

% Grp_UNIV_id
tff(fact_6409_list_Orel__Grp,axiom,
    ! [B: $tType,A: $tType,A4: set(A),F: fun(A,B)] : list_all2(A,B,bNF_Grp(A,B,A4,F)) = bNF_Grp(list(A),list(B),aa(fun(list(A),$o),set(list(A)),collect(list(A)),aTP_Lamp_aic(set(A),fun(list(A),$o),A4)),map(A,B,F)) ).

% list.rel_Grp
tff(fact_6410_relcompp__relcomp__eq,axiom,
    ! [A: $tType,B: $tType,C: $tType,R2: set(product_prod(A,C)),S2: set(product_prod(C,B)),X3: A,Xa3: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(fun(C,fun(B,$o)),fun(A,fun(B,$o)),aa(fun(A,fun(C,$o)),fun(fun(C,fun(B,$o)),fun(A,fun(B,$o))),relcompp(A,C,B),aTP_Lamp_ape(set(product_prod(A,C)),fun(A,fun(C,$o)),R2)),aTP_Lamp_apf(set(product_prod(C,B)),fun(C,fun(B,$o)),S2)),X3),Xa3)
    <=> aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Xa3)),relcomp(A,C,B,R2,S2)) ) ).

% relcompp_relcomp_eq
tff(fact_6411_Grp__def,axiom,
    ! [B: $tType,A: $tType,A4: set(A),F: fun(A,B),X3: A,Xa3: B] :
      ( aa(B,$o,aa(A,fun(B,$o),bNF_Grp(A,B,A4,F),X3),Xa3)
    <=> ( ( Xa3 = aa(A,B,F,X3) )
        & aa(set(A),$o,member(A,X3),A4) ) ) ).

% Grp_def
tff(fact_6412_OO__def,axiom,
    ! [A: $tType,C: $tType,B: $tType,R: fun(A,fun(C,$o)),S: fun(C,fun(B,$o)),X3: A,Xa3: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(fun(C,fun(B,$o)),fun(A,fun(B,$o)),aa(fun(A,fun(C,$o)),fun(fun(C,fun(B,$o)),fun(A,fun(B,$o))),relcompp(A,C,B),R),S),X3),Xa3)
    <=> ? [Y3: C] :
          ( aa(C,$o,aa(A,fun(C,$o),R,X3),Y3)
          & aa(B,$o,aa(C,fun(B,$o),S,Y3),Xa3) ) ) ).

% OO_def
tff(fact_6413_Grp__UNIV__idI,axiom,
    ! [A: $tType,X: A,Y: A] :
      ( ( X = Y )
     => aa(A,$o,aa(A,fun(A,$o),bNF_Grp(A,A,top_top(set(A)),id(A)),X),Y) ) ).

% Grp_UNIV_idI
tff(fact_6414_eq__alt,axiom,
    ! [A: $tType] : fequal(A) = bNF_Grp(A,A,top_top(set(A)),id(A)) ).

% eq_alt
tff(fact_6415_rel__filter__Grp,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B)] : rel_filter(A,B,bNF_Grp(A,B,top_top(set(A)),F)) = bNF_Grp(filter(A),filter(B),top_top(set(filter(A))),aa(fun(A,B),fun(filter(A),filter(B)),filtermap(A,B),F)) ).

% rel_filter_Grp
tff(fact_6416_fun_Orel__Grp,axiom,
    ! [A: $tType,C: $tType,B: $tType,A4: set(B),F: fun(B,C)] : bNF_rel_fun(A,A,B,C,fequal(A),bNF_Grp(B,C,A4,F)) = bNF_Grp(fun(A,B),fun(A,C),aa(fun(fun(A,B),$o),set(fun(A,B)),collect(fun(A,B)),aTP_Lamp_amy(set(B),fun(fun(A,B),$o),A4)),comp(B,C,A,F)) ).

% fun.rel_Grp
tff(fact_6417_relcompp__SUP__distrib,axiom,
    ! [C: $tType,B: $tType,A: $tType,D: $tType,S2: fun(A,fun(C,$o)),R2: fun(D,fun(C,fun(B,$o))),I4: set(D)] : aa(fun(C,fun(B,$o)),fun(A,fun(B,$o)),aa(fun(A,fun(C,$o)),fun(fun(C,fun(B,$o)),fun(A,fun(B,$o))),relcompp(A,C,B),S2),aa(set(fun(C,fun(B,$o))),fun(C,fun(B,$o)),complete_Sup_Sup(fun(C,fun(B,$o))),aa(set(D),set(fun(C,fun(B,$o))),image2(D,fun(C,fun(B,$o)),R2),I4))) = aa(set(fun(A,fun(B,$o))),fun(A,fun(B,$o)),complete_Sup_Sup(fun(A,fun(B,$o))),aa(set(D),set(fun(A,fun(B,$o))),image2(D,fun(A,fun(B,$o)),aa(fun(D,fun(C,fun(B,$o))),fun(D,fun(A,fun(B,$o))),aTP_Lamp_apg(fun(A,fun(C,$o)),fun(fun(D,fun(C,fun(B,$o))),fun(D,fun(A,fun(B,$o)))),S2),R2)),I4)) ).

% relcompp_SUP_distrib
tff(fact_6418_relcompp__SUP__distrib2,axiom,
    ! [C: $tType,B: $tType,A: $tType,D: $tType,R2: fun(D,fun(A,fun(C,$o))),I4: set(D),S2: fun(C,fun(B,$o))] : aa(fun(C,fun(B,$o)),fun(A,fun(B,$o)),aa(fun(A,fun(C,$o)),fun(fun(C,fun(B,$o)),fun(A,fun(B,$o))),relcompp(A,C,B),aa(set(fun(A,fun(C,$o))),fun(A,fun(C,$o)),complete_Sup_Sup(fun(A,fun(C,$o))),aa(set(D),set(fun(A,fun(C,$o))),image2(D,fun(A,fun(C,$o)),R2),I4))),S2) = aa(set(fun(A,fun(B,$o))),fun(A,fun(B,$o)),complete_Sup_Sup(fun(A,fun(B,$o))),aa(set(D),set(fun(A,fun(B,$o))),image2(D,fun(A,fun(B,$o)),aa(fun(C,fun(B,$o)),fun(D,fun(A,fun(B,$o))),aTP_Lamp_aph(fun(D,fun(A,fun(C,$o))),fun(fun(C,fun(B,$o)),fun(D,fun(A,fun(B,$o)))),R2),S2)),I4)) ).

% relcompp_SUP_distrib2
tff(fact_6419_prod_Orel__Grp,axiom,
    ! [A: $tType,C: $tType,D: $tType,B: $tType,A15: set(A),F1: fun(A,C),A24: set(B),F22: fun(B,D)] : basic_rel_prod(A,C,B,D,bNF_Grp(A,C,A15,F1),bNF_Grp(B,D,A24,F22)) = bNF_Grp(product_prod(A,B),product_prod(C,D),aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(set(B),fun(product_prod(A,B),$o),aTP_Lamp_api(set(A),fun(set(B),fun(product_prod(A,B),$o)),A15),A24)),product_map_prod(A,C,B,D,F1,F22)) ).

% prod.rel_Grp
tff(fact_6420_relcomp__def,axiom,
    ! [A: $tType,C: $tType,B: $tType,X3: set(product_prod(A,B)),Xa3: set(product_prod(B,C))] : relcomp(A,B,C,X3,Xa3) = aa(fun(product_prod(A,C),$o),set(product_prod(A,C)),collect(product_prod(A,C)),aa(fun(A,fun(C,$o)),fun(product_prod(A,C),$o),product_case_prod(A,C,$o),aa(fun(B,fun(C,$o)),fun(A,fun(C,$o)),aa(fun(A,fun(B,$o)),fun(fun(B,fun(C,$o)),fun(A,fun(C,$o))),relcompp(A,B,C),aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_as(set(product_prod(A,B)),fun(A,fun(B,$o))),X3)),aTP_Lamp_apj(set(product_prod(B,C)),fun(B,fun(C,$o)),Xa3)))) ).

% relcomp_def
tff(fact_6421_type__copy__vimage2p__Grp__Abs,axiom,
    ! [A: $tType,B: $tType,D: $tType,C: $tType,Rep: fun(A,B),Abs: fun(B,A),G: fun(C,D),P: fun(D,$o),H2: fun(D,A)] :
      ( type_definition(A,B,Rep,Abs,top_top(set(B)))
     => ( bNF_vimage2p(C,D,B,A,$o,G,Abs,bNF_Grp(D,A,aa(fun(D,$o),set(D),collect(D),P),H2)) = bNF_Grp(C,B,aa(fun(C,$o),set(C),collect(C),aa(fun(D,$o),fun(C,$o),aTP_Lamp_apk(fun(C,D),fun(fun(D,$o),fun(C,$o)),G),P)),aa(fun(C,D),fun(C,B),comp(D,B,C,aa(fun(D,A),fun(D,B),comp(A,B,D,Rep),H2)),G)) ) ) ).

% type_copy_vimage2p_Grp_Abs
tff(fact_6422_type__copy__vimage2p__Grp__Rep,axiom,
    ! [B: $tType,A: $tType,D: $tType,C: $tType,Rep: fun(A,B),Abs: fun(B,A),F: fun(C,D),P: fun(D,$o),H2: fun(D,B)] :
      ( type_definition(A,B,Rep,Abs,top_top(set(B)))
     => ( bNF_vimage2p(C,D,A,B,$o,F,Rep,bNF_Grp(D,B,aa(fun(D,$o),set(D),collect(D),P),H2)) = bNF_Grp(C,A,aa(fun(C,$o),set(C),collect(C),aa(fun(D,$o),fun(C,$o),aTP_Lamp_apk(fun(C,D),fun(fun(D,$o),fun(C,$o)),F),P)),aa(fun(C,D),fun(C,A),comp(D,A,C,aa(fun(D,B),fun(D,A),comp(B,A,D,Abs),H2)),F)) ) ) ).

% type_copy_vimage2p_Grp_Rep
tff(fact_6423_vimage2p__def,axiom,
    ! [A: $tType,D: $tType,C: $tType,E: $tType,B: $tType,F: fun(A,D),G: fun(B,E),R: fun(D,fun(E,C)),X3: A,Xa3: B] : aa(B,C,aa(A,fun(B,C),bNF_vimage2p(A,D,B,E,C,F,G,R),X3),Xa3) = aa(E,C,aa(D,fun(E,C),R,aa(A,D,F,X3)),aa(B,E,G,Xa3)) ).

% vimage2p_def
tff(fact_6424_list_Orel__compp__Grp,axiom,
    ! [B: $tType,A: $tType,R: fun(A,fun(B,$o))] : list_all2(A,B,R) = aa(fun(list(product_prod(A,B)),fun(list(B),$o)),fun(list(A),fun(list(B),$o)),aa(fun(list(A),fun(list(product_prod(A,B)),$o)),fun(fun(list(product_prod(A,B)),fun(list(B),$o)),fun(list(A),fun(list(B),$o))),relcompp(list(A),list(product_prod(A,B)),list(B)),conversep(list(product_prod(A,B)),list(A),bNF_Grp(list(product_prod(A,B)),list(A),aa(fun(list(product_prod(A,B)),$o),set(list(product_prod(A,B))),collect(list(product_prod(A,B))),aTP_Lamp_ait(fun(A,fun(B,$o)),fun(list(product_prod(A,B)),$o),R)),map(product_prod(A,B),A,product_fst(A,B))))),bNF_Grp(list(product_prod(A,B)),list(B),aa(fun(list(product_prod(A,B)),$o),set(list(product_prod(A,B))),collect(list(product_prod(A,B))),aTP_Lamp_ait(fun(A,fun(B,$o)),fun(list(product_prod(A,B)),$o),R)),map(product_prod(A,B),B,product_snd(A,B)))) ).

% list.rel_compp_Grp
tff(fact_6425_Abs__transfer,axiom,
    ! [B: $tType,A: $tType,C: $tType,D: $tType,Rep1: fun(A,B),Abs1: fun(B,A),Rep22: fun(C,D),Abs22: fun(D,C),R: fun(B,fun(D,$o))] :
      ( type_definition(A,B,Rep1,Abs1,top_top(set(B)))
     => ( type_definition(C,D,Rep22,Abs22,top_top(set(D)))
       => aa(fun(D,C),$o,aa(fun(B,A),fun(fun(D,C),$o),bNF_rel_fun(B,D,A,C,R,bNF_vimage2p(A,B,C,D,$o,Rep1,Rep22,R)),Abs1),Abs22) ) ) ).

% Abs_transfer
tff(fact_6426_fun_Orel__compp__Grp,axiom,
    ! [A: $tType,C: $tType,B: $tType,R: fun(B,fun(C,$o))] : bNF_rel_fun(A,A,B,C,fequal(A),R) = aa(fun(fun(A,product_prod(B,C)),fun(fun(A,C),$o)),fun(fun(A,B),fun(fun(A,C),$o)),aa(fun(fun(A,B),fun(fun(A,product_prod(B,C)),$o)),fun(fun(fun(A,product_prod(B,C)),fun(fun(A,C),$o)),fun(fun(A,B),fun(fun(A,C),$o))),relcompp(fun(A,B),fun(A,product_prod(B,C)),fun(A,C)),conversep(fun(A,product_prod(B,C)),fun(A,B),bNF_Grp(fun(A,product_prod(B,C)),fun(A,B),aa(fun(fun(A,product_prod(B,C)),$o),set(fun(A,product_prod(B,C))),collect(fun(A,product_prod(B,C))),aTP_Lamp_wd(fun(B,fun(C,$o)),fun(fun(A,product_prod(B,C)),$o),R)),comp(product_prod(B,C),B,A,product_fst(B,C))))),bNF_Grp(fun(A,product_prod(B,C)),fun(A,C),aa(fun(fun(A,product_prod(B,C)),$o),set(fun(A,product_prod(B,C))),collect(fun(A,product_prod(B,C))),aTP_Lamp_wd(fun(B,fun(C,$o)),fun(fun(A,product_prod(B,C)),$o),R)),comp(product_prod(B,C),C,A,product_snd(B,C)))) ).

% fun.rel_compp_Grp
tff(fact_6427_vimage2p__Grp,axiom,
    ! [A: $tType,C: $tType,D: $tType,B: $tType,F: fun(A,C),G: fun(B,D),P: fun(C,fun(D,$o))] : bNF_vimage2p(A,C,B,D,$o,F,G,P) = aa(fun(C,fun(B,$o)),fun(A,fun(B,$o)),aa(fun(A,fun(C,$o)),fun(fun(C,fun(B,$o)),fun(A,fun(B,$o))),relcompp(A,C,B),bNF_Grp(A,C,top_top(set(A)),F)),aa(fun(D,fun(B,$o)),fun(C,fun(B,$o)),aa(fun(C,fun(D,$o)),fun(fun(D,fun(B,$o)),fun(C,fun(B,$o))),relcompp(C,D,B),P),conversep(B,D,bNF_Grp(B,D,top_top(set(B)),G)))) ).

% vimage2p_Grp
tff(fact_6428_vimage2p__relcompp__converse,axiom,
    ! [E: $tType,C: $tType,D: $tType,A: $tType,F3: $tType,B: $tType,Rep: fun(A,B),Abs: fun(B,A),F: fun(C,E),G: fun(D,F3),R: fun(B,fun(E,$o)),S: fun(B,fun(F3,$o))] :
      ( type_definition(A,B,Rep,Abs,top_top(set(B)))
     => ( bNF_vimage2p(C,E,D,F3,$o,F,G,aa(fun(B,fun(F3,$o)),fun(E,fun(F3,$o)),aa(fun(E,fun(B,$o)),fun(fun(B,fun(F3,$o)),fun(E,fun(F3,$o))),relcompp(E,B,F3),conversep(B,E,R)),S)) = aa(fun(A,fun(D,$o)),fun(C,fun(D,$o)),aa(fun(C,fun(A,$o)),fun(fun(A,fun(D,$o)),fun(C,fun(D,$o))),relcompp(C,A,D),conversep(A,C,bNF_vimage2p(A,B,C,E,$o,Rep,F,R))),bNF_vimage2p(A,B,D,F3,$o,Rep,G,S)) ) ) ).

% vimage2p_relcompp_converse
tff(fact_6429_Quotient__alt__def5,axiom,
    ! [B: $tType,A: $tType,R: fun(A,fun(A,$o)),Abs: fun(A,B),Rep: fun(B,A),T2: fun(A,fun(B,$o))] :
      ( quotient(A,B,R,Abs,Rep,T2)
    <=> ( aa(fun(A,fun(B,$o)),$o,aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),$o),ord_less_eq(fun(A,fun(B,$o))),T2),bNF_Grp(A,B,top_top(set(A)),Abs))
        & aa(fun(B,fun(A,$o)),$o,aa(fun(B,fun(A,$o)),fun(fun(B,fun(A,$o)),$o),ord_less_eq(fun(B,fun(A,$o))),bNF_Grp(B,A,top_top(set(B)),Rep)),conversep(A,B,T2))
        & ( R = aa(fun(B,fun(A,$o)),fun(A,fun(A,$o)),aa(fun(A,fun(B,$o)),fun(fun(B,fun(A,$o)),fun(A,fun(A,$o))),relcompp(A,B,A),T2),conversep(A,B,T2)) ) ) ) ).

% Quotient_alt_def5
tff(fact_6430_acyclic__insert,axiom,
    ! [A: $tType,Y: A,X: A,R2: set(product_prod(A,A))] :
      ( transitive_acyclic(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(product_prod(A,A),fun(set(product_prod(A,A)),set(product_prod(A,A))),insert2(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),X)),R2))
    <=> ( transitive_acyclic(A,R2)
        & ~ aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),transitive_rtrancl(A,R2)) ) ) ).

% acyclic_insert
tff(fact_6431_acyclic__empty,axiom,
    ! [A: $tType] : transitive_acyclic(A,bot_bot(set(product_prod(A,A)))) ).

% acyclic_empty
tff(fact_6432_acyclicP__converse,axiom,
    ! [A: $tType,R2: fun(A,fun(A,$o))] :
      ( transitive_acyclic(A,aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),conversep(A,A,R2))))
    <=> transitive_acyclic(A,aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),R2))) ) ).

% acyclicP_converse
tff(fact_6433_UNIV__typedef__to__Quotient,axiom,
    ! [A: $tType,B: $tType,Rep: fun(A,B),Abs: fun(B,A),T2: fun(B,fun(A,$o))] :
      ( type_definition(A,B,Rep,Abs,top_top(set(B)))
     => ( ! [X2: B,Xa4: A] :
            ( aa(A,$o,aa(B,fun(A,$o),T2,X2),Xa4)
          <=> ( X2 = aa(A,B,Rep,Xa4) ) )
       => quotient(B,A,fequal(B),Abs,Rep,T2) ) ) ).

% UNIV_typedef_to_Quotient
tff(fact_6434_acyclicI__order,axiom,
    ! [B: $tType,A: $tType] :
      ( preorder(B)
     => ! [R2: set(product_prod(A,A)),F: fun(A,B)] :
          ( ! [A6: A,B5: A] :
              ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A6),B5)),R2)
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F,B5)),aa(A,B,F,A6)) )
         => transitive_acyclic(A,R2) ) ) ).

% acyclicI_order
tff(fact_6435_Quotient__cr__rel,axiom,
    ! [A: $tType,B: $tType,R: fun(A,fun(A,$o)),Abs: fun(A,B),Rep: fun(B,A),T2: fun(A,fun(B,$o))] :
      ( quotient(A,B,R,Abs,Rep,T2)
     => ! [X3: A,Xa3: B] :
          ( aa(B,$o,aa(A,fun(B,$o),T2,X3),Xa3)
        <=> ( aa(A,$o,aa(A,fun(A,$o),R,X3),X3)
            & ( aa(A,B,Abs,X3) = Xa3 ) ) ) ) ).

% Quotient_cr_rel
tff(fact_6436_Quotient__def,axiom,
    ! [A: $tType,B: $tType,R: fun(A,fun(A,$o)),Abs: fun(A,B),Rep: fun(B,A),T2: fun(A,fun(B,$o))] :
      ( quotient(A,B,R,Abs,Rep,T2)
    <=> ( ! [A10: B] : aa(A,B,Abs,aa(B,A,Rep,A10)) = A10
        & ! [A10: B] : aa(A,$o,aa(A,fun(A,$o),R,aa(B,A,Rep,A10)),aa(B,A,Rep,A10))
        & ! [R5: A,S6: A] :
            ( aa(A,$o,aa(A,fun(A,$o),R,R5),S6)
          <=> ( aa(A,$o,aa(A,fun(A,$o),R,R5),R5)
              & aa(A,$o,aa(A,fun(A,$o),R,S6),S6)
              & ( aa(A,B,Abs,R5) = aa(A,B,Abs,S6) ) ) )
        & ! [X4: A,Xa2: B] :
            ( aa(B,$o,aa(A,fun(B,$o),T2,X4),Xa2)
          <=> ( aa(A,$o,aa(A,fun(A,$o),R,X4),X4)
              & ( aa(A,B,Abs,X4) = Xa2 ) ) ) ) ) ).

% Quotient_def
tff(fact_6437_QuotientI,axiom,
    ! [A: $tType,B: $tType,Abs: fun(B,A),Rep: fun(A,B),R: fun(B,fun(B,$o)),T2: fun(B,fun(A,$o))] :
      ( ! [A6: A] : aa(B,A,Abs,aa(A,B,Rep,A6)) = A6
     => ( ! [A6: A] : aa(B,$o,aa(B,fun(B,$o),R,aa(A,B,Rep,A6)),aa(A,B,Rep,A6))
       => ( ! [R4: B,S7: B] :
              ( aa(B,$o,aa(B,fun(B,$o),R,R4),S7)
            <=> ( aa(B,$o,aa(B,fun(B,$o),R,R4),R4)
                & aa(B,$o,aa(B,fun(B,$o),R,S7),S7)
                & ( aa(B,A,Abs,R4) = aa(B,A,Abs,S7) ) ) )
         => ( ! [X2: B,Xa4: A] :
                ( aa(A,$o,aa(B,fun(A,$o),T2,X2),Xa4)
              <=> ( aa(B,$o,aa(B,fun(B,$o),R,X2),X2)
                  & ( aa(B,A,Abs,X2) = Xa4 ) ) )
           => quotient(B,A,R,Abs,Rep,T2) ) ) ) ) ).

% QuotientI
tff(fact_6438_acyclic__union_I2_J,axiom,
    ! [A: $tType,A4: set(product_prod(A,A)),B3: set(product_prod(A,A))] :
      ( transitive_acyclic(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))),A4),B3))
     => transitive_acyclic(A,B3) ) ).

% acyclic_union(2)
tff(fact_6439_acyclic__union_I1_J,axiom,
    ! [A: $tType,A4: set(product_prod(A,A)),B3: set(product_prod(A,A))] :
      ( transitive_acyclic(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))),A4),B3))
     => transitive_acyclic(A,A4) ) ).

% acyclic_union(1)
tff(fact_6440_cyclicE,axiom,
    ! [A: $tType,G: set(product_prod(A,A))] :
      ( ~ transitive_acyclic(A,G)
     => ~ ! [X2: A] : ~ aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),X2)),transitive_trancl(A,G)) ) ).

% cyclicE
tff(fact_6441_acyclic__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( transitive_acyclic(A,R2)
    <=> ! [X4: A] : ~ aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),X4)),transitive_trancl(A,R2)) ) ).

% acyclic_def
tff(fact_6442_acyclicI,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( ! [X2: A] : ~ aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),X2)),transitive_trancl(A,R2))
     => transitive_acyclic(A,R2) ) ).

% acyclicI
tff(fact_6443_acyclic__insert__cyclic,axiom,
    ! [A: $tType,G: set(product_prod(A,A)),X: A,Y: A] :
      ( transitive_acyclic(A,G)
     => ( ~ transitive_acyclic(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(product_prod(A,A),fun(set(product_prod(A,A)),set(product_prod(A,A))),insert2(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),G))
       => aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),X)),transitive_rtrancl(A,G)) ) ) ).

% acyclic_insert_cyclic
tff(fact_6444_Quotient__eq__onp__type__copy,axiom,
    ! [B: $tType,A: $tType,Abs: fun(A,B),Rep: fun(B,A),Cr: fun(A,fun(B,$o))] :
      ( quotient(A,B,fequal(A),Abs,Rep,Cr)
     => type_definition(B,A,Rep,Abs,top_top(set(A))) ) ).

% Quotient_eq_onp_type_copy
tff(fact_6445_Quotient__crel__typecopy,axiom,
    ! [A: $tType,B: $tType,Abs: fun(A,B),Rep: fun(B,A),T2: fun(A,fun(B,$o))] :
      ( quotient(A,B,fequal(A),Abs,Rep,T2)
     => ! [X3: A,Xa3: B] :
          ( aa(B,$o,aa(A,fun(B,$o),T2,X3),Xa3)
        <=> ( X3 = aa(B,A,Rep,Xa3) ) ) ) ).

% Quotient_crel_typecopy
tff(fact_6446_su__rel__fun_Of__def,axiom,
    ! [B: $tType,A: $tType,F4: set(product_prod(A,B)),F: fun(A,B),A4: A] :
      ( su_rel_fun(A,B,F4,F)
     => ( aa(A,B,F,A4) = the(B,aa(A,fun(B,$o),aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_as(set(product_prod(A,B)),fun(A,fun(B,$o))),F4),A4)) ) ) ).

% su_rel_fun.f_def
tff(fact_6447_su__rel__fun_Ointro,axiom,
    ! [B: $tType,A: $tType,F4: set(product_prod(A,B)),F: fun(A,B)] :
      ( ! [A8: A,B10: B,B7: B] :
          ( aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A8),B10)),F4)
         => ( aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A8),B7)),F4)
           => ( B10 = B7 ) ) )
     => ( ! [A8: A,P2: $o] :
            ( ! [B16: B] :
                ( aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A8),B16)),F4)
               => (P2) )
           => (P2) )
       => ( ! [A8: A] : aa(A,B,F,A8) = the(B,aa(A,fun(B,$o),aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_as(set(product_prod(A,B)),fun(A,fun(B,$o))),F4),A8))
         => su_rel_fun(A,B,F4,F) ) ) ) ).

% su_rel_fun.intro
tff(fact_6448_su__rel__fun_Osurjective,axiom,
    ! [B: $tType,A: $tType,F4: set(product_prod(A,B)),F: fun(A,B),A4: A] :
      ( su_rel_fun(A,B,F4,F)
     => ~ ! [B10: B] : ~ aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B10)),F4) ) ).

% su_rel_fun.surjective
tff(fact_6449_su__rel__fun_Ounique,axiom,
    ! [A: $tType,B: $tType,F4: set(product_prod(A,B)),F: fun(A,B),A4: A,B3: B,B14: B] :
      ( su_rel_fun(A,B,F4,F)
     => ( aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3)),F4)
       => ( aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B14)),F4)
         => ( B3 = B14 ) ) ) ) ).

% su_rel_fun.unique
tff(fact_6450_su__rel__fun_Orepr2,axiom,
    ! [B: $tType,A: $tType,F4: set(product_prod(A,B)),F: fun(A,B),A4: A,B3: B] :
      ( su_rel_fun(A,B,F4,F)
     => ( aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3)),F4)
       => ( B3 = aa(A,B,F,A4) ) ) ) ).

% su_rel_fun.repr2
tff(fact_6451_su__rel__fun_Orepr1,axiom,
    ! [B: $tType,A: $tType,F4: set(product_prod(A,B)),F: fun(A,B),A4: A] :
      ( su_rel_fun(A,B,F4,F)
     => aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),aa(A,B,F,A4))),F4) ) ).

% su_rel_fun.repr1
tff(fact_6452_su__rel__fun_Orepr,axiom,
    ! [B: $tType,A: $tType,F4: set(product_prod(A,B)),F: fun(A,B),A4: A,B3: B] :
      ( su_rel_fun(A,B,F4,F)
     => ( ( aa(A,B,F,A4) = B3 )
      <=> aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3)),F4) ) ) ).

% su_rel_fun.repr
tff(fact_6453_su__rel__fun__def,axiom,
    ! [B: $tType,A: $tType,F4: set(product_prod(A,B)),F: fun(A,B)] :
      ( su_rel_fun(A,B,F4,F)
    <=> ( ! [A9: A,B9: B,B17: B] :
            ( aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A9),B9)),F4)
           => ( aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A9),B17)),F4)
             => ( B9 = B17 ) ) )
        & ! [A9: A,P5: $o] :
            ( ! [B9: B] :
                ( aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A9),B9)),F4)
               => (P5) )
           => (P5) )
        & ! [A9: A] : aa(A,B,F,A9) = the(B,aa(A,fun(B,$o),aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_as(set(product_prod(A,B)),fun(A,fun(B,$o))),F4),A9)) ) ) ).

% su_rel_fun_def
tff(fact_6454_rotate__length01,axiom,
    ! [A: $tType,Xs: list(A),N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat))
     => ( rotate(A,N,Xs) = Xs ) ) ).

% rotate_length01
tff(fact_6455_aboveS__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A3: A] : order_aboveS(A,R2,A3) = aa(fun(A,$o),set(A),collect(A),aa(A,fun(A,$o),aTP_Lamp_apl(set(product_prod(A,A)),fun(A,fun(A,$o)),R2),A3)) ).

% aboveS_def
tff(fact_6456_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_6457_mult__cancel,axiom,
    ! [A: $tType,S2: set(product_prod(A,A)),X5: multiset(A),Z5: multiset(A),Y4: multiset(A)] :
      ( trans(A,S2)
     => ( irrefl(A,S2)
       => ( aa(set(product_prod(multiset(A),multiset(A))),$o,member(product_prod(multiset(A),multiset(A)),aa(multiset(A),product_prod(multiset(A),multiset(A)),aa(multiset(A),fun(multiset(A),product_prod(multiset(A),multiset(A))),product_Pair(multiset(A),multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),X5),Z5)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),Y4),Z5))),mult(A,S2))
        <=> aa(set(product_prod(multiset(A),multiset(A))),$o,member(product_prod(multiset(A),multiset(A)),aa(multiset(A),product_prod(multiset(A),multiset(A)),aa(multiset(A),fun(multiset(A),product_prod(multiset(A),multiset(A))),product_Pair(multiset(A),multiset(A)),X5),Y4)),mult(A,S2)) ) ) ) ).

% mult_cancel
tff(fact_6458_mult__cancel__max,axiom,
    ! [A: $tType,S2: set(product_prod(A,A)),X5: multiset(A),Y4: multiset(A)] :
      ( trans(A,S2)
     => ( irrefl(A,S2)
       => ( aa(set(product_prod(multiset(A),multiset(A))),$o,member(product_prod(multiset(A),multiset(A)),aa(multiset(A),product_prod(multiset(A),multiset(A)),aa(multiset(A),fun(multiset(A),product_prod(multiset(A),multiset(A))),product_Pair(multiset(A),multiset(A)),X5),Y4)),mult(A,S2))
        <=> aa(set(product_prod(multiset(A),multiset(A))),$o,member(product_prod(multiset(A),multiset(A)),aa(multiset(A),product_prod(multiset(A),multiset(A)),aa(multiset(A),fun(multiset(A),product_prod(multiset(A),multiset(A))),product_Pair(multiset(A),multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),X5),Y4)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),Y4),X5))),mult(A,S2)) ) ) ) ).

% mult_cancel_max
tff(fact_6459_mult__cancel__add__mset,axiom,
    ! [A: $tType,S2: set(product_prod(A,A)),Uu: A,X5: multiset(A),Y4: multiset(A)] :
      ( trans(A,S2)
     => ( irrefl(A,S2)
       => ( aa(set(product_prod(multiset(A),multiset(A))),$o,member(product_prod(multiset(A),multiset(A)),aa(multiset(A),product_prod(multiset(A),multiset(A)),aa(multiset(A),fun(multiset(A),product_prod(multiset(A),multiset(A))),product_Pair(multiset(A),multiset(A)),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),Uu),X5)),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),Uu),Y4))),mult(A,S2))
        <=> aa(set(product_prod(multiset(A),multiset(A))),$o,member(product_prod(multiset(A),multiset(A)),aa(multiset(A),product_prod(multiset(A),multiset(A)),aa(multiset(A),fun(multiset(A),product_prod(multiset(A),multiset(A))),product_Pair(multiset(A),multiset(A)),X5),Y4)),mult(A,S2)) ) ) ) ).

% mult_cancel_add_mset
tff(fact_6460_multp__code__iff__mult,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),P: fun(A,fun(A,$o)),N4: multiset(A),M4: multiset(A)] :
      ( irrefl(A,R)
     => ( trans(A,R)
       => ( ! [X2: A,Y2: A] :
              ( aa(A,$o,aa(A,fun(A,$o),P,X2),Y2)
            <=> aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y2)),R) )
         => ( multp_code(A,P,N4,M4)
          <=> aa(set(product_prod(multiset(A),multiset(A))),$o,member(product_prod(multiset(A),multiset(A)),aa(multiset(A),product_prod(multiset(A),multiset(A)),aa(multiset(A),fun(multiset(A),product_prod(multiset(A),multiset(A))),product_Pair(multiset(A),multiset(A)),N4),M4)),mult(A,R)) ) ) ) ) ).

% multp_code_iff_mult
tff(fact_6461_multeqp__code__iff__reflcl__mult,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),P: fun(A,fun(A,$o)),N4: multiset(A),M4: multiset(A)] :
      ( irrefl(A,R)
     => ( trans(A,R)
       => ( ! [X2: A,Y2: A] :
              ( aa(A,$o,aa(A,fun(A,$o),P,X2),Y2)
            <=> aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y2)),R) )
         => ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),multeqp_code(A,P),N4),M4)
          <=> aa(set(product_prod(multiset(A),multiset(A))),$o,member(product_prod(multiset(A),multiset(A)),aa(multiset(A),product_prod(multiset(A),multiset(A)),aa(multiset(A),fun(multiset(A),product_prod(multiset(A),multiset(A))),product_Pair(multiset(A),multiset(A)),N4),M4)),aa(set(product_prod(multiset(A),multiset(A))),set(product_prod(multiset(A),multiset(A))),aa(set(product_prod(multiset(A),multiset(A))),fun(set(product_prod(multiset(A),multiset(A))),set(product_prod(multiset(A),multiset(A)))),sup_sup(set(product_prod(multiset(A),multiset(A)))),mult(A,R)),id2(multiset(A)))) ) ) ) ) ).

% multeqp_code_iff_reflcl_mult
tff(fact_6462_mult__implies__one__step,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),M4: multiset(A),N4: multiset(A)] :
      ( trans(A,R2)
     => ( aa(set(product_prod(multiset(A),multiset(A))),$o,member(product_prod(multiset(A),multiset(A)),aa(multiset(A),product_prod(multiset(A),multiset(A)),aa(multiset(A),fun(multiset(A),product_prod(multiset(A),multiset(A))),product_Pair(multiset(A),multiset(A)),M4),N4)),mult(A,R2))
       => ? [I6: multiset(A),J5: multiset(A)] :
            ( ( N4 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),I6),J5) )
            & ? [K7: multiset(A)] :
                ( ( M4 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),I6),K7) )
                & ( J5 != zero_zero(multiset(A)) )
                & ! [X3: A] :
                    ( aa(set(A),$o,member(A,X3),aa(multiset(A),set(A),set_mset(A),K7))
                   => ? [Xa4: A] :
                        ( aa(set(A),$o,member(A,Xa4),aa(multiset(A),set(A),set_mset(A),J5))
                        & aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Xa4)),R2) ) ) ) ) ) ) ).

% mult_implies_one_step
tff(fact_6463_at__most__one__mset__mset__diff,axiom,
    ! [A: $tType,A3: A,M4: multiset(A)] :
      ( ~ aa(set(A),$o,member(A,A3),aa(multiset(A),set(A),set_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),M4),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A3),zero_zero(multiset(A))))))
     => ( aa(multiset(A),set(A),set_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),M4),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A3),zero_zero(multiset(A))))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(multiset(A),set(A),set_mset(A),M4)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A)))) ) ) ).

% at_most_one_mset_mset_diff
tff(fact_6464_set__mset__empty,axiom,
    ! [A: $tType] : aa(multiset(A),set(A),set_mset(A),zero_zero(multiset(A))) = bot_bot(set(A)) ).

% set_mset_empty
tff(fact_6465_set__mset__eq__empty__iff,axiom,
    ! [A: $tType,M4: multiset(A)] :
      ( ( aa(multiset(A),set(A),set_mset(A),M4) = bot_bot(set(A)) )
    <=> ( M4 = zero_zero(multiset(A)) ) ) ).

% set_mset_eq_empty_iff
tff(fact_6466_set__mset__union,axiom,
    ! [A: $tType,M4: multiset(A),N4: multiset(A)] : aa(multiset(A),set(A),set_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M4),N4)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(multiset(A),set(A),set_mset(A),M4)),aa(multiset(A),set(A),set_mset(A),N4)) ).

% set_mset_union
tff(fact_6467_in__Inf__multiset__iff,axiom,
    ! [A: $tType,A4: set(multiset(A)),X: A] :
      ( ( A4 != bot_bot(set(multiset(A))) )
     => ( aa(set(A),$o,member(A,X),aa(multiset(A),set(A),set_mset(A),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),A4)))
      <=> ! [X4: multiset(A)] :
            ( aa(set(multiset(A)),$o,member(multiset(A),X4),A4)
           => aa(set(A),$o,member(A,X),aa(multiset(A),set(A),set_mset(A),X4)) ) ) ) ).

% in_Inf_multiset_iff
tff(fact_6468_image__mset__cong,axiom,
    ! [B: $tType,A: $tType,M4: multiset(A),F: fun(A,B),G: fun(A,B)] :
      ( ! [X2: A] :
          ( aa(set(A),$o,member(A,X2),aa(multiset(A),set(A),set_mset(A),M4))
         => ( aa(A,B,F,X2) = aa(A,B,G,X2) ) )
     => ( aa(multiset(A),multiset(B),image_mset(A,B,F),M4) = aa(multiset(A),multiset(B),image_mset(A,B,G),M4) ) ) ).

% image_mset_cong
tff(fact_6469_in__image__mset,axiom,
    ! [A: $tType,B: $tType,Y: A,F: fun(B,A),M4: multiset(B)] :
      ( aa(set(A),$o,member(A,Y),aa(multiset(A),set(A),set_mset(A),aa(multiset(B),multiset(A),image_mset(B,A,F),M4)))
    <=> aa(set(A),$o,member(A,Y),aa(set(B),set(A),image2(B,A,F),aa(multiset(B),set(B),set_mset(B),M4))) ) ).

% in_image_mset
tff(fact_6470_image__mset__cong__pair,axiom,
    ! [C: $tType,B: $tType,A: $tType,M4: multiset(product_prod(A,B)),F: fun(A,fun(B,C)),G: fun(A,fun(B,C))] :
      ( ! [X2: A,Y2: B] :
          ( aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Y2)),aa(multiset(product_prod(A,B)),set(product_prod(A,B)),set_mset(product_prod(A,B)),M4))
         => ( aa(B,C,aa(A,fun(B,C),F,X2),Y2) = aa(B,C,aa(A,fun(B,C),G,X2),Y2) ) )
     => ( aa(multiset(product_prod(A,B)),multiset(C),image_mset(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F)),M4) = aa(multiset(product_prod(A,B)),multiset(C),image_mset(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),G)),M4) ) ) ).

% image_mset_cong_pair
tff(fact_6471_infinite__set__mset__mset__set,axiom,
    ! [A: $tType,A4: set(A)] :
      ( ~ aa(set(A),$o,finite_finite2(A),A4)
     => ( aa(multiset(A),set(A),set_mset(A),mset_set(A,A4)) = bot_bot(set(A)) ) ) ).

% infinite_set_mset_mset_set
tff(fact_6472_set__mset__Inf,axiom,
    ! [A: $tType,A4: set(multiset(A))] :
      ( ( A4 != bot_bot(set(multiset(A))) )
     => ( aa(multiset(A),set(A),set_mset(A),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),A4)) = aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(multiset(A)),set(set(A)),image2(multiset(A),set(A),set_mset(A)),A4)) ) ) ).

% set_mset_Inf
tff(fact_6473_set__mset__single,axiom,
    ! [A: $tType,B2: A] : aa(multiset(A),set(A),set_mset(A),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),B2),zero_zero(multiset(A)))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A))) ).

% set_mset_single
tff(fact_6474_one__step__implies__mult,axiom,
    ! [A: $tType,J3: multiset(A),K5: multiset(A),R2: set(product_prod(A,A)),I4: multiset(A)] :
      ( ( J3 != zero_zero(multiset(A)) )
     => ( ! [X2: A] :
            ( aa(set(A),$o,member(A,X2),aa(multiset(A),set(A),set_mset(A),K5))
           => ? [Xa3: A] :
                ( aa(set(A),$o,member(A,Xa3),aa(multiset(A),set(A),set_mset(A),J3))
                & aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Xa3)),R2) ) )
       => aa(set(product_prod(multiset(A),multiset(A))),$o,member(product_prod(multiset(A),multiset(A)),aa(multiset(A),product_prod(multiset(A),multiset(A)),aa(multiset(A),fun(multiset(A),product_prod(multiset(A),multiset(A))),product_Pair(multiset(A),multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),I4),K5)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),I4),J3))),mult(A,R2)) ) ) ).

% one_step_implies_mult
tff(fact_6475_mult1__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : mult1(A,R2) = aa(fun(product_prod(multiset(A),multiset(A)),$o),set(product_prod(multiset(A),multiset(A))),collect(product_prod(multiset(A),multiset(A))),aa(fun(multiset(A),fun(multiset(A),$o)),fun(product_prod(multiset(A),multiset(A)),$o),product_case_prod(multiset(A),multiset(A),$o),aTP_Lamp_apm(set(product_prod(A,A)),fun(multiset(A),fun(multiset(A),$o)),R2))) ).

% mult1_def
tff(fact_6476_less__add,axiom,
    ! [A: $tType,N4: multiset(A),A3: A,M0: multiset(A),R2: set(product_prod(A,A))] :
      ( aa(set(product_prod(multiset(A),multiset(A))),$o,member(product_prod(multiset(A),multiset(A)),aa(multiset(A),product_prod(multiset(A),multiset(A)),aa(multiset(A),fun(multiset(A),product_prod(multiset(A),multiset(A))),product_Pair(multiset(A),multiset(A)),N4),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A3),M0))),mult1(A,R2))
     => ( ? [M7: multiset(A)] :
            ( aa(set(product_prod(multiset(A),multiset(A))),$o,member(product_prod(multiset(A),multiset(A)),aa(multiset(A),product_prod(multiset(A),multiset(A)),aa(multiset(A),fun(multiset(A),product_prod(multiset(A),multiset(A))),product_Pair(multiset(A),multiset(A)),M7),M0)),mult1(A,R2))
            & ( N4 = aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A3),M7) ) )
        | ? [K7: multiset(A)] :
            ( ! [B12: A] :
                ( aa(set(A),$o,member(A,B12),aa(multiset(A),set(A),set_mset(A),K7))
               => aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B12),A3)),R2) )
            & ( N4 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M0),K7) ) ) ) ) ).

% less_add
tff(fact_6477_not__less__empty,axiom,
    ! [A: $tType,M4: multiset(A),R2: set(product_prod(A,A))] : ~ aa(set(product_prod(multiset(A),multiset(A))),$o,member(product_prod(multiset(A),multiset(A)),aa(multiset(A),product_prod(multiset(A),multiset(A)),aa(multiset(A),fun(multiset(A),product_prod(multiset(A),multiset(A))),product_Pair(multiset(A),multiset(A)),M4),zero_zero(multiset(A)))),mult1(A,R2)) ).

% not_less_empty
tff(fact_6478_mult1__union,axiom,
    ! [A: $tType,B3: multiset(A),D4: multiset(A),R2: set(product_prod(A,A)),C3: multiset(A)] :
      ( aa(set(product_prod(multiset(A),multiset(A))),$o,member(product_prod(multiset(A),multiset(A)),aa(multiset(A),product_prod(multiset(A),multiset(A)),aa(multiset(A),fun(multiset(A),product_prod(multiset(A),multiset(A))),product_Pair(multiset(A),multiset(A)),B3),D4)),mult1(A,R2))
     => aa(set(product_prod(multiset(A),multiset(A))),$o,member(product_prod(multiset(A),multiset(A)),aa(multiset(A),product_prod(multiset(A),multiset(A)),aa(multiset(A),fun(multiset(A),product_prod(multiset(A),multiset(A))),product_Pair(multiset(A),multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),C3),B3)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),C3),D4))),mult1(A,R2)) ) ).

% mult1_union
tff(fact_6479_mult1E,axiom,
    ! [A: $tType,N4: multiset(A),M4: multiset(A),R2: set(product_prod(A,A))] :
      ( aa(set(product_prod(multiset(A),multiset(A))),$o,member(product_prod(multiset(A),multiset(A)),aa(multiset(A),product_prod(multiset(A),multiset(A)),aa(multiset(A),fun(multiset(A),product_prod(multiset(A),multiset(A))),product_Pair(multiset(A),multiset(A)),N4),M4)),mult1(A,R2))
     => ~ ! [A6: A,M02: multiset(A)] :
            ( ( M4 = aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A6),M02) )
           => ! [K7: multiset(A)] :
                ( ( N4 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M02),K7) )
               => ~ ! [B12: A] :
                      ( aa(set(A),$o,member(A,B12),aa(multiset(A),set(A),set_mset(A),K7))
                     => aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B12),A6)),R2) ) ) ) ) ).

% mult1E
tff(fact_6480_mult1I,axiom,
    ! [A: $tType,M4: multiset(A),A3: A,M0: multiset(A),N4: multiset(A),K5: multiset(A),R2: set(product_prod(A,A))] :
      ( ( M4 = aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A3),M0) )
     => ( ( N4 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M0),K5) )
       => ( ! [B5: A] :
              ( aa(set(A),$o,member(A,B5),aa(multiset(A),set(A),set_mset(A),K5))
             => aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B5),A3)),R2) )
         => aa(set(product_prod(multiset(A),multiset(A))),$o,member(product_prod(multiset(A),multiset(A)),aa(multiset(A),product_prod(multiset(A),multiset(A)),aa(multiset(A),fun(multiset(A),product_prod(multiset(A),multiset(A))),product_Pair(multiset(A),multiset(A)),N4),M4)),mult1(A,R2)) ) ) ) ).

% mult1I
tff(fact_6481_mult1__lessE,axiom,
    ! [A: $tType,N4: multiset(A),M4: multiset(A),R2: fun(A,fun(A,$o))] :
      ( aa(set(product_prod(multiset(A),multiset(A))),$o,member(product_prod(multiset(A),multiset(A)),aa(multiset(A),product_prod(multiset(A),multiset(A)),aa(multiset(A),fun(multiset(A),product_prod(multiset(A),multiset(A))),product_Pair(multiset(A),multiset(A)),N4),M4)),mult1(A,aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),R2))))
     => ( asymp(A,R2)
       => ~ ! [A6: A,M02: multiset(A)] :
              ( ( M4 = aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A6),M02) )
             => ! [K7: multiset(A)] :
                  ( ( N4 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M02),K7) )
                 => ( ~ aa(set(A),$o,member(A,A6),aa(multiset(A),set(A),set_mset(A),K7))
                   => ~ ! [B12: A] :
                          ( aa(set(A),$o,member(A,B12),aa(multiset(A),set(A),set_mset(A),K7))
                         => aa(A,$o,aa(A,fun(A,$o),R2,B12),A6) ) ) ) ) ) ) ).

% mult1_lessE
tff(fact_6482_sum__wcount__Int,axiom,
    ! [A: $tType,A4: set(A),F: fun(A,nat),N4: multiset(A)] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),wcount(A,F,N4)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),aa(multiset(A),set(A),set_mset(A),N4))) = aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),wcount(A,F,N4)),A4) ) ) ).

% sum_wcount_Int
tff(fact_6483_asymp__greater,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => asymp(A,aTP_Lamp_sz(A,fun(A,$o))) ) ).

% asymp_greater
tff(fact_6484_multiset_Oin__rel,axiom,
    ! [A: $tType,B: $tType,R: fun(A,fun(B,$o)),A3: multiset(A),B2: multiset(B)] :
      ( aa(multiset(B),$o,aa(multiset(A),fun(multiset(B),$o),rel_mset(A,B,R),A3),B2)
    <=> ? [Z4: multiset(product_prod(A,B))] :
          ( aa(set(multiset(product_prod(A,B))),$o,member(multiset(product_prod(A,B)),Z4),aa(fun(multiset(product_prod(A,B)),$o),set(multiset(product_prod(A,B))),collect(multiset(product_prod(A,B))),aTP_Lamp_apn(fun(A,fun(B,$o)),fun(multiset(product_prod(A,B)),$o),R)))
          & ( aa(multiset(product_prod(A,B)),multiset(A),image_mset(product_prod(A,B),A,product_fst(A,B)),Z4) = A3 )
          & ( aa(multiset(product_prod(A,B)),multiset(B),image_mset(product_prod(A,B),B,product_snd(A,B)),Z4) = B2 ) ) ) ).

% multiset.in_rel
tff(fact_6485_multiset_Orel__compp__Grp,axiom,
    ! [B: $tType,A: $tType,R: fun(A,fun(B,$o))] : rel_mset(A,B,R) = aa(fun(multiset(product_prod(A,B)),fun(multiset(B),$o)),fun(multiset(A),fun(multiset(B),$o)),aa(fun(multiset(A),fun(multiset(product_prod(A,B)),$o)),fun(fun(multiset(product_prod(A,B)),fun(multiset(B),$o)),fun(multiset(A),fun(multiset(B),$o))),relcompp(multiset(A),multiset(product_prod(A,B)),multiset(B)),conversep(multiset(product_prod(A,B)),multiset(A),bNF_Grp(multiset(product_prod(A,B)),multiset(A),aa(fun(multiset(product_prod(A,B)),$o),set(multiset(product_prod(A,B))),collect(multiset(product_prod(A,B))),aTP_Lamp_apn(fun(A,fun(B,$o)),fun(multiset(product_prod(A,B)),$o),R)),image_mset(product_prod(A,B),A,product_fst(A,B))))),bNF_Grp(multiset(product_prod(A,B)),multiset(B),aa(fun(multiset(product_prod(A,B)),$o),set(multiset(product_prod(A,B))),collect(multiset(product_prod(A,B))),aTP_Lamp_apn(fun(A,fun(B,$o)),fun(multiset(product_prod(A,B)),$o),R)),image_mset(product_prod(A,B),B,product_snd(A,B)))) ).

% multiset.rel_compp_Grp
tff(fact_6486_multiset_Orel__map_I2_J,axiom,
    ! [A: $tType,B: $tType,C: $tType,Sa: fun(A,fun(B,$o)),X: multiset(A),G: fun(C,B),Y: multiset(C)] :
      ( aa(multiset(B),$o,aa(multiset(A),fun(multiset(B),$o),rel_mset(A,B,Sa),X),aa(multiset(C),multiset(B),image_mset(C,B,G),Y))
    <=> aa(multiset(C),$o,aa(multiset(A),fun(multiset(C),$o),rel_mset(A,C,aa(fun(C,B),fun(A,fun(C,$o)),aTP_Lamp_aiu(fun(A,fun(B,$o)),fun(fun(C,B),fun(A,fun(C,$o))),Sa),G)),X),Y) ) ).

% multiset.rel_map(2)
tff(fact_6487_multiset_Orel__map_I1_J,axiom,
    ! [C: $tType,A: $tType,B: $tType,Sb: fun(A,fun(B,$o)),I: fun(C,A),X: multiset(C),Y: multiset(B)] :
      ( aa(multiset(B),$o,aa(multiset(A),fun(multiset(B),$o),rel_mset(A,B,Sb),aa(multiset(C),multiset(A),image_mset(C,A,I),X)),Y)
    <=> aa(multiset(B),$o,aa(multiset(C),fun(multiset(B),$o),rel_mset(C,B,aa(fun(C,A),fun(C,fun(B,$o)),aTP_Lamp_aiv(fun(A,fun(B,$o)),fun(fun(C,A),fun(C,fun(B,$o))),Sb),I)),X),Y) ) ).

% multiset.rel_map(1)
tff(fact_6488_multiset_Orel__Grp,axiom,
    ! [B: $tType,A: $tType,A4: set(A),F: fun(A,B)] : rel_mset(A,B,bNF_Grp(A,B,A4,F)) = bNF_Grp(multiset(A),multiset(B),aa(fun(multiset(A),$o),set(multiset(A)),collect(multiset(A)),aTP_Lamp_apo(set(A),fun(multiset(A),$o),A4)),image_mset(A,B,F)) ).

% multiset.rel_Grp
tff(fact_6489_smsI,axiom,
    ! [A4: multiset(product_prod(nat,nat)),B3: multiset(product_prod(nat,nat)),Z5: multiset(product_prod(nat,nat))] :
      ( aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),$o,member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),set(product_prod(nat,nat)),set_mset(product_prod(nat,nat)),A4)),aa(multiset(product_prod(nat,nat)),set(product_prod(nat,nat)),set_mset(product_prod(nat,nat)),B3))),fun_max_strict)
     => aa(set(product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)))),$o,member(product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),fun(multiset(product_prod(nat,nat)),product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)))),product_Pair(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(multiset(product_prod(nat,nat)),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),plus_plus(multiset(product_prod(nat,nat))),Z5),A4)),aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(multiset(product_prod(nat,nat)),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),plus_plus(multiset(product_prod(nat,nat))),Z5),B3))),ms_strict) ) ).

% smsI
tff(fact_6490_wmsI,axiom,
    ! [A4: multiset(product_prod(nat,nat)),B3: multiset(product_prod(nat,nat)),Z5: multiset(product_prod(nat,nat))] :
      ( ( aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),$o,member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),set(product_prod(nat,nat)),set_mset(product_prod(nat,nat)),A4)),aa(multiset(product_prod(nat,nat)),set(product_prod(nat,nat)),set_mset(product_prod(nat,nat)),B3))),fun_max_strict)
        | ( ( A4 = zero_zero(multiset(product_prod(nat,nat))) )
          & ( B3 = zero_zero(multiset(product_prod(nat,nat))) ) ) )
     => aa(set(product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)))),$o,member(product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),fun(multiset(product_prod(nat,nat)),product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)))),product_Pair(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(multiset(product_prod(nat,nat)),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),plus_plus(multiset(product_prod(nat,nat))),Z5),A4)),aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(multiset(product_prod(nat,nat)),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),plus_plus(multiset(product_prod(nat,nat))),Z5),B3))),ms_weak) ) ).

% wmsI
tff(fact_6491_ms__weak__def,axiom,
    ms_weak = aa(set(product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)))),set(product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)))),aa(set(product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)))),fun(set(product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)))),set(product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))))),sup_sup(set(product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))))),ms_strict),id2(multiset(product_prod(nat,nat)))) ).

% ms_weak_def
tff(fact_6492_ms__reduction__pair,axiom,
    fun_reduction_pair(multiset(product_prod(nat,nat)),aa(set(product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)))),product_prod(set(product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)))),set(product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))))),aa(set(product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)))),fun(set(product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)))),product_prod(set(product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)))),set(product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)))))),product_Pair(set(product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)))),set(product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))))),ms_strict),ms_weak)) ).

% ms_reduction_pair
tff(fact_6493_ms__strictI,axiom,
    ! [Z5: multiset(product_prod(nat,nat)),Z10: multiset(product_prod(nat,nat)),A4: multiset(product_prod(nat,nat)),B3: multiset(product_prod(nat,nat))] :
      ( pw_leq(Z5,Z10)
     => ( aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),$o,member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),set(product_prod(nat,nat)),set_mset(product_prod(nat,nat)),A4)),aa(multiset(product_prod(nat,nat)),set(product_prod(nat,nat)),set_mset(product_prod(nat,nat)),B3))),fun_max_strict)
       => aa(set(product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)))),$o,member(product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),fun(multiset(product_prod(nat,nat)),product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)))),product_Pair(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(multiset(product_prod(nat,nat)),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),plus_plus(multiset(product_prod(nat,nat))),Z5),A4)),aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(multiset(product_prod(nat,nat)),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),plus_plus(multiset(product_prod(nat,nat))),Z10),B3))),ms_strict) ) ) ).

% ms_strictI
tff(fact_6494_ms__weakI1,axiom,
    ! [Z5: multiset(product_prod(nat,nat)),Z10: multiset(product_prod(nat,nat)),A4: multiset(product_prod(nat,nat)),B3: multiset(product_prod(nat,nat))] :
      ( pw_leq(Z5,Z10)
     => ( aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),$o,member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),set(product_prod(nat,nat)),set_mset(product_prod(nat,nat)),A4)),aa(multiset(product_prod(nat,nat)),set(product_prod(nat,nat)),set_mset(product_prod(nat,nat)),B3))),fun_max_strict)
       => aa(set(product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)))),$o,member(product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),fun(multiset(product_prod(nat,nat)),product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)))),product_Pair(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(multiset(product_prod(nat,nat)),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),plus_plus(multiset(product_prod(nat,nat))),Z5),A4)),aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(multiset(product_prod(nat,nat)),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),plus_plus(multiset(product_prod(nat,nat))),Z10),B3))),ms_weak) ) ) ).

% ms_weakI1
tff(fact_6495_pw__leq__lstep,axiom,
    ! [X: product_prod(nat,nat),Y: product_prod(nat,nat)] :
      ( aa(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),$o,member(product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),X),Y)),fun_pair_leq)
     => pw_leq(aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(product_prod(nat,nat),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),add_mset(product_prod(nat,nat)),X),zero_zero(multiset(product_prod(nat,nat)))),aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(product_prod(nat,nat),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),add_mset(product_prod(nat,nat)),Y),zero_zero(multiset(product_prod(nat,nat))))) ) ).

% pw_leq_lstep
tff(fact_6496_pw__leq_Ocases,axiom,
    ! [A1: multiset(product_prod(nat,nat)),A22: multiset(product_prod(nat,nat))] :
      ( pw_leq(A1,A22)
     => ( ( ( A1 = zero_zero(multiset(product_prod(nat,nat))) )
         => ( A22 != zero_zero(multiset(product_prod(nat,nat))) ) )
       => ~ ! [X2: product_prod(nat,nat),Y2: product_prod(nat,nat),X6: multiset(product_prod(nat,nat))] :
              ( ( A1 = aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(multiset(product_prod(nat,nat)),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),plus_plus(multiset(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(product_prod(nat,nat),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),add_mset(product_prod(nat,nat)),X2),zero_zero(multiset(product_prod(nat,nat))))),X6) )
             => ! [Y9: multiset(product_prod(nat,nat))] :
                  ( ( A22 = aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(multiset(product_prod(nat,nat)),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),plus_plus(multiset(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(product_prod(nat,nat),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),add_mset(product_prod(nat,nat)),Y2),zero_zero(multiset(product_prod(nat,nat))))),Y9) )
                 => ( aa(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),$o,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)),X2),Y2)),fun_pair_leq)
                   => ~ pw_leq(X6,Y9) ) ) ) ) ) ).

% pw_leq.cases
tff(fact_6497_pw__leq_Osimps,axiom,
    ! [A1: multiset(product_prod(nat,nat)),A22: multiset(product_prod(nat,nat))] :
      ( pw_leq(A1,A22)
    <=> ( ( ( A1 = zero_zero(multiset(product_prod(nat,nat))) )
          & ( A22 = zero_zero(multiset(product_prod(nat,nat))) ) )
        | ? [X4: product_prod(nat,nat),Y3: product_prod(nat,nat),X8: multiset(product_prod(nat,nat)),Y8: multiset(product_prod(nat,nat))] :
            ( ( A1 = aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(multiset(product_prod(nat,nat)),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),plus_plus(multiset(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(product_prod(nat,nat),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),add_mset(product_prod(nat,nat)),X4),zero_zero(multiset(product_prod(nat,nat))))),X8) )
            & ( A22 = aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(multiset(product_prod(nat,nat)),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),plus_plus(multiset(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(product_prod(nat,nat),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),add_mset(product_prod(nat,nat)),Y3),zero_zero(multiset(product_prod(nat,nat))))),Y8) )
            & aa(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),$o,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)),X4),Y3)),fun_pair_leq)
            & pw_leq(X8,Y8) ) ) ) ).

% pw_leq.simps
tff(fact_6498_pw__leq__step,axiom,
    ! [X: product_prod(nat,nat),Y: product_prod(nat,nat),X5: multiset(product_prod(nat,nat)),Y4: multiset(product_prod(nat,nat))] :
      ( aa(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),$o,member(product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),X),Y)),fun_pair_leq)
     => ( pw_leq(X5,Y4)
       => pw_leq(aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(multiset(product_prod(nat,nat)),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),plus_plus(multiset(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(product_prod(nat,nat),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),add_mset(product_prod(nat,nat)),X),zero_zero(multiset(product_prod(nat,nat))))),X5),aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(multiset(product_prod(nat,nat)),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),plus_plus(multiset(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(product_prod(nat,nat),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),add_mset(product_prod(nat,nat)),Y),zero_zero(multiset(product_prod(nat,nat))))),Y4)) ) ) ).

% pw_leq_step
tff(fact_6499_ms__weakI2,axiom,
    ! [Z5: multiset(product_prod(nat,nat)),Z10: multiset(product_prod(nat,nat))] :
      ( pw_leq(Z5,Z10)
     => aa(set(product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)))),$o,member(product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),fun(multiset(product_prod(nat,nat)),product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)))),product_Pair(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(multiset(product_prod(nat,nat)),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),plus_plus(multiset(product_prod(nat,nat))),Z5),zero_zero(multiset(product_prod(nat,nat))))),aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(multiset(product_prod(nat,nat)),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),plus_plus(multiset(product_prod(nat,nat))),Z10),zero_zero(multiset(product_prod(nat,nat)))))),ms_weak) ) ).

% ms_weakI2
tff(fact_6500_pw__leq__split,axiom,
    ! [X5: multiset(product_prod(nat,nat)),Y4: multiset(product_prod(nat,nat))] :
      ( pw_leq(X5,Y4)
     => ? [A8: multiset(product_prod(nat,nat)),B10: multiset(product_prod(nat,nat)),Z9: multiset(product_prod(nat,nat))] :
          ( ( X5 = aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(multiset(product_prod(nat,nat)),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),plus_plus(multiset(product_prod(nat,nat))),A8),Z9) )
          & ( Y4 = aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(multiset(product_prod(nat,nat)),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),plus_plus(multiset(product_prod(nat,nat))),B10),Z9) )
          & ( aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),$o,member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),set(product_prod(nat,nat)),set_mset(product_prod(nat,nat)),A8)),aa(multiset(product_prod(nat,nat)),set(product_prod(nat,nat)),set_mset(product_prod(nat,nat)),B10))),fun_max_strict)
            | ( ( B10 = zero_zero(multiset(product_prod(nat,nat))) )
              & ( A8 = zero_zero(multiset(product_prod(nat,nat))) ) ) ) ) ) ).

% pw_leq_split
tff(fact_6501_multp__code__def,axiom,
    ! [A: $tType,P: fun(A,fun(A,$o)),N4: multiset(A),M4: multiset(A)] :
      ( multp_code(A,P,N4,M4)
    <=> $let(
          z: multiset(A),
          z:= aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),M4),N4),
          $let(
            x2: multiset(A),
            x2:= aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),M4),z),
            ( ( x2 != zero_zero(multiset(A)) )
            & ! [X4: A] :
                ( aa(set(A),$o,member(A,X4),aa(multiset(A),set(A),set_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),N4),z)))
               => ? [Y3: A] :
                    ( aa(set(A),$o,member(A,Y3),aa(multiset(A),set(A),set_mset(A),x2))
                    & aa(A,$o,aa(A,fun(A,$o),P,X4),Y3) ) ) ) ) ) ) ).

% multp_code_def
tff(fact_6502_count__image__mset,axiom,
    ! [A: $tType,B: $tType,F: fun(B,A),A4: multiset(B),X: A] : aa(A,nat,aa(multiset(A),fun(A,nat),count(A),aa(multiset(B),multiset(A),image_mset(B,A,F),A4)),X) = aa(set(B),nat,aa(fun(B,nat),fun(set(B),nat),groups7311177749621191930dd_sum(B,nat),aa(multiset(B),fun(B,nat),count(B),A4)),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(set(A),set(B),aa(fun(B,A),fun(set(A),set(B)),vimage(B,A),F),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))))),aa(multiset(B),set(B),set_mset(B),A4))) ).

% count_image_mset
tff(fact_6503_set__mset__inter,axiom,
    ! [A: $tType,A4: multiset(A),B3: multiset(A)] : aa(multiset(A),set(A),set_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),A4),B3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(multiset(A),set(A),set_mset(A),A4)),aa(multiset(A),set(A),set_mset(A),B3)) ).

% set_mset_inter
tff(fact_6504_count__greater__eq__one__iff,axiom,
    ! [A: $tType,M4: multiset(A),X: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),aa(A,nat,aa(multiset(A),fun(A,nat),count(A),M4),X))
    <=> aa(set(A),$o,member(A,X),aa(multiset(A),set(A),set_mset(A),M4)) ) ).

% count_greater_eq_one_iff
tff(fact_6505_count__mset__set_I1_J,axiom,
    ! [A: $tType,A4: set(A),X: A] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( aa(set(A),$o,member(A,X),A4)
       => ( aa(A,nat,aa(multiset(A),fun(A,nat),count(A),mset_set(A,A4)),X) = one_one(nat) ) ) ) ).

% count_mset_set(1)
tff(fact_6506_count__sum,axiom,
    ! [A: $tType,B: $tType,F: fun(B,multiset(A)),A4: set(B),X: A] : aa(A,nat,aa(multiset(A),fun(A,nat),count(A),aa(set(B),multiset(A),aa(fun(B,multiset(A)),fun(set(B),multiset(A)),groups7311177749621191930dd_sum(B,multiset(A)),F),A4)),X) = aa(set(B),nat,aa(fun(B,nat),fun(set(B),nat),groups7311177749621191930dd_sum(B,nat),aa(A,fun(B,nat),aTP_Lamp_app(fun(B,multiset(A)),fun(A,fun(B,nat)),F),X)),A4) ).

% count_sum
tff(fact_6507_set__mset__def,axiom,
    ! [A: $tType,M4: multiset(A)] : aa(multiset(A),set(A),set_mset(A),M4) = aa(fun(A,$o),set(A),collect(A),aTP_Lamp_apq(multiset(A),fun(A,$o),M4)) ).

% set_mset_def
tff(fact_6508_set__mset__diff,axiom,
    ! [A: $tType,M4: multiset(A),N4: multiset(A)] : aa(multiset(A),set(A),set_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),M4),N4)) = aa(fun(A,$o),set(A),collect(A),aa(multiset(A),fun(A,$o),aTP_Lamp_apr(multiset(A),fun(multiset(A),fun(A,$o)),M4),N4)) ).

% set_mset_diff
tff(fact_6509_count__mset,axiom,
    ! [A: $tType,Xs: list(A),X: A] : aa(A,nat,aa(multiset(A),fun(A,nat),count(A),aa(list(A),multiset(A),mset(A),Xs)),X) = aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),filter2(A,aa(A,fun(A,$o),fequal(A),X)),Xs)) ).

% count_mset
tff(fact_6510_count__image__mset_H,axiom,
    ! [B: $tType,A: $tType,F: fun(B,A),X5: multiset(B),Y: A] : aa(A,nat,aa(multiset(A),fun(A,nat),count(A),aa(multiset(B),multiset(A),image_mset(B,A,F),X5)),Y) = aa(set(B),nat,aa(fun(B,nat),fun(set(B),nat),groups7311177749621191930dd_sum(B,nat),aa(multiset(B),fun(B,nat),count(B),X5)),aa(fun(B,$o),set(B),collect(B),aa(A,fun(B,$o),aa(multiset(B),fun(A,fun(B,$o)),aTP_Lamp_aps(fun(B,A),fun(multiset(B),fun(A,fun(B,$o))),F),X5),Y))) ).

% count_image_mset'
tff(fact_6511_count__induct,axiom,
    ! [A: $tType,Y: fun(A,nat),P: fun(fun(A,nat),$o)] :
      ( aa(set(fun(A,nat)),$o,member(fun(A,nat),Y),aa(fun(fun(A,nat),$o),set(fun(A,nat)),collect(fun(A,nat)),aTP_Lamp_apt(fun(A,nat),$o)))
     => ( ! [X2: multiset(A)] : aa(fun(A,nat),$o,P,aa(multiset(A),fun(A,nat),count(A),X2))
       => aa(fun(A,nat),$o,P,Y) ) ) ).

% count_induct
tff(fact_6512_count__cases,axiom,
    ! [A: $tType,Y: fun(A,nat)] :
      ( aa(set(fun(A,nat)),$o,member(fun(A,nat),Y),aa(fun(fun(A,nat),$o),set(fun(A,nat)),collect(fun(A,nat)),aTP_Lamp_apt(fun(A,nat),$o)))
     => ~ ! [X2: multiset(A)] : Y != aa(multiset(A),fun(A,nat),count(A),X2) ) ).

% count_cases
tff(fact_6513_count,axiom,
    ! [A: $tType,X: multiset(A)] : aa(set(fun(A,nat)),$o,member(fun(A,nat),aa(multiset(A),fun(A,nat),count(A),X)),aa(fun(fun(A,nat),$o),set(fun(A,nat)),collect(fun(A,nat)),aTP_Lamp_apt(fun(A,nat),$o))) ).

% count
tff(fact_6514_Inf__multiset_Orep__eq,axiom,
    ! [A: $tType,X: set(multiset(A)),X3: A] :
      aa(A,nat,aa(multiset(A),fun(A,nat),count(A),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),X)),X3) = $ite(aa(set(multiset(A)),set(fun(A,nat)),image2(multiset(A),fun(A,nat),count(A)),X) = bot_bot(set(fun(A,nat))),zero_zero(nat),aa(set(nat),nat,complete_Inf_Inf(nat),aa(set(fun(A,nat)),set(nat),image2(fun(A,nat),nat,aTP_Lamp_amb(A,fun(fun(A,nat),nat),X3)),aa(set(multiset(A)),set(fun(A,nat)),image2(multiset(A),fun(A,nat),count(A)),X)))) ).

% Inf_multiset.rep_eq
tff(fact_6515_count__Inf__multiset__nonempty,axiom,
    ! [A: $tType,A4: set(multiset(A)),X: A] :
      ( ( A4 != bot_bot(set(multiset(A))) )
     => ( aa(A,nat,aa(multiset(A),fun(A,nat),count(A),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),A4)),X) = aa(set(nat),nat,complete_Inf_Inf(nat),aa(set(multiset(A)),set(nat),image2(multiset(A),nat,aTP_Lamp_apu(A,fun(multiset(A),nat),X)),A4)) ) ) ).

% count_Inf_multiset_nonempty
tff(fact_6516_count__single,axiom,
    ! [A: $tType,B2: A,A3: A] :
      aa(A,nat,aa(multiset(A),fun(A,nat),count(A),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),B2),zero_zero(multiset(A)))),A3) = $ite(B2 = A3,one_one(nat),zero_zero(nat)) ).

% count_single
tff(fact_6517_count__mset__set__finite__iff,axiom,
    ! [A: $tType,S: set(A),A3: A] :
      ( aa(set(A),$o,finite_finite2(A),S)
     => ( aa(A,nat,aa(multiset(A),fun(A,nat),count(A),mset_set(A,S)),A3) = $ite(aa(set(A),$o,member(A,A3),S),one_one(nat),zero_zero(nat)) ) ) ).

% count_mset_set_finite_iff
tff(fact_6518_count__mset__set_H,axiom,
    ! [A: $tType,A4: set(A),X: A] :
      aa(A,nat,aa(multiset(A),fun(A,nat),count(A),mset_set(A,A4)),X) = $ite(
        ( aa(set(A),$o,finite_finite2(A),A4)
        & aa(set(A),$o,member(A,X),A4) ),
        one_one(nat),
        zero_zero(nat) ) ).

% count_mset_set'
tff(fact_6519_fold__mset__def,axiom,
    ! [A: $tType,B: $tType,F: fun(B,fun(A,A)),S2: A,M4: multiset(B)] : fold_mset(B,A,F,S2,M4) = finite_fold(B,A,aa(multiset(B),fun(B,fun(A,A)),aTP_Lamp_apv(fun(B,fun(A,A)),fun(multiset(B),fun(B,fun(A,A))),F),M4),S2,aa(multiset(B),set(B),set_mset(B),M4)) ).

% fold_mset_def
tff(fact_6520_distinct__count__atmost__1,axiom,
    ! [A: $tType,X: list(A)] :
      ( distinct(A,X)
    <=> ! [A10: A] :
          aa(A,nat,aa(multiset(A),fun(A,nat),count(A),aa(list(A),multiset(A),mset(A),X)),A10) = $ite(aa(set(A),$o,member(A,A10),aa(list(A),set(A),set2(A),X)),one_one(nat),zero_zero(nat)) ) ).

% distinct_count_atmost_1
tff(fact_6521_mult__cancel__max0,axiom,
    ! [A: $tType,S2: set(product_prod(A,A)),X5: multiset(A),Y4: multiset(A)] :
      ( trans(A,S2)
     => ( irrefl(A,S2)
       => ( aa(set(product_prod(multiset(A),multiset(A))),$o,member(product_prod(multiset(A),multiset(A)),aa(multiset(A),product_prod(multiset(A),multiset(A)),aa(multiset(A),fun(multiset(A),product_prod(multiset(A),multiset(A))),product_Pair(multiset(A),multiset(A)),X5),Y4)),mult(A,S2))
        <=> aa(set(product_prod(multiset(A),multiset(A))),$o,member(product_prod(multiset(A),multiset(A)),aa(multiset(A),product_prod(multiset(A),multiset(A)),aa(multiset(A),fun(multiset(A),product_prod(multiset(A),multiset(A))),product_Pair(multiset(A),multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),X5),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),X5),Y4))),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),Y4),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),X5),Y4)))),mult(A,S2)) ) ) ) ).

% mult_cancel_max0
tff(fact_6522_multeqp__code__def,axiom,
    ! [A: $tType,P: fun(A,fun(A,$o)),N4: multiset(A),M4: multiset(A)] :
      ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),multeqp_code(A,P),N4),M4)
    <=> $let(
          z: multiset(A),
          z:= aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),M4),N4),
          ! [X4: A] :
            ( aa(set(A),$o,member(A,X4),aa(multiset(A),set(A),set_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),N4),z)))
           => ? [Y3: A] :
                ( aa(set(A),$o,member(A,Y3),aa(multiset(A),set(A),set_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),M4),z)))
                & aa(A,$o,aa(A,fun(A,$o),P,X4),Y3) ) ) ) ) ).

% multeqp_code_def
tff(fact_6523_size__multiset__eq,axiom,
    ! [A: $tType,F: fun(A,nat),M4: multiset(A)] : aa(multiset(A),nat,size_multiset(A,F),M4) = aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aa(multiset(A),fun(A,nat),aTP_Lamp_apw(fun(A,nat),fun(multiset(A),fun(A,nat)),F),M4)),aa(multiset(A),set(A),set_mset(A),M4)) ).

% size_multiset_eq
tff(fact_6524_set__mset__replicate__mset__subset,axiom,
    ! [A: $tType,N: nat,X: A] :
      aa(multiset(A),set(A),set_mset(A),replicate_mset(A,N,X)) = $ite(N = zero_zero(nat),bot_bot(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) ).

% set_mset_replicate_mset_subset
tff(fact_6525_Inf__multiset__def,axiom,
    ! [A: $tType] : complete_Inf_Inf(multiset(A)) = aa(fun(set(fun(A,nat)),fun(A,nat)),fun(set(multiset(A)),multiset(A)),map_fun(set(multiset(A)),set(fun(A,nat)),fun(A,nat),multiset(A),image2(multiset(A),fun(A,nat),count(A)),abs_multiset(A)),aTP_Lamp_apx(set(fun(A,nat)),fun(A,nat))) ).

% Inf_multiset_def
tff(fact_6526_count__Abs__multiset,axiom,
    ! [A: $tType,F: fun(A,nat)] :
      ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_it(fun(A,nat),fun(A,$o),F)))
     => ( aa(multiset(A),fun(A,nat),count(A),aa(fun(A,nat),multiset(A),abs_multiset(A),F)) = F ) ) ).

% count_Abs_multiset
tff(fact_6527_zero__multiset__def,axiom,
    ! [A: $tType] : zero_zero(multiset(A)) = aa(fun(A,nat),multiset(A),abs_multiset(A),aTP_Lamp_apy(A,nat)) ).

% zero_multiset_def
tff(fact_6528_Abs__multiset__inject,axiom,
    ! [A: $tType,X: fun(A,nat),Y: fun(A,nat)] :
      ( aa(set(fun(A,nat)),$o,member(fun(A,nat),X),aa(fun(fun(A,nat),$o),set(fun(A,nat)),collect(fun(A,nat)),aTP_Lamp_apt(fun(A,nat),$o)))
     => ( aa(set(fun(A,nat)),$o,member(fun(A,nat),Y),aa(fun(fun(A,nat),$o),set(fun(A,nat)),collect(fun(A,nat)),aTP_Lamp_apt(fun(A,nat),$o)))
       => ( ( aa(fun(A,nat),multiset(A),abs_multiset(A),X) = aa(fun(A,nat),multiset(A),abs_multiset(A),Y) )
        <=> ( X = Y ) ) ) ) ).

% Abs_multiset_inject
tff(fact_6529_Abs__multiset__induct,axiom,
    ! [A: $tType,P: fun(multiset(A),$o),X: multiset(A)] :
      ( ! [Y2: fun(A,nat)] :
          ( aa(set(fun(A,nat)),$o,member(fun(A,nat),Y2),aa(fun(fun(A,nat),$o),set(fun(A,nat)),collect(fun(A,nat)),aTP_Lamp_apt(fun(A,nat),$o)))
         => aa(multiset(A),$o,P,aa(fun(A,nat),multiset(A),abs_multiset(A),Y2)) )
     => aa(multiset(A),$o,P,X) ) ).

% Abs_multiset_induct
tff(fact_6530_Abs__multiset__cases,axiom,
    ! [A: $tType,X: multiset(A)] :
      ~ ! [Y2: fun(A,nat)] :
          ( ( X = aa(fun(A,nat),multiset(A),abs_multiset(A),Y2) )
         => ~ aa(set(fun(A,nat)),$o,member(fun(A,nat),Y2),aa(fun(fun(A,nat),$o),set(fun(A,nat)),collect(fun(A,nat)),aTP_Lamp_apt(fun(A,nat),$o))) ) ).

% Abs_multiset_cases
tff(fact_6531_plus__multiset__def,axiom,
    ! [A: $tType] : plus_plus(multiset(A)) = aa(fun(fun(A,nat),fun(fun(A,nat),fun(A,nat))),fun(multiset(A),fun(multiset(A),multiset(A))),map_fun(multiset(A),fun(A,nat),fun(fun(A,nat),fun(A,nat)),fun(multiset(A),multiset(A)),count(A),map_fun(multiset(A),fun(A,nat),fun(A,nat),multiset(A),count(A),abs_multiset(A))),aTP_Lamp_apz(fun(A,nat),fun(fun(A,nat),fun(A,nat)))) ).

% plus_multiset_def
tff(fact_6532_minus__multiset__def,axiom,
    ! [A: $tType] : minus_minus(multiset(A)) = aa(fun(fun(A,nat),fun(fun(A,nat),fun(A,nat))),fun(multiset(A),fun(multiset(A),multiset(A))),map_fun(multiset(A),fun(A,nat),fun(fun(A,nat),fun(A,nat)),fun(multiset(A),multiset(A)),count(A),map_fun(multiset(A),fun(A,nat),fun(A,nat),multiset(A),count(A),abs_multiset(A))),aTP_Lamp_aqa(fun(A,nat),fun(fun(A,nat),fun(A,nat)))) ).

% minus_multiset_def
tff(fact_6533_Abs__multiset__inverse,axiom,
    ! [A: $tType,Y: fun(A,nat)] :
      ( aa(set(fun(A,nat)),$o,member(fun(A,nat),Y),aa(fun(fun(A,nat),$o),set(fun(A,nat)),collect(fun(A,nat)),aTP_Lamp_apt(fun(A,nat),$o)))
     => ( aa(multiset(A),fun(A,nat),count(A),aa(fun(A,nat),multiset(A),abs_multiset(A),Y)) = Y ) ) ).

% Abs_multiset_inverse
tff(fact_6534_type__definition__multiset,axiom,
    ! [A: $tType] : type_definition(multiset(A),fun(A,nat),count(A),abs_multiset(A),aa(fun(fun(A,nat),$o),set(fun(A,nat)),collect(fun(A,nat)),aTP_Lamp_apt(fun(A,nat),$o))) ).

% type_definition_multiset
tff(fact_6535_add__mset__def,axiom,
    ! [A: $tType] : add_mset(A) = aa(fun(A,fun(fun(A,nat),fun(A,nat))),fun(A,fun(multiset(A),multiset(A))),map_fun(A,A,fun(fun(A,nat),fun(A,nat)),fun(multiset(A),multiset(A)),id(A),map_fun(multiset(A),fun(A,nat),fun(A,nat),multiset(A),count(A),abs_multiset(A))),aTP_Lamp_aqb(A,fun(fun(A,nat),fun(A,nat)))) ).

% add_mset_def
tff(fact_6536_size__Diff__singleton,axiom,
    ! [A: $tType,X: A,M4: multiset(A)] :
      ( aa(set(A),$o,member(A,X),aa(multiset(A),set(A),set_mset(A),M4))
     => ( aa(multiset(A),nat,size_size(multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),M4),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),X),zero_zero(multiset(A))))) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(multiset(A),nat,size_size(multiset(A)),M4)),one_one(nat)) ) ) ).

% size_Diff_singleton
tff(fact_6537_size__Diff__singleton__if,axiom,
    ! [A: $tType,A4: multiset(A),X: A] :
      aa(multiset(A),nat,size_size(multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),A4),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),X),zero_zero(multiset(A))))) = $ite(aa(set(A),$o,member(A,X),aa(multiset(A),set(A),set_mset(A),A4)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(multiset(A),nat,size_size(multiset(A)),A4)),one_one(nat)),aa(multiset(A),nat,size_size(multiset(A)),A4)) ).

% size_Diff_singleton_if
tff(fact_6538_image__mset__const__eq,axiom,
    ! [B: $tType,A: $tType,C2: A,M4: multiset(B)] : aa(multiset(B),multiset(A),image_mset(B,A,aa(A,fun(B,A),aTP_Lamp_kf(A,fun(B,A)),C2)),M4) = replicate_mset(A,aa(multiset(B),nat,size_size(multiset(B)),M4),C2) ).

% image_mset_const_eq
tff(fact_6539_size__multiset__overloaded__def,axiom,
    ! [A: $tType] : size_size(multiset(A)) = size_multiset(A,aTP_Lamp_apy(A,nat)) ).

% size_multiset_overloaded_def
tff(fact_6540_size__1__singleton__mset,axiom,
    ! [A: $tType,M4: multiset(A)] :
      ( ( aa(multiset(A),nat,size_size(multiset(A)),M4) = one_one(nat) )
     => ? [A6: A] : M4 = aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A6),zero_zero(multiset(A))) ) ).

% size_1_singleton_mset
tff(fact_6541_size__single,axiom,
    ! [A: $tType,B2: A] : aa(multiset(A),nat,size_size(multiset(A)),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),B2),zero_zero(multiset(A)))) = one_one(nat) ).

% size_single
tff(fact_6542_size__multiset__overloaded__eq,axiom,
    ! [A: $tType,X: multiset(A)] : aa(multiset(A),nat,size_size(multiset(A)),X) = aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aa(multiset(A),fun(A,nat),count(A),X)),aa(multiset(A),set(A),set_mset(A),X)) ).

% size_multiset_overloaded_eq
tff(fact_6543_mset__size1elem,axiom,
    ! [A: $tType,P: multiset(A),Q3: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(multiset(A),nat,size_size(multiset(A)),P)),one_one(nat))
     => ( aa(set(A),$o,member(A,Q3),aa(multiset(A),set(A),set_mset(A),P))
       => ( P = aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),Q3),zero_zero(multiset(A))) ) ) ) ).

% mset_size1elem
tff(fact_6544_size__diff__se,axiom,
    ! [A: $tType,T3: A,S: multiset(A)] :
      ( aa(set(A),$o,member(A,T3),aa(multiset(A),set(A),set_mset(A),S))
     => ( aa(multiset(A),nat,size_size(multiset(A)),S) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(multiset(A),nat,size_size(multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),S),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),T3),zero_zero(multiset(A)))))),one_one(nat)) ) ) ).

% size_diff_se
tff(fact_6545_repeat__mset__def,axiom,
    ! [A: $tType] : repeat_mset(A) = aa(fun(nat,fun(fun(A,nat),fun(A,nat))),fun(nat,fun(multiset(A),multiset(A))),map_fun(nat,nat,fun(fun(A,nat),fun(A,nat)),fun(multiset(A),multiset(A)),id(nat),map_fun(multiset(A),fun(A,nat),fun(A,nat),multiset(A),count(A),abs_multiset(A))),aTP_Lamp_aqc(nat,fun(fun(A,nat),fun(A,nat)))) ).

% repeat_mset_def
tff(fact_6546_sum__mset__constant,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(A)
     => ! [Y: A,A4: multiset(B)] : comm_m7189776963980413722m_mset(A,aa(multiset(B),multiset(A),image_mset(B,A,aTP_Lamp_ja(A,fun(B,A),Y)),A4)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(multiset(B),nat,size_size(multiset(B)),A4))),Y) ) ).

% sum_mset_constant
tff(fact_6547_Union__image__single__mset,axiom,
    ! [A: $tType,M: multiset(A)] : comm_m7189776963980413722m_mset(multiset(A),aa(multiset(A),multiset(multiset(A)),image_mset(A,multiset(A),aTP_Lamp_ajv(A,multiset(A))),M)) = M ).

% Union_image_single_mset
tff(fact_6548_sum__mset_Oneutral__const,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [A4: multiset(B)] : comm_m7189776963980413722m_mset(A,aa(multiset(B),multiset(A),image_mset(B,A,aTP_Lamp_cu(B,A)),A4)) = zero_zero(A) ) ).

% sum_mset.neutral_const
tff(fact_6549_set__mset__Union__mset,axiom,
    ! [A: $tType,MM: multiset(multiset(A))] : aa(multiset(A),set(A),set_mset(A),comm_m7189776963980413722m_mset(multiset(A),MM)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(multiset(A)),set(set(A)),image2(multiset(A),set(A),set_mset(A)),aa(multiset(multiset(A)),set(multiset(A)),set_mset(multiset(A)),MM))) ).

% set_mset_Union_mset
tff(fact_6550_sum__mset__replicate__mset,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [N: nat,A3: A] : comm_m7189776963980413722m_mset(A,replicate_mset(A,N,A3)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),A3) ) ).

% sum_mset_replicate_mset
tff(fact_6551_sum__mset_Oswap,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(B,fun(C,A)),B3: multiset(C),A4: multiset(B)] : comm_m7189776963980413722m_mset(A,aa(multiset(B),multiset(A),image_mset(B,A,aa(multiset(C),fun(B,A),aTP_Lamp_aqd(fun(B,fun(C,A)),fun(multiset(C),fun(B,A)),G),B3)),A4)) = comm_m7189776963980413722m_mset(A,aa(multiset(C),multiset(A),image_mset(C,A,aa(multiset(B),fun(C,A),aTP_Lamp_aqe(fun(B,fun(C,A)),fun(multiset(B),fun(C,A)),G),A4)),B3)) ) ).

% sum_mset.swap
tff(fact_6552_sum__mset_Odistrib,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(B,A),H2: fun(B,A),A4: multiset(B)] : comm_m7189776963980413722m_mset(A,aa(multiset(B),multiset(A),image_mset(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_cy(fun(B,A),fun(fun(B,A),fun(B,A)),G),H2)),A4)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),comm_m7189776963980413722m_mset(A,aa(multiset(B),multiset(A),image_mset(B,A,G),A4))),comm_m7189776963980413722m_mset(A,aa(multiset(B),multiset(A),image_mset(B,A,H2),A4))) ) ).

% sum_mset.distrib
tff(fact_6553_sum__mset__distrib__left,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_0(A)
     => ! [C2: A,F: fun(B,A),M4: multiset(B)] : aa(A,A,aa(A,fun(A,A),times_times(A),C2),comm_m7189776963980413722m_mset(A,aa(multiset(B),multiset(A),image_mset(B,A,F),M4))) = comm_m7189776963980413722m_mset(A,aa(multiset(B),multiset(A),image_mset(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_dc(A,fun(fun(B,A),fun(B,A)),C2),F)),M4)) ) ).

% sum_mset_distrib_left
tff(fact_6554_sum__mset__distrib__right,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_0(A)
     => ! [F: fun(B,A),M4: multiset(B),C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),comm_m7189776963980413722m_mset(A,aa(multiset(B),multiset(A),image_mset(B,A,F),M4))),C2) = comm_m7189776963980413722m_mset(A,aa(multiset(B),multiset(A),image_mset(B,A,aa(A,fun(B,A),aTP_Lamp_db(fun(B,A),fun(A,fun(B,A)),F),C2)),M4)) ) ).

% sum_mset_distrib_right
tff(fact_6555_sum__mset__product,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( ( comm_monoid_add(B)
        & times(B)
        & semiring_0(A) )
     => ! [F: fun(B,A),A4: multiset(B),G: fun(C,A),B3: multiset(C)] : aa(A,A,aa(A,fun(A,A),times_times(A),comm_m7189776963980413722m_mset(A,aa(multiset(B),multiset(A),image_mset(B,A,F),A4))),comm_m7189776963980413722m_mset(A,aa(multiset(C),multiset(A),image_mset(C,A,G),B3))) = comm_m7189776963980413722m_mset(A,aa(multiset(B),multiset(A),image_mset(B,A,aa(multiset(C),fun(B,A),aa(fun(C,A),fun(multiset(C),fun(B,A)),aTP_Lamp_aqg(fun(B,A),fun(fun(C,A),fun(multiset(C),fun(B,A))),F),G),B3)),A4)) ) ).

% sum_mset_product
tff(fact_6556_size__eq__sum__mset,axiom,
    ! [A: $tType,M4: multiset(A)] : aa(multiset(A),nat,size_size(multiset(A)),M4) = comm_m7189776963980413722m_mset(nat,aa(multiset(A),multiset(nat),image_mset(A,nat,aTP_Lamp_jf(A,nat)),M4)) ).

% size_eq_sum_mset
tff(fact_6557_sum__mset__delta,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_1(A)
     => ! [Y: B,C2: A,A4: multiset(B)] : comm_m7189776963980413722m_mset(A,aa(multiset(B),multiset(A),image_mset(B,A,aa(A,fun(B,A),aTP_Lamp_aqh(B,fun(A,fun(B,A)),Y),C2)),A4)) = aa(A,A,aa(A,fun(A,A),times_times(A),C2),aa(nat,A,semiring_1_of_nat(A),aa(B,nat,aa(multiset(B),fun(B,nat),count(B),A4),Y))) ) ).

% sum_mset_delta
tff(fact_6558_sum__mset__delta_H,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_1(A)
     => ! [Y: B,C2: A,A4: multiset(B)] : comm_m7189776963980413722m_mset(A,aa(multiset(B),multiset(A),image_mset(B,A,aa(A,fun(B,A),aTP_Lamp_aqi(B,fun(A,fun(B,A)),Y),C2)),A4)) = aa(A,A,aa(A,fun(A,A),times_times(A),C2),aa(nat,A,semiring_1_of_nat(A),aa(B,nat,aa(multiset(B),fun(B,nat),count(B),A4),Y))) ) ).

% sum_mset_delta'
tff(fact_6559_repeat__mset_Oabs__eq,axiom,
    ! [A: $tType,X: fun(A,nat),Xa: nat] :
      ( aa(fun(A,nat),$o,aa(fun(A,nat),fun(fun(A,nat),$o),bNF_eq_onp(fun(A,nat),aTP_Lamp_apt(fun(A,nat),$o)),X),X)
     => ( aa(multiset(A),multiset(A),aa(nat,fun(multiset(A),multiset(A)),repeat_mset(A),Xa),aa(fun(A,nat),multiset(A),abs_multiset(A),X)) = aa(fun(A,nat),multiset(A),abs_multiset(A),aa(nat,fun(A,nat),aTP_Lamp_aqj(fun(A,nat),fun(nat,fun(A,nat)),X),Xa)) ) ) ).

% repeat_mset.abs_eq
tff(fact_6560_filter__mset__def,axiom,
    ! [A: $tType] : filter_mset(A) = aa(fun(fun(A,$o),fun(fun(A,nat),fun(A,nat))),fun(fun(A,$o),fun(multiset(A),multiset(A))),map_fun(fun(A,$o),fun(A,$o),fun(fun(A,nat),fun(A,nat)),fun(multiset(A),multiset(A)),id(fun(A,$o)),map_fun(multiset(A),fun(A,nat),fun(A,nat),multiset(A),count(A),abs_multiset(A))),aTP_Lamp_aqk(fun(A,$o),fun(fun(A,nat),fun(A,nat)))) ).

% filter_mset_def
tff(fact_6561_filter__mset__True,axiom,
    ! [A: $tType,M4: multiset(A)] : aa(multiset(A),multiset(A),aa(fun(A,$o),fun(multiset(A),multiset(A)),filter_mset(A),aTP_Lamp_ar(A,$o)),M4) = M4 ).

% filter_mset_True
tff(fact_6562_filter__mset__False,axiom,
    ! [A: $tType,M4: multiset(A)] : aa(multiset(A),multiset(A),aa(fun(A,$o),fun(multiset(A),multiset(A)),filter_mset(A),aTP_Lamp_ak(A,$o)),M4) = zero_zero(multiset(A)) ).

% filter_mset_False
tff(fact_6563_set__mset__filter,axiom,
    ! [A: $tType,P: fun(A,$o),M4: multiset(A)] : aa(multiset(A),set(A),set_mset(A),aa(multiset(A),multiset(A),aa(fun(A,$o),fun(multiset(A),multiset(A)),filter_mset(A),P),M4)) = aa(fun(A,$o),set(A),collect(A),aa(multiset(A),fun(A,$o),aTP_Lamp_aql(fun(A,$o),fun(multiset(A),fun(A,$o)),P),M4)) ).

% set_mset_filter
tff(fact_6564_filter__mset__mset__set,axiom,
    ! [A: $tType,A4: set(A),P: fun(A,$o)] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( aa(multiset(A),multiset(A),aa(fun(A,$o),fun(multiset(A),multiset(A)),filter_mset(A),P),mset_set(A,A4)) = mset_set(A,aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_bi(set(A),fun(fun(A,$o),fun(A,$o)),A4),P))) ) ) ).

% filter_mset_mset_set
tff(fact_6565_mset__filter,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] : aa(list(A),multiset(A),mset(A),aa(list(A),list(A),filter2(A,P),Xs)) = aa(multiset(A),multiset(A),aa(fun(A,$o),fun(multiset(A),multiset(A)),filter_mset(A),P),aa(list(A),multiset(A),mset(A),Xs)) ).

% mset_filter
tff(fact_6566_filter__mset_Oabs__eq,axiom,
    ! [A: $tType,X: fun(A,nat),Xa: fun(A,$o)] :
      ( aa(fun(A,nat),$o,aa(fun(A,nat),fun(fun(A,nat),$o),bNF_eq_onp(fun(A,nat),aTP_Lamp_apt(fun(A,nat),$o)),X),X)
     => ( aa(multiset(A),multiset(A),aa(fun(A,$o),fun(multiset(A),multiset(A)),filter_mset(A),Xa),aa(fun(A,nat),multiset(A),abs_multiset(A),X)) = aa(fun(A,nat),multiset(A),abs_multiset(A),aa(fun(A,$o),fun(A,nat),aTP_Lamp_aqm(fun(A,nat),fun(fun(A,$o),fun(A,nat)),X),Xa)) ) ) ).

% filter_mset.abs_eq
tff(fact_6567_multiset__partition,axiom,
    ! [A: $tType,M4: multiset(A),P: fun(A,$o)] : M4 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(fun(A,$o),fun(multiset(A),multiset(A)),filter_mset(A),P),M4)),aa(multiset(A),multiset(A),aa(fun(A,$o),fun(multiset(A),multiset(A)),filter_mset(A),aTP_Lamp_aj(fun(A,$o),fun(A,$o),P)),M4)) ).

% multiset_partition
tff(fact_6568_rel__fun__eq__eq__onp,axiom,
    ! [A: $tType,B: $tType,P: fun(B,$o)] : bNF_rel_fun(A,A,B,B,fequal(A),bNF_eq_onp(B,P)) = bNF_eq_onp(fun(A,B),aTP_Lamp_aqn(fun(B,$o),fun(fun(A,B),$o),P)) ).

% rel_fun_eq_eq_onp
tff(fact_6569_eq__onp__def,axiom,
    ! [A: $tType,R: fun(A,$o),X3: A,Xa3: A] :
      ( aa(A,$o,aa(A,fun(A,$o),bNF_eq_onp(A,R),X3),Xa3)
    <=> ( aa(A,$o,R,X3)
        & ( X3 = Xa3 ) ) ) ).

% eq_onp_def
tff(fact_6570_eq__onp__True,axiom,
    ! [A: $tType] : bNF_eq_onp(A,aTP_Lamp_ar(A,$o)) = fequal(A) ).

% eq_onp_True
tff(fact_6571_filter__filter__mset,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o),M4: multiset(A)] : aa(multiset(A),multiset(A),aa(fun(A,$o),fun(multiset(A),multiset(A)),filter_mset(A),P),aa(multiset(A),multiset(A),aa(fun(A,$o),fun(multiset(A),multiset(A)),filter_mset(A),Q),M4)) = aa(multiset(A),multiset(A),aa(fun(A,$o),fun(multiset(A),multiset(A)),filter_mset(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_yx(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q)),M4) ).

% filter_filter_mset
tff(fact_6572_eq__onp__top__eq__eq,axiom,
    ! [A: $tType] : bNF_eq_onp(A,top_top(fun(A,$o))) = fequal(A) ).

% eq_onp_top_eq_eq
tff(fact_6573_Quotient__crel__typedef,axiom,
    ! [A: $tType,B: $tType,P: fun(A,$o),Abs: fun(A,B),Rep: fun(B,A),T2: fun(A,fun(B,$o))] :
      ( quotient(A,B,bNF_eq_onp(A,P),Abs,Rep,T2)
     => ! [X3: A,Xa3: B] :
          ( aa(B,$o,aa(A,fun(B,$o),T2,X3),Xa3)
        <=> ( X3 = aa(B,A,Rep,Xa3) ) ) ) ).

% Quotient_crel_typedef
tff(fact_6574_Quotient__eq__onp__typedef,axiom,
    ! [B: $tType,A: $tType,P: fun(A,$o),Abs: fun(A,B),Rep: fun(B,A),Cr: fun(A,fun(B,$o))] :
      ( quotient(A,B,bNF_eq_onp(A,P),Abs,Rep,Cr)
     => type_definition(B,A,Rep,Abs,aa(fun(A,$o),set(A),collect(A),P)) ) ).

% Quotient_eq_onp_typedef
tff(fact_6575_open__typedef__to__Quotient,axiom,
    ! [A: $tType,B: $tType,Rep: fun(A,B),Abs: fun(B,A),P: fun(B,$o),T2: fun(B,fun(A,$o))] :
      ( type_definition(A,B,Rep,Abs,aa(fun(B,$o),set(B),collect(B),P))
     => ( ! [X2: B,Xa4: A] :
            ( aa(A,$o,aa(B,fun(A,$o),T2,X2),Xa4)
          <=> ( X2 = aa(A,B,Rep,Xa4) ) )
       => quotient(B,A,bNF_eq_onp(B,P),Abs,Rep,T2) ) ) ).

% open_typedef_to_Quotient
tff(fact_6576_typedef__to__Quotient,axiom,
    ! [A: $tType,B: $tType,Rep: fun(A,B),Abs: fun(B,A),S: set(B),T2: fun(B,fun(A,$o))] :
      ( type_definition(A,B,Rep,Abs,S)
     => ( ! [X2: B,Xa4: A] :
            ( aa(A,$o,aa(B,fun(A,$o),T2,X2),Xa4)
          <=> ( X2 = aa(A,B,Rep,Xa4) ) )
       => quotient(B,A,bNF_eq_onp(B,aTP_Lamp_aks(set(B),fun(B,$o),S)),Abs,Rep,T2) ) ) ).

% typedef_to_Quotient
tff(fact_6577_image__mset__If,axiom,
    ! [A: $tType,B: $tType,P: fun(B,$o),F: fun(B,A),G: fun(B,A),A4: multiset(B)] : aa(multiset(B),multiset(A),image_mset(B,A,aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_kc(fun(B,$o),fun(fun(B,A),fun(fun(B,A),fun(B,A))),P),F),G)),A4) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(B),multiset(A),image_mset(B,A,F),aa(multiset(B),multiset(B),aa(fun(B,$o),fun(multiset(B),multiset(B)),filter_mset(B),P),A4))),aa(multiset(B),multiset(A),image_mset(B,A,G),aa(multiset(B),multiset(B),aa(fun(B,$o),fun(multiset(B),multiset(B)),filter_mset(B),aTP_Lamp_kd(fun(B,$o),fun(B,$o),P)),A4))) ).

% image_mset_If
tff(fact_6578_rel__fun__eq__onp__rel,axiom,
    ! [B: $tType,C: $tType,A: $tType,R: fun(A,$o),S: fun(B,fun(C,$o)),X3: fun(A,B),Xa3: fun(A,C)] :
      ( aa(fun(A,C),$o,aa(fun(A,B),fun(fun(A,C),$o),bNF_rel_fun(A,A,B,C,bNF_eq_onp(A,R),S),X3),Xa3)
    <=> ! [Xb4: A] :
          ( aa(A,$o,R,Xb4)
         => aa(C,$o,aa(B,fun(C,$o),S,aa(A,B,X3,Xb4)),aa(A,C,Xa3,Xb4)) ) ) ).

% rel_fun_eq_onp_rel
tff(fact_6579_filter__eq__replicate__mset,axiom,
    ! [A: $tType,X: A,D4: multiset(A)] : aa(multiset(A),multiset(A),aa(fun(A,$o),fun(multiset(A),multiset(A)),filter_mset(A),aTP_Lamp_cf(A,fun(A,$o),X)),D4) = replicate_mset(A,aa(A,nat,aa(multiset(A),fun(A,nat),count(A),D4),X),X) ).

% filter_eq_replicate_mset
tff(fact_6580_filter__mset_Orsp,axiom,
    ! [A: $tType] : aa(fun(fun(A,$o),fun(fun(A,nat),fun(A,nat))),$o,aa(fun(fun(A,$o),fun(fun(A,nat),fun(A,nat))),fun(fun(fun(A,$o),fun(fun(A,nat),fun(A,nat))),$o),bNF_rel_fun(fun(A,$o),fun(A,$o),fun(fun(A,nat),fun(A,nat)),fun(fun(A,nat),fun(A,nat)),fequal(fun(A,$o)),bNF_rel_fun(fun(A,nat),fun(A,nat),fun(A,nat),fun(A,nat),bNF_eq_onp(fun(A,nat),aTP_Lamp_apt(fun(A,nat),$o)),bNF_eq_onp(fun(A,nat),aTP_Lamp_apt(fun(A,nat),$o)))),aTP_Lamp_aqk(fun(A,$o),fun(fun(A,nat),fun(A,nat)))),aTP_Lamp_aqk(fun(A,$o),fun(fun(A,nat),fun(A,nat)))) ).

% filter_mset.rsp
tff(fact_6581_zero__multiset_Orsp,axiom,
    ! [A: $tType] : aa(fun(A,nat),$o,aa(fun(A,nat),fun(fun(A,nat),$o),bNF_eq_onp(fun(A,nat),aTP_Lamp_apt(fun(A,nat),$o)),aTP_Lamp_apy(A,nat)),aTP_Lamp_apy(A,nat)) ).

% zero_multiset.rsp
tff(fact_6582_add__mset_Orsp,axiom,
    ! [A: $tType] : aa(fun(A,fun(fun(A,nat),fun(A,nat))),$o,aa(fun(A,fun(fun(A,nat),fun(A,nat))),fun(fun(A,fun(fun(A,nat),fun(A,nat))),$o),bNF_rel_fun(A,A,fun(fun(A,nat),fun(A,nat)),fun(fun(A,nat),fun(A,nat)),fequal(A),bNF_rel_fun(fun(A,nat),fun(A,nat),fun(A,nat),fun(A,nat),bNF_eq_onp(fun(A,nat),aTP_Lamp_apt(fun(A,nat),$o)),bNF_eq_onp(fun(A,nat),aTP_Lamp_apt(fun(A,nat),$o)))),aTP_Lamp_aqb(A,fun(fun(A,nat),fun(A,nat)))),aTP_Lamp_aqb(A,fun(fun(A,nat),fun(A,nat)))) ).

% add_mset.rsp
tff(fact_6583_plus__multiset_Orsp,axiom,
    ! [A: $tType] : aa(fun(fun(A,nat),fun(fun(A,nat),fun(A,nat))),$o,aa(fun(fun(A,nat),fun(fun(A,nat),fun(A,nat))),fun(fun(fun(A,nat),fun(fun(A,nat),fun(A,nat))),$o),bNF_rel_fun(fun(A,nat),fun(A,nat),fun(fun(A,nat),fun(A,nat)),fun(fun(A,nat),fun(A,nat)),bNF_eq_onp(fun(A,nat),aTP_Lamp_apt(fun(A,nat),$o)),bNF_rel_fun(fun(A,nat),fun(A,nat),fun(A,nat),fun(A,nat),bNF_eq_onp(fun(A,nat),aTP_Lamp_apt(fun(A,nat),$o)),bNF_eq_onp(fun(A,nat),aTP_Lamp_apt(fun(A,nat),$o)))),aTP_Lamp_apz(fun(A,nat),fun(fun(A,nat),fun(A,nat)))),aTP_Lamp_apz(fun(A,nat),fun(fun(A,nat),fun(A,nat)))) ).

% plus_multiset.rsp
tff(fact_6584_minus__multiset_Orsp,axiom,
    ! [A: $tType] : aa(fun(fun(A,nat),fun(fun(A,nat),fun(A,nat))),$o,aa(fun(fun(A,nat),fun(fun(A,nat),fun(A,nat))),fun(fun(fun(A,nat),fun(fun(A,nat),fun(A,nat))),$o),bNF_rel_fun(fun(A,nat),fun(A,nat),fun(fun(A,nat),fun(A,nat)),fun(fun(A,nat),fun(A,nat)),bNF_eq_onp(fun(A,nat),aTP_Lamp_apt(fun(A,nat),$o)),bNF_rel_fun(fun(A,nat),fun(A,nat),fun(A,nat),fun(A,nat),bNF_eq_onp(fun(A,nat),aTP_Lamp_apt(fun(A,nat),$o)),bNF_eq_onp(fun(A,nat),aTP_Lamp_apt(fun(A,nat),$o)))),aTP_Lamp_aqa(fun(A,nat),fun(fun(A,nat),fun(A,nat)))),aTP_Lamp_aqa(fun(A,nat),fun(fun(A,nat),fun(A,nat)))) ).

% minus_multiset.rsp
tff(fact_6585_repeat__mset_Orsp,axiom,
    ! [A: $tType] : aa(fun(nat,fun(fun(A,nat),fun(A,nat))),$o,aa(fun(nat,fun(fun(A,nat),fun(A,nat))),fun(fun(nat,fun(fun(A,nat),fun(A,nat))),$o),bNF_rel_fun(nat,nat,fun(fun(A,nat),fun(A,nat)),fun(fun(A,nat),fun(A,nat)),fequal(nat),bNF_rel_fun(fun(A,nat),fun(A,nat),fun(A,nat),fun(A,nat),bNF_eq_onp(fun(A,nat),aTP_Lamp_apt(fun(A,nat),$o)),bNF_eq_onp(fun(A,nat),aTP_Lamp_apt(fun(A,nat),$o)))),aTP_Lamp_aqc(nat,fun(fun(A,nat),fun(A,nat)))),aTP_Lamp_aqc(nat,fun(fun(A,nat),fun(A,nat)))) ).

% repeat_mset.rsp
tff(fact_6586_add__mset_Oabs__eq,axiom,
    ! [A: $tType,X: fun(A,nat),Xa: A] :
      ( aa(fun(A,nat),$o,aa(fun(A,nat),fun(fun(A,nat),$o),bNF_eq_onp(fun(A,nat),aTP_Lamp_apt(fun(A,nat),$o)),X),X)
     => ( aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),Xa),aa(fun(A,nat),multiset(A),abs_multiset(A),X)) = aa(fun(A,nat),multiset(A),abs_multiset(A),aa(A,fun(A,nat),aTP_Lamp_aqo(fun(A,nat),fun(A,fun(A,nat)),X),Xa)) ) ) ).

% add_mset.abs_eq
tff(fact_6587_plus__multiset_Oabs__eq,axiom,
    ! [A: $tType,Xa: fun(A,nat),X: fun(A,nat)] :
      ( aa(fun(A,nat),$o,aa(fun(A,nat),fun(fun(A,nat),$o),bNF_eq_onp(fun(A,nat),aTP_Lamp_apt(fun(A,nat),$o)),Xa),Xa)
     => ( aa(fun(A,nat),$o,aa(fun(A,nat),fun(fun(A,nat),$o),bNF_eq_onp(fun(A,nat),aTP_Lamp_apt(fun(A,nat),$o)),X),X)
       => ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(fun(A,nat),multiset(A),abs_multiset(A),Xa)),aa(fun(A,nat),multiset(A),abs_multiset(A),X)) = aa(fun(A,nat),multiset(A),abs_multiset(A),aa(fun(A,nat),fun(A,nat),aa(fun(A,nat),fun(fun(A,nat),fun(A,nat)),aTP_Lamp_apz(fun(A,nat),fun(fun(A,nat),fun(A,nat))),Xa),X)) ) ) ) ).

% plus_multiset.abs_eq
tff(fact_6588_minus__multiset_Oabs__eq,axiom,
    ! [A: $tType,Xa: fun(A,nat),X: fun(A,nat)] :
      ( aa(fun(A,nat),$o,aa(fun(A,nat),fun(fun(A,nat),$o),bNF_eq_onp(fun(A,nat),aTP_Lamp_apt(fun(A,nat),$o)),Xa),Xa)
     => ( aa(fun(A,nat),$o,aa(fun(A,nat),fun(fun(A,nat),$o),bNF_eq_onp(fun(A,nat),aTP_Lamp_apt(fun(A,nat),$o)),X),X)
       => ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),aa(fun(A,nat),multiset(A),abs_multiset(A),Xa)),aa(fun(A,nat),multiset(A),abs_multiset(A),X)) = aa(fun(A,nat),multiset(A),abs_multiset(A),aa(fun(A,nat),fun(A,nat),aa(fun(A,nat),fun(fun(A,nat),fun(A,nat)),aTP_Lamp_aqa(fun(A,nat),fun(fun(A,nat),fun(A,nat))),Xa),X)) ) ) ) ).

% minus_multiset.abs_eq
tff(fact_6589_Quotient__multiset,axiom,
    ! [A: $tType] : quotient(fun(A,nat),multiset(A),bNF_eq_onp(fun(A,nat),aTP_Lamp_apt(fun(A,nat),$o)),abs_multiset(A),count(A),cr_multiset(A)) ).

% Quotient_multiset
tff(fact_6590_Inf__multiset_Oabs__eq,axiom,
    ! [A: $tType,X: set(fun(A,nat))] :
      ( aa(set(fun(A,nat)),$o,aa(set(fun(A,nat)),fun(set(fun(A,nat)),$o),bNF_rel_set(fun(A,nat),fun(A,nat),bNF_eq_onp(fun(A,nat),aTP_Lamp_apt(fun(A,nat),$o))),X),X)
     => ( aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),aa(set(fun(A,nat)),set(multiset(A)),image2(fun(A,nat),multiset(A),abs_multiset(A)),X)) = aa(fun(A,nat),multiset(A),abs_multiset(A),aa(set(fun(A,nat)),fun(A,nat),aTP_Lamp_apx(set(fun(A,nat)),fun(A,nat)),X)) ) ) ).

% Inf_multiset.abs_eq
tff(fact_6591_rel__set__def,axiom,
    ! [A: $tType,B: $tType,R: fun(A,fun(B,$o)),X3: set(A),Xa3: set(B)] :
      ( aa(set(B),$o,aa(set(A),fun(set(B),$o),bNF_rel_set(A,B,R),X3),Xa3)
    <=> ( ! [Xb4: A] :
            ( aa(set(A),$o,member(A,Xb4),X3)
           => ? [Xc2: B] :
                ( aa(set(B),$o,member(B,Xc2),Xa3)
                & aa(B,$o,aa(A,fun(B,$o),R,Xb4),Xc2) ) )
        & ! [Xb4: B] :
            ( aa(set(B),$o,member(B,Xb4),Xa3)
           => ? [Xc2: A] :
                ( aa(set(A),$o,member(A,Xc2),X3)
                & aa(B,$o,aa(A,fun(B,$o),R,Xc2),Xb4) ) ) ) ) ).

% rel_set_def
tff(fact_6592_fun_Oset__transfer,axiom,
    ! [B: $tType,C: $tType,A: $tType,R: fun(B,fun(C,$o))] : aa(fun(fun(A,C),set(C)),$o,aa(fun(fun(A,B),set(B)),fun(fun(fun(A,C),set(C)),$o),bNF_rel_fun(fun(A,B),fun(A,C),set(B),set(C),bNF_rel_fun(A,A,B,C,fequal(A),R),bNF_rel_set(B,C,R)),aTP_Lamp_aqp(fun(A,B),set(B))),aTP_Lamp_aqq(fun(A,C),set(C))) ).

% fun.set_transfer
tff(fact_6593_cr__multiset__def,axiom,
    ! [A: $tType,X3: fun(A,nat),Xa3: multiset(A)] :
      ( aa(multiset(A),$o,aa(fun(A,nat),fun(multiset(A),$o),cr_multiset(A),X3),Xa3)
    <=> ( X3 = aa(multiset(A),fun(A,nat),count(A),Xa3) ) ) ).

% cr_multiset_def
tff(fact_6594_Inf__multiset_Orsp,axiom,
    ! [A: $tType] : aa(fun(set(fun(A,nat)),fun(A,nat)),$o,aa(fun(set(fun(A,nat)),fun(A,nat)),fun(fun(set(fun(A,nat)),fun(A,nat)),$o),bNF_rel_fun(set(fun(A,nat)),set(fun(A,nat)),fun(A,nat),fun(A,nat),bNF_rel_set(fun(A,nat),fun(A,nat),bNF_eq_onp(fun(A,nat),aTP_Lamp_apt(fun(A,nat),$o))),bNF_eq_onp(fun(A,nat),aTP_Lamp_apt(fun(A,nat),$o))),aTP_Lamp_apx(set(fun(A,nat)),fun(A,nat))),aTP_Lamp_apx(set(fun(A,nat)),fun(A,nat))) ).

% Inf_multiset.rsp
tff(fact_6595_INF__parametric,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( complete_Inf(C)
     => ! [A4: fun(A,fun(B,$o))] : aa(fun(set(B),fun(fun(B,C),C)),$o,aa(fun(set(A),fun(fun(A,C),C)),fun(fun(set(B),fun(fun(B,C),C)),$o),bNF_rel_fun(set(A),set(B),fun(fun(A,C),C),fun(fun(B,C),C),bNF_rel_set(A,B,A4),bNF_rel_fun(fun(A,C),fun(B,C),C,C,bNF_rel_fun(A,B,C,C,A4,fequal(C)),fequal(C))),aTP_Lamp_aqr(set(A),fun(fun(A,C),C))),aTP_Lamp_aqs(set(B),fun(fun(B,C),C))) ) ).

% INF_parametric
tff(fact_6596_SUP__parametric,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( complete_Sup(C)
     => ! [R: fun(A,fun(B,$o))] : aa(fun(set(B),fun(fun(B,C),C)),$o,aa(fun(set(A),fun(fun(A,C),C)),fun(fun(set(B),fun(fun(B,C),C)),$o),bNF_rel_fun(set(A),set(B),fun(fun(A,C),C),fun(fun(B,C),C),bNF_rel_set(A,B,R),bNF_rel_fun(fun(A,C),fun(B,C),C,C,bNF_rel_fun(A,B,C,C,R,fequal(C)),fequal(C))),aTP_Lamp_aqt(set(A),fun(fun(A,C),C))),aTP_Lamp_aqu(set(B),fun(fun(B,C),C))) ) ).

% SUP_parametric
tff(fact_6597_empty__transfer,axiom,
    ! [A: $tType,B: $tType,A4: fun(A,fun(B,$o))] : aa(set(B),$o,aa(set(A),fun(set(B),$o),bNF_rel_set(A,B,A4),bot_bot(set(A))),bot_bot(set(B))) ).

% empty_transfer
tff(fact_6598_union__transfer,axiom,
    ! [A: $tType,B: $tType,A4: fun(A,fun(B,$o))] : aa(fun(set(B),fun(set(B),set(B))),$o,aa(fun(set(A),fun(set(A),set(A))),fun(fun(set(B),fun(set(B),set(B))),$o),bNF_rel_fun(set(A),set(B),fun(set(A),set(A)),fun(set(B),set(B)),bNF_rel_set(A,B,A4),bNF_rel_fun(set(A),set(B),set(A),set(B),bNF_rel_set(A,B,A4),bNF_rel_set(A,B,A4))),sup_sup(set(A))),sup_sup(set(B))) ).

% union_transfer
tff(fact_6599_Union__transfer,axiom,
    ! [A: $tType,B: $tType,A4: fun(A,fun(B,$o))] : aa(fun(set(set(B)),set(B)),$o,aa(fun(set(set(A)),set(A)),fun(fun(set(set(B)),set(B)),$o),bNF_rel_fun(set(set(A)),set(set(B)),set(A),set(B),bNF_rel_set(set(A),set(B),bNF_rel_set(A,B,A4)),bNF_rel_set(A,B,A4)),complete_Sup_Sup(set(A))),complete_Sup_Sup(set(B))) ).

% Union_transfer
tff(fact_6600_set__relator__eq__onp,axiom,
    ! [A: $tType,P: fun(A,$o)] : bNF_rel_set(A,A,bNF_eq_onp(A,P)) = bNF_eq_onp(set(A),aTP_Lamp_aqv(fun(A,$o),fun(set(A),$o),P)) ).

% set_relator_eq_onp
tff(fact_6601_UNION__transfer,axiom,
    ! [A: $tType,C: $tType,D: $tType,B: $tType,A4: fun(A,fun(B,$o)),B3: fun(C,fun(D,$o))] : aa(fun(set(B),fun(fun(B,set(D)),set(D))),$o,aa(fun(set(A),fun(fun(A,set(C)),set(C))),fun(fun(set(B),fun(fun(B,set(D)),set(D))),$o),bNF_rel_fun(set(A),set(B),fun(fun(A,set(C)),set(C)),fun(fun(B,set(D)),set(D)),bNF_rel_set(A,B,A4),bNF_rel_fun(fun(A,set(C)),fun(B,set(D)),set(C),set(D),bNF_rel_fun(A,B,set(C),set(D),A4,bNF_rel_set(C,D,B3)),bNF_rel_set(C,D,B3))),aTP_Lamp_aqw(set(A),fun(fun(A,set(C)),set(C)))),aTP_Lamp_aqx(set(B),fun(fun(B,set(D)),set(D)))) ).

% UNION_transfer
tff(fact_6602_Inf__multiset_Otransfer,axiom,
    ! [A: $tType] : aa(fun(set(multiset(A)),multiset(A)),$o,aa(fun(set(fun(A,nat)),fun(A,nat)),fun(fun(set(multiset(A)),multiset(A)),$o),bNF_rel_fun(set(fun(A,nat)),set(multiset(A)),fun(A,nat),multiset(A),bNF_rel_set(fun(A,nat),multiset(A),pcr_multiset(A,A,fequal(A))),pcr_multiset(A,A,fequal(A))),aTP_Lamp_apx(set(fun(A,nat)),fun(A,nat))),complete_Inf_Inf(multiset(A))) ).

% Inf_multiset.transfer
tff(fact_6603_multp__def,axiom,
    ! [A: $tType,R2: fun(A,fun(A,$o)),M4: multiset(A),N4: multiset(A)] :
      ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),multp(A,R2),M4),N4)
    <=> aa(set(product_prod(multiset(A),multiset(A))),$o,member(product_prod(multiset(A),multiset(A)),aa(multiset(A),product_prod(multiset(A),multiset(A)),aa(multiset(A),fun(multiset(A),product_prod(multiset(A),multiset(A))),product_Pair(multiset(A),multiset(A)),M4),N4)),mult(A,aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),R2)))) ) ).

% multp_def
tff(fact_6604_zero__multiset_Otransfer,axiom,
    ! [A: $tType] : aa(multiset(A),$o,aa(fun(A,nat),fun(multiset(A),$o),pcr_multiset(A,A,fequal(A)),aTP_Lamp_apy(A,nat)),zero_zero(multiset(A))) ).

% zero_multiset.transfer
tff(fact_6605_multiset_Orep__transfer,axiom,
    ! [A: $tType,B: $tType,T2: fun(A,fun(B,$o))] : aa(fun(multiset(B),fun(B,nat)),$o,aa(fun(fun(A,nat),fun(A,nat)),fun(fun(multiset(B),fun(B,nat)),$o),bNF_rel_fun(fun(A,nat),multiset(B),fun(A,nat),fun(B,nat),pcr_multiset(A,B,T2),bNF_rel_fun(A,B,nat,nat,T2,fequal(nat))),aTP_Lamp_aqy(fun(A,nat),fun(A,nat))),count(B)) ).

% multiset.rep_transfer
tff(fact_6606_add__mset_Otransfer,axiom,
    ! [A: $tType] : aa(fun(A,fun(multiset(A),multiset(A))),$o,aa(fun(A,fun(fun(A,nat),fun(A,nat))),fun(fun(A,fun(multiset(A),multiset(A))),$o),bNF_rel_fun(A,A,fun(fun(A,nat),fun(A,nat)),fun(multiset(A),multiset(A)),fequal(A),bNF_rel_fun(fun(A,nat),multiset(A),fun(A,nat),multiset(A),pcr_multiset(A,A,fequal(A)),pcr_multiset(A,A,fequal(A)))),aTP_Lamp_aqb(A,fun(fun(A,nat),fun(A,nat)))),add_mset(A)) ).

% add_mset.transfer
tff(fact_6607_plus__multiset_Otransfer,axiom,
    ! [A: $tType] : aa(fun(multiset(A),fun(multiset(A),multiset(A))),$o,aa(fun(fun(A,nat),fun(fun(A,nat),fun(A,nat))),fun(fun(multiset(A),fun(multiset(A),multiset(A))),$o),bNF_rel_fun(fun(A,nat),multiset(A),fun(fun(A,nat),fun(A,nat)),fun(multiset(A),multiset(A)),pcr_multiset(A,A,fequal(A)),bNF_rel_fun(fun(A,nat),multiset(A),fun(A,nat),multiset(A),pcr_multiset(A,A,fequal(A)),pcr_multiset(A,A,fequal(A)))),aTP_Lamp_apz(fun(A,nat),fun(fun(A,nat),fun(A,nat)))),plus_plus(multiset(A))) ).

% plus_multiset.transfer
tff(fact_6608_minus__multiset_Otransfer,axiom,
    ! [A: $tType] : aa(fun(multiset(A),fun(multiset(A),multiset(A))),$o,aa(fun(fun(A,nat),fun(fun(A,nat),fun(A,nat))),fun(fun(multiset(A),fun(multiset(A),multiset(A))),$o),bNF_rel_fun(fun(A,nat),multiset(A),fun(fun(A,nat),fun(A,nat)),fun(multiset(A),multiset(A)),pcr_multiset(A,A,fequal(A)),bNF_rel_fun(fun(A,nat),multiset(A),fun(A,nat),multiset(A),pcr_multiset(A,A,fequal(A)),pcr_multiset(A,A,fequal(A)))),aTP_Lamp_aqa(fun(A,nat),fun(fun(A,nat),fun(A,nat)))),minus_minus(multiset(A))) ).

% minus_multiset.transfer
tff(fact_6609_filter__mset_Otransfer,axiom,
    ! [A: $tType] : aa(fun(fun(A,$o),fun(multiset(A),multiset(A))),$o,aa(fun(fun(A,$o),fun(fun(A,nat),fun(A,nat))),fun(fun(fun(A,$o),fun(multiset(A),multiset(A))),$o),bNF_rel_fun(fun(A,$o),fun(A,$o),fun(fun(A,nat),fun(A,nat)),fun(multiset(A),multiset(A)),fequal(fun(A,$o)),bNF_rel_fun(fun(A,nat),multiset(A),fun(A,nat),multiset(A),pcr_multiset(A,A,fequal(A)),pcr_multiset(A,A,fequal(A)))),aTP_Lamp_aqk(fun(A,$o),fun(fun(A,nat),fun(A,nat)))),filter_mset(A)) ).

% filter_mset.transfer
tff(fact_6610_repeat__mset_Otransfer,axiom,
    ! [A: $tType] : aa(fun(nat,fun(multiset(A),multiset(A))),$o,aa(fun(nat,fun(fun(A,nat),fun(A,nat))),fun(fun(nat,fun(multiset(A),multiset(A))),$o),bNF_rel_fun(nat,nat,fun(fun(A,nat),fun(A,nat)),fun(multiset(A),multiset(A)),fequal(nat),bNF_rel_fun(fun(A,nat),multiset(A),fun(A,nat),multiset(A),pcr_multiset(A,A,fequal(A)),pcr_multiset(A,A,fequal(A)))),aTP_Lamp_aqc(nat,fun(fun(A,nat),fun(A,nat)))),repeat_mset(A)) ).

% repeat_mset.transfer
tff(fact_6611_multiset_Odomain,axiom,
    ! [A: $tType,B: $tType,T2: fun(A,fun(B,$o)),X3: fun(A,nat)] :
      ( aa(fun(A,nat),$o,aa(fun(fun(A,nat),fun(multiset(B),$o)),fun(fun(A,nat),$o),domainp(fun(A,nat),multiset(B)),pcr_multiset(A,B,T2)),X3)
    <=> ? [Y3: fun(B,nat)] :
          ( aa(fun(B,nat),$o,aa(fun(A,nat),fun(fun(B,nat),$o),bNF_rel_fun(A,B,nat,nat,T2,fequal(nat)),X3),Y3)
          & aa(set(B),$o,finite_finite2(B),aa(fun(B,$o),set(B),collect(B),aTP_Lamp_aqz(fun(B,nat),fun(B,$o),Y3))) ) ) ).

% multiset.domain
tff(fact_6612_prod__mset_Oremove,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [X: A,A4: multiset(A)] :
          ( aa(set(A),$o,member(A,X),aa(multiset(A),set(A),set_mset(A),A4))
         => ( aa(multiset(A),A,comm_m9189036328036947845d_mset(A),A4) = aa(A,A,aa(A,fun(A,A),times_times(A),X),aa(multiset(A),A,comm_m9189036328036947845d_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),A4),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),X),zero_zero(multiset(A)))))) ) ) ) ).

% prod_mset.remove
tff(fact_6613_prod__mset__empty,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ( aa(multiset(A),A,comm_m9189036328036947845d_mset(A),zero_zero(multiset(A))) = one_one(A) ) ) ).

% prod_mset_empty
tff(fact_6614_prod__mset_Oadd__mset,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [X: A,N4: multiset(A)] : aa(multiset(A),A,comm_m9189036328036947845d_mset(A),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),X),N4)) = aa(A,A,aa(A,fun(A,A),times_times(A),X),aa(multiset(A),A,comm_m9189036328036947845d_mset(A),N4)) ) ).

% prod_mset.add_mset
tff(fact_6615_prod__mset__Un,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A4: multiset(A),B3: multiset(A)] : aa(multiset(A),A,comm_m9189036328036947845d_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A4),B3)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(multiset(A),A,comm_m9189036328036947845d_mset(A),A4)),aa(multiset(A),A,comm_m9189036328036947845d_mset(A),B3)) ) ).

% prod_mset_Un
tff(fact_6616_prod__mset_Ounion,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M4: multiset(A),N4: multiset(A)] : aa(multiset(A),A,comm_m9189036328036947845d_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M4),N4)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(multiset(A),A,comm_m9189036328036947845d_mset(A),M4)),aa(multiset(A),A,comm_m9189036328036947845d_mset(A),N4)) ) ).

% prod_mset.union
tff(fact_6617_prod__mset_Oneutral__const,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A4: multiset(B)] : aa(multiset(A),A,comm_m9189036328036947845d_mset(A),aa(multiset(B),multiset(A),image_mset(B,A,aTP_Lamp_fb(B,A)),A4)) = one_one(A) ) ).

% prod_mset.neutral_const
tff(fact_6618_prod__mset_Oinsert,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(B,A),X: B,A4: multiset(B)] : aa(multiset(A),A,comm_m9189036328036947845d_mset(A),aa(multiset(B),multiset(A),image_mset(B,A,G),aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),X),A4))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,G,X)),aa(multiset(A),A,comm_m9189036328036947845d_mset(A),aa(multiset(B),multiset(A),image_mset(B,A,G),A4))) ) ).

% prod_mset.insert
tff(fact_6619_prod__mset__constant,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [C2: A,A4: multiset(B)] : aa(multiset(A),A,comm_m9189036328036947845d_mset(A),aa(multiset(B),multiset(A),image_mset(B,A,aTP_Lamp_iz(A,fun(B,A),C2)),A4)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),C2),aa(multiset(B),nat,size_size(multiset(B)),A4)) ) ).

% prod_mset_constant
tff(fact_6620_Domain__eq__top,axiom,
    ! [A: $tType] : aa(fun(A,fun(A,$o)),fun(A,$o),domainp(A,A),fequal(A)) = top_top(fun(A,$o)) ).

% Domain_eq_top
tff(fact_6621_pcr__Domainp,axiom,
    ! [C: $tType,B: $tType,A: $tType,B3: fun(A,fun(B,$o)),P: fun(A,$o),A4: fun(C,fun(A,$o))] :
      ( ( aa(fun(A,fun(B,$o)),fun(A,$o),domainp(A,B),B3) = P )
     => ! [X3: C] :
          ( aa(C,$o,aa(fun(C,fun(B,$o)),fun(C,$o),domainp(C,B),aa(fun(A,fun(B,$o)),fun(C,fun(B,$o)),aa(fun(C,fun(A,$o)),fun(fun(A,fun(B,$o)),fun(C,fun(B,$o))),relcompp(C,A,B),A4),B3)),X3)
        <=> ? [Y3: A] :
              ( aa(A,$o,aa(C,fun(A,$o),A4,X3),Y3)
              & aa(A,$o,P,Y3) ) ) ) ).

% pcr_Domainp
tff(fact_6622_prod__mset_Oswap,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(B,fun(C,A)),B3: multiset(C),A4: multiset(B)] : aa(multiset(A),A,comm_m9189036328036947845d_mset(A),aa(multiset(B),multiset(A),image_mset(B,A,aa(multiset(C),fun(B,A),aTP_Lamp_ara(fun(B,fun(C,A)),fun(multiset(C),fun(B,A)),G),B3)),A4)) = aa(multiset(A),A,comm_m9189036328036947845d_mset(A),aa(multiset(C),multiset(A),image_mset(C,A,aa(multiset(B),fun(C,A),aTP_Lamp_arb(fun(B,fun(C,A)),fun(multiset(B),fun(C,A)),G),A4)),B3)) ) ).

% prod_mset.swap
tff(fact_6623_prod__mset_Oneutral,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A4: multiset(A)] :
          ( ! [X2: A] :
              ( aa(set(A),$o,member(A,X2),aa(multiset(A),set(A),set_mset(A),A4))
             => ( X2 = one_one(A) ) )
         => ( aa(multiset(A),A,comm_m9189036328036947845d_mset(A),A4) = one_one(A) ) ) ) ).

% prod_mset.neutral
tff(fact_6624_prod__mset_Odistrib,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(B,A),H2: fun(B,A),A4: multiset(B)] : aa(multiset(A),A,comm_m9189036328036947845d_mset(A),aa(multiset(B),multiset(A),image_mset(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_fh(fun(B,A),fun(fun(B,A),fun(B,A)),G),H2)),A4)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(multiset(A),A,comm_m9189036328036947845d_mset(A),aa(multiset(B),multiset(A),image_mset(B,A,G),A4))),aa(multiset(A),A,comm_m9189036328036947845d_mset(A),aa(multiset(B),multiset(A),image_mset(B,A,H2),A4))) ) ).

% prod_mset.distrib
tff(fact_6625_is__unit__prod__mset__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A4: multiset(A)] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(multiset(A),A,comm_m9189036328036947845d_mset(A),A4)),one_one(A))
        <=> ! [X4: A] :
              ( aa(set(A),$o,member(A,X4),aa(multiset(A),set(A),set_mset(A),A4))
             => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),X4),one_one(A)) ) ) ) ).

% is_unit_prod_mset_iff
tff(fact_6626_pcr__Domainp__par,axiom,
    ! [A: $tType,B: $tType,C: $tType,B3: fun(A,fun(B,$o)),P23: fun(A,$o),A4: fun(C,fun(A,$o)),P12: fun(C,$o),P24: fun(C,$o)] :
      ( ( aa(fun(A,fun(B,$o)),fun(A,$o),domainp(A,B),B3) = P23 )
     => ( ( aa(fun(C,fun(A,$o)),fun(C,$o),domainp(C,A),A4) = P12 )
       => ( aa(fun(A,$o),$o,aa(fun(C,$o),fun(fun(A,$o),$o),bNF_rel_fun(C,A,$o,$o,A4,fequal($o)),P24),P23)
         => ( aa(fun(C,fun(B,$o)),fun(C,$o),domainp(C,B),aa(fun(A,fun(B,$o)),fun(C,fun(B,$o)),aa(fun(C,fun(A,$o)),fun(fun(A,fun(B,$o)),fun(C,fun(B,$o))),relcompp(C,A,B),A4),B3)) = aa(fun(C,$o),fun(C,$o),aa(fun(C,$o),fun(fun(C,$o),fun(C,$o)),inf_inf(fun(C,$o)),P12),P24) ) ) ) ) ).

% pcr_Domainp_par
tff(fact_6627_prod__mset_Oeq__fold,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M4: multiset(A)] : aa(multiset(A),A,comm_m9189036328036947845d_mset(A),M4) = fold_mset(A,A,times_times(A),one_one(A),M4) ) ).

% prod_mset.eq_fold
tff(fact_6628_multiset_Odomain__eq,axiom,
    ! [A: $tType,X3: fun(A,nat)] :
      ( aa(fun(A,nat),$o,aa(fun(fun(A,nat),fun(multiset(A),$o)),fun(fun(A,nat),$o),domainp(fun(A,nat),multiset(A)),pcr_multiset(A,A,fequal(A))),X3)
    <=> aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_it(fun(A,nat),fun(A,$o),X3))) ) ).

% multiset.domain_eq
tff(fact_6629_prod__mset__delta,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [Y: B,C2: A,A4: multiset(B)] : aa(multiset(A),A,comm_m9189036328036947845d_mset(A),aa(multiset(B),multiset(A),image_mset(B,A,aa(A,fun(B,A),aTP_Lamp_arc(B,fun(A,fun(B,A)),Y),C2)),A4)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),C2),aa(B,nat,aa(multiset(B),fun(B,nat),count(B),A4),Y)) ) ).

% prod_mset_delta
tff(fact_6630_prod__mset__delta_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [Y: B,C2: A,A4: multiset(B)] : aa(multiset(A),A,comm_m9189036328036947845d_mset(A),aa(multiset(B),multiset(A),image_mset(B,A,aa(A,fun(B,A),aTP_Lamp_ard(B,fun(A,fun(B,A)),Y),C2)),A4)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),C2),aa(B,nat,aa(multiset(B),fun(B,nat),count(B),A4),Y)) ) ).

% prod_mset_delta'
tff(fact_6631_prod__mset__multiplicity,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M4: multiset(A)] : aa(multiset(A),A,comm_m9189036328036947845d_mset(A),M4) = aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7121269368397514597t_prod(A,A),aTP_Lamp_are(multiset(A),fun(A,A),M4)),aa(multiset(A),set(A),set_mset(A),M4)) ) ).

% prod_mset_multiplicity
tff(fact_6632_multiset_Odomain__par__left__total,axiom,
    ! [B: $tType,A: $tType,T2: fun(A,fun(B,$o)),P4: fun(fun(A,nat),$o)] :
      ( left_total(fun(A,nat),fun(B,nat),bNF_rel_fun(A,B,nat,nat,T2,fequal(nat)))
     => ( aa(fun(fun(B,nat),$o),$o,aa(fun(fun(A,nat),$o),fun(fun(fun(B,nat),$o),$o),bNF_rel_fun(fun(A,nat),fun(B,nat),$o,$o,bNF_rel_fun(A,B,nat,nat,T2,fequal(nat)),fequal($o)),P4),aTP_Lamp_arf(fun(B,nat),$o))
       => ( aa(fun(fun(A,nat),fun(multiset(B),$o)),fun(fun(A,nat),$o),domainp(fun(A,nat),multiset(B)),pcr_multiset(A,B,T2)) = P4 ) ) ) ).

% multiset.domain_par_left_total
tff(fact_6633_composed__equiv__rel__eq__onp,axiom,
    ! [B: $tType,A: $tType,R: fun(A,fun(B,$o)),P: fun(A,$o),P4: fun(B,$o),P8: fun(A,$o)] :
      ( left_unique(A,B,R)
     => ( aa(fun(B,$o),$o,aa(fun(A,$o),fun(fun(B,$o),$o),bNF_rel_fun(A,B,$o,$o,R,fequal($o)),P),P4)
       => ( ( aa(fun(A,fun(B,$o)),fun(A,$o),domainp(A,B),R) = P8 )
         => ( aa(fun(B,fun(A,$o)),fun(A,fun(A,$o)),aa(fun(A,fun(B,$o)),fun(fun(B,fun(A,$o)),fun(A,fun(A,$o))),relcompp(A,B,A),R),aa(fun(B,fun(A,$o)),fun(B,fun(A,$o)),aa(fun(B,fun(B,$o)),fun(fun(B,fun(A,$o)),fun(B,fun(A,$o))),relcompp(B,B,A),bNF_eq_onp(B,P4)),conversep(A,B,R))) = bNF_eq_onp(A,aa(fun(A,$o),fun(A,$o),aa(fun(A,$o),fun(fun(A,$o),fun(A,$o)),inf_inf(fun(A,$o)),P8),P)) ) ) ) ) ).

% composed_equiv_rel_eq_onp
tff(fact_6634_typedef__left__unique,axiom,
    ! [A: $tType,B: $tType,Rep: fun(A,B),Abs: fun(B,A),A4: set(B),T2: fun(B,fun(A,$o))] :
      ( type_definition(A,B,Rep,Abs,A4)
     => ( ! [X2: B,Xa4: A] :
            ( aa(A,$o,aa(B,fun(A,$o),T2,X2),Xa4)
          <=> ( X2 = aa(A,B,Rep,Xa4) ) )
       => left_unique(B,A,T2) ) ) ).

% typedef_left_unique
tff(fact_6635_left__unique__iff,axiom,
    ! [A: $tType,B: $tType,R: fun(A,fun(B,$o))] :
      ( left_unique(A,B,R)
    <=> ! [Z4: B] : uniq(A,aa(B,fun(A,$o),aTP_Lamp_afh(fun(A,fun(B,$o)),fun(B,fun(A,$o)),R),Z4)) ) ).

% left_unique_iff
tff(fact_6636_rat_Odomain,axiom,
    ! [X3: product_prod(int,int)] :
      ( aa(product_prod(int,int),$o,aa(fun(product_prod(int,int),fun(rat,$o)),fun(product_prod(int,int),$o),domainp(product_prod(int,int),rat),pcr_rat),X3)
    <=> ? [Y3: product_prod(int,int)] :
          ( aa(product_prod(int,int),$o,aa(product_prod(int,int),fun(product_prod(int,int),$o),basic_rel_prod(int,int,int,int,fequal(int),fequal(int)),X3),Y3)
          & aa(product_prod(int,int),$o,aa(product_prod(int,int),fun(product_prod(int,int),$o),ratrel,Y3),Y3) ) ) ).

% rat.domain
tff(fact_6637_multiset_Odomain__par,axiom,
    ! [B: $tType,A: $tType,T2: fun(A,fun(B,$o)),DT: fun(A,$o),DS: fun(nat,$o),P24: fun(fun(A,nat),$o)] :
      ( ( aa(fun(A,fun(B,$o)),fun(A,$o),domainp(A,B),T2) = DT )
     => ( ( aa(fun(nat,fun(nat,$o)),fun(nat,$o),domainp(nat,nat),fequal(nat)) = DS )
       => ( left_unique(A,B,T2)
         => ( aa(fun(fun(B,nat),$o),$o,aa(fun(fun(A,nat),$o),fun(fun(fun(B,nat),$o),$o),bNF_rel_fun(fun(A,nat),fun(B,nat),$o,$o,bNF_rel_fun(A,B,nat,nat,T2,fequal(nat)),fequal($o)),P24),aTP_Lamp_arf(fun(B,nat),$o))
           => ( aa(fun(fun(A,nat),fun(multiset(B),$o)),fun(fun(A,nat),$o),domainp(fun(A,nat),multiset(B)),pcr_multiset(A,B,T2)) = aa(fun(fun(A,nat),$o),fun(fun(A,nat),$o),aa(fun(fun(A,nat),$o),fun(fun(fun(A,nat),$o),fun(fun(A,nat),$o)),inf_inf(fun(fun(A,nat),$o)),aa(fun(nat,$o),fun(fun(A,nat),$o),basic_pred_fun(A,nat,DT),DS)),P24) ) ) ) ) ) ).

% multiset.domain_par
tff(fact_6638_right__total__relcompp__transfer,axiom,
    ! [A: $tType,E: $tType,C: $tType,F3: $tType,B: $tType,D: $tType,B3: fun(A,fun(B,$o)),A4: fun(C,fun(D,$o)),C3: fun(E,fun(F3,$o))] :
      ( right_total(A,B,B3)
     => aa(fun(fun(D,fun(B,$o)),fun(fun(B,fun(F3,$o)),fun(D,fun(F3,$o)))),$o,aa(fun(fun(C,fun(A,$o)),fun(fun(A,fun(E,$o)),fun(C,fun(E,$o)))),fun(fun(fun(D,fun(B,$o)),fun(fun(B,fun(F3,$o)),fun(D,fun(F3,$o)))),$o),bNF_rel_fun(fun(C,fun(A,$o)),fun(D,fun(B,$o)),fun(fun(A,fun(E,$o)),fun(C,fun(E,$o))),fun(fun(B,fun(F3,$o)),fun(D,fun(F3,$o))),bNF_rel_fun(C,D,fun(A,$o),fun(B,$o),A4,bNF_rel_fun(A,B,$o,$o,B3,fequal($o))),bNF_rel_fun(fun(A,fun(E,$o)),fun(B,fun(F3,$o)),fun(C,fun(E,$o)),fun(D,fun(F3,$o)),bNF_rel_fun(A,B,fun(E,$o),fun(F3,$o),B3,bNF_rel_fun(E,F3,$o,$o,C3,fequal($o))),bNF_rel_fun(C,D,fun(E,$o),fun(F3,$o),A4,bNF_rel_fun(E,F3,$o,$o,C3,fequal($o))))),aTP_Lamp_arg(fun(A,fun(B,$o)),fun(fun(C,fun(A,$o)),fun(fun(A,fun(E,$o)),fun(C,fun(E,$o)))),B3)),relcompp(D,B,F3)) ) ).

% right_total_relcompp_transfer
tff(fact_6639_pred__fun__def,axiom,
    ! [B: $tType,A: $tType,A4: fun(A,$o),B3: fun(B,$o),X3: fun(A,B)] :
      ( aa(fun(A,B),$o,aa(fun(B,$o),fun(fun(A,B),$o),basic_pred_fun(A,B,A4),B3),X3)
    <=> ! [Xa2: A] :
          ( aa(A,$o,A4,Xa2)
         => aa(B,$o,B3,aa(A,B,X3,Xa2)) ) ) ).

% pred_fun_def
tff(fact_6640_fun_Omap__cong__pred,axiom,
    ! [C: $tType,B: $tType,A: $tType,X: fun(A,B),Ya: fun(A,B),F: fun(B,C),G: fun(B,C)] :
      ( ( X = Ya )
     => ( aa(fun(A,B),$o,aa(fun(B,$o),fun(fun(A,B),$o),basic_pred_fun(A,B,aTP_Lamp_ar(A,$o)),aa(fun(B,C),fun(B,$o),aTP_Lamp_arh(fun(B,C),fun(fun(B,C),fun(B,$o)),F),G)),Ya)
       => ( aa(fun(A,B),fun(A,C),comp(B,C,A,F),X) = aa(fun(A,B),fun(A,C),comp(B,C,A,G),Ya) ) ) ) ).

% fun.map_cong_pred
tff(fact_6641_fun_Opred__True,axiom,
    ! [B: $tType,A: $tType,X3: fun(A,B)] : aa(fun(A,B),$o,aa(fun(B,$o),fun(fun(A,B),$o),basic_pred_fun(A,B,aTP_Lamp_ar(A,$o)),aTP_Lamp_ari(B,$o)),X3) ).

% fun.pred_True
tff(fact_6642_fun_Opred__mono,axiom,
    ! [B: $tType,A: $tType,P: fun(A,$o),Pa: fun(A,$o)] :
      ( aa(fun(A,$o),$o,aa(fun(A,$o),fun(fun(A,$o),$o),ord_less_eq(fun(A,$o)),P),Pa)
     => aa(fun(fun(B,A),$o),$o,aa(fun(fun(B,A),$o),fun(fun(fun(B,A),$o),$o),ord_less_eq(fun(fun(B,A),$o)),aa(fun(A,$o),fun(fun(B,A),$o),basic_pred_fun(B,A,aTP_Lamp_ari(B,$o)),P)),aa(fun(A,$o),fun(fun(B,A),$o),basic_pred_fun(B,A,aTP_Lamp_ari(B,$o)),Pa)) ) ).

% fun.pred_mono
tff(fact_6643_typedef__right__total,axiom,
    ! [A: $tType,B: $tType,Rep: fun(A,B),Abs: fun(B,A),A4: set(B),T2: fun(B,fun(A,$o))] :
      ( type_definition(A,B,Rep,Abs,A4)
     => ( ! [X2: B,Xa4: A] :
            ( aa(A,$o,aa(B,fun(A,$o),T2,X2),Xa4)
          <=> ( X2 = aa(A,B,Rep,Xa4) ) )
       => right_total(B,A,T2) ) ) ).

% typedef_right_total
tff(fact_6644_pred__fun__True__id,axiom,
    ! [A: $tType,B: $tType,C: $tType,P3: fun(B,$o),F: fun(C,B)] :
      ( nO_MATCH(fun(A,A),fun(B,$o),id(A),P3)
     => ( aa(fun(C,B),$o,aa(fun(B,$o),fun(fun(C,B),$o),basic_pred_fun(C,B,aTP_Lamp_arj(C,$o)),P3),F)
      <=> aa(fun(C,$o),$o,aa(fun($o,$o),fun(fun(C,$o),$o),basic_pred_fun(C,$o,aTP_Lamp_arj(C,$o)),id($o)),aa(fun(C,B),fun(C,$o),comp(B,$o,C,P3),F)) ) ) ).

% pred_fun_True_id
tff(fact_6645_right__total__UNIV__transfer,axiom,
    ! [A: $tType,B: $tType,A4: fun(A,fun(B,$o))] :
      ( right_total(A,B,A4)
     => aa(set(B),$o,aa(set(A),fun(set(B),$o),bNF_rel_set(A,B,A4),aa(fun(A,$o),set(A),collect(A),aa(fun(A,fun(B,$o)),fun(A,$o),domainp(A,B),A4))),top_top(set(B))) ) ).

% right_total_UNIV_transfer
tff(fact_6646_fun_Opred__mono__strong,axiom,
    ! [B: $tType,A: $tType,P: fun(B,$o),X: fun(A,B),Pa: fun(B,$o)] :
      ( aa(fun(A,B),$o,aa(fun(B,$o),fun(fun(A,B),$o),basic_pred_fun(A,B,aTP_Lamp_ar(A,$o)),P),X)
     => ( ! [Z3: B] :
            ( aa(set(B),$o,member(B,Z3),aa(set(A),set(B),image2(A,B,X),top_top(set(A))))
           => ( aa(B,$o,P,Z3)
             => aa(B,$o,Pa,Z3) ) )
       => aa(fun(A,B),$o,aa(fun(B,$o),fun(fun(A,B),$o),basic_pred_fun(A,B,aTP_Lamp_ar(A,$o)),Pa),X) ) ) ).

% fun.pred_mono_strong
tff(fact_6647_fun_Opred__cong,axiom,
    ! [B: $tType,A: $tType,X: fun(A,B),Ya: fun(A,B),P: fun(B,$o),Pa: fun(B,$o)] :
      ( ( X = Ya )
     => ( ! [Z3: B] :
            ( aa(set(B),$o,member(B,Z3),aa(set(A),set(B),image2(A,B,Ya),top_top(set(A))))
           => ( aa(B,$o,P,Z3)
            <=> aa(B,$o,Pa,Z3) ) )
       => ( aa(fun(A,B),$o,aa(fun(B,$o),fun(fun(A,B),$o),basic_pred_fun(A,B,aTP_Lamp_ar(A,$o)),P),X)
        <=> aa(fun(A,B),$o,aa(fun(B,$o),fun(fun(A,B),$o),basic_pred_fun(A,B,aTP_Lamp_ar(A,$o)),Pa),Ya) ) ) ) ).

% fun.pred_cong
tff(fact_6648_fun_Opred__rel,axiom,
    ! [B: $tType,A: $tType,P: fun(B,$o),X: fun(A,B)] :
      ( aa(fun(A,B),$o,aa(fun(B,$o),fun(fun(A,B),$o),basic_pred_fun(A,B,aTP_Lamp_ar(A,$o)),P),X)
    <=> aa(fun(A,B),$o,aa(fun(A,B),fun(fun(A,B),$o),bNF_rel_fun(A,A,B,B,fequal(A),bNF_eq_onp(B,P)),X),X) ) ).

% fun.pred_rel
tff(fact_6649_fun_ODomainp__rel,axiom,
    ! [A: $tType,C: $tType,B: $tType,R: fun(B,fun(C,$o))] : aa(fun(fun(A,B),fun(fun(A,C),$o)),fun(fun(A,B),$o),domainp(fun(A,B),fun(A,C)),bNF_rel_fun(A,A,B,C,fequal(A),R)) = aa(fun(B,$o),fun(fun(A,B),$o),basic_pred_fun(A,B,aTP_Lamp_ar(A,$o)),aa(fun(B,fun(C,$o)),fun(B,$o),domainp(B,C),R)) ).

% fun.Domainp_rel
tff(fact_6650_fun_Opred__map,axiom,
    ! [B: $tType,C: $tType,A: $tType,Q: fun(B,$o),F: fun(C,B),X: fun(A,C)] :
      ( aa(fun(A,B),$o,aa(fun(B,$o),fun(fun(A,B),$o),basic_pred_fun(A,B,aTP_Lamp_ar(A,$o)),Q),aa(fun(A,C),fun(A,B),comp(C,B,A,F),X))
    <=> aa(fun(A,C),$o,aa(fun(C,$o),fun(fun(A,C),$o),basic_pred_fun(A,C,aTP_Lamp_ar(A,$o)),aa(fun(C,B),fun(C,$o),comp(B,$o,C,Q),F)),X) ) ).

% fun.pred_map
tff(fact_6651_right__total__Collect__transfer,axiom,
    ! [A: $tType,B: $tType,A4: fun(A,fun(B,$o))] :
      ( right_total(A,B,A4)
     => aa(fun(fun(B,$o),set(B)),$o,aa(fun(fun(A,$o),set(A)),fun(fun(fun(B,$o),set(B)),$o),bNF_rel_fun(fun(A,$o),fun(B,$o),set(A),set(B),bNF_rel_fun(A,B,$o,$o,A4,fequal($o)),bNF_rel_set(A,B,A4)),aTP_Lamp_arl(fun(A,fun(B,$o)),fun(fun(A,$o),set(A)),A4)),collect(B)) ) ).

% right_total_Collect_transfer
tff(fact_6652_fun_Opred__set,axiom,
    ! [A: $tType,B: $tType,P: fun(B,$o),X3: fun(A,B)] :
      ( aa(fun(A,B),$o,aa(fun(B,$o),fun(fun(A,B),$o),basic_pred_fun(A,B,aTP_Lamp_ar(A,$o)),P),X3)
    <=> ! [Xa2: B] :
          ( aa(set(B),$o,member(B,Xa2),aa(set(A),set(B),image2(A,B,X3),top_top(set(A))))
         => aa(B,$o,P,Xa2) ) ) ).

% fun.pred_set
tff(fact_6653_fun_Opred__transfer,axiom,
    ! [A: $tType,B: $tType,C: $tType,R: fun(A,fun(B,$o))] : aa(fun(fun(B,$o),fun(fun(C,B),$o)),$o,aa(fun(fun(A,$o),fun(fun(C,A),$o)),fun(fun(fun(B,$o),fun(fun(C,B),$o)),$o),bNF_rel_fun(fun(A,$o),fun(B,$o),fun(fun(C,A),$o),fun(fun(C,B),$o),bNF_rel_fun(A,B,$o,$o,R,fequal($o)),bNF_rel_fun(fun(C,A),fun(C,B),$o,$o,bNF_rel_fun(C,C,A,B,fequal(C),R),fequal($o))),basic_pred_fun(C,A,aTP_Lamp_arj(C,$o))),basic_pred_fun(C,B,aTP_Lamp_arj(C,$o))) ).

% fun.pred_transfer
tff(fact_6654_fun_Orel__eq__onp,axiom,
    ! [A: $tType,B: $tType,P: fun(B,$o)] : bNF_rel_fun(A,A,B,B,fequal(A),bNF_eq_onp(B,P)) = bNF_eq_onp(fun(A,B),aa(fun(B,$o),fun(fun(A,B),$o),basic_pred_fun(A,B,aTP_Lamp_ar(A,$o)),P)) ).

% fun.rel_eq_onp
tff(fact_6655_right__total__Domainp__transfer,axiom,
    ! [A: $tType,C: $tType,B: $tType,D: $tType,B3: fun(A,fun(B,$o)),A4: fun(C,fun(D,$o))] :
      ( right_total(A,B,B3)
     => aa(fun(fun(D,fun(B,$o)),fun(D,$o)),$o,aa(fun(fun(C,fun(A,$o)),fun(C,$o)),fun(fun(fun(D,fun(B,$o)),fun(D,$o)),$o),bNF_rel_fun(fun(C,fun(A,$o)),fun(D,fun(B,$o)),fun(C,$o),fun(D,$o),bNF_rel_fun(C,D,fun(A,$o),fun(B,$o),A4,bNF_rel_fun(A,B,$o,$o,B3,fequal($o))),bNF_rel_fun(C,D,$o,$o,A4,fequal($o))),aTP_Lamp_arm(fun(A,fun(B,$o)),fun(fun(C,fun(A,$o)),fun(C,$o)),B3)),domainp(D,B)) ) ).

% right_total_Domainp_transfer
tff(fact_6656_rat_Odomain__par,axiom,
    ! [DR1: fun(int,$o),DR2: fun(int,$o),P24: fun(product_prod(int,int),$o)] :
      ( ( aa(fun(int,fun(int,$o)),fun(int,$o),domainp(int,int),fequal(int)) = DR1 )
     => ( ( aa(fun(int,fun(int,$o)),fun(int,$o),domainp(int,int),fequal(int)) = DR2 )
       => ( aa(fun(product_prod(int,int),$o),$o,aa(fun(product_prod(int,int),$o),fun(fun(product_prod(int,int),$o),$o),bNF_rel_fun(product_prod(int,int),product_prod(int,int),$o,$o,basic_rel_prod(int,int,int,int,fequal(int),fequal(int)),fequal($o)),P24),aTP_Lamp_arn(product_prod(int,int),$o))
         => ( aa(fun(product_prod(int,int),fun(rat,$o)),fun(product_prod(int,int),$o),domainp(product_prod(int,int),rat),pcr_rat) = aa(fun(product_prod(int,int),$o),fun(product_prod(int,int),$o),aa(fun(product_prod(int,int),$o),fun(fun(product_prod(int,int),$o),fun(product_prod(int,int),$o)),inf_inf(fun(product_prod(int,int),$o)),basic_pred_prod(int,int,DR1,DR2)),P24) ) ) ) ) ).

% rat.domain_par
tff(fact_6657_rat_Odomain__par__left__total,axiom,
    ! [P4: fun(product_prod(int,int),$o)] :
      ( left_total(product_prod(int,int),product_prod(int,int),basic_rel_prod(int,int,int,int,fequal(int),fequal(int)))
     => ( aa(fun(product_prod(int,int),$o),$o,aa(fun(product_prod(int,int),$o),fun(fun(product_prod(int,int),$o),$o),bNF_rel_fun(product_prod(int,int),product_prod(int,int),$o,$o,basic_rel_prod(int,int,int,int,fequal(int),fequal(int)),fequal($o)),P4),aTP_Lamp_arn(product_prod(int,int),$o))
       => ( aa(fun(product_prod(int,int),fun(rat,$o)),fun(product_prod(int,int),$o),domainp(product_prod(int,int),rat),pcr_rat) = P4 ) ) ) ).

% rat.domain_par_left_total
tff(fact_6658_pred__prod__inject,axiom,
    ! [A: $tType,B: $tType,P12: fun(A,$o),P23: fun(B,$o),A3: A,B2: B] :
      ( aa(product_prod(A,B),$o,basic_pred_prod(A,B,P12,P23),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2))
    <=> ( aa(A,$o,P12,A3)
        & aa(B,$o,P23,B2) ) ) ).

% pred_prod_inject
tff(fact_6659_pred__prod__split,axiom,
    ! [A: $tType,B: $tType,P: fun($o,$o),Q: fun(A,$o),R: fun(B,$o),Xy: product_prod(A,B)] :
      ( aa($o,$o,P,aa(product_prod(A,B),$o,basic_pred_prod(A,B,Q,R),Xy))
    <=> ! [X4: A,Y3: B] :
          ( ( Xy = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Y3) )
         => aa($o,$o,P,
              ( aa(A,$o,Q,X4)
              & aa(B,$o,R,Y3) )) ) ) ).

% pred_prod_split
tff(fact_6660_pred__prod_Ointros,axiom,
    ! [A: $tType,B: $tType,P12: fun(A,$o),A3: A,P23: fun(B,$o),B2: B] :
      ( aa(A,$o,P12,A3)
     => ( aa(B,$o,P23,B2)
       => aa(product_prod(A,B),$o,basic_pred_prod(A,B,P12,P23),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)) ) ) ).

% pred_prod.intros
tff(fact_6661_pred__prod_Osimps,axiom,
    ! [A: $tType,B: $tType,P12: fun(A,$o),P23: fun(B,$o),A3: product_prod(A,B)] :
      ( aa(product_prod(A,B),$o,basic_pred_prod(A,B,P12,P23),A3)
    <=> ? [A10: A,B6: B] :
          ( ( A3 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A10),B6) )
          & aa(A,$o,P12,A10)
          & aa(B,$o,P23,B6) ) ) ).

% pred_prod.simps
tff(fact_6662_pred__prod_Ocases,axiom,
    ! [A: $tType,B: $tType,P12: fun(A,$o),P23: fun(B,$o),A3: product_prod(A,B)] :
      ( aa(product_prod(A,B),$o,basic_pred_prod(A,B,P12,P23),A3)
     => ~ ! [A6: A,B5: B] :
            ( ( A3 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A6),B5) )
           => ( aa(A,$o,P12,A6)
             => ~ aa(B,$o,P23,B5) ) ) ) ).

% pred_prod.cases
tff(fact_6663_prod_Omap__cong__pred,axiom,
    ! [D: $tType,C: $tType,B: $tType,A: $tType,X: product_prod(A,B),Ya: product_prod(A,B),F1: fun(A,C),G1: fun(A,C),F22: fun(B,D),G22: fun(B,D)] :
      ( ( X = Ya )
     => ( aa(product_prod(A,B),$o,basic_pred_prod(A,B,aa(fun(A,C),fun(A,$o),aTP_Lamp_aro(fun(A,C),fun(fun(A,C),fun(A,$o)),F1),G1),aa(fun(B,D),fun(B,$o),aTP_Lamp_arp(fun(B,D),fun(fun(B,D),fun(B,$o)),F22),G22)),Ya)
       => ( aa(product_prod(A,B),product_prod(C,D),product_map_prod(A,C,B,D,F1,F22),X) = aa(product_prod(A,B),product_prod(C,D),product_map_prod(A,C,B,D,G1,G22),Ya) ) ) ) ).

% prod.map_cong_pred
tff(fact_6664_prod_Opred__True,axiom,
    ! [B: $tType,A: $tType,X3: product_prod(A,B)] : aa(product_prod(A,B),$o,basic_pred_prod(A,B,aTP_Lamp_ar(A,$o),aTP_Lamp_ari(B,$o)),X3) ).

% prod.pred_True
tff(fact_6665_right__total__Inter__transfer,axiom,
    ! [A: $tType,B: $tType,A4: fun(A,fun(B,$o))] :
      ( bi_unique(A,B,A4)
     => ( right_total(A,B,A4)
       => aa(fun(set(set(B)),set(B)),$o,aa(fun(set(set(A)),set(A)),fun(fun(set(set(B)),set(B)),$o),bNF_rel_fun(set(set(A)),set(set(B)),set(A),set(B),bNF_rel_set(set(A),set(B),bNF_rel_set(A,B,A4)),bNF_rel_set(A,B,A4)),aTP_Lamp_arq(fun(A,fun(B,$o)),fun(set(set(A)),set(A)),A4)),complete_Inf_Inf(set(B))) ) ) ).

% right_total_Inter_transfer
tff(fact_6666_vimage__right__total__transfer,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType,B3: fun(A,fun(B,$o)),A4: fun(C,fun(D,$o))] :
      ( bi_unique(A,B,B3)
     => ( right_total(C,D,A4)
       => aa(fun(fun(D,B),fun(set(B),set(D))),$o,aa(fun(fun(C,A),fun(set(A),set(C))),fun(fun(fun(D,B),fun(set(B),set(D))),$o),bNF_rel_fun(fun(C,A),fun(D,B),fun(set(A),set(C)),fun(set(B),set(D)),bNF_rel_fun(C,D,A,B,A4,B3),bNF_rel_fun(set(A),set(B),set(C),set(D),bNF_rel_set(A,B,B3),bNF_rel_set(C,D,A4))),aTP_Lamp_arr(fun(C,fun(D,$o)),fun(fun(C,A),fun(set(A),set(C))),A4)),vimage(D,B)) ) ) ).

% vimage_right_total_transfer
tff(fact_6667_bi__unique__iff,axiom,
    ! [B: $tType,A: $tType,R: fun(A,fun(B,$o))] :
      ( bi_unique(A,B,R)
    <=> ( ! [Z4: B] : uniq(A,aa(B,fun(A,$o),aTP_Lamp_afh(fun(A,fun(B,$o)),fun(B,fun(A,$o)),R),Z4))
        & ! [Z4: A] : uniq(B,aa(A,fun(B,$o),R,Z4)) ) ) ).

% bi_unique_iff
tff(fact_6668_typedef__bi__unique,axiom,
    ! [A: $tType,B: $tType,Rep: fun(A,B),Abs: fun(B,A),A4: set(B),T2: fun(B,fun(A,$o))] :
      ( type_definition(A,B,Rep,Abs,A4)
     => ( ! [X2: B,Xa4: A] :
            ( aa(A,$o,aa(B,fun(A,$o),T2,X2),Xa4)
          <=> ( X2 = aa(A,B,Rep,Xa4) ) )
       => bi_unique(B,A,T2) ) ) ).

% typedef_bi_unique
tff(fact_6669_inter__transfer,axiom,
    ! [A: $tType,B: $tType,A4: fun(A,fun(B,$o))] :
      ( bi_unique(A,B,A4)
     => aa(fun(set(B),fun(set(B),set(B))),$o,aa(fun(set(A),fun(set(A),set(A))),fun(fun(set(B),fun(set(B),set(B))),$o),bNF_rel_fun(set(A),set(B),fun(set(A),set(A)),fun(set(B),set(B)),bNF_rel_set(A,B,A4),bNF_rel_fun(set(A),set(B),set(A),set(B),bNF_rel_set(A,B,A4),bNF_rel_set(A,B,A4))),inf_inf(set(A))),inf_inf(set(B))) ) ).

% inter_transfer
tff(fact_6670_right__total__Compl__transfer,axiom,
    ! [A: $tType,B: $tType,A4: fun(A,fun(B,$o))] :
      ( bi_unique(A,B,A4)
     => ( right_total(A,B,A4)
       => aa(fun(set(B),set(B)),$o,aa(fun(set(A),set(A)),fun(fun(set(B),set(B)),$o),bNF_rel_fun(set(A),set(B),set(A),set(B),bNF_rel_set(A,B,A4),bNF_rel_set(A,B,A4)),aTP_Lamp_ars(fun(A,fun(B,$o)),fun(set(A),set(A)),A4)),uminus_uminus(set(B))) ) ) ).

% right_total_Compl_transfer
tff(fact_6671_right__total__fun__eq__transfer,axiom,
    ! [A: $tType,C: $tType,D: $tType,B: $tType,A4: fun(A,fun(B,$o)),B3: fun(C,fun(D,$o))] :
      ( right_total(A,B,A4)
     => ( bi_unique(C,D,B3)
       => aa(fun(fun(B,D),fun(fun(B,D),$o)),$o,aa(fun(fun(A,C),fun(fun(A,C),$o)),fun(fun(fun(B,D),fun(fun(B,D),$o)),$o),bNF_rel_fun(fun(A,C),fun(B,D),fun(fun(A,C),$o),fun(fun(B,D),$o),bNF_rel_fun(A,B,C,D,A4,B3),bNF_rel_fun(fun(A,C),fun(B,D),$o,$o,bNF_rel_fun(A,B,C,D,A4,B3),fequal($o))),aTP_Lamp_art(fun(A,fun(B,$o)),fun(fun(A,C),fun(fun(A,C),$o)),A4)),fequal(fun(B,D))) ) ) ).

% right_total_fun_eq_transfer
tff(fact_6672_Inter__transfer,axiom,
    ! [A: $tType,B: $tType,A4: fun(A,fun(B,$o))] :
      ( bi_unique(A,B,A4)
     => ( bi_total(A,B,A4)
       => aa(fun(set(set(B)),set(B)),$o,aa(fun(set(set(A)),set(A)),fun(fun(set(set(B)),set(B)),$o),bNF_rel_fun(set(set(A)),set(set(B)),set(A),set(B),bNF_rel_set(set(A),set(B),bNF_rel_set(A,B,A4)),bNF_rel_set(A,B,A4)),complete_Inf_Inf(set(A))),complete_Inf_Inf(set(B))) ) ) ).

% Inter_transfer
tff(fact_6673_inf__filter__parametric,axiom,
    ! [A: $tType,B: $tType,A4: fun(A,fun(B,$o))] :
      ( bi_unique(A,B,A4)
     => ( bi_total(A,B,A4)
       => aa(fun(filter(B),fun(filter(B),filter(B))),$o,aa(fun(filter(A),fun(filter(A),filter(A))),fun(fun(filter(B),fun(filter(B),filter(B))),$o),bNF_rel_fun(filter(A),filter(B),fun(filter(A),filter(A)),fun(filter(B),filter(B)),rel_filter(A,B,A4),bNF_rel_fun(filter(A),filter(B),filter(A),filter(B),rel_filter(A,B,A4),rel_filter(A,B,A4))),inf_inf(filter(A))),inf_inf(filter(B))) ) ) ).

% inf_filter_parametric
tff(fact_6674_UNIV__transfer,axiom,
    ! [A: $tType,B: $tType,A4: fun(A,fun(B,$o))] :
      ( bi_total(A,B,A4)
     => aa(set(B),$o,aa(set(A),fun(set(B),$o),bNF_rel_set(A,B,A4),top_top(set(A))),top_top(set(B))) ) ).

% UNIV_transfer
tff(fact_6675_top__filter__parametric,axiom,
    ! [A: $tType,B: $tType,A4: fun(A,fun(B,$o))] :
      ( bi_total(A,B,A4)
     => aa(filter(B),$o,aa(filter(A),fun(filter(B),$o),rel_filter(A,B,A4),top_top(filter(A))),top_top(filter(B))) ) ).

% top_filter_parametric
tff(fact_6676_Compl__transfer,axiom,
    ! [A: $tType,B: $tType,A4: fun(A,fun(B,$o))] :
      ( bi_unique(A,B,A4)
     => ( bi_total(A,B,A4)
       => aa(fun(set(B),set(B)),$o,aa(fun(set(A),set(A)),fun(fun(set(B),set(B)),$o),bNF_rel_fun(set(A),set(B),set(A),set(B),bNF_rel_set(A,B,A4),bNF_rel_set(A,B,A4)),uminus_uminus(set(A))),uminus_uminus(set(B))) ) ) ).

% Compl_transfer
tff(fact_6677_Abs__rat__inject,axiom,
    ! [X: set(product_prod(int,int)),Y: set(product_prod(int,int))] :
      ( aa(set(set(product_prod(int,int))),$o,member(set(product_prod(int,int)),X),aa(fun(set(product_prod(int,int)),$o),set(set(product_prod(int,int))),collect(set(product_prod(int,int))),aTP_Lamp_aru(set(product_prod(int,int)),$o)))
     => ( aa(set(set(product_prod(int,int))),$o,member(set(product_prod(int,int)),Y),aa(fun(set(product_prod(int,int)),$o),set(set(product_prod(int,int))),collect(set(product_prod(int,int))),aTP_Lamp_aru(set(product_prod(int,int)),$o)))
       => ( ( aa(set(product_prod(int,int)),rat,abs_rat,X) = aa(set(product_prod(int,int)),rat,abs_rat,Y) )
        <=> ( X = Y ) ) ) ) ).

% Abs_rat_inject
tff(fact_6678_Abs__rat__induct,axiom,
    ! [P: fun(rat,$o),X: rat] :
      ( ! [Y2: set(product_prod(int,int))] :
          ( aa(set(set(product_prod(int,int))),$o,member(set(product_prod(int,int)),Y2),aa(fun(set(product_prod(int,int)),$o),set(set(product_prod(int,int))),collect(set(product_prod(int,int))),aTP_Lamp_aru(set(product_prod(int,int)),$o)))
         => aa(rat,$o,P,aa(set(product_prod(int,int)),rat,abs_rat,Y2)) )
     => aa(rat,$o,P,X) ) ).

% Abs_rat_induct
tff(fact_6679_Abs__rat__cases,axiom,
    ! [X: rat] :
      ~ ! [Y2: set(product_prod(int,int))] :
          ( ( X = aa(set(product_prod(int,int)),rat,abs_rat,Y2) )
         => ~ aa(set(set(product_prod(int,int))),$o,member(set(product_prod(int,int)),Y2),aa(fun(set(product_prod(int,int)),$o),set(set(product_prod(int,int))),collect(set(product_prod(int,int))),aTP_Lamp_aru(set(product_prod(int,int)),$o))) ) ).

% Abs_rat_cases
tff(fact_6680_type__definition__rat,axiom,
    type_definition(rat,set(product_prod(int,int)),rep_rat,abs_rat,aa(fun(set(product_prod(int,int)),$o),set(set(product_prod(int,int))),collect(set(product_prod(int,int))),aTP_Lamp_aru(set(product_prod(int,int)),$o))) ).

% type_definition_rat
tff(fact_6681_Abs__rat__inverse,axiom,
    ! [Y: set(product_prod(int,int))] :
      ( aa(set(set(product_prod(int,int))),$o,member(set(product_prod(int,int)),Y),aa(fun(set(product_prod(int,int)),$o),set(set(product_prod(int,int))),collect(set(product_prod(int,int))),aTP_Lamp_aru(set(product_prod(int,int)),$o)))
     => ( aa(rat,set(product_prod(int,int)),rep_rat,aa(set(product_prod(int,int)),rat,abs_rat,Y)) = Y ) ) ).

% Abs_rat_inverse
tff(fact_6682_Rep__rat__induct,axiom,
    ! [Y: set(product_prod(int,int)),P: fun(set(product_prod(int,int)),$o)] :
      ( aa(set(set(product_prod(int,int))),$o,member(set(product_prod(int,int)),Y),aa(fun(set(product_prod(int,int)),$o),set(set(product_prod(int,int))),collect(set(product_prod(int,int))),aTP_Lamp_aru(set(product_prod(int,int)),$o)))
     => ( ! [X2: rat] : aa(set(product_prod(int,int)),$o,P,aa(rat,set(product_prod(int,int)),rep_rat,X2))
       => aa(set(product_prod(int,int)),$o,P,Y) ) ) ).

% Rep_rat_induct
tff(fact_6683_Rep__rat__cases,axiom,
    ! [Y: set(product_prod(int,int))] :
      ( aa(set(set(product_prod(int,int))),$o,member(set(product_prod(int,int)),Y),aa(fun(set(product_prod(int,int)),$o),set(set(product_prod(int,int))),collect(set(product_prod(int,int))),aTP_Lamp_aru(set(product_prod(int,int)),$o)))
     => ~ ! [X2: rat] : Y != aa(rat,set(product_prod(int,int)),rep_rat,X2) ) ).

% Rep_rat_cases
tff(fact_6684_Rep__rat,axiom,
    ! [X: rat] : aa(set(set(product_prod(int,int))),$o,member(set(product_prod(int,int)),aa(rat,set(product_prod(int,int)),rep_rat,X)),aa(fun(set(product_prod(int,int)),$o),set(set(product_prod(int,int))),collect(set(product_prod(int,int))),aTP_Lamp_aru(set(product_prod(int,int)),$o))) ).

% Rep_rat
tff(fact_6685_wfP__SUP,axiom,
    ! [B: $tType,A: $tType,R2: fun(A,fun(B,fun(B,$o)))] :
      ( ! [I2: A] : wfP(B,aa(A,fun(B,fun(B,$o)),R2,I2))
     => ( ! [I2: A,J2: A] :
            ( ( aa(A,fun(B,fun(B,$o)),R2,I2) != aa(A,fun(B,fun(B,$o)),R2,J2) )
           => ( aa(fun(B,$o),fun(B,$o),aa(fun(B,$o),fun(fun(B,$o),fun(B,$o)),inf_inf(fun(B,$o)),aa(fun(B,fun(B,$o)),fun(B,$o),domainp(B,B),aa(A,fun(B,fun(B,$o)),R2,I2))),rangep(B,B,aa(A,fun(B,fun(B,$o)),R2,J2))) = bot_bot(fun(B,$o)) ) )
       => wfP(B,aa(set(fun(B,fun(B,$o))),fun(B,fun(B,$o)),complete_Sup_Sup(fun(B,fun(B,$o))),aa(set(A),set(fun(B,fun(B,$o))),image2(A,fun(B,fun(B,$o)),R2),top_top(set(A))))) ) ) ).

% wfP_SUP
tff(fact_6686_prod__mset_Ounion__disjoint,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A4: multiset(A),B3: multiset(A),G: fun(A,B)] :
          ( ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),A4),B3) = zero_zero(multiset(A)) )
         => ( aa(multiset(B),B,comm_m9189036328036947845d_mset(B),aa(multiset(A),multiset(B),image_mset(A,B,G),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),union_mset(A),A4),B3))) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(multiset(B),B,comm_m9189036328036947845d_mset(B),aa(multiset(A),multiset(B),image_mset(A,B,G),A4))),aa(multiset(B),B,comm_m9189036328036947845d_mset(B),aa(multiset(A),multiset(B),image_mset(A,B,G),B3))) ) ) ) ).

% prod_mset.union_disjoint
tff(fact_6687_wfP__empty,axiom,
    ! [A: $tType] : wfP(A,aTP_Lamp_arv(A,fun(A,$o))) ).

% wfP_empty
tff(fact_6688_set__mset__sup,axiom,
    ! [A: $tType,A4: multiset(A),B3: multiset(A)] : aa(multiset(A),set(A),set_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),union_mset(A),A4),B3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(multiset(A),set(A),set_mset(A),A4)),aa(multiset(A),set(A),set_mset(A),B3)) ).

% set_mset_sup
tff(fact_6689_subset__mset_Ocomm__monoid__axioms,axiom,
    ! [A: $tType] : comm_monoid(multiset(A),union_mset(A),zero_zero(multiset(A))) ).

% subset_mset.comm_monoid_axioms
tff(fact_6690_subset__mset_Osemilattice__neutr__axioms,axiom,
    ! [A: $tType] : semilattice_neutr(multiset(A),union_mset(A),zero_zero(multiset(A))) ).

% subset_mset.semilattice_neutr_axioms
tff(fact_6691_subset__mset_Omonoid__axioms,axiom,
    ! [A: $tType] : monoid(multiset(A),union_mset(A),zero_zero(multiset(A))) ).

% subset_mset.monoid_axioms
tff(fact_6692_wfP__wf__eq,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( wfP(A,aTP_Lamp_ux(set(product_prod(A,A)),fun(A,fun(A,$o)),R2))
    <=> wf(A,R2) ) ).

% wfP_wf_eq
tff(fact_6693_wfP__def,axiom,
    ! [A: $tType,R2: fun(A,fun(A,$o))] :
      ( wfP(A,R2)
    <=> wf(A,aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),R2))) ) ).

% wfP_def
tff(fact_6694_wfP__acyclicP,axiom,
    ! [A: $tType,R2: fun(A,fun(A,$o))] :
      ( wfP(A,R2)
     => transitive_acyclic(A,aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),R2))) ) ).

% wfP_acyclicP
tff(fact_6695_subset__mset_Oinf__Sup1__distrib,axiom,
    ! [A: $tType,A4: set(multiset(A)),X: multiset(A)] :
      ( aa(set(multiset(A)),$o,finite_finite2(multiset(A)),A4)
     => ( ( A4 != bot_bot(set(multiset(A))) )
       => ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),X),lattic4630905495605216202up_fin(multiset(A),union_mset(A),A4)) = lattic4630905495605216202up_fin(multiset(A),union_mset(A),aa(fun(multiset(A),$o),set(multiset(A)),collect(multiset(A)),aa(multiset(A),fun(multiset(A),$o),aTP_Lamp_arw(set(multiset(A)),fun(multiset(A),fun(multiset(A),$o)),A4),X))) ) ) ) ).

% subset_mset.inf_Sup1_distrib
tff(fact_6696_subset__mset_Oinf__Sup2__distrib,axiom,
    ! [A: $tType,A4: set(multiset(A)),B3: set(multiset(A))] :
      ( aa(set(multiset(A)),$o,finite_finite2(multiset(A)),A4)
     => ( ( A4 != bot_bot(set(multiset(A))) )
       => ( aa(set(multiset(A)),$o,finite_finite2(multiset(A)),B3)
         => ( ( B3 != bot_bot(set(multiset(A))) )
           => ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),lattic4630905495605216202up_fin(multiset(A),union_mset(A),A4)),lattic4630905495605216202up_fin(multiset(A),union_mset(A),B3)) = lattic4630905495605216202up_fin(multiset(A),union_mset(A),aa(fun(multiset(A),$o),set(multiset(A)),collect(multiset(A)),aa(set(multiset(A)),fun(multiset(A),$o),aTP_Lamp_arx(set(multiset(A)),fun(set(multiset(A)),fun(multiset(A),$o)),A4),B3))) ) ) ) ) ) ).

% subset_mset.inf_Sup2_distrib
tff(fact_6697_subset__mset_OSup__fin_Osingleton,axiom,
    ! [A: $tType,X: multiset(A)] : lattic4630905495605216202up_fin(multiset(A),union_mset(A),aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert2(multiset(A)),X),bot_bot(set(multiset(A))))) = X ).

% subset_mset.Sup_fin.singleton
tff(fact_6698_subset__mset_OSup__fin_Oinsert,axiom,
    ! [A: $tType,A4: set(multiset(A)),X: multiset(A)] :
      ( aa(set(multiset(A)),$o,finite_finite2(multiset(A)),A4)
     => ( ( A4 != bot_bot(set(multiset(A))) )
       => ( lattic4630905495605216202up_fin(multiset(A),union_mset(A),aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert2(multiset(A)),X),A4)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),union_mset(A),X),lattic4630905495605216202up_fin(multiset(A),union_mset(A),A4)) ) ) ) ).

% subset_mset.Sup_fin.insert
tff(fact_6699_subset__mset_OSup__fin_Osubset,axiom,
    ! [A: $tType,A4: set(multiset(A)),B3: set(multiset(A))] :
      ( aa(set(multiset(A)),$o,finite_finite2(multiset(A)),A4)
     => ( ( B3 != bot_bot(set(multiset(A))) )
       => ( aa(set(multiset(A)),$o,aa(set(multiset(A)),fun(set(multiset(A)),$o),ord_less_eq(set(multiset(A))),B3),A4)
         => ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),union_mset(A),lattic4630905495605216202up_fin(multiset(A),union_mset(A),B3)),lattic4630905495605216202up_fin(multiset(A),union_mset(A),A4)) = lattic4630905495605216202up_fin(multiset(A),union_mset(A),A4) ) ) ) ) ).

% subset_mset.Sup_fin.subset
tff(fact_6700_subset__mset_OSup__fin_Ohom__commute,axiom,
    ! [A: $tType,H2: fun(multiset(A),multiset(A)),N4: set(multiset(A))] :
      ( ! [X2: multiset(A),Y2: multiset(A)] : aa(multiset(A),multiset(A),H2,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),union_mset(A),X2),Y2)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),union_mset(A),aa(multiset(A),multiset(A),H2,X2)),aa(multiset(A),multiset(A),H2,Y2))
     => ( aa(set(multiset(A)),$o,finite_finite2(multiset(A)),N4)
       => ( ( N4 != bot_bot(set(multiset(A))) )
         => ( aa(multiset(A),multiset(A),H2,lattic4630905495605216202up_fin(multiset(A),union_mset(A),N4)) = lattic4630905495605216202up_fin(multiset(A),union_mset(A),aa(set(multiset(A)),set(multiset(A)),image2(multiset(A),multiset(A),H2),N4)) ) ) ) ) ).

% subset_mset.Sup_fin.hom_commute
tff(fact_6701_subset__mset_OSup__fin_Oeq__fold_H,axiom,
    ! [A: $tType,A4: set(multiset(A))] : lattic4630905495605216202up_fin(multiset(A),union_mset(A),A4) = aa(option(multiset(A)),multiset(A),the2(multiset(A)),finite_fold(multiset(A),option(multiset(A)),aTP_Lamp_ary(multiset(A),fun(option(multiset(A)),option(multiset(A)))),none(multiset(A)),A4)) ).

% subset_mset.Sup_fin.eq_fold'
tff(fact_6702_subset__mset_OSup__fin_Ounion,axiom,
    ! [A: $tType,A4: set(multiset(A)),B3: set(multiset(A))] :
      ( aa(set(multiset(A)),$o,finite_finite2(multiset(A)),A4)
     => ( ( A4 != bot_bot(set(multiset(A))) )
       => ( aa(set(multiset(A)),$o,finite_finite2(multiset(A)),B3)
         => ( ( B3 != bot_bot(set(multiset(A))) )
           => ( lattic4630905495605216202up_fin(multiset(A),union_mset(A),aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),sup_sup(set(multiset(A))),A4),B3)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),union_mset(A),lattic4630905495605216202up_fin(multiset(A),union_mset(A),A4)),lattic4630905495605216202up_fin(multiset(A),union_mset(A),B3)) ) ) ) ) ) ).

% subset_mset.Sup_fin.union
tff(fact_6703_subset__mset_OcSup__eq__Sup__fin,axiom,
    ! [A: $tType,X5: set(multiset(A))] :
      ( aa(set(multiset(A)),$o,finite_finite2(multiset(A)),X5)
     => ( ( X5 != bot_bot(set(multiset(A))) )
       => ( aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),X5) = lattic4630905495605216202up_fin(multiset(A),union_mset(A),X5) ) ) ) ).

% subset_mset.cSup_eq_Sup_fin
tff(fact_6704_subset__mset_OSup__fin_Oinsert__not__elem,axiom,
    ! [A: $tType,A4: set(multiset(A)),X: multiset(A)] :
      ( aa(set(multiset(A)),$o,finite_finite2(multiset(A)),A4)
     => ( ~ aa(set(multiset(A)),$o,member(multiset(A),X),A4)
       => ( ( A4 != bot_bot(set(multiset(A))) )
         => ( lattic4630905495605216202up_fin(multiset(A),union_mset(A),aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert2(multiset(A)),X),A4)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),union_mset(A),X),lattic4630905495605216202up_fin(multiset(A),union_mset(A),A4)) ) ) ) ) ).

% subset_mset.Sup_fin.insert_not_elem
tff(fact_6705_subset__mset_OSup__fin_Oclosed,axiom,
    ! [A: $tType,A4: set(multiset(A))] :
      ( aa(set(multiset(A)),$o,finite_finite2(multiset(A)),A4)
     => ( ( A4 != bot_bot(set(multiset(A))) )
       => ( ! [X2: multiset(A),Y2: multiset(A)] : aa(set(multiset(A)),$o,member(multiset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),union_mset(A),X2),Y2)),aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert2(multiset(A)),X2),aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert2(multiset(A)),Y2),bot_bot(set(multiset(A))))))
         => aa(set(multiset(A)),$o,member(multiset(A),lattic4630905495605216202up_fin(multiset(A),union_mset(A),A4)),A4) ) ) ) ).

% subset_mset.Sup_fin.closed
tff(fact_6706_subset__mset_OSup__fin_Oinsert__remove,axiom,
    ! [A: $tType,A4: set(multiset(A)),X: multiset(A)] :
      ( aa(set(multiset(A)),$o,finite_finite2(multiset(A)),A4)
     => ( lattic4630905495605216202up_fin(multiset(A),union_mset(A),aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert2(multiset(A)),X),A4)) = $ite(aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),minus_minus(set(multiset(A))),A4),aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert2(multiset(A)),X),bot_bot(set(multiset(A))))) = bot_bot(set(multiset(A))),X,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),union_mset(A),X),lattic4630905495605216202up_fin(multiset(A),union_mset(A),aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),minus_minus(set(multiset(A))),A4),aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert2(multiset(A)),X),bot_bot(set(multiset(A)))))))) ) ) ).

% subset_mset.Sup_fin.insert_remove
tff(fact_6707_subset__mset_OSup__fin_Oremove,axiom,
    ! [A: $tType,A4: set(multiset(A)),X: multiset(A)] :
      ( aa(set(multiset(A)),$o,finite_finite2(multiset(A)),A4)
     => ( aa(set(multiset(A)),$o,member(multiset(A),X),A4)
       => ( lattic4630905495605216202up_fin(multiset(A),union_mset(A),A4) = $ite(aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),minus_minus(set(multiset(A))),A4),aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert2(multiset(A)),X),bot_bot(set(multiset(A))))) = bot_bot(set(multiset(A))),X,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),union_mset(A),X),lattic4630905495605216202up_fin(multiset(A),union_mset(A),aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),minus_minus(set(multiset(A))),A4),aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert2(multiset(A)),X),bot_bot(set(multiset(A)))))))) ) ) ) ).

% subset_mset.Sup_fin.remove
tff(fact_6708_subset__mset_Osup__Inf2__distrib,axiom,
    ! [A: $tType,A4: set(multiset(A)),B3: set(multiset(A))] :
      ( aa(set(multiset(A)),$o,finite_finite2(multiset(A)),A4)
     => ( ( A4 != bot_bot(set(multiset(A))) )
       => ( aa(set(multiset(A)),$o,finite_finite2(multiset(A)),B3)
         => ( ( B3 != bot_bot(set(multiset(A))) )
           => ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),union_mset(A),lattic8678736583308907530nf_fin(multiset(A),inter_mset(A),A4)),lattic8678736583308907530nf_fin(multiset(A),inter_mset(A),B3)) = lattic8678736583308907530nf_fin(multiset(A),inter_mset(A),aa(fun(multiset(A),$o),set(multiset(A)),collect(multiset(A)),aa(set(multiset(A)),fun(multiset(A),$o),aTP_Lamp_arz(set(multiset(A)),fun(set(multiset(A)),fun(multiset(A),$o)),A4),B3))) ) ) ) ) ) ).

% subset_mset.sup_Inf2_distrib
tff(fact_6709_subset__mset_Osup__Inf1__distrib,axiom,
    ! [A: $tType,A4: set(multiset(A)),X: multiset(A)] :
      ( aa(set(multiset(A)),$o,finite_finite2(multiset(A)),A4)
     => ( ( A4 != bot_bot(set(multiset(A))) )
       => ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),union_mset(A),X),lattic8678736583308907530nf_fin(multiset(A),inter_mset(A),A4)) = lattic8678736583308907530nf_fin(multiset(A),inter_mset(A),aa(fun(multiset(A),$o),set(multiset(A)),collect(multiset(A)),aa(multiset(A),fun(multiset(A),$o),aTP_Lamp_asa(set(multiset(A)),fun(multiset(A),fun(multiset(A),$o)),A4),X))) ) ) ) ).

% subset_mset.sup_Inf1_distrib
tff(fact_6710_subset__mset_OInf__fin_Osingleton,axiom,
    ! [A: $tType,X: multiset(A)] : lattic8678736583308907530nf_fin(multiset(A),inter_mset(A),aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert2(multiset(A)),X),bot_bot(set(multiset(A))))) = X ).

% subset_mset.Inf_fin.singleton
tff(fact_6711_subset__mset_OInf__fin_Oinsert,axiom,
    ! [A: $tType,A4: set(multiset(A)),X: multiset(A)] :
      ( aa(set(multiset(A)),$o,finite_finite2(multiset(A)),A4)
     => ( ( A4 != bot_bot(set(multiset(A))) )
       => ( lattic8678736583308907530nf_fin(multiset(A),inter_mset(A),aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert2(multiset(A)),X),A4)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),X),lattic8678736583308907530nf_fin(multiset(A),inter_mset(A),A4)) ) ) ) ).

% subset_mset.Inf_fin.insert
tff(fact_6712_subset__mset_OInf__fin_Oeq__fold_H,axiom,
    ! [A: $tType,A4: set(multiset(A))] : lattic8678736583308907530nf_fin(multiset(A),inter_mset(A),A4) = aa(option(multiset(A)),multiset(A),the2(multiset(A)),finite_fold(multiset(A),option(multiset(A)),aTP_Lamp_asb(multiset(A),fun(option(multiset(A)),option(multiset(A)))),none(multiset(A)),A4)) ).

% subset_mset.Inf_fin.eq_fold'
tff(fact_6713_subset__mset_OInf__fin_Osubset,axiom,
    ! [A: $tType,A4: set(multiset(A)),B3: set(multiset(A))] :
      ( aa(set(multiset(A)),$o,finite_finite2(multiset(A)),A4)
     => ( ( B3 != bot_bot(set(multiset(A))) )
       => ( aa(set(multiset(A)),$o,aa(set(multiset(A)),fun(set(multiset(A)),$o),ord_less_eq(set(multiset(A))),B3),A4)
         => ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),lattic8678736583308907530nf_fin(multiset(A),inter_mset(A),B3)),lattic8678736583308907530nf_fin(multiset(A),inter_mset(A),A4)) = lattic8678736583308907530nf_fin(multiset(A),inter_mset(A),A4) ) ) ) ) ).

% subset_mset.Inf_fin.subset
tff(fact_6714_subset__mset_OInf__fin_Ohom__commute,axiom,
    ! [A: $tType,H2: fun(multiset(A),multiset(A)),N4: set(multiset(A))] :
      ( ! [X2: multiset(A),Y2: multiset(A)] : aa(multiset(A),multiset(A),H2,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),X2),Y2)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),aa(multiset(A),multiset(A),H2,X2)),aa(multiset(A),multiset(A),H2,Y2))
     => ( aa(set(multiset(A)),$o,finite_finite2(multiset(A)),N4)
       => ( ( N4 != bot_bot(set(multiset(A))) )
         => ( aa(multiset(A),multiset(A),H2,lattic8678736583308907530nf_fin(multiset(A),inter_mset(A),N4)) = lattic8678736583308907530nf_fin(multiset(A),inter_mset(A),aa(set(multiset(A)),set(multiset(A)),image2(multiset(A),multiset(A),H2),N4)) ) ) ) ) ).

% subset_mset.Inf_fin.hom_commute
tff(fact_6715_subset__mset_OInf__fin_Ounion,axiom,
    ! [A: $tType,A4: set(multiset(A)),B3: set(multiset(A))] :
      ( aa(set(multiset(A)),$o,finite_finite2(multiset(A)),A4)
     => ( ( A4 != bot_bot(set(multiset(A))) )
       => ( aa(set(multiset(A)),$o,finite_finite2(multiset(A)),B3)
         => ( ( B3 != bot_bot(set(multiset(A))) )
           => ( lattic8678736583308907530nf_fin(multiset(A),inter_mset(A),aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),sup_sup(set(multiset(A))),A4),B3)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),lattic8678736583308907530nf_fin(multiset(A),inter_mset(A),A4)),lattic8678736583308907530nf_fin(multiset(A),inter_mset(A),B3)) ) ) ) ) ) ).

% subset_mset.Inf_fin.union
tff(fact_6716_subset__mset_OcInf__eq__Inf__fin,axiom,
    ! [A: $tType,X5: set(multiset(A))] :
      ( aa(set(multiset(A)),$o,finite_finite2(multiset(A)),X5)
     => ( ( X5 != bot_bot(set(multiset(A))) )
       => ( aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),X5) = lattic8678736583308907530nf_fin(multiset(A),inter_mset(A),X5) ) ) ) ).

% subset_mset.cInf_eq_Inf_fin
tff(fact_6717_subset__mset_OInf__fin_Oinsert__not__elem,axiom,
    ! [A: $tType,A4: set(multiset(A)),X: multiset(A)] :
      ( aa(set(multiset(A)),$o,finite_finite2(multiset(A)),A4)
     => ( ~ aa(set(multiset(A)),$o,member(multiset(A),X),A4)
       => ( ( A4 != bot_bot(set(multiset(A))) )
         => ( lattic8678736583308907530nf_fin(multiset(A),inter_mset(A),aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert2(multiset(A)),X),A4)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),X),lattic8678736583308907530nf_fin(multiset(A),inter_mset(A),A4)) ) ) ) ) ).

% subset_mset.Inf_fin.insert_not_elem
tff(fact_6718_subset__mset_OInf__fin_Oclosed,axiom,
    ! [A: $tType,A4: set(multiset(A))] :
      ( aa(set(multiset(A)),$o,finite_finite2(multiset(A)),A4)
     => ( ( A4 != bot_bot(set(multiset(A))) )
       => ( ! [X2: multiset(A),Y2: multiset(A)] : aa(set(multiset(A)),$o,member(multiset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),X2),Y2)),aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert2(multiset(A)),X2),aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert2(multiset(A)),Y2),bot_bot(set(multiset(A))))))
         => aa(set(multiset(A)),$o,member(multiset(A),lattic8678736583308907530nf_fin(multiset(A),inter_mset(A),A4)),A4) ) ) ) ).

% subset_mset.Inf_fin.closed
tff(fact_6719_subset__mset_OInf__fin_Oinsert__remove,axiom,
    ! [A: $tType,A4: set(multiset(A)),X: multiset(A)] :
      ( aa(set(multiset(A)),$o,finite_finite2(multiset(A)),A4)
     => ( lattic8678736583308907530nf_fin(multiset(A),inter_mset(A),aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert2(multiset(A)),X),A4)) = $ite(aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),minus_minus(set(multiset(A))),A4),aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert2(multiset(A)),X),bot_bot(set(multiset(A))))) = bot_bot(set(multiset(A))),X,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),X),lattic8678736583308907530nf_fin(multiset(A),inter_mset(A),aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),minus_minus(set(multiset(A))),A4),aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert2(multiset(A)),X),bot_bot(set(multiset(A)))))))) ) ) ).

% subset_mset.Inf_fin.insert_remove
tff(fact_6720_subset__mset_OInf__fin_Oremove,axiom,
    ! [A: $tType,A4: set(multiset(A)),X: multiset(A)] :
      ( aa(set(multiset(A)),$o,finite_finite2(multiset(A)),A4)
     => ( aa(set(multiset(A)),$o,member(multiset(A),X),A4)
       => ( lattic8678736583308907530nf_fin(multiset(A),inter_mset(A),A4) = $ite(aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),minus_minus(set(multiset(A))),A4),aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert2(multiset(A)),X),bot_bot(set(multiset(A))))) = bot_bot(set(multiset(A))),X,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),X),lattic8678736583308907530nf_fin(multiset(A),inter_mset(A),aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),minus_minus(set(multiset(A))),A4),aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert2(multiset(A)),X),bot_bot(set(multiset(A)))))))) ) ) ) ).

% subset_mset.Inf_fin.remove
tff(fact_6721_subset__mset_OInf__fin__le__Sup__fin,axiom,
    ! [A: $tType,A4: set(multiset(A))] :
      ( aa(set(multiset(A)),$o,finite_finite2(multiset(A)),A4)
     => ( ( A4 != bot_bot(set(multiset(A))) )
       => aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),lattic8678736583308907530nf_fin(multiset(A),inter_mset(A),A4)),lattic4630905495605216202up_fin(multiset(A),union_mset(A),A4)) ) ) ).

% subset_mset.Inf_fin_le_Sup_fin
tff(fact_6722_subset__mset_OSup__fin_Osubset__imp,axiom,
    ! [A: $tType,A4: set(multiset(A)),B3: set(multiset(A))] :
      ( aa(set(multiset(A)),$o,aa(set(multiset(A)),fun(set(multiset(A)),$o),ord_less_eq(set(multiset(A))),A4),B3)
     => ( ( A4 != bot_bot(set(multiset(A))) )
       => ( aa(set(multiset(A)),$o,finite_finite2(multiset(A)),B3)
         => aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),lattic4630905495605216202up_fin(multiset(A),union_mset(A),A4)),lattic4630905495605216202up_fin(multiset(A),union_mset(A),B3)) ) ) ) ).

% subset_mset.Sup_fin.subset_imp
tff(fact_6723_subset__mset_OcInf__eq__non__empty,axiom,
    ! [A: $tType,X5: set(multiset(A)),A3: multiset(A)] :
      ( ( X5 != bot_bot(set(multiset(A))) )
     => ( ! [X2: multiset(A)] :
            ( aa(set(multiset(A)),$o,member(multiset(A),X2),X5)
           => aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),A3),X2) )
       => ( ! [Y2: multiset(A)] :
              ( ! [X3: multiset(A)] :
                  ( aa(set(multiset(A)),$o,member(multiset(A),X3),X5)
                 => aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),Y2),X3) )
             => aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),Y2),A3) )
         => ( aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),X5) = A3 ) ) ) ) ).

% subset_mset.cInf_eq_non_empty
tff(fact_6724_subset__mset_OcInf__greatest,axiom,
    ! [A: $tType,X5: set(multiset(A)),Z2: multiset(A)] :
      ( ( X5 != bot_bot(set(multiset(A))) )
     => ( ! [X2: multiset(A)] :
            ( aa(set(multiset(A)),$o,member(multiset(A),X2),X5)
           => aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),Z2),X2) )
       => aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),Z2),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),X5)) ) ) ).

% subset_mset.cInf_greatest
tff(fact_6725_subset__mset_OcSup__least,axiom,
    ! [A: $tType,X5: set(multiset(A)),Z2: multiset(A)] :
      ( ( X5 != bot_bot(set(multiset(A))) )
     => ( ! [X2: multiset(A)] :
            ( aa(set(multiset(A)),$o,member(multiset(A),X2),X5)
           => aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),X2),Z2) )
       => aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),X5)),Z2) ) ) ).

% subset_mset.cSup_least
tff(fact_6726_subset__mset_OcSup__eq__non__empty,axiom,
    ! [A: $tType,X5: set(multiset(A)),A3: multiset(A)] :
      ( ( X5 != bot_bot(set(multiset(A))) )
     => ( ! [X2: multiset(A)] :
            ( aa(set(multiset(A)),$o,member(multiset(A),X2),X5)
           => aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),X2),A3) )
       => ( ! [Y2: multiset(A)] :
              ( ! [X3: multiset(A)] :
                  ( aa(set(multiset(A)),$o,member(multiset(A),X3),X5)
                 => aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),X3),Y2) )
             => aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),A3),Y2) )
         => ( aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),X5) = A3 ) ) ) ) ).

% subset_mset.cSup_eq_non_empty
tff(fact_6727_subset__mset_OLeast__def,axiom,
    ! [A: $tType,P: fun(multiset(A),$o)] : aa(fun(multiset(A),$o),multiset(A),least(multiset(A),subseteq_mset(A)),P) = the(multiset(A),aTP_Lamp_asc(fun(multiset(A),$o),fun(multiset(A),$o),P)) ).

% subset_mset.Least_def
tff(fact_6728_subset__mset_Ofinite__has__minimal,axiom,
    ! [A: $tType,A4: set(multiset(A))] :
      ( aa(set(multiset(A)),$o,finite_finite2(multiset(A)),A4)
     => ( ( A4 != bot_bot(set(multiset(A))) )
       => ? [X2: multiset(A)] :
            ( aa(set(multiset(A)),$o,member(multiset(A),X2),A4)
            & ! [Xa3: multiset(A)] :
                ( aa(set(multiset(A)),$o,member(multiset(A),Xa3),A4)
               => ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),Xa3),X2)
                 => ( X2 = Xa3 ) ) ) ) ) ) ).

% subset_mset.finite_has_minimal
tff(fact_6729_subset__mset_Ofinite__has__maximal,axiom,
    ! [A: $tType,A4: set(multiset(A))] :
      ( aa(set(multiset(A)),$o,finite_finite2(multiset(A)),A4)
     => ( ( A4 != bot_bot(set(multiset(A))) )
       => ? [X2: multiset(A)] :
            ( aa(set(multiset(A)),$o,member(multiset(A),X2),A4)
            & ! [Xa3: multiset(A)] :
                ( aa(set(multiset(A)),$o,member(multiset(A),Xa3),A4)
               => ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),X2),Xa3)
                 => ( X2 = Xa3 ) ) ) ) ) ) ).

% subset_mset.finite_has_maximal
tff(fact_6730_subset__mset_OcINF__greatest,axiom,
    ! [B: $tType,A: $tType,A4: set(A),M: multiset(B),F: fun(A,multiset(B))] :
      ( ( A4 != bot_bot(set(A)) )
     => ( ! [X2: A] :
            ( aa(set(A),$o,member(A,X2),A4)
           => aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),M),aa(A,multiset(B),F,X2)) )
       => aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),M),aa(set(multiset(B)),multiset(B),complete_Inf_Inf(multiset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),F),A4))) ) ) ).

% subset_mset.cINF_greatest
tff(fact_6731_subset__mset_OcSUP__least,axiom,
    ! [A: $tType,B: $tType,A4: set(A),F: fun(A,multiset(B)),M4: multiset(B)] :
      ( ( A4 != bot_bot(set(A)) )
     => ( ! [X2: A] :
            ( aa(set(A),$o,member(A,X2),A4)
           => aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),aa(A,multiset(B),F,X2)),M4) )
       => aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),aa(set(multiset(B)),multiset(B),complete_Sup_Sup(multiset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),F),A4))),M4) ) ) ).

% subset_mset.cSUP_least
tff(fact_6732_subset__mset_OInf__fin_Obounded__iff,axiom,
    ! [A: $tType,A4: set(multiset(A)),X: multiset(A)] :
      ( aa(set(multiset(A)),$o,finite_finite2(multiset(A)),A4)
     => ( ( A4 != bot_bot(set(multiset(A))) )
       => ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),X),lattic8678736583308907530nf_fin(multiset(A),inter_mset(A),A4))
        <=> ! [X4: multiset(A)] :
              ( aa(set(multiset(A)),$o,member(multiset(A),X4),A4)
             => aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),X),X4) ) ) ) ) ).

% subset_mset.Inf_fin.bounded_iff
tff(fact_6733_subset__mset_OInf__fin_OboundedI,axiom,
    ! [A: $tType,A4: set(multiset(A)),X: multiset(A)] :
      ( aa(set(multiset(A)),$o,finite_finite2(multiset(A)),A4)
     => ( ( A4 != bot_bot(set(multiset(A))) )
       => ( ! [A6: multiset(A)] :
              ( aa(set(multiset(A)),$o,member(multiset(A),A6),A4)
             => aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),X),A6) )
         => aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),X),lattic8678736583308907530nf_fin(multiset(A),inter_mset(A),A4)) ) ) ) ).

% subset_mset.Inf_fin.boundedI
tff(fact_6734_subset__mset_OInf__fin_OboundedE,axiom,
    ! [A: $tType,A4: set(multiset(A)),X: multiset(A)] :
      ( aa(set(multiset(A)),$o,finite_finite2(multiset(A)),A4)
     => ( ( A4 != bot_bot(set(multiset(A))) )
       => ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),X),lattic8678736583308907530nf_fin(multiset(A),inter_mset(A),A4))
         => ! [A11: multiset(A)] :
              ( aa(set(multiset(A)),$o,member(multiset(A),A11),A4)
             => aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),X),A11) ) ) ) ) ).

% subset_mset.Inf_fin.boundedE
tff(fact_6735_subset__mset_OSup__fin_OboundedE,axiom,
    ! [A: $tType,A4: set(multiset(A)),X: multiset(A)] :
      ( aa(set(multiset(A)),$o,finite_finite2(multiset(A)),A4)
     => ( ( A4 != bot_bot(set(multiset(A))) )
       => ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),lattic4630905495605216202up_fin(multiset(A),union_mset(A),A4)),X)
         => ! [A11: multiset(A)] :
              ( aa(set(multiset(A)),$o,member(multiset(A),A11),A4)
             => aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),A11),X) ) ) ) ) ).

% subset_mset.Sup_fin.boundedE
tff(fact_6736_subset__mset_OSup__fin_OboundedI,axiom,
    ! [A: $tType,A4: set(multiset(A)),X: multiset(A)] :
      ( aa(set(multiset(A)),$o,finite_finite2(multiset(A)),A4)
     => ( ( A4 != bot_bot(set(multiset(A))) )
       => ( ! [A6: multiset(A)] :
              ( aa(set(multiset(A)),$o,member(multiset(A),A6),A4)
             => aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),A6),X) )
         => aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),lattic4630905495605216202up_fin(multiset(A),union_mset(A),A4)),X) ) ) ) ).

% subset_mset.Sup_fin.boundedI
tff(fact_6737_subset__mset_OSup__fin_Obounded__iff,axiom,
    ! [A: $tType,A4: set(multiset(A)),X: multiset(A)] :
      ( aa(set(multiset(A)),$o,finite_finite2(multiset(A)),A4)
     => ( ( A4 != bot_bot(set(multiset(A))) )
       => ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),lattic4630905495605216202up_fin(multiset(A),union_mset(A),A4)),X)
        <=> ! [X4: multiset(A)] :
              ( aa(set(multiset(A)),$o,member(multiset(A),X4),A4)
             => aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),X4),X) ) ) ) ) ).

% subset_mset.Sup_fin.bounded_iff
tff(fact_6738_subset__mset_OInf__fin_Osubset__imp,axiom,
    ! [A: $tType,A4: set(multiset(A)),B3: set(multiset(A))] :
      ( aa(set(multiset(A)),$o,aa(set(multiset(A)),fun(set(multiset(A)),$o),ord_less_eq(set(multiset(A))),A4),B3)
     => ( ( A4 != bot_bot(set(multiset(A))) )
       => ( aa(set(multiset(A)),$o,finite_finite2(multiset(A)),B3)
         => aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),lattic8678736583308907530nf_fin(multiset(A),inter_mset(A),B3)),lattic8678736583308907530nf_fin(multiset(A),inter_mset(A),A4)) ) ) ) ).

% subset_mset.Inf_fin.subset_imp
tff(fact_6739_Sup__multiset__def,axiom,
    ! [A: $tType,A4: set(multiset(A))] :
      aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),A4) = $ite(
        ( ( A4 != bot_bot(set(multiset(A))) )
        & condit8047198070973881523_above(multiset(A),subseteq_mset(A),A4) ),
        aa(fun(A,nat),multiset(A),abs_multiset(A),aTP_Lamp_asd(set(multiset(A)),fun(A,nat),A4)),
        zero_zero(multiset(A)) ) ).

% Sup_multiset_def
tff(fact_6740_Sup__multiset__in__multiset,axiom,
    ! [A: $tType,A4: set(multiset(A))] :
      ( ( A4 != bot_bot(set(multiset(A))) )
     => ( condit8047198070973881523_above(multiset(A),subseteq_mset(A),A4)
       => aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_ase(set(multiset(A)),fun(A,$o),A4))) ) ) ).

% Sup_multiset_in_multiset
tff(fact_6741_subset__mset_Obdd__above__empty,axiom,
    ! [A: $tType] : condit8047198070973881523_above(multiset(A),subseteq_mset(A),bot_bot(set(multiset(A)))) ).

% subset_mset.bdd_above_empty
tff(fact_6742_subset__mset_Obdd__above__UN,axiom,
    ! [B: $tType,A: $tType,I4: set(A),A4: fun(A,set(multiset(B)))] :
      ( aa(set(A),$o,finite_finite2(A),I4)
     => ( condit8047198070973881523_above(multiset(B),subseteq_mset(B),aa(set(set(multiset(B))),set(multiset(B)),complete_Sup_Sup(set(multiset(B))),aa(set(A),set(set(multiset(B))),image2(A,set(multiset(B)),A4),I4)))
      <=> ! [X4: A] :
            ( aa(set(A),$o,member(A,X4),I4)
           => condit8047198070973881523_above(multiset(B),subseteq_mset(B),aa(A,set(multiset(B)),A4,X4)) ) ) ) ).

% subset_mset.bdd_above_UN
tff(fact_6743_subset__mset_Obdd__above__Un,axiom,
    ! [A: $tType,A4: set(multiset(A)),B3: set(multiset(A))] :
      ( condit8047198070973881523_above(multiset(A),subseteq_mset(A),aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),sup_sup(set(multiset(A))),A4),B3))
    <=> ( condit8047198070973881523_above(multiset(A),subseteq_mset(A),A4)
        & condit8047198070973881523_above(multiset(A),subseteq_mset(A),B3) ) ) ).

% subset_mset.bdd_above_Un
tff(fact_6744_subset__mset_Obdd__above__image__sup,axiom,
    ! [A: $tType,B: $tType,F: fun(B,multiset(A)),G: fun(B,multiset(A)),A4: set(B)] :
      ( condit8047198070973881523_above(multiset(A),subseteq_mset(A),aa(set(B),set(multiset(A)),image2(B,multiset(A),aa(fun(B,multiset(A)),fun(B,multiset(A)),aTP_Lamp_asf(fun(B,multiset(A)),fun(fun(B,multiset(A)),fun(B,multiset(A))),F),G)),A4))
    <=> ( condit8047198070973881523_above(multiset(A),subseteq_mset(A),aa(set(B),set(multiset(A)),image2(B,multiset(A),F),A4))
        & condit8047198070973881523_above(multiset(A),subseteq_mset(A),aa(set(B),set(multiset(A)),image2(B,multiset(A),G),A4)) ) ) ).

% subset_mset.bdd_above_image_sup
tff(fact_6745_subset__mset_Obdd__above__Int2,axiom,
    ! [A: $tType,B3: set(multiset(A)),A4: set(multiset(A))] :
      ( condit8047198070973881523_above(multiset(A),subseteq_mset(A),B3)
     => condit8047198070973881523_above(multiset(A),subseteq_mset(A),aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),inf_inf(set(multiset(A))),A4),B3)) ) ).

% subset_mset.bdd_above_Int2
tff(fact_6746_subset__mset_Obdd__above__Int1,axiom,
    ! [A: $tType,A4: set(multiset(A)),B3: set(multiset(A))] :
      ( condit8047198070973881523_above(multiset(A),subseteq_mset(A),A4)
     => condit8047198070973881523_above(multiset(A),subseteq_mset(A),aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),inf_inf(set(multiset(A))),A4),B3)) ) ).

% subset_mset.bdd_above_Int1
tff(fact_6747_subset__mset_OcSup__le__iff,axiom,
    ! [A: $tType,S: set(multiset(A)),A3: multiset(A)] :
      ( ( S != bot_bot(set(multiset(A))) )
     => ( condit8047198070973881523_above(multiset(A),subseteq_mset(A),S)
       => ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),S)),A3)
        <=> ! [X4: multiset(A)] :
              ( aa(set(multiset(A)),$o,member(multiset(A),X4),S)
             => aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),X4),A3) ) ) ) ) ).

% subset_mset.cSup_le_iff
tff(fact_6748_subset__mset_OcSup__mono,axiom,
    ! [A: $tType,B3: set(multiset(A)),A4: set(multiset(A))] :
      ( ( B3 != bot_bot(set(multiset(A))) )
     => ( condit8047198070973881523_above(multiset(A),subseteq_mset(A),A4)
       => ( ! [B5: multiset(A)] :
              ( aa(set(multiset(A)),$o,member(multiset(A),B5),B3)
             => ? [X3: multiset(A)] :
                  ( aa(set(multiset(A)),$o,member(multiset(A),X3),A4)
                  & aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),B5),X3) ) )
         => aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),B3)),aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),A4)) ) ) ) ).

% subset_mset.cSup_mono
tff(fact_6749_bdd__above__multiset__imp__bdd__above__count,axiom,
    ! [A: $tType,A4: set(multiset(A)),X: A] :
      ( condit8047198070973881523_above(multiset(A),subseteq_mset(A),A4)
     => condit941137186595557371_above(nat,aa(set(multiset(A)),set(nat),image2(multiset(A),nat,aTP_Lamp_apu(A,fun(multiset(A),nat),X)),A4)) ) ).

% bdd_above_multiset_imp_bdd_above_count
tff(fact_6750_subset__mset_OcSUP__mono,axiom,
    ! [A: $tType,B: $tType,C: $tType,A4: set(A),G: fun(C,multiset(B)),B3: set(C),F: fun(A,multiset(B))] :
      ( ( A4 != bot_bot(set(A)) )
     => ( condit8047198070973881523_above(multiset(B),subseteq_mset(B),aa(set(C),set(multiset(B)),image2(C,multiset(B),G),B3))
       => ( ! [N3: A] :
              ( aa(set(A),$o,member(A,N3),A4)
             => ? [X3: C] :
                  ( aa(set(C),$o,member(C,X3),B3)
                  & aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),aa(A,multiset(B),F,N3)),aa(C,multiset(B),G,X3)) ) )
         => aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),aa(set(multiset(B)),multiset(B),complete_Sup_Sup(multiset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),F),A4))),aa(set(multiset(B)),multiset(B),complete_Sup_Sup(multiset(B)),aa(set(C),set(multiset(B)),image2(C,multiset(B),G),B3))) ) ) ) ).

% subset_mset.cSUP_mono
tff(fact_6751_subset__mset_OcSUP__le__iff,axiom,
    ! [A: $tType,B: $tType,A4: set(A),F: fun(A,multiset(B)),U: multiset(B)] :
      ( ( A4 != bot_bot(set(A)) )
     => ( condit8047198070973881523_above(multiset(B),subseteq_mset(B),aa(set(A),set(multiset(B)),image2(A,multiset(B),F),A4))
       => ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),aa(set(multiset(B)),multiset(B),complete_Sup_Sup(multiset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),F),A4))),U)
        <=> ! [X4: A] :
              ( aa(set(A),$o,member(A,X4),A4)
             => aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),aa(A,multiset(B),F,X4)),U) ) ) ) ) ).

% subset_mset.cSUP_le_iff
tff(fact_6752_subset__mset_OcSup__inter__less__eq,axiom,
    ! [A: $tType,A4: set(multiset(A)),B3: set(multiset(A))] :
      ( condit8047198070973881523_above(multiset(A),subseteq_mset(A),A4)
     => ( condit8047198070973881523_above(multiset(A),subseteq_mset(A),B3)
       => ( ( aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),inf_inf(set(multiset(A))),A4),B3) != bot_bot(set(multiset(A))) )
         => aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),inf_inf(set(multiset(A))),A4),B3))),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),union_mset(A),aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),A4)),aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),B3))) ) ) ) ).

% subset_mset.cSup_inter_less_eq
tff(fact_6753_subset__mset_OcSup__subset__mono,axiom,
    ! [A: $tType,A4: set(multiset(A)),B3: set(multiset(A))] :
      ( ( A4 != bot_bot(set(multiset(A))) )
     => ( condit8047198070973881523_above(multiset(A),subseteq_mset(A),B3)
       => ( aa(set(multiset(A)),$o,aa(set(multiset(A)),fun(set(multiset(A)),$o),ord_less_eq(set(multiset(A))),A4),B3)
         => aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),A4)),aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),B3)) ) ) ) ).

% subset_mset.cSup_subset_mono
tff(fact_6754_subset__mset_OcSup__cInf,axiom,
    ! [A: $tType,S: set(multiset(A))] :
      ( ( S != bot_bot(set(multiset(A))) )
     => ( condit8047198070973881523_above(multiset(A),subseteq_mset(A),S)
       => ( aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),S) = aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),aa(fun(multiset(A),$o),set(multiset(A)),collect(multiset(A)),aTP_Lamp_asg(set(multiset(A)),fun(multiset(A),$o),S))) ) ) ) ).

% subset_mset.cSup_cInf
tff(fact_6755_subset__mset_OcSUP__subset__mono,axiom,
    ! [B: $tType,A: $tType,A4: set(A),G: fun(A,multiset(B)),B3: set(A),F: fun(A,multiset(B))] :
      ( ( A4 != bot_bot(set(A)) )
     => ( condit8047198070973881523_above(multiset(B),subseteq_mset(B),aa(set(A),set(multiset(B)),image2(A,multiset(B),G),B3))
       => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B3)
         => ( ! [X2: A] :
                ( aa(set(A),$o,member(A,X2),A4)
               => aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),aa(A,multiset(B),F,X2)),aa(A,multiset(B),G,X2)) )
           => aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),aa(set(multiset(B)),multiset(B),complete_Sup_Sup(multiset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),F),A4))),aa(set(multiset(B)),multiset(B),complete_Sup_Sup(multiset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),G),B3))) ) ) ) ) ).

% subset_mset.cSUP_subset_mono
tff(fact_6756_subset__mset_OcSup__insert__If,axiom,
    ! [A: $tType,X5: set(multiset(A)),A3: multiset(A)] :
      ( condit8047198070973881523_above(multiset(A),subseteq_mset(A),X5)
     => ( aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert2(multiset(A)),A3),X5)) = $ite(X5 = bot_bot(set(multiset(A))),A3,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),union_mset(A),A3),aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),X5))) ) ) ).

% subset_mset.cSup_insert_If
tff(fact_6757_subset__mset_OcSup__insert,axiom,
    ! [A: $tType,X5: set(multiset(A)),A3: multiset(A)] :
      ( ( X5 != bot_bot(set(multiset(A))) )
     => ( condit8047198070973881523_above(multiset(A),subseteq_mset(A),X5)
       => ( aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert2(multiset(A)),A3),X5)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),union_mset(A),A3),aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),X5)) ) ) ) ).

% subset_mset.cSup_insert
tff(fact_6758_subset__mset_OSUP__sup__distrib,axiom,
    ! [B: $tType,A: $tType,A4: set(A),F: fun(A,multiset(B)),G: fun(A,multiset(B))] :
      ( ( A4 != bot_bot(set(A)) )
     => ( condit8047198070973881523_above(multiset(B),subseteq_mset(B),aa(set(A),set(multiset(B)),image2(A,multiset(B),F),A4))
       => ( condit8047198070973881523_above(multiset(B),subseteq_mset(B),aa(set(A),set(multiset(B)),image2(A,multiset(B),G),A4))
         => ( aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),union_mset(B),aa(set(multiset(B)),multiset(B),complete_Sup_Sup(multiset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),F),A4))),aa(set(multiset(B)),multiset(B),complete_Sup_Sup(multiset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),G),A4))) = aa(set(multiset(B)),multiset(B),complete_Sup_Sup(multiset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),aa(fun(A,multiset(B)),fun(A,multiset(B)),aTP_Lamp_ash(fun(A,multiset(B)),fun(fun(A,multiset(B)),fun(A,multiset(B))),F),G)),A4)) ) ) ) ) ).

% subset_mset.SUP_sup_distrib
tff(fact_6759_subset__mset_OcSup__union__distrib,axiom,
    ! [A: $tType,A4: set(multiset(A)),B3: set(multiset(A))] :
      ( ( A4 != bot_bot(set(multiset(A))) )
     => ( condit8047198070973881523_above(multiset(A),subseteq_mset(A),A4)
       => ( ( B3 != bot_bot(set(multiset(A))) )
         => ( condit8047198070973881523_above(multiset(A),subseteq_mset(A),B3)
           => ( aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),sup_sup(set(multiset(A))),A4),B3)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),union_mset(A),aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),A4)),aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),B3)) ) ) ) ) ) ).

% subset_mset.cSup_union_distrib
tff(fact_6760_subset__mset_OcSUP__insert,axiom,
    ! [B: $tType,A: $tType,A4: set(A),F: fun(A,multiset(B)),A3: A] :
      ( ( A4 != bot_bot(set(A)) )
     => ( condit8047198070973881523_above(multiset(B),subseteq_mset(B),aa(set(A),set(multiset(B)),image2(A,multiset(B),F),A4))
       => ( aa(set(multiset(B)),multiset(B),complete_Sup_Sup(multiset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),F),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),A4))) = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),union_mset(B),aa(A,multiset(B),F,A3)),aa(set(multiset(B)),multiset(B),complete_Sup_Sup(multiset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),F),A4))) ) ) ) ).

% subset_mset.cSUP_insert
tff(fact_6761_subset__mset_OcSUP__union,axiom,
    ! [B: $tType,A: $tType,A4: set(A),F: fun(A,multiset(B)),B3: set(A)] :
      ( ( A4 != bot_bot(set(A)) )
     => ( condit8047198070973881523_above(multiset(B),subseteq_mset(B),aa(set(A),set(multiset(B)),image2(A,multiset(B),F),A4))
       => ( ( B3 != bot_bot(set(A)) )
         => ( condit8047198070973881523_above(multiset(B),subseteq_mset(B),aa(set(A),set(multiset(B)),image2(A,multiset(B),F),B3))
           => ( aa(set(multiset(B)),multiset(B),complete_Sup_Sup(multiset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),F),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3))) = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),union_mset(B),aa(set(multiset(B)),multiset(B),complete_Sup_Sup(multiset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),F),A4))),aa(set(multiset(B)),multiset(B),complete_Sup_Sup(multiset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),F),B3))) ) ) ) ) ) ).

% subset_mset.cSUP_union
tff(fact_6762_set__mset__Sup,axiom,
    ! [A: $tType,A4: set(multiset(A))] :
      ( condit8047198070973881523_above(multiset(A),subseteq_mset(A),A4)
     => ( aa(multiset(A),set(A),set_mset(A),aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),A4)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(multiset(A)),set(set(A)),image2(multiset(A),set(A),set_mset(A)),A4)) ) ) ).

% set_mset_Sup
tff(fact_6763_count__Sup__multiset__nonempty,axiom,
    ! [A: $tType,A4: set(multiset(A)),X: A] :
      ( ( A4 != bot_bot(set(multiset(A))) )
     => ( condit8047198070973881523_above(multiset(A),subseteq_mset(A),A4)
       => ( aa(A,nat,aa(multiset(A),fun(A,nat),count(A),aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),A4)),X) = aa(set(nat),nat,complete_Sup_Sup(nat),aa(set(multiset(A)),set(nat),image2(multiset(A),nat,aTP_Lamp_apu(A,fun(multiset(A),nat),X)),A4)) ) ) ) ).

% count_Sup_multiset_nonempty
tff(fact_6764_bdd__above__multiset__imp__finite__support,axiom,
    ! [A: $tType,A4: set(multiset(A))] :
      ( ( A4 != bot_bot(set(multiset(A))) )
     => ( condit8047198070973881523_above(multiset(A),subseteq_mset(A),A4)
       => aa(set(A),$o,finite_finite2(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(multiset(A)),set(set(A)),image2(multiset(A),set(A),aTP_Lamp_asi(multiset(A),set(A))),A4))) ) ) ).

% bdd_above_multiset_imp_finite_support
tff(fact_6765_subset__mset_Omono__cSUP,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [F: fun(multiset(A),B),A4: fun(C,multiset(A)),I4: set(C)] :
          ( aa(fun(multiset(A),B),$o,mono(multiset(A),B,subseteq_mset(A)),F)
         => ( condit8047198070973881523_above(multiset(A),subseteq_mset(A),aa(set(C),set(multiset(A)),image2(C,multiset(A),A4),I4))
           => ( ( I4 != bot_bot(set(C)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(C),set(B),image2(C,B,aa(fun(C,multiset(A)),fun(C,B),aTP_Lamp_asj(fun(multiset(A),B),fun(fun(C,multiset(A)),fun(C,B)),F),A4)),I4))),aa(multiset(A),B,F,aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),aa(set(C),set(multiset(A)),image2(C,multiset(A),A4),I4)))) ) ) ) ) ).

% subset_mset.mono_cSUP
tff(fact_6766_subset__mset_Omono__cSup,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [F: fun(multiset(A),B),A4: set(multiset(A))] :
          ( aa(fun(multiset(A),B),$o,mono(multiset(A),B,subseteq_mset(A)),F)
         => ( condit8047198070973881523_above(multiset(A),subseteq_mset(A),A4)
           => ( ( A4 != bot_bot(set(multiset(A))) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(multiset(A)),set(B),image2(multiset(A),B,F),A4))),aa(multiset(A),B,F,aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),A4))) ) ) ) ) ).

% subset_mset.mono_cSup
tff(fact_6767_order_Omono_Ocong,axiom,
    ! [B: $tType,A: $tType] :
      ( order(B)
     => ! [Less_eq: fun(A,fun(A,$o))] : mono(A,B,Less_eq) = mono(A,B,Less_eq) ) ).

% order.mono.cong
tff(fact_6768_subset__mset_Omono__inf,axiom,
    ! [B: $tType,A: $tType] :
      ( semilattice_inf(B)
     => ! [F: fun(multiset(A),B),A4: multiset(A),B3: multiset(A)] :
          ( aa(fun(multiset(A),B),$o,mono(multiset(A),B,subseteq_mset(A)),F)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(multiset(A),B,F,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),A4),B3))),aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(multiset(A),B,F,A4)),aa(multiset(A),B,F,B3))) ) ) ).

% subset_mset.mono_inf
tff(fact_6769_subset__mset_Omono__sup,axiom,
    ! [B: $tType,A: $tType] :
      ( semilattice_sup(B)
     => ! [F: fun(multiset(A),B),A4: multiset(A),B3: multiset(A)] :
          ( aa(fun(multiset(A),B),$o,mono(multiset(A),B,subseteq_mset(A)),F)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),sup_sup(B),aa(multiset(A),B,F,A4)),aa(multiset(A),B,F,B3))),aa(multiset(A),B,F,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),union_mset(A),A4),B3))) ) ) ).

% subset_mset.mono_sup
tff(fact_6770_subset__mset_OGreatest__def,axiom,
    ! [A: $tType,P: fun(multiset(A),$o)] : aa(fun(multiset(A),$o),multiset(A),greatest(multiset(A),subseteq_mset(A)),P) = the(multiset(A),aTP_Lamp_ask(fun(multiset(A),$o),fun(multiset(A),$o),P)) ).

% subset_mset.Greatest_def
tff(fact_6771_subset__mset_Omono__cINF,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [F: fun(multiset(A),B),A4: fun(C,multiset(A)),I4: set(C)] :
          ( aa(fun(multiset(A),B),$o,mono(multiset(A),B,subseteq_mset(A)),F)
         => ( aa(set(multiset(A)),$o,condit8119078960628432327_below(multiset(A),subseteq_mset(A)),aa(set(C),set(multiset(A)),image2(C,multiset(A),A4),I4))
           => ( ( I4 != bot_bot(set(C)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(multiset(A),B,F,aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),aa(set(C),set(multiset(A)),image2(C,multiset(A),A4),I4)))),aa(set(B),B,complete_Inf_Inf(B),aa(set(C),set(B),image2(C,B,aa(fun(C,multiset(A)),fun(C,B),aTP_Lamp_asj(fun(multiset(A),B),fun(fun(C,multiset(A)),fun(C,B)),F),A4)),I4))) ) ) ) ) ).

% subset_mset.mono_cINF
tff(fact_6772_subset__mset_Obdd__below__empty,axiom,
    ! [A: $tType] : aa(set(multiset(A)),$o,condit8119078960628432327_below(multiset(A),subseteq_mset(A)),bot_bot(set(multiset(A)))) ).

% subset_mset.bdd_below_empty
tff(fact_6773_subset__mset_Obdd__below__Un,axiom,
    ! [A: $tType,A4: set(multiset(A)),B3: set(multiset(A))] :
      ( aa(set(multiset(A)),$o,condit8119078960628432327_below(multiset(A),subseteq_mset(A)),aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),sup_sup(set(multiset(A))),A4),B3))
    <=> ( aa(set(multiset(A)),$o,condit8119078960628432327_below(multiset(A),subseteq_mset(A)),A4)
        & aa(set(multiset(A)),$o,condit8119078960628432327_below(multiset(A),subseteq_mset(A)),B3) ) ) ).

% subset_mset.bdd_below_Un
tff(fact_6774_subset__mset_Obdd__below__image__inf,axiom,
    ! [A: $tType,B: $tType,F: fun(B,multiset(A)),G: fun(B,multiset(A)),A4: set(B)] :
      ( aa(set(multiset(A)),$o,condit8119078960628432327_below(multiset(A),subseteq_mset(A)),aa(set(B),set(multiset(A)),image2(B,multiset(A),aa(fun(B,multiset(A)),fun(B,multiset(A)),aTP_Lamp_asl(fun(B,multiset(A)),fun(fun(B,multiset(A)),fun(B,multiset(A))),F),G)),A4))
    <=> ( aa(set(multiset(A)),$o,condit8119078960628432327_below(multiset(A),subseteq_mset(A)),aa(set(B),set(multiset(A)),image2(B,multiset(A),F),A4))
        & aa(set(multiset(A)),$o,condit8119078960628432327_below(multiset(A),subseteq_mset(A)),aa(set(B),set(multiset(A)),image2(B,multiset(A),G),A4)) ) ) ).

% subset_mset.bdd_below_image_inf
tff(fact_6775_subset__mset_Obdd__below__UN,axiom,
    ! [B: $tType,A: $tType,I4: set(A),A4: fun(A,set(multiset(B)))] :
      ( aa(set(A),$o,finite_finite2(A),I4)
     => ( aa(set(multiset(B)),$o,condit8119078960628432327_below(multiset(B),subseteq_mset(B)),aa(set(set(multiset(B))),set(multiset(B)),complete_Sup_Sup(set(multiset(B))),aa(set(A),set(set(multiset(B))),image2(A,set(multiset(B)),A4),I4)))
      <=> ! [X4: A] :
            ( aa(set(A),$o,member(A,X4),I4)
           => aa(set(multiset(B)),$o,condit8119078960628432327_below(multiset(B),subseteq_mset(B)),aa(A,set(multiset(B)),A4,X4)) ) ) ) ).

% subset_mset.bdd_below_UN
tff(fact_6776_subset__mset_Obdd__below__Int1,axiom,
    ! [A: $tType,A4: set(multiset(A)),B3: set(multiset(A))] :
      ( aa(set(multiset(A)),$o,condit8119078960628432327_below(multiset(A),subseteq_mset(A)),A4)
     => aa(set(multiset(A)),$o,condit8119078960628432327_below(multiset(A),subseteq_mset(A)),aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),inf_inf(set(multiset(A))),A4),B3)) ) ).

% subset_mset.bdd_below_Int1
tff(fact_6777_subset__mset_Obdd__below__Int2,axiom,
    ! [A: $tType,B3: set(multiset(A)),A4: set(multiset(A))] :
      ( aa(set(multiset(A)),$o,condit8119078960628432327_below(multiset(A),subseteq_mset(A)),B3)
     => aa(set(multiset(A)),$o,condit8119078960628432327_below(multiset(A),subseteq_mset(A)),aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),inf_inf(set(multiset(A))),A4),B3)) ) ).

% subset_mset.bdd_below_Int2
tff(fact_6778_order_OGreatest_Ocong,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,$o))] : greatest(A,Less_eq) = greatest(A,Less_eq) ).

% order.Greatest.cong
tff(fact_6779_subset__mset_Ole__cInf__iff,axiom,
    ! [A: $tType,S: set(multiset(A)),A3: multiset(A)] :
      ( ( S != bot_bot(set(multiset(A))) )
     => ( aa(set(multiset(A)),$o,condit8119078960628432327_below(multiset(A),subseteq_mset(A)),S)
       => ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),A3),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),S))
        <=> ! [X4: multiset(A)] :
              ( aa(set(multiset(A)),$o,member(multiset(A),X4),S)
             => aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),A3),X4) ) ) ) ) ).

% subset_mset.le_cInf_iff
tff(fact_6780_subset__mset_OcInf__mono,axiom,
    ! [A: $tType,B3: set(multiset(A)),A4: set(multiset(A))] :
      ( ( B3 != bot_bot(set(multiset(A))) )
     => ( aa(set(multiset(A)),$o,condit8119078960628432327_below(multiset(A),subseteq_mset(A)),A4)
       => ( ! [B5: multiset(A)] :
              ( aa(set(multiset(A)),$o,member(multiset(A),B5),B3)
             => ? [X3: multiset(A)] :
                  ( aa(set(multiset(A)),$o,member(multiset(A),X3),A4)
                  & aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),X3),B5) ) )
         => aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),A4)),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),B3)) ) ) ) ).

% subset_mset.cInf_mono
tff(fact_6781_subset__mset_Ole__cINF__iff,axiom,
    ! [B: $tType,A: $tType,A4: set(A),F: fun(A,multiset(B)),U: multiset(B)] :
      ( ( A4 != bot_bot(set(A)) )
     => ( aa(set(multiset(B)),$o,condit8119078960628432327_below(multiset(B),subseteq_mset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),F),A4))
       => ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),U),aa(set(multiset(B)),multiset(B),complete_Inf_Inf(multiset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),F),A4)))
        <=> ! [X4: A] :
              ( aa(set(A),$o,member(A,X4),A4)
             => aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),U),aa(A,multiset(B),F,X4)) ) ) ) ) ).

% subset_mset.le_cINF_iff
tff(fact_6782_subset__mset_OcINF__mono,axiom,
    ! [C: $tType,B: $tType,A: $tType,B3: set(A),F: fun(C,multiset(B)),A4: set(C),G: fun(A,multiset(B))] :
      ( ( B3 != bot_bot(set(A)) )
     => ( aa(set(multiset(B)),$o,condit8119078960628432327_below(multiset(B),subseteq_mset(B)),aa(set(C),set(multiset(B)),image2(C,multiset(B),F),A4))
       => ( ! [M3: A] :
              ( aa(set(A),$o,member(A,M3),B3)
             => ? [X3: C] :
                  ( aa(set(C),$o,member(C,X3),A4)
                  & aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),aa(C,multiset(B),F,X3)),aa(A,multiset(B),G,M3)) ) )
         => aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),aa(set(multiset(B)),multiset(B),complete_Inf_Inf(multiset(B)),aa(set(C),set(multiset(B)),image2(C,multiset(B),F),A4))),aa(set(multiset(B)),multiset(B),complete_Inf_Inf(multiset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),G),B3))) ) ) ) ).

% subset_mset.cINF_mono
tff(fact_6783_subset__mset_OcInf__superset__mono,axiom,
    ! [A: $tType,A4: set(multiset(A)),B3: set(multiset(A))] :
      ( ( A4 != bot_bot(set(multiset(A))) )
     => ( aa(set(multiset(A)),$o,condit8119078960628432327_below(multiset(A),subseteq_mset(A)),B3)
       => ( aa(set(multiset(A)),$o,aa(set(multiset(A)),fun(set(multiset(A)),$o),ord_less_eq(set(multiset(A))),A4),B3)
         => aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),B3)),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),A4)) ) ) ) ).

% subset_mset.cInf_superset_mono
tff(fact_6784_subset__mset_OcINF__superset__mono,axiom,
    ! [B: $tType,A: $tType,A4: set(A),G: fun(A,multiset(B)),B3: set(A),F: fun(A,multiset(B))] :
      ( ( A4 != bot_bot(set(A)) )
     => ( aa(set(multiset(B)),$o,condit8119078960628432327_below(multiset(B),subseteq_mset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),G),B3))
       => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B3)
         => ( ! [X2: A] :
                ( aa(set(A),$o,member(A,X2),B3)
               => aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),aa(A,multiset(B),G,X2)),aa(A,multiset(B),F,X2)) )
           => aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),aa(set(multiset(B)),multiset(B),complete_Inf_Inf(multiset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),G),B3))),aa(set(multiset(B)),multiset(B),complete_Inf_Inf(multiset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),F),A4))) ) ) ) ) ).

% subset_mset.cINF_superset_mono
tff(fact_6785_subset__mset_OcInf__insert,axiom,
    ! [A: $tType,X5: set(multiset(A)),A3: multiset(A)] :
      ( ( X5 != bot_bot(set(multiset(A))) )
     => ( aa(set(multiset(A)),$o,condit8119078960628432327_below(multiset(A),subseteq_mset(A)),X5)
       => ( aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert2(multiset(A)),A3),X5)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),A3),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),X5)) ) ) ) ).

% subset_mset.cInf_insert
tff(fact_6786_subset__mset_OcInf__insert__If,axiom,
    ! [A: $tType,X5: set(multiset(A)),A3: multiset(A)] :
      ( aa(set(multiset(A)),$o,condit8119078960628432327_below(multiset(A),subseteq_mset(A)),X5)
     => ( aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert2(multiset(A)),A3),X5)) = $ite(X5 = bot_bot(set(multiset(A))),A3,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),A3),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),X5))) ) ) ).

% subset_mset.cInf_insert_If
tff(fact_6787_subset__mset_OcInf__le__cSup,axiom,
    ! [A: $tType,A4: set(multiset(A))] :
      ( ( A4 != bot_bot(set(multiset(A))) )
     => ( condit8047198070973881523_above(multiset(A),subseteq_mset(A),A4)
       => ( aa(set(multiset(A)),$o,condit8119078960628432327_below(multiset(A),subseteq_mset(A)),A4)
         => aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),A4)),aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),A4)) ) ) ) ).

% subset_mset.cInf_le_cSup
tff(fact_6788_subset__mset_Oless__eq__cInf__inter,axiom,
    ! [A: $tType,A4: set(multiset(A)),B3: set(multiset(A))] :
      ( aa(set(multiset(A)),$o,condit8119078960628432327_below(multiset(A),subseteq_mset(A)),A4)
     => ( aa(set(multiset(A)),$o,condit8119078960628432327_below(multiset(A),subseteq_mset(A)),B3)
       => ( ( aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),inf_inf(set(multiset(A))),A4),B3) != bot_bot(set(multiset(A))) )
         => aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),A4)),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),B3))),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),inf_inf(set(multiset(A))),A4),B3))) ) ) ) ).

% subset_mset.less_eq_cInf_inter
tff(fact_6789_subset__mset_OcINF__inf__distrib,axiom,
    ! [B: $tType,A: $tType,A4: set(A),F: fun(A,multiset(B)),G: fun(A,multiset(B))] :
      ( ( A4 != bot_bot(set(A)) )
     => ( aa(set(multiset(B)),$o,condit8119078960628432327_below(multiset(B),subseteq_mset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),F),A4))
       => ( aa(set(multiset(B)),$o,condit8119078960628432327_below(multiset(B),subseteq_mset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),G),A4))
         => ( aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),inter_mset(B),aa(set(multiset(B)),multiset(B),complete_Inf_Inf(multiset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),F),A4))),aa(set(multiset(B)),multiset(B),complete_Inf_Inf(multiset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),G),A4))) = aa(set(multiset(B)),multiset(B),complete_Inf_Inf(multiset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),aa(fun(A,multiset(B)),fun(A,multiset(B)),aTP_Lamp_asm(fun(A,multiset(B)),fun(fun(A,multiset(B)),fun(A,multiset(B))),F),G)),A4)) ) ) ) ) ).

% subset_mset.cINF_inf_distrib
tff(fact_6790_subset__mset_OcInf__union__distrib,axiom,
    ! [A: $tType,A4: set(multiset(A)),B3: set(multiset(A))] :
      ( ( A4 != bot_bot(set(multiset(A))) )
     => ( aa(set(multiset(A)),$o,condit8119078960628432327_below(multiset(A),subseteq_mset(A)),A4)
       => ( ( B3 != bot_bot(set(multiset(A))) )
         => ( aa(set(multiset(A)),$o,condit8119078960628432327_below(multiset(A),subseteq_mset(A)),B3)
           => ( aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),sup_sup(set(multiset(A))),A4),B3)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),A4)),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),B3)) ) ) ) ) ) ).

% subset_mset.cInf_union_distrib
tff(fact_6791_subset__mset_OcInf__cSup,axiom,
    ! [A: $tType,S: set(multiset(A))] :
      ( ( S != bot_bot(set(multiset(A))) )
     => ( aa(set(multiset(A)),$o,condit8119078960628432327_below(multiset(A),subseteq_mset(A)),S)
       => ( aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),S) = aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),aa(fun(multiset(A),$o),set(multiset(A)),collect(multiset(A)),aTP_Lamp_asn(set(multiset(A)),fun(multiset(A),$o),S))) ) ) ) ).

% subset_mset.cInf_cSup
tff(fact_6792_subset__mset_OcINF__insert,axiom,
    ! [B: $tType,A: $tType,A4: set(A),F: fun(A,multiset(B)),A3: A] :
      ( ( A4 != bot_bot(set(A)) )
     => ( aa(set(multiset(B)),$o,condit8119078960628432327_below(multiset(B),subseteq_mset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),F),A4))
       => ( aa(set(multiset(B)),multiset(B),complete_Inf_Inf(multiset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),F),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),A4))) = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),inter_mset(B),aa(A,multiset(B),F,A3)),aa(set(multiset(B)),multiset(B),complete_Inf_Inf(multiset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),F),A4))) ) ) ) ).

% subset_mset.cINF_insert
tff(fact_6793_subset__mset_OcINF__union,axiom,
    ! [B: $tType,A: $tType,A4: set(A),F: fun(A,multiset(B)),B3: set(A)] :
      ( ( A4 != bot_bot(set(A)) )
     => ( aa(set(multiset(B)),$o,condit8119078960628432327_below(multiset(B),subseteq_mset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),F),A4))
       => ( ( B3 != bot_bot(set(A)) )
         => ( aa(set(multiset(B)),$o,condit8119078960628432327_below(multiset(B),subseteq_mset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),F),B3))
           => ( aa(set(multiset(B)),multiset(B),complete_Inf_Inf(multiset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),F),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3))) = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),inter_mset(B),aa(set(multiset(B)),multiset(B),complete_Inf_Inf(multiset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),F),A4))),aa(set(multiset(B)),multiset(B),complete_Inf_Inf(multiset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),F),B3))) ) ) ) ) ) ).

% subset_mset.cINF_union
tff(fact_6794_subset__mset_Omono__cInf,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [F: fun(multiset(A),B),A4: set(multiset(A))] :
          ( aa(fun(multiset(A),B),$o,mono(multiset(A),B,subseteq_mset(A)),F)
         => ( aa(set(multiset(A)),$o,condit8119078960628432327_below(multiset(A),subseteq_mset(A)),A4)
           => ( ( A4 != bot_bot(set(multiset(A))) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(multiset(A),B,F,aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),A4))),aa(set(B),B,complete_Inf_Inf(B),aa(set(multiset(A)),set(B),image2(multiset(A),B,F),A4))) ) ) ) ) ).

% subset_mset.mono_cInf
tff(fact_6795_subset__mset_Osum__nonneg__0,axiom,
    ! [A: $tType,B: $tType,S2: set(A),F: fun(A,multiset(B)),I: A] :
      ( aa(set(A),$o,finite_finite2(A),S2)
     => ( ! [I2: A] :
            ( aa(set(A),$o,member(A,I2),S2)
           => aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),zero_zero(multiset(B))),aa(A,multiset(B),F,I2)) )
       => ( ( groups3894954378712506084id_sum(multiset(B),A,plus_plus(multiset(B)),zero_zero(multiset(B)),F,S2) = zero_zero(multiset(B)) )
         => ( aa(set(A),$o,member(A,I),S2)
           => ( aa(A,multiset(B),F,I) = zero_zero(multiset(B)) ) ) ) ) ) ).

% subset_mset.sum_nonneg_0
tff(fact_6796_subset__mset_Osum__nonneg__leq__bound,axiom,
    ! [A: $tType,B: $tType,S2: set(A),F: fun(A,multiset(B)),B3: multiset(B),I: A] :
      ( aa(set(A),$o,finite_finite2(A),S2)
     => ( ! [I2: A] :
            ( aa(set(A),$o,member(A,I2),S2)
           => aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),zero_zero(multiset(B))),aa(A,multiset(B),F,I2)) )
       => ( ( groups3894954378712506084id_sum(multiset(B),A,plus_plus(multiset(B)),zero_zero(multiset(B)),F,S2) = B3 )
         => ( aa(set(A),$o,member(A,I),S2)
           => aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),aa(A,multiset(B),F,I)),B3) ) ) ) ) ).

% subset_mset.sum_nonneg_leq_bound
tff(fact_6797_subset__mset_Osum__mono,axiom,
    ! [B: $tType,A: $tType,K5: set(A),F: fun(A,multiset(B)),G: fun(A,multiset(B))] :
      ( ! [I2: A] :
          ( aa(set(A),$o,member(A,I2),K5)
         => aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),aa(A,multiset(B),F,I2)),aa(A,multiset(B),G,I2)) )
     => aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),groups3894954378712506084id_sum(multiset(B),A,plus_plus(multiset(B)),zero_zero(multiset(B)),F,K5)),groups3894954378712506084id_sum(multiset(B),A,plus_plus(multiset(B)),zero_zero(multiset(B)),G,K5)) ) ).

% subset_mset.sum_mono
tff(fact_6798_subset__mset_OatLeast__eq__UNIV__iff,axiom,
    ! [A: $tType,X: multiset(A)] :
      ( ( set_atLeast(multiset(A),subseteq_mset(A),X) = top_top(set(multiset(A))) )
    <=> ( X = zero_zero(multiset(A)) ) ) ).

% subset_mset.atLeast_eq_UNIV_iff
tff(fact_6799_subset__mset_Osum__strict__mono,axiom,
    ! [B: $tType,A: $tType,A4: set(A),F: fun(A,multiset(B)),G: fun(A,multiset(B))] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( ( A4 != bot_bot(set(A)) )
       => ( ! [X2: A] :
              ( aa(set(A),$o,member(A,X2),A4)
             => aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subset_mset(B),aa(A,multiset(B),F,X2)),aa(A,multiset(B),G,X2)) )
         => aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subset_mset(B),groups3894954378712506084id_sum(multiset(B),A,plus_plus(multiset(B)),zero_zero(multiset(B)),F,A4)),groups3894954378712506084id_sum(multiset(B),A,plus_plus(multiset(B)),zero_zero(multiset(B)),G,A4)) ) ) ) ).

% subset_mset.sum_strict_mono
tff(fact_6800_subset__mset_OSup__fin_Osemilattice__order__set__axioms,axiom,
    ! [A: $tType] : lattic4895041142388067077er_set(multiset(A),union_mset(A),aTP_Lamp_aso(multiset(A),fun(multiset(A),$o)),aTP_Lamp_asp(multiset(A),fun(multiset(A),$o))) ).

% subset_mset.Sup_fin.semilattice_order_set_axioms
tff(fact_6801_subset__mset_OatLeast__def,axiom,
    ! [A: $tType,L: multiset(A)] : set_atLeast(multiset(A),subseteq_mset(A),L) = aa(fun(multiset(A),$o),set(multiset(A)),collect(multiset(A)),aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),L)) ).

% subset_mset.atLeast_def
tff(fact_6802_subset__mset_Oordering__top__axioms,axiom,
    ! [A: $tType] : ordering_top(multiset(A),aTP_Lamp_aso(multiset(A),fun(multiset(A),$o)),aTP_Lamp_asp(multiset(A),fun(multiset(A),$o)),zero_zero(multiset(A))) ).

% subset_mset.ordering_top_axioms
tff(fact_6803_wf__subset__mset__rel,axiom,
    ! [A: $tType] : wf(multiset(A),aa(fun(product_prod(multiset(A),multiset(A)),$o),set(product_prod(multiset(A),multiset(A))),collect(product_prod(multiset(A),multiset(A))),aa(fun(multiset(A),fun(multiset(A),$o)),fun(product_prod(multiset(A),multiset(A)),$o),product_case_prod(multiset(A),multiset(A),$o),subset_mset(A)))) ).

% wf_subset_mset_rel
tff(fact_6804_subset__mset_Oasymp__greater,axiom,
    ! [A: $tType] : asymp(multiset(A),aTP_Lamp_asp(multiset(A),fun(multiset(A),$o))) ).

% subset_mset.asymp_greater
tff(fact_6805_ord_OatLeast__def,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,$o)),L: A] : set_atLeast(A,Less_eq,L) = aa(fun(A,$o),set(A),collect(A),aa(A,fun(A,$o),Less_eq,L)) ).

% ord.atLeast_def
tff(fact_6806_subset__implies__mult,axiom,
    ! [A: $tType,A4: multiset(A),B3: multiset(A),R2: set(product_prod(A,A))] :
      ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subset_mset(A),A4),B3)
     => aa(set(product_prod(multiset(A),multiset(A))),$o,member(product_prod(multiset(A),multiset(A)),aa(multiset(A),product_prod(multiset(A),multiset(A)),aa(multiset(A),fun(multiset(A),product_prod(multiset(A),multiset(A))),product_Pair(multiset(A),multiset(A)),A4),B3)),mult(A,R2)) ) ).

% subset_implies_mult
tff(fact_6807_subset__mset_OacyclicI__order,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,A)),F: fun(A,multiset(B))] :
      ( ! [A6: A,B5: A] :
          ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A6),B5)),R2)
         => aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subset_mset(B),aa(A,multiset(B),F,B5)),aa(A,multiset(B),F,A6)) )
     => transitive_acyclic(A,R2) ) ).

% subset_mset.acyclicI_order
tff(fact_6808_subset__mset_Olexordp_Omono,axiom,
    ! [A: $tType] : order_mono(fun(list(multiset(A)),fun(list(multiset(A)),$o)),fun(list(multiset(A)),fun(list(multiset(A)),$o)),aTP_Lamp_asq(fun(list(multiset(A)),fun(list(multiset(A)),$o)),fun(list(multiset(A)),fun(list(multiset(A)),$o)))) ).

% subset_mset.lexordp.mono
tff(fact_6809_subset__mset_Olexordp__def,axiom,
    ! [A: $tType] : lexordp2(multiset(A),subset_mset(A)) = complete_lattice_lfp(fun(list(multiset(A)),fun(list(multiset(A)),$o)),aTP_Lamp_asq(fun(list(multiset(A)),fun(list(multiset(A)),$o)),fun(list(multiset(A)),fun(list(multiset(A)),$o)))) ).

% subset_mset.lexordp_def
tff(fact_6810_subset__mset_Onot__empty__eq__Ici__eq__empty,axiom,
    ! [A: $tType,L: multiset(A)] : bot_bot(set(multiset(A))) != set_atLeast(multiset(A),subseteq_mset(A),L) ).

% subset_mset.not_empty_eq_Ici_eq_empty
tff(fact_6811_subset__mset_Osemilattice__neutr__order__axioms,axiom,
    ! [A: $tType] : semila1105856199041335345_order(multiset(A),union_mset(A),zero_zero(multiset(A)),aTP_Lamp_aso(multiset(A),fun(multiset(A),$o)),aTP_Lamp_asp(multiset(A),fun(multiset(A),$o))) ).

% subset_mset.semilattice_neutr_order_axioms
tff(fact_6812_subset__mset_OcSUP__lessD,axiom,
    ! [B: $tType,A: $tType,F: fun(B,multiset(A)),A4: set(B),Y: multiset(A),I: B] :
      ( condit8047198070973881523_above(multiset(A),subseteq_mset(A),aa(set(B),set(multiset(A)),image2(B,multiset(A),F),A4))
     => ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subset_mset(A),aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),aa(set(B),set(multiset(A)),image2(B,multiset(A),F),A4))),Y)
       => ( aa(set(B),$o,member(B,I),A4)
         => aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subset_mset(A),aa(B,multiset(A),F,I)),Y) ) ) ) ).

% subset_mset.cSUP_lessD
tff(fact_6813_subset__mset_Oless__cINF__D,axiom,
    ! [A: $tType,B: $tType,F: fun(B,multiset(A)),A4: set(B),Y: multiset(A),I: B] :
      ( aa(set(multiset(A)),$o,condit8119078960628432327_below(multiset(A),subseteq_mset(A)),aa(set(B),set(multiset(A)),image2(B,multiset(A),F),A4))
     => ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subset_mset(A),Y),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),aa(set(B),set(multiset(A)),image2(B,multiset(A),F),A4)))
       => ( aa(set(B),$o,member(B,I),A4)
         => aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subset_mset(A),Y),aa(B,multiset(A),F,I)) ) ) ) ).

% subset_mset.less_cINF_D
tff(fact_6814_subset__mset_Osum__pos,axiom,
    ! [B: $tType,A: $tType,I4: set(A),F: fun(A,multiset(B))] :
      ( aa(set(A),$o,finite_finite2(A),I4)
     => ( ( I4 != bot_bot(set(A)) )
       => ( ! [I2: A] :
              ( aa(set(A),$o,member(A,I2),I4)
             => aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subset_mset(B),zero_zero(multiset(B))),aa(A,multiset(B),F,I2)) )
         => aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subset_mset(B),zero_zero(multiset(B))),groups3894954378712506084id_sum(multiset(B),A,plus_plus(multiset(B)),zero_zero(multiset(B)),F,I4)) ) ) ) ).

% subset_mset.sum_pos
tff(fact_6815_subset__mset_OgreaterThanLessThan__empty,axiom,
    ! [A: $tType,L: multiset(A),K: multiset(A)] :
      ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),L),K)
     => ( set_gr287244882034783167ssThan(multiset(A),subset_mset(A),K,L) = bot_bot(set(multiset(A))) ) ) ).

% subset_mset.greaterThanLessThan_empty
tff(fact_6816_subset__mset_OIio__Int__singleton,axiom,
    ! [A: $tType,K: multiset(A),X: multiset(A)] :
      aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),inf_inf(set(multiset(A))),set_lessThan(multiset(A),subset_mset(A),K)),aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert2(multiset(A)),X),bot_bot(set(multiset(A))))) = $ite(aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subset_mset(A),X),K),aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert2(multiset(A)),X),bot_bot(set(multiset(A)))),bot_bot(set(multiset(A)))) ).

% subset_mset.Iio_Int_singleton
tff(fact_6817_subset__mset_OlessThan__def,axiom,
    ! [A: $tType,U: multiset(A)] : set_lessThan(multiset(A),subset_mset(A),U) = aa(fun(multiset(A),$o),set(multiset(A)),collect(multiset(A)),aa(multiset(A),fun(multiset(A),$o),aTP_Lamp_asp(multiset(A),fun(multiset(A),$o)),U)) ).

% subset_mset.lessThan_def
tff(fact_6818_ord_OlessThan__def,axiom,
    ! [A: $tType,Less: fun(A,fun(A,$o)),U: A] : set_lessThan(A,Less,U) = aa(fun(A,$o),set(A),collect(A),aa(A,fun(A,$o),aTP_Lamp_aak(fun(A,fun(A,$o)),fun(A,fun(A,$o)),Less),U)) ).

% ord.lessThan_def
tff(fact_6819_ord_OatLeastLessThan__def,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),L: A,U: A] : set_atLeastLessThan(A,Less_eq,Less,L,U) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_atLeast(A,Less_eq,L)),set_lessThan(A,Less,U)) ).

% ord.atLeastLessThan_def
tff(fact_6820_subset__mset_OatLeastLessThan__def,axiom,
    ! [A: $tType,L: multiset(A),U: multiset(A)] : set_atLeastLessThan(multiset(A),subseteq_mset(A),subset_mset(A),L,U) = aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),inf_inf(set(multiset(A))),set_atLeast(multiset(A),subseteq_mset(A),L)),set_lessThan(multiset(A),subset_mset(A),U)) ).

% subset_mset.atLeastLessThan_def
tff(fact_6821_subset__mset_OatLeastLessThan__empty,axiom,
    ! [A: $tType,B2: multiset(A),A3: multiset(A)] :
      ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),B2),A3)
     => ( set_atLeastLessThan(multiset(A),subseteq_mset(A),subset_mset(A),A3,B2) = bot_bot(set(multiset(A))) ) ) ).

% subset_mset.atLeastLessThan_empty
tff(fact_6822_subset__mset_OatLeastLessThan__empty__iff,axiom,
    ! [A: $tType,A3: multiset(A),B2: multiset(A)] :
      ( ( set_atLeastLessThan(multiset(A),subseteq_mset(A),subset_mset(A),A3,B2) = bot_bot(set(multiset(A))) )
    <=> ~ aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subset_mset(A),A3),B2) ) ).

% subset_mset.atLeastLessThan_empty_iff
tff(fact_6823_subset__mset_OatLeastLessThan__empty__iff2,axiom,
    ! [A: $tType,A3: multiset(A),B2: multiset(A)] :
      ( ( bot_bot(set(multiset(A))) = set_atLeastLessThan(multiset(A),subseteq_mset(A),subset_mset(A),A3,B2) )
    <=> ~ aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subset_mset(A),A3),B2) ) ).

% subset_mset.atLeastLessThan_empty_iff2
tff(fact_6824_subset__mset_OatLeastLessThan__eq__atLeastAtMost__diff,axiom,
    ! [A: $tType,A3: multiset(A),B2: multiset(A)] : set_atLeastLessThan(multiset(A),subseteq_mset(A),subset_mset(A),A3,B2) = aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),minus_minus(set(multiset(A))),set_atLeastAtMost(multiset(A),subseteq_mset(A),A3,B2)),aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert2(multiset(A)),B2),bot_bot(set(multiset(A))))) ).

% subset_mset.atLeastLessThan_eq_atLeastAtMost_diff
tff(fact_6825_subset__mset_OgreaterThanLessThan__eq,axiom,
    ! [A: $tType,A3: multiset(A),B2: multiset(A)] : set_gr287244882034783167ssThan(multiset(A),subset_mset(A),A3,B2) = aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),inf_inf(set(multiset(A))),set_greaterThan(multiset(A),subset_mset(A),A3)),set_lessThan(multiset(A),subset_mset(A),B2)) ).

% subset_mset.greaterThanLessThan_eq
tff(fact_6826_subset__mset_OatLeastatMost__empty__iff,axiom,
    ! [A: $tType,A3: multiset(A),B2: multiset(A)] :
      ( ( set_atLeastAtMost(multiset(A),subseteq_mset(A),A3,B2) = bot_bot(set(multiset(A))) )
    <=> ~ aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),A3),B2) ) ).

% subset_mset.atLeastatMost_empty_iff
tff(fact_6827_subset__mset_OatLeastatMost__empty__iff2,axiom,
    ! [A: $tType,A3: multiset(A),B2: multiset(A)] :
      ( ( bot_bot(set(multiset(A))) = set_atLeastAtMost(multiset(A),subseteq_mset(A),A3,B2) )
    <=> ~ aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),A3),B2) ) ).

% subset_mset.atLeastatMost_empty_iff2
tff(fact_6828_subset__mset_OatLeastatMost__empty,axiom,
    ! [A: $tType,B2: multiset(A),A3: multiset(A)] :
      ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subset_mset(A),B2),A3)
     => ( set_atLeastAtMost(multiset(A),subseteq_mset(A),A3,B2) = bot_bot(set(multiset(A))) ) ) ).

% subset_mset.atLeastatMost_empty
tff(fact_6829_subset__mset_OatLeastAtMost__singleton,axiom,
    ! [A: $tType,A3: multiset(A)] : set_atLeastAtMost(multiset(A),subseteq_mset(A),A3,A3) = aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert2(multiset(A)),A3),bot_bot(set(multiset(A)))) ).

% subset_mset.atLeastAtMost_singleton
tff(fact_6830_subset__mset_OatLeastAtMost__singleton__iff,axiom,
    ! [A: $tType,A3: multiset(A),B2: multiset(A),C2: multiset(A)] :
      ( ( set_atLeastAtMost(multiset(A),subseteq_mset(A),A3,B2) = aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert2(multiset(A)),C2),bot_bot(set(multiset(A)))) )
    <=> ( ( A3 = B2 )
        & ( B2 = C2 ) ) ) ).

% subset_mset.atLeastAtMost_singleton_iff
tff(fact_6831_subset__mset_OgreaterThan__def,axiom,
    ! [A: $tType,L: multiset(A)] : set_greaterThan(multiset(A),subset_mset(A),L) = aa(fun(multiset(A),$o),set(multiset(A)),collect(multiset(A)),aa(multiset(A),fun(multiset(A),$o),subset_mset(A),L)) ).

% subset_mset.greaterThan_def
tff(fact_6832_ord_OgreaterThan__def,axiom,
    ! [A: $tType,Less: fun(A,fun(A,$o)),L: A] : set_greaterThan(A,Less,L) = aa(fun(A,$o),set(A),collect(A),aa(A,fun(A,$o),Less,L)) ).

% ord.greaterThan_def
tff(fact_6833_subset__mset_OatLeastAtMost__singleton_H,axiom,
    ! [A: $tType,A3: multiset(A),B2: multiset(A)] :
      ( ( A3 = B2 )
     => ( set_atLeastAtMost(multiset(A),subseteq_mset(A),A3,B2) = aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert2(multiset(A)),A3),bot_bot(set(multiset(A)))) ) ) ).

% subset_mset.atLeastAtMost_singleton'
tff(fact_6834_ord_OgreaterThanLessThan__eq,axiom,
    ! [A: $tType,Less: fun(A,fun(A,$o)),A3: A,B2: A] : set_gr287244882034783167ssThan(A,Less,A3,B2) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_greaterThan(A,Less,A3)),set_lessThan(A,Less,B2)) ).

% ord.greaterThanLessThan_eq
tff(fact_6835_ord_OgreaterThanLessThan__def,axiom,
    ! [A: $tType,Less: fun(A,fun(A,$o)),L: A,U: A] : set_gr287244882034783167ssThan(A,Less,L,U) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_greaterThan(A,Less,L)),set_lessThan(A,Less,U)) ).

% ord.greaterThanLessThan_def
tff(fact_6836_subset__mset_OgreaterThanLessThan__def,axiom,
    ! [A: $tType,L: multiset(A),U: multiset(A)] : set_gr287244882034783167ssThan(multiset(A),subset_mset(A),L,U) = aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),inf_inf(set(multiset(A))),set_greaterThan(multiset(A),subset_mset(A),L)),set_lessThan(multiset(A),subset_mset(A),U)) ).

% subset_mset.greaterThanLessThan_def
tff(fact_6837_subset__mset_OgreaterThanAtMost__eq__atLeastAtMost__diff,axiom,
    ! [A: $tType,A3: multiset(A),B2: multiset(A)] : set_gr3752724095348155675AtMost(multiset(A),subseteq_mset(A),subset_mset(A),A3,B2) = aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),minus_minus(set(multiset(A))),set_atLeastAtMost(multiset(A),subseteq_mset(A),A3,B2)),aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert2(multiset(A)),A3),bot_bot(set(multiset(A))))) ).

% subset_mset.greaterThanAtMost_eq_atLeastAtMost_diff
tff(fact_6838_ord_OatLeastAtMost__def,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,$o)),L: A,U: A] : set_atLeastAtMost(A,Less_eq,L,U) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_atLeast(A,Less_eq,L)),set_atMost(A,Less_eq,U)) ).

% ord.atLeastAtMost_def
tff(fact_6839_subset__mset_OgreaterThanAtMost__empty,axiom,
    ! [A: $tType,L: multiset(A),K: multiset(A)] :
      ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),L),K)
     => ( set_gr3752724095348155675AtMost(multiset(A),subseteq_mset(A),subset_mset(A),K,L) = bot_bot(set(multiset(A))) ) ) ).

% subset_mset.greaterThanAtMost_empty
tff(fact_6840_subset__mset_OgreaterThanAtMost__empty__iff,axiom,
    ! [A: $tType,K: multiset(A),L: multiset(A)] :
      ( ( set_gr3752724095348155675AtMost(multiset(A),subseteq_mset(A),subset_mset(A),K,L) = bot_bot(set(multiset(A))) )
    <=> ~ aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subset_mset(A),K),L) ) ).

% subset_mset.greaterThanAtMost_empty_iff
tff(fact_6841_subset__mset_OgreaterThanAtMost__empty__iff2,axiom,
    ! [A: $tType,K: multiset(A),L: multiset(A)] :
      ( ( bot_bot(set(multiset(A))) = set_gr3752724095348155675AtMost(multiset(A),subseteq_mset(A),subset_mset(A),K,L) )
    <=> ~ aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subset_mset(A),K),L) ) ).

% subset_mset.greaterThanAtMost_empty_iff2
tff(fact_6842_subset__mset_OatMost__def,axiom,
    ! [A: $tType,U: multiset(A)] : set_atMost(multiset(A),subseteq_mset(A),U) = aa(fun(multiset(A),$o),set(multiset(A)),collect(multiset(A)),aa(multiset(A),fun(multiset(A),$o),aTP_Lamp_aso(multiset(A),fun(multiset(A),$o)),U)) ).

% subset_mset.atMost_def
tff(fact_6843_ord_OatMost__def,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,$o)),U: A] : set_atMost(A,Less_eq,U) = aa(fun(A,$o),set(A),collect(A),aa(A,fun(A,$o),aTP_Lamp_aak(fun(A,fun(A,$o)),fun(A,fun(A,$o)),Less_eq),U)) ).

% ord.atMost_def
tff(fact_6844_ord_OgreaterThanAtMost__def,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),L: A,U: A] : set_gr3752724095348155675AtMost(A,Less_eq,Less,L,U) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_greaterThan(A,Less,L)),set_atMost(A,Less_eq,U)) ).

% ord.greaterThanAtMost_def
tff(fact_6845_subset__mset_Onot__empty__eq__Iic__eq__empty,axiom,
    ! [A: $tType,H2: multiset(A)] : bot_bot(set(multiset(A))) != set_atMost(multiset(A),subseteq_mset(A),H2) ).

% subset_mset.not_empty_eq_Iic_eq_empty
tff(fact_6846_subset__mset_OgreaterThanAtMost__def,axiom,
    ! [A: $tType,L: multiset(A),U: multiset(A)] : set_gr3752724095348155675AtMost(multiset(A),subseteq_mset(A),subset_mset(A),L,U) = aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),inf_inf(set(multiset(A))),set_greaterThan(multiset(A),subset_mset(A),L)),set_atMost(multiset(A),subseteq_mset(A),U)) ).

% subset_mset.greaterThanAtMost_def
tff(fact_6847_subset__mset_OatLeastAtMost__def,axiom,
    ! [A: $tType,L: multiset(A),U: multiset(A)] : set_atLeastAtMost(multiset(A),subseteq_mset(A),L,U) = aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),inf_inf(set(multiset(A))),set_atLeast(multiset(A),subseteq_mset(A),L)),set_atMost(multiset(A),subseteq_mset(A),U)) ).

% subset_mset.atLeastAtMost_def
tff(fact_6848_folding__def_H,axiom,
    ! [B: $tType,A: $tType,F: fun(A,fun(B,B))] :
      ( finite_folding(A,B,F)
    <=> finite_folding_on(A,B,top_top(set(A)),F) ) ).

% folding_def'
tff(fact_6849_scomp__unfold,axiom,
    ! [D: $tType,B: $tType,C: $tType,A: $tType,X3: fun(A,product_prod(B,C)),Xa3: fun(B,fun(C,D)),Xb2: A] : aa(A,D,product_scomp(A,B,C,D,X3,Xa3),Xb2) = aa(C,D,aa(B,fun(C,D),Xa3,aa(product_prod(B,C),B,product_fst(B,C),aa(A,product_prod(B,C),X3,Xb2))),aa(product_prod(B,C),C,product_snd(B,C),aa(A,product_prod(B,C),X3,Xb2))) ).

% scomp_unfold
tff(fact_6850_Pair__scomp,axiom,
    ! [A: $tType,B: $tType,C: $tType,X: C,F: fun(C,fun(A,B))] : product_scomp(A,C,A,B,aa(C,fun(A,product_prod(C,A)),product_Pair(C,A),X),F) = aa(C,fun(A,B),F,X) ).

% Pair_scomp
tff(fact_6851_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_6852_scomp__scomp,axiom,
    ! [A: $tType,E: $tType,F3: $tType,B: $tType,D: $tType,C: $tType,F: fun(A,product_prod(E,F3)),G: fun(E,fun(F3,product_prod(C,D))),H2: fun(C,fun(D,B))] : product_scomp(A,C,D,B,product_scomp(A,E,F3,product_prod(C,D),F,G),H2) = product_scomp(A,E,F3,B,F,aa(fun(C,fun(D,B)),fun(E,fun(F3,B)),aTP_Lamp_asr(fun(E,fun(F3,product_prod(C,D))),fun(fun(C,fun(D,B)),fun(E,fun(F3,B))),G),H2)) ).

% scomp_scomp
tff(fact_6853_card_Ofolding__axioms,axiom,
    ! [A: $tType] : finite_folding(A,nat,aTP_Lamp_oo(A,fun(nat,nat))) ).

% card.folding_axioms
tff(fact_6854_semilattice__order__set_Osubset__imp,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A4: set(A),B3: set(A)] :
      ( lattic4895041142388067077er_set(A,F,Less_eq,Less)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B3)
       => ( ( A4 != bot_bot(set(A)) )
         => ( aa(set(A),$o,finite_finite2(A),B3)
           => aa(A,$o,aa(A,fun(A,$o),Less_eq,aa(set(A),A,lattic1715443433743089157tice_F(A,F),B3)),aa(set(A),A,lattic1715443433743089157tice_F(A,F),A4)) ) ) ) ) ).

% semilattice_order_set.subset_imp
tff(fact_6855_cr__int__def,axiom,
    ! [X3: product_prod(nat,nat)] : cr_int(X3) = aa(int,fun(int,$o),fequal(int),aa(product_prod(nat,nat),int,abs_Integ,X3)) ).

% cr_int_def
tff(fact_6856_Inf__fin__def,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ( lattic7752659483105999362nf_fin(A) = lattic1715443433743089157tice_F(A,inf_inf(A)) ) ) ).

% Inf_fin_def
tff(fact_6857_Sup__fin__def,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ( lattic5882676163264333800up_fin(A) = lattic1715443433743089157tice_F(A,sup_sup(A)) ) ) ).

% Sup_fin_def
tff(fact_6858_semilattice__order__set_Obounded__iff,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A4: set(A),X: A] :
      ( lattic4895041142388067077er_set(A,F,Less_eq,Less)
     => ( aa(set(A),$o,finite_finite2(A),A4)
       => ( ( A4 != bot_bot(set(A)) )
         => ( aa(A,$o,aa(A,fun(A,$o),Less_eq,X),aa(set(A),A,lattic1715443433743089157tice_F(A,F),A4))
          <=> ! [X4: A] :
                ( aa(set(A),$o,member(A,X4),A4)
               => aa(A,$o,aa(A,fun(A,$o),Less_eq,X),X4) ) ) ) ) ) ).

% semilattice_order_set.bounded_iff
tff(fact_6859_semilattice__order__set_OboundedI,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A4: set(A),X: A] :
      ( lattic4895041142388067077er_set(A,F,Less_eq,Less)
     => ( aa(set(A),$o,finite_finite2(A),A4)
       => ( ( A4 != bot_bot(set(A)) )
         => ( ! [A6: A] :
                ( aa(set(A),$o,member(A,A6),A4)
               => aa(A,$o,aa(A,fun(A,$o),Less_eq,X),A6) )
           => aa(A,$o,aa(A,fun(A,$o),Less_eq,X),aa(set(A),A,lattic1715443433743089157tice_F(A,F),A4)) ) ) ) ) ).

% semilattice_order_set.boundedI
tff(fact_6860_semilattice__order__set_OboundedE,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A4: set(A),X: A] :
      ( lattic4895041142388067077er_set(A,F,Less_eq,Less)
     => ( aa(set(A),$o,finite_finite2(A),A4)
       => ( ( A4 != bot_bot(set(A)) )
         => ( aa(A,$o,aa(A,fun(A,$o),Less_eq,X),aa(set(A),A,lattic1715443433743089157tice_F(A,F),A4))
           => ! [A11: A] :
                ( aa(set(A),$o,member(A,A11),A4)
               => aa(A,$o,aa(A,fun(A,$o),Less_eq,X),A11) ) ) ) ) ) ).

% semilattice_order_set.boundedE
tff(fact_6861_semilattice__set_Oeq__fold_H,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),A4: set(A)] :
      ( lattic149705377957585745ce_set(A,F)
     => ( aa(set(A),A,lattic1715443433743089157tice_F(A,F),A4) = aa(option(A),A,the2(A),finite_fold(A,option(A),aTP_Lamp_ass(fun(A,fun(A,A)),fun(A,fun(option(A),option(A))),F),none(A),A4)) ) ) ).

% semilattice_set.eq_fold'
tff(fact_6862_semilattice__set_Oremove,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),A4: set(A),X: A] :
      ( lattic149705377957585745ce_set(A,F)
     => ( aa(set(A),$o,finite_finite2(A),A4)
       => ( aa(set(A),$o,member(A,X),A4)
         => ( aa(set(A),A,lattic1715443433743089157tice_F(A,F),A4) = $ite(aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) = bot_bot(set(A)),X,aa(A,A,aa(A,fun(A,A),F,X),aa(set(A),A,lattic1715443433743089157tice_F(A,F),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))))))) ) ) ) ) ).

% semilattice_set.remove
tff(fact_6863_Sup__fin_Osemilattice__set__axioms,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => lattic149705377957585745ce_set(A,sup_sup(A)) ) ).

% Sup_fin.semilattice_set_axioms
tff(fact_6864_Inf__fin_Osemilattice__set__axioms,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => lattic149705377957585745ce_set(A,inf_inf(A)) ) ).

% Inf_fin.semilattice_set_axioms
tff(fact_6865_semilattice__set_Osingleton,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),X: A] :
      ( lattic149705377957585745ce_set(A,F)
     => ( aa(set(A),A,lattic1715443433743089157tice_F(A,F),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) = X ) ) ).

% semilattice_set.singleton
tff(fact_6866_semilattice__set_Ohom__commute,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),H2: fun(A,A),N4: set(A)] :
      ( lattic149705377957585745ce_set(A,F)
     => ( ! [X2: A,Y2: A] : aa(A,A,H2,aa(A,A,aa(A,fun(A,A),F,X2),Y2)) = aa(A,A,aa(A,fun(A,A),F,aa(A,A,H2,X2)),aa(A,A,H2,Y2))
       => ( aa(set(A),$o,finite_finite2(A),N4)
         => ( ( N4 != bot_bot(set(A)) )
           => ( aa(A,A,H2,aa(set(A),A,lattic1715443433743089157tice_F(A,F),N4)) = aa(set(A),A,lattic1715443433743089157tice_F(A,F),aa(set(A),set(A),image2(A,A,H2),N4)) ) ) ) ) ) ).

% semilattice_set.hom_commute
tff(fact_6867_semilattice__set_Osubset,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),A4: set(A),B3: set(A)] :
      ( lattic149705377957585745ce_set(A,F)
     => ( aa(set(A),$o,finite_finite2(A),A4)
       => ( ( B3 != bot_bot(set(A)) )
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),A4)
           => ( aa(A,A,aa(A,fun(A,A),F,aa(set(A),A,lattic1715443433743089157tice_F(A,F),B3)),aa(set(A),A,lattic1715443433743089157tice_F(A,F),A4)) = aa(set(A),A,lattic1715443433743089157tice_F(A,F),A4) ) ) ) ) ) ).

% semilattice_set.subset
tff(fact_6868_semilattice__set_Oinsert__not__elem,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),A4: set(A),X: A] :
      ( lattic149705377957585745ce_set(A,F)
     => ( aa(set(A),$o,finite_finite2(A),A4)
       => ( ~ aa(set(A),$o,member(A,X),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( aa(set(A),A,lattic1715443433743089157tice_F(A,F),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A4)) = aa(A,A,aa(A,fun(A,A),F,X),aa(set(A),A,lattic1715443433743089157tice_F(A,F),A4)) ) ) ) ) ) ).

% semilattice_set.insert_not_elem
tff(fact_6869_semilattice__set_Oinsert,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),A4: set(A),X: A] :
      ( lattic149705377957585745ce_set(A,F)
     => ( aa(set(A),$o,finite_finite2(A),A4)
       => ( ( A4 != bot_bot(set(A)) )
         => ( aa(set(A),A,lattic1715443433743089157tice_F(A,F),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A4)) = aa(A,A,aa(A,fun(A,A),F,X),aa(set(A),A,lattic1715443433743089157tice_F(A,F),A4)) ) ) ) ) ).

% semilattice_set.insert
tff(fact_6870_semilattice__set_Oclosed,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),A4: set(A)] :
      ( lattic149705377957585745ce_set(A,F)
     => ( aa(set(A),$o,finite_finite2(A),A4)
       => ( ( A4 != bot_bot(set(A)) )
         => ( ! [X2: A,Y2: A] : aa(set(A),$o,member(A,aa(A,A,aa(A,fun(A,A),F,X2),Y2)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y2),bot_bot(set(A)))))
           => aa(set(A),$o,member(A,aa(set(A),A,lattic1715443433743089157tice_F(A,F),A4)),A4) ) ) ) ) ).

% semilattice_set.closed
tff(fact_6871_semilattice__set_Ounion,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),A4: set(A),B3: set(A)] :
      ( lattic149705377957585745ce_set(A,F)
     => ( aa(set(A),$o,finite_finite2(A),A4)
       => ( ( A4 != bot_bot(set(A)) )
         => ( aa(set(A),$o,finite_finite2(A),B3)
           => ( ( B3 != bot_bot(set(A)) )
             => ( aa(set(A),A,lattic1715443433743089157tice_F(A,F),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3)) = aa(A,A,aa(A,fun(A,A),F,aa(set(A),A,lattic1715443433743089157tice_F(A,F),A4)),aa(set(A),A,lattic1715443433743089157tice_F(A,F),B3)) ) ) ) ) ) ) ).

% semilattice_set.union
tff(fact_6872_semilattice__set_Oinsert__remove,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),A4: set(A),X: A] :
      ( lattic149705377957585745ce_set(A,F)
     => ( aa(set(A),$o,finite_finite2(A),A4)
       => ( aa(set(A),A,lattic1715443433743089157tice_F(A,F),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A4)) = $ite(aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) = bot_bot(set(A)),X,aa(A,A,aa(A,fun(A,A),F,X),aa(set(A),A,lattic1715443433743089157tice_F(A,F),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))))))) ) ) ) ).

% semilattice_set.insert_remove
tff(fact_6873_drop__bit__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,N: nat] :
          bit_se4197421643247451524op_bit(A,M,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)) = aa(A,A,
            aa(A,fun(A,A),times_times(A),
              aa($o,A,zero_neq_one_of_bool(A),
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),N)
                & bit_se6407376104438227557le_bit(A,type2(A),N) ))),
            aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M))) ) ).

% drop_bit_exp_eq
tff(fact_6874_antisymp__antisym__eq,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( antisymp(A,aTP_Lamp_ux(set(product_prod(A,A)),fun(A,fun(A,$o)),R2))
    <=> antisym(A,R2) ) ).

% antisymp_antisym_eq
tff(fact_6875_bit__minus__1__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,uminus_uminus(A),one_one(A))),N)
        <=> bit_se6407376104438227557le_bit(A,type2(A),N) ) ) ).

% bit_minus_1_iff
tff(fact_6876_bit__minus__2__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),N)
        <=> ( bit_se6407376104438227557le_bit(A,type2(A),N)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N) ) ) ) ).

% bit_minus_2_iff
tff(fact_6877_bit__2__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [N: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)
        <=> ( bit_se6407376104438227557le_bit(A,type2(A),one_one(nat))
            & ( N = one_one(nat) ) ) ) ) ).

% bit_2_iff
tff(fact_6878_bit__minus__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A3: A,N: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,uminus_uminus(A),A3)),N)
        <=> ( bit_se6407376104438227557le_bit(A,type2(A),N)
            & ~ aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),one_one(A))),N) ) ) ) ).

% bit_minus_iff
tff(fact_6879_antisym__bot,axiom,
    ! [A: $tType] : antisymp(A,bot_bot(fun(A,fun(A,$o)))) ).

% antisym_bot
tff(fact_6880_bit__minus__exp__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: nat,N: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,uminus_uminus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),M))),N)
        <=> ( bit_se6407376104438227557le_bit(A,type2(A),N)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),N) ) ) ) ).

% bit_minus_exp_iff
tff(fact_6881_bit__mask__sub__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M: nat,N: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),M)),one_one(A))),N)
        <=> ( bit_se6407376104438227557le_bit(A,type2(A),N)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),M) ) ) ) ).

% bit_mask_sub_iff
tff(fact_6882_bit__double__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,N: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A3)),N)
        <=> ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A3),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)))
            & ( N != zero_zero(nat) )
            & bit_se6407376104438227557le_bit(A,type2(A),N) ) ) ) ).

% bit_double_iff
tff(fact_6883_insort__insert__insort,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Xs: list(A)] :
          ( ~ aa(set(A),$o,member(A,X),aa(list(A),set(A),set2(A),Xs))
         => ( linord329482645794927042rt_key(A,A,aTP_Lamp_yq(A,A),X,Xs) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_yq(A,A)),X),Xs) ) ) ) ).

% insort_insert_insort
tff(fact_6884_in__range_Osimps,axiom,
    ! [H2: heap_ext(product_unit),As: set(nat)] :
      ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As))
    <=> ! [X4: nat] :
          ( aa(set(nat),$o,member(nat,X4),As)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X4),lim(product_unit,H2)) ) ) ).

% in_range.simps
tff(fact_6885_insort__insert__triv,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Xs: list(A)] :
          ( aa(set(A),$o,member(A,X),aa(list(A),set(A),set2(A),Xs))
         => ( linord329482645794927042rt_key(A,A,aTP_Lamp_yq(A,A),X,Xs) = Xs ) ) ) ).

% insort_insert_triv
tff(fact_6886_set__insort__insert,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Xs: list(A)] : aa(list(A),set(A),set2(A),linord329482645794927042rt_key(A,A,aTP_Lamp_yq(A,A),X,Xs)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),aa(list(A),set(A),set2(A),Xs)) ) ).

% set_insort_insert
tff(fact_6887_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_yq(A,A),X,Xs)) ) ) ).

% sorted_insort_insert
tff(fact_6888_in__range_Oelims_I3_J,axiom,
    ! [X: product_prod(heap_ext(product_unit),set(nat))] :
      ( ~ aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,X)
     => ~ ! [H: heap_ext(product_unit),As4: set(nat)] :
            ( ( X = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As4) )
           => ! [X2: nat] :
                ( aa(set(nat),$o,member(nat,X2),As4)
               => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X2),lim(product_unit,H)) ) ) ) ).

% in_range.elims(3)
tff(fact_6889_in__range_Oelims_I2_J,axiom,
    ! [X: product_prod(heap_ext(product_unit),set(nat))] :
      ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,X)
     => ~ ! [H: heap_ext(product_unit),As4: set(nat)] :
            ( ( X = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As4) )
           => ~ ! [X3: nat] :
                  ( aa(set(nat),$o,member(nat,X3),As4)
                 => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X3),lim(product_unit,H)) ) ) ) ).

% in_range.elims(2)
tff(fact_6890_in__range_Oelims_I1_J,axiom,
    ! [X: product_prod(heap_ext(product_unit),set(nat)),Y: $o] :
      ( ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,X)
      <=> (Y) )
     => ~ ! [H: heap_ext(product_unit),As4: set(nat)] :
            ( ( X = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As4) )
           => ( (Y)
            <=> ~ ! [X4: nat] :
                    ( aa(set(nat),$o,member(nat,X4),As4)
                   => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X4),lim(product_unit,H)) ) ) ) ) ).

% in_range.elims(1)
tff(fact_6891_bind__singleton__conv__image,axiom,
    ! [A: $tType,B: $tType,A4: set(B),F: fun(B,A)] : bind3(B,A,A4,aTP_Lamp_md(fun(B,A),fun(B,set(A)),F)) = aa(set(B),set(A),image2(B,A,F),A4) ).

% bind_singleton_conv_image
tff(fact_6892_surj__prod__encode,axiom,
    aa(set(product_prod(nat,nat)),set(nat),image2(product_prod(nat,nat),nat,nat_prod_encode),top_top(set(product_prod(nat,nat)))) = top_top(set(nat)) ).

% surj_prod_encode
tff(fact_6893_empty__bind,axiom,
    ! [B: $tType,A: $tType,F: fun(B,set(A))] : bind3(B,A,bot_bot(set(B)),F) = bot_bot(set(A)) ).

% empty_bind
tff(fact_6894_Set_Obind__bind,axiom,
    ! [C: $tType,A: $tType,B: $tType,A4: set(C),B3: fun(C,set(B)),C3: fun(B,set(A))] : bind3(B,A,bind3(C,B,A4,B3),C3) = bind3(C,A,A4,aa(fun(B,set(A)),fun(C,set(A)),aTP_Lamp_ast(fun(C,set(B)),fun(fun(B,set(A)),fun(C,set(A))),B3),C3)) ).

% Set.bind_bind
tff(fact_6895_nonempty__bind__const,axiom,
    ! [A: $tType,B: $tType,A4: set(A),B3: set(B)] :
      ( ( A4 != bot_bot(set(A)) )
     => ( bind3(A,B,A4,aTP_Lamp_qz(set(B),fun(A,set(B)),B3)) = B3 ) ) ).

% nonempty_bind_const
tff(fact_6896_bind__const,axiom,
    ! [B: $tType,A: $tType,A4: set(B),B3: set(A)] :
      bind3(B,A,A4,aTP_Lamp_ke(set(A),fun(B,set(A)),B3)) = $ite(A4 = bot_bot(set(B)),bot_bot(set(A)),B3) ).

% bind_const
tff(fact_6897_le__prod__encode__1,axiom,
    ! [A3: nat,B2: nat] : aa(nat,$o,aa(nat,fun(nat,$o),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_6898_le__prod__encode__2,axiom,
    ! [B2: nat,A3: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),B2),aa(product_prod(nat,nat),nat,nat_prod_encode,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),A3),B2))) ).

% le_prod_encode_2
tff(fact_6899_Set_Obind__def,axiom,
    ! [A: $tType,B: $tType,A4: set(B),F: fun(B,set(A))] : bind3(B,A,A4,F) = aa(fun(A,$o),set(A),collect(A),aa(fun(B,set(A)),fun(A,$o),aTP_Lamp_asu(set(B),fun(fun(B,set(A)),fun(A,$o)),A4),F)) ).

% Set.bind_def
tff(fact_6900_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_6901_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) ) )
       => ~ ! [X2: nat,Xs3: list(nat)] :
              ( ( X = aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),X2),Xs3) )
             => ( 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),X2),aa(list(nat),nat,nat_list_encode,Xs3)))) ) ) ) ) ).

% list_encode.elims
tff(fact_6902_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_asv(nat,fun(nat,nat))) ).

% prod_encode_def
tff(fact_6903_surj__list__encode,axiom,
    aa(set(list(nat)),set(nat),image2(list(nat),nat,nat_list_encode),top_top(set(list(nat)))) = top_top(set(nat)) ).

% surj_list_encode
tff(fact_6904_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_6905_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_6906_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)) ) )
         => ~ ! [X2: nat,Xs3: list(nat)] :
                ( ( X = aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),X2),Xs3) )
               => ( ( 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),X2),aa(list(nat),nat,nat_list_encode,Xs3)))) )
                 => ~ accp(list(nat),nat_list_encode_rel,aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),X2),Xs3)) ) ) ) ) ) ).

% list_encode.pelims
tff(fact_6907_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_yq(A,A)) ) ).

% sorted_list_of_set.folding_insort_key_axioms
tff(fact_6908_folding__insort__key_Osorted__key__list__of__set__remove,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),S: set(B),F: fun(B,A),X: B,A4: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S,F)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),X),A4)),S)
       => ( aa(set(B),$o,finite_finite2(B),A4)
         => ( sorted8670434370408473282of_set(A,B,Less_eq,F,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A4),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),X),bot_bot(set(B))))) = remove1(B,X,sorted8670434370408473282of_set(A,B,Less_eq,F,A4)) ) ) ) ) ).

% folding_insort_key.sorted_key_list_of_set_remove
tff(fact_6909_folding__insort__key_Osorted__key__list__of__set__eq__Nil__iff,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),S: set(B),F: fun(B,A),A4: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S,F)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A4),S)
       => ( aa(set(B),$o,finite_finite2(B),A4)
         => ( ( sorted8670434370408473282of_set(A,B,Less_eq,F,A4) = nil(B) )
          <=> ( A4 = bot_bot(set(B)) ) ) ) ) ) ).

% folding_insort_key.sorted_key_list_of_set_eq_Nil_iff
tff(fact_6910_folding__insort__key_Osorted__key__list__of__set__empty,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),S: set(B),F: fun(B,A)] :
      ( folding_insort_key(A,B,Less_eq,Less,S,F)
     => ( sorted8670434370408473282of_set(A,B,Less_eq,F,bot_bot(set(B))) = nil(B) ) ) ).

% folding_insort_key.sorted_key_list_of_set_empty
tff(fact_6911_folding__insort__key_Osorted__key__list__of__set__insert__remove,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),S: set(B),F: fun(B,A),X: B,A4: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S,F)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),X),A4)),S)
       => ( aa(set(B),$o,finite_finite2(B),A4)
         => ( sorted8670434370408473282of_set(A,B,Less_eq,F,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),X),A4)) = insort_key(A,B,Less_eq,F,X,sorted8670434370408473282of_set(A,B,Less_eq,F,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A4),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),X),bot_bot(set(B)))))) ) ) ) ) ).

% folding_insort_key.sorted_key_list_of_set_insert_remove
tff(fact_6912_String_Oless__literal__def,axiom,
    ord_less(literal) = aa(fun(list(char),fun(list(char),$o)),fun(literal,fun(literal,$o)),map_fun(literal,list(char),fun(list(char),$o),fun(literal,$o),explode,map_fun(literal,list(char),$o,$o,explode,id($o))),lexordp2(char,aTP_Lamp_asw(char,fun(char,$o)))) ).

% String.less_literal_def
tff(fact_6913_less__literal_Orep__eq,axiom,
    ! [X: literal,Xa: literal] :
      ( aa(literal,$o,aa(literal,fun(literal,$o),ord_less(literal),X),Xa)
    <=> aa(list(char),$o,aa(list(char),fun(list(char),$o),lexordp2(char,aTP_Lamp_asw(char,fun(char,$o))),aa(literal,list(char),explode,X)),aa(literal,list(char),explode,Xa)) ) ).

% less_literal.rep_eq
tff(fact_6914_String_Oless__eq__literal__def,axiom,
    ord_less_eq(literal) = aa(fun(list(char),fun(list(char),$o)),fun(literal,fun(literal,$o)),map_fun(literal,list(char),fun(list(char),$o),fun(literal,$o),explode,map_fun(literal,list(char),$o,$o,explode,id($o))),lexordp_eq(char,aTP_Lamp_asw(char,fun(char,$o)))) ).

% String.less_eq_literal_def
tff(fact_6915_less__literal_Otransfer,axiom,
    aa(fun(literal,fun(literal,$o)),$o,aa(fun(list(char),fun(list(char),$o)),fun(fun(literal,fun(literal,$o)),$o),bNF_rel_fun(list(char),literal,fun(list(char),$o),fun(literal,$o),pcr_literal,bNF_rel_fun(list(char),literal,$o,$o,pcr_literal,fequal($o))),lexordp2(char,aTP_Lamp_asw(char,fun(char,$o)))),ord_less(literal)) ).

% less_literal.transfer
tff(fact_6916_literal_Orep__transfer,axiom,
    aa(fun(literal,list(char)),$o,aa(fun(list(char),list(char)),fun(fun(literal,list(char)),$o),bNF_rel_fun(list(char),literal,list(char),list(char),pcr_literal,list_all2(char,char,fequal(char))),aTP_Lamp_asx(list(char),list(char))),explode) ).

% literal.rep_transfer
tff(fact_6917_less__eq__literal_Otransfer,axiom,
    aa(fun(literal,fun(literal,$o)),$o,aa(fun(list(char),fun(list(char),$o)),fun(fun(literal,fun(literal,$o)),$o),bNF_rel_fun(list(char),literal,fun(list(char),$o),fun(literal,$o),pcr_literal,bNF_rel_fun(list(char),literal,$o,$o,pcr_literal,fequal($o))),lexordp_eq(char,aTP_Lamp_asw(char,fun(char,$o)))),ord_less_eq(literal)) ).

% less_eq_literal.transfer
tff(fact_6918_less__eq__literal_Orep__eq,axiom,
    ! [X: literal,Xa: literal] :
      ( aa(literal,$o,aa(literal,fun(literal,$o),ord_less_eq(literal),X),Xa)
    <=> aa(list(char),$o,aa(list(char),fun(list(char),$o),lexordp_eq(char,aTP_Lamp_asw(char,fun(char,$o))),aa(literal,list(char),explode,X)),aa(literal,list(char),explode,Xa)) ) ).

% less_eq_literal.rep_eq
tff(fact_6919_String_Ocr__literal__def,axiom,
    ! [X3: list(char),Xa3: literal] :
      ( aa(literal,$o,aa(list(char),fun(literal,$o),cr_literal,X3),Xa3)
    <=> ( X3 = aa(literal,list(char),explode,Xa3) ) ) ).

% String.cr_literal_def
tff(fact_6920_MOST__eq_I2_J,axiom,
    ! [A: $tType,A3: A] :
      ( eventually(A,aa(A,fun(A,$o),fequal(A),A3),cofinite(A))
    <=> aa(set(A),$o,finite_finite2(A),top_top(set(A))) ) ).

% MOST_eq(2)
tff(fact_6921_cofinite__bot,axiom,
    ! [A: $tType] :
      ( ( cofinite(A) = bot_bot(filter(A)) )
    <=> aa(set(A),$o,finite_finite2(A),top_top(set(A))) ) ).

% cofinite_bot
tff(fact_6922_MOST__const,axiom,
    ! [A: $tType,P: $o] :
      ( eventually(A,aTP_Lamp_ah($o,fun(A,$o),(P)),cofinite(A))
    <=> ( (P)
        | aa(set(A),$o,finite_finite2(A),top_top(set(A))) ) ) ).

% MOST_const
tff(fact_6923_MOST__eq_I1_J,axiom,
    ! [A: $tType,A3: A] :
      ( eventually(A,aTP_Lamp_cf(A,fun(A,$o),A3),cofinite(A))
    <=> aa(set(A),$o,finite_finite2(A),top_top(set(A))) ) ).

% MOST_eq(1)
tff(fact_6924_MOST__conj__distrib,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] :
      ( eventually(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_am(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q),cofinite(A))
    <=> ( eventually(A,P,cofinite(A))
        & eventually(A,Q,cofinite(A)) ) ) ).

% MOST_conj_distrib
tff(fact_6925_MOST__imp__iff,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] :
      ( eventually(A,P,cofinite(A))
     => ( eventually(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ao(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q),cofinite(A))
      <=> eventually(A,Q,cofinite(A)) ) ) ).

% MOST_imp_iff
tff(fact_6926_MOST__rev__mp,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] :
      ( eventually(A,P,cofinite(A))
     => ( eventually(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ao(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q),cofinite(A))
       => eventually(A,Q,cofinite(A)) ) ) ).

% MOST_rev_mp
tff(fact_6927_MOST__conjI,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] :
      ( eventually(A,P,cofinite(A))
     => ( eventually(A,Q,cofinite(A))
       => eventually(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_am(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q),cofinite(A)) ) ) ).

% MOST_conjI
tff(fact_6928_MOST__mono,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] :
      ( eventually(A,P,cofinite(A))
     => ( ! [X2: A] :
            ( aa(A,$o,P,X2)
           => aa(A,$o,Q,X2) )
       => eventually(A,Q,cofinite(A)) ) ) ).

% MOST_mono
tff(fact_6929_ALL__MOST,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( ! [X_1: A] : aa(A,$o,P,X_1)
     => eventually(A,P,cofinite(A)) ) ).

% ALL_MOST
tff(fact_6930_MOST__I,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( ! [X2: A] : aa(A,$o,P,X2)
     => eventually(A,P,cofinite(A)) ) ).

% MOST_I
tff(fact_6931_MOST__eq__imp_I1_J,axiom,
    ! [A: $tType,A3: A,P: fun(A,$o)] : eventually(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_asy(A,fun(fun(A,$o),fun(A,$o)),A3),P),cofinite(A)) ).

% MOST_eq_imp(1)
tff(fact_6932_MOST__eq__imp_I2_J,axiom,
    ! [A: $tType,A3: A,P: fun(A,$o)] : eventually(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_asz(A,fun(fun(A,$o),fun(A,$o)),A3),P),cofinite(A)) ).

% MOST_eq_imp(2)
tff(fact_6933_MOST__neq_I1_J,axiom,
    ! [A: $tType,A3: A] : eventually(A,aa(A,fun(A,$o),aTP_Lamp_ze(A,fun(A,$o)),A3),cofinite(A)) ).

% MOST_neq(1)
tff(fact_6934_MOST__neq_I2_J,axiom,
    ! [A: $tType,A3: A] : eventually(A,aa(A,fun(A,$o),aTP_Lamp_za(A,fun(A,$o)),A3),cofinite(A)) ).

% MOST_neq(2)
tff(fact_6935_MOST__iff__finiteNeg,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( eventually(A,P,cofinite(A))
    <=> aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_aj(fun(A,$o),fun(A,$o),P))) ) ).

% MOST_iff_finiteNeg
tff(fact_6936_eventually__cofinite,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( eventually(A,P,cofinite(A))
    <=> aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_aj(fun(A,$o),fun(A,$o),P))) ) ).

% eventually_cofinite
tff(fact_6937_MOST__SucD,axiom,
    ! [P: fun(nat,$o)] :
      ( eventually(nat,aTP_Lamp_vf(fun(nat,$o),fun(nat,$o),P),cofinite(nat))
     => eventually(nat,P,cofinite(nat)) ) ).

% MOST_SucD
tff(fact_6938_MOST__SucI,axiom,
    ! [P: fun(nat,$o)] :
      ( eventually(nat,P,cofinite(nat))
     => eventually(nat,aTP_Lamp_vf(fun(nat,$o),fun(nat,$o),P),cofinite(nat)) ) ).

% MOST_SucI
tff(fact_6939_MOST__Suc__iff,axiom,
    ! [P: fun(nat,$o)] :
      ( eventually(nat,aTP_Lamp_vf(fun(nat,$o),fun(nat,$o),P),cofinite(nat))
    <=> eventually(nat,P,cofinite(nat)) ) ).

% MOST_Suc_iff
tff(fact_6940_MOST__nat,axiom,
    ! [P: fun(nat,$o)] :
      ( eventually(nat,P,cofinite(nat))
    <=> ? [M2: nat] :
        ! [N2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),N2)
         => aa(nat,$o,P,N2) ) ) ).

% MOST_nat
tff(fact_6941_MOST__ge__nat,axiom,
    ! [M: nat] : eventually(nat,aa(nat,fun(nat,$o),ord_less_eq(nat),M),cofinite(nat)) ).

% MOST_ge_nat
tff(fact_6942_MOST__nat__le,axiom,
    ! [P: fun(nat,$o)] :
      ( eventually(nat,P,cofinite(nat))
    <=> ? [M2: nat] :
        ! [N2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),N2)
         => aa(nat,$o,P,N2) ) ) ).

% MOST_nat_le
tff(fact_6943_cofinite__def,axiom,
    ! [A: $tType] : cofinite(A) = abs_filter(A,aTP_Lamp_ata(fun(A,$o),$o)) ).

% cofinite_def
tff(fact_6944_MOST__finite__Ball__distrib,axiom,
    ! [A: $tType,B: $tType,A4: set(A),P: fun(A,fun(B,$o))] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( eventually(B,aa(fun(A,fun(B,$o)),fun(B,$o),aTP_Lamp_atb(set(A),fun(fun(A,fun(B,$o)),fun(B,$o)),A4),P),cofinite(B))
      <=> ! [X4: A] :
            ( aa(set(A),$o,member(A,X4),A4)
           => eventually(B,aa(A,fun(B,$o),P,X4),cofinite(B)) ) ) ) ).

% MOST_finite_Ball_distrib
tff(fact_6945_MOST__inj,axiom,
    ! [A: $tType,B: $tType,P: fun(A,$o),F: fun(B,A)] :
      ( eventually(A,P,cofinite(A))
     => ( inj_on(B,A,F,top_top(set(B)))
       => eventually(B,aa(fun(B,A),fun(B,$o),aTP_Lamp_akb(fun(A,$o),fun(fun(B,A),fun(B,$o)),P),F),cofinite(B)) ) ) ).

% MOST_inj
tff(fact_6946_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_6947_less__literal_Orsp,axiom,
    aa(fun(list(char),fun(list(char),$o)),$o,aa(fun(list(char),fun(list(char),$o)),fun(fun(list(char),fun(list(char),$o)),$o),bNF_rel_fun(list(char),list(char),fun(list(char),$o),fun(list(char),$o),bNF_eq_onp(list(char),aTP_Lamp_atc(list(char),$o)),bNF_rel_fun(list(char),list(char),$o,$o,bNF_eq_onp(list(char),aTP_Lamp_atc(list(char),$o)),fequal($o))),lexordp2(char,aTP_Lamp_asw(char,fun(char,$o)))),lexordp2(char,aTP_Lamp_asw(char,fun(char,$o)))) ).

% less_literal.rsp
tff(fact_6948_plus__literal_Orsp,axiom,
    aa(fun(list(char),fun(list(char),list(char))),$o,aa(fun(list(char),fun(list(char),list(char))),fun(fun(list(char),fun(list(char),list(char))),$o),bNF_rel_fun(list(char),list(char),fun(list(char),list(char)),fun(list(char),list(char)),bNF_eq_onp(list(char),aTP_Lamp_atc(list(char),$o)),bNF_rel_fun(list(char),list(char),list(char),list(char),bNF_eq_onp(list(char),aTP_Lamp_atc(list(char),$o)),bNF_eq_onp(list(char),aTP_Lamp_atc(list(char),$o)))),append(char)),append(char)) ).

% plus_literal.rsp
tff(fact_6949_asciis__of__literal_Orsp,axiom,
    aa(fun(list(char),list(code_integer)),$o,aa(fun(list(char),list(code_integer)),fun(fun(list(char),list(code_integer)),$o),bNF_rel_fun(list(char),list(char),list(code_integer),list(code_integer),bNF_eq_onp(list(char),aTP_Lamp_atc(list(char),$o)),fequal(list(code_integer))),map(char,code_integer,comm_s6883823935334413003f_char(code_integer))),map(char,code_integer,comm_s6883823935334413003f_char(code_integer))) ).

% asciis_of_literal.rsp
tff(fact_6950_size__literal_Orsp,axiom,
    aa(fun(list(char),nat),$o,aa(fun(list(char),nat),fun(fun(list(char),nat),$o),bNF_rel_fun(list(char),list(char),nat,nat,bNF_eq_onp(list(char),aTP_Lamp_atc(list(char),$o)),fequal(nat)),size_size(list(char))),size_size(list(char))) ).

% size_literal.rsp
tff(fact_6951_Literal_Orsp,axiom,
    aa(fun($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun(list(char),list(char))))))))),$o,aa(fun($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun(list(char),list(char))))))))),fun(fun($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun(list(char),list(char))))))))),$o),bNF_rel_fun($o,$o,fun($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun(list(char),list(char)))))))),fun($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun(list(char),list(char)))))))),fequal($o),bNF_rel_fun($o,$o,fun($o,fun($o,fun($o,fun($o,fun($o,fun(list(char),list(char))))))),fun($o,fun($o,fun($o,fun($o,fun($o,fun(list(char),list(char))))))),fequal($o),bNF_rel_fun($o,$o,fun($o,fun($o,fun($o,fun($o,fun(list(char),list(char)))))),fun($o,fun($o,fun($o,fun($o,fun(list(char),list(char)))))),fequal($o),bNF_rel_fun($o,$o,fun($o,fun($o,fun($o,fun(list(char),list(char))))),fun($o,fun($o,fun($o,fun(list(char),list(char))))),fequal($o),bNF_rel_fun($o,$o,fun($o,fun($o,fun(list(char),list(char)))),fun($o,fun($o,fun(list(char),list(char)))),fequal($o),bNF_rel_fun($o,$o,fun($o,fun(list(char),list(char))),fun($o,fun(list(char),list(char))),fequal($o),bNF_rel_fun($o,$o,fun(list(char),list(char)),fun(list(char),list(char)),fequal($o),bNF_rel_fun(list(char),list(char),list(char),list(char),bNF_eq_onp(list(char),aTP_Lamp_atc(list(char),$o)),bNF_eq_onp(list(char),aTP_Lamp_atc(list(char),$o)))))))))),aTP_Lamp_atd($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun(list(char),list(char)))))))))),aTP_Lamp_atd($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun(list(char),list(char)))))))))) ).

% Literal.rsp
tff(fact_6952_literal_Oexplode,axiom,
    ! [X: literal] : aa(set(list(char)),$o,member(list(char),aa(literal,list(char),explode,X)),aa(fun(list(char),$o),set(list(char)),collect(list(char)),aTP_Lamp_atc(list(char),$o))) ).

% literal.explode
tff(fact_6953_literal_Oexplode__cases,axiom,
    ! [Y: list(char)] :
      ( aa(set(list(char)),$o,member(list(char),Y),aa(fun(list(char),$o),set(list(char)),collect(list(char)),aTP_Lamp_atc(list(char),$o)))
     => ~ ! [X2: literal] : Y != aa(literal,list(char),explode,X2) ) ).

% literal.explode_cases
tff(fact_6954_literal_Oexplode__induct,axiom,
    ! [Y: list(char),P: fun(list(char),$o)] :
      ( aa(set(list(char)),$o,member(list(char),Y),aa(fun(list(char),$o),set(list(char)),collect(list(char)),aTP_Lamp_atc(list(char),$o)))
     => ( ! [X2: literal] : aa(list(char),$o,P,aa(literal,list(char),explode,X2))
       => aa(list(char),$o,P,Y) ) ) ).

% literal.explode_induct
tff(fact_6955_group_Oleft__cancel,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A),A3: A,B2: A,C2: A] :
      ( group(A,F,Z2,Inverse)
     => ( ( aa(A,A,aa(A,fun(A,A),F,A3),B2) = aa(A,A,aa(A,fun(A,A),F,A3),C2) )
      <=> ( B2 = C2 ) ) ) ).

% group.left_cancel
tff(fact_6956_group_Oleft__inverse,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A),A3: A] :
      ( group(A,F,Z2,Inverse)
     => ( aa(A,A,aa(A,fun(A,A),F,aa(A,A,Inverse,A3)),A3) = Z2 ) ) ).

% group.left_inverse
tff(fact_6957_group_Oright__cancel,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A),B2: A,A3: A,C2: A] :
      ( group(A,F,Z2,Inverse)
     => ( ( aa(A,A,aa(A,fun(A,A),F,B2),A3) = aa(A,A,aa(A,fun(A,A),F,C2),A3) )
      <=> ( B2 = C2 ) ) ) ).

% group.right_cancel
tff(fact_6958_group_Oright__inverse,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A),A3: A] :
      ( group(A,F,Z2,Inverse)
     => ( aa(A,A,aa(A,fun(A,A),F,A3),aa(A,A,Inverse,A3)) = Z2 ) ) ).

% group.right_inverse
tff(fact_6959_group_Oinverse__unique,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A),A3: A,B2: A] :
      ( group(A,F,Z2,Inverse)
     => ( ( aa(A,A,aa(A,fun(A,A),F,A3),B2) = Z2 )
       => ( aa(A,A,Inverse,A3) = B2 ) ) ) ).

% group.inverse_unique
tff(fact_6960_group_Oinverse__inverse,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A),A3: A] :
      ( group(A,F,Z2,Inverse)
     => ( aa(A,A,Inverse,aa(A,A,Inverse,A3)) = A3 ) ) ).

% group.inverse_inverse
tff(fact_6961_group_Oinverse__neutral,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A)] :
      ( group(A,F,Z2,Inverse)
     => ( aa(A,A,Inverse,Z2) = Z2 ) ) ).

% group.inverse_neutral
tff(fact_6962_group_Ogroup__left__neutral,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A),A3: A] :
      ( group(A,F,Z2,Inverse)
     => ( aa(A,A,aa(A,fun(A,A),F,Z2),A3) = A3 ) ) ).

% group.group_left_neutral
tff(fact_6963_group_Oinverse__distrib__swap,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A),A3: A,B2: A] :
      ( group(A,F,Z2,Inverse)
     => ( aa(A,A,Inverse,aa(A,A,aa(A,fun(A,A),F,A3),B2)) = aa(A,A,aa(A,fun(A,A),F,aa(A,A,Inverse,B2)),aa(A,A,Inverse,A3)) ) ) ).

% group.inverse_distrib_swap
tff(fact_6964_zero__literal_Orsp,axiom,
    aa(list(char),$o,aa(list(char),fun(list(char),$o),bNF_eq_onp(list(char),aTP_Lamp_atc(list(char),$o)),nil(char)),nil(char)) ).

% zero_literal.rsp
tff(fact_6965_literal_Odomain__eq,axiom,
    ! [X3: list(char)] :
      ( aa(list(char),$o,aa(fun(list(char),fun(literal,$o)),fun(list(char),$o),domainp(list(char),literal),pcr_literal),X3)
    <=> ! [Xa2: char] :
          ( aa(set(char),$o,member(char,Xa2),aa(list(char),set(char),set2(char),X3))
         => ~ digit7(Xa2) ) ) ).

% literal.domain_eq
tff(fact_6966_literal_Odomain,axiom,
    ! [X3: list(char)] :
      ( aa(list(char),$o,aa(fun(list(char),fun(literal,$o)),fun(list(char),$o),domainp(list(char),literal),pcr_literal),X3)
    <=> ? [Y3: list(char)] :
          ( aa(list(char),$o,aa(list(char),fun(list(char),$o),list_all2(char,char,fequal(char)),X3),Y3)
          & ! [Xa2: char] :
              ( aa(set(char),$o,member(char,Xa2),aa(list(char),set(char),set2(char),Y3))
             => ~ digit7(Xa2) ) ) ) ).

% literal.domain
tff(fact_6967_less__eq__literal_Orsp,axiom,
    aa(fun(list(char),fun(list(char),$o)),$o,aa(fun(list(char),fun(list(char),$o)),fun(fun(list(char),fun(list(char),$o)),$o),bNF_rel_fun(list(char),list(char),fun(list(char),$o),fun(list(char),$o),bNF_eq_onp(list(char),aTP_Lamp_atc(list(char),$o)),bNF_rel_fun(list(char),list(char),$o,$o,bNF_eq_onp(list(char),aTP_Lamp_atc(list(char),$o)),fequal($o))),lexordp_eq(char,aTP_Lamp_asw(char,fun(char,$o)))),lexordp_eq(char,aTP_Lamp_asw(char,fun(char,$o)))) ).

% less_eq_literal.rsp
tff(fact_6968_less__eq__literal_Oabs__eq,axiom,
    ! [Xa: list(char),X: list(char)] :
      ( aa(list(char),$o,aa(list(char),fun(list(char),$o),bNF_eq_onp(list(char),aTP_Lamp_atc(list(char),$o)),Xa),Xa)
     => ( aa(list(char),$o,aa(list(char),fun(list(char),$o),bNF_eq_onp(list(char),aTP_Lamp_atc(list(char),$o)),X),X)
       => ( aa(literal,$o,aa(literal,fun(literal,$o),ord_less_eq(literal),aa(list(char),literal,abs_literal,Xa)),aa(list(char),literal,abs_literal,X))
        <=> aa(list(char),$o,aa(list(char),fun(list(char),$o),lexordp_eq(char,aTP_Lamp_asw(char,fun(char,$o))),Xa),X) ) ) ) ).

% less_eq_literal.abs_eq
tff(fact_6969_less__literal_Oabs__eq,axiom,
    ! [Xa: list(char),X: list(char)] :
      ( aa(list(char),$o,aa(list(char),fun(list(char),$o),bNF_eq_onp(list(char),aTP_Lamp_atc(list(char),$o)),Xa),Xa)
     => ( aa(list(char),$o,aa(list(char),fun(list(char),$o),bNF_eq_onp(list(char),aTP_Lamp_atc(list(char),$o)),X),X)
       => ( aa(literal,$o,aa(literal,fun(literal,$o),ord_less(literal),aa(list(char),literal,abs_literal,Xa)),aa(list(char),literal,abs_literal,X))
        <=> aa(list(char),$o,aa(list(char),fun(list(char),$o),lexordp2(char,aTP_Lamp_asw(char,fun(char,$o))),Xa),X) ) ) ) ).

% less_literal.abs_eq
tff(fact_6970_literal_OAbs__literal__cases,axiom,
    ! [X: literal] :
      ~ ! [Y2: list(char)] :
          ( ( X = aa(list(char),literal,abs_literal,Y2) )
         => ~ aa(set(list(char)),$o,member(list(char),Y2),aa(fun(list(char),$o),set(list(char)),collect(list(char)),aTP_Lamp_atc(list(char),$o))) ) ).

% literal.Abs_literal_cases
tff(fact_6971_literal_OAbs__literal__induct,axiom,
    ! [P: fun(literal,$o),X: literal] :
      ( ! [Y2: list(char)] :
          ( aa(set(list(char)),$o,member(list(char),Y2),aa(fun(list(char),$o),set(list(char)),collect(list(char)),aTP_Lamp_atc(list(char),$o)))
         => aa(literal,$o,P,aa(list(char),literal,abs_literal,Y2)) )
     => aa(literal,$o,P,X) ) ).

% literal.Abs_literal_induct
tff(fact_6972_literal_OAbs__literal__inject,axiom,
    ! [X: list(char),Y: list(char)] :
      ( aa(set(list(char)),$o,member(list(char),X),aa(fun(list(char),$o),set(list(char)),collect(list(char)),aTP_Lamp_atc(list(char),$o)))
     => ( aa(set(list(char)),$o,member(list(char),Y),aa(fun(list(char),$o),set(list(char)),collect(list(char)),aTP_Lamp_atc(list(char),$o)))
       => ( ( aa(list(char),literal,abs_literal,X) = aa(list(char),literal,abs_literal,Y) )
        <=> ( X = Y ) ) ) ) ).

% literal.Abs_literal_inject
tff(fact_6973_literal_Otype__definition__literal,axiom,
    type_definition(literal,list(char),explode,abs_literal,aa(fun(list(char),$o),set(list(char)),collect(list(char)),aTP_Lamp_atc(list(char),$o))) ).

% literal.type_definition_literal
tff(fact_6974_literal_OAbs__literal__inverse,axiom,
    ! [Y: list(char)] :
      ( aa(set(list(char)),$o,member(list(char),Y),aa(fun(list(char),$o),set(list(char)),collect(list(char)),aTP_Lamp_atc(list(char),$o)))
     => ( aa(literal,list(char),explode,aa(list(char),literal,abs_literal,Y)) = Y ) ) ).

% literal.Abs_literal_inverse
tff(fact_6975_plus__literal_Oabs__eq,axiom,
    ! [Xa: list(char),X: list(char)] :
      ( aa(list(char),$o,aa(list(char),fun(list(char),$o),bNF_eq_onp(list(char),aTP_Lamp_atc(list(char),$o)),Xa),Xa)
     => ( aa(list(char),$o,aa(list(char),fun(list(char),$o),bNF_eq_onp(list(char),aTP_Lamp_atc(list(char),$o)),X),X)
       => ( aa(literal,literal,aa(literal,fun(literal,literal),plus_plus(literal),aa(list(char),literal,abs_literal,Xa)),aa(list(char),literal,abs_literal,X)) = aa(list(char),literal,abs_literal,aa(list(char),list(char),aa(list(char),fun(list(char),list(char)),append(char),Xa),X)) ) ) ) ).

% plus_literal.abs_eq
tff(fact_6976_String_OQuotient__literal,axiom,
    quotient(list(char),literal,bNF_eq_onp(list(char),aTP_Lamp_atc(list(char),$o)),abs_literal,explode,cr_literal) ).

% String.Quotient_literal
tff(fact_6977_asciis__of__literal_Oabs__eq,axiom,
    ! [X: list(char)] :
      ( aa(list(char),$o,aa(list(char),fun(list(char),$o),bNF_eq_onp(list(char),aTP_Lamp_atc(list(char),$o)),X),X)
     => ( asciis_of_literal(aa(list(char),literal,abs_literal,X)) = aa(list(char),list(code_integer),map(char,code_integer,comm_s6883823935334413003f_char(code_integer)),X) ) ) ).

% asciis_of_literal.abs_eq
tff(fact_6978_size__literal_Oabs__eq,axiom,
    ! [X: list(char)] :
      ( aa(list(char),$o,aa(list(char),fun(list(char),$o),bNF_eq_onp(list(char),aTP_Lamp_atc(list(char),$o)),X),X)
     => ( aa(literal,nat,size_size(literal),aa(list(char),literal,abs_literal,X)) = aa(list(char),nat,size_size(list(char)),X) ) ) ).

% size_literal.abs_eq
tff(fact_6979_literal_Odomain__par__left__total,axiom,
    ! [P4: fun(list(char),$o)] :
      ( left_total(list(char),list(char),list_all2(char,char,fequal(char)))
     => ( aa(fun(list(char),$o),$o,aa(fun(list(char),$o),fun(fun(list(char),$o),$o),bNF_rel_fun(list(char),list(char),$o,$o,list_all2(char,char,fequal(char)),fequal($o)),P4),aTP_Lamp_atc(list(char),$o))
       => ( aa(fun(list(char),fun(literal,$o)),fun(list(char),$o),domainp(list(char),literal),pcr_literal) = P4 ) ) ) ).

% literal.domain_par_left_total
tff(fact_6980_literal_Odomain__par,axiom,
    ! [DR: fun(char,$o),P24: fun(list(char),$o)] :
      ( ( aa(fun(char,fun(char,$o)),fun(char,$o),domainp(char,char),fequal(char)) = DR )
     => ( aa(fun(list(char),$o),$o,aa(fun(list(char),$o),fun(fun(list(char),$o),$o),bNF_rel_fun(list(char),list(char),$o,$o,list_all2(char,char,fequal(char)),fequal($o)),P24),aTP_Lamp_atc(list(char),$o))
       => ( aa(fun(list(char),fun(literal,$o)),fun(list(char),$o),domainp(list(char),literal),pcr_literal) = aa(fun(list(char),$o),fun(list(char),$o),aa(fun(list(char),$o),fun(fun(list(char),$o),fun(list(char),$o)),inf_inf(fun(list(char),$o)),list_all(char,DR)),P24) ) ) ) ).

% literal.domain_par
tff(fact_6981_list_Opred__True,axiom,
    ! [A: $tType,X3: list(A)] : aa(list(A),$o,list_all(A,aTP_Lamp_ar(A,$o)),X3) ).

% list.pred_True
tff(fact_6982_list_Omap__cong__pred,axiom,
    ! [B: $tType,A: $tType,X: list(A),Ya: list(A),F: fun(A,B),G: fun(A,B)] :
      ( ( X = Ya )
     => ( aa(list(A),$o,list_all(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_ck(fun(A,B),fun(fun(A,B),fun(A,$o)),F),G)),Ya)
       => ( aa(list(A),list(B),map(A,B,F),X) = aa(list(A),list(B),map(A,B,G),Ya) ) ) ) ).

% list.map_cong_pred
tff(fact_6983_implode_Orsp,axiom,
    aa(fun(list(char),list(char)),$o,aa(fun(list(char),list(char)),fun(fun(list(char),list(char)),$o),bNF_rel_fun(list(char),list(char),list(char),list(char),fequal(list(char)),bNF_eq_onp(list(char),aTP_Lamp_atc(list(char),$o))),map(char,char,ascii_of)),map(char,char,ascii_of)) ).

% implode.rsp
tff(fact_6984_list__ex1__simps_I2_J,axiom,
    ! [A: $tType,P: fun(A,$o),X: A,Xs: list(A)] :
      ( list_ex1(A,P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs))
    <=> $ite(aa(A,$o,P,X),aa(list(A),$o,list_all(A,aa(A,fun(A,$o),aTP_Lamp_ate(fun(A,$o),fun(A,fun(A,$o)),P),X)),Xs),list_ex1(A,P,Xs)) ) ).

% list_ex1_simps(2)
tff(fact_6985_literal__of__asciis_Orsp,axiom,
    aa(fun(list(code_integer),list(char)),$o,aa(fun(list(code_integer),list(char)),fun(fun(list(code_integer),list(char)),$o),bNF_rel_fun(list(code_integer),list(code_integer),list(char),list(char),fequal(list(code_integer)),bNF_eq_onp(list(char),aTP_Lamp_atc(list(char),$o))),map(code_integer,char,aa(fun(code_integer,char),fun(code_integer,char),comp(char,char,code_integer,ascii_of),unique5772411509450598832har_of(code_integer)))),map(code_integer,char,aa(fun(code_integer,char),fun(code_integer,char),comp(char,char,code_integer,ascii_of),unique5772411509450598832har_of(code_integer)))) ).

% literal_of_asciis.rsp
tff(fact_6986_Literal_Oabs__eq,axiom,
    ! [X: list(char),Xg: $o,Xf: $o,Xe: $o,Xd: $o,Xc: $o,Xb: $o,Xa: $o] :
      ( aa(list(char),$o,aa(list(char),fun(list(char),$o),bNF_eq_onp(list(char),aTP_Lamp_atc(list(char),$o)),X),X)
     => ( aa(literal,literal,aa($o,fun(literal,literal),aa($o,fun($o,fun(literal,literal)),aa($o,fun($o,fun($o,fun(literal,literal))),aa($o,fun($o,fun($o,fun($o,fun(literal,literal)))),aa($o,fun($o,fun($o,fun($o,fun($o,fun(literal,literal))))),aa($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun(literal,literal)))))),aa($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun(literal,literal))))))),literal2,(Xg)),(Xf)),(Xe)),(Xd)),(Xc)),(Xb)),(Xa)),aa(list(char),literal,abs_literal,X)) = aa(list(char),literal,abs_literal,aa(list(char),list(char),aa(char,fun(list(char),list(char)),cons(char),aa($o,char,char2((Xg),(Xf),(Xe),(Xd),(Xc),(Xb),(Xa)),$false)),X)) ) ) ).

% Literal.abs_eq
tff(fact_6987_String_OLiteral__def,axiom,
    literal2 = aa(fun($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun(list(char),list(char))))))))),fun($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun(literal,literal)))))))),map_fun($o,$o,fun($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun(list(char),list(char)))))))),fun($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun(literal,literal))))))),id($o),map_fun($o,$o,fun($o,fun($o,fun($o,fun($o,fun($o,fun(list(char),list(char))))))),fun($o,fun($o,fun($o,fun($o,fun($o,fun(literal,literal)))))),id($o),map_fun($o,$o,fun($o,fun($o,fun($o,fun($o,fun(list(char),list(char)))))),fun($o,fun($o,fun($o,fun($o,fun(literal,literal))))),id($o),map_fun($o,$o,fun($o,fun($o,fun($o,fun(list(char),list(char))))),fun($o,fun($o,fun($o,fun(literal,literal)))),id($o),map_fun($o,$o,fun($o,fun($o,fun(list(char),list(char)))),fun($o,fun($o,fun(literal,literal))),id($o),map_fun($o,$o,fun($o,fun(list(char),list(char))),fun($o,fun(literal,literal)),id($o),map_fun($o,$o,fun(list(char),list(char)),fun(literal,literal),id($o),map_fun(literal,list(char),list(char),literal,explode,abs_literal)))))))),aTP_Lamp_atd($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun(list(char),list(char)))))))))) ).

% String.Literal_def
tff(fact_6988_Literal_Otransfer,axiom,
    aa(fun($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun(literal,literal)))))))),$o,aa(fun($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun(list(char),list(char))))))))),fun(fun($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun(literal,literal)))))))),$o),bNF_rel_fun($o,$o,fun($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun(list(char),list(char)))))))),fun($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun(literal,literal))))))),fequal($o),bNF_rel_fun($o,$o,fun($o,fun($o,fun($o,fun($o,fun($o,fun(list(char),list(char))))))),fun($o,fun($o,fun($o,fun($o,fun($o,fun(literal,literal)))))),fequal($o),bNF_rel_fun($o,$o,fun($o,fun($o,fun($o,fun($o,fun(list(char),list(char)))))),fun($o,fun($o,fun($o,fun($o,fun(literal,literal))))),fequal($o),bNF_rel_fun($o,$o,fun($o,fun($o,fun($o,fun(list(char),list(char))))),fun($o,fun($o,fun($o,fun(literal,literal)))),fequal($o),bNF_rel_fun($o,$o,fun($o,fun($o,fun(list(char),list(char)))),fun($o,fun($o,fun(literal,literal))),fequal($o),bNF_rel_fun($o,$o,fun($o,fun(list(char),list(char))),fun($o,fun(literal,literal)),fequal($o),bNF_rel_fun($o,$o,fun(list(char),list(char)),fun(literal,literal),fequal($o),bNF_rel_fun(list(char),literal,list(char),literal,pcr_literal,pcr_literal)))))))),aTP_Lamp_atd($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun(list(char),list(char)))))))))),literal2) ).

% Literal.transfer
tff(fact_6989_listsp__inf__eq,axiom,
    ! [A: $tType,A4: fun(A,$o),B3: fun(A,$o)] : listsp(A,aa(fun(A,$o),fun(A,$o),aa(fun(A,$o),fun(fun(A,$o),fun(A,$o)),inf_inf(fun(A,$o)),A4),B3)) = aa(fun(list(A),$o),fun(list(A),$o),aa(fun(list(A),$o),fun(fun(list(A),$o),fun(list(A),$o)),inf_inf(fun(list(A),$o)),listsp(A,A4)),listsp(A,B3)) ).

% listsp_inf_eq
tff(fact_6990_inv__o__cancel,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B)] :
      ( inj_on(A,B,F,top_top(set(A)))
     => ( aa(fun(A,B),fun(A,A),comp(B,A,A,hilbert_inv_into(A,B,top_top(set(A)),F)),F) = id(A) ) ) ).

% inv_o_cancel
tff(fact_6991_listsp__conj__eq,axiom,
    ! [A: $tType,A4: fun(A,$o),B3: fun(A,$o),X3: list(A)] :
      ( aa(list(A),$o,listsp(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_am(fun(A,$o),fun(fun(A,$o),fun(A,$o)),A4),B3)),X3)
    <=> ( aa(list(A),$o,listsp(A,A4),X3)
        & aa(list(A),$o,listsp(A,B3),X3) ) ) ).

% listsp_conj_eq
tff(fact_6992_inv__identity,axiom,
    ! [A: $tType,X3: A] : aa(A,A,hilbert_inv_into(A,A,top_top(set(A)),aTP_Lamp_au(A,A)),X3) = X3 ).

% inv_identity
tff(fact_6993_inv__id,axiom,
    ! [A: $tType] : hilbert_inv_into(A,A,top_top(set(A)),id(A)) = id(A) ).

% inv_id
tff(fact_6994_o__inv__o__cancel,axiom,
    ! [B: $tType,C: $tType,A: $tType,F: fun(A,B),G: fun(A,C)] :
      ( inj_on(A,B,F,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,G),hilbert_inv_into(A,B,top_top(set(A)),F))),F) = G ) ) ).

% o_inv_o_cancel
tff(fact_6995_inj__map__inv__f,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),L: list(A)] :
      ( inj_on(A,B,F,top_top(set(A)))
     => ( aa(list(B),list(A),map(B,A,hilbert_inv_into(A,B,top_top(set(A)),F)),aa(list(A),list(B),map(A,B,F),L)) = L ) ) ).

% inj_map_inv_f
tff(fact_6996_listsp__infI,axiom,
    ! [A: $tType,A4: fun(A,$o),L: list(A),B3: fun(A,$o)] :
      ( aa(list(A),$o,listsp(A,A4),L)
     => ( aa(list(A),$o,listsp(A,B3),L)
       => aa(list(A),$o,listsp(A,aa(fun(A,$o),fun(A,$o),aa(fun(A,$o),fun(fun(A,$o),fun(A,$o)),inf_inf(fun(A,$o)),A4),B3)),L) ) ) ).

% listsp_infI
tff(fact_6997_bij__imp__bij__inv,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B)] :
      ( bij_betw(A,B,F,top_top(set(A)),top_top(set(B)))
     => bij_betw(B,A,hilbert_inv_into(A,B,top_top(set(A)),F),top_top(set(B)),top_top(set(A))) ) ).

% bij_imp_bij_inv
tff(fact_6998_bij__inv__eq__iff,axiom,
    ! [A: $tType,B: $tType,P3: fun(A,B),X: A,Y: B] :
      ( bij_betw(A,B,P3,top_top(set(A)),top_top(set(B)))
     => ( ( X = aa(B,A,hilbert_inv_into(A,B,top_top(set(A)),P3),Y) )
      <=> ( aa(A,B,P3,X) = Y ) ) ) ).

% bij_inv_eq_iff
tff(fact_6999_inv__inv__eq,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B)] :
      ( bij_betw(A,B,F,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)),F)) = F ) ) ).

% inv_inv_eq
tff(fact_7000_surj__f__inv__f,axiom,
    ! [B: $tType,A: $tType,F: fun(B,A),Y: A] :
      ( ( aa(set(B),set(A),image2(B,A,F),top_top(set(B))) = top_top(set(A)) )
     => ( aa(B,A,F,aa(A,B,hilbert_inv_into(B,A,top_top(set(B)),F),Y)) = Y ) ) ).

% surj_f_inv_f
tff(fact_7001_surj__iff__all,axiom,
    ! [B: $tType,A: $tType,F: fun(B,A)] :
      ( ( aa(set(B),set(A),image2(B,A,F),top_top(set(B))) = top_top(set(A)) )
    <=> ! [X4: A] : aa(B,A,F,aa(A,B,hilbert_inv_into(B,A,top_top(set(B)),F),X4)) = X4 ) ).

% surj_iff_all
tff(fact_7002_image__f__inv__f,axiom,
    ! [B: $tType,A: $tType,F: fun(B,A),A4: set(A)] :
      ( ( aa(set(B),set(A),image2(B,A,F),top_top(set(B))) = top_top(set(A)) )
     => ( aa(set(B),set(A),image2(B,A,F),aa(set(A),set(B),image2(A,B,hilbert_inv_into(B,A,top_top(set(B)),F)),A4)) = A4 ) ) ).

% image_f_inv_f
tff(fact_7003_surj__imp__inv__eq,axiom,
    ! [B: $tType,A: $tType,F: fun(B,A),G: fun(A,B)] :
      ( ( aa(set(B),set(A),image2(B,A,F),top_top(set(B))) = top_top(set(A)) )
     => ( ! [X2: B] : aa(A,B,G,aa(B,A,F,X2)) = X2
       => ( hilbert_inv_into(B,A,top_top(set(B)),F) = G ) ) ) ).

% surj_imp_inv_eq
tff(fact_7004_inv__def,axiom,
    ! [B: $tType,A: $tType,F: fun(B,A),X3: A] : aa(A,B,hilbert_inv_into(B,A,top_top(set(B)),F),X3) = fChoice(B,aa(A,fun(B,$o),aTP_Lamp_atf(fun(B,A),fun(A,fun(B,$o)),F),X3)) ).

% inv_def
tff(fact_7005_inv__equality,axiom,
    ! [A: $tType,B: $tType,G: fun(B,A),F: fun(A,B)] :
      ( ! [X2: A] : aa(B,A,G,aa(A,B,F,X2)) = X2
     => ( ! [Y2: B] : aa(A,B,F,aa(B,A,G,Y2)) = Y2
       => ( hilbert_inv_into(A,B,top_top(set(A)),F) = G ) ) ) ).

% inv_equality
tff(fact_7006_inv__f__f,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),X: A] :
      ( inj_on(A,B,F,top_top(set(A)))
     => ( aa(B,A,hilbert_inv_into(A,B,top_top(set(A)),F),aa(A,B,F,X)) = X ) ) ).

% inv_f_f
tff(fact_7007_inv__f__eq,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),X: A,Y: B] :
      ( inj_on(A,B,F,top_top(set(A)))
     => ( ( aa(A,B,F,X) = Y )
       => ( aa(B,A,hilbert_inv_into(A,B,top_top(set(A)),F),Y) = X ) ) ) ).

% inv_f_eq
tff(fact_7008_inj__imp__inv__eq,axiom,
    ! [A: $tType,B: $tType,F: fun(A,B),G: fun(B,A)] :
      ( inj_on(A,B,F,top_top(set(A)))
     => ( ! [X2: B] : aa(A,B,F,aa(B,A,G,X2)) = X2
       => ( hilbert_inv_into(A,B,top_top(set(A)),F) = G ) ) ) ).

% inj_imp_inv_eq
tff(fact_7009_lists__def,axiom,
    ! [A: $tType,X3: set(A)] : lists(A,X3) = aa(fun(list(A),$o),set(list(A)),collect(list(A)),listsp(A,aa(set(A),fun(A,$o),aTP_Lamp_ag(set(A),fun(A,$o)),X3))) ).

% lists_def
tff(fact_7010_listsp__lists__eq,axiom,
    ! [A: $tType,A4: set(A),X3: list(A)] :
      ( aa(list(A),$o,listsp(A,aa(set(A),fun(A,$o),aTP_Lamp_ag(set(A),fun(A,$o)),A4)),X3)
    <=> aa(set(list(A)),$o,member(list(A),X3),lists(A,A4)) ) ).

% listsp_lists_eq
tff(fact_7011_inv__into__def2,axiom,
    ! [A: $tType,B: $tType,A4: set(A),F: fun(A,B),X: B] : aa(B,A,hilbert_inv_into(A,B,A4,F),X) = fChoice(A,aa(B,fun(A,$o),aa(fun(A,B),fun(B,fun(A,$o)),aTP_Lamp_mt(set(A),fun(fun(A,B),fun(B,fun(A,$o))),A4),F),X)) ).

% inv_into_def2
tff(fact_7012_inv__into__def,axiom,
    ! [B: $tType,A: $tType,A4: set(B),F: fun(B,A),X3: A] : aa(A,B,hilbert_inv_into(B,A,A4,F),X3) = fChoice(B,aa(A,fun(B,$o),aa(fun(B,A),fun(A,fun(B,$o)),aTP_Lamp_acy(set(B),fun(fun(B,A),fun(A,fun(B,$o))),A4),F),X3)) ).

% inv_into_def
tff(fact_7013_inj__transfer,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),P: fun(A,$o),X: A] :
      ( inj_on(A,B,F,top_top(set(A)))
     => ( ! [Y2: B] :
            ( aa(set(B),$o,member(B,Y2),aa(set(A),set(B),image2(A,B,F),top_top(set(A))))
           => aa(A,$o,P,aa(B,A,hilbert_inv_into(A,B,top_top(set(A)),F),Y2)) )
       => aa(A,$o,P,X) ) ) ).

% inj_transfer
tff(fact_7014_image__inv__f__f,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),A4: set(A)] :
      ( inj_on(A,B,F,top_top(set(A)))
     => ( aa(set(B),set(A),image2(B,A,hilbert_inv_into(A,B,top_top(set(A)),F)),aa(set(A),set(B),image2(A,B,F),A4)) = A4 ) ) ).

% image_inv_f_f
tff(fact_7015_inj__imp__surj__inv,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B)] :
      ( inj_on(A,B,F,top_top(set(A)))
     => ( aa(set(B),set(A),image2(B,A,hilbert_inv_into(A,B,top_top(set(A)),F)),top_top(set(B))) = top_top(set(A)) ) ) ).

% inj_imp_surj_inv
tff(fact_7016_surj__imp__inj__inv,axiom,
    ! [B: $tType,A: $tType,F: fun(B,A)] :
      ( ( aa(set(B),set(A),image2(B,A,F),top_top(set(B))) = top_top(set(A)) )
     => inj_on(A,B,hilbert_inv_into(B,A,top_top(set(B)),F),top_top(set(A))) ) ).

% surj_imp_inj_inv
tff(fact_7017_inv__unique__comp,axiom,
    ! [B: $tType,A: $tType,F: fun(B,A),G: fun(A,B)] :
      ( ( aa(fun(A,B),fun(A,A),comp(B,A,A,F),G) = id(A) )
     => ( ( aa(fun(B,A),fun(B,B),comp(A,B,B,G),F) = id(B) )
       => ( hilbert_inv_into(B,A,top_top(set(B)),F) = G ) ) ) ).

% inv_unique_comp
tff(fact_7018_o__inv__distrib,axiom,
    ! [C: $tType,B: $tType,A: $tType,F: fun(A,B),G: fun(C,A)] :
      ( bij_betw(A,B,F,top_top(set(A)),top_top(set(B)))
     => ( bij_betw(C,A,G,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,F),G)) = aa(fun(B,A),fun(B,C),comp(A,C,B,hilbert_inv_into(C,A,top_top(set(C)),G)),hilbert_inv_into(A,B,top_top(set(A)),F)) ) ) ) ).

% o_inv_distrib
tff(fact_7019_inv__fn,axiom,
    ! [A: $tType,F: fun(A,A),N: nat] :
      ( bij_betw(A,A,F,top_top(set(A)),top_top(set(A)))
     => ( hilbert_inv_into(A,A,top_top(set(A)),compow(fun(A,A),N,F)) = compow(fun(A,A),N,hilbert_inv_into(A,A,top_top(set(A)),F)) ) ) ).

% inv_fn
tff(fact_7020_mono__inv,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & linorder(B) )
     => ! [F: fun(A,B)] :
          ( order_mono(A,B,F)
         => ( bij_betw(A,B,F,top_top(set(A)),top_top(set(B)))
           => order_mono(B,A,hilbert_inv_into(A,B,top_top(set(A)),F)) ) ) ) ).

% mono_inv
tff(fact_7021_iso__backward,axiom,
    ! [A: $tType,B: $tType,X: A,Y: A,R3: set(product_prod(A,A)),R2: set(product_prod(B,B)),F: fun(B,A)] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R3)
     => ( bNF_Wellorder_iso(B,A,R2,R3,F)
       => aa(set(product_prod(B,B)),$o,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,hilbert_inv_into(B,A,field2(B,R2),F),X)),aa(A,B,hilbert_inv_into(B,A,field2(B,R2),F),Y))),R2) ) ) ).

% iso_backward
tff(fact_7022_bij__image__Collect__eq,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),P: fun(A,$o)] :
      ( bij_betw(A,B,F,top_top(set(A)),top_top(set(B)))
     => ( aa(set(A),set(B),image2(A,B,F),aa(fun(A,$o),set(A),collect(A),P)) = aa(fun(B,$o),set(B),collect(B),aa(fun(A,$o),fun(B,$o),aTP_Lamp_atg(fun(A,B),fun(fun(A,$o),fun(B,$o)),F),P)) ) ) ).

% bij_image_Collect_eq
tff(fact_7023_from__nat__def,axiom,
    ! [A: $tType] :
      ( countable(A)
     => ( from_nat(A) = hilbert_inv_into(A,nat,top_top(set(A)),to_nat(A)) ) ) ).

% from_nat_def
tff(fact_7024_surj__iff,axiom,
    ! [B: $tType,A: $tType,F: fun(B,A)] :
      ( ( aa(set(B),set(A),image2(B,A,F),top_top(set(B))) = top_top(set(A)) )
    <=> ( aa(fun(A,B),fun(A,A),comp(B,A,A,F),hilbert_inv_into(B,A,top_top(set(B)),F)) = id(A) ) ) ).

% surj_iff
tff(fact_7025_inj__imp__bij__betw__inv,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),M4: set(A)] :
      ( inj_on(A,B,F,top_top(set(A)))
     => bij_betw(B,A,hilbert_inv_into(A,B,top_top(set(A)),F),aa(set(A),set(B),image2(A,B,F),M4),M4) ) ).

% inj_imp_bij_betw_inv
tff(fact_7026_inj__iff,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B)] :
      ( inj_on(A,B,F,top_top(set(A)))
    <=> ( aa(fun(A,B),fun(A,A),comp(B,A,A,hilbert_inv_into(A,B,top_top(set(A)),F)),F) = id(A) ) ) ).

% inj_iff
tff(fact_7027_bij__vimage__eq__inv__image,axiom,
    ! [A: $tType,B: $tType,F: fun(A,B),A4: set(B)] :
      ( bij_betw(A,B,F,top_top(set(A)),top_top(set(B)))
     => ( aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F),A4) = aa(set(B),set(A),image2(B,A,hilbert_inv_into(A,B,top_top(set(A)),F)),A4) ) ) ).

% bij_vimage_eq_inv_image
tff(fact_7028_fn__o__inv__fn__is__id,axiom,
    ! [A: $tType,F: fun(A,A),N: nat] :
      ( bij_betw(A,A,F,top_top(set(A)),top_top(set(A)))
     => ! [X3: A] : aa(A,A,aa(fun(A,A),fun(A,A),comp(A,A,A,compow(fun(A,A),N,F)),compow(fun(A,A),N,hilbert_inv_into(A,A,top_top(set(A)),F))),X3) = X3 ) ).

% fn_o_inv_fn_is_id
tff(fact_7029_inv__fn__o__fn__is__id,axiom,
    ! [A: $tType,F: fun(A,A),N: nat] :
      ( bij_betw(A,A,F,top_top(set(A)),top_top(set(A)))
     => ! [X3: A] : aa(A,A,aa(fun(A,A),fun(A,A),comp(A,A,A,compow(fun(A,A),N,hilbert_inv_into(A,A,top_top(set(A)),F))),compow(fun(A,A),N,F)),X3) = X3 ) ).

% inv_fn_o_fn_is_id
tff(fact_7030_bijection_Oinv__comp__left,axiom,
    ! [A: $tType,F: fun(A,A)] :
      ( hilbert_bijection(A,F)
     => ( aa(fun(A,A),fun(A,A),comp(A,A,A,hilbert_inv_into(A,A,top_top(set(A)),F)),F) = id(A) ) ) ).

% bijection.inv_comp_left
tff(fact_7031_bijection_Oinv__comp__right,axiom,
    ! [A: $tType,F: fun(A,A)] :
      ( hilbert_bijection(A,F)
     => ( aa(fun(A,A),fun(A,A),comp(A,A,A,F),hilbert_inv_into(A,A,top_top(set(A)),F)) = id(A) ) ) ).

% bijection.inv_comp_right
tff(fact_7032_bijection_Osurj,axiom,
    ! [A: $tType,F: fun(A,A)] :
      ( hilbert_bijection(A,F)
     => ( aa(set(A),set(A),image2(A,A,F),top_top(set(A))) = top_top(set(A)) ) ) ).

% bijection.surj
tff(fact_7033_bijection_Oinj,axiom,
    ! [A: $tType,F: fun(A,A)] :
      ( hilbert_bijection(A,F)
     => inj_on(A,A,F,top_top(set(A))) ) ).

% bijection.inj
tff(fact_7034_bijection__def,axiom,
    ! [A: $tType,F: fun(A,A)] :
      ( hilbert_bijection(A,F)
    <=> bij_betw(A,A,F,top_top(set(A)),top_top(set(A))) ) ).

% bijection_def
tff(fact_7035_bijection_Ointro,axiom,
    ! [A: $tType,F: fun(A,A)] :
      ( bij_betw(A,A,F,top_top(set(A)),top_top(set(A)))
     => hilbert_bijection(A,F) ) ).

% bijection.intro
tff(fact_7036_bijection_Obij,axiom,
    ! [A: $tType,F: fun(A,A)] :
      ( hilbert_bijection(A,F)
     => bij_betw(A,A,F,top_top(set(A)),top_top(set(A))) ) ).

% bijection.bij
tff(fact_7037_bijection_Oeq__invI,axiom,
    ! [A: $tType,F: fun(A,A),A3: A,B2: A] :
      ( hilbert_bijection(A,F)
     => ( ( aa(A,A,hilbert_inv_into(A,A,top_top(set(A)),F),A3) = aa(A,A,hilbert_inv_into(A,A,top_top(set(A)),F),B2) )
       => ( A3 = B2 ) ) ) ).

% bijection.eq_invI
tff(fact_7038_bijection_Oinv__left,axiom,
    ! [A: $tType,F: fun(A,A),A3: A] :
      ( hilbert_bijection(A,F)
     => ( aa(A,A,hilbert_inv_into(A,A,top_top(set(A)),F),aa(A,A,F,A3)) = A3 ) ) ).

% bijection.inv_left
tff(fact_7039_bijection_Oinv__right,axiom,
    ! [A: $tType,F: fun(A,A),A3: A] :
      ( hilbert_bijection(A,F)
     => ( aa(A,A,F,aa(A,A,hilbert_inv_into(A,A,top_top(set(A)),F),A3)) = A3 ) ) ).

% bijection.inv_right
tff(fact_7040_bijection_Oeq__inv__iff,axiom,
    ! [A: $tType,F: fun(A,A),A3: A,B2: A] :
      ( hilbert_bijection(A,F)
     => ( ( aa(A,A,hilbert_inv_into(A,A,top_top(set(A)),F),A3) = aa(A,A,hilbert_inv_into(A,A,top_top(set(A)),F),B2) )
      <=> ( A3 = B2 ) ) ) ).

% bijection.eq_inv_iff
tff(fact_7041_bijection_Oinv__left__eq__iff,axiom,
    ! [A: $tType,F: fun(A,A),A3: A,B2: A] :
      ( hilbert_bijection(A,F)
     => ( ( aa(A,A,hilbert_inv_into(A,A,top_top(set(A)),F),A3) = B2 )
      <=> ( aa(A,A,F,B2) = A3 ) ) ) ).

% bijection.inv_left_eq_iff
tff(fact_7042_bijection_Oinv__right__eq__iff,axiom,
    ! [A: $tType,F: fun(A,A),B2: A,A3: A] :
      ( hilbert_bijection(A,F)
     => ( ( B2 = aa(A,A,hilbert_inv_into(A,A,top_top(set(A)),F),A3) )
      <=> ( aa(A,A,F,B2) = A3 ) ) ) ).

% bijection.inv_right_eq_iff
tff(fact_7043_bijection_Osurj__inv,axiom,
    ! [A: $tType,F: fun(A,A)] :
      ( hilbert_bijection(A,F)
     => ( aa(set(A),set(A),image2(A,A,hilbert_inv_into(A,A,top_top(set(A)),F)),top_top(set(A))) = top_top(set(A)) ) ) ).

% bijection.surj_inv
tff(fact_7044_bijection_Oinj__inv,axiom,
    ! [A: $tType,F: fun(A,A)] :
      ( hilbert_bijection(A,F)
     => inj_on(A,A,hilbert_inv_into(A,A,top_top(set(A)),F),top_top(set(A))) ) ).

% bijection.inj_inv
tff(fact_7045_bijection_Obij__inv,axiom,
    ! [A: $tType,F: fun(A,A)] :
      ( hilbert_bijection(A,F)
     => bij_betw(A,A,hilbert_inv_into(A,A,top_top(set(A)),F),top_top(set(A)),top_top(set(A))) ) ).

% bijection.bij_inv
tff(fact_7046_cr__integer__def,axiom,
    ! [X3: int,Xa3: code_integer] :
      ( code_cr_integer(X3,Xa3)
    <=> ( X3 = aa(code_integer,int,code_int_of_integer,Xa3) ) ) ).

% cr_integer_def
tff(fact_7047_subseqs_Osimps_I2_J,axiom,
    ! [A: $tType,X: A,Xs: list(A)] :
      subseqs(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = $let(
        xss: list(list(A)),
        xss:= subseqs(A,Xs),
        aa(list(list(A)),list(list(A)),aa(list(list(A)),fun(list(list(A)),list(list(A))),append(list(A)),aa(list(list(A)),list(list(A)),map(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X)),xss)),xss) ) ).

% subseqs.simps(2)
tff(fact_7048_ndepth__Push__Node__aux,axiom,
    ! [A: $tType,I: nat,F: fun(nat,sum_sum(A,nat)),K: nat] :
      ( ( aa(nat,sum_sum(A,nat),aa(fun(nat,sum_sum(A,nat)),fun(nat,sum_sum(A,nat)),aa(sum_sum(A,nat),fun(fun(nat,sum_sum(A,nat)),fun(nat,sum_sum(A,nat))),case_nat(sum_sum(A,nat)),aa(nat,sum_sum(A,nat),sum_Inr(nat,A),aa(nat,nat,suc,I))),F),K) = aa(nat,sum_sum(A,nat),sum_Inr(nat,A),zero_zero(nat)) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,ord_Least(nat,aTP_Lamp_ath(fun(nat,sum_sum(A,nat)),fun(nat,$o),F)))),K) ) ).

% ndepth_Push_Node_aux
tff(fact_7049_Powp__Pow__eq,axiom,
    ! [A: $tType,A4: set(A),X3: set(A)] :
      ( powp(A,aa(set(A),fun(A,$o),aTP_Lamp_ag(set(A),fun(A,$o)),A4),X3)
    <=> aa(set(set(A)),$o,member(set(A),X3),pow2(A,A4)) ) ).

% Powp_Pow_eq
tff(fact_7050_Powp__def,axiom,
    ! [A: $tType,A4: fun(A,$o),X3: set(A)] :
      ( powp(A,A4,X3)
    <=> ! [Xa2: A] :
          ( aa(set(A),$o,member(A,Xa2),X3)
         => aa(A,$o,A4,Xa2) ) ) ).

% Powp_def
tff(fact_7051_Union__sum,axiom,
    ! [B: $tType,A: $tType,C: $tType,F: fun(sum_sum(B,C),set(A))] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(sum_sum(B,C)),set(set(A)),image2(sum_sum(B,C),set(A),F),top_top(set(sum_sum(B,C))))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aTP_Lamp_ati(fun(sum_sum(B,C),set(A)),fun(B,set(A)),F)),top_top(set(B))))),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(C),set(set(A)),image2(C,set(A),aTP_Lamp_atj(fun(sum_sum(B,C),set(A)),fun(C,set(A)),F)),top_top(set(C))))) ).

% Union_sum
tff(fact_7052_Node__def,axiom,
    ! [B: $tType,A: $tType] : old_Node(A,B) = aa(fun(product_prod(fun(nat,sum_sum(A,nat)),sum_sum(B,nat)),$o),set(product_prod(fun(nat,sum_sum(A,nat)),sum_sum(B,nat))),collect(product_prod(fun(nat,sum_sum(A,nat)),sum_sum(B,nat))),aTP_Lamp_atk(product_prod(fun(nat,sum_sum(A,nat)),sum_sum(B,nat)),$o)) ).

% Node_def
tff(fact_7053_Plus__def,axiom,
    ! [A: $tType,B: $tType,A4: set(A),B3: set(B)] : sum_Plus(A,B,A4,B3) = aa(set(sum_sum(A,B)),set(sum_sum(A,B)),aa(set(sum_sum(A,B)),fun(set(sum_sum(A,B)),set(sum_sum(A,B))),sup_sup(set(sum_sum(A,B))),aa(set(A),set(sum_sum(A,B)),image2(A,sum_sum(A,B),sum_Inl(A,B)),A4)),aa(set(B),set(sum_sum(A,B)),image2(B,sum_sum(A,B),sum_Inr(B,A)),B3)) ).

% Plus_def
tff(fact_7054_UNIV__sum,axiom,
    ! [A: $tType,B: $tType] : top_top(set(sum_sum(A,B))) = aa(set(sum_sum(A,B)),set(sum_sum(A,B)),aa(set(sum_sum(A,B)),fun(set(sum_sum(A,B)),set(sum_sum(A,B))),sup_sup(set(sum_sum(A,B))),aa(set(A),set(sum_sum(A,B)),image2(A,sum_sum(A,B),sum_Inl(A,B)),top_top(set(A)))),aa(set(B),set(sum_sum(A,B)),image2(B,sum_sum(A,B),sum_Inr(B,A)),top_top(set(B)))) ).

% UNIV_sum
tff(fact_7055_Node__K0__I,axiom,
    ! [B: $tType,A: $tType,A3: sum_sum(B,nat)] : aa(set(product_prod(fun(nat,sum_sum(A,nat)),sum_sum(B,nat))),$o,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_atl(nat,sum_sum(A,nat))),A3)),old_Node(A,B)) ).

% Node_K0_I
tff(fact_7056_Field__csum,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(A,A)),S2: set(product_prod(B,B))] : field2(sum_sum(A,B),bNF_Cardinal_csum(A,B,R2,S2)) = aa(set(sum_sum(A,B)),set(sum_sum(A,B)),aa(set(sum_sum(A,B)),fun(set(sum_sum(A,B)),set(sum_sum(A,B))),sup_sup(set(sum_sum(A,B))),aa(set(A),set(sum_sum(A,B)),image2(A,sum_sum(A,B),sum_Inl(A,B)),field2(A,R2))),aa(set(B),set(sum_sum(A,B)),image2(B,sum_sum(A,B),sum_Inr(B,A)),field2(B,S2))) ).

% Field_csum
tff(fact_7057_prod_OPlus,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [A4: set(A),B3: set(B),G: fun(sum_sum(A,B),C)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(B),$o,finite_finite2(B),B3)
           => ( aa(set(sum_sum(A,B)),C,aa(fun(sum_sum(A,B),C),fun(set(sum_sum(A,B)),C),groups7121269368397514597t_prod(sum_sum(A,B),C),G),sum_Plus(A,B,A4,B3)) = aa(C,C,aa(C,fun(C,C),times_times(C),aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7121269368397514597t_prod(A,C),aa(fun(A,sum_sum(A,B)),fun(A,C),comp(sum_sum(A,B),C,A,G),sum_Inl(A,B))),A4)),aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),aa(fun(B,sum_sum(A,B)),fun(B,C),comp(sum_sum(A,B),C,B,G),sum_Inr(B,A))),B3)) ) ) ) ) ).

% prod.Plus
tff(fact_7058_Union__plus,axiom,
    ! [B: $tType,A: $tType,C: $tType,F: fun(sum_sum(B,C),set(A)),A4: set(B),B3: set(C)] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(sum_sum(B,C)),set(set(A)),image2(sum_sum(B,C),set(A),F),sum_Plus(B,C,A4,B3))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aTP_Lamp_ati(fun(sum_sum(B,C),set(A)),fun(B,set(A)),F)),A4))),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(C),set(set(A)),image2(C,set(A),aTP_Lamp_atj(fun(sum_sum(B,C),set(A)),fun(C,set(A)),F)),B3))) ).

% Union_plus
tff(fact_7059_int__encode__def,axiom,
    ! [I: int] :
      aa(int,nat,nat_int_encode,I) = aa(sum_sum(nat,nat),nat,nat_sum_encode,
        $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),I),aa(nat,sum_sum(nat,nat),sum_Inl(nat,nat),aa(int,nat,nat2,I)),aa(nat,sum_sum(nat,nat),sum_Inr(nat,nat),aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,uminus_uminus(int),I)),one_one(int)))))) ).

% int_encode_def
tff(fact_7060_sum__set__simps_I1_J,axiom,
    ! [B: $tType,A: $tType,X: A] : aa(sum_sum(A,B),set(A),basic_setl(A,B),aa(A,sum_sum(A,B),sum_Inl(A,B),X)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))) ).

% sum_set_simps(1)
tff(fact_7061_sum__set__simps_I2_J,axiom,
    ! [B: $tType,A: $tType,X: B] : aa(sum_sum(A,B),set(A),basic_setl(A,B),aa(B,sum_sum(A,B),sum_Inr(B,A),X)) = bot_bot(set(A)) ).

% sum_set_simps(2)
tff(fact_7062_surj__sum__encode,axiom,
    aa(set(sum_sum(nat,nat)),set(nat),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_7063_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_7064_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),aa(num,num,bit0,one2))),aTP_Lamp_atm(nat,nat)),X) ).

% sum_encode_def
tff(fact_7065_case__sum__KK,axiom,
    ! [A: $tType,B: $tType,C: $tType,A3: C,X3: sum_sum(A,B)] : aa(sum_sum(A,B),C,sum_case_sum(A,C,B,aTP_Lamp_atn(C,fun(A,C),A3),aTP_Lamp_ato(C,fun(B,C),A3)),X3) = A3 ).

% case_sum_KK
tff(fact_7066_surj__sum__decode,axiom,
    aa(set(nat),set(sum_sum(nat,nat)),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_7067_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_7068_sum_Ocase__distrib,axiom,
    ! [B: $tType,A: $tType,D: $tType,C: $tType,H2: fun(B,A),F1: fun(C,B),F22: fun(D,B),Sum: sum_sum(C,D)] : aa(B,A,H2,aa(sum_sum(C,D),B,sum_case_sum(C,B,D,F1,F22),Sum)) = aa(sum_sum(C,D),A,sum_case_sum(C,A,D,aa(fun(C,B),fun(C,A),aTP_Lamp_ll(fun(B,A),fun(fun(C,B),fun(C,A)),H2),F1),aa(fun(D,B),fun(D,A),aTP_Lamp_atp(fun(B,A),fun(fun(D,B),fun(D,A)),H2),F22)),Sum) ).

% sum.case_distrib
tff(fact_7069_disjE__realizer,axiom,
    ! [C: $tType,B: $tType,A: $tType,P: fun(A,$o),Q: fun(B,$o),X: sum_sum(A,B),R: fun(C,$o),F: fun(A,C),G: fun(B,C)] :
      ( aa(sum_sum(A,B),$o,sum_case_sum(A,$o,B,P,Q),X)
     => ( ! [P9: A] :
            ( aa(A,$o,P,P9)
           => aa(C,$o,R,aa(A,C,F,P9)) )
       => ( ! [Q5: B] :
              ( aa(B,$o,Q,Q5)
             => aa(C,$o,R,aa(B,C,G,Q5)) )
         => aa(C,$o,R,aa(sum_sum(A,B),C,sum_case_sum(A,C,B,F,G),X)) ) ) ) ).

% disjE_realizer
tff(fact_7070_surjective__sum,axiom,
    ! [C: $tType,B: $tType,A: $tType,F: fun(sum_sum(A,B),C)] : sum_case_sum(A,C,B,aTP_Lamp_atq(fun(sum_sum(A,B),C),fun(A,C),F),aTP_Lamp_atr(fun(sum_sum(A,B),C),fun(B,C),F)) = F ).

% surjective_sum
tff(fact_7071_nth__item_Opinduct,axiom,
    ! [A0: nat,P: fun(nat,$o)] :
      ( accp(nat,nth_item_rel,A0)
     => ( ( accp(nat,nth_item_rel,zero_zero(nat))
         => aa(nat,$o,P,zero_zero(nat)) )
       => ( ! [N3: nat] :
              ( accp(nat,nth_item_rel,aa(nat,nat,suc,N3))
             => ( ! [A11: nat,Aa3: nat] :
                    ( ( aa(nat,sum_sum(nat,nat),nat_sum_decode,N3) = aa(nat,sum_sum(nat,nat),sum_Inl(nat,nat),A11) )
                   => ( ( aa(nat,sum_sum(nat,nat),nat_sum_decode,A11) = aa(nat,sum_sum(nat,nat),sum_Inl(nat,nat),Aa3) )
                     => aa(nat,$o,P,Aa3) ) )
               => ( ! [A11: nat,B12: nat] :
                      ( ( aa(nat,sum_sum(nat,nat),nat_sum_decode,N3) = aa(nat,sum_sum(nat,nat),sum_Inl(nat,nat),A11) )
                     => ( ( aa(nat,sum_sum(nat,nat),nat_sum_decode,A11) = aa(nat,sum_sum(nat,nat),sum_Inr(nat,nat),B12) )
                       => aa(nat,$o,P,B12) ) )
                 => ( ! [B12: nat,Ba2: nat,X3: nat,Y5: nat] :
                        ( ( aa(nat,sum_sum(nat,nat),nat_sum_decode,N3) = aa(nat,sum_sum(nat,nat),sum_Inr(nat,nat),B12) )
                       => ( ( aa(nat,sum_sum(nat,nat),nat_sum_decode,B12) = 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),X3),Y5) = aa(nat,product_prod(nat,nat),nat_prod_decode,Ba2) )
                           => aa(nat,$o,P,X3) ) ) )
                   => ( ! [B12: nat,Ba2: nat,X3: nat,Y5: nat] :
                          ( ( aa(nat,sum_sum(nat,nat),nat_sum_decode,N3) = aa(nat,sum_sum(nat,nat),sum_Inr(nat,nat),B12) )
                         => ( ( aa(nat,sum_sum(nat,nat),nat_sum_decode,B12) = aa(nat,sum_sum(nat,nat),sum_Inr(nat,nat),Ba2) )
                           => ( ( aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X3),Y5) = aa(nat,product_prod(nat,nat),nat_prod_decode,Ba2) )
                             => aa(nat,$o,P,Y5) ) ) )
                     => aa(nat,$o,P,aa(nat,nat,suc,N3)) ) ) ) ) )
         => aa(nat,$o,P,A0) ) ) ) ).

% nth_item.pinduct
tff(fact_7072_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_ats(nat,int)),aa(nat,sum_sum(nat,nat),nat_sum_decode,N)) ).

% int_decode_def
tff(fact_7073_sum__set__defs_I1_J,axiom,
    ! [A: $tType,B: $tType] : basic_setl(A,B) = sum_case_sum(A,set(A),B,aTP_Lamp_kg(A,set(A)),aTP_Lamp_mo(B,set(A))) ).

% sum_set_defs(1)
tff(fact_7074_sum__set__simps_I4_J,axiom,
    ! [B: $tType,A: $tType,X: A] : aa(sum_sum(B,A),set(A),basic_setr(B,A),aa(A,sum_sum(B,A),sum_Inr(A,B),X)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))) ).

% sum_set_simps(4)
tff(fact_7075_card__order__csum__cone__cexp__def,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(A,A)),A15: set(B)] :
      ( aa(set(product_prod(A,A)),$o,bNF_Ca8970107618336181345der_on(A,top_top(set(A))),R2)
     => ( bNF_Cardinal_cexp(sum_sum(B,product_unit),A,bNF_Cardinal_csum(B,product_unit,bNF_Ca6860139660246222851ard_of(B,A15),bNF_Cardinal_cone),R2) = bNF_Ca6860139660246222851ard_of(fun(A,sum_sum(B,product_unit)),bNF_Wellorder_Func(A,sum_sum(B,product_unit),top_top(set(A)),aa(set(sum_sum(B,product_unit)),set(sum_sum(B,product_unit)),aa(set(sum_sum(B,product_unit)),fun(set(sum_sum(B,product_unit)),set(sum_sum(B,product_unit))),sup_sup(set(sum_sum(B,product_unit))),aa(set(B),set(sum_sum(B,product_unit)),image2(B,sum_sum(B,product_unit),sum_Inl(B,product_unit)),A15)),aa(set(sum_sum(B,product_unit)),set(sum_sum(B,product_unit)),aa(sum_sum(B,product_unit),fun(set(sum_sum(B,product_unit)),set(sum_sum(B,product_unit))),insert2(sum_sum(B,product_unit)),aa(product_unit,sum_sum(B,product_unit),sum_Inr(product_unit,B),product_Unity)),bot_bot(set(sum_sum(B,product_unit))))))) ) ) ).

% card_order_csum_cone_cexp_def
tff(fact_7076_unit__abs__eta__conv,axiom,
    ! [A: $tType,F: fun(product_unit,A)] : aTP_Lamp_att(fun(product_unit,A),fun(product_unit,A),F) = F ).

% unit_abs_eta_conv
tff(fact_7077_sum__set__simps_I3_J,axiom,
    ! [B: $tType,A: $tType,X: B] : aa(sum_sum(B,A),set(A),basic_setr(B,A),aa(B,sum_sum(B,A),sum_Inl(B,A),X)) = bot_bot(set(A)) ).

% sum_set_simps(3)
tff(fact_7078_sup__unit__def,axiom,
    ! [Uu: product_unit,Uv: product_unit] : aa(product_unit,product_unit,aa(product_unit,fun(product_unit,product_unit),sup_sup(product_unit),Uu),Uv) = product_Unity ).

% sup_unit_def
tff(fact_7079_cone__def,axiom,
    bNF_Cardinal_cone = bNF_Ca6860139660246222851ard_of(product_unit,aa(set(product_unit),set(product_unit),aa(product_unit,fun(set(product_unit),set(product_unit)),insert2(product_unit),product_Unity),bot_bot(set(product_unit)))) ).

% cone_def
tff(fact_7080_bot__unit__def,axiom,
    bot_bot(product_unit) = product_Unity ).

% bot_unit_def
tff(fact_7081_uminus__unit__def,axiom,
    ! [Uu: product_unit] : aa(product_unit,product_unit,uminus_uminus(product_unit),Uu) = product_Unity ).

% uminus_unit_def
tff(fact_7082_inf__unit__def,axiom,
    ! [Uu: product_unit,Uv: product_unit] : aa(product_unit,product_unit,aa(product_unit,fun(product_unit,product_unit),inf_inf(product_unit),Uu),Uv) = product_Unity ).

% inf_unit_def
tff(fact_7083_top__unit__def,axiom,
    top_top(product_unit) = product_Unity ).

% top_unit_def
tff(fact_7084_UNIV__unit,axiom,
    top_top(set(product_unit)) = aa(set(product_unit),set(product_unit),aa(product_unit,fun(set(product_unit),set(product_unit)),insert2(product_unit),product_Unity),bot_bot(set(product_unit))) ).

% UNIV_unit
tff(fact_7085_sum__set__defs_I2_J,axiom,
    ! [A: $tType,B: $tType] : basic_setr(A,B) = sum_case_sum(A,set(B),B,aTP_Lamp_ra(A,set(B)),aTP_Lamp_atu(B,set(B))) ).

% sum_set_defs(2)
tff(fact_7086_Push__def,axiom,
    ! [A: $tType] : old_Push(A) = case_nat(sum_sum(A,nat)) ).

% Push_def
tff(fact_7087_natural__zero__minus__one,axiom,
    aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),minus_minus(code_natural),zero_zero(code_natural)),one_one(code_natural)) = zero_zero(code_natural) ).

% natural_zero_minus_one
tff(fact_7088_log_Oelims,axiom,
    ! [X: code_natural,Xa: code_natural,Y: code_natural] :
      ( ( log(X,Xa) = Y )
     => ( Y = $ite(
            ( aa(code_natural,$o,aa(code_natural,fun(code_natural,$o),ord_less_eq(code_natural),X),one_one(code_natural))
            | aa(code_natural,$o,aa(code_natural,fun(code_natural,$o),ord_less(code_natural),Xa),X) ),
            one_one(code_natural),
            aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),plus_plus(code_natural),one_one(code_natural)),log(X,aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),divide_divide(code_natural),Xa),X))) ) ) ) ).

% log.elims
tff(fact_7089_log_Osimps,axiom,
    ! [B2: code_natural,I: code_natural] :
      log(B2,I) = $ite(
        ( aa(code_natural,$o,aa(code_natural,fun(code_natural,$o),ord_less_eq(code_natural),B2),one_one(code_natural))
        | aa(code_natural,$o,aa(code_natural,fun(code_natural,$o),ord_less(code_natural),I),B2) ),
        one_one(code_natural),
        aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),plus_plus(code_natural),one_one(code_natural)),log(B2,aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),divide_divide(code_natural),I),B2))) ) ).

% log.simps
tff(fact_7090_iterate_Osimps,axiom,
    ! [A: $tType,B: $tType,K: code_natural,F: fun(B,fun(A,product_prod(B,A))),X: B] :
      aa(B,fun(A,product_prod(B,A)),iterate(B,A,K,F),X) = $ite(K = zero_zero(code_natural),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X),product_scomp(A,B,A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),F,X),iterate(B,A,aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),minus_minus(code_natural),K),one_one(code_natural)),F))) ).

% iterate.simps
tff(fact_7091_iterate_Oelims,axiom,
    ! [A: $tType,B: $tType,X: code_natural,Xa: 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,Xa),Xb) = Y )
     => ( Y = $ite(X = zero_zero(code_natural),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Xb),product_scomp(A,B,A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),Xa,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)),Xa))) ) ) ).

% iterate.elims
tff(fact_7092_Random_Orange__def,axiom,
    ! [K: code_natural] : range(K) = 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,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,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))))))))))))))))))))))))))))))),K),aTP_Lamp_atw(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_atx(code_natural,fun(code_natural,fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural)))),K)) ).

% Random.range_def
tff(fact_7093_next_Osimps,axiom,
    ! [V: 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),V),W2)) = $let(
        v: code_natural,
        v:= minus_shift(aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,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,V,aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,one2)))))))))))))))))),aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,one2))))))))))))))))),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),times_times(code_natural),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),divide_divide(code_natural),V),aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,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,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,one2)))))))))))))))),
        $let(
          w2: code_natural,
          w2:= minus_shift(aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,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),aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,one2)))))))))))))))))),aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,one2))))))))))))))))),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),times_times(code_natural),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),divide_divide(code_natural),W2),aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,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,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,one2)))))))))))))),
          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),aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,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))))))))))))))))))))))))))))))),v,aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),plus_plus(code_natural),w2),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),v),w2)) ) ) ).

% next.simps
tff(fact_7094_log_Ocases,axiom,
    ! [X: product_prod(code_natural,code_natural)] :
      ~ ! [B5: code_natural,I2: code_natural] : X != 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),B5),I2) ).

% log.cases
tff(fact_7095_full__exhaustive__natural_H_Ocases,axiom,
    ! [X: product_prod(fun(product_prod(code_natural,fun(product_unit,code_term)),option(product_prod($o,list(code_term)))),product_prod(code_natural,code_natural))] :
      ~ ! [F2: fun(product_prod(code_natural,fun(product_unit,code_term)),option(product_prod($o,list(code_term)))),D2: code_natural,I2: code_natural] : X != aa(product_prod(code_natural,code_natural),product_prod(fun(product_prod(code_natural,fun(product_unit,code_term)),option(product_prod($o,list(code_term)))),product_prod(code_natural,code_natural)),aa(fun(product_prod(code_natural,fun(product_unit,code_term)),option(product_prod($o,list(code_term)))),fun(product_prod(code_natural,code_natural),product_prod(fun(product_prod(code_natural,fun(product_unit,code_term)),option(product_prod($o,list(code_term)))),product_prod(code_natural,code_natural))),product_Pair(fun(product_prod(code_natural,fun(product_unit,code_term)),option(product_prod($o,list(code_term)))),product_prod(code_natural,code_natural)),F2),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),D2),I2)) ).

% full_exhaustive_natural'.cases
tff(fact_7096_full__exhaustive__fun_H_Ocases,axiom,
    ! [A: $tType,B: $tType] :
      ( ( quickc3360725361186068524ustive(B)
        & cl_HOL_Oequal(A)
        & quickc3360725361186068524ustive(A) )
     => ! [X: product_prod(fun(product_prod(fun(A,B),fun(product_unit,code_term)),option(product_prod($o,list(code_term)))),product_prod(code_natural,code_natural))] :
          ~ ! [F2: fun(product_prod(fun(A,B),fun(product_unit,code_term)),option(product_prod($o,list(code_term)))),I2: code_natural,D2: code_natural] : X != aa(product_prod(code_natural,code_natural),product_prod(fun(product_prod(fun(A,B),fun(product_unit,code_term)),option(product_prod($o,list(code_term)))),product_prod(code_natural,code_natural)),aa(fun(product_prod(fun(A,B),fun(product_unit,code_term)),option(product_prod($o,list(code_term)))),fun(product_prod(code_natural,code_natural),product_prod(fun(product_prod(fun(A,B),fun(product_unit,code_term)),option(product_prod($o,list(code_term)))),product_prod(code_natural,code_natural))),product_Pair(fun(product_prod(fun(A,B),fun(product_unit,code_term)),option(product_prod($o,list(code_term)))),product_prod(code_natural,code_natural)),F2),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),I2),D2)) ) ).

% full_exhaustive_fun'.cases
tff(fact_7097_exhaustive__natural_H_Ocases,axiom,
    ! [X: product_prod(fun(code_natural,option(product_prod($o,list(code_term)))),product_prod(code_natural,code_natural))] :
      ~ ! [F2: fun(code_natural,option(product_prod($o,list(code_term)))),D2: code_natural,I2: code_natural] : X != aa(product_prod(code_natural,code_natural),product_prod(fun(code_natural,option(product_prod($o,list(code_term)))),product_prod(code_natural,code_natural)),aa(fun(code_natural,option(product_prod($o,list(code_term)))),fun(product_prod(code_natural,code_natural),product_prod(fun(code_natural,option(product_prod($o,list(code_term)))),product_prod(code_natural,code_natural))),product_Pair(fun(code_natural,option(product_prod($o,list(code_term)))),product_prod(code_natural,code_natural)),F2),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),D2),I2)) ).

% exhaustive_natural'.cases
tff(fact_7098_exhaustive__fun_H_Ocases,axiom,
    ! [A: $tType,B: $tType] :
      ( ( quickc658316121487927005ustive(B)
        & cl_HOL_Oequal(A)
        & quickc658316121487927005ustive(A) )
     => ! [X: product_prod(fun(fun(A,B),option(product_prod($o,list(code_term)))),product_prod(code_natural,code_natural))] :
          ~ ! [F2: fun(fun(A,B),option(product_prod($o,list(code_term)))),I2: code_natural,D2: code_natural] : X != aa(product_prod(code_natural,code_natural),product_prod(fun(fun(A,B),option(product_prod($o,list(code_term)))),product_prod(code_natural,code_natural)),aa(fun(fun(A,B),option(product_prod($o,list(code_term)))),fun(product_prod(code_natural,code_natural),product_prod(fun(fun(A,B),option(product_prod($o,list(code_term)))),product_prod(code_natural,code_natural))),product_Pair(fun(fun(A,B),option(product_prod($o,list(code_term)))),product_prod(code_natural,code_natural)),F2),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),I2),D2)) ) ).

% exhaustive_fun'.cases
tff(fact_7099_Lazy__Sequence_Oiterate__upto_Ocases,axiom,
    ! [A: $tType,X: product_prod(fun(code_natural,A),product_prod(code_natural,code_natural))] :
      ~ ! [F2: fun(code_natural,A),N3: code_natural,M3: code_natural] : X != aa(product_prod(code_natural,code_natural),product_prod(fun(code_natural,A),product_prod(code_natural,code_natural)),aa(fun(code_natural,A),fun(product_prod(code_natural,code_natural),product_prod(fun(code_natural,A),product_prod(code_natural,code_natural))),product_Pair(fun(code_natural,A),product_prod(code_natural,code_natural)),F2),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),N3),M3)) ).

% Lazy_Sequence.iterate_upto.cases
tff(fact_7100_iter_Ocases,axiom,
    ! [A: $tType,X: product_prod(fun(product_prod(code_natural,code_natural),product_prod(product_prod(A,fun(product_unit,code_term)),product_prod(code_natural,code_natural))),product_prod(code_natural,product_prod(code_natural,code_natural)))] :
      ~ ! [Random: fun(product_prod(code_natural,code_natural),product_prod(product_prod(A,fun(product_unit,code_term)),product_prod(code_natural,code_natural))),Nrandom: code_natural,Seed: product_prod(code_natural,code_natural)] : X != aa(product_prod(code_natural,product_prod(code_natural,code_natural)),product_prod(fun(product_prod(code_natural,code_natural),product_prod(product_prod(A,fun(product_unit,code_term)),product_prod(code_natural,code_natural))),product_prod(code_natural,product_prod(code_natural,code_natural))),aa(fun(product_prod(code_natural,code_natural),product_prod(product_prod(A,fun(product_unit,code_term)),product_prod(code_natural,code_natural))),fun(product_prod(code_natural,product_prod(code_natural,code_natural)),product_prod(fun(product_prod(code_natural,code_natural),product_prod(product_prod(A,fun(product_unit,code_term)),product_prod(code_natural,code_natural))),product_prod(code_natural,product_prod(code_natural,code_natural)))),product_Pair(fun(product_prod(code_natural,code_natural),product_prod(product_prod(A,fun(product_unit,code_term)),product_prod(code_natural,code_natural))),product_prod(code_natural,product_prod(code_natural,code_natural))),Random),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)),Nrandom),Seed)) ).

% iter.cases
tff(fact_7101_split__seed__def,axiom,
    ! [S2: product_prod(code_natural,code_natural)] : split_seed(S2) = aa(product_prod(code_natural,code_natural),product_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural)),aa(fun(code_natural,fun(code_natural,product_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural)))),fun(product_prod(code_natural,code_natural),product_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural))),product_case_prod(code_natural,code_natural,product_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural))),aTP_Lamp_atz(product_prod(code_natural,code_natural),fun(code_natural,fun(code_natural,product_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural)))),S2)),S2) ).

% split_seed_def
tff(fact_7102_inc__shift__def,axiom,
    ! [V: code_natural,K: code_natural] :
      inc_shift(V,K) = $ite(V = K,one_one(code_natural),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),plus_plus(code_natural),K),one_one(code_natural))) ).

% inc_shift_def
tff(fact_7103_Predicate_Oiterate__upto_Opinduct,axiom,
    ! [A: $tType,A0: fun(code_natural,A),A1: code_natural,A22: code_natural,P: fun(fun(code_natural,A),fun(code_natural,fun(code_natural,$o)))] :
      ( accp(product_prod(fun(code_natural,A),product_prod(code_natural,code_natural)),iterate_upto_rel(A),aa(product_prod(code_natural,code_natural),product_prod(fun(code_natural,A),product_prod(code_natural,code_natural)),aa(fun(code_natural,A),fun(product_prod(code_natural,code_natural),product_prod(fun(code_natural,A),product_prod(code_natural,code_natural))),product_Pair(fun(code_natural,A),product_prod(code_natural,code_natural)),A0),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),A1),A22)))
     => ( ! [F2: fun(code_natural,A),N3: code_natural,M3: code_natural] :
            ( accp(product_prod(fun(code_natural,A),product_prod(code_natural,code_natural)),iterate_upto_rel(A),aa(product_prod(code_natural,code_natural),product_prod(fun(code_natural,A),product_prod(code_natural,code_natural)),aa(fun(code_natural,A),fun(product_prod(code_natural,code_natural),product_prod(fun(code_natural,A),product_prod(code_natural,code_natural))),product_Pair(fun(code_natural,A),product_prod(code_natural,code_natural)),F2),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),N3),M3)))
           => ( ! [X3: product_unit] :
                  ( ~ aa(code_natural,$o,aa(code_natural,fun(code_natural,$o),ord_less(code_natural),M3),N3)
                 => aa(code_natural,$o,aa(code_natural,fun(code_natural,$o),aa(fun(code_natural,A),fun(code_natural,fun(code_natural,$o)),P,F2),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),plus_plus(code_natural),N3),one_one(code_natural))),M3) )
             => aa(code_natural,$o,aa(code_natural,fun(code_natural,$o),aa(fun(code_natural,A),fun(code_natural,fun(code_natural,$o)),P,F2),N3),M3) ) )
       => aa(code_natural,$o,aa(code_natural,fun(code_natural,$o),aa(fun(code_natural,A),fun(code_natural,fun(code_natural,$o)),P,A0),A1),A22) ) ) ).

% Predicate.iterate_upto.pinduct
tff(fact_7104_log_Opelims,axiom,
    ! [X: code_natural,Xa: code_natural,Y: code_natural] :
      ( ( log(X,Xa) = Y )
     => ( accp(product_prod(code_natural,code_natural),log_rel,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),X),Xa))
       => ~ ( ( Y = $ite(
                  ( aa(code_natural,$o,aa(code_natural,fun(code_natural,$o),ord_less_eq(code_natural),X),one_one(code_natural))
                  | aa(code_natural,$o,aa(code_natural,fun(code_natural,$o),ord_less(code_natural),Xa),X) ),
                  one_one(code_natural),
                  aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),plus_plus(code_natural),one_one(code_natural)),log(X,aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),divide_divide(code_natural),Xa),X))) ) )
           => ~ accp(product_prod(code_natural,code_natural),log_rel,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),X),Xa)) ) ) ) ).

% log.pelims
tff(fact_7105_iterate_Opelims,axiom,
    ! [A: $tType,B: $tType,X: code_natural,Xa: 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,Xa),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),Xa),Xb)))
       => ~ ( ( Y = $ite(X = zero_zero(code_natural),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Xb),product_scomp(A,B,A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),Xa,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)),Xa))) )
           => ~ 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),Xa),Xb))) ) ) ) ).

% iterate.pelims
tff(fact_7106_pick__drop__zero,axiom,
    ! [A: $tType,Xs: list(product_prod(code_natural,A))] : pick(A,aa(list(product_prod(code_natural,A)),list(product_prod(code_natural,A)),filter2(product_prod(code_natural,A),aa(fun(code_natural,fun(A,$o)),fun(product_prod(code_natural,A),$o),product_case_prod(code_natural,A,$o),aTP_Lamp_aua(code_natural,fun(A,$o)))),Xs)) = pick(A,Xs) ).

% pick_drop_zero
tff(fact_7107_iterate_Ocases,axiom,
    ! [B: $tType,A: $tType,X: product_prod(code_natural,product_prod(fun(A,fun(B,product_prod(A,B))),A))] :
      ~ ! [K2: code_natural,F2: fun(A,fun(B,product_prod(A,B))),X2: A] : X != aa(product_prod(fun(A,fun(B,product_prod(A,B))),A),product_prod(code_natural,product_prod(fun(A,fun(B,product_prod(A,B))),A)),aa(code_natural,fun(product_prod(fun(A,fun(B,product_prod(A,B))),A),product_prod(code_natural,product_prod(fun(A,fun(B,product_prod(A,B))),A))),product_Pair(code_natural,product_prod(fun(A,fun(B,product_prod(A,B))),A)),K2),aa(A,product_prod(fun(A,fun(B,product_prod(A,B))),A),aa(fun(A,fun(B,product_prod(A,B))),fun(A,product_prod(fun(A,fun(B,product_prod(A,B))),A)),product_Pair(fun(A,fun(B,product_prod(A,B))),A),F2),X2)) ).

% iterate.cases
tff(fact_7108_select__weight__drop__zero,axiom,
    ! [A: $tType,Xs: list(product_prod(code_natural,A))] : select_weight(A,aa(list(product_prod(code_natural,A)),list(product_prod(code_natural,A)),filter2(product_prod(code_natural,A),aa(fun(code_natural,fun(A,$o)),fun(product_prod(code_natural,A),$o),product_case_prod(code_natural,A,$o),aTP_Lamp_aua(code_natural,fun(A,$o)))),Xs)) = select_weight(A,Xs) ).

% select_weight_drop_zero
tff(fact_7109_select__weight__def,axiom,
    ! [A: $tType,Xs: list(product_prod(code_natural,A))] : select_weight(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(groups8242544230860333062m_list(code_natural,aa(list(product_prod(code_natural,A)),list(code_natural),map(product_prod(code_natural,A),code_natural,product_fst(code_natural,A)),Xs))),aTP_Lamp_aub(list(product_prod(code_natural,A)),fun(code_natural,fun(product_prod(code_natural,code_natural),product_prod(A,product_prod(code_natural,code_natural)))),Xs)) ).

% select_weight_def
tff(fact_7110_select__weight__cons__zero,axiom,
    ! [A: $tType,X: A,Xs: list(product_prod(code_natural,A))] : select_weight(A,aa(list(product_prod(code_natural,A)),list(product_prod(code_natural,A)),aa(product_prod(code_natural,A),fun(list(product_prod(code_natural,A)),list(product_prod(code_natural,A))),cons(product_prod(code_natural,A)),aa(A,product_prod(code_natural,A),aa(code_natural,fun(A,product_prod(code_natural,A)),product_Pair(code_natural,A),zero_zero(code_natural)),X)),Xs)) = select_weight(A,Xs) ).

% select_weight_cons_zero
tff(fact_7111_select__weight__select,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( Xs != nil(A) )
     => ( select_weight(A,aa(list(A),list(product_prod(code_natural,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)) = select(A,Xs) ) ) ).

% select_weight_select
tff(fact_7112_pick__same,axiom,
    ! [A: $tType,L: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),L),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(code_natural,A,pick(A,aa(list(A),list(product_prod(code_natural,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_7113_one__natural__def,axiom,
    one_one(code_natural) = aa(nat,code_natural,code_natural_of_nat,one_one(nat)) ).

% one_natural_def
tff(fact_7114_natural__of__nat__cases,axiom,
    ! [X: code_natural] :
      ~ ! [Y2: nat] :
          ( ( X = aa(nat,code_natural,code_natural_of_nat,Y2) )
         => ~ aa(set(nat),$o,member(nat,Y2),top_top(set(nat))) ) ).

% natural_of_nat_cases
tff(fact_7115_natural__of__nat__induct,axiom,
    ! [P: fun(code_natural,$o),X: code_natural] :
      ( ! [Y2: nat] :
          ( aa(set(nat),$o,member(nat,Y2),top_top(set(nat)))
         => aa(code_natural,$o,P,aa(nat,code_natural,code_natural_of_nat,Y2)) )
     => aa(code_natural,$o,P,X) ) ).

% natural_of_nat_induct
tff(fact_7116_natural__of__nat__inject,axiom,
    ! [X: nat,Y: nat] :
      ( aa(set(nat),$o,member(nat,X),top_top(set(nat)))
     => ( aa(set(nat),$o,member(nat,Y),top_top(set(nat)))
       => ( ( aa(nat,code_natural,code_natural_of_nat,X) = aa(nat,code_natural,code_natural_of_nat,Y) )
        <=> ( X = Y ) ) ) ) ).

% natural_of_nat_inject
tff(fact_7117_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_auc(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_7118_iter_H_Ocases,axiom,
    ! [A: $tType] :
      ( quickcheck_random(A)
     => ! [X: product_prod(itself(A),product_prod(code_natural,product_prod(code_natural,product_prod(code_natural,code_natural))))] :
          ~ ! [T5: itself(A),Nrandom: code_natural,Sz: code_natural,Seed: product_prod(code_natural,code_natural)] : X != aa(product_prod(code_natural,product_prod(code_natural,product_prod(code_natural,code_natural))),product_prod(itself(A),product_prod(code_natural,product_prod(code_natural,product_prod(code_natural,code_natural)))),aa(itself(A),fun(product_prod(code_natural,product_prod(code_natural,product_prod(code_natural,code_natural))),product_prod(itself(A),product_prod(code_natural,product_prod(code_natural,product_prod(code_natural,code_natural))))),product_Pair(itself(A),product_prod(code_natural,product_prod(code_natural,product_prod(code_natural,code_natural)))),T5),aa(product_prod(code_natural,product_prod(code_natural,code_natural)),product_prod(code_natural,product_prod(code_natural,product_prod(code_natural,code_natural))),aa(code_natural,fun(product_prod(code_natural,product_prod(code_natural,code_natural)),product_prod(code_natural,product_prod(code_natural,product_prod(code_natural,code_natural)))),product_Pair(code_natural,product_prod(code_natural,product_prod(code_natural,code_natural))),Nrandom),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)),Sz),Seed))) ) ).

% iter'.cases
tff(fact_7119_one__natural_Orep__eq,axiom,
    aa(code_natural,nat,code_nat_of_natural,one_one(code_natural)) = one_one(nat) ).

% one_natural.rep_eq
tff(fact_7120_type__definition__natural,axiom,
    type_definition(code_natural,nat,code_nat_of_natural,code_natural_of_nat,top_top(set(nat))) ).

% type_definition_natural
tff(fact_7121_natural__of__nat__inverse,axiom,
    ! [Y: nat] :
      ( aa(set(nat),$o,member(nat,Y),top_top(set(nat)))
     => ( aa(code_natural,nat,code_nat_of_natural,aa(nat,code_natural,code_natural_of_nat,Y)) = Y ) ) ).

% natural_of_nat_inverse
tff(fact_7122_nat__of__natural__induct,axiom,
    ! [Y: nat,P: fun(nat,$o)] :
      ( aa(set(nat),$o,member(nat,Y),top_top(set(nat)))
     => ( ! [X2: code_natural] : aa(nat,$o,P,aa(code_natural,nat,code_nat_of_natural,X2))
       => aa(nat,$o,P,Y) ) ) ).

% nat_of_natural_induct
tff(fact_7123_nat__of__natural__cases,axiom,
    ! [Y: nat] :
      ( aa(set(nat),$o,member(nat,Y),top_top(set(nat)))
     => ~ ! [X2: code_natural] : Y != aa(code_natural,nat,code_nat_of_natural,X2) ) ).

% nat_of_natural_cases
tff(fact_7124_nat__of__natural,axiom,
    ! [X: code_natural] : aa(set(nat),$o,member(nat,aa(code_natural,nat,code_nat_of_natural,X)),top_top(set(nat))) ).

% nat_of_natural
tff(fact_7125_cr__natural__def,axiom,
    ! [X3: nat,Xa3: code_natural] :
      ( code_cr_natural(X3,Xa3)
    <=> ( X3 = aa(code_natural,nat,code_nat_of_natural,Xa3) ) ) ).

% cr_natural_def
tff(fact_7126_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_aue(fun(nat,sum_sum(B,nat)),fun(sum_sum(A,nat),nat))),old_Rep_Node(A,B,N)) ).

% ndepth_def
tff(fact_7127_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_auf(nat,sum_sum(B,nat))),X))) = zero_zero(nat) ).

% ndepth_K0
tff(fact_7128_mergesort__by__rel__split_Opelims,axiom,
    ! [A: $tType,X: product_prod(list(A),list(A)),Xa: list(A),Y: product_prod(list(A),list(A))] :
      ( ( merges295452479951948502_split(A,X,Xa) = Y )
     => ( accp(product_prod(product_prod(list(A),list(A)),list(A)),merges7066485432131860899it_rel(A),aa(list(A),product_prod(product_prod(list(A),list(A)),list(A)),aa(product_prod(list(A),list(A)),fun(list(A),product_prod(product_prod(list(A),list(A)),list(A))),product_Pair(product_prod(list(A),list(A)),list(A)),X),Xa))
       => ( ! [Xs12: list(A),Xs22: list(A)] :
              ( ( X = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs12),Xs22) )
             => ( ( Xa = nil(A) )
               => ( ( Y = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs12),Xs22) )
                 => ~ accp(product_prod(product_prod(list(A),list(A)),list(A)),merges7066485432131860899it_rel(A),aa(list(A),product_prod(product_prod(list(A),list(A)),list(A)),aa(product_prod(list(A),list(A)),fun(list(A),product_prod(product_prod(list(A),list(A)),list(A))),product_Pair(product_prod(list(A),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)),Xs12),Xs22)),nil(A))) ) ) )
         => ( ! [Xs12: list(A),Xs22: list(A)] :
                ( ( X = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs12),Xs22) )
               => ! [X2: A] :
                    ( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),nil(A)) )
                   => ( ( Y = aa(list(A),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),X2),Xs12)),Xs22) )
                     => ~ accp(product_prod(product_prod(list(A),list(A)),list(A)),merges7066485432131860899it_rel(A),aa(list(A),product_prod(product_prod(list(A),list(A)),list(A)),aa(product_prod(list(A),list(A)),fun(list(A),product_prod(product_prod(list(A),list(A)),list(A))),product_Pair(product_prod(list(A),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)),Xs12),Xs22)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),nil(A)))) ) ) )
           => ~ ! [Xs12: list(A),Xs22: list(A)] :
                  ( ( X = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs12),Xs22) )
                 => ! [X12: A,X23: A,Xs3: list(A)] :
                      ( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X12),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X23),Xs3)) )
                     => ( ( Y = merges295452479951948502_split(A,aa(list(A),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),X12),Xs12)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X23),Xs22)),Xs3) )
                       => ~ accp(product_prod(product_prod(list(A),list(A)),list(A)),merges7066485432131860899it_rel(A),aa(list(A),product_prod(product_prod(list(A),list(A)),list(A)),aa(product_prod(list(A),list(A)),fun(list(A),product_prod(product_prod(list(A),list(A)),list(A))),product_Pair(product_prod(list(A),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)),Xs12),Xs22)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X12),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X23),Xs3)))) ) ) ) ) ) ) ) ).

% mergesort_by_rel_split.pelims
tff(fact_7129_Atom__def,axiom,
    ! [B: $tType,A: $tType,X3: sum_sum(A,nat)] : aa(sum_sum(A,nat),set(old_node(A,B)),old_Atom(A,B),X3) = aa(set(old_node(A,B)),set(old_node(A,B)),aa(old_node(A,B),fun(set(old_node(A,B)),set(old_node(A,B))),insert2(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_auf(nat,sum_sum(B,nat))),X3))),bot_bot(set(old_node(A,B)))) ).

% Atom_def
tff(fact_7130_Push__Node__def,axiom,
    ! [A: $tType,B: $tType,X3: sum_sum(A,nat),Xa3: old_node(B,A)] : aa(old_node(B,A),old_node(B,A),old_Push_Node(A,B,X3),Xa3) = old_Abs_Node(A,B,aa(product_prod(fun(nat,sum_sum(A,nat)),sum_sum(B,nat)),product_prod(fun(nat,sum_sum(A,nat)),sum_sum(B,nat)),product_apfst(fun(nat,sum_sum(A,nat)),fun(nat,sum_sum(A,nat)),sum_sum(B,nat),aa(sum_sum(A,nat),fun(fun(nat,sum_sum(A,nat)),fun(nat,sum_sum(A,nat))),old_Push(A),X3)),old_Rep_Node(B,A,Xa3))) ).

% Push_Node_def
tff(fact_7131_inj__Atom,axiom,
    ! [B: $tType,A: $tType] : inj_on(sum_sum(A,nat),set(old_node(A,B)),old_Atom(A,B),top_top(set(sum_sum(A,nat)))) ).

% inj_Atom
tff(fact_7132_Lim__def,axiom,
    ! [A: $tType,B: $tType,F: fun(B,set(old_node(A,B)))] : old_Lim(B,A,F) = aa(set(set(old_node(A,B))),set(old_node(A,B)),complete_Sup_Sup(set(old_node(A,B))),aa(fun(set(old_node(A,B)),$o),set(set(old_node(A,B))),collect(set(old_node(A,B))),aTP_Lamp_aug(fun(B,set(old_node(A,B))),fun(set(old_node(A,B)),$o),F))) ).

% Lim_def
tff(fact_7133_Scons__def,axiom,
    ! [B: $tType,A: $tType,M4: set(old_node(A,B)),N4: set(old_node(A,B))] : old_Scons(A,B,M4,N4) = 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))),aa(set(old_node(A,B)),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)))),M4)),aa(set(old_node(A,B)),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),aa(nat,nat,suc,one_one(nat))))),N4)) ).

% Scons_def
tff(fact_7134_Scons__UN1__y,axiom,
    ! [A: $tType,B: $tType,C: $tType,M4: set(old_node(A,B)),F: fun(C,set(old_node(A,B)))] : old_Scons(A,B,M4,aa(set(set(old_node(A,B))),set(old_node(A,B)),complete_Sup_Sup(set(old_node(A,B))),aa(set(C),set(set(old_node(A,B))),image2(C,set(old_node(A,B)),F),top_top(set(C))))) = aa(set(set(old_node(A,B))),set(old_node(A,B)),complete_Sup_Sup(set(old_node(A,B))),aa(set(C),set(set(old_node(A,B))),image2(C,set(old_node(A,B)),aa(fun(C,set(old_node(A,B))),fun(C,set(old_node(A,B))),aTP_Lamp_auh(set(old_node(A,B)),fun(fun(C,set(old_node(A,B))),fun(C,set(old_node(A,B)))),M4),F)),top_top(set(C)))) ).

% Scons_UN1_y
tff(fact_7135_Scons__UN1__x,axiom,
    ! [A: $tType,B: $tType,C: $tType,F: fun(C,set(old_node(A,B))),M4: set(old_node(A,B))] : old_Scons(A,B,aa(set(set(old_node(A,B))),set(old_node(A,B)),complete_Sup_Sup(set(old_node(A,B))),aa(set(C),set(set(old_node(A,B))),image2(C,set(old_node(A,B)),F),top_top(set(C)))),M4) = aa(set(set(old_node(A,B))),set(old_node(A,B)),complete_Sup_Sup(set(old_node(A,B))),aa(set(C),set(set(old_node(A,B))),image2(C,set(old_node(A,B)),aa(set(old_node(A,B)),fun(C,set(old_node(A,B))),aTP_Lamp_aui(fun(C,set(old_node(A,B))),fun(set(old_node(A,B)),fun(C,set(old_node(A,B)))),F),M4)),top_top(set(C)))) ).

% Scons_UN1_x
tff(fact_7136_Split__def,axiom,
    ! [A: $tType,C: $tType,B: $tType,C2: fun(set(old_node(B,C)),fun(set(old_node(B,C)),A)),M4: set(old_node(B,C))] : aa(set(old_node(B,C)),A,old_Split(B,C,A,C2),M4) = the(A,aa(set(old_node(B,C)),fun(A,$o),aTP_Lamp_auj(fun(set(old_node(B,C)),fun(set(old_node(B,C)),A)),fun(set(old_node(B,C)),fun(A,$o)),C2),M4)) ).

% Split_def
tff(fact_7137_inj__Leaf,axiom,
    ! [B: $tType,A: $tType] : inj_on(A,set(old_node(A,B)),old_Leaf(A,B),top_top(set(A))) ).

% inj_Leaf
tff(fact_7138_ntrunc__UN1,axiom,
    ! [A: $tType,B: $tType,C: $tType,K: nat,F: fun(C,set(old_node(A,B)))] : old_ntrunc(A,B,K,aa(set(set(old_node(A,B))),set(old_node(A,B)),complete_Sup_Sup(set(old_node(A,B))),aa(set(C),set(set(old_node(A,B))),image2(C,set(old_node(A,B)),F),top_top(set(C))))) = aa(set(set(old_node(A,B))),set(old_node(A,B)),complete_Sup_Sup(set(old_node(A,B))),aa(set(C),set(set(old_node(A,B))),image2(C,set(old_node(A,B)),aa(fun(C,set(old_node(A,B))),fun(C,set(old_node(A,B))),aTP_Lamp_auk(nat,fun(fun(C,set(old_node(A,B))),fun(C,set(old_node(A,B)))),K),F)),top_top(set(C)))) ).

% ntrunc_UN1
tff(fact_7139_In1__UN1,axiom,
    ! [A: $tType,B: $tType,C: $tType,F: fun(C,set(old_node(A,B)))] : aa(set(old_node(A,B)),set(old_node(A,B)),old_In1(A,B),aa(set(set(old_node(A,B))),set(old_node(A,B)),complete_Sup_Sup(set(old_node(A,B))),aa(set(C),set(set(old_node(A,B))),image2(C,set(old_node(A,B)),F),top_top(set(C))))) = aa(set(set(old_node(A,B))),set(old_node(A,B)),complete_Sup_Sup(set(old_node(A,B))),aa(set(C),set(set(old_node(A,B))),image2(C,set(old_node(A,B)),aTP_Lamp_aul(fun(C,set(old_node(A,B))),fun(C,set(old_node(A,B))),F)),top_top(set(C)))) ).

% In1_UN1
tff(fact_7140_ntrunc__0,axiom,
    ! [B: $tType,A: $tType,M4: set(old_node(A,B))] : old_ntrunc(A,B,zero_zero(nat),M4) = bot_bot(set(old_node(A,B))) ).

% ntrunc_0
tff(fact_7141_ntrunc__one__In1,axiom,
    ! [B: $tType,A: $tType,M4: set(old_node(A,B))] : old_ntrunc(A,B,aa(nat,nat,suc,zero_zero(nat)),aa(set(old_node(A,B)),set(old_node(A,B)),old_In1(A,B),M4)) = bot_bot(set(old_node(A,B))) ).

% ntrunc_one_In1
tff(fact_7142_inj__In1,axiom,
    ! [B: $tType,A: $tType] : inj_on(set(old_node(A,B)),set(old_node(A,B)),old_In1(A,B),top_top(set(set(old_node(A,B))))) ).

% inj_In1
tff(fact_7143_ntrunc__def,axiom,
    ! [B: $tType,A: $tType,K: nat,N4: set(old_node(A,B))] : old_ntrunc(A,B,K,N4) = aa(fun(old_node(A,B),$o),set(old_node(A,B)),collect(old_node(A,B)),aa(set(old_node(A,B)),fun(old_node(A,B),$o),aTP_Lamp_aum(nat,fun(set(old_node(A,B)),fun(old_node(A,B),$o)),K),N4)) ).

% ntrunc_def
tff(fact_7144_ntrunc__one__In0,axiom,
    ! [B: $tType,A: $tType,M4: set(old_node(A,B))] : old_ntrunc(A,B,aa(nat,nat,suc,zero_zero(nat)),aa(set(old_node(A,B)),set(old_node(A,B)),old_In0(A,B),M4)) = bot_bot(set(old_node(A,B))) ).

% ntrunc_one_In0
tff(fact_7145_In1__def,axiom,
    ! [B: $tType,A: $tType,M4: set(old_node(A,B))] : aa(set(old_node(A,B)),set(old_node(A,B)),old_In1(A,B),M4) = old_Scons(A,B,aa(nat,set(old_node(A,B)),old_Numb(A,B),one_one(nat)),M4) ).

% In1_def
tff(fact_7146_inj__Numb,axiom,
    ! [A: $tType,B: $tType] : inj_on(nat,set(old_node(A,B)),old_Numb(A,B),top_top(set(nat))) ).

% inj_Numb
tff(fact_7147_inj__In0,axiom,
    ! [B: $tType,A: $tType] : inj_on(set(old_node(A,B)),set(old_node(A,B)),old_In0(A,B),top_top(set(set(old_node(A,B))))) ).

% inj_In0
tff(fact_7148_In0__UN1,axiom,
    ! [A: $tType,B: $tType,C: $tType,F: fun(C,set(old_node(A,B)))] : aa(set(old_node(A,B)),set(old_node(A,B)),old_In0(A,B),aa(set(set(old_node(A,B))),set(old_node(A,B)),complete_Sup_Sup(set(old_node(A,B))),aa(set(C),set(set(old_node(A,B))),image2(C,set(old_node(A,B)),F),top_top(set(C))))) = aa(set(set(old_node(A,B))),set(old_node(A,B)),complete_Sup_Sup(set(old_node(A,B))),aa(set(C),set(set(old_node(A,B))),image2(C,set(old_node(A,B)),aTP_Lamp_aun(fun(C,set(old_node(A,B))),fun(C,set(old_node(A,B))),F)),top_top(set(C)))) ).

% In0_UN1
tff(fact_7149_Case__def,axiom,
    ! [A: $tType,C: $tType,B: $tType,C2: fun(set(old_node(B,C)),A),D3: fun(set(old_node(B,C)),A),M4: set(old_node(B,C))] : old_Case(B,C,A,C2,D3,M4) = the(A,aa(set(old_node(B,C)),fun(A,$o),aa(fun(set(old_node(B,C)),A),fun(set(old_node(B,C)),fun(A,$o)),aTP_Lamp_auo(fun(set(old_node(B,C)),A),fun(fun(set(old_node(B,C)),A),fun(set(old_node(B,C)),fun(A,$o))),C2),D3),M4)) ).

% Case_def
tff(fact_7150_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_aur(nat,set(old_node(A,product_unit))),aTP_Lamp_auv(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_7151_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_aur(nat,set(old_node(A,product_unit))),aTP_Lamp_auv(nat,set(old_node(A,product_unit)))),aa(nat,sum_sum(nat,nat),nat_sum_decode,N)) ) ).

% nth_item.simps(2)
tff(fact_7152_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_aur(nat,set(old_node(A,product_unit))),aTP_Lamp_auv(nat,set(old_node(A,product_unit)))),aa(nat,sum_sum(nat,nat),nat_sum_decode,N3)) ) ) ) ) ) ).

% nth_item.elims
tff(fact_7153_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_aur(nat,set(old_node(A,product_unit))),aTP_Lamp_auv(nat,set(old_node(A,product_unit)))),aa(nat,sum_sum(nat,nat),nat_sum_decode,N)) ) ) ) ).

% nth_item.psimps(2)
tff(fact_7154_dsum__def,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),S2: set(product_prod(set(old_node(A,B)),set(old_node(A,B))))] : old_dsum(A,B,R2,S2) = aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),fun(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),sup_sup(set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),aa(set(set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),complete_Sup_Sup(set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),set(set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),image2(product_prod(set(old_node(A,B)),set(old_node(A,B))),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),aa(fun(set(old_node(A,B)),fun(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))))),fun(product_prod(set(old_node(A,B)),set(old_node(A,B))),set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),product_case_prod(set(old_node(A,B)),set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),aTP_Lamp_auw(set(old_node(A,B)),fun(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))))))),R2))),aa(set(set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),complete_Sup_Sup(set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),set(set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),image2(product_prod(set(old_node(A,B)),set(old_node(A,B))),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),aa(fun(set(old_node(A,B)),fun(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))))),fun(product_prod(set(old_node(A,B)),set(old_node(A,B))),set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),product_case_prod(set(old_node(A,B)),set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),aTP_Lamp_aux(set(old_node(A,B)),fun(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))))))),S2))) ).

% dsum_def
tff(fact_7155_dsumE,axiom,
    ! [B: $tType,A: $tType,W2: product_prod(set(old_node(A,B)),set(old_node(A,B))),R2: set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),S2: set(product_prod(set(old_node(A,B)),set(old_node(A,B))))] :
      ( aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),$o,member(product_prod(set(old_node(A,B)),set(old_node(A,B))),W2),old_dsum(A,B,R2,S2))
     => ( ! [X2: set(old_node(A,B)),X9: set(old_node(A,B))] :
            ( aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),$o,member(product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),fun(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B)))),product_Pair(set(old_node(A,B)),set(old_node(A,B))),X2),X9)),R2)
           => ( W2 != aa(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),fun(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B)))),product_Pair(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),set(old_node(A,B)),old_In0(A,B),X2)),aa(set(old_node(A,B)),set(old_node(A,B)),old_In0(A,B),X9)) ) )
       => ~ ! [Y2: set(old_node(A,B)),Y7: set(old_node(A,B))] :
              ( aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),$o,member(product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),fun(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B)))),product_Pair(set(old_node(A,B)),set(old_node(A,B))),Y2),Y7)),S2)
             => ( W2 != aa(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),fun(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B)))),product_Pair(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),set(old_node(A,B)),old_In1(A,B),Y2)),aa(set(old_node(A,B)),set(old_node(A,B)),old_In1(A,B),Y7)) ) ) ) ) ).

% dsumE
tff(fact_7156_dsum__In0I,axiom,
    ! [B: $tType,A: $tType,M4: set(old_node(A,B)),M8: set(old_node(A,B)),R2: set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),S2: set(product_prod(set(old_node(A,B)),set(old_node(A,B))))] :
      ( aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),$o,member(product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),fun(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B)))),product_Pair(set(old_node(A,B)),set(old_node(A,B))),M4),M8)),R2)
     => aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),$o,member(product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),fun(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B)))),product_Pair(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),set(old_node(A,B)),old_In0(A,B),M4)),aa(set(old_node(A,B)),set(old_node(A,B)),old_In0(A,B),M8))),old_dsum(A,B,R2,S2)) ) ).

% dsum_In0I
tff(fact_7157_dsum__In1I,axiom,
    ! [B: $tType,A: $tType,N4: set(old_node(A,B)),N10: set(old_node(A,B)),S2: set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),R2: set(product_prod(set(old_node(A,B)),set(old_node(A,B))))] :
      ( aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),$o,member(product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),fun(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B)))),product_Pair(set(old_node(A,B)),set(old_node(A,B))),N4),N10)),S2)
     => aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),$o,member(product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),fun(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B)))),product_Pair(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),set(old_node(A,B)),old_In1(A,B),N4)),aa(set(old_node(A,B)),set(old_node(A,B)),old_In1(A,B),N10))),old_dsum(A,B,R2,S2)) ) ).

% dsum_In1I
tff(fact_7158_dprod__def,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),S2: set(product_prod(set(old_node(A,B)),set(old_node(A,B))))] : old_dprod(A,B,R2,S2) = aa(set(set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),complete_Sup_Sup(set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),set(set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),image2(product_prod(set(old_node(A,B)),set(old_node(A,B))),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),aa(fun(set(old_node(A,B)),fun(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))))),fun(product_prod(set(old_node(A,B)),set(old_node(A,B))),set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),product_case_prod(set(old_node(A,B)),set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),aTP_Lamp_auz(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),fun(set(old_node(A,B)),fun(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))))),S2))),R2)) ).

% dprod_def
tff(fact_7159_dprodI,axiom,
    ! [B: $tType,A: $tType,M4: set(old_node(A,B)),M8: set(old_node(A,B)),R2: set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),N4: set(old_node(A,B)),N10: set(old_node(A,B)),S2: set(product_prod(set(old_node(A,B)),set(old_node(A,B))))] :
      ( aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),$o,member(product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),fun(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B)))),product_Pair(set(old_node(A,B)),set(old_node(A,B))),M4),M8)),R2)
     => ( aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),$o,member(product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),fun(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B)))),product_Pair(set(old_node(A,B)),set(old_node(A,B))),N4),N10)),S2)
       => aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),$o,member(product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),fun(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B)))),product_Pair(set(old_node(A,B)),set(old_node(A,B))),old_Scons(A,B,M4,N4)),old_Scons(A,B,M8,N10))),old_dprod(A,B,R2,S2)) ) ) ).

% dprodI
tff(fact_7160_dprodE,axiom,
    ! [B: $tType,A: $tType,C2: product_prod(set(old_node(A,B)),set(old_node(A,B))),R2: set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),S2: set(product_prod(set(old_node(A,B)),set(old_node(A,B))))] :
      ( aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),$o,member(product_prod(set(old_node(A,B)),set(old_node(A,B))),C2),old_dprod(A,B,R2,S2))
     => ~ ! [X2: set(old_node(A,B)),Y2: set(old_node(A,B)),X9: set(old_node(A,B))] :
            ( aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),$o,member(product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),fun(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B)))),product_Pair(set(old_node(A,B)),set(old_node(A,B))),X2),X9)),R2)
           => ! [Y7: set(old_node(A,B))] :
                ( aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),$o,member(product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),fun(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B)))),product_Pair(set(old_node(A,B)),set(old_node(A,B))),Y2),Y7)),S2)
               => ( C2 != aa(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),fun(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B)))),product_Pair(set(old_node(A,B)),set(old_node(A,B))),old_Scons(A,B,X2,Y2)),old_Scons(A,B,X9,Y7)) ) ) ) ) ).

% dprodE
tff(fact_7161_usum__def,axiom,
    ! [B: $tType,A: $tType,A4: set(set(old_node(A,B))),B3: set(set(old_node(A,B)))] : old_usum(A,B,A4,B3) = aa(set(set(old_node(A,B))),set(set(old_node(A,B))),aa(set(set(old_node(A,B))),fun(set(set(old_node(A,B))),set(set(old_node(A,B)))),sup_sup(set(set(old_node(A,B)))),aa(set(set(old_node(A,B))),set(set(old_node(A,B))),image2(set(old_node(A,B)),set(old_node(A,B)),old_In0(A,B)),A4)),aa(set(set(old_node(A,B))),set(set(old_node(A,B))),image2(set(old_node(A,B)),set(old_node(A,B)),old_In1(A,B)),B3)) ).

% usum_def
tff(fact_7162_MOST__INFM,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( ~ aa(set(A),$o,finite_finite2(A),top_top(set(A)))
     => ( eventually(A,P,cofinite(A))
       => frequently(A,P,cofinite(A)) ) ) ).

% MOST_INFM
tff(fact_7163_frequently__const,axiom,
    ! [A: $tType,F4: filter(A),P: $o] :
      ( ( F4 != bot_bot(filter(A)) )
     => ( frequently(A,aTP_Lamp_ah($o,fun(A,$o),(P)),F4)
      <=> (P) ) ) ).

% frequently_const
tff(fact_7164_not__MOST,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( ~ eventually(A,P,cofinite(A))
    <=> frequently(A,aTP_Lamp_aj(fun(A,$o),fun(A,$o),P),cofinite(A)) ) ).

% not_MOST
tff(fact_7165_not__INFM,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( ~ frequently(A,P,cofinite(A))
    <=> eventually(A,aTP_Lamp_aj(fun(A,$o),fun(A,$o),P),cofinite(A)) ) ).

% not_INFM
tff(fact_7166_INFM__neq_I2_J,axiom,
    ! [A: $tType,A3: A] :
      ( frequently(A,aa(A,fun(A,$o),aTP_Lamp_za(A,fun(A,$o)),A3),cofinite(A))
    <=> ~ aa(set(A),$o,finite_finite2(A),top_top(set(A))) ) ).

% INFM_neq(2)
tff(fact_7167_INFM__neq_I1_J,axiom,
    ! [A: $tType,A3: A] :
      ( frequently(A,aa(A,fun(A,$o),aTP_Lamp_ze(A,fun(A,$o)),A3),cofinite(A))
    <=> ~ aa(set(A),$o,finite_finite2(A),top_top(set(A))) ) ).

% INFM_neq(1)
tff(fact_7168_INFM__const,axiom,
    ! [A: $tType,P: $o] :
      ( frequently(A,aTP_Lamp_ah($o,fun(A,$o),(P)),cofinite(A))
    <=> ( (P)
        & ~ aa(set(A),$o,finite_finite2(A),top_top(set(A))) ) ) ).

% INFM_const
tff(fact_7169_INFM__nat,axiom,
    ! [P: fun(nat,$o)] :
      ( frequently(nat,P,cofinite(nat))
    <=> ! [M2: nat] :
        ? [N2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),N2)
          & aa(nat,$o,P,N2) ) ) ).

% INFM_nat
tff(fact_7170_INFM__nat__le,axiom,
    ! [P: fun(nat,$o)] :
      ( frequently(nat,P,cofinite(nat))
    <=> ! [M2: nat] :
        ? [N2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),N2)
          & aa(nat,$o,P,N2) ) ) ).

% INFM_nat_le
tff(fact_7171_frequently__cofinite,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( frequently(A,P,cofinite(A))
    <=> ~ aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),P)) ) ).

% frequently_cofinite
tff(fact_7172_INFM__iff__infinite,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( frequently(A,P,cofinite(A))
    <=> ~ aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),P)) ) ).

% INFM_iff_infinite
tff(fact_7173_not__INFM__eq_I2_J,axiom,
    ! [A: $tType,A3: A] : ~ frequently(A,aa(A,fun(A,$o),fequal(A),A3),cofinite(A)) ).

% not_INFM_eq(2)
tff(fact_7174_not__INFM__eq_I1_J,axiom,
    ! [A: $tType,A3: A] : ~ frequently(A,aTP_Lamp_cf(A,fun(A,$o),A3),cofinite(A)) ).

% not_INFM_eq(1)
tff(fact_7175_INFM__E,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( frequently(A,P,cofinite(A))
     => ~ ! [X2: A] : ~ aa(A,$o,P,X2) ) ).

% INFM_E
tff(fact_7176_INFM__EX,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( frequently(A,P,cofinite(A))
     => ? [X_1: A] : aa(A,$o,P,X_1) ) ).

% INFM_EX
tff(fact_7177_INFM__mono,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] :
      ( frequently(A,P,cofinite(A))
     => ( ! [X2: A] :
            ( aa(A,$o,P,X2)
           => aa(A,$o,Q,X2) )
       => frequently(A,Q,cofinite(A)) ) ) ).

% INFM_mono
tff(fact_7178_INFM__disj__distrib,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] :
      ( frequently(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ap(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q),cofinite(A))
    <=> ( frequently(A,P,cofinite(A))
        | frequently(A,Q,cofinite(A)) ) ) ).

% INFM_disj_distrib
tff(fact_7179_INFM__imp__distrib,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] :
      ( frequently(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ao(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q),cofinite(A))
    <=> ( eventually(A,P,cofinite(A))
       => frequently(A,Q,cofinite(A)) ) ) ).

% INFM_imp_distrib
tff(fact_7180_Alm__all__def,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( eventually(A,P,cofinite(A))
    <=> ~ frequently(A,aTP_Lamp_aj(fun(A,$o),fun(A,$o),P),cofinite(A)) ) ).

% Alm_all_def
tff(fact_7181_INFM__conjI,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] :
      ( frequently(A,P,cofinite(A))
     => ( eventually(A,Q,cofinite(A))
       => frequently(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_am(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q),cofinite(A)) ) ) ).

% INFM_conjI
tff(fact_7182_frequently__bex__finite,axiom,
    ! [A: $tType,B: $tType,A4: set(A),P: fun(B,fun(A,$o)),F4: filter(B)] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( frequently(B,aa(fun(B,fun(A,$o)),fun(B,$o),aTP_Lamp_aht(set(A),fun(fun(B,fun(A,$o)),fun(B,$o)),A4),P),F4)
       => ? [X2: A] :
            ( aa(set(A),$o,member(A,X2),A4)
            & frequently(B,aa(A,fun(B,$o),aTP_Lamp_aei(fun(B,fun(A,$o)),fun(A,fun(B,$o)),P),X2),F4) ) ) ) ).

% frequently_bex_finite
tff(fact_7183_frequently__bex__finite__distrib,axiom,
    ! [A: $tType,B: $tType,A4: set(A),P: fun(B,fun(A,$o)),F4: filter(B)] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( frequently(B,aa(fun(B,fun(A,$o)),fun(B,$o),aTP_Lamp_aht(set(A),fun(fun(B,fun(A,$o)),fun(B,$o)),A4),P),F4)
      <=> ? [X4: A] :
            ( aa(set(A),$o,member(A,X4),A4)
            & frequently(B,aa(A,fun(B,$o),aTP_Lamp_aei(fun(B,fun(A,$o)),fun(A,fun(B,$o)),P),X4),F4) ) ) ) ).

% frequently_bex_finite_distrib
tff(fact_7184_eventually__frequently__const__simps_I2_J,axiom,
    ! [A: $tType,C3: $o,P: fun(A,$o),F4: filter(A)] :
      ( frequently(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ava($o,fun(fun(A,$o),fun(A,$o)),(C3)),P),F4)
    <=> ( (C3)
        & frequently(A,P,F4) ) ) ).

% eventually_frequently_const_simps(2)
tff(fact_7185_eventually__frequently__const__simps_I1_J,axiom,
    ! [A: $tType,P: fun(A,$o),C3: $o,F4: filter(A)] :
      ( frequently(A,aa($o,fun(A,$o),aTP_Lamp_avb(fun(A,$o),fun($o,fun(A,$o)),P),(C3)),F4)
    <=> ( frequently(A,P,F4)
        & (C3) ) ) ).

% eventually_frequently_const_simps(1)
tff(fact_7186_frequently__ex,axiom,
    ! [A: $tType,P: fun(A,$o),F4: filter(A)] :
      ( frequently(A,P,F4)
     => ? [X_1: A] : aa(A,$o,P,X_1) ) ).

% frequently_ex
tff(fact_7187_frequently__disj,axiom,
    ! [A: $tType,P: fun(A,$o),F4: filter(A),Q: fun(A,$o)] :
      ( frequently(A,P,F4)
     => ( frequently(A,Q,F4)
       => frequently(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ap(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q),F4) ) ) ).

% frequently_disj
tff(fact_7188_frequently__elim1,axiom,
    ! [A: $tType,P: fun(A,$o),F4: filter(A),Q: fun(A,$o)] :
      ( frequently(A,P,F4)
     => ( ! [I2: A] :
            ( aa(A,$o,P,I2)
           => aa(A,$o,Q,I2) )
       => frequently(A,Q,F4) ) ) ).

% frequently_elim1
tff(fact_7189_frequently__disj__iff,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o),F4: filter(A)] :
      ( frequently(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ap(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q),F4)
    <=> ( frequently(A,P,F4)
        | frequently(A,Q,F4) ) ) ).

% frequently_disj_iff
tff(fact_7190_not__frequently__False,axiom,
    ! [A: $tType,F4: filter(A)] : ~ frequently(A,aTP_Lamp_ak(A,$o),F4) ).

% not_frequently_False
tff(fact_7191_frequently__imp__iff,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o),F4: filter(A)] :
      ( frequently(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ao(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q),F4)
    <=> ( eventually(A,P,F4)
       => frequently(A,Q,F4) ) ) ).

% frequently_imp_iff
tff(fact_7192_frequently__rev__mp,axiom,
    ! [A: $tType,P: fun(A,$o),F4: filter(A),Q: fun(A,$o)] :
      ( frequently(A,P,F4)
     => ( eventually(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ao(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q),F4)
       => frequently(A,Q,F4) ) ) ).

% frequently_rev_mp
tff(fact_7193_not__frequently,axiom,
    ! [A: $tType,P: fun(A,$o),F4: filter(A)] :
      ( ~ frequently(A,P,F4)
    <=> eventually(A,aTP_Lamp_aj(fun(A,$o),fun(A,$o),P),F4) ) ).

% not_frequently
tff(fact_7194_not__eventually,axiom,
    ! [A: $tType,P: fun(A,$o),F4: filter(A)] :
      ( ~ eventually(A,P,F4)
    <=> frequently(A,aTP_Lamp_aj(fun(A,$o),fun(A,$o),P),F4) ) ).

% not_eventually
tff(fact_7195_frequently__def,axiom,
    ! [A: $tType,P: fun(A,$o),F4: filter(A)] :
      ( frequently(A,P,F4)
    <=> ~ eventually(A,aTP_Lamp_aj(fun(A,$o),fun(A,$o),P),F4) ) ).

% frequently_def
tff(fact_7196_frequently__mp,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o),F4: filter(A)] :
      ( eventually(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ao(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q),F4)
     => ( frequently(A,P,F4)
       => frequently(A,Q,F4) ) ) ).

% frequently_mp
tff(fact_7197_eventually__frequently__const__simps_I5_J,axiom,
    ! [A: $tType,P: fun(A,$o),C3: $o,F4: filter(A)] :
      ( eventually(A,aa($o,fun(A,$o),aTP_Lamp_avc(fun(A,$o),fun($o,fun(A,$o)),P),(C3)),F4)
    <=> ( frequently(A,P,F4)
       => (C3) ) ) ).

% eventually_frequently_const_simps(5)
tff(fact_7198_frequently__all,axiom,
    ! [B: $tType,A: $tType,P: fun(A,fun(B,$o)),F4: filter(A)] :
      ( frequently(A,aTP_Lamp_ain(fun(A,fun(B,$o)),fun(A,$o),P),F4)
    <=> ! [Y8: fun(A,B)] : frequently(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_aka(fun(A,fun(B,$o)),fun(fun(A,B),fun(A,$o)),P),Y8),F4) ) ).

% frequently_all
tff(fact_7199_eventually__frequentlyE,axiom,
    ! [A: $tType,P: fun(A,$o),F4: filter(A),Q: fun(A,$o)] :
      ( eventually(A,P,F4)
     => ( eventually(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_avd(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q),F4)
       => ( ( F4 != bot_bot(filter(A)) )
         => frequently(A,Q,F4) ) ) ) ).

% eventually_frequentlyE
tff(fact_7200_eventually__frequently,axiom,
    ! [A: $tType,F4: filter(A),P: fun(A,$o)] :
      ( ( F4 != bot_bot(filter(A)) )
     => ( eventually(A,P,F4)
       => frequently(A,P,F4) ) ) ).

% eventually_frequently
tff(fact_7201_frequently__const__iff,axiom,
    ! [A: $tType,P: $o,F4: filter(A)] :
      ( frequently(A,aTP_Lamp_ah($o,fun(A,$o),(P)),F4)
    <=> ( (P)
        & ( F4 != bot_bot(filter(A)) ) ) ) ).

% frequently_const_iff
tff(fact_7202_dsum__Sigma,axiom,
    ! [B: $tType,A: $tType,A4: set(set(old_node(A,B))),B3: set(set(old_node(A,B))),C3: set(set(old_node(A,B))),D4: set(set(old_node(A,B)))] : aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),$o,aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),fun(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),$o),ord_less_eq(set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),old_dsum(A,B,product_Sigma(set(old_node(A,B)),set(old_node(A,B)),A4,aTP_Lamp_ave(set(set(old_node(A,B))),fun(set(old_node(A,B)),set(set(old_node(A,B)))),B3)),product_Sigma(set(old_node(A,B)),set(old_node(A,B)),C3,aTP_Lamp_ave(set(set(old_node(A,B))),fun(set(old_node(A,B)),set(set(old_node(A,B)))),D4)))),product_Sigma(set(old_node(A,B)),set(old_node(A,B)),old_usum(A,B,A4,C3),aa(set(set(old_node(A,B))),fun(set(old_node(A,B)),set(set(old_node(A,B)))),aTP_Lamp_avf(set(set(old_node(A,B))),fun(set(set(old_node(A,B))),fun(set(old_node(A,B)),set(set(old_node(A,B))))),B3),D4))) ).

% dsum_Sigma
tff(fact_7203_dsum__subset__Sigma,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),A4: set(set(old_node(A,B))),B3: set(set(old_node(A,B))),S2: set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),C3: set(set(old_node(A,B))),D4: set(set(old_node(A,B)))] :
      ( aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),$o,aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),fun(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),$o),ord_less_eq(set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),R2),product_Sigma(set(old_node(A,B)),set(old_node(A,B)),A4,aTP_Lamp_ave(set(set(old_node(A,B))),fun(set(old_node(A,B)),set(set(old_node(A,B)))),B3)))
     => ( aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),$o,aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),fun(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),$o),ord_less_eq(set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),S2),product_Sigma(set(old_node(A,B)),set(old_node(A,B)),C3,aTP_Lamp_ave(set(set(old_node(A,B))),fun(set(old_node(A,B)),set(set(old_node(A,B)))),D4)))
       => aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),$o,aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),fun(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),$o),ord_less_eq(set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),old_dsum(A,B,R2,S2)),product_Sigma(set(old_node(A,B)),set(old_node(A,B)),old_usum(A,B,A4,C3),aa(set(set(old_node(A,B))),fun(set(old_node(A,B)),set(set(old_node(A,B)))),aTP_Lamp_avf(set(set(old_node(A,B))),fun(set(set(old_node(A,B))),fun(set(old_node(A,B)),set(set(old_node(A,B))))),B3),D4))) ) ) ).

% dsum_subset_Sigma
tff(fact_7204_INFM__inj,axiom,
    ! [A: $tType,B: $tType,P: fun(B,$o),F: fun(A,B)] :
      ( frequently(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_avg(fun(B,$o),fun(fun(A,B),fun(A,$o)),P),F),cofinite(A))
     => ( inj_on(A,B,F,top_top(set(A)))
       => frequently(B,P,cofinite(B)) ) ) ).

% INFM_inj
tff(fact_7205_INFM__finite__Bex__distrib,axiom,
    ! [A: $tType,B: $tType,A4: set(A),P: fun(A,fun(B,$o))] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( frequently(B,aa(fun(A,fun(B,$o)),fun(B,$o),aTP_Lamp_avh(set(A),fun(fun(A,fun(B,$o)),fun(B,$o)),A4),P),cofinite(B))
      <=> ? [X4: A] :
            ( aa(set(A),$o,member(A,X4),A4)
            & frequently(B,aa(A,fun(B,$o),P,X4),cofinite(B)) ) ) ) ).

% INFM_finite_Bex_distrib
tff(fact_7206_INFM__nat__inductI,axiom,
    ! [P: fun(nat,$o),Q: fun(nat,$o)] :
      ( aa(nat,$o,P,zero_zero(nat))
     => ( ! [I2: nat] :
            ( aa(nat,$o,P,I2)
           => ? [J6: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),J6)
                & aa(nat,$o,P,J6)
                & aa(nat,$o,Q,J6) ) )
       => frequently(nat,Q,cofinite(nat)) ) ) ).

% INFM_nat_inductI
tff(fact_7207_integer__of__nat__1,axiom,
    code_integer_of_nat(one_one(nat)) = one_one(code_integer) ).

% integer_of_nat_1
tff(fact_7208_irreflp__irrefl__eq,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( irreflp(A,aTP_Lamp_ux(set(product_prod(A,A)),fun(A,fun(A,$o)),R))
    <=> irrefl(A,R) ) ).

% irreflp_irrefl_eq
tff(fact_7209_subset__mset_Oirreflp__greater,axiom,
    ! [A: $tType] : irreflp(multiset(A),aTP_Lamp_asp(multiset(A),fun(multiset(A),$o))) ).

% subset_mset.irreflp_greater
tff(fact_7210_irreflp__greater,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => irreflp(A,aTP_Lamp_sz(A,fun(A,$o))) ) ).

% irreflp_greater
tff(fact_7211_dprod__subset__Sigma2,axiom,
    ! [B: $tType,A: $tType,A4: set(set(old_node(A,B))),B3: fun(set(old_node(A,B)),set(set(old_node(A,B)))),C3: set(set(old_node(A,B))),D4: fun(set(old_node(A,B)),set(set(old_node(A,B))))] : aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),$o,aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),fun(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),$o),ord_less_eq(set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),old_dprod(A,B,product_Sigma(set(old_node(A,B)),set(old_node(A,B)),A4,B3),product_Sigma(set(old_node(A,B)),set(old_node(A,B)),C3,D4))),product_Sigma(set(old_node(A,B)),set(old_node(A,B)),old_uprod(A,B,A4,C3),old_Split(A,B,set(set(old_node(A,B))),aa(fun(set(old_node(A,B)),set(set(old_node(A,B)))),fun(set(old_node(A,B)),fun(set(old_node(A,B)),set(set(old_node(A,B))))),aTP_Lamp_avi(fun(set(old_node(A,B)),set(set(old_node(A,B)))),fun(fun(set(old_node(A,B)),set(set(old_node(A,B)))),fun(set(old_node(A,B)),fun(set(old_node(A,B)),set(set(old_node(A,B)))))),B3),D4)))) ).

% dprod_subset_Sigma2
tff(fact_7212_dprod__subset__Sigma,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),A4: set(set(old_node(A,B))),B3: set(set(old_node(A,B))),S2: set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),C3: set(set(old_node(A,B))),D4: set(set(old_node(A,B)))] :
      ( aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),$o,aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),fun(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),$o),ord_less_eq(set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),R2),product_Sigma(set(old_node(A,B)),set(old_node(A,B)),A4,aTP_Lamp_ave(set(set(old_node(A,B))),fun(set(old_node(A,B)),set(set(old_node(A,B)))),B3)))
     => ( aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),$o,aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),fun(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),$o),ord_less_eq(set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),S2),product_Sigma(set(old_node(A,B)),set(old_node(A,B)),C3,aTP_Lamp_ave(set(set(old_node(A,B))),fun(set(old_node(A,B)),set(set(old_node(A,B)))),D4)))
       => aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),$o,aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),fun(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),$o),ord_less_eq(set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),old_dprod(A,B,R2,S2)),product_Sigma(set(old_node(A,B)),set(old_node(A,B)),old_uprod(A,B,A4,C3),aa(set(set(old_node(A,B))),fun(set(old_node(A,B)),set(set(old_node(A,B)))),aTP_Lamp_avj(set(set(old_node(A,B))),fun(set(set(old_node(A,B))),fun(set(old_node(A,B)),set(set(old_node(A,B))))),B3),D4))) ) ) ).

% dprod_subset_Sigma
tff(fact_7213_uprod__def,axiom,
    ! [B: $tType,A: $tType,A4: set(set(old_node(A,B))),B3: set(set(old_node(A,B)))] : old_uprod(A,B,A4,B3) = aa(set(set(set(old_node(A,B)))),set(set(old_node(A,B))),complete_Sup_Sup(set(set(old_node(A,B)))),aa(set(set(old_node(A,B))),set(set(set(old_node(A,B)))),image2(set(old_node(A,B)),set(set(old_node(A,B))),aTP_Lamp_avl(set(set(old_node(A,B))),fun(set(old_node(A,B)),set(set(old_node(A,B)))),B3)),A4)) ).

% uprod_def
tff(fact_7214_dprod__Sigma,axiom,
    ! [B: $tType,A: $tType,A4: set(set(old_node(A,B))),B3: set(set(old_node(A,B))),C3: set(set(old_node(A,B))),D4: set(set(old_node(A,B)))] : aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),$o,aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),fun(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),$o),ord_less_eq(set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),old_dprod(A,B,product_Sigma(set(old_node(A,B)),set(old_node(A,B)),A4,aTP_Lamp_ave(set(set(old_node(A,B))),fun(set(old_node(A,B)),set(set(old_node(A,B)))),B3)),product_Sigma(set(old_node(A,B)),set(old_node(A,B)),C3,aTP_Lamp_ave(set(set(old_node(A,B))),fun(set(old_node(A,B)),set(set(old_node(A,B)))),D4)))),product_Sigma(set(old_node(A,B)),set(old_node(A,B)),old_uprod(A,B,A4,C3),aa(set(set(old_node(A,B))),fun(set(old_node(A,B)),set(set(old_node(A,B)))),aTP_Lamp_avj(set(set(old_node(A,B))),fun(set(set(old_node(A,B))),fun(set(old_node(A,B)),set(set(old_node(A,B))))),B3),D4))) ).

% dprod_Sigma
tff(fact_7215_irreflp__multp,axiom,
    ! [A: $tType,R2: fun(A,fun(A,$o))] :
      ( transp(A,R2)
     => ( irreflp(A,R2)
       => irreflp(multiset(A),multp(A,R2)) ) ) ).

% irreflp_multp
tff(fact_7216_multeqp__code__eq__reflclp__multp,axiom,
    ! [A: $tType,R2: fun(A,fun(A,$o))] :
      ( irreflp(A,R2)
     => ( transp(A,R2)
       => ( multeqp_code(A,R2) = aa(fun(multiset(A),fun(multiset(A),$o)),fun(multiset(A),fun(multiset(A),$o)),aa(fun(multiset(A),fun(multiset(A),$o)),fun(fun(multiset(A),fun(multiset(A),$o)),fun(multiset(A),fun(multiset(A),$o))),sup_sup(fun(multiset(A),fun(multiset(A),$o))),multp(A,R2)),fequal(multiset(A))) ) ) ) ).

% multeqp_code_eq_reflclp_multp
tff(fact_7217_transp__trans,axiom,
    ! [A: $tType,R2: fun(A,fun(A,$o))] :
      ( transp(A,R2)
    <=> trans(A,aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),R2))) ) ).

% transp_trans
tff(fact_7218_transp__trans__eq,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( transp(A,aTP_Lamp_ux(set(product_prod(A,A)),fun(A,fun(A,$o)),R2))
    <=> trans(A,R2) ) ).

% transp_trans_eq
tff(fact_7219_transp__inf,axiom,
    ! [A: $tType,R2: fun(A,fun(A,$o)),S2: fun(A,fun(A,$o))] :
      ( transp(A,R2)
     => ( transp(A,S2)
       => transp(A,aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(fun(A,fun(A,$o)),fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),inf_inf(fun(A,fun(A,$o))),R2),S2)) ) ) ).

% transp_inf
tff(fact_7220_subset__mset_Otransp__gr,axiom,
    ! [A: $tType] : transp(multiset(A),aTP_Lamp_asp(multiset(A),fun(multiset(A),$o))) ).

% subset_mset.transp_gr
tff(fact_7221_transp__singleton,axiom,
    ! [A: $tType,A3: A] : transp(A,aTP_Lamp_avm(A,fun(A,fun(A,$o)),A3)) ).

% transp_singleton
tff(fact_7222_transp__empty,axiom,
    ! [A: $tType] : transp(A,aTP_Lamp_arv(A,fun(A,$o))) ).

% transp_empty
tff(fact_7223_subset__mset_Otransp__ge,axiom,
    ! [A: $tType] : transp(multiset(A),aTP_Lamp_aso(multiset(A),fun(multiset(A),$o))) ).

% subset_mset.transp_ge
tff(fact_7224_transp__gr,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => transp(A,aTP_Lamp_sz(A,fun(A,$o))) ) ).

% transp_gr
tff(fact_7225_transp__ge,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => transp(A,aTP_Lamp_avn(A,fun(A,$o))) ) ).

% transp_ge
tff(fact_7226_multp__cancel__add__mset,axiom,
    ! [A: $tType,R2: fun(A,fun(A,$o)),Uu: A,X5: multiset(A),Y4: multiset(A)] :
      ( transp(A,R2)
     => ( irreflp(A,R2)
       => ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),multp(A,R2),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),Uu),X5)),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),Uu),Y4))
        <=> aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),multp(A,R2),X5),Y4) ) ) ) ).

% multp_cancel_add_mset
tff(fact_7227_multp__cancel,axiom,
    ! [A: $tType,R2: fun(A,fun(A,$o)),X5: multiset(A),Z5: multiset(A),Y4: multiset(A)] :
      ( transp(A,R2)
     => ( irreflp(A,R2)
       => ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),multp(A,R2),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),X5),Z5)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),Y4),Z5))
        <=> aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),multp(A,R2),X5),Y4) ) ) ) ).

% multp_cancel
tff(fact_7228_multp__cancel__max,axiom,
    ! [A: $tType,R2: fun(A,fun(A,$o)),X5: multiset(A),Y4: multiset(A)] :
      ( transp(A,R2)
     => ( irreflp(A,R2)
       => ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),multp(A,R2),X5),Y4)
        <=> aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),multp(A,R2),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),X5),Y4)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),Y4),X5)) ) ) ) ).

% multp_cancel_max
tff(fact_7229_equivp__equiv,axiom,
    ! [A: $tType,A4: set(product_prod(A,A))] :
      ( equiv_equiv(A,top_top(set(A)),A4)
    <=> equiv_equivp(A,aTP_Lamp_ux(set(product_prod(A,A)),fun(A,fun(A,$o)),A4)) ) ).

% equivp_equiv
tff(fact_7230_asymp__asym__eq,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( asymp(A,aTP_Lamp_ux(set(product_prod(A,A)),fun(A,fun(A,$o)),R))
    <=> asym(A,R) ) ).

% asymp_asym_eq
tff(fact_7231_Quotient__crel__quotient,axiom,
    ! [B: $tType,A: $tType,R: fun(A,fun(A,$o)),Abs: fun(A,B),Rep: fun(B,A),T2: fun(A,fun(B,$o))] :
      ( quotient(A,B,R,Abs,Rep,T2)
     => ( equiv_equivp(A,R)
       => ! [X3: A] : aa(A,fun(B,$o),T2,X3) = aa(B,fun(B,$o),fequal(B),aa(A,B,Abs,X3)) ) ) ).

% Quotient_crel_quotient
tff(fact_7232_UNIV__typedef__to__equivp,axiom,
    ! [A: $tType,B: $tType,Rep: fun(A,B),Abs: fun(B,A)] :
      ( type_definition(A,B,Rep,Abs,top_top(set(B)))
     => equiv_equivp(B,fequal(B)) ) ).

% UNIV_typedef_to_equivp
tff(fact_7233_asym__iff,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( asym(A,R)
    <=> ! [X4: A,Y3: A] :
          ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Y3)),R)
         => ~ aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),X4)),R) ) ) ).

% asym_iff
tff(fact_7234_asymD,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),X: A,Y: A] :
      ( asym(A,R)
     => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R)
       => ~ aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),X)),R) ) ) ).

% asymD
tff(fact_7235_asym_Ointros,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( ! [A6: A,B5: A] :
          ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A6),B5)),R)
         => ~ aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B5),A6)),R) )
     => asym(A,R) ) ).

% asym.intros
tff(fact_7236_asym_Osimps,axiom,
    ! [A: $tType,A3: set(product_prod(A,A))] :
      ( asym(A,A3)
    <=> ? [R7: set(product_prod(A,A))] :
          ( ( A3 = R7 )
          & ! [X4: A,Xa2: A] :
              ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Xa2)),R7)
             => ~ aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa2),X4)),R7) ) ) ) ).

% asym.simps
tff(fact_7237_asym_Ocases,axiom,
    ! [A: $tType,A3: set(product_prod(A,A))] :
      ( asym(A,A3)
     => ! [A11: A,B12: A] :
          ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A11),B12)),A3)
         => ~ aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B12),A11)),A3) ) ) ).

% asym.cases
tff(fact_7238_lexord__asymmetric,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A3: list(A),B2: list(A)] :
      ( asym(A,R)
     => ( aa(set(product_prod(list(A),list(A))),$o,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)),A3),B2)),lexord(A,R))
       => ~ aa(set(product_prod(list(A),list(A))),$o,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)),B2),A3)),lexord(A,R)) ) ) ).

% lexord_asymmetric
tff(fact_7239_Range__insert,axiom,
    ! [A: $tType,B: $tType,A3: B,B2: A,R2: set(product_prod(B,A))] : aa(set(product_prod(B,A)),set(A),range2(B,A),aa(set(product_prod(B,A)),set(product_prod(B,A)),aa(product_prod(B,A),fun(set(product_prod(B,A)),set(product_prod(B,A))),insert2(product_prod(B,A)),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),A3),B2)),R2)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),aa(set(product_prod(B,A)),set(A),range2(B,A),R2)) ).

% Range_insert
tff(fact_7240_Shift__def,axiom,
    ! [A: $tType,Kl: set(list(A)),K: A] : bNF_Greatest_Shift(A,Kl,K) = aa(fun(list(A),$o),set(list(A)),collect(list(A)),aa(A,fun(list(A),$o),aTP_Lamp_avo(set(list(A)),fun(A,fun(list(A),$o)),Kl),K)) ).

% Shift_def
tff(fact_7241_Range__rtrancl,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] : aa(set(product_prod(A,A)),set(A),range2(A,A),transitive_rtrancl(A,R)) = top_top(set(A)) ).

% Range_rtrancl
tff(fact_7242_Range__empty,axiom,
    ! [B: $tType,A: $tType] : aa(set(product_prod(B,A)),set(A),range2(B,A),bot_bot(set(product_prod(B,A)))) = bot_bot(set(A)) ).

% Range_empty
tff(fact_7243_Range__Id,axiom,
    ! [A: $tType] : aa(set(product_prod(A,A)),set(A),range2(A,A),id2(A)) = top_top(set(A)) ).

% Range_Id
tff(fact_7244_Range__Collect__case__prod,axiom,
    ! [A: $tType,B: $tType,P: fun(B,fun(A,$o))] : aa(set(product_prod(B,A)),set(A),range2(B,A),aa(fun(product_prod(B,A),$o),set(product_prod(B,A)),collect(product_prod(B,A)),aa(fun(B,fun(A,$o)),fun(product_prod(B,A),$o),product_case_prod(B,A,$o),P))) = aa(fun(A,$o),set(A),collect(A),aTP_Lamp_avp(fun(B,fun(A,$o)),fun(A,$o),P)) ).

% Range_Collect_case_prod
tff(fact_7245_Range_Ocases,axiom,
    ! [A: $tType,B: $tType,A3: A,R2: set(product_prod(B,A))] :
      ( aa(set(A),$o,member(A,A3),aa(set(product_prod(B,A)),set(A),range2(B,A),R2))
     => ~ ! [A6: B] : ~ aa(set(product_prod(B,A)),$o,member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),A6),A3)),R2) ) ).

% Range.cases
tff(fact_7246_Range_Osimps,axiom,
    ! [A: $tType,B: $tType,A3: A,R2: set(product_prod(B,A))] :
      ( aa(set(A),$o,member(A,A3),aa(set(product_prod(B,A)),set(A),range2(B,A),R2))
    <=> ? [A10: B,B6: A] :
          ( ( A3 = B6 )
          & aa(set(product_prod(B,A)),$o,member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),A10),B6)),R2) ) ) ).

% Range.simps
tff(fact_7247_Range_Ointros,axiom,
    ! [B: $tType,A: $tType,A3: A,B2: B,R2: set(product_prod(A,B))] :
      ( aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),R2)
     => aa(set(B),$o,member(B,B2),aa(set(product_prod(A,B)),set(B),range2(A,B),R2)) ) ).

% Range.intros
tff(fact_7248_RangeE,axiom,
    ! [A: $tType,B: $tType,B2: A,R2: set(product_prod(B,A))] :
      ( aa(set(A),$o,member(A,B2),aa(set(product_prod(B,A)),set(A),range2(B,A),R2))
     => ~ ! [A6: B] : ~ aa(set(product_prod(B,A)),$o,member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),A6),B2)),R2) ) ).

% RangeE
tff(fact_7249_Range__iff,axiom,
    ! [A: $tType,B: $tType,A3: A,R2: set(product_prod(B,A))] :
      ( aa(set(A),$o,member(A,A3),aa(set(product_prod(B,A)),set(A),range2(B,A),R2))
    <=> ? [Y3: B] : aa(set(product_prod(B,A)),$o,member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Y3),A3)),R2) ) ).

% Range_iff
tff(fact_7250_Range__empty__iff,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(B,A))] :
      ( ( aa(set(product_prod(B,A)),set(A),range2(B,A),R2) = bot_bot(set(A)) )
    <=> ( R2 = bot_bot(set(product_prod(B,A))) ) ) ).

% Range_empty_iff
tff(fact_7251_Range__Un__eq,axiom,
    ! [A: $tType,B: $tType,A4: set(product_prod(B,A)),B3: set(product_prod(B,A))] : aa(set(product_prod(B,A)),set(A),range2(B,A),aa(set(product_prod(B,A)),set(product_prod(B,A)),aa(set(product_prod(B,A)),fun(set(product_prod(B,A)),set(product_prod(B,A))),sup_sup(set(product_prod(B,A))),A4),B3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(product_prod(B,A)),set(A),range2(B,A),A4)),aa(set(product_prod(B,A)),set(A),range2(B,A),B3)) ).

% Range_Un_eq
tff(fact_7252_Range__Union,axiom,
    ! [A: $tType,B: $tType,S: set(set(product_prod(B,A)))] : aa(set(product_prod(B,A)),set(A),range2(B,A),aa(set(set(product_prod(B,A))),set(product_prod(B,A)),complete_Sup_Sup(set(product_prod(B,A))),S)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(set(product_prod(B,A))),set(set(A)),image2(set(product_prod(B,A)),set(A),range2(B,A)),S)) ).

% Range_Union
tff(fact_7253_Rangep__Range__eq,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(B,A)),X3: A] :
      ( aa(A,$o,rangep(B,A,aTP_Lamp_wv(set(product_prod(B,A)),fun(B,fun(A,$o)),R2)),X3)
    <=> aa(set(A),$o,member(A,X3),aa(set(product_prod(B,A)),set(A),range2(B,A),R2)) ) ).

% Rangep_Range_eq
tff(fact_7254_Range__def,axiom,
    ! [B: $tType,A: $tType,X3: set(product_prod(A,B))] : aa(set(product_prod(A,B)),set(B),range2(A,B),X3) = aa(fun(B,$o),set(B),collect(B),rangep(A,B,aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_as(set(product_prod(A,B)),fun(A,fun(B,$o))),X3))) ).

% Range_def
tff(fact_7255_Range__Int__subset,axiom,
    ! [A: $tType,B: $tType,A4: set(product_prod(B,A)),B3: set(product_prod(B,A))] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(product_prod(B,A)),set(A),range2(B,A),aa(set(product_prod(B,A)),set(product_prod(B,A)),aa(set(product_prod(B,A)),fun(set(product_prod(B,A)),set(product_prod(B,A))),inf_inf(set(product_prod(B,A))),A4),B3))),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(product_prod(B,A)),set(A),range2(B,A),A4)),aa(set(product_prod(B,A)),set(A),range2(B,A),B3))) ).

% Range_Int_subset
tff(fact_7256_wf__UN,axiom,
    ! [B: $tType,A: $tType,I4: set(A),R2: fun(A,set(product_prod(B,B)))] :
      ( ! [I2: A] :
          ( aa(set(A),$o,member(A,I2),I4)
         => wf(B,aa(A,set(product_prod(B,B)),R2,I2)) )
     => ( ! [I2: A,J2: A] :
            ( aa(set(A),$o,member(A,I2),I4)
           => ( aa(set(A),$o,member(A,J2),I4)
             => ( ( aa(A,set(product_prod(B,B)),R2,I2) != aa(A,set(product_prod(B,B)),R2,J2) )
               => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(set(product_prod(B,B)),set(B),domain(B,B),aa(A,set(product_prod(B,B)),R2,I2))),aa(set(product_prod(B,B)),set(B),range2(B,B),aa(A,set(product_prod(B,B)),R2,J2))) = bot_bot(set(B)) ) ) ) )
       => wf(B,aa(set(set(product_prod(B,B))),set(product_prod(B,B)),complete_Sup_Sup(set(product_prod(B,B))),aa(set(A),set(set(product_prod(B,B))),image2(A,set(product_prod(B,B)),R2),I4))) ) ) ).

% wf_UN
tff(fact_7257_dom__ran__disj__comp,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(product_prod(A,A)),set(A),domain(A,A),R)),aa(set(product_prod(A,A)),set(A),range2(A,A),R)) = bot_bot(set(A)) )
     => ( relcomp(A,A,A,R,R) = bot_bot(set(product_prod(A,A))) ) ) ).

% dom_ran_disj_comp
tff(fact_7258_Domain__rtrancl,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] : aa(set(product_prod(A,A)),set(A),domain(A,A),transitive_rtrancl(A,R)) = top_top(set(A)) ).

% Domain_rtrancl
tff(fact_7259_Domain__empty,axiom,
    ! [B: $tType,A: $tType] : aa(set(product_prod(A,B)),set(A),domain(A,B),bot_bot(set(product_prod(A,B)))) = bot_bot(set(A)) ).

% Domain_empty
tff(fact_7260_Domain__Id,axiom,
    ! [A: $tType] : aa(set(product_prod(A,A)),set(A),domain(A,A),id2(A)) = top_top(set(A)) ).

% Domain_Id
tff(fact_7261_Domain__Collect__case__prod,axiom,
    ! [B: $tType,A: $tType,P: fun(A,fun(B,$o))] : aa(set(product_prod(A,B)),set(A),domain(A,B),aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),P))) = aa(fun(A,$o),set(A),collect(A),aTP_Lamp_afv(fun(A,fun(B,$o)),fun(A,$o),P)) ).

% Domain_Collect_case_prod
tff(fact_7262_Domain__insert,axiom,
    ! [B: $tType,A: $tType,A3: A,B2: B,R2: set(product_prod(A,B))] : aa(set(product_prod(A,B)),set(A),domain(A,B),aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(product_prod(A,B),fun(set(product_prod(A,B)),set(product_prod(A,B))),insert2(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),R2)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),aa(set(product_prod(A,B)),set(A),domain(A,B),R2)) ).

% Domain_insert
tff(fact_7263_Domain__Union,axiom,
    ! [B: $tType,A: $tType,S: set(set(product_prod(A,B)))] : aa(set(product_prod(A,B)),set(A),domain(A,B),aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),S)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(set(product_prod(A,B))),set(set(A)),image2(set(product_prod(A,B)),set(A),domain(A,B)),S)) ).

% Domain_Union
tff(fact_7264_Domain__Un__eq,axiom,
    ! [B: $tType,A: $tType,A4: set(product_prod(A,B)),B3: set(product_prod(A,B))] : aa(set(product_prod(A,B)),set(A),domain(A,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))),sup_sup(set(product_prod(A,B))),A4),B3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(product_prod(A,B)),set(A),domain(A,B),A4)),aa(set(product_prod(A,B)),set(A),domain(A,B),B3)) ).

% Domain_Un_eq
tff(fact_7265_Domain__empty__iff,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,B))] :
      ( ( aa(set(product_prod(A,B)),set(A),domain(A,B),R2) = bot_bot(set(A)) )
    <=> ( R2 = bot_bot(set(product_prod(A,B))) ) ) ).

% Domain_empty_iff
tff(fact_7266_Domain__unfold,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,B))] : aa(set(product_prod(A,B)),set(A),domain(A,B),R2) = aa(fun(A,$o),set(A),collect(A),aTP_Lamp_avq(set(product_prod(A,B)),fun(A,$o),R2)) ).

% Domain_unfold
tff(fact_7267_Not__Domain__rtrancl,axiom,
    ! [A: $tType,X: A,R: set(product_prod(A,A)),Y: A] :
      ( ~ aa(set(A),$o,member(A,X),aa(set(product_prod(A,A)),set(A),domain(A,A),R))
     => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),transitive_rtrancl(A,R))
      <=> ( X = Y ) ) ) ).

% Not_Domain_rtrancl
tff(fact_7268_Domain_Ocases,axiom,
    ! [B: $tType,A: $tType,A3: A,R2: set(product_prod(A,B))] :
      ( aa(set(A),$o,member(A,A3),aa(set(product_prod(A,B)),set(A),domain(A,B),R2))
     => ~ ! [B5: B] : ~ aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B5)),R2) ) ).

% Domain.cases
tff(fact_7269_Domain_Osimps,axiom,
    ! [B: $tType,A: $tType,A3: A,R2: set(product_prod(A,B))] :
      ( aa(set(A),$o,member(A,A3),aa(set(product_prod(A,B)),set(A),domain(A,B),R2))
    <=> ? [A10: A,B6: B] :
          ( ( A3 = A10 )
          & aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A10),B6)),R2) ) ) ).

% Domain.simps
tff(fact_7270_Domain_ODomainI,axiom,
    ! [B: $tType,A: $tType,A3: A,B2: B,R2: set(product_prod(A,B))] :
      ( aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),R2)
     => aa(set(A),$o,member(A,A3),aa(set(product_prod(A,B)),set(A),domain(A,B),R2)) ) ).

% Domain.DomainI
tff(fact_7271_DomainE,axiom,
    ! [B: $tType,A: $tType,A3: A,R2: set(product_prod(A,B))] :
      ( aa(set(A),$o,member(A,A3),aa(set(product_prod(A,B)),set(A),domain(A,B),R2))
     => ~ ! [B5: B] : ~ aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B5)),R2) ) ).

% DomainE
tff(fact_7272_Domain__iff,axiom,
    ! [B: $tType,A: $tType,A3: A,R2: set(product_prod(A,B))] :
      ( aa(set(A),$o,member(A,A3),aa(set(product_prod(A,B)),set(A),domain(A,B),R2))
    <=> ? [Y3: B] : aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),Y3)),R2) ) ).

% Domain_iff
tff(fact_7273_Domain__def,axiom,
    ! [B: $tType,A: $tType,X3: set(product_prod(A,B))] : aa(set(product_prod(A,B)),set(A),domain(A,B),X3) = aa(fun(A,$o),set(A),collect(A),aa(fun(A,fun(B,$o)),fun(A,$o),domainp(A,B),aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_as(set(product_prod(A,B)),fun(A,fun(B,$o))),X3))) ).

% Domain_def
tff(fact_7274_Domainp__Domain__eq,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,B)),X3: A] :
      ( aa(A,$o,aa(fun(A,fun(B,$o)),fun(A,$o),domainp(A,B),aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_as(set(product_prod(A,B)),fun(A,fun(B,$o))),R2)),X3)
    <=> aa(set(A),$o,member(A,X3),aa(set(product_prod(A,B)),set(A),domain(A,B),R2)) ) ).

% Domainp_Domain_eq
tff(fact_7275_Domain__Int__subset,axiom,
    ! [B: $tType,A: $tType,A4: set(product_prod(A,B)),B3: set(product_prod(A,B))] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(product_prod(A,B)),set(A),domain(A,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))),A4),B3))),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(product_prod(A,B)),set(A),domain(A,B),A4)),aa(set(product_prod(A,B)),set(A),domain(A,B),B3))) ).

% Domain_Int_subset
tff(fact_7276_for__in__RI,axiom,
    ! [B: $tType,A: $tType,X: A,R: set(product_prod(A,B))] :
      ( aa(set(A),$o,member(A,X),aa(set(product_prod(A,B)),set(A),domain(A,B),R))
     => aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),fun_of_rel(A,B,R,X))),R) ) ).

% for_in_RI
tff(fact_7277_Field__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : field2(A,R2) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(product_prod(A,A)),set(A),domain(A,A),R2)),aa(set(product_prod(A,A)),set(A),range2(A,A),R2)) ).

% Field_def
tff(fact_7278_wf__no__path,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(product_prod(A,A)),set(A),domain(A,A),R)),aa(set(product_prod(A,A)),set(A),range2(A,A),R)) = bot_bot(set(A)) )
     => wf(A,R) ) ).

% wf_no_path
tff(fact_7279_wf__min,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( wf(A,R)
     => ( ( R != bot_bot(set(product_prod(A,A))) )
       => ~ ! [M3: A] : ~ aa(set(A),$o,member(A,M3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(product_prod(A,A)),set(A),domain(A,A),R)),aa(set(product_prod(A,A)),set(A),range2(A,A),R))) ) ) ).

% wf_min
tff(fact_7280_wf__Un,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),S2: set(product_prod(A,A))] :
      ( wf(A,R2)
     => ( wf(A,S2)
       => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(product_prod(A,A)),set(A),domain(A,A),R2)),aa(set(product_prod(A,A)),set(A),range2(A,A),S2)) = bot_bot(set(A)) )
         => wf(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),R2),S2)) ) ) ) ).

% wf_Un
tff(fact_7281_wf__Union,axiom,
    ! [A: $tType,R: set(set(product_prod(A,A)))] :
      ( ! [X2: set(product_prod(A,A))] :
          ( aa(set(set(product_prod(A,A))),$o,member(set(product_prod(A,A)),X2),R)
         => wf(A,X2) )
     => ( ! [X2: set(product_prod(A,A))] :
            ( aa(set(set(product_prod(A,A))),$o,member(set(product_prod(A,A)),X2),R)
           => ! [Xa4: set(product_prod(A,A))] :
                ( aa(set(set(product_prod(A,A))),$o,member(set(product_prod(A,A)),Xa4),R)
               => ( ( X2 != Xa4 )
                 => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(product_prod(A,A)),set(A),domain(A,A),X2)),aa(set(product_prod(A,A)),set(A),range2(A,A),Xa4)) = bot_bot(set(A)) ) ) ) )
       => wf(A,aa(set(set(product_prod(A,A))),set(product_prod(A,A)),complete_Sup_Sup(set(product_prod(A,A))),R)) ) ) ).

% wf_Union
tff(fact_7282_wf__max,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( wf(A,converse(A,A,R))
     => ( ( R != bot_bot(set(product_prod(A,A))) )
       => ~ ! [M3: A] : ~ aa(set(A),$o,member(A,M3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(product_prod(A,A)),set(A),range2(A,A),R)),aa(set(product_prod(A,A)),set(A),domain(A,A),R))) ) ) ).

% wf_max
tff(fact_7283_projl__def,axiom,
    ! [B: $tType,A: $tType,Sum: sum_sum(A,B)] : sum_projl(A,B,Sum) = aa(sum_sum(A,B),A,sum_case_sum(A,A,B,aTP_Lamp_au(A,A),aTP_Lamp_avr(B,A)),Sum) ).

% projl_def
tff(fact_7284_projr__def,axiom,
    ! [A: $tType,B: $tType,Sum: sum_sum(B,A)] : sum_projr(B,A,Sum) = aa(sum_sum(B,A),A,sum_case_sum(B,A,A,aTP_Lamp_avr(B,A),aTP_Lamp_au(A,A)),Sum) ).

% projr_def
tff(fact_7285_cut__def,axiom,
    ! [A: $tType,B: $tType,F: fun(A,B),R: set(product_prod(A,A)),X: A,X3: A] :
      aa(A,B,cut(A,B,F,R,X),X3) = $ite(aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),X)),R),aa(A,B,F,X3),undefined(B)) ).

% cut_def
tff(fact_7286_old_Orec__unit__def,axiom,
    ! [A: $tType,X3: A,Xa3: product_unit] : product_rec_unit(A,X3,Xa3) = the(A,product_rec_set_unit(A,X3,Xa3)) ).

% old.rec_unit_def
tff(fact_7287_cuts__eq,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),R: set(product_prod(A,A)),X: A,G: fun(A,B)] :
      ( ( cut(A,B,F,R,X) = cut(A,B,G,R,X) )
    <=> ! [Y3: A] :
          ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),X)),R)
         => ( aa(A,B,F,Y3) = aa(A,B,G,Y3) ) ) ) ).

% cuts_eq
tff(fact_7288_cut__apply,axiom,
    ! [B: $tType,A: $tType,X: A,A3: A,R: set(product_prod(A,A)),F: fun(A,B)] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),A3)),R)
     => ( aa(A,B,cut(A,B,F,R,A3),X) = aa(A,B,F,X) ) ) ).

% cut_apply
tff(fact_7289_adm__lemma,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,A)),F4: fun(fun(A,B),fun(A,B))] : adm_wf(A,B,R,aa(fun(fun(A,B),fun(A,B)),fun(fun(A,B),fun(A,B)),aTP_Lamp_avs(set(product_prod(A,A)),fun(fun(fun(A,B),fun(A,B)),fun(fun(A,B),fun(A,B))),R),F4)) ).

% adm_lemma
tff(fact_7290_Lcm__Gcd,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A4: set(A)] : gcd_Lcm(A,A4) = gcd_Gcd(A,aa(fun(A,$o),set(A),collect(A),aTP_Lamp_avt(set(A),fun(A,$o),A4))) ) ).

% Lcm_Gcd
tff(fact_7291_Lcm__abs__eq,axiom,
    ! [K5: set(int)] : gcd_Lcm(int,aa(set(int),set(int),image2(int,int,abs_abs(int)),K5)) = gcd_Lcm(int,K5) ).

% Lcm_abs_eq
tff(fact_7292_Lcm__int__eq,axiom,
    ! [N4: set(nat)] : gcd_Lcm(int,aa(set(nat),set(int),image2(nat,int,semiring_1_of_nat(int)),N4)) = aa(nat,int,semiring_1_of_nat(int),gcd_Lcm(nat,N4)) ).

% Lcm_int_eq
tff(fact_7293_Lcm__UNIV,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ( gcd_Lcm(A,top_top(set(A))) = zero_zero(A) ) ) ).

% Lcm_UNIV
tff(fact_7294_Lcm__empty,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ( gcd_Lcm(A,bot_bot(set(A))) = one_one(A) ) ) ).

% Lcm_empty
tff(fact_7295_Lcm__1__iff,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A4: set(A)] :
          ( ( gcd_Lcm(A,A4) = one_one(A) )
        <=> ! [X4: A] :
              ( aa(set(A),$o,member(A,X4),A4)
             => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),X4),one_one(A)) ) ) ) ).

% Lcm_1_iff
tff(fact_7296_Lcm__nat__abs__eq,axiom,
    ! [K5: set(int)] : gcd_Lcm(nat,aa(set(int),set(nat),image2(int,nat,aTP_Lamp_so(int,nat)),K5)) = aa(int,nat,nat2,gcd_Lcm(int,K5)) ).

% Lcm_nat_abs_eq
tff(fact_7297_Lcm__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_Gcd(B)
     => ! [A4: set(A),F: fun(A,B),G: fun(A,B)] :
          ( ! [X2: A] :
              ( aa(set(A),$o,member(A,X2),A4)
             => aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(A,B,F,X2)),aa(A,B,G,X2)) )
         => aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),gcd_Lcm(B,aa(set(A),set(B),image2(A,B,F),A4))),gcd_Lcm(B,aa(set(A),set(B),image2(A,B,G),A4))) ) ) ).

% Lcm_mono
tff(fact_7298_adm__wf__def,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,A)),F4: fun(fun(A,B),fun(A,B))] :
      ( adm_wf(A,B,R,F4)
    <=> ! [F6: fun(A,B),G5: fun(A,B),X4: A] :
          ( ! [Z4: A] :
              ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z4),X4)),R)
             => ( aa(A,B,F6,Z4) = aa(A,B,G5,Z4) ) )
         => ( aa(A,B,aa(fun(A,B),fun(A,B),F4,F6),X4) = aa(A,B,aa(fun(A,B),fun(A,B),F4,G5),X4) ) ) ) ).

% adm_wf_def
tff(fact_7299_Lcm__nat__empty,axiom,
    gcd_Lcm(nat,bot_bot(set(nat))) = one_one(nat) ).

% Lcm_nat_empty
tff(fact_7300_Gcd__nat__def,axiom,
    ! [M4: set(nat)] : gcd_Gcd(nat,M4) = gcd_Lcm(nat,aa(fun(nat,$o),set(nat),collect(nat),aTP_Lamp_avu(set(nat),fun(nat,$o),M4))) ).

% Gcd_nat_def
tff(fact_7301_Lcm__int__def,axiom,
    ! [K5: set(int)] : gcd_Lcm(int,K5) = aa(nat,int,semiring_1_of_nat(int),gcd_Lcm(nat,aa(set(int),set(nat),image2(int,nat,aa(fun(int,int),fun(int,nat),comp(int,nat,int,nat2),abs_abs(int))),K5))) ).

% Lcm_int_def
tff(fact_7302_Lcm__no__units,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A4: set(A)] : gcd_Lcm(A,A4) = gcd_Lcm(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_avv(A,$o)))) ) ).

% Lcm_no_units
tff(fact_7303_Gcd__Lcm,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A4: set(A)] : gcd_Gcd(A,A4) = gcd_Lcm(A,aa(fun(A,$o),set(A),collect(A),aTP_Lamp_avw(set(A),fun(A,$o),A4))) ) ).

% Gcd_Lcm
tff(fact_7304_Lcm__coprime_H,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A4: set(A)] :
          ( ( aa(set(A),nat,finite_card(A),A4) != zero_zero(nat) )
         => ( ! [A6: A,B5: A] :
                ( aa(set(A),$o,member(A,A6),A4)
               => ( aa(set(A),$o,member(A,B5),A4)
                 => ( ( A6 != B5 )
                   => algebr8660921524188924756oprime(A,A6,B5) ) ) )
           => ( gcd_Lcm(A,A4) = aa(A,A,normal6383669964737779283malize(A),aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7121269368397514597t_prod(A,A),aTP_Lamp_avx(A,A)),A4)) ) ) ) ) ).

% Lcm_coprime'
tff(fact_7305_Lcm__coprime,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A4: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( ! [A6: A,B5: A] :
                  ( aa(set(A),$o,member(A,A6),A4)
                 => ( aa(set(A),$o,member(A,B5),A4)
                   => ( ( A6 != B5 )
                     => algebr8660921524188924756oprime(A,A6,B5) ) ) )
             => ( gcd_Lcm(A,A4) = aa(A,A,normal6383669964737779283malize(A),aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7121269368397514597t_prod(A,A),aTP_Lamp_avx(A,A)),A4)) ) ) ) ) ) ).

% Lcm_coprime
tff(fact_7306_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_7307_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_7308_normalize__1,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ( aa(A,A,normal6383669964737779283malize(A),one_one(A)) = one_one(A) ) ) ).

% normalize_1
tff(fact_7309_gcd_Onormalize__bottom,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ( aa(A,A,normal6383669964737779283malize(A),one_one(A)) = one_one(A) ) ) ).

% gcd.normalize_bottom
tff(fact_7310_normalize__mult__unit__left,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),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_7311_normalize__mult__unit__right,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [B2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),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_7312_Lcm__singleton,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A3: A] : gcd_Lcm(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A)))) = aa(A,A,normal6383669964737779283malize(A),A3) ) ).

% Lcm_singleton
tff(fact_7313_Gcd__singleton,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A3: A] : gcd_Gcd(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A)))) = aa(A,A,normal6383669964737779283malize(A),A3) ) ).

% Gcd_singleton
tff(fact_7314_coprime__crossproduct,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,D3: A,B2: A,C2: A] :
          ( algebr8660921524188924756oprime(A,A3,D3)
         => ( algebr8660921524188924756oprime(A,B2,C2)
           => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,normal6383669964737779283malize(A),A3)),aa(A,A,normal6383669964737779283malize(A),C2)) = 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),C2) = aa(A,A,normal6383669964737779283malize(A),D3) ) ) ) ) ) ) ).

% coprime_crossproduct
tff(fact_7315_gcd__mult__left,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [C2: 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),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) = aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2))) ) ).

% gcd_mult_left
tff(fact_7316_gcd__mult__right,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,C2: 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),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = 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)),C2)) ) ).

% gcd_mult_right
tff(fact_7317_gcd__mult__distrib_H,axiom,
    ! [A: $tType] :
      ( semiri6843258321239162965malize(A)
     => ! [C2: A,A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,normal6383669964737779283malize(A),C2)),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),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) ) ).

% gcd_mult_distrib'
tff(fact_7318_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_7319_normalize__idem__imp__is__unit__iff,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A] :
          ( ( aa(A,A,normal6383669964737779283malize(A),A3) = A3 )
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A3),one_one(A))
          <=> ( A3 = one_one(A) ) ) ) ) ).

% normalize_idem_imp_is_unit_iff
tff(fact_7320_is__unit__normalize,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A3),one_one(A))
         => ( aa(A,A,normal6383669964737779283malize(A),A3) = one_one(A) ) ) ) ).

% is_unit_normalize
tff(fact_7321_normalize__1__iff,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A] :
          ( ( aa(A,A,normal6383669964737779283malize(A),A3) = one_one(A) )
        <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A3),one_one(A)) ) ) ).

% normalize_1_iff
tff(fact_7322_associated__unit,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A,B2: A] :
          ( ( aa(A,A,normal6383669964737779283malize(A),A3) = aa(A,A,normal6383669964737779283malize(A),B2) )
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A3),one_one(A))
           => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),one_one(A)) ) ) ) ).

% associated_unit
tff(fact_7323_Gcd__mult,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [C2: A,A4: set(A)] : gcd_Gcd(A,aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),times_times(A),C2)),A4)) = aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),gcd_Gcd(A,A4))) ) ).

% Gcd_mult
tff(fact_7324_Lcm__mult,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A4: set(A),C2: A] :
          ( ( A4 != bot_bot(set(A)) )
         => ( gcd_Lcm(A,aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),times_times(A),C2)),A4)) = aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),gcd_Lcm(A,A4))) ) ) ) ).

% Lcm_mult
tff(fact_7325_Gcd__fin__mult,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A4: set(A),B2: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),A,semiring_gcd_Gcd_fin(A),aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),times_times(A),B2)),A4)) = 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),A4))) ) ) ) ).

% Gcd_fin_mult
tff(fact_7326_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_7327_Lcm__eq__Max__nat,axiom,
    ! [M4: set(nat)] :
      ( aa(set(nat),$o,finite_finite2(nat),M4)
     => ( ( M4 != bot_bot(set(nat)) )
       => ( ~ aa(set(nat),$o,member(nat,zero_zero(nat)),M4)
         => ( ! [M3: nat,N3: nat] :
                ( aa(set(nat),$o,member(nat,M3),M4)
               => ( aa(set(nat),$o,member(nat,N3),M4)
                 => aa(set(nat),$o,member(nat,aa(nat,nat,aa(nat,fun(nat,nat),gcd_lcm(nat),M3),N3)),M4) ) )
           => ( gcd_Lcm(nat,M4) = aa(set(nat),nat,lattic643756798349783984er_Max(nat),M4) ) ) ) ) ) ).

% Lcm_eq_Max_nat
tff(fact_7328_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_7329_lcm__neg1,axiom,
    ! [A: $tType] :
      ( ring_gcd(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),gcd_lcm(A),aa(A,A,uminus_uminus(A),A3)),B2) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B2) ) ).

% lcm_neg1
tff(fact_7330_lcm__neg2,axiom,
    ! [A: $tType] :
      ( ring_gcd(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),aa(A,A,uminus_uminus(A),B2)) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B2) ) ).

% lcm_neg2
tff(fact_7331_lcm__neg__numeral__2,axiom,
    ! [A: $tType] :
      ( ring_gcd(A)
     => ! [A3: A,N: num] : aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),aa(num,A,numeral_numeral(A),N)) ) ).

% lcm_neg_numeral_2
tff(fact_7332_lcm__neg__numeral__1,axiom,
    ! [A: $tType] :
      ( ring_gcd(A)
     => ! [N: num,A3: A] : aa(A,A,aa(A,fun(A,A),gcd_lcm(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))),A3) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),aa(num,A,numeral_numeral(A),N)),A3) ) ).

% lcm_neg_numeral_1
tff(fact_7333_lcm__eq__1__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B2) = one_one(A) )
        <=> ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A3),one_one(A))
            & aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),one_one(A)) ) ) ) ).

% lcm_eq_1_iff
tff(fact_7334_lcm_Otop__left__normalize,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),gcd_lcm(A),one_one(A)),A3) = aa(A,A,normal6383669964737779283malize(A),A3) ) ).

% lcm.top_left_normalize
tff(fact_7335_lcm_Otop__right__normalize,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),one_one(A)) = aa(A,A,normal6383669964737779283malize(A),A3) ) ).

% lcm.top_right_normalize
tff(fact_7336_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_7337_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_7338_Lcm__2,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A3: A,B2: A] : gcd_Lcm(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A))))) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B2) ) ).

% Lcm_2
tff(fact_7339_Lcm__nat__set__eq__fold,axiom,
    ! [Xs: list(nat)] : gcd_Lcm(nat,aa(list(nat),set(nat),set2(nat),Xs)) = aa(nat,nat,fold(nat,nat,gcd_lcm(nat),Xs),one_one(nat)) ).

% Lcm_nat_set_eq_fold
tff(fact_7340_Lcm__Un,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A4: set(A),B3: set(A)] : gcd_Lcm(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3)) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),gcd_Lcm(A,A4)),gcd_Lcm(A,B3)) ) ).

% Lcm_Un
tff(fact_7341_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_7342_lcm__mult__distrib_H,axiom,
    ! [A: $tType] :
      ( semiri6843258321239162965malize(A)
     => ! [C2: A,A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,normal6383669964737779283malize(A),C2)),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),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) ) ).

% lcm_mult_distrib'
tff(fact_7343_lcm__mult__right,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,C2: 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),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = 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)),C2)) ) ).

% lcm_mult_right
tff(fact_7344_lcm__mult__left,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [C2: 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),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) = aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B2))) ) ).

% lcm_mult_left
tff(fact_7345_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_7346_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_7347_lcm__div__unit2,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),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),divide_divide(A),C2),A3)) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),B2),C2) ) ) ) ).

% lcm_div_unit2
tff(fact_7348_lcm__div__unit1,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),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),divide_divide(A),B2),A3)),C2) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),B2),C2) ) ) ) ).

% lcm_div_unit1
tff(fact_7349_lcm__mult__unit2,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),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),C2),A3)) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),B2),C2) ) ) ) ).

% lcm_mult_unit2
tff(fact_7350_lcm__mult__unit1,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),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)),C2) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),B2),C2) ) ) ) ).

% lcm_mult_unit1
tff(fact_7351_Lcm__in__lcm__closed__set__nat,axiom,
    ! [M4: set(nat)] :
      ( aa(set(nat),$o,finite_finite2(nat),M4)
     => ( ( M4 != bot_bot(set(nat)) )
       => ( ! [M3: nat,N3: nat] :
              ( aa(set(nat),$o,member(nat,M3),M4)
             => ( aa(set(nat),$o,member(nat,N3),M4)
               => aa(set(nat),$o,member(nat,aa(nat,nat,aa(nat,fun(nat,nat),gcd_lcm(nat),M3),N3)),M4) ) )
         => aa(set(nat),$o,member(nat,gcd_Lcm(nat,M4)),M4) ) ) ) ).

% Lcm_in_lcm_closed_set_nat
tff(fact_7352_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),aa(A,A,aa(A,fun(A,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_7353_Lcm__set__eq__fold,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [Xs: list(A)] : gcd_Lcm(A,aa(list(A),set(A),set2(A),Xs)) = aa(A,A,fold(A,A,gcd_lcm(A),Xs),one_one(A)) ) ).

% Lcm_set_eq_fold
tff(fact_7354_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_7355_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),aa(A,A,aa(A,fun(A,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_7356_Lcm__nat__def,axiom,
    ! [M4: set(nat)] :
      gcd_Lcm(nat,M4) = $ite(aa(set(nat),$o,finite_finite2(nat),M4),lattic5214292709420241887eutr_F(nat,gcd_lcm(nat),one_one(nat),M4),zero_zero(nat)) ).

% Lcm_nat_def
tff(fact_7357_Lcm__fin_Oeq__fold,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A4: set(A)] :
          aa(set(A),A,semiring_gcd_Lcm_fin(A),A4) = $ite(aa(set(A),$o,finite_finite2(A),A4),finite_fold(A,A,gcd_lcm(A),one_one(A),A4),zero_zero(A)) ) ).

% Lcm_fin.eq_fold
tff(fact_7358_lcm__1__iff__int,axiom,
    ! [M: int,N: int] :
      ( ( aa(int,int,aa(int,fun(int,int),gcd_lcm(int),M),N) = one_one(int) )
    <=> ( ( ( M = one_one(int) )
          | ( M = aa(int,int,uminus_uminus(int),one_one(int)) ) )
        & ( ( N = one_one(int) )
          | ( N = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ) ) ).

% lcm_1_iff_int
tff(fact_7359_Lcm__fin_Oempty,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ( aa(set(A),A,semiring_gcd_Lcm_fin(A),bot_bot(set(A))) = one_one(A) ) ) ).

% Lcm_fin.empty
tff(fact_7360_is__unit__Lcm__fin__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A4: set(A)] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(set(A),A,semiring_gcd_Lcm_fin(A),A4)),one_one(A))
        <=> ( aa(set(A),A,semiring_gcd_Lcm_fin(A),A4) = one_one(A) ) ) ) ).

% is_unit_Lcm_fin_iff
tff(fact_7361_lcm__neg1__int,axiom,
    ! [X: int,Y: int] : aa(int,int,aa(int,fun(int,int),gcd_lcm(int),aa(int,int,uminus_uminus(int),X)),Y) = aa(int,int,aa(int,fun(int,int),gcd_lcm(int),X),Y) ).

% lcm_neg1_int
tff(fact_7362_lcm__neg2__int,axiom,
    ! [X: int,Y: int] : aa(int,int,aa(int,fun(int,int),gcd_lcm(int),X),aa(int,int,uminus_uminus(int),Y)) = aa(int,int,aa(int,fun(int,int),gcd_lcm(int),X),Y) ).

% lcm_neg2_int
tff(fact_7363_lcm__cases__int,axiom,
    ! [X: int,Y: int,P: fun(int,$o)] :
      ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),X)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Y)
         => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),gcd_lcm(int),X),Y)) ) )
     => ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),X)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Y),zero_zero(int))
           => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),gcd_lcm(int),X),aa(int,int,uminus_uminus(int),Y))) ) )
       => ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),X),zero_zero(int))
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Y)
             => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),gcd_lcm(int),aa(int,int,uminus_uminus(int),X)),Y)) ) )
         => ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),X),zero_zero(int))
             => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Y),zero_zero(int))
               => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),gcd_lcm(int),aa(int,int,uminus_uminus(int),X)),aa(int,int,uminus_uminus(int),Y))) ) )
           => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),gcd_lcm(int),X),Y)) ) ) ) ) ).

% lcm_cases_int
tff(fact_7364_Lcm__fin_Ounion,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A4: set(A),B3: set(A)] : aa(set(A),A,semiring_gcd_Lcm_fin(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3)) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),aa(set(A),A,semiring_gcd_Lcm_fin(A),A4)),aa(set(A),A,semiring_gcd_Lcm_fin(A),B3)) ) ).

% Lcm_fin.union
tff(fact_7365_Lcm__fin__1__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A4: set(A)] :
          ( ( aa(set(A),A,semiring_gcd_Lcm_fin(A),A4) = one_one(A) )
        <=> ( ! [X4: A] :
                ( aa(set(A),$o,member(A,X4),A4)
               => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),X4),one_one(A)) )
            & aa(set(A),$o,finite_finite2(A),A4) ) ) ) ).

% Lcm_fin_1_iff
tff(fact_7366_Lcm__int__set__eq__fold,axiom,
    ! [Xs: list(int)] : gcd_Lcm(int,aa(list(int),set(int),set2(int),Xs)) = aa(int,int,fold(int,int,gcd_lcm(int),Xs),one_one(int)) ).

% Lcm_int_set_eq_fold
tff(fact_7367_Lcm__fin__mult,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A4: set(A),B2: A] :
          ( ( A4 != bot_bot(set(A)) )
         => ( aa(set(A),A,semiring_gcd_Lcm_fin(A),aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),times_times(A),B2)),A4)) = 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),A4))) ) ) ) ).

% Lcm_fin_mult
tff(fact_7368_Lcm__fin_Oinsert__remove,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,A4: set(A)] : aa(set(A),A,semiring_gcd_Lcm_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),A4)) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),aa(set(A),A,semiring_gcd_Lcm_fin(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A)))))) ) ).

% Lcm_fin.insert_remove
tff(fact_7369_Lcm__fin_Oremove,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,A4: set(A)] :
          ( aa(set(A),$o,member(A,A3),A4)
         => ( aa(set(A),A,semiring_gcd_Lcm_fin(A),A4) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),aa(set(A),A,semiring_gcd_Lcm_fin(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A)))))) ) ) ) ).

% Lcm_fin.remove
tff(fact_7370_Lcm__fin_Oset__eq__fold,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [Xs: list(A)] : aa(set(A),A,semiring_gcd_Lcm_fin(A),aa(list(A),set(A),set2(A),Xs)) = aa(A,A,fold(A,A,gcd_lcm(A),Xs),one_one(A)) ) ).

% Lcm_fin.set_eq_fold
tff(fact_7371_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_7372_semilattice__neutr__set_Oremove,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z2: A,A4: set(A),X: A] :
      ( lattic5652469242046573047tr_set(A,F,Z2)
     => ( aa(set(A),$o,finite_finite2(A),A4)
       => ( aa(set(A),$o,member(A,X),A4)
         => ( lattic5214292709420241887eutr_F(A,F,Z2,A4) = aa(A,A,aa(A,fun(A,A),F,X),lattic5214292709420241887eutr_F(A,F,Z2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))))) ) ) ) ) ).

% semilattice_neutr_set.remove
tff(fact_7373_semilattice__neutr__set_Oinsert__remove,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z2: A,A4: set(A),X: A] :
      ( lattic5652469242046573047tr_set(A,F,Z2)
     => ( aa(set(A),$o,finite_finite2(A),A4)
       => ( lattic5214292709420241887eutr_F(A,F,Z2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A4)) = aa(A,A,aa(A,fun(A,A),F,X),lattic5214292709420241887eutr_F(A,F,Z2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))))) ) ) ) ).

% semilattice_neutr_set.insert_remove
tff(fact_7374_semilattice__neutr__set__def,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
      ( lattic5652469242046573047tr_set(A,F,Z2)
    <=> semilattice_neutr(A,F,Z2) ) ).

% semilattice_neutr_set_def
tff(fact_7375_semilattice__neutr__set_Oaxioms,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
      ( lattic5652469242046573047tr_set(A,F,Z2)
     => semilattice_neutr(A,F,Z2) ) ).

% semilattice_neutr_set.axioms
tff(fact_7376_semilattice__neutr__set_Ointro,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
      ( semilattice_neutr(A,F,Z2)
     => lattic5652469242046573047tr_set(A,F,Z2) ) ).

% semilattice_neutr_set.intro
tff(fact_7377_semilattice__neutr__set_Oempty,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
      ( lattic5652469242046573047tr_set(A,F,Z2)
     => ( lattic5214292709420241887eutr_F(A,F,Z2,bot_bot(set(A))) = Z2 ) ) ).

% semilattice_neutr_set.empty
tff(fact_7378_semilattice__neutr__set_Ounion,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z2: A,A4: set(A),B3: set(A)] :
      ( lattic5652469242046573047tr_set(A,F,Z2)
     => ( aa(set(A),$o,finite_finite2(A),A4)
       => ( aa(set(A),$o,finite_finite2(A),B3)
         => ( lattic5214292709420241887eutr_F(A,F,Z2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3)) = aa(A,A,aa(A,fun(A,A),F,lattic5214292709420241887eutr_F(A,F,Z2,A4)),lattic5214292709420241887eutr_F(A,F,Z2,B3)) ) ) ) ) ).

% semilattice_neutr_set.union
tff(fact_7379_semilattice__neutr__set_Oclosed,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z2: A,A4: set(A)] :
      ( lattic5652469242046573047tr_set(A,F,Z2)
     => ( aa(set(A),$o,finite_finite2(A),A4)
       => ( ( A4 != bot_bot(set(A)) )
         => ( ! [X2: A,Y2: A] : aa(set(A),$o,member(A,aa(A,A,aa(A,fun(A,A),F,X2),Y2)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y2),bot_bot(set(A)))))
           => aa(set(A),$o,member(A,lattic5214292709420241887eutr_F(A,F,Z2,A4)),A4) ) ) ) ) ).

% semilattice_neutr_set.closed
tff(fact_7380_combine__options__def,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),X: option(A),Y: option(A)] : combine_options(A,F,X,Y) = case_option(option(A),A,Y,aa(option(A),fun(A,option(A)),aTP_Lamp_avz(fun(A,fun(A,A)),fun(option(A),fun(A,option(A))),F),Y),X) ).

% combine_options_def
tff(fact_7381_normalize__div,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,normal6383669964737779283malize(A),A3)),A3) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),unit_f5069060285200089521factor(A,A3)) ) ).

% normalize_div
tff(fact_7382_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_7383_unit__factor__1,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ( unit_f5069060285200089521factor(A,one_one(A)) = one_one(A) ) ) ).

% unit_factor_1
tff(fact_7384_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_7385_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_7386_inv__unit__factor__eq__0__iff,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A] :
          ( ( aa(A,A,aa(A,fun(A,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_7387_unit__factor__mult__unit__left,axiom,
    ! [A: $tType] :
      ( semido2269285787275462019factor(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),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_7388_unit__factor__mult__unit__right,axiom,
    ! [A: $tType] :
      ( semido2269285787275462019factor(A)
     => ! [A3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),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_7389_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),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),unit_f5069060285200089521factor(A,B2))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),unit_f5069060285200089521factor(A,B2)) ) ).

% mult_one_div_unit_factor
tff(fact_7390_unit__factor__lcm,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A] :
          unit_f5069060285200089521factor(A,aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B2)) = $ite(
            ( ( A3 = zero_zero(A) )
            | ( B2 = zero_zero(A) ) ),
            zero_zero(A),
            one_one(A) ) ) ).

% unit_factor_lcm
tff(fact_7391_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_7392_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_7393_mult__lcm__left,axiom,
    ! [A: $tType] :
      ( semiri6843258321239162965malize(A)
     => ! [C2: A,A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),C2),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,C2)),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))) ) ).

% mult_lcm_left
tff(fact_7394_mult__lcm__right,axiom,
    ! [A: $tType] :
      ( semiri6843258321239162965malize(A)
     => ! [A3: A,B2: A,C2: 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)),C2) = 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),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))),unit_f5069060285200089521factor(A,C2)) ) ).

% mult_lcm_right
tff(fact_7395_lcm__mult__distrib,axiom,
    ! [A: $tType] :
      ( semiri6843258321239162965malize(A)
     => ! [K: A,A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),K),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),K),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),K),B2))),unit_f5069060285200089521factor(A,K)) ) ).

% lcm_mult_distrib
tff(fact_7396_mult__gcd__left,axiom,
    ! [A: $tType] :
      ( semiri6843258321239162965malize(A)
     => ! [C2: A,A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),C2),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,C2)),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))) ) ).

% mult_gcd_left
tff(fact_7397_mult__gcd__right,axiom,
    ! [A: $tType] :
      ( semiri6843258321239162965malize(A)
     => ! [A3: A,B2: A,C2: 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)),C2) = 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),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))),unit_f5069060285200089521factor(A,C2)) ) ).

% mult_gcd_right
tff(fact_7398_gcd__mult__distrib,axiom,
    ! [A: $tType] :
      ( semiri6843258321239162965malize(A)
     => ! [K: A,A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),K),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),K),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),K),B2))),unit_f5069060285200089521factor(A,K)) ) ).

% gcd_mult_distrib
tff(fact_7399_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_7400_is__unit__unit__factor,axiom,
    ! [A: $tType] :
      ( semido2269285787275462019factor(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A3),one_one(A))
         => ( unit_f5069060285200089521factor(A,A3) = A3 ) ) ) ).

% is_unit_unit_factor
tff(fact_7401_unit__factor__nat__def,axiom,
    ! [N: nat] :
      unit_f5069060285200089521factor(nat,N) = $ite(N = zero_zero(nat),zero_zero(nat),one_one(nat)) ).

% unit_factor_nat_def
tff(fact_7402_unit__factor__is__unit,axiom,
    ! [A: $tType] :
      ( semido2269285787275462019factor(A)
     => ! [A3: A] :
          ( ( A3 != zero_zero(A) )
         => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),unit_f5069060285200089521factor(A,A3)),one_one(A)) ) ) ).

% unit_factor_is_unit
tff(fact_7403_unit__factor__gcd,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A] :
          unit_f5069060285200089521factor(A,aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2)) = $ite(
            ( ( A3 = zero_zero(A) )
            & ( B2 = zero_zero(A) ) ),
            zero_zero(A),
            one_one(A) ) ) ).

% unit_factor_gcd
tff(fact_7404_coprime__crossproduct_H,axiom,
    ! [A: $tType] :
      ( semiri6843258321239162965malize(A)
     => ! [B2: A,D3: A,A3: A,C2: A] :
          ( ( B2 != zero_zero(A) )
         => ( ( unit_f5069060285200089521factor(A,B2) = unit_f5069060285200089521factor(A,D3) )
           => ( algebr8660921524188924756oprime(A,A3,B2)
             => ( algebr8660921524188924756oprime(A,C2,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),C2) )
                <=> ( ( A3 = C2 )
                    & ( B2 = D3 ) ) ) ) ) ) ) ) ).

% coprime_crossproduct'
tff(fact_7405_unit__factor__Lcm,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A4: set(A)] :
          unit_f5069060285200089521factor(A,gcd_Lcm(A,A4)) = $ite(gcd_Lcm(A,A4) = zero_zero(A),zero_zero(A),one_one(A)) ) ).

% unit_factor_Lcm
tff(fact_7406_unit__factor__Gcd,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A4: set(A)] :
          unit_f5069060285200089521factor(A,gcd_Gcd(A,A4)) = $ite(gcd_Gcd(A,A4) = zero_zero(A),zero_zero(A),one_one(A)) ) ).

% unit_factor_Gcd
tff(fact_7407_old_Orec__sum__def,axiom,
    ! [B: $tType,C: $tType,A: $tType,X3: fun(A,B),Xa3: fun(C,B),Xb2: sum_sum(A,C)] : sum_rec_sum(A,B,C,X3,Xa3,Xb2) = the(B,sum_rec_set_sum(A,B,C,X3,Xa3,Xb2)) ).

% old.rec_sum_def
tff(fact_7408_mk__less__def,axiom,
    ! [A: $tType,R: fun(A,fun(A,$o)),X3: A,Xa3: A] :
      ( partial_mk_less(A,R,X3,Xa3)
    <=> ( aa(A,$o,aa(A,fun(A,$o),R,X3),Xa3)
        & ~ aa(A,$o,aa(A,fun(A,$o),R,Xa3),X3) ) ) ).

% mk_less_def
tff(fact_7409_Random__Pred_Ounion__def,axiom,
    ! [A: $tType,R1: fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))),R22: fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))),X3: product_prod(code_natural,code_natural)] : random_union(A,R1,R22,X3) = aa(product_prod(pred(A),product_prod(code_natural,code_natural)),product_prod(pred(A),product_prod(code_natural,code_natural)),aa(fun(pred(A),fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural)))),fun(product_prod(pred(A),product_prod(code_natural,code_natural)),product_prod(pred(A),product_prod(code_natural,code_natural))),product_case_prod(pred(A),product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))),aTP_Lamp_awb(fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))),fun(pred(A),fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural)))),R22)),aa(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural)),R1,X3)) ).

% Random_Pred.union_def
tff(fact_7410_img__ord__def,axiom,
    ! [B: $tType,C: $tType,A: $tType,F: fun(A,C),Ord: fun(C,fun(C,B)),X3: A,Xa3: A] : partial_img_ord(A,C,B,F,Ord,X3,Xa3) = aa(C,B,aa(C,fun(C,B),Ord,aa(A,C,F,X3)),aa(A,C,F,Xa3)) ).

% img_ord_def
tff(fact_7411_Random__Pred_Oempty__def,axiom,
    ! [A: $tType] : random_empty(A) = aa(pred(A),fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))),product_Pair(pred(A),product_prod(code_natural,code_natural)),bot_bot(pred(A))) ).

% Random_Pred.empty_def
tff(fact_7412_Random__Pred_Obind__def,axiom,
    ! [A: $tType,B: $tType,R: fun(product_prod(code_natural,code_natural),product_prod(pred(B),product_prod(code_natural,code_natural))),F: fun(B,fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural)))),X3: product_prod(code_natural,code_natural)] : random_bind(B,A,R,F,X3) = aa(product_prod(pred(B),product_prod(code_natural,code_natural)),product_prod(pred(A),product_prod(code_natural,code_natural)),aa(fun(pred(B),fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural)))),fun(product_prod(pred(B),product_prod(code_natural,code_natural)),product_prod(pred(A),product_prod(code_natural,code_natural))),product_case_prod(pred(B),product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))),aTP_Lamp_awe(fun(B,fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural)))),fun(pred(B),fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural)))),F)),aa(product_prod(code_natural,code_natural),product_prod(pred(B),product_prod(code_natural,code_natural)),R,X3)) ).

% Random_Pred.bind_def
tff(fact_7413_sup__bind,axiom,
    ! [A: $tType,B: $tType,P: pred(B),Q: pred(B),R: fun(B,pred(A))] : bind2(B,A,aa(pred(B),pred(B),aa(pred(B),fun(pred(B),pred(B)),sup_sup(pred(B)),P),Q),R) = aa(pred(A),pred(A),aa(pred(A),fun(pred(A),pred(A)),sup_sup(pred(A)),bind2(B,A,P,R)),bind2(B,A,Q,R)) ).

% sup_bind
tff(fact_7414_Sup__bind,axiom,
    ! [A: $tType,B: $tType,A4: set(pred(B)),F: fun(B,pred(A))] : bind2(B,A,aa(set(pred(B)),pred(B),complete_Sup_Sup(pred(B)),A4),F) = aa(set(pred(A)),pred(A),complete_Sup_Sup(pred(A)),aa(set(pred(B)),set(pred(A)),image2(pred(B),pred(A),aTP_Lamp_awf(fun(B,pred(A)),fun(pred(B),pred(A)),F)),A4)) ).

% Sup_bind
tff(fact_7415_Predicate_Obind__bind,axiom,
    ! [C: $tType,A: $tType,B: $tType,P: pred(C),Q: fun(C,pred(B)),R: fun(B,pred(A))] : bind2(B,A,bind2(C,B,P,Q),R) = bind2(C,A,P,aa(fun(B,pred(A)),fun(C,pred(A)),aTP_Lamp_awg(fun(C,pred(B)),fun(fun(B,pred(A)),fun(C,pred(A))),Q),R)) ).

% Predicate.bind_bind
tff(fact_7416_bottom__bind,axiom,
    ! [B: $tType,A: $tType,P: fun(B,pred(A))] : bind2(B,A,bot_bot(pred(B)),P) = bot_bot(pred(A)) ).

% bottom_bind
tff(fact_7417_singleton__sup,axiom,
    ! [A: $tType,Default: fun(product_unit,A),A4: pred(A),B3: pred(A)] :
      singleton(A,Default,aa(pred(A),pred(A),aa(pred(A),fun(pred(A),pred(A)),sup_sup(pred(A)),A4),B3)) = $ite(
        A4 = bot_bot(pred(A)),
        singleton(A,Default,B3),
        $ite(
          B3 = bot_bot(pred(A)),
          singleton(A,Default,A4),
          $ite(singleton(A,Default,A4) = singleton(A,Default,B3),singleton(A,Default,A4),aa(product_unit,A,Default,product_Unity)) ) ) ).

% singleton_sup
tff(fact_7418_singleton__bot,axiom,
    ! [A: $tType,Default: fun(product_unit,A)] : singleton(A,Default,bot_bot(pred(A))) = aa(product_unit,A,Default,product_Unity) ).

% singleton_bot
tff(fact_7419_subset_Osuc__Union__closed__empty,axiom,
    ! [A: $tType,A4: set(set(A))] : aa(set(set(set(A))),$o,member(set(set(A)),bot_bot(set(set(A)))),pred_s596693808085603175closed(set(A),A4,ord_less(set(A)))) ).

% subset.suc_Union_closed_empty
tff(fact_7420_pred__on_Osuc__Union__closed__empty,axiom,
    ! [A: $tType,A4: set(A),P: fun(A,fun(A,$o))] : aa(set(set(A)),$o,member(set(A),bot_bot(set(A))),pred_s596693808085603175closed(A,A4,P)) ).

% pred_on.suc_Union_closed_empty
tff(fact_7421_pred__on_Osuc__Union__closed__def,axiom,
    ! [A: $tType,A4: set(A),P: fun(A,fun(A,$o))] : pred_s596693808085603175closed(A,A4,P) = aa(fun(set(A),$o),set(set(A)),collect(set(A)),pred_s7749564232668923593losedp(A,A4,P)) ).

% pred_on.suc_Union_closed_def
tff(fact_7422_subset_Osuc__Union__closed__def,axiom,
    ! [A: $tType,A4: set(set(A))] : pred_s596693808085603175closed(set(A),A4,ord_less(set(A))) = aa(fun(set(set(A)),$o),set(set(set(A))),collect(set(set(A))),pred_s7749564232668923593losedp(set(A),A4,ord_less(set(A)))) ).

% subset.suc_Union_closed_def
tff(fact_7423_eval__bind,axiom,
    ! [A: $tType,B: $tType,P: pred(B),F: fun(B,pred(A))] : eval(A,bind2(B,A,P,F)) = eval(A,aa(set(pred(A)),pred(A),complete_Sup_Sup(pred(A)),aa(set(B),set(pred(A)),image2(B,pred(A),F),aa(fun(B,$o),set(B),collect(B),eval(B,P))))) ).

% eval_bind
tff(fact_7424_singleton__sup__aux,axiom,
    ! [A: $tType,Default: fun(product_unit,A),A4: pred(A),B3: pred(A)] :
      singleton(A,Default,aa(pred(A),pred(A),aa(pred(A),fun(pred(A),pred(A)),sup_sup(pred(A)),A4),B3)) = $ite(
        A4 = bot_bot(pred(A)),
        singleton(A,Default,B3),
        $ite(B3 = bot_bot(pred(A)),singleton(A,Default,A4),singleton(A,Default,aa(pred(A),pred(A),aa(pred(A),fun(pred(A),pred(A)),sup_sup(pred(A)),aa(A,pred(A),single(A),singleton(A,Default,A4))),aa(A,pred(A),single(A),singleton(A,Default,B3))))) ) ).

% singleton_sup_aux
tff(fact_7425_eval__inf,axiom,
    ! [A: $tType,P: pred(A),Q: pred(A)] : eval(A,aa(pred(A),pred(A),aa(pred(A),fun(pred(A),pred(A)),inf_inf(pred(A)),P),Q)) = aa(fun(A,$o),fun(A,$o),aa(fun(A,$o),fun(fun(A,$o),fun(A,$o)),inf_inf(fun(A,$o)),eval(A,P)),eval(A,Q)) ).

% eval_inf
tff(fact_7426_eval__top,axiom,
    ! [A: $tType] : eval(A,top_top(pred(A))) = top_top(fun(A,$o)) ).

% eval_top
tff(fact_7427_eval__compl,axiom,
    ! [A: $tType,P: pred(A)] : eval(A,aa(pred(A),pred(A),uminus_uminus(pred(A)),P)) = aa(fun(A,$o),fun(A,$o),uminus_uminus(fun(A,$o)),eval(A,P)) ).

% eval_compl
tff(fact_7428_eval__bot,axiom,
    ! [A: $tType] : eval(A,bot_bot(pred(A))) = bot_bot(fun(A,$o)) ).

% eval_bot
tff(fact_7429_eval__sup,axiom,
    ! [A: $tType,P: pred(A),Q: pred(A)] : eval(A,aa(pred(A),pred(A),aa(pred(A),fun(pred(A),pred(A)),sup_sup(pred(A)),P),Q)) = aa(fun(A,$o),fun(A,$o),aa(fun(A,$o),fun(fun(A,$o),fun(A,$o)),sup_sup(fun(A,$o)),eval(A,P)),eval(A,Q)) ).

% eval_sup
tff(fact_7430_supE,axiom,
    ! [A: $tType,A4: pred(A),B3: pred(A),X: A] :
      ( aa(A,$o,eval(A,aa(pred(A),pred(A),aa(pred(A),fun(pred(A),pred(A)),sup_sup(pred(A)),A4),B3)),X)
     => ( ~ aa(A,$o,eval(A,A4),X)
       => aa(A,$o,eval(A,B3),X) ) ) ).

% supE
tff(fact_7431_supI1,axiom,
    ! [A: $tType,A4: pred(A),X: A,B3: pred(A)] :
      ( aa(A,$o,eval(A,A4),X)
     => aa(A,$o,eval(A,aa(pred(A),pred(A),aa(pred(A),fun(pred(A),pred(A)),sup_sup(pred(A)),A4),B3)),X) ) ).

% supI1
tff(fact_7432_supI2,axiom,
    ! [A: $tType,B3: pred(A),X: A,A4: pred(A)] :
      ( aa(A,$o,eval(A,B3),X)
     => aa(A,$o,eval(A,aa(pred(A),pred(A),aa(pred(A),fun(pred(A),pred(A)),sup_sup(pred(A)),A4),B3)),X) ) ).

% supI2
tff(fact_7433_botE,axiom,
    ! [A: $tType,X: A] : ~ aa(A,$o,eval(A,bot_bot(pred(A))),X) ).

% botE
tff(fact_7434_not__bot,axiom,
    ! [A: $tType,A4: pred(A)] :
      ( ( A4 != bot_bot(pred(A)) )
     => ~ ! [X2: A] : ~ aa(A,$o,eval(A,A4),X2) ) ).

% not_bot
tff(fact_7435_single__not__bot,axiom,
    ! [A: $tType,X: A] : aa(A,pred(A),single(A),X) != bot_bot(pred(A)) ).

% single_not_bot
tff(fact_7436_unit__pred__cases,axiom,
    ! [P: fun(pred(product_unit),$o),Q: pred(product_unit)] :
      ( aa(pred(product_unit),$o,P,bot_bot(pred(product_unit)))
     => ( aa(pred(product_unit),$o,P,aa(product_unit,pred(product_unit),single(product_unit),product_Unity))
       => aa(pred(product_unit),$o,P,Q) ) ) ).

% unit_pred_cases
tff(fact_7437_Predicate_Obind__def,axiom,
    ! [A: $tType,B: $tType,P: pred(B),F: fun(B,pred(A))] : bind2(B,A,P,F) = aa(set(pred(A)),pred(A),complete_Sup_Sup(pred(A)),aa(set(B),set(pred(A)),image2(B,pred(A),F),aa(fun(B,$o),set(B),collect(B),eval(B,P)))) ).

% Predicate.bind_def
tff(fact_7438_singleton__sup__single__single,axiom,
    ! [A: $tType,Default: fun(product_unit,A),X: A,Y: A] :
      singleton(A,Default,aa(pred(A),pred(A),aa(pred(A),fun(pred(A),pred(A)),sup_sup(pred(A)),aa(A,pred(A),single(A),X)),aa(A,pred(A),single(A),Y))) = $ite(X = Y,X,aa(product_unit,A,Default,product_Unity)) ).

% singleton_sup_single_single
tff(fact_7439_Random__Pred_ORandom__def,axiom,
    ! [A: $tType,G: fun(product_prod(code_natural,code_natural),product_prod(product_prod(A,fun(product_unit,code_term)),product_prod(code_natural,code_natural)))] : random_Random(A,G) = product_scomp(product_prod(code_natural,code_natural),product_prod(A,fun(product_unit,code_term)),product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural)),G,aa(fun(product_prod(A,fun(product_unit,code_term)),pred(A)),fun(product_prod(A,fun(product_unit,code_term)),fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural)))),comp(pred(A),fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))),product_prod(A,fun(product_unit,code_term)),product_Pair(pred(A),product_prod(code_natural,code_natural))),aa(fun(product_prod(A,fun(product_unit,code_term)),A),fun(product_prod(A,fun(product_unit,code_term)),pred(A)),comp(A,pred(A),product_prod(A,fun(product_unit,code_term)),single(A)),product_fst(A,fun(product_unit,code_term))))) ).

% Random_Pred.Random_def
tff(fact_7440_pred__of__set__fold__sup,axiom,
    ! [A: $tType,A4: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( aa(set(A),pred(A),pred_of_set(A),A4) = finite_fold(pred(A),pred(A),sup_sup(pred(A)),bot_bot(pred(A)),aa(set(A),set(pred(A)),image2(A,pred(A),single(A)),A4)) ) ) ).

% pred_of_set_fold_sup
tff(fact_7441_pred__of__set__set__fold__sup,axiom,
    ! [A: $tType,Xs: list(A)] : aa(set(A),pred(A),pred_of_set(A),aa(list(A),set(A),set2(A),Xs)) = aa(pred(A),pred(A),fold(pred(A),pred(A),sup_sup(pred(A)),aa(list(A),list(pred(A)),map(A,pred(A),single(A)),Xs)),bot_bot(pred(A))) ).

% pred_of_set_set_fold_sup
tff(fact_7442_pred__of__set__set__foldr__sup,axiom,
    ! [A: $tType,Xs: list(A)] : aa(set(A),pred(A),pred_of_set(A),aa(list(A),set(A),set2(A),Xs)) = aa(pred(A),pred(A),foldr(pred(A),pred(A),sup_sup(pred(A)),aa(list(A),list(pred(A)),map(A,pred(A),single(A)),Xs)),bot_bot(pred(A))) ).

% pred_of_set_set_foldr_sup
tff(fact_7443_Random__Pred_Onot__randompred__def,axiom,
    ! [P: fun(product_prod(code_natural,code_natural),product_prod(pred(product_unit),product_prod(code_natural,code_natural))),X3: product_prod(code_natural,code_natural)] : random6974930770145893639ompred(P,X3) = aa(product_prod(pred(product_unit),product_prod(code_natural,code_natural)),product_prod(pred(product_unit),product_prod(code_natural,code_natural)),aa(fun(pred(product_unit),fun(product_prod(code_natural,code_natural),product_prod(pred(product_unit),product_prod(code_natural,code_natural)))),fun(product_prod(pred(product_unit),product_prod(code_natural,code_natural)),product_prod(pred(product_unit),product_prod(code_natural,code_natural))),product_case_prod(pred(product_unit),product_prod(code_natural,code_natural),product_prod(pred(product_unit),product_prod(code_natural,code_natural))),aTP_Lamp_awh(pred(product_unit),fun(product_prod(code_natural,code_natural),product_prod(pred(product_unit),product_prod(code_natural,code_natural))))),aa(product_prod(code_natural,code_natural),product_prod(pred(product_unit),product_prod(code_natural,code_natural)),P,X3)) ).

% Random_Pred.not_randompred_def
tff(fact_7444_Random__Pred_Osingle__def,axiom,
    ! [A: $tType,X: A] : random_single(A,X) = aa(pred(A),fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))),product_Pair(pred(A),product_prod(code_natural,code_natural)),aa(A,pred(A),single(A),X)) ).

% Random_Pred.single_def
tff(fact_7445_not__pred__eq,axiom,
    ! [P: pred(product_unit)] :
      not_pred(P) = $ite(aa(product_unit,$o,eval(product_unit,P),product_Unity),bot_bot(pred(product_unit)),aa(product_unit,pred(product_unit),single(product_unit),product_Unity)) ).

% not_pred_eq
tff(fact_7446_if__pred__eq,axiom,
    ! [B2: $o] :
      if_pred((B2)) = $ite((B2),aa(product_unit,pred(product_unit),single(product_unit),product_Unity),bot_bot(pred(product_unit))) ).

% if_pred_eq
tff(fact_7447_eval__map,axiom,
    ! [A: $tType,B: $tType,F: fun(B,A),P: pred(B)] : eval(A,aa(pred(B),pred(A),map2(B,A,F),P)) = aa(set(fun(A,$o)),fun(A,$o),complete_Sup_Sup(fun(A,$o)),aa(set(B),set(fun(A,$o)),image2(B,fun(A,$o),aTP_Lamp_awi(fun(B,A),fun(B,fun(A,$o)),F)),aa(fun(B,$o),set(B),collect(B),eval(B,P)))) ).

% eval_map
tff(fact_7448_the__eqI,axiom,
    ! [A: $tType,P: pred(A),X: A] :
      ( ( the(A,eval(A,P)) = X )
     => ( the3(A,P) = X ) ) ).

% the_eqI
tff(fact_7449_Predicate_Omap_Oidentity,axiom,
    ! [A: $tType] : map2(A,A,aTP_Lamp_au(A,A)) = id(pred(A)) ).

% Predicate.map.identity
tff(fact_7450_Predicate_Othe__def,axiom,
    ! [A: $tType,A4: pred(A)] : the3(A,A4) = the(A,eval(A,A4)) ).

% Predicate.the_def
tff(fact_7451_the__eq,axiom,
    ! [A: $tType,A4: pred(A)] : the3(A,A4) = singleton(A,aTP_Lamp_awk(pred(A),fun(product_unit,A),A4),A4) ).

% the_eq
tff(fact_7452_uminus__pred__def,axiom,
    ! [A: $tType,P: pred(A)] : aa(pred(A),pred(A),uminus_uminus(pred(A)),P) = aa(fun(A,$o),pred(A),pred3(A),aa(fun(A,$o),fun(A,$o),uminus_uminus(fun(A,$o)),eval(A,P))) ).

% uminus_pred_def
tff(fact_7453_pred__of__set__def,axiom,
    ! [A: $tType] : pred_of_set(A) = aa(fun(set(A),fun(A,$o)),fun(set(A),pred(A)),comp(fun(A,$o),pred(A),set(A),pred3(A)),aTP_Lamp_ag(set(A),fun(A,$o))) ).

% pred_of_set_def
tff(fact_7454_bot__pred__def,axiom,
    ! [A: $tType] : bot_bot(pred(A)) = aa(fun(A,$o),pred(A),pred3(A),bot_bot(fun(A,$o))) ).

% bot_pred_def
tff(fact_7455_not__predE,axiom,
    ! [P: $o,X: product_unit] :
      ( aa(product_unit,$o,eval(product_unit,not_pred(aa(fun(product_unit,$o),pred(product_unit),pred3(product_unit),aTP_Lamp_awl($o,fun(product_unit,$o),(P))))),X)
     => ~ (P) ) ).

% not_predE
tff(fact_7456_top__pred__def,axiom,
    ! [A: $tType] : top_top(pred(A)) = aa(fun(A,$o),pred(A),pred3(A),top_top(fun(A,$o))) ).

% top_pred_def
tff(fact_7457_sup__pred__def,axiom,
    ! [A: $tType,P: pred(A),Q: pred(A)] : aa(pred(A),pred(A),aa(pred(A),fun(pred(A),pred(A)),sup_sup(pred(A)),P),Q) = aa(fun(A,$o),pred(A),pred3(A),aa(fun(A,$o),fun(A,$o),aa(fun(A,$o),fun(fun(A,$o),fun(A,$o)),sup_sup(fun(A,$o)),eval(A,P)),eval(A,Q))) ).

% sup_pred_def
tff(fact_7458_not__predI,axiom,
    ! [P: $o] :
      ( ~ (P)
     => aa(product_unit,$o,eval(product_unit,not_pred(aa(fun(product_unit,$o),pred(product_unit),pred3(product_unit),aTP_Lamp_awl($o,fun(product_unit,$o),(P))))),product_Unity) ) ).

% not_predI
tff(fact_7459_inf__pred__def,axiom,
    ! [A: $tType,P: pred(A),Q: pred(A)] : aa(pred(A),pred(A),aa(pred(A),fun(pred(A),pred(A)),inf_inf(pred(A)),P),Q) = aa(fun(A,$o),pred(A),pred3(A),aa(fun(A,$o),fun(A,$o),aa(fun(A,$o),fun(fun(A,$o),fun(A,$o)),inf_inf(fun(A,$o)),eval(A,P)),eval(A,Q))) ).

% inf_pred_def
tff(fact_7460_contains__pred__def,axiom,
    ! [A: $tType,A4: set(A),X: A] :
      predic7144156976422707464s_pred(A,A4,X) = $ite(aa(set(A),$o,member(A,X),A4),aa(product_unit,pred(product_unit),single(product_unit),product_Unity),bot_bot(pred(product_unit))) ).

% contains_pred_def
tff(fact_7461_singleton__def,axiom,
    ! [A: $tType,Default: fun(product_unit,A),A4: pred(A)] :
      singleton(A,Default,A4) = $ite(
        ? [X4: A] :
          ( aa(A,$o,eval(A,A4),X4)
          & ! [Y3: A] :
              ( aa(A,$o,eval(A,A4),Y3)
             => ( Y3 = X4 ) ) ),
        the(A,eval(A,A4)),
        aa(product_unit,A,Default,product_Unity) ) ).

% singleton_def
tff(fact_7462_Nitpick_OEx1__unfold,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( ? [X4: A] :
          ( aa(A,$o,P,X4)
          & ! [Y3: A] :
              ( aa(A,$o,P,Y3)
             => ( Y3 = X4 ) ) )
    <=> ? [X4: A] : aa(fun(A,$o),set(A),collect(A),P) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X4),bot_bot(set(A))) ) ).

% Nitpick.Ex1_unfold
tff(fact_7463_Predicate__Compile_Ocontains__pred__eq,axiom,
    ! [A: $tType,X3: set(A),Xa3: A] : predic7144156976422707464s_pred(A,X3,Xa3) = aa(fun(product_unit,$o),pred(product_unit),pred3(product_unit),aa(A,fun(product_unit,$o),aTP_Lamp_awm(set(A),fun(A,fun(product_unit,$o)),X3),Xa3)) ).

% Predicate_Compile.contains_pred_eq
tff(fact_7464_THE__default__def,axiom,
    ! [A: $tType,D3: A,P: fun(A,$o)] :
      fun_THE_default(A,D3,P) = $ite(
        ? [X4: A] :
          ( aa(A,$o,P,X4)
          & ! [Y3: A] :
              ( aa(A,$o,P,Y3)
             => ( Y3 = X4 ) ) ),
        the(A,P),
        D3 ) ).

% THE_default_def
tff(fact_7465_fundef__ex1__uniqueness,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),D3: fun(A,B),G4: fun(A,fun(B,$o)),X: A,H2: fun(A,B)] :
      ( ! [X2: A] : aa(A,B,F,X2) = fun_THE_default(B,aa(A,B,D3,X2),aa(A,fun(B,$o),G4,X2))
     => ( ? [X3: B] :
            ( aa(B,$o,aa(A,fun(B,$o),G4,X),X3)
            & ! [Y2: B] :
                ( aa(B,$o,aa(A,fun(B,$o),G4,X),Y2)
               => ( Y2 = X3 ) ) )
       => ( aa(B,$o,aa(A,fun(B,$o),G4,X),aa(A,B,H2,X))
         => ( aa(A,B,H2,X) = aa(A,B,F,X) ) ) ) ) ).

% fundef_ex1_uniqueness
tff(fact_7466_fundef__ex1__existence,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),D3: fun(A,B),G4: fun(A,fun(B,$o)),X: A] :
      ( ! [X2: A] : aa(A,B,F,X2) = fun_THE_default(B,aa(A,B,D3,X2),aa(A,fun(B,$o),G4,X2))
     => ( ? [X3: B] :
            ( aa(B,$o,aa(A,fun(B,$o),G4,X),X3)
            & ! [Y2: B] :
                ( aa(B,$o,aa(A,fun(B,$o),G4,X),Y2)
               => ( Y2 = X3 ) ) )
       => aa(B,$o,aa(A,fun(B,$o),G4,X),aa(A,B,F,X)) ) ) ).

% fundef_ex1_existence
tff(fact_7467_fundef__default__value,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),D3: fun(A,B),G4: fun(A,fun(B,$o)),D4: fun(A,$o),X: A] :
      ( ! [X2: A] : aa(A,B,F,X2) = fun_THE_default(B,aa(A,B,D3,X2),aa(A,fun(B,$o),G4,X2))
     => ( ! [X2: A,Y2: B] :
            ( aa(B,$o,aa(A,fun(B,$o),G4,X2),Y2)
           => aa(A,$o,D4,X2) )
       => ( ~ aa(A,$o,D4,X)
         => ( aa(A,B,F,X) = aa(A,B,D3,X) ) ) ) ) ).

% fundef_default_value
tff(fact_7468_fundef__ex1__iff,axiom,
    ! [A: $tType,B: $tType,F: fun(A,B),D3: fun(A,B),G4: fun(A,fun(B,$o)),X: A,Y: B] :
      ( ! [X2: A] : aa(A,B,F,X2) = fun_THE_default(B,aa(A,B,D3,X2),aa(A,fun(B,$o),G4,X2))
     => ( ? [X3: B] :
            ( aa(B,$o,aa(A,fun(B,$o),G4,X),X3)
            & ! [Y2: B] :
                ( aa(B,$o,aa(A,fun(B,$o),G4,X),Y2)
               => ( Y2 = X3 ) ) )
       => ( aa(B,$o,aa(A,fun(B,$o),G4,X),Y)
        <=> ( aa(A,B,F,X) = Y ) ) ) ) ).

% fundef_ex1_iff
tff(fact_7469_is__num_Ocases,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [A3: A] :
          ( neg_numeral_is_num(A,A3)
         => ( ( A3 != one_one(A) )
           => ( ! [X2: A] :
                  ( ( A3 = aa(A,A,uminus_uminus(A),X2) )
                 => ~ neg_numeral_is_num(A,X2) )
             => ~ ! [X2: A,Y2: A] :
                    ( ( A3 = aa(A,A,aa(A,fun(A,A),plus_plus(A),X2),Y2) )
                   => ( neg_numeral_is_num(A,X2)
                     => ~ neg_numeral_is_num(A,Y2) ) ) ) ) ) ) ).

% is_num.cases
tff(fact_7470_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,Y3: A] :
                ( ( A3 = aa(A,A,aa(A,fun(A,A),plus_plus(A),X4),Y3) )
                & neg_numeral_is_num(A,X4)
                & neg_numeral_is_num(A,Y3) ) ) ) ) ).

% is_num.simps
tff(fact_7471_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_7472_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_7473_open__typedef__to__part__equivp,axiom,
    ! [A: $tType,B: $tType,Rep: fun(A,B),Abs: fun(B,A),P: fun(B,$o)] :
      ( type_definition(A,B,Rep,Abs,aa(fun(B,$o),set(B),collect(B),P))
     => equiv_part_equivp(B,bNF_eq_onp(B,P)) ) ).

% open_typedef_to_part_equivp
tff(fact_7474_typedef__to__part__equivp,axiom,
    ! [A: $tType,B: $tType,Rep: fun(A,B),Abs: fun(B,A),S: set(B)] :
      ( type_definition(A,B,Rep,Abs,S)
     => equiv_part_equivp(B,bNF_eq_onp(B,aTP_Lamp_aks(set(B),fun(B,$o),S))) ) ).

% typedef_to_part_equivp
tff(fact_7475_part__equivp__typedef,axiom,
    ! [A: $tType,R: fun(A,fun(A,$o))] :
      ( equiv_part_equivp(A,R)
     => ? [D2: set(A)] : aa(set(set(A)),$o,member(set(A),D2),aa(fun(set(A),$o),set(set(A)),collect(set(A)),aTP_Lamp_awn(fun(A,fun(A,$o)),fun(set(A),$o),R))) ) ).

% part_equivp_typedef
tff(fact_7476_snds__def,axiom,
    ! [B: $tType,A: $tType,X3: product_prod(A,B)] : basic_snds(A,B,X3) = aa(fun(B,$o),set(B),collect(B),basic_sndsp(A,B,X3)) ).

% snds_def
tff(fact_7477_nat__of__num__code_I3_J,axiom,
    ! [N: num] :
      nat_of_num(aa(num,num,bit1,N)) = $let(
        m: nat,
        m:= nat_of_num(N),
        aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),m),m)) ) ).

% nat_of_num_code(3)
tff(fact_7478_nat__of__num__code_I1_J,axiom,
    nat_of_num(one2) = one_one(nat) ).

% nat_of_num_code(1)
tff(fact_7479_nat__of__num__code_I2_J,axiom,
    ! [N: num] :
      nat_of_num(aa(num,num,bit0,N)) = $let(
        m: nat,
        m:= nat_of_num(N),
        aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),m),m) ) ).

% nat_of_num_code(2)
tff(fact_7480_setr__def,axiom,
    ! [B: $tType,A: $tType,X3: sum_sum(A,B)] : aa(sum_sum(A,B),set(B),basic_setr(A,B),X3) = aa(fun(B,$o),set(B),collect(B),basic_setrp(A,B,X3)) ).

% setr_def
tff(fact_7481_setl__def,axiom,
    ! [B: $tType,A: $tType,X3: sum_sum(A,B)] : aa(sum_sum(A,B),set(A),basic_setl(A,B),X3) = aa(fun(A,$o),set(A),collect(A),basic_setlp(A,B,X3)) ).

% setl_def
tff(fact_7482_fsts__def,axiom,
    ! [B: $tType,A: $tType,X3: product_prod(A,B)] : basic_fsts(A,B,X3) = aa(fun(A,$o),set(A),collect(A),basic_fstsp(A,B,X3)) ).

% fsts_def
tff(fact_7483_pick__middlep__def,axiom,
    ! [B: $tType,A: $tType,C: $tType,P: fun(B,fun(A,$o)),Q: fun(A,fun(C,$o)),A3: B,C2: C] : bNF_pick_middlep(B,A,C,P,Q,A3,C2) = fChoice(A,aa(C,fun(A,$o),aa(B,fun(C,fun(A,$o)),aa(fun(A,fun(C,$o)),fun(B,fun(C,fun(A,$o))),aTP_Lamp_awo(fun(B,fun(A,$o)),fun(fun(A,fun(C,$o)),fun(B,fun(C,fun(A,$o)))),P),Q),A3),C2)) ).

% pick_middlep_def
tff(fact_7484_sndOp__def,axiom,
    ! [A: $tType,B: $tType,C: $tType,P: fun(C,fun(A,$o)),Q: fun(A,fun(B,$o)),Ac: product_prod(C,B)] : bNF_sndOp(C,A,B,P,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,P,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_7485_fstOp__def,axiom,
    ! [B: $tType,C: $tType,A: $tType,P: fun(A,fun(B,$o)),Q: fun(B,fun(C,$o)),Ac: product_prod(A,C)] : bNF_fstOp(A,B,C,P,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,P,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_7486_inv__on__def,axiom,
    ! [A: $tType,B: $tType,F: fun(A,B),A4: set(A),X: B] : inv_on(A,B,F,A4,X) = fChoice(A,aa(B,fun(A,$o),aa(set(A),fun(B,fun(A,$o)),aTP_Lamp_acw(fun(A,B),fun(set(A),fun(B,fun(A,$o))),F),A4),X)) ).

% inv_on_def
tff(fact_7487_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_7488_prod_H__def,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ( groups1962203154675924110t_prod(A,B) = groups_comm_monoid_G(B,A,times_times(B),one_one(B)) ) ) ).

% prod'_def
tff(fact_7489_subset__mset_Osemilattice__order__axioms,axiom,
    ! [A: $tType] : semilattice_order(multiset(A),union_mset(A),aTP_Lamp_aso(multiset(A),fun(multiset(A),$o)),aTP_Lamp_asp(multiset(A),fun(multiset(A),$o))) ).

% subset_mset.semilattice_order_axioms
tff(fact_7490_semilattice__neutr__order_Ointro,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z2: A,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o))] :
      ( semilattice_neutr(A,F,Z2)
     => ( semilattice_order(A,F,Less_eq,Less)
       => semila1105856199041335345_order(A,F,Z2,Less_eq,Less) ) ) ).

% semilattice_neutr_order.intro
tff(fact_7491_semilattice__neutr__order__def,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z2: A,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o))] :
      ( semila1105856199041335345_order(A,F,Z2,Less_eq,Less)
    <=> ( semilattice_neutr(A,F,Z2)
        & semilattice_order(A,F,Less_eq,Less) ) ) ).

% semilattice_neutr_order_def
tff(fact_7492_semilattice__neutr__order_Oaxioms_I2_J,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z2: A,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o))] :
      ( semila1105856199041335345_order(A,F,Z2,Less_eq,Less)
     => semilattice_order(A,F,Less_eq,Less) ) ).

% semilattice_neutr_order.axioms(2)
tff(fact_7493_semilattice__order_Ostrict__coboundedI2,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),B2: A,C2: A,A3: A] :
      ( semilattice_order(A,F,Less_eq,Less)
     => ( aa(A,$o,aa(A,fun(A,$o),Less,B2),C2)
       => aa(A,$o,aa(A,fun(A,$o),Less,aa(A,A,aa(A,fun(A,A),F,A3),B2)),C2) ) ) ).

% semilattice_order.strict_coboundedI2
tff(fact_7494_semilattice__order_Ostrict__coboundedI1,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A3: A,C2: A,B2: A] :
      ( semilattice_order(A,F,Less_eq,Less)
     => ( aa(A,$o,aa(A,fun(A,$o),Less,A3),C2)
       => aa(A,$o,aa(A,fun(A,$o),Less,aa(A,A,aa(A,fun(A,A),F,A3),B2)),C2) ) ) ).

% semilattice_order.strict_coboundedI1
tff(fact_7495_semilattice__order_Ostrict__order__iff,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A3: A,B2: A] :
      ( semilattice_order(A,F,Less_eq,Less)
     => ( aa(A,$o,aa(A,fun(A,$o),Less,A3),B2)
      <=> ( ( A3 = aa(A,A,aa(A,fun(A,A),F,A3),B2) )
          & ( A3 != B2 ) ) ) ) ).

% semilattice_order.strict_order_iff
tff(fact_7496_semilattice__order_Ostrict__boundedE,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A3: A,B2: A,C2: A] :
      ( semilattice_order(A,F,Less_eq,Less)
     => ( aa(A,$o,aa(A,fun(A,$o),Less,A3),aa(A,A,aa(A,fun(A,A),F,B2),C2))
       => ~ ( aa(A,$o,aa(A,fun(A,$o),Less,A3),B2)
           => ~ aa(A,$o,aa(A,fun(A,$o),Less,A3),C2) ) ) ) ).

% semilattice_order.strict_boundedE
tff(fact_7497_semilattice__order_OcoboundedI2,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),B2: A,C2: A,A3: A] :
      ( semilattice_order(A,F,Less_eq,Less)
     => ( aa(A,$o,aa(A,fun(A,$o),Less_eq,B2),C2)
       => aa(A,$o,aa(A,fun(A,$o),Less_eq,aa(A,A,aa(A,fun(A,A),F,A3),B2)),C2) ) ) ).

% semilattice_order.coboundedI2
tff(fact_7498_semilattice__order_OcoboundedI1,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A3: A,C2: A,B2: A] :
      ( semilattice_order(A,F,Less_eq,Less)
     => ( aa(A,$o,aa(A,fun(A,$o),Less_eq,A3),C2)
       => aa(A,$o,aa(A,fun(A,$o),Less_eq,aa(A,A,aa(A,fun(A,A),F,A3),B2)),C2) ) ) ).

% semilattice_order.coboundedI1
tff(fact_7499_semilattice__order_Obounded__iff,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A3: A,B2: A,C2: A] :
      ( semilattice_order(A,F,Less_eq,Less)
     => ( aa(A,$o,aa(A,fun(A,$o),Less_eq,A3),aa(A,A,aa(A,fun(A,A),F,B2),C2))
      <=> ( aa(A,$o,aa(A,fun(A,$o),Less_eq,A3),B2)
          & aa(A,$o,aa(A,fun(A,$o),Less_eq,A3),C2) ) ) ) ).

% semilattice_order.bounded_iff
tff(fact_7500_semilattice__order_Oabsorb__iff2,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),B2: A,A3: A] :
      ( semilattice_order(A,F,Less_eq,Less)
     => ( aa(A,$o,aa(A,fun(A,$o),Less_eq,B2),A3)
      <=> ( aa(A,A,aa(A,fun(A,A),F,A3),B2) = B2 ) ) ) ).

% semilattice_order.absorb_iff2
tff(fact_7501_semilattice__order_Oabsorb__iff1,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A3: A,B2: A] :
      ( semilattice_order(A,F,Less_eq,Less)
     => ( aa(A,$o,aa(A,fun(A,$o),Less_eq,A3),B2)
      <=> ( aa(A,A,aa(A,fun(A,A),F,A3),B2) = A3 ) ) ) ).

% semilattice_order.absorb_iff1
tff(fact_7502_semilattice__order_Ocobounded2,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A3: A,B2: A] :
      ( semilattice_order(A,F,Less_eq,Less)
     => aa(A,$o,aa(A,fun(A,$o),Less_eq,aa(A,A,aa(A,fun(A,A),F,A3),B2)),B2) ) ).

% semilattice_order.cobounded2
tff(fact_7503_semilattice__order_Ocobounded1,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A3: A,B2: A] :
      ( semilattice_order(A,F,Less_eq,Less)
     => aa(A,$o,aa(A,fun(A,$o),Less_eq,aa(A,A,aa(A,fun(A,A),F,A3),B2)),A3) ) ).

% semilattice_order.cobounded1
tff(fact_7504_semilattice__order_Oorder__iff,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A3: A,B2: A] :
      ( semilattice_order(A,F,Less_eq,Less)
     => ( aa(A,$o,aa(A,fun(A,$o),Less_eq,A3),B2)
      <=> ( A3 = aa(A,A,aa(A,fun(A,A),F,A3),B2) ) ) ) ).

% semilattice_order.order_iff
tff(fact_7505_semilattice__order_OboundedI,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A3: A,B2: A,C2: A] :
      ( semilattice_order(A,F,Less_eq,Less)
     => ( aa(A,$o,aa(A,fun(A,$o),Less_eq,A3),B2)
       => ( aa(A,$o,aa(A,fun(A,$o),Less_eq,A3),C2)
         => aa(A,$o,aa(A,fun(A,$o),Less_eq,A3),aa(A,A,aa(A,fun(A,A),F,B2),C2)) ) ) ) ).

% semilattice_order.boundedI
tff(fact_7506_semilattice__order_OboundedE,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A3: A,B2: A,C2: A] :
      ( semilattice_order(A,F,Less_eq,Less)
     => ( aa(A,$o,aa(A,fun(A,$o),Less_eq,A3),aa(A,A,aa(A,fun(A,A),F,B2),C2))
       => ~ ( aa(A,$o,aa(A,fun(A,$o),Less_eq,A3),B2)
           => ~ aa(A,$o,aa(A,fun(A,$o),Less_eq,A3),C2) ) ) ) ).

% semilattice_order.boundedE
tff(fact_7507_semilattice__order_Oabsorb4,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),B2: A,A3: A] :
      ( semilattice_order(A,F,Less_eq,Less)
     => ( aa(A,$o,aa(A,fun(A,$o),Less,B2),A3)
       => ( aa(A,A,aa(A,fun(A,A),F,A3),B2) = B2 ) ) ) ).

% semilattice_order.absorb4
tff(fact_7508_semilattice__order_Oabsorb3,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A3: A,B2: A] :
      ( semilattice_order(A,F,Less_eq,Less)
     => ( aa(A,$o,aa(A,fun(A,$o),Less,A3),B2)
       => ( aa(A,A,aa(A,fun(A,A),F,A3),B2) = A3 ) ) ) ).

% semilattice_order.absorb3
tff(fact_7509_semilattice__order_Oabsorb2,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),B2: A,A3: A] :
      ( semilattice_order(A,F,Less_eq,Less)
     => ( aa(A,$o,aa(A,fun(A,$o),Less_eq,B2),A3)
       => ( aa(A,A,aa(A,fun(A,A),F,A3),B2) = B2 ) ) ) ).

% semilattice_order.absorb2
tff(fact_7510_semilattice__order_Oabsorb1,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A3: A,B2: A] :
      ( semilattice_order(A,F,Less_eq,Less)
     => ( aa(A,$o,aa(A,fun(A,$o),Less_eq,A3),B2)
       => ( aa(A,A,aa(A,fun(A,A),F,A3),B2) = A3 ) ) ) ).

% semilattice_order.absorb1
tff(fact_7511_semilattice__order_OorderI,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A3: A,B2: A] :
      ( semilattice_order(A,F,Less_eq,Less)
     => ( ( A3 = aa(A,A,aa(A,fun(A,A),F,A3),B2) )
       => aa(A,$o,aa(A,fun(A,$o),Less_eq,A3),B2) ) ) ).

% semilattice_order.orderI
tff(fact_7512_semilattice__order_OorderE,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A3: A,B2: A] :
      ( semilattice_order(A,F,Less_eq,Less)
     => ( aa(A,$o,aa(A,fun(A,$o),Less_eq,A3),B2)
       => ( A3 = aa(A,A,aa(A,fun(A,A),F,A3),B2) ) ) ) ).

% semilattice_order.orderE
tff(fact_7513_semilattice__order_Omono,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A3: A,C2: A,B2: A,D3: A] :
      ( semilattice_order(A,F,Less_eq,Less)
     => ( aa(A,$o,aa(A,fun(A,$o),Less_eq,A3),C2)
       => ( aa(A,$o,aa(A,fun(A,$o),Less_eq,B2),D3)
         => aa(A,$o,aa(A,fun(A,$o),Less_eq,aa(A,A,aa(A,fun(A,A),F,A3),B2)),aa(A,A,aa(A,fun(A,A),F,C2),D3)) ) ) ) ).

% semilattice_order.mono
tff(fact_7514_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_7515_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_7516_sup_Osemilattice__order__axioms,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => semilattice_order(A,sup_sup(A),aTP_Lamp_aob(A,fun(A,$o)),aTP_Lamp_aoc(A,fun(A,$o))) ) ).

% sup.semilattice_order_axioms
tff(fact_7517_max_Osemilattice__order__axioms,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => semilattice_order(A,ord_max(A),aTP_Lamp_tm(A,fun(A,$o)),aTP_Lamp_ano(A,fun(A,$o))) ) ).

% max.semilattice_order_axioms
tff(fact_7518_Suc__natural__minus__one,axiom,
    ! [N: code_natural] : aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),minus_minus(code_natural),code_Suc(N)),one_one(code_natural)) = N ).

% Suc_natural_minus_one
tff(fact_7519_comp__fun__idem__def_H,axiom,
    ! [B: $tType,A: $tType,F: fun(A,fun(B,B))] :
      ( finite_comp_fun_idem(A,B,F)
    <=> finite673082921795544331dem_on(A,B,top_top(set(A)),F) ) ).

% comp_fun_idem_def'
tff(fact_7520_gcd__nat_Osemilattice__order__axioms,axiom,
    semilattice_order(nat,gcd_gcd(nat),dvd_dvd(nat),aTP_Lamp_cc(nat,fun(nat,$o))) ).

% gcd_nat.semilattice_order_axioms
tff(fact_7521_comp__fun__idem__inf,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => finite_comp_fun_idem(A,A,inf_inf(A)) ) ).

% comp_fun_idem_inf
tff(fact_7522_comp__fun__idem__sup,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => finite_comp_fun_idem(A,A,sup_sup(A)) ) ).

% comp_fun_idem_sup
tff(fact_7523_random__aux__set_Ocases,axiom,
    ! [X: product_prod(code_natural,code_natural)] :
      ( ! [J2: code_natural] : X != 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),zero_zero(code_natural)),J2)
     => ~ ! [I2: code_natural,J2: code_natural] : X != 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),code_Suc(I2)),J2) ) ).

% random_aux_set.cases
tff(fact_7524_prod__mset__def,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ( comm_m9189036328036947845d_mset(A) = comm_monoid_F(A,times_times(A),one_one(A)) ) ) ).

% prod_mset_def
tff(fact_7525_comm__monoid__mset_Oneutral__const,axiom,
    ! [B: $tType,A: $tType,F: fun(A,fun(A,A)),Z2: A,A4: multiset(B)] :
      ( comm_monoid_mset(A,F,Z2)
     => ( aa(multiset(A),A,comm_monoid_F(A,F,Z2),aa(multiset(B),multiset(A),image_mset(B,A,aa(A,fun(B,A),aTP_Lamp_kf(A,fun(B,A)),Z2)),A4)) = Z2 ) ) ).

% comm_monoid_mset.neutral_const
tff(fact_7526_comm__monoid__mset_Odistrib,axiom,
    ! [A: $tType,B: $tType,F: fun(A,fun(A,A)),Z2: A,G: fun(B,A),H2: fun(B,A),A4: multiset(B)] :
      ( comm_monoid_mset(A,F,Z2)
     => ( aa(multiset(A),A,comm_monoid_F(A,F,Z2),aa(multiset(B),multiset(A),image_mset(B,A,aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_awp(fun(A,fun(A,A)),fun(fun(B,A),fun(fun(B,A),fun(B,A))),F),G),H2)),A4)) = aa(A,A,aa(A,fun(A,A),F,aa(multiset(A),A,comm_monoid_F(A,F,Z2),aa(multiset(B),multiset(A),image_mset(B,A,G),A4))),aa(multiset(A),A,comm_monoid_F(A,F,Z2),aa(multiset(B),multiset(A),image_mset(B,A,H2),A4))) ) ) ).

% comm_monoid_mset.distrib
tff(fact_7527_comm__monoid__mset__def,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
      ( comm_monoid_mset(A,F,Z2)
    <=> comm_monoid(A,F,Z2) ) ).

% comm_monoid_mset_def
tff(fact_7528_comm__monoid__mset_Oaxioms,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
      ( comm_monoid_mset(A,F,Z2)
     => comm_monoid(A,F,Z2) ) ).

% comm_monoid_mset.axioms
tff(fact_7529_comm__monoid__mset_Ointro,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
      ( comm_monoid(A,F,Z2)
     => comm_monoid_mset(A,F,Z2) ) ).

% comm_monoid_mset.intro
tff(fact_7530_prod__mset_Ocomm__monoid__mset__axioms,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => comm_monoid_mset(A,times_times(A),one_one(A)) ) ).

% prod_mset.comm_monoid_mset_axioms
tff(fact_7531_comm__monoid__mset_Oswap,axiom,
    ! [B: $tType,A: $tType,C: $tType,F: fun(A,fun(A,A)),Z2: A,G: fun(B,fun(C,A)),B3: multiset(C),A4: multiset(B)] :
      ( comm_monoid_mset(A,F,Z2)
     => ( aa(multiset(A),A,comm_monoid_F(A,F,Z2),aa(multiset(B),multiset(A),image_mset(B,A,aa(multiset(C),fun(B,A),aa(fun(B,fun(C,A)),fun(multiset(C),fun(B,A)),aa(A,fun(fun(B,fun(C,A)),fun(multiset(C),fun(B,A))),aTP_Lamp_awq(fun(A,fun(A,A)),fun(A,fun(fun(B,fun(C,A)),fun(multiset(C),fun(B,A)))),F),Z2),G),B3)),A4)) = aa(multiset(A),A,comm_monoid_F(A,F,Z2),aa(multiset(C),multiset(A),image_mset(C,A,aa(multiset(B),fun(C,A),aa(fun(B,fun(C,A)),fun(multiset(B),fun(C,A)),aa(A,fun(fun(B,fun(C,A)),fun(multiset(B),fun(C,A))),aTP_Lamp_aws(fun(A,fun(A,A)),fun(A,fun(fun(B,fun(C,A)),fun(multiset(B),fun(C,A)))),F),Z2),G),A4)),B3)) ) ) ).

% comm_monoid_mset.swap
tff(fact_7532_or__num_Opelims,axiom,
    ! [X: num,Xa: num,Y: num] :
      ( ( bit_un6697907153464112080or_num(X,Xa) = Y )
     => ( accp(product_prod(num,num),bit_un4773296044027857193um_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),X),Xa))
       => ( ( ( X = one2 )
           => ( ( Xa = one2 )
             => ( ( Y = one2 )
               => ~ accp(product_prod(num,num),bit_un4773296044027857193um_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),one2)) ) ) )
         => ( ( ( X = one2 )
             => ! [N3: num] :
                  ( ( Xa = aa(num,num,bit0,N3) )
                 => ( ( Y = aa(num,num,bit1,N3) )
                   => ~ accp(product_prod(num,num),bit_un4773296044027857193um_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),aa(num,num,bit0,N3))) ) ) )
           => ( ( ( X = one2 )
               => ! [N3: num] :
                    ( ( Xa = aa(num,num,bit1,N3) )
                   => ( ( Y = aa(num,num,bit1,N3) )
                     => ~ accp(product_prod(num,num),bit_un4773296044027857193um_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),aa(num,num,bit1,N3))) ) ) )
             => ( ! [M3: num] :
                    ( ( X = aa(num,num,bit0,M3) )
                   => ( ( Xa = one2 )
                     => ( ( Y = aa(num,num,bit1,M3) )
                       => ~ accp(product_prod(num,num),bit_un4773296044027857193um_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit0,M3)),one2)) ) ) )
               => ( ! [M3: num] :
                      ( ( X = aa(num,num,bit0,M3) )
                     => ! [N3: num] :
                          ( ( Xa = aa(num,num,bit0,N3) )
                         => ( ( Y = aa(num,num,bit0,bit_un6697907153464112080or_num(M3,N3)) )
                           => ~ accp(product_prod(num,num),bit_un4773296044027857193um_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit0,M3)),aa(num,num,bit0,N3))) ) ) )
                 => ( ! [M3: num] :
                        ( ( X = aa(num,num,bit0,M3) )
                       => ! [N3: num] :
                            ( ( Xa = aa(num,num,bit1,N3) )
                           => ( ( Y = aa(num,num,bit1,bit_un6697907153464112080or_num(M3,N3)) )
                             => ~ accp(product_prod(num,num),bit_un4773296044027857193um_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit0,M3)),aa(num,num,bit1,N3))) ) ) )
                   => ( ! [M3: num] :
                          ( ( X = aa(num,num,bit1,M3) )
                         => ( ( Xa = one2 )
                           => ( ( Y = aa(num,num,bit1,M3) )
                             => ~ accp(product_prod(num,num),bit_un4773296044027857193um_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M3)),one2)) ) ) )
                     => ( ! [M3: num] :
                            ( ( X = aa(num,num,bit1,M3) )
                           => ! [N3: num] :
                                ( ( Xa = aa(num,num,bit0,N3) )
                               => ( ( Y = aa(num,num,bit1,bit_un6697907153464112080or_num(M3,N3)) )
                                 => ~ accp(product_prod(num,num),bit_un4773296044027857193um_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M3)),aa(num,num,bit0,N3))) ) ) )
                       => ~ ! [M3: num] :
                              ( ( X = aa(num,num,bit1,M3) )
                             => ! [N3: num] :
                                  ( ( Xa = aa(num,num,bit1,N3) )
                                 => ( ( Y = aa(num,num,bit1,bit_un6697907153464112080or_num(M3,N3)) )
                                   => ~ accp(product_prod(num,num),bit_un4773296044027857193um_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M3)),aa(num,num,bit1,N3))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% or_num.pelims
tff(fact_7533_bdd__below__primitive__def,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ( condit1013018076250108175_below(A) = condit16957441358409770ng_bdd(A,aTP_Lamp_avn(A,fun(A,$o))) ) ) ).

% bdd_below_primitive_def
tff(fact_7534_subset__mset_Obdd__below__primitive__def,axiom,
    ! [A: $tType] : condit8119078960628432327_below(multiset(A),subseteq_mset(A)) = condit16957441358409770ng_bdd(multiset(A),aTP_Lamp_aso(multiset(A),fun(multiset(A),$o))) ).

% subset_mset.bdd_below_primitive_def
tff(fact_7535_option_Osimps_I15_J,axiom,
    ! [A: $tType,X22: A] : set_option(A,aa(A,option(A),some(A),X22)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X22),bot_bot(set(A))) ).

% option.simps(15)
tff(fact_7536_those_Osimps_I2_J,axiom,
    ! [A: $tType,X: option(A),Xs: list(option(A))] : those(A,aa(list(option(A)),list(option(A)),aa(option(A),fun(list(option(A)),list(option(A))),cons(option(A)),X),Xs)) = case_option(option(list(A)),A,none(list(A)),aTP_Lamp_awt(list(option(A)),fun(A,option(list(A))),Xs),X) ).

% those.simps(2)
tff(fact_7537_set__empty__eq,axiom,
    ! [A: $tType,Xo: option(A)] :
      ( ( set_option(A,Xo) = bot_bot(set(A)) )
    <=> ( Xo = none(A) ) ) ).

% set_empty_eq
tff(fact_7538_option_Osimps_I14_J,axiom,
    ! [A: $tType] : set_option(A,none(A)) = bot_bot(set(A)) ).

% option.simps(14)
tff(fact_7539_option_Orel__Grp,axiom,
    ! [B: $tType,A: $tType,A4: set(A),F: fun(A,B)] : rel_option(A,B,bNF_Grp(A,B,A4,F)) = bNF_Grp(option(A),option(B),aa(fun(option(A),$o),set(option(A)),collect(option(A)),aTP_Lamp_awu(set(A),fun(option(A),$o),A4)),map_option(A,B,F)) ).

% option.rel_Grp
tff(fact_7540_option_Oin__rel,axiom,
    ! [A: $tType,B: $tType,R: fun(A,fun(B,$o)),A3: option(A),B2: option(B)] :
      ( aa(option(B),$o,aa(option(A),fun(option(B),$o),rel_option(A,B,R),A3),B2)
    <=> ? [Z4: option(product_prod(A,B))] :
          ( aa(set(option(product_prod(A,B))),$o,member(option(product_prod(A,B)),Z4),aa(fun(option(product_prod(A,B)),$o),set(option(product_prod(A,B))),collect(option(product_prod(A,B))),aTP_Lamp_awv(fun(A,fun(B,$o)),fun(option(product_prod(A,B)),$o),R)))
          & ( aa(option(product_prod(A,B)),option(A),map_option(product_prod(A,B),A,product_fst(A,B)),Z4) = A3 )
          & ( aa(option(product_prod(A,B)),option(B),map_option(product_prod(A,B),B,product_snd(A,B)),Z4) = B2 ) ) ) ).

% option.in_rel
tff(fact_7541_option_Orel__map_I2_J,axiom,
    ! [A: $tType,B: $tType,C: $tType,Sa: fun(A,fun(B,$o)),X: option(A),G: fun(C,B),Y: option(C)] :
      ( aa(option(B),$o,aa(option(A),fun(option(B),$o),rel_option(A,B,Sa),X),aa(option(C),option(B),map_option(C,B,G),Y))
    <=> aa(option(C),$o,aa(option(A),fun(option(C),$o),rel_option(A,C,aa(fun(C,B),fun(A,fun(C,$o)),aTP_Lamp_aiu(fun(A,fun(B,$o)),fun(fun(C,B),fun(A,fun(C,$o))),Sa),G)),X),Y) ) ).

% option.rel_map(2)
tff(fact_7542_option_Orel__map_I1_J,axiom,
    ! [C: $tType,A: $tType,B: $tType,Sb: fun(A,fun(B,$o)),I: fun(C,A),X: option(C),Y: option(B)] :
      ( aa(option(B),$o,aa(option(A),fun(option(B),$o),rel_option(A,B,Sb),aa(option(C),option(A),map_option(C,A,I),X)),Y)
    <=> aa(option(B),$o,aa(option(C),fun(option(B),$o),rel_option(C,B,aa(fun(C,A),fun(C,fun(B,$o)),aTP_Lamp_aiv(fun(A,fun(B,$o)),fun(fun(C,A),fun(C,fun(B,$o))),Sb),I)),X),Y) ) ).

% option.rel_map(1)
tff(fact_7543_option_Odisc__transfer_I2_J,axiom,
    ! [A: $tType,B: $tType,R: fun(A,fun(B,$o))] : aa(fun(option(B),$o),$o,aa(fun(option(A),$o),fun(fun(option(B),$o),$o),bNF_rel_fun(option(A),option(B),$o,$o,rel_option(A,B,R),fequal($o)),aTP_Lamp_aww(option(A),$o)),aTP_Lamp_awx(option(B),$o)) ).

% option.disc_transfer(2)
tff(fact_7544_option_Odisc__transfer_I1_J,axiom,
    ! [A: $tType,B: $tType,R: fun(A,fun(B,$o))] : aa(fun(option(B),$o),$o,aa(fun(option(A),$o),fun(fun(option(B),$o),$o),bNF_rel_fun(option(A),option(B),$o,$o,rel_option(A,B,R),fequal($o)),aTP_Lamp_awy(option(A),$o)),aTP_Lamp_awz(option(B),$o)) ).

% option.disc_transfer(1)
tff(fact_7545_rel__option__iff,axiom,
    ! [A: $tType,B: $tType,R: fun(A,fun(B,$o)),X: option(A),Y: option(B)] :
      ( aa(option(B),$o,aa(option(A),fun(option(B),$o),rel_option(A,B,R),X),Y)
    <=> aa(product_prod(option(A),option(B)),$o,aa(fun(option(A),fun(option(B),$o)),fun(product_prod(option(A),option(B)),$o),product_case_prod(option(A),option(B),$o),aTP_Lamp_axc(fun(A,fun(B,$o)),fun(option(A),fun(option(B),$o)),R)),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_7546_rel__option__inf,axiom,
    ! [B: $tType,A: $tType,A4: fun(A,fun(B,$o)),B3: fun(A,fun(B,$o))] : aa(fun(option(A),fun(option(B),$o)),fun(option(A),fun(option(B),$o)),aa(fun(option(A),fun(option(B),$o)),fun(fun(option(A),fun(option(B),$o)),fun(option(A),fun(option(B),$o))),inf_inf(fun(option(A),fun(option(B),$o))),rel_option(A,B,A4)),rel_option(A,B,B3)) = rel_option(A,B,aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),inf_inf(fun(A,fun(B,$o))),A4),B3)) ).

% rel_option_inf
tff(fact_7547_option_Orel__compp__Grp,axiom,
    ! [B: $tType,A: $tType,R: fun(A,fun(B,$o))] : rel_option(A,B,R) = aa(fun(option(product_prod(A,B)),fun(option(B),$o)),fun(option(A),fun(option(B),$o)),aa(fun(option(A),fun(option(product_prod(A,B)),$o)),fun(fun(option(product_prod(A,B)),fun(option(B),$o)),fun(option(A),fun(option(B),$o))),relcompp(option(A),option(product_prod(A,B)),option(B)),conversep(option(product_prod(A,B)),option(A),bNF_Grp(option(product_prod(A,B)),option(A),aa(fun(option(product_prod(A,B)),$o),set(option(product_prod(A,B))),collect(option(product_prod(A,B))),aTP_Lamp_awv(fun(A,fun(B,$o)),fun(option(product_prod(A,B)),$o),R)),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)),$o),set(option(product_prod(A,B))),collect(option(product_prod(A,B))),aTP_Lamp_awv(fun(A,fun(B,$o)),fun(option(product_prod(A,B)),$o),R)),map_option(product_prod(A,B),B,product_snd(A,B)))) ).

% option.rel_compp_Grp
tff(fact_7548_preordering__bdd_OInt1,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A4: set(A),B3: set(A)] :
      ( condit622319405099724424ng_bdd(A,Less_eq,Less)
     => ( aa(set(A),$o,condit16957441358409770ng_bdd(A,Less_eq),A4)
       => aa(set(A),$o,condit16957441358409770ng_bdd(A,Less_eq),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3)) ) ) ).

% preordering_bdd.Int1
tff(fact_7549_preordering__bdd_OInt2,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),B3: set(A),A4: set(A)] :
      ( condit622319405099724424ng_bdd(A,Less_eq,Less)
     => ( aa(set(A),$o,condit16957441358409770ng_bdd(A,Less_eq),B3)
       => aa(set(A),$o,condit16957441358409770ng_bdd(A,Less_eq),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3)) ) ) ).

% preordering_bdd.Int2
tff(fact_7550_subset__mset_Obdd__below_Opreordering__bdd__axioms,axiom,
    ! [A: $tType] : condit622319405099724424ng_bdd(multiset(A),aTP_Lamp_aso(multiset(A),fun(multiset(A),$o)),aTP_Lamp_asp(multiset(A),fun(multiset(A),$o))) ).

% subset_mset.bdd_below.preordering_bdd_axioms
tff(fact_7551_preordering__bdd_Oempty,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o))] :
      ( condit622319405099724424ng_bdd(A,Less_eq,Less)
     => aa(set(A),$o,condit16957441358409770ng_bdd(A,Less_eq),bot_bot(set(A))) ) ).

% preordering_bdd.empty
tff(fact_7552_bdd__below_Opreordering__bdd__axioms,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => condit622319405099724424ng_bdd(A,aTP_Lamp_avn(A,fun(A,$o)),aTP_Lamp_sz(A,fun(A,$o))) ) ).

% bdd_below.preordering_bdd_axioms
tff(fact_7553_numeral__sqr,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [K: num] : aa(num,A,numeral_numeral(A),sqr(K)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),K)),aa(num,A,numeral_numeral(A),K)) ) ).

% numeral_sqr
tff(fact_7554_Nitpick_Otranclp__unfold,axiom,
    ! [A: $tType,R2: fun(A,fun(A,$o)),A3: A,B2: A] :
      ( aa(A,$o,aa(A,fun(A,$o),transitive_tranclp(A,R2),A3),B2)
    <=> aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_trancl(A,aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),R2)))) ) ).

% Nitpick.tranclp_unfold
tff(fact_7555_tranclp__trancl__eq,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),X3: A,Xa3: A] :
      ( aa(A,$o,aa(A,fun(A,$o),transitive_tranclp(A,aTP_Lamp_ux(set(product_prod(A,A)),fun(A,fun(A,$o)),R2)),X3),Xa3)
    <=> aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Xa3)),transitive_trancl(A,R2)) ) ).

% tranclp_trancl_eq
tff(fact_7556_tranclp__induct2,axiom,
    ! [A: $tType,B: $tType,R2: fun(product_prod(A,B),fun(product_prod(A,B),$o)),Ax: A,Ay: B,Bx: A,By: B,P: fun(A,fun(B,$o))] :
      ( aa(product_prod(A,B),$o,aa(product_prod(A,B),fun(product_prod(A,B),$o),transitive_tranclp(product_prod(A,B),R2),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Bx),By))
     => ( ! [A6: A,B5: B] :
            ( aa(product_prod(A,B),$o,aa(product_prod(A,B),fun(product_prod(A,B),$o),R2,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A6),B5))
           => aa(B,$o,aa(A,fun(B,$o),P,A6),B5) )
       => ( ! [A6: A,B5: B,Aa2: A,Ba: B] :
              ( aa(product_prod(A,B),$o,aa(product_prod(A,B),fun(product_prod(A,B),$o),transitive_tranclp(product_prod(A,B),R2),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A6),B5))
             => ( aa(product_prod(A,B),$o,aa(product_prod(A,B),fun(product_prod(A,B),$o),R2,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A6),B5)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa2),Ba))
               => ( aa(B,$o,aa(A,fun(B,$o),P,A6),B5)
                 => aa(B,$o,aa(A,fun(B,$o),P,Aa2),Ba) ) ) )
         => aa(B,$o,aa(A,fun(B,$o),P,Bx),By) ) ) ) ).

% tranclp_induct2
tff(fact_7557_less__nat__rel,axiom,
    ord_less(nat) = transitive_tranclp(nat,aTP_Lamp_sh(nat,fun(nat,$o))) ).

% less_nat_rel
tff(fact_7558_tranclp__def,axiom,
    ! [A: $tType,X3: fun(A,fun(A,$o))] : transitive_tranclp(A,X3) = complete_lattice_lfp(fun(A,fun(A,$o)),aTP_Lamp_aey(fun(A,fun(A,$o)),fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),X3)) ).

% tranclp_def
tff(fact_7559_trancl__def,axiom,
    ! [A: $tType,X3: set(product_prod(A,A))] : transitive_trancl(A,X3) = aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),transitive_tranclp(A,aTP_Lamp_ux(set(product_prod(A,A)),fun(A,fun(A,$o)),X3)))) ).

% trancl_def
tff(fact_7560_reflp__refl__eq,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( reflp(A,aTP_Lamp_ux(set(product_prod(A,A)),fun(A,fun(A,$o)),R2))
    <=> refl_on(A,top_top(set(A)),R2) ) ).

% reflp_refl_eq
tff(fact_7561_image2p__def,axiom,
    ! [A: $tType,C: $tType,D: $tType,B: $tType,F: fun(C,A),G: fun(D,B),R: fun(C,fun(D,$o)),X3: A,Xa3: B] :
      ( bNF_Greatest_image2p(C,A,D,B,F,G,R,X3,Xa3)
    <=> ? [X10: C,Y10: D] :
          ( aa(D,$o,aa(C,fun(D,$o),R,X10),Y10)
          & ( aa(C,A,F,X10) = X3 )
          & ( aa(D,B,G,Y10) = Xa3 ) ) ) ).

% image2p_def
tff(fact_7562_reflp__sup,axiom,
    ! [A: $tType,R2: fun(A,fun(A,$o)),S2: fun(A,fun(A,$o))] :
      ( reflp(A,R2)
     => ( reflp(A,S2)
       => reflp(A,aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(fun(A,fun(A,$o)),fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),sup_sup(fun(A,fun(A,$o))),R2),S2)) ) ) ).

% reflp_sup
tff(fact_7563_Quotient__id__abs__transfer,axiom,
    ! [B: $tType,A: $tType,R: fun(A,fun(A,$o)),Abs: fun(A,B),Rep: fun(B,A),T2: fun(A,fun(B,$o))] :
      ( quotient(A,B,R,Abs,Rep,T2)
     => ( reflp(A,R)
       => aa(fun(A,B),$o,aa(fun(A,A),fun(fun(A,B),$o),bNF_rel_fun(A,A,A,B,fequal(A),T2),aTP_Lamp_au(A,A)),Abs) ) ) ).

% Quotient_id_abs_transfer
tff(fact_7564_reflp__inf,axiom,
    ! [A: $tType,R2: fun(A,fun(A,$o)),S2: fun(A,fun(A,$o))] :
      ( reflp(A,R2)
     => ( reflp(A,S2)
       => reflp(A,aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(fun(A,fun(A,$o)),fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),inf_inf(fun(A,fun(A,$o))),R2),S2)) ) ) ).

% reflp_inf
tff(fact_7565_sub_Otransfer,axiom,
    aa(fun(num,fun(num,code_integer)),$o,aa(fun(num,fun(num,int)),fun(fun(num,fun(num,code_integer)),$o),bNF_rel_fun(num,num,fun(num,int),fun(num,code_integer),fequal(num),bNF_rel_fun(num,num,int,code_integer,fequal(num),code_pcr_integer)),aTP_Lamp_uk(num,fun(num,int))),code_sub) ).

% sub.transfer
tff(fact_7566_of__int__code_I1_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [K: num] : aa(int,A,ring_1_of_int(A),neg(K)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),K)) ) ).

% of_int_code(1)
tff(fact_7567_uminus__integer_Otransfer,axiom,
    aa(fun(code_integer,code_integer),$o,aa(fun(int,int),fun(fun(code_integer,code_integer),$o),bNF_rel_fun(int,code_integer,int,code_integer,code_pcr_integer,code_pcr_integer),uminus_uminus(int)),uminus_uminus(code_integer)) ).

% uminus_integer.transfer
tff(fact_7568_integer_Oid__abs__transfer,axiom,
    aa(fun(int,code_integer),$o,aa(fun(int,int),fun(fun(int,code_integer),$o),bNF_rel_fun(int,int,int,code_integer,fequal(int),code_pcr_integer),aTP_Lamp_dl(int,int)),code_integer_of_int) ).

% integer.id_abs_transfer
tff(fact_7569_integer_Orep__transfer,axiom,
    aa(fun(code_integer,int),$o,aa(fun(int,int),fun(fun(code_integer,int),$o),bNF_rel_fun(int,code_integer,int,int,code_pcr_integer,fequal(int)),aTP_Lamp_dl(int,int)),code_int_of_integer) ).

% integer.rep_transfer
tff(fact_7570_dup_Otransfer,axiom,
    aa(fun(code_integer,code_integer),$o,aa(fun(int,int),fun(fun(code_integer,code_integer),$o),bNF_rel_fun(int,code_integer,int,code_integer,code_pcr_integer,code_pcr_integer),aTP_Lamp_ug(int,int)),code_dup) ).

% dup.transfer
tff(fact_7571_one__integer_Otransfer,axiom,
    aa(code_integer,$o,aa(int,fun(code_integer,$o),code_pcr_integer,one_one(int)),one_one(code_integer)) ).

% one_integer.transfer
tff(fact_7572_rtranclp__imp__Sup__relpowp,axiom,
    ! [A: $tType,P: fun(A,fun(A,$o)),X: A,Y: A] :
      ( aa(A,$o,aa(A,fun(A,$o),transitive_rtranclp(A,P),X),Y)
     => aa(A,$o,aa(A,fun(A,$o),aa(set(fun(A,fun(A,$o))),fun(A,fun(A,$o)),complete_Sup_Sup(fun(A,fun(A,$o))),aa(set(nat),set(fun(A,fun(A,$o))),image2(nat,fun(A,fun(A,$o)),aTP_Lamp_axd(fun(A,fun(A,$o)),fun(nat,fun(A,fun(A,$o))),P)),top_top(set(nat)))),X),Y) ) ).

% rtranclp_imp_Sup_relpowp
tff(fact_7573_rtranclp__is__Sup__relpowp,axiom,
    ! [A: $tType,P: fun(A,fun(A,$o))] : transitive_rtranclp(A,P) = aa(set(fun(A,fun(A,$o))),fun(A,fun(A,$o)),complete_Sup_Sup(fun(A,fun(A,$o))),aa(set(nat),set(fun(A,fun(A,$o))),image2(nat,fun(A,fun(A,$o)),aTP_Lamp_axd(fun(A,fun(A,$o)),fun(nat,fun(A,fun(A,$o))),P)),top_top(set(nat)))) ).

% rtranclp_is_Sup_relpowp
tff(fact_7574_rtranclp__reflclp,axiom,
    ! [A: $tType,R: fun(A,fun(A,$o))] : transitive_rtranclp(A,aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(fun(A,fun(A,$o)),fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),sup_sup(fun(A,fun(A,$o))),R),fequal(A))) = transitive_rtranclp(A,R) ).

% rtranclp_reflclp
tff(fact_7575_rtranclp__reflclp__absorb,axiom,
    ! [A: $tType,R: fun(A,fun(A,$o))] : aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(fun(A,fun(A,$o)),fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),sup_sup(fun(A,fun(A,$o))),transitive_rtranclp(A,R)),fequal(A)) = transitive_rtranclp(A,R) ).

% rtranclp_reflclp_absorb
tff(fact_7576_reflclp__tranclp,axiom,
    ! [A: $tType,R2: fun(A,fun(A,$o))] : aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(fun(A,fun(A,$o)),fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),sup_sup(fun(A,fun(A,$o))),transitive_tranclp(A,R2)),fequal(A)) = transitive_rtranclp(A,R2) ).

% reflclp_tranclp
tff(fact_7577_rtranclp__sup__rtranclp,axiom,
    ! [A: $tType,R: fun(A,fun(A,$o)),S: fun(A,fun(A,$o))] : transitive_rtranclp(A,aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(fun(A,fun(A,$o)),fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),sup_sup(fun(A,fun(A,$o))),transitive_rtranclp(A,R)),transitive_rtranclp(A,S))) = transitive_rtranclp(A,aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(fun(A,fun(A,$o)),fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),sup_sup(fun(A,fun(A,$o))),R),S)) ).

% rtranclp_sup_rtranclp
tff(fact_7578_Enum_Ortranclp__rtrancl__eq,axiom,
    ! [A: $tType,R2: fun(A,fun(A,$o)),X: A,Y: A] :
      ( aa(A,$o,aa(A,fun(A,$o),transitive_rtranclp(A,R2),X),Y)
    <=> aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),transitive_rtrancl(A,aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),R2)))) ) ).

% Enum.rtranclp_rtrancl_eq
tff(fact_7579_rtrancl__def,axiom,
    ! [A: $tType,X3: set(product_prod(A,A))] : transitive_rtrancl(A,X3) = aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),transitive_rtranclp(A,aTP_Lamp_ux(set(product_prod(A,A)),fun(A,fun(A,$o)),X3)))) ).

% rtrancl_def
tff(fact_7580_Transitive__Closure_Ortranclp__rtrancl__eq,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),X3: A,Xa3: A] :
      ( aa(A,$o,aa(A,fun(A,$o),transitive_rtranclp(A,aTP_Lamp_ux(set(product_prod(A,A)),fun(A,fun(A,$o)),R2)),X3),Xa3)
    <=> aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Xa3)),transitive_rtrancl(A,R2)) ) ).

% Transitive_Closure.rtranclp_rtrancl_eq
tff(fact_7581_converse__rtranclp__induct2,axiom,
    ! [A: $tType,B: $tType,R2: fun(product_prod(A,B),fun(product_prod(A,B),$o)),Ax: A,Ay: B,Bx: A,By: B,P: fun(A,fun(B,$o))] :
      ( aa(product_prod(A,B),$o,aa(product_prod(A,B),fun(product_prod(A,B),$o),transitive_rtranclp(product_prod(A,B),R2),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Bx),By))
     => ( aa(B,$o,aa(A,fun(B,$o),P,Bx),By)
       => ( ! [A6: A,B5: B,Aa2: A,Ba: B] :
              ( aa(product_prod(A,B),$o,aa(product_prod(A,B),fun(product_prod(A,B),$o),R2,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A6),B5)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa2),Ba))
             => ( aa(product_prod(A,B),$o,aa(product_prod(A,B),fun(product_prod(A,B),$o),transitive_rtranclp(product_prod(A,B),R2),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa2),Ba)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Bx),By))
               => ( aa(B,$o,aa(A,fun(B,$o),P,Aa2),Ba)
                 => aa(B,$o,aa(A,fun(B,$o),P,A6),B5) ) ) )
         => aa(B,$o,aa(A,fun(B,$o),P,Ax),Ay) ) ) ) ).

% converse_rtranclp_induct2
tff(fact_7582_converse__rtranclpE2,axiom,
    ! [A: $tType,B: $tType,R2: fun(product_prod(A,B),fun(product_prod(A,B),$o)),Xa: A,Xb: B,Za2: A,Zb: B] :
      ( aa(product_prod(A,B),$o,aa(product_prod(A,B),fun(product_prod(A,B),$o),transitive_rtranclp(product_prod(A,B),R2),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa),Xb)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Za2),Zb))
     => ( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa),Xb) != aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Za2),Zb) )
       => ~ ! [A6: A,B5: B] :
              ( aa(product_prod(A,B),$o,aa(product_prod(A,B),fun(product_prod(A,B),$o),R2,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa),Xb)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A6),B5))
             => ~ aa(product_prod(A,B),$o,aa(product_prod(A,B),fun(product_prod(A,B),$o),transitive_rtranclp(product_prod(A,B),R2),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A6),B5)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Za2),Zb)) ) ) ) ).

% converse_rtranclpE2
tff(fact_7583_rtranclp__induct2,axiom,
    ! [A: $tType,B: $tType,R2: fun(product_prod(A,B),fun(product_prod(A,B),$o)),Ax: A,Ay: B,Bx: A,By: B,P: fun(A,fun(B,$o))] :
      ( aa(product_prod(A,B),$o,aa(product_prod(A,B),fun(product_prod(A,B),$o),transitive_rtranclp(product_prod(A,B),R2),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Bx),By))
     => ( aa(B,$o,aa(A,fun(B,$o),P,Ax),Ay)
       => ( ! [A6: A,B5: B,Aa2: A,Ba: B] :
              ( aa(product_prod(A,B),$o,aa(product_prod(A,B),fun(product_prod(A,B),$o),transitive_rtranclp(product_prod(A,B),R2),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A6),B5))
             => ( aa(product_prod(A,B),$o,aa(product_prod(A,B),fun(product_prod(A,B),$o),R2,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A6),B5)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa2),Ba))
               => ( aa(B,$o,aa(A,fun(B,$o),P,A6),B5)
                 => aa(B,$o,aa(A,fun(B,$o),P,Aa2),Ba) ) ) )
         => aa(B,$o,aa(A,fun(B,$o),P,Bx),By) ) ) ) ).

% rtranclp_induct2
tff(fact_7584_rtranclp__r__diff__Id,axiom,
    ! [A: $tType,R2: fun(A,fun(A,$o))] : transitive_rtranclp(A,aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(fun(A,fun(A,$o)),fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),inf_inf(fun(A,fun(A,$o))),R2),aTP_Lamp_za(A,fun(A,$o)))) = transitive_rtranclp(A,R2) ).

% rtranclp_r_diff_Id
tff(fact_7585_rtranclp__def,axiom,
    ! [A: $tType,X3: fun(A,fun(A,$o))] : transitive_rtranclp(A,X3) = complete_lattice_lfp(fun(A,fun(A,$o)),aTP_Lamp_aex(fun(A,fun(A,$o)),fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),X3)) ).

% rtranclp_def
tff(fact_7586_old_Orec__bool__def,axiom,
    ! [A: $tType,X3: A,Xa3: A,Xb2: $o] : product_rec_bool(A,X3,Xa3,(Xb2)) = the(A,product_rec_set_bool(A,X3,Xa3,(Xb2))) ).

% old.rec_bool_def
tff(fact_7587_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_7588_comm__monoid__list_Oaxioms_I1_J,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
      ( groups1828464146339083142d_list(A,F,Z2)
     => comm_monoid(A,F,Z2) ) ).

% comm_monoid_list.axioms(1)
tff(fact_7589_comm__monoid__list_Ointro,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
      ( comm_monoid(A,F,Z2)
     => ( groups_monoid_list(A,F,Z2)
       => groups1828464146339083142d_list(A,F,Z2) ) ) ).

% comm_monoid_list.intro
tff(fact_7590_comm__monoid__list__def,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
      ( groups1828464146339083142d_list(A,F,Z2)
    <=> ( comm_monoid(A,F,Z2)
        & groups_monoid_list(A,F,Z2) ) ) ).

% comm_monoid_list_def
tff(fact_7591_monoid__list_Ointro,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
      ( monoid(A,F,Z2)
     => groups_monoid_list(A,F,Z2) ) ).

% monoid_list.intro
tff(fact_7592_monoid__list_Oaxioms,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
      ( groups_monoid_list(A,F,Z2)
     => monoid(A,F,Z2) ) ).

% monoid_list.axioms
tff(fact_7593_monoid__list__def,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
      ( groups_monoid_list(A,F,Z2)
    <=> monoid(A,F,Z2) ) ).

% monoid_list_def
tff(fact_7594_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_7595_laz__weak__Pa,axiom,
    ! [B: $tType,A: $tType,P: fun(A,$o),A4: list(A),B3: list(B)] :
      ( list_all_zip(A,B,aTP_Lamp_axe(fun(A,$o),fun(A,fun(B,$o)),P),A4,B3)
    <=> ( ( aa(list(A),nat,size_size(list(A)),A4) = aa(list(B),nat,size_size(list(B)),B3) )
        & ! [X4: A] :
            ( aa(set(A),$o,member(A,X4),aa(list(A),set(A),set2(A),A4))
           => aa(A,$o,P,X4) ) ) ) ).

% laz_weak_Pa
tff(fact_7596_laz__weak__Pb,axiom,
    ! [A: $tType,B: $tType,P: fun(B,$o),A4: list(A),B3: list(B)] :
      ( list_all_zip(A,B,aTP_Lamp_anw(fun(B,$o),fun(A,fun(B,$o)),P),A4,B3)
    <=> ( ( aa(list(A),nat,size_size(list(A)),A4) = aa(list(B),nat,size_size(list(B)),B3) )
        & ! [X4: B] :
            ( aa(set(B),$o,member(B,X4),aa(list(B),set(B),set2(B),B3))
           => aa(B,$o,P,X4) ) ) ) ).

% laz_weak_Pb
tff(fact_7597_laz__conj,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,$o)),Q: fun(A,fun(B,$o)),A3: list(A),B2: list(B)] :
      ( list_all_zip(A,B,aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aTP_Lamp_axf(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),P),Q),A3,B2)
    <=> ( list_all_zip(A,B,P,A3,B2)
        & list_all_zip(A,B,Q,A3,B2) ) ) ).

% laz_conj
tff(fact_7598_list__all__zip__map2,axiom,
    ! [A: $tType,B: $tType,C: $tType,P: fun(A,fun(B,$o)),As: list(A),F: fun(C,B),Bs: list(C)] :
      ( list_all_zip(A,B,P,As,aa(list(C),list(B),map(C,B,F),Bs))
    <=> list_all_zip(A,C,aa(fun(C,B),fun(A,fun(C,$o)),aTP_Lamp_aiu(fun(A,fun(B,$o)),fun(fun(C,B),fun(A,fun(C,$o))),P),F),As,Bs) ) ).

% list_all_zip_map2
tff(fact_7599_list__all__zip__map1,axiom,
    ! [C: $tType,A: $tType,B: $tType,P: fun(A,fun(B,$o)),F: fun(C,A),As: list(C),Bs: list(B)] :
      ( list_all_zip(A,B,P,aa(list(C),list(A),map(C,A,F),As),Bs)
    <=> list_all_zip(C,B,aa(fun(C,A),fun(C,fun(B,$o)),aTP_Lamp_aiv(fun(A,fun(B,$o)),fun(fun(C,A),fun(C,fun(B,$o))),P),F),As,Bs) ) ).

% list_all_zip_map1
tff(fact_7600_laz__swap__ex,axiom,
    ! [B: $tType,A: $tType,C: $tType,P: fun(A,fun(B,fun(C,$o))),A4: list(A),B3: list(B)] :
      ( list_all_zip(A,B,aTP_Lamp_axg(fun(A,fun(B,fun(C,$o))),fun(A,fun(B,$o)),P),A4,B3)
     => ~ ! [C7: list(C)] :
            ( list_all_zip(A,C,aTP_Lamp_axh(fun(A,fun(B,fun(C,$o))),fun(A,fun(C,$o)),P),A4,C7)
           => ~ list_all_zip(B,C,aTP_Lamp_axi(fun(A,fun(B,fun(C,$o))),fun(B,fun(C,$o)),P),B3,C7) ) ) ).

% laz_swap_ex
tff(fact_7601_list__all__zip_Opelims_I1_J,axiom,
    ! [A: $tType,B: $tType,X: fun(A,fun(B,$o)),Xa: list(A),Xb: list(B),Y: $o] :
      ( ( list_all_zip(A,B,X,Xa,Xb)
      <=> (Y) )
     => ( accp(product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),list_all_zip_rel(A,B),aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),aa(fun(A,fun(B,$o)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,$o)),product_prod(list(A),list(B))),X),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)),Xa),Xb)))
       => ( ( ( Xa = nil(A) )
           => ( ( Xb = nil(B) )
             => ( (Y)
               => ~ accp(product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),list_all_zip_rel(A,B),aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),aa(fun(A,fun(B,$o)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,$o)),product_prod(list(A),list(B))),X),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),nil(A)),nil(B)))) ) ) )
         => ( ! [A6: A,As4: list(A)] :
                ( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A6),As4) )
               => ! [B5: B,Bs2: list(B)] :
                    ( ( Xb = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B5),Bs2) )
                   => ( ( (Y)
                      <=> ( aa(B,$o,aa(A,fun(B,$o),X,A6),B5)
                          & list_all_zip(A,B,X,As4,Bs2) ) )
                     => ~ accp(product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),list_all_zip_rel(A,B),aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),aa(fun(A,fun(B,$o)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,$o)),product_prod(list(A),list(B))),X),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),A6),As4)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B5),Bs2)))) ) ) )
           => ( ! [V2: A,Va: list(A)] :
                  ( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V2),Va) )
                 => ( ( Xb = nil(B) )
                   => ( ~ (Y)
                     => ~ accp(product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),list_all_zip_rel(A,B),aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),aa(fun(A,fun(B,$o)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,$o)),product_prod(list(A),list(B))),X),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V2),Va)),nil(B)))) ) ) )
             => ~ ( ( Xa = nil(A) )
                 => ! [V2: B,Va: list(B)] :
                      ( ( Xb = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),V2),Va) )
                     => ( ~ (Y)
                       => ~ accp(product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),list_all_zip_rel(A,B),aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),aa(fun(A,fun(B,$o)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,$o)),product_prod(list(A),list(B))),X),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),nil(A)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),V2),Va)))) ) ) ) ) ) ) ) ) ).

% list_all_zip.pelims(1)
tff(fact_7602_list__all__zip_Opelims_I2_J,axiom,
    ! [A: $tType,B: $tType,X: fun(A,fun(B,$o)),Xa: list(A),Xb: list(B)] :
      ( list_all_zip(A,B,X,Xa,Xb)
     => ( accp(product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),list_all_zip_rel(A,B),aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),aa(fun(A,fun(B,$o)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,$o)),product_prod(list(A),list(B))),X),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xa),Xb)))
       => ( ( ( Xa = nil(A) )
           => ( ( Xb = nil(B) )
             => ~ accp(product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),list_all_zip_rel(A,B),aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),aa(fun(A,fun(B,$o)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,$o)),product_prod(list(A),list(B))),X),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),nil(A)),nil(B)))) ) )
         => ~ ! [A6: A,As4: list(A)] :
                ( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A6),As4) )
               => ! [B5: B,Bs2: list(B)] :
                    ( ( Xb = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B5),Bs2) )
                   => ( accp(product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),list_all_zip_rel(A,B),aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),aa(fun(A,fun(B,$o)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,$o)),product_prod(list(A),list(B))),X),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A6),As4)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B5),Bs2))))
                     => ~ ( aa(B,$o,aa(A,fun(B,$o),X,A6),B5)
                          & list_all_zip(A,B,X,As4,Bs2) ) ) ) ) ) ) ) ).

% list_all_zip.pelims(2)
tff(fact_7603_list__all__zip_Opelims_I3_J,axiom,
    ! [A: $tType,B: $tType,X: fun(A,fun(B,$o)),Xa: list(A),Xb: list(B)] :
      ( ~ list_all_zip(A,B,X,Xa,Xb)
     => ( accp(product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),list_all_zip_rel(A,B),aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),aa(fun(A,fun(B,$o)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,$o)),product_prod(list(A),list(B))),X),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xa),Xb)))
       => ( ! [A6: A,As4: list(A)] :
              ( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A6),As4) )
             => ! [B5: B,Bs2: list(B)] :
                  ( ( Xb = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B5),Bs2) )
                 => ( accp(product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),list_all_zip_rel(A,B),aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),aa(fun(A,fun(B,$o)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,$o)),product_prod(list(A),list(B))),X),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),A6),As4)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B5),Bs2))))
                   => ( aa(B,$o,aa(A,fun(B,$o),X,A6),B5)
                      & list_all_zip(A,B,X,As4,Bs2) ) ) ) )
         => ( ! [V2: A,Va: list(A)] :
                ( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V2),Va) )
               => ( ( Xb = nil(B) )
                 => ~ accp(product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),list_all_zip_rel(A,B),aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),aa(fun(A,fun(B,$o)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,$o)),product_prod(list(A),list(B))),X),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V2),Va)),nil(B)))) ) )
           => ~ ( ( Xa = nil(A) )
               => ! [V2: B,Va: list(B)] :
                    ( ( Xb = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),V2),Va) )
                   => ~ accp(product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),list_all_zip_rel(A,B),aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),aa(fun(A,fun(B,$o)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,$o)),product_prod(list(A),list(B))),X),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),nil(A)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),V2),Va)))) ) ) ) ) ) ) ).

% list_all_zip.pelims(3)
tff(fact_7604_option_Orec__o__map,axiom,
    ! [B: $tType,C: $tType,A: $tType,G: B,Ga: fun(C,B),F: fun(A,C)] : aa(fun(option(A),option(C)),fun(option(A),B),comp(option(C),B,option(A),rec_option(B,C,G,Ga)),map_option(A,C,F)) = rec_option(B,A,G,aa(fun(A,C),fun(A,B),aTP_Lamp_axj(fun(C,B),fun(fun(A,C),fun(A,B)),Ga),F)) ).

% option.rec_o_map
tff(fact_7605_group_Oaxioms_I2_J,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A)] :
      ( group(A,F,Z2,Inverse)
     => group_axioms(A,F,Z2,Inverse) ) ).

% group.axioms(2)
tff(fact_7606_group__axioms__def,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A)] :
      ( group_axioms(A,F,Z2,Inverse)
    <=> ( ! [A10: A] : aa(A,A,aa(A,fun(A,A),F,Z2),A10) = A10
        & ! [A10: A] : aa(A,A,aa(A,fun(A,A),F,aa(A,A,Inverse,A10)),A10) = Z2 ) ) ).

% group_axioms_def
tff(fact_7607_group__axioms_Ointro,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A)] :
      ( ! [A6: A] : aa(A,A,aa(A,fun(A,A),F,Z2),A6) = A6
     => ( ! [A6: A] : aa(A,A,aa(A,fun(A,A),F,aa(A,A,Inverse,A6)),A6) = Z2
       => group_axioms(A,F,Z2,Inverse) ) ) ).

% group_axioms.intro
tff(fact_7608_group_Ointro,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A)] :
      ( semigroup(A,F)
     => ( group_axioms(A,F,Z2,Inverse)
       => group(A,F,Z2,Inverse) ) ) ).

% group.intro
tff(fact_7609_group__def,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A)] :
      ( group(A,F,Z2,Inverse)
    <=> ( semigroup(A,F)
        & group_axioms(A,F,Z2,Inverse) ) ) ).

% group_def
tff(fact_7610_sup_Osemigroup__axioms,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => semigroup(A,sup_sup(A)) ) ).

% sup.semigroup_axioms
tff(fact_7611_min_Osemigroup__axioms,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => semigroup(A,ord_min(A)) ) ).

% min.semigroup_axioms
tff(fact_7612_max_Osemigroup__axioms,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => semigroup(A,ord_max(A)) ) ).

% max.semigroup_axioms
tff(fact_7613_semigroup_Oassoc,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),A3: A,B2: A,C2: A] :
      ( semigroup(A,F)
     => ( aa(A,A,aa(A,fun(A,A),F,aa(A,A,aa(A,fun(A,A),F,A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),F,A3),aa(A,A,aa(A,fun(A,A),F,B2),C2)) ) ) ).

% semigroup.assoc
tff(fact_7614_semigroup_Ointro,axiom,
    ! [A: $tType,F: fun(A,fun(A,A))] :
      ( ! [A6: A,B5: A,C4: A] : aa(A,A,aa(A,fun(A,A),F,aa(A,A,aa(A,fun(A,A),F,A6),B5)),C4) = aa(A,A,aa(A,fun(A,A),F,A6),aa(A,A,aa(A,fun(A,A),F,B5),C4))
     => semigroup(A,F) ) ).

% semigroup.intro
tff(fact_7615_semigroup__def,axiom,
    ! [A: $tType,F: fun(A,fun(A,A))] :
      ( semigroup(A,F)
    <=> ! [A10: A,B6: A,C5: A] : aa(A,A,aa(A,fun(A,A),F,aa(A,A,aa(A,fun(A,A),F,A10),B6)),C5) = aa(A,A,aa(A,fun(A,A),F,A10),aa(A,A,aa(A,fun(A,A),F,B6),C5)) ) ).

% semigroup_def
tff(fact_7616_monoid_Oaxioms_I1_J,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
      ( monoid(A,F,Z2)
     => semigroup(A,F) ) ).

% monoid.axioms(1)
tff(fact_7617_group_Oaxioms_I1_J,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A)] :
      ( group(A,F,Z2,Inverse)
     => semigroup(A,F) ) ).

% group.axioms(1)
tff(fact_7618_add_Osemigroup__axioms,axiom,
    ! [A: $tType] :
      ( semigroup_add(A)
     => semigroup(A,plus_plus(A)) ) ).

% add.semigroup_axioms
tff(fact_7619_mult_Osemigroup__axioms,axiom,
    ! [A: $tType] :
      ( semigroup_mult(A)
     => semigroup(A,times_times(A)) ) ).

% mult.semigroup_axioms
tff(fact_7620_inf_Osemigroup__axioms,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => semigroup(A,inf_inf(A)) ) ).

% inf.semigroup_axioms
tff(fact_7621_monoid_Ointro,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
      ( semigroup(A,F)
     => ( monoid_axioms(A,F,Z2)
       => monoid(A,F,Z2) ) ) ).

% monoid.intro
tff(fact_7622_monoid__def,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
      ( monoid(A,F,Z2)
    <=> ( semigroup(A,F)
        & monoid_axioms(A,F,Z2) ) ) ).

% monoid_def
tff(fact_7623_monoid__axioms__def,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
      ( monoid_axioms(A,F,Z2)
    <=> ( ! [A10: A] : aa(A,A,aa(A,fun(A,A),F,Z2),A10) = A10
        & ! [A10: A] : aa(A,A,aa(A,fun(A,A),F,A10),Z2) = A10 ) ) ).

% monoid_axioms_def
tff(fact_7624_monoid__axioms_Ointro,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
      ( ! [A6: A] : aa(A,A,aa(A,fun(A,A),F,Z2),A6) = A6
     => ( ! [A6: A] : aa(A,A,aa(A,fun(A,A),F,A6),Z2) = A6
       => monoid_axioms(A,F,Z2) ) ) ).

% monoid_axioms.intro
tff(fact_7625_monoid_Oaxioms_I2_J,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
      ( monoid(A,F,Z2)
     => monoid_axioms(A,F,Z2) ) ).

% monoid.axioms(2)
tff(fact_7626_ordering__top_Oaxioms_I2_J,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),Top: A] :
      ( ordering_top(A,Less_eq,Less,Top)
     => ordering_top_axioms(A,Less_eq,Top) ) ).

% ordering_top.axioms(2)
tff(fact_7627_sum_Oinj__map,axiom,
    ! [D: $tType,B: $tType,C: $tType,A: $tType,F1: fun(A,B),F22: fun(C,D)] :
      ( inj_on(A,B,F1,top_top(set(A)))
     => ( inj_on(C,D,F22,top_top(set(C)))
       => inj_on(sum_sum(A,C),sum_sum(B,D),sum_map_sum(A,B,C,D,F1,F22),top_top(set(sum_sum(A,C)))) ) ) ).

% sum.inj_map
tff(fact_7628_map__sum_Oidentity,axiom,
    ! [B: $tType,A: $tType] : sum_map_sum(A,A,B,B,aTP_Lamp_au(A,A),aTP_Lamp_oc(B,B)) = id(sum_sum(A,B)) ).

% map_sum.identity
tff(fact_7629_sum_Omap__ident,axiom,
    ! [B: $tType,A: $tType,T3: sum_sum(A,B)] : aa(sum_sum(A,B),sum_sum(A,B),sum_map_sum(A,A,B,B,aTP_Lamp_au(A,A),aTP_Lamp_oc(B,B)),T3) = T3 ).

% sum.map_ident
tff(fact_7630_ordering__top__axioms_Ointro,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,$o)),Top: A] :
      ( ! [A6: A] : aa(A,$o,aa(A,fun(A,$o),Less_eq,A6),Top)
     => ordering_top_axioms(A,Less_eq,Top) ) ).

% ordering_top_axioms.intro
tff(fact_7631_ordering__top__axioms__def,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,$o)),Top: A] :
      ( ordering_top_axioms(A,Less_eq,Top)
    <=> ! [A10: A] : aa(A,$o,aa(A,fun(A,$o),Less_eq,A10),Top) ) ).

% ordering_top_axioms_def
tff(fact_7632_sum_Orel__Grp,axiom,
    ! [A: $tType,C: $tType,D: $tType,B: $tType,A15: set(A),F1: fun(A,C),A24: set(B),F22: fun(B,D)] : bNF_rel_sum(A,C,B,D,bNF_Grp(A,C,A15,F1),bNF_Grp(B,D,A24,F22)) = bNF_Grp(sum_sum(A,B),sum_sum(C,D),aa(fun(sum_sum(A,B),$o),set(sum_sum(A,B)),collect(sum_sum(A,B)),aa(set(B),fun(sum_sum(A,B),$o),aTP_Lamp_axk(set(A),fun(set(B),fun(sum_sum(A,B),$o)),A15),A24)),sum_map_sum(A,C,B,D,F1,F22)) ).

% sum.rel_Grp
tff(fact_7633_sum_Oin__rel,axiom,
    ! [A: $tType,C: $tType,D: $tType,B: $tType,R1: fun(A,fun(B,$o)),R22: fun(C,fun(D,$o)),A3: sum_sum(A,C),B2: sum_sum(B,D)] :
      ( aa(sum_sum(B,D),$o,aa(sum_sum(A,C),fun(sum_sum(B,D),$o),bNF_rel_sum(A,B,C,D,R1,R22),A3),B2)
    <=> ? [Z4: sum_sum(product_prod(A,B),product_prod(C,D))] :
          ( aa(set(sum_sum(product_prod(A,B),product_prod(C,D))),$o,member(sum_sum(product_prod(A,B),product_prod(C,D)),Z4),aa(fun(sum_sum(product_prod(A,B),product_prod(C,D)),$o),set(sum_sum(product_prod(A,B),product_prod(C,D))),collect(sum_sum(product_prod(A,B),product_prod(C,D))),aa(fun(C,fun(D,$o)),fun(sum_sum(product_prod(A,B),product_prod(C,D)),$o),aTP_Lamp_axl(fun(A,fun(B,$o)),fun(fun(C,fun(D,$o)),fun(sum_sum(product_prod(A,B),product_prod(C,D)),$o)),R1),R22)))
          & ( aa(sum_sum(product_prod(A,B),product_prod(C,D)),sum_sum(A,C),sum_map_sum(product_prod(A,B),A,product_prod(C,D),C,product_fst(A,B),product_fst(C,D)),Z4) = A3 )
          & ( aa(sum_sum(product_prod(A,B),product_prod(C,D)),sum_sum(B,D),sum_map_sum(product_prod(A,B),B,product_prod(C,D),D,product_snd(A,B),product_snd(C,D)),Z4) = B2 ) ) ) ).

% sum.in_rel
tff(fact_7634_sum_Orel__map_I1_J,axiom,
    ! [E: $tType,F3: $tType,A: $tType,C: $tType,D: $tType,B: $tType,S1b: fun(A,fun(B,$o)),S2b: fun(C,fun(D,$o)),I1: fun(E,A),I22: fun(F3,C),X: sum_sum(E,F3),Y: sum_sum(B,D)] :
      ( aa(sum_sum(B,D),$o,aa(sum_sum(A,C),fun(sum_sum(B,D),$o),bNF_rel_sum(A,B,C,D,S1b,S2b),aa(sum_sum(E,F3),sum_sum(A,C),sum_map_sum(E,A,F3,C,I1,I22),X)),Y)
    <=> aa(sum_sum(B,D),$o,aa(sum_sum(E,F3),fun(sum_sum(B,D),$o),bNF_rel_sum(E,B,F3,D,aa(fun(E,A),fun(E,fun(B,$o)),aTP_Lamp_aoy(fun(A,fun(B,$o)),fun(fun(E,A),fun(E,fun(B,$o))),S1b),I1),aa(fun(F3,C),fun(F3,fun(D,$o)),aTP_Lamp_aoz(fun(C,fun(D,$o)),fun(fun(F3,C),fun(F3,fun(D,$o))),S2b),I22)),X),Y) ) ).

% sum.rel_map(1)
tff(fact_7635_sum_Orel__map_I2_J,axiom,
    ! [A: $tType,C: $tType,B: $tType,D: $tType,F3: $tType,E: $tType,S1a: fun(A,fun(B,$o)),S2a: fun(C,fun(D,$o)),X: sum_sum(A,C),G1: fun(E,B),G22: fun(F3,D),Y: sum_sum(E,F3)] :
      ( aa(sum_sum(B,D),$o,aa(sum_sum(A,C),fun(sum_sum(B,D),$o),bNF_rel_sum(A,B,C,D,S1a,S2a),X),aa(sum_sum(E,F3),sum_sum(B,D),sum_map_sum(E,B,F3,D,G1,G22),Y))
    <=> aa(sum_sum(E,F3),$o,aa(sum_sum(A,C),fun(sum_sum(E,F3),$o),bNF_rel_sum(A,E,C,F3,aa(fun(E,B),fun(A,fun(E,$o)),aTP_Lamp_aow(fun(A,fun(B,$o)),fun(fun(E,B),fun(A,fun(E,$o))),S1a),G1),aa(fun(F3,D),fun(C,fun(F3,$o)),aTP_Lamp_aox(fun(C,fun(D,$o)),fun(fun(F3,D),fun(C,fun(F3,$o))),S2a),G22)),X),Y) ) ).

% sum.rel_map(2)
tff(fact_7636_sum_Orel__compp__Grp,axiom,
    ! [A: $tType,C: $tType,D: $tType,B: $tType,R1: fun(A,fun(C,$o)),R22: fun(B,fun(D,$o))] : bNF_rel_sum(A,C,B,D,R1,R22) = aa(fun(sum_sum(product_prod(A,C),product_prod(B,D)),fun(sum_sum(C,D),$o)),fun(sum_sum(A,B),fun(sum_sum(C,D),$o)),aa(fun(sum_sum(A,B),fun(sum_sum(product_prod(A,C),product_prod(B,D)),$o)),fun(fun(sum_sum(product_prod(A,C),product_prod(B,D)),fun(sum_sum(C,D),$o)),fun(sum_sum(A,B),fun(sum_sum(C,D),$o))),relcompp(sum_sum(A,B),sum_sum(product_prod(A,C),product_prod(B,D)),sum_sum(C,D)),conversep(sum_sum(product_prod(A,C),product_prod(B,D)),sum_sum(A,B),bNF_Grp(sum_sum(product_prod(A,C),product_prod(B,D)),sum_sum(A,B),aa(fun(sum_sum(product_prod(A,C),product_prod(B,D)),$o),set(sum_sum(product_prod(A,C),product_prod(B,D))),collect(sum_sum(product_prod(A,C),product_prod(B,D))),aa(fun(B,fun(D,$o)),fun(sum_sum(product_prod(A,C),product_prod(B,D)),$o),aTP_Lamp_axm(fun(A,fun(C,$o)),fun(fun(B,fun(D,$o)),fun(sum_sum(product_prod(A,C),product_prod(B,D)),$o)),R1),R22)),sum_map_sum(product_prod(A,C),A,product_prod(B,D),B,product_fst(A,C),product_fst(B,D))))),bNF_Grp(sum_sum(product_prod(A,C),product_prod(B,D)),sum_sum(C,D),aa(fun(sum_sum(product_prod(A,C),product_prod(B,D)),$o),set(sum_sum(product_prod(A,C),product_prod(B,D))),collect(sum_sum(product_prod(A,C),product_prod(B,D))),aa(fun(B,fun(D,$o)),fun(sum_sum(product_prod(A,C),product_prod(B,D)),$o),aTP_Lamp_axm(fun(A,fun(C,$o)),fun(fun(B,fun(D,$o)),fun(sum_sum(product_prod(A,C),product_prod(B,D)),$o)),R1),R22)),sum_map_sum(product_prod(A,C),C,product_prod(B,D),D,product_snd(A,C),product_snd(B,D)))) ).

% sum.rel_compp_Grp
tff(fact_7637_ordering__top__def,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),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_7638_ordering__top_Ointro,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),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_7639_ordering_Oeq__iff,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A3: A,B2: A] :
      ( ordering(A,Less_eq,Less)
     => ( ( A3 = B2 )
      <=> ( aa(A,$o,aa(A,fun(A,$o),Less_eq,A3),B2)
          & aa(A,$o,aa(A,fun(A,$o),Less_eq,B2),A3) ) ) ) ).

% ordering.eq_iff
tff(fact_7640_ordering_Oantisym,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A3: A,B2: A] :
      ( ordering(A,Less_eq,Less)
     => ( aa(A,$o,aa(A,fun(A,$o),Less_eq,A3),B2)
       => ( aa(A,$o,aa(A,fun(A,$o),Less_eq,B2),A3)
         => ( A3 = B2 ) ) ) ) ).

% ordering.antisym
tff(fact_7641_ordering_Oorder__iff__strict,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A3: A,B2: A] :
      ( ordering(A,Less_eq,Less)
     => ( aa(A,$o,aa(A,fun(A,$o),Less_eq,A3),B2)
      <=> ( aa(A,$o,aa(A,fun(A,$o),Less,A3),B2)
          | ( A3 = B2 ) ) ) ) ).

% ordering.order_iff_strict
tff(fact_7642_ordering_Ostrict__iff__order,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A3: A,B2: A] :
      ( ordering(A,Less_eq,Less)
     => ( aa(A,$o,aa(A,fun(A,$o),Less,A3),B2)
      <=> ( aa(A,$o,aa(A,fun(A,$o),Less_eq,A3),B2)
          & ( A3 != B2 ) ) ) ) ).

% ordering.strict_iff_order
tff(fact_7643_ordering_Ostrict__implies__not__eq,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A3: A,B2: A] :
      ( ordering(A,Less_eq,Less)
     => ( aa(A,$o,aa(A,fun(A,$o),Less,A3),B2)
       => ( A3 != B2 ) ) ) ).

% ordering.strict_implies_not_eq
tff(fact_7644_ordering_Onot__eq__order__implies__strict,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A3: A,B2: A] :
      ( ordering(A,Less_eq,Less)
     => ( ( A3 != B2 )
       => ( aa(A,$o,aa(A,fun(A,$o),Less_eq,A3),B2)
         => aa(A,$o,aa(A,fun(A,$o),Less,A3),B2) ) ) ) ).

% ordering.not_eq_order_implies_strict
tff(fact_7645_ordering__strictI,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o))] :
      ( ! [A6: A,B5: A] :
          ( aa(A,$o,aa(A,fun(A,$o),Less_eq,A6),B5)
        <=> ( aa(A,$o,aa(A,fun(A,$o),Less,A6),B5)
            | ( A6 = B5 ) ) )
     => ( ! [A6: A,B5: A] :
            ( aa(A,$o,aa(A,fun(A,$o),Less,A6),B5)
           => ~ aa(A,$o,aa(A,fun(A,$o),Less,B5),A6) )
       => ( ! [A6: A] : ~ aa(A,$o,aa(A,fun(A,$o),Less,A6),A6)
         => ( ! [A6: A,B5: A,C4: A] :
                ( aa(A,$o,aa(A,fun(A,$o),Less,A6),B5)
               => ( aa(A,$o,aa(A,fun(A,$o),Less,B5),C4)
                 => aa(A,$o,aa(A,fun(A,$o),Less,A6),C4) ) )
           => ordering(A,Less_eq,Less) ) ) ) ) ).

% ordering_strictI
tff(fact_7646_ordering__dualI,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o))] :
      ( ordering(A,aTP_Lamp_aak(fun(A,fun(A,$o)),fun(A,fun(A,$o)),Less_eq),aTP_Lamp_aak(fun(A,fun(A,$o)),fun(A,fun(A,$o)),Less))
     => ordering(A,Less_eq,Less) ) ).

% ordering_dualI
tff(fact_7647_gcd__nat_Oordering__axioms,axiom,
    ordering(nat,dvd_dvd(nat),aTP_Lamp_cc(nat,fun(nat,$o))) ).

% gcd_nat.ordering_axioms
tff(fact_7648_ordering__top_Oaxioms_I1_J,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),Top: A] :
      ( ordering_top(A,Less_eq,Less,Top)
     => ordering(A,Less_eq,Less) ) ).

% ordering_top.axioms(1)
tff(fact_7649_order_Oordering__axioms,axiom,
    ! [A: $tType] :
      ( order(A)
     => ordering(A,ord_less_eq(A),ord_less(A)) ) ).

% order.ordering_axioms
tff(fact_7650_dual__order_Oordering__axioms,axiom,
    ! [A: $tType] :
      ( order(A)
     => ordering(A,aTP_Lamp_axn(A,fun(A,$o)),aTP_Lamp_axo(A,fun(A,$o))) ) ).

% dual_order.ordering_axioms
tff(fact_7651_subset__mset_Odual__order_Oordering__axioms,axiom,
    ! [A: $tType] : ordering(multiset(A),aTP_Lamp_aso(multiset(A),fun(multiset(A),$o)),aTP_Lamp_asp(multiset(A),fun(multiset(A),$o))) ).

% subset_mset.dual_order.ordering_axioms
tff(fact_7652_strict__mono__inv,axiom,
    ! [A: $tType,B: $tType] :
      ( ( linorder(B)
        & linorder(A) )
     => ! [F: fun(A,B),G: fun(B,A)] :
          ( order_strict_mono(A,B,F)
         => ( ( aa(set(A),set(B),image2(A,B,F),top_top(set(A))) = top_top(set(B)) )
           => ( ! [X2: A] : aa(B,A,G,aa(A,B,F,X2)) = X2
             => order_strict_mono(B,A,G) ) ) ) ) ).

% strict_mono_inv
tff(fact_7653_Predicate_Oiterate__upto_Opsimps,axiom,
    ! [A: $tType,F: fun(code_natural,A),N: code_natural,M: code_natural] :
      ( accp(product_prod(fun(code_natural,A),product_prod(code_natural,code_natural)),iterate_upto_rel(A),aa(product_prod(code_natural,code_natural),product_prod(fun(code_natural,A),product_prod(code_natural,code_natural)),aa(fun(code_natural,A),fun(product_prod(code_natural,code_natural),product_prod(fun(code_natural,A),product_prod(code_natural,code_natural))),product_Pair(fun(code_natural,A),product_prod(code_natural,code_natural)),F),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),N),M)))
     => ( iterate_upto(A,F,N,M) = seq2(A,aa(code_natural,fun(product_unit,seq(A)),aa(code_natural,fun(code_natural,fun(product_unit,seq(A))),aTP_Lamp_axp(fun(code_natural,A),fun(code_natural,fun(code_natural,fun(product_unit,seq(A)))),F),N),M)) ) ) ).

% Predicate.iterate_upto.psimps
tff(fact_7654_strict__mono__less__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & order(B) )
     => ! [F: fun(A,B),X: A,Y: A] :
          ( order_strict_mono(A,B,F)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,X)),aa(A,B,F,Y))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y) ) ) ) ).

% strict_mono_less_eq
tff(fact_7655_Predicate_Oiterate__upto_Oelims,axiom,
    ! [A: $tType,X: fun(code_natural,A),Xa: code_natural,Xb: code_natural,Y: pred(A)] :
      ( ( iterate_upto(A,X,Xa,Xb) = Y )
     => ( Y = seq2(A,aa(code_natural,fun(product_unit,seq(A)),aa(code_natural,fun(code_natural,fun(product_unit,seq(A))),aTP_Lamp_axp(fun(code_natural,A),fun(code_natural,fun(code_natural,fun(product_unit,seq(A)))),X),Xa),Xb)) ) ) ).

% Predicate.iterate_upto.elims
tff(fact_7656_Predicate_Oiterate__upto_Osimps,axiom,
    ! [A: $tType,F: fun(code_natural,A),N: code_natural,M: code_natural] : iterate_upto(A,F,N,M) = seq2(A,aa(code_natural,fun(product_unit,seq(A)),aa(code_natural,fun(code_natural,fun(product_unit,seq(A))),aTP_Lamp_axp(fun(code_natural,A),fun(code_natural,fun(code_natural,fun(product_unit,seq(A)))),F),N),M)) ).

% Predicate.iterate_upto.simps
tff(fact_7657_strict__monoD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F: fun(A,B),X: A,Y: A] :
          ( order_strict_mono(A,B,F)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
           => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F,X)),aa(A,B,F,Y)) ) ) ) ).

% strict_monoD
tff(fact_7658_strict__monoI,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F: fun(A,B)] :
          ( ! [X2: A,Y2: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X2),Y2)
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F,X2)),aa(A,B,F,Y2)) )
         => order_strict_mono(A,B,F) ) ) ).

% strict_monoI
tff(fact_7659_strict__mono__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F: fun(A,B)] :
          ( order_strict_mono(A,B,F)
        <=> ! [X4: A,Y3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Y3)
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F,X4)),aa(A,B,F,Y3)) ) ) ) ).

% strict_mono_def
tff(fact_7660_strict__mono__less,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & order(B) )
     => ! [F: fun(A,B),X: A,Y: A] :
          ( order_strict_mono(A,B,F)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F,X)),aa(A,B,F,Y))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y) ) ) ) ).

% strict_mono_less
tff(fact_7661_strict__mono__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F: fun(A,B)] :
          ( order_strict_mono(A,B,F)
         => order_mono(A,B,F) ) ) ).

% strict_mono_mono
tff(fact_7662_strict__mono__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & order(B) )
     => ! [F: fun(A,B),X: A,Y: A] :
          ( order_strict_mono(A,B,F)
         => ( ( aa(A,B,F,X) = aa(A,B,F,Y) )
          <=> ( X = Y ) ) ) ) ).

% strict_mono_eq
tff(fact_7663_bot__set__code,axiom,
    ! [A: $tType] : bot_bot(pred(A)) = seq2(A,aTP_Lamp_axq(product_unit,seq(A))) ).

% bot_set_code
tff(fact_7664_Predicate_Osingle__code,axiom,
    ! [A: $tType,X: A] : aa(A,pred(A),single(A),X) = seq2(A,aTP_Lamp_axr(A,fun(product_unit,seq(A)),X)) ).

% Predicate.single_code
tff(fact_7665_Predicate_Oiterate__upto_Opelims,axiom,
    ! [A: $tType,X: fun(code_natural,A),Xa: code_natural,Xb: code_natural,Y: pred(A)] :
      ( ( iterate_upto(A,X,Xa,Xb) = Y )
     => ( accp(product_prod(fun(code_natural,A),product_prod(code_natural,code_natural)),iterate_upto_rel(A),aa(product_prod(code_natural,code_natural),product_prod(fun(code_natural,A),product_prod(code_natural,code_natural)),aa(fun(code_natural,A),fun(product_prod(code_natural,code_natural),product_prod(fun(code_natural,A),product_prod(code_natural,code_natural))),product_Pair(fun(code_natural,A),product_prod(code_natural,code_natural)),X),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),Xa),Xb)))
       => ~ ( ( Y = seq2(A,aa(code_natural,fun(product_unit,seq(A)),aa(code_natural,fun(code_natural,fun(product_unit,seq(A))),aTP_Lamp_axp(fun(code_natural,A),fun(code_natural,fun(code_natural,fun(product_unit,seq(A)))),X),Xa),Xb)) )
           => ~ accp(product_prod(fun(code_natural,A),product_prod(code_natural,code_natural)),iterate_upto_rel(A),aa(product_prod(code_natural,code_natural),product_prod(fun(code_natural,A),product_prod(code_natural,code_natural)),aa(fun(code_natural,A),fun(product_prod(code_natural,code_natural),product_prod(fun(code_natural,A),product_prod(code_natural,code_natural))),product_Pair(fun(code_natural,A),product_prod(code_natural,code_natural)),X),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),Xa),Xb))) ) ) ) ).

% Predicate.iterate_upto.pelims
tff(fact_7666_Random__Pred_Oiterate__upto__def,axiom,
    ! [A: $tType,F: fun(code_natural,A),N: code_natural,M: code_natural] : random_iterate_upto(A,F,N,M) = aa(pred(A),fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))),product_Pair(pred(A),product_prod(code_natural,code_natural)),iterate_upto(A,F,N,M)) ).

% Random_Pred.iterate_upto_def
tff(fact_7667_bind__code,axiom,
    ! [A: $tType,B: $tType,G: fun(product_unit,seq(B)),F: fun(B,pred(A))] : bind2(B,A,seq2(B,G),F) = seq2(A,aa(fun(B,pred(A)),fun(product_unit,seq(A)),aTP_Lamp_axs(fun(product_unit,seq(B)),fun(fun(B,pred(A)),fun(product_unit,seq(A))),G),F)) ).

% bind_code
tff(fact_7668_strict__mono__inv__on__range,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & order(B) )
     => ! [F: fun(A,B)] :
          ( order_strict_mono(A,B,F)
         => strict_mono_on(B,A,hilbert_inv_into(A,B,top_top(set(A)),F),aa(set(A),set(B),image2(A,B,F),top_top(set(A)))) ) ) ).

% strict_mono_inv_on_range
tff(fact_7669_pred__of__seq_Osimps_I2_J,axiom,
    ! [A: $tType,X: A,P: pred(A)] : pred_of_seq(A,insert(A,X,P)) = aa(pred(A),pred(A),aa(pred(A),fun(pred(A),pred(A)),sup_sup(pred(A)),aa(A,pred(A),single(A),X)),P) ).

% pred_of_seq.simps(2)
tff(fact_7670_pred__of__seq_Osimps_I1_J,axiom,
    ! [A: $tType] : pred_of_seq(A,empty(A)) = bot_bot(pred(A)) ).

% pred_of_seq.simps(1)
tff(fact_7671_adjunct__sup,axiom,
    ! [A: $tType,P: pred(A),Xq: seq(A)] : pred_of_seq(A,adjunct(A,P,Xq)) = aa(pred(A),pred(A),aa(pred(A),fun(pred(A),pred(A)),sup_sup(pred(A)),P),pred_of_seq(A,Xq)) ).

% adjunct_sup
tff(fact_7672_adjunct_Osimps_I2_J,axiom,
    ! [A: $tType,P: pred(A),X: A,Q: pred(A)] : adjunct(A,P,insert(A,X,Q)) = insert(A,X,aa(pred(A),pred(A),aa(pred(A),fun(pred(A),pred(A)),sup_sup(pred(A)),Q),P)) ).

% adjunct.simps(2)
tff(fact_7673_sup__code,axiom,
    ! [A: $tType,F: fun(product_unit,seq(A)),G: fun(product_unit,seq(A))] : aa(pred(A),pred(A),aa(pred(A),fun(pred(A),pred(A)),sup_sup(pred(A)),seq2(A,F)),seq2(A,G)) = seq2(A,aa(fun(product_unit,seq(A)),fun(product_unit,seq(A)),aTP_Lamp_axv(fun(product_unit,seq(A)),fun(fun(product_unit,seq(A)),fun(product_unit,seq(A))),F),G)) ).

% sup_code
tff(fact_7674_of__seq__code_I1_J,axiom,
    ! [A: $tType] : set_of_seq(A,empty(A)) = bot_bot(set(A)) ).

% of_seq_code(1)
tff(fact_7675_seq_Ocase__distrib,axiom,
    ! [B: $tType,A: $tType,C: $tType,H2: fun(B,A),F1: B,F22: fun(C,fun(pred(C),B)),F32: fun(pred(C),fun(seq(C),B)),Seq: seq(C)] : aa(B,A,H2,case_seq(B,C,F1,F22,F32,Seq)) = case_seq(A,C,aa(B,A,H2,F1),aa(fun(C,fun(pred(C),B)),fun(C,fun(pred(C),A)),aTP_Lamp_axw(fun(B,A),fun(fun(C,fun(pred(C),B)),fun(C,fun(pred(C),A))),H2),F22),aa(fun(pred(C),fun(seq(C),B)),fun(pred(C),fun(seq(C),A)),aTP_Lamp_axx(fun(B,A),fun(fun(pred(C),fun(seq(C),B)),fun(pred(C),fun(seq(C),A))),H2),F32),Seq) ).

% seq.case_distrib
tff(fact_7676_pred__of__seq_Osimps_I3_J,axiom,
    ! [A: $tType,P: pred(A),Xq: seq(A)] : pred_of_seq(A,join(A,P,Xq)) = aa(pred(A),pred(A),aa(pred(A),fun(pred(A),pred(A)),sup_sup(pred(A)),P),pred_of_seq(A,Xq)) ).

% pred_of_seq.simps(3)
tff(fact_7677_of__seq__code_I3_J,axiom,
    ! [A: $tType,P: pred(A),Xq: seq(A)] : set_of_seq(A,join(A,P,Xq)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_of_pred(A,P)),set_of_seq(A,Xq)) ).

% of_seq_code(3)
tff(fact_7678_of__pred__code,axiom,
    ! [A: $tType,F: fun(product_unit,seq(A))] : set_of_pred(A,seq2(A,F)) = case_seq(set(A),A,bot_bot(set(A)),aTP_Lamp_axy(A,fun(pred(A),set(A))),aTP_Lamp_axz(pred(A),fun(seq(A),set(A))),aa(product_unit,seq(A),F,product_Unity)) ).

% of_pred_code
tff(fact_7679_less__eq__pred__code,axiom,
    ! [A: $tType,F: fun(product_unit,seq(A)),Q: pred(A)] :
      ( aa(pred(A),$o,aa(pred(A),fun(pred(A),$o),ord_less_eq(pred(A)),seq2(A,F)),Q)
    <=> case_seq($o,A,$true,aTP_Lamp_aya(pred(A),fun(A,fun(pred(A),$o)),Q),aTP_Lamp_ayb(pred(A),fun(pred(A),fun(seq(A),$o)),Q),aa(product_unit,seq(A),F,product_Unity)) ) ).

% less_eq_pred_code
tff(fact_7680_rec__natural__def,axiom,
    ! [A: $tType,X3: A,Xa3: fun(code_natural,fun(A,A)),Xb2: code_natural] : aa(code_natural,A,code_rec_natural(A,X3,Xa3),Xb2) = the(A,code_rec_set_natural(A,X3,Xa3,Xb2)) ).

% rec_natural_def
tff(fact_7681_case__natural__def,axiom,
    ! [A: $tType,X3: A,Xa3: fun(code_natural,A)] : code_case_natural(A,X3,Xa3) = code_rec_natural(A,X3,aTP_Lamp_ayc(fun(code_natural,A),fun(code_natural,fun(A,A)),Xa3)) ).

% case_natural_def
tff(fact_7682_the__only_Osimps_I2_J,axiom,
    ! [A: $tType,Default: fun(product_unit,A),X: A,P: pred(A)] :
      the_only(A,Default,insert(A,X,P)) = $ite(
        is_empty(A,P),
        X,
        $ite(X = singleton(A,Default,P),X,aa(product_unit,A,Default,product_Unity)) ) ).

% the_only.simps(2)
tff(fact_7683_Predicate_Ois__empty__def,axiom,
    ! [A: $tType,A4: pred(A)] :
      ( is_empty(A,A4)
    <=> ( A4 = bot_bot(pred(A)) ) ) ).

% Predicate.is_empty_def
tff(fact_7684_is__empty__bot,axiom,
    ! [A: $tType] : is_empty(A,bot_bot(pred(A))) ).

% is_empty_bot
tff(fact_7685_is__empty__sup,axiom,
    ! [A: $tType,A4: pred(A),B3: pred(A)] :
      ( is_empty(A,aa(pred(A),pred(A),aa(pred(A),fun(pred(A),pred(A)),sup_sup(pred(A)),A4),B3))
    <=> ( is_empty(A,A4)
        & is_empty(A,B3) ) ) ).

% is_empty_sup
tff(fact_7686_singleton__code,axiom,
    ! [A: $tType,Default: fun(product_unit,A),F: fun(product_unit,seq(A))] : singleton(A,Default,seq2(A,F)) = case_seq(A,A,aa(product_unit,A,Default,product_Unity),aTP_Lamp_ayd(fun(product_unit,A),fun(A,fun(pred(A),A)),Default),aTP_Lamp_aye(fun(product_unit,A),fun(pred(A),fun(seq(A),A)),Default),aa(product_unit,seq(A),F,product_Unity)) ).

% singleton_code
tff(fact_7687_the__only_Osimps_I3_J,axiom,
    ! [A: $tType,Default: fun(product_unit,A),P: pred(A),Xq: seq(A)] :
      the_only(A,Default,join(A,P,Xq)) = $ite(
        is_empty(A,P),
        the_only(A,Default,Xq),
        $ite(
          null(A,Xq),
          singleton(A,Default,P),
          $let(
            x3: A,
            x3:= singleton(A,Default,P),
            $ite(x3 = the_only(A,Default,Xq),x3,aa(product_unit,A,Default,product_Unity)) ) ) ) ).

% the_only.simps(3)
tff(fact_7688_Abs__int__cases,axiom,
    ! [X: int] :
      ~ ! [Y2: set(product_prod(nat,nat))] :
          ( ( X = aa(set(product_prod(nat,nat)),int,abs_int,Y2) )
         => ~ aa(set(set(product_prod(nat,nat))),$o,member(set(product_prod(nat,nat)),Y2),aa(fun(set(product_prod(nat,nat)),$o),set(set(product_prod(nat,nat))),collect(set(product_prod(nat,nat))),aTP_Lamp_ayf(set(product_prod(nat,nat)),$o))) ) ).

% Abs_int_cases
tff(fact_7689_Abs__int__induct,axiom,
    ! [P: fun(int,$o),X: int] :
      ( ! [Y2: set(product_prod(nat,nat))] :
          ( aa(set(set(product_prod(nat,nat))),$o,member(set(product_prod(nat,nat)),Y2),aa(fun(set(product_prod(nat,nat)),$o),set(set(product_prod(nat,nat))),collect(set(product_prod(nat,nat))),aTP_Lamp_ayf(set(product_prod(nat,nat)),$o)))
         => aa(int,$o,P,aa(set(product_prod(nat,nat)),int,abs_int,Y2)) )
     => aa(int,$o,P,X) ) ).

% Abs_int_induct
tff(fact_7690_Abs__int__inject,axiom,
    ! [X: set(product_prod(nat,nat)),Y: set(product_prod(nat,nat))] :
      ( aa(set(set(product_prod(nat,nat))),$o,member(set(product_prod(nat,nat)),X),aa(fun(set(product_prod(nat,nat)),$o),set(set(product_prod(nat,nat))),collect(set(product_prod(nat,nat))),aTP_Lamp_ayf(set(product_prod(nat,nat)),$o)))
     => ( aa(set(set(product_prod(nat,nat))),$o,member(set(product_prod(nat,nat)),Y),aa(fun(set(product_prod(nat,nat)),$o),set(set(product_prod(nat,nat))),collect(set(product_prod(nat,nat))),aTP_Lamp_ayf(set(product_prod(nat,nat)),$o)))
       => ( ( aa(set(product_prod(nat,nat)),int,abs_int,X) = aa(set(product_prod(nat,nat)),int,abs_int,Y) )
        <=> ( X = Y ) ) ) ) ).

% Abs_int_inject
tff(fact_7691_Abs__int__inverse,axiom,
    ! [Y: set(product_prod(nat,nat))] :
      ( aa(set(set(product_prod(nat,nat))),$o,member(set(product_prod(nat,nat)),Y),aa(fun(set(product_prod(nat,nat)),$o),set(set(product_prod(nat,nat))),collect(set(product_prod(nat,nat))),aTP_Lamp_ayf(set(product_prod(nat,nat)),$o)))
     => ( aa(int,set(product_prod(nat,nat)),rep_int,aa(set(product_prod(nat,nat)),int,abs_int,Y)) = Y ) ) ).

% Abs_int_inverse
tff(fact_7692_type__definition__int,axiom,
    type_definition(int,set(product_prod(nat,nat)),rep_int,abs_int,aa(fun(set(product_prod(nat,nat)),$o),set(set(product_prod(nat,nat))),collect(set(product_prod(nat,nat))),aTP_Lamp_ayf(set(product_prod(nat,nat)),$o))) ).

% type_definition_int
tff(fact_7693_Rep__int__induct,axiom,
    ! [Y: set(product_prod(nat,nat)),P: fun(set(product_prod(nat,nat)),$o)] :
      ( aa(set(set(product_prod(nat,nat))),$o,member(set(product_prod(nat,nat)),Y),aa(fun(set(product_prod(nat,nat)),$o),set(set(product_prod(nat,nat))),collect(set(product_prod(nat,nat))),aTP_Lamp_ayf(set(product_prod(nat,nat)),$o)))
     => ( ! [X2: int] : aa(set(product_prod(nat,nat)),$o,P,aa(int,set(product_prod(nat,nat)),rep_int,X2))
       => aa(set(product_prod(nat,nat)),$o,P,Y) ) ) ).

% Rep_int_induct
tff(fact_7694_Rep__int__cases,axiom,
    ! [Y: set(product_prod(nat,nat))] :
      ( aa(set(set(product_prod(nat,nat))),$o,member(set(product_prod(nat,nat)),Y),aa(fun(set(product_prod(nat,nat)),$o),set(set(product_prod(nat,nat))),collect(set(product_prod(nat,nat))),aTP_Lamp_ayf(set(product_prod(nat,nat)),$o)))
     => ~ ! [X2: int] : Y != aa(int,set(product_prod(nat,nat)),rep_int,X2) ) ).

% Rep_int_cases
tff(fact_7695_Rep__int,axiom,
    ! [X: int] : aa(set(set(product_prod(nat,nat))),$o,member(set(product_prod(nat,nat)),aa(int,set(product_prod(nat,nat)),rep_int,X)),aa(fun(set(product_prod(nat,nat)),$o),set(set(product_prod(nat,nat))),collect(set(product_prod(nat,nat))),aTP_Lamp_ayf(set(product_prod(nat,nat)),$o))) ).

% Rep_int
tff(fact_7696_ordering_Oaxioms_I2_J,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o))] :
      ( ordering(A,Less_eq,Less)
     => ordering_axioms(A,Less_eq,Less) ) ).

% ordering.axioms(2)
tff(fact_7697_Int_Osub__code_I9_J,axiom,
    ! [M: num,N: num] : sub(aa(num,num,bit0,M),aa(num,num,bit1,N)) = aa(int,int,aa(int,fun(int,int),minus_minus(int),dup(sub(M,N))),one_one(int)) ).

% Int.sub_code(9)
tff(fact_7698_ordering__axioms_Ointro,axiom,
    ! [A: $tType,Less: fun(A,fun(A,$o)),Less_eq: fun(A,fun(A,$o))] :
      ( ! [A6: A,B5: A] :
          ( aa(A,$o,aa(A,fun(A,$o),Less,A6),B5)
        <=> ( aa(A,$o,aa(A,fun(A,$o),Less_eq,A6),B5)
            & ( A6 != B5 ) ) )
     => ( ! [A6: A,B5: A] :
            ( aa(A,$o,aa(A,fun(A,$o),Less_eq,A6),B5)
           => ( aa(A,$o,aa(A,fun(A,$o),Less_eq,B5),A6)
             => ( A6 = B5 ) ) )
       => ordering_axioms(A,Less_eq,Less) ) ) ).

% ordering_axioms.intro
tff(fact_7699_ordering__axioms__def,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o))] :
      ( ordering_axioms(A,Less_eq,Less)
    <=> ( ! [A10: A,B6: A] :
            ( aa(A,$o,aa(A,fun(A,$o),Less,A10),B6)
          <=> ( aa(A,$o,aa(A,fun(A,$o),Less_eq,A10),B6)
              & ( A10 != B6 ) ) )
        & ! [A10: A,B6: A] :
            ( aa(A,$o,aa(A,fun(A,$o),Less_eq,A10),B6)
           => ( aa(A,$o,aa(A,fun(A,$o),Less_eq,B6),A10)
             => ( A10 = B6 ) ) ) ) ) ).

% ordering_axioms_def
tff(fact_7700_Int_Osub__code_I8_J,axiom,
    ! [M: num,N: num] : sub(aa(num,num,bit1,M),aa(num,num,bit0,N)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),dup(sub(M,N))),one_one(int)) ).

% Int.sub_code(8)
tff(fact_7701_ordering_Ointro,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o))] :
      ( partial_preordering(A,Less_eq)
     => ( ordering_axioms(A,Less_eq,Less)
       => ordering(A,Less_eq,Less) ) ) ).

% ordering.intro
tff(fact_7702_ordering__def,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o))] :
      ( ordering(A,Less_eq,Less)
    <=> ( partial_preordering(A,Less_eq)
        & ordering_axioms(A,Less_eq,Less) ) ) ).

% ordering_def
tff(fact_7703_subset__mset_Odual__order_Opartial__preordering__axioms,axiom,
    ! [A: $tType] : partial_preordering(multiset(A),aTP_Lamp_aso(multiset(A),fun(multiset(A),$o))) ).

% subset_mset.dual_order.partial_preordering_axioms
tff(fact_7704_partial__preordering__def,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,$o))] :
      ( partial_preordering(A,Less_eq)
    <=> ( ! [A10: A] : aa(A,$o,aa(A,fun(A,$o),Less_eq,A10),A10)
        & ! [A10: A,B6: A,C5: A] :
            ( aa(A,$o,aa(A,fun(A,$o),Less_eq,A10),B6)
           => ( aa(A,$o,aa(A,fun(A,$o),Less_eq,B6),C5)
             => aa(A,$o,aa(A,fun(A,$o),Less_eq,A10),C5) ) ) ) ) ).

% partial_preordering_def
tff(fact_7705_partial__preordering_Otrans,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,$o)),A3: A,B2: A,C2: A] :
      ( partial_preordering(A,Less_eq)
     => ( aa(A,$o,aa(A,fun(A,$o),Less_eq,A3),B2)
       => ( aa(A,$o,aa(A,fun(A,$o),Less_eq,B2),C2)
         => aa(A,$o,aa(A,fun(A,$o),Less_eq,A3),C2) ) ) ) ).

% partial_preordering.trans
tff(fact_7706_partial__preordering_Ointro,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,$o))] :
      ( ! [A6: A] : aa(A,$o,aa(A,fun(A,$o),Less_eq,A6),A6)
     => ( ! [A6: A,B5: A,C4: A] :
            ( aa(A,$o,aa(A,fun(A,$o),Less_eq,A6),B5)
           => ( aa(A,$o,aa(A,fun(A,$o),Less_eq,B5),C4)
             => aa(A,$o,aa(A,fun(A,$o),Less_eq,A6),C4) ) )
       => partial_preordering(A,Less_eq) ) ) ).

% partial_preordering.intro
tff(fact_7707_partial__preordering_Orefl,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,$o)),A3: A] :
      ( partial_preordering(A,Less_eq)
     => aa(A,$o,aa(A,fun(A,$o),Less_eq,A3),A3) ) ).

% partial_preordering.refl
tff(fact_7708_ordering_Oaxioms_I1_J,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o))] :
      ( ordering(A,Less_eq,Less)
     => partial_preordering(A,Less_eq) ) ).

% ordering.axioms(1)
tff(fact_7709_dual__order_Opartial__preordering__axioms,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => partial_preordering(A,aTP_Lamp_avn(A,fun(A,$o))) ) ).

% dual_order.partial_preordering_axioms
tff(fact_7710_order_Opartial__preordering__axioms,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => partial_preordering(A,ord_less_eq(A)) ) ).

% order.partial_preordering_axioms
tff(fact_7711_semilattice__neutr__def,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
      ( semilattice_neutr(A,F,Z2)
    <=> ( semilattice(A,F)
        & comm_monoid(A,F,Z2) ) ) ).

% semilattice_neutr_def
tff(fact_7712_semilattice__neutr_Ointro,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
      ( semilattice(A,F)
     => ( comm_monoid(A,F,Z2)
       => semilattice_neutr(A,F,Z2) ) ) ).

% semilattice_neutr.intro
tff(fact_7713_semilattice__order_Oaxioms_I1_J,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o))] :
      ( semilattice_order(A,F,Less_eq,Less)
     => semilattice(A,F) ) ).

% semilattice_order.axioms(1)
tff(fact_7714_sup_Osemilattice__axioms,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => semilattice(A,sup_sup(A)) ) ).

% sup.semilattice_axioms
tff(fact_7715_min_Osemilattice__axioms,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => semilattice(A,ord_min(A)) ) ).

% min.semilattice_axioms
tff(fact_7716_max_Osemilattice__axioms,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => semilattice(A,ord_max(A)) ) ).

% max.semilattice_axioms
tff(fact_7717_semilattice_Oidem,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),A3: A] :
      ( semilattice(A,F)
     => ( aa(A,A,aa(A,fun(A,A),F,A3),A3) = A3 ) ) ).

% semilattice.idem
tff(fact_7718_semilattice_Oleft__idem,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),A3: A,B2: A] :
      ( semilattice(A,F)
     => ( aa(A,A,aa(A,fun(A,A),F,A3),aa(A,A,aa(A,fun(A,A),F,A3),B2)) = aa(A,A,aa(A,fun(A,A),F,A3),B2) ) ) ).

% semilattice.left_idem
tff(fact_7719_semilattice_Oright__idem,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),A3: A,B2: A] :
      ( semilattice(A,F)
     => ( aa(A,A,aa(A,fun(A,A),F,aa(A,A,aa(A,fun(A,A),F,A3),B2)),B2) = aa(A,A,aa(A,fun(A,A),F,A3),B2) ) ) ).

% semilattice.right_idem
tff(fact_7720_semilattice__neutr_Oaxioms_I1_J,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
      ( semilattice_neutr(A,F,Z2)
     => semilattice(A,F) ) ).

% semilattice_neutr.axioms(1)
tff(fact_7721_semilattice__map2,axiom,
    ! [A: $tType,F: fun(A,fun(A,A))] :
      ( semilattice(A,F)
     => semilattice(list(A),aTP_Lamp_ayg(fun(A,fun(A,A)),fun(list(A),fun(list(A),list(A))),F)) ) ).

% semilattice_map2
tff(fact_7722_inf_Osemilattice__axioms,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => semilattice(A,inf_inf(A)) ) ).

% inf.semilattice_axioms
tff(fact_7723_semilattice__order__def,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o))] :
      ( semilattice_order(A,F,Less_eq,Less)
    <=> ( semilattice(A,F)
        & semila6385135966242565138axioms(A,F,Less_eq,Less) ) ) ).

% semilattice_order_def
tff(fact_7724_semilattice__order_Ointro,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o))] :
      ( semilattice(A,F)
     => ( semila6385135966242565138axioms(A,F,Less_eq,Less)
       => semilattice_order(A,F,Less_eq,Less) ) ) ).

% semilattice_order.intro
tff(fact_7725_semilattice__order__axioms__def,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o))] :
      ( semila6385135966242565138axioms(A,F,Less_eq,Less)
    <=> ( ! [A10: A,B6: A] :
            ( aa(A,$o,aa(A,fun(A,$o),Less_eq,A10),B6)
          <=> ( A10 = aa(A,A,aa(A,fun(A,A),F,A10),B6) ) )
        & ! [A10: A,B6: A] :
            ( aa(A,$o,aa(A,fun(A,$o),Less,A10),B6)
          <=> ( ( A10 = aa(A,A,aa(A,fun(A,A),F,A10),B6) )
              & ( A10 != B6 ) ) ) ) ) ).

% semilattice_order_axioms_def
tff(fact_7726_semilattice__order__axioms_Ointro,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,$o)),F: fun(A,fun(A,A)),Less: fun(A,fun(A,$o))] :
      ( ! [A6: A,B5: A] :
          ( aa(A,$o,aa(A,fun(A,$o),Less_eq,A6),B5)
        <=> ( A6 = aa(A,A,aa(A,fun(A,A),F,A6),B5) ) )
     => ( ! [A6: A,B5: A] :
            ( aa(A,$o,aa(A,fun(A,$o),Less,A6),B5)
          <=> ( ( A6 = aa(A,A,aa(A,fun(A,A),F,A6),B5) )
              & ( A6 != B5 ) ) )
       => semila6385135966242565138axioms(A,F,Less_eq,Less) ) ) ).

% semilattice_order_axioms.intro
tff(fact_7727_semilattice__order_Oaxioms_I2_J,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o))] :
      ( semilattice_order(A,F,Less_eq,Less)
     => semila6385135966242565138axioms(A,F,Less_eq,Less) ) ).

% semilattice_order.axioms(2)
tff(fact_7728_subset__mset_Odual__order_Opreordering__axioms,axiom,
    ! [A: $tType] : preordering(multiset(A),aTP_Lamp_aso(multiset(A),fun(multiset(A),$o)),aTP_Lamp_asp(multiset(A),fun(multiset(A),$o))) ).

% subset_mset.dual_order.preordering_axioms
tff(fact_7729_comm__monoid_Oaxioms_I2_J,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
      ( comm_monoid(A,F,Z2)
     => comm_monoid_axioms(A,F,Z2) ) ).

% comm_monoid.axioms(2)
tff(fact_7730_order_Opreordering__axioms,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => preordering(A,ord_less_eq(A),ord_less(A)) ) ).

% order.preordering_axioms
tff(fact_7731_comm__monoid__axioms_Ointro,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
      ( ! [A6: A] : aa(A,A,aa(A,fun(A,A),F,A6),Z2) = A6
     => comm_monoid_axioms(A,F,Z2) ) ).

% comm_monoid_axioms.intro
tff(fact_7732_comm__monoid__axioms__def,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
      ( comm_monoid_axioms(A,F,Z2)
    <=> ! [A10: A] : aa(A,A,aa(A,fun(A,A),F,A10),Z2) = A10 ) ).

% comm_monoid_axioms_def
tff(fact_7733_preordering_Oasym,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A3: A,B2: A] :
      ( preordering(A,Less_eq,Less)
     => ( aa(A,$o,aa(A,fun(A,$o),Less,A3),B2)
       => ~ aa(A,$o,aa(A,fun(A,$o),Less,B2),A3) ) ) ).

% preordering.asym
tff(fact_7734_preordering_Oirrefl,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A3: A] :
      ( preordering(A,Less_eq,Less)
     => ~ aa(A,$o,aa(A,fun(A,$o),Less,A3),A3) ) ).

% preordering.irrefl
tff(fact_7735_preordering_Ostrict__trans,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A3: A,B2: A,C2: A] :
      ( preordering(A,Less_eq,Less)
     => ( aa(A,$o,aa(A,fun(A,$o),Less,A3),B2)
       => ( aa(A,$o,aa(A,fun(A,$o),Less,B2),C2)
         => aa(A,$o,aa(A,fun(A,$o),Less,A3),C2) ) ) ) ).

% preordering.strict_trans
tff(fact_7736_preordering_Ostrict__trans1,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A3: A,B2: A,C2: A] :
      ( preordering(A,Less_eq,Less)
     => ( aa(A,$o,aa(A,fun(A,$o),Less_eq,A3),B2)
       => ( aa(A,$o,aa(A,fun(A,$o),Less,B2),C2)
         => aa(A,$o,aa(A,fun(A,$o),Less,A3),C2) ) ) ) ).

% preordering.strict_trans1
tff(fact_7737_preordering_Ostrict__trans2,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A3: A,B2: A,C2: A] :
      ( preordering(A,Less_eq,Less)
     => ( aa(A,$o,aa(A,fun(A,$o),Less,A3),B2)
       => ( aa(A,$o,aa(A,fun(A,$o),Less_eq,B2),C2)
         => aa(A,$o,aa(A,fun(A,$o),Less,A3),C2) ) ) ) ).

% preordering.strict_trans2
tff(fact_7738_preordering_Ostrict__iff__not,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A3: A,B2: A] :
      ( preordering(A,Less_eq,Less)
     => ( aa(A,$o,aa(A,fun(A,$o),Less,A3),B2)
      <=> ( aa(A,$o,aa(A,fun(A,$o),Less_eq,A3),B2)
          & ~ aa(A,$o,aa(A,fun(A,$o),Less_eq,B2),A3) ) ) ) ).

% preordering.strict_iff_not
tff(fact_7739_preordering_Ostrict__implies__order,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A3: A,B2: A] :
      ( preordering(A,Less_eq,Less)
     => ( aa(A,$o,aa(A,fun(A,$o),Less,A3),B2)
       => aa(A,$o,aa(A,fun(A,$o),Less_eq,A3),B2) ) ) ).

% preordering.strict_implies_order
tff(fact_7740_preordering__strictI,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o))] :
      ( ! [A6: A,B5: A] :
          ( aa(A,$o,aa(A,fun(A,$o),Less_eq,A6),B5)
        <=> ( aa(A,$o,aa(A,fun(A,$o),Less,A6),B5)
            | ( A6 = B5 ) ) )
     => ( ! [A6: A,B5: A] :
            ( aa(A,$o,aa(A,fun(A,$o),Less,A6),B5)
           => ~ aa(A,$o,aa(A,fun(A,$o),Less,B5),A6) )
       => ( ! [A6: A] : ~ aa(A,$o,aa(A,fun(A,$o),Less,A6),A6)
         => ( ! [A6: A,B5: A,C4: A] :
                ( aa(A,$o,aa(A,fun(A,$o),Less,A6),B5)
               => ( aa(A,$o,aa(A,fun(A,$o),Less,B5),C4)
                 => aa(A,$o,aa(A,fun(A,$o),Less,A6),C4) ) )
           => preordering(A,Less_eq,Less) ) ) ) ) ).

% preordering_strictI
tff(fact_7741_preordering__dualI,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o))] :
      ( preordering(A,aTP_Lamp_aak(fun(A,fun(A,$o)),fun(A,fun(A,$o)),Less_eq),aTP_Lamp_aak(fun(A,fun(A,$o)),fun(A,fun(A,$o)),Less))
     => preordering(A,Less_eq,Less) ) ).

% preordering_dualI
tff(fact_7742_preordering_Oaxioms_I1_J,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o))] :
      ( preordering(A,Less_eq,Less)
     => partial_preordering(A,Less_eq) ) ).

% preordering.axioms(1)
tff(fact_7743_dual__order_Opreordering__axioms,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => preordering(A,aTP_Lamp_avn(A,fun(A,$o)),aTP_Lamp_sz(A,fun(A,$o))) ) ).

% dual_order.preordering_axioms
tff(fact_7744_preordering_Ointro,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o))] :
      ( partial_preordering(A,Less_eq)
     => ( preordering_axioms(A,Less_eq,Less)
       => preordering(A,Less_eq,Less) ) ) ).

% preordering.intro
tff(fact_7745_preordering__def,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o))] :
      ( preordering(A,Less_eq,Less)
    <=> ( partial_preordering(A,Less_eq)
        & preordering_axioms(A,Less_eq,Less) ) ) ).

% preordering_def
tff(fact_7746_gcd__nat_Opreordering__axioms,axiom,
    preordering(nat,dvd_dvd(nat),aTP_Lamp_cc(nat,fun(nat,$o))) ).

% gcd_nat.preordering_axioms
tff(fact_7747_preordering__axioms__def,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o))] :
      ( preordering_axioms(A,Less_eq,Less)
    <=> ! [A10: A,B6: A] :
          ( aa(A,$o,aa(A,fun(A,$o),Less,A10),B6)
        <=> ( aa(A,$o,aa(A,fun(A,$o),Less_eq,A10),B6)
            & ~ aa(A,$o,aa(A,fun(A,$o),Less_eq,B6),A10) ) ) ) ).

% preordering_axioms_def
tff(fact_7748_preordering__axioms_Ointro,axiom,
    ! [A: $tType,Less: fun(A,fun(A,$o)),Less_eq: fun(A,fun(A,$o))] :
      ( ! [A6: A,B5: A] :
          ( aa(A,$o,aa(A,fun(A,$o),Less,A6),B5)
        <=> ( aa(A,$o,aa(A,fun(A,$o),Less_eq,A6),B5)
            & ~ aa(A,$o,aa(A,fun(A,$o),Less_eq,B5),A6) ) )
     => preordering_axioms(A,Less_eq,Less) ) ).

% preordering_axioms.intro
tff(fact_7749_preordering_Oaxioms_I2_J,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o))] :
      ( preordering(A,Less_eq,Less)
     => preordering_axioms(A,Less_eq,Less) ) ).

% preordering.axioms(2)
tff(fact_7750_comm__monoid__def,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
      ( comm_monoid(A,F,Z2)
    <=> ( abel_semigroup(A,F)
        & comm_monoid_axioms(A,F,Z2) ) ) ).

% comm_monoid_def
tff(fact_7751_comm__monoid_Ointro,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
      ( abel_semigroup(A,F)
     => ( comm_monoid_axioms(A,F,Z2)
       => comm_monoid(A,F,Z2) ) ) ).

% comm_monoid.intro
tff(fact_7752_semilattice_Oaxioms_I1_J,axiom,
    ! [A: $tType,F: fun(A,fun(A,A))] :
      ( semilattice(A,F)
     => abel_semigroup(A,F) ) ).

% semilattice.axioms(1)
tff(fact_7753_sup_Oabel__semigroup__axioms,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => abel_semigroup(A,sup_sup(A)) ) ).

% sup.abel_semigroup_axioms
tff(fact_7754_abstract__boolean__algebra_Oaxioms_I1_J,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A] :
      ( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
     => abel_semigroup(A,Conj) ) ).

% abstract_boolean_algebra.axioms(1)
tff(fact_7755_abstract__boolean__algebra_Oaxioms_I2_J,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A] :
      ( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
     => abel_semigroup(A,Disj) ) ).

% abstract_boolean_algebra.axioms(2)
tff(fact_7756_min_Oabel__semigroup__axioms,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => abel_semigroup(A,ord_min(A)) ) ).

% min.abel_semigroup_axioms
tff(fact_7757_max_Oabel__semigroup__axioms,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => abel_semigroup(A,ord_max(A)) ) ).

% max.abel_semigroup_axioms
tff(fact_7758_abel__semigroup_Ocommute,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),A3: A,B2: A] :
      ( abel_semigroup(A,F)
     => ( aa(A,A,aa(A,fun(A,A),F,A3),B2) = aa(A,A,aa(A,fun(A,A),F,B2),A3) ) ) ).

% abel_semigroup.commute
tff(fact_7759_abel__semigroup_Oleft__commute,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),B2: A,A3: A,C2: A] :
      ( abel_semigroup(A,F)
     => ( aa(A,A,aa(A,fun(A,A),F,B2),aa(A,A,aa(A,fun(A,A),F,A3),C2)) = aa(A,A,aa(A,fun(A,A),F,A3),aa(A,A,aa(A,fun(A,A),F,B2),C2)) ) ) ).

% abel_semigroup.left_commute
tff(fact_7760_comm__monoid_Oaxioms_I1_J,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
      ( comm_monoid(A,F,Z2)
     => abel_semigroup(A,F) ) ).

% comm_monoid.axioms(1)
tff(fact_7761_add_Oabel__semigroup__axioms,axiom,
    ! [A: $tType] :
      ( ab_semigroup_add(A)
     => abel_semigroup(A,plus_plus(A)) ) ).

% add.abel_semigroup_axioms
tff(fact_7762_mult_Oabel__semigroup__axioms,axiom,
    ! [A: $tType] :
      ( ab_semigroup_mult(A)
     => abel_semigroup(A,times_times(A)) ) ).

% mult.abel_semigroup_axioms
tff(fact_7763_inf_Oabel__semigroup__axioms,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => abel_semigroup(A,inf_inf(A)) ) ).

% inf.abel_semigroup_axioms
tff(fact_7764_abel__semigroup_Oaxioms_I1_J,axiom,
    ! [A: $tType,F: fun(A,fun(A,A))] :
      ( abel_semigroup(A,F)
     => semigroup(A,F) ) ).

% abel_semigroup.axioms(1)
tff(fact_7765_abel__semigroup__def,axiom,
    ! [A: $tType,F: fun(A,fun(A,A))] :
      ( abel_semigroup(A,F)
    <=> ( semigroup(A,F)
        & abel_s757365448890700780axioms(A,F) ) ) ).

% abel_semigroup_def
tff(fact_7766_abel__semigroup_Ointro,axiom,
    ! [A: $tType,F: fun(A,fun(A,A))] :
      ( semigroup(A,F)
     => ( abel_s757365448890700780axioms(A,F)
       => abel_semigroup(A,F) ) ) ).

% abel_semigroup.intro
tff(fact_7767_abel__semigroup_Oaxioms_I2_J,axiom,
    ! [A: $tType,F: fun(A,fun(A,A))] :
      ( abel_semigroup(A,F)
     => abel_s757365448890700780axioms(A,F) ) ).

% abel_semigroup.axioms(2)
tff(fact_7768_abel__semigroup__axioms__def,axiom,
    ! [A: $tType,F: fun(A,fun(A,A))] :
      ( abel_s757365448890700780axioms(A,F)
    <=> ! [A10: A,B6: A] : aa(A,A,aa(A,fun(A,A),F,A10),B6) = aa(A,A,aa(A,fun(A,A),F,B6),A10) ) ).

% abel_semigroup_axioms_def
tff(fact_7769_abel__semigroup__axioms_Ointro,axiom,
    ! [A: $tType,F: fun(A,fun(A,A))] :
      ( ! [A6: A,B5: A] : aa(A,A,aa(A,fun(A,A),F,A6),B5) = aa(A,A,aa(A,fun(A,A),F,B5),A6)
     => abel_s757365448890700780axioms(A,F) ) ).

% abel_semigroup_axioms.intro
tff(fact_7770_semilattice__def,axiom,
    ! [A: $tType,F: fun(A,fun(A,A))] :
      ( semilattice(A,F)
    <=> ( abel_semigroup(A,F)
        & semilattice_axioms(A,F) ) ) ).

% semilattice_def
tff(fact_7771_semilattice_Ointro,axiom,
    ! [A: $tType,F: fun(A,fun(A,A))] :
      ( abel_semigroup(A,F)
     => ( semilattice_axioms(A,F)
       => semilattice(A,F) ) ) ).

% semilattice.intro
tff(fact_7772_semilattice_Oaxioms_I2_J,axiom,
    ! [A: $tType,F: fun(A,fun(A,A))] :
      ( semilattice(A,F)
     => semilattice_axioms(A,F) ) ).

% semilattice.axioms(2)
tff(fact_7773_semilattice__axioms_Ointro,axiom,
    ! [A: $tType,F: fun(A,fun(A,A))] :
      ( ! [A6: A] : aa(A,A,aa(A,fun(A,A),F,A6),A6) = A6
     => semilattice_axioms(A,F) ) ).

% semilattice_axioms.intro
tff(fact_7774_semilattice__axioms__def,axiom,
    ! [A: $tType,F: fun(A,fun(A,A))] :
      ( semilattice_axioms(A,F)
    <=> ! [A10: A] : aa(A,A,aa(A,fun(A,A),F,A10),A10) = A10 ) ).

% semilattice_axioms_def
tff(fact_7775_abstract__boolean__algebra__def,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A] :
      ( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
    <=> ( abel_semigroup(A,Conj)
        & abel_semigroup(A,Disj)
        & boolea6902313364301356556axioms(A,Conj,Disj,Compl,Zero,One) ) ) ).

% abstract_boolean_algebra_def
tff(fact_7776_abstract__boolean__algebra_Ointro,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A] :
      ( abel_semigroup(A,Conj)
     => ( abel_semigroup(A,Disj)
       => ( boolea6902313364301356556axioms(A,Conj,Disj,Compl,Zero,One)
         => boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One) ) ) ) ).

% abstract_boolean_algebra.intro
tff(fact_7777_abstract__boolean__algebra__axioms__def,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A] :
      ( boolea6902313364301356556axioms(A,Conj,Disj,Compl,Zero,One)
    <=> ( ! [X4: A,Y3: A,Z4: A] : aa(A,A,aa(A,fun(A,A),Conj,X4),aa(A,A,aa(A,fun(A,A),Disj,Y3),Z4)) = aa(A,A,aa(A,fun(A,A),Disj,aa(A,A,aa(A,fun(A,A),Conj,X4),Y3)),aa(A,A,aa(A,fun(A,A),Conj,X4),Z4))
        & ! [X4: A,Y3: A,Z4: A] : aa(A,A,aa(A,fun(A,A),Disj,X4),aa(A,A,aa(A,fun(A,A),Conj,Y3),Z4)) = aa(A,A,aa(A,fun(A,A),Conj,aa(A,A,aa(A,fun(A,A),Disj,X4),Y3)),aa(A,A,aa(A,fun(A,A),Disj,X4),Z4))
        & ! [X4: A] : aa(A,A,aa(A,fun(A,A),Conj,X4),One) = X4
        & ! [X4: A] : aa(A,A,aa(A,fun(A,A),Disj,X4),Zero) = X4
        & ! [X4: A] : aa(A,A,aa(A,fun(A,A),Conj,X4),aa(A,A,Compl,X4)) = Zero
        & ! [X4: A] : aa(A,A,aa(A,fun(A,A),Disj,X4),aa(A,A,Compl,X4)) = One ) ) ).

% abstract_boolean_algebra_axioms_def
tff(fact_7778_abstract__boolean__algebra__axioms_Ointro,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),One: A,Zero: A,Compl: fun(A,A)] :
      ( ! [X2: A,Y2: A,Z3: A] : aa(A,A,aa(A,fun(A,A),Conj,X2),aa(A,A,aa(A,fun(A,A),Disj,Y2),Z3)) = aa(A,A,aa(A,fun(A,A),Disj,aa(A,A,aa(A,fun(A,A),Conj,X2),Y2)),aa(A,A,aa(A,fun(A,A),Conj,X2),Z3))
     => ( ! [X2: A,Y2: A,Z3: A] : aa(A,A,aa(A,fun(A,A),Disj,X2),aa(A,A,aa(A,fun(A,A),Conj,Y2),Z3)) = aa(A,A,aa(A,fun(A,A),Conj,aa(A,A,aa(A,fun(A,A),Disj,X2),Y2)),aa(A,A,aa(A,fun(A,A),Disj,X2),Z3))
       => ( ! [X2: A] : aa(A,A,aa(A,fun(A,A),Conj,X2),One) = X2
         => ( ! [X2: A] : aa(A,A,aa(A,fun(A,A),Disj,X2),Zero) = X2
           => ( ! [X2: A] : aa(A,A,aa(A,fun(A,A),Conj,X2),aa(A,A,Compl,X2)) = Zero
             => ( ! [X2: A] : aa(A,A,aa(A,fun(A,A),Disj,X2),aa(A,A,Compl,X2)) = One
               => boolea6902313364301356556axioms(A,Conj,Disj,Compl,Zero,One) ) ) ) ) ) ) ).

% abstract_boolean_algebra_axioms.intro
tff(fact_7779_abstract__boolean__algebra_Oaxioms_I3_J,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A] :
      ( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
     => boolea6902313364301356556axioms(A,Conj,Disj,Compl,Zero,One) ) ).

% abstract_boolean_algebra.axioms(3)
tff(fact_7780_ATP_Olambda__1,axiom,
    ! [Uu: product_prod(int,int)] :
      aa(product_prod(int,int),product_prod(int,int),aTP_Lamp_wm(product_prod(int,int),product_prod(int,int)),Uu) = $ite(aa(product_prod(int,int),int,product_fst(int,int),Uu) = zero_zero(int),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int)),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(product_prod(int,int),int,product_snd(int,int),Uu)),aa(product_prod(int,int),int,product_fst(int,int),Uu))) ).

% ATP.lambda_1
tff(fact_7781_ATP_Olambda__2,axiom,
    ! [A: $tType] :
      ( countable(A)
     => ! [Uu: nat] : aa(nat,set(old_node(A,product_unit)),aTP_Lamp_auv(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_aus(nat,set(old_node(A,product_unit))),aTP_Lamp_auu(nat,set(old_node(A,product_unit)))),aa(nat,sum_sum(nat,nat),nat_sum_decode,Uu)) ) ).

% ATP.lambda_2
tff(fact_7782_ATP_Olambda__3,axiom,
    ! [A: $tType] :
      ( countable(A)
     => ! [Uu: nat] : aa(nat,set(old_node(A,product_unit)),aTP_Lamp_aur(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_aup(nat,set(old_node(A,product_unit))),aTP_Lamp_auq(nat,set(old_node(A,product_unit)))),aa(nat,sum_sum(nat,nat),nat_sum_decode,Uu)) ) ).

% ATP.lambda_3
tff(fact_7783_ATP_Olambda__4,axiom,
    ! [A: $tType,Uu: set(set(A))] : aa(set(set(A)),int,aTP_Lamp_jq(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_4
tff(fact_7784_ATP_Olambda__5,axiom,
    ! [A: $tType,Uu: A] : aa(A,set(product_prod(A,A)),aTP_Lamp_nw(A,set(product_prod(A,A))),Uu) = aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(product_prod(A,A),fun(set(product_prod(A,A)),set(product_prod(A,A))),insert2(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uu),Uu)),bot_bot(set(product_prod(A,A)))) ).

% ATP.lambda_5
tff(fact_7785_ATP_Olambda__6,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Uu: A] :
          ( aa(A,$o,aTP_Lamp_sn(A,$o),Uu)
        <=> ( aa(set(A),$o,member(A,Uu),ring_1_Ints(A))
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Uu) ) ) ) ).

% ATP.lambda_6
tff(fact_7786_ATP_Olambda__7,axiom,
    ! [Uu: product_prod(heap_ext(product_unit),set(nat))] :
      ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aTP_Lamp_aa(product_prod(heap_ext(product_unit),set(nat)),$o),Uu)
    <=> ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,y),Uu)
        & aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,x),Uu) ) ) ).

% ATP.lambda_7
tff(fact_7787_ATP_Olambda__8,axiom,
    ! [Uu: product_prod(heap_ext(product_unit),set(nat))] :
      ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aTP_Lamp_a(product_prod(heap_ext(product_unit),set(nat)),$o),Uu)
    <=> ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,x),Uu)
        & aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,y),Uu) ) ) ).

% ATP.lambda_8
tff(fact_7788_ATP_Olambda__9,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: product_prod(int,int)] : aa(product_prod(int,int),A,aTP_Lamp_wp(product_prod(int,int),A),Uu) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(int,A,ring_1_of_int(A),aa(product_prod(int,int),int,product_fst(int,int),Uu))),aa(int,A,ring_1_of_int(A),aa(product_prod(int,int),int,product_snd(int,int),Uu))) ) ).

% ATP.lambda_9
tff(fact_7789_ATP_Olambda__10,axiom,
    ! [Uu: product_prod(int,int)] : aa(product_prod(int,int),product_prod(int,int),aTP_Lamp_wk(product_prod(int,int),product_prod(int,int)),Uu) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,uminus_uminus(int),aa(product_prod(int,int),int,product_fst(int,int),Uu))),aa(product_prod(int,int),int,product_snd(int,int),Uu)) ).

% ATP.lambda_10
tff(fact_7790_ATP_Olambda__11,axiom,
    ! [Uu: nat] : aa(nat,int,aTP_Lamp_ats(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_7791_ATP_Olambda__12,axiom,
    ! [A: $tType,Uu: product_prod(A,A)] :
      ( aa(product_prod(A,A),$o,aTP_Lamp_vw(product_prod(A,A),$o),Uu)
    <=> ( aa(product_prod(A,A),A,product_fst(A,A),Uu) = aa(product_prod(A,A),A,product_snd(A,A),Uu) ) ) ).

% ATP.lambda_12
tff(fact_7792_ATP_Olambda__13,axiom,
    ! [A: $tType,Uu: list(A)] : aa(list(A),product_prod(nat,list(A)),aTP_Lamp_alx(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_13
tff(fact_7793_ATP_Olambda__14,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Uu: nat] :
          ( aa(nat,$o,aTP_Lamp_sq(nat,$o),Uu)
        <=> ( aa(nat,A,semiring_1_of_nat(A),Uu) = zero_zero(A) ) ) ) ).

% ATP.lambda_14
tff(fact_7794_ATP_Olambda__15,axiom,
    ! [Uu: nat] : aa(nat,nat,aTP_Lamp_jn(nat,nat),Uu) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu),aa(nat,nat,suc,zero_zero(nat))) ).

% ATP.lambda_15
tff(fact_7795_ATP_Olambda__16,axiom,
    ! [Uu: int] : aa(int,int,aTP_Lamp_ug(int,int),Uu) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Uu),Uu) ).

% ATP.lambda_16
tff(fact_7796_ATP_Olambda__17,axiom,
    ! [B: $tType,Uu: B] : aa(B,product_prod(B,B),aTP_Lamp_vt(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_17
tff(fact_7797_ATP_Olambda__18,axiom,
    ! [A: $tType,Uu: A] : aa(A,product_prod(A,A),aTP_Lamp_vl(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_18
tff(fact_7798_ATP_Olambda__19,axiom,
    ! [Uu: product_prod(int,int)] :
      ( aa(product_prod(int,int),$o,aTP_Lamp_arn(product_prod(int,int),$o),Uu)
    <=> aa(product_prod(int,int),$o,aa(product_prod(int,int),fun(product_prod(int,int),$o),ratrel,Uu),Uu) ) ).

% ATP.lambda_19
tff(fact_7799_ATP_Olambda__20,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_ec(A,A),Uu) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),one_one(A)) ) ).

% ATP.lambda_20
tff(fact_7800_ATP_Olambda__21,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [Uu: A] :
          ( aa(A,$o,aTP_Lamp_avv(A,$o),Uu)
        <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),Uu),one_one(A)) ) ) ).

% ATP.lambda_21
tff(fact_7801_ATP_Olambda__22,axiom,
    ! [Uu: nat] : aa(nat,product_prod(nat,nat),aTP_Lamp_wo(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_22
tff(fact_7802_ATP_Olambda__23,axiom,
    ! [A: $tType,Uu: A] : aa(A,multiset(A),aTP_Lamp_ajv(A,multiset(A)),Uu) = aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),Uu),zero_zero(multiset(A))) ).

% ATP.lambda_23
tff(fact_7803_ATP_Olambda__24,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A] : aa(A,list(A),aTP_Lamp_abt(A,list(A)),Uu) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uu),nil(A)) ) ).

% ATP.lambda_24
tff(fact_7804_ATP_Olambda__25,axiom,
    ! [A: $tType,Uu: A] : aa(A,list(A),aTP_Lamp_xq(A,list(A)),Uu) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uu),nil(A)) ).

% ATP.lambda_25
tff(fact_7805_ATP_Olambda__26,axiom,
    ! [B: $tType,Uu: B] : aa(B,set(B),aTP_Lamp_atu(B,set(B)),Uu) = aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),Uu),bot_bot(set(B))) ).

% ATP.lambda_26
tff(fact_7806_ATP_Olambda__27,axiom,
    ! [A: $tType,Uu: A] : aa(A,set(A),aTP_Lamp_kg(A,set(A)),Uu) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Uu),bot_bot(set(A))) ).

% ATP.lambda_27
tff(fact_7807_ATP_Olambda__28,axiom,
    ! [A: $tType] :
      ( countable(A)
     => ! [Uu: fun(A,nat)] :
          ( aa(fun(A,nat),$o,aTP_Lamp_aor(fun(A,nat),$o),Uu)
        <=> inj_on(A,nat,Uu,top_top(set(A))) ) ) ).

% ATP.lambda_28
tff(fact_7808_ATP_Olambda__29,axiom,
    ! [C: $tType,A: $tType,Uu: fun(A,C)] : aa(fun(A,C),set(C),aTP_Lamp_aqq(fun(A,C),set(C)),Uu) = aa(set(A),set(C),image2(A,C,Uu),top_top(set(A))) ).

% ATP.lambda_29
tff(fact_7809_ATP_Olambda__30,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B)] : aa(fun(A,B),set(B),aTP_Lamp_aqp(fun(A,B),set(B)),Uu) = aa(set(A),set(B),image2(A,B,Uu),top_top(set(A))) ).

% ATP.lambda_30
tff(fact_7810_ATP_Olambda__31,axiom,
    ! [B: $tType,Uu: option(B)] :
      ( aa(option(B),$o,aTP_Lamp_awz(option(B),$o),Uu)
    <=> ( Uu = none(B) ) ) ).

% ATP.lambda_31
tff(fact_7811_ATP_Olambda__32,axiom,
    ! [A: $tType,Uu: option(A)] :
      ( aa(option(A),$o,aTP_Lamp_awy(option(A),$o),Uu)
    <=> ( Uu = none(A) ) ) ).

% ATP.lambda_32
tff(fact_7812_ATP_Olambda__33,axiom,
    ! [B: $tType,Uu: list(B)] :
      ( aa(list(B),$o,aTP_Lamp_aix(list(B),$o),Uu)
    <=> ( Uu = nil(B) ) ) ).

% ATP.lambda_33
tff(fact_7813_ATP_Olambda__34,axiom,
    ! [A: $tType,Uu: list(A)] :
      ( aa(list(A),$o,aTP_Lamp_aiw(list(A),$o),Uu)
    <=> ( Uu = nil(A) ) ) ).

% ATP.lambda_34
tff(fact_7814_ATP_Olambda__35,axiom,
    ! [A: $tType] :
      ( countable(A)
     => ! [Uu: nat] : aa(nat,set(old_node(A,product_unit)),aTP_Lamp_auu(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_aut(nat,fun(nat,set(old_node(A,product_unit))))),aa(nat,product_prod(nat,nat),nat_prod_decode,Uu)) ) ).

% ATP.lambda_35
tff(fact_7815_ATP_Olambda__36,axiom,
    ! [Uu: product_prod(int,int)] :
      ( aa(product_prod(int,int),$o,aTP_Lamp_wn(product_prod(int,int),$o),Uu)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),Uu)),aa(product_prod(int,int),int,product_snd(int,int),Uu))) ) ).

% ATP.lambda_36
tff(fact_7816_ATP_Olambda__37,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_atw(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_atv(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_37
tff(fact_7817_ATP_Olambda__38,axiom,
    ! [Uu: nat] : aa(nat,nat,aTP_Lamp_atm(nat,nat),Uu) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Uu)) ).

% ATP.lambda_38
tff(fact_7818_ATP_Olambda__39,axiom,
    ! [B: $tType,Uu: list(B)] : aa(list(B),fun(nat,nat),aTP_Lamp_zo(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_39
tff(fact_7819_ATP_Olambda__40,axiom,
    ! [A: $tType,Uu: A] : aa(A,fun(set(product_prod(A,A)),set(product_prod(A,A))),aTP_Lamp_ol(A,fun(set(product_prod(A,A)),set(product_prod(A,A)))),Uu) = aa(product_prod(A,A),fun(set(product_prod(A,A)),set(product_prod(A,A))),insert2(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uu),Uu)) ).

% ATP.lambda_40
tff(fact_7820_ATP_Olambda__41,axiom,
    ! [B: $tType,Uu: option(B)] :
      ( aa(option(B),$o,aTP_Lamp_awx(option(B),$o),Uu)
    <=> ( Uu != none(B) ) ) ).

% ATP.lambda_41
tff(fact_7821_ATP_Olambda__42,axiom,
    ! [A: $tType,Uu: option(A)] :
      ( aa(option(A),$o,aTP_Lamp_aww(option(A),$o),Uu)
    <=> ( Uu != none(A) ) ) ).

% ATP.lambda_42
tff(fact_7822_ATP_Olambda__43,axiom,
    ! [B: $tType,Uu: list(B)] :
      ( aa(list(B),$o,aTP_Lamp_zp(list(B),$o),Uu)
    <=> ( Uu != nil(B) ) ) ).

% ATP.lambda_43
tff(fact_7823_ATP_Olambda__44,axiom,
    ! [A: $tType,Uu: list(A)] :
      ( aa(list(A),$o,aTP_Lamp_zt(list(A),$o),Uu)
    <=> ( Uu != nil(A) ) ) ).

% ATP.lambda_44
tff(fact_7824_ATP_Olambda__45,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_anq(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_anp(A,fun(B,fun(C,product_prod(product_prod(A,B),C))),Uu)) ).

% ATP.lambda_45
tff(fact_7825_ATP_Olambda__46,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_adt(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_ads(fun(A,option(B)),fun(A,fun(B,fun(A,option(B)))),Uu)) ).

% ATP.lambda_46
tff(fact_7826_ATP_Olambda__47,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_ado(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_adn(B,fun(A,fun(C,product_prod(A,product_prod(B,C)))),Uu)) ).

% ATP.lambda_47
tff(fact_7827_ATP_Olambda__48,axiom,
    ! [A: $tType,Uu: multiset(A)] : aa(multiset(A),set(A),aTP_Lamp_asi(multiset(A),set(A)),Uu) = aa(fun(A,$o),set(A),collect(A),aTP_Lamp_apq(multiset(A),fun(A,$o),Uu)) ).

% ATP.lambda_48
tff(fact_7828_ATP_Olambda__49,axiom,
    ! [Uu: nat] : aa(nat,set(nat),aTP_Lamp_sp(nat,set(nat)),Uu) = aa(fun(nat,$o),set(nat),collect(nat),aTP_Lamp_hx(nat,fun(nat,$o),Uu)) ).

% ATP.lambda_49
tff(fact_7829_ATP_Olambda__50,axiom,
    ! [B: $tType,Uu: fun(B,nat)] :
      ( aa(fun(B,nat),$o,aTP_Lamp_arf(fun(B,nat),$o),Uu)
    <=> aa(set(B),$o,finite_finite2(B),aa(fun(B,$o),set(B),collect(B),aTP_Lamp_aqz(fun(B,nat),fun(B,$o),Uu))) ) ).

% ATP.lambda_50
tff(fact_7830_ATP_Olambda__51,axiom,
    ! [A: $tType,Uu: fun(A,nat)] :
      ( aa(fun(A,nat),$o,aTP_Lamp_apt(fun(A,nat),$o),Uu)
    <=> aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_it(fun(A,nat),fun(A,$o),Uu))) ) ).

% ATP.lambda_51
tff(fact_7831_ATP_Olambda__52,axiom,
    ! [A: $tType,Uu: fun(A,$o)] :
      ( aa(fun(A,$o),$o,aTP_Lamp_ata(fun(A,$o),$o),Uu)
    <=> aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_aj(fun(A,$o),fun(A,$o),Uu))) ) ).

% ATP.lambda_52
tff(fact_7832_ATP_Olambda__53,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B)] : aa(fun(A,B),set(product_prod(A,B)),aTP_Lamp_acm(fun(A,B),set(product_prod(A,B))),Uu) = aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aTP_Lamp_acl(fun(A,B),fun(A,fun(B,$o)),Uu))) ).

% ATP.lambda_53
tff(fact_7833_ATP_Olambda__54,axiom,
    ! [B: $tType,Uu: list(B)] : aa(list(B),fun(nat,nat),aTP_Lamp_zn(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_54
tff(fact_7834_ATP_Olambda__55,axiom,
    ! [A: $tType,Uu: list(A)] : aa(list(A),fun(nat,nat),aTP_Lamp_zs(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_55
tff(fact_7835_ATP_Olambda__56,axiom,
    ! [Uu: num] : aa(num,option(num),aTP_Lamp_st(num,option(num)),Uu) = aa(num,option(num),some(num),aa(num,num,bit1,Uu)) ).

% ATP.lambda_56
tff(fact_7836_ATP_Olambda__57,axiom,
    ! [Uu: num] : aa(num,option(num),aTP_Lamp_te(num,option(num)),Uu) = aa(num,option(num),some(num),aa(num,num,bit0,Uu)) ).

% ATP.lambda_57
tff(fact_7837_ATP_Olambda__58,axiom,
    ! [Uu: int] : aa(int,fun(int,product_prod(int,int)),aTP_Lamp_pg(int,fun(int,product_prod(int,int))),Uu) = aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,uminus_uminus(int),Uu)) ).

% ATP.lambda_58
tff(fact_7838_ATP_Olambda__59,axiom,
    ! [Uu: int] : aa(int,fun(int,product_prod(int,int)),aTP_Lamp_ph(int,fun(int,product_prod(int,int))),Uu) = aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,abs_abs(int),Uu)) ).

% ATP.lambda_59
tff(fact_7839_ATP_Olambda__60,axiom,
    ! [Uu: nat] : aa(nat,fun(nat,product_prod(nat,nat)),aTP_Lamp_by(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_60
tff(fact_7840_ATP_Olambda__61,axiom,
    ! [A: $tType] :
      ( countable(A)
     => ! [Uu: nat] : aa(nat,set(old_node(A,product_unit)),aTP_Lamp_aus(nat,set(old_node(A,product_unit))),Uu) = aa(A,set(old_node(A,product_unit)),old_Leaf(A,product_unit),aa(nat,A,from_nat(A),Uu)) ) ).

% ATP.lambda_61
tff(fact_7841_ATP_Olambda__62,axiom,
    ! [A: $tType] :
      ( countable(A)
     => ! [Uu: nat] : aa(nat,set(old_node(A,product_unit)),aTP_Lamp_auq(nat,set(old_node(A,product_unit))),Uu) = aa(set(old_node(A,product_unit)),set(old_node(A,product_unit)),old_In1(A,product_unit),nth_item(A,Uu)) ) ).

% ATP.lambda_62
tff(fact_7842_ATP_Olambda__63,axiom,
    ! [A: $tType] :
      ( countable(A)
     => ! [Uu: nat] : aa(nat,set(old_node(A,product_unit)),aTP_Lamp_aup(nat,set(old_node(A,product_unit))),Uu) = aa(set(old_node(A,product_unit)),set(old_node(A,product_unit)),old_In0(A,product_unit),nth_item(A,Uu)) ) ).

% ATP.lambda_63
tff(fact_7843_ATP_Olambda__64,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A] : aa(A,filter(A),aTP_Lamp_alj(A,filter(A)),Uu) = principal(A,aa(A,set(A),set_ord_atLeast(A),Uu)) ) ).

% ATP.lambda_64
tff(fact_7844_ATP_Olambda__65,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Uu: A] : aa(A,filter(A),aTP_Lamp_alh(A,filter(A)),Uu) = principal(A,aa(A,set(A),set_ord_atLeast(A),Uu)) ) ).

% ATP.lambda_65
tff(fact_7845_ATP_Olambda__66,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A] : aa(A,filter(A),aTP_Lamp_tw(A,filter(A)),Uu) = principal(A,aa(A,set(A),set_ord_atMost(A),Uu)) ) ).

% ATP.lambda_66
tff(fact_7846_ATP_Olambda__67,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Uu: A] : aa(A,filter(A),aTP_Lamp_tv(A,filter(A)),Uu) = principal(A,aa(A,set(A),set_ord_atMost(A),Uu)) ) ).

% ATP.lambda_67
tff(fact_7847_ATP_Olambda__68,axiom,
    ! [Uu: int] : aa(int,nat,aTP_Lamp_so(int,nat),Uu) = aa(int,nat,nat2,aa(int,int,abs_abs(int),Uu)) ).

% ATP.lambda_68
tff(fact_7848_ATP_Olambda__69,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Uu: A] :
          ( aa(A,$o,aTP_Lamp_aha(A,$o),Uu)
        <=> ? [N2: int] :
              ( ( Uu = aa(int,A,ring_1_of_int(A),N2) )
              & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),N2) ) ) ) ).

% ATP.lambda_69
tff(fact_7849_ATP_Olambda__70,axiom,
    ! [Uu: list(char)] :
      ( aa(list(char),$o,aTP_Lamp_atc(list(char),$o),Uu)
    <=> ! [X4: char] :
          ( aa(set(char),$o,member(char,X4),aa(list(char),set(char),set2(char),Uu))
         => ~ digit7(X4) ) ) ).

% ATP.lambda_70
tff(fact_7850_ATP_Olambda__71,axiom,
    ! [Uu: set(product_prod(int,int))] :
      ( aa(set(product_prod(int,int)),$o,aTP_Lamp_aru(set(product_prod(int,int)),$o),Uu)
    <=> ? [X4: product_prod(int,int)] :
          ( aa(product_prod(int,int),$o,aa(product_prod(int,int),fun(product_prod(int,int),$o),ratrel,X4),X4)
          & ( Uu = aa(fun(product_prod(int,int),$o),set(product_prod(int,int)),collect(product_prod(int,int)),aa(product_prod(int,int),fun(product_prod(int,int),$o),ratrel,X4)) ) ) ) ).

% ATP.lambda_71
tff(fact_7851_ATP_Olambda__72,axiom,
    ! [Uu: set(product_prod(nat,nat))] :
      ( aa(set(product_prod(nat,nat)),$o,aTP_Lamp_ayf(set(product_prod(nat,nat)),$o),Uu)
    <=> ? [X4: product_prod(nat,nat)] :
          ( aa(product_prod(nat,nat),$o,aa(product_prod(nat,nat),fun(product_prod(nat,nat),$o),intrel,X4),X4)
          & ( Uu = aa(fun(product_prod(nat,nat),$o),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(product_prod(nat,nat),fun(product_prod(nat,nat),$o),intrel,X4)) ) ) ) ).

% ATP.lambda_72
tff(fact_7852_ATP_Olambda__73,axiom,
    ! [A: $tType,Uu: product_prod(A,A)] :
      ( aa(product_prod(A,A),$o,aTP_Lamp_afj(product_prod(A,A),$o),Uu)
    <=> ? [X4: A] : Uu = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),X4) ) ).

% ATP.lambda_73
tff(fact_7853_ATP_Olambda__74,axiom,
    ! [A: $tType,B: $tType,Uu: product_prod(product_prod($o,A),product_prod($o,B))] :
      ( aa(product_prod(product_prod($o,A),product_prod($o,B)),$o,aTP_Lamp_ahf(product_prod(product_prod($o,A),product_prod($o,B)),$o),Uu)
    <=> ? [X4: A,Y3: B] : Uu = aa(product_prod($o,B),product_prod(product_prod($o,A),product_prod($o,B)),aa(product_prod($o,A),fun(product_prod($o,B),product_prod(product_prod($o,A),product_prod($o,B))),product_Pair(product_prod($o,A),product_prod($o,B)),aa(A,product_prod($o,A),aa($o,fun(A,product_prod($o,A)),product_Pair($o,A),$true),X4)),aa(B,product_prod($o,B),aa($o,fun(B,product_prod($o,B)),product_Pair($o,B),$false),Y3)) ) ).

% ATP.lambda_74
tff(fact_7854_ATP_Olambda__75,axiom,
    ! [B: $tType,A: $tType,Uu: product_prod(fun(nat,sum_sum(A,nat)),sum_sum(B,nat))] :
      ( aa(product_prod(fun(nat,sum_sum(A,nat)),sum_sum(B,nat)),$o,aTP_Lamp_atk(product_prod(fun(nat,sum_sum(A,nat)),sum_sum(B,nat)),$o),Uu)
    <=> ? [F6: fun(nat,sum_sum(A,nat)),X4: sum_sum(B,nat),K4: nat] :
          ( ( Uu = 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)),F6),X4) )
          & ( aa(nat,sum_sum(A,nat),F6,K4) = aa(nat,sum_sum(A,nat),sum_Inr(nat,A),zero_zero(nat)) ) ) ) ).

% ATP.lambda_75
tff(fact_7855_ATP_Olambda__76,axiom,
    ! [B: $tType,Uu: nat] : aa(nat,sum_sum(B,nat),aTP_Lamp_auf(nat,sum_sum(B,nat)),Uu) = aa(nat,sum_sum(B,nat),sum_Inr(nat,B),zero_zero(nat)) ).

% ATP.lambda_76
tff(fact_7856_ATP_Olambda__77,axiom,
    ! [A: $tType,Uu: nat] : aa(nat,sum_sum(A,nat),aTP_Lamp_atl(nat,sum_sum(A,nat)),Uu) = aa(nat,sum_sum(A,nat),sum_Inr(nat,A),zero_zero(nat)) ).

% ATP.lambda_77
tff(fact_7857_ATP_Olambda__78,axiom,
    ! [Uu: num,Uua: nat] : aa(nat,option(num),aTP_Lamp_tk(num,fun(nat,option(num)),Uu),Uua) = case_num(option(num),aa(num,option(num),some(num),one2),aTP_Lamp_ti(nat,fun(num,option(num)),Uua),aTP_Lamp_tj(nat,fun(num,option(num)),Uua),Uu) ).

% ATP.lambda_78
tff(fact_7858_ATP_Olambda__79,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: nat,Uua: nat] :
          aa(nat,A,aTP_Lamp_ey(nat,fun(nat,A),Uu),Uua) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Uua),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Uu),Uua)),zero_zero(A)) ) ).

% ATP.lambda_79
tff(fact_7859_ATP_Olambda__80,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_aap(product_prod(A,C),fun(product_prod(C,B),list(product_prod(A,B))),Uu),Uua) = $ite(aa(product_prod(A,C),C,product_snd(A,C),Uu) = aa(product_prod(C,B),C,product_fst(C,B),Uua),aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(product_prod(A,B),fun(list(product_prod(A,B)),list(product_prod(A,B))),cons(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(product_prod(A,C),A,product_fst(A,C),Uu)),aa(product_prod(C,B),B,product_snd(C,B),Uua))),nil(product_prod(A,B))),nil(product_prod(A,B))) ).

% ATP.lambda_80
tff(fact_7860_ATP_Olambda__81,axiom,
    ! [A: $tType,Uu: list(A),Uua: nat] :
      aa(nat,option(A),aTP_Lamp_xr(list(A),fun(nat,option(A)),Uu),Uua) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uua),aa(list(A),nat,size_size(list(A)),Uu)),aa(A,option(A),some(A),aa(nat,A,nth(A,Uu),Uua)),none(A)) ).

% ATP.lambda_81
tff(fact_7861_ATP_Olambda__82,axiom,
    ! [Uu: pred(product_unit),Uua: product_prod(code_natural,code_natural)] :
      aa(product_prod(code_natural,code_natural),product_prod(pred(product_unit),product_prod(code_natural,code_natural)),aa(pred(product_unit),fun(product_prod(code_natural,code_natural),product_prod(pred(product_unit),product_prod(code_natural,code_natural))),aTP_Lamp_awh(pred(product_unit),fun(product_prod(code_natural,code_natural),product_prod(pred(product_unit),product_prod(code_natural,code_natural)))),Uu),Uua) = $ite(aa(product_unit,$o,eval(product_unit,Uu),product_Unity),aa(product_prod(code_natural,code_natural),product_prod(pred(product_unit),product_prod(code_natural,code_natural)),aa(pred(product_unit),fun(product_prod(code_natural,code_natural),product_prod(pred(product_unit),product_prod(code_natural,code_natural))),product_Pair(pred(product_unit),product_prod(code_natural,code_natural)),bot_bot(pred(product_unit))),Uua),aa(product_prod(code_natural,code_natural),product_prod(pred(product_unit),product_prod(code_natural,code_natural)),aa(pred(product_unit),fun(product_prod(code_natural,code_natural),product_prod(pred(product_unit),product_prod(code_natural,code_natural))),product_Pair(pred(product_unit),product_prod(code_natural,code_natural)),aa(product_unit,pred(product_unit),single(product_unit),product_Unity)),Uua)) ).

% ATP.lambda_82
tff(fact_7862_ATP_Olambda__83,axiom,
    ! [Uu: int,Uua: int] :
      aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_qe(int,fun(int,product_prod(int,int))),Uu),Uua) = $ite(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_83
tff(fact_7863_ATP_Olambda__84,axiom,
    ! [Uu: int,Uua: int] :
      aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_wj(int,fun(int,product_prod(int,int))),Uu),Uua) = $ite(Uua = zero_zero(int),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int)),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Uu),Uua)) ).

% ATP.lambda_84
tff(fact_7864_ATP_Olambda__85,axiom,
    ! [A: $tType,Uu: set(fun(A,nat)),Uua: A] :
      aa(A,nat,aa(set(fun(A,nat)),fun(A,nat),aTP_Lamp_apx(set(fun(A,nat)),fun(A,nat)),Uu),Uua) = $ite(Uu = bot_bot(set(fun(A,nat))),zero_zero(nat),aa(set(nat),nat,complete_Inf_Inf(nat),aa(set(fun(A,nat)),set(nat),image2(fun(A,nat),nat,aTP_Lamp_amb(A,fun(fun(A,nat),nat),Uua)),Uu))) ).

% ATP.lambda_85
tff(fact_7865_ATP_Olambda__86,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: nat,Uua: nat] :
          aa(nat,A,aTP_Lamp_ex(nat,fun(nat,A),Uu),Uua) = $ite(~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Uua),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Uu),Uua)),zero_zero(A)) ) ).

% ATP.lambda_86
tff(fact_7866_ATP_Olambda__87,axiom,
    ! [A: $tType] :
      ( ( lattice(A)
        & order_top(A) )
     => ! [Uu: fun(A,A),Uua: nat] : aa(nat,A,aTP_Lamp_ail(fun(A,A),fun(nat,A),Uu),Uua) = aa(A,A,compow(fun(A,A),Uua,Uu),top_top(A)) ) ).

% ATP.lambda_87
tff(fact_7867_ATP_Olambda__88,axiom,
    ! [A: $tType] :
      ( ( lattice(A)
        & order_bot(A) )
     => ! [Uu: fun(A,A),Uua: nat] : aa(nat,A,aTP_Lamp_yg(fun(A,A),fun(nat,A),Uu),Uua) = aa(A,A,compow(fun(A,A),Uua,Uu),bot_bot(A)) ) ).

% ATP.lambda_88
tff(fact_7868_ATP_Olambda__89,axiom,
    ! [A: $tType,Uu: list(list(A)),Uua: list(A)] :
      ( aa(list(A),$o,aTP_Lamp_aiz(list(list(A)),fun(list(A),$o),Uu),Uua)
    <=> aa(list(list(A)),$o,aa(list(A),fun(list(list(A)),$o),list_all2(A,list(A),aTP_Lamp_aiy(A,fun(list(A),$o))),Uua),Uu) ) ).

% ATP.lambda_89
tff(fact_7869_ATP_Olambda__90,axiom,
    ! [A: $tType] :
      ( inf(A)
     => ! [Uu: option(A),Uua: A] : aa(A,option(A),aTP_Lamp_th(option(A),fun(A,option(A)),Uu),Uua) = case_option(option(A),A,none(A),aTP_Lamp_tg(A,fun(A,option(A)),Uua),Uu) ) ).

% ATP.lambda_90
tff(fact_7870_ATP_Olambda__91,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Uu: option(A),Uua: A] :
          ( aa(A,$o,aTP_Lamp_ta(option(A),fun(A,$o),Uu),Uua)
        <=> case_option($o,A,$true,aa(A,fun(A,$o),aTP_Lamp_sz(A,fun(A,$o)),Uua),Uu) ) ) ).

% ATP.lambda_91
tff(fact_7871_ATP_Olambda__92,axiom,
    ! [Uu: nat,Uua: num] : aa(num,option(num),aa(nat,fun(num,option(num)),aTP_Lamp_tl(nat,fun(num,option(num))),Uu),Uua) = aa(nat,option(num),aa(fun(nat,option(num)),fun(nat,option(num)),aa(option(num),fun(fun(nat,option(num)),fun(nat,option(num))),case_nat(option(num)),none(num)),aTP_Lamp_tk(num,fun(nat,option(num)),Uua)),Uu) ).

% ATP.lambda_92
tff(fact_7872_ATP_Olambda__93,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Uu: option(A),Uua: A] :
          ( aa(A,$o,aTP_Lamp_sy(option(A),fun(A,$o),Uu),Uua)
        <=> case_option($o,A,$false,aa(A,fun(A,$o),ord_less_eq(A),Uua),Uu) ) ) ).

% ATP.lambda_93
tff(fact_7873_ATP_Olambda__94,axiom,
    ! [Uu: nat,Uua: num] : aa(num,option(num),aTP_Lamp_ti(nat,fun(num,option(num)),Uu),Uua) = case_option(option(num),num,none(num),aTP_Lamp_te(num,option(num)),bit_take_bit_num(Uu,Uua)) ).

% ATP.lambda_94
tff(fact_7874_ATP_Olambda__95,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,fun(B,$o)),Uua: product_prod(A,B)] :
      ( aa(product_prod(A,B),$o,aTP_Lamp_vy(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),Uu),Uua)
    <=> aa(B,$o,aa(A,fun(B,$o),Uu,aa(product_prod(A,B),A,product_fst(A,B),Uua)),aa(product_prod(A,B),B,product_snd(A,B),Uua)) ) ).

% ATP.lambda_95
tff(fact_7875_ATP_Olambda__96,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: fun(A,fun(A,A)),Uua: A] : aa(A,A,aTP_Lamp_ahm(fun(A,fun(A,A)),fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),Uu,Uua),Uua) ) ).

% ATP.lambda_96
tff(fact_7876_ATP_Olambda__97,axiom,
    ! [A: $tType,Uu: pred(A),Uua: product_unit] : aa(product_unit,A,aTP_Lamp_awk(pred(A),fun(product_unit,A),Uu),Uua) = abort(A,aa(literal,literal,aa($o,fun(literal,literal),aa($o,fun($o,fun(literal,literal)),aa($o,fun($o,fun($o,fun(literal,literal))),aa($o,fun($o,fun($o,fun($o,fun(literal,literal)))),aa($o,fun($o,fun($o,fun($o,fun($o,fun(literal,literal))))),aa($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun(literal,literal)))))),aa($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun(literal,literal))))))),literal2,$false),$true),$true),$true),$false),$true),$true),aa(literal,literal,aa($o,fun(literal,literal),aa($o,fun($o,fun(literal,literal)),aa($o,fun($o,fun($o,fun(literal,literal))),aa($o,fun($o,fun($o,fun($o,fun(literal,literal)))),aa($o,fun($o,fun($o,fun($o,fun($o,fun(literal,literal))))),aa($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun(literal,literal)))))),aa($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun(literal,literal))))))),literal2,$true),$true),$true),$true),$false),$true),$true),aa(literal,literal,aa($o,fun(literal,literal),aa($o,fun($o,fun(literal,literal)),aa($o,fun($o,fun($o,fun(literal,literal))),aa($o,fun($o,fun($o,fun($o,fun(literal,literal)))),aa($o,fun($o,fun($o,fun($o,fun($o,fun(literal,literal))))),aa($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun(literal,literal)))))),aa($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun(literal,literal))))))),literal2,$false),$false),$true),$false),$true),$true),$true),aa(literal,literal,aa($o,fun(literal,literal),aa($o,fun($o,fun(literal,literal)),aa($o,fun($o,fun($o,fun(literal,literal))),aa($o,fun($o,fun($o,fun($o,fun(literal,literal)))),aa($o,fun($o,fun($o,fun($o,fun($o,fun(literal,literal))))),aa($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun(literal,literal)))))),aa($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun(literal,literal))))))),literal2,$true),$true),$true),$true),$true),$false),$true),aa(literal,literal,aa($o,fun(literal,literal),aa($o,fun($o,fun(literal,literal)),aa($o,fun($o,fun($o,fun(literal,literal))),aa($o,fun($o,fun($o,fun($o,fun(literal,literal)))),aa($o,fun($o,fun($o,fun($o,fun($o,fun(literal,literal))))),aa($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun(literal,literal)))))),aa($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun(literal,literal))))))),literal2,$true),$false),$true),$false),$true),$true),$true),aa(literal,literal,aa($o,fun(literal,literal),aa($o,fun($o,fun(literal,literal)),aa($o,fun($o,fun($o,fun(literal,literal))),aa($o,fun($o,fun($o,fun($o,fun(literal,literal)))),aa($o,fun($o,fun($o,fun($o,fun($o,fun(literal,literal))))),aa($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun(literal,literal)))))),aa($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun(literal,literal))))))),literal2,$false),$true),$true),$true),$false),$true),$true),aa(literal,literal,aa($o,fun(literal,literal),aa($o,fun($o,fun(literal,literal)),aa($o,fun($o,fun($o,fun(literal,literal))),aa($o,fun($o,fun($o,fun($o,fun(literal,literal)))),aa($o,fun($o,fun($o,fun($o,fun($o,fun(literal,literal))))),aa($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun(literal,literal)))))),aa($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun(literal,literal))))))),literal2,$true),$false),$false),$true),$false),$true),$true),aa(literal,literal,aa($o,fun(literal,literal),aa($o,fun($o,fun(literal,literal)),aa($o,fun($o,fun($o,fun(literal,literal))),aa($o,fun($o,fun($o,fun($o,fun(literal,literal)))),aa($o,fun($o,fun($o,fun($o,fun($o,fun(literal,literal))))),aa($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun(literal,literal)))))),aa($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun(literal,literal))))))),literal2,$true),$false),$false),$false),$true),$true),$true),aa(literal,literal,aa($o,fun(literal,literal),aa($o,fun($o,fun(literal,literal)),aa($o,fun($o,fun($o,fun(literal,literal))),aa($o,fun($o,fun($o,fun($o,fun(literal,literal)))),aa($o,fun($o,fun($o,fun($o,fun($o,fun(literal,literal))))),aa($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun(literal,literal)))))),aa($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun(literal,literal))))))),literal2,$true),$false),$true),$false),$true),$true),$true),aa(literal,literal,aa($o,fun(literal,literal),aa($o,fun($o,fun(literal,literal)),aa($o,fun($o,fun($o,fun(literal,literal))),aa($o,fun($o,fun($o,fun($o,fun(literal,literal)))),aa($o,fun($o,fun($o,fun($o,fun($o,fun(literal,literal))))),aa($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun(literal,literal)))))),aa($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun(literal,literal))))))),literal2,$true),$false),$true),$false),$false),$true),$true),zero_zero(literal))))))))))),aTP_Lamp_awj(pred(A),fun(product_unit,A),Uu)) ).

% ATP.lambda_97
tff(fact_7877_ATP_Olambda__98,axiom,
    ! [A: $tType,Uu: list(list(A)),Uua: nat] : aa(nat,list(A),aTP_Lamp_aba(list(list(A)),fun(nat,list(A)),Uu),Uua) = aa(list(nat),list(A),map(nat,A,aa(nat,fun(nat,A),aTP_Lamp_aaz(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_98
tff(fact_7878_ATP_Olambda__99,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_hs(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_hr(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua)),aa(nat,set(nat),set_ord_atMost(nat),Uua)) ) ).

% ATP.lambda_99
tff(fact_7879_ATP_Olambda__100,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_es(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_er(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua)),aa(nat,set(nat),set_ord_atMost(nat),Uua)) ) ).

% ATP.lambda_100
tff(fact_7880_ATP_Olambda__101,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_wh(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_101
tff(fact_7881_ATP_Olambda__102,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_gm(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_102
tff(fact_7882_ATP_Olambda__103,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_eo(nat,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu),Uua))),Uua)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Uua)) ) ).

% ATP.lambda_103
tff(fact_7883_ATP_Olambda__104,axiom,
    ! [Uu: code_integer,Uua: code_integer] :
      aa(code_integer,int,aa(code_integer,fun(code_integer,int),aTP_Lamp_co(code_integer,fun(code_integer,int)),Uu),Uua) = $let(
        l2: int,
        l2:= aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(code_integer,int,code_int_of_integer,Uu)),
        $ite(Uua = zero_zero(code_integer),l2,aa(int,int,aa(int,fun(int,int),plus_plus(int),l2),one_one(int))) ) ).

% ATP.lambda_104
tff(fact_7884_ATP_Olambda__105,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_wl(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_105
tff(fact_7885_ATP_Olambda__106,axiom,
    ! [B: $tType,A: $tType,Uu: set(old_node(A,B)),Uua: set(old_node(A,B))] : aa(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),aa(set(old_node(A,B)),fun(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),aTP_Lamp_aux(set(old_node(A,B)),fun(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))))),Uu),Uua) = aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),aa(product_prod(set(old_node(A,B)),set(old_node(A,B))),fun(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),insert2(product_prod(set(old_node(A,B)),set(old_node(A,B)))),aa(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),fun(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B)))),product_Pair(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),set(old_node(A,B)),old_In1(A,B),Uu)),aa(set(old_node(A,B)),set(old_node(A,B)),old_In1(A,B),Uua))),bot_bot(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))))) ).

% ATP.lambda_106
tff(fact_7886_ATP_Olambda__107,axiom,
    ! [B: $tType,A: $tType,Uu: set(old_node(A,B)),Uua: set(old_node(A,B))] : aa(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),aa(set(old_node(A,B)),fun(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),aTP_Lamp_auw(set(old_node(A,B)),fun(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))))),Uu),Uua) = aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),aa(product_prod(set(old_node(A,B)),set(old_node(A,B))),fun(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),insert2(product_prod(set(old_node(A,B)),set(old_node(A,B)))),aa(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),fun(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B)))),product_Pair(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),set(old_node(A,B)),old_In0(A,B),Uu)),aa(set(old_node(A,B)),set(old_node(A,B)),old_In0(A,B),Uua))),bot_bot(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))))) ).

% ATP.lambda_107
tff(fact_7887_ATP_Olambda__108,axiom,
    ! [B: $tType,A: $tType,Uu: set(product_prod(A,A)),Uua: set(product_prod(B,B))] :
      ( aa(set(product_prod(B,B)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),$o),aTP_Lamp_amo(set(product_prod(A,A)),fun(set(product_prod(B,B)),$o)),Uu),Uua)
    <=> ( order_well_order_on(A,field2(A,Uu),Uu)
        & order_well_order_on(B,field2(B,Uua),Uua)
        & ? [X_12: fun(A,B)] : bNF_Wellorder_embedS(A,B,Uu,Uua,X_12) ) ) ).

% ATP.lambda_108
tff(fact_7888_ATP_Olambda__109,axiom,
    ! [B: $tType,A: $tType,Uu: set(product_prod(A,A)),Uua: set(product_prod(B,B))] :
      ( aa(set(product_prod(B,B)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),$o),aTP_Lamp_amr(set(product_prod(A,A)),fun(set(product_prod(B,B)),$o)),Uu),Uua)
    <=> ( order_well_order_on(A,field2(A,Uu),Uu)
        & order_well_order_on(B,field2(B,Uua),Uua)
        & ? [X_12: fun(A,B)] : aa(fun(A,B),$o,bNF_Wellorder_embed(A,B,Uu,Uua),X_12) ) ) ).

% ATP.lambda_109
tff(fact_7889_ATP_Olambda__110,axiom,
    ! [B: $tType,A: $tType,Uu: set(product_prod(A,A)),Uua: set(product_prod(B,B))] :
      ( aa(set(product_prod(B,B)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),$o),aTP_Lamp_amq(set(product_prod(A,A)),fun(set(product_prod(B,B)),$o)),Uu),Uua)
    <=> ( order_well_order_on(A,field2(A,Uu),Uu)
        & order_well_order_on(B,field2(B,Uua),Uua)
        & ? [X_12: fun(A,B)] : bNF_Wellorder_iso(A,B,Uu,Uua,X_12) ) ) ).

% ATP.lambda_110
tff(fact_7890_ATP_Olambda__111,axiom,
    ! [Uu: rat,Uua: int] :
      ( aa(int,$o,aTP_Lamp_qr(rat,fun(int,$o),Uu),Uua)
    <=> ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),aa(int,rat,ring_1_of_int(rat),Uua)),Uu)
        & aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),Uu),aa(int,rat,ring_1_of_int(rat),aa(int,int,aa(int,fun(int,int),plus_plus(int),Uua),one_one(int)))) ) ) ).

% ATP.lambda_111
tff(fact_7891_ATP_Olambda__112,axiom,
    ! [A: $tType,Uu: set(A),Uua: list(A)] :
      ( aa(list(A),$o,aTP_Lamp_xf(set(A),fun(list(A),$o),Uu),Uua)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uua)),Uu)
        & distinct(A,Uua) ) ) ).

% ATP.lambda_112
tff(fact_7892_ATP_Olambda__113,axiom,
    ! [A: $tType,Uu: set(A),Uua: list(A)] :
      ( aa(list(A),$o,aTP_Lamp_xe(set(A),fun(list(A),$o),Uu),Uua)
    <=> ( ( aa(list(A),set(A),set2(A),Uua) = Uu )
        & distinct(A,Uua) ) ) ).

% ATP.lambda_113
tff(fact_7893_ATP_Olambda__114,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_fa(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_114
tff(fact_7894_ATP_Olambda__115,axiom,
    ! [Uu: rat,Uua: product_prod(int,int)] :
      ( aa(product_prod(int,int),$o,aTP_Lamp_wt(rat,fun(product_prod(int,int),$o),Uu),Uua)
    <=> ( ( Uu = aa(int,rat,aa(int,fun(int,rat),fract,aa(product_prod(int,int),int,product_fst(int,int),Uua)),aa(product_prod(int,int),int,product_snd(int,int),Uua)) )
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),aa(product_prod(int,int),int,product_snd(int,int),Uua))
        & algebr8660921524188924756oprime(int,aa(product_prod(int,int),int,product_fst(int,int),Uua),aa(product_prod(int,int),int,product_snd(int,int),Uua)) ) ) ).

% ATP.lambda_115
tff(fact_7895_ATP_Olambda__116,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: A] : aa(A,set(set(A)),aTP_Lamp_wy(set(product_prod(A,A)),fun(A,set(set(A))),Uu),Uua) = aa(set(set(A)),set(set(A)),aa(set(A),fun(set(set(A)),set(set(A))),insert2(set(A)),aa(set(A),set(A),image(A,A,Uu),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Uua),bot_bot(set(A))))),bot_bot(set(set(A)))) ).

% ATP.lambda_116
tff(fact_7896_ATP_Olambda__117,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_eh(A,fun(nat,A),Uu),Uua) = aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),aa(nat,A,semiring_1_of_nat(A),Uua))),Uua) ) ).

% ATP.lambda_117
tff(fact_7897_ATP_Olambda__118,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_ee(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),aa(A,A,aa(A,fun(A,A),divide_divide(A),Uu),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),aa(nat,A,semiring_1_of_nat(A),Uua))) ) ).

% ATP.lambda_118
tff(fact_7898_ATP_Olambda__119,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_ek(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_119
tff(fact_7899_ATP_Olambda__120,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_cc(nat,fun(nat,$o)),Uu),Uua)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Uu),Uua)
        & ( Uu != Uua ) ) ) ).

% ATP.lambda_120
tff(fact_7900_ATP_Olambda__121,axiom,
    ! [A: $tType,Uu: set(set(A)),Uua: set(set(A))] :
      ( aa(set(set(A)),$o,aTP_Lamp_jr(set(set(A)),fun(set(set(A)),$o),Uu),Uua)
    <=> ( aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),Uua),Uu)
        & ( Uua != bot_bot(set(set(A))) ) ) ) ).

% ATP.lambda_121
tff(fact_7901_ATP_Olambda__122,axiom,
    ! [A: $tType,Uu: set(option(A)),Uua: option(A)] :
      ( aa(option(A),$o,aTP_Lamp_to(set(option(A)),fun(option(A),$o),Uu),Uua)
    <=> ( aa(set(option(A)),$o,member(option(A),Uua),Uu)
        & ( Uua != none(A) ) ) ) ).

% ATP.lambda_122
tff(fact_7902_ATP_Olambda__123,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_gj(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),aa(num,num,bit0,one2))) ).

% ATP.lambda_123
tff(fact_7903_ATP_Olambda__124,axiom,
    ! [Uu: set(int),Uua: int] :
      ( aa(int,$o,aTP_Lamp_ahb(set(int),fun(int,$o),Uu),Uua)
    <=> ( aa(set(int),$o,member(int,Uua),Uu)
        & ! [X4: int] :
            ( aa(set(int),$o,member(int,X4),Uu)
           => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),X4),Uua) ) ) ) ).

% ATP.lambda_124
tff(fact_7904_ATP_Olambda__125,axiom,
    ! [A: $tType,Uu: set(set(A)),Uua: set(A)] :
      ( aa(set(A),$o,aTP_Lamp_agy(set(set(A)),fun(set(A),$o),Uu),Uua)
    <=> ( aa(set(set(A)),$o,member(set(A),Uua),Uu)
        & ! [X4: set(A)] :
            ( aa(set(set(A)),$o,member(set(A),X4),Uu)
           => ~ aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),Uua),X4) ) ) ) ).

% ATP.lambda_125
tff(fact_7905_ATP_Olambda__126,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),aTP_Lamp_ajq(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o)),Uu),Uua)
    <=> ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),Uu),Uua)
        & ! [A10: A,B6: A,C5: A] :
            ( ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A10),B6)),Uua)
              & aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B6),C5)),Uu) )
           => aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A10),B6)),Uu) ) ) ) ).

% ATP.lambda_126
tff(fact_7906_ATP_Olambda__127,axiom,
    ! [A: $tType,Uu: set(set(A)),Uua: set(set(A))] :
      ( aa(set(set(A)),$o,aTP_Lamp_wc(set(set(A)),fun(set(set(A)),$o),Uu),Uua)
    <=> ( aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),Uua),Uu)
        & chain_subset(A,Uua) ) ) ).

% ATP.lambda_127
tff(fact_7907_ATP_Olambda__128,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A)] :
      ( aa(set(A),$o,aTP_Lamp_oz(set(A),fun(set(A),$o),Uu),Uua)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Uua),Uu)
        & aa(set(A),$o,finite_finite2(A),Uua) ) ) ).

% ATP.lambda_128
tff(fact_7908_ATP_Olambda__129,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),aTP_Lamp_ov(set(A),fun(set(A),$o)),Uu),Uua)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),Uu),Uua)
        & aa(set(A),$o,finite_finite2(A),Uua) ) ) ).

% ATP.lambda_129
tff(fact_7909_ATP_Olambda__130,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_ts(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_130
tff(fact_7910_ATP_Olambda__131,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_fz(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu),Uua)),Uu) ).

% ATP.lambda_131
tff(fact_7911_ATP_Olambda__132,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_fy(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_132
tff(fact_7912_ATP_Olambda__133,axiom,
    ! [B: $tType,A: $tType,Uu: set(old_node(A,B)),Uua: set(old_node(A,B))] : aa(set(old_node(A,B)),set(set(old_node(A,B))),aTP_Lamp_avk(set(old_node(A,B)),fun(set(old_node(A,B)),set(set(old_node(A,B)))),Uu),Uua) = aa(set(set(old_node(A,B))),set(set(old_node(A,B))),aa(set(old_node(A,B)),fun(set(set(old_node(A,B))),set(set(old_node(A,B)))),insert2(set(old_node(A,B))),old_Scons(A,B,Uu,Uua)),bot_bot(set(set(old_node(A,B))))) ).

% ATP.lambda_133
tff(fact_7913_ATP_Olambda__134,axiom,
    ! [A: $tType,B: $tType,Uu: B,Uua: A] : aa(A,set(product_prod(B,A)),aTP_Lamp_oi(B,fun(A,set(product_prod(B,A))),Uu),Uua) = aa(set(product_prod(B,A)),set(product_prod(B,A)),aa(product_prod(B,A),fun(set(product_prod(B,A)),set(product_prod(B,A))),insert2(product_prod(B,A)),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Uu),Uua)),bot_bot(set(product_prod(B,A)))) ).

% ATP.lambda_134
tff(fact_7914_ATP_Olambda__135,axiom,
    ! [B: $tType,A: $tType,Uu: A,Uua: B] : aa(B,set(product_prod(A,B)),aTP_Lamp_sb(A,fun(B,set(product_prod(A,B))),Uu),Uua) = aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(product_prod(A,B),fun(set(product_prod(A,B)),set(product_prod(A,B))),insert2(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uu),Uua)),bot_bot(set(product_prod(A,B)))) ).

% ATP.lambda_135
tff(fact_7915_ATP_Olambda__136,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_sm(A,fun(A,$o),Uu),Uua)
        <=> ( aa(set(A),$o,member(A,Uua),ring_1_Ints(A))
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),Uua)),Uu) ) ) ) ).

% ATP.lambda_136
tff(fact_7916_ATP_Olambda__137,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,fun(B,$o)),Uua: product_prod(A,B)] :
      ( aa(product_prod(A,B),$o,aTP_Lamp_wb(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),Uu),Uua)
    <=> aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(product_prod(A,B),fun(set(product_prod(A,B)),set(product_prod(A,B))),insert2(product_prod(A,B)),Uua),bot_bot(set(product_prod(A,B))))),aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),Uu))) ) ).

% ATP.lambda_137
tff(fact_7917_ATP_Olambda__138,axiom,
    ! [A: $tType,C: $tType,B: $tType,Uu: fun(B,fun(C,$o)),Uua: fun(A,product_prod(B,C))] :
      ( aa(fun(A,product_prod(B,C)),$o,aTP_Lamp_wd(fun(B,fun(C,$o)),fun(fun(A,product_prod(B,C)),$o),Uu),Uua)
    <=> aa(set(product_prod(B,C)),$o,aa(set(product_prod(B,C)),fun(set(product_prod(B,C)),$o),ord_less_eq(set(product_prod(B,C))),aa(set(A),set(product_prod(B,C)),image2(A,product_prod(B,C),Uua),top_top(set(A)))),aa(fun(product_prod(B,C),$o),set(product_prod(B,C)),collect(product_prod(B,C)),aa(fun(B,fun(C,$o)),fun(product_prod(B,C),$o),product_case_prod(B,C,$o),Uu))) ) ).

% ATP.lambda_138
tff(fact_7918_ATP_Olambda__139,axiom,
    ! [A: $tType,Uu: fun(set(A),$o),Uua: set(A)] :
      ( aa(set(A),$o,aa(fun(set(A),$o),fun(set(A),$o),aTP_Lamp_aeq(fun(set(A),$o),fun(set(A),$o)),Uu),Uua)
    <=> ( ( Uua = bot_bot(set(A)) )
        | ? [A9: set(A),A10: A] :
            ( ( Uua = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A10),A9) )
            & aa(set(A),$o,Uu,A9) ) ) ) ).

% ATP.lambda_139
tff(fact_7919_ATP_Olambda__140,axiom,
    ! [A: $tType,B: $tType,Uu: set(B),Uua: fun(A,B)] :
      ( aa(fun(A,B),$o,aTP_Lamp_amy(set(B),fun(fun(A,B),$o),Uu),Uua)
    <=> aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,Uua),top_top(set(A)))),Uu) ) ).

% ATP.lambda_140
tff(fact_7920_ATP_Olambda__141,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: A] : aa(A,set(set(A)),aTP_Lamp_acr(set(product_prod(A,A)),fun(A,set(set(A))),Uu),Uua) = equiv_quotient(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Uua),bot_bot(set(A))),Uu) ).

% ATP.lambda_141
tff(fact_7921_ATP_Olambda__142,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: set(A),Uua: list(A)] :
          ( aa(list(A),$o,aTP_Lamp_aof(set(A),fun(list(A),$o),Uu),Uua)
        <=> ( sorted_wrt(A,ord_less(A),Uua)
            & ( aa(list(A),set(A),set2(A),Uua) = Uu ) ) ) ) ).

% ATP.lambda_142
tff(fact_7922_ATP_Olambda__143,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_or(set(product_prod(A,A)),fun(nat,$o),Uu),Uua)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Uua)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Uua),aa(set(product_prod(A,A)),nat,finite_card(product_prod(A,A)),Uu)) ) ) ).

% ATP.lambda_143
tff(fact_7923_ATP_Olambda__144,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_ow(nat,fun(nat,$o),Uu),Uua)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Uua)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Uua),aa(nat,nat,suc,Uu)) ) ) ).

% ATP.lambda_144
tff(fact_7924_ATP_Olambda__145,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: fun(C,product_prod(A,B)),Uua: A] : aa(A,set(B),aTP_Lamp_wa(fun(C,product_prod(A,B)),fun(A,set(B)),Uu),Uua) = aa(set(C),set(B),image2(C,B,aa(fun(C,product_prod(A,B)),fun(C,B),comp(product_prod(A,B),B,C,product_snd(A,B)),Uu)),top_top(set(C))) ).

% ATP.lambda_145
tff(fact_7925_ATP_Olambda__146,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_gf(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),aa(num,num,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),aa(num,num,bit0,one2))),Uua)))) ) ).

% ATP.lambda_146
tff(fact_7926_ATP_Olambda__147,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_dx(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,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),aa(num,num,bit0,one2))),Uua)))) ) ).

% ATP.lambda_147
tff(fact_7927_ATP_Olambda__148,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_ds(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_148
tff(fact_7928_ATP_Olambda__149,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_fu(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_149
tff(fact_7929_ATP_Olambda__150,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_dm(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_150
tff(fact_7930_ATP_Olambda__151,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_gw(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_151
tff(fact_7931_ATP_Olambda__152,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_dw(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_152
tff(fact_7932_ATP_Olambda__153,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_dv(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_153
tff(fact_7933_ATP_Olambda__154,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_ep(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_154
tff(fact_7934_ATP_Olambda__155,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu: fun(A,$o),Uua: A] :
          ( aa(A,$o,aTP_Lamp_aio(fun(A,$o),fun(A,$o),Uu),Uua)
        <=> ( aa(A,$o,Uu,Uua)
            & ! [Y3: A] :
                ( aa(A,$o,Uu,Y3)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),Y3) ) ) ) ) ).

% ATP.lambda_155
tff(fact_7935_ATP_Olambda__156,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Uu: fun(A,$o),Uua: A] :
          ( aa(A,$o,aTP_Lamp_aiq(fun(A,$o),fun(A,$o),Uu),Uua)
        <=> ( aa(A,$o,Uu,Uua)
            & ! [Y3: A] :
                ( aa(A,$o,Uu,Y3)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y3),Uua) ) ) ) ) ).

% ATP.lambda_156
tff(fact_7936_ATP_Olambda__157,axiom,
    ! [A: $tType,Uu: fun(multiset(A),$o),Uua: multiset(A)] :
      ( aa(multiset(A),$o,aTP_Lamp_asc(fun(multiset(A),$o),fun(multiset(A),$o),Uu),Uua)
    <=> ( aa(multiset(A),$o,Uu,Uua)
        & ! [Y3: multiset(A)] :
            ( aa(multiset(A),$o,Uu,Y3)
           => aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),Uua),Y3) ) ) ) ).

% ATP.lambda_157
tff(fact_7937_ATP_Olambda__158,axiom,
    ! [A: $tType,Uu: fun(multiset(A),$o),Uua: multiset(A)] :
      ( aa(multiset(A),$o,aTP_Lamp_ask(fun(multiset(A),$o),fun(multiset(A),$o),Uu),Uua)
    <=> ( aa(multiset(A),$o,Uu,Uua)
        & ! [Y3: multiset(A)] :
            ( aa(multiset(A),$o,Uu,Y3)
           => aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),Y3),Uua) ) ) ) ).

% ATP.lambda_158
tff(fact_7938_ATP_Olambda__159,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_ic(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_159
tff(fact_7939_ATP_Olambda__160,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_ib(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_160
tff(fact_7940_ATP_Olambda__161,axiom,
    ! [B: $tType,Uu: fun(nat,sum_sum(B,nat)),Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_aud(fun(nat,sum_sum(B,nat)),fun(nat,$o),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_161
tff(fact_7941_ATP_Olambda__162,axiom,
    ! [A: $tType,Uu: fun(nat,sum_sum(A,nat)),Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_ath(fun(nat,sum_sum(A,nat)),fun(nat,$o),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_162
tff(fact_7942_ATP_Olambda__163,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: fun(A,A),Uua: A] :
          ( aa(A,$o,aTP_Lamp_ahq(fun(A,A),fun(A,$o),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,Uu,Uua)),Uua) ) ) ).

% ATP.lambda_163
tff(fact_7943_ATP_Olambda__164,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: A] : aa(A,product_prod(nat,A),aTP_Lamp_ame(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_164
tff(fact_7944_ATP_Olambda__165,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: A] : aa(A,product_prod(B,A),aTP_Lamp_acg(fun(A,B),fun(A,product_prod(B,A)),Uu),Uua) = aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),aa(A,B,Uu,Uua)),Uua) ).

% ATP.lambda_165
tff(fact_7945_ATP_Olambda__166,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,A),Uua: B] : aa(B,list(A),aTP_Lamp_zy(fun(B,A),fun(B,list(A)),Uu),Uua) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(B,A,Uu,Uua)),nil(A)) ).

% ATP.lambda_166
tff(fact_7946_ATP_Olambda__167,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,A),Uua: B] : aa(B,set(A),aTP_Lamp_md(fun(B,A),fun(B,set(A)),Uu),Uua) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),aa(B,A,Uu,Uua)),bot_bot(set(A))) ).

% ATP.lambda_167
tff(fact_7947_ATP_Olambda__168,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: fun(A,B),Uua: A] :
          ( aa(A,$o,aTP_Lamp_hu(fun(A,B),fun(A,$o),Uu),Uua)
        <=> ( aa(A,B,Uu,Uua) = zero_zero(B) ) ) ) ).

% ATP.lambda_168
tff(fact_7948_ATP_Olambda__169,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: fun(A,B),Uua: A] :
          ( aa(A,$o,aTP_Lamp_hw(fun(A,B),fun(A,$o),Uu),Uua)
        <=> ( aa(A,B,Uu,Uua) = one_one(B) ) ) ) ).

% ATP.lambda_169
tff(fact_7949_ATP_Olambda__170,axiom,
    ! [Uu: code_integer,Uua: $o] : aa($o,char,aa(code_integer,fun($o,char),aTP_Lamp_qq(code_integer,fun($o,char)),Uu),(Uua)) = aa(product_prod(code_integer,$o),char,aa(fun(code_integer,fun($o,char)),fun(product_prod(code_integer,$o),char),product_case_prod(code_integer,$o,char),aTP_Lamp_qp($o,fun(code_integer,fun($o,char)),(Uua))),code_bit_cut_integer(Uu)) ).

% ATP.lambda_170
tff(fact_7950_ATP_Olambda__171,axiom,
    ! [A: $tType,Uu: list(fun(A,nat)),Uua: A] : aa(A,list(nat),aTP_Lamp_amc(list(fun(A,nat)),fun(A,list(nat)),Uu),Uua) = aa(list(fun(A,nat)),list(nat),map(fun(A,nat),nat,aTP_Lamp_amb(A,fun(fun(A,nat),nat),Uua)),Uu) ).

% ATP.lambda_171
tff(fact_7951_ATP_Olambda__172,axiom,
    ! [A: $tType,Uu: list(A),Uua: list(A)] : aa(list(A),list(list(A)),aTP_Lamp_aao(list(A),fun(list(A),list(list(A))),Uu),Uua) = aa(list(A),list(list(A)),map(A,list(A),aa(list(A),fun(A,list(A)),aTP_Lamp_xo(list(A),fun(A,list(A))),Uua)),Uu) ).

% ATP.lambda_172
tff(fact_7952_ATP_Olambda__173,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_asv(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_173
tff(fact_7953_ATP_Olambda__174,axiom,
    ! [Uu: num,Uua: num] : aa(num,int,aTP_Lamp_si(num,fun(num,int),Uu),Uua) = bit_se2584673776208193580ke_bit(int,aa(num,nat,numeral_numeral(nat),Uu),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(num,nat,numeral_numeral(nat),Uu))),aa(num,int,numeral_numeral(int),Uua))) ).

% ATP.lambda_174
tff(fact_7954_ATP_Olambda__175,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A)] :
      ( aa(set(A),$o,aTP_Lamp_tz(set(A),fun(set(A),$o),Uu),Uua)
    <=> ( aa(set(A),$o,finite_finite2(A),Uua)
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Uua),Uu) ) ) ).

% ATP.lambda_175
tff(fact_7955_ATP_Olambda__176,axiom,
    ! [Uu: assn,Uua: product_prod(heap_ext(product_unit),set(nat))] :
      ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aTP_Lamp_az(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),Uu),Uua)
    <=> ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,Uua)
        & ~ aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,Uu),Uua) ) ) ).

% ATP.lambda_176
tff(fact_7956_ATP_Olambda__177,axiom,
    ! [Uu: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Uua: product_prod(heap_ext(product_unit),set(nat))] :
      ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aTP_Lamp_aw(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(heap_ext(product_unit),set(nat)),$o),Uu),Uua)
    <=> ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,Uua)
        & ~ aa(product_prod(heap_ext(product_unit),set(nat)),$o,Uu,Uua) ) ) ).

% ATP.lambda_177
tff(fact_7957_ATP_Olambda__178,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,fun(B,$o)),Uua: option(product_prod(A,B))] :
      ( aa(option(product_prod(A,B)),$o,aTP_Lamp_awv(fun(A,fun(B,$o)),fun(option(product_prod(A,B)),$o),Uu),Uua)
    <=> aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),set_option(product_prod(A,B),Uua)),aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),Uu))) ) ).

% ATP.lambda_178
tff(fact_7958_ATP_Olambda__179,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,fun(B,$o)),Uua: multiset(product_prod(A,B))] :
      ( aa(multiset(product_prod(A,B)),$o,aTP_Lamp_apn(fun(A,fun(B,$o)),fun(multiset(product_prod(A,B)),$o),Uu),Uua)
    <=> aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),aa(multiset(product_prod(A,B)),set(product_prod(A,B)),set_mset(product_prod(A,B)),Uua)),aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),Uu))) ) ).

% ATP.lambda_179
tff(fact_7959_ATP_Olambda__180,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,fun(B,$o)),Uua: list(product_prod(A,B))] :
      ( aa(list(product_prod(A,B)),$o,aTP_Lamp_ait(fun(A,fun(B,$o)),fun(list(product_prod(A,B)),$o),Uu),Uua)
    <=> aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Uua)),aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),Uu))) ) ).

% ATP.lambda_180
tff(fact_7960_ATP_Olambda__181,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,fun(B,$o)),Uua: set(set(A))] : aa(set(set(A)),set(A),aTP_Lamp_arq(fun(A,fun(B,$o)),fun(set(set(A)),set(A)),Uu),Uua) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),Uua)),aa(fun(A,$o),set(A),collect(A),aa(fun(A,fun(B,$o)),fun(A,$o),domainp(A,B),Uu))) ).

% ATP.lambda_181
tff(fact_7961_ATP_Olambda__182,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,fun(B,$o)),Uua: set(A)] : aa(set(A),set(A),aTP_Lamp_ars(fun(A,fun(B,$o)),fun(set(A),set(A)),Uu),Uua) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),Uua)),aa(fun(A,$o),set(A),collect(A),aa(fun(A,fun(B,$o)),fun(A,$o),domainp(A,B),Uu))) ).

% ATP.lambda_182
tff(fact_7962_ATP_Olambda__183,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(A)
     => ! [Uu: product_prod(A,B),Uua: product_prod(A,B)] :
          ( aa(product_prod(A,B),$o,aa(product_prod(A,B),fun(product_prod(A,B),$o),aTP_Lamp_aah(product_prod(A,B),fun(product_prod(A,B),$o)),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(product_prod(A,B),A,product_fst(A,B),Uu)),aa(product_prod(A,B),A,product_fst(A,B),Uua)) ) ) ).

% ATP.lambda_183
tff(fact_7963_ATP_Olambda__184,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(A)
     => ! [Uu: product_prod(A,B),Uua: product_prod(A,B)] :
          ( aa(product_prod(A,B),$o,aa(product_prod(A,B),fun(product_prod(A,B),$o),aTP_Lamp_aai(product_prod(A,B),fun(product_prod(A,B),$o)),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(product_prod(A,B),A,product_fst(A,B),Uua)),aa(product_prod(A,B),A,product_fst(A,B),Uu)) ) ) ).

% ATP.lambda_184
tff(fact_7964_ATP_Olambda__185,axiom,
    ! [Uu: char,Uua: char] :
      ( aa(char,$o,aa(char,fun(char,$o),aTP_Lamp_asw(char,fun(char,$o)),Uu),Uua)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(char,nat,comm_s6883823935334413003f_char(nat),Uu)),aa(char,nat,comm_s6883823935334413003f_char(nat),Uua)) ) ).

% ATP.lambda_185
tff(fact_7965_ATP_Olambda__186,axiom,
    ! [A: $tType,Uu: list(option(A)),Uua: A] : aa(A,option(list(A)),aTP_Lamp_awt(list(option(A)),fun(A,option(list(A))),Uu),Uua) = aa(option(list(A)),option(list(A)),map_option(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uua)),those(A,Uu)) ).

% ATP.lambda_186
tff(fact_7966_ATP_Olambda__187,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_pn(nat,fun(nat,A)),Uu),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,semiring_1_of_nat(A),Uu)),aa(nat,A,semiring_1_of_nat(A),Uua)) ) ).

% ATP.lambda_187
tff(fact_7967_ATP_Olambda__188,axiom,
    ! [Uu: num,Uua: num] : aa(num,int,aa(num,fun(num,int),aTP_Lamp_uk(num,fun(num,int)),Uu),Uua) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(num,int,numeral_numeral(int),Uu)),aa(num,int,numeral_numeral(int),Uua)) ).

% ATP.lambda_188
tff(fact_7968_ATP_Olambda__189,axiom,
    ! [A: $tType,Uu: pred(A),Uua: seq(A)] : aa(seq(A),set(A),aa(pred(A),fun(seq(A),set(A)),aTP_Lamp_axz(pred(A),fun(seq(A),set(A))),Uu),Uua) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_of_pred(A,Uu)),set_of_seq(A,Uua)) ).

% ATP.lambda_189
tff(fact_7969_ATP_Olambda__190,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_aut(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_190
tff(fact_7970_ATP_Olambda__191,axiom,
    ! [A: $tType,Uu: list(list(A)),Uua: A] : aa(A,list(list(A)),aTP_Lamp_aas(list(list(A)),fun(A,list(list(A))),Uu),Uua) = aa(list(list(A)),list(list(A)),map(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uua)),product_lists(A,Uu)) ).

% ATP.lambda_191
tff(fact_7971_ATP_Olambda__192,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,rat,aa(nat,fun(nat,rat),aTP_Lamp_ajf(nat,fun(nat,rat)),Uu),Uua) = aa(int,rat,aa(int,fun(int,rat),fract,aa(nat,int,nat_int_decode,Uu)),aa(nat,int,nat_int_decode,Uua)) ).

% ATP.lambda_192
tff(fact_7972_ATP_Olambda__193,axiom,
    ! [A: $tType,Uu: list(A),Uua: list(A)] :
      ( aa(list(A),$o,aa(list(A),fun(list(A),$o),aTP_Lamp_agg(list(A),fun(list(A),$o)),Uu),Uua)
    <=> ( aa(list(A),nat,size_size(list(A)),Uu) = aa(list(A),nat,size_size(list(A)),Uua) ) ) ).

% ATP.lambda_193
tff(fact_7973_ATP_Olambda__194,axiom,
    ! [A: $tType,Uu: list(A),Uua: list(A)] :
      ( aa(list(A),$o,aTP_Lamp_ajd(list(A),fun(list(A),$o),Uu),Uua)
    <=> ( aa(list(A),multiset(A),mset(A),Uua) = aa(list(A),multiset(A),mset(A),Uu) ) ) ).

% ATP.lambda_194
tff(fact_7974_ATP_Olambda__195,axiom,
    ! [Uu: code_integer,Uua: code_integer] :
      aa(code_integer,num,aa(code_integer,fun(code_integer,num),aTP_Lamp_cl(code_integer,fun(code_integer,num)),Uu),Uua) = $let(
        l2: num,
        l2:= code_num_of_integer(Uu),
        $let(
          l3: num,
          l3:= aa(num,num,aa(num,fun(num,num),plus_plus(num),l2),l2),
          $ite(Uua = zero_zero(code_integer),l3,aa(num,num,aa(num,fun(num,num),plus_plus(num),l3),one2)) ) ) ).

% ATP.lambda_195
tff(fact_7975_ATP_Olambda__196,axiom,
    ! [Uu: code_integer,Uua: code_integer] :
      aa(code_integer,nat,aa(code_integer,fun(code_integer,nat),aTP_Lamp_cn(code_integer,fun(code_integer,nat)),Uu),Uua) = $let(
        l2: nat,
        l2:= code_nat_of_integer(Uu),
        $let(
          l3: nat,
          l3:= aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),l2),l2),
          $ite(Uua = zero_zero(code_integer),l3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),l3),one_one(nat))) ) ) ).

% ATP.lambda_196
tff(fact_7976_ATP_Olambda__197,axiom,
    ! [A: $tType,Uu: set(A),Uua: option(A)] :
      ( aa(option(A),$o,aTP_Lamp_awu(set(A),fun(option(A),$o),Uu),Uua)
    <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_option(A,Uua)),Uu) ) ).

% ATP.lambda_197
tff(fact_7977_ATP_Olambda__198,axiom,
    ! [A: $tType,Uu: set(A),Uua: multiset(A)] :
      ( aa(multiset(A),$o,aTP_Lamp_apo(set(A),fun(multiset(A),$o),Uu),Uua)
    <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(multiset(A),set(A),set_mset(A),Uua)),Uu) ) ).

% ATP.lambda_198
tff(fact_7978_ATP_Olambda__199,axiom,
    ! [A: $tType,Uu: set(A),Uua: list(A)] :
      ( aa(list(A),$o,aTP_Lamp_aic(set(A),fun(list(A),$o),Uu),Uua)
    <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uua)),Uu) ) ).

% ATP.lambda_199
tff(fact_7979_ATP_Olambda__200,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_et(nat,fun(nat,A),Uu),Uua) = aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),Uua)),Uu) ) ).

% ATP.lambda_200
tff(fact_7980_ATP_Olambda__201,axiom,
    ! [A: $tType,B: $tType,Uu: list(B),Uua: A] : aa(A,list(product_prod(A,B)),aTP_Lamp_aaj(list(B),fun(A,list(product_prod(A,B))),Uu),Uua) = aa(list(B),list(product_prod(A,B)),map(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua)),Uu) ).

% ATP.lambda_201
tff(fact_7981_ATP_Olambda__202,axiom,
    ! [A: $tType,Uu: set(nat),Uua: product_prod(A,nat)] :
      ( aa(product_prod(A,nat),$o,aTP_Lamp_adu(set(nat),fun(product_prod(A,nat),$o),Uu),Uua)
    <=> aa(set(nat),$o,member(nat,aa(product_prod(A,nat),nat,product_snd(A,nat),Uua)),Uu) ) ).

% ATP.lambda_202
tff(fact_7982_ATP_Olambda__203,axiom,
    ! [Uu: set(nat),Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_aca(set(nat),fun(nat,$o),Uu),Uua)
    <=> aa(set(nat),$o,member(nat,aa(nat,nat,suc,Uua)),Uu) ) ).

% ATP.lambda_203
tff(fact_7983_ATP_Olambda__204,axiom,
    ! [A: $tType,Uu: nat,Uua: list(A)] :
      ( aa(list(A),$o,aTP_Lamp_aii(nat,fun(list(A),$o),Uu),Uua)
    <=> ( aa(list(A),nat,size_size(list(A)),Uua) = Uu ) ) ).

% ATP.lambda_204
tff(fact_7984_ATP_Olambda__205,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: set(product_prod(A,A))] : aa(set(product_prod(A,A)),set(set(product_prod(A,A))),aTP_Lamp_aom(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(set(product_prod(A,A)))),Uu),Uua) = aa(set(set(product_prod(A,A))),set(set(product_prod(A,A))),image(set(product_prod(A,A)),set(product_prod(A,A)),converse(set(product_prod(A,A)),set(product_prod(A,A)),bNF_We4044943003108391690rdLess(A,A))),aa(set(set(product_prod(A,A))),set(set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(set(product_prod(A,A))),set(set(product_prod(A,A)))),insert2(set(product_prod(A,A))),Uu),bot_bot(set(set(product_prod(A,A)))))) ).

% ATP.lambda_205
tff(fact_7985_ATP_Olambda__206,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_bl(A,fun(A,product_prod(A,A))),Uu),Uua) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uu),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Uua)),one_one(A))) ) ).

% ATP.lambda_206
tff(fact_7986_ATP_Olambda__207,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_bn(A,fun(A,product_prod(A,A))),Uu),Uua) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uu),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Uua)) ) ).

% ATP.lambda_207
tff(fact_7987_ATP_Olambda__208,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_ew(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,semiring_1_of_nat(A),Uua)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))) ) ).

% ATP.lambda_208
tff(fact_7988_ATP_Olambda__209,axiom,
    ! [A: $tType,Uu: A,Uua: set(set(A))] : aa(set(set(A)),set(set(A)),aa(A,fun(set(set(A)),set(set(A))),aTP_Lamp_op(A,fun(set(set(A)),set(set(A)))),Uu),Uua) = aa(set(set(A)),set(set(A)),aa(set(set(A)),fun(set(set(A)),set(set(A))),sup_sup(set(set(A))),Uua),aa(set(set(A)),set(set(A)),image2(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Uu)),Uua)) ).

% ATP.lambda_209
tff(fact_7989_ATP_Olambda__210,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: list(A),Uua: A] :
          ( aa(A,$o,aTP_Lamp_ys(list(A),fun(A,$o),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Uua),aa(nat,A,nth(A,Uu),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(list(A),nat,size_size(list(A)),Uu)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ) ) ).

% ATP.lambda_210
tff(fact_7990_ATP_Olambda__211,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: list(A),Uua: A] :
          ( aa(A,$o,aTP_Lamp_yt(list(A),fun(A,$o),Uu),Uua)
        <=> ( Uua = aa(nat,A,nth(A,Uu),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(list(A),nat,size_size(list(A)),Uu)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ) ) ) ).

% ATP.lambda_211
tff(fact_7991_ATP_Olambda__212,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_gn(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_212
tff(fact_7992_ATP_Olambda__213,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: multiset(A),Uua: A] : aa(A,A,aTP_Lamp_are(multiset(A),fun(A,A),Uu),Uua) = aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),aa(A,nat,aa(multiset(A),fun(A,nat),count(A),Uu),Uua)) ) ).

% ATP.lambda_213
tff(fact_7993_ATP_Olambda__214,axiom,
    ! [A: $tType,B: $tType,Uu: set(product_prod(B,A)),Uua: B] : aa(B,set(A),aTP_Lamp_ww(set(product_prod(B,A)),fun(B,set(A)),Uu),Uua) = aa(set(B),set(A),image(B,A,Uu),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),Uua),bot_bot(set(B)))) ).

% ATP.lambda_214
tff(fact_7994_ATP_Olambda__215,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,B),Uua: B] : aa(B,set(A),aTP_Lamp_my(fun(A,B),fun(B,set(A)),Uu),Uua) = aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),Uu),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),Uua),bot_bot(set(B)))) ).

% ATP.lambda_215
tff(fact_7995_ATP_Olambda__216,axiom,
    ! [A: $tType,Uu: set(A),Uua: A] :
      ( aa(A,$o,aTP_Lamp_qs(set(A),fun(A,$o),Uu),Uua)
    <=> ( Uu = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Uua),bot_bot(set(A))) ) ) ).

% ATP.lambda_216
tff(fact_7996_ATP_Olambda__217,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: fun(A,A),Uua: A] :
          ( aa(A,$o,aTP_Lamp_alp(fun(A,A),fun(A,$o),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),aa(A,A,Uu,Uua)) ) ) ).

% ATP.lambda_217
tff(fact_7997_ATP_Olambda__218,axiom,
    ! [A: $tType,Uu: fun(nat,A),Uua: nat] : aa(nat,product_prod(nat,A),aTP_Lamp_abc(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_218
tff(fact_7998_ATP_Olambda__219,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: A] : aa(A,product_prod(A,B),aTP_Lamp_abn(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_219
tff(fact_7999_ATP_Olambda__220,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_qa(A,fun(nat,A),Uu),Uua) = bit_se4730199178511100633sh_bit(A,Uua,aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,bit_se5641148757651400278ts_bit(A,Uu),Uua))) ) ).

% ATP.lambda_220
tff(fact_8000_ATP_Olambda__221,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,option(B)),Uua: A] : aa(A,product_prod(A,B),aTP_Lamp_adj(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_221
tff(fact_8001_ATP_Olambda__222,axiom,
    ! [Uu: int,Uua: int] : aa(int,int,aa(int,fun(int,int),aTP_Lamp_cd(int,fun(int,int)),Uu),Uua) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Uu),aa($o,int,zero_neq_one_of_bool(int),Uua != zero_zero(int))) ).

% ATP.lambda_222
tff(fact_8002_ATP_Olambda__223,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_ou(set(product_prod(A,A)),fun(nat,$o),Uu),Uua)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Uua),aa(set(product_prod(A,A)),nat,finite_card(product_prod(A,A)),Uu)) ) ).

% ATP.lambda_223
tff(fact_8003_ATP_Olambda__224,axiom,
    ! [A: $tType,Uu: nat,Uua: list(A)] :
      ( aa(list(A),$o,aTP_Lamp_zx(nat,fun(list(A),$o),Uu),Uua)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uu),aa(list(A),nat,size_size(list(A)),Uua)) ) ).

% ATP.lambda_224
tff(fact_8004_ATP_Olambda__225,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_fq(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_225
tff(fact_8005_ATP_Olambda__226,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_fp(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_226
tff(fact_8006_ATP_Olambda__227,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_ga(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_227
tff(fact_8007_ATP_Olambda__228,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,list(nat),aa(nat,fun(nat,list(nat)),aTP_Lamp_ajh(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_228
tff(fact_8008_ATP_Olambda__229,axiom,
    ! [B: $tType,A: $tType,Uu: list(product_prod(A,B)),Uua: fun(A,option(B))] : aa(fun(A,option(B)),fun(A,option(B)),aa(list(product_prod(A,B)),fun(fun(A,option(B)),fun(A,option(B))),aTP_Lamp_abw(list(product_prod(A,B)),fun(fun(A,option(B)),fun(A,option(B)))),Uu),Uua) = map_add(A,B,Uua,map_of(A,B,Uu)) ).

% ATP.lambda_229
tff(fact_8009_ATP_Olambda__230,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] :
      ( aa(list(A),$o,aa(A,fun(list(A),$o),aTP_Lamp_aiy(A,fun(list(A),$o)),Uu),Uua)
    <=> aa(set(A),$o,member(A,Uu),aa(list(A),set(A),set2(A),Uua)) ) ).

% ATP.lambda_230
tff(fact_8010_ATP_Olambda__231,axiom,
    ! [A: $tType,Uu: list(set(A)),Uua: set(A)] :
      ( aa(set(A),$o,aTP_Lamp_xk(list(set(A)),fun(set(A),$o),Uu),Uua)
    <=> aa(set(set(A)),$o,member(set(A),Uua),aa(list(set(A)),set(set(A)),set2(set(A)),Uu)) ) ).

% ATP.lambda_231
tff(fact_8011_ATP_Olambda__232,axiom,
    ! [A: $tType,Uu: list(A),Uua: A] :
      ( aa(A,$o,aTP_Lamp_zf(list(A),fun(A,$o),Uu),Uua)
    <=> aa(set(A),$o,member(A,Uua),aa(list(A),set(A),set2(A),Uu)) ) ).

% ATP.lambda_232
tff(fact_8012_ATP_Olambda__233,axiom,
    ! [A: $tType,Uu: A,Uua: pred(A)] : aa(pred(A),set(A),aa(A,fun(pred(A),set(A)),aTP_Lamp_axy(A,fun(pred(A),set(A))),Uu),Uua) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Uu),set_of_pred(A,Uua)) ).

% ATP.lambda_233
tff(fact_8013_ATP_Olambda__234,axiom,
    ! [A: $tType,Uu: nat,Uua: list(A)] :
      ( aa(list(A),$o,aTP_Lamp_aih(nat,fun(list(A),$o),Uu),Uua)
    <=> ( Uu = aa(list(A),nat,size_size(list(A)),Uua) ) ) ).

% ATP.lambda_234
tff(fact_8014_ATP_Olambda__235,axiom,
    ! [A: $tType,Uu: list(A),Uua: A] :
      ( aa(A,$o,aTP_Lamp_abr(list(A),fun(A,$o),Uu),Uua)
    <=> ( Uua = aa(list(A),A,hd(A),Uu) ) ) ).

% ATP.lambda_235
tff(fact_8015_ATP_Olambda__236,axiom,
    ! [A: $tType,Uu: list(A),Uua: A] :
      ( aa(A,$o,aTP_Lamp_aed(list(A),fun(A,$o),Uu),Uua)
    <=> ( Uua = last(A,Uu) ) ) ).

% ATP.lambda_236
tff(fact_8016_ATP_Olambda__237,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_sh(nat,fun(nat,$o)),Uu),Uua)
    <=> ( Uua = aa(nat,nat,suc,Uu) ) ) ).

% ATP.lambda_237
tff(fact_8017_ATP_Olambda__238,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A)] :
      ( aa(set(A),$o,aTP_Lamp_in(set(A),fun(set(A),$o),Uu),Uua)
    <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Uua),Uu) ) ).

% ATP.lambda_238
tff(fact_8018_ATP_Olambda__239,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_bj(nat,fun(nat,$o)),Uu),Uua)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Uua),Uu) ) ).

% ATP.lambda_239
tff(fact_8019_ATP_Olambda__240,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_bd(A,fun(A,$o)),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),Uu) ) ) ).

% ATP.lambda_240
tff(fact_8020_ATP_Olambda__241,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_aob(A,fun(A,$o)),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),Uu) ) ) ).

% ATP.lambda_241
tff(fact_8021_ATP_Olambda__242,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_bf(A,fun(A,$o)),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),Uu) ) ) ).

% ATP.lambda_242
tff(fact_8022_ATP_Olambda__243,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_avn(A,fun(A,$o)),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),Uu) ) ) ).

% ATP.lambda_243
tff(fact_8023_ATP_Olambda__244,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_tm(A,fun(A,$o)),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),Uu) ) ) ).

% ATP.lambda_244
tff(fact_8024_ATP_Olambda__245,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_axn(A,fun(A,$o)),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),Uu) ) ) ).

% ATP.lambda_245
tff(fact_8025_ATP_Olambda__246,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_eg(A,fun(A,$o),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),Uu) ) ) ).

% ATP.lambda_246
tff(fact_8026_ATP_Olambda__247,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_oa(nat,fun(nat,nat),Uu),Uua) = modulo_modulo(nat,Uua,Uu) ).

% ATP.lambda_247
tff(fact_8027_ATP_Olambda__248,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_kl(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),divide_divide(A),Uua),Uu) ) ).

% ATP.lambda_248
tff(fact_8028_ATP_Olambda__249,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_acc(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),divide_divide(A),Uua),Uu) ) ).

% ATP.lambda_249
tff(fact_8029_ATP_Olambda__250,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_bk(nat,fun(nat,$o)),Uu),Uua)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uua),Uu) ) ).

% ATP.lambda_250
tff(fact_8030_ATP_Olambda__251,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_be(A,fun(A,$o)),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Uua),Uu) ) ) ).

% ATP.lambda_251
tff(fact_8031_ATP_Olambda__252,axiom,
    ! [A: $tType] :
      ( unboun7993243217541854897norder(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_akl(A,fun(A,$o),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Uua),Uu) ) ) ).

% ATP.lambda_252
tff(fact_8032_ATP_Olambda__253,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_aoc(A,fun(A,$o)),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Uua),Uu) ) ) ).

% ATP.lambda_253
tff(fact_8033_ATP_Olambda__254,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_bg(A,fun(A,$o)),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Uua),Uu) ) ) ).

% ATP.lambda_254
tff(fact_8034_ATP_Olambda__255,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_sz(A,fun(A,$o)),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Uua),Uu) ) ) ).

% ATP.lambda_255
tff(fact_8035_ATP_Olambda__256,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_ano(A,fun(A,$o)),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Uua),Uu) ) ) ).

% ATP.lambda_256
tff(fact_8036_ATP_Olambda__257,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_axo(A,fun(A,$o)),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Uua),Uu) ) ) ).

% ATP.lambda_257
tff(fact_8037_ATP_Olambda__258,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_hi(A,fun(A,$o),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Uua),Uu) ) ) ).

% ATP.lambda_258
tff(fact_8038_ATP_Olambda__259,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_ne(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),Uua),Uu) ) ).

% ATP.lambda_259
tff(fact_8039_ATP_Olambda__260,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_ob(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),Uu) ).

% ATP.lambda_260
tff(fact_8040_ATP_Olambda__261,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_ka(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),Uua),Uu) ) ).

% ATP.lambda_261
tff(fact_8041_ATP_Olambda__262,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_nl(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),Uua),Uu) ) ).

% ATP.lambda_262
tff(fact_8042_ATP_Olambda__263,axiom,
    ! [A: $tType,Uu: A,Uua: multiset(A)] : aa(multiset(A),nat,aTP_Lamp_apu(A,fun(multiset(A),nat),Uu),Uua) = aa(A,nat,aa(multiset(A),fun(A,nat),count(A),Uua),Uu) ).

% ATP.lambda_263
tff(fact_8043_ATP_Olambda__264,axiom,
    ! [A: $tType,Uu: multiset(A),Uua: multiset(A)] :
      ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),aTP_Lamp_aso(multiset(A),fun(multiset(A),$o)),Uu),Uua)
    <=> aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),Uua),Uu) ) ).

% ATP.lambda_264
tff(fact_8044_ATP_Olambda__265,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_lh(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Uua),Uu) ) ).

% ATP.lambda_265
tff(fact_8045_ATP_Olambda__266,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A)] : aa(set(A),set(A),aTP_Lamp_nv(set(A),fun(set(A),set(A)),Uu),Uua) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Uua),Uu) ).

% ATP.lambda_266
tff(fact_8046_ATP_Olambda__267,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_mg(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Uua),Uu) ) ).

% ATP.lambda_267
tff(fact_8047_ATP_Olambda__268,axiom,
    ! [Uu: code_integer,Uua: code_integer] : aa(code_integer,code_integer,aTP_Lamp_nz(code_integer,fun(code_integer,code_integer),Uu),Uua) = aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),Uua),Uu) ).

% ATP.lambda_268
tff(fact_8048_ATP_Olambda__269,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_abb(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uu) ).

% ATP.lambda_269
tff(fact_8049_ATP_Olambda__270,axiom,
    ! [Uu: int,Uua: int] : aa(int,int,aTP_Lamp_ny(int,fun(int,int),Uu),Uua) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Uua),Uu) ).

% ATP.lambda_270
tff(fact_8050_ATP_Olambda__271,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_ace(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uu) ) ).

% ATP.lambda_271
tff(fact_8051_ATP_Olambda__272,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_jz(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uu) ) ).

% ATP.lambda_272
tff(fact_8052_ATP_Olambda__273,axiom,
    ! [A: $tType,Uu: multiset(A),Uua: multiset(A)] :
      ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),aTP_Lamp_asp(multiset(A),fun(multiset(A),$o)),Uu),Uua)
    <=> aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subset_mset(A),Uua),Uu) ) ).

% ATP.lambda_273
tff(fact_8053_ATP_Olambda__274,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_hx(nat,fun(nat,$o),Uu),Uua)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Uua),Uu) ) ).

% ATP.lambda_274
tff(fact_8054_ATP_Olambda__275,axiom,
    ! [Uu: int,Uua: int] :
      ( aa(int,$o,aTP_Lamp_is(int,fun(int,$o),Uu),Uua)
    <=> aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),Uua),Uu) ) ).

% ATP.lambda_275
tff(fact_8055_ATP_Olambda__276,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_cb(A,fun(A,$o),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),Uua),Uu) ) ) ).

% ATP.lambda_276
tff(fact_8056_ATP_Olambda__277,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,set(B)),Uua: set(A)] : aa(set(A),set(product_prod(A,B)),aTP_Lamp_rn(fun(A,set(B)),fun(set(A),set(product_prod(A,B))),Uu),Uua) = product_Sigma(A,B,Uua,Uu) ).

% ATP.lambda_277
tff(fact_8057_ATP_Olambda__278,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),aTP_Lamp_pm(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_278
tff(fact_8058_ATP_Olambda__279,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_ru(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_279
tff(fact_8059_ATP_Olambda__280,axiom,
    ! [A: $tType,Uu: A,Uua: nat] : aa(nat,product_prod(nat,A),aTP_Lamp_aby(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_280
tff(fact_8060_ATP_Olambda__281,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_vx(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_281
tff(fact_8061_ATP_Olambda__282,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_fs(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,binomial(Uua),Uu) ).

% ATP.lambda_282
tff(fact_8062_ATP_Olambda__283,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,pred(A)),Uua: pred(B)] : aa(pred(B),pred(A),aTP_Lamp_awf(fun(B,pred(A)),fun(pred(B),pred(A)),Uu),Uua) = bind2(B,A,Uua,Uu) ).

% ATP.lambda_283
tff(fact_8063_ATP_Olambda__284,axiom,
    ! [A: $tType,Uu: list(A),Uua: A] : aa(A,list(A),aa(list(A),fun(A,list(A)),aTP_Lamp_xo(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_284
tff(fact_8064_ATP_Olambda__285,axiom,
    ! [Uu: set(nat),Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_vd(set(nat),fun(nat,$o),Uu),Uua)
    <=> aa(set(nat),$o,member(nat,Uua),Uu) ) ).

% ATP.lambda_285
tff(fact_8065_ATP_Olambda__286,axiom,
    ! [B: $tType,Uu: set(B),Uua: B] :
      ( aa(B,$o,aTP_Lamp_aks(set(B),fun(B,$o),Uu),Uua)
    <=> aa(set(B),$o,member(B,Uua),Uu) ) ).

% ATP.lambda_286
tff(fact_8066_ATP_Olambda__287,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [Uu: set(A),Uua: A] :
          ( aa(A,$o,aTP_Lamp_vc(set(A),fun(A,$o),Uu),Uua)
        <=> aa(set(A),$o,member(A,Uua),Uu) ) ) ).

% ATP.lambda_287
tff(fact_8067_ATP_Olambda__288,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Uu: set(A),Uua: A] :
          ( aa(A,$o,aTP_Lamp_vj(set(A),fun(A,$o),Uu),Uua)
        <=> aa(set(A),$o,member(A,Uua),Uu) ) ) ).

% ATP.lambda_288
tff(fact_8068_ATP_Olambda__289,axiom,
    ! [A: $tType,Uu: set(A),Uua: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_ag(set(A),fun(A,$o)),Uu),Uua)
    <=> aa(set(A),$o,member(A,Uua),Uu) ) ).

% ATP.lambda_289
tff(fact_8069_ATP_Olambda__290,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: nat] : aa(nat,set(product_prod(A,A)),aTP_Lamp_oq(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),Uu),Uua) = compow(set(product_prod(A,A)),Uua,Uu) ).

% ATP.lambda_290
tff(fact_8070_ATP_Olambda__291,axiom,
    ! [A: $tType,Uu: fun(A,fun(A,$o)),Uua: nat] : aa(nat,fun(A,fun(A,$o)),aTP_Lamp_axd(fun(A,fun(A,$o)),fun(nat,fun(A,fun(A,$o))),Uu),Uua) = compow(fun(A,fun(A,$o)),Uua,Uu) ).

% ATP.lambda_291
tff(fact_8071_ATP_Olambda__292,axiom,
    ! [A: $tType,Uu: list(A),Uua: nat] : aa(nat,list(A),aTP_Lamp_xz(list(A),fun(nat,list(A)),Uu),Uua) = drop(A,Uua,Uu) ).

% ATP.lambda_292
tff(fact_8072_ATP_Olambda__293,axiom,
    ! [A: $tType,Uu: nat,Uua: list(A)] : aa(list(A),A,aTP_Lamp_zw(nat,fun(list(A),A),Uu),Uua) = aa(nat,A,nth(A,Uua),Uu) ).

% ATP.lambda_293
tff(fact_8073_ATP_Olambda__294,axiom,
    ! [B: $tType,C: $tType,A: $tType,Uu: fun(A,C),Uua: fun(C,set(B))] : aa(fun(C,set(B)),fun(A,set(B)),aTP_Lamp_ue(fun(A,C),fun(fun(C,set(B)),fun(A,set(B))),Uu),Uua) = aa(fun(A,C),fun(A,set(B)),comp(C,set(B),A,Uua),Uu) ).

% ATP.lambda_294
tff(fact_8074_ATP_Olambda__295,axiom,
    ! [A: $tType,Uu: A,Uua: A] :
      ( aa(A,$o,aTP_Lamp_cf(A,fun(A,$o),Uu),Uua)
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_295
tff(fact_8075_ATP_Olambda__296,axiom,
    ! [A: $tType,Uu: A,Uua: product_unit] : aa(product_unit,seq(A),aTP_Lamp_axr(A,fun(product_unit,seq(A)),Uu),Uua) = insert(A,Uu,bot_bot(pred(A))) ).

% ATP.lambda_296
tff(fact_8076_ATP_Olambda__297,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] : aa(list(A),list(A),aa(A,fun(list(A),list(A)),aTP_Lamp_aac(A,fun(list(A),list(A))),Uu),Uua) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uu),nil(A)) ).

% ATP.lambda_297
tff(fact_8077_ATP_Olambda__298,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] : aa(list(A),list(list(A)),aa(A,fun(list(A),list(list(A))),aTP_Lamp_aad(A,fun(list(A),list(list(A)))),Uu),Uua) = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),Uua),nil(list(A))) ).

% ATP.lambda_298
tff(fact_8078_ATP_Olambda__299,axiom,
    ! [A: $tType,Uu: A,Uua: A] : aa(A,set(A),aTP_Lamp_sf(A,fun(A,set(A)),Uu),Uua) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Uu),bot_bot(set(A))) ).

% ATP.lambda_299
tff(fact_8079_ATP_Olambda__300,axiom,
    ! [A: $tType,Uu: multiset(A),Uua: A] :
      ( aa(A,$o,aTP_Lamp_apq(multiset(A),fun(A,$o),Uu),Uua)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(A,nat,aa(multiset(A),fun(A,nat),count(A),Uu),Uua)) ) ).

% ATP.lambda_300
tff(fact_8080_ATP_Olambda__301,axiom,
    ! [B: $tType,Uu: fun(B,nat),Uua: B] :
      ( aa(B,$o,aTP_Lamp_aqz(fun(B,nat),fun(B,$o),Uu),Uua)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(B,nat,Uu,Uua)) ) ).

% ATP.lambda_301
tff(fact_8081_ATP_Olambda__302,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: A] :
      ( aa(A,$o,aTP_Lamp_it(fun(A,nat),fun(A,$o),Uu),Uua)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(A,nat,Uu,Uua)) ) ).

% ATP.lambda_302
tff(fact_8082_ATP_Olambda__303,axiom,
    ! [A: $tType,Uu: set(multiset(A)),Uua: A] :
      ( aa(A,$o,aTP_Lamp_ase(set(multiset(A)),fun(A,$o),Uu),Uua)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(set(nat),nat,complete_Sup_Sup(nat),aa(set(multiset(A)),set(nat),image2(multiset(A),nat,aTP_Lamp_apu(A,fun(multiset(A),nat),Uua)),Uu))) ) ).

% ATP.lambda_303
tff(fact_8083_ATP_Olambda__304,axiom,
    ! [A: $tType,Uu: code_natural,Uua: A] :
      ( aa(A,$o,aa(code_natural,fun(A,$o),aTP_Lamp_aua(code_natural,fun(A,$o)),Uu),Uua)
    <=> aa(code_natural,$o,aa(code_natural,fun(code_natural,$o),ord_less(code_natural),zero_zero(code_natural)),Uu) ) ).

% ATP.lambda_304
tff(fact_8084_ATP_Olambda__305,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,set(nat),aTP_Lamp_ani(nat,fun(nat,set(nat)),Uu),Uua) = order_underS(nat,bNF_Ca8665028551170535155natLeq,Uu) ).

% ATP.lambda_305
tff(fact_8085_ATP_Olambda__306,axiom,
    ! [B: $tType,A: $tType,Uu: set(product_prod(A,B)),Uua: A] : aa(A,set(B),aTP_Lamp_vz(set(product_prod(A,B)),fun(A,set(B)),Uu),Uua) = aa(set(product_prod(A,B)),set(B),image2(product_prod(A,B),B,product_snd(A,B)),Uu) ).

% ATP.lambda_306
tff(fact_8086_ATP_Olambda__307,axiom,
    ! [A: $tType,Uu: fun(nat,$o),Uua: product_prod(A,nat)] :
      ( aa(product_prod(A,nat),$o,aTP_Lamp_adp(fun(nat,$o),fun(product_prod(A,nat),$o),Uu),Uua)
    <=> aa(nat,$o,Uu,aa(nat,nat,suc,aa(product_prod(A,nat),nat,product_snd(A,nat),Uua))) ) ).

% ATP.lambda_307
tff(fact_8087_ATP_Olambda__308,axiom,
    ! [A: $tType,Uu: fun(nat,$o),Uua: product_prod(A,nat)] :
      ( aa(product_prod(A,nat),$o,aTP_Lamp_adq(fun(nat,$o),fun(product_prod(A,nat),$o),Uu),Uua)
    <=> aa(nat,$o,Uu,aa(product_prod(A,nat),nat,product_snd(A,nat),Uua)) ) ).

% ATP.lambda_308
tff(fact_8088_ATP_Olambda__309,axiom,
    ! [C: $tType,B: $tType,A: $tType,Uu: fun(B,C),Uua: product_prod(A,B)] : aa(product_prod(A,B),C,aTP_Lamp_vr(fun(B,C),fun(product_prod(A,B),C),Uu),Uua) = aa(B,C,Uu,aa(product_prod(A,B),B,product_snd(A,B),Uua)) ).

% ATP.lambda_309
tff(fact_8089_ATP_Olambda__310,axiom,
    ! [C: $tType,B: $tType,A: $tType,Uu: fun(A,C),Uua: product_prod(A,B)] : aa(product_prod(A,B),C,aTP_Lamp_vm(fun(A,C),fun(product_prod(A,B),C),Uu),Uua) = aa(A,C,Uu,aa(product_prod(A,B),A,product_fst(A,B),Uua)) ).

% ATP.lambda_310
tff(fact_8090_ATP_Olambda__311,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: fun(sum_sum(B,C),set(A)),Uua: C] : aa(C,set(A),aTP_Lamp_atj(fun(sum_sum(B,C),set(A)),fun(C,set(A)),Uu),Uua) = aa(sum_sum(B,C),set(A),Uu,aa(C,sum_sum(B,C),sum_Inr(C,B),Uua)) ).

% ATP.lambda_311
tff(fact_8091_ATP_Olambda__312,axiom,
    ! [C: $tType,A: $tType,B: $tType,Uu: fun(sum_sum(A,B),C),Uua: B] : aa(B,C,aTP_Lamp_atr(fun(sum_sum(A,B),C),fun(B,C),Uu),Uua) = aa(sum_sum(A,B),C,Uu,aa(B,sum_sum(A,B),sum_Inr(B,A),Uua)) ).

% ATP.lambda_312
tff(fact_8092_ATP_Olambda__313,axiom,
    ! [A: $tType,C: $tType,B: $tType,Uu: fun(sum_sum(B,C),set(A)),Uua: B] : aa(B,set(A),aTP_Lamp_ati(fun(sum_sum(B,C),set(A)),fun(B,set(A)),Uu),Uua) = aa(sum_sum(B,C),set(A),Uu,aa(B,sum_sum(B,C),sum_Inl(B,C),Uua)) ).

% ATP.lambda_313
tff(fact_8093_ATP_Olambda__314,axiom,
    ! [C: $tType,B: $tType,A: $tType,Uu: fun(sum_sum(A,B),C),Uua: A] : aa(A,C,aTP_Lamp_atq(fun(sum_sum(A,B),C),fun(A,C),Uu),Uua) = aa(sum_sum(A,B),C,Uu,aa(A,sum_sum(A,B),sum_Inl(A,B),Uua)) ).

% ATP.lambda_314
tff(fact_8094_ATP_Olambda__315,axiom,
    ! [Uu: fun(nat,$o),Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_vf(fun(nat,$o),fun(nat,$o),Uu),Uua)
    <=> aa(nat,$o,Uu,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_315
tff(fact_8095_ATP_Olambda__316,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_fl(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_316
tff(fact_8096_ATP_Olambda__317,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_dq(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_317
tff(fact_8097_ATP_Olambda__318,axiom,
    ! [A: $tType,Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_abd(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ).

% ATP.lambda_318
tff(fact_8098_ATP_Olambda__319,axiom,
    ! [A: $tType,C: $tType,Uu: C,Uua: fun(C,set(set(A)))] : aa(fun(C,set(set(A))),set(set(A)),aTP_Lamp_amt(C,fun(fun(C,set(set(A))),set(set(A))),Uu),Uua) = aa(C,set(set(A)),Uua,Uu) ).

% ATP.lambda_319
tff(fact_8099_ATP_Olambda__320,axiom,
    ! [A: $tType,B: $tType,Uu: B,Uua: fun(B,set(A))] : aa(fun(B,set(A)),set(A),aTP_Lamp_uf(B,fun(fun(B,set(A)),set(A)),Uu),Uua) = aa(B,set(A),Uua,Uu) ).

% ATP.lambda_320
tff(fact_8100_ATP_Olambda__321,axiom,
    ! [A: $tType,B: $tType] :
      ( complete_Sup(A)
     => ! [Uu: B,Uua: fun(B,A)] : aa(fun(B,A),A,aTP_Lamp_nn(B,fun(fun(B,A),A),Uu),Uua) = aa(B,A,Uua,Uu) ) ).

% ATP.lambda_321
tff(fact_8101_ATP_Olambda__322,axiom,
    ! [A: $tType,B: $tType] :
      ( complete_Inf(A)
     => ! [Uu: B,Uua: fun(B,A)] : aa(fun(B,A),A,aTP_Lamp_no(B,fun(fun(B,A),A),Uu),Uua) = aa(B,A,Uua,Uu) ) ).

% ATP.lambda_322
tff(fact_8102_ATP_Olambda__323,axiom,
    ! [A: $tType,Uu: A,Uua: fun(A,nat)] : aa(fun(A,nat),nat,aTP_Lamp_amb(A,fun(fun(A,nat),nat),Uu),Uua) = aa(A,nat,Uua,Uu) ).

% ATP.lambda_323
tff(fact_8103_ATP_Olambda__324,axiom,
    ! [B: $tType,A: $tType] :
      ( complete_Sup(B)
     => ! [Uu: A,Uua: fun(A,B)] : aa(fun(A,B),B,aTP_Lamp_ns(A,fun(fun(A,B),B),Uu),Uua) = aa(A,B,Uua,Uu) ) ).

% ATP.lambda_324
tff(fact_8104_ATP_Olambda__325,axiom,
    ! [B: $tType,A: $tType] :
      ( complete_Inf(B)
     => ! [Uu: A,Uua: fun(A,B)] : aa(fun(A,B),B,aTP_Lamp_nt(A,fun(fun(A,B),B),Uu),Uua) = aa(A,B,Uua,Uu) ) ).

% ATP.lambda_325
tff(fact_8105_ATP_Olambda__326,axiom,
    ! [A: $tType,Uu: fun(product_unit,A),Uua: product_unit] : aa(product_unit,A,aTP_Lamp_att(fun(product_unit,A),fun(product_unit,A),Uu),Uua) = aa(product_unit,A,Uu,product_Unity) ).

% ATP.lambda_326
tff(fact_8106_ATP_Olambda__327,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A,Uua: option(A)] : aa(option(A),option(A),aa(A,fun(option(A),option(A)),aTP_Lamp_uj(A,fun(option(A),option(A))),Uu),Uua) = aa(A,option(A),some(A),case_option(A,A,Uu,aa(A,fun(A,A),ord_min(A),Uu),Uua)) ) ).

% ATP.lambda_327
tff(fact_8107_ATP_Olambda__328,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A,Uua: option(A)] : aa(option(A),option(A),aa(A,fun(option(A),option(A)),aTP_Lamp_ub(A,fun(option(A),option(A))),Uu),Uua) = aa(A,option(A),some(A),case_option(A,A,Uu,aa(A,fun(A,A),ord_max(A),Uu),Uua)) ) ).

% ATP.lambda_328
tff(fact_8108_ATP_Olambda__329,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_uh(A,fun(option(A),option(A))),Uu),Uua) = aa(A,option(A),some(A),case_option(A,A,Uu,aa(A,fun(A,A),sup_sup(A),Uu),Uua)) ) ).

% ATP.lambda_329
tff(fact_8109_ATP_Olambda__330,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_ui(A,fun(option(A),option(A))),Uu),Uua) = aa(A,option(A),some(A),case_option(A,A,Uu,aa(A,fun(A,A),inf_inf(A),Uu),Uua)) ) ).

% ATP.lambda_330
tff(fact_8110_ATP_Olambda__331,axiom,
    ! [A: $tType,Uu: multiset(A),Uua: option(multiset(A))] : aa(option(multiset(A)),option(multiset(A)),aa(multiset(A),fun(option(multiset(A)),option(multiset(A))),aTP_Lamp_ary(multiset(A),fun(option(multiset(A)),option(multiset(A)))),Uu),Uua) = aa(multiset(A),option(multiset(A)),some(multiset(A)),case_option(multiset(A),multiset(A),Uu,aa(multiset(A),fun(multiset(A),multiset(A)),union_mset(A),Uu),Uua)) ).

% ATP.lambda_331
tff(fact_8111_ATP_Olambda__332,axiom,
    ! [A: $tType,Uu: multiset(A),Uua: option(multiset(A))] : aa(option(multiset(A)),option(multiset(A)),aa(multiset(A),fun(option(multiset(A)),option(multiset(A))),aTP_Lamp_asb(multiset(A),fun(option(multiset(A)),option(multiset(A)))),Uu),Uua) = aa(multiset(A),option(multiset(A)),some(multiset(A)),case_option(multiset(A),multiset(A),Uu,aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),Uu),Uua)) ).

% ATP.lambda_332
tff(fact_8112_ATP_Olambda__333,axiom,
    ! [Uu: nat,Uua: num] : aa(num,option(num),aTP_Lamp_tj(nat,fun(num,option(num)),Uu),Uua) = aa(num,option(num),some(num),case_option(num,num,one2,bit1,bit_take_bit_num(Uu,Uua))) ).

% ATP.lambda_333
tff(fact_8113_ATP_Olambda__334,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: fun(C,fun(A,C)),Uua: C] : aa(C,fun(product_prod(B,A),C),aTP_Lamp_adz(fun(C,fun(A,C)),fun(C,fun(product_prod(B,A),C)),Uu),Uua) = aa(fun(B,fun(A,C)),fun(product_prod(B,A),C),product_case_prod(B,A,C),aa(C,fun(B,fun(A,C)),aTP_Lamp_ady(fun(C,fun(A,C)),fun(C,fun(B,fun(A,C))),Uu),Uua)) ).

% ATP.lambda_334
tff(fact_8114_ATP_Olambda__335,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Uu: fun(A,B),Uua: list(A)] : aa(list(A),fun(product_prod(list(A),list(A)),list(A)),aTP_Lamp_yl(fun(A,B),fun(list(A),fun(product_prod(list(A),list(A)),list(A))),Uu),Uua) = aa(fun(list(A),fun(list(A),list(A))),fun(product_prod(list(A),list(A)),list(A)),product_case_prod(list(A),list(A),list(A)),aa(list(A),fun(list(A),fun(list(A),list(A))),aTP_Lamp_yk(fun(A,B),fun(list(A),fun(list(A),fun(list(A),list(A)))),Uu),Uua)) ) ).

% ATP.lambda_335
tff(fact_8115_ATP_Olambda__336,axiom,
    ! [A: $tType,Uu: list(A),Uua: list(A)] : aa(list(A),fun(product_prod(A,list(A)),option($o)),aTP_Lamp_yi(list(A),fun(list(A),fun(product_prod(A,list(A)),option($o))),Uu),Uua) = aa(fun(A,fun(list(A),option($o))),fun(product_prod(A,list(A)),option($o)),product_case_prod(A,list(A),option($o)),aa(list(A),fun(A,fun(list(A),option($o))),aTP_Lamp_yh(list(A),fun(list(A),fun(A,fun(list(A),option($o)))),Uu),Uua)) ).

% ATP.lambda_336
tff(fact_8116_ATP_Olambda__337,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] : aa(list(A),fun(product_prod(A,list(A)),option(product_prod(list(A),product_prod(A,list(A))))),aTP_Lamp_yc(A,fun(list(A),fun(product_prod(A,list(A)),option(product_prod(list(A),product_prod(A,list(A)))))),Uu),Uua) = aa(fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A)))))),fun(product_prod(A,list(A)),option(product_prod(list(A),product_prod(A,list(A))))),product_case_prod(A,list(A),option(product_prod(list(A),product_prod(A,list(A))))),aa(list(A),fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A)))))),aTP_Lamp_yb(A,fun(list(A),fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A))))))),Uu),Uua)) ).

% ATP.lambda_337
tff(fact_8117_ATP_Olambda__338,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,fun(product_prod(nat,nat),$o),aa(nat,fun(nat,fun(product_prod(nat,nat),$o)),aTP_Lamp_ws(nat,fun(nat,fun(product_prod(nat,nat),$o))),Uu),Uua) = aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aa(nat,fun(nat,fun(nat,$o)),aTP_Lamp_wr(nat,fun(nat,fun(nat,fun(nat,$o))),Uu),Uua)) ).

% ATP.lambda_338
tff(fact_8118_ATP_Olambda__339,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_pv(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_pu(nat,fun(nat,fun(nat,fun(nat,product_prod(nat,nat)))),Uu),Uua)) ).

% ATP.lambda_339
tff(fact_8119_ATP_Olambda__340,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_pt(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_ps(nat,fun(nat,fun(nat,fun(nat,product_prod(nat,nat)))),Uu),Uua)) ).

% ATP.lambda_340
tff(fact_8120_ATP_Olambda__341,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,fun(product_prod(nat,nat),$o),aa(nat,fun(nat,fun(product_prod(nat,nat),$o)),aTP_Lamp_pr(nat,fun(nat,fun(product_prod(nat,nat),$o))),Uu),Uua) = aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aa(nat,fun(nat,fun(nat,$o)),aTP_Lamp_pq(nat,fun(nat,fun(nat,fun(nat,$o))),Uu),Uua)) ).

% ATP.lambda_341
tff(fact_8121_ATP_Olambda__342,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,fun(product_prod(nat,nat),$o),aa(nat,fun(nat,fun(product_prod(nat,nat),$o)),aTP_Lamp_pp(nat,fun(nat,fun(product_prod(nat,nat),$o))),Uu),Uua) = aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aa(nat,fun(nat,fun(nat,$o)),aTP_Lamp_po(nat,fun(nat,fun(nat,fun(nat,$o))),Uu),Uua)) ).

% ATP.lambda_342
tff(fact_8122_ATP_Olambda__343,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_pb(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_pa(nat,fun(nat,fun(nat,fun(nat,product_prod(nat,nat)))),Uu),Uua)) ).

% ATP.lambda_343
tff(fact_8123_ATP_Olambda__344,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,fun(B,$o)),Uua: fun(A,$o)] : aa(fun(A,$o),set(A),aTP_Lamp_arl(fun(A,fun(B,$o)),fun(fun(A,$o),set(A)),Uu),Uua) = aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_ark(fun(A,fun(B,$o)),fun(fun(A,$o),fun(A,$o)),Uu),Uua)) ).

% ATP.lambda_344
tff(fact_8124_ATP_Olambda__345,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,fun(B,$o)),Uua: B] : aa(B,set(A),aTP_Lamp_afi(fun(A,fun(B,$o)),fun(B,set(A)),Uu),Uua) = aa(fun(A,$o),set(A),collect(A),aa(B,fun(A,$o),aTP_Lamp_afh(fun(A,fun(B,$o)),fun(B,fun(A,$o)),Uu),Uua)) ).

% ATP.lambda_345
tff(fact_8125_ATP_Olambda__346,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_lw(fun(C,fun(B,A)),fun(fun(B,C),A),Uu),Uua) = aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,aa(fun(B,C),fun(B,A),aTP_Lamp_lt(fun(C,fun(B,A)),fun(fun(B,C),fun(B,A)),Uu),Uua)),top_top(set(B)))) ) ).

% ATP.lambda_346
tff(fact_8126_ATP_Olambda__347,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: fun(C,fun(B,A)),Uua: B] : aa(B,A,aTP_Lamp_ls(fun(C,fun(B,A)),fun(B,A),Uu),Uua) = aa(set(A),A,complete_Sup_Sup(A),aa(set(C),set(A),image2(C,A,aa(B,fun(C,A),aTP_Lamp_lr(fun(C,fun(B,A)),fun(B,fun(C,A)),Uu),Uua)),top_top(set(C)))) ) ).

% ATP.lambda_347
tff(fact_8127_ATP_Olambda__348,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_lu(fun(C,fun(B,A)),fun(fun(B,C),A),Uu),Uua) = aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,aa(fun(B,C),fun(B,A),aTP_Lamp_lt(fun(C,fun(B,A)),fun(fun(B,C),fun(B,A)),Uu),Uua)),top_top(set(B)))) ) ).

% ATP.lambda_348
tff(fact_8128_ATP_Olambda__349,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: fun(C,fun(B,A)),Uua: B] : aa(B,A,aTP_Lamp_lv(fun(C,fun(B,A)),fun(B,A),Uu),Uua) = aa(set(A),A,complete_Inf_Inf(A),aa(set(C),set(A),image2(C,A,aa(B,fun(C,A),aTP_Lamp_lr(fun(C,fun(B,A)),fun(B,fun(C,A)),Uu),Uua)),top_top(set(C)))) ) ).

% ATP.lambda_349
tff(fact_8129_ATP_Olambda__350,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,option(A)),Uua: B] :
      ( aa(B,$o,aTP_Lamp_aag(fun(B,option(A)),fun(B,$o),Uu),Uua)
    <=> ( aa(B,option(A),Uu,Uua) != none(A) ) ) ).

% ATP.lambda_350
tff(fact_8130_ATP_Olambda__351,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,option(B)),Uua: A] :
      ( aa(A,$o,aTP_Lamp_adf(fun(A,option(B)),fun(A,$o),Uu),Uua)
    <=> ( aa(A,option(B),Uu,Uua) != none(B) ) ) ).

% ATP.lambda_351
tff(fact_8131_ATP_Olambda__352,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,set(B)),Uua: A] : aa(A,set(product_prod(A,B)),aTP_Lamp_sc(fun(A,set(B)),fun(A,set(product_prod(A,B))),Uu),Uua) = aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),aa(set(B),set(set(product_prod(A,B))),image2(B,set(product_prod(A,B)),aTP_Lamp_sb(A,fun(B,set(product_prod(A,B))),Uua)),aa(A,set(B),Uu,Uua))) ).

% ATP.lambda_352
tff(fact_8132_ATP_Olambda__353,axiom,
    ! [B: $tType,A: $tType,Uu: set(set(old_node(A,B))),Uua: set(old_node(A,B))] : aa(set(old_node(A,B)),set(set(old_node(A,B))),aTP_Lamp_avl(set(set(old_node(A,B))),fun(set(old_node(A,B)),set(set(old_node(A,B)))),Uu),Uua) = aa(set(set(set(old_node(A,B)))),set(set(old_node(A,B))),complete_Sup_Sup(set(set(old_node(A,B)))),aa(set(set(old_node(A,B))),set(set(set(old_node(A,B)))),image2(set(old_node(A,B)),set(set(old_node(A,B))),aTP_Lamp_avk(set(old_node(A,B)),fun(set(old_node(A,B)),set(set(old_node(A,B)))),Uua)),Uu)) ).

% ATP.lambda_353
tff(fact_8133_ATP_Olambda__354,axiom,
    ! [A: $tType,Uu: set(multiset(A)),Uua: A] : aa(A,nat,aTP_Lamp_asd(set(multiset(A)),fun(A,nat),Uu),Uua) = aa(set(nat),nat,complete_Sup_Sup(nat),aa(set(multiset(A)),set(nat),image2(multiset(A),nat,aTP_Lamp_apu(A,fun(multiset(A),nat),Uua)),Uu)) ).

% ATP.lambda_354
tff(fact_8134_ATP_Olambda__355,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_aaq(list(product_prod(C,B)),fun(product_prod(A,C),list(product_prod(A,B))),Uu),Uua) = concat(product_prod(A,B),aa(list(product_prod(C,B)),list(list(product_prod(A,B))),map(product_prod(C,B),list(product_prod(A,B)),aTP_Lamp_aap(product_prod(A,C),fun(product_prod(C,B),list(product_prod(A,B))),Uua)),Uu)) ).

% ATP.lambda_355
tff(fact_8135_ATP_Olambda__356,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_sr(nat,fun(nat,$o),Uu),Uua)
    <=> ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uu),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Uua))) ) ).

% ATP.lambda_356
tff(fact_8136_ATP_Olambda__357,axiom,
    ! [A: $tType,Uu: set(set(A)),Uua: A] :
      ( aa(A,$o,aTP_Lamp_nu(set(set(A)),fun(A,$o),Uu),Uua)
    <=> aa(set($o),$o,complete_Sup_Sup($o),aa(set(set(A)),set($o),image2(set(A),$o,member(A,Uua)),Uu)) ) ).

% ATP.lambda_357
tff(fact_8137_ATP_Olambda__358,axiom,
    ! [A: $tType,Uu: set(set(A)),Uua: A] :
      ( aa(A,$o,aTP_Lamp_nq(set(set(A)),fun(A,$o),Uu),Uua)
    <=> aa(set($o),$o,complete_Inf_Inf($o),aa(set(set(A)),set($o),image2(set(A),$o,member(A,Uua)),Uu)) ) ).

% ATP.lambda_358
tff(fact_8138_ATP_Olambda__359,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_atv(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,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,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_359
tff(fact_8139_ATP_Olambda__360,axiom,
    ! [A: $tType,Uu: fun(nat,set(A)),Uua: nat] : aa(nat,set(A),aTP_Lamp_mz(fun(nat,set(A)),fun(nat,set(A)),Uu),Uua) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),Uu),set_or7035219750837199246ssThan(nat,zero_zero(nat),Uua))) ).

% ATP.lambda_360
tff(fact_8140_ATP_Olambda__361,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_ez(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_361
tff(fact_8141_ATP_Olambda__362,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: A,Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_ed(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_362
tff(fact_8142_ATP_Olambda__363,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_auc(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_363
tff(fact_8143_ATP_Olambda__364,axiom,
    ! [A: $tType,Uu: list(A),Uua: A] :
      ( aa(A,$o,aTP_Lamp_zg(list(A),fun(A,$o),Uu),Uua)
    <=> ~ aa(set(A),$o,member(A,Uua),aa(list(A),set(A),set2(A),Uu)) ) ).

% ATP.lambda_364
tff(fact_8144_ATP_Olambda__365,axiom,
    ! [A: $tType,Uu: list(A),Uua: A] :
      ( aa(A,$o,aTP_Lamp_akq(list(A),fun(A,$o),Uu),Uua)
    <=> ( Uua != last(A,Uu) ) ) ).

% ATP.lambda_365
tff(fact_8145_ATP_Olambda__366,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,set(A)),Uua: set(B)] : aa(set(B),set(A),aTP_Lamp_kr(fun(B,set(A)),fun(set(B),set(A)),Uu),Uua) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),Uu),Uua)) ).

% ATP.lambda_366
tff(fact_8146_ATP_Olambda__367,axiom,
    ! [A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: fun(B,A),Uua: set(B)] : aa(set(B),A,aTP_Lamp_agu(fun(B,A),fun(set(B),A),Uu),Uua) = aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,Uu),Uua)) ) ).

% ATP.lambda_367
tff(fact_8147_ATP_Olambda__368,axiom,
    ! [D: $tType,B: $tType,Uu: set(B),Uua: fun(B,set(D))] : aa(fun(B,set(D)),set(D),aa(set(B),fun(fun(B,set(D)),set(D)),aTP_Lamp_aqx(set(B),fun(fun(B,set(D)),set(D))),Uu),Uua) = aa(set(set(D)),set(D),complete_Sup_Sup(set(D)),aa(set(B),set(set(D)),image2(B,set(D),Uua),Uu)) ).

% ATP.lambda_368
tff(fact_8148_ATP_Olambda__369,axiom,
    ! [C: $tType,B: $tType] :
      ( complete_Sup(C)
     => ! [Uu: set(B),Uua: fun(B,C)] : aa(fun(B,C),C,aa(set(B),fun(fun(B,C),C),aTP_Lamp_aqu(set(B),fun(fun(B,C),C)),Uu),Uua) = aa(set(C),C,complete_Sup_Sup(C),aa(set(B),set(C),image2(B,C,Uua),Uu)) ) ).

% ATP.lambda_369
tff(fact_8149_ATP_Olambda__370,axiom,
    ! [A: $tType,Uu: set(A),Uua: fun(A,$o)] :
      ( aa(fun(A,$o),$o,aa(set(A),fun(fun(A,$o),$o),aTP_Lamp_ahv(set(A),fun(fun(A,$o),$o)),Uu),Uua)
    <=> aa(set($o),$o,complete_Sup_Sup($o),aa(set(A),set($o),image2(A,$o,Uua),Uu)) ) ).

% ATP.lambda_370
tff(fact_8150_ATP_Olambda__371,axiom,
    ! [C: $tType,A: $tType,Uu: set(A),Uua: fun(A,set(C))] : aa(fun(A,set(C)),set(C),aa(set(A),fun(fun(A,set(C)),set(C)),aTP_Lamp_aqw(set(A),fun(fun(A,set(C)),set(C))),Uu),Uua) = aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(A),set(set(C)),image2(A,set(C),Uua),Uu)) ).

% ATP.lambda_371
tff(fact_8151_ATP_Olambda__372,axiom,
    ! [C: $tType,A: $tType] :
      ( complete_Sup(C)
     => ! [Uu: set(A),Uua: fun(A,C)] : aa(fun(A,C),C,aa(set(A),fun(fun(A,C),C),aTP_Lamp_aqt(set(A),fun(fun(A,C),C)),Uu),Uua) = aa(set(C),C,complete_Sup_Sup(C),aa(set(A),set(C),image2(A,C,Uua),Uu)) ) ).

% ATP.lambda_372
tff(fact_8152_ATP_Olambda__373,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,set(A)),Uua: set(B)] : aa(set(B),set(A),aTP_Lamp_kt(fun(B,set(A)),fun(set(B),set(A)),Uu),Uua) = aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),Uu),Uua)) ).

% ATP.lambda_373
tff(fact_8153_ATP_Olambda__374,axiom,
    ! [A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: fun(B,A),Uua: set(B)] : aa(set(B),A,aTP_Lamp_agt(fun(B,A),fun(set(B),A),Uu),Uua) = aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,Uu),Uua)) ) ).

% ATP.lambda_374
tff(fact_8154_ATP_Olambda__375,axiom,
    ! [C: $tType,B: $tType] :
      ( complete_Inf(C)
     => ! [Uu: set(B),Uua: fun(B,C)] : aa(fun(B,C),C,aa(set(B),fun(fun(B,C),C),aTP_Lamp_aqs(set(B),fun(fun(B,C),C)),Uu),Uua) = aa(set(C),C,complete_Inf_Inf(C),aa(set(B),set(C),image2(B,C,Uua),Uu)) ) ).

% ATP.lambda_375
tff(fact_8155_ATP_Olambda__376,axiom,
    ! [A: $tType,Uu: set(A),Uua: fun(A,$o)] :
      ( aa(fun(A,$o),$o,aa(set(A),fun(fun(A,$o),$o),aTP_Lamp_ael(set(A),fun(fun(A,$o),$o)),Uu),Uua)
    <=> aa(set($o),$o,complete_Inf_Inf($o),aa(set(A),set($o),image2(A,$o,Uua),Uu)) ) ).

% ATP.lambda_376
tff(fact_8156_ATP_Olambda__377,axiom,
    ! [C: $tType,A: $tType] :
      ( complete_Inf(C)
     => ! [Uu: set(A),Uua: fun(A,C)] : aa(fun(A,C),C,aa(set(A),fun(fun(A,C),C),aTP_Lamp_aqr(set(A),fun(fun(A,C),C)),Uu),Uua) = aa(set(C),C,complete_Inf_Inf(C),aa(set(A),set(C),image2(A,C,Uua),Uu)) ) ).

% ATP.lambda_377
tff(fact_8157_ATP_Olambda__378,axiom,
    ! [A: $tType] :
      ( sup(A)
     => ! [Uu: A,Uua: A] : aa(A,option(A),aTP_Lamp_tb(A,fun(A,option(A)),Uu),Uua) = aa(A,option(A),some(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Uu),Uua)) ) ).

% ATP.lambda_378
tff(fact_8158_ATP_Olambda__379,axiom,
    ! [A: $tType] :
      ( inf(A)
     => ! [Uu: A,Uua: A] : aa(A,option(A),aTP_Lamp_tg(A,fun(A,option(A)),Uu),Uua) = aa(A,option(A),some(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Uu),Uua)) ) ).

% ATP.lambda_379
tff(fact_8159_ATP_Olambda__380,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_atx(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_380
tff(fact_8160_ATP_Olambda__381,axiom,
    ! [A: $tType,Uu: pred(A),Uua: pred(A)] : aa(pred(A),fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))),aTP_Lamp_awa(pred(A),fun(pred(A),fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural)))),Uu),Uua) = aa(pred(A),fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))),product_Pair(pred(A),product_prod(code_natural,code_natural)),aa(pred(A),pred(A),aa(pred(A),fun(pred(A),pred(A)),sup_sup(pred(A)),Uu),Uua)) ).

% ATP.lambda_381
tff(fact_8161_ATP_Olambda__382,axiom,
    ! [C: $tType,A: $tType,B: $tType,Uu: A,Uua: B] : aa(B,fun(C,product_prod(product_prod(A,B),C)),aTP_Lamp_anp(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_382
tff(fact_8162_ATP_Olambda__383,axiom,
    ! [A: $tType,Uu: list(product_prod(code_natural,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_aub(list(product_prod(code_natural,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(code_natural,A,pick(A,Uu),Uua)) ).

% ATP.lambda_383
tff(fact_8163_ATP_Olambda__384,axiom,
    ! [B: $tType,A: $tType,Uu: list(product_prod(A,B)),Uua: A] : aa(A,B,aTP_Lamp_aje(list(product_prod(A,B)),fun(A,B),Uu),Uua) = aa(option(B),B,the2(B),aa(A,option(B),map_of(A,B,Uu),Uua)) ).

% ATP.lambda_384
tff(fact_8164_ATP_Olambda__385,axiom,
    ! [B: $tType,A: $tType,Uu: B,Uua: A] : aa(A,fun(set(product_prod(B,A)),set(product_prod(B,A))),aTP_Lamp_oj(B,fun(A,fun(set(product_prod(B,A)),set(product_prod(B,A)))),Uu),Uua) = aa(product_prod(B,A),fun(set(product_prod(B,A)),set(product_prod(B,A))),insert2(product_prod(B,A)),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Uu),Uua)) ).

% ATP.lambda_385
tff(fact_8165_ATP_Olambda__386,axiom,
    ! [A: $tType,B: $tType,Uu: A,Uua: B] : aa(B,fun(set(product_prod(A,B)),set(product_prod(A,B))),aTP_Lamp_sd(A,fun(B,fun(set(product_prod(A,B)),set(product_prod(A,B)))),Uu),Uua) = aa(product_prod(A,B),fun(set(product_prod(A,B)),set(product_prod(A,B))),insert2(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uu),Uua)) ).

% ATP.lambda_386
tff(fact_8166_ATP_Olambda__387,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_ud(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_387
tff(fact_8167_ATP_Olambda__388,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_uc(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_388
tff(fact_8168_ATP_Olambda__389,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_jp(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_389
tff(fact_8169_ATP_Olambda__390,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_jo(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_390
tff(fact_8170_ATP_Olambda__391,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_zl(A,fun(A,$o),Uu),Uua)
        <=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uu),Uua) ) ) ).

% ATP.lambda_391
tff(fact_8171_ATP_Olambda__392,axiom,
    ! [A: $tType,Uu: set(A),Uua: A] :
      ( aa(A,$o,aTP_Lamp_ai(set(A),fun(A,$o),Uu),Uua)
    <=> ~ aa(set(A),$o,member(A,Uua),Uu) ) ).

% ATP.lambda_392
tff(fact_8172_ATP_Olambda__393,axiom,
    ! [A: $tType,Uu: A,Uua: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_za(A,fun(A,$o)),Uu),Uua)
    <=> ( Uu != Uua ) ) ).

% ATP.lambda_393
tff(fact_8173_ATP_Olambda__394,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_top(A) )
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_alc(A,fun(A,$o),Uu),Uua)
        <=> ( Uua != Uu ) ) ) ).

% ATP.lambda_394
tff(fact_8174_ATP_Olambda__395,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_bot(A) )
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_akf(A,fun(A,$o),Uu),Uua)
        <=> ( Uua != Uu ) ) ) ).

% ATP.lambda_395
tff(fact_8175_ATP_Olambda__396,axiom,
    ! [A: $tType,Uu: A,Uua: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_ze(A,fun(A,$o)),Uu),Uua)
    <=> ( Uua != Uu ) ) ).

% ATP.lambda_396
tff(fact_8176_ATP_Olambda__397,axiom,
    ! [A: $tType,Uu: A,Uua: A] : aa(A,set(A),aTP_Lamp_sg(A,fun(A,set(A)),Uu),Uua) = aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Uu),bot_bot(set(A)))) ).

% ATP.lambda_397
tff(fact_8177_ATP_Olambda__398,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: fun(A,fun(A,A)),Uua: A] : aa(A,A,aTP_Lamp_ahl(fun(A,fun(A,A)),fun(A,A),Uu),Uua) = complete_lattice_lfp(A,aa(A,fun(A,A),Uu,Uua)) ) ).

% ATP.lambda_398
tff(fact_8178_ATP_Olambda__399,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: fun(A,fun(A,A)),Uua: A] : aa(A,A,aTP_Lamp_alo(fun(A,fun(A,A)),fun(A,A),Uu),Uua) = complete_lattice_gfp(A,aa(A,fun(A,A),Uu,Uua)) ) ).

% ATP.lambda_399
tff(fact_8179_ATP_Olambda__400,axiom,
    ! [C: $tType,B: $tType,Uu: fun(B,set(C)),Uua: B] : aa(B,set(product_prod(C,C)),aTP_Lamp_anf(fun(B,set(C)),fun(B,set(product_prod(C,C))),Uu),Uua) = bNF_Ca6860139660246222851ard_of(C,aa(B,set(C),Uu,Uua)) ).

% ATP.lambda_400
tff(fact_8180_ATP_Olambda__401,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,set(B)),Uua: A] : aa(A,set(product_prod(B,B)),aTP_Lamp_anb(fun(A,set(B)),fun(A,set(product_prod(B,B))),Uu),Uua) = bNF_Ca6860139660246222851ard_of(B,aa(A,set(B),Uu,Uua)) ).

% ATP.lambda_401
tff(fact_8181_ATP_Olambda__402,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(B)
     => ! [Uu: fun(A,$o),Uua: A] : aa(A,B,aTP_Lamp_jb(fun(A,$o),fun(A,B),Uu),Uua) = aa($o,B,zero_neq_one_of_bool(B),aa(A,$o,Uu,Uua)) ) ).

% ATP.lambda_402
tff(fact_8182_ATP_Olambda__403,axiom,
    ! [A: $tType,B: $tType] :
      ( field_char_0(A)
     => ! [Uu: fun(B,rat),Uua: B] : aa(B,A,aTP_Lamp_wq(fun(B,rat),fun(B,A),Uu),Uua) = aa(rat,A,field_char_0_of_rat(A),aa(B,rat,Uu,Uua)) ) ).

% ATP.lambda_403
tff(fact_8183_ATP_Olambda__404,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: fun(B,nat),Uua: B] : aa(B,A,aTP_Lamp_fc(fun(B,nat),fun(B,A),Uu),Uua) = aa(nat,A,semiring_1_of_nat(A),aa(B,nat,Uu,Uua)) ) ).

% ATP.lambda_404
tff(fact_8184_ATP_Olambda__405,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_1(A)
     => ! [Uu: fun(B,nat),Uua: B] : aa(B,A,aTP_Lamp_ea(fun(B,nat),fun(B,A),Uu),Uua) = aa(nat,A,semiring_1_of_nat(A),aa(B,nat,Uu,Uua)) ) ).

% ATP.lambda_405
tff(fact_8185_ATP_Olambda__406,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: A] : aa(A,int,aTP_Lamp_eb(fun(A,nat),fun(A,int),Uu),Uua) = aa(nat,int,semiring_1_of_nat(int),aa(A,nat,Uu,Uua)) ).

% ATP.lambda_406
tff(fact_8186_ATP_Olambda__407,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,set(A)),Uua: B] : aa(B,set(A),aTP_Lamp_kk(fun(B,set(A)),fun(B,set(A)),Uu),Uua) = aa(set(A),set(A),uminus_uminus(set(A)),aa(B,set(A),Uu,Uua)) ).

% ATP.lambda_407
tff(fact_8187_ATP_Olambda__408,axiom,
    ! [A: $tType,B: $tType] :
      ( comple489889107523837845lgebra(A)
     => ! [Uu: fun(B,A),Uua: B] : aa(B,A,aTP_Lamp_lp(fun(B,A),fun(B,A),Uu),Uua) = aa(A,A,uminus_uminus(A),aa(B,A,Uu,Uua)) ) ).

% ATP.lambda_408
tff(fact_8188_ATP_Olambda__409,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(B,A),Uua: B] : aa(B,A,aTP_Lamp_de(fun(B,A),fun(B,A),Uu),Uua) = aa(A,A,uminus_uminus(A),aa(B,A,Uu,Uua)) ) ).

% ATP.lambda_409
tff(fact_8189_ATP_Olambda__410,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,set(product_prod(B,B))),Uua: A] : aa(A,set(product_prod(B,B)),aTP_Lamp_qd(fun(A,set(product_prod(B,B))),fun(A,set(product_prod(B,B))),Uu),Uua) = transitive_trancl(B,aa(A,set(product_prod(B,B)),Uu,Uua)) ).

% ATP.lambda_410
tff(fact_8190_ATP_Olambda__411,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_fx(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_411
tff(fact_8191_ATP_Olambda__412,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: fun(B,int),Uua: B] : aa(B,A,aTP_Lamp_fd(fun(B,int),fun(B,A),Uu),Uua) = aa(int,A,ring_1_of_int(A),aa(B,int,Uu,Uua)) ) ).

% ATP.lambda_412
tff(fact_8192_ATP_Olambda__413,axiom,
    ! [A: $tType,B: $tType] :
      ( ring_1(A)
     => ! [Uu: fun(B,int),Uua: B] : aa(B,A,aTP_Lamp_cp(fun(B,int),fun(B,A),Uu),Uua) = aa(int,A,ring_1_of_int(A),aa(B,int,Uu,Uua)) ) ).

% ATP.lambda_413
tff(fact_8193_ATP_Olambda__414,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_du(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_414
tff(fact_8194_ATP_Olambda__415,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [Uu: fun(B,A),Uua: B] : aa(B,A,aTP_Lamp_cs(fun(B,A),fun(B,A),Uu),Uua) = aa(A,A,abs_abs(A),aa(B,A,Uu,Uua)) ) ).

% ATP.lambda_415
tff(fact_8195_ATP_Olambda__416,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_field(A)
     => ! [Uu: fun(B,A),Uua: B] : aa(B,A,aTP_Lamp_fk(fun(B,A),fun(B,A),Uu),Uua) = aa(A,A,abs_abs(A),aa(B,A,Uu,Uua)) ) ).

% ATP.lambda_416
tff(fact_8196_ATP_Olambda__417,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: fun(B,A),Uua: B] : aa(B,option(A),aTP_Lamp_td(fun(B,A),fun(B,option(A)),Uu),Uua) = aa(A,option(A),some(A),aa(B,A,Uu,Uua)) ) ).

% ATP.lambda_417
tff(fact_8197_ATP_Olambda__418,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: B] : aa(B,option(A),aTP_Lamp_tr(fun(B,A),fun(B,option(A)),Uu),Uua) = aa(A,option(A),some(A),aa(B,A,Uu,Uua)) ).

% ATP.lambda_418
tff(fact_8198_ATP_Olambda__419,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [Uu: fun(A,B),Uua: A] : aa(A,option(B),aTP_Lamp_sw(fun(A,B),fun(A,option(B)),Uu),Uua) = aa(B,option(B),some(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_419
tff(fact_8199_ATP_Olambda__420,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: A] : aa(A,option(B),aTP_Lamp_add(fun(A,B),fun(A,option(B)),Uu),Uua) = aa(B,option(B),some(B),aa(A,B,Uu,Uua)) ).

% ATP.lambda_420
tff(fact_8200_ATP_Olambda__421,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu: fun(C,set(product_prod(B,A))),Uua: C] : aa(C,set(product_prod(A,B)),aTP_Lamp_aoi(fun(C,set(product_prod(B,A))),fun(C,set(product_prod(A,B))),Uu),Uua) = converse(B,A,aa(C,set(product_prod(B,A)),Uu,Uua)) ).

% ATP.lambda_421
tff(fact_8201_ATP_Olambda__422,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu: fun(C,A),Uua: C] : aa(C,fun(B,product_prod(A,B)),aTP_Lamp_aea(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_422
tff(fact_8202_ATP_Olambda__423,axiom,
    ! [C: $tType,B: $tType,A: $tType,Uu: fun(A,B),Uua: A] : aa(A,fun(C,product_prod(B,C)),aTP_Lamp_acb(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_423
tff(fact_8203_ATP_Olambda__424,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: fun(C,set(old_node(A,B))),Uua: C] : aa(C,set(old_node(A,B)),aTP_Lamp_aul(fun(C,set(old_node(A,B))),fun(C,set(old_node(A,B))),Uu),Uua) = aa(set(old_node(A,B)),set(old_node(A,B)),old_In1(A,B),aa(C,set(old_node(A,B)),Uu,Uua)) ).

% ATP.lambda_424
tff(fact_8204_ATP_Olambda__425,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: fun(C,set(old_node(A,B))),Uua: C] : aa(C,set(old_node(A,B)),aTP_Lamp_aun(fun(C,set(old_node(A,B))),fun(C,set(old_node(A,B))),Uu),Uua) = aa(set(old_node(A,B)),set(old_node(A,B)),old_In0(A,B),aa(C,set(old_node(A,B)),Uu,Uua)) ).

% ATP.lambda_425
tff(fact_8205_ATP_Olambda__426,axiom,
    ! [C: $tType,D: $tType,Uu: fun(D,set(C)),Uua: D] : aa(D,filter(C),aTP_Lamp_aji(fun(D,set(C)),fun(D,filter(C)),Uu),Uua) = principal(C,aa(D,set(C),Uu,Uua)) ).

% ATP.lambda_426
tff(fact_8206_ATP_Olambda__427,axiom,
    ! [D: $tType,C: $tType,Uu: fun(C,set(D)),Uua: C] : aa(C,filter(D),aTP_Lamp_ajm(fun(C,set(D)),fun(C,filter(D)),Uu),Uua) = principal(D,aa(C,set(D),Uu,Uua)) ).

% ATP.lambda_427
tff(fact_8207_ATP_Olambda__428,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,set(A)),Uua: B] : aa(B,filter(A),aTP_Lamp_tu(fun(B,set(A)),fun(B,filter(A)),Uu),Uua) = principal(A,aa(B,set(A),Uu,Uua)) ).

% ATP.lambda_428
tff(fact_8208_ATP_Olambda__429,axiom,
    ! [E: $tType,A: $tType,Uu: fun(A,set(E)),Uua: A] : aa(A,filter(E),aTP_Lamp_ajl(fun(A,set(E)),fun(A,filter(E)),Uu),Uua) = principal(E,aa(A,set(E),Uu,Uua)) ).

% ATP.lambda_429
tff(fact_8209_ATP_Olambda__430,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,set(B)),Uua: A] : aa(A,filter(B),aTP_Lamp_tt(fun(A,set(B)),fun(A,filter(B)),Uu),Uua) = principal(B,aa(A,set(B),Uu,Uua)) ).

% ATP.lambda_430
tff(fact_8210_ATP_Olambda__431,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,set(B)),Uua: A] : aa(A,nat,aTP_Lamp_nd(fun(A,set(B)),fun(A,nat),Uu),Uua) = aa(set(B),nat,finite_card(B),aa(A,set(B),Uu,Uua)) ).

% ATP.lambda_431
tff(fact_8211_ATP_Olambda__432,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,set(product_prod(B,B))),Uua: A] : aa(A,set(B),aTP_Lamp_ane(fun(A,set(product_prod(B,B))),fun(A,set(B)),Uu),Uua) = field2(B,aa(A,set(product_prod(B,B)),Uu,Uua)) ).

% ATP.lambda_432
tff(fact_8212_ATP_Olambda__433,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,list(A)),Uua: B] : aa(B,set(A),aTP_Lamp_xg(fun(B,list(A)),fun(B,set(A)),Uu),Uua) = aa(list(A),set(A),set2(A),aa(B,list(A),Uu,Uua)) ).

% ATP.lambda_433
tff(fact_8213_ATP_Olambda__434,axiom,
    ! [A: $tType,Uu: fun(set(A),fun(A,$o)),Uua: set(A)] : aa(set(A),set(A),aTP_Lamp_alq(fun(set(A),fun(A,$o)),fun(set(A),set(A)),Uu),Uua) = aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),Uu,Uua)) ).

% ATP.lambda_434
tff(fact_8214_ATP_Olambda__435,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,fun(B,$o)),Uua: A] : aa(A,set(B),aTP_Lamp_ri(fun(A,fun(B,$o)),fun(A,set(B)),Uu),Uua) = aa(fun(B,$o),set(B),collect(B),aa(A,fun(B,$o),Uu,Uua)) ).

% ATP.lambda_435
tff(fact_8215_ATP_Olambda__436,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: A] : aa(A,fun(set(B),set(B)),aTP_Lamp_ok(fun(A,B),fun(A,fun(set(B),set(B))),Uu),Uua) = aa(B,fun(set(B),set(B)),insert2(B),aa(A,B,Uu,Uua)) ).

% ATP.lambda_436
tff(fact_8216_ATP_Olambda__437,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,set(A)),Uua: B] : aa(B,set(set(A)),aTP_Lamp_os(fun(B,set(A)),fun(B,set(set(A))),Uu),Uua) = pow2(A,aa(B,set(A),Uu,Uua)) ).

% ATP.lambda_437
tff(fact_8217_ATP_Olambda__438,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: A] : aa(A,nat,aTP_Lamp_jj(fun(A,nat),fun(A,nat),Uu),Uua) = aa(nat,nat,suc,aa(A,nat,Uu,Uua)) ).

% ATP.lambda_438
tff(fact_8218_ATP_Olambda__439,axiom,
    ! [B: $tType,Uu: fun(B,$o),Uua: B] :
      ( aa(B,$o,aTP_Lamp_kd(fun(B,$o),fun(B,$o),Uu),Uua)
    <=> ~ aa(B,$o,Uu,Uua) ) ).

% ATP.lambda_439
tff(fact_8219_ATP_Olambda__440,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: A] :
      ( aa(A,$o,aTP_Lamp_aj(fun(A,$o),fun(A,$o),Uu),Uua)
    <=> ~ aa(A,$o,Uu,Uua) ) ).

% ATP.lambda_440
tff(fact_8220_ATP_Olambda__441,axiom,
    ! [B: $tType,A: $tType] :
      ( finite_finite(A)
     => ! [Uu: fun(B,fun(A,$o)),Uua: B] :
          ( aa(B,$o,aTP_Lamp_ake(fun(B,fun(A,$o)),fun(B,$o),Uu),Uua)
        <=> ! [X_12: A] : aa(A,$o,aa(B,fun(A,$o),Uu,Uua),X_12) ) ) ).

% ATP.lambda_441
tff(fact_8221_ATP_Olambda__442,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,fun(B,$o)),Uua: A] :
      ( aa(A,$o,aTP_Lamp_ain(fun(A,fun(B,$o)),fun(A,$o),Uu),Uua)
    <=> ! [X_12: B] : aa(B,$o,aa(A,fun(B,$o),Uu,Uua),X_12) ) ).

% ATP.lambda_442
tff(fact_8222_ATP_Olambda__443,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: B] : aa(B,fun(A,$o),aTP_Lamp_awi(fun(B,A),fun(B,fun(A,$o)),Uu),Uua) = aa(A,fun(A,$o),fequal(A),aa(B,A,Uu,Uua)) ).

% ATP.lambda_443
tff(fact_8223_ATP_Olambda__444,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,fun(B,$o)),Uua: A] :
      ( aa(A,$o,aTP_Lamp_afv(fun(A,fun(B,$o)),fun(A,$o),Uu),Uua)
    <=> ? [X_12: B] : aa(B,$o,aa(A,fun(B,$o),Uu,Uua),X_12) ) ).

% ATP.lambda_444
tff(fact_8224_ATP_Olambda__445,axiom,
    ! [A: $tType,B: $tType,Uu: fun(nat,sum_sum(B,nat)),Uua: sum_sum(A,nat)] : aa(sum_sum(A,nat),nat,aa(fun(nat,sum_sum(B,nat)),fun(sum_sum(A,nat),nat),aTP_Lamp_aue(fun(nat,sum_sum(B,nat)),fun(sum_sum(A,nat),nat)),Uu),Uua) = ord_Least(nat,aTP_Lamp_aud(fun(nat,sum_sum(B,nat)),fun(nat,$o),Uu)) ).

% ATP.lambda_445
tff(fact_8225_ATP_Olambda__446,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,set(nat),aTP_Lamp_anj(nat,fun(nat,set(nat)),Uu),Uua) = aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_bk(nat,fun(nat,$o)),Uu)) ).

% ATP.lambda_446
tff(fact_8226_ATP_Olambda__447,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A)] : aa(set(A),filter(set(A)),aTP_Lamp_ty(set(A),fun(set(A),filter(set(A))),Uu),Uua) = principal(set(A),aa(fun(set(A),$o),set(set(A)),collect(set(A)),aa(set(A),fun(set(A),$o),aTP_Lamp_tx(set(A),fun(set(A),fun(set(A),$o)),Uu),Uua))) ).

% ATP.lambda_447
tff(fact_8227_ATP_Olambda__448,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,$o),Uua: fun(B,$o)] : aa(fun(B,$o),filter(product_prod(A,B)),aa(fun(A,$o),fun(fun(B,$o),filter(product_prod(A,B))),aTP_Lamp_ans(fun(A,$o),fun(fun(B,$o),filter(product_prod(A,B)))),Uu),Uua) = principal(product_prod(A,B),aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aa(fun(B,$o),fun(A,fun(B,$o)),aTP_Lamp_qx(fun(A,$o),fun(fun(B,$o),fun(A,fun(B,$o))),Uu),Uua)))) ).

% ATP.lambda_448
tff(fact_8228_ATP_Olambda__449,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,fun(A,$o)),Uua: A] :
      ( aa(A,$o,aTP_Lamp_avp(fun(B,fun(A,$o)),fun(A,$o),Uu),Uua)
    <=> ? [X4: B] : aa(A,$o,aa(B,fun(A,$o),Uu,X4),Uua) ) ).

% ATP.lambda_449
tff(fact_8229_ATP_Olambda__450,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,fun(B,$o)),Uua: B] :
      ( aa(B,$o,aTP_Lamp_aop(fun(A,fun(B,$o)),fun(B,$o),Uu),Uua)
    <=> ? [A10: A] : aa(B,$o,aa(A,fun(B,$o),Uu,A10),Uua) ) ).

% ATP.lambda_450
tff(fact_8230_ATP_Olambda__451,axiom,
    ! [A: $tType,Uu: fun(A,fun(A,$o)),Uua: set(A)] :
      ( aa(set(A),$o,aTP_Lamp_awn(fun(A,fun(A,$o)),fun(set(A),$o),Uu),Uua)
    <=> ? [X4: A] :
          ( aa(A,$o,aa(A,fun(A,$o),Uu,X4),X4)
          & ( Uua = aa(fun(A,$o),set(A),collect(A),aa(A,fun(A,$o),Uu,X4)) ) ) ) ).

% ATP.lambda_451
tff(fact_8231_ATP_Olambda__452,axiom,
    ! [A: $tType,Uu: list(A),Uua: A] :
      ( aa(A,$o,aTP_Lamp_agb(list(A),fun(A,$o),Uu),Uua)
    <=> ? [I3: nat] :
          ( ( Uua = aa(nat,A,nth(A,Uu),I3) )
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),Uu)) ) ) ).

% ATP.lambda_452
tff(fact_8232_ATP_Olambda__453,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: set(set(option(A))),Uua: set(option(A))] :
          ( aa(set(option(A)),$o,aTP_Lamp_agw(set(set(option(A))),fun(set(option(A)),$o),Uu),Uua)
        <=> ? [F6: fun(set(option(A)),option(A))] :
              ( ( Uua = aa(set(set(option(A))),set(option(A)),image2(set(option(A)),option(A),F6),Uu) )
              & ! [X4: set(option(A))] :
                  ( aa(set(set(option(A))),$o,member(set(option(A)),X4),Uu)
                 => aa(set(option(A)),$o,member(option(A),aa(set(option(A)),option(A),F6,X4)),X4) ) ) ) ) ).

% ATP.lambda_453
tff(fact_8233_ATP_Olambda__454,axiom,
    ! [B: $tType,Uu: set(set(B)),Uua: set(B)] :
      ( aa(set(B),$o,aTP_Lamp_agv(set(set(B)),fun(set(B),$o),Uu),Uua)
    <=> ? [F6: fun(set(B),B)] :
          ( ( Uua = aa(set(set(B)),set(B),image2(set(B),B,F6),Uu) )
          & ! [X4: set(B)] :
              ( aa(set(set(B)),$o,member(set(B),X4),Uu)
             => aa(set(B),$o,member(B,aa(set(B),B,F6,X4)),X4) ) ) ) ).

% ATP.lambda_454
tff(fact_8234_ATP_Olambda__455,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: set(set(A)),Uua: set(A)] :
          ( aa(set(A),$o,aTP_Lamp_ago(set(set(A)),fun(set(A),$o),Uu),Uua)
        <=> ? [F6: fun(set(A),A)] :
              ( ( Uua = aa(set(set(A)),set(A),image2(set(A),A,F6),Uu) )
              & ! [X4: set(A)] :
                  ( aa(set(set(A)),$o,member(set(A),X4),Uu)
                 => aa(set(A),$o,member(A,aa(set(A),A,F6,X4)),X4) ) ) ) ) ).

% ATP.lambda_455
tff(fact_8235_ATP_Olambda__456,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: set(set(A)),Uua: set(A)] :
          ( aa(set(A),$o,aTP_Lamp_agr(set(set(A)),fun(set(A),$o),Uu),Uua)
        <=> ? [F6: fun(set(A),A)] :
              ( ( Uua = aa(set(set(A)),set(A),image2(set(A),A,F6),Uu) )
              & ! [X4: set(A)] :
                  ( aa(set(set(A)),$o,member(set(A),X4),Uu)
                 => aa(set(A),$o,member(A,aa(set(A),A,F6,X4)),X4) ) ) ) ) ).

% ATP.lambda_456
tff(fact_8236_ATP_Olambda__457,axiom,
    ! [A: $tType] :
      ( finite8700451911770168679attice(A)
     => ! [Uu: set(set(A)),Uua: set(A)] :
          ( aa(set(A),$o,aTP_Lamp_ags(set(set(A)),fun(set(A),$o),Uu),Uua)
        <=> ? [F6: fun(set(A),A)] :
              ( ( Uua = aa(set(set(A)),set(A),image2(set(A),A,F6),Uu) )
              & ! [X4: set(A)] :
                  ( aa(set(set(A)),$o,member(set(A),X4),Uu)
                 => aa(set(A),$o,member(A,aa(set(A),A,F6,X4)),X4) ) ) ) ) ).

% ATP.lambda_457
tff(fact_8237_ATP_Olambda__458,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A)] :
      ( aa(set(A),$o,aTP_Lamp_agx(set(A),fun(set(A),$o),Uu),Uua)
    <=> ? [B9: set(A)] :
          ( ( Uua = aa(set(A),set(A),uminus_uminus(set(A)),B9) )
          & aa(set(set(A)),$o,member(set(A),Uu),pow2(A,B9)) ) ) ).

% ATP.lambda_458
tff(fact_8238_ATP_Olambda__459,axiom,
    ! [A: $tType,Uu: set(filter(A)),Uua: filter(A)] :
      ( aa(filter(A),$o,aTP_Lamp_agz(set(filter(A)),fun(filter(A),$o),Uu),Uua)
    <=> ! [X4: filter(A)] :
          ( aa(set(filter(A)),$o,member(filter(A),X4),Uu)
         => aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),Uua),X4) ) ) ).

% ATP.lambda_459
tff(fact_8239_ATP_Olambda__460,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Uu: set(A),Uua: A] :
          ( aa(A,$o,aTP_Lamp_amm(set(A),fun(A,$o),Uu),Uua)
        <=> ! [X4: A] :
              ( aa(set(A),$o,member(A,X4),Uu)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),X4) ) ) ) ).

% ATP.lambda_460
tff(fact_8240_ATP_Olambda__461,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: set(A),Uua: A] :
          ( aa(A,$o,aTP_Lamp_aft(set(A),fun(A,$o),Uu),Uua)
        <=> ! [X4: A] :
              ( aa(set(A),$o,member(A,X4),Uu)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),X4) ) ) ) ).

% ATP.lambda_461
tff(fact_8241_ATP_Olambda__462,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Uu: set(A),Uua: A] :
          ( aa(A,$o,aTP_Lamp_amf(set(A),fun(A,$o),Uu),Uua)
        <=> ! [X4: A] :
              ( aa(set(A),$o,member(A,X4),Uu)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),Uua) ) ) ) ).

% ATP.lambda_462
tff(fact_8242_ATP_Olambda__463,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: set(A),Uua: A] :
          ( aa(A,$o,aTP_Lamp_afs(set(A),fun(A,$o),Uu),Uua)
        <=> ! [X4: A] :
              ( aa(set(A),$o,member(A,X4),Uu)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),Uua) ) ) ) ).

% ATP.lambda_463
tff(fact_8243_ATP_Olambda__464,axiom,
    ! [A: $tType,Uu: set(multiset(A)),Uua: multiset(A)] :
      ( aa(multiset(A),$o,aTP_Lamp_asn(set(multiset(A)),fun(multiset(A),$o),Uu),Uua)
    <=> ! [X4: multiset(A)] :
          ( aa(set(multiset(A)),$o,member(multiset(A),X4),Uu)
         => aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),Uua),X4) ) ) ).

% ATP.lambda_464
tff(fact_8244_ATP_Olambda__465,axiom,
    ! [A: $tType,Uu: set(multiset(A)),Uua: multiset(A)] :
      ( aa(multiset(A),$o,aTP_Lamp_asg(set(multiset(A)),fun(multiset(A),$o),Uu),Uua)
    <=> ! [X4: multiset(A)] :
          ( aa(set(multiset(A)),$o,member(multiset(A),X4),Uu)
         => aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),X4),Uua) ) ) ).

% ATP.lambda_465
tff(fact_8245_ATP_Olambda__466,axiom,
    ! [Uu: set(nat),Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_avu(set(nat),fun(nat,$o),Uu),Uua)
    <=> ! [X4: nat] :
          ( aa(set(nat),$o,member(nat,X4),Uu)
         => aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Uua),X4) ) ) ).

% ATP.lambda_466
tff(fact_8246_ATP_Olambda__467,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [Uu: set(A),Uua: A] :
          ( aa(A,$o,aTP_Lamp_avw(set(A),fun(A,$o),Uu),Uua)
        <=> ! [X4: A] :
              ( aa(set(A),$o,member(A,X4),Uu)
             => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),Uua),X4) ) ) ) ).

% ATP.lambda_467
tff(fact_8247_ATP_Olambda__468,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [Uu: set(A),Uua: A] :
          ( aa(A,$o,aTP_Lamp_avt(set(A),fun(A,$o),Uu),Uua)
        <=> ! [X4: A] :
              ( aa(set(A),$o,member(A,X4),Uu)
             => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),X4),Uua) ) ) ) ).

% ATP.lambda_468
tff(fact_8248_ATP_Olambda__469,axiom,
    ! [A: $tType,Uu: set(filter(A)),Uua: fun(A,$o)] :
      ( aa(fun(A,$o),$o,aTP_Lamp_akt(set(filter(A)),fun(fun(A,$o),$o),Uu),Uua)
    <=> ! [X4: filter(A)] :
          ( aa(set(filter(A)),$o,member(filter(A),X4),Uu)
         => eventually(A,Uua,X4) ) ) ).

% ATP.lambda_469
tff(fact_8249_ATP_Olambda__470,axiom,
    ! [A: $tType,Uu: set(set(A)),Uua: A] :
      ( aa(A,$o,aTP_Lamp_agp(set(set(A)),fun(A,$o),Uu),Uua)
    <=> ! [X4: set(A)] :
          ( aa(set(set(A)),$o,member(set(A),X4),Uu)
         => aa(set(A),$o,member(A,Uua),X4) ) ) ).

% ATP.lambda_470
tff(fact_8250_ATP_Olambda__471,axiom,
    ! [A: $tType,Uu: set(set(A)),Uua: A] :
      ( aa(A,$o,aTP_Lamp_ahx(set(set(A)),fun(A,$o),Uu),Uua)
    <=> ? [X4: set(A)] :
          ( aa(set(set(A)),$o,member(set(A),X4),Uu)
          & aa(set(A),$o,member(A,Uua),X4) ) ) ).

% ATP.lambda_471
tff(fact_8251_ATP_Olambda__472,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: fun(A,$o),Uua: A] :
          ( aa(A,$o,aTP_Lamp_ald(fun(A,$o),fun(A,$o),Uu),Uua)
        <=> ! [Y3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),Y3)
             => aa(A,$o,Uu,Y3) ) ) ) ).

% ATP.lambda_472
tff(fact_8252_ATP_Olambda__473,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: set(A)] :
      ( aa(set(A),$o,aTP_Lamp_aqv(fun(A,$o),fun(set(A),$o),Uu),Uua)
    <=> ! [X4: A] :
          ( aa(set(A),$o,member(A,X4),Uua)
         => aa(A,$o,Uu,X4) ) ) ).

% ATP.lambda_473
tff(fact_8253_ATP_Olambda__474,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: set(A)] :
      ( aa(set(A),$o,aTP_Lamp_ajt(set(product_prod(A,A)),fun(set(A),$o),Uu),Uua)
    <=> ! [X4: A] :
          ( aa(set(A),$o,member(A,X4),Uua)
         => ! [Xa2: A] :
              ( aa(set(A),$o,member(A,Xa2),Uua)
             => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Xa2)),Uu)
                | aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa2),X4)),Uu) ) ) ) ) ).

% ATP.lambda_474
tff(fact_8254_ATP_Olambda__475,axiom,
    ! [B: $tType,A: $tType,Uu: set(product_prod(A,B)),Uua: A] :
      ( aa(A,$o,aTP_Lamp_avq(set(product_prod(A,B)),fun(A,$o),Uu),Uua)
    <=> ? [Y3: B] : aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),Y3)),Uu) ) ).

% ATP.lambda_475
tff(fact_8255_ATP_Olambda__476,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,option(A)),Uua: A] :
      ( aa(A,$o,aTP_Lamp_afl(fun(B,option(A)),fun(A,$o),Uu),Uua)
    <=> ? [A10: B] : aa(B,option(A),Uu,A10) = aa(A,option(A),some(A),Uua) ) ).

% ATP.lambda_476
tff(fact_8256_ATP_Olambda__477,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,set(old_node(A,B))),Uua: set(old_node(A,B))] :
      ( aa(set(old_node(A,B)),$o,aTP_Lamp_aug(fun(B,set(old_node(A,B))),fun(set(old_node(A,B)),$o),Uu),Uua)
    <=> ? [X4: B] : Uua = aa(set(old_node(A,B)),set(old_node(A,B)),image2(old_node(A,B),old_node(A,B),old_Push_Node(B,A,aa(B,sum_sum(B,nat),sum_Inl(B,nat),X4))),aa(B,set(old_node(A,B)),Uu,X4)) ) ).

% ATP.lambda_477
tff(fact_8257_ATP_Olambda__478,axiom,
    ! [A: $tType,B: $tType,Uu: A,Uua: product_prod(A,B)] :
      ( aa(product_prod(A,B),$o,aTP_Lamp_ahd(A,fun(product_prod(A,B),$o),Uu),Uua)
    <=> ? [V5: B] : Uua = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uu),V5) ) ).

% ATP.lambda_478
tff(fact_8258_ATP_Olambda__479,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: A] :
      ( aa(A,$o,aTP_Lamp_afc(fun(B,A),fun(A,$o),Uu),Uua)
    <=> ? [X4: B] : Uua = aa(B,A,Uu,X4) ) ).

% ATP.lambda_479
tff(fact_8259_ATP_Olambda__480,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,$o),Uua: fun(A,B)] :
      ( aa(fun(A,B),$o,aTP_Lamp_aqn(fun(B,$o),fun(fun(A,B),$o),Uu),Uua)
    <=> ! [X4: A] : aa(B,$o,Uu,aa(A,B,Uua,X4)) ) ).

% ATP.lambda_480
tff(fact_8260_ATP_Olambda__481,axiom,
    ! [B: $tType,A: $tType,Uu: set(product_prod(A,B)),Uua: product_prod(product_prod($o,A),product_prod($o,B))] :
      ( aa(product_prod(product_prod($o,A),product_prod($o,B)),$o,aTP_Lamp_ahe(set(product_prod(A,B)),fun(product_prod(product_prod($o,A),product_prod($o,B)),$o),Uu),Uua)
    <=> ? [X4: A,Y3: B] :
          ( ( Uua = aa(product_prod($o,B),product_prod(product_prod($o,A),product_prod($o,B)),aa(product_prod($o,A),fun(product_prod($o,B),product_prod(product_prod($o,A),product_prod($o,B))),product_Pair(product_prod($o,A),product_prod($o,B)),aa(A,product_prod($o,A),aa($o,fun(A,product_prod($o,A)),product_Pair($o,A),$false),X4)),aa(B,product_prod($o,B),aa($o,fun(B,product_prod($o,B)),product_Pair($o,B),$false),Y3)) )
          & aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Y3)),Uu) ) ) ).

% ATP.lambda_481
tff(fact_8261_ATP_Olambda__482,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,option(B)),Uua: product_prod(A,B)] :
      ( aa(product_prod(A,B),$o,aTP_Lamp_afy(fun(A,option(B)),fun(product_prod(A,B),$o),Uu),Uua)
    <=> ? [A10: A,B6: B] :
          ( ( Uua = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A10),B6) )
          & ( aa(A,option(B),Uu,A10) = aa(B,option(B),some(B),B6) ) ) ) ).

% ATP.lambda_482
tff(fact_8262_ATP_Olambda__483,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: product_prod(set(A),set(A))] :
      ( aa(product_prod(set(A),set(A)),$o,aTP_Lamp_ahs(set(product_prod(A,A)),fun(product_prod(set(A),set(A)),$o),Uu),Uua)
    <=> ? [X8: set(A),Y8: set(A)] :
          ( ( Uua = aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),X8),Y8) )
          & ( X8 != bot_bot(set(A)) )
          & ! [X4: A] :
              ( aa(set(A),$o,member(A,X4),Y8)
             => ? [Xa2: A] :
                  ( aa(set(A),$o,member(A,Xa2),X8)
                  & aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa2),X4)),Uu) ) ) ) ) ).

% ATP.lambda_483
tff(fact_8263_ATP_Olambda__484,axiom,
    ! [A: $tType,B: $tType,Uu: set(B),Uua: A] : aa(A,set(B),aTP_Lamp_rc(set(B),fun(A,set(B)),Uu),Uua) = aa(set(B),set(B),uminus_uminus(set(B)),Uu) ).

% ATP.lambda_484
tff(fact_8264_ATP_Olambda__485,axiom,
    ! [A: $tType,Uu: set(A),Uua: A] : aa(A,set(A),aTP_Lamp_rv(set(A),fun(A,set(A)),Uu),Uua) = aa(set(A),set(A),uminus_uminus(set(A)),Uu) ).

% ATP.lambda_485
tff(fact_8265_ATP_Olambda__486,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] : aa(list(A),option(A),aa(A,fun(list(A),option(A)),aTP_Lamp_aay(A,fun(list(A),option(A))),Uu),Uua) = aa(A,option(A),some(A),Uu) ).

% ATP.lambda_486
tff(fact_8266_ATP_Olambda__487,axiom,
    ! [C: $tType,B: $tType,Uu: set(product_prod(B,B)),Uua: C] : aa(C,set(B),aTP_Lamp_ana(set(product_prod(B,B)),fun(C,set(B)),Uu),Uua) = field2(B,Uu) ).

% ATP.lambda_487
tff(fact_8267_ATP_Olambda__488,axiom,
    ! [A: $tType,B: $tType,Uu: set(product_prod(B,B)),Uua: A] : aa(A,set(B),aTP_Lamp_and(set(product_prod(B,B)),fun(A,set(B)),Uu),Uua) = field2(B,Uu) ).

% ATP.lambda_488
tff(fact_8268_ATP_Olambda__489,axiom,
    ! [C: $tType,A: $tType,Uu: set(product_prod(A,A)),Uua: C] : aa(C,set(A),aTP_Lamp_amz(set(product_prod(A,A)),fun(C,set(A)),Uu),Uua) = field2(A,Uu) ).

% ATP.lambda_489
tff(fact_8269_ATP_Olambda__490,axiom,
    ! [B: $tType,A: $tType,Uu: set(product_prod(A,A)),Uua: B] : aa(B,set(A),aTP_Lamp_anc(set(product_prod(A,A)),fun(B,set(A)),Uu),Uua) = field2(A,Uu) ).

% ATP.lambda_490
tff(fact_8270_ATP_Olambda__491,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: A] : aa(A,set(A),aTP_Lamp_ut(set(product_prod(A,A)),fun(A,set(A)),Uu),Uua) = field2(A,Uu) ).

% ATP.lambda_491
tff(fact_8271_ATP_Olambda__492,axiom,
    ! [A: $tType,Uu: pred(A),Uua: product_unit] : aa(product_unit,A,aTP_Lamp_awj(pred(A),fun(product_unit,A),Uu),Uua) = the3(A,Uu) ).

% ATP.lambda_492
tff(fact_8272_ATP_Olambda__493,axiom,
    ! [A: $tType,B: $tType,Uu: list(B),Uua: A] : aa(A,set(B),aTP_Lamp_wz(list(B),fun(A,set(B)),Uu),Uua) = aa(list(B),set(B),set2(B),Uu) ).

% ATP.lambda_493
tff(fact_8273_ATP_Olambda__494,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,$o),Uua: A] : aa(A,set(B),aTP_Lamp_qy(fun(B,$o),fun(A,set(B)),Uu),Uua) = aa(fun(B,$o),set(B),collect(B),Uu) ).

% ATP.lambda_494
tff(fact_8274_ATP_Olambda__495,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] : aa(list(A),fun(set(A),set(A)),aa(A,fun(list(A),fun(set(A),set(A))),aTP_Lamp_ajp(A,fun(list(A),fun(set(A),set(A)))),Uu),Uua) = aa(A,fun(set(A),set(A)),insert2(A),Uu) ).

% ATP.lambda_495
tff(fact_8275_ATP_Olambda__496,axiom,
    ! [A: $tType,Uu: set(A),Uua: list(A)] : aa(list(A),set(list(A)),aTP_Lamp_aig(set(A),fun(list(A),set(list(A))),Uu),Uua) = lists(A,Uu) ).

% ATP.lambda_496
tff(fact_8276_ATP_Olambda__497,axiom,
    ! [A: $tType,Uu: nat,Uua: A] : aa(A,nat,aa(nat,fun(A,nat),aTP_Lamp_xh(nat,fun(A,nat)),Uu),Uua) = aa(nat,nat,suc,Uu) ).

% ATP.lambda_497
tff(fact_8277_ATP_Olambda__498,axiom,
    ! [A: $tType,Uu: fun(product_unit,seq(A)),Uua: fun(product_unit,seq(A)),Uub: product_unit] : aa(product_unit,seq(A),aa(fun(product_unit,seq(A)),fun(product_unit,seq(A)),aTP_Lamp_axv(fun(product_unit,seq(A)),fun(fun(product_unit,seq(A)),fun(product_unit,seq(A))),Uu),Uua),Uub) = case_seq(seq(A),A,aa(product_unit,seq(A),Uua,product_Unity),aTP_Lamp_axt(fun(product_unit,seq(A)),fun(A,fun(pred(A),seq(A))),Uua),aTP_Lamp_axu(fun(product_unit,seq(A)),fun(pred(A),fun(seq(A),seq(A))),Uua),aa(product_unit,seq(A),Uu,product_Unity)) ).

% ATP.lambda_498
tff(fact_8278_ATP_Olambda__499,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,fun(B,$o)),Uua: option(A),Uub: option(B)] :
      ( aa(option(B),$o,aa(option(A),fun(option(B),$o),aTP_Lamp_axc(fun(A,fun(B,$o)),fun(option(A),fun(option(B),$o)),Uu),Uua),Uub)
    <=> case_option($o,A,case_option($o,B,$true,aTP_Lamp_axa(B,$o),Uub),aa(option(B),fun(A,$o),aTP_Lamp_axb(fun(A,fun(B,$o)),fun(option(B),fun(A,$o)),Uu),Uub),Uua) ) ).

% ATP.lambda_499
tff(fact_8279_ATP_Olambda__500,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_ce(num,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),Uu),Uua),Uub) = $ite(aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less_eq(code_integer),aa(num,code_integer,numeral_numeral(code_integer),Uu)),Uub),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2))),Uua)),one_one(code_integer))),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),Uub),aa(num,code_integer,numeral_numeral(code_integer),Uu))),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2))),Uua)),Uub)) ).

% ATP.lambda_500
tff(fact_8280_ATP_Olambda__501,axiom,
    ! [Uu: num,Uua: nat,Uub: nat] :
      aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),aTP_Lamp_bs(num,fun(nat,fun(nat,product_prod(nat,nat))),Uu),Uua),Uub) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),Uu)),Uub),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Uua)),one_one(nat))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),aa(num,nat,numeral_numeral(nat),Uu))),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Uua)),Uub)) ).

% ATP.lambda_501
tff(fact_8281_ATP_Olambda__502,axiom,
    ! [Uu: num,Uua: int,Uub: int] :
      aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_br(num,fun(int,fun(int,product_prod(int,int))),Uu),Uua),Uub) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(num,int,numeral_numeral(int),Uu)),Uub),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Uua)),one_one(int))),aa(int,int,aa(int,fun(int,int),minus_minus(int),Uub),aa(num,int,numeral_numeral(int),Uu))),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Uua)),Uub)) ).

% ATP.lambda_502
tff(fact_8282_ATP_Olambda__503,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_bm(num,fun(A,fun(A,product_prod(A,A))),Uu),Uua),Uub) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),Uu)),Uub),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Uua)),one_one(A))),aa(A,A,aa(A,fun(A,A),minus_minus(A),Uub),aa(num,A,numeral_numeral(A),Uu))),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Uua)),Uub)) ) ).

% ATP.lambda_503
tff(fact_8283_ATP_Olambda__504,axiom,
    ! [A: $tType,Uu: A,Uua: A,Uub: list(A)] :
      aa(list(A),list(A),aa(A,fun(list(A),list(A)),aTP_Lamp_yr(A,fun(A,fun(list(A),list(A))),Uu),Uua),Uub) = $ite(Uu = Uua,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uua),Uub),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uu),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uua),Uub))) ).

% ATP.lambda_504
tff(fact_8284_ATP_Olambda__505,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: fun(A,B),Uua: set(A),Uub: A] :
          aa(A,B,aa(set(A),fun(A,B),aTP_Lamp_ht(fun(A,B),fun(set(A),fun(A,B)),Uu),Uua),Uub) = $ite(aa(set(A),$o,member(A,Uub),Uua),aa(A,B,Uu,Uub),zero_zero(B)) ) ).

% ATP.lambda_505
tff(fact_8285_ATP_Olambda__506,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: fun(A,B),Uua: set(A),Uub: A] :
          aa(A,B,aa(set(A),fun(A,B),aTP_Lamp_hv(fun(A,B),fun(set(A),fun(A,B)),Uu),Uua),Uub) = $ite(aa(set(A),$o,member(A,Uub),Uua),aa(A,B,Uu,Uub),one_one(B)) ) ).

% ATP.lambda_506
tff(fact_8286_ATP_Olambda__507,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: A,Uua: fun(A,B),Uub: A] :
          aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_gq(A,fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = $ite(Uu = Uub,aa(A,B,Uua,Uub),zero_zero(B)) ) ).

% ATP.lambda_507
tff(fact_8287_ATP_Olambda__508,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: A,Uua: fun(A,B),Uub: A] :
          aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_gt(A,fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = $ite(Uu = Uub,aa(A,B,Uua,Uub),one_one(B)) ) ).

% ATP.lambda_508
tff(fact_8288_ATP_Olambda__509,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: A,Uua: fun(A,B),Uub: A] :
          aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_gr(A,fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = $ite(Uub = Uu,aa(A,B,Uua,Uub),zero_zero(B)) ) ).

% ATP.lambda_509
tff(fact_8289_ATP_Olambda__510,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: A,Uua: fun(A,B),Uub: A] :
          aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_gs(A,fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = $ite(Uub = Uu,aa(A,B,Uua,Uub),one_one(B)) ) ).

% ATP.lambda_510
tff(fact_8290_ATP_Olambda__511,axiom,
    ! [A: $tType,Uu: A,Uua: fun(A,nat),Uub: A] :
      aa(A,nat,aa(fun(A,nat),fun(A,nat),aa(A,fun(fun(A,nat),fun(A,nat)),aTP_Lamp_aqb(A,fun(fun(A,nat),fun(A,nat))),Uu),Uua),Uub) = $ite(Uub = Uu,aa(nat,nat,suc,aa(A,nat,Uua,Uub)),aa(A,nat,Uua,Uub)) ).

% ATP.lambda_511
tff(fact_8291_ATP_Olambda__512,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: A,Uub: A] :
      aa(A,nat,aa(A,fun(A,nat),aTP_Lamp_aqo(fun(A,nat),fun(A,fun(A,nat)),Uu),Uua),Uub) = $ite(Uub = Uua,aa(nat,nat,suc,aa(A,nat,Uu,Uub)),aa(A,nat,Uu,Uub)) ).

% ATP.lambda_512
tff(fact_8292_ATP_Olambda__513,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(A)
     => ! [Uu: B,Uua: A,Uub: B] :
          aa(B,A,aa(A,fun(B,A),aTP_Lamp_aqi(B,fun(A,fun(B,A)),Uu),Uua),Uub) = $ite(Uu = Uub,Uua,zero_zero(A)) ) ).

% ATP.lambda_513
tff(fact_8293_ATP_Olambda__514,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: B,Uua: A,Uub: B] :
          aa(B,A,aa(A,fun(B,A),aTP_Lamp_ard(B,fun(A,fun(B,A)),Uu),Uua),Uub) = $ite(Uu = Uub,Uua,one_one(A)) ) ).

% ATP.lambda_514
tff(fact_8294_ATP_Olambda__515,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(A)
     => ! [Uu: B,Uua: A,Uub: B] :
          aa(B,A,aa(A,fun(B,A),aTP_Lamp_aqh(B,fun(A,fun(B,A)),Uu),Uua),Uub) = $ite(Uub = Uu,Uua,zero_zero(A)) ) ).

% ATP.lambda_515
tff(fact_8295_ATP_Olambda__516,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: B,Uua: A,Uub: B] :
          aa(B,A,aa(A,fun(B,A),aTP_Lamp_arc(B,fun(A,fun(B,A)),Uu),Uua),Uub) = $ite(Uub = Uu,Uua,one_one(A)) ) ).

% ATP.lambda_516
tff(fact_8296_ATP_Olambda__517,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_di(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),Uu),Uua),Uub) = $ite(Uub = zero_zero(code_integer),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),zero_zero(code_integer)),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),one_one(code_integer))),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),aa(code_integer,code_integer,uminus_uminus(code_integer),Uu)),Uub))) ).

% ATP.lambda_517
tff(fact_8297_ATP_Olambda__518,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_jl(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),Uu),Uua),Uub) = $ite(Uub = zero_zero(code_integer),aa(code_integer,product_prod(code_integer,code_integer),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_518
tff(fact_8298_ATP_Olambda__519,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_dh(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),Uu),Uua),Uub) = $ite(Uub = zero_zero(code_integer),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),zero_zero(code_integer)),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),one_one(code_integer))),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),Uu),Uub))) ).

% ATP.lambda_519
tff(fact_8299_ATP_Olambda__520,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: A,Uub: list(A)] :
      aa(list(A),list(A),aa(A,fun(list(A),list(A)),aTP_Lamp_zu(fun(A,$o),fun(A,fun(list(A),list(A))),Uu),Uua),Uub) = $ite(aa(A,$o,Uu,Uua),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uua),Uub),Uub) ).

% ATP.lambda_520
tff(fact_8300_ATP_Olambda__521,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: A,Uub: set(A)] :
      aa(set(A),set(A),aa(A,fun(set(A),set(A)),aTP_Lamp_om(fun(A,$o),fun(A,fun(set(A),set(A))),Uu),Uua),Uub) = $ite(aa(A,$o,Uu,Uua),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Uua),Uub),Uub) ).

% ATP.lambda_521
tff(fact_8301_ATP_Olambda__522,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: fun(A,nat),Uub: A] :
      aa(A,nat,aa(fun(A,nat),fun(A,nat),aa(fun(A,$o),fun(fun(A,nat),fun(A,nat)),aTP_Lamp_aqk(fun(A,$o),fun(fun(A,nat),fun(A,nat))),Uu),Uua),Uub) = $ite(aa(A,$o,Uu,Uub),aa(A,nat,Uua,Uub),zero_zero(nat)) ).

% ATP.lambda_522
tff(fact_8302_ATP_Olambda__523,axiom,
    ! [B: $tType,A: $tType] :
      ( monoid_add(A)
     => ! [Uu: fun(B,A),Uua: fun(B,$o),Uub: B] :
          aa(B,A,aa(fun(B,$o),fun(B,A),aTP_Lamp_abg(fun(B,A),fun(fun(B,$o),fun(B,A)),Uu),Uua),Uub) = $ite(aa(B,$o,Uua,Uub),aa(B,A,Uu,Uub),zero_zero(A)) ) ).

% ATP.lambda_523
tff(fact_8303_ATP_Olambda__524,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: fun(A,$o),Uub: A] :
      aa(A,nat,aa(fun(A,$o),fun(A,nat),aTP_Lamp_aqm(fun(A,nat),fun(fun(A,$o),fun(A,nat)),Uu),Uua),Uub) = $ite(aa(A,$o,Uua,Uub),aa(A,nat,Uu,Uub),zero_zero(nat)) ).

% ATP.lambda_524
tff(fact_8304_ATP_Olambda__525,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: fun(A,B),Uua: fun(A,$o),Uub: A] :
          aa(A,B,aa(fun(A,$o),fun(A,B),aTP_Lamp_hn(fun(A,B),fun(fun(A,$o),fun(A,B)),Uu),Uua),Uub) = $ite(aa(A,$o,Uua,Uub),aa(A,B,Uu,Uub),zero_zero(B)) ) ).

% ATP.lambda_525
tff(fact_8305_ATP_Olambda__526,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: fun(A,B),Uua: fun(A,$o),Uub: A] :
          aa(A,B,aa(fun(A,$o),fun(A,B),aTP_Lamp_ho(fun(A,B),fun(fun(A,$o),fun(A,B)),Uu),Uua),Uub) = $ite(aa(A,$o,Uua,Uub),aa(A,B,Uu,Uub),one_one(B)) ) ).

% ATP.lambda_526
tff(fact_8306_ATP_Olambda__527,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,fun(A,A)),Uua: fun(B,$o),Uub: B] :
      aa(B,fun(A,A),aa(fun(B,$o),fun(B,fun(A,A)),aTP_Lamp_zq(fun(B,fun(A,A)),fun(fun(B,$o),fun(B,fun(A,A))),Uu),Uua),Uub) = $ite(aa(B,$o,Uua,Uub),aa(B,fun(A,A),Uu,Uub),id(A)) ).

% ATP.lambda_527
tff(fact_8307_ATP_Olambda__528,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,A),Uua: fun(B,$o),Uub: B] :
      aa(B,option(A),aa(fun(B,$o),fun(B,option(A)),aTP_Lamp_aar(fun(B,A),fun(fun(B,$o),fun(B,option(A))),Uu),Uua),Uub) = $ite(aa(B,$o,Uua,Uub),aa(A,option(A),some(A),aa(B,A,Uu,Uub)),none(A)) ).

% ATP.lambda_528
tff(fact_8308_ATP_Olambda__529,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_se(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_sd(A,fun(B,fun(set(product_prod(A,B)),set(product_prod(A,B)))),Uua),Uub,Uu) ).

% ATP.lambda_529
tff(fact_8309_ATP_Olambda__530,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_ul(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_oj(B,fun(A,fun(set(product_prod(B,A)),set(product_prod(B,A)))),Uua),Uub,Uu) ).

% ATP.lambda_530
tff(fact_8310_ATP_Olambda__531,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_avz(fun(A,fun(A,A)),fun(option(A),fun(A,option(A))),Uu),Uua),Uub) = case_option(option(A),A,aa(A,option(A),some(A),Uub),aa(A,fun(A,option(A)),aTP_Lamp_avy(fun(A,fun(A,A)),fun(A,fun(A,option(A))),Uu),Uub),Uua) ).

% ATP.lambda_531
tff(fact_8311_ATP_Olambda__532,axiom,
    ! [Uu: fun(product_unit,a),Uua: pred(a),Uub: seq(a)] :
      aa(seq(a),a,aa(pred(a),fun(seq(a),a),aTP_Lamp_aye(fun(product_unit,a),fun(pred(a),fun(seq(a),a)),Uu),Uua),Uub) = $ite(
        is_empty(a,Uua),
        the_only(a,Uu,Uub),
        $ite(
          null(a,Uub),
          singleton(a,Uu,Uua),
          $let(
            x3: a,
            x3:= singleton(a,Uu,Uua),
            $ite(x3 = the_only(a,Uu,Uub),x3,aa(product_unit,a,Uu,product_Unity)) ) ) ) ).

% ATP.lambda_532
tff(fact_8312_ATP_Olambda__533,axiom,
    ! [A: $tType,Uu: fun(product_unit,A),Uua: A,Uub: pred(A)] :
      aa(pred(A),A,aa(A,fun(pred(A),A),aTP_Lamp_ayd(fun(product_unit,A),fun(A,fun(pred(A),A)),Uu),Uua),Uub) = $ite(
        is_empty(A,Uub),
        Uua,
        $ite(Uua = singleton(A,Uu,Uub),Uua,aa(product_unit,A,Uu,product_Unity)) ) ).

% ATP.lambda_533
tff(fact_8313_ATP_Olambda__534,axiom,
    ! [A: $tType,Uu: fun(A,fun(A,$o)),Uua: list(A),Uub: list(A)] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),aTP_Lamp_xl(fun(A,fun(A,$o)),fun(list(A),fun(list(A),list(A))),Uu),Uua),Uub) = merges9089515139780605204_merge(A,Uu,aa(list(A),list(A),mergesort_by_rel(A,Uu),Uua),aa(list(A),list(A),mergesort_by_rel(A,Uu),Uub)) ).

% ATP.lambda_534
tff(fact_8314_ATP_Olambda__535,axiom,
    ! [A: $tType] :
      ( sup(A)
     => ! [Uu: option(A),Uua: option(A),Uub: A] : aa(A,option(A),aa(option(A),fun(A,option(A)),aTP_Lamp_tc(option(A),fun(option(A),fun(A,option(A))),Uu),Uua),Uub) = case_option(option(A),A,Uu,aTP_Lamp_tb(A,fun(A,option(A)),Uub),Uua) ) ).

% ATP.lambda_535
tff(fact_8315_ATP_Olambda__536,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_ads(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_536
tff(fact_8316_ATP_Olambda__537,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_abv(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_537
tff(fact_8317_ATP_Olambda__538,axiom,
    ! [A: $tType,Uu: fun(A,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_acq(fun(A,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,compow(fun(A,A),Uub,Uu),Uua) ).

% ATP.lambda_538
tff(fact_8318_ATP_Olambda__539,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A),Uub: $o] :
      aa($o,set(A),aa(set(A),fun($o,set(A)),aTP_Lamp_nx(set(A),fun(set(A),fun($o,set(A))),Uu),Uua),(Uub)) = $ite((Uub),Uu,Uua) ).

% ATP.lambda_539
tff(fact_8319_ATP_Olambda__540,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,fun(B,$o)),Uua: option(B),Uub: A] :
      ( aa(A,$o,aa(option(B),fun(A,$o),aTP_Lamp_axb(fun(A,fun(B,$o)),fun(option(B),fun(A,$o)),Uu),Uua),Uub)
    <=> case_option($o,B,$false,aa(A,fun(B,$o),Uu,Uub),Uua) ) ).

% ATP.lambda_540
tff(fact_8320_ATP_Olambda__541,axiom,
    ! [D: $tType,C: $tType,B: $tType,A: $tType,Uu: fun(D,fun(B,C)),Uua: fun(A,D),Uub: product_prod(A,B)] : aa(product_prod(A,B),C,aa(fun(A,D),fun(product_prod(A,B),C),aTP_Lamp_vu(fun(D,fun(B,C)),fun(fun(A,D),fun(product_prod(A,B),C)),Uu),Uua),Uub) = aa(B,C,aa(D,fun(B,C),Uu,aa(A,D,Uua,aa(product_prod(A,B),A,product_fst(A,B),Uub))),aa(product_prod(A,B),B,product_snd(A,B),Uub)) ).

% ATP.lambda_541
tff(fact_8321_ATP_Olambda__542,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_lt(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_542
tff(fact_8322_ATP_Olambda__543,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_hr(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_543
tff(fact_8323_ATP_Olambda__544,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_er(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_544
tff(fact_8324_ATP_Olambda__545,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,fun(B,$o)),Uua: fun(A,B),Uub: A] :
      ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_aka(fun(A,fun(B,$o)),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
    <=> aa(B,$o,aa(A,fun(B,$o),Uu,Uub),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_545
tff(fact_8325_ATP_Olambda__546,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_fv(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_546
tff(fact_8326_ATP_Olambda__547,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_dn(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_547
tff(fact_8327_ATP_Olambda__548,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_lr(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_548
tff(fact_8328_ATP_Olambda__549,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( complete_Sup(A)
     => ! [Uu: fun(C,fun(B,A)),Uua: B,Uub: C] : aa(C,A,aa(B,fun(C,A),aTP_Lamp_js(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_549
tff(fact_8329_ATP_Olambda__550,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( complete_Inf(A)
     => ! [Uu: fun(C,fun(B,A)),Uua: B,Uub: C] : aa(C,A,aa(B,fun(C,A),aTP_Lamp_ju(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_550
tff(fact_8330_ATP_Olambda__551,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_alu(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_551
tff(fact_8331_ATP_Olambda__552,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: fun(B,fun(C,A)),Uua: C,Uub: B] : aa(B,A,aa(C,fun(B,A),aTP_Lamp_lc(fun(B,fun(C,A)),fun(C,fun(B,A)),Uu),Uua),Uub) = aa(C,A,aa(B,fun(C,A),Uu,Uub),Uua) ) ).

% ATP.lambda_552
tff(fact_8332_ATP_Olambda__553,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(B,fun(C,A)),Uua: C,Uub: B] : aa(B,A,aa(C,fun(B,A),aTP_Lamp_ff(fun(B,fun(C,A)),fun(C,fun(B,A)),Uu),Uua),Uub) = aa(C,A,aa(B,fun(C,A),Uu,Uub),Uua) ) ).

% ATP.lambda_553
tff(fact_8333_ATP_Olambda__554,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(B,fun(C,A)),Uua: C,Uub: B] : aa(B,A,aa(C,fun(B,A),aTP_Lamp_cw(fun(B,fun(C,A)),fun(C,fun(B,A)),Uu),Uua),Uub) = aa(C,A,aa(B,fun(C,A),Uu,Uub),Uua) ) ).

% ATP.lambda_554
tff(fact_8334_ATP_Olambda__555,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu: fun(B,fun(C,A)),Uua: C,Uub: B] : aa(B,A,aa(C,fun(B,A),aTP_Lamp_awr(fun(B,fun(C,A)),fun(C,fun(B,A)),Uu),Uua),Uub) = aa(C,A,aa(B,fun(C,A),Uu,Uub),Uua) ).

% ATP.lambda_555
tff(fact_8335_ATP_Olambda__556,axiom,
    ! [B: $tType,A: $tType] :
      ( finite_finite(A)
     => ! [Uu: fun(B,fun(A,$o)),Uua: A,Uub: B] :
          ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_akd(fun(B,fun(A,$o)),fun(A,fun(B,$o)),Uu),Uua),Uub)
        <=> aa(A,$o,aa(B,fun(A,$o),Uu,Uub),Uua) ) ) ).

% ATP.lambda_556
tff(fact_8336_ATP_Olambda__557,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,fun(A,$o)),Uua: A,Uub: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_aei(fun(B,fun(A,$o)),fun(A,fun(B,$o)),Uu),Uua),Uub)
    <=> aa(A,$o,aa(B,fun(A,$o),Uu,Uub),Uua) ) ).

% ATP.lambda_557
tff(fact_8337_ATP_Olambda__558,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,fun(A,A)),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_aam(fun(B,fun(A,A)),fun(A,fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(B,fun(A,A),Uu,Uub),Uua) ).

% ATP.lambda_558
tff(fact_8338_ATP_Olambda__559,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,fun(B,$o)),Uua: B,Uub: A] :
      ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_afh(fun(A,fun(B,$o)),fun(B,fun(A,$o)),Uu),Uua),Uub)
    <=> aa(B,$o,aa(A,fun(B,$o),Uu,Uub),Uua) ) ).

% ATP.lambda_559
tff(fact_8339_ATP_Olambda__560,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [Uu: fun(A,fun(B,C)),Uua: B,Uub: A] : aa(A,C,aa(B,fun(A,C),aTP_Lamp_hg(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_560
tff(fact_8340_ATP_Olambda__561,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [Uu: fun(A,fun(B,C)),Uua: B,Uub: A] : aa(A,C,aa(B,fun(A,C),aTP_Lamp_hc(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_561
tff(fact_8341_ATP_Olambda__562,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,fun(B,B)),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_xi(fun(A,fun(B,B)),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(A,fun(B,B),Uu,Uub),Uua) ).

% ATP.lambda_562
tff(fact_8342_ATP_Olambda__563,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,fun(B,A)),Uua: B,Uub: A] : aa(A,A,aa(B,fun(A,A),aTP_Lamp_aal(fun(A,fun(B,A)),fun(B,fun(A,A)),Uu),Uua),Uub) = aa(B,A,aa(A,fun(B,A),Uu,Uub),Uua) ).

% ATP.lambda_563
tff(fact_8343_ATP_Olambda__564,axiom,
    ! [A: $tType,Uu: fun(A,fun(A,$o)),Uua: A,Uub: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_aak(fun(A,fun(A,$o)),fun(A,fun(A,$o)),Uu),Uua),Uub)
    <=> aa(A,$o,aa(A,fun(A,$o),Uu,Uub),Uua) ) ).

% ATP.lambda_564
tff(fact_8344_ATP_Olambda__565,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] :
      aa(code_integer,product_prod(code_integer,$o),aa(code_integer,fun(code_integer,product_prod(code_integer,$o)),aTP_Lamp_cm(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,$o))),Uu),Uua),Uub) = aa($o,product_prod(code_integer,$o),
        aa(code_integer,fun($o,product_prod(code_integer,$o)),product_Pair(code_integer,$o),
          $ite(aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),zero_zero(code_integer)),Uu),Uua,aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),Uub))),
        Uub = one_one(code_integer)) ).

% ATP.lambda_565
tff(fact_8345_ATP_Olambda__566,axiom,
    ! [A: $tType,Uu: set(set(A)),Uua: set(set(A)),Uub: set(set(A))] :
      ( aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),aTP_Lamp_aou(set(set(A)),fun(set(set(A)),fun(set(set(A)),$o)),Uu),Uua),Uub)
    <=> ( aa(set(set(A)),$o,pred_chain(set(A),Uu,ord_less(set(A))),Uub)
        & aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less(set(set(A))),Uua),Uub) ) ) ).

% ATP.lambda_566
tff(fact_8346_ATP_Olambda__567,axiom,
    ! [A: $tType,Uu: fun(A,fun(A,$o)),Uua: A,Uub: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_bh(fun(A,fun(A,$o)),fun(A,fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(A,$o,aa(A,fun(A,$o),Uu,Uua),Uub)
        | ( Uua = Uub ) ) ) ).

% ATP.lambda_567
tff(fact_8347_ATP_Olambda__568,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_fw(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_fv(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uub)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Uub),Uua)) ) ).

% ATP.lambda_568
tff(fact_8348_ATP_Olambda__569,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_do(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_dn(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uub)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Uub),Uua)) ) ).

% ATP.lambda_569
tff(fact_8349_ATP_Olambda__570,axiom,
    ! [Uu: product_prod(code_natural,code_natural),Uua: code_natural,Uub: code_natural] : aa(code_natural,product_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural)),aa(code_natural,fun(code_natural,product_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural))),aTP_Lamp_atz(product_prod(code_natural,code_natural),fun(code_natural,fun(code_natural,product_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural)))),Uu),Uua),Uub) = aa(product_prod(code_natural,code_natural),product_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural)),aa(fun(code_natural,fun(code_natural,product_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural)))),fun(product_prod(code_natural,code_natural),product_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural))),product_case_prod(code_natural,code_natural,product_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural))),aa(code_natural,fun(code_natural,fun(code_natural,product_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural)))),aTP_Lamp_aty(code_natural,fun(code_natural,fun(code_natural,fun(code_natural,product_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural))))),Uua),Uub)),aa(product_prod(code_natural,product_prod(code_natural,code_natural)),product_prod(code_natural,code_natural),product_snd(code_natural,product_prod(code_natural,code_natural)),aa(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural)),next,Uu))) ).

% ATP.lambda_570
tff(fact_8350_ATP_Olambda__571,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural)))),Uua: pred(B),Uub: product_prod(code_natural,code_natural)] : aa(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural)),aa(pred(B),fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))),aTP_Lamp_awe(fun(B,fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural)))),fun(pred(B),fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural)))),Uu),Uua),Uub) = aa(product_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural)),product_prod(pred(A),product_prod(code_natural,code_natural)),aa(fun(product_prod(code_natural,code_natural),fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural)))),fun(product_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural)),product_prod(pred(A),product_prod(code_natural,code_natural))),product_case_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))),aa(pred(B),fun(product_prod(code_natural,code_natural),fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural)))),aTP_Lamp_awd(fun(B,fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural)))),fun(pred(B),fun(product_prod(code_natural,code_natural),fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))))),Uu),Uua)),split_seed(Uub)) ).

% ATP.lambda_571
tff(fact_8351_ATP_Olambda__572,axiom,
    ! [Uu: $o,Uua: code_integer,Uub: $o] : aa($o,char,aa(code_integer,fun($o,char),aTP_Lamp_qp($o,fun(code_integer,fun($o,char)),(Uu)),Uua),(Uub)) = aa(product_prod(code_integer,$o),char,aa(fun(code_integer,fun($o,char)),fun(product_prod(code_integer,$o),char),product_case_prod(code_integer,$o,char),aa($o,fun(code_integer,fun($o,char)),aTP_Lamp_qo($o,fun($o,fun(code_integer,fun($o,char))),(Uu)),(Uub))),code_bit_cut_integer(Uua)) ).

% ATP.lambda_572
tff(fact_8352_ATP_Olambda__573,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_qc(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_qb(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_573
tff(fact_8353_ATP_Olambda__574,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_pz(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_py(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_574
tff(fact_8354_ATP_Olambda__575,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_px(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_pw(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_575
tff(fact_8355_ATP_Olambda__576,axiom,
    ! [Uu: rat,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_pl(rat,fun(int,fun(int,$o)),Uu),Uua),Uub)
    <=> aa(product_prod(int,int),$o,aa(fun(int,fun(int,$o)),fun(product_prod(int,int),$o),product_case_prod(int,int,$o),aa(int,fun(int,fun(int,$o)),aTP_Lamp_pk(int,fun(int,fun(int,fun(int,$o))),Uua),Uub)),quotient_of(Uu)) ) ).

% ATP.lambda_576
tff(fact_8356_ATP_Olambda__577,axiom,
    ! [Uu: rat,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_pj(rat,fun(int,fun(int,$o)),Uu),Uua),Uub)
    <=> aa(product_prod(int,int),$o,aa(fun(int,fun(int,$o)),fun(product_prod(int,int),$o),product_case_prod(int,int,$o),aa(int,fun(int,fun(int,$o)),aTP_Lamp_pi(int,fun(int,fun(int,fun(int,$o))),Uua),Uub)),quotient_of(Uu)) ) ).

% ATP.lambda_577
tff(fact_8357_ATP_Olambda__578,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_pd(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_pc(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_578
tff(fact_8358_ATP_Olambda__579,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(B,fun(C,A)),Uua: set(B),Uub: C] : aa(C,A,aa(set(B),fun(C,A),aTP_Lamp_fg(fun(B,fun(C,A)),fun(set(B),fun(C,A)),Uu),Uua),Uub) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(C,fun(B,A),aTP_Lamp_ff(fun(B,fun(C,A)),fun(C,fun(B,A)),Uu),Uub)),Uua) ) ).

% ATP.lambda_579
tff(fact_8359_ATP_Olambda__580,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(B,fun(C,A)),Uua: set(B),Uub: C] : aa(C,A,aa(set(B),fun(C,A),aTP_Lamp_cx(fun(B,fun(C,A)),fun(set(B),fun(C,A)),Uu),Uua),Uub) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(C,fun(B,A),aTP_Lamp_cw(fun(B,fun(C,A)),fun(C,fun(B,A)),Uu),Uub)),Uua) ) ).

% ATP.lambda_580
tff(fact_8360_ATP_Olambda__581,axiom,
    ! [D: $tType,E: $tType,A: $tType,C: $tType,B: $tType,Uu: fun(B,fun(C,fun(D,fun(E,set(A))))),Uua: product_prod(B,C),Uub: product_prod(D,E)] : aa(product_prod(D,E),set(A),aa(product_prod(B,C),fun(product_prod(D,E),set(A)),aTP_Lamp_of(fun(B,fun(C,fun(D,fun(E,set(A))))),fun(product_prod(B,C),fun(product_prod(D,E),set(A))),Uu),Uua),Uub) = aa(product_prod(B,C),set(A),aa(fun(B,fun(C,set(A))),fun(product_prod(B,C),set(A)),product_case_prod(B,C,set(A)),aa(product_prod(D,E),fun(B,fun(C,set(A))),aTP_Lamp_oe(fun(B,fun(C,fun(D,fun(E,set(A))))),fun(product_prod(D,E),fun(B,fun(C,set(A)))),Uu),Uub)),Uua) ).

% ATP.lambda_581
tff(fact_8361_ATP_Olambda__582,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: filter(B),Uub: fun(A,$o)] :
      ( aa(fun(A,$o),$o,aa(filter(B),fun(fun(A,$o),$o),aTP_Lamp_ann(fun(B,A),fun(filter(B),fun(fun(A,$o),$o)),Uu),Uua),Uub)
    <=> eventually(B,aa(fun(A,$o),fun(B,$o),aTP_Lamp_anm(fun(B,A),fun(fun(A,$o),fun(B,$o)),Uu),Uub),Uua) ) ).

% ATP.lambda_582
tff(fact_8362_ATP_Olambda__583,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_gp(fun(nat,A),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),Uub))),aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_583
tff(fact_8363_ATP_Olambda__584,axiom,
    ! [A: $tType,Uu: fun(set(A),set(A)),Uua: set(A),Uub: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),aTP_Lamp_aln(fun(set(A),set(A)),fun(set(A),fun(set(A),set(A))),Uu),Uua),Uub) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),Uu,Uub)),Uua)),complete_lattice_gfp(set(A),Uu)) ).

% ATP.lambda_584
tff(fact_8364_ATP_Olambda__585,axiom,
    ! [A: $tType,Uu: set(A),Uua: fun(set(A),set(A)),Uub: set(A)] : aa(set(A),set(A),aa(fun(set(A),set(A)),fun(set(A),set(A)),aTP_Lamp_alr(set(A),fun(fun(set(A),set(A)),fun(set(A),set(A))),Uu),Uua),Uub) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),Uua,Uub)),Uu)),complete_lattice_gfp(set(A),Uua)) ).

% ATP.lambda_585
tff(fact_8365_ATP_Olambda__586,axiom,
    ! [Uu: nat,Uua: nat,Uub: list(nat)] :
      ( aa(list(nat),$o,aa(nat,fun(list(nat),$o),aTP_Lamp_abj(nat,fun(nat,fun(list(nat),$o)),Uu),Uua),Uub)
    <=> ( ( aa(list(nat),nat,size_size(list(nat)),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu),one_one(nat)) )
        & ( groups8242544230860333062m_list(nat,Uub) = Uua ) ) ) ).

% ATP.lambda_586
tff(fact_8366_ATP_Olambda__587,axiom,
    ! [A: $tType,C: $tType,D: $tType,B: $tType,Uu: fun(A,fun(C,$o)),Uua: fun(B,fun(D,$o)),Uub: sum_sum(product_prod(A,C),product_prod(B,D))] :
      ( aa(sum_sum(product_prod(A,C),product_prod(B,D)),$o,aa(fun(B,fun(D,$o)),fun(sum_sum(product_prod(A,C),product_prod(B,D)),$o),aTP_Lamp_axm(fun(A,fun(C,$o)),fun(fun(B,fun(D,$o)),fun(sum_sum(product_prod(A,C),product_prod(B,D)),$o)),Uu),Uua),Uub)
    <=> ( aa(set(product_prod(A,C)),$o,aa(set(product_prod(A,C)),fun(set(product_prod(A,C)),$o),ord_less_eq(set(product_prod(A,C))),aa(sum_sum(product_prod(A,C),product_prod(B,D)),set(product_prod(A,C)),basic_setl(product_prod(A,C),product_prod(B,D)),Uub)),aa(fun(product_prod(A,C),$o),set(product_prod(A,C)),collect(product_prod(A,C)),aa(fun(A,fun(C,$o)),fun(product_prod(A,C),$o),product_case_prod(A,C,$o),Uu)))
        & aa(set(product_prod(B,D)),$o,aa(set(product_prod(B,D)),fun(set(product_prod(B,D)),$o),ord_less_eq(set(product_prod(B,D))),aa(sum_sum(product_prod(A,C),product_prod(B,D)),set(product_prod(B,D)),basic_setr(product_prod(A,C),product_prod(B,D)),Uub)),aa(fun(product_prod(B,D),$o),set(product_prod(B,D)),collect(product_prod(B,D)),aa(fun(B,fun(D,$o)),fun(product_prod(B,D),$o),product_case_prod(B,D,$o),Uua))) ) ) ).

% ATP.lambda_587
tff(fact_8367_ATP_Olambda__588,axiom,
    ! [A: $tType,B: $tType,D: $tType,C: $tType,Uu: fun(A,fun(B,$o)),Uua: fun(C,fun(D,$o)),Uub: sum_sum(product_prod(A,B),product_prod(C,D))] :
      ( aa(sum_sum(product_prod(A,B),product_prod(C,D)),$o,aa(fun(C,fun(D,$o)),fun(sum_sum(product_prod(A,B),product_prod(C,D)),$o),aTP_Lamp_axl(fun(A,fun(B,$o)),fun(fun(C,fun(D,$o)),fun(sum_sum(product_prod(A,B),product_prod(C,D)),$o)),Uu),Uua),Uub)
    <=> ( aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),aa(sum_sum(product_prod(A,B),product_prod(C,D)),set(product_prod(A,B)),basic_setl(product_prod(A,B),product_prod(C,D)),Uub)),aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),Uu)))
        & aa(set(product_prod(C,D)),$o,aa(set(product_prod(C,D)),fun(set(product_prod(C,D)),$o),ord_less_eq(set(product_prod(C,D))),aa(sum_sum(product_prod(A,B),product_prod(C,D)),set(product_prod(C,D)),basic_setr(product_prod(A,B),product_prod(C,D)),Uub)),aa(fun(product_prod(C,D),$o),set(product_prod(C,D)),collect(product_prod(C,D)),aa(fun(C,fun(D,$o)),fun(product_prod(C,D),$o),product_case_prod(C,D,$o),Uua))) ) ) ).

% ATP.lambda_588
tff(fact_8368_ATP_Olambda__589,axiom,
    ! [A: $tType,C: $tType,D: $tType,B: $tType,Uu: fun(A,fun(C,$o)),Uua: fun(B,fun(D,$o)),Uub: product_prod(product_prod(A,C),product_prod(B,D))] :
      ( aa(product_prod(product_prod(A,C),product_prod(B,D)),$o,aa(fun(B,fun(D,$o)),fun(product_prod(product_prod(A,C),product_prod(B,D)),$o),aTP_Lamp_apd(fun(A,fun(C,$o)),fun(fun(B,fun(D,$o)),fun(product_prod(product_prod(A,C),product_prod(B,D)),$o)),Uu),Uua),Uub)
    <=> ( aa(set(product_prod(A,C)),$o,aa(set(product_prod(A,C)),fun(set(product_prod(A,C)),$o),ord_less_eq(set(product_prod(A,C))),basic_fsts(product_prod(A,C),product_prod(B,D),Uub)),aa(fun(product_prod(A,C),$o),set(product_prod(A,C)),collect(product_prod(A,C)),aa(fun(A,fun(C,$o)),fun(product_prod(A,C),$o),product_case_prod(A,C,$o),Uu)))
        & aa(set(product_prod(B,D)),$o,aa(set(product_prod(B,D)),fun(set(product_prod(B,D)),$o),ord_less_eq(set(product_prod(B,D))),basic_snds(product_prod(A,C),product_prod(B,D),Uub)),aa(fun(product_prod(B,D),$o),set(product_prod(B,D)),collect(product_prod(B,D)),aa(fun(B,fun(D,$o)),fun(product_prod(B,D),$o),product_case_prod(B,D,$o),Uua))) ) ) ).

% ATP.lambda_589
tff(fact_8369_ATP_Olambda__590,axiom,
    ! [A: $tType,B: $tType,D: $tType,C: $tType,Uu: fun(A,fun(B,$o)),Uua: fun(C,fun(D,$o)),Uub: product_prod(product_prod(A,B),product_prod(C,D))] :
      ( aa(product_prod(product_prod(A,B),product_prod(C,D)),$o,aa(fun(C,fun(D,$o)),fun(product_prod(product_prod(A,B),product_prod(C,D)),$o),aTP_Lamp_aov(fun(A,fun(B,$o)),fun(fun(C,fun(D,$o)),fun(product_prod(product_prod(A,B),product_prod(C,D)),$o)),Uu),Uua),Uub)
    <=> ( aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),basic_fsts(product_prod(A,B),product_prod(C,D),Uub)),aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),Uu)))
        & aa(set(product_prod(C,D)),$o,aa(set(product_prod(C,D)),fun(set(product_prod(C,D)),$o),ord_less_eq(set(product_prod(C,D))),basic_snds(product_prod(A,B),product_prod(C,D),Uub)),aa(fun(product_prod(C,D),$o),set(product_prod(C,D)),collect(product_prod(C,D)),aa(fun(C,fun(D,$o)),fun(product_prod(C,D),$o),product_case_prod(C,D,$o),Uua))) ) ) ).

% ATP.lambda_590
tff(fact_8370_ATP_Olambda__591,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: list(A),Uub: list(A)] :
      ( aa(list(A),$o,aa(list(A),fun(list(A),$o),aTP_Lamp_zv(set(product_prod(A,A)),fun(list(A),fun(list(A),$o)),Uu),Uua),Uub)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),Uua)),aa(list(A),nat,size_size(list(A)),Uub))
        | ( ( aa(list(A),nat,size_size(list(A)),Uua) = aa(list(A),nat,size_size(list(A)),Uub) )
          & aa(set(product_prod(list(A),list(A))),$o,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_591
tff(fact_8371_ATP_Olambda__592,axiom,
    ! [A: $tType,Uu: nat,Uua: list(A),Uub: list(A)] :
      ( aa(list(A),$o,aa(list(A),fun(list(A),$o),aTP_Lamp_aef(nat,fun(list(A),fun(list(A),$o)),Uu),Uua),Uub)
    <=> ( ( aa(list(A),nat,size_size(list(A)),Uua) = aa(nat,nat,suc,Uu) )
        & ( aa(list(A),nat,size_size(list(A)),Uub) = aa(nat,nat,suc,Uu) ) ) ) ).

% ATP.lambda_592
tff(fact_8372_ATP_Olambda__593,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: list(A),Uub: list(A)] :
      ( aa(list(A),$o,aa(list(A),fun(list(A),$o),aTP_Lamp_afu(set(product_prod(A,A)),fun(list(A),fun(list(A),$o)),Uu),Uua),Uub)
    <=> ( ( aa(list(A),nat,size_size(list(A)),Uua) = aa(list(A),nat,size_size(list(A)),Uub) )
        & ? [Xys2: list(A),X4: A,Y3: A,Xs6: list(A),Ys5: list(A)] :
            ( ( Uua = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xys2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs6)) )
            & ( Uub = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xys2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys5)) )
            & aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Y3)),Uu) ) ) ) ).

% ATP.lambda_593
tff(fact_8373_ATP_Olambda__594,axiom,
    ! [Uu: nat,Uua: nat,Uub: list(nat)] :
      ( aa(list(nat),$o,aa(nat,fun(list(nat),$o),aTP_Lamp_abk(nat,fun(nat,fun(list(nat),$o)),Uu),Uua),Uub)
    <=> ( ( aa(list(nat),nat,size_size(list(nat)),Uub) = Uu )
        & ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),groups8242544230860333062m_list(nat,Uub)),one_one(nat)) = Uua ) ) ) ).

% ATP.lambda_594
tff(fact_8374_ATP_Olambda__595,axiom,
    ! [A: $tType,Uu: nat,Uua: set(A),Uub: list(A)] :
      ( aa(list(A),$o,aa(set(A),fun(list(A),$o),aTP_Lamp_xd(nat,fun(set(A),fun(list(A),$o)),Uu),Uua),Uub)
    <=> ( ( aa(list(A),nat,size_size(list(A)),Uub) = Uu )
        & distinct(A,Uub)
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uub)),Uua) ) ) ).

% ATP.lambda_595
tff(fact_8375_ATP_Olambda__596,axiom,
    ! [A: $tType,Uu: set(A),Uua: nat,Uub: list(A)] :
      ( aa(list(A),$o,aa(nat,fun(list(A),$o),aTP_Lamp_xc(set(A),fun(nat,fun(list(A),$o)),Uu),Uua),Uub)
    <=> ( ( aa(list(A),nat,size_size(list(A)),Uub) = Uua )
        & distinct(A,Uub)
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uub)),Uu) ) ) ).

% ATP.lambda_596
tff(fact_8376_ATP_Olambda__597,axiom,
    ! [A: $tType,Uu: set(A),Uua: nat,Uub: list(A)] :
      ( aa(list(A),$o,aa(nat,fun(list(A),$o),aTP_Lamp_xn(set(A),fun(nat,fun(list(A),$o)),Uu),Uua),Uub)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uub)),Uu)
        & ( aa(list(A),nat,size_size(list(A)),Uub) = aa(nat,nat,suc,Uua) ) ) ) ).

% ATP.lambda_597
tff(fact_8377_ATP_Olambda__598,axiom,
    ! [A: $tType,Uu: nat,Uua: list(A),Uub: list(A)] :
      ( aa(list(A),$o,aa(list(A),fun(list(A),$o),aTP_Lamp_xm(nat,fun(list(A),fun(list(A),$o)),Uu),Uua),Uub)
    <=> ( ( aa(list(A),nat,size_size(list(A)),Uub) = Uu )
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uub)),aa(list(A),set(A),set2(A),Uua)) ) ) ).

% ATP.lambda_598
tff(fact_8378_ATP_Olambda__599,axiom,
    ! [A: $tType,B: $tType,Uu: set(A),Uua: set(B),Uub: sum_sum(A,B)] :
      ( aa(sum_sum(A,B),$o,aa(set(B),fun(sum_sum(A,B),$o),aTP_Lamp_axk(set(A),fun(set(B),fun(sum_sum(A,B),$o)),Uu),Uua),Uub)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(sum_sum(A,B),set(A),basic_setl(A,B),Uub)),Uu)
        & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(sum_sum(A,B),set(B),basic_setr(A,B),Uub)),Uua) ) ) ).

% ATP.lambda_599
tff(fact_8379_ATP_Olambda__600,axiom,
    ! [A: $tType,B: $tType,Uu: set(A),Uua: set(B),Uub: product_prod(A,B)] :
      ( aa(product_prod(A,B),$o,aa(set(B),fun(product_prod(A,B),$o),aTP_Lamp_api(set(A),fun(set(B),fun(product_prod(A,B),$o)),Uu),Uua),Uub)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),basic_fsts(A,B,Uub)),Uu)
        & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),basic_snds(A,B,Uub)),Uua) ) ) ).

% ATP.lambda_600
tff(fact_8380_ATP_Olambda__601,axiom,
    ! [A: $tType,Uu: set(A),Uua: nat,Uub: list(A)] :
      ( aa(list(A),$o,aa(nat,fun(list(A),$o),aTP_Lamp_xa(set(A),fun(nat,fun(list(A),$o)),Uu),Uua),Uub)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uub)),Uu)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Uub)),Uua) ) ) ).

% ATP.lambda_601
tff(fact_8381_ATP_Olambda__602,axiom,
    ! [A: $tType,Uu: set(A),Uua: nat,Uub: list(A)] :
      ( aa(list(A),$o,aa(nat,fun(list(A),$o),aTP_Lamp_xb(set(A),fun(nat,fun(list(A),$o)),Uu),Uua),Uub)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uub)),Uu)
        & ( aa(list(A),nat,size_size(list(A)),Uub) = Uua ) ) ) ).

% ATP.lambda_602
tff(fact_8382_ATP_Olambda__603,axiom,
    ! [Uu: nat,Uua: nat,Uub: list(nat)] :
      ( aa(list(nat),$o,aa(nat,fun(list(nat),$o),aTP_Lamp_abi(nat,fun(nat,fun(list(nat),$o)),Uu),Uua),Uub)
    <=> ( ( aa(list(nat),nat,size_size(list(nat)),Uub) = Uu )
        & ( groups8242544230860333062m_list(nat,Uub) = Uua ) ) ) ).

% ATP.lambda_603
tff(fact_8383_ATP_Olambda__604,axiom,
    ! [A: $tType,B: $tType,Uu: set(A),Uua: set(B),Uub: fun(A,option(B))] :
      ( aa(fun(A,option(B)),$o,aa(set(B),fun(fun(A,option(B)),$o),aTP_Lamp_adh(set(A),fun(set(B),fun(fun(A,option(B)),$o)),Uu),Uua),Uub)
    <=> ( ( dom(A,B,Uub) = Uu )
        & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),ran(A,B,Uub)),Uua) ) ) ).

% ATP.lambda_604
tff(fact_8384_ATP_Olambda__605,axiom,
    ! [Uu: set(nat),Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_ji(set(nat),fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> ( aa(set(nat),$o,member(nat,aa(nat,nat,suc,Uub)),Uu)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uub),Uua) ) ) ).

% ATP.lambda_605
tff(fact_8385_ATP_Olambda__606,axiom,
    ! [A: $tType,Uu: set(nat),Uua: nat,Uub: product_prod(A,nat)] :
      ( aa(product_prod(A,nat),$o,aa(nat,fun(product_prod(A,nat),$o),aTP_Lamp_adv(set(nat),fun(nat,fun(product_prod(A,nat),$o)),Uu),Uua),Uub)
    <=> aa(set(nat),$o,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_606
tff(fact_8386_ATP_Olambda__607,axiom,
    ! [A: $tType,Uu: set(list(A)),Uua: list(A),Uub: A] :
      ( aa(A,$o,aa(list(A),fun(A,$o),aTP_Lamp_yj(set(list(A)),fun(list(A),fun(A,$o)),Uu),Uua),Uub)
    <=> aa(set(list(A)),$o,member(list(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Uua),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uub),nil(A)))),Uu) ) ).

% ATP.lambda_607
tff(fact_8387_ATP_Olambda__608,axiom,
    ! [Uu: nat,Uua: nat,Uub: set(nat)] :
      ( aa(set(nat),$o,aa(nat,fun(set(nat),$o),aTP_Lamp_ot(nat,fun(nat,fun(set(nat),$o)),Uu),Uua),Uub)
    <=> ( aa(set(set(nat)),$o,member(set(nat),Uub),pow2(nat,set_or7035219750837199246ssThan(nat,zero_zero(nat),Uu)))
        & ( aa(set(nat),nat,finite_card(nat),Uub) = Uua ) ) ) ).

% ATP.lambda_608
tff(fact_8388_ATP_Olambda__609,axiom,
    ! [A: $tType,Uu: list(A),Uua: set(nat),Uub: nat] :
      ( aa(nat,$o,aa(set(nat),fun(nat,$o),aTP_Lamp_acu(list(A),fun(set(nat),fun(nat,$o)),Uu),Uua),Uub)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uub),aa(list(A),nat,size_size(list(A)),Uu))
        & aa(set(nat),$o,member(nat,Uub),Uua) ) ) ).

% ATP.lambda_609
tff(fact_8389_ATP_Olambda__610,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: list(A),Uub: nat] :
      ( aa(nat,$o,aa(list(A),fun(nat,$o),aTP_Lamp_zh(fun(A,$o),fun(list(A),fun(nat,$o)),Uu),Uua),Uub)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uub),aa(list(A),nat,size_size(list(A)),Uua))
        & aa(A,$o,Uu,aa(nat,A,nth(A,Uua),Uub)) ) ) ).

% ATP.lambda_610
tff(fact_8390_ATP_Olambda__611,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: multiset(A),Uub: A] :
      ( aa(A,$o,aa(multiset(A),fun(A,$o),aTP_Lamp_aql(fun(A,$o),fun(multiset(A),fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(set(A),$o,member(A,Uub),aa(multiset(A),set(A),set_mset(A),Uua))
        & aa(A,$o,Uu,Uub) ) ) ).

% ATP.lambda_611
tff(fact_8391_ATP_Olambda__612,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: list(A),Uua: fun(A,$o),Uub: A] :
          ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ya(list(A),fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(set(A),$o,member(A,Uub),aa(list(A),set(A),set2(A),Uu))
            & aa(A,$o,Uua,Uub) ) ) ) ).

% ATP.lambda_612
tff(fact_8392_ATP_Olambda__613,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: list(A),Uub: A] :
      ( aa(A,$o,aa(list(A),fun(A,$o),aTP_Lamp_yy(fun(A,$o),fun(list(A),fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(set(A),$o,member(A,Uub),aa(list(A),set(A),set2(A),Uua))
        & aa(A,$o,Uu,Uub) ) ) ).

% ATP.lambda_613
tff(fact_8393_ATP_Olambda__614,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,option(B)),Uua: A,Uub: product_prod(A,B)] :
      ( aa(product_prod(A,B),$o,aa(A,fun(product_prod(A,B),$o),aTP_Lamp_vo(fun(A,option(B)),fun(A,fun(product_prod(A,B),$o)),Uu),Uua),Uub)
    <=> ( aa(set(product_prod(A,B)),$o,member(product_prod(A,B),Uub),graph(A,B,Uu))
        & ( aa(product_prod(A,B),A,product_fst(A,B),Uub) != Uua ) ) ) ).

% ATP.lambda_614
tff(fact_8394_ATP_Olambda__615,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_gg(A,fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),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_615
tff(fact_8395_ATP_Olambda__616,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: set(A),Uub: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_agn(set(product_prod(A,A)),fun(set(A),fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(set(A),$o,member(A,Uub),field2(A,Uu))
        & ! [X4: A] :
            ( aa(set(A),$o,member(A,X4),Uua)
           => aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),X4)),Uu) ) ) ) ).

% ATP.lambda_616
tff(fact_8396_ATP_Olambda__617,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: set(A),Uub: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_ahc(set(product_prod(A,A)),fun(set(A),fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(set(A),$o,member(A,Uub),field2(A,Uu))
        & ! [X4: A] :
            ( aa(set(A),$o,member(A,X4),Uua)
           => aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Uub)),Uu) ) ) ) ).

% ATP.lambda_617
tff(fact_8397_ATP_Olambda__618,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: set(A),Uub: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_agm(set(product_prod(A,A)),fun(set(A),fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(set(A),$o,member(A,Uub),field2(A,Uu))
        & ! [X4: A] :
            ( aa(set(A),$o,member(A,X4),Uua)
           => ( ( Uub != X4 )
              & aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),X4)),Uu) ) ) ) ) ).

% ATP.lambda_618
tff(fact_8398_ATP_Olambda__619,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: set(A),Uub: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_alt(set(product_prod(A,A)),fun(set(A),fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(set(A),$o,member(A,Uub),field2(A,Uu))
        & ! [X4: A] :
            ( aa(set(A),$o,member(A,X4),Uua)
           => ( ( Uub != X4 )
              & aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Uub)),Uu) ) ) ) ) ).

% ATP.lambda_619
tff(fact_8399_ATP_Olambda__620,axiom,
    ! [A: $tType,Uu: list(A),Uua: set(nat),Uub: nat] :
      ( aa(nat,$o,aa(set(nat),fun(nat,$o),aTP_Lamp_acs(list(A),fun(set(nat),fun(nat,$o)),Uu),Uua),Uub)
    <=> aa(set(nat),$o,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_620
tff(fact_8400_ATP_Olambda__621,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_uw(nat,fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uua),Uu)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uub),Uu)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Uua),Uub) ) ) ).

% ATP.lambda_621
tff(fact_8401_ATP_Olambda__622,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [Uu: set(A),Uua: nat,Uub: A] :
          ( aa(A,$o,aa(nat,fun(A,$o),aTP_Lamp_vb(set(A),fun(nat,fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(set(A),$o,member(A,Uub),Uu)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,infini527867602293511546merate(A,Uu),Uua)),Uub) ) ) ) ).

% ATP.lambda_622
tff(fact_8402_ATP_Olambda__623,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_aon(set(A),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(set(A),$o,member(A,Uub),Uu)
            & ( aa(A,B,Uua,Uub) = aa(set(B),B,lattic643756798350308766er_Min(B),aa(set(A),set(B),image2(A,B,Uua),Uu)) ) ) ) ) ).

% ATP.lambda_623
tff(fact_8403_ATP_Olambda__624,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: set(A),Uub: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),aTP_Lamp_amh(set(product_prod(A,A)),fun(set(A),fun(set(A),$o)),Uu),Uua),Uub)
    <=> ( order_ofilter(A,Uu,Uua)
        & ( Uua != field2(A,Uu) )
        & order_ofilter(A,Uu,Uub)
        & ( Uub != field2(A,Uu) )
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),Uua),Uub) ) ) ).

% ATP.lambda_624
tff(fact_8404_ATP_Olambda__625,axiom,
    ! [Uu: set(nat),Uua: set(nat),Uub: nat] :
      ( aa(nat,$o,aa(set(nat),fun(nat,$o),aTP_Lamp_aif(set(nat),fun(set(nat),fun(nat,$o)),Uu),Uua),Uub)
    <=> ( aa(set(nat),$o,member(nat,Uub),Uu)
        & aa(set(nat),$o,member(nat,aa(set(nat),nat,finite_card(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_aie(set(nat),fun(nat,fun(nat,$o)),Uu),Uub)))),Uua) ) ) ).

% ATP.lambda_625
tff(fact_8405_ATP_Olambda__626,axiom,
    ! [A: $tType,Uu: set(A),Uua: nat,Uub: set(A)] :
      ( aa(set(A),$o,aa(nat,fun(set(A),$o),aTP_Lamp_jc(set(A),fun(nat,fun(set(A),$o)),Uu),Uua),Uub)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Uub),Uu)
        & ( aa(set(A),nat,finite_card(A),Uub) = Uua ) ) ) ).

% ATP.lambda_626
tff(fact_8406_ATP_Olambda__627,axiom,
    ! [A: $tType,B: $tType,Uu: nat,Uua: set(old_node(A,B)),Uub: old_node(A,B)] :
      ( aa(old_node(A,B),$o,aa(set(old_node(A,B)),fun(old_node(A,B),$o),aTP_Lamp_aum(nat,fun(set(old_node(A,B)),fun(old_node(A,B),$o)),Uu),Uua),Uub)
    <=> ( aa(set(old_node(A,B)),$o,member(old_node(A,B),Uub),Uua)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),old_ndepth(A,B,Uub)),Uu) ) ) ).

% ATP.lambda_627
tff(fact_8407_ATP_Olambda__628,axiom,
    ! [Uu: set(nat),Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_jh(set(nat),fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> ( aa(set(nat),$o,member(nat,Uub),Uu)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uub),aa(nat,nat,suc,Uua)) ) ) ).

% ATP.lambda_628
tff(fact_8408_ATP_Olambda__629,axiom,
    ! [A: $tType,Uu: multiset(A),Uua: multiset(A),Uub: A] :
      ( aa(A,$o,aa(multiset(A),fun(A,$o),aTP_Lamp_apr(multiset(A),fun(multiset(A),fun(A,$o)),Uu),Uua),Uub)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,aa(multiset(A),fun(A,nat),count(A),Uua),Uub)),aa(A,nat,aa(multiset(A),fun(A,nat),count(A),Uu),Uub)) ) ).

% ATP.lambda_629
tff(fact_8409_ATP_Olambda__630,axiom,
    ! [A: $tType,B: $tType,Uu: set(product_prod(A,A)),Uua: set(product_prod(A,A)),Uub: fun(B,A)] : aa(fun(B,A),product_prod(set(product_prod(B,B)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(fun(B,A),product_prod(set(product_prod(B,B)),set(product_prod(B,B)))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),fun(fun(B,A),product_prod(set(product_prod(B,B)),set(product_prod(B,B))))),aTP_Lamp_alw(set(product_prod(A,A)),fun(set(product_prod(A,A)),fun(fun(B,A),product_prod(set(product_prod(B,B)),set(product_prod(B,B)))))),Uu),Uua),Uub) = aa(set(product_prod(B,B)),product_prod(set(product_prod(B,B)),set(product_prod(B,B))),aa(set(product_prod(B,B)),fun(set(product_prod(B,B)),product_prod(set(product_prod(B,B)),set(product_prod(B,B)))),product_Pair(set(product_prod(B,B)),set(product_prod(B,B))),inv_image(A,B,Uu,Uub)),inv_image(A,B,Uua,Uub)) ).

% ATP.lambda_630
tff(fact_8410_ATP_Olambda__631,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_gb(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_631
tff(fact_8411_ATP_Olambda__632,axiom,
    ! [Uu: assn,Uua: assn,Uub: product_prod(heap_ext(product_unit),set(nat))] :
      ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),aTP_Lamp_ac(assn,fun(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o)),Uu),Uua),Uub)
    <=> ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,Uu),Uub)
        | aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,Uua),Uub) ) ) ).

% ATP.lambda_632
tff(fact_8412_ATP_Olambda__633,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A),Uub: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_an(set(A),fun(set(A),fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(set(A),$o,member(A,Uub),Uu)
        | aa(set(A),$o,member(A,Uub),Uua) ) ) ).

% ATP.lambda_633
tff(fact_8413_ATP_Olambda__634,axiom,
    ! [A: $tType,Uu: A,Uua: set(A),Uub: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_ch(A,fun(set(A),fun(A,$o)),Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        | aa(set(A),$o,member(A,Uub),Uua) ) ) ).

% ATP.lambda_634
tff(fact_8414_ATP_Olambda__635,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_ca(int,fun(int,fun(int,$o)),Uu),Uua),Uub)
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Uu),Uua)
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),Uua),Uub) ) ) ).

% ATP.lambda_635
tff(fact_8415_ATP_Olambda__636,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_io(int,fun(int,fun(int,$o)),Uu),Uua),Uub)
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Uu),Uub)
        & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Uub),Uua) ) ) ).

% ATP.lambda_636
tff(fact_8416_ATP_Olambda__637,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_bz(int,fun(int,fun(int,$o)),Uu),Uua),Uub)
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Uu),Uub)
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),Uua),Uub) ) ) ).

% ATP.lambda_637
tff(fact_8417_ATP_Olambda__638,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_iq(int,fun(int,fun(int,$o)),Uu),Uua),Uub)
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Uu),Uub)
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),Uub),Uua) ) ) ).

% ATP.lambda_638
tff(fact_8418_ATP_Olambda__639,axiom,
    ! [A: $tType,Uu: pred(A),Uua: pred(A),Uub: seq(A)] :
      ( aa(seq(A),$o,aa(pred(A),fun(seq(A),$o),aTP_Lamp_ayb(pred(A),fun(pred(A),fun(seq(A),$o)),Uu),Uua),Uub)
    <=> ( aa(pred(A),$o,aa(pred(A),fun(pred(A),$o),ord_less_eq(pred(A)),Uua),Uu)
        & contained(A,Uub,Uu) ) ) ).

% ATP.lambda_639
tff(fact_8419_ATP_Olambda__640,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_ir(int,fun(int,fun(int,$o)),Uu),Uua),Uub)
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Uu),Uub)
        & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Uub),Uua) ) ) ).

% ATP.lambda_640
tff(fact_8420_ATP_Olambda__641,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_ip(int,fun(int,fun(int,$o)),Uu),Uua),Uub)
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Uu),Uub)
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),Uub),Uua) ) ) ).

% ATP.lambda_641
tff(fact_8421_ATP_Olambda__642,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A),Uub: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),aTP_Lamp_vh(set(A),fun(set(A),fun(set(A),$o)),Uu),Uua),Uub)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),Uua),Uub)
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Uub),Uu) ) ) ).

% ATP.lambda_642
tff(fact_8422_ATP_Olambda__643,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A),Uub: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),aTP_Lamp_vg(set(A),fun(set(A),fun(set(A),$o)),Uu),Uua),Uub)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),Uub),Uua)
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Uua),Uu) ) ) ).

% ATP.lambda_643
tff(fact_8423_ATP_Olambda__644,axiom,
    ! [Uu: assn,Uua: assn,Uub: product_prod(heap_ext(product_unit),set(nat))] :
      ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),aTP_Lamp_ab(assn,fun(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o)),Uu),Uua),Uub)
    <=> ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,Uu),Uub)
        & aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,Uua),Uub) ) ) ).

% ATP.lambda_644
tff(fact_8424_ATP_Olambda__645,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_wi(nat,fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Uub),Uua)
        & aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Uub),Uu) ) ) ).

% ATP.lambda_645
tff(fact_8425_ATP_Olambda__646,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_wg(int,fun(int,fun(int,$o)),Uu),Uua),Uub)
    <=> ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),Uub),Uua)
        & aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),Uub),Uu) ) ) ).

% ATP.lambda_646
tff(fact_8426_ATP_Olambda__647,axiom,
    ! [A: $tType,Uu: pred(A),Uua: A,Uub: pred(A)] :
      ( aa(pred(A),$o,aa(A,fun(pred(A),$o),aTP_Lamp_aya(pred(A),fun(A,fun(pred(A),$o)),Uu),Uua),Uub)
    <=> ( aa(A,$o,eval(A,Uu),Uua)
        & aa(pred(A),$o,aa(pred(A),fun(pred(A),$o),ord_less_eq(pred(A)),Uub),Uu) ) ) ).

% ATP.lambda_647
tff(fact_8427_ATP_Olambda__648,axiom,
    ! [A: $tType,Uu: filter(A),Uua: filter(A),Uub: fun(A,$o)] :
      ( aa(fun(A,$o),$o,aa(filter(A),fun(fun(A,$o),$o),aTP_Lamp_aky(filter(A),fun(filter(A),fun(fun(A,$o),$o)),Uu),Uua),Uub)
    <=> ( eventually(A,Uub,Uu)
        & eventually(A,Uub,Uua) ) ) ).

% ATP.lambda_648
tff(fact_8428_ATP_Olambda__649,axiom,
    ! [A: $tType,Uu: set(A),Uua: A,Uub: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_ais(set(A),fun(A,fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(set(A),$o,member(A,Uua),Uu)
        & aa(set(A),$o,member(A,Uub),Uu) ) ) ).

% ATP.lambda_649
tff(fact_8429_ATP_Olambda__650,axiom,
    ! [Uu: set(nat),Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_aie(set(nat),fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> ( aa(set(nat),$o,member(nat,Uub),Uu)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uub),Uua) ) ) ).

% ATP.lambda_650
tff(fact_8430_ATP_Olambda__651,axiom,
    ! [A: $tType,Uu: set(set(A)),Uua: A,Uub: set(A)] :
      ( aa(set(A),$o,aa(A,fun(set(A),$o),aTP_Lamp_aoe(set(set(A)),fun(A,fun(set(A),$o)),Uu),Uua),Uub)
    <=> ( aa(set(set(A)),$o,member(set(A),Uub),Uu)
        & aa(set(A),$o,member(A,Uua),Uub) ) ) ).

% ATP.lambda_651
tff(fact_8431_ATP_Olambda__652,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A),Uub: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_al(set(A),fun(set(A),fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(set(A),$o,member(A,Uub),Uu)
        & aa(set(A),$o,member(A,Uub),Uua) ) ) ).

% ATP.lambda_652
tff(fact_8432_ATP_Olambda__653,axiom,
    ! [A: $tType,Uu: A,Uua: A,Uub: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_avm(A,fun(A,fun(A,$o)),Uu),Uua),Uub)
    <=> ( ( Uua = Uu )
        & ( Uub = Uu ) ) ) ).

% ATP.lambda_653
tff(fact_8433_ATP_Olambda__654,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A),Uub: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),aTP_Lamp_gx(set(A),fun(set(A),fun(set(A),$o)),Uu),Uua),Uub)
    <=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Uub),Uu) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Uua),Uu) ) ) ).

% ATP.lambda_654
tff(fact_8434_ATP_Olambda__655,axiom,
    ! [A: $tType,Uu: list(A),Uua: fun(A,nat),Uub: A] : aa(A,nat,aa(fun(A,nat),fun(A,nat),aTP_Lamp_abo(list(A),fun(fun(A,nat),fun(A,nat)),Uu),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),count_list(A,Uu,Uub)),aa(A,nat,Uua,Uub)) ).

% ATP.lambda_655
tff(fact_8435_ATP_Olambda__656,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: list(A),Uub: A] : aa(A,nat,aa(list(A),fun(A,nat),aTP_Lamp_abp(fun(A,nat),fun(list(A),fun(A,nat)),Uu),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),count_list(A,Uua,Uub)),aa(A,nat,Uu,Uub)) ).

% ATP.lambda_656
tff(fact_8436_ATP_Olambda__657,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: set(A),Uub: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_akm(fun(A,$o),fun(set(A),fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(set(A),$o,member(A,Uub),Uua)
       => aa(A,$o,Uu,Uub) ) ) ).

% ATP.lambda_657
tff(fact_8437_ATP_Olambda__658,axiom,
    ! [A: $tType,Uu: A,Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_asz(A,fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> ( ( Uu = Uub )
       => aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_658
tff(fact_8438_ATP_Olambda__659,axiom,
    ! [A: $tType,Uu: A,Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_asy(A,fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
       => aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_659
tff(fact_8439_ATP_Olambda__660,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,fun(A,A)),Uua: multiset(B),Uub: B] : aa(B,fun(A,A),aa(multiset(B),fun(B,fun(A,A)),aTP_Lamp_apv(fun(B,fun(A,A)),fun(multiset(B),fun(B,fun(A,A))),Uu),Uua),Uub) = compow(fun(A,A),aa(B,nat,aa(multiset(B),fun(B,nat),count(B),Uua),Uub),aa(B,fun(A,A),Uu,Uub)) ).

% ATP.lambda_660
tff(fact_8440_ATP_Olambda__661,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu: set(A),Uua: fun(A,$o),Uub: A] :
          ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_vk(set(A),fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(set(A),$o,member(A,Uub),Uu)
            & aa(A,$o,Uua,Uub) ) ) ) ).

% ATP.lambda_661
tff(fact_8441_ATP_Olambda__662,axiom,
    ! [A: $tType,Uu: set(A),Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_bi(set(A),fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(set(A),$o,member(A,Uub),Uu)
        & aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_662
tff(fact_8442_ATP_Olambda__663,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: set(A),Uub: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_on(fun(A,$o),fun(set(A),fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(set(A),$o,member(A,Uub),Uua)
        & aa(A,$o,Uu,Uub) ) ) ).

% ATP.lambda_663
tff(fact_8443_ATP_Olambda__664,axiom,
    ! [A: $tType,Uu: A,Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ci(A,fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> ( ( Uu = Uub )
        & aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_664
tff(fact_8444_ATP_Olambda__665,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: A,Uub: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_bu(fun(A,$o),fun(A,fun(A,$o)),Uu),Uua),Uub)
    <=> ( ( Uua = Uub )
        & aa(A,$o,Uu,Uua) ) ) ).

% ATP.lambda_665
tff(fact_8445_ATP_Olambda__666,axiom,
    ! [A: $tType,Uu: A,Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_cj(A,fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        & aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_666
tff(fact_8446_ATP_Olambda__667,axiom,
    ! [A: $tType,B: $tType,Uu: set(A),Uua: fun(A,set(B)),Uub: A] :
      ( aa(A,$o,aa(fun(A,set(B)),fun(A,$o),aTP_Lamp_ro(set(A),fun(fun(A,set(B)),fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(set(A),$o,member(A,Uub),Uu)
        & ( aa(A,set(B),Uua,Uub) != bot_bot(set(B)) ) ) ) ).

% ATP.lambda_667
tff(fact_8447_ATP_Olambda__668,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_hj(set(A),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(set(A),$o,member(A,Uub),Uu)
            & ( aa(A,B,Uua,Uub) != zero_zero(B) ) ) ) ) ).

% ATP.lambda_668
tff(fact_8448_ATP_Olambda__669,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_add(B)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_ix(set(A),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(set(A),$o,member(A,Uub),Uu)
            & ( aa(A,B,Uua,Uub) != zero_zero(B) ) ) ) ) ).

% ATP.lambda_669
tff(fact_8449_ATP_Olambda__670,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_hl(set(A),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(set(A),$o,member(A,Uub),Uu)
            & ( aa(A,B,Uua,Uub) != one_one(B) ) ) ) ) ).

% ATP.lambda_670
tff(fact_8450_ATP_Olambda__671,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(B,A),Uua: set(B),Uub: B] :
          ( aa(B,$o,aa(set(B),fun(B,$o),aTP_Lamp_jd(fun(B,A),fun(set(B),fun(B,$o)),Uu),Uua),Uub)
        <=> ( aa(set(B),$o,member(B,Uub),Uua)
            & ( aa(B,A,Uu,Uub) != zero_zero(A) ) ) ) ) ).

% ATP.lambda_671
tff(fact_8451_ATP_Olambda__672,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(B,A),Uua: set(B),Uub: B] :
          ( aa(B,$o,aa(set(B),fun(B,$o),aTP_Lamp_sj(fun(B,A),fun(set(B),fun(B,$o)),Uu),Uua),Uub)
        <=> ( aa(set(B),$o,member(B,Uub),Uua)
            & ( aa(B,A,Uu,Uub) != one_one(A) ) ) ) ) ).

% ATP.lambda_672
tff(fact_8452_ATP_Olambda__673,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_parity(B)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_iy(set(A),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(set(A),$o,member(A,Uub),Uu)
            & ~ aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),aa(A,B,Uua,Uub)) ) ) ) ).

% ATP.lambda_673
tff(fact_8453_ATP_Olambda__674,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A),Uub: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_bc(set(A),fun(set(A),fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(set(A),$o,member(A,Uub),Uu)
        & ~ aa(set(A),$o,member(A,Uub),Uua) ) ) ).

% ATP.lambda_674
tff(fact_8454_ATP_Olambda__675,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: multiset(A),Uub: A] : aa(A,nat,aa(multiset(A),fun(A,nat),aTP_Lamp_apw(fun(A,nat),fun(multiset(A),fun(A,nat)),Uu),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(A,nat,aa(multiset(A),fun(A,nat),count(A),Uua),Uub)),aa(nat,nat,suc,aa(A,nat,Uu,Uub))) ).

% ATP.lambda_675
tff(fact_8455_ATP_Olambda__676,axiom,
    ! [A: $tType,D: $tType,C: $tType,Uu: fun(C,fun(D,$o)),Uua: fun(C,A),Uub: set(A)] : aa(set(A),set(C),aa(fun(C,A),fun(set(A),set(C)),aTP_Lamp_arr(fun(C,fun(D,$o)),fun(fun(C,A),fun(set(A),set(C))),Uu),Uua),Uub) = aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),inf_inf(set(C)),aa(set(A),set(C),aa(fun(C,A),fun(set(A),set(C)),vimage(C,A),Uua),Uub)),aa(fun(C,$o),set(C),collect(C),aa(fun(C,fun(D,$o)),fun(C,$o),domainp(C,D),Uu))) ).

% ATP.lambda_676
tff(fact_8456_ATP_Olambda__677,axiom,
    ! [A: $tType,B: $tType,Uu: list(product_prod(A,B)),Uua: A,Uub: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_abm(list(product_prod(A,B)),fun(A,fun(B,$o)),Uu),Uua),Uub)
    <=> ( aa(A,option(B),map_of(A,B,Uu),Uua) = aa(B,option(B),some(B),Uub) ) ) ).

% ATP.lambda_677
tff(fact_8457_ATP_Olambda__678,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_sv(nat,fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua)),Uu) ) ).

% ATP.lambda_678
tff(fact_8458_ATP_Olambda__679,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_eq(nat,fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uub)),Uu) ) ).

% ATP.lambda_679
tff(fact_8459_ATP_Olambda__680,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_go(nat,fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uub)),Uu) ) ).

% ATP.lambda_680
tff(fact_8460_ATP_Olambda__681,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_ni(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),divide_divide(A),Uub),Uu)),Uua) ) ).

% ATP.lambda_681
tff(fact_8461_ATP_Olambda__682,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_ng(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_682
tff(fact_8462_ATP_Olambda__683,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_nh(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),divide_divide(A),Uub),Uu)),Uua) ) ).

% ATP.lambda_683
tff(fact_8463_ATP_Olambda__684,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_nf(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_684
tff(fact_8464_ATP_Olambda__685,axiom,
    ! [B: $tType,C: $tType,Uu: set(product_prod(C,B)),Uua: C,Uub: B] :
      ( aa(B,$o,aa(C,fun(B,$o),aTP_Lamp_apf(set(product_prod(C,B)),fun(C,fun(B,$o)),Uu),Uua),Uub)
    <=> aa(set(product_prod(C,B)),$o,member(product_prod(C,B),aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),Uua),Uub)),Uu) ) ).

% ATP.lambda_685
tff(fact_8465_ATP_Olambda__686,axiom,
    ! [C: $tType,B: $tType,Uu: set(product_prod(B,C)),Uua: B,Uub: C] :
      ( aa(C,$o,aa(B,fun(C,$o),aTP_Lamp_apj(set(product_prod(B,C)),fun(B,fun(C,$o)),Uu),Uua),Uub)
    <=> aa(set(product_prod(B,C)),$o,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_686
tff(fact_8466_ATP_Olambda__687,axiom,
    ! [A: $tType,B: $tType,Uu: set(product_prod(B,A)),Uua: B,Uub: A] :
      ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_wv(set(product_prod(B,A)),fun(B,fun(A,$o)),Uu),Uua),Uub)
    <=> aa(set(product_prod(B,A)),$o,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_687
tff(fact_8467_ATP_Olambda__688,axiom,
    ! [C: $tType,A: $tType,Uu: set(product_prod(A,C)),Uua: A,Uub: C] :
      ( aa(C,$o,aa(A,fun(C,$o),aTP_Lamp_ape(set(product_prod(A,C)),fun(A,fun(C,$o)),Uu),Uua),Uub)
    <=> aa(set(product_prod(A,C)),$o,member(product_prod(A,C),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),Uua),Uub)),Uu) ) ).

% ATP.lambda_688
tff(fact_8468_ATP_Olambda__689,axiom,
    ! [B: $tType,A: $tType,Uu: set(product_prod(A,B)),Uua: A,Uub: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_as(set(product_prod(A,B)),fun(A,fun(B,$o))),Uu),Uua),Uub)
    <=> aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),Uub)),Uu) ) ).

% ATP.lambda_689
tff(fact_8469_ATP_Olambda__690,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_ux(set(product_prod(A,A)),fun(A,fun(A,$o)),Uu),Uua),Uub)
    <=> aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uua),Uub)),Uu) ) ).

% ATP.lambda_690
tff(fact_8470_ATP_Olambda__691,axiom,
    ! [A: $tType,B: $tType,Uu: set(product_prod(B,A)),Uua: A,Uub: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_aoh(set(product_prod(B,A)),fun(A,fun(B,$o)),Uu),Uua),Uub)
    <=> aa(set(product_prod(B,A)),$o,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_691
tff(fact_8471_ATP_Olambda__692,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_amk(set(product_prod(A,A)),fun(A,fun(A,$o)),Uu),Uua),Uub)
    <=> aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),Uua)),Uu) ) ).

% ATP.lambda_692
tff(fact_8472_ATP_Olambda__693,axiom,
    ! [A: $tType,Uu: set(list(A)),Uua: A,Uub: list(A)] :
      ( aa(list(A),$o,aa(A,fun(list(A),$o),aTP_Lamp_avo(set(list(A)),fun(A,fun(list(A),$o)),Uu),Uua),Uub)
    <=> aa(set(list(A)),$o,member(list(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uua),Uub)),Uu) ) ).

% ATP.lambda_693
tff(fact_8473_ATP_Olambda__694,axiom,
    ! [A: $tType,Uu: list(list(A)),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_aaz(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_694
tff(fact_8474_ATP_Olambda__695,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Uu: A,Uua: A,Uub: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_sk(A,fun(A,fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(set(A),$o,member(A,Uub),ring_1_Ints(A))
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uu),Uub)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uub),Uua) ) ) ) ).

% ATP.lambda_695
tff(fact_8475_ATP_Olambda__696,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Uu: fun(A,B),Uua: list(A),Uub: A] :
          ( aa(A,$o,aa(list(A),fun(A,$o),aTP_Lamp_yw(fun(A,B),fun(list(A),fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,Uu,aa(nat,A,nth(A,Uua),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(list(A),nat,size_size(list(A)),Uua)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),aa(A,B,Uu,Uub)) ) ) ).

% ATP.lambda_696
tff(fact_8476_ATP_Olambda__697,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: fun(B,A),Uub: A] :
      ( aa(A,$o,aa(fun(B,A),fun(A,$o),aTP_Lamp_ank(fun(A,B),fun(fun(B,A),fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(B,A,Uua,aa(A,B,Uu,Uub)) = Uub ) ) ).

% ATP.lambda_697
tff(fact_8477_ATP_Olambda__698,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,$o),Uua: fun(B,$o),Uub: product_prod(A,B)] :
      ( aa(product_prod(A,B),$o,aa(fun(B,$o),fun(product_prod(A,B),$o),aTP_Lamp_anu(fun(A,$o),fun(fun(B,$o),fun(product_prod(A,B),$o)),Uu),Uua),Uub)
    <=> ( aa(A,$o,Uu,aa(product_prod(A,B),A,product_fst(A,B),Uub))
        & aa(B,$o,Uua,aa(product_prod(A,B),B,product_snd(A,B),Uub)) ) ) ).

% ATP.lambda_698
tff(fact_8478_ATP_Olambda__699,axiom,
    ! [Uu: fun(nat,$o),Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_gy(fun(nat,$o),fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> ( aa(nat,$o,Uu,Uub)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uub),Uua) ) ) ).

% ATP.lambda_699
tff(fact_8479_ATP_Olambda__700,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,fun(B,$o)),Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ark(fun(A,fun(B,$o)),fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(A,$o,Uua,Uub)
        & aa(A,$o,aa(fun(A,fun(B,$o)),fun(A,$o),domainp(A,B),Uu),Uub) ) ) ).

% ATP.lambda_700
tff(fact_8480_ATP_Olambda__701,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Uu: fun(A,B),Uua: list(A),Uub: A] :
          ( aa(A,$o,aa(list(A),fun(A,$o),aTP_Lamp_yu(fun(A,B),fun(list(A),fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,Uu,Uub)),aa(A,B,Uu,aa(nat,A,nth(A,Uua),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(list(A),nat,size_size(list(A)),Uua)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ) ) ).

% ATP.lambda_701
tff(fact_8481_ATP_Olambda__702,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Uu: fun(A,B),Uua: list(A),Uub: A] :
          ( aa(A,$o,aa(list(A),fun(A,$o),aTP_Lamp_yv(fun(A,B),fun(list(A),fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(A,B,Uu,Uub) = aa(A,B,Uu,aa(nat,A,nth(A,Uua),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(list(A),nat,size_size(list(A)),Uua)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ) ) ) ).

% ATP.lambda_702
tff(fact_8482_ATP_Olambda__703,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( comm_monoid_mult(C)
     => ! [Uu: fun(A,set(B)),Uua: fun(A,fun(B,C)),Uub: A] : aa(A,C,aa(fun(A,fun(B,C)),fun(A,C),aTP_Lamp_sa(fun(A,set(B)),fun(fun(A,fun(B,C)),fun(A,C)),Uu),Uua),Uub) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),aa(A,fun(B,C),Uua,Uub)),aa(A,set(B),Uu,Uub)) ) ).

% ATP.lambda_703
tff(fact_8483_ATP_Olambda__704,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( comm_monoid_add(C)
     => ! [Uu: fun(A,set(B)),Uua: fun(A,fun(B,C)),Uub: A] : aa(A,C,aa(fun(A,fun(B,C)),fun(A,C),aTP_Lamp_rz(fun(A,set(B)),fun(fun(A,fun(B,C)),fun(A,C)),Uu),Uua),Uub) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),aa(A,fun(B,C),Uua,Uub)),aa(A,set(B),Uu,Uub)) ) ).

% ATP.lambda_704
tff(fact_8484_ATP_Olambda__705,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [Uu: fun(B,A),Uua: B,Uub: B] :
          ( aa(B,$o,aa(B,fun(B,$o),aTP_Lamp_aab(fun(B,A),fun(B,fun(B,$o)),Uu),Uua),Uub)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,Uu,Uua)),aa(B,A,Uu,Uub)) ) ) ).

% ATP.lambda_705
tff(fact_8485_ATP_Olambda__706,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_alf(fun(A,B),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ) ).

% ATP.lambda_706
tff(fact_8486_ATP_Olambda__707,axiom,
    ! [A: $tType,B: $tType] :
      ( field(A)
     => ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_fi(fun(B,A),fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(B,A,Uu,Uub)),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_707
tff(fact_8487_ATP_Olambda__708,axiom,
    ! [B: $tType,C: $tType,A: $tType,Uu: fun(A,fun(C,B)),Uua: fun(A,option(C)),Uub: A] : aa(A,option(B),aa(fun(A,option(C)),fun(A,option(B)),aTP_Lamp_ade(fun(A,fun(C,B)),fun(fun(A,option(C)),fun(A,option(B))),Uu),Uua),Uub) = aa(option(C),option(B),map_option(C,B,aa(A,fun(C,B),Uu,Uub)),aa(A,option(C),Uua,Uub)) ).

% ATP.lambda_708
tff(fact_8488_ATP_Olambda__709,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_fh(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_709
tff(fact_8489_ATP_Olambda__710,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_sl(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_710
tff(fact_8490_ATP_Olambda__711,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_ct(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_711
tff(fact_8491_ATP_Olambda__712,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_dd(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_712
tff(fact_8492_ATP_Olambda__713,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_rg(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_713
tff(fact_8493_ATP_Olambda__714,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: fun(A,nat),Uub: A] : aa(A,nat,aa(fun(A,nat),fun(A,nat),aa(fun(A,nat),fun(fun(A,nat),fun(A,nat)),aTP_Lamp_aqa(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,Uu,Uub)),aa(A,nat,Uua,Uub)) ).

% ATP.lambda_714
tff(fact_8494_ATP_Olambda__715,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_hy(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_715
tff(fact_8495_ATP_Olambda__716,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_dg(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_716
tff(fact_8496_ATP_Olambda__717,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,set(A)),Uua: fun(B,set(A)),Uub: B] : aa(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_ms(fun(B,set(A)),fun(fun(B,set(A)),fun(B,set(A))),Uu),Uua),Uub) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(B,set(A),Uu,Uub)),aa(B,set(A),Uua,Uub)) ).

% ATP.lambda_717
tff(fact_8497_ATP_Olambda__718,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_mm(fun(B,A),fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(B,A,Uu,Uub)),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_718
tff(fact_8498_ATP_Olambda__719,axiom,
    ! [A: $tType,B: $tType] :
      ( lattice(A)
     => ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_ama(fun(B,A),fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(B,A,Uu,Uub)),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_719
tff(fact_8499_ATP_Olambda__720,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_rm(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_720
tff(fact_8500_ATP_Olambda__721,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_amd(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),sup_sup(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_721
tff(fact_8501_ATP_Olambda__722,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,set(A)),Uua: fun(B,set(A)),Uub: B] : aa(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_kz(fun(B,set(A)),fun(fun(B,set(A)),fun(B,set(A))),Uu),Uua),Uub) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(B,set(A),Uu,Uub)),aa(B,set(A),Uua,Uub)) ).

% ATP.lambda_722
tff(fact_8502_ATP_Olambda__723,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_la(fun(B,A),fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(B,A,Uu,Uub)),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_723
tff(fact_8503_ATP_Olambda__724,axiom,
    ! [A: $tType,B: $tType] :
      ( lattice(A)
     => ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_amj(fun(B,A),fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(B,A,Uu,Uub)),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_724
tff(fact_8504_ATP_Olambda__725,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_rk(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_725
tff(fact_8505_ATP_Olambda__726,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_aml(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_726
tff(fact_8506_ATP_Olambda__727,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_cy(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_727
tff(fact_8507_ATP_Olambda__728,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: fun(A,nat),Uub: A] : aa(A,nat,aa(fun(A,nat),fun(A,nat),aa(fun(A,nat),fun(fun(A,nat),fun(A,nat)),aTP_Lamp_apz(fun(A,nat),fun(fun(A,nat),fun(A,nat))),Uu),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,Uu,Uub)),aa(A,nat,Uua,Uub)) ).

% ATP.lambda_728
tff(fact_8508_ATP_Olambda__729,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_jg(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_729
tff(fact_8509_ATP_Olambda__730,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,multiset(A)),Uua: fun(B,multiset(A)),Uub: B] : aa(B,multiset(A),aa(fun(B,multiset(A)),fun(B,multiset(A)),aTP_Lamp_asf(fun(B,multiset(A)),fun(fun(B,multiset(A)),fun(B,multiset(A))),Uu),Uua),Uub) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),union_mset(A),aa(B,multiset(A),Uu,Uub)),aa(B,multiset(A),Uua,Uub)) ).

% ATP.lambda_730
tff(fact_8510_ATP_Olambda__731,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,multiset(B)),Uua: fun(A,multiset(B)),Uub: A] : aa(A,multiset(B),aa(fun(A,multiset(B)),fun(A,multiset(B)),aTP_Lamp_ash(fun(A,multiset(B)),fun(fun(A,multiset(B)),fun(A,multiset(B))),Uu),Uua),Uub) = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),union_mset(B),aa(A,multiset(B),Uu,Uub)),aa(A,multiset(B),Uua,Uub)) ).

% ATP.lambda_731
tff(fact_8511_ATP_Olambda__732,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,multiset(A)),Uua: fun(B,multiset(A)),Uub: B] : aa(B,multiset(A),aa(fun(B,multiset(A)),fun(B,multiset(A)),aTP_Lamp_asl(fun(B,multiset(A)),fun(fun(B,multiset(A)),fun(B,multiset(A))),Uu),Uua),Uub) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),aa(B,multiset(A),Uu,Uub)),aa(B,multiset(A),Uua,Uub)) ).

% ATP.lambda_732
tff(fact_8512_ATP_Olambda__733,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,multiset(B)),Uua: fun(A,multiset(B)),Uub: A] : aa(A,multiset(B),aa(fun(A,multiset(B)),fun(A,multiset(B)),aTP_Lamp_asm(fun(A,multiset(B)),fun(fun(A,multiset(B)),fun(A,multiset(B))),Uu),Uua),Uub) = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),inter_mset(B),aa(A,multiset(B),Uu,Uub)),aa(A,multiset(B),Uua,Uub)) ).

% ATP.lambda_733
tff(fact_8513_ATP_Olambda__734,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_qv(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_734
tff(fact_8514_ATP_Olambda__735,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_anr(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_735
tff(fact_8515_ATP_Olambda__736,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ao(fun(A,$o),fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(A,$o,Uu,Uub)
       => aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_736
tff(fact_8516_ATP_Olambda__737,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,set(B)),Uua: A,Uub: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_ajy(fun(A,set(B)),fun(A,fun(A,$o)),Uu),Uua),Uub)
    <=> aa(set(B),$o,aa(set(B),fun(set(B),$o),disjnt(B),aa(A,set(B),Uu,Uua)),aa(A,set(B),Uu,Uub)) ) ).

% ATP.lambda_737
tff(fact_8517_ATP_Olambda__738,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,fun(B,B)),Uua: fun(A,nat),Uub: A] : aa(A,fun(B,B),aa(fun(A,nat),fun(A,fun(B,B)),aTP_Lamp_yf(fun(A,fun(B,B)),fun(fun(A,nat),fun(A,fun(B,B))),Uu),Uua),Uub) = compow(fun(B,B),aa(A,nat,Uua,Uub),aa(A,fun(B,B),Uu,Uub)) ).

% ATP.lambda_738
tff(fact_8518_ATP_Olambda__739,axiom,
    ! [Uu: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Uua: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Uub: product_prod(heap_ext(product_unit),set(nat))] :
      ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(heap_ext(product_unit),set(nat)),$o),aTP_Lamp_ae(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(heap_ext(product_unit),set(nat)),$o)),Uu),Uua),Uub)
    <=> ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,Uu,Uub)
        | aa(product_prod(heap_ext(product_unit),set(nat)),$o,Uua,Uub) ) ) ).

% ATP.lambda_739
tff(fact_8519_ATP_Olambda__740,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ap(fun(A,$o),fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(A,$o,Uu,Uub)
        | aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_740
tff(fact_8520_ATP_Olambda__741,axiom,
    ! [Uu: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Uua: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Uub: product_prod(heap_ext(product_unit),set(nat))] :
      ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(heap_ext(product_unit),set(nat)),$o),aTP_Lamp_af(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(heap_ext(product_unit),set(nat)),$o)),Uu),Uua),Uub)
    <=> ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,Uu,Uub)
        & aa(product_prod(heap_ext(product_unit),set(nat)),$o,Uua,Uub) ) ) ).

% ATP.lambda_741
tff(fact_8521_ATP_Olambda__742,axiom,
    ! [B: $tType,Uu: fun(B,$o),Uua: fun(B,$o),Uub: B] :
      ( aa(B,$o,aa(fun(B,$o),fun(B,$o),aTP_Lamp_aax(fun(B,$o),fun(fun(B,$o),fun(B,$o)),Uu),Uua),Uub)
    <=> ( aa(B,$o,Uu,Uub)
        & aa(B,$o,Uua,Uub) ) ) ).

% ATP.lambda_742
tff(fact_8522_ATP_Olambda__743,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_am(fun(A,$o),fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(A,$o,Uu,Uub)
        & aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_743
tff(fact_8523_ATP_Olambda__744,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_yx(fun(A,$o),fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(A,$o,Uua,Uub)
        & aa(A,$o,Uu,Uub) ) ) ).

% ATP.lambda_744
tff(fact_8524_ATP_Olambda__745,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Uu: fun(A,B),Uua: A,Uub: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_ajc(fun(A,B),fun(A,fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(A,B,Uu,Uua) = aa(A,B,Uu,Uub) ) ) ) ).

% ATP.lambda_745
tff(fact_8525_ATP_Olambda__746,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: A,Uub: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_acn(fun(A,B),fun(A,fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(A,B,Uu,Uua) = aa(A,B,Uu,Uub) ) ) ).

% ATP.lambda_746
tff(fact_8526_ATP_Olambda__747,axiom,
    ! [D: $tType,B: $tType,Uu: fun(B,D),Uua: fun(B,D),Uub: B] :
      ( aa(B,$o,aa(fun(B,D),fun(B,$o),aTP_Lamp_arp(fun(B,D),fun(fun(B,D),fun(B,$o)),Uu),Uua),Uub)
    <=> ( aa(B,D,Uu,Uub) = aa(B,D,Uua,Uub) ) ) ).

% ATP.lambda_747
tff(fact_8527_ATP_Olambda__748,axiom,
    ! [C: $tType,B: $tType,Uu: fun(B,C),Uua: fun(B,C),Uub: B] :
      ( aa(B,$o,aa(fun(B,C),fun(B,$o),aTP_Lamp_arh(fun(B,C),fun(fun(B,C),fun(B,$o)),Uu),Uua),Uub)
    <=> ( aa(B,C,Uu,Uub) = aa(B,C,Uua,Uub) ) ) ).

% ATP.lambda_748
tff(fact_8528_ATP_Olambda__749,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: fun(B,A),Uub: B] :
      ( aa(B,$o,aa(fun(B,A),fun(B,$o),aTP_Lamp_akc(fun(B,A),fun(fun(B,A),fun(B,$o)),Uu),Uua),Uub)
    <=> ( aa(B,A,Uu,Uub) = aa(B,A,Uua,Uub) ) ) ).

% ATP.lambda_749
tff(fact_8529_ATP_Olambda__750,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_akj(fun(A,$o),fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(A,$o,Uu,Uub)
      <=> aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_750
tff(fact_8530_ATP_Olambda__751,axiom,
    ! [C: $tType,A: $tType,Uu: fun(A,C),Uua: fun(A,C),Uub: A] :
      ( aa(A,$o,aa(fun(A,C),fun(A,$o),aTP_Lamp_aro(fun(A,C),fun(fun(A,C),fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(A,C,Uu,Uub) = aa(A,C,Uua,Uub) ) ) ).

% ATP.lambda_751
tff(fact_8531_ATP_Olambda__752,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: fun(A,B),Uub: A] :
      ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_ck(fun(A,B),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(A,B,Uu,Uub) = aa(A,B,Uua,Uub) ) ) ).

% ATP.lambda_752
tff(fact_8532_ATP_Olambda__753,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Uu: A,Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_abs(A,fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(A,B,Uua,Uu) = aa(A,B,Uua,Uub) ) ) ) ).

% ATP.lambda_753
tff(fact_8533_ATP_Olambda__754,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(B)
     => ! [Uu: fun(A,B),Uua: fun(A,$o),Uub: A] : aa(A,B,aa(fun(A,$o),fun(A,B),aTP_Lamp_gu(fun(A,B),fun(fun(A,$o),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),aa($o,B,zero_neq_one_of_bool(B),aa(A,$o,Uua,Uub))) ) ).

% ATP.lambda_754
tff(fact_8534_ATP_Olambda__755,axiom,
    ! [A: $tType,Uu: fun(A,fun(A,$o)),Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_air(fun(A,fun(A,$o)),fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(A,$o,Uua,Uub)
        & ! [Y3: A] :
            ( aa(A,$o,Uua,Y3)
           => aa(A,$o,aa(A,fun(A,$o),Uu,Uub),Y3) ) ) ) ).

% ATP.lambda_755
tff(fact_8535_ATP_Olambda__756,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,option(B)),Uua: A,Uub: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_ahg(fun(A,option(B)),fun(A,fun(B,$o)),Uu),Uua),Uub)
    <=> ( aa(A,option(B),Uu,Uua) = aa(B,option(B),some(B),Uub) ) ) ).

% ATP.lambda_756
tff(fact_8536_ATP_Olambda__757,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(B,fun(C,A)),Uua: set(C),Uub: B] : aa(B,A,aa(set(C),fun(B,A),aTP_Lamp_fe(fun(B,fun(C,A)),fun(set(C),fun(B,A)),Uu),Uua),Uub) = aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7121269368397514597t_prod(C,A),aa(B,fun(C,A),Uu,Uub)),Uua) ) ).

% ATP.lambda_757
tff(fact_8537_ATP_Olambda__758,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(B,fun(C,A)),Uua: set(C),Uub: B] : aa(B,A,aa(set(C),fun(B,A),aTP_Lamp_cv(fun(B,fun(C,A)),fun(set(C),fun(B,A)),Uu),Uua),Uub) = aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7311177749621191930dd_sum(C,A),aa(B,fun(C,A),Uu,Uub)),Uua) ) ).

% ATP.lambda_758
tff(fact_8538_ATP_Olambda__759,axiom,
    ! [Uu: fun(nat,nat),Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_gz(fun(nat,nat),fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,Uu,Uub)),Uua) ) ).

% ATP.lambda_759
tff(fact_8539_ATP_Olambda__760,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,set(A)),Uua: set(A),Uub: B] :
      ( aa(B,$o,aa(set(A),fun(B,$o),aTP_Lamp_mp(fun(B,set(A)),fun(set(A),fun(B,$o)),Uu),Uua),Uub)
    <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(B,set(A),Uu,Uub)),Uua) ) ).

% ATP.lambda_760
tff(fact_8540_ATP_Olambda__761,axiom,
    ! [A: $tType,B: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_akr(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,Uu,Uub)),Uua) ) ) ).

% ATP.lambda_761
tff(fact_8541_ATP_Olambda__762,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_ako(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,Uu,Uub)),Uua) ) ) ).

% ATP.lambda_762
tff(fact_8542_ATP_Olambda__763,axiom,
    ! [B: $tType,A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_cq(fun(B,A),fun(A,fun(B,A)),Uu),Uua),Uub) = modulo_modulo(A,aa(B,A,Uu,Uub),Uua) ) ).

% ATP.lambda_763
tff(fact_8543_ATP_Olambda__764,axiom,
    ! [B: $tType,A: $tType] :
      ( field(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_df(fun(B,A),fun(A,fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(B,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_764
tff(fact_8544_ATP_Olambda__765,axiom,
    ! [A: $tType,B: $tType] :
      ( euclid4440199948858584721cancel(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_ig(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_765
tff(fact_8545_ATP_Olambda__766,axiom,
    ! [A: $tType,B: $tType] :
      ( ( dense_linorder(B)
        & no_bot(B) )
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_akp(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,Uu,Uub)),Uua) ) ) ).

% ATP.lambda_766
tff(fact_8546_ATP_Olambda__767,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_zi(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,Uu,Uub)),Uua) ) ) ).

% ATP.lambda_767
tff(fact_8547_ATP_Olambda__768,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_db(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_768
tff(fact_8548_ATP_Olambda__769,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,set(A)),Uua: set(A),Uub: B] : aa(B,set(A),aa(set(A),fun(B,set(A)),aTP_Lamp_ko(fun(B,set(A)),fun(set(A),fun(B,set(A))),Uu),Uua),Uub) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(B,set(A),Uu,Uub)),Uua) ).

% ATP.lambda_769
tff(fact_8549_ATP_Olambda__770,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: fun(B,A),Uua: nat,Uub: B] : aa(B,A,aa(nat,fun(B,A),aTP_Lamp_fj(fun(B,A),fun(nat,fun(B,A)),Uu),Uua),Uub) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(B,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_770
tff(fact_8550_ATP_Olambda__771,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,multiset(A)),Uua: A,Uub: B] : aa(B,nat,aa(A,fun(B,nat),aTP_Lamp_app(fun(B,multiset(A)),fun(A,fun(B,nat)),Uu),Uua),Uub) = aa(A,nat,aa(multiset(A),fun(A,nat),count(A),aa(B,multiset(A),Uu,Uub)),Uua) ).

% ATP.lambda_771
tff(fact_8551_ATP_Olambda__772,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,set(A)),Uua: set(A),Uub: B] : aa(B,set(A),aa(set(A),fun(B,set(A)),aTP_Lamp_kj(fun(B,set(A)),fun(set(A),fun(B,set(A))),Uu),Uua),Uub) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(B,set(A),Uu,Uub)),Uua) ).

% ATP.lambda_772
tff(fact_8552_ATP_Olambda__773,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_li(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_773
tff(fact_8553_ATP_Olambda__774,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,set(A)),Uua: set(A),Uub: B] : aa(B,set(A),aa(set(A),fun(B,set(A)),aTP_Lamp_km(fun(B,set(A)),fun(set(A),fun(B,set(A))),Uu),Uua),Uub) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(B,set(A),Uu,Uub)),Uua) ).

% ATP.lambda_774
tff(fact_8554_ATP_Olambda__775,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_mf(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_775
tff(fact_8555_ATP_Olambda__776,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_lo(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_776
tff(fact_8556_ATP_Olambda__777,axiom,
    ! [A: $tType,B: $tType] :
      ( linord4140545234300271783up_add(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_su(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_777
tff(fact_8557_ATP_Olambda__778,axiom,
    ! [E: $tType,F3: $tType,B: $tType,D: $tType,C: $tType,Uu: fun(E,fun(F3,product_prod(C,D))),Uua: fun(C,fun(D,B)),Uub: E] : aa(E,fun(F3,B),aa(fun(C,fun(D,B)),fun(E,fun(F3,B)),aTP_Lamp_asr(fun(E,fun(F3,product_prod(C,D))),fun(fun(C,fun(D,B)),fun(E,fun(F3,B))),Uu),Uua),Uub) = product_scomp(F3,C,D,B,aa(E,fun(F3,product_prod(C,D)),Uu,Uub),Uua) ).

% ATP.lambda_778
tff(fact_8558_ATP_Olambda__779,axiom,
    ! [C: $tType,B: $tType,A: $tType,Uu: fun(C,set(old_node(A,B))),Uua: set(old_node(A,B)),Uub: C] : aa(C,set(old_node(A,B)),aa(set(old_node(A,B)),fun(C,set(old_node(A,B))),aTP_Lamp_aui(fun(C,set(old_node(A,B))),fun(set(old_node(A,B)),fun(C,set(old_node(A,B)))),Uu),Uua),Uub) = old_Scons(A,B,aa(C,set(old_node(A,B)),Uu,Uub),Uua) ).

% ATP.lambda_779
tff(fact_8559_ATP_Olambda__780,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: fun(A,filter(B)),Uua: filter(C),Uub: A] : aa(A,filter(product_prod(B,C)),aa(filter(C),fun(A,filter(product_prod(B,C))),aTP_Lamp_any(fun(A,filter(B)),fun(filter(C),fun(A,filter(product_prod(B,C)))),Uu),Uua),Uub) = prod_filter(B,C,aa(A,filter(B),Uu,Uub),Uua) ).

% ATP.lambda_780
tff(fact_8560_ATP_Olambda__781,axiom,
    ! [D: $tType,A: $tType,B: $tType,C: $tType,Uu: fun(D,fun(A,fun(C,$o))),Uua: fun(C,fun(B,$o)),Uub: D] : aa(D,fun(A,fun(B,$o)),aa(fun(C,fun(B,$o)),fun(D,fun(A,fun(B,$o))),aTP_Lamp_aph(fun(D,fun(A,fun(C,$o))),fun(fun(C,fun(B,$o)),fun(D,fun(A,fun(B,$o)))),Uu),Uua),Uub) = aa(fun(C,fun(B,$o)),fun(A,fun(B,$o)),aa(fun(A,fun(C,$o)),fun(fun(C,fun(B,$o)),fun(A,fun(B,$o))),relcompp(A,C,B),aa(D,fun(A,fun(C,$o)),Uu,Uub)),Uua) ).

% ATP.lambda_781
tff(fact_8561_ATP_Olambda__782,axiom,
    ! [D: $tType,A: $tType,B: $tType,C: $tType,Uu: fun(D,set(product_prod(A,C))),Uua: set(product_prod(C,B)),Uub: D] : aa(D,set(product_prod(A,B)),aa(set(product_prod(C,B)),fun(D,set(product_prod(A,B))),aTP_Lamp_uv(fun(D,set(product_prod(A,C))),fun(set(product_prod(C,B)),fun(D,set(product_prod(A,B)))),Uu),Uua),Uub) = relcomp(A,C,B,aa(D,set(product_prod(A,C)),Uu,Uub),Uua) ).

% ATP.lambda_782
tff(fact_8562_ATP_Olambda__783,axiom,
    ! [C: $tType,A: $tType,B: $tType,Uu: fun(C,set(product_prod(B,A))),Uua: set(B),Uub: C] : aa(C,set(A),aa(set(B),fun(C,set(A)),aTP_Lamp_wx(fun(C,set(product_prod(B,A))),fun(set(B),fun(C,set(A))),Uu),Uua),Uub) = aa(set(B),set(A),image(B,A,aa(C,set(product_prod(B,A)),Uu,Uub)),Uua) ).

% ATP.lambda_783
tff(fact_8563_ATP_Olambda__784,axiom,
    ! [C: $tType,A: $tType,B: $tType,Uu: fun(C,pred(B)),Uua: fun(B,pred(A)),Uub: C] : aa(C,pred(A),aa(fun(B,pred(A)),fun(C,pred(A)),aTP_Lamp_awg(fun(C,pred(B)),fun(fun(B,pred(A)),fun(C,pred(A))),Uu),Uua),Uub) = bind2(B,A,aa(C,pred(B),Uu,Uub),Uua) ).

% ATP.lambda_784
tff(fact_8564_ATP_Olambda__785,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: $o,Uub: A] :
      ( aa(A,$o,aa($o,fun(A,$o),aTP_Lamp_avc(fun(A,$o),fun($o,fun(A,$o)),Uu),(Uua)),Uub)
    <=> ( aa(A,$o,Uu,Uub)
       => (Uua) ) ) ).

% ATP.lambda_785
tff(fact_8565_ATP_Olambda__786,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,B),Uua: set(B),Uub: A] :
      ( aa(A,$o,aa(set(B),fun(A,$o),aTP_Lamp_ay(fun(A,B),fun(set(B),fun(A,$o)),Uu),Uua),Uub)
    <=> aa(set(B),$o,member(B,aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_786
tff(fact_8566_ATP_Olambda__787,axiom,
    ! [B: $tType,A: $tType,Uu: set(A),Uua: fun(B,A),Uub: B] :
      ( aa(B,$o,aa(fun(B,A),fun(B,$o),aTP_Lamp_ach(set(A),fun(fun(B,A),fun(B,$o)),Uu),Uua),Uub)
    <=> aa(set(A),$o,member(A,aa(B,A,Uua,Uub)),Uu) ) ).

% ATP.lambda_787
tff(fact_8567_ATP_Olambda__788,axiom,
    ! [C: $tType,A: $tType,B: $tType,Uu: fun(C,set(B)),Uua: fun(B,set(A)),Uub: C] : aa(C,set(A),aa(fun(B,set(A)),fun(C,set(A)),aTP_Lamp_ast(fun(C,set(B)),fun(fun(B,set(A)),fun(C,set(A))),Uu),Uua),Uub) = bind3(B,A,aa(C,set(B),Uu,Uub),Uua) ).

% ATP.lambda_788
tff(fact_8568_ATP_Olambda__789,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: $o,Uub: A] :
      ( aa(A,$o,aa($o,fun(A,$o),aTP_Lamp_aki(fun(A,$o),fun($o,fun(A,$o)),Uu),(Uua)),Uub)
    <=> ( aa(A,$o,Uu,Uub)
        | (Uua) ) ) ).

% ATP.lambda_789
tff(fact_8569_ATP_Olambda__790,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: $o,Uub: A] :
      ( aa(A,$o,aa($o,fun(A,$o),aTP_Lamp_avb(fun(A,$o),fun($o,fun(A,$o)),Uu),(Uua)),Uub)
    <=> ( aa(A,$o,Uu,Uub)
        & (Uua) ) ) ).

% ATP.lambda_790
tff(fact_8570_ATP_Olambda__791,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,A),Uua: A,Uub: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_atf(fun(B,A),fun(A,fun(B,$o)),Uu),Uua),Uub)
    <=> ( aa(B,A,Uu,Uub) = Uua ) ) ).

% ATP.lambda_791
tff(fact_8571_ATP_Olambda__792,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_yz(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(A,B,Uu,Uub) = Uua ) ) ) ).

% ATP.lambda_792
tff(fact_8572_ATP_Olambda__793,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,B),Uua: B,Uub: A] :
      ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_acv(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(A,B,Uu,Uub) = Uua ) ) ).

% ATP.lambda_793
tff(fact_8573_ATP_Olambda__794,axiom,
    ! [A: $tType,Uu: fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))),Uua: pred(A),Uub: product_prod(code_natural,code_natural)] : aa(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural)),aa(pred(A),fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))),aTP_Lamp_awb(fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))),fun(pred(A),fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural)))),Uu),Uua),Uub) = aa(product_prod(pred(A),product_prod(code_natural,code_natural)),product_prod(pred(A),product_prod(code_natural,code_natural)),aa(fun(pred(A),fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural)))),fun(product_prod(pred(A),product_prod(code_natural,code_natural)),product_prod(pred(A),product_prod(code_natural,code_natural))),product_case_prod(pred(A),product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))),aTP_Lamp_awa(pred(A),fun(pred(A),fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural)))),Uua)),aa(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural)),Uu,Uub)) ).

% ATP.lambda_794
tff(fact_8574_ATP_Olambda__795,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_apl(set(product_prod(A,A)),fun(A,fun(A,$o)),Uu),Uua),Uub)
    <=> ( ( Uub != Uua )
        & aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uua),Uub)),Uu) ) ) ).

% ATP.lambda_795
tff(fact_8575_ATP_Olambda__796,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_amn(set(product_prod(A,A)),fun(A,fun(A,$o)),Uu),Uua),Uub)
    <=> ( ( Uub != Uua )
        & aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),Uua)),Uu) ) ) ).

% ATP.lambda_796
tff(fact_8576_ATP_Olambda__797,axiom,
    ! [A: $tType,Uu: A,Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_cg(A,fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> ( ( Uub != Uu )
       => aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_797
tff(fact_8577_ATP_Olambda__798,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: nat,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_fo(nat,fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),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_798
tff(fact_8578_ATP_Olambda__799,axiom,
    ! [A: $tType,Uu: A,Uua: A,Uub: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_afe(A,fun(A,fun(A,$o)),Uu),Uua),Uub)
    <=> ( ( Uua != Uu )
        & ( Uub != Uu ) ) ) ).

% ATP.lambda_799
tff(fact_8579_ATP_Olambda__800,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: A,Uub: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_ate(fun(A,$o),fun(A,fun(A,$o)),Uu),Uua),Uub)
    <=> ( ~ aa(A,$o,Uu,Uub)
        | ( Uua = Uub ) ) ) ).

% ATP.lambda_800
tff(fact_8580_ATP_Olambda__801,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(B)
     => ! [Uu: fun(A,$o),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_gv(fun(A,$o),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa($o,B,zero_neq_one_of_bool(B),aa(A,$o,Uu,Uub))),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_801
tff(fact_8581_ATP_Olambda__802,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_avd(fun(A,$o),fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> ( ~ aa(A,$o,Uu,Uub)
        | aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_802
tff(fact_8582_ATP_Olambda__803,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [Uu: fun(A,A),Uua: fun(A,$o),Uub: A] :
          ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ahh(fun(A,A),fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
        <=> ( ? [X4: A] :
                ( ( Uub = aa(A,A,Uu,X4) )
                & aa(A,$o,Uua,X4) )
            | ? [M9: set(A)] :
                ( ( Uub = aa(set(A),A,complete_Sup_Sup(A),M9) )
                & comple1602240252501008431_chain(A,ord_less_eq(A),M9)
                & ! [X4: A] :
                    ( aa(set(A),$o,member(A,X4),M9)
                   => aa(A,$o,Uua,X4) ) ) ) ) ) ).

% ATP.lambda_803
tff(fact_8583_ATP_Olambda__804,axiom,
    ! [A: $tType,B: $tType,Uu: set(A),Uua: set(B),Uub: fun(A,B)] :
      ( aa(fun(A,B),$o,aa(set(B),fun(fun(A,B),$o),aTP_Lamp_aip(set(A),fun(set(B),fun(fun(A,B),$o)),Uu),Uua),Uub)
    <=> ( ! [X4: A] :
            ( aa(set(A),$o,member(A,X4),Uu)
           => aa(set(B),$o,member(B,aa(A,B,Uub,X4)),Uua) )
        & ! [A10: A] :
            ( ~ aa(set(A),$o,member(A,A10),Uu)
           => ( aa(A,B,Uub,A10) = undefined(B) ) ) ) ) ).

% ATP.lambda_804
tff(fact_8584_ATP_Olambda__805,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu: fun(list(A),fun(list(A),$o)),Uua: list(A),Uub: list(A)] :
          ( aa(list(A),$o,aa(list(A),fun(list(A),$o),aa(fun(list(A),fun(list(A),$o)),fun(list(A),fun(list(A),$o)),aTP_Lamp_aez(fun(list(A),fun(list(A),$o)),fun(list(A),fun(list(A),$o))),Uu),Uua),Uub)
        <=> ( ? [Y3: A,Ys3: list(A)] :
                ( ( Uua = nil(A) )
                & ( Uub = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys3) ) )
            | ? [X4: A,Y3: A,Xs4: list(A),Ys3: list(A)] :
                ( ( Uua = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs4) )
                & ( Uub = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys3) )
                & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Y3) )
            | ? [X4: A,Y3: A,Xs4: list(A),Ys3: list(A)] :
                ( ( Uua = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs4) )
                & ( Uub = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys3) )
                & ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Y3)
                & ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y3),X4)
                & aa(list(A),$o,aa(list(A),fun(list(A),$o),Uu,Xs4),Ys3) ) ) ) ) ).

% ATP.lambda_805
tff(fact_8585_ATP_Olambda__806,axiom,
    ! [A: $tType,Uu: fun(list(multiset(A)),fun(list(multiset(A)),$o)),Uua: list(multiset(A)),Uub: list(multiset(A))] :
      ( aa(list(multiset(A)),$o,aa(list(multiset(A)),fun(list(multiset(A)),$o),aa(fun(list(multiset(A)),fun(list(multiset(A)),$o)),fun(list(multiset(A)),fun(list(multiset(A)),$o)),aTP_Lamp_asq(fun(list(multiset(A)),fun(list(multiset(A)),$o)),fun(list(multiset(A)),fun(list(multiset(A)),$o))),Uu),Uua),Uub)
    <=> ( ? [Y3: multiset(A),Ys3: list(multiset(A))] :
            ( ( Uua = nil(multiset(A)) )
            & ( Uub = aa(list(multiset(A)),list(multiset(A)),aa(multiset(A),fun(list(multiset(A)),list(multiset(A))),cons(multiset(A)),Y3),Ys3) ) )
        | ? [X4: multiset(A),Y3: multiset(A),Xs4: list(multiset(A)),Ys3: list(multiset(A))] :
            ( ( Uua = aa(list(multiset(A)),list(multiset(A)),aa(multiset(A),fun(list(multiset(A)),list(multiset(A))),cons(multiset(A)),X4),Xs4) )
            & ( Uub = aa(list(multiset(A)),list(multiset(A)),aa(multiset(A),fun(list(multiset(A)),list(multiset(A))),cons(multiset(A)),Y3),Ys3) )
            & aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subset_mset(A),X4),Y3) )
        | ? [X4: multiset(A),Y3: multiset(A),Xs4: list(multiset(A)),Ys3: list(multiset(A))] :
            ( ( Uua = aa(list(multiset(A)),list(multiset(A)),aa(multiset(A),fun(list(multiset(A)),list(multiset(A))),cons(multiset(A)),X4),Xs4) )
            & ( Uub = aa(list(multiset(A)),list(multiset(A)),aa(multiset(A),fun(list(multiset(A)),list(multiset(A))),cons(multiset(A)),Y3),Ys3) )
            & ~ aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subset_mset(A),X4),Y3)
            & ~ aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subset_mset(A),Y3),X4)
            & aa(list(multiset(A)),$o,aa(list(multiset(A)),fun(list(multiset(A)),$o),Uu,Xs4),Ys3) ) ) ) ).

% ATP.lambda_806
tff(fact_8586_ATP_Olambda__807,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A),Uub: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),aTP_Lamp_tx(set(A),fun(set(A),fun(set(A),$o)),Uu),Uua),Uub)
    <=> ( aa(set(A),$o,finite_finite2(A),Uub)
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Uua),Uub)
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Uub),Uu) ) ) ).

% ATP.lambda_807
tff(fact_8587_ATP_Olambda__808,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: set(A),Uub: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),aTP_Lamp_aik(set(product_prod(A,A)),fun(set(A),fun(set(A),$o)),Uu),Uua),Uub)
    <=> ( aa(set(A),$o,finite_finite2(A),Uua)
        & aa(set(A),$o,finite_finite2(A),Uub)
        & ( Uub != bot_bot(set(A)) )
        & ! [X4: A] :
            ( aa(set(A),$o,member(A,X4),Uua)
           => ? [Xa2: A] :
                ( aa(set(A),$o,member(A,Xa2),Uub)
                & aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Xa2)),Uu) ) ) ) ) ).

% ATP.lambda_808
tff(fact_8588_ATP_Olambda__809,axiom,
    ! [A: $tType,Uu: fun(product_unit,seq(A)),Uua: pred(A),Uub: seq(A)] : aa(seq(A),seq(A),aa(pred(A),fun(seq(A),seq(A)),aTP_Lamp_axu(fun(product_unit,seq(A)),fun(pred(A),fun(seq(A),seq(A))),Uu),Uua),Uub) = adjunct(A,seq2(A,Uu),join(A,Uua,Uub)) ).

% ATP.lambda_809
tff(fact_8589_ATP_Olambda__810,axiom,
    ! [A: $tType,Uu: fun(A,fun(A,A)),Uua: list(A),Uub: list(A)] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),aTP_Lamp_ayg(fun(A,fun(A,A)),fun(list(A),fun(list(A),list(A))),Uu),Uua),Uub) = aa(list(product_prod(A,A)),list(A),map(product_prod(A,A),A,aa(fun(A,fun(A,A)),fun(product_prod(A,A),A),product_case_prod(A,A,A),Uu)),zip(A,A,Uua,Uub)) ).

% ATP.lambda_810
tff(fact_8590_ATP_Olambda__811,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_aoj(set(product_prod(A,A)),fun(A,fun(A,set(A))),Uu),Uua),Uub) = aa(set(A),set(A),image(A,A,converse(A,A,Uu)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Uua),bot_bot(set(A)))) ).

% ATP.lambda_811
tff(fact_8591_ATP_Olambda__812,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Uu: fun(A,B),Uua: list(A),Uub: A] : aa(A,fun(list(A),list(A)),aa(list(A),fun(A,fun(list(A),list(A))),aTP_Lamp_yo(fun(A,B),fun(list(A),fun(A,fun(list(A),list(A)))),Uu),Uua),Uub) = case_list(list(A),A,Uua,aa(A,fun(A,fun(list(A),list(A))),aa(list(A),fun(A,fun(A,fun(list(A),list(A)))),aTP_Lamp_yn(fun(A,B),fun(list(A),fun(A,fun(A,fun(list(A),list(A))))),Uu),Uua),Uub)) ) ).

% ATP.lambda_812
tff(fact_8592_ATP_Olambda__813,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_ia(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_813
tff(fact_8593_ATP_Olambda__814,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_hz(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_814
tff(fact_8594_ATP_Olambda__815,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,A),Uua: list(A),Uub: B] : aa(B,list(A),aa(list(A),fun(B,list(A)),aTP_Lamp_aae(fun(B,A),fun(list(A),fun(B,list(A))),Uu),Uua),Uub) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Uua),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(B,A,Uu,Uub)),nil(A))) ).

% ATP.lambda_815
tff(fact_8595_ATP_Olambda__816,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Uu: A,Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_dy(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_816
tff(fact_8596_ATP_Olambda__817,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: A,Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_dz(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_817
tff(fact_8597_ATP_Olambda__818,axiom,
    ! [A: $tType,Uu: A,Uua: set(A),Uub: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_ahr(A,fun(set(A),fun(A,$o)),Uu),Uua),Uub)
    <=> aa(set(A),$o,member(A,Uub),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Uua),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Uu),bot_bot(set(A))))) ) ).

% ATP.lambda_818
tff(fact_8598_ATP_Olambda__819,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_im(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_819
tff(fact_8599_ATP_Olambda__820,axiom,
    ! [A: $tType,Uu: fun(product_unit,seq(A)),Uua: A,Uub: pred(A)] : aa(pred(A),seq(A),aa(A,fun(pred(A),seq(A)),aTP_Lamp_axt(fun(product_unit,seq(A)),fun(A,fun(pred(A),seq(A))),Uu),Uua),Uub) = insert(A,Uua,aa(pred(A),pred(A),aa(pred(A),fun(pred(A),pred(A)),sup_sup(pred(A)),Uub),seq2(A,Uu))) ).

% ATP.lambda_820
tff(fact_8600_ATP_Olambda__821,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,option(B)),Uua: set(A),Uub: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_adk(fun(A,option(B)),fun(set(A),fun(A,$o)),Uu),Uua),Uub)
    <=> aa(set(A),$o,member(A,Uub),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Uua),dom(A,B,Uu))) ) ).

% ATP.lambda_821
tff(fact_8601_ATP_Olambda__822,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,option(B)),Uua: set(A),Uub: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_adl(fun(A,option(B)),fun(set(A),fun(A,$o)),Uu),Uua),Uub)
    <=> aa(set(A),$o,member(A,Uub),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Uua),dom(A,B,Uu))) ) ).

% ATP.lambda_822
tff(fact_8602_ATP_Olambda__823,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_cr(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_823
tff(fact_8603_ATP_Olambda__824,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_dk(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_824
tff(fact_8604_ATP_Olambda__825,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_adm(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_825
tff(fact_8605_ATP_Olambda__826,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_adn(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_826
tff(fact_8606_ATP_Olambda__827,axiom,
    ! [A: $tType,Uu: A,Uua: list(A),Uub: nat] : aa(nat,list(A),aa(list(A),fun(nat,list(A)),aTP_Lamp_ada(A,fun(list(A),fun(nat,list(A))),Uu),Uua),Uub) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uu),take(A,Uub,Uua)) ).

% ATP.lambda_827
tff(fact_8607_ATP_Olambda__828,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,option(A)),Uua: list(B),Uub: A] : aa(A,list(A),aa(list(B),fun(A,list(A)),aTP_Lamp_aan(fun(B,option(A)),fun(list(B),fun(A,list(A))),Uu),Uua),Uub) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uub),map_filter(B,A,Uu,Uua)) ).

% ATP.lambda_828
tff(fact_8608_ATP_Olambda__829,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [Uu: fun(A,B),Uua: set(A),Uub: B] :
          ( aa(B,$o,aa(set(A),fun(B,$o),aTP_Lamp_vi(fun(A,B),fun(set(A),fun(B,$o)),Uu),Uua),Uub)
        <=> aa(set(B),$o,member(B,Uub),aa(set(A),set(B),image2(A,B,Uu),Uua)) ) ) ).

% ATP.lambda_829
tff(fact_8609_ATP_Olambda__830,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( comm_monoid_mult(C)
     => ! [Uu: fun(A,set(B)),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_nc(fun(A,set(B)),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),Uua),aa(A,set(B),Uu,Uub)) ) ).

% ATP.lambda_830
tff(fact_8610_ATP_Olambda__831,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( comm_monoid_add(C)
     => ! [Uu: fun(A,set(B)),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_nb(fun(A,set(B)),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),Uua),aa(A,set(B),Uu,Uub)) ) ).

% ATP.lambda_831
tff(fact_8611_ATP_Olambda__832,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_ala(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Uua),aa(A,B,Uu,Uub)) ) ) ).

% ATP.lambda_832
tff(fact_8612_ATP_Olambda__833,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_ale(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Uua),aa(A,B,Uu,Uub)) ) ) ).

% ATP.lambda_833
tff(fact_8613_ATP_Olambda__834,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] :
          ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_aav(fun(B,A),fun(A,fun(B,$o)),Uu),Uua),Uub)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Uua),aa(B,A,Uu,Uub)) ) ) ).

% ATP.lambda_834
tff(fact_8614_ATP_Olambda__835,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_alg(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),Uua),aa(A,B,Uu,Uub)) ) ) ).

% ATP.lambda_835
tff(fact_8615_ATP_Olambda__836,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_zj(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),Uua),aa(A,B,Uu,Uub)) ) ) ).

% ATP.lambda_836
tff(fact_8616_ATP_Olambda__837,axiom,
    ! [A: $tType,C: $tType,B: $tType,Uu: fun(C,A),Uua: fun(B,option(C)),Uub: B] : aa(B,option(A),aa(fun(B,option(C)),fun(B,option(A)),aTP_Lamp_xu(fun(C,A),fun(fun(B,option(C)),fun(B,option(A))),Uu),Uua),Uub) = aa(option(C),option(A),map_option(C,A,Uu),aa(B,option(C),Uua,Uub)) ).

% ATP.lambda_837
tff(fact_8617_ATP_Olambda__838,axiom,
    ! [A: $tType,Uu: nat,Uua: fun(A,nat),Uub: A] : aa(A,nat,aa(fun(A,nat),fun(A,nat),aa(nat,fun(fun(A,nat),fun(A,nat)),aTP_Lamp_aqc(nat,fun(fun(A,nat),fun(A,nat))),Uu),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uu),aa(A,nat,Uua,Uub)) ).

% ATP.lambda_838
tff(fact_8618_ATP_Olambda__839,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_dc(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_839
tff(fact_8619_ATP_Olambda__840,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: nat,Uub: A] : aa(A,nat,aa(nat,fun(A,nat),aTP_Lamp_aqj(fun(A,nat),fun(nat,fun(A,nat)),Uu),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),aa(A,nat,Uu,Uub)) ).

% ATP.lambda_840
tff(fact_8620_ATP_Olambda__841,axiom,
    ! [A: $tType,B: $tType,Uu: set(A),Uua: fun(B,set(A)),Uub: B] : aa(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_kp(set(A),fun(fun(B,set(A)),fun(B,set(A))),Uu),Uua),Uub) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Uu),aa(B,set(A),Uua,Uub)) ).

% ATP.lambda_841
tff(fact_8621_ATP_Olambda__842,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: A,Uua: fun(B,nat),Uub: B] : aa(B,A,aa(fun(B,nat),fun(B,A),aTP_Lamp_fm(A,fun(fun(B,nat),fun(B,A)),Uu),Uua),Uub) = aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),aa(B,nat,Uua,Uub)) ) ).

% ATP.lambda_842
tff(fact_8622_ATP_Olambda__843,axiom,
    ! [A: $tType,B: $tType,Uu: set(A),Uua: fun(B,set(A)),Uub: B] : aa(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_ki(set(A),fun(fun(B,set(A)),fun(B,set(A))),Uu),Uua),Uub) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Uu),aa(B,set(A),Uua,Uub)) ).

% ATP.lambda_843
tff(fact_8623_ATP_Olambda__844,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_lg(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_844
tff(fact_8624_ATP_Olambda__845,axiom,
    ! [A: $tType,B: $tType,Uu: set(A),Uua: fun(B,set(A)),Uub: B] : aa(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_kn(set(A),fun(fun(B,set(A)),fun(B,set(A))),Uu),Uua),Uub) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Uu),aa(B,set(A),Uua,Uub)) ).

% ATP.lambda_845
tff(fact_8625_ATP_Olambda__846,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [Uu: B,Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ln(B,fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),inf_inf(B),Uu),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_846
tff(fact_8626_ATP_Olambda__847,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_mh(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_847
tff(fact_8627_ATP_Olambda__848,axiom,
    ! [B: $tType,A: $tType] :
      ( euclid4440199948858584721cancel(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_ih(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),Uua),aa(A,B,Uu,Uub)) ) ) ).

% ATP.lambda_848
tff(fact_8628_ATP_Olambda__849,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: nat,Uua: fun(C,set(old_node(A,B))),Uub: C] : aa(C,set(old_node(A,B)),aa(fun(C,set(old_node(A,B))),fun(C,set(old_node(A,B))),aTP_Lamp_auk(nat,fun(fun(C,set(old_node(A,B))),fun(C,set(old_node(A,B)))),Uu),Uua),Uub) = old_ntrunc(A,B,Uu,aa(C,set(old_node(A,B)),Uua,Uub)) ).

% ATP.lambda_849
tff(fact_8629_ATP_Olambda__850,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: set(old_node(A,B)),Uua: fun(C,set(old_node(A,B))),Uub: C] : aa(C,set(old_node(A,B)),aa(fun(C,set(old_node(A,B))),fun(C,set(old_node(A,B))),aTP_Lamp_auh(set(old_node(A,B)),fun(fun(C,set(old_node(A,B))),fun(C,set(old_node(A,B)))),Uu),Uua),Uub) = old_Scons(A,B,Uu,aa(C,set(old_node(A,B)),Uua,Uub)) ).

% ATP.lambda_850
tff(fact_8630_ATP_Olambda__851,axiom,
    ! [B: $tType,C: $tType,A: $tType,Uu: filter(B),Uua: fun(A,filter(C)),Uub: A] : aa(A,filter(product_prod(B,C)),aa(fun(A,filter(C)),fun(A,filter(product_prod(B,C))),aTP_Lamp_anx(filter(B),fun(fun(A,filter(C)),fun(A,filter(product_prod(B,C)))),Uu),Uua),Uub) = prod_filter(B,C,Uu,aa(A,filter(C),Uua,Uub)) ).

% ATP.lambda_851
tff(fact_8631_ATP_Olambda__852,axiom,
    ! [A: $tType,C: $tType,B: $tType,Uu: fun(A,C),Uua: fun(B,filter(C)),Uub: B] : aa(B,filter(A),aa(fun(B,filter(C)),fun(B,filter(A)),aTP_Lamp_alm(fun(A,C),fun(fun(B,filter(C)),fun(B,filter(A))),Uu),Uua),Uub) = filtercomap(A,C,Uu,aa(B,filter(C),Uua,Uub)) ).

% ATP.lambda_852
tff(fact_8632_ATP_Olambda__853,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: fun(A,B),Uua: fun(C,filter(B)),Uub: C] : aa(C,filter(A),aa(fun(C,filter(B)),fun(C,filter(A)),aTP_Lamp_all(fun(A,B),fun(fun(C,filter(B)),fun(C,filter(A))),Uu),Uua),Uub) = filtercomap(A,B,Uu,aa(C,filter(B),Uua,Uub)) ).

% ATP.lambda_853
tff(fact_8633_ATP_Olambda__854,axiom,
    ! [A: $tType,C: $tType,B: $tType,D: $tType,Uu: fun(A,fun(C,$o)),Uua: fun(D,fun(C,fun(B,$o))),Uub: D] : aa(D,fun(A,fun(B,$o)),aa(fun(D,fun(C,fun(B,$o))),fun(D,fun(A,fun(B,$o))),aTP_Lamp_apg(fun(A,fun(C,$o)),fun(fun(D,fun(C,fun(B,$o))),fun(D,fun(A,fun(B,$o)))),Uu),Uua),Uub) = aa(fun(C,fun(B,$o)),fun(A,fun(B,$o)),aa(fun(A,fun(C,$o)),fun(fun(C,fun(B,$o)),fun(A,fun(B,$o))),relcompp(A,C,B),Uu),aa(D,fun(C,fun(B,$o)),Uua,Uub)) ).

% ATP.lambda_854
tff(fact_8634_ATP_Olambda__855,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_abl(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_855
tff(fact_8635_ATP_Olambda__856,axiom,
    ! [A: $tType,C: $tType,B: $tType,D: $tType,Uu: set(product_prod(A,C)),Uua: fun(D,set(product_prod(C,B))),Uub: D] : aa(D,set(product_prod(A,B)),aa(fun(D,set(product_prod(C,B))),fun(D,set(product_prod(A,B))),aTP_Lamp_uu(set(product_prod(A,C)),fun(fun(D,set(product_prod(C,B))),fun(D,set(product_prod(A,B)))),Uu),Uua),Uub) = relcomp(A,C,B,Uu,aa(D,set(product_prod(C,B)),Uua,Uub)) ).

% ATP.lambda_856
tff(fact_8636_ATP_Olambda__857,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: fun(B,A),Uua: fun(C,filter(B)),Uub: C] : aa(C,filter(A),aa(fun(C,filter(B)),fun(C,filter(A)),aTP_Lamp_anl(fun(B,A),fun(fun(C,filter(B)),fun(C,filter(A))),Uu),Uua),Uub) = aa(filter(B),filter(A),aa(fun(B,A),fun(filter(B),filter(A)),filtermap(B,A),Uu),aa(C,filter(B),Uua,Uub)) ).

% ATP.lambda_857
tff(fact_8637_ATP_Olambda__858,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: set(product_prod(B,A)),Uua: fun(C,set(B)),Uub: C] : aa(C,set(A),aa(fun(C,set(B)),fun(C,set(A)),aTP_Lamp_wu(set(product_prod(B,A)),fun(fun(C,set(B)),fun(C,set(A))),Uu),Uua),Uub) = aa(set(B),set(A),image(B,A,Uu),aa(C,set(B),Uua,Uub)) ).

% ATP.lambda_858
tff(fact_8638_ATP_Olambda__859,axiom,
    ! [A: $tType,Uu: $o,Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_akg($o,fun(fun(A,$o),fun(A,$o)),(Uu)),Uua),Uub)
    <=> ( (Uu)
       => aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_859
tff(fact_8639_ATP_Olambda__860,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: fun(A,B),Uua: fun(C,set(B)),Uub: C] : aa(C,set(A),aa(fun(C,set(B)),fun(C,set(A)),aTP_Lamp_kw(fun(A,B),fun(fun(C,set(B)),fun(C,set(A))),Uu),Uua),Uub) = aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),Uu),aa(C,set(B),Uua,Uub)) ).

% ATP.lambda_860
tff(fact_8640_ATP_Olambda__861,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,set(A)),Uua: B,Uub: A] :
      ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_nr(fun(B,set(A)),fun(B,fun(A,$o)),Uu),Uua),Uub)
    <=> aa(set(A),$o,member(A,Uub),aa(B,set(A),Uu,Uua)) ) ).

% ATP.lambda_861
tff(fact_8641_ATP_Olambda__862,axiom,
    ! [B: $tType,A: $tType,Uu: B,Uua: fun(A,set(B)),Uub: A] : aa(A,set(B),aa(fun(A,set(B)),fun(A,set(B)),aTP_Lamp_kv(B,fun(fun(A,set(B)),fun(A,set(B))),Uu),Uua),Uub) = aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),Uu),aa(A,set(B),Uua,Uub)) ).

% ATP.lambda_862
tff(fact_8642_ATP_Olambda__863,axiom,
    ! [A: $tType,B: $tType,Uu: A,Uua: fun(B,set(A)),Uub: B] : aa(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_kh(A,fun(fun(B,set(A)),fun(B,set(A))),Uu),Uua),Uub) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Uu),aa(B,set(A),Uua,Uub)) ).

% ATP.lambda_863
tff(fact_8643_ATP_Olambda__864,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: fun(B,A),Uua: fun(C,set(B)),Uub: C] : aa(C,set(A),aa(fun(C,set(B)),fun(C,set(A)),aTP_Lamp_ku(fun(B,A),fun(fun(C,set(B)),fun(C,set(A))),Uu),Uua),Uub) = aa(set(B),set(A),image2(B,A,Uu),aa(C,set(B),Uua,Uub)) ).

% ATP.lambda_864
tff(fact_8644_ATP_Olambda__865,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu: fun(A,B),Uua: fun(C,set(A)),Uub: C] : aa(C,set(B),aa(fun(C,set(A)),fun(C,set(B)),aTP_Lamp_acp(fun(A,B),fun(fun(C,set(A)),fun(C,set(B))),Uu),Uua),Uub) = aa(set(A),set(B),image2(A,B,Uu),aa(C,set(A),Uua,Uub)) ).

% ATP.lambda_865
tff(fact_8645_ATP_Olambda__866,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( condit1219197933456340205attice(C)
     => ! [Uu: fun(A,set(B)),Uua: fun(B,C),Uub: A] : aa(A,set(C),aa(fun(B,C),fun(A,set(C)),aTP_Lamp_aly(fun(A,set(B)),fun(fun(B,C),fun(A,set(C))),Uu),Uua),Uub) = aa(set(B),set(C),image2(B,C,Uua),aa(A,set(B),Uu,Uub)) ) ).

% ATP.lambda_866
tff(fact_8646_ATP_Olambda__867,axiom,
    ! [A: $tType,Uu: $o,Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_akh($o,fun(fun(A,$o),fun(A,$o)),(Uu)),Uua),Uub)
    <=> ( (Uu)
        | aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_867
tff(fact_8647_ATP_Olambda__868,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: A,Uub: $o] :
      ( aa($o,$o,aa(A,fun($o,$o),aTP_Lamp_aid(fun(A,$o),fun(A,fun($o,$o)),Uu),Uua),(Uub))
    <=> ( (Uub)
        | aa(A,$o,Uu,Uua) ) ) ).

% ATP.lambda_868
tff(fact_8648_ATP_Olambda__869,axiom,
    ! [A: $tType,Uu: $o,Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ava($o,fun(fun(A,$o),fun(A,$o)),(Uu)),Uua),Uub)
    <=> ( (Uu)
        & aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_869
tff(fact_8649_ATP_Olambda__870,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: A,Uub: $o] :
      ( aa($o,$o,aa(A,fun($o,$o),aTP_Lamp_aep(fun(A,$o),fun(A,fun($o,$o)),Uu),Uua),(Uub))
    <=> ( (Uub)
        & aa(A,$o,Uu,Uua) ) ) ).

% ATP.lambda_870
tff(fact_8650_ATP_Olambda__871,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: fun(list(A),A),Uua: list(A),Uub: A] :
          ( aa(A,$o,aa(list(A),fun(A,$o),aTP_Lamp_zd(fun(list(A),A),fun(list(A),fun(A,$o)),Uu),Uua),Uub)
        <=> ( Uub = aa(list(A),A,Uu,Uua) ) ) ) ).

% ATP.lambda_871
tff(fact_8651_ATP_Olambda__872,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: A,Uub: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_acl(fun(A,B),fun(A,fun(B,$o)),Uu),Uua),Uub)
    <=> ( Uub = aa(A,B,Uu,Uua) ) ) ).

% ATP.lambda_872
tff(fact_8652_ATP_Olambda__873,axiom,
    ! [A: $tType,B: $tType,Uu: fun(product_unit,seq(B)),Uua: fun(B,pred(A)),Uub: product_unit] : aa(product_unit,seq(A),aa(fun(B,pred(A)),fun(product_unit,seq(A)),aTP_Lamp_axs(fun(product_unit,seq(B)),fun(fun(B,pred(A)),fun(product_unit,seq(A))),Uu),Uua),Uub) = apply(B,A,Uua,aa(product_unit,seq(B),Uu,product_Unity)) ).

% ATP.lambda_873
tff(fact_8653_ATP_Olambda__874,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: A,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_gd(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_874
tff(fact_8654_ATP_Olambda__875,axiom,
    ! [A: $tType,Uu: list(A),Uua: set(nat),Uub: A] :
      ( aa(A,$o,aa(set(nat),fun(A,$o),aTP_Lamp_act(list(A),fun(set(nat),fun(A,$o)),Uu),Uua),Uub)
    <=> aa(set(A),$o,member(A,Uub),aa(list(A),set(A),set2(A),nths(A,Uu,Uua))) ) ).

% ATP.lambda_875
tff(fact_8655_ATP_Olambda__876,axiom,
    ! [A: $tType,B: $tType,Uu: set(product_prod(B,B)),Uua: fun(B,A),Uub: A] : aa(A,set(A),aa(fun(B,A),fun(A,set(A)),aTP_Lamp_ams(set(product_prod(B,B)),fun(fun(B,A),fun(A,set(A))),Uu),Uua),Uub) = aa(set(B),set(A),image2(B,A,Uua),field2(B,Uu)) ).

% ATP.lambda_876
tff(fact_8656_ATP_Olambda__877,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_amw(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_877
tff(fact_8657_ATP_Olambda__878,axiom,
    ! [A: $tType,Uu: set(A),Uua: A,Uub: product_unit] :
      ( aa(product_unit,$o,aa(A,fun(product_unit,$o),aTP_Lamp_awm(set(A),fun(A,fun(product_unit,$o)),Uu),Uua),Uub)
    <=> predicate_contains(A,Uu,Uua) ) ).

% ATP.lambda_878
tff(fact_8658_ATP_Olambda__879,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A),Uub: A] : aa(A,set(A),aa(set(A),fun(A,set(A)),aTP_Lamp_rt(set(A),fun(set(A),fun(A,set(A))),Uu),Uua),Uub) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Uu),Uua) ).

% ATP.lambda_879
tff(fact_8659_ATP_Olambda__880,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_rj(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_880
tff(fact_8660_ATP_Olambda__881,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_amp(set(product_prod(A,A)),fun(A,fun(A,set(A))),Uu),Uua),Uub) = order_underS(A,Uu,Uua) ).

% ATP.lambda_881
tff(fact_8661_ATP_Olambda__882,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_va(set(product_prod(A,A)),fun(A,fun(A,set(A))),Uu),Uua),Uub) = order_above(A,Uu,Uua) ).

% ATP.lambda_882
tff(fact_8662_ATP_Olambda__883,axiom,
    ! [B: $tType,A: $tType,Uu: set(set(old_node(A,B))),Uua: set(set(old_node(A,B))),Uub: set(old_node(A,B))] : aa(set(old_node(A,B)),set(set(old_node(A,B))),aa(set(set(old_node(A,B))),fun(set(old_node(A,B)),set(set(old_node(A,B)))),aTP_Lamp_avj(set(set(old_node(A,B))),fun(set(set(old_node(A,B))),fun(set(old_node(A,B)),set(set(old_node(A,B))))),Uu),Uua),Uub) = old_uprod(A,B,Uu,Uua) ).

% ATP.lambda_883
tff(fact_8663_ATP_Olambda__884,axiom,
    ! [B: $tType,A: $tType,Uu: set(set(old_node(A,B))),Uua: set(set(old_node(A,B))),Uub: set(old_node(A,B))] : aa(set(old_node(A,B)),set(set(old_node(A,B))),aa(set(set(old_node(A,B))),fun(set(old_node(A,B)),set(set(old_node(A,B)))),aTP_Lamp_avf(set(set(old_node(A,B))),fun(set(set(old_node(A,B))),fun(set(old_node(A,B)),set(set(old_node(A,B))))),Uu),Uua),Uub) = old_usum(A,B,Uu,Uua) ).

% ATP.lambda_884
tff(fact_8664_ATP_Olambda__885,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_anh(set(B),fun(set(C),fun(A,set(sum_sum(B,C)))),Uu),Uua),Uub) = sum_Plus(B,C,Uu,Uua) ).

% ATP.lambda_885
tff(fact_8665_ATP_Olambda__886,axiom,
    ! [A: $tType,Uu: list(A),Uua: A,Uub: list(A)] : aa(list(A),list(A),aa(A,fun(list(A),list(A)),aTP_Lamp_aau(list(A),fun(A,fun(list(A),list(A))),Uu),Uua),Uub) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Uub),Uu) ).

% ATP.lambda_886
tff(fact_8666_ATP_Olambda__887,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_rf(B,fun(set(B),fun(A,set(B))),Uu),Uua),Uub) = aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),Uu),Uua) ).

% ATP.lambda_887
tff(fact_8667_ATP_Olambda__888,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_ry(fun(D,B),fun(set(D),fun(A,set(B))),Uu),Uua),Uub) = aa(set(D),set(B),image2(D,B,Uu),Uua) ).

% ATP.lambda_888
tff(fact_8668_ATP_Olambda__889,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: set(B),Uub: A] : aa(A,set(A),aa(set(B),fun(A,set(A)),aTP_Lamp_re(fun(B,A),fun(set(B),fun(A,set(A))),Uu),Uua),Uub) = aa(set(B),set(A),image2(B,A,Uu),Uua) ).

% ATP.lambda_889
tff(fact_8669_ATP_Olambda__890,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: A,Uub: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_iv(fun(A,nat),fun(A,fun(A,$o)),Uu),Uua),Uub)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),
          $ite(Uub = Uua,aa(nat,nat,suc,aa(A,nat,Uu,Uub)),aa(A,nat,Uu,Uub))) ) ).

% ATP.lambda_890
tff(fact_8670_ATP_Olambda__891,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_iw(fun(A,nat),fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),
          $ite(aa(A,$o,Uua,Uub),aa(A,nat,Uu,Uub),zero_zero(nat))) ) ).

% ATP.lambda_891
tff(fact_8671_ATP_Olambda__892,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: fun(A,nat),Uub: A] :
      ( aa(A,$o,aa(fun(A,nat),fun(A,$o),aTP_Lamp_iu(fun(A,nat),fun(fun(A,nat),fun(A,$o)),Uu),Uua),Uub)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(A,nat,Uu,Uub)),aa(A,nat,Uua,Uub))) ) ).

% ATP.lambda_892
tff(fact_8672_ATP_Olambda__893,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,B),Uua: fun(A,$o),Uub: B] :
      ( aa(B,$o,aa(fun(A,$o),fun(B,$o),aTP_Lamp_atg(fun(A,B),fun(fun(A,$o),fun(B,$o)),Uu),Uua),Uub)
    <=> aa(A,$o,Uua,aa(B,A,hilbert_inv_into(A,B,top_top(set(A)),Uu),Uub)) ) ).

% ATP.lambda_893
tff(fact_8673_ATP_Olambda__894,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_hq(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_894
tff(fact_8674_ATP_Olambda__895,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_hp(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_895
tff(fact_8675_ATP_Olambda__896,axiom,
    ! [Uu: fun(nat,$o),Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_alb(fun(nat,$o),fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> aa(nat,$o,Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)) ) ).

% ATP.lambda_896
tff(fact_8676_ATP_Olambda__897,axiom,
    ! [A: $tType,Uu: fun(nat,set(A)),Uua: nat,Uub: nat] : aa(nat,set(A),aa(nat,fun(nat,set(A)),aTP_Lamp_na(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_897
tff(fact_8677_ATP_Olambda__898,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_fn(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_898
tff(fact_8678_ATP_Olambda__899,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_dr(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_899
tff(fact_8679_ATP_Olambda__900,axiom,
    ! [A: $tType,B: $tType,Uu: fun(product_prod(A,B),$o),Uua: A,Uub: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_aoo(fun(product_prod(A,B),$o),fun(A,fun(B,$o)),Uu),Uua),Uub)
    <=> aa(product_prod(A,B),$o,Uu,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),Uub)) ) ).

% ATP.lambda_900
tff(fact_8680_ATP_Olambda__901,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_bq(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_901
tff(fact_8681_ATP_Olambda__902,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: list(A),Uub: nat] :
      ( aa(nat,$o,aa(list(A),fun(nat,$o),aTP_Lamp_abe(fun(A,$o),fun(list(A),fun(nat,$o)),Uu),Uua),Uub)
    <=> aa(A,$o,Uu,aa(nat,A,nth(A,Uua),Uub)) ) ).

% ATP.lambda_902
tff(fact_8682_ATP_Olambda__903,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [Uu: fun(multiset(A),B),Uua: fun(C,multiset(A)),Uub: C] : aa(C,B,aa(fun(C,multiset(A)),fun(C,B),aTP_Lamp_asj(fun(multiset(A),B),fun(fun(C,multiset(A)),fun(C,B)),Uu),Uua),Uub) = aa(multiset(A),B,Uu,aa(C,multiset(A),Uua,Uub)) ) ).

% ATP.lambda_903
tff(fact_8683_ATP_Olambda__904,axiom,
    ! [B: $tType,C: $tType,D: $tType,A: $tType,Uu: fun(D,fun(B,C)),Uua: fun(A,D),Uub: A] : aa(A,fun(B,C),aa(fun(A,D),fun(A,fun(B,C)),aTP_Lamp_vv(fun(D,fun(B,C)),fun(fun(A,D),fun(A,fun(B,C))),Uu),Uua),Uub) = aa(D,fun(B,C),Uu,aa(A,D,Uua,Uub)) ).

% ATP.lambda_904
tff(fact_8684_ATP_Olambda__905,axiom,
    ! [A: $tType,C: $tType,B: $tType,Uu: fun(C,set(A)),Uua: fun(B,C),Uub: B] : aa(B,set(A),aa(fun(B,C),fun(B,set(A)),aTP_Lamp_kq(fun(C,set(A)),fun(fun(B,C),fun(B,set(A))),Uu),Uua),Uub) = aa(C,set(A),Uu,aa(B,C,Uua,Uub)) ).

% ATP.lambda_905
tff(fact_8685_ATP_Olambda__906,axiom,
    ! [D: $tType,C: $tType,F3: $tType,Uu: fun(C,fun(D,$o)),Uua: fun(F3,C),Uub: F3] : aa(F3,fun(D,$o),aa(fun(F3,C),fun(F3,fun(D,$o)),aTP_Lamp_aoz(fun(C,fun(D,$o)),fun(fun(F3,C),fun(F3,fun(D,$o))),Uu),Uua),Uub) = aa(C,fun(D,$o),Uu,aa(F3,C,Uua,Uub)) ).

% ATP.lambda_906
tff(fact_8686_ATP_Olambda__907,axiom,
    ! [B: $tType,C: $tType,A: $tType,Uu: fun(C,B),Uua: fun(A,C),Uub: A] : aa(A,B,aa(fun(A,C),fun(A,B),aTP_Lamp_axj(fun(C,B),fun(fun(A,C),fun(A,B)),Uu),Uua),Uub) = aa(C,B,Uu,aa(A,C,Uua,Uub)) ).

% ATP.lambda_907
tff(fact_8687_ATP_Olambda__908,axiom,
    ! [A: $tType,C: $tType,B: $tType,Uu: fun(C,A),Uua: fun(B,C),Uub: B] : aa(B,A,aa(fun(B,C),fun(B,A),aTP_Lamp_ly(fun(C,A),fun(fun(B,C),fun(B,A)),Uu),Uua),Uub) = aa(C,A,Uu,aa(B,C,Uua,Uub)) ).

% ATP.lambda_908
tff(fact_8688_ATP_Olambda__909,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,$o),Uua: fun(A,B),Uub: A] :
      ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_avg(fun(B,$o),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
    <=> aa(B,$o,Uu,aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_909
tff(fact_8689_ATP_Olambda__910,axiom,
    ! [C: $tType,B: $tType,D: $tType,Uu: fun(B,fun(C,$o)),Uua: fun(D,B),Uub: D] : aa(D,fun(C,$o),aa(fun(D,B),fun(D,fun(C,$o)),aTP_Lamp_wf(fun(B,fun(C,$o)),fun(fun(D,B),fun(D,fun(C,$o))),Uu),Uua),Uub) = aa(B,fun(C,$o),Uu,aa(D,B,Uua,Uub)) ).

% ATP.lambda_910
tff(fact_8690_ATP_Olambda__911,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: fun(num,B),Uub: num] : aa(num,A,aa(fun(num,B),fun(num,A),aTP_Lamp_tn(fun(B,A),fun(fun(num,B),fun(num,A)),Uu),Uua),Uub) = aa(B,A,Uu,aa(num,B,Uua,Uub)) ).

% ATP.lambda_911
tff(fact_8691_ATP_Olambda__912,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: fun(nat,B),Uub: nat] : aa(nat,A,aa(fun(nat,B),fun(nat,A),aTP_Lamp_ef(fun(B,A),fun(fun(nat,B),fun(nat,A)),Uu),Uua),Uub) = aa(B,A,Uu,aa(nat,B,Uua,Uub)) ).

% ATP.lambda_912
tff(fact_8692_ATP_Olambda__913,axiom,
    ! [A: $tType,B: $tType,D: $tType,Uu: fun(B,A),Uua: fun(D,B),Uub: D] : aa(D,A,aa(fun(D,B),fun(D,A),aTP_Lamp_atp(fun(B,A),fun(fun(D,B),fun(D,A)),Uu),Uua),Uub) = aa(B,A,Uu,aa(D,B,Uua,Uub)) ).

% ATP.lambda_913
tff(fact_8693_ATP_Olambda__914,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: fun(B,A),Uua: fun(C,B),Uub: C] : aa(C,A,aa(fun(C,B),fun(C,A),aTP_Lamp_ll(fun(B,A),fun(fun(C,B),fun(C,A)),Uu),Uua),Uub) = aa(B,A,Uu,aa(C,B,Uua,Uub)) ).

% ATP.lambda_914
tff(fact_8694_ATP_Olambda__915,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_amg(fun(A,B),fun(fun(C,A),fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_915
tff(fact_8695_ATP_Olambda__916,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_tq(fun(A,B),fun(fun(C,A),fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_916
tff(fact_8696_ATP_Olambda__917,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comple6319245703460814977attice(B)
        & comple6319245703460814977attice(A) )
     => ! [Uu: fun(A,B),Uua: fun(B,A),Uub: B] : aa(B,B,aa(fun(B,A),fun(B,B),aTP_Lamp_ahp(fun(A,B),fun(fun(B,A),fun(B,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_917
tff(fact_8697_ATP_Olambda__918,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,$o),Uua: fun(B,A),Uub: B] :
      ( aa(B,$o,aa(fun(B,A),fun(B,$o),aTP_Lamp_akb(fun(A,$o),fun(fun(B,A),fun(B,$o)),Uu),Uua),Uub)
    <=> aa(A,$o,Uu,aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_918
tff(fact_8698_ATP_Olambda__919,axiom,
    ! [B: $tType,A: $tType,E: $tType,Uu: fun(A,fun(B,$o)),Uua: fun(E,A),Uub: E] : aa(E,fun(B,$o),aa(fun(E,A),fun(E,fun(B,$o)),aTP_Lamp_aoy(fun(A,fun(B,$o)),fun(fun(E,A),fun(E,fun(B,$o))),Uu),Uua),Uub) = aa(A,fun(B,$o),Uu,aa(E,A,Uua,Uub)) ).

% ATP.lambda_919
tff(fact_8699_ATP_Olambda__920,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu: fun(A,fun(B,$o)),Uua: fun(C,A),Uub: C] : aa(C,fun(B,$o),aa(fun(C,A),fun(C,fun(B,$o)),aTP_Lamp_aiv(fun(A,fun(B,$o)),fun(fun(C,A),fun(C,fun(B,$o))),Uu),Uua),Uub) = aa(A,fun(B,$o),Uu,aa(C,A,Uua,Uub)) ).

% ATP.lambda_920
tff(fact_8700_ATP_Olambda__921,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu: fun(A,B),Uua: fun(C,A),Uub: C] : aa(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_ajj(fun(A,B),fun(fun(C,A),fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(C,A,Uua,Uub)) ).

% ATP.lambda_921
tff(fact_8701_ATP_Olambda__922,axiom,
    ! [D: $tType,C: $tType,Uu: fun(C,D),Uua: fun(D,$o),Uub: C] :
      ( aa(C,$o,aa(fun(D,$o),fun(C,$o),aTP_Lamp_apk(fun(C,D),fun(fun(D,$o),fun(C,$o)),Uu),Uua),Uub)
    <=> aa(D,$o,Uua,aa(C,D,Uu,Uub)) ) ).

% ATP.lambda_922
tff(fact_8702_ATP_Olambda__923,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: fun(A,$o),Uub: B] :
      ( aa(B,$o,aa(fun(A,$o),fun(B,$o),aTP_Lamp_anm(fun(B,A),fun(fun(A,$o),fun(B,$o)),Uu),Uua),Uub)
    <=> aa(A,$o,Uua,aa(B,A,Uu,Uub)) ) ).

% ATP.lambda_923
tff(fact_8703_ATP_Olambda__924,axiom,
    ! [C: $tType,A: $tType,B: $tType,Uu: fun(B,A),Uua: fun(A,C),Uub: B] : aa(B,C,aa(fun(A,C),fun(B,C),aTP_Lamp_me(fun(B,A),fun(fun(A,C),fun(B,C)),Uu),Uua),Uub) = aa(A,C,Uua,aa(B,A,Uu,Uub)) ).

% ATP.lambda_924
tff(fact_8704_ATP_Olambda__925,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comple6319245703460814977attice(A)
        & comple6319245703460814977attice(B) )
     => ! [Uu: fun(A,B),Uua: fun(B,A),Uub: A] : aa(A,A,aa(fun(B,A),fun(A,A),aTP_Lamp_aho(fun(A,B),fun(fun(B,A),fun(A,A)),Uu),Uua),Uub) = aa(B,A,Uua,aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_925
tff(fact_8705_ATP_Olambda__926,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: fun(B,$o),Uub: A] :
      ( aa(A,$o,aa(fun(B,$o),fun(A,$o),aTP_Lamp_av(fun(A,B),fun(fun(B,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> aa(B,$o,Uua,aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_926
tff(fact_8706_ATP_Olambda__927,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( comm_monoid_mult(C)
     => ! [Uu: fun(A,B),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_ajs(fun(A,B),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_927
tff(fact_8707_ATP_Olambda__928,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( comm_monoid_add(C)
     => ! [Uu: fun(A,B),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_ajr(fun(A,B),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_928
tff(fact_8708_ATP_Olambda__929,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_nj(fun(A,B),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_929
tff(fact_8709_ATP_Olambda__930,axiom,
    ! [C: $tType,B: $tType,A: $tType,Uu: fun(A,B),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_alk(fun(A,B),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu,Uub)) ).

% ATP.lambda_930
tff(fact_8710_ATP_Olambda__931,axiom,
    ! [A: $tType,Uu: fun(code_natural,A),Uua: code_natural,Uub: A] : aa(A,A,aa(code_natural,fun(A,A),aTP_Lamp_ayc(fun(code_natural,A),fun(code_natural,fun(A,A)),Uu),Uua),Uub) = aa(code_natural,A,Uu,Uua) ).

% ATP.lambda_931
tff(fact_8711_ATP_Olambda__932,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_rp(fun(D,set(B)),fun(D,fun(A,set(B))),Uu),Uua),Uub) = aa(D,set(B),Uu,Uua) ).

% ATP.lambda_932
tff(fact_8712_ATP_Olambda__933,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu: fun(C,fun(A,C)),Uua: C,Uub: B] : aa(B,fun(A,C),aa(C,fun(B,fun(A,C)),aTP_Lamp_ady(fun(C,fun(A,C)),fun(C,fun(B,fun(A,C))),Uu),Uua),Uub) = aa(C,fun(A,C),Uu,Uua) ).

% ATP.lambda_933
tff(fact_8713_ATP_Olambda__934,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,$o),Uua: B,Uub: A] :
      ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_anv(fun(B,$o),fun(B,fun(A,$o)),Uu),Uua),Uub)
    <=> aa(B,$o,Uu,Uua) ) ).

% ATP.lambda_934
tff(fact_8714_ATP_Olambda__935,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,$o),Uua: A,Uub: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_axe(fun(A,$o),fun(A,fun(B,$o)),Uu),Uua),Uub)
    <=> aa(A,$o,Uu,Uua) ) ).

% ATP.lambda_935
tff(fact_8715_ATP_Olambda__936,axiom,
    ! [B: $tType,C: $tType,A: $tType,Uu: fun(A,C),Uua: A,Uub: B] : aa(B,C,aa(A,fun(B,C),aTP_Lamp_vn(fun(A,C),fun(A,fun(B,C)),Uu),Uua),Uub) = aa(A,C,Uu,Uua) ).

% ATP.lambda_936
tff(fact_8716_ATP_Olambda__937,axiom,
    ! [C: $tType,B: $tType,A: $tType,Uu: fun(A,B),Uua: A,Uub: C] : aa(C,B,aa(A,fun(C,B),aTP_Lamp_acd(fun(A,B),fun(A,fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,Uua) ).

% ATP.lambda_937
tff(fact_8717_ATP_Olambda__938,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A] : aa(A,fun(product_prod(A,A),$o),aa(A,fun(A,fun(product_prod(A,A),$o)),aTP_Lamp_uz(set(product_prod(A,A)),fun(A,fun(A,fun(product_prod(A,A),$o))),Uu),Uua),Uub) = aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),aa(A,fun(A,fun(A,$o)),aa(A,fun(A,fun(A,fun(A,$o))),aTP_Lamp_uy(set(product_prod(A,A)),fun(A,fun(A,fun(A,fun(A,$o)))),Uu),Uua),Uub)) ).

% ATP.lambda_938
tff(fact_8718_ATP_Olambda__939,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_ass(fun(A,fun(A,A)),fun(A,fun(option(A),option(A))),Uu),Uua),Uub) = aa(A,option(A),some(A),case_option(A,A,Uua,aa(A,fun(A,A),Uu,Uua),Uub)) ).

% ATP.lambda_939
tff(fact_8719_ATP_Olambda__940,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural)))),Uua: product_prod(code_natural,code_natural),Uub: B] : aa(B,pred(A),aa(product_prod(code_natural,code_natural),fun(B,pred(A)),aTP_Lamp_awc(fun(B,fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural)))),fun(product_prod(code_natural,code_natural),fun(B,pred(A))),Uu),Uua),Uub) = aa(product_prod(pred(A),product_prod(code_natural,code_natural)),pred(A),product_fst(pred(A),product_prod(code_natural,code_natural)),aa(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural)),aa(B,fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))),Uu,Uub),Uua)) ).

% ATP.lambda_940
tff(fact_8720_ATP_Olambda__941,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_avy(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_941
tff(fact_8721_ATP_Olambda__942,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu: fun(A,fun(B,fun(C,$o))),Uua: A,Uub: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_axg(fun(A,fun(B,fun(C,$o))),fun(A,fun(B,$o)),Uu),Uua),Uub)
    <=> ? [X_12: C] : aa(C,$o,aa(B,fun(C,$o),aa(A,fun(B,fun(C,$o)),Uu,Uua),Uub),X_12) ) ).

% ATP.lambda_942
tff(fact_8722_ATP_Olambda__943,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(B,fun(C,A)),Uua: multiset(B),Uub: C] : aa(C,A,aa(multiset(B),fun(C,A),aTP_Lamp_arb(fun(B,fun(C,A)),fun(multiset(B),fun(C,A)),Uu),Uua),Uub) = aa(multiset(A),A,comm_m9189036328036947845d_mset(A),aa(multiset(B),multiset(A),image_mset(B,A,aa(C,fun(B,A),aTP_Lamp_ff(fun(B,fun(C,A)),fun(C,fun(B,A)),Uu),Uub)),Uua)) ) ).

% ATP.lambda_943
tff(fact_8723_ATP_Olambda__944,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(B,fun(C,A)),Uua: multiset(B),Uub: C] : aa(C,A,aa(multiset(B),fun(C,A),aTP_Lamp_aqe(fun(B,fun(C,A)),fun(multiset(B),fun(C,A)),Uu),Uua),Uub) = comm_m7189776963980413722m_mset(A,aa(multiset(B),multiset(A),image_mset(B,A,aa(C,fun(B,A),aTP_Lamp_cw(fun(B,fun(C,A)),fun(C,fun(B,A)),Uu),Uub)),Uua)) ) ).

% ATP.lambda_944
tff(fact_8724_ATP_Olambda__945,axiom,
    ! [B: $tType,C: $tType,A: $tType,E: $tType,D: $tType,Uu: fun(B,fun(C,fun(D,fun(E,set(A))))),Uua: set(product_prod(D,E)),Uub: product_prod(B,C)] : aa(product_prod(B,C),set(A),aa(set(product_prod(D,E)),fun(product_prod(B,C),set(A)),aTP_Lamp_og(fun(B,fun(C,fun(D,fun(E,set(A))))),fun(set(product_prod(D,E)),fun(product_prod(B,C),set(A))),Uu),Uua),Uub) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(product_prod(D,E)),set(set(A)),image2(product_prod(D,E),set(A),aa(product_prod(B,C),fun(product_prod(D,E),set(A)),aTP_Lamp_of(fun(B,fun(C,fun(D,fun(E,set(A))))),fun(product_prod(B,C),fun(product_prod(D,E),set(A))),Uu),Uub)),Uua)) ).

% ATP.lambda_945
tff(fact_8725_ATP_Olambda__946,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: fun(B,fun(C,A)),Uua: set(B),Uub: C] : aa(C,A,aa(set(B),fun(C,A),aTP_Lamp_ma(fun(B,fun(C,A)),fun(set(B),fun(C,A)),Uu),Uua),Uub) = aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,aa(C,fun(B,A),aTP_Lamp_lc(fun(B,fun(C,A)),fun(C,fun(B,A)),Uu),Uub)),Uua)) ) ).

% ATP.lambda_946
tff(fact_8726_ATP_Olambda__947,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: fun(B,fun(C,A)),Uua: set(B),Uub: C] : aa(C,A,aa(set(B),fun(C,A),aTP_Lamp_ld(fun(B,fun(C,A)),fun(set(B),fun(C,A)),Uu),Uua),Uub) = aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,aa(C,fun(B,A),aTP_Lamp_lc(fun(B,fun(C,A)),fun(C,fun(B,A)),Uu),Uub)),Uua)) ) ).

% ATP.lambda_947
tff(fact_8727_ATP_Olambda__948,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(B,fun(C,A)),Uua: multiset(C),Uub: B] : aa(B,A,aa(multiset(C),fun(B,A),aTP_Lamp_ara(fun(B,fun(C,A)),fun(multiset(C),fun(B,A)),Uu),Uua),Uub) = aa(multiset(A),A,comm_m9189036328036947845d_mset(A),aa(multiset(C),multiset(A),image_mset(C,A,aa(B,fun(C,A),Uu,Uub)),Uua)) ) ).

% ATP.lambda_948
tff(fact_8728_ATP_Olambda__949,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(B,fun(C,A)),Uua: multiset(C),Uub: B] : aa(B,A,aa(multiset(C),fun(B,A),aTP_Lamp_aqd(fun(B,fun(C,A)),fun(multiset(C),fun(B,A)),Uu),Uua),Uub) = comm_m7189776963980413722m_mset(A,aa(multiset(C),multiset(A),image_mset(C,A,aa(B,fun(C,A),Uu,Uub)),Uua)) ) ).

% ATP.lambda_949
tff(fact_8729_ATP_Olambda__950,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: fun(B,fun(C,A)),Uua: set(C),Uub: B] : aa(B,A,aa(set(C),fun(B,A),aTP_Lamp_lz(fun(B,fun(C,A)),fun(set(C),fun(B,A)),Uu),Uua),Uub) = aa(set(A),A,complete_Sup_Sup(A),aa(set(C),set(A),image2(C,A,aa(B,fun(C,A),Uu,Uub)),Uua)) ) ).

% ATP.lambda_950
tff(fact_8730_ATP_Olambda__951,axiom,
    ! [A: $tType,C: $tType,B: $tType,Uu: fun(A,fun(B,set(C))),Uua: set(B),Uub: A] : aa(A,set(C),aa(set(B),fun(A,set(C)),aTP_Lamp_ajx(fun(A,fun(B,set(C))),fun(set(B),fun(A,set(C))),Uu),Uua),Uub) = aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(B),set(set(C)),image2(B,set(C),aa(A,fun(B,set(C)),Uu,Uub)),Uua)) ).

% ATP.lambda_951
tff(fact_8731_ATP_Olambda__952,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: fun(B,fun(C,A)),Uua: set(C),Uub: B] : aa(B,A,aa(set(C),fun(B,A),aTP_Lamp_lb(fun(B,fun(C,A)),fun(set(C),fun(B,A)),Uu),Uua),Uub) = aa(set(A),A,complete_Inf_Inf(A),aa(set(C),set(A),image2(C,A,aa(B,fun(C,A),Uu,Uub)),Uua)) ) ).

% ATP.lambda_952
tff(fact_8732_ATP_Olambda__953,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,B),Uua: B,Uub: A] :
      ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_pe(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(A,B,Uu,Uub) != Uua ) ) ).

% ATP.lambda_953
tff(fact_8733_ATP_Olambda__954,axiom,
    ! [A: $tType,B: $tType,Uu: B,Uua: fun(A,B),Uub: A] :
      ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_pf(B,fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(A,B,Uua,Uub) != Uu ) ) ).

% ATP.lambda_954
tff(fact_8734_ATP_Olambda__955,axiom,
    ! [B: $tType,A: $tType,Uu: set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),Uua: set(old_node(A,B)),Uub: set(old_node(A,B))] : aa(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),aa(set(old_node(A,B)),fun(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),aTP_Lamp_auz(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),fun(set(old_node(A,B)),fun(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))))),Uu),Uua),Uub) = aa(set(set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),complete_Sup_Sup(set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),set(set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),image2(product_prod(set(old_node(A,B)),set(old_node(A,B))),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),aa(fun(set(old_node(A,B)),fun(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))))),fun(product_prod(set(old_node(A,B)),set(old_node(A,B))),set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),product_case_prod(set(old_node(A,B)),set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),aa(set(old_node(A,B)),fun(set(old_node(A,B)),fun(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))))),aTP_Lamp_auy(set(old_node(A,B)),fun(set(old_node(A,B)),fun(set(old_node(A,B)),fun(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B))))))),Uua),Uub))),Uu)) ).

% ATP.lambda_955
tff(fact_8735_ATP_Olambda__956,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural)))),Uua: pred(B),Uub: product_prod(code_natural,code_natural)] : aa(product_prod(code_natural,code_natural),fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))),aa(pred(B),fun(product_prod(code_natural,code_natural),fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural)))),aTP_Lamp_awd(fun(B,fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural)))),fun(pred(B),fun(product_prod(code_natural,code_natural),fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))))),Uu),Uua),Uub) = aa(pred(A),fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))),product_Pair(pred(A),product_prod(code_natural,code_natural)),bind2(B,A,Uua,aa(product_prod(code_natural,code_natural),fun(B,pred(A)),aTP_Lamp_awc(fun(B,fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural)))),fun(product_prod(code_natural,code_natural),fun(B,pred(A))),Uu),Uub))) ).

% ATP.lambda_956
tff(fact_8736_ATP_Olambda__957,axiom,
    ! [A: $tType,C: $tType,B: $tType,Uu: fun(C,set(A)),Uua: fun(B,set(C)),Uub: B] : aa(B,set(A),aa(fun(B,set(C)),fun(B,set(A)),aTP_Lamp_mc(fun(C,set(A)),fun(fun(B,set(C)),fun(B,set(A))),Uu),Uua),Uub) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(C),set(set(A)),image2(C,set(A),Uu),aa(B,set(C),Uua,Uub))) ).

% ATP.lambda_957
tff(fact_8737_ATP_Olambda__958,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: fun(B,set(A)),Uua: fun(C,set(B)),Uub: C] : aa(C,set(A),aa(fun(C,set(B)),fun(C,set(A)),aTP_Lamp_mb(fun(B,set(A)),fun(fun(C,set(B)),fun(C,set(A))),Uu),Uua),Uub) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),Uu),aa(C,set(B),Uua,Uub))) ).

% ATP.lambda_958
tff(fact_8738_ATP_Olambda__959,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: fun(B,A),Uua: fun(C,set(B)),Uub: C] : aa(C,A,aa(fun(C,set(B)),fun(C,A),aTP_Lamp_lq(fun(B,A),fun(fun(C,set(B)),fun(C,A)),Uu),Uua),Uub) = aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,Uu),aa(C,set(B),Uua,Uub))) ) ).

% ATP.lambda_959
tff(fact_8739_ATP_Olambda__960,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( condit1219197933456340205attice(C)
     => ! [Uu: fun(A,set(B)),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_alz(fun(A,set(B)),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(set(C),C,complete_Sup_Sup(C),aa(set(B),set(C),image2(B,C,Uua),aa(A,set(B),Uu,Uub))) ) ).

% ATP.lambda_960
tff(fact_8740_ATP_Olambda__961,axiom,
    ! [A: $tType,C: $tType,B: $tType,Uu: fun(C,set(A)),Uua: fun(B,set(C)),Uub: B] : aa(B,set(A),aa(fun(B,set(C)),fun(B,set(A)),aTP_Lamp_ks(fun(C,set(A)),fun(fun(B,set(C)),fun(B,set(A))),Uu),Uua),Uub) = aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(C),set(set(A)),image2(C,set(A),Uu),aa(B,set(C),Uua,Uub))) ).

% ATP.lambda_961
tff(fact_8741_ATP_Olambda__962,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( condit1219197933456340205attice(C)
     => ! [Uu: fun(A,set(B)),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_ami(fun(A,set(B)),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(set(C),C,complete_Inf_Inf(C),aa(set(B),set(C),image2(B,C,Uua),aa(A,set(B),Uu,Uub))) ) ).

% ATP.lambda_962
tff(fact_8742_ATP_Olambda__963,axiom,
    ! [B: $tType,A: $tType] :
      ( euclid4440199948858584721cancel(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_ii(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> ~ aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),Uua),aa(A,B,Uu,Uub)) ) ) ).

% ATP.lambda_963
tff(fact_8743_ATP_Olambda__964,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_rs(fun(D,set(B)),fun(set(D),fun(A,set(B))),Uu),Uua),Uub) = aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(D),set(set(B)),image2(D,set(B),Uu),Uua)) ).

% ATP.lambda_964
tff(fact_8744_ATP_Olambda__965,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: B,Uub: list(B)] : aa(list(B),fun(list(A),list(A)),aa(B,fun(list(B),fun(list(A),list(A))),aTP_Lamp_ajn(fun(B,A),fun(B,fun(list(B),fun(list(A),list(A)))),Uu),Uua),Uub) = aa(A,fun(list(A),list(A)),cons(A),aa(B,A,Uu,Uua)) ).

% ATP.lambda_965
tff(fact_8745_ATP_Olambda__966,axiom,
    ! [B: $tType,A: $tType,Uu: set(B),Uua: fun(A,fun(B,$o)),Uub: A] : aa(A,nat,aa(fun(A,fun(B,$o)),fun(A,nat),aTP_Lamp_je(set(B),fun(fun(A,fun(B,$o)),fun(A,nat)),Uu),Uua),Uub) = aa(set(B),nat,finite_card(B),aa(fun(B,$o),set(B),collect(B),aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aTP_Lamp_ha(set(B),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),Uu),Uua),Uub))) ).

% ATP.lambda_966
tff(fact_8746_ATP_Olambda__967,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu: fun(A,fun(B,fun(C,$o))),Uua: A,Uub: C] :
      ( aa(C,$o,aa(A,fun(C,$o),aTP_Lamp_axh(fun(A,fun(B,fun(C,$o))),fun(A,fun(C,$o)),Uu),Uua),Uub)
    <=> ? [B6: B] : aa(C,$o,aa(B,fun(C,$o),aa(A,fun(B,fun(C,$o)),Uu,Uua),B6),Uub) ) ).

% ATP.lambda_967
tff(fact_8747_ATP_Olambda__968,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu: fun(A,fun(B,fun(C,$o))),Uua: B,Uub: C] :
      ( aa(C,$o,aa(B,fun(C,$o),aTP_Lamp_axi(fun(A,fun(B,fun(C,$o))),fun(B,fun(C,$o)),Uu),Uua),Uub)
    <=> ? [A10: A] : aa(C,$o,aa(B,fun(C,$o),aa(A,fun(B,fun(C,$o)),Uu,A10),Uua),Uub) ) ).

% ATP.lambda_968
tff(fact_8748_ATP_Olambda__969,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu: fun(A,fun(B,C)),Uua: set(C),Uub: A] :
      ( aa(A,$o,aa(set(C),fun(A,$o),aTP_Lamp_aes(fun(A,fun(B,C)),fun(set(C),fun(A,$o)),Uu),Uua),Uub)
    <=> ? [B6: B] : aa(set(C),$o,member(C,aa(B,C,aa(A,fun(B,C),Uu,Uub),B6)),Uua) ) ).

% ATP.lambda_969
tff(fact_8749_ATP_Olambda__970,axiom,
    ! [A: $tType,B: $tType,Uu: list(A),Uua: list(B),Uub: product_prod(A,B)] :
      ( aa(product_prod(A,B),$o,aa(list(B),fun(product_prod(A,B),$o),aTP_Lamp_aeg(list(A),fun(list(B),fun(product_prod(A,B),$o)),Uu),Uua),Uub)
    <=> ? [I3: 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),I3)),aa(nat,B,nth(B,Uua),I3)) )
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),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_970
tff(fact_8750_ATP_Olambda__971,axiom,
    ! [B: $tType,A: $tType,Uu: set(A),Uua: fun(A,B),Uub: product_prod(A,B)] :
      ( aa(product_prod(A,B),$o,aa(fun(A,B),fun(product_prod(A,B),$o),aTP_Lamp_afm(set(A),fun(fun(A,B),fun(product_prod(A,B),$o)),Uu),Uua),Uub)
    <=> ? [A10: A] :
          ( ( Uub = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A10),aa(A,B,Uua,A10)) )
          & aa(set(A),$o,member(A,A10),Uu) ) ) ).

% ATP.lambda_971
tff(fact_8751_ATP_Olambda__972,axiom,
    ! [A: $tType,Uu: list(A),Uua: set(nat),Uub: A] :
      ( aa(A,$o,aa(set(nat),fun(A,$o),aTP_Lamp_agh(list(A),fun(set(nat),fun(A,$o)),Uu),Uua),Uub)
    <=> ? [I3: nat] :
          ( ( Uub = aa(nat,A,nth(A,Uu),I3) )
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),Uu))
          & aa(set(nat),$o,member(nat,I3),Uua) ) ) ).

% ATP.lambda_972
tff(fact_8752_ATP_Olambda__973,axiom,
    ! [A: $tType,Uu: nat,Uua: list(A),Uub: A] :
      ( aa(A,$o,aa(list(A),fun(A,$o),aTP_Lamp_agj(nat,fun(list(A),fun(A,$o)),Uu),Uua),Uub)
    <=> ? [I3: nat] :
          ( ( Uub = aa(nat,A,nth(A,Uua),I3) )
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Uu),I3)
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),Uua)) ) ) ).

% ATP.lambda_973
tff(fact_8753_ATP_Olambda__974,axiom,
    ! [B: $tType,A: $tType,Uu: set(B),Uua: fun(B,set(A)),Uub: A] :
      ( aa(A,$o,aa(fun(B,set(A)),fun(A,$o),aTP_Lamp_asu(set(B),fun(fun(B,set(A)),fun(A,$o)),Uu),Uua),Uub)
    <=> ? [X4: set(A)] :
          ( aa(set(set(A)),$o,member(set(A),X4),aa(set(B),set(set(A)),image2(B,set(A),Uua),Uu))
          & aa(set(A),$o,member(A,Uub),X4) ) ) ).

% ATP.lambda_974
tff(fact_8754_ATP_Olambda__975,axiom,
    ! [A: $tType,Uu: set(set(product_prod(A,A))),Uua: set(product_prod(A,A)),Uub: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),aTP_Lamp_aju(set(set(product_prod(A,A))),fun(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o)),Uu),Uua),Uub)
    <=> ? [R5: set(product_prod(A,A))] :
          ( ( Uub = 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))),R5),Uua) )
          & aa(set(set(product_prod(A,A))),$o,member(set(product_prod(A,A)),R5),Uu) ) ) ).

% ATP.lambda_975
tff(fact_8755_ATP_Olambda__976,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [Uu: set(A),Uua: A,Uub: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_agf(set(A),fun(A,fun(A,$o)),Uu),Uua),Uub)
        <=> ? [A10: A] :
              ( ( Uub = aa(A,A,aa(A,fun(A,A),sup_sup(A),Uua),A10) )
              & aa(set(A),$o,member(A,A10),Uu) ) ) ) ).

% ATP.lambda_976
tff(fact_8756_ATP_Olambda__977,axiom,
    ! [A: $tType] :
      ( finite8700451911770168679attice(A)
     => ! [Uu: A,Uua: set(A),Uub: A] :
          ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_afd(A,fun(set(A),fun(A,$o)),Uu),Uua),Uub)
        <=> ? [B6: A] :
              ( ( Uub = aa(A,A,aa(A,fun(A,A),inf_inf(A),Uu),B6) )
              & aa(set(A),$o,member(A,B6),Uua) ) ) ) ).

% ATP.lambda_977
tff(fact_8757_ATP_Olambda__978,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [Uu: set(A),Uua: A,Uub: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_agd(set(A),fun(A,fun(A,$o)),Uu),Uua),Uub)
        <=> ? [A10: A] :
              ( ( Uub = aa(A,A,aa(A,fun(A,A),inf_inf(A),Uua),A10) )
              & aa(set(A),$o,member(A,A10),Uu) ) ) ) ).

% ATP.lambda_978
tff(fact_8758_ATP_Olambda__979,axiom,
    ! [A: $tType,Uu: set(multiset(A)),Uua: multiset(A),Uub: multiset(A)] :
      ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),aTP_Lamp_asa(set(multiset(A)),fun(multiset(A),fun(multiset(A),$o)),Uu),Uua),Uub)
    <=> ? [A10: multiset(A)] :
          ( ( Uub = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),union_mset(A),Uua),A10) )
          & aa(set(multiset(A)),$o,member(multiset(A),A10),Uu) ) ) ).

% ATP.lambda_979
tff(fact_8759_ATP_Olambda__980,axiom,
    ! [A: $tType,Uu: set(multiset(A)),Uua: multiset(A),Uub: multiset(A)] :
      ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),aTP_Lamp_arw(set(multiset(A)),fun(multiset(A),fun(multiset(A),$o)),Uu),Uua),Uub)
    <=> ? [A10: multiset(A)] :
          ( ( Uub = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),Uua),A10) )
          & aa(set(multiset(A)),$o,member(multiset(A),A10),Uu) ) ) ).

% ATP.lambda_980
tff(fact_8760_ATP_Olambda__981,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,set(A)),Uua: list(B),Uub: set(A)] :
      ( aa(set(A),$o,aa(list(B),fun(set(A),$o),aTP_Lamp_ahi(fun(B,set(A)),fun(list(B),fun(set(A),$o)),Uu),Uua),Uub)
    <=> ? [I3: nat] :
          ( ( Uub = aa(B,set(A),Uu,aa(nat,B,nth(B,Uua),I3)) )
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(list(B),nat,size_size(list(B)),Uua)) ) ) ).

% ATP.lambda_981
tff(fact_8761_ATP_Olambda__982,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,set(A)),Uua: list(B),Uub: set(A)] :
      ( aa(set(A),$o,aa(list(B),fun(set(A),$o),aTP_Lamp_ahj(fun(B,set(A)),fun(list(B),fun(set(A),$o)),Uu),Uua),Uub)
    <=> ? [A10: B] :
          ( ( Uub = aa(B,set(A),Uu,A10) )
          & aa(set(B),$o,member(B,A10),aa(list(B),set(B),set2(B),Uua)) ) ) ).

% ATP.lambda_982
tff(fact_8762_ATP_Olambda__983,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: set(B),Uub: A] :
      ( aa(A,$o,aa(set(B),fun(A,$o),aTP_Lamp_aek(fun(B,A),fun(set(B),fun(A,$o)),Uu),Uua),Uub)
    <=> ? [L4: B] :
          ( ( Uub = aa(B,A,Uu,L4) )
          & aa(set(B),$o,member(B,L4),Uua) ) ) ).

% ATP.lambda_983
tff(fact_8763_ATP_Olambda__984,axiom,
    ! [B: $tType,C: $tType,Uu: fun(B,set(C)),Uua: fun(C,$o),Uub: B] :
      ( aa(B,$o,aa(fun(C,$o),fun(B,$o),aTP_Lamp_afn(fun(B,set(C)),fun(fun(C,$o),fun(B,$o)),Uu),Uua),Uub)
    <=> ! [X4: C] :
          ( aa(set(C),$o,member(C,X4),aa(B,set(C),Uu,Uub))
         => aa(C,$o,Uua,X4) ) ) ).

% ATP.lambda_984
tff(fact_8764_ATP_Olambda__985,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,set(A)),Uua: fun(B,$o),Uub: set(A)] :
      ( aa(set(A),$o,aa(fun(B,$o),fun(set(A),$o),aTP_Lamp_aev(fun(B,set(A)),fun(fun(B,$o),fun(set(A),$o)),Uu),Uua),Uub)
    <=> ? [X4: B] :
          ( ( Uub = aa(B,set(A),Uu,X4) )
          & aa(B,$o,Uua,X4) ) ) ).

% ATP.lambda_985
tff(fact_8765_ATP_Olambda__986,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: fun(B,$o),Uub: A] :
      ( aa(A,$o,aa(fun(B,$o),fun(A,$o),aTP_Lamp_aet(fun(B,A),fun(fun(B,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> ? [X4: B] :
          ( ( Uub = aa(B,A,Uu,X4) )
          & aa(B,$o,Uua,X4) ) ) ).

% ATP.lambda_986
tff(fact_8766_ATP_Olambda__987,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,$o),Uua: fun(A,B),Uub: B] :
      ( aa(B,$o,aa(fun(A,B),fun(B,$o),aTP_Lamp_aeo(fun(A,$o),fun(fun(A,B),fun(B,$o)),Uu),Uua),Uub)
    <=> ? [X4: A] :
          ( ( Uub = aa(A,B,Uua,X4) )
          & aa(A,$o,Uu,X4) ) ) ).

% ATP.lambda_987
tff(fact_8767_ATP_Olambda__988,axiom,
    ! [B: $tType,C: $tType,A: $tType,Uu: fun(A,fun(B,$o)),Uua: fun(C,fun(A,$o)),Uub: C] :
      ( aa(C,$o,aa(fun(C,fun(A,$o)),fun(C,$o),aTP_Lamp_arm(fun(A,fun(B,$o)),fun(fun(C,fun(A,$o)),fun(C,$o)),Uu),Uua),Uub)
    <=> ? [X4: A] :
          ( aa(set(A),$o,member(A,X4),aa(fun(A,$o),set(A),collect(A),aa(fun(A,fun(B,$o)),fun(A,$o),domainp(A,B),Uu)))
          & aa(A,$o,aa(C,fun(A,$o),Uua,Uub),X4) ) ) ).

% ATP.lambda_988
tff(fact_8768_ATP_Olambda__989,axiom,
    ! [B: $tType,C: $tType,A: $tType,Uu: fun(A,fun(B,$o)),Uua: fun(A,C),Uub: fun(A,C)] :
      ( aa(fun(A,C),$o,aa(fun(A,C),fun(fun(A,C),$o),aTP_Lamp_art(fun(A,fun(B,$o)),fun(fun(A,C),fun(fun(A,C),$o)),Uu),Uua),Uub)
    <=> ! [X4: A] :
          ( aa(set(A),$o,member(A,X4),aa(fun(A,$o),set(A),collect(A),aa(fun(A,fun(B,$o)),fun(A,$o),domainp(A,B),Uu)))
         => ( aa(A,C,Uua,X4) = aa(A,C,Uub,X4) ) ) ) ).

% ATP.lambda_989
tff(fact_8769_ATP_Olambda__990,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,option(A)),Uua: list(B),Uub: A] :
      ( aa(A,$o,aa(list(B),fun(A,$o),aTP_Lamp_aga(fun(B,option(A)),fun(list(B),fun(A,$o)),Uu),Uua),Uub)
    <=> ? [X4: B] :
          ( aa(set(B),$o,member(B,X4),aa(list(B),set(B),set2(B),Uua))
          & ( aa(B,option(A),Uu,X4) = aa(A,option(A),some(A),Uub) ) ) ) ).

% ATP.lambda_990
tff(fact_8770_ATP_Olambda__991,axiom,
    ! [A: $tType,B: $tType,Uu: set(B),Uua: fun(A,fun(B,$o)),Uub: A] :
      ( aa(A,$o,aa(fun(A,fun(B,$o)),fun(A,$o),aTP_Lamp_afg(set(B),fun(fun(A,fun(B,$o)),fun(A,$o)),Uu),Uua),Uub)
    <=> ! [X4: B] :
          ( aa(set(B),$o,member(B,X4),Uu)
         => aa(B,$o,aa(A,fun(B,$o),Uua,Uub),X4) ) ) ).

% ATP.lambda_991
tff(fact_8771_ATP_Olambda__992,axiom,
    ! [B: $tType,A: $tType,Uu: set(A),Uua: fun(B,fun(A,$o)),Uub: B] :
      ( aa(B,$o,aa(fun(B,fun(A,$o)),fun(B,$o),aTP_Lamp_akk(set(A),fun(fun(B,fun(A,$o)),fun(B,$o)),Uu),Uua),Uub)
    <=> ! [X4: A] :
          ( aa(set(A),$o,member(A,X4),Uu)
         => aa(A,$o,aa(B,fun(A,$o),Uua,Uub),X4) ) ) ).

% ATP.lambda_992
tff(fact_8772_ATP_Olambda__993,axiom,
    ! [A: $tType,B: $tType,Uu: set(A),Uua: fun(A,fun(B,$o)),Uub: B] :
      ( aa(B,$o,aa(fun(A,fun(B,$o)),fun(B,$o),aTP_Lamp_atb(set(A),fun(fun(A,fun(B,$o)),fun(B,$o)),Uu),Uua),Uub)
    <=> ! [X4: A] :
          ( aa(set(A),$o,member(A,X4),Uu)
         => aa(B,$o,aa(A,fun(B,$o),Uua,X4),Uub) ) ) ).

% ATP.lambda_993
tff(fact_8773_ATP_Olambda__994,axiom,
    ! [A: $tType,B: $tType,Uu: set(B),Uua: fun(A,fun(B,$o)),Uub: A] :
      ( aa(A,$o,aa(fun(A,fun(B,$o)),fun(A,$o),aTP_Lamp_aia(set(B),fun(fun(A,fun(B,$o)),fun(A,$o)),Uu),Uua),Uub)
    <=> ? [X4: B] :
          ( aa(set(B),$o,member(B,X4),Uu)
          & aa(B,$o,aa(A,fun(B,$o),Uua,Uub),X4) ) ) ).

% ATP.lambda_994
tff(fact_8774_ATP_Olambda__995,axiom,
    ! [B: $tType,A: $tType,Uu: set(A),Uua: fun(B,fun(A,$o)),Uub: B] :
      ( aa(B,$o,aa(fun(B,fun(A,$o)),fun(B,$o),aTP_Lamp_aht(set(A),fun(fun(B,fun(A,$o)),fun(B,$o)),Uu),Uua),Uub)
    <=> ? [X4: A] :
          ( aa(set(A),$o,member(A,X4),Uu)
          & aa(A,$o,aa(B,fun(A,$o),Uua,Uub),X4) ) ) ).

% ATP.lambda_995
tff(fact_8775_ATP_Olambda__996,axiom,
    ! [A: $tType,B: $tType,Uu: set(A),Uua: fun(A,fun(B,$o)),Uub: B] :
      ( aa(B,$o,aa(fun(A,fun(B,$o)),fun(B,$o),aTP_Lamp_avh(set(A),fun(fun(A,fun(B,$o)),fun(B,$o)),Uu),Uua),Uub)
    <=> ? [X4: A] :
          ( aa(set(A),$o,member(A,X4),Uu)
          & aa(B,$o,aa(A,fun(B,$o),Uua,X4),Uub) ) ) ).

% ATP.lambda_996
tff(fact_8776_ATP_Olambda__997,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,fun(A,$o)),Uua: set(B),Uub: A] :
      ( aa(A,$o,aa(set(B),fun(A,$o),aTP_Lamp_ahu(fun(B,fun(A,$o)),fun(set(B),fun(A,$o)),Uu),Uua),Uub)
    <=> ? [X4: B] :
          ( aa(set(B),$o,member(B,X4),Uua)
          & aa(A,$o,aa(B,fun(A,$o),Uu,X4),Uub) ) ) ).

% ATP.lambda_997
tff(fact_8777_ATP_Olambda__998,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_aol(set(product_prod(A,A)),fun(A,fun(A,$o)),Uu),Uua),Uub)
    <=> ! [X4: product_prod(A,A)] :
          ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),X4),Uu)
         => aa(product_prod(A,A),$o,aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),aa(A,fun(A,fun(A,$o)),aa(A,fun(A,fun(A,fun(A,$o))),aTP_Lamp_aok(set(product_prod(A,A)),fun(A,fun(A,fun(A,fun(A,$o)))),Uu),Uua),Uub)),X4) ) ) ).

% ATP.lambda_998
tff(fact_8778_ATP_Olambda__999,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: fun(A,fun(B,C)),Uua: set(C),Uub: product_prod(A,B)] :
      ( aa(product_prod(A,B),$o,aa(set(C),fun(product_prod(A,B),$o),aTP_Lamp_aer(fun(A,fun(B,C)),fun(set(C),fun(product_prod(A,B),$o)),Uu),Uua),Uub)
    <=> ? [P6: product_prod(A,B)] :
          ( ( Uub = P6 )
          & aa(set(C),$o,member(C,aa(B,C,aa(A,fun(B,C),Uu,aa(product_prod(A,B),A,product_fst(A,B),P6)),aa(product_prod(A,B),B,product_snd(A,B),P6))),Uua) ) ) ).

% ATP.lambda_999
tff(fact_8779_ATP_Olambda__1000,axiom,
    ! [A: $tType,B: $tType,Uu: set(product_prod(B,A)),Uua: set(B),Uub: A] :
      ( aa(A,$o,aa(set(B),fun(A,$o),aTP_Lamp_ahy(set(product_prod(B,A)),fun(set(B),fun(A,$o)),Uu),Uua),Uub)
    <=> ? [X4: B] :
          ( aa(set(B),$o,member(B,X4),Uua)
          & aa(set(product_prod(B,A)),$o,member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X4),Uub)),Uu) ) ) ).

% ATP.lambda_1000
tff(fact_8780_ATP_Olambda__1001,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: set(A),Uub: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_aib(fun(A,B),fun(set(A),fun(A,$o)),Uu),Uua),Uub)
    <=> ? [X4: A] :
          ( aa(set(A),$o,member(A,X4),Uua)
          & ( aa(A,B,Uu,X4) = aa(A,B,Uu,Uub) ) ) ) ).

% ATP.lambda_1001
tff(fact_8781_ATP_Olambda__1002,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,option(A)),Uua: set(B),Uub: A] :
      ( aa(A,$o,aa(set(B),fun(A,$o),aTP_Lamp_aij(fun(B,option(A)),fun(set(B),fun(A,$o)),Uu),Uua),Uub)
    <=> ? [X4: B] :
          ( aa(set(B),$o,member(B,X4),Uua)
          & ( aa(B,option(A),Uu,X4) = aa(A,option(A),some(A),Uub) ) ) ) ).

% ATP.lambda_1002
tff(fact_8782_ATP_Olambda__1003,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,set(A)),Uua: set(B),Uub: A] :
      ( aa(A,$o,aa(set(B),fun(A,$o),aTP_Lamp_aff(fun(B,set(A)),fun(set(B),fun(A,$o)),Uu),Uua),Uub)
    <=> ! [X4: B] :
          ( aa(set(B),$o,member(B,X4),Uua)
         => aa(set(A),$o,member(A,Uub),aa(B,set(A),Uu,X4)) ) ) ).

% ATP.lambda_1003
tff(fact_8783_ATP_Olambda__1004,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,set(A)),Uua: set(B),Uub: A] :
      ( aa(A,$o,aa(set(B),fun(A,$o),aTP_Lamp_ahz(fun(B,set(A)),fun(set(B),fun(A,$o)),Uu),Uua),Uub)
    <=> ? [X4: B] :
          ( aa(set(B),$o,member(B,X4),Uua)
          & aa(set(A),$o,member(A,Uub),aa(B,set(A),Uu,X4)) ) ) ).

% ATP.lambda_1004
tff(fact_8784_ATP_Olambda__1005,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: set(B),Uub: A] :
      ( aa(A,$o,aa(set(B),fun(A,$o),aTP_Lamp_ahw(fun(B,A),fun(set(B),fun(A,$o)),Uu),Uua),Uub)
    <=> ? [X4: B] :
          ( aa(set(B),$o,member(B,X4),Uua)
          & ( Uub = aa(B,A,Uu,X4) ) ) ) ).

% ATP.lambda_1005
tff(fact_8785_ATP_Olambda__1006,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: filter(B),Uub: fun(A,$o)] :
      ( aa(fun(A,$o),$o,aa(filter(B),fun(fun(A,$o),$o),aTP_Lamp_ali(fun(A,B),fun(filter(B),fun(fun(A,$o),$o)),Uu),Uua),Uub)
    <=> ? [Q7: fun(B,$o)] :
          ( eventually(B,Q7,Uua)
          & ! [X4: A] :
              ( aa(B,$o,Q7,aa(A,B,Uu,X4))
             => aa(A,$o,Uub,X4) ) ) ) ).

% ATP.lambda_1006
tff(fact_8786_ATP_Olambda__1007,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,$o),Uua: fun(B,fun(A,$o)),Uub: B] :
      ( aa(B,$o,aa(fun(B,fun(A,$o)),fun(B,$o),aTP_Lamp_aeh(fun(A,$o),fun(fun(B,fun(A,$o)),fun(B,$o)),Uu),Uua),Uub)
    <=> ? [Y3: A] :
          ( aa(A,$o,Uu,Y3)
          & aa(A,$o,aa(B,fun(A,$o),Uua,Uub),Y3) ) ) ).

% ATP.lambda_1007
tff(fact_8787_ATP_Olambda__1008,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,set(A)),Uua: fun(B,$o),Uub: A] :
      ( aa(A,$o,aa(fun(B,$o),fun(A,$o),aTP_Lamp_aew(fun(B,set(A)),fun(fun(B,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> ? [X4: B] :
          ( aa(B,$o,Uua,X4)
          & aa(set(A),$o,member(A,Uub),aa(B,set(A),Uu,X4)) ) ) ).

% ATP.lambda_1008
tff(fact_8788_ATP_Olambda__1009,axiom,
    ! [A: $tType,Uu: fun(A,A),Uua: A,Uub: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_agi(fun(A,A),fun(A,fun(A,$o)),Uu),Uua),Uub)
    <=> ? [N2: nat] : Uub = aa(A,A,compow(fun(A,A),N2,Uu),Uua) ) ).

% ATP.lambda_1009
tff(fact_8789_ATP_Olambda__1010,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: fun(B,fun(B,$o)),Uub: product_prod(A,A)] :
      ( aa(product_prod(A,A),$o,aa(fun(B,fun(B,$o)),fun(product_prod(A,A),$o),aTP_Lamp_aem(fun(B,A),fun(fun(B,fun(B,$o)),fun(product_prod(A,A),$o)),Uu),Uua),Uub)
    <=> ? [A10: B,B6: B] :
          ( ( Uub = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(B,A,Uu,A10)),aa(B,A,Uu,B6)) )
          & aa(B,$o,aa(B,fun(B,$o),Uua,A10),B6) ) ) ).

% ATP.lambda_1010
tff(fact_8790_ATP_Olambda__1011,axiom,
    ! [A: $tType,B: $tType,Uu: set(product_prod(B,B)),Uua: fun(B,A),Uub: product_prod(A,A)] :
      ( aa(product_prod(A,A),$o,aa(fun(B,A),fun(product_prod(A,A),$o),aTP_Lamp_amx(set(product_prod(B,B)),fun(fun(B,A),fun(product_prod(A,A),$o)),Uu),Uua),Uub)
    <=> ? [A10: B,B6: B] :
          ( ( Uub = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(B,A,Uua,A10)),aa(B,A,Uua,B6)) )
          & aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),A10),B6)),Uu) ) ) ).

% ATP.lambda_1011
tff(fact_8791_ATP_Olambda__1012,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: list(A),Uub: list(A)] :
      ( aa(list(A),$o,aa(list(A),fun(list(A),$o),aTP_Lamp_afp(set(product_prod(A,A)),fun(list(A),fun(list(A),$o)),Uu),Uua),Uub)
    <=> ? [A10: A,V4: list(A)] :
          ( ( Uub = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Uua),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A10),V4)) )
          | ? [U4: list(A),Aa4: A,B6: A,Va2: list(A),W3: list(A)] :
              ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Aa4),B6)),Uu)
              & ( Uua = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),U4),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Aa4),Va2)) )
              & ( Uub = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),U4),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),B6),W3)) ) ) ) ) ).

% ATP.lambda_1012
tff(fact_8792_ATP_Olambda__1013,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [Uu: set(A),Uua: set(A),Uub: A] :
          ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_age(set(A),fun(set(A),fun(A,$o)),Uu),Uua),Uub)
        <=> ? [A10: A,B6: A] :
              ( ( Uub = aa(A,A,aa(A,fun(A,A),sup_sup(A),A10),B6) )
              & aa(set(A),$o,member(A,A10),Uu)
              & aa(set(A),$o,member(A,B6),Uua) ) ) ) ).

% ATP.lambda_1013
tff(fact_8793_ATP_Olambda__1014,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [Uu: set(A),Uua: set(A),Uub: A] :
          ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_agc(set(A),fun(set(A),fun(A,$o)),Uu),Uua),Uub)
        <=> ? [A10: A,B6: A] :
              ( ( Uub = aa(A,A,aa(A,fun(A,A),inf_inf(A),A10),B6) )
              & aa(set(A),$o,member(A,A10),Uu)
              & aa(set(A),$o,member(A,B6),Uua) ) ) ) ).

% ATP.lambda_1014
tff(fact_8794_ATP_Olambda__1015,axiom,
    ! [A: $tType,Uu: set(multiset(A)),Uua: set(multiset(A)),Uub: multiset(A)] :
      ( aa(multiset(A),$o,aa(set(multiset(A)),fun(multiset(A),$o),aTP_Lamp_arz(set(multiset(A)),fun(set(multiset(A)),fun(multiset(A),$o)),Uu),Uua),Uub)
    <=> ? [A10: multiset(A),B6: multiset(A)] :
          ( ( Uub = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),union_mset(A),A10),B6) )
          & aa(set(multiset(A)),$o,member(multiset(A),A10),Uu)
          & aa(set(multiset(A)),$o,member(multiset(A),B6),Uua) ) ) ).

% ATP.lambda_1015
tff(fact_8795_ATP_Olambda__1016,axiom,
    ! [A: $tType,Uu: set(multiset(A)),Uua: set(multiset(A)),Uub: multiset(A)] :
      ( aa(multiset(A),$o,aa(set(multiset(A)),fun(multiset(A),$o),aTP_Lamp_arx(set(multiset(A)),fun(set(multiset(A)),fun(multiset(A),$o)),Uu),Uua),Uub)
    <=> ? [A10: multiset(A),B6: multiset(A)] :
          ( ( Uub = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),A10),B6) )
          & aa(set(multiset(A)),$o,member(multiset(A),A10),Uu)
          & aa(set(multiset(A)),$o,member(multiset(A),B6),Uua) ) ) ).

% ATP.lambda_1016
tff(fact_8796_ATP_Olambda__1017,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(list(A)),Uub: list(A)] :
      ( aa(list(A),$o,aa(set(list(A)),fun(list(A),$o),aTP_Lamp_afa(set(A),fun(set(list(A)),fun(list(A),$o)),Uu),Uua),Uub)
    <=> ? [X4: A,Xs4: list(A)] :
          ( ( Uub = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs4) )
          & aa(set(A),$o,member(A,X4),Uu)
          & aa(set(list(A)),$o,member(list(A),Xs4),Uua) ) ) ).

% ATP.lambda_1017
tff(fact_8797_ATP_Olambda__1018,axiom,
    ! [A: $tType,C: $tType,B: $tType,Uu: fun(set(old_node(B,C)),fun(set(old_node(B,C)),A)),Uua: set(old_node(B,C)),Uub: A] :
      ( aa(A,$o,aa(set(old_node(B,C)),fun(A,$o),aTP_Lamp_auj(fun(set(old_node(B,C)),fun(set(old_node(B,C)),A)),fun(set(old_node(B,C)),fun(A,$o)),Uu),Uua),Uub)
    <=> ? [X4: set(old_node(B,C)),Y3: set(old_node(B,C))] :
          ( ( Uua = old_Scons(B,C,X4,Y3) )
          & ( Uub = aa(set(old_node(B,C)),A,aa(set(old_node(B,C)),fun(set(old_node(B,C)),A),Uu,X4),Y3) ) ) ) ).

% ATP.lambda_1018
tff(fact_8798_ATP_Olambda__1019,axiom,
    ! [A: $tType,Uu: filter(A),Uua: filter(A),Uub: fun(A,$o)] :
      ( aa(fun(A,$o),$o,aa(filter(A),fun(fun(A,$o),$o),aTP_Lamp_akz(filter(A),fun(filter(A),fun(fun(A,$o),$o)),Uu),Uua),Uub)
    <=> ? [Q7: fun(A,$o),R7: fun(A,$o)] :
          ( eventually(A,Q7,Uu)
          & eventually(A,R7,Uua)
          & ! [X4: A] :
              ( ( aa(A,$o,Q7,X4)
                & aa(A,$o,R7,X4) )
             => aa(A,$o,Uub,X4) ) ) ) ).

% ATP.lambda_1019
tff(fact_8799_ATP_Olambda__1020,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: multiset(A),Uub: multiset(A)] :
      ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),aTP_Lamp_apm(set(product_prod(A,A)),fun(multiset(A),fun(multiset(A),$o)),Uu),Uua),Uub)
    <=> ? [A10: A,M03: multiset(A),K6: multiset(A)] :
          ( ( Uub = aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A10),M03) )
          & ( Uua = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M03),K6) )
          & ! [B6: A] :
              ( aa(set(A),$o,member(A,B6),aa(multiset(A),set(A),set_mset(A),K6))
             => aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B6),A10)),Uu) ) ) ) ).

% ATP.lambda_1020
tff(fact_8800_ATP_Olambda__1021,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: list(A),Uub: list(A)] :
      ( aa(list(A),$o,aa(list(A),fun(list(A),$o),aTP_Lamp_afo(set(product_prod(A,A)),fun(list(A),fun(list(A),$o)),Uu),Uua),Uub)
    <=> ? [Us3: list(A),Z4: A,Z11: A,Vs3: list(A)] :
          ( ( Uua = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Z4),Vs3)) )
          & aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z4),Z11)),Uu)
          & ( Uub = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Z11),Vs3)) ) ) ) ).

% ATP.lambda_1021
tff(fact_8801_ATP_Olambda__1022,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType,Uu: fun(B,fun(C,fun(D,A))),Uua: D,Uub: B,Uuc: C] : aa(C,A,aa(B,fun(C,A),aa(D,fun(B,fun(C,A)),aTP_Lamp_bp(fun(B,fun(C,fun(D,A))),fun(D,fun(B,fun(C,A))),Uu),Uua),Uub),Uuc) = aa(D,A,aa(C,fun(D,A),aa(B,fun(C,fun(D,A)),Uu,Uub),Uuc),Uua) ).

% ATP.lambda_1022
tff(fact_8802_ATP_Olambda__1023,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_vp(product_prod(C,A),fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B))))),Uu),Uua),Uub),Uuc) = $ite(aa(product_prod(C,A),A,product_snd(C,A),Uu) = Uua,aa(set(product_prod(C,B)),set(product_prod(C,B)),aa(product_prod(C,B),fun(set(product_prod(C,B)),set(product_prod(C,B))),insert2(product_prod(C,B)),aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),aa(product_prod(C,A),C,product_fst(C,A),Uu)),Uub)),Uuc),Uuc) ).

% ATP.lambda_1023
tff(fact_8803_ATP_Olambda__1024,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: set(A),Uub: A,Uuc: B] :
      aa(B,A,aa(A,fun(B,A),aa(set(A),fun(A,fun(B,A)),aTP_Lamp_adb(fun(A,B),fun(set(A),fun(A,fun(B,A))),Uu),Uua),Uub),Uuc) = $ite(aa(set(B),$o,member(B,Uuc),aa(set(A),set(B),image2(A,B,Uu),Uua)),aa(B,A,the_inv_into(A,B,Uua,Uu),Uuc),Uub) ).

% ATP.lambda_1024
tff(fact_8804_ATP_Olambda__1025,axiom,
    ! [B: $tType,A: $tType,Uu: set(A),Uua: fun(A,B),Uub: A,Uuc: B] :
      aa(B,A,aa(A,fun(B,A),aa(fun(A,B),fun(A,fun(B,A)),aTP_Lamp_acx(set(A),fun(fun(A,B),fun(A,fun(B,A))),Uu),Uua),Uub),Uuc) = $ite(aa(set(B),$o,member(B,Uuc),aa(set(A),set(B),image2(A,B,Uua),Uu)),aa(B,A,the_inv_into(A,B,Uu,Uua),Uuc),Uub) ).

% ATP.lambda_1025
tff(fact_8805_ATP_Olambda__1026,axiom,
    ! [A: $tType,B: $tType,Uu: set(A),Uua: A,Uub: B,Uuc: set(B)] :
      aa(set(B),set(B),aa(B,fun(set(B),set(B)),aa(A,fun(B,fun(set(B),set(B))),aTP_Lamp_up(set(A),fun(A,fun(B,fun(set(B),set(B)))),Uu),Uua),Uub),Uuc) = $ite(aa(set(A),$o,member(A,Uua),Uu),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),Uub),Uuc),Uuc) ).

% ATP.lambda_1026
tff(fact_8806_ATP_Olambda__1027,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_eu(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = $ite(
            aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uuc),Uu),
            aa(nat,A,Uua,Uuc),
            $ite(Uuc = Uu,zero_zero(A),aa(nat,A,Uub,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uuc),aa(nat,nat,suc,zero_zero(nat))))) ) ) ).

% ATP.lambda_1027
tff(fact_8807_ATP_Olambda__1028,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_gk(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = $ite(
            aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uuc),Uu),
            aa(nat,A,Uua,Uuc),
            $ite(Uuc = Uu,one_one(A),aa(nat,A,Uub,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uuc),aa(nat,nat,suc,zero_zero(nat))))) ) ) ).

% ATP.lambda_1028
tff(fact_8808_ATP_Olambda__1029,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: nat,Uua: fun(nat,A),Uub: fun(nat,A),Uuc: nat] :
          aa(nat,A,aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_gl(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uuc),Uu),aa(nat,A,Uua,Uuc),aa(nat,A,Uub,Uuc)) ) ).

% ATP.lambda_1029
tff(fact_8809_ATP_Olambda__1030,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_ev(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uuc),Uu),aa(nat,A,Uua,Uuc),aa(nat,A,Uub,Uuc)) ) ).

% ATP.lambda_1030
tff(fact_8810_ATP_Olambda__1031,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: set(A),Uub: fun(A,B),Uuc: A] :
      aa(A,B,aa(fun(A,B),fun(A,B),aa(set(A),fun(fun(A,B),fun(A,B)),aTP_Lamp_aco(fun(A,B),fun(set(A),fun(fun(A,B),fun(A,B))),Uu),Uua),Uub),Uuc) = $ite(aa(set(A),$o,member(A,Uuc),Uua),aa(A,B,Uu,Uuc),aa(A,B,Uub,Uuc)) ).

% ATP.lambda_1031
tff(fact_8811_ATP_Olambda__1032,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: A,Uua: fun(A,B),Uub: fun(A,B),Uuc: A] :
          aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_ij(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa(A,B,Uua,Uuc),aa(A,B,Uub,Uuc)) ) ).

% ATP.lambda_1032
tff(fact_8812_ATP_Olambda__1033,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: A,Uua: fun(A,B),Uub: fun(A,B),Uuc: A] :
          aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_if(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa(A,B,Uua,Uuc),aa(A,B,Uub,Uuc)) ) ).

% ATP.lambda_1033
tff(fact_8813_ATP_Olambda__1034,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: A,Uua: fun(A,B),Uub: B,Uuc: A] :
          aa(A,B,aa(B,fun(A,B),aa(fun(A,B),fun(B,fun(A,B)),aTP_Lamp_jk(A,fun(fun(A,B),fun(B,fun(A,B))),Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa(A,B,Uua,Uuc),Uub) ) ).

% ATP.lambda_1034
tff(fact_8814_ATP_Olambda__1035,axiom,
    ! [B: $tType,A: $tType,Uu: A,Uua: B,Uub: fun(A,option(B)),Uuc: A] :
      aa(A,option(B),aa(fun(A,option(B)),fun(A,option(B)),aa(B,fun(fun(A,option(B)),fun(A,option(B))),aTP_Lamp_adc(A,fun(B,fun(fun(A,option(B)),fun(A,option(B)))),Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa(B,option(B),some(B),Uua),aa(A,option(B),Uub,Uuc)) ).

% ATP.lambda_1035
tff(fact_8815_ATP_Olambda__1036,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,option(A)),Uua: B,Uub: A,Uuc: B] :
      aa(B,option(A),aa(A,fun(B,option(A)),aa(B,fun(A,fun(B,option(A))),aTP_Lamp_xt(fun(B,option(A)),fun(B,fun(A,fun(B,option(A)))),Uu),Uua),Uub),Uuc) = $ite(Uuc = Uua,aa(A,option(A),some(A),Uub),aa(B,option(A),Uu,Uuc)) ).

% ATP.lambda_1036
tff(fact_8816_ATP_Olambda__1037,axiom,
    ! [A: $tType,B: $tType,Uu: set(A),Uua: B,Uub: B,Uuc: A] :
      aa(A,B,aa(B,fun(A,B),aa(B,fun(B,fun(A,B)),aTP_Lamp_at(set(A),fun(B,fun(B,fun(A,B))),Uu),Uua),Uub),Uuc) = $ite(aa(set(A),$o,member(A,Uuc),Uu),Uua,Uub) ).

% ATP.lambda_1037
tff(fact_8817_ATP_Olambda__1038,axiom,
    ! [A: $tType,Uu: fun(code_natural,A),Uua: code_natural,Uub: code_natural,Uuc: product_unit] :
      aa(product_unit,seq(A),aa(code_natural,fun(product_unit,seq(A)),aa(code_natural,fun(code_natural,fun(product_unit,seq(A))),aTP_Lamp_axp(fun(code_natural,A),fun(code_natural,fun(code_natural,fun(product_unit,seq(A)))),Uu),Uua),Uub),Uuc) = $ite(aa(code_natural,$o,aa(code_natural,fun(code_natural,$o),ord_less(code_natural),Uub),Uua),empty(A),insert(A,aa(code_natural,A,Uu,Uua),iterate_upto(A,Uu,aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),plus_plus(code_natural),Uua),one_one(code_natural)),Uub))) ).

% ATP.lambda_1038
tff(fact_8818_ATP_Olambda__1039,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: A,Uub: list(A),Uuc: list(A)] :
      aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),aa(A,fun(list(A),fun(list(A),product_prod(list(A),list(A)))),aTP_Lamp_abu(fun(A,$o),fun(A,fun(list(A),fun(list(A),product_prod(list(A),list(A))))),Uu),Uua),Uub),Uuc) = $ite(aa(A,$o,Uu,Uua),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uua),Uub)),Uuc),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Uub),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uua),Uuc))) ).

% ATP.lambda_1039
tff(fact_8819_ATP_Olambda__1040,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,$o),Uua: fun(B,A),Uub: fun(B,A),Uuc: B] :
      aa(B,A,aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_kc(fun(B,$o),fun(fun(B,A),fun(fun(B,A),fun(B,A))),Uu),Uua),Uub),Uuc) = $ite(aa(B,$o,Uu,Uuc),aa(B,A,Uua,Uuc),aa(B,A,Uub,Uuc)) ).

% ATP.lambda_1040
tff(fact_8820_ATP_Olambda__1041,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,$o),Uua: fun(A,option(B)),Uub: fun(A,option(B)),Uuc: A] :
      aa(A,option(B),aa(fun(A,option(B)),fun(A,option(B)),aa(fun(A,option(B)),fun(fun(A,option(B)),fun(A,option(B))),aTP_Lamp_adg(fun(A,$o),fun(fun(A,option(B)),fun(fun(A,option(B)),fun(A,option(B)))),Uu),Uua),Uub),Uuc) = $ite(aa(A,$o,Uu,Uuc),aa(A,option(B),Uua,Uuc),aa(A,option(B),Uub,Uuc)) ).

% ATP.lambda_1041
tff(fact_8821_ATP_Olambda__1042,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: fun(A,$o),Uua: fun(A,B),Uub: fun(A,B),Uuc: A] :
          aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_ie(fun(A,$o),fun(fun(A,B),fun(fun(A,B),fun(A,B))),Uu),Uua),Uub),Uuc) = $ite(aa(A,$o,Uu,Uuc),aa(A,B,Uua,Uuc),aa(A,B,Uub,Uuc)) ) ).

% ATP.lambda_1042
tff(fact_8822_ATP_Olambda__1043,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: fun(A,$o),Uua: fun(A,B),Uub: fun(A,B),Uuc: A] :
          aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_id(fun(A,$o),fun(fun(A,B),fun(fun(A,B),fun(A,B))),Uu),Uua),Uub),Uuc) = $ite(aa(A,$o,Uu,Uuc),aa(A,B,Uua,Uuc),aa(A,B,Uub,Uuc)) ) ).

% ATP.lambda_1043
tff(fact_8823_ATP_Olambda__1044,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: fun(A,$o),Uub: fun(A,B),Uuc: A] :
      aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,$o),fun(fun(A,B),fun(A,B)),aTP_Lamp_ajk(fun(A,B),fun(fun(A,$o),fun(fun(A,B),fun(A,B))),Uu),Uua),Uub),Uuc) = $ite(aa(A,$o,Uua,Uuc),aa(A,B,Uu,Uuc),aa(A,B,Uub,Uuc)) ).

% ATP.lambda_1044
tff(fact_8824_ATP_Olambda__1045,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_ur(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_uq(A,fun(B,fun(B,fun(C,fun(set(product_prod(A,C)),set(product_prod(A,C)))))),Uua),Uub)),Uuc,Uu) ).

% ATP.lambda_1045
tff(fact_8825_ATP_Olambda__1046,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_uo(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_un(C,fun(A,fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B)))))),Uua),Uub)),Uuc,Uu) ).

% ATP.lambda_1046
tff(fact_8826_ATP_Olambda__1047,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_ye(fun(B,A),fun(A,fun(list(B),fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(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_1047
tff(fact_8827_ATP_Olambda__1048,axiom,
    ! [B: $tType,A: $tType,Uu: set(product_prod(A,A)),Uua: fun(fun(A,B),fun(A,B)),Uub: fun(A,B),Uuc: A] : aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(fun(A,B),fun(A,B)),fun(fun(A,B),fun(A,B)),aTP_Lamp_avs(set(product_prod(A,A)),fun(fun(fun(A,B),fun(A,B)),fun(fun(A,B),fun(A,B))),Uu),Uua),Uub),Uuc) = aa(A,B,aa(fun(A,B),fun(A,B),Uua,cut(A,B,Uub,Uu,Uuc)),Uuc) ).

% ATP.lambda_1048
tff(fact_8828_ATP_Olambda__1049,axiom,
    ! [B: $tType,C: $tType,A: $tType,Uu: fun(C,fun(list(C),fun(B,B))),Uua: fun(A,C),Uub: A,Uuc: list(A)] : aa(list(A),fun(B,B),aa(A,fun(list(A),fun(B,B)),aa(fun(A,C),fun(A,fun(list(A),fun(B,B))),aTP_Lamp_ajo(fun(C,fun(list(C),fun(B,B))),fun(fun(A,C),fun(A,fun(list(A),fun(B,B)))),Uu),Uua),Uub),Uuc) = aa(list(C),fun(B,B),aa(C,fun(list(C),fun(B,B)),Uu,aa(A,C,Uua,Uub)),aa(list(A),list(C),map(A,C,Uua),Uuc)) ).

% ATP.lambda_1049
tff(fact_8829_ATP_Olambda__1050,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,fun(A,$o)),Uua: fun(B,A),Uub: B,Uuc: B] :
      ( aa(B,$o,aa(B,fun(B,$o),aa(fun(B,A),fun(B,fun(B,$o)),aTP_Lamp_zz(fun(A,fun(A,$o)),fun(fun(B,A),fun(B,fun(B,$o))),Uu),Uua),Uub),Uuc)
    <=> aa(A,$o,aa(A,fun(A,$o),Uu,aa(B,A,Uua,Uub)),aa(B,A,Uua,Uuc)) ) ).

% ATP.lambda_1050
tff(fact_8830_ATP_Olambda__1051,axiom,
    ! [B: $tType,A: $tType,C: $tType,D: $tType,Uu: fun(B,fun(C,A)),Uua: fun(D,B),Uub: fun(D,C),Uuc: D] : aa(D,A,aa(fun(D,C),fun(D,A),aa(fun(D,B),fun(fun(D,C),fun(D,A)),aTP_Lamp_adr(fun(B,fun(C,A)),fun(fun(D,B),fun(fun(D,C),fun(D,A))),Uu),Uua),Uub),Uuc) = aa(C,A,aa(B,fun(C,A),Uu,aa(D,B,Uua,Uuc)),aa(D,C,Uub,Uuc)) ).

% ATP.lambda_1051
tff(fact_8831_ATP_Olambda__1052,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,fun(A,A)),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_awp(fun(A,fun(A,A)),fun(fun(B,A),fun(fun(B,A),fun(B,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),Uu,aa(B,A,Uua,Uuc)),aa(B,A,Uub,Uuc)) ).

% ATP.lambda_1052
tff(fact_8832_ATP_Olambda__1053,axiom,
    ! [C: $tType,D: $tType,F3: $tType,Uu: fun(C,fun(D,$o)),Uua: fun(F3,D),Uub: C,Uuc: F3] :
      ( aa(F3,$o,aa(C,fun(F3,$o),aa(fun(F3,D),fun(C,fun(F3,$o)),aTP_Lamp_aox(fun(C,fun(D,$o)),fun(fun(F3,D),fun(C,fun(F3,$o))),Uu),Uua),Uub),Uuc)
    <=> aa(D,$o,aa(C,fun(D,$o),Uu,Uub),aa(F3,D,Uua,Uuc)) ) ).

% ATP.lambda_1053
tff(fact_8833_ATP_Olambda__1054,axiom,
    ! [B: $tType,C: $tType,D: $tType,Uu: fun(B,fun(C,$o)),Uua: fun(D,C),Uub: B,Uuc: D] :
      ( aa(D,$o,aa(B,fun(D,$o),aa(fun(D,C),fun(B,fun(D,$o)),aTP_Lamp_we(fun(B,fun(C,$o)),fun(fun(D,C),fun(B,fun(D,$o))),Uu),Uua),Uub),Uuc)
    <=> aa(C,$o,aa(B,fun(C,$o),Uu,Uub),aa(D,C,Uua,Uuc)) ) ).

% ATP.lambda_1054
tff(fact_8834_ATP_Olambda__1055,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_agq(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_1055
tff(fact_8835_ATP_Olambda__1056,axiom,
    ! [A: $tType,B: $tType,E: $tType,Uu: fun(A,fun(B,$o)),Uua: fun(E,B),Uub: A,Uuc: E] :
      ( aa(E,$o,aa(A,fun(E,$o),aa(fun(E,B),fun(A,fun(E,$o)),aTP_Lamp_aow(fun(A,fun(B,$o)),fun(fun(E,B),fun(A,fun(E,$o))),Uu),Uua),Uub),Uuc)
    <=> aa(B,$o,aa(A,fun(B,$o),Uu,Uub),aa(E,B,Uua,Uuc)) ) ).

% ATP.lambda_1056
tff(fact_8836_ATP_Olambda__1057,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: fun(A,fun(B,$o)),Uua: fun(C,B),Uub: A,Uuc: C] :
      ( aa(C,$o,aa(A,fun(C,$o),aa(fun(C,B),fun(A,fun(C,$o)),aTP_Lamp_aiu(fun(A,fun(B,$o)),fun(fun(C,B),fun(A,fun(C,$o))),Uu),Uua),Uub),Uuc)
    <=> aa(B,$o,aa(A,fun(B,$o),Uu,Uub),aa(C,B,Uua,Uuc)) ) ).

% ATP.lambda_1057
tff(fact_8837_ATP_Olambda__1058,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: fun(A,fun(B,A)),Uua: fun(C,B),Uub: A,Uuc: C] : aa(C,A,aa(A,fun(C,A),aa(fun(C,B),fun(A,fun(C,A)),aTP_Lamp_aaa(fun(A,fun(B,A)),fun(fun(C,B),fun(A,fun(C,A))),Uu),Uua),Uub),Uuc) = aa(B,A,aa(A,fun(B,A),Uu,Uub),aa(C,B,Uua,Uuc)) ).

% ATP.lambda_1058
tff(fact_8838_ATP_Olambda__1059,axiom,
    ! [Uu: $o,Uua: $o,Uub: code_integer,Uuc: $o] : aa($o,char,aa(code_integer,fun($o,char),aa($o,fun(code_integer,fun($o,char)),aTP_Lamp_qo($o,fun($o,fun(code_integer,fun($o,char))),(Uu)),(Uua)),Uub),(Uuc)) = aa(product_prod(code_integer,$o),char,aa(fun(code_integer,fun($o,char)),fun(product_prod(code_integer,$o),char),product_case_prod(code_integer,$o,char),aa($o,fun(code_integer,fun($o,char)),aa($o,fun($o,fun(code_integer,fun($o,char))),aTP_Lamp_qn($o,fun($o,fun($o,fun(code_integer,fun($o,char)))),(Uu)),(Uua)),(Uuc))),code_bit_cut_integer(Uub)) ).

% ATP.lambda_1059
tff(fact_8839_ATP_Olambda__1060,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( semiring_0(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_da(fun(B,A),fun(fun(C,A),fun(set(C),fun(B,A))),Uu),Uua),Uub),Uuc) = aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7311177749621191930dd_sum(C,A),aa(B,fun(C,A),aa(fun(C,A),fun(B,fun(C,A)),aTP_Lamp_cz(fun(B,A),fun(fun(C,A),fun(B,fun(C,A))),Uu),Uua),Uuc)),Uub) ) ).

% ATP.lambda_1060
tff(fact_8840_ATP_Olambda__1061,axiom,
    ! [A: $tType,B: $tType,Uu: set(B),Uua: fun(B,A),Uub: filter(B),Uuc: fun(A,$o)] :
      ( aa(fun(A,$o),$o,aa(filter(B),fun(fun(A,$o),$o),aa(fun(B,A),fun(filter(B),fun(fun(A,$o),$o)),aTP_Lamp_akv(set(B),fun(fun(B,A),fun(filter(B),fun(fun(A,$o),$o))),Uu),Uua),Uub),Uuc)
    <=> eventually(B,aa(fun(A,$o),fun(B,$o),aa(fun(B,A),fun(fun(A,$o),fun(B,$o)),aTP_Lamp_aku(set(B),fun(fun(B,A),fun(fun(A,$o),fun(B,$o))),Uu),Uua),Uuc),Uub) ) ).

% ATP.lambda_1061
tff(fact_8841_ATP_Olambda__1062,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Uu: fun(A,B),Uua: list(A),Uub: A,Uuc: A] :
          aa(A,fun(list(A),list(A)),aa(A,fun(A,fun(list(A),list(A))),aa(list(A),fun(A,fun(A,fun(list(A),list(A)))),aTP_Lamp_yn(fun(A,B),fun(list(A),fun(A,fun(A,fun(list(A),list(A))))),Uu),Uua),Uub),Uuc) = case_list(list(A),A,
            $ite(aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,Uu,Uub)),aa(A,B,Uu,Uuc)),Uua,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uuc),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uub),nil(A)))),
            aa(list(A),fun(A,fun(list(A),list(A))),aTP_Lamp_ym(fun(A,B),fun(list(A),fun(A,fun(list(A),list(A)))),Uu),Uua)) ) ).

% ATP.lambda_1062
tff(fact_8842_ATP_Olambda__1063,axiom,
    ! [A: $tType,Uu: set(A),Uua: fun(A,fun(A,$o)),Uub: set(A),Uuc: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),aa(fun(A,fun(A,$o)),fun(set(A),fun(set(A),$o)),aTP_Lamp_aos(set(A),fun(fun(A,fun(A,$o)),fun(set(A),fun(set(A),$o))),Uu),Uua),Uub),Uuc)
    <=> ( aa(set(A),$o,pred_chain(A,Uu,Uua),Uuc)
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),Uub),Uuc) ) ) ).

% ATP.lambda_1063
tff(fact_8843_ATP_Olambda__1064,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,fun(B,$o)),Uua: fun(A,fun(B,$o)),Uub: A,Uuc: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aTP_Lamp_axf(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),Uu),Uua),Uub),Uuc)
    <=> ( aa(B,$o,aa(A,fun(B,$o),Uu,Uub),Uuc)
        & aa(B,$o,aa(A,fun(B,$o),Uua,Uub),Uuc) ) ) ).

% ATP.lambda_1064
tff(fact_8844_ATP_Olambda__1065,axiom,
    ! [C: $tType,A: $tType,B: $tType,E: $tType,D: $tType,Uu: fun(B,fun(C,fun(D,fun(E,set(A))))),Uua: product_prod(D,E),Uub: B,Uuc: C] : aa(C,set(A),aa(B,fun(C,set(A)),aa(product_prod(D,E),fun(B,fun(C,set(A))),aTP_Lamp_oe(fun(B,fun(C,fun(D,fun(E,set(A))))),fun(product_prod(D,E),fun(B,fun(C,set(A)))),Uu),Uua),Uub),Uuc) = aa(product_prod(D,E),set(A),aa(fun(D,fun(E,set(A))),fun(product_prod(D,E),set(A)),product_case_prod(D,E,set(A)),aa(C,fun(D,fun(E,set(A))),aa(B,fun(C,fun(D,fun(E,set(A)))),Uu,Uub),Uuc)),Uua) ).

% ATP.lambda_1065
tff(fact_8845_ATP_Olambda__1066,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [Uu: set(A),Uua: fun(A,fun(B,C)),Uub: fun(A,fun(B,$o)),Uuc: B] : aa(B,C,aa(fun(A,fun(B,$o)),fun(B,C),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(B,C)),aTP_Lamp_hh(set(A),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(B,C))),Uu),Uua),Uub),Uuc) = aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7121269368397514597t_prod(A,C),aa(B,fun(A,C),aTP_Lamp_hg(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uuc)),aa(fun(A,$o),set(A),collect(A),aa(B,fun(A,$o),aa(fun(A,fun(B,$o)),fun(B,fun(A,$o)),aTP_Lamp_hd(set(A),fun(fun(A,fun(B,$o)),fun(B,fun(A,$o))),Uu),Uub),Uuc))) ) ).

% ATP.lambda_1066
tff(fact_8846_ATP_Olambda__1067,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [Uu: set(A),Uua: fun(A,fun(B,C)),Uub: fun(A,fun(B,$o)),Uuc: B] : aa(B,C,aa(fun(A,fun(B,$o)),fun(B,C),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(B,C)),aTP_Lamp_he(set(A),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(B,C))),Uu),Uua),Uub),Uuc) = aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7311177749621191930dd_sum(A,C),aa(B,fun(A,C),aTP_Lamp_hc(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uuc)),aa(fun(A,$o),set(A),collect(A),aa(B,fun(A,$o),aa(fun(A,fun(B,$o)),fun(B,fun(A,$o)),aTP_Lamp_hd(set(A),fun(fun(A,fun(B,$o)),fun(B,fun(A,$o))),Uu),Uub),Uuc))) ) ).

% ATP.lambda_1067
tff(fact_8847_ATP_Olambda__1068,axiom,
    ! [Uu: code_natural,Uua: code_natural,Uub: code_natural,Uuc: code_natural] : aa(code_natural,product_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural)),aa(code_natural,fun(code_natural,product_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural))),aa(code_natural,fun(code_natural,fun(code_natural,product_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural)))),aTP_Lamp_aty(code_natural,fun(code_natural,fun(code_natural,fun(code_natural,product_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural))))),Uu),Uua),Uub),Uuc) = aa(product_prod(code_natural,code_natural),product_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural)),aa(product_prod(code_natural,code_natural),fun(product_prod(code_natural,code_natural),product_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural))),product_Pair(product_prod(code_natural,code_natural),product_prod(code_natural,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),inc_shift(aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,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))))))))))))))))))))))))))))))),Uu)),Uuc)),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),Uub),inc_shift(aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,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))))))))))))))))))))))))))))))),Uua))) ).

% ATP.lambda_1068
tff(fact_8848_ATP_Olambda__1069,axiom,
    ! [A: $tType,Uu: fun(A,fun(A,$o)),Uua: set(A),Uub: A,Uuc: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aa(set(A),fun(A,fun(A,$o)),aTP_Lamp_ajg(fun(A,fun(A,$o)),fun(set(A),fun(A,fun(A,$o))),Uu),Uua),Uub),Uuc)
    <=> ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),Uuc)),product_Sigma(A,A,Uua,aTP_Lamp_rl(set(A),fun(A,set(A)),Uua)))
        & aa(A,$o,aa(A,fun(A,$o),Uu,Uub),Uuc) ) ) ).

% ATP.lambda_1069
tff(fact_8849_ATP_Olambda__1070,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_pa(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_1070
tff(fact_8850_ATP_Olambda__1071,axiom,
    ! [B: $tType,A: $tType,Uu: set(old_node(A,B)),Uua: set(old_node(A,B)),Uub: set(old_node(A,B)),Uuc: set(old_node(A,B))] : aa(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),aa(set(old_node(A,B)),fun(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),aa(set(old_node(A,B)),fun(set(old_node(A,B)),fun(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))))),aTP_Lamp_auy(set(old_node(A,B)),fun(set(old_node(A,B)),fun(set(old_node(A,B)),fun(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B))))))),Uu),Uua),Uub),Uuc) = aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),aa(product_prod(set(old_node(A,B)),set(old_node(A,B))),fun(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),insert2(product_prod(set(old_node(A,B)),set(old_node(A,B)))),aa(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),fun(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B)))),product_Pair(set(old_node(A,B)),set(old_node(A,B))),old_Scons(A,B,Uu,Uub)),old_Scons(A,B,Uua,Uuc))),bot_bot(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))))) ).

% ATP.lambda_1071
tff(fact_8851_ATP_Olambda__1072,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: set(A),Uub: A,Uuc: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aa(set(A),fun(A,fun(A,$o)),aTP_Lamp_bv(set(product_prod(A,A)),fun(set(A),fun(A,fun(A,$o))),Uu),Uua),Uub),Uuc)
    <=> ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),Uuc)),Uu)
        & ~ aa(set(A),$o,member(A,Uub),Uua)
        & ~ aa(set(A),$o,member(A,Uuc),Uua) ) ) ).

% ATP.lambda_1072
tff(fact_8852_ATP_Olambda__1073,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: set(product_prod(A,A)),Uub: A,Uuc: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aa(set(product_prod(A,A)),fun(A,fun(A,$o)),aTP_Lamp_oh(fun(A,nat),fun(set(product_prod(A,A)),fun(A,fun(A,$o))),Uu),Uua),Uub),Uuc)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,Uu,Uub)),aa(A,nat,Uu,Uuc))
        | ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,Uu,Uub)),aa(A,nat,Uu,Uuc))
          & aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),Uuc)),Uua) ) ) ) ).

% ATP.lambda_1073
tff(fact_8853_ATP_Olambda__1074,axiom,
    ! [A: $tType,B: $tType,Uu: set(product_prod(B,B)),Uua: fun(A,B),Uub: A,Uuc: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aa(fun(A,B),fun(A,fun(A,$o)),aTP_Lamp_alv(set(product_prod(B,B)),fun(fun(A,B),fun(A,fun(A,$o))),Uu),Uua),Uub),Uuc)
    <=> aa(set(product_prod(B,B)),$o,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_1074
tff(fact_8854_ATP_Olambda__1075,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,option(B)),Uua: fun(product_prod(A,B),$o),Uub: A,Uuc: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(fun(product_prod(A,B),$o),fun(A,fun(B,$o)),aTP_Lamp_sx(fun(A,option(B)),fun(fun(product_prod(A,B),$o),fun(A,fun(B,$o))),Uu),Uua),Uub),Uuc)
    <=> ( ( aa(A,option(B),Uu,Uub) = aa(B,option(B),some(B),Uuc) )
        & aa(product_prod(A,B),$o,Uua,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uub),Uuc)) ) ) ).

% ATP.lambda_1075
tff(fact_8855_ATP_Olambda__1076,axiom,
    ! [A: $tType,B: $tType] :
      ( order(A)
     => ! [Uu: fun(A,set(B)),Uua: set(B),Uub: set(B),Uuc: A] : aa(A,set(B),aa(set(B),fun(A,set(B)),aa(set(B),fun(set(B),fun(A,set(B))),aTP_Lamp_ua(fun(A,set(B)),fun(set(B),fun(set(B),fun(A,set(B)))),Uu),Uua),Uub),Uuc) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),aa(A,set(B),Uu,Uuc)),Uua)),Uub) ) ).

% ATP.lambda_1076
tff(fact_8856_ATP_Olambda__1077,axiom,
    ! [A: $tType,Uu: set(A),Uua: fun(set(A),set(A)),Uub: set(A),Uuc: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),aa(fun(set(A),set(A)),fun(set(A),fun(set(A),set(A))),aTP_Lamp_als(set(A),fun(fun(set(A),set(A)),fun(set(A),fun(set(A),set(A)))),Uu),Uua),Uub),Uuc) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),Uua,Uuc)),Uub)),Uu) ).

% ATP.lambda_1077
tff(fact_8857_ATP_Olambda__1078,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_gi(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_1078
tff(fact_8858_ATP_Olambda__1079,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_ge(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_1079
tff(fact_8859_ATP_Olambda__1080,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_gh(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_1080
tff(fact_8860_ATP_Olambda__1081,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: nat,Uub: list(A),Uuc: list(A)] :
      ( aa(list(A),$o,aa(list(A),fun(list(A),$o),aa(nat,fun(list(A),fun(list(A),$o)),aTP_Lamp_afw(set(product_prod(A,A)),fun(nat,fun(list(A),fun(list(A),$o))),Uu),Uua),Uub),Uuc)
    <=> ( ( aa(list(A),nat,size_size(list(A)),Uub) = Uua )
        & ( aa(list(A),nat,size_size(list(A)),Uuc) = Uua )
        & ? [Xys2: list(A),X4: A,Y3: A,Xs6: list(A),Ys5: list(A)] :
            ( ( Uub = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xys2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs6)) )
            & ( Uuc = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xys2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys5)) )
            & aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Y3)),Uu) ) ) ) ).

% ATP.lambda_1081
tff(fact_8861_ATP_Olambda__1082,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_ik(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_1082
tff(fact_8862_ATP_Olambda__1083,axiom,
    ! [A: $tType,Uu: $o,Uua: A,Uub: A,Uuc: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aa(A,fun(A,fun(A,$o)),aTP_Lamp_qu($o,fun(A,fun(A,fun(A,$o))),(Uu)),Uua),Uub),Uuc)
    <=> ( ( (Uu)
         => ( Uuc = Uua ) )
        & ( ~ (Uu)
         => ( Uuc = Uub ) ) ) ) ).

% ATP.lambda_1083
tff(fact_8863_ATP_Olambda__1084,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,A),Uua: set(B),Uub: fun(A,$o),Uuc: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aa(set(B),fun(fun(A,$o),fun(A,$o)),aTP_Lamp_lj(fun(B,A),fun(set(B),fun(fun(A,$o),fun(A,$o))),Uu),Uua),Uub),Uuc)
    <=> ( aa(set(A),$o,member(A,Uuc),aa(set(B),set(A),image2(B,A,Uu),Uua))
        & aa(A,$o,Uub,Uuc) ) ) ).

% ATP.lambda_1084
tff(fact_8864_ATP_Olambda__1085,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: multiset(B),Uub: A,Uuc: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(multiset(B),fun(A,fun(B,$o)),aTP_Lamp_aps(fun(B,A),fun(multiset(B),fun(A,fun(B,$o))),Uu),Uua),Uub),Uuc)
    <=> ( aa(set(B),$o,member(B,Uuc),aa(multiset(B),set(B),set_mset(B),Uua))
        & ( Uub = aa(B,A,Uu,Uuc) ) ) ) ).

% ATP.lambda_1085
tff(fact_8865_ATP_Olambda__1086,axiom,
    ! [A: $tType,B: $tType,Uu: set(B),Uua: fun(A,fun(B,$o)),Uub: A,Uuc: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aTP_Lamp_ha(set(B),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),Uu),Uua),Uub),Uuc)
    <=> ( aa(set(B),$o,member(B,Uuc),Uu)
        & aa(B,$o,aa(A,fun(B,$o),Uua,Uub),Uuc) ) ) ).

% ATP.lambda_1086
tff(fact_8866_ATP_Olambda__1087,axiom,
    ! [A: $tType,B: $tType,Uu: set(A),Uua: fun(A,fun(B,$o)),Uub: B,Uuc: A] :
      ( aa(A,$o,aa(B,fun(A,$o),aa(fun(A,fun(B,$o)),fun(B,fun(A,$o)),aTP_Lamp_hd(set(A),fun(fun(A,fun(B,$o)),fun(B,fun(A,$o))),Uu),Uua),Uub),Uuc)
    <=> ( aa(set(A),$o,member(A,Uuc),Uu)
        & aa(B,$o,aa(A,fun(B,$o),Uua,Uuc),Uub) ) ) ).

% ATP.lambda_1087
tff(fact_8867_ATP_Olambda__1088,axiom,
    ! [A: $tType,B: $tType,Uu: set(A),Uua: set(product_prod(B,B)),Uub: B,Uuc: A] :
      ( aa(A,$o,aa(B,fun(A,$o),aa(set(product_prod(B,B)),fun(B,fun(A,$o)),aTP_Lamp_aoq(set(A),fun(set(product_prod(B,B)),fun(B,fun(A,$o))),Uu),Uua),Uub),Uuc)
    <=> ( aa(set(A),$o,member(A,Uuc),Uu)
        & ( aa(A,B,bNF_Greatest_toCard(A,B,Uu,Uua),Uuc) = Uub ) ) ) ).

% ATP.lambda_1088
tff(fact_8868_ATP_Olambda__1089,axiom,
    ! [B: $tType,A: $tType,Uu: set(A),Uua: fun(A,B),Uub: A,Uuc: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aa(fun(A,B),fun(A,fun(A,$o)),aTP_Lamp_lx(set(A),fun(fun(A,B),fun(A,fun(A,$o))),Uu),Uua),Uub),Uuc)
    <=> ( aa(set(A),$o,member(A,Uuc),Uu)
        & ( aa(A,B,Uua,Uuc) = aa(A,B,Uua,Uub) ) ) ) ).

% ATP.lambda_1089
tff(fact_8869_ATP_Olambda__1090,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: set(A),Uub: A,Uuc: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aa(set(A),fun(A,fun(A,$o)),aTP_Lamp_aod(fun(A,B),fun(set(A),fun(A,fun(A,$o))),Uu),Uua),Uub),Uuc)
    <=> ( aa(set(A),$o,member(A,Uuc),Uua)
        & ( aa(A,B,Uu,Uub) = aa(A,B,Uu,Uuc) ) ) ) ).

% ATP.lambda_1090
tff(fact_8870_ATP_Olambda__1091,axiom,
    ! [B: $tType,A: $tType,Uu: set(B),Uua: fun(B,A),Uub: A,Uuc: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(fun(B,A),fun(A,fun(B,$o)),aTP_Lamp_acy(set(B),fun(fun(B,A),fun(A,fun(B,$o))),Uu),Uua),Uub),Uuc)
    <=> ( aa(set(B),$o,member(B,Uuc),Uu)
        & ( aa(B,A,Uua,Uuc) = Uub ) ) ) ).

% ATP.lambda_1091
tff(fact_8871_ATP_Olambda__1092,axiom,
    ! [A: $tType,C: $tType,Uu: set(A),Uua: fun(A,C),Uub: C,Uuc: A] :
      ( aa(A,$o,aa(C,fun(A,$o),aa(fun(A,C),fun(C,fun(A,$o)),aTP_Lamp_mk(set(A),fun(fun(A,C),fun(C,fun(A,$o))),Uu),Uua),Uub),Uuc)
    <=> ( aa(set(A),$o,member(A,Uuc),Uu)
        & ( aa(A,C,Uua,Uuc) = Uub ) ) ) ).

% ATP.lambda_1092
tff(fact_8872_ATP_Olambda__1093,axiom,
    ! [A: $tType,B: $tType,Uu: set(A),Uua: fun(A,B),Uub: B,Uuc: A] :
      ( aa(A,$o,aa(B,fun(A,$o),aa(fun(A,B),fun(B,fun(A,$o)),aTP_Lamp_mt(set(A),fun(fun(A,B),fun(B,fun(A,$o))),Uu),Uua),Uub),Uuc)
    <=> ( aa(set(A),$o,member(A,Uuc),Uu)
        & ( aa(A,B,Uua,Uuc) = Uub ) ) ) ).

% ATP.lambda_1093
tff(fact_8873_ATP_Olambda__1094,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,B),Uua: set(A),Uub: B,Uuc: A] :
      ( aa(A,$o,aa(B,fun(A,$o),aa(set(A),fun(B,fun(A,$o)),aTP_Lamp_acw(fun(A,B),fun(set(A),fun(B,fun(A,$o))),Uu),Uua),Uub),Uuc)
    <=> ( aa(set(A),$o,member(A,Uuc),Uua)
        & ( aa(A,B,Uu,Uuc) = Uub ) ) ) ).

% ATP.lambda_1094
tff(fact_8874_ATP_Olambda__1095,axiom,
    ! [B: $tType,A: $tType,Uu: set(A),Uua: fun(B,A),Uub: set(B),Uuc: B] :
      ( aa(B,$o,aa(set(B),fun(B,$o),aa(fun(B,A),fun(set(B),fun(B,$o)),aTP_Lamp_acf(set(A),fun(fun(B,A),fun(set(B),fun(B,$o))),Uu),Uua),Uub),Uuc)
    <=> ( aa(set(B),$o,member(B,Uuc),Uub)
        & aa(set(A),$o,member(A,aa(B,A,Uua,Uuc)),Uu) ) ) ).

% ATP.lambda_1095
tff(fact_8875_ATP_Olambda__1096,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_il(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_1096
tff(fact_8876_ATP_Olambda__1097,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_gc(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_1097
tff(fact_8877_ATP_Olambda__1098,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Uu: fun(A,B),Uua: list(A),Uub: list(A),Uuc: list(A)] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),aa(list(A),fun(list(A),fun(list(A),list(A))),aTP_Lamp_yk(fun(A,B),fun(list(A),fun(list(A),fun(list(A),list(A)))),Uu),Uua),Uub),Uuc) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),linorder_sort_key(A,B,Uu),Uua)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Uub),aa(list(A),list(A),linorder_sort_key(A,B,Uu),Uuc))) ) ).

% ATP.lambda_1098
tff(fact_8878_ATP_Olambda__1099,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aa(int,fun(int,fun(int,$o)),aTP_Lamp_pi(int,fun(int,fun(int,fun(int,$o))),Uu),Uua),Uub),Uuc)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),times_times(int),Uu),Uuc)),aa(int,int,aa(int,fun(int,int),times_times(int),Uua),Uub)) ) ).

% ATP.lambda_1099
tff(fact_8879_ATP_Olambda__1100,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aa(nat,fun(nat,fun(nat,$o)),aTP_Lamp_pq(nat,fun(nat,fun(nat,fun(nat,$o))),Uu),Uua),Uub),Uuc)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu),Uuc)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)) ) ).

% ATP.lambda_1100
tff(fact_8880_ATP_Olambda__1101,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aa(int,fun(int,fun(int,$o)),aTP_Lamp_pk(int,fun(int,fun(int,fun(int,$o))),Uu),Uua),Uub),Uuc)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),times_times(int),Uu),Uuc)),aa(int,int,aa(int,fun(int,int),times_times(int),Uua),Uub)) ) ).

% ATP.lambda_1101
tff(fact_8881_ATP_Olambda__1102,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aa(nat,fun(nat,fun(nat,$o)),aTP_Lamp_po(nat,fun(nat,fun(nat,fun(nat,$o))),Uu),Uua),Uub),Uuc)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu),Uuc)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)) ) ).

% ATP.lambda_1102
tff(fact_8882_ATP_Olambda__1103,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_ps(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_1103
tff(fact_8883_ATP_Olambda__1104,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_pu(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_1104
tff(fact_8884_ATP_Olambda__1105,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_aec(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_1105
tff(fact_8885_ATP_Olambda__1106,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_aeb(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_1106
tff(fact_8886_ATP_Olambda__1107,axiom,
    ! [A: $tType,B: $tType,Uu: filter(A),Uua: filter(B),Uub: fun(A,$o),Uuc: fun(B,$o)] :
      ( aa(fun(B,$o),$o,aa(fun(A,$o),fun(fun(B,$o),$o),aa(filter(B),fun(fun(A,$o),fun(fun(B,$o),$o)),aTP_Lamp_ant(filter(A),fun(filter(B),fun(fun(A,$o),fun(fun(B,$o),$o))),Uu),Uua),Uub),Uuc)
    <=> ( eventually(A,Uub,Uu)
        & eventually(B,Uuc,Uua) ) ) ).

% ATP.lambda_1107
tff(fact_8887_ATP_Olambda__1108,axiom,
    ! [A: $tType,B: $tType,Uu: A,Uua: B,Uub: A,Uuc: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(B,fun(A,fun(B,$o)),aTP_Lamp_qt(A,fun(B,fun(A,fun(B,$o))),Uu),Uua),Uub),Uuc)
    <=> ( ( Uu = Uub )
        & ( Uua = Uuc ) ) ) ).

% ATP.lambda_1108
tff(fact_8888_ATP_Olambda__1109,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aa(nat,fun(nat,fun(nat,$o)),aTP_Lamp_wr(nat,fun(nat,fun(nat,fun(nat,$o))),Uu),Uua),Uub),Uuc)
    <=> ( 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_1109
tff(fact_8889_ATP_Olambda__1110,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: set(B),Uub: fun(A,$o),Uuc: B] :
      ( aa(B,$o,aa(fun(A,$o),fun(B,$o),aa(set(B),fun(fun(A,$o),fun(B,$o)),aTP_Lamp_lk(fun(B,A),fun(set(B),fun(fun(A,$o),fun(B,$o))),Uu),Uua),Uub),Uuc)
    <=> ( aa(set(B),$o,member(B,Uuc),Uua)
        & aa(A,$o,Uub,aa(B,A,Uu,Uuc)) ) ) ).

% ATP.lambda_1110
tff(fact_8890_ATP_Olambda__1111,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: fun(C,set(product_prod(A,B))),Uua: C,Uub: A,Uuc: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(C,fun(A,fun(B,$o)),aTP_Lamp_np(fun(C,set(product_prod(A,B))),fun(C,fun(A,fun(B,$o))),Uu),Uua),Uub),Uuc)
    <=> aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uub),Uuc)),aa(C,set(product_prod(A,B)),Uu,Uua)) ) ).

% ATP.lambda_1111
tff(fact_8891_ATP_Olambda__1112,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: fun(A,B),Uuc: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aa(fun(A,B),fun(fun(A,B),fun(A,$o)),aTP_Lamp_hm(set(A),fun(fun(A,B),fun(fun(A,B),fun(A,$o))),Uu),Uua),Uub),Uuc)
        <=> ( aa(set(A),$o,member(A,Uuc),Uu)
            & ( aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uua,Uuc)),aa(A,B,Uub,Uuc)) != one_one(B) ) ) ) ) ).

% ATP.lambda_1112
tff(fact_8892_ATP_Olambda__1113,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: fun(A,B),Uuc: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aa(fun(A,B),fun(fun(A,B),fun(A,$o)),aTP_Lamp_hk(set(A),fun(fun(A,B),fun(fun(A,B),fun(A,$o))),Uu),Uua),Uub),Uuc)
        <=> ( aa(set(A),$o,member(A,Uuc),Uu)
            & ( aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,Uua,Uuc)),aa(A,B,Uub,Uuc)) != zero_zero(B) ) ) ) ) ).

% ATP.lambda_1113
tff(fact_8893_ATP_Olambda__1114,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_xv(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_1114
tff(fact_8894_ATP_Olambda__1115,axiom,
    ! [B: $tType,A: $tType,Uu: set(A),Uua: fun(B,$o),Uub: fun(A,B),Uuc: A] :
      ( aa(A,$o,aa(fun(A,B),fun(A,$o),aa(fun(B,$o),fun(fun(A,B),fun(A,$o)),aTP_Lamp_akx(set(A),fun(fun(B,$o),fun(fun(A,B),fun(A,$o))),Uu),Uua),Uub),Uuc)
    <=> ( aa(B,$o,Uua,aa(A,B,Uub,Uuc))
        & aa(set(A),$o,member(A,Uuc),Uu) ) ) ).

% ATP.lambda_1115
tff(fact_8895_ATP_Olambda__1116,axiom,
    ! [A: $tType,B: $tType,Uu: set(B),Uua: fun(B,A),Uub: fun(A,$o),Uuc: B] :
      ( aa(B,$o,aa(fun(A,$o),fun(B,$o),aa(fun(B,A),fun(fun(A,$o),fun(B,$o)),aTP_Lamp_aku(set(B),fun(fun(B,A),fun(fun(A,$o),fun(B,$o))),Uu),Uua),Uub),Uuc)
    <=> ( aa(A,$o,Uub,aa(B,A,Uua,Uuc))
        & aa(set(B),$o,member(B,Uuc),Uu) ) ) ).

% ATP.lambda_1116
tff(fact_8896_ATP_Olambda__1117,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,$o),Uua: fun(A,fun(B,$o)),Uub: A,Uuc: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aTP_Lamp_rh(fun(A,$o),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),Uu),Uua),Uub),Uuc)
    <=> ( aa(A,$o,Uu,Uub)
        & aa(B,$o,aa(A,fun(B,$o),Uua,Uub),Uuc) ) ) ).

% ATP.lambda_1117
tff(fact_8897_ATP_Olambda__1118,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_rq(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_rp(fun(D,set(B)),fun(D,fun(A,set(B))),Uua),Uuc)) ).

% ATP.lambda_1118
tff(fact_8898_ATP_Olambda__1119,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: fun(A,A),Uub: A,Uuc: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aa(fun(A,A),fun(A,fun(A,$o)),aTP_Lamp_ve(fun(A,$o),fun(fun(A,A),fun(A,fun(A,$o))),Uu),Uua),Uub),Uuc)
    <=> ( aa(A,$o,Uu,Uuc)
        & ( Uub = aa(A,A,Uua,Uuc) ) ) ) ).

% ATP.lambda_1119
tff(fact_8899_ATP_Olambda__1120,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_zr(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_1120
tff(fact_8900_ATP_Olambda__1121,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( semiring_0(A)
        & comm_monoid_add(B)
        & times(B) )
     => ! [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_aqf(fun(B,A),fun(fun(C,A),fun(B,fun(C,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,Uu,Uub)),aa(C,A,Uua,Uuc)) ) ).

% ATP.lambda_1121
tff(fact_8901_ATP_Olambda__1122,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( semiring_0(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_cz(fun(B,A),fun(fun(C,A),fun(B,fun(C,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,Uu,Uub)),aa(C,A,Uua,Uuc)) ) ).

% ATP.lambda_1122
tff(fact_8902_ATP_Olambda__1123,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_ack(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_1123
tff(fact_8903_ATP_Olambda__1124,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( semiring_0(C)
     => ! [Uu: fun(A,C),Uua: fun(B,C),Uub: A,Uuc: B] : aa(B,C,aa(A,fun(B,C),aa(fun(B,C),fun(A,fun(B,C)),aTP_Lamp_agk(fun(A,C),fun(fun(B,C),fun(A,fun(B,C))),Uu),Uua),Uub),Uuc) = aa(C,C,aa(C,fun(C,C),times_times(C),aa(A,C,Uu,Uub)),aa(B,C,Uua,Uuc)) ) ).

% ATP.lambda_1124
tff(fact_8904_ATP_Olambda__1125,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu: fun(B,set(A)),Uua: fun(C,set(A)),Uub: B,Uuc: C] : aa(C,set(A),aa(B,fun(C,set(A)),aa(fun(C,set(A)),fun(B,fun(C,set(A))),aTP_Lamp_kx(fun(B,set(A)),fun(fun(C,set(A)),fun(B,fun(C,set(A)))),Uu),Uua),Uub),Uuc) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(B,set(A),Uu,Uub)),aa(C,set(A),Uua,Uuc)) ).

% ATP.lambda_1125
tff(fact_8905_ATP_Olambda__1126,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_le(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_1126
tff(fact_8906_ATP_Olambda__1127,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu: fun(B,set(A)),Uua: fun(C,set(A)),Uub: B,Uuc: C] : aa(C,set(A),aa(B,fun(C,set(A)),aa(fun(C,set(A)),fun(B,fun(C,set(A))),aTP_Lamp_mq(fun(B,set(A)),fun(fun(C,set(A)),fun(B,fun(C,set(A)))),Uu),Uua),Uub),Uuc) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(B,set(A),Uu,Uub)),aa(C,set(A),Uua,Uuc)) ).

% ATP.lambda_1127
tff(fact_8907_ATP_Olambda__1128,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_mi(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_1128
tff(fact_8908_ATP_Olambda__1129,axiom,
    ! [B: $tType,A: $tType,Uu: fun(set(old_node(A,B)),set(set(old_node(A,B)))),Uua: fun(set(old_node(A,B)),set(set(old_node(A,B)))),Uub: set(old_node(A,B)),Uuc: set(old_node(A,B))] : aa(set(old_node(A,B)),set(set(old_node(A,B))),aa(set(old_node(A,B)),fun(set(old_node(A,B)),set(set(old_node(A,B)))),aa(fun(set(old_node(A,B)),set(set(old_node(A,B)))),fun(set(old_node(A,B)),fun(set(old_node(A,B)),set(set(old_node(A,B))))),aTP_Lamp_avi(fun(set(old_node(A,B)),set(set(old_node(A,B)))),fun(fun(set(old_node(A,B)),set(set(old_node(A,B)))),fun(set(old_node(A,B)),fun(set(old_node(A,B)),set(set(old_node(A,B)))))),Uu),Uua),Uub),Uuc) = old_uprod(A,B,aa(set(old_node(A,B)),set(set(old_node(A,B))),Uu,Uub),aa(set(old_node(A,B)),set(set(old_node(A,B))),Uua,Uuc)) ).

% ATP.lambda_1129
tff(fact_8909_ATP_Olambda__1130,axiom,
    ! [A: $tType,C: $tType,D: $tType,B: $tType,Uu: fun(A,filter(C)),Uua: fun(B,filter(D)),Uub: A,Uuc: B] : aa(B,filter(product_prod(C,D)),aa(A,fun(B,filter(product_prod(C,D))),aa(fun(B,filter(D)),fun(A,fun(B,filter(product_prod(C,D)))),aTP_Lamp_anz(fun(A,filter(C)),fun(fun(B,filter(D)),fun(A,fun(B,filter(product_prod(C,D))))),Uu),Uua),Uub),Uuc) = prod_filter(C,D,aa(A,filter(C),Uu,Uub),aa(B,filter(D),Uua,Uuc)) ).

% ATP.lambda_1130
tff(fact_8910_ATP_Olambda__1131,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_rx(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_1131
tff(fact_8911_ATP_Olambda__1132,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_afk(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_1132
tff(fact_8912_ATP_Olambda__1133,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,$o),Uua: fun(B,$o),Uub: A,Uuc: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(fun(B,$o),fun(A,fun(B,$o)),aTP_Lamp_qx(fun(A,$o),fun(fun(B,$o),fun(A,fun(B,$o))),Uu),Uua),Uub),Uuc)
    <=> ( aa(A,$o,Uu,Uub)
        & aa(B,$o,Uua,Uuc) ) ) ).

% ATP.lambda_1133
tff(fact_8913_ATP_Olambda__1134,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [Uu: fun(B,A),Uua: fun(list(B),A),Uub: list(B),Uuc: B] :
          ( aa(B,$o,aa(list(B),fun(B,$o),aa(fun(list(B),A),fun(list(B),fun(B,$o)),aTP_Lamp_aaf(fun(B,A),fun(fun(list(B),A),fun(list(B),fun(B,$o))),Uu),Uua),Uub),Uuc)
        <=> ( aa(B,A,Uu,Uuc) = aa(list(B),A,Uua,Uub) ) ) ) ).

% ATP.lambda_1134
tff(fact_8914_ATP_Olambda__1135,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( comm_monoid_mult(C)
     => ! [Uu: set(B),Uua: fun(A,fun(B,C)),Uub: fun(A,fun(B,$o)),Uuc: A] : aa(A,C,aa(fun(A,fun(B,$o)),fun(A,C),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(A,C)),aTP_Lamp_hf(set(B),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(A,C))),Uu),Uua),Uub),Uuc) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),aa(A,fun(B,C),Uua,Uuc)),aa(fun(B,$o),set(B),collect(B),aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aTP_Lamp_ha(set(B),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),Uu),Uub),Uuc))) ) ).

% ATP.lambda_1135
tff(fact_8915_ATP_Olambda__1136,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( comm_monoid_add(C)
     => ! [Uu: set(B),Uua: fun(A,fun(B,C)),Uub: fun(A,fun(B,$o)),Uuc: A] : aa(A,C,aa(fun(A,fun(B,$o)),fun(A,C),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(A,C)),aTP_Lamp_hb(set(B),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(A,C))),Uu),Uua),Uub),Uuc) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),aa(A,fun(B,C),Uua,Uuc)),aa(fun(B,$o),set(B),collect(B),aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aTP_Lamp_ha(set(B),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),Uu),Uub),Uuc))) ) ).

% ATP.lambda_1136
tff(fact_8916_ATP_Olambda__1137,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Uu: fun(A,B),Uua: list(A),Uub: A,Uuc: list(A)] : aa(list(A),list(A),aa(A,fun(list(A),list(A)),aa(list(A),fun(A,fun(list(A),list(A))),aTP_Lamp_ym(fun(A,B),fun(list(A),fun(A,fun(list(A),list(A)))),Uu),Uua),Uub),Uuc) = aa(product_prod(list(A),product_prod(list(A),list(A))),list(A),aa(fun(list(A),fun(product_prod(list(A),list(A)),list(A))),fun(product_prod(list(A),product_prod(list(A),list(A))),list(A)),product_case_prod(list(A),product_prod(list(A),list(A)),list(A)),aTP_Lamp_yl(fun(A,B),fun(list(A),fun(product_prod(list(A),list(A)),list(A))),Uu)),linorder_part(A,B,Uu,aa(A,B,Uu,aa(nat,A,nth(A,Uua),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(list(A),nat,size_size(list(A)),Uua)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),Uua)) ) ).

% ATP.lambda_1137
tff(fact_8917_ATP_Olambda__1138,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,fun(A,$o)),Uua: fun(B,A),Uub: B,Uuc: B] :
      ( aa(B,$o,aa(B,fun(B,$o),aa(fun(B,A),fun(B,fun(B,$o)),aTP_Lamp_ajz(fun(A,fun(A,$o)),fun(fun(B,A),fun(B,fun(B,$o))),Uu),Uua),Uub),Uuc)
    <=> ( ( aa(B,A,Uua,Uub) != aa(B,A,Uua,Uuc) )
       => aa(A,$o,aa(A,fun(A,$o),Uu,aa(B,A,Uua,Uub)),aa(B,A,Uua,Uuc)) ) ) ).

% ATP.lambda_1138
tff(fact_8918_ATP_Olambda__1139,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,set(B)),Uua: set(B),Uub: set(B),Uuc: A] :
      ( aa(A,$o,aa(set(B),fun(A,$o),aa(set(B),fun(set(B),fun(A,$o)),aTP_Lamp_akn(fun(A,set(B)),fun(set(B),fun(set(B),fun(A,$o))),Uu),Uua),Uub),Uuc)
    <=> ( aa(set(B),$o,finite_finite2(B),aa(A,set(B),Uu,Uuc))
        & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),Uub),aa(A,set(B),Uu,Uuc))
        & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(A,set(B),Uu,Uuc)),Uua) ) ) ).

% ATP.lambda_1139
tff(fact_8919_ATP_Olambda__1140,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( semiring_1(C)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: fun(B,C),Uuc: B] : aa(B,C,aa(fun(B,C),fun(B,C),aa(fun(A,B),fun(fun(B,C),fun(B,C)),aTP_Lamp_nk(set(A),fun(fun(A,B),fun(fun(B,C),fun(B,C))),Uu),Uua),Uub),Uuc) = aa(C,C,aa(C,fun(C,C),times_times(C),aa(nat,C,semiring_1_of_nat(C),aa(set(A),nat,finite_card(A),aa(fun(A,$o),set(A),collect(A),aa(B,fun(A,$o),aa(fun(A,B),fun(B,fun(A,$o)),aTP_Lamp_mt(set(A),fun(fun(A,B),fun(B,fun(A,$o))),Uu),Uua),Uuc))))),aa(B,C,Uub,Uuc)) ) ).

% ATP.lambda_1140
tff(fact_8920_ATP_Olambda__1141,axiom,
    ! [A: $tType,C: $tType,B: $tType,Uu: fun(set(old_node(B,C)),A),Uua: fun(set(old_node(B,C)),A),Uub: set(old_node(B,C)),Uuc: A] :
      ( aa(A,$o,aa(set(old_node(B,C)),fun(A,$o),aa(fun(set(old_node(B,C)),A),fun(set(old_node(B,C)),fun(A,$o)),aTP_Lamp_auo(fun(set(old_node(B,C)),A),fun(fun(set(old_node(B,C)),A),fun(set(old_node(B,C)),fun(A,$o))),Uu),Uua),Uub),Uuc)
    <=> ( ? [X4: set(old_node(B,C))] :
            ( ( Uub = aa(set(old_node(B,C)),set(old_node(B,C)),old_In0(B,C),X4) )
            & ( Uuc = aa(set(old_node(B,C)),A,Uu,X4) ) )
        | ? [Y3: set(old_node(B,C))] :
            ( ( Uub = aa(set(old_node(B,C)),set(old_node(B,C)),old_In1(B,C),Y3) )
            & ( Uuc = aa(set(old_node(B,C)),A,Uua,Y3) ) ) ) ) ).

% ATP.lambda_1141
tff(fact_8921_ATP_Olambda__1142,axiom,
    ! [A: $tType,Uu: fun(A,fun(A,$o)),Uua: fun(A,fun(A,$o)),Uub: A,Uuc: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aTP_Lamp_aex(fun(A,fun(A,$o)),fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),Uu),Uua),Uub),Uuc)
    <=> ( ? [A10: A] :
            ( ( Uub = A10 )
            & ( Uuc = A10 ) )
        | ? [A10: A,B6: A,C5: A] :
            ( ( Uub = A10 )
            & ( Uuc = C5 )
            & aa(A,$o,aa(A,fun(A,$o),Uua,A10),B6)
            & aa(A,$o,aa(A,fun(A,$o),Uu,B6),C5) ) ) ) ).

% ATP.lambda_1142
tff(fact_8922_ATP_Olambda__1143,axiom,
    ! [A: $tType,Uu: fun(A,fun(A,$o)),Uua: fun(A,fun(A,$o)),Uub: A,Uuc: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aTP_Lamp_aey(fun(A,fun(A,$o)),fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),Uu),Uua),Uub),Uuc)
    <=> ( ? [A10: A,B6: A] :
            ( ( Uub = A10 )
            & ( Uuc = B6 )
            & aa(A,$o,aa(A,fun(A,$o),Uu,A10),B6) )
        | ? [A10: A,B6: A,C5: A] :
            ( ( Uub = A10 )
            & ( Uuc = C5 )
            & aa(A,$o,aa(A,fun(A,$o),Uua,A10),B6)
            & aa(A,$o,aa(A,fun(A,$o),Uu,B6),C5) ) ) ) ).

% ATP.lambda_1143
tff(fact_8923_ATP_Olambda__1144,axiom,
    ! [A: $tType,Uu: fun(A,fun(A,$o)),Uua: fun(list(A),fun(list(A),$o)),Uub: list(A),Uuc: list(A)] :
      ( aa(list(A),$o,aa(list(A),fun(list(A),$o),aa(fun(list(A),fun(list(A),$o)),fun(list(A),fun(list(A),$o)),aTP_Lamp_afb(fun(A,fun(A,$o)),fun(fun(list(A),fun(list(A),$o)),fun(list(A),fun(list(A),$o))),Uu),Uua),Uub),Uuc)
    <=> ( ? [Y3: A,Ys3: list(A)] :
            ( ( Uub = nil(A) )
            & ( Uuc = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys3) ) )
        | ? [X4: A,Y3: A,Xs4: list(A),Ys3: list(A)] :
            ( ( Uub = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs4) )
            & ( Uuc = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys3) )
            & aa(A,$o,aa(A,fun(A,$o),Uu,X4),Y3) )
        | ? [X4: A,Y3: A,Xs4: list(A),Ys3: list(A)] :
            ( ( Uub = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs4) )
            & ( Uuc = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys3) )
            & ~ aa(A,$o,aa(A,fun(A,$o),Uu,X4),Y3)
            & ~ aa(A,$o,aa(A,fun(A,$o),Uu,Y3),X4)
            & aa(list(A),$o,aa(list(A),fun(list(A),$o),Uua,Xs4),Ys3) ) ) ) ).

% ATP.lambda_1144
tff(fact_8924_ATP_Olambda__1145,axiom,
    ! [A: $tType,Uu: A,Uua: list(A),Uub: A,Uuc: nat] : aa(nat,list(A),aa(A,fun(nat,list(A)),aa(list(A),fun(A,fun(nat,list(A))),aTP_Lamp_yd(A,fun(list(A),fun(A,fun(nat,list(A)))),Uu),Uua),Uub),Uuc) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uu),list_update(A,Uua,Uuc,Uub)) ).

% ATP.lambda_1145
tff(fact_8925_ATP_Olambda__1146,axiom,
    ! [A: $tType,B: $tType,Uu: $o,Uua: fun(A,fun(B,$o)),Uub: A,Uuc: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aTP_Lamp_bw($o,fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),(Uu)),Uua),Uub),Uuc)
    <=> ( (Uu)
        & aa(B,$o,aa(A,fun(B,$o),Uua,Uub),Uuc) ) ) ).

% ATP.lambda_1146
tff(fact_8926_ATP_Olambda__1147,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_qw(fun(C,A),fun(fun(C,B),fun(set(C),fun(A,set(B)))),Uu),Uua),Uub),Uuc) = aa(set(C),set(B),image2(C,B,Uua),aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),inf_inf(set(C)),aa(set(A),set(C),aa(fun(C,A),fun(set(A),set(C)),vimage(C,A),Uu),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Uuc),bot_bot(set(A))))),Uub)) ).

% ATP.lambda_1147
tff(fact_8927_ATP_Olambda__1148,axiom,
    ! [A: $tType,Uu: list(A),Uua: list(A),Uub: A,Uuc: list(A)] : aa(list(A),option($o),aa(A,fun(list(A),option($o)),aa(list(A),fun(A,fun(list(A),option($o))),aTP_Lamp_yh(list(A),fun(list(A),fun(A,fun(list(A),option($o)))),Uu),Uua),Uub),Uuc) = subset_eq_mset_impl(A,Uu,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Uua),Uuc)) ).

% ATP.lambda_1148
tff(fact_8928_ATP_Olambda__1149,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: fun(A,C),Uuc: C] : aa(C,B,aa(fun(A,C),fun(C,B),aa(fun(A,B),fun(fun(A,C),fun(C,B)),aTP_Lamp_mn(set(A),fun(fun(A,B),fun(fun(A,C),fun(C,B))),Uu),Uua),Uub),Uuc) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),Uua),aa(fun(A,$o),set(A),collect(A),aa(C,fun(A,$o),aa(fun(A,C),fun(C,fun(A,$o)),aTP_Lamp_mk(set(A),fun(fun(A,C),fun(C,fun(A,$o))),Uu),Uub),Uuc))) ) ).

% ATP.lambda_1149
tff(fact_8929_ATP_Olambda__1150,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: fun(A,C),Uuc: B] : aa(B,C,aa(fun(A,C),fun(B,C),aa(fun(A,B),fun(fun(A,C),fun(B,C)),aTP_Lamp_mv(set(A),fun(fun(A,B),fun(fun(A,C),fun(B,C))),Uu),Uua),Uub),Uuc) = aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7121269368397514597t_prod(A,C),Uub),aa(fun(A,$o),set(A),collect(A),aa(B,fun(A,$o),aa(fun(A,B),fun(B,fun(A,$o)),aTP_Lamp_mt(set(A),fun(fun(A,B),fun(B,fun(A,$o))),Uu),Uua),Uuc))) ) ).

% ATP.lambda_1150
tff(fact_8930_ATP_Olambda__1151,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: fun(A,C),Uuc: C] : aa(C,B,aa(fun(A,C),fun(C,B),aa(fun(A,B),fun(fun(A,C),fun(C,B)),aTP_Lamp_ml(set(A),fun(fun(A,B),fun(fun(A,C),fun(C,B))),Uu),Uua),Uub),Uuc) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),Uua),aa(fun(A,$o),set(A),collect(A),aa(C,fun(A,$o),aa(fun(A,C),fun(C,fun(A,$o)),aTP_Lamp_mk(set(A),fun(fun(A,C),fun(C,fun(A,$o))),Uu),Uub),Uuc))) ) ).

% ATP.lambda_1151
tff(fact_8931_ATP_Olambda__1152,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: fun(A,C),Uuc: B] : aa(B,C,aa(fun(A,C),fun(B,C),aa(fun(A,B),fun(fun(A,C),fun(B,C)),aTP_Lamp_mu(set(A),fun(fun(A,B),fun(fun(A,C),fun(B,C))),Uu),Uua),Uub),Uuc) = aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7311177749621191930dd_sum(A,C),Uub),aa(fun(A,$o),set(A),collect(A),aa(B,fun(A,$o),aa(fun(A,B),fun(B,fun(A,$o)),aTP_Lamp_mt(set(A),fun(fun(A,B),fun(B,fun(A,$o))),Uu),Uua),Uuc))) ) ).

% ATP.lambda_1152
tff(fact_8932_ATP_Olambda__1153,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu: A,Uua: list(A),Uub: A,Uuc: list(A)] : aa(list(A),A,aa(A,fun(list(A),A),aa(list(A),fun(A,fun(list(A),A)),aTP_Lamp_zm(A,fun(list(A),fun(A,fun(list(A),A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),ord_min(A),Uu),min_list(A,Uua)) ) ).

% ATP.lambda_1153
tff(fact_8933_ATP_Olambda__1154,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: fun(B,A),Uua: fun(pred(C),fun(seq(C),B)),Uub: pred(C),Uuc: seq(C)] : aa(seq(C),A,aa(pred(C),fun(seq(C),A),aa(fun(pred(C),fun(seq(C),B)),fun(pred(C),fun(seq(C),A)),aTP_Lamp_axx(fun(B,A),fun(fun(pred(C),fun(seq(C),B)),fun(pred(C),fun(seq(C),A))),Uu),Uua),Uub),Uuc) = aa(B,A,Uu,aa(seq(C),B,aa(pred(C),fun(seq(C),B),Uua,Uub),Uuc)) ).

% ATP.lambda_1154
tff(fact_8934_ATP_Olambda__1155,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: fun(B,A),Uua: fun(C,fun(pred(C),B)),Uub: C,Uuc: pred(C)] : aa(pred(C),A,aa(C,fun(pred(C),A),aa(fun(C,fun(pred(C),B)),fun(C,fun(pred(C),A)),aTP_Lamp_axw(fun(B,A),fun(fun(C,fun(pred(C),B)),fun(C,fun(pred(C),A))),Uu),Uua),Uub),Uuc) = aa(B,A,Uu,aa(pred(C),B,aa(C,fun(pred(C),B),Uua,Uub),Uuc)) ).

% ATP.lambda_1155
tff(fact_8935_ATP_Olambda__1156,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: fun(B,A),Uua: fun(C,fun(list(C),B)),Uub: C,Uuc: list(C)] : aa(list(C),A,aa(C,fun(list(C),A),aa(fun(C,fun(list(C),B)),fun(C,fun(list(C),A)),aTP_Lamp_yp(fun(B,A),fun(fun(C,fun(list(C),B)),fun(C,fun(list(C),A))),Uu),Uua),Uub),Uuc) = aa(B,A,Uu,aa(list(C),B,aa(C,fun(list(C),B),Uua,Uub),Uuc)) ).

% ATP.lambda_1156
tff(fact_8936_ATP_Olambda__1157,axiom,
    ! [A: $tType,C: $tType,B: $tType,D: $tType,Uu: fun(B,A),Uua: fun(C,fun(D,B)),Uub: C,Uuc: D] : aa(D,A,aa(C,fun(D,A),aa(fun(C,fun(D,B)),fun(C,fun(D,A)),aTP_Lamp_bo(fun(B,A),fun(fun(C,fun(D,B)),fun(C,fun(D,A))),Uu),Uua),Uub),Uuc) = aa(B,A,Uu,aa(D,B,aa(C,fun(D,B),Uua,Uub),Uuc)) ).

% ATP.lambda_1157
tff(fact_8937_ATP_Olambda__1158,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_ft(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_1158
tff(fact_8938_ATP_Olambda__1159,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_dt(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_1159
tff(fact_8939_ATP_Olambda__1160,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_fr(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_1160
tff(fact_8940_ATP_Olambda__1161,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_dp(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_1161
tff(fact_8941_ATP_Olambda__1162,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: nat,Uub: list(A),Uuc: nat] :
      ( aa(nat,$o,aa(list(A),fun(nat,$o),aa(nat,fun(list(A),fun(nat,$o)),aTP_Lamp_acz(fun(A,$o),fun(nat,fun(list(A),fun(nat,$o))),Uu),Uua),Uub),Uuc)
    <=> aa(A,$o,Uu,aa(nat,A,nth(A,take(A,Uua,Uub)),Uuc)) ) ).

% ATP.lambda_1162
tff(fact_8942_ATP_Olambda__1163,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_adx(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_1163
tff(fact_8943_ATP_Olambda__1164,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_adw(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_1164
tff(fact_8944_ATP_Olambda__1165,axiom,
    ! [A: $tType,C: $tType,D: $tType,B: $tType,Uu: fun(A,fun(C,$o)),Uua: fun(B,fun(D,$o)),Uub: A,Uuc: B] : aa(B,fun(product_prod(C,D),$o),aa(A,fun(B,fun(product_prod(C,D),$o)),aa(fun(B,fun(D,$o)),fun(A,fun(B,fun(product_prod(C,D),$o))),aTP_Lamp_apb(fun(A,fun(C,$o)),fun(fun(B,fun(D,$o)),fun(A,fun(B,fun(product_prod(C,D),$o)))),Uu),Uua),Uub),Uuc) = aa(fun(C,fun(D,$o)),fun(product_prod(C,D),$o),product_case_prod(C,D,$o),aa(B,fun(C,fun(D,$o)),aa(A,fun(B,fun(C,fun(D,$o))),aa(fun(B,fun(D,$o)),fun(A,fun(B,fun(C,fun(D,$o)))),aTP_Lamp_apa(fun(A,fun(C,$o)),fun(fun(B,fun(D,$o)),fun(A,fun(B,fun(C,fun(D,$o))))),Uu),Uua),Uub),Uuc)) ).

% ATP.lambda_1165
tff(fact_8945_ATP_Olambda__1166,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A,Uuc: list(A)] : aa(list(A),fun(product_prod(list(A),list(A)),product_prod(list(A),product_prod(list(A),list(A)))),aa(A,fun(list(A),fun(product_prod(list(A),list(A)),product_prod(list(A),product_prod(list(A),list(A))))),aa(B,fun(A,fun(list(A),fun(product_prod(list(A),list(A)),product_prod(list(A),product_prod(list(A),list(A)))))),aTP_Lamp_xx(fun(A,B),fun(B,fun(A,fun(list(A),fun(product_prod(list(A),list(A)),product_prod(list(A),product_prod(list(A),list(A))))))),Uu),Uua),Uub),Uuc) = aa(fun(list(A),fun(list(A),product_prod(list(A),product_prod(list(A),list(A))))),fun(product_prod(list(A),list(A)),product_prod(list(A),product_prod(list(A),list(A)))),product_case_prod(list(A),list(A),product_prod(list(A),product_prod(list(A),list(A)))),aa(list(A),fun(list(A),fun(list(A),product_prod(list(A),product_prod(list(A),list(A))))),aa(A,fun(list(A),fun(list(A),fun(list(A),product_prod(list(A),product_prod(list(A),list(A)))))),aa(B,fun(A,fun(list(A),fun(list(A),fun(list(A),product_prod(list(A),product_prod(list(A),list(A))))))),aTP_Lamp_xw(fun(A,B),fun(B,fun(A,fun(list(A),fun(list(A),fun(list(A),product_prod(list(A),product_prod(list(A),list(A)))))))),Uu),Uua),Uub),Uuc)) ) ).

% ATP.lambda_1166
tff(fact_8946_ATP_Olambda__1167,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,$o),Uua: fun(A,set(product_prod(B,B))),Uub: A,Uuc: B] : aa(B,fun(product_prod(A,B),$o),aa(A,fun(B,fun(product_prod(A,B),$o)),aa(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(product_prod(A,B),$o))),aTP_Lamp_qi(fun(A,$o),fun(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(product_prod(A,B),$o)))),Uu),Uua),Uub),Uuc) = aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aa(B,fun(A,fun(B,$o)),aa(A,fun(B,fun(A,fun(B,$o))),aa(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(A,fun(B,$o)))),aTP_Lamp_qh(fun(A,$o),fun(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(A,fun(B,$o))))),Uu),Uua),Uub),Uuc)) ).

% ATP.lambda_1167
tff(fact_8947_ATP_Olambda__1168,axiom,
    ! [A: $tType,B: $tType,Uu: set(product_prod(A,A)),Uua: set(product_prod(B,B)),Uub: A,Uuc: B] : aa(B,fun(product_prod(A,B),$o),aa(A,fun(B,fun(product_prod(A,B),$o)),aa(set(product_prod(B,B)),fun(A,fun(B,fun(product_prod(A,B),$o))),aTP_Lamp_qg(set(product_prod(A,A)),fun(set(product_prod(B,B)),fun(A,fun(B,fun(product_prod(A,B),$o)))),Uu),Uua),Uub),Uuc) = aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aa(B,fun(A,fun(B,$o)),aa(A,fun(B,fun(A,fun(B,$o))),aa(set(product_prod(B,B)),fun(A,fun(B,fun(A,fun(B,$o)))),aTP_Lamp_qf(set(product_prod(A,A)),fun(set(product_prod(B,B)),fun(A,fun(B,fun(A,fun(B,$o))))),Uu),Uua),Uub),Uuc)) ).

% ATP.lambda_1168
tff(fact_8948_ATP_Olambda__1169,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( semiring_0(A)
        & comm_monoid_add(B)
        & times(B) )
     => ! [Uu: fun(B,A),Uua: fun(C,A),Uub: multiset(C),Uuc: B] : aa(B,A,aa(multiset(C),fun(B,A),aa(fun(C,A),fun(multiset(C),fun(B,A)),aTP_Lamp_aqg(fun(B,A),fun(fun(C,A),fun(multiset(C),fun(B,A))),Uu),Uua),Uub),Uuc) = comm_m7189776963980413722m_mset(A,aa(multiset(C),multiset(A),image_mset(C,A,aa(B,fun(C,A),aa(fun(C,A),fun(B,fun(C,A)),aTP_Lamp_aqf(fun(B,A),fun(fun(C,A),fun(B,fun(C,A))),Uu),Uua),Uuc)),Uub)) ) ).

% ATP.lambda_1169
tff(fact_8949_ATP_Olambda__1170,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu: fun(B,set(A)),Uua: fun(C,set(A)),Uub: set(C),Uuc: B] : aa(B,set(A),aa(set(C),fun(B,set(A)),aa(fun(C,set(A)),fun(set(C),fun(B,set(A))),aTP_Lamp_mr(fun(B,set(A)),fun(fun(C,set(A)),fun(set(C),fun(B,set(A)))),Uu),Uua),Uub),Uuc) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(C),set(set(A)),image2(C,set(A),aa(B,fun(C,set(A)),aa(fun(C,set(A)),fun(B,fun(C,set(A))),aTP_Lamp_mq(fun(B,set(A)),fun(fun(C,set(A)),fun(B,fun(C,set(A)))),Uu),Uua),Uuc)),Uub)) ).

% ATP.lambda_1170
tff(fact_8950_ATP_Olambda__1171,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_mj(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),aa(set(C),set(A),image2(C,A,aa(B,fun(C,A),aa(fun(C,A),fun(B,fun(C,A)),aTP_Lamp_mi(fun(B,A),fun(fun(C,A),fun(B,fun(C,A))),Uu),Uua),Uuc)),Uub)) ) ).

% ATP.lambda_1171
tff(fact_8951_ATP_Olambda__1172,axiom,
    ! [A: $tType,C: $tType,D: $tType,B: $tType,Uu: set(B),Uua: fun(A,filter(C)),Uub: fun(B,filter(D)),Uuc: A] : aa(A,filter(product_prod(C,D)),aa(fun(B,filter(D)),fun(A,filter(product_prod(C,D))),aa(fun(A,filter(C)),fun(fun(B,filter(D)),fun(A,filter(product_prod(C,D)))),aTP_Lamp_aoa(set(B),fun(fun(A,filter(C)),fun(fun(B,filter(D)),fun(A,filter(product_prod(C,D))))),Uu),Uua),Uub),Uuc) = aa(set(filter(product_prod(C,D))),filter(product_prod(C,D)),complete_Inf_Inf(filter(product_prod(C,D))),aa(set(B),set(filter(product_prod(C,D))),image2(B,filter(product_prod(C,D)),aa(A,fun(B,filter(product_prod(C,D))),aa(fun(B,filter(D)),fun(A,fun(B,filter(product_prod(C,D)))),aTP_Lamp_anz(fun(A,filter(C)),fun(fun(B,filter(D)),fun(A,fun(B,filter(product_prod(C,D))))),Uua),Uub),Uuc)),Uu)) ).

% ATP.lambda_1172
tff(fact_8952_ATP_Olambda__1173,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_lf(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),aa(set(C),set(A),image2(C,A,aa(B,fun(C,A),aa(fun(C,A),fun(B,fun(C,A)),aTP_Lamp_le(fun(B,A),fun(fun(C,A),fun(B,fun(C,A))),Uu),Uua),Uuc)),Uub)) ) ).

% ATP.lambda_1173
tff(fact_8953_ATP_Olambda__1174,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu: fun(B,set(A)),Uua: fun(C,set(A)),Uub: set(C),Uuc: B] : aa(B,set(A),aa(set(C),fun(B,set(A)),aa(fun(C,set(A)),fun(set(C),fun(B,set(A))),aTP_Lamp_ky(fun(B,set(A)),fun(fun(C,set(A)),fun(set(C),fun(B,set(A)))),Uu),Uua),Uub),Uuc) = aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(C),set(set(A)),image2(C,set(A),aa(B,fun(C,set(A)),aa(fun(C,set(A)),fun(B,fun(C,set(A))),aTP_Lamp_kx(fun(B,set(A)),fun(fun(C,set(A)),fun(B,fun(C,set(A)))),Uu),Uua),Uuc)),Uub)) ).

% ATP.lambda_1174
tff(fact_8954_ATP_Olambda__1175,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_pc(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_1175
tff(fact_8955_ATP_Olambda__1176,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_qb(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_1176
tff(fact_8956_ATP_Olambda__1177,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: list(A),Uub: A,Uuc: list(A)] : aa(list(A),option(product_prod(list(A),product_prod(A,list(A)))),aa(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A))))),aa(list(A),fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A)))))),aTP_Lamp_aaw(fun(A,$o),fun(list(A),fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A))))))),Uu),Uua),Uub),Uuc) = aa(product_prod(list(A),product_prod(A,list(A))),option(product_prod(list(A),product_prod(A,list(A)))),some(product_prod(list(A),product_prod(A,list(A)))),aa(product_prod(A,list(A)),product_prod(list(A),product_prod(A,list(A))),aa(list(A),fun(product_prod(A,list(A)),product_prod(list(A),product_prod(A,list(A)))),product_Pair(list(A),product_prod(A,list(A))),takeWhile(A,aa(fun(A,$o),fun(A,$o),comp($o,$o,A,fNot),Uu),Uua)),aa(list(A),product_prod(A,list(A)),aa(A,fun(list(A),product_prod(A,list(A))),product_Pair(A,list(A)),Uub),Uuc))) ).

% ATP.lambda_1177
tff(fact_8957_ATP_Olambda__1178,axiom,
    ! [A: $tType,Uu: A,Uua: list(A),Uub: A,Uuc: list(A)] : aa(list(A),option(product_prod(list(A),product_prod(A,list(A)))),aa(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A))))),aa(list(A),fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A)))))),aTP_Lamp_yb(A,fun(list(A),fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A))))))),Uu),Uua),Uub),Uuc) = aa(product_prod(list(A),product_prod(A,list(A))),option(product_prod(list(A),product_prod(A,list(A)))),some(product_prod(list(A),product_prod(A,list(A)))),aa(product_prod(A,list(A)),product_prod(list(A),product_prod(A,list(A))),aa(list(A),fun(product_prod(A,list(A)),product_prod(list(A),product_prod(A,list(A)))),product_Pair(list(A),product_prod(A,list(A))),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uu),Uua)),aa(list(A),product_prod(A,list(A)),aa(A,fun(list(A),product_prod(A,list(A))),product_Pair(A,list(A)),Uub),Uuc))) ).

% ATP.lambda_1178
tff(fact_8958_ATP_Olambda__1179,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_py(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_1179
tff(fact_8959_ATP_Olambda__1180,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_pw(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_1180
tff(fact_8960_ATP_Olambda__1181,axiom,
    ! [A: $tType,C: $tType,B: $tType,Uu: set(product_prod(B,B)),Uua: fun(A,fun(B,set(C))),Uub: B,Uuc: A] : aa(A,set(C),aa(B,fun(A,set(C)),aa(fun(A,fun(B,set(C))),fun(B,fun(A,set(C))),aTP_Lamp_ajw(set(product_prod(B,B)),fun(fun(A,fun(B,set(C))),fun(B,fun(A,set(C)))),Uu),Uua),Uub),Uuc) = aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(B),set(set(C)),image2(B,set(C),aa(A,fun(B,set(C)),Uua,Uuc)),aa(set(B),set(B),image(B,B,Uu),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),Uub),bot_bot(set(B)))))) ).

% ATP.lambda_1181
tff(fact_8961_ATP_Olambda__1182,axiom,
    ! [C: $tType,A: $tType,B: $tType,E: $tType,D: $tType,Uu: fun(B,fun(C,fun(D,fun(E,set(A))))),Uua: set(product_prod(D,E)),Uub: B,Uuc: C] : aa(C,set(A),aa(B,fun(C,set(A)),aa(set(product_prod(D,E)),fun(B,fun(C,set(A))),aTP_Lamp_od(fun(B,fun(C,fun(D,fun(E,set(A))))),fun(set(product_prod(D,E)),fun(B,fun(C,set(A)))),Uu),Uua),Uub),Uuc) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(product_prod(D,E)),set(set(A)),image2(product_prod(D,E),set(A),aa(fun(D,fun(E,set(A))),fun(product_prod(D,E),set(A)),product_case_prod(D,E,set(A)),aa(C,fun(D,fun(E,set(A))),aa(B,fun(C,fun(D,fun(E,set(A)))),Uu,Uub),Uuc))),Uua)) ).

% ATP.lambda_1182
tff(fact_8962_ATP_Olambda__1183,axiom,
    ! [A: $tType,B: $tType,Uu: set(A),Uua: set(B),Uub: B,Uuc: fun(A,B)] :
      ( aa(fun(A,B),$o,aa(B,fun(fun(A,B),$o),aa(set(B),fun(B,fun(fun(A,B),$o)),aTP_Lamp_aim(set(A),fun(set(B),fun(B,fun(fun(A,B),$o))),Uu),Uua),Uub),Uuc)
    <=> ! [X4: A] :
          ( ( aa(set(A),$o,member(A,X4),Uu)
           => aa(set(B),$o,member(B,aa(A,B,Uuc,X4)),Uua) )
          & ( ~ aa(set(A),$o,member(A,X4),Uu)
           => ( aa(A,B,Uuc,X4) = Uub ) ) ) ) ).

% ATP.lambda_1183
tff(fact_8963_ATP_Olambda__1184,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: set(product_prod(A,C)),Uua: set(product_prod(C,B)),Uub: A,Uuc: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(set(product_prod(C,B)),fun(A,fun(B,$o)),aTP_Lamp_afx(set(product_prod(A,C)),fun(set(product_prod(C,B)),fun(A,fun(B,$o))),Uu),Uua),Uub),Uuc)
    <=> ? [Y3: C] :
          ( aa(set(product_prod(A,C)),$o,member(product_prod(A,C),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),Uub),Y3)),Uu)
          & aa(set(product_prod(C,B)),$o,member(product_prod(C,B),aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),Y3),Uuc)),Uua) ) ) ).

% ATP.lambda_1184
tff(fact_8964_ATP_Olambda__1185,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu: set(C),Uua: fun(C,A),Uub: fun(C,B),Uuc: product_prod(A,B)] :
      ( aa(product_prod(A,B),$o,aa(fun(C,B),fun(product_prod(A,B),$o),aa(fun(C,A),fun(fun(C,B),fun(product_prod(A,B),$o)),aTP_Lamp_ahk(set(C),fun(fun(C,A),fun(fun(C,B),fun(product_prod(A,B),$o))),Uu),Uua),Uub),Uuc)
    <=> ? [A10: C] :
          ( ( Uuc = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,Uua,A10)),aa(C,B,Uub,A10)) )
          & aa(set(C),$o,member(C,A10),Uu) ) ) ).

% ATP.lambda_1185
tff(fact_8965_ATP_Olambda__1186,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: B,Uub: set(B),Uuc: A] :
      ( aa(A,$o,aa(set(B),fun(A,$o),aa(B,fun(set(B),fun(A,$o)),aTP_Lamp_aej(fun(B,A),fun(B,fun(set(B),fun(A,$o))),Uu),Uua),Uub),Uuc)
    <=> ? [L4: B] :
          ( ( Uuc = aa(B,A,Uu,L4) )
          & ( ( L4 = Uua )
            | aa(set(B),$o,member(B,L4),Uub) ) ) ) ).

% ATP.lambda_1186
tff(fact_8966_ATP_Olambda__1187,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(A)
     => ! [Uu: A,Uua: A,Uub: fun(A,B),Uuc: B] :
          ( aa(B,$o,aa(fun(A,B),fun(B,$o),aa(A,fun(fun(A,B),fun(B,$o)),aTP_Lamp_afr(A,fun(A,fun(fun(A,B),fun(B,$o))),Uu),Uua),Uub),Uuc)
        <=> ? [I3: A] :
              ( ( Uuc = aa(A,B,Uub,I3) )
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uu),I3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),I3),Uua) ) ) ) ).

% ATP.lambda_1187
tff(fact_8967_ATP_Olambda__1188,axiom,
    ! [A: $tType,C: $tType,B: $tType,Uu: set(product_prod(B,B)),Uua: fun(A,fun(B,C)),Uub: A,Uuc: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aa(fun(A,fun(B,C)),fun(A,fun(A,$o)),aTP_Lamp_ajb(set(product_prod(B,B)),fun(fun(A,fun(B,C)),fun(A,fun(A,$o))),Uu),Uua),Uub),Uuc)
    <=> ! [X4: product_prod(B,B)] :
          ( aa(set(product_prod(B,B)),$o,member(product_prod(B,B),X4),Uu)
         => aa(product_prod(B,B),$o,aa(fun(B,fun(B,$o)),fun(product_prod(B,B),$o),product_case_prod(B,B,$o),aa(A,fun(B,fun(B,$o)),aa(A,fun(A,fun(B,fun(B,$o))),aTP_Lamp_aja(fun(A,fun(B,C)),fun(A,fun(A,fun(B,fun(B,$o)))),Uua),Uub),Uuc)),X4) ) ) ).

% ATP.lambda_1188
tff(fact_8968_ATP_Olambda__1189,axiom,
    ! [A: $tType,C: $tType,B: $tType,Uu: fun(A,$o),Uua: fun(B,$o),Uub: fun(A,fun(B,C)),Uuc: C] :
      ( aa(C,$o,aa(fun(A,fun(B,C)),fun(C,$o),aa(fun(B,$o),fun(fun(A,fun(B,C)),fun(C,$o)),aTP_Lamp_aen(fun(A,$o),fun(fun(B,$o),fun(fun(A,fun(B,C)),fun(C,$o))),Uu),Uua),Uub),Uuc)
    <=> ? [X4: A,Y3: B] :
          ( ( Uuc = aa(B,C,aa(A,fun(B,C),Uub,X4),Y3) )
          & aa(A,$o,Uu,X4)
          & aa(B,$o,Uua,Y3) ) ) ).

% ATP.lambda_1189
tff(fact_8969_ATP_Olambda__1190,axiom,
    ! [A: $tType,B: $tType,Uu: set(A),Uua: set(product_prod(B,B)),Uub: fun(A,B),Uuc: product_prod(A,A)] :
      ( aa(product_prod(A,A),$o,aa(fun(A,B),fun(product_prod(A,A),$o),aa(set(product_prod(B,B)),fun(fun(A,B),fun(product_prod(A,A),$o)),aTP_Lamp_aog(set(A),fun(set(product_prod(B,B)),fun(fun(A,B),fun(product_prod(A,A),$o))),Uu),Uua),Uub),Uuc)
    <=> ? [A17: A,A25: A] :
          ( ( Uuc = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A17),A25) )
          & aa(set(A),$o,member(A,A17),Uu)
          & aa(set(A),$o,member(A,A25),Uu)
          & aa(set(product_prod(B,B)),$o,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,A25))),Uua) ) ) ).

% ATP.lambda_1190
tff(fact_8970_ATP_Olambda__1191,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_un(C,fun(A,fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B)))))),Uu),Uua),Uub),Uuc),Uud) = $ite(Uua = Uub,aa(set(product_prod(C,B)),set(product_prod(C,B)),aa(product_prod(C,B),fun(set(product_prod(C,B)),set(product_prod(C,B))),insert2(product_prod(C,B)),aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),Uu),Uuc)),Uud),Uud) ).

% ATP.lambda_1191
tff(fact_8971_ATP_Olambda__1192,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_uq(A,fun(B,fun(B,fun(C,fun(set(product_prod(A,C)),set(product_prod(A,C)))))),Uu),Uua),Uub),Uuc),Uud) = $ite(Uua = Uub,aa(set(product_prod(A,C)),set(product_prod(A,C)),aa(product_prod(A,C),fun(set(product_prod(A,C)),set(product_prod(A,C))),insert2(product_prod(A,C)),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),Uu),Uuc)),Uud),Uud) ).

% ATP.lambda_1192
tff(fact_8972_ATP_Olambda__1193,axiom,
    ! [A: $tType,Uu: fun(A,fun(A,$o)),Uua: list(A),Uub: A,Uuc: list(A),Uud: list(A)] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),aa(A,fun(list(A),fun(list(A),list(A))),aa(list(A),fun(A,fun(list(A),fun(list(A),list(A)))),aTP_Lamp_abx(fun(A,fun(A,$o)),fun(list(A),fun(A,fun(list(A),fun(list(A),list(A))))),Uu),Uua),Uub),Uuc),Uud) = aa(list(A),list(A),quicksort_by_rel(A,Uu,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uub),aa(list(A),list(A),quicksort_by_rel(A,Uu,Uua),Uud))),Uuc) ).

% ATP.lambda_1193
tff(fact_8973_ATP_Olambda__1194,axiom,
    ! [C: $tType,A: $tType,B: $tType,Uu: fun(A,fun(A,A)),Uua: A,Uub: fun(B,fun(C,A)),Uuc: multiset(B),Uud: C] : aa(C,A,aa(multiset(B),fun(C,A),aa(fun(B,fun(C,A)),fun(multiset(B),fun(C,A)),aa(A,fun(fun(B,fun(C,A)),fun(multiset(B),fun(C,A))),aTP_Lamp_aws(fun(A,fun(A,A)),fun(A,fun(fun(B,fun(C,A)),fun(multiset(B),fun(C,A)))),Uu),Uua),Uub),Uuc),Uud) = aa(multiset(A),A,comm_monoid_F(A,Uu,Uua),aa(multiset(B),multiset(A),image_mset(B,A,aa(C,fun(B,A),aTP_Lamp_awr(fun(B,fun(C,A)),fun(C,fun(B,A)),Uub),Uud)),Uuc)) ).

% ATP.lambda_1194
tff(fact_8974_ATP_Olambda__1195,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu: fun(A,fun(A,A)),Uua: A,Uub: fun(B,fun(C,A)),Uuc: multiset(C),Uud: B] : aa(B,A,aa(multiset(C),fun(B,A),aa(fun(B,fun(C,A)),fun(multiset(C),fun(B,A)),aa(A,fun(fun(B,fun(C,A)),fun(multiset(C),fun(B,A))),aTP_Lamp_awq(fun(A,fun(A,A)),fun(A,fun(fun(B,fun(C,A)),fun(multiset(C),fun(B,A)))),Uu),Uua),Uub),Uuc),Uud) = aa(multiset(A),A,comm_monoid_F(A,Uu,Uua),aa(multiset(C),multiset(A),image_mset(C,A,aa(B,fun(C,A),Uub,Uud)),Uuc)) ).

% ATP.lambda_1195
tff(fact_8975_ATP_Olambda__1196,axiom,
    ! [A: $tType,D: $tType,C: $tType,E: $tType,B: $tType,Uu: fun(D,fun(E,C)),Uua: fun(A,D),Uub: fun(B,E),Uuc: A,Uud: B] : aa(B,C,aa(A,fun(B,C),aa(fun(B,E),fun(A,fun(B,C)),aa(fun(A,D),fun(fun(B,E),fun(A,fun(B,C))),aTP_Lamp_afq(fun(D,fun(E,C)),fun(fun(A,D),fun(fun(B,E),fun(A,fun(B,C)))),Uu),Uua),Uub),Uuc),Uud) = aa(E,C,aa(D,fun(E,C),Uu,aa(A,D,Uua,Uuc)),aa(B,E,Uub,Uud)) ).

% ATP.lambda_1196
tff(fact_8976_ATP_Olambda__1197,axiom,
    ! [D: $tType,B: $tType,A: $tType,C: $tType,E: $tType,Uu: fun(B,fun(C,A)),Uua: fun(D,B),Uub: fun(E,C),Uuc: D,Uud: E] : aa(E,A,aa(D,fun(E,A),aa(fun(E,C),fun(D,fun(E,A)),aa(fun(D,B),fun(fun(E,C),fun(D,fun(E,A))),aTP_Lamp_aeu(fun(B,fun(C,A)),fun(fun(D,B),fun(fun(E,C),fun(D,fun(E,A)))),Uu),Uua),Uub),Uuc),Uud) = aa(C,A,aa(B,fun(C,A),Uu,aa(D,B,Uua,Uuc)),aa(E,C,Uub,Uud)) ).

% ATP.lambda_1197
tff(fact_8977_ATP_Olambda__1198,axiom,
    ! [Uu: $o,Uua: $o,Uub: $o,Uuc: code_integer,Uud: $o] : aa($o,char,aa(code_integer,fun($o,char),aa($o,fun(code_integer,fun($o,char)),aa($o,fun($o,fun(code_integer,fun($o,char))),aTP_Lamp_qn($o,fun($o,fun($o,fun(code_integer,fun($o,char)))),(Uu)),(Uua)),(Uub)),Uuc),(Uud)) = aa(product_prod(code_integer,$o),char,aa(fun(code_integer,fun($o,char)),fun(product_prod(code_integer,$o),char),product_case_prod(code_integer,$o,char),aa($o,fun(code_integer,fun($o,char)),aa($o,fun($o,fun(code_integer,fun($o,char))),aa($o,fun($o,fun($o,fun(code_integer,fun($o,char)))),aTP_Lamp_qm($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char))))),(Uu)),(Uua)),(Uub)),(Uud))),code_bit_cut_integer(Uuc)) ).

% ATP.lambda_1198
tff(fact_8978_ATP_Olambda__1199,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu: fun(A,fun(C,A)),Uua: fun(A,fun(B,fun(C,B))),Uub: A,Uuc: B,Uud: C] : aa(C,product_prod(A,B),aa(B,fun(C,product_prod(A,B)),aa(A,fun(B,fun(C,product_prod(A,B))),aa(fun(A,fun(B,fun(C,B))),fun(A,fun(B,fun(C,product_prod(A,B)))),aTP_Lamp_xj(fun(A,fun(C,A)),fun(fun(A,fun(B,fun(C,B))),fun(A,fun(B,fun(C,product_prod(A,B))))),Uu),Uua),Uub),Uuc),Uud) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,aa(A,fun(C,A),Uu,Uub),Uud)),aa(C,B,aa(B,fun(C,B),aa(A,fun(B,fun(C,B)),Uua,Uub),Uuc),Uud)) ).

% ATP.lambda_1199
tff(fact_8979_ATP_Olambda__1200,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu: fun(B,fun(A,$o)),Uua: fun(A,fun(C,$o)),Uub: B,Uuc: C,Uud: A] :
      ( aa(A,$o,aa(C,fun(A,$o),aa(B,fun(C,fun(A,$o)),aa(fun(A,fun(C,$o)),fun(B,fun(C,fun(A,$o))),aTP_Lamp_awo(fun(B,fun(A,$o)),fun(fun(A,fun(C,$o)),fun(B,fun(C,fun(A,$o)))),Uu),Uua),Uub),Uuc),Uud)
    <=> ( aa(A,$o,aa(B,fun(A,$o),Uu,Uub),Uud)
        & aa(C,$o,aa(A,fun(C,$o),Uua,Uud),Uuc) ) ) ).

% ATP.lambda_1200
tff(fact_8980_ATP_Olambda__1201,axiom,
    ! [A: $tType,C: $tType,B: $tType,Uu: fun(A,fun(B,C)),Uua: A,Uub: A,Uuc: B,Uud: B] :
      ( aa(B,$o,aa(B,fun(B,$o),aa(A,fun(B,fun(B,$o)),aa(A,fun(A,fun(B,fun(B,$o))),aTP_Lamp_aja(fun(A,fun(B,C)),fun(A,fun(A,fun(B,fun(B,$o)))),Uu),Uua),Uub),Uuc),Uud)
    <=> ( aa(B,C,aa(A,fun(B,C),Uu,Uua),Uuc) = aa(B,C,aa(A,fun(B,C),Uu,Uub),Uud) ) ) ).

% ATP.lambda_1201
tff(fact_8981_ATP_Olambda__1202,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_en(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_1202
tff(fact_8982_ATP_Olambda__1203,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_el(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_1203
tff(fact_8983_ATP_Olambda__1204,axiom,
    ! [A: $tType,Uu: A,Uua: A,Uub: set(product_prod(A,A)),Uuc: A,Uud: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aa(set(product_prod(A,A)),fun(A,fun(A,$o)),aa(A,fun(set(product_prod(A,A)),fun(A,fun(A,$o))),aTP_Lamp_oy(A,fun(A,fun(set(product_prod(A,A)),fun(A,fun(A,$o)))),Uu),Uua),Uub),Uuc),Uud)
    <=> ( ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uuc),Uu)),transitive_trancl(A,Uub))
          | ( Uuc = Uu ) )
        & ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uua),Uud)),transitive_trancl(A,Uub))
          | ( Uud = Uua ) ) ) ) ).

% ATP.lambda_1204
tff(fact_8984_ATP_Olambda__1205,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_em(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_1205
tff(fact_8985_ATP_Olambda__1206,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A,Uuc: A,Uud: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aa(A,fun(A,fun(A,$o)),aa(A,fun(A,fun(A,fun(A,$o))),aTP_Lamp_uy(set(product_prod(A,A)),fun(A,fun(A,fun(A,fun(A,$o)))),Uu),Uua),Uub),Uuc),Uud)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Uua),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Uub),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Uuc),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Uud),bot_bot(set(A))))))),field2(A,Uu))
        & ( ( ( Uua = Uuc )
            & ( Uub = Uud ) )
          | aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),bNF_We1388413361240627857o_max2(A,Uu,Uua,Uub)),bNF_We1388413361240627857o_max2(A,Uu,Uuc,Uud))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),minus_minus(set(product_prod(A,A))),Uu),id2(A)))
          | ( ( bNF_We1388413361240627857o_max2(A,Uu,Uua,Uub) = bNF_We1388413361240627857o_max2(A,Uu,Uuc,Uud) )
            & aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uua),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 )
            & aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),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_1206
tff(fact_8986_ATP_Olambda__1207,axiom,
    ! [A: $tType,Uu: A,Uua: A,Uub: set(product_prod(A,A)),Uuc: A,Uud: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aa(set(product_prod(A,A)),fun(A,fun(A,$o)),aa(A,fun(set(product_prod(A,A)),fun(A,fun(A,$o))),aTP_Lamp_ox(A,fun(A,fun(set(product_prod(A,A)),fun(A,fun(A,$o)))),Uu),Uua),Uub),Uuc),Uud)
    <=> ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uuc),Uu)),transitive_rtrancl(A,Uub))
        & aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uua),Uud)),transitive_rtrancl(A,Uub)) ) ) ).

% ATP.lambda_1207
tff(fact_8987_ATP_Olambda__1208,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A,Uuc: A,Uud: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aa(A,fun(A,fun(A,$o)),aa(A,fun(A,fun(A,fun(A,$o))),aTP_Lamp_aok(set(product_prod(A,A)),fun(A,fun(A,fun(A,fun(A,$o)))),Uu),Uua),Uub),Uuc),Uud)
    <=> ( ( Uub = Uuc )
       => aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uua),Uud)),Uu) ) ) ).

% ATP.lambda_1208
tff(fact_8988_ATP_Olambda__1209,axiom,
    ! [C: $tType,B: $tType,A: $tType,E: $tType,Uu: fun(A,fun(B,$o)),Uua: fun(C,fun(A,$o)),Uub: fun(A,fun(E,$o)),Uuc: C,Uud: E] :
      ( aa(E,$o,aa(C,fun(E,$o),aa(fun(A,fun(E,$o)),fun(C,fun(E,$o)),aa(fun(C,fun(A,$o)),fun(fun(A,fun(E,$o)),fun(C,fun(E,$o))),aTP_Lamp_arg(fun(A,fun(B,$o)),fun(fun(C,fun(A,$o)),fun(fun(A,fun(E,$o)),fun(C,fun(E,$o)))),Uu),Uua),Uub),Uuc),Uud)
    <=> ? [X4: A] :
          ( aa(set(A),$o,member(A,X4),aa(fun(A,$o),set(A),collect(A),aa(fun(A,fun(B,$o)),fun(A,$o),domainp(A,B),Uu)))
          & aa(A,$o,aa(C,fun(A,$o),Uua,Uuc),X4)
          & aa(E,$o,aa(A,fun(E,$o),Uub,X4),Uud) ) ) ).

% ATP.lambda_1209
tff(fact_8989_ATP_Olambda__1210,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( semiring_0(C)
     => ! [Uu: fun(A,C),Uua: fun(B,C),Uub: set(A),Uuc: set(B),Uud: C] :
          ( aa(C,$o,aa(set(B),fun(C,$o),aa(set(A),fun(set(B),fun(C,$o)),aa(fun(B,C),fun(set(A),fun(set(B),fun(C,$o))),aTP_Lamp_agl(fun(A,C),fun(fun(B,C),fun(set(A),fun(set(B),fun(C,$o)))),Uu),Uua),Uub),Uuc),Uud)
        <=> ? [A10: A,B6: B] :
              ( ( Uud = aa(C,C,aa(C,fun(C,C),times_times(C),aa(A,C,Uu,A10)),aa(B,C,Uua,B6)) )
              & aa(set(A),$o,member(A,A10),Uub)
              & aa(set(B),$o,member(B,B6),Uuc) ) ) ) ).

% ATP.lambda_1210
tff(fact_8990_ATP_Olambda__1211,axiom,
    ! [Uu: $o,Uua: $o,Uub: $o,Uuc: $o,Uud: code_integer,Uue: $o] : aa($o,char,aa(code_integer,fun($o,char),aa($o,fun(code_integer,fun($o,char)),aa($o,fun($o,fun(code_integer,fun($o,char))),aa($o,fun($o,fun($o,fun(code_integer,fun($o,char)))),aTP_Lamp_qm($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char))))),(Uu)),(Uua)),(Uub)),(Uuc)),Uud),(Uue)) = aa(product_prod(code_integer,$o),char,aa(fun(code_integer,fun($o,char)),fun(product_prod(code_integer,$o),char),product_case_prod(code_integer,$o,char),aa($o,fun(code_integer,fun($o,char)),aa($o,fun($o,fun(code_integer,fun($o,char))),aa($o,fun($o,fun($o,fun(code_integer,fun($o,char)))),aa($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char))))),aTP_Lamp_ql($o,fun($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char)))))),(Uu)),(Uua)),(Uub)),(Uuc)),(Uue))),code_bit_cut_integer(Uud)) ).

% ATP.lambda_1211
tff(fact_8991_ATP_Olambda__1212,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType,Uu: fun(A,fun(C,$o)),Uua: fun(B,fun(D,$o)),Uub: A,Uuc: B,Uud: C,Uue: D] :
      ( aa(D,$o,aa(C,fun(D,$o),aa(B,fun(C,fun(D,$o)),aa(A,fun(B,fun(C,fun(D,$o))),aa(fun(B,fun(D,$o)),fun(A,fun(B,fun(C,fun(D,$o)))),aTP_Lamp_apa(fun(A,fun(C,$o)),fun(fun(B,fun(D,$o)),fun(A,fun(B,fun(C,fun(D,$o))))),Uu),Uua),Uub),Uuc),Uud),Uue)
    <=> ( aa(C,$o,aa(A,fun(C,$o),Uu,Uub),Uud)
        & aa(D,$o,aa(B,fun(D,$o),Uua,Uuc),Uue) ) ) ).

% ATP.lambda_1212
tff(fact_8992_ATP_Olambda__1213,axiom,
    ! [A: $tType,B: $tType,Uu: set(product_prod(A,A)),Uua: set(product_prod(B,B)),Uub: A,Uuc: B,Uud: A,Uue: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(B,fun(A,fun(B,$o)),aa(A,fun(B,fun(A,fun(B,$o))),aa(set(product_prod(B,B)),fun(A,fun(B,fun(A,fun(B,$o)))),aTP_Lamp_qf(set(product_prod(A,A)),fun(set(product_prod(B,B)),fun(A,fun(B,fun(A,fun(B,$o))))),Uu),Uua),Uub),Uuc),Uud),Uue)
    <=> ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),Uud)),Uu)
        | ( ( Uub = Uud )
          & aa(set(product_prod(B,B)),$o,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_1213
tff(fact_8993_ATP_Olambda__1214,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,$o),Uua: fun(A,set(product_prod(B,B))),Uub: A,Uuc: B,Uud: A,Uue: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(B,fun(A,fun(B,$o)),aa(A,fun(B,fun(A,fun(B,$o))),aa(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(A,fun(B,$o)))),aTP_Lamp_qh(fun(A,$o),fun(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(A,fun(B,$o))))),Uu),Uua),Uub),Uuc),Uud),Uue)
    <=> ( ( Uub = Uud )
        & aa(A,$o,Uu,Uud)
        & aa(set(product_prod(B,B)),$o,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_1214
tff(fact_8994_ATP_Olambda__1215,axiom,
    ! [Uu: fun(a,b),Uua: b,Uub: a,Uuc: list(a),Uud: list(a),Uue: list(a)] :
      aa(list(a),product_prod(list(a),product_prod(list(a),list(a))),aa(list(a),fun(list(a),product_prod(list(a),product_prod(list(a),list(a)))),aa(list(a),fun(list(a),fun(list(a),product_prod(list(a),product_prod(list(a),list(a))))),aa(a,fun(list(a),fun(list(a),fun(list(a),product_prod(list(a),product_prod(list(a),list(a)))))),aa(b,fun(a,fun(list(a),fun(list(a),fun(list(a),product_prod(list(a),product_prod(list(a),list(a))))))),aTP_Lamp_xw(fun(a,b),fun(b,fun(a,fun(list(a),fun(list(a),fun(list(a),product_prod(list(a),product_prod(list(a),list(a)))))))),Uu),Uua),Uub),Uuc),Uud),Uue) = $let(
        x4: b,
        x4:= aa(a,b,Uu,Uub),
        $ite(
          aa(b,$o,aa(b,fun(b,$o),ord_less(b),x4),Uua),
          aa(product_prod(list(a),list(a)),product_prod(list(a),product_prod(list(a),list(a))),aa(list(a),fun(product_prod(list(a),list(a)),product_prod(list(a),product_prod(list(a),list(a)))),product_Pair(list(a),product_prod(list(a),list(a))),aa(list(a),list(a),aa(a,fun(list(a),list(a)),cons(a),Uub),Uuc)),aa(list(a),product_prod(list(a),list(a)),aa(list(a),fun(list(a),product_prod(list(a),list(a))),product_Pair(list(a),list(a)),Uud),Uue)),
          $ite(aa(b,$o,aa(b,fun(b,$o),ord_less(b),Uua),x4),aa(product_prod(list(a),list(a)),product_prod(list(a),product_prod(list(a),list(a))),aa(list(a),fun(product_prod(list(a),list(a)),product_prod(list(a),product_prod(list(a),list(a)))),product_Pair(list(a),product_prod(list(a),list(a))),Uuc),aa(list(a),product_prod(list(a),list(a)),aa(list(a),fun(list(a),product_prod(list(a),list(a))),product_Pair(list(a),list(a)),Uud),aa(list(a),list(a),aa(a,fun(list(a),list(a)),cons(a),Uub),Uue))),aa(product_prod(list(a),list(a)),product_prod(list(a),product_prod(list(a),list(a))),aa(list(a),fun(product_prod(list(a),list(a)),product_prod(list(a),product_prod(list(a),list(a)))),product_Pair(list(a),product_prod(list(a),list(a))),Uuc),aa(list(a),product_prod(list(a),list(a)),aa(list(a),fun(list(a),product_prod(list(a),list(a))),product_Pair(list(a),list(a)),aa(list(a),list(a),aa(a,fun(list(a),list(a)),cons(a),Uub),Uud)),Uue))) ) ) ).

% ATP.lambda_1215
tff(fact_8995_ATP_Olambda__1216,axiom,
    ! [Uu: $o,Uua: $o,Uub: $o,Uuc: $o,Uud: $o,Uue: code_integer,Uuf: $o] : aa($o,char,aa(code_integer,fun($o,char),aa($o,fun(code_integer,fun($o,char)),aa($o,fun($o,fun(code_integer,fun($o,char))),aa($o,fun($o,fun($o,fun(code_integer,fun($o,char)))),aa($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char))))),aTP_Lamp_ql($o,fun($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char)))))),(Uu)),(Uua)),(Uub)),(Uuc)),(Uud)),Uue),(Uuf)) = aa(product_prod(code_integer,$o),char,aa(fun(code_integer,fun($o,char)),fun(product_prod(code_integer,$o),char),product_case_prod(code_integer,$o,char),aa($o,fun(code_integer,fun($o,char)),aa($o,fun($o,fun(code_integer,fun($o,char))),aa($o,fun($o,fun($o,fun(code_integer,fun($o,char)))),aa($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char))))),aa($o,fun($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char)))))),aTP_Lamp_qk($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char))))))),(Uu)),(Uua)),(Uub)),(Uuc)),(Uud)),(Uuf))),code_bit_cut_integer(Uue)) ).

% ATP.lambda_1216
tff(fact_8996_ATP_Olambda__1217,axiom,
    ! [Uu: $o,Uua: $o,Uub: $o,Uuc: $o,Uud: $o,Uue: $o,Uuf: $o] : aa($o,fun(list(char),list(char)),aa($o,fun($o,fun(list(char),list(char))),aa($o,fun($o,fun($o,fun(list(char),list(char)))),aa($o,fun($o,fun($o,fun($o,fun(list(char),list(char))))),aa($o,fun($o,fun($o,fun($o,fun($o,fun(list(char),list(char)))))),aa($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun(list(char),list(char))))))),aa($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun(list(char),list(char)))))))),aTP_Lamp_atd($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun(list(char),list(char))))))))),(Uu)),(Uua)),(Uub)),(Uuc)),(Uud)),(Uue)),(Uuf)) = aa(char,fun(list(char),list(char)),cons(char),aa($o,char,char2((Uu),(Uua),(Uub),(Uuc),(Uud),(Uue),(Uuf)),$false)) ).

% ATP.lambda_1217
tff(fact_8997_ATP_Olambda__1218,axiom,
    ! [Uu: $o,Uua: $o,Uub: $o,Uuc: $o,Uud: $o,Uue: $o,Uuf: $o,Uug: code_integer] : aa(code_integer,fun($o,char),aa($o,fun(code_integer,fun($o,char)),aa($o,fun($o,fun(code_integer,fun($o,char))),aa($o,fun($o,fun($o,fun(code_integer,fun($o,char)))),aa($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char))))),aa($o,fun($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char)))))),aa($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char))))))),aTP_Lamp_qj($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char)))))))),(Uu)),(Uua)),(Uub)),(Uuc)),(Uud)),(Uue)),(Uuf)),Uug) = char2((Uu),(Uua),(Uub),(Uuc),(Uud),(Uue),(Uuf)) ).

% ATP.lambda_1218
tff(fact_8998_ATP_Olambda__1219,axiom,
    ! [Uu: $o,Uua: $o,Uub: $o,Uuc: $o,Uud: $o,Uue: $o,Uuf: code_integer,Uug: $o] : aa($o,char,aa(code_integer,fun($o,char),aa($o,fun(code_integer,fun($o,char)),aa($o,fun($o,fun(code_integer,fun($o,char))),aa($o,fun($o,fun($o,fun(code_integer,fun($o,char)))),aa($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char))))),aa($o,fun($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char)))))),aTP_Lamp_qk($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char))))))),(Uu)),(Uua)),(Uub)),(Uuc)),(Uud)),(Uue)),Uuf),(Uug)) = aa(product_prod(code_integer,$o),char,aa(fun(code_integer,fun($o,char)),fun(product_prod(code_integer,$o),char),product_case_prod(code_integer,$o,char),aa($o,fun(code_integer,fun($o,char)),aa($o,fun($o,fun(code_integer,fun($o,char))),aa($o,fun($o,fun($o,fun(code_integer,fun($o,char)))),aa($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char))))),aa($o,fun($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char)))))),aa($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char))))))),aTP_Lamp_qj($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char)))))))),(Uu)),(Uua)),(Uub)),(Uuc)),(Uud)),(Uue)),(Uug))),code_bit_cut_integer(Uuf)) ).

% ATP.lambda_1219
tff(fact_8999_ATP_Olambda__1220,axiom,
    ! [B: $tType,A: $tType,Uu: $o,Uua: A,Uub: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_bt($o,fun(A,fun(B,$o)),(Uu)),Uua),Uub)
    <=> (Uu) ) ).

% ATP.lambda_1220
tff(fact_9000_ATP_Olambda__1221,axiom,
    ! [A: $tType,B: $tType,Uu: multiset(B),Uua: A] : aa(A,multiset(B),aTP_Lamp_nm(multiset(B),fun(A,multiset(B)),Uu),Uua) = Uu ).

% ATP.lambda_1221
tff(fact_9001_ATP_Olambda__1222,axiom,
    ! [Uu: $o,Uua: product_unit] :
      ( aa(product_unit,$o,aTP_Lamp_awl($o,fun(product_unit,$o),(Uu)),Uua)
    <=> (Uu) ) ).

% ATP.lambda_1222
tff(fact_9002_ATP_Olambda__1223,axiom,
    ! [A: $tType,Uu: $o,Uua: A] :
      ( aa(A,$o,aTP_Lamp_ah($o,fun(A,$o),(Uu)),Uua)
    <=> (Uu) ) ).

% ATP.lambda_1223
tff(fact_9003_ATP_Olambda__1224,axiom,
    ! [B: $tType,A: $tType,Uu: set(set(old_node(A,B))),Uua: set(old_node(A,B))] : aa(set(old_node(A,B)),set(set(old_node(A,B))),aTP_Lamp_ave(set(set(old_node(A,B))),fun(set(old_node(A,B)),set(set(old_node(A,B)))),Uu),Uua) = Uu ).

% ATP.lambda_1224
tff(fact_9004_ATP_Olambda__1225,axiom,
    ! [C: $tType,D: $tType,Uu: set(D),Uua: C] : aa(C,set(D),aTP_Lamp_rr(set(D),fun(C,set(D)),Uu),Uua) = Uu ).

% ATP.lambda_1225
tff(fact_9005_ATP_Olambda__1226,axiom,
    ! [B: $tType,D: $tType,Uu: set(D),Uua: B] : aa(B,set(D),aTP_Lamp_afz(set(D),fun(B,set(D)),Uu),Uua) = Uu ).

% ATP.lambda_1226
tff(fact_9006_ATP_Olambda__1227,axiom,
    ! [B: $tType,C: $tType,Uu: set(C),Uua: B] : aa(B,set(C),aTP_Lamp_rw(set(C),fun(B,set(C)),Uu),Uua) = Uu ).

% ATP.lambda_1227
tff(fact_9007_ATP_Olambda__1228,axiom,
    ! [A: $tType,C: $tType,Uu: set(C),Uua: A] : aa(A,set(C),aTP_Lamp_us(set(C),fun(A,set(C)),Uu),Uua) = Uu ).

% ATP.lambda_1228
tff(fact_9008_ATP_Olambda__1229,axiom,
    ! [C: $tType,B: $tType,Uu: set(B),Uua: C] : aa(C,set(B),aTP_Lamp_amv(set(B),fun(C,set(B)),Uu),Uua) = Uu ).

% ATP.lambda_1229
tff(fact_9009_ATP_Olambda__1230,axiom,
    ! [B: $tType,Uu: set(B),Uua: B] : aa(B,set(B),aTP_Lamp_rd(set(B),fun(B,set(B)),Uu),Uua) = Uu ).

% ATP.lambda_1230
tff(fact_9010_ATP_Olambda__1231,axiom,
    ! [A: $tType,B: $tType,Uu: set(B),Uua: A] : aa(A,set(B),aTP_Lamp_qz(set(B),fun(A,set(B)),Uu),Uua) = Uu ).

% ATP.lambda_1231
tff(fact_9011_ATP_Olambda__1232,axiom,
    ! [A: $tType,Uu: set(A),Uua: list(A)] : aa(list(A),set(A),aTP_Lamp_xp(set(A),fun(list(A),set(A)),Uu),Uua) = Uu ).

% ATP.lambda_1232
tff(fact_9012_ATP_Olambda__1233,axiom,
    ! [C: $tType,A: $tType,Uu: set(A),Uua: C] : aa(C,set(A),aTP_Lamp_amu(set(A),fun(C,set(A)),Uu),Uua) = Uu ).

% ATP.lambda_1233
tff(fact_9013_ATP_Olambda__1234,axiom,
    ! [B: $tType,A: $tType,Uu: set(A),Uua: B] : aa(B,set(A),aTP_Lamp_ke(set(A),fun(B,set(A)),Uu),Uua) = Uu ).

% ATP.lambda_1234
tff(fact_9014_ATP_Olambda__1235,axiom,
    ! [A: $tType,Uu: set(A),Uua: A] : aa(A,set(A),aTP_Lamp_rl(set(A),fun(A,set(A)),Uu),Uua) = Uu ).

% ATP.lambda_1235
tff(fact_9015_ATP_Olambda__1236,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,$o),Uua: A] : aa(A,fun(B,$o),aTP_Lamp_anw(fun(B,$o),fun(A,fun(B,$o)),Uu),Uua) = Uu ).

% ATP.lambda_1236
tff(fact_9016_ATP_Olambda__1237,axiom,
    ! [A: $tType,C: $tType,B: $tType,Uu: fun(B,C),Uua: A] : aa(A,fun(B,C),aTP_Lamp_vs(fun(B,C),fun(A,fun(B,C)),Uu),Uua) = Uu ).

% ATP.lambda_1237
tff(fact_9017_ATP_Olambda__1238,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,B),Uua: A] : aa(A,fun(B,B),aTP_Lamp_um(fun(B,B),fun(A,fun(B,B)),Uu),Uua) = Uu ).

% ATP.lambda_1238
tff(fact_9018_ATP_Olambda__1239,axiom,
    ! [B: $tType,C: $tType,A: $tType,Uu: fun(A,fun(C,C)),Uua: B] : aa(B,fun(A,fun(C,C)),aTP_Lamp_aee(fun(A,fun(C,C)),fun(B,fun(A,fun(C,C))),Uu),Uua) = Uu ).

% ATP.lambda_1239
tff(fact_9019_ATP_Olambda__1240,axiom,
    ! [B: $tType,C: $tType,Uu: C,Uua: B] : aa(B,C,aTP_Lamp_ato(C,fun(B,C),Uu),Uua) = Uu ).

% ATP.lambda_1240
tff(fact_9020_ATP_Olambda__1241,axiom,
    ! [A: $tType,C: $tType,Uu: C,Uua: A] : aa(A,C,aTP_Lamp_atn(C,fun(A,C),Uu),Uua) = Uu ).

% ATP.lambda_1241
tff(fact_9021_ATP_Olambda__1242,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [Uu: B,Uua: A] : aa(A,B,aTP_Lamp_jx(B,fun(A,B),Uu),Uua) = Uu ) ).

% ATP.lambda_1242
tff(fact_9022_ATP_Olambda__1243,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [Uu: B,Uua: A] : aa(A,B,aTP_Lamp_jy(B,fun(A,B),Uu),Uua) = Uu ) ).

% ATP.lambda_1243
tff(fact_9023_ATP_Olambda__1244,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(B)
     => ! [Uu: B,Uua: A] : aa(A,B,aTP_Lamp_ss(B,fun(A,B),Uu),Uua) = Uu ) ).

% ATP.lambda_1244
tff(fact_9024_ATP_Olambda__1245,axiom,
    ! [C: $tType,B: $tType,Uu: B,Uua: C] : aa(C,B,aTP_Lamp_jm(B,fun(C,B),Uu),Uua) = Uu ).

% ATP.lambda_1245
tff(fact_9025_ATP_Olambda__1246,axiom,
    ! [A: $tType,B: $tType,Uu: B,Uua: A] : aa(A,B,aTP_Lamp_ax(B,fun(A,B),Uu),Uua) = Uu ).

% ATP.lambda_1246
tff(fact_9026_ATP_Olambda__1247,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: A,Uua: B] : aa(B,A,aTP_Lamp_lm(A,fun(B,A),Uu),Uua) = Uu ) ).

% ATP.lambda_1247
tff(fact_9027_ATP_Olambda__1248,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_ahn(A,fun(A,A),Uu),Uua) = Uu ) ).

% ATP.lambda_1248
tff(fact_9028_ATP_Olambda__1249,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: A,Uua: B] : aa(B,A,aTP_Lamp_iz(A,fun(B,A),Uu),Uua) = Uu ) ).

% ATP.lambda_1249
tff(fact_9029_ATP_Olambda__1250,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A,Uua: list(A)] : aa(list(A),A,aa(A,fun(list(A),A),aTP_Lamp_zk(A,fun(list(A),A)),Uu),Uua) = Uu ) ).

% ATP.lambda_1250
tff(fact_9030_ATP_Olambda__1251,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(A)
     => ! [Uu: A,Uua: B] : aa(B,A,aTP_Lamp_ja(A,fun(B,A),Uu),Uua) = Uu ) ).

% ATP.lambda_1251
tff(fact_9031_ATP_Olambda__1252,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] : aa(list(A),A,aa(A,fun(list(A),A),aTP_Lamp_abq(A,fun(list(A),A)),Uu),Uua) = Uu ).

% ATP.lambda_1252
tff(fact_9032_ATP_Olambda__1253,axiom,
    ! [A: $tType,Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_abz(A,fun(nat,A),Uu),Uua) = Uu ).

% ATP.lambda_1253
tff(fact_9033_ATP_Olambda__1254,axiom,
    ! [B: $tType,A: $tType,Uu: A,Uua: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_kf(A,fun(B,A)),Uu),Uua) = Uu ).

% ATP.lambda_1254
tff(fact_9034_ATP_Olambda__1255,axiom,
    ! [B: $tType,A: $tType,Uu: B,Uua: A] : aa(A,A,aa(B,fun(A,A),aTP_Lamp_vq(B,fun(A,A)),Uu),Uua) = Uua ).

% ATP.lambda_1255
tff(fact_9035_ATP_Olambda__1256,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] : aa(list(A),list(A),aa(A,fun(list(A),list(A)),aTP_Lamp_aat(A,fun(list(A),list(A))),Uu),Uua) = Uua ).

% ATP.lambda_1256
tff(fact_9036_ATP_Olambda__1257,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] :
      ( aa(list(A),$o,aa(A,fun(list(A),$o),aTP_Lamp_zc(A,fun(list(A),$o)),Uu),Uua)
    <=> $false ) ).

% ATP.lambda_1257
tff(fact_9037_ATP_Olambda__1258,axiom,
    ! [A: $tType,Uu: A,Uua: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_arv(A,fun(A,$o)),Uu),Uua)
    <=> $false ) ).

% ATP.lambda_1258
tff(fact_9038_ATP_Olambda__1259,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] :
      ( aa(list(A),$o,aa(A,fun(list(A),$o),aTP_Lamp_zb(A,fun(list(A),$o)),Uu),Uua)
    <=> $true ) ).

% ATP.lambda_1259
tff(fact_9039_ATP_Olambda__1260,axiom,
    ! [B: $tType,A: $tType,Uu: A,Uua: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_bx(A,fun(B,$o)),Uu),Uua)
    <=> $true ) ).

% ATP.lambda_1260
tff(fact_9040_ATP_Olambda__1261,axiom,
    ! [A: $tType,Uu: A,Uua: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_xy(A,fun(A,$o)),Uu),Uua)
    <=> $true ) ).

% ATP.lambda_1261
tff(fact_9041_ATP_Olambda__1262,axiom,
    ! [Uu: product_prod(nat,nat)] : aa(product_prod(nat,nat),product_prod(nat,nat),aTP_Lamp_apc(product_prod(nat,nat),product_prod(nat,nat)),Uu) = Uu ).

% ATP.lambda_1262
tff(fact_9042_ATP_Olambda__1263,axiom,
    ! [Uu: list(char)] : aa(list(char),list(char),aTP_Lamp_asx(list(char),list(char)),Uu) = Uu ).

% ATP.lambda_1263
tff(fact_9043_ATP_Olambda__1264,axiom,
    ! [Uu: nat] : aa(nat,nat,aTP_Lamp_dj(nat,nat),Uu) = Uu ).

% ATP.lambda_1264
tff(fact_9044_ATP_Olambda__1265,axiom,
    ! [Uu: int] : aa(int,int,aTP_Lamp_dl(int,int),Uu) = Uu ).

% ATP.lambda_1265
tff(fact_9045_ATP_Olambda__1266,axiom,
    ! [A: $tType,Uu: fun(A,nat)] : aa(fun(A,nat),fun(A,nat),aTP_Lamp_aqy(fun(A,nat),fun(A,nat)),Uu) = Uu ).

% ATP.lambda_1266
tff(fact_9046_ATP_Olambda__1267,axiom,
    ! [B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: B] : aa(B,B,aTP_Lamp_acj(B,B),Uu) = Uu ) ).

% ATP.lambda_1267
tff(fact_9047_ATP_Olambda__1268,axiom,
    ! [B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: B] : aa(B,B,aTP_Lamp_aci(B,B),Uu) = Uu ) ).

% ATP.lambda_1268
tff(fact_9048_ATP_Olambda__1269,axiom,
    ! [B: $tType,Uu: B] : aa(B,B,aTP_Lamp_oc(B,B),Uu) = Uu ).

% ATP.lambda_1269
tff(fact_9049_ATP_Olambda__1270,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_abh(A,A),Uu) = Uu ) ).

% ATP.lambda_1270
tff(fact_9050_ATP_Olambda__1271,axiom,
    ! [A: $tType] :
      ( complete_Sup(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_jt(A,A),Uu) = Uu ) ).

% ATP.lambda_1271
tff(fact_9051_ATP_Olambda__1272,axiom,
    ! [A: $tType] :
      ( complete_Inf(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_jv(A,A),Uu) = Uu ) ).

% ATP.lambda_1272
tff(fact_9052_ATP_Olambda__1273,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_yq(A,A),Uu) = Uu ) ).

% ATP.lambda_1273
tff(fact_9053_ATP_Olambda__1274,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_aq(A,A),Uu) = Uu ) ).

% ATP.lambda_1274
tff(fact_9054_ATP_Olambda__1275,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_avx(A,A),Uu) = Uu ) ).

% ATP.lambda_1275
tff(fact_9055_ATP_Olambda__1276,axiom,
    ! [A: $tType,Uu: A] : aa(A,A,aTP_Lamp_au(A,A),Uu) = Uu ).

% ATP.lambda_1276
tff(fact_9056_ATP_Olambda__1277,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: B] : aa(B,A,aTP_Lamp_kb(B,A),Uu) = top_top(A) ) ).

% ATP.lambda_1277
tff(fact_9057_ATP_Olambda__1278,axiom,
    ! [A: $tType,Uu: A] : aa(A,set($o),aTP_Lamp_ang(A,set($o)),Uu) = top_top(set($o)) ).

% ATP.lambda_1278
tff(fact_9058_ATP_Olambda__1279,axiom,
    ! [A: $tType,B: $tType,Uu: A] : aa(A,set(B),aTP_Lamp_rb(A,set(B)),Uu) = top_top(set(B)) ).

% ATP.lambda_1279
tff(fact_9059_ATP_Olambda__1280,axiom,
    ! [C: $tType,B: $tType,Uu: C] : aa(C,set(B),aTP_Lamp_mw(C,set(B)),Uu) = bot_bot(set(B)) ).

% ATP.lambda_1280
tff(fact_9060_ATP_Olambda__1281,axiom,
    ! [B: $tType,A: $tType,Uu: B] : aa(B,set(A),aTP_Lamp_mo(B,set(A)),Uu) = bot_bot(set(A)) ).

% ATP.lambda_1281
tff(fact_9061_ATP_Olambda__1282,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: B] : aa(B,A,aTP_Lamp_jw(B,A),Uu) = bot_bot(A) ) ).

% ATP.lambda_1282
tff(fact_9062_ATP_Olambda__1283,axiom,
    ! [A: $tType,D: $tType,Uu: A] : aa(A,set(D),aTP_Lamp_mx(A,set(D)),Uu) = bot_bot(set(D)) ).

% ATP.lambda_1283
tff(fact_9063_ATP_Olambda__1284,axiom,
    ! [A: $tType,B: $tType,Uu: A] : aa(A,set(B),aTP_Lamp_ra(A,set(B)),Uu) = bot_bot(set(B)) ).

% ATP.lambda_1284
tff(fact_9064_ATP_Olambda__1285,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: B] : aa(B,A,aTP_Lamp_cu(B,A),Uu) = zero_zero(A) ) ).

% ATP.lambda_1285
tff(fact_9065_ATP_Olambda__1286,axiom,
    ! [B: $tType,A: $tType] :
      ( monoid_add(A)
     => ! [Uu: B] : aa(B,A,aTP_Lamp_abf(B,A),Uu) = zero_zero(A) ) ).

% ATP.lambda_1286
tff(fact_9066_ATP_Olambda__1287,axiom,
    ! [A: $tType] :
      ( mult_zero(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_bb(A,A),Uu) = zero_zero(A) ) ).

% ATP.lambda_1287
tff(fact_9067_ATP_Olambda__1288,axiom,
    ! [A: $tType,Uu: A] : aa(A,nat,aTP_Lamp_apy(A,nat),Uu) = zero_zero(nat) ).

% ATP.lambda_1288
tff(fact_9068_ATP_Olambda__1289,axiom,
    ! [A: $tType,B: $tType] :
      ( zero(B)
     => ! [Uu: A] : aa(A,B,aTP_Lamp_ba(A,B),Uu) = zero_zero(B) ) ).

% ATP.lambda_1289
tff(fact_9069_ATP_Olambda__1290,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: B] : aa(B,A,aTP_Lamp_fb(B,A),Uu) = one_one(A) ) ).

% ATP.lambda_1290
tff(fact_9070_ATP_Olambda__1291,axiom,
    ! [A: $tType,Uu: A] : aa(A,nat,aTP_Lamp_jf(A,nat),Uu) = one_one(nat) ).

% ATP.lambda_1291
tff(fact_9071_ATP_Olambda__1292,axiom,
    ! [A: $tType,Uu: product_unit] : aa(product_unit,seq(A),aTP_Lamp_axq(product_unit,seq(A)),Uu) = empty(A) ).

% ATP.lambda_1292
tff(fact_9072_ATP_Olambda__1293,axiom,
    ! [C: $tType,B: $tType,Uu: C] : aa(C,option(B),aTP_Lamp_aot(C,option(B)),Uu) = none(B) ).

% ATP.lambda_1293
tff(fact_9073_ATP_Olambda__1294,axiom,
    ! [B: $tType,A: $tType,Uu: B] : aa(B,option(A),aTP_Lamp_xs(B,option(A)),Uu) = none(A) ).

% ATP.lambda_1294
tff(fact_9074_ATP_Olambda__1295,axiom,
    ! [A: $tType,C: $tType,Uu: A] : aa(A,option(C),aTP_Lamp_tp(A,option(C)),Uu) = none(C) ).

% ATP.lambda_1295
tff(fact_9075_ATP_Olambda__1296,axiom,
    ! [A: $tType,B: $tType,Uu: A] : aa(A,option(B),aTP_Lamp_tf(A,option(B)),Uu) = none(B) ).

% ATP.lambda_1296
tff(fact_9076_ATP_Olambda__1297,axiom,
    ! [B: $tType,A: $tType,Uu: B] : aa(B,A,aTP_Lamp_avr(B,A),Uu) = undefined(A) ).

% ATP.lambda_1297
tff(fact_9077_ATP_Olambda__1298,axiom,
    ! [A: $tType,B: $tType,Uu: A] : aa(A,B,aTP_Lamp_adi(A,B),Uu) = undefined(B) ).

% ATP.lambda_1298
tff(fact_9078_ATP_Olambda__1299,axiom,
    ! [Uu: product_prod(heap_ext(product_unit),set(nat))] :
      ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aTP_Lamp_ad(product_prod(heap_ext(product_unit),set(nat)),$o),Uu)
    <=> $false ) ).

% ATP.lambda_1299
tff(fact_9079_ATP_Olambda__1300,axiom,
    ! [Uu: nat] :
      ( aa(nat,$o,aTP_Lamp_ej(nat,$o),Uu)
    <=> $false ) ).

% ATP.lambda_1300
tff(fact_9080_ATP_Olambda__1301,axiom,
    ! [B: $tType,Uu: B] :
      ( aa(B,$o,aTP_Lamp_axa(B,$o),Uu)
    <=> $false ) ).

% ATP.lambda_1301
tff(fact_9081_ATP_Olambda__1302,axiom,
    ! [A: $tType,Uu: A] :
      ( aa(A,$o,aTP_Lamp_ak(A,$o),Uu)
    <=> $false ) ).

% ATP.lambda_1302
tff(fact_9082_ATP_Olambda__1303,axiom,
    ! [Uu: nat] :
      ( aa(nat,$o,aTP_Lamp_ei(nat,$o),Uu)
    <=> $true ) ).

% ATP.lambda_1303
tff(fact_9083_ATP_Olambda__1304,axiom,
    ! [A: $tType,Uu: fun(A,$o)] :
      ( aa(fun(A,$o),$o,aTP_Lamp_akw(fun(A,$o),$o),Uu)
    <=> $true ) ).

% ATP.lambda_1304
tff(fact_9084_ATP_Olambda__1305,axiom,
    ! [C: $tType,Uu: C] :
      ( aa(C,$o,aTP_Lamp_arj(C,$o),Uu)
    <=> $true ) ).

% ATP.lambda_1305
tff(fact_9085_ATP_Olambda__1306,axiom,
    ! [B: $tType,Uu: B] :
      ( aa(B,$o,aTP_Lamp_ari(B,$o),Uu)
    <=> $true ) ).

% ATP.lambda_1306
tff(fact_9086_ATP_Olambda__1307,axiom,
    ! [A: $tType,Uu: A] :
      ( aa(A,$o,aTP_Lamp_ar(A,$o),Uu)
    <=> $true ) ).

% ATP.lambda_1307
tff(fact_9087_ATP_Olambda__1308,axiom,
    ! [A: $tType,Uu: A] : aa(A,fun(nat,nat),aTP_Lamp_oo(A,fun(nat,nat)),Uu) = suc ).

% ATP.lambda_1308

% Type constructors (836)
tff(tcon_Code__Numeral_Onatural___Code__Evaluation_Oterm__of,axiom,
    code_term_of(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Typerep_Otyperep,axiom,
    typerep(code_natural) ).

tff(tcon_Code__Numeral_Ointeger___Code__Evaluation_Oterm__of_1,axiom,
    code_term_of(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Typerep_Otyperep_2,axiom,
    typerep(code_integer) ).

tff(tcon_Code__Evaluation_Oterm___Code__Evaluation_Oterm__of_3,axiom,
    code_term_of(code_term) ).

tff(tcon_Code__Evaluation_Oterm___Typerep_Otyperep_4,axiom,
    typerep(code_term) ).

tff(tcon_Heap_Oheap_Oheap__ext___Code__Evaluation_Oterm__of_5,axiom,
    ! [A18: $tType] :
      ( typerep(A18)
     => code_term_of(heap_ext(A18)) ) ).

tff(tcon_Heap_Oheap_Oheap__ext___Typerep_Otyperep_6,axiom,
    ! [A18: $tType] :
      ( typerep(A18)
     => typerep(heap_ext(A18)) ) ).

tff(tcon_Product__Type_Ounit___Code__Evaluation_Oterm__of_7,axiom,
    code_term_of(product_unit) ).

tff(tcon_Product__Type_Ounit___Enum_Oenum,axiom,
    enum(product_unit) ).

tff(tcon_Product__Type_Ounit___Typerep_Otyperep_8,axiom,
    typerep(product_unit) ).

tff(tcon_Product__Type_Oprod___Code__Evaluation_Oterm__of_9,axiom,
    ! [A18: $tType,A19: $tType] :
      ( ( typerep(A18)
        & typerep(A19) )
     => code_term_of(product_prod(A18,A19)) ) ).

tff(tcon_Product__Type_Oprod___Enum_Oenum_10,axiom,
    ! [A18: $tType,A19: $tType] :
      ( ( enum(A18)
        & enum(A19) )
     => enum(product_prod(A18,A19)) ) ).

tff(tcon_Product__Type_Oprod___Typerep_Otyperep_11,axiom,
    ! [A18: $tType,A19: $tType] :
      ( ( typerep(A18)
        & typerep(A19) )
     => typerep(product_prod(A18,A19)) ) ).

tff(tcon_Old__Datatype_Onode___Typerep_Otyperep_12,axiom,
    ! [A18: $tType,A19: $tType] :
      ( ( typerep(A18)
        & typerep(A19) )
     => typerep(old_node(A18,A19)) ) ).

tff(tcon_Multiset_Omultiset___Code__Evaluation_Oterm__of_13,axiom,
    ! [A18: $tType] :
      ( typerep(A18)
     => code_term_of(multiset(A18)) ) ).

tff(tcon_Multiset_Omultiset___Typerep_Otyperep_14,axiom,
    ! [A18: $tType] :
      ( typerep(A18)
     => typerep(multiset(A18)) ) ).

tff(tcon_Assertions_Oassn___Typerep_Otyperep_15,axiom,
    typerep(assn) ).

tff(tcon_String_Oliteral___Code__Evaluation_Oterm__of_16,axiom,
    code_term_of(literal) ).

tff(tcon_String_Oliteral___Typerep_Otyperep_17,axiom,
    typerep(literal) ).

tff(tcon_Predicate_Opred___Code__Evaluation_Oterm__of_18,axiom,
    ! [A18: $tType] :
      ( typerep(A18)
     => code_term_of(pred(A18)) ) ).

tff(tcon_Predicate_Opred___Typerep_Otyperep_19,axiom,
    ! [A18: $tType] :
      ( typerep(A18)
     => typerep(pred(A18)) ) ).

tff(tcon_Predicate_Oseq___Code__Evaluation_Oterm__of_20,axiom,
    ! [A18: $tType] :
      ( typerep(A18)
     => code_term_of(seq(A18)) ) ).

tff(tcon_Predicate_Oseq___Typerep_Otyperep_21,axiom,
    ! [A18: $tType] :
      ( typerep(A18)
     => typerep(seq(A18)) ) ).

tff(tcon_Option_Ooption___Code__Evaluation_Oterm__of_22,axiom,
    ! [A18: $tType] :
      ( typerep(A18)
     => code_term_of(option(A18)) ) ).

tff(tcon_Option_Ooption___Enum_Oenum_23,axiom,
    ! [A18: $tType] :
      ( enum(A18)
     => enum(option(A18)) ) ).

tff(tcon_Option_Ooption___Typerep_Otyperep_24,axiom,
    ! [A18: $tType] :
      ( typerep(A18)
     => typerep(option(A18)) ) ).

tff(tcon_Filter_Ofilter___Code__Evaluation_Oterm__of_25,axiom,
    ! [A18: $tType] :
      ( typerep(A18)
     => code_term_of(filter(A18)) ) ).

tff(tcon_Filter_Ofilter___Typerep_Otyperep_26,axiom,
    ! [A18: $tType] :
      ( typerep(A18)
     => typerep(filter(A18)) ) ).

tff(tcon_Sum__Type_Osum___Code__Evaluation_Oterm__of_27,axiom,
    ! [A18: $tType,A19: $tType] :
      ( ( typerep(A18)
        & typerep(A19) )
     => code_term_of(sum_sum(A18,A19)) ) ).

tff(tcon_Sum__Type_Osum___Enum_Oenum_28,axiom,
    ! [A18: $tType,A19: $tType] :
      ( ( enum(A18)
        & enum(A19) )
     => enum(sum_sum(A18,A19)) ) ).

tff(tcon_Sum__Type_Osum___Typerep_Otyperep_29,axiom,
    ! [A18: $tType,A19: $tType] :
      ( ( typerep(A18)
        & typerep(A19) )
     => typerep(sum_sum(A18,A19)) ) ).

tff(tcon_String_Ochar___Code__Evaluation_Oterm__of_30,axiom,
    code_term_of(char) ).

tff(tcon_String_Ochar___Enum_Oenum_31,axiom,
    enum(char) ).

tff(tcon_String_Ochar___Typerep_Otyperep_32,axiom,
    typerep(char) ).

tff(tcon_List_Olist___Code__Evaluation_Oterm__of_33,axiom,
    ! [A18: $tType] :
      ( typerep(A18)
     => code_term_of(list(A18)) ) ).

tff(tcon_List_Olist___Typerep_Otyperep_34,axiom,
    ! [A18: $tType] :
      ( typerep(A18)
     => typerep(list(A18)) ) ).

tff(tcon_HOL_Obool___Code__Evaluation_Oterm__of_35,axiom,
    code_term_of($o) ).

tff(tcon_HOL_Obool___Enum_Oenum_36,axiom,
    enum($o) ).

tff(tcon_HOL_Obool___Typerep_Otyperep_37,axiom,
    typerep($o) ).

tff(tcon_Set_Oset___Code__Evaluation_Oterm__of_38,axiom,
    ! [A18: $tType] :
      ( typerep(A18)
     => code_term_of(set(A18)) ) ).

tff(tcon_Set_Oset___Enum_Oenum_39,axiom,
    ! [A18: $tType] :
      ( enum(A18)
     => enum(set(A18)) ) ).

tff(tcon_Set_Oset___Typerep_Otyperep_40,axiom,
    ! [A18: $tType] :
      ( typerep(A18)
     => typerep(set(A18)) ) ).

tff(tcon_Rat_Orat___Code__Evaluation_Oterm__of_41,axiom,
    code_term_of(rat) ).

tff(tcon_Rat_Orat___Typerep_Otyperep_42,axiom,
    typerep(rat) ).

tff(tcon_Num_Onum___Code__Evaluation_Oterm__of_43,axiom,
    code_term_of(num) ).

tff(tcon_Num_Onum___Typerep_Otyperep_44,axiom,
    typerep(num) ).

tff(tcon_Nat_Onat___Code__Evaluation_Oterm__of_45,axiom,
    code_term_of(nat) ).

tff(tcon_Nat_Onat___Typerep_Otyperep_46,axiom,
    typerep(nat) ).

tff(tcon_Int_Oint___Code__Evaluation_Oterm__of_47,axiom,
    code_term_of(int) ).

tff(tcon_Int_Oint___Typerep_Otyperep_48,axiom,
    typerep(int) ).

tff(tcon_itself___Code__Evaluation_Oterm__of_49,axiom,
    ! [A18: $tType] :
      ( typerep(A18)
     => code_term_of(itself(A18)) ) ).

tff(tcon_itself___Typerep_Otyperep_50,axiom,
    ! [A18: $tType] :
      ( typerep(A18)
     => typerep(itself(A18)) ) ).

tff(tcon_fun___Code__Evaluation_Oterm__of_51,axiom,
    ! [A18: $tType,A19: $tType] :
      ( ( typerep(A18)
        & typerep(A19) )
     => code_term_of(fun(A18,A19)) ) ).

tff(tcon_fun___Enum_Oenum_52,axiom,
    ! [A18: $tType,A19: $tType] :
      ( ( enum(A18)
        & enum(A19) )
     => enum(fun(A18,A19)) ) ).

tff(tcon_fun___Typerep_Otyperep_53,axiom,
    ! [A18: $tType,A19: $tType] :
      ( ( typerep(A18)
        & typerep(A19) )
     => typerep(fun(A18,A19)) ) ).

tff(tcon_fun___Conditionally__Complete__Lattices_Oconditionally__complete__lattice,axiom,
    ! [A18: $tType,A19: $tType] :
      ( comple6319245703460814977attice(A19)
     => condit1219197933456340205attice(fun(A18,A19)) ) ).

tff(tcon_fun___Complete__Lattices_Ocomplete__distrib__lattice,axiom,
    ! [A18: $tType,A19: $tType] :
      ( comple592849572758109894attice(A19)
     => comple592849572758109894attice(fun(A18,A19)) ) ).

tff(tcon_fun___Complete__Lattices_Ocomplete__boolean__algebra,axiom,
    ! [A18: $tType,A19: $tType] :
      ( comple489889107523837845lgebra(A19)
     => comple489889107523837845lgebra(fun(A18,A19)) ) ).

tff(tcon_fun___Quickcheck__Exhaustive_Ofull__exhaustive,axiom,
    ! [A18: $tType,A19: $tType] :
      ( ( cl_HOL_Oequal(A18)
        & quickc3360725361186068524ustive(A18)
        & quickc3360725361186068524ustive(A19) )
     => quickc3360725361186068524ustive(fun(A18,A19)) ) ).

tff(tcon_fun___Lattices_Obounded__semilattice__sup__bot,axiom,
    ! [A18: $tType,A19: $tType] :
      ( bounded_lattice(A19)
     => bounde4967611905675639751up_bot(fun(A18,A19)) ) ).

tff(tcon_fun___Lattices_Obounded__semilattice__inf__top,axiom,
    ! [A18: $tType,A19: $tType] :
      ( bounded_lattice(A19)
     => bounde4346867609351753570nf_top(fun(A18,A19)) ) ).

tff(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
    ! [A18: $tType,A19: $tType] :
      ( comple6319245703460814977attice(A19)
     => comple6319245703460814977attice(fun(A18,A19)) ) ).

tff(tcon_fun___Quickcheck__Exhaustive_Oexhaustive,axiom,
    ! [A18: $tType,A19: $tType] :
      ( ( cl_HOL_Oequal(A18)
        & quickc658316121487927005ustive(A18)
        & quickc658316121487927005ustive(A19) )
     => quickc658316121487927005ustive(fun(A18,A19)) ) ).

tff(tcon_fun___Boolean__Algebras_Oboolean__algebra,axiom,
    ! [A18: $tType,A19: $tType] :
      ( boolea8198339166811842893lgebra(A19)
     => boolea8198339166811842893lgebra(fun(A18,A19)) ) ).

tff(tcon_fun___Lattices_Obounded__lattice__top,axiom,
    ! [A18: $tType,A19: $tType] :
      ( bounded_lattice(A19)
     => bounded_lattice_top(fun(A18,A19)) ) ).

tff(tcon_fun___Lattices_Obounded__lattice__bot,axiom,
    ! [A18: $tType,A19: $tType] :
      ( bounded_lattice(A19)
     => bounded_lattice_bot(fun(A18,A19)) ) ).

tff(tcon_fun___Complete__Partial__Order_Occpo,axiom,
    ! [A18: $tType,A19: $tType] :
      ( comple6319245703460814977attice(A19)
     => comple9053668089753744459l_ccpo(fun(A18,A19)) ) ).

tff(tcon_fun___Quickcheck__Random_Orandom,axiom,
    ! [A18: $tType,A19: $tType] :
      ( ( code_term_of(A18)
        & cl_HOL_Oequal(A18)
        & quickcheck_random(A19) )
     => quickcheck_random(fun(A18,A19)) ) ).

tff(tcon_fun___Lattices_Osemilattice__sup,axiom,
    ! [A18: $tType,A19: $tType] :
      ( semilattice_sup(A19)
     => semilattice_sup(fun(A18,A19)) ) ).

tff(tcon_fun___Lattices_Osemilattice__inf,axiom,
    ! [A18: $tType,A19: $tType] :
      ( semilattice_inf(A19)
     => semilattice_inf(fun(A18,A19)) ) ).

tff(tcon_fun___Lattices_Odistrib__lattice,axiom,
    ! [A18: $tType,A19: $tType] :
      ( distrib_lattice(A19)
     => distrib_lattice(fun(A18,A19)) ) ).

tff(tcon_fun___Lattices_Obounded__lattice,axiom,
    ! [A18: $tType,A19: $tType] :
      ( bounded_lattice(A19)
     => bounded_lattice(fun(A18,A19)) ) ).

tff(tcon_fun___Complete__Lattices_OSup,axiom,
    ! [A18: $tType,A19: $tType] :
      ( complete_Sup(A19)
     => complete_Sup(fun(A18,A19)) ) ).

tff(tcon_fun___Complete__Lattices_OInf,axiom,
    ! [A18: $tType,A19: $tType] :
      ( complete_Inf(A19)
     => complete_Inf(fun(A18,A19)) ) ).

tff(tcon_fun___Orderings_Oorder__top,axiom,
    ! [A18: $tType,A19: $tType] :
      ( order_top(A19)
     => order_top(fun(A18,A19)) ) ).

tff(tcon_fun___Orderings_Oorder__bot,axiom,
    ! [A18: $tType,A19: $tType] :
      ( order_bot(A19)
     => order_bot(fun(A18,A19)) ) ).

tff(tcon_fun___Countable_Ocountable,axiom,
    ! [A18: $tType,A19: $tType] :
      ( ( finite_finite(A18)
        & countable(A19) )
     => countable(fun(A18,A19)) ) ).

tff(tcon_fun___Orderings_Opreorder,axiom,
    ! [A18: $tType,A19: $tType] :
      ( preorder(A19)
     => preorder(fun(A18,A19)) ) ).

tff(tcon_fun___Finite__Set_Ofinite,axiom,
    ! [A18: $tType,A19: $tType] :
      ( ( finite_finite(A18)
        & finite_finite(A19) )
     => finite_finite(fun(A18,A19)) ) ).

tff(tcon_fun___Lattices_Olattice,axiom,
    ! [A18: $tType,A19: $tType] :
      ( lattice(A19)
     => lattice(fun(A18,A19)) ) ).

tff(tcon_fun___Orderings_Oorder,axiom,
    ! [A18: $tType,A19: $tType] :
      ( order(A19)
     => order(fun(A18,A19)) ) ).

tff(tcon_fun___Orderings_Otop,axiom,
    ! [A18: $tType,A19: $tType] :
      ( top(A19)
     => top(fun(A18,A19)) ) ).

tff(tcon_fun___Orderings_Oord,axiom,
    ! [A18: $tType,A19: $tType] :
      ( ord(A19)
     => ord(fun(A18,A19)) ) ).

tff(tcon_fun___Orderings_Obot,axiom,
    ! [A18: $tType,A19: $tType] :
      ( bot(A19)
     => bot(fun(A18,A19)) ) ).

tff(tcon_fun___Groups_Ouminus,axiom,
    ! [A18: $tType,A19: $tType] :
      ( uminus(A19)
     => uminus(fun(A18,A19)) ) ).

tff(tcon_fun___Lattices_Osup,axiom,
    ! [A18: $tType,A19: $tType] :
      ( semilattice_sup(A19)
     => sup(fun(A18,A19)) ) ).

tff(tcon_fun___Lattices_Oinf,axiom,
    ! [A18: $tType,A19: $tType] :
      ( semilattice_inf(A19)
     => inf(fun(A18,A19)) ) ).

tff(tcon_fun___Groups_Ominus,axiom,
    ! [A18: $tType,A19: $tType] :
      ( minus(A19)
     => minus(fun(A18,A19)) ) ).

tff(tcon_fun___HOL_Oequal,axiom,
    ! [A18: $tType,A19: $tType] :
      ( ( enum(A18)
        & cl_HOL_Oequal(A19) )
     => cl_HOL_Oequal(fun(A18,A19)) ) ).

tff(tcon_itself___Quickcheck__Random_Orandom_54,axiom,
    ! [A18: $tType] :
      ( typerep(A18)
     => quickcheck_random(itself(A18)) ) ).

tff(tcon_itself___HOL_Oequal_55,axiom,
    ! [A18: $tType] : cl_HOL_Oequal(itself(A18)) ).

tff(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
    condit6923001295902523014norder(int) ).

tff(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_56,axiom,
    condit1219197933456340205attice(int) ).

tff(tcon_Int_Oint___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations,axiom,
    bit_un5681908812861735899ations(int) ).

tff(tcon_Int_Oint___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,axiom,
    semiri1453513574482234551roduct(int) ).

tff(tcon_Int_Oint___Euclidean__Division_Ounique__euclidean__semiring__with__nat,axiom,
    euclid5411537665997757685th_nat(int) ).

tff(tcon_Int_Oint___Euclidean__Division_Ounique__euclidean__ring__with__nat,axiom,
    euclid8789492081693882211th_nat(int) ).

tff(tcon_Int_Oint___Groups_Oordered__ab__semigroup__monoid__add__imp__le,axiom,
    ordere1937475149494474687imp_le(int) ).

tff(tcon_Int_Oint___Euclidean__Division_Ounique__euclidean__semiring,axiom,
    euclid3128863361964157862miring(int) ).

tff(tcon_Int_Oint___Euclidean__Division_Oeuclidean__semiring__cancel,axiom,
    euclid4440199948858584721cancel(int) ).

tff(tcon_Int_Oint___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___Quickcheck__Exhaustive_Ofull__exhaustive_57,axiom,
    quickc3360725361186068524ustive(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___Rings_Olinordered__semiring__1__strict,axiom,
    linord715952674999750819strict(int) ).

tff(tcon_Int_Oint___Groups_Olinordered__ab__semigroup__add,axiom,
    linord4140545234300271783up_add(int) ).

tff(tcon_Int_Oint___Bit__Operations_Oring__bit__operations,axiom,
    bit_ri3973907225187159222ations(int) ).

tff(tcon_Int_Oint___Rings_Olinordered__nonzero__semiring,axiom,
    linord181362715937106298miring(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___Quickcheck__Exhaustive_Oexhaustive_58,axiom,
    quickc658316121487927005ustive(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___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___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___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___Quickcheck__Random_Orandom_59,axiom,
    quickcheck_random(int) ).

tff(tcon_Int_Oint___Lattices_Osemilattice__sup_60,axiom,
    semilattice_sup(int) ).

tff(tcon_Int_Oint___Lattices_Osemilattice__inf_61,axiom,
    semilattice_inf(int) ).

tff(tcon_Int_Oint___Lattices_Odistrib__lattice_62,axiom,
    distrib_lattice(int) ).

tff(tcon_Int_Oint___Groups_Oab__semigroup__mult,axiom,
    ab_semigroup_mult(int) ).

tff(tcon_Int_Oint___Rings_Oalgebraic__semidom,axiom,
    algebraic_semidom(int) ).

tff(tcon_Int_Oint___Groups_Ocomm__monoid__mult,axiom,
    comm_monoid_mult(int) ).

tff(tcon_Int_Oint___Groups_Oab__semigroup__add,axiom,
    ab_semigroup_add(int) ).

tff(tcon_Int_Oint___Rings_Oordered__semiring,axiom,
    ordered_semiring(int) ).

tff(tcon_Int_Oint___Rings_Oordered__ring__abs,axiom,
    ordered_ring_abs(int) ).

tff(tcon_Int_Oint___Parity_Osemiring__parity,axiom,
    semiring_parity(int) ).

tff(tcon_Int_Oint___Groups_Ocomm__monoid__add,axiom,
    comm_monoid_add(int) ).

tff(tcon_Int_Oint___Rings_Osemiring__modulo,axiom,
    semiring_modulo(int) ).

tff(tcon_Int_Oint___Rings_Olinordered__ring,axiom,
    linordered_ring(int) ).

tff(tcon_Int_Oint___Rings_Olinordered__idom,axiom,
    linordered_idom(int) ).

tff(tcon_Int_Oint___Rings_Ocomm__semiring__1,axiom,
    comm_semiring_1(int) ).

tff(tcon_Int_Oint___Rings_Ocomm__semiring__0,axiom,
    comm_semiring_0(int) ).

tff(tcon_Int_Oint___Groups_Osemigroup__mult,axiom,
    semigroup_mult(int) ).

tff(tcon_Int_Oint___Complete__Lattices_OSup_63,axiom,
    complete_Sup(int) ).

tff(tcon_Int_Oint___Complete__Lattices_OInf_64,axiom,
    complete_Inf(int) ).

tff(tcon_Int_Oint___Rings_Osemidom__modulo,axiom,
    semidom_modulo(int) ).

tff(tcon_Int_Oint___Rings_Osemidom__divide,axiom,
    semidom_divide(int) ).

tff(tcon_Int_Oint___Num_Osemiring__numeral,axiom,
    semiring_numeral(int) ).

tff(tcon_Int_Oint___Groups_Osemigroup__add,axiom,
    semigroup_add(int) ).

tff(tcon_Int_Oint___Rings_Ozero__less__one,axiom,
    zero_less_one(int) ).

tff(tcon_Int_Oint___Rings_Ocomm__semiring,axiom,
    comm_semiring(int) ).

tff(tcon_Int_Oint___Nat_Osemiring__char__0,axiom,
    semiring_char_0(int) ).

tff(tcon_Int_Oint___Groups_Oab__group__add,axiom,
    ab_group_add(int) ).

tff(tcon_Int_Oint___Countable_Ocountable_65,axiom,
    countable(int) ).

tff(tcon_Int_Oint___Rings_Ozero__neq__one,axiom,
    zero_neq_one(int) ).

tff(tcon_Int_Oint___Rings_Oordered__ring,axiom,
    ordered_ring(int) ).

tff(tcon_Int_Oint___Rings_Oidom__abs__sgn,axiom,
    idom_abs_sgn(int) ).

tff(tcon_Int_Oint___Parity_Oring__parity,axiom,
    ring_parity(int) ).

tff(tcon_Int_Oint___Orderings_Opreorder_66,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__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_67,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_68,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___Orderings_Oord_69,axiom,
    ord(int) ).

tff(tcon_Int_Oint___Groups_Ouminus_70,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___Lattices_Osup_71,axiom,
    sup(int) ).

tff(tcon_Int_Oint___Lattices_Oinf_72,axiom,
    inf(int) ).

tff(tcon_Int_Oint___Groups_Otimes,axiom,
    times(int) ).

tff(tcon_Int_Oint___Groups_Ominus_73,axiom,
    minus(int) ).

tff(tcon_Int_Oint___GCD_Oring__gcd,axiom,
    ring_gcd(int) ).

tff(tcon_Int_Oint___Power_Opower,axiom,
    power(int) ).

tff(tcon_Int_Oint___Num_Onumeral,axiom,
    numeral(int) ).

tff(tcon_Int_Oint___Groups_Ozero,axiom,
    zero(int) ).

tff(tcon_Int_Oint___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_Int_Oint___HOL_Oequal_74,axiom,
    cl_HOL_Oequal(int) ).

tff(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_75,axiom,
    condit6923001295902523014norder(nat) ).

tff(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_76,axiom,
    condit1219197933456340205attice(nat) ).

tff(tcon_Nat_Onat___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_77,axiom,
    bit_un5681908812861735899ations(nat) ).

tff(tcon_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_78,axiom,
    semiri1453513574482234551roduct(nat) ).

tff(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring__with__nat_79,axiom,
    euclid5411537665997757685th_nat(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_80,axiom,
    ordere1937475149494474687imp_le(nat) ).

tff(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring_81,axiom,
    euclid3128863361964157862miring(nat) ).

tff(tcon_Nat_Onat___Euclidean__Division_Oeuclidean__semiring__cancel_82,axiom,
    euclid4440199948858584721cancel(nat) ).

tff(tcon_Nat_Onat___Rings_Onormalization__semidom__multiplicative_83,axiom,
    normal6328177297339901930cative(nat) ).

tff(tcon_Nat_Onat___Divides_Ounique__euclidean__semiring__numeral_84,axiom,
    unique1627219031080169319umeral(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors__cancel_85,axiom,
    semiri6575147826004484403cancel(nat) ).

tff(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add_86,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_87,axiom,
    ordere580206878836729694up_add(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le_88,axiom,
    ordere2412721322843649153imp_le(nat) ).

tff(tcon_Nat_Onat___Bit__Operations_Osemiring__bit__operations_89,axiom,
    bit_se359711467146920520ations(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__comm__semiring__strict_90,axiom,
    linord2810124833399127020strict(nat) ).

tff(tcon_Nat_Onat___Quickcheck__Exhaustive_Ofull__exhaustive_91,axiom,
    quickc3360725361186068524ustive(nat) ).

tff(tcon_Nat_Onat___Groups_Ostrict__ordered__comm__monoid__add_92,axiom,
    strict7427464778891057005id_add(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__cancel__comm__monoid__add_93,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_94,axiom,
    euclid3725896446679973847miring(nat) ).

tff(tcon_Nat_Onat___Groups_Olinordered__ab__semigroup__add_95,axiom,
    linord4140545234300271783up_add(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring_96,axiom,
    linord181362715937106298miring(nat) ).

tff(tcon_Nat_Onat___Rings_Osemidom__divide__unit__factor_97,axiom,
    semido2269285787275462019factor(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__semiring__strict_98,axiom,
    linord8928482502909563296strict(nat) ).

tff(tcon_Nat_Onat___Quickcheck__Exhaustive_Oexhaustive_99,axiom,
    quickc658316121487927005ustive(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors_100,axiom,
    semiri3467727345109120633visors(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add_101,axiom,
    ordere6658533253407199908up_add(nat) ).

tff(tcon_Nat_Onat___GCD_Osemiring__gcd__mult__normalize_102,axiom,
    semiri6843258321239162965malize(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__comm__monoid__add_103,axiom,
    ordere6911136660526730532id_add(nat) ).

tff(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add_104,axiom,
    cancel2418104881723323429up_add(nat) ).

tff(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add_105,axiom,
    cancel1802427076303600483id_add(nat) ).

tff(tcon_Nat_Onat___Rings_Ocomm__semiring__1__cancel_106,axiom,
    comm_s4317794764714335236cancel(nat) ).

tff(tcon_Nat_Onat___Bit__Operations_Osemiring__bits_107,axiom,
    bit_semiring_bits(nat) ).

tff(tcon_Nat_Onat___Rings_Oordered__comm__semiring_108,axiom,
    ordere2520102378445227354miring(nat) ).

tff(tcon_Nat_Onat___Rings_Onormalization__semidom_109,axiom,
    normal8620421768224518004emidom(nat) ).

tff(tcon_Nat_Onat___Groups_Ocancel__semigroup__add_110,axiom,
    cancel_semigroup_add(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__semiring_111,axiom,
    linordered_semiring(nat) ).

tff(tcon_Nat_Onat___Rings_Oordered__semiring__0_112,axiom,
    ordered_semiring_0(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__semidom_113,axiom,
    linordered_semidom(nat) ).

tff(tcon_Nat_Onat___Quickcheck__Random_Orandom_114,axiom,
    quickcheck_random(nat) ).

tff(tcon_Nat_Onat___Lattices_Osemilattice__sup_115,axiom,
    semilattice_sup(nat) ).

tff(tcon_Nat_Onat___Lattices_Osemilattice__inf_116,axiom,
    semilattice_inf(nat) ).

tff(tcon_Nat_Onat___Lattices_Odistrib__lattice_117,axiom,
    distrib_lattice(nat) ).

tff(tcon_Nat_Onat___Groups_Oab__semigroup__mult_118,axiom,
    ab_semigroup_mult(nat) ).

tff(tcon_Nat_Onat___Rings_Oalgebraic__semidom_119,axiom,
    algebraic_semidom(nat) ).

tff(tcon_Nat_Onat___Groups_Ocomm__monoid__mult_120,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_121,axiom,
    ab_semigroup_add(nat) ).

tff(tcon_Nat_Onat___Rings_Oordered__semiring_122,axiom,
    ordered_semiring(nat) ).

tff(tcon_Nat_Onat___Parity_Osemiring__parity_123,axiom,
    semiring_parity(nat) ).

tff(tcon_Nat_Onat___Groups_Ocomm__monoid__add_124,axiom,
    comm_monoid_add(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__modulo_125,axiom,
    semiring_modulo(nat) ).

tff(tcon_Nat_Onat___Rings_Ocomm__semiring__1_126,axiom,
    comm_semiring_1(nat) ).

tff(tcon_Nat_Onat___Rings_Ocomm__semiring__0_127,axiom,
    comm_semiring_0(nat) ).

tff(tcon_Nat_Onat___Groups_Osemigroup__mult_128,axiom,
    semigroup_mult(nat) ).

tff(tcon_Nat_Onat___Complete__Lattices_OSup_129,axiom,
    complete_Sup(nat) ).

tff(tcon_Nat_Onat___Complete__Lattices_OInf_130,axiom,
    complete_Inf(nat) ).

tff(tcon_Nat_Onat___Rings_Osemidom__modulo_131,axiom,
    semidom_modulo(nat) ).

tff(tcon_Nat_Onat___Rings_Osemidom__divide_132,axiom,
    semidom_divide(nat) ).

tff(tcon_Nat_Onat___Num_Osemiring__numeral_133,axiom,
    semiring_numeral(nat) ).

tff(tcon_Nat_Onat___Groups_Osemigroup__add_134,axiom,
    semigroup_add(nat) ).

tff(tcon_Nat_Onat___Rings_Ozero__less__one_135,axiom,
    zero_less_one(nat) ).

tff(tcon_Nat_Onat___Rings_Ocomm__semiring_136,axiom,
    comm_semiring(nat) ).

tff(tcon_Nat_Onat___Orderings_Owellorder,axiom,
    wellorder(nat) ).

tff(tcon_Nat_Onat___Orderings_Oorder__bot_137,axiom,
    order_bot(nat) ).

tff(tcon_Nat_Onat___Nat_Osemiring__char__0_138,axiom,
    semiring_char_0(nat) ).

tff(tcon_Nat_Onat___Countable_Ocountable_139,axiom,
    countable(nat) ).

tff(tcon_Nat_Onat___Rings_Ozero__neq__one_140,axiom,
    zero_neq_one(nat) ).

tff(tcon_Nat_Onat___Orderings_Opreorder_141,axiom,
    preorder(nat) ).

tff(tcon_Nat_Onat___Orderings_Olinorder_142,axiom,
    linorder(nat) ).

tff(tcon_Nat_Onat___Groups_Omonoid__mult_143,axiom,
    monoid_mult(nat) ).

tff(tcon_Nat_Onat___Groups_Omonoid__add_144,axiom,
    monoid_add(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__1_145,axiom,
    semiring_1(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__0_146,axiom,
    semiring_0(nat) ).

tff(tcon_Nat_Onat___Orderings_Ono__top_147,axiom,
    no_top(nat) ).

tff(tcon_Nat_Onat___Lattices_Olattice_148,axiom,
    lattice(nat) ).

tff(tcon_Nat_Onat___GCD_Osemiring__gcd_149,axiom,
    semiring_gcd(nat) ).

tff(tcon_Nat_Onat___GCD_Osemiring__Gcd_150,axiom,
    semiring_Gcd(nat) ).

tff(tcon_Nat_Onat___Rings_Omult__zero_151,axiom,
    mult_zero(nat) ).

tff(tcon_Nat_Onat___Orderings_Oorder_152,axiom,
    order(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring_153,axiom,
    semiring(nat) ).

tff(tcon_Nat_Onat___Orderings_Oord_154,axiom,
    ord(nat) ).

tff(tcon_Nat_Onat___Orderings_Obot_155,axiom,
    bot(nat) ).

tff(tcon_Nat_Onat___Lattices_Osup_156,axiom,
    sup(nat) ).

tff(tcon_Nat_Onat___Lattices_Oinf_157,axiom,
    inf(nat) ).

tff(tcon_Nat_Onat___Groups_Otimes_158,axiom,
    times(nat) ).

tff(tcon_Nat_Onat___Groups_Ominus_159,axiom,
    minus(nat) ).

tff(tcon_Nat_Onat___Power_Opower_160,axiom,
    power(nat) ).

tff(tcon_Nat_Onat___Num_Onumeral_161,axiom,
    numeral(nat) ).

tff(tcon_Nat_Onat___Groups_Ozero_162,axiom,
    zero(nat) ).

tff(tcon_Nat_Onat___Groups_Oone_163,axiom,
    one(nat) ).

tff(tcon_Nat_Onat___Rings_Odvd_164,axiom,
    dvd(nat) ).

tff(tcon_Nat_Onat___HOL_Oequal_165,axiom,
    cl_HOL_Oequal(nat) ).

tff(tcon_Num_Onum___Quickcheck__Exhaustive_Ofull__exhaustive_166,axiom,
    quickc3360725361186068524ustive(num) ).

tff(tcon_Num_Onum___Quickcheck__Random_Orandom_167,axiom,
    quickcheck_random(num) ).

tff(tcon_Num_Onum___Orderings_Opreorder_168,axiom,
    preorder(num) ).

tff(tcon_Num_Onum___Orderings_Olinorder_169,axiom,
    linorder(num) ).

tff(tcon_Num_Onum___Orderings_Oorder_170,axiom,
    order(num) ).

tff(tcon_Num_Onum___Orderings_Oord_171,axiom,
    ord(num) ).

tff(tcon_Num_Onum___Groups_Otimes_172,axiom,
    times(num) ).

tff(tcon_Num_Onum___HOL_Oequal_173,axiom,
    cl_HOL_Oequal(num) ).

tff(tcon_Rat_Orat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_174,axiom,
    semiri1453513574482234551roduct(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_175,axiom,
    ordere1937475149494474687imp_le(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors__cancel_176,axiom,
    semiri6575147826004484403cancel(rat) ).

tff(tcon_Rat_Orat___Groups_Ostrict__ordered__ab__semigroup__add_177,axiom,
    strict9044650504122735259up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__cancel__ab__semigroup__add_178,axiom,
    ordere580206878836729694up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add__imp__le_179,axiom,
    ordere2412721322843649153imp_le(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__comm__semiring__strict_180,axiom,
    linord2810124833399127020strict(rat) ).

tff(tcon_Rat_Orat___Quickcheck__Exhaustive_Ofull__exhaustive_181,axiom,
    quickc3360725361186068524ustive(rat) ).

tff(tcon_Rat_Orat___Groups_Ostrict__ordered__comm__monoid__add_182,axiom,
    strict7427464778891057005id_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__cancel__comm__monoid__add_183,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_184,axiom,
    linord715952674999750819strict(rat) ).

tff(tcon_Rat_Orat___Orderings_Ounbounded__dense__linorder,axiom,
    unboun7993243217541854897norder(rat) ).

tff(tcon_Rat_Orat___Groups_Olinordered__ab__semigroup__add_185,axiom,
    linord4140545234300271783up_add(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__nonzero__semiring_186,axiom,
    linord181362715937106298miring(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semiring__strict_187,axiom,
    linord8928482502909563296strict(rat) ).

tff(tcon_Rat_Orat___Quickcheck__Exhaustive_Oexhaustive_188,axiom,
    quickc658316121487927005ustive(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors_189,axiom,
    semiri3467727345109120633visors(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add_190,axiom,
    ordere6658533253407199908up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__group__add__abs_191,axiom,
    ordere166539214618696060dd_abs(rat) ).

tff(tcon_Rat_Orat___Archimedean__Field_Ofloor__ceiling,axiom,
    archim2362893244070406136eiling(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__comm__monoid__add_192,axiom,
    ordere6911136660526730532id_add(rat) ).

tff(tcon_Rat_Orat___Groups_Olinordered__ab__group__add_193,axiom,
    linord5086331880401160121up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Ocancel__ab__semigroup__add_194,axiom,
    cancel2418104881723323429up_add(rat) ).

tff(tcon_Rat_Orat___Rings_Oring__1__no__zero__divisors_195,axiom,
    ring_15535105094025558882visors(rat) ).

tff(tcon_Rat_Orat___Groups_Ocancel__comm__monoid__add_196,axiom,
    cancel1802427076303600483id_add(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__ring__strict_197,axiom,
    linord4710134922213307826strict(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring__1__cancel_198,axiom,
    comm_s4317794764714335236cancel(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__comm__semiring_199,axiom,
    ordere2520102378445227354miring(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semiring__1_200,axiom,
    linord6961819062388156250ring_1(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__group__add_201,axiom,
    ordered_ab_group_add(rat) ).

tff(tcon_Rat_Orat___Groups_Ocancel__semigroup__add_202,axiom,
    cancel_semigroup_add(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semiring_203,axiom,
    linordered_semiring(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__semiring__0_204,axiom,
    ordered_semiring_0(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semidom_205,axiom,
    linordered_semidom(rat) ).

tff(tcon_Rat_Orat___Quickcheck__Random_Orandom_206,axiom,
    quickcheck_random(rat) ).

tff(tcon_Rat_Orat___Orderings_Odense__linorder,axiom,
    dense_linorder(rat) ).

tff(tcon_Rat_Orat___Lattices_Osemilattice__sup_207,axiom,
    semilattice_sup(rat) ).

tff(tcon_Rat_Orat___Lattices_Osemilattice__inf_208,axiom,
    semilattice_inf(rat) ).

tff(tcon_Rat_Orat___Lattices_Odistrib__lattice_209,axiom,
    distrib_lattice(rat) ).

tff(tcon_Rat_Orat___Groups_Oab__semigroup__mult_210,axiom,
    ab_semigroup_mult(rat) ).

tff(tcon_Rat_Orat___Groups_Ocomm__monoid__mult_211,axiom,
    comm_monoid_mult(rat) ).

tff(tcon_Rat_Orat___Groups_Oab__semigroup__add_212,axiom,
    ab_semigroup_add(rat) ).

tff(tcon_Rat_Orat___Fields_Olinordered__field,axiom,
    linordered_field(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__semiring_213,axiom,
    ordered_semiring(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__ring__abs_214,axiom,
    ordered_ring_abs(rat) ).

tff(tcon_Rat_Orat___Groups_Ocomm__monoid__add_215,axiom,
    comm_monoid_add(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__ring_216,axiom,
    linordered_ring(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__idom_217,axiom,
    linordered_idom(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring__1_218,axiom,
    comm_semiring_1(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring__0_219,axiom,
    comm_semiring_0(rat) ).

tff(tcon_Rat_Orat___Orderings_Odense__order,axiom,
    dense_order(rat) ).

tff(tcon_Rat_Orat___Groups_Osemigroup__mult_220,axiom,
    semigroup_mult(rat) ).

tff(tcon_Rat_Orat___Rings_Osemidom__divide_221,axiom,
    semidom_divide(rat) ).

tff(tcon_Rat_Orat___Num_Osemiring__numeral_222,axiom,
    semiring_numeral(rat) ).

tff(tcon_Rat_Orat___Groups_Osemigroup__add_223,axiom,
    semigroup_add(rat) ).

tff(tcon_Rat_Orat___Fields_Odivision__ring,axiom,
    division_ring(rat) ).

tff(tcon_Rat_Orat___Rings_Ozero__less__one_224,axiom,
    zero_less_one(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring_225,axiom,
    comm_semiring(rat) ).

tff(tcon_Rat_Orat___Nat_Osemiring__char__0_226,axiom,
    semiring_char_0(rat) ).

tff(tcon_Rat_Orat___Groups_Oab__group__add_227,axiom,
    ab_group_add(rat) ).

tff(tcon_Rat_Orat___Fields_Ofield__char__0,axiom,
    field_char_0(rat) ).

tff(tcon_Rat_Orat___Countable_Ocountable_228,axiom,
    countable(rat) ).

tff(tcon_Rat_Orat___Rings_Ozero__neq__one_229,axiom,
    zero_neq_one(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__ring_230,axiom,
    ordered_ring(rat) ).

tff(tcon_Rat_Orat___Rings_Oidom__abs__sgn_231,axiom,
    idom_abs_sgn(rat) ).

tff(tcon_Rat_Orat___Orderings_Opreorder_232,axiom,
    preorder(rat) ).

tff(tcon_Rat_Orat___Orderings_Olinorder_233,axiom,
    linorder(rat) ).

tff(tcon_Rat_Orat___Groups_Omonoid__mult_234,axiom,
    monoid_mult(rat) ).

tff(tcon_Rat_Orat___Rings_Oidom__divide_235,axiom,
    idom_divide(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__ring__1_236,axiom,
    comm_ring_1(rat) ).

tff(tcon_Rat_Orat___Groups_Omonoid__add_237,axiom,
    monoid_add(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__1_238,axiom,
    semiring_1(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__0_239,axiom,
    semiring_0(rat) ).

tff(tcon_Rat_Orat___Orderings_Ono__top_240,axiom,
    no_top(rat) ).

tff(tcon_Rat_Orat___Orderings_Ono__bot_241,axiom,
    no_bot(rat) ).

tff(tcon_Rat_Orat___Lattices_Olattice_242,axiom,
    lattice(rat) ).

tff(tcon_Rat_Orat___Groups_Ogroup__add_243,axiom,
    group_add(rat) ).

tff(tcon_Rat_Orat___Rings_Omult__zero_244,axiom,
    mult_zero(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__ring_245,axiom,
    comm_ring(rat) ).

tff(tcon_Rat_Orat___Orderings_Oorder_246,axiom,
    order(rat) ).

tff(tcon_Rat_Orat___Num_Oneg__numeral_247,axiom,
    neg_numeral(rat) ).

tff(tcon_Rat_Orat___Nat_Oring__char__0_248,axiom,
    ring_char_0(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring_249,axiom,
    semiring(rat) ).

tff(tcon_Rat_Orat___Fields_Oinverse,axiom,
    inverse(rat) ).

tff(tcon_Rat_Orat___Orderings_Oord_250,axiom,
    ord(rat) ).

tff(tcon_Rat_Orat___Groups_Ouminus_251,axiom,
    uminus(rat) ).

tff(tcon_Rat_Orat___Rings_Oring__1_252,axiom,
    ring_1(rat) ).

tff(tcon_Rat_Orat___Rings_Oabs__if_253,axiom,
    abs_if(rat) ).

tff(tcon_Rat_Orat___Lattices_Osup_254,axiom,
    sup(rat) ).

tff(tcon_Rat_Orat___Lattices_Oinf_255,axiom,
    inf(rat) ).

tff(tcon_Rat_Orat___Groups_Otimes_256,axiom,
    times(rat) ).

tff(tcon_Rat_Orat___Groups_Ominus_257,axiom,
    minus(rat) ).

tff(tcon_Rat_Orat___Fields_Ofield,axiom,
    field(rat) ).

tff(tcon_Rat_Orat___Power_Opower_258,axiom,
    power(rat) ).

tff(tcon_Rat_Orat___Num_Onumeral_259,axiom,
    numeral(rat) ).

tff(tcon_Rat_Orat___Groups_Ozero_260,axiom,
    zero(rat) ).

tff(tcon_Rat_Orat___Rings_Oring_261,axiom,
    ring(rat) ).

tff(tcon_Rat_Orat___Rings_Oidom_262,axiom,
    idom(rat) ).

tff(tcon_Rat_Orat___Groups_Oone_263,axiom,
    one(rat) ).

tff(tcon_Rat_Orat___Rings_Odvd_264,axiom,
    dvd(rat) ).

tff(tcon_Rat_Orat___HOL_Oequal_265,axiom,
    cl_HOL_Oequal(rat) ).

tff(tcon_Set_Oset___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_266,axiom,
    ! [A18: $tType] : condit1219197933456340205attice(set(A18)) ).

tff(tcon_Set_Oset___Complete__Lattices_Ocomplete__distrib__lattice_267,axiom,
    ! [A18: $tType] : comple592849572758109894attice(set(A18)) ).

tff(tcon_Set_Oset___Complete__Lattices_Ocomplete__boolean__algebra_268,axiom,
    ! [A18: $tType] : comple489889107523837845lgebra(set(A18)) ).

tff(tcon_Set_Oset___Quickcheck__Exhaustive_Ofull__exhaustive_269,axiom,
    ! [A18: $tType] :
      ( quickc3360725361186068524ustive(A18)
     => quickc3360725361186068524ustive(set(A18)) ) ).

tff(tcon_Set_Oset___Lattices_Obounded__semilattice__sup__bot_270,axiom,
    ! [A18: $tType] : bounde4967611905675639751up_bot(set(A18)) ).

tff(tcon_Set_Oset___Lattices_Obounded__semilattice__inf__top_271,axiom,
    ! [A18: $tType] : bounde4346867609351753570nf_top(set(A18)) ).

tff(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_272,axiom,
    ! [A18: $tType] : comple6319245703460814977attice(set(A18)) ).

tff(tcon_Set_Oset___Quickcheck__Exhaustive_Oexhaustive_273,axiom,
    ! [A18: $tType] :
      ( quickc658316121487927005ustive(A18)
     => quickc658316121487927005ustive(set(A18)) ) ).

tff(tcon_Set_Oset___Boolean__Algebras_Oboolean__algebra_274,axiom,
    ! [A18: $tType] : boolea8198339166811842893lgebra(set(A18)) ).

tff(tcon_Set_Oset___Lattices_Obounded__lattice__top_275,axiom,
    ! [A18: $tType] : bounded_lattice_top(set(A18)) ).

tff(tcon_Set_Oset___Lattices_Obounded__lattice__bot_276,axiom,
    ! [A18: $tType] : bounded_lattice_bot(set(A18)) ).

tff(tcon_Set_Oset___Complete__Partial__Order_Occpo_277,axiom,
    ! [A18: $tType] : comple9053668089753744459l_ccpo(set(A18)) ).

tff(tcon_Set_Oset___Quickcheck__Random_Orandom_278,axiom,
    ! [A18: $tType] :
      ( quickcheck_random(A18)
     => quickcheck_random(set(A18)) ) ).

tff(tcon_Set_Oset___Lattices_Osemilattice__sup_279,axiom,
    ! [A18: $tType] : semilattice_sup(set(A18)) ).

tff(tcon_Set_Oset___Lattices_Osemilattice__inf_280,axiom,
    ! [A18: $tType] : semilattice_inf(set(A18)) ).

tff(tcon_Set_Oset___Lattices_Odistrib__lattice_281,axiom,
    ! [A18: $tType] : distrib_lattice(set(A18)) ).

tff(tcon_Set_Oset___Lattices_Obounded__lattice_282,axiom,
    ! [A18: $tType] : bounded_lattice(set(A18)) ).

tff(tcon_Set_Oset___Complete__Lattices_OSup_283,axiom,
    ! [A18: $tType] : complete_Sup(set(A18)) ).

tff(tcon_Set_Oset___Complete__Lattices_OInf_284,axiom,
    ! [A18: $tType] : complete_Inf(set(A18)) ).

tff(tcon_Set_Oset___Orderings_Oorder__top_285,axiom,
    ! [A18: $tType] : order_top(set(A18)) ).

tff(tcon_Set_Oset___Orderings_Oorder__bot_286,axiom,
    ! [A18: $tType] : order_bot(set(A18)) ).

tff(tcon_Set_Oset___Countable_Ocountable_287,axiom,
    ! [A18: $tType] :
      ( finite_finite(A18)
     => countable(set(A18)) ) ).

tff(tcon_Set_Oset___Orderings_Opreorder_288,axiom,
    ! [A18: $tType] : preorder(set(A18)) ).

tff(tcon_Set_Oset___Finite__Set_Ofinite_289,axiom,
    ! [A18: $tType] :
      ( finite_finite(A18)
     => finite_finite(set(A18)) ) ).

tff(tcon_Set_Oset___Lattices_Olattice_290,axiom,
    ! [A18: $tType] : lattice(set(A18)) ).

tff(tcon_Set_Oset___Orderings_Oorder_291,axiom,
    ! [A18: $tType] : order(set(A18)) ).

tff(tcon_Set_Oset___Orderings_Otop_292,axiom,
    ! [A18: $tType] : top(set(A18)) ).

tff(tcon_Set_Oset___Orderings_Oord_293,axiom,
    ! [A18: $tType] : ord(set(A18)) ).

tff(tcon_Set_Oset___Orderings_Obot_294,axiom,
    ! [A18: $tType] : bot(set(A18)) ).

tff(tcon_Set_Oset___Groups_Ouminus_295,axiom,
    ! [A18: $tType] : uminus(set(A18)) ).

tff(tcon_Set_Oset___Lattices_Osup_296,axiom,
    ! [A18: $tType] : sup(set(A18)) ).

tff(tcon_Set_Oset___Lattices_Oinf_297,axiom,
    ! [A18: $tType] : inf(set(A18)) ).

tff(tcon_Set_Oset___Groups_Ominus_298,axiom,
    ! [A18: $tType] : minus(set(A18)) ).

tff(tcon_Set_Oset___HOL_Oequal_299,axiom,
    ! [A18: $tType] :
      ( cl_HOL_Oequal(A18)
     => cl_HOL_Oequal(set(A18)) ) ).

tff(tcon_HOL_Obool___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_300,axiom,
    condit1219197933456340205attice($o) ).

tff(tcon_HOL_Obool___Complete__Lattices_Ocomplete__distrib__lattice_301,axiom,
    comple592849572758109894attice($o) ).

tff(tcon_HOL_Obool___Complete__Lattices_Ocomplete__boolean__algebra_302,axiom,
    comple489889107523837845lgebra($o) ).

tff(tcon_HOL_Obool___Quickcheck__Exhaustive_Ofull__exhaustive_303,axiom,
    quickc3360725361186068524ustive($o) ).

tff(tcon_HOL_Obool___Lattices_Obounded__semilattice__sup__bot_304,axiom,
    bounde4967611905675639751up_bot($o) ).

tff(tcon_HOL_Obool___Lattices_Obounded__semilattice__inf__top_305,axiom,
    bounde4346867609351753570nf_top($o) ).

tff(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_306,axiom,
    comple6319245703460814977attice($o) ).

tff(tcon_HOL_Obool___Boolean__Algebras_Oboolean__algebra_307,axiom,
    boolea8198339166811842893lgebra($o) ).

tff(tcon_HOL_Obool___Lattices_Obounded__lattice__top_308,axiom,
    bounded_lattice_top($o) ).

tff(tcon_HOL_Obool___Lattices_Obounded__lattice__bot_309,axiom,
    bounded_lattice_bot($o) ).

tff(tcon_HOL_Obool___Complete__Partial__Order_Occpo_310,axiom,
    comple9053668089753744459l_ccpo($o) ).

tff(tcon_HOL_Obool___Quickcheck__Random_Orandom_311,axiom,
    quickcheck_random($o) ).

tff(tcon_HOL_Obool___Lattices_Osemilattice__sup_312,axiom,
    semilattice_sup($o) ).

tff(tcon_HOL_Obool___Lattices_Osemilattice__inf_313,axiom,
    semilattice_inf($o) ).

tff(tcon_HOL_Obool___Lattices_Odistrib__lattice_314,axiom,
    distrib_lattice($o) ).

tff(tcon_HOL_Obool___Lattices_Obounded__lattice_315,axiom,
    bounded_lattice($o) ).

tff(tcon_HOL_Obool___Complete__Lattices_OSup_316,axiom,
    complete_Sup($o) ).

tff(tcon_HOL_Obool___Complete__Lattices_OInf_317,axiom,
    complete_Inf($o) ).

tff(tcon_HOL_Obool___Orderings_Oorder__top_318,axiom,
    order_top($o) ).

tff(tcon_HOL_Obool___Orderings_Oorder__bot_319,axiom,
    order_bot($o) ).

tff(tcon_HOL_Obool___Countable_Ocountable_320,axiom,
    countable($o) ).

tff(tcon_HOL_Obool___Orderings_Opreorder_321,axiom,
    preorder($o) ).

tff(tcon_HOL_Obool___Orderings_Olinorder_322,axiom,
    linorder($o) ).

tff(tcon_HOL_Obool___Finite__Set_Ofinite_323,axiom,
    finite_finite($o) ).

tff(tcon_HOL_Obool___Lattices_Olattice_324,axiom,
    lattice($o) ).

tff(tcon_HOL_Obool___Orderings_Oorder_325,axiom,
    order($o) ).

tff(tcon_HOL_Obool___Orderings_Otop_326,axiom,
    top($o) ).

tff(tcon_HOL_Obool___Orderings_Oord_327,axiom,
    ord($o) ).

tff(tcon_HOL_Obool___Orderings_Obot_328,axiom,
    bot($o) ).

tff(tcon_HOL_Obool___Groups_Ouminus_329,axiom,
    uminus($o) ).

tff(tcon_HOL_Obool___Lattices_Osup_330,axiom,
    sup($o) ).

tff(tcon_HOL_Obool___Lattices_Oinf_331,axiom,
    inf($o) ).

tff(tcon_HOL_Obool___Groups_Ominus_332,axiom,
    minus($o) ).

tff(tcon_HOL_Obool___HOL_Oequal_333,axiom,
    cl_HOL_Oequal($o) ).

tff(tcon_List_Olist___Quickcheck__Exhaustive_Ofull__exhaustive_334,axiom,
    ! [A18: $tType] :
      ( quickc3360725361186068524ustive(A18)
     => quickc3360725361186068524ustive(list(A18)) ) ).

tff(tcon_List_Olist___Quickcheck__Random_Orandom_335,axiom,
    ! [A18: $tType] :
      ( quickcheck_random(A18)
     => quickcheck_random(list(A18)) ) ).

tff(tcon_List_Olist___Countable_Ocountable_336,axiom,
    ! [A18: $tType] :
      ( countable(A18)
     => countable(list(A18)) ) ).

tff(tcon_List_Olist___HOL_Oequal_337,axiom,
    ! [A18: $tType] : cl_HOL_Oequal(list(A18)) ).

tff(tcon_String_Ochar___Quickcheck__Exhaustive_Ofull__exhaustive_338,axiom,
    quickc3360725361186068524ustive(char) ).

tff(tcon_String_Ochar___Quickcheck__Random_Orandom_339,axiom,
    quickcheck_random(char) ).

tff(tcon_String_Ochar___Countable_Ocountable_340,axiom,
    countable(char) ).

tff(tcon_String_Ochar___Finite__Set_Ofinite_341,axiom,
    finite_finite(char) ).

tff(tcon_String_Ochar___HOL_Oequal_342,axiom,
    cl_HOL_Oequal(char) ).

tff(tcon_Sum__Type_Osum___Quickcheck__Exhaustive_Ofull__exhaustive_343,axiom,
    ! [A18: $tType,A19: $tType] :
      ( ( quickc3360725361186068524ustive(A18)
        & quickc3360725361186068524ustive(A19) )
     => quickc3360725361186068524ustive(sum_sum(A18,A19)) ) ).

tff(tcon_Sum__Type_Osum___Quickcheck__Random_Orandom_344,axiom,
    ! [A18: $tType,A19: $tType] :
      ( ( quickcheck_random(A18)
        & quickcheck_random(A19) )
     => quickcheck_random(sum_sum(A18,A19)) ) ).

tff(tcon_Sum__Type_Osum___Countable_Ocountable_345,axiom,
    ! [A18: $tType,A19: $tType] :
      ( ( countable(A18)
        & countable(A19) )
     => countable(sum_sum(A18,A19)) ) ).

tff(tcon_Sum__Type_Osum___Finite__Set_Ofinite_346,axiom,
    ! [A18: $tType,A19: $tType] :
      ( ( finite_finite(A18)
        & finite_finite(A19) )
     => finite_finite(sum_sum(A18,A19)) ) ).

tff(tcon_Sum__Type_Osum___HOL_Oequal_347,axiom,
    ! [A18: $tType,A19: $tType] : cl_HOL_Oequal(sum_sum(A18,A19)) ).

tff(tcon_Filter_Ofilter___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_348,axiom,
    ! [A18: $tType] : condit1219197933456340205attice(filter(A18)) ).

tff(tcon_Filter_Ofilter___Lattices_Obounded__semilattice__sup__bot_349,axiom,
    ! [A18: $tType] : bounde4967611905675639751up_bot(filter(A18)) ).

tff(tcon_Filter_Ofilter___Lattices_Obounded__semilattice__inf__top_350,axiom,
    ! [A18: $tType] : bounde4346867609351753570nf_top(filter(A18)) ).

tff(tcon_Filter_Ofilter___Complete__Lattices_Ocomplete__lattice_351,axiom,
    ! [A18: $tType] : comple6319245703460814977attice(filter(A18)) ).

tff(tcon_Filter_Ofilter___Lattices_Obounded__lattice__top_352,axiom,
    ! [A18: $tType] : bounded_lattice_top(filter(A18)) ).

tff(tcon_Filter_Ofilter___Lattices_Obounded__lattice__bot_353,axiom,
    ! [A18: $tType] : bounded_lattice_bot(filter(A18)) ).

tff(tcon_Filter_Ofilter___Complete__Partial__Order_Occpo_354,axiom,
    ! [A18: $tType] : comple9053668089753744459l_ccpo(filter(A18)) ).

tff(tcon_Filter_Ofilter___Lattices_Osemilattice__sup_355,axiom,
    ! [A18: $tType] : semilattice_sup(filter(A18)) ).

tff(tcon_Filter_Ofilter___Lattices_Osemilattice__inf_356,axiom,
    ! [A18: $tType] : semilattice_inf(filter(A18)) ).

tff(tcon_Filter_Ofilter___Lattices_Odistrib__lattice_357,axiom,
    ! [A18: $tType] : distrib_lattice(filter(A18)) ).

tff(tcon_Filter_Ofilter___Lattices_Obounded__lattice_358,axiom,
    ! [A18: $tType] : bounded_lattice(filter(A18)) ).

tff(tcon_Filter_Ofilter___Complete__Lattices_OSup_359,axiom,
    ! [A18: $tType] : complete_Sup(filter(A18)) ).

tff(tcon_Filter_Ofilter___Complete__Lattices_OInf_360,axiom,
    ! [A18: $tType] : complete_Inf(filter(A18)) ).

tff(tcon_Filter_Ofilter___Orderings_Oorder__top_361,axiom,
    ! [A18: $tType] : order_top(filter(A18)) ).

tff(tcon_Filter_Ofilter___Orderings_Oorder__bot_362,axiom,
    ! [A18: $tType] : order_bot(filter(A18)) ).

tff(tcon_Filter_Ofilter___Orderings_Opreorder_363,axiom,
    ! [A18: $tType] : preorder(filter(A18)) ).

tff(tcon_Filter_Ofilter___Lattices_Olattice_364,axiom,
    ! [A18: $tType] : lattice(filter(A18)) ).

tff(tcon_Filter_Ofilter___Orderings_Oorder_365,axiom,
    ! [A18: $tType] : order(filter(A18)) ).

tff(tcon_Filter_Ofilter___Orderings_Otop_366,axiom,
    ! [A18: $tType] : top(filter(A18)) ).

tff(tcon_Filter_Ofilter___Orderings_Oord_367,axiom,
    ! [A18: $tType] : ord(filter(A18)) ).

tff(tcon_Filter_Ofilter___Orderings_Obot_368,axiom,
    ! [A18: $tType] : bot(filter(A18)) ).

tff(tcon_Filter_Ofilter___Lattices_Osup_369,axiom,
    ! [A18: $tType] : sup(filter(A18)) ).

tff(tcon_Filter_Ofilter___Lattices_Oinf_370,axiom,
    ! [A18: $tType] : inf(filter(A18)) ).

tff(tcon_Filter_Ofilter___HOL_Oequal_371,axiom,
    ! [A18: $tType] :
      ( cl_HOL_Oequal(A18)
     => cl_HOL_Oequal(filter(A18)) ) ).

tff(tcon_Option_Ooption___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_372,axiom,
    ! [A18: $tType] :
      ( comple5582772986160207858norder(A18)
     => condit6923001295902523014norder(option(A18)) ) ).

tff(tcon_Option_Ooption___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_373,axiom,
    ! [A18: $tType] :
      ( comple6319245703460814977attice(A18)
     => condit1219197933456340205attice(option(A18)) ) ).

tff(tcon_Option_Ooption___Complete__Lattices_Ocomplete__distrib__lattice_374,axiom,
    ! [A18: $tType] :
      ( comple592849572758109894attice(A18)
     => comple592849572758109894attice(option(A18)) ) ).

tff(tcon_Option_Ooption___Quickcheck__Exhaustive_Ofull__exhaustive_375,axiom,
    ! [A18: $tType] :
      ( quickc3360725361186068524ustive(A18)
     => quickc3360725361186068524ustive(option(A18)) ) ).

tff(tcon_Option_Ooption___Lattices_Obounded__semilattice__sup__bot_376,axiom,
    ! [A18: $tType] :
      ( lattice(A18)
     => bounde4967611905675639751up_bot(option(A18)) ) ).

tff(tcon_Option_Ooption___Lattices_Obounded__semilattice__inf__top_377,axiom,
    ! [A18: $tType] :
      ( bounded_lattice_top(A18)
     => bounde4346867609351753570nf_top(option(A18)) ) ).

tff(tcon_Option_Ooption___Complete__Lattices_Ocomplete__linorder,axiom,
    ! [A18: $tType] :
      ( comple5582772986160207858norder(A18)
     => comple5582772986160207858norder(option(A18)) ) ).

tff(tcon_Option_Ooption___Complete__Lattices_Ocomplete__lattice_378,axiom,
    ! [A18: $tType] :
      ( comple6319245703460814977attice(A18)
     => comple6319245703460814977attice(option(A18)) ) ).

tff(tcon_Option_Ooption___Lattices_Obounded__lattice__top_379,axiom,
    ! [A18: $tType] :
      ( bounded_lattice_top(A18)
     => bounded_lattice_top(option(A18)) ) ).

tff(tcon_Option_Ooption___Lattices_Obounded__lattice__bot_380,axiom,
    ! [A18: $tType] :
      ( lattice(A18)
     => bounded_lattice_bot(option(A18)) ) ).

tff(tcon_Option_Ooption___Complete__Partial__Order_Occpo_381,axiom,
    ! [A18: $tType] :
      ( comple6319245703460814977attice(A18)
     => comple9053668089753744459l_ccpo(option(A18)) ) ).

tff(tcon_Option_Ooption___Quickcheck__Random_Orandom_382,axiom,
    ! [A18: $tType] :
      ( quickcheck_random(A18)
     => quickcheck_random(option(A18)) ) ).

tff(tcon_Option_Ooption___Lattices_Osemilattice__sup_383,axiom,
    ! [A18: $tType] :
      ( semilattice_sup(A18)
     => semilattice_sup(option(A18)) ) ).

tff(tcon_Option_Ooption___Lattices_Osemilattice__inf_384,axiom,
    ! [A18: $tType] :
      ( semilattice_inf(A18)
     => semilattice_inf(option(A18)) ) ).

tff(tcon_Option_Ooption___Lattices_Odistrib__lattice_385,axiom,
    ! [A18: $tType] :
      ( distrib_lattice(A18)
     => distrib_lattice(option(A18)) ) ).

tff(tcon_Option_Ooption___Lattices_Obounded__lattice_386,axiom,
    ! [A18: $tType] :
      ( bounded_lattice_top(A18)
     => bounded_lattice(option(A18)) ) ).

tff(tcon_Option_Ooption___Complete__Lattices_OSup_387,axiom,
    ! [A18: $tType] :
      ( comple6319245703460814977attice(A18)
     => complete_Sup(option(A18)) ) ).

tff(tcon_Option_Ooption___Complete__Lattices_OInf_388,axiom,
    ! [A18: $tType] :
      ( comple6319245703460814977attice(A18)
     => complete_Inf(option(A18)) ) ).

tff(tcon_Option_Ooption___Orderings_Owellorder_389,axiom,
    ! [A18: $tType] :
      ( wellorder(A18)
     => wellorder(option(A18)) ) ).

tff(tcon_Option_Ooption___Orderings_Oorder__top_390,axiom,
    ! [A18: $tType] :
      ( order_top(A18)
     => order_top(option(A18)) ) ).

tff(tcon_Option_Ooption___Orderings_Oorder__bot_391,axiom,
    ! [A18: $tType] :
      ( order(A18)
     => order_bot(option(A18)) ) ).

tff(tcon_Option_Ooption___Countable_Ocountable_392,axiom,
    ! [A18: $tType] :
      ( countable(A18)
     => countable(option(A18)) ) ).

tff(tcon_Option_Ooption___Orderings_Opreorder_393,axiom,
    ! [A18: $tType] :
      ( preorder(A18)
     => preorder(option(A18)) ) ).

tff(tcon_Option_Ooption___Orderings_Olinorder_394,axiom,
    ! [A18: $tType] :
      ( linorder(A18)
     => linorder(option(A18)) ) ).

tff(tcon_Option_Ooption___Finite__Set_Ofinite_395,axiom,
    ! [A18: $tType] :
      ( finite_finite(A18)
     => finite_finite(option(A18)) ) ).

tff(tcon_Option_Ooption___Lattices_Olattice_396,axiom,
    ! [A18: $tType] :
      ( lattice(A18)
     => lattice(option(A18)) ) ).

tff(tcon_Option_Ooption___Orderings_Oorder_397,axiom,
    ! [A18: $tType] :
      ( order(A18)
     => order(option(A18)) ) ).

tff(tcon_Option_Ooption___Orderings_Otop_398,axiom,
    ! [A18: $tType] :
      ( order_top(A18)
     => top(option(A18)) ) ).

tff(tcon_Option_Ooption___Orderings_Oord_399,axiom,
    ! [A18: $tType] :
      ( preorder(A18)
     => ord(option(A18)) ) ).

tff(tcon_Option_Ooption___Orderings_Obot_400,axiom,
    ! [A18: $tType] :
      ( order(A18)
     => bot(option(A18)) ) ).

tff(tcon_Option_Ooption___Lattices_Osup_401,axiom,
    ! [A18: $tType] :
      ( sup(A18)
     => sup(option(A18)) ) ).

tff(tcon_Option_Ooption___Lattices_Oinf_402,axiom,
    ! [A18: $tType] :
      ( inf(A18)
     => inf(option(A18)) ) ).

tff(tcon_Option_Ooption___HOL_Oequal_403,axiom,
    ! [A18: $tType] : cl_HOL_Oequal(option(A18)) ).

tff(tcon_Predicate_Oseq___HOL_Oequal_404,axiom,
    ! [A18: $tType] : cl_HOL_Oequal(seq(A18)) ).

tff(tcon_Predicate_Opred___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_405,axiom,
    ! [A18: $tType] : condit1219197933456340205attice(pred(A18)) ).

tff(tcon_Predicate_Opred___Complete__Lattices_Ocomplete__distrib__lattice_406,axiom,
    ! [A18: $tType] : comple592849572758109894attice(pred(A18)) ).

tff(tcon_Predicate_Opred___Complete__Lattices_Ocomplete__boolean__algebra_407,axiom,
    ! [A18: $tType] : comple489889107523837845lgebra(pred(A18)) ).

tff(tcon_Predicate_Opred___Lattices_Obounded__semilattice__sup__bot_408,axiom,
    ! [A18: $tType] : bounde4967611905675639751up_bot(pred(A18)) ).

tff(tcon_Predicate_Opred___Lattices_Obounded__semilattice__inf__top_409,axiom,
    ! [A18: $tType] : bounde4346867609351753570nf_top(pred(A18)) ).

tff(tcon_Predicate_Opred___Complete__Lattices_Ocomplete__lattice_410,axiom,
    ! [A18: $tType] : comple6319245703460814977attice(pred(A18)) ).

tff(tcon_Predicate_Opred___Boolean__Algebras_Oboolean__algebra_411,axiom,
    ! [A18: $tType] : boolea8198339166811842893lgebra(pred(A18)) ).

tff(tcon_Predicate_Opred___Lattices_Obounded__lattice__top_412,axiom,
    ! [A18: $tType] : bounded_lattice_top(pred(A18)) ).

tff(tcon_Predicate_Opred___Lattices_Obounded__lattice__bot_413,axiom,
    ! [A18: $tType] : bounded_lattice_bot(pred(A18)) ).

tff(tcon_Predicate_Opred___Complete__Partial__Order_Occpo_414,axiom,
    ! [A18: $tType] : comple9053668089753744459l_ccpo(pred(A18)) ).

tff(tcon_Predicate_Opred___Lattices_Osemilattice__sup_415,axiom,
    ! [A18: $tType] : semilattice_sup(pred(A18)) ).

tff(tcon_Predicate_Opred___Lattices_Osemilattice__inf_416,axiom,
    ! [A18: $tType] : semilattice_inf(pred(A18)) ).

tff(tcon_Predicate_Opred___Lattices_Odistrib__lattice_417,axiom,
    ! [A18: $tType] : distrib_lattice(pred(A18)) ).

tff(tcon_Predicate_Opred___Lattices_Obounded__lattice_418,axiom,
    ! [A18: $tType] : bounded_lattice(pred(A18)) ).

tff(tcon_Predicate_Opred___Complete__Lattices_OSup_419,axiom,
    ! [A18: $tType] : complete_Sup(pred(A18)) ).

tff(tcon_Predicate_Opred___Complete__Lattices_OInf_420,axiom,
    ! [A18: $tType] : complete_Inf(pred(A18)) ).

tff(tcon_Predicate_Opred___Orderings_Oorder__top_421,axiom,
    ! [A18: $tType] : order_top(pred(A18)) ).

tff(tcon_Predicate_Opred___Orderings_Oorder__bot_422,axiom,
    ! [A18: $tType] : order_bot(pred(A18)) ).

tff(tcon_Predicate_Opred___Orderings_Opreorder_423,axiom,
    ! [A18: $tType] : preorder(pred(A18)) ).

tff(tcon_Predicate_Opred___Lattices_Olattice_424,axiom,
    ! [A18: $tType] : lattice(pred(A18)) ).

tff(tcon_Predicate_Opred___Orderings_Oorder_425,axiom,
    ! [A18: $tType] : order(pred(A18)) ).

tff(tcon_Predicate_Opred___Orderings_Otop_426,axiom,
    ! [A18: $tType] : top(pred(A18)) ).

tff(tcon_Predicate_Opred___Orderings_Oord_427,axiom,
    ! [A18: $tType] : ord(pred(A18)) ).

tff(tcon_Predicate_Opred___Orderings_Obot_428,axiom,
    ! [A18: $tType] : bot(pred(A18)) ).

tff(tcon_Predicate_Opred___Groups_Ouminus_429,axiom,
    ! [A18: $tType] : uminus(pred(A18)) ).

tff(tcon_Predicate_Opred___Lattices_Osup_430,axiom,
    ! [A18: $tType] : sup(pred(A18)) ).

tff(tcon_Predicate_Opred___Lattices_Oinf_431,axiom,
    ! [A18: $tType] : inf(pred(A18)) ).

tff(tcon_Predicate_Opred___Groups_Ominus_432,axiom,
    ! [A18: $tType] : minus(pred(A18)) ).

tff(tcon_Predicate_Opred___HOL_Oequal_433,axiom,
    ! [A18: $tType] : cl_HOL_Oequal(pred(A18)) ).

tff(tcon_String_Oliteral___Quickcheck__Random_Orandom_434,axiom,
    quickcheck_random(literal) ).

tff(tcon_String_Oliteral___Groups_Osemigroup__add_435,axiom,
    semigroup_add(literal) ).

tff(tcon_String_Oliteral___Countable_Ocountable_436,axiom,
    countable(literal) ).

tff(tcon_String_Oliteral___Orderings_Opreorder_437,axiom,
    preorder(literal) ).

tff(tcon_String_Oliteral___Orderings_Olinorder_438,axiom,
    linorder(literal) ).

tff(tcon_String_Oliteral___Groups_Omonoid__add_439,axiom,
    monoid_add(literal) ).

tff(tcon_String_Oliteral___Orderings_Oorder_440,axiom,
    order(literal) ).

tff(tcon_String_Oliteral___Orderings_Oord_441,axiom,
    ord(literal) ).

tff(tcon_String_Oliteral___Groups_Ozero_442,axiom,
    zero(literal) ).

tff(tcon_String_Oliteral___HOL_Oequal_443,axiom,
    cl_HOL_Oequal(literal) ).

tff(tcon_Assertions_Oassn___Groups_Oab__semigroup__mult_444,axiom,
    ab_semigroup_mult(assn) ).

tff(tcon_Assertions_Oassn___Groups_Ocomm__monoid__mult_445,axiom,
    comm_monoid_mult(assn) ).

tff(tcon_Assertions_Oassn___Groups_Osemigroup__mult_446,axiom,
    semigroup_mult(assn) ).

tff(tcon_Assertions_Oassn___Groups_Omonoid__mult_447,axiom,
    monoid_mult(assn) ).

tff(tcon_Assertions_Oassn___Groups_Otimes_448,axiom,
    times(assn) ).

tff(tcon_Assertions_Oassn___Power_Opower_449,axiom,
    power(assn) ).

tff(tcon_Assertions_Oassn___Groups_Oone_450,axiom,
    one(assn) ).

tff(tcon_Assertions_Oassn___Rings_Odvd_451,axiom,
    dvd(assn) ).

tff(tcon_Multiset_Omultiset___Quickcheck__Exhaustive_Ofull__exhaustive_452,axiom,
    ! [A18: $tType] :
      ( quickc3360725361186068524ustive(A18)
     => quickc3360725361186068524ustive(multiset(A18)) ) ).

tff(tcon_Multiset_Omultiset___Groups_Oordered__ab__semigroup__add_453,axiom,
    ! [A18: $tType] :
      ( preorder(A18)
     => ordere6658533253407199908up_add(multiset(A18)) ) ).

tff(tcon_Multiset_Omultiset___Groups_Ocancel__ab__semigroup__add_454,axiom,
    ! [A18: $tType] : cancel2418104881723323429up_add(multiset(A18)) ).

tff(tcon_Multiset_Omultiset___Groups_Ocancel__comm__monoid__add_455,axiom,
    ! [A18: $tType] : cancel1802427076303600483id_add(multiset(A18)) ).

tff(tcon_Multiset_Omultiset___Groups_Ocancel__semigroup__add_456,axiom,
    ! [A18: $tType] : cancel_semigroup_add(multiset(A18)) ).

tff(tcon_Multiset_Omultiset___Quickcheck__Random_Orandom_457,axiom,
    ! [A18: $tType] :
      ( quickcheck_random(A18)
     => quickcheck_random(multiset(A18)) ) ).

tff(tcon_Multiset_Omultiset___Groups_Ocomm__monoid__diff_458,axiom,
    ! [A18: $tType] : comm_monoid_diff(multiset(A18)) ).

tff(tcon_Multiset_Omultiset___Groups_Oab__semigroup__add_459,axiom,
    ! [A18: $tType] : ab_semigroup_add(multiset(A18)) ).

tff(tcon_Multiset_Omultiset___Groups_Ocomm__monoid__add_460,axiom,
    ! [A18: $tType] : comm_monoid_add(multiset(A18)) ).

tff(tcon_Multiset_Omultiset___Complete__Lattices_OSup_461,axiom,
    ! [A18: $tType] : complete_Sup(multiset(A18)) ).

tff(tcon_Multiset_Omultiset___Complete__Lattices_OInf_462,axiom,
    ! [A18: $tType] : complete_Inf(multiset(A18)) ).

tff(tcon_Multiset_Omultiset___Groups_Osemigroup__add_463,axiom,
    ! [A18: $tType] : semigroup_add(multiset(A18)) ).

tff(tcon_Multiset_Omultiset___Orderings_Opreorder_464,axiom,
    ! [A18: $tType] :
      ( preorder(A18)
     => preorder(multiset(A18)) ) ).

tff(tcon_Multiset_Omultiset___Groups_Omonoid__add_465,axiom,
    ! [A18: $tType] : monoid_add(multiset(A18)) ).

tff(tcon_Multiset_Omultiset___Orderings_Oorder_466,axiom,
    ! [A18: $tType] :
      ( preorder(A18)
     => order(multiset(A18)) ) ).

tff(tcon_Multiset_Omultiset___Orderings_Oord_467,axiom,
    ! [A18: $tType] :
      ( preorder(A18)
     => ord(multiset(A18)) ) ).

tff(tcon_Multiset_Omultiset___Groups_Ominus_468,axiom,
    ! [A18: $tType] : minus(multiset(A18)) ).

tff(tcon_Multiset_Omultiset___Groups_Ozero_469,axiom,
    ! [A18: $tType] : zero(multiset(A18)) ).

tff(tcon_Multiset_Omultiset___HOL_Oequal_470,axiom,
    ! [A18: $tType] :
      ( cl_HOL_Oequal(A18)
     => cl_HOL_Oequal(multiset(A18)) ) ).

tff(tcon_Product__Type_Oprod___Quickcheck__Exhaustive_Ofull__exhaustive_471,axiom,
    ! [A18: $tType,A19: $tType] :
      ( ( quickc3360725361186068524ustive(A18)
        & quickc3360725361186068524ustive(A19) )
     => quickc3360725361186068524ustive(product_prod(A18,A19)) ) ).

tff(tcon_Product__Type_Oprod___Quickcheck__Exhaustive_Oexhaustive_472,axiom,
    ! [A18: $tType,A19: $tType] :
      ( ( quickc658316121487927005ustive(A18)
        & quickc658316121487927005ustive(A19) )
     => quickc658316121487927005ustive(product_prod(A18,A19)) ) ).

tff(tcon_Product__Type_Oprod___Quickcheck__Random_Orandom_473,axiom,
    ! [A18: $tType,A19: $tType] :
      ( ( quickcheck_random(A18)
        & quickcheck_random(A19) )
     => quickcheck_random(product_prod(A18,A19)) ) ).

tff(tcon_Product__Type_Oprod___Countable_Ocountable_474,axiom,
    ! [A18: $tType,A19: $tType] :
      ( ( countable(A18)
        & countable(A19) )
     => countable(product_prod(A18,A19)) ) ).

tff(tcon_Product__Type_Oprod___Finite__Set_Ofinite_475,axiom,
    ! [A18: $tType,A19: $tType] :
      ( ( finite_finite(A18)
        & finite_finite(A19) )
     => finite_finite(product_prod(A18,A19)) ) ).

tff(tcon_Product__Type_Oprod___HOL_Oequal_476,axiom,
    ! [A18: $tType,A19: $tType] : cl_HOL_Oequal(product_prod(A18,A19)) ).

tff(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_477,axiom,
    condit6923001295902523014norder(product_unit) ).

tff(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_478,axiom,
    condit1219197933456340205attice(product_unit) ).

tff(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__distrib__lattice_479,axiom,
    comple592849572758109894attice(product_unit) ).

tff(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__boolean__algebra_480,axiom,
    comple489889107523837845lgebra(product_unit) ).

tff(tcon_Product__Type_Ounit___Quickcheck__Exhaustive_Ofull__exhaustive_481,axiom,
    quickc3360725361186068524ustive(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Obounded__semilattice__sup__bot_482,axiom,
    bounde4967611905675639751up_bot(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Obounded__semilattice__inf__top_483,axiom,
    bounde4346867609351753570nf_top(product_unit) ).

tff(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__linorder_484,axiom,
    comple5582772986160207858norder(product_unit) ).

tff(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__lattice_485,axiom,
    comple6319245703460814977attice(product_unit) ).

tff(tcon_Product__Type_Ounit___Boolean__Algebras_Oboolean__algebra_486,axiom,
    boolea8198339166811842893lgebra(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Obounded__lattice__top_487,axiom,
    bounded_lattice_top(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Obounded__lattice__bot_488,axiom,
    bounded_lattice_bot(product_unit) ).

tff(tcon_Product__Type_Ounit___Complete__Partial__Order_Occpo_489,axiom,
    comple9053668089753744459l_ccpo(product_unit) ).

tff(tcon_Product__Type_Ounit___Quickcheck__Random_Orandom_490,axiom,
    quickcheck_random(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Osemilattice__sup_491,axiom,
    semilattice_sup(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Osemilattice__inf_492,axiom,
    semilattice_inf(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Odistrib__lattice_493,axiom,
    distrib_lattice(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Obounded__lattice_494,axiom,
    bounded_lattice(product_unit) ).

tff(tcon_Product__Type_Ounit___Complete__Lattices_OSup_495,axiom,
    complete_Sup(product_unit) ).

tff(tcon_Product__Type_Ounit___Complete__Lattices_OInf_496,axiom,
    complete_Inf(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Owellorder_497,axiom,
    wellorder(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Oorder__top_498,axiom,
    order_top(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Oorder__bot_499,axiom,
    order_bot(product_unit) ).

tff(tcon_Product__Type_Ounit___Countable_Ocountable_500,axiom,
    countable(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Opreorder_501,axiom,
    preorder(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Olinorder_502,axiom,
    linorder(product_unit) ).

tff(tcon_Product__Type_Ounit___Finite__Set_Ofinite_503,axiom,
    finite_finite(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Olattice_504,axiom,
    lattice(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Oorder_505,axiom,
    order(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Otop_506,axiom,
    top(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Oord_507,axiom,
    ord(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Obot_508,axiom,
    bot(product_unit) ).

tff(tcon_Product__Type_Ounit___Groups_Ouminus_509,axiom,
    uminus(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Osup_510,axiom,
    sup(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Oinf_511,axiom,
    inf(product_unit) ).

tff(tcon_Product__Type_Ounit___Groups_Ominus_512,axiom,
    minus(product_unit) ).

tff(tcon_Product__Type_Ounit___HOL_Oequal_513,axiom,
    cl_HOL_Oequal(product_unit) ).

tff(tcon_Heap_Oheap_Oheap__ext___Quickcheck__Random_Orandom_514,axiom,
    ! [A18: $tType] :
      ( quickcheck_random(A18)
     => quickcheck_random(heap_ext(A18)) ) ).

tff(tcon_Heap_Oheap_Oheap__ext___HOL_Oequal_515,axiom,
    ! [A18: $tType] : cl_HOL_Oequal(heap_ext(A18)) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_516,axiom,
    bit_un5681908812861735899ations(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_517,axiom,
    semiri1453513574482234551roduct(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring__with__nat_518,axiom,
    euclid5411537665997757685th_nat(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__ring__with__nat_519,axiom,
    euclid8789492081693882211th_nat(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__monoid__add__imp__le_520,axiom,
    ordere1937475149494474687imp_le(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring_521,axiom,
    euclid3128863361964157862miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring__cancel_522,axiom,
    euclid4440199948858584721cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Divides_Ounique__euclidean__semiring__numeral_523,axiom,
    unique1627219031080169319umeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__ring__cancel_524,axiom,
    euclid8851590272496341667cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors__cancel_525,axiom,
    semiri6575147826004484403cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__ab__semigroup__add_526,axiom,
    strict9044650504122735259up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__ab__semigroup__add_527,axiom,
    ordere580206878836729694up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add__imp__le_528,axiom,
    ordere2412721322843649153imp_le(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bit__operations_529,axiom,
    bit_se359711467146920520ations(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__comm__semiring__strict_530,axiom,
    linord2810124833399127020strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Quickcheck__Exhaustive_Ofull__exhaustive_531,axiom,
    quickc3360725361186068524ustive(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__comm__monoid__add_532,axiom,
    strict7427464778891057005id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__comm__monoid__add_533,axiom,
    ordere8940638589300402666id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring_534,axiom,
    euclid3725896446679973847miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1__strict_535,axiom,
    linord715952674999750819strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__semigroup__add_536,axiom,
    linord4140545234300271783up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Oring__bit__operations_537,axiom,
    bit_ri3973907225187159222ations(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__nonzero__semiring_538,axiom,
    linord181362715937106298miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__strict_539,axiom,
    linord8928482502909563296strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Quickcheck__Exhaustive_Oexhaustive_540,axiom,
    quickc658316121487927005ustive(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors_541,axiom,
    semiri3467727345109120633visors(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add_542,axiom,
    ordere6658533253407199908up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add__abs_543,axiom,
    ordere166539214618696060dd_abs(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__comm__monoid__add_544,axiom,
    ordere6911136660526730532id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__group__add_545,axiom,
    linord5086331880401160121up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocancel__ab__semigroup__add_546,axiom,
    cancel2418104881723323429up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oring__1__no__zero__divisors_547,axiom,
    ring_15535105094025558882visors(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocancel__comm__monoid__add_548,axiom,
    cancel1802427076303600483id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring__strict_549,axiom,
    linord4710134922213307826strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1__cancel_550,axiom,
    comm_s4317794764714335236cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bits_551,axiom,
    bit_semiring_bits(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__comm__semiring_552,axiom,
    ordere2520102378445227354miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1_553,axiom,
    linord6961819062388156250ring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add_554,axiom,
    ordered_ab_group_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocancel__semigroup__add_555,axiom,
    cancel_semigroup_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring_556,axiom,
    linordered_semiring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring__0_557,axiom,
    ordered_semiring_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semidom_558,axiom,
    linordered_semidom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Quickcheck__Random_Orandom_559,axiom,
    quickcheck_random(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__mult_560,axiom,
    ab_semigroup_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oalgebraic__semidom_561,axiom,
    algebraic_semidom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__mult_562,axiom,
    comm_monoid_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__add_563,axiom,
    ab_semigroup_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring_564,axiom,
    ordered_semiring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring__abs_565,axiom,
    ordered_ring_abs(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Parity_Osemiring__parity_566,axiom,
    semiring_parity(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__add_567,axiom,
    comm_monoid_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__modulo_568,axiom,
    semiring_modulo(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring_569,axiom,
    linordered_ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__idom_570,axiom,
    linordered_idom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1_571,axiom,
    comm_semiring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__0_572,axiom,
    comm_semiring_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__mult_573,axiom,
    semigroup_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemidom__modulo_574,axiom,
    semidom_modulo(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemidom__divide_575,axiom,
    semidom_divide(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Num_Osemiring__numeral_576,axiom,
    semiring_numeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__add_577,axiom,
    semigroup_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ozero__less__one_578,axiom,
    zero_less_one(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring_579,axiom,
    comm_semiring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Nat_Osemiring__char__0_580,axiom,
    semiring_char_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oab__group__add_581,axiom,
    ab_group_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ozero__neq__one_582,axiom,
    zero_neq_one(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring_583,axiom,
    ordered_ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oidom__abs__sgn_584,axiom,
    idom_abs_sgn(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Parity_Oring__parity_585,axiom,
    ring_parity(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Opreorder_586,axiom,
    preorder(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Olinorder_587,axiom,
    linorder(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Omonoid__mult_588,axiom,
    monoid_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oidom__divide_589,axiom,
    idom_divide(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring__1_590,axiom,
    comm_ring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Omonoid__add_591,axiom,
    monoid_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1_592,axiom,
    semiring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__0_593,axiom,
    semiring_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ogroup__add_594,axiom,
    group_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Omult__zero_595,axiom,
    mult_zero(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring_596,axiom,
    comm_ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Oorder_597,axiom,
    order(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Num_Oneg__numeral_598,axiom,
    neg_numeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Nat_Oring__char__0_599,axiom,
    ring_char_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring_600,axiom,
    semiring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Oord_601,axiom,
    ord(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ouminus_602,axiom,
    uminus(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oring__1_603,axiom,
    ring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oabs__if_604,axiom,
    abs_if(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Otimes_605,axiom,
    times(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ominus_606,axiom,
    minus(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Power_Opower_607,axiom,
    power(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Num_Onumeral_608,axiom,
    numeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ozero_609,axiom,
    zero(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oring_610,axiom,
    ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oidom_611,axiom,
    idom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oone_612,axiom,
    one(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Odvd_613,axiom,
    dvd(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___HOL_Oequal_614,axiom,
    cl_HOL_Oequal(code_integer) ).

tff(tcon_Code__Numeral_Onatural___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_615,axiom,
    bit_un5681908812861735899ations(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Euclidean__Division_Ounique__euclidean__semiring__with__nat_616,axiom,
    euclid5411537665997757685th_nat(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__monoid__add__imp__le_617,axiom,
    ordere1937475149494474687imp_le(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Euclidean__Division_Ounique__euclidean__semiring_618,axiom,
    euclid3128863361964157862miring(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Euclidean__Division_Oeuclidean__semiring__cancel_619,axiom,
    euclid4440199948858584721cancel(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Osemiring__no__zero__divisors__cancel_620,axiom,
    semiri6575147826004484403cancel(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Ostrict__ordered__ab__semigroup__add_621,axiom,
    strict9044650504122735259up_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Oordered__cancel__ab__semigroup__add_622,axiom,
    ordere580206878836729694up_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__add__imp__le_623,axiom,
    ordere2412721322843649153imp_le(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Bit__Operations_Osemiring__bit__operations_624,axiom,
    bit_se359711467146920520ations(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Olinordered__comm__semiring__strict_625,axiom,
    linord2810124833399127020strict(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Quickcheck__Exhaustive_Ofull__exhaustive_626,axiom,
    quickc3360725361186068524ustive(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Ostrict__ordered__comm__monoid__add_627,axiom,
    strict7427464778891057005id_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Oordered__cancel__comm__monoid__add_628,axiom,
    ordere8940638589300402666id_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Euclidean__Division_Oeuclidean__semiring_629,axiom,
    euclid3725896446679973847miring(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Olinordered__ab__semigroup__add_630,axiom,
    linord4140545234300271783up_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Olinordered__nonzero__semiring_631,axiom,
    linord181362715937106298miring(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Olinordered__semiring__strict_632,axiom,
    linord8928482502909563296strict(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Quickcheck__Exhaustive_Oexhaustive_633,axiom,
    quickc658316121487927005ustive(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Osemiring__no__zero__divisors_634,axiom,
    semiri3467727345109120633visors(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__add_635,axiom,
    ordere6658533253407199908up_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Oordered__comm__monoid__add_636,axiom,
    ordere6911136660526730532id_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Ocancel__ab__semigroup__add_637,axiom,
    cancel2418104881723323429up_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Ocancel__comm__monoid__add_638,axiom,
    cancel1802427076303600483id_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__1__cancel_639,axiom,
    comm_s4317794764714335236cancel(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Bit__Operations_Osemiring__bits_640,axiom,
    bit_semiring_bits(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Oordered__comm__semiring_641,axiom,
    ordere2520102378445227354miring(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Ocancel__semigroup__add_642,axiom,
    cancel_semigroup_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Olinordered__semiring_643,axiom,
    linordered_semiring(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Oordered__semiring__0_644,axiom,
    ordered_semiring_0(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Olinordered__semidom_645,axiom,
    linordered_semidom(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Quickcheck__Random_Orandom_646,axiom,
    quickcheck_random(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Oab__semigroup__mult_647,axiom,
    ab_semigroup_mult(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Oalgebraic__semidom_648,axiom,
    algebraic_semidom(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__mult_649,axiom,
    comm_monoid_mult(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__diff_650,axiom,
    comm_monoid_diff(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Oab__semigroup__add_651,axiom,
    ab_semigroup_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Oordered__semiring_652,axiom,
    ordered_semiring(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Parity_Osemiring__parity_653,axiom,
    semiring_parity(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__add_654,axiom,
    comm_monoid_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Osemiring__modulo_655,axiom,
    semiring_modulo(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__1_656,axiom,
    comm_semiring_1(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__0_657,axiom,
    comm_semiring_0(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Osemigroup__mult_658,axiom,
    semigroup_mult(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Osemidom__modulo_659,axiom,
    semidom_modulo(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Osemidom__divide_660,axiom,
    semidom_divide(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Num_Osemiring__numeral_661,axiom,
    semiring_numeral(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Osemigroup__add_662,axiom,
    semigroup_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Ozero__less__one_663,axiom,
    zero_less_one(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring_664,axiom,
    comm_semiring(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Nat_Osemiring__char__0_665,axiom,
    semiring_char_0(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Ozero__neq__one_666,axiom,
    zero_neq_one(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Orderings_Opreorder_667,axiom,
    preorder(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Orderings_Olinorder_668,axiom,
    linorder(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Omonoid__mult_669,axiom,
    monoid_mult(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Omonoid__add_670,axiom,
    monoid_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Osemiring__1_671,axiom,
    semiring_1(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Osemiring__0_672,axiom,
    semiring_0(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Omult__zero_673,axiom,
    mult_zero(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Orderings_Oorder_674,axiom,
    order(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Osemiring_675,axiom,
    semiring(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Orderings_Oord_676,axiom,
    ord(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Otimes_677,axiom,
    times(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Ominus_678,axiom,
    minus(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Power_Opower_679,axiom,
    power(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Num_Onumeral_680,axiom,
    numeral(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Ozero_681,axiom,
    zero(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Oone_682,axiom,
    one(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Odvd_683,axiom,
    dvd(code_natural) ).

tff(tcon_Code__Numeral_Onatural___HOL_Oequal_684,axiom,
    cl_HOL_Oequal(code_natural) ).

% Helper facts (6)
tff(help_fAll_1_1_U,axiom,
    ! [A: $tType,P: fun(A,$o),X: A] :
      ( ~ aa(fun(A,$o),$o,fAll(A),P)
      | aa(A,$o,P,X) ) ).

tff(help_fNot_2_1_U,axiom,
    ! [P: $o] :
      ( (P)
      | aa($o,$o,fNot,(P)) ) ).

tff(help_fNot_1_1_U,axiom,
    ! [P: $o] :
      ( ~ aa($o,$o,fNot,(P))
      | ~ (P) ) ).

tff(help_fequal_2_1_T,axiom,
    ! [A: $tType,X: A,Y: A] :
      ( ( X != Y )
      | aa(A,$o,aa(A,fun(A,$o),fequal(A),X),Y) ) ).

tff(help_fequal_1_1_T,axiom,
    ! [A: $tType,X: A,Y: A] :
      ( ~ aa(A,$o,aa(A,fun(A,$o),fequal(A),X),Y)
      | ( X = Y ) ) ).

tff(help_fChoice_1_1_T,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( aa(A,$o,P,fChoice(A,P))
      = ( ? [X8: A] : aa(A,$o,P,X8) ) ) ).

% Free types (1)
tff(tfree_0,hypothesis,
    linorder(b) ).

% Conjectures (1)
tff(conj_0,conjecture,
    ( ( ( x = aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),assn,abs_assn,aTP_Lamp_a(product_prod(heap_ext(product_unit),set(nat)),$o)) )
      & ( x != y ) )
  <=> ( ( x = aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),assn,abs_assn,aTP_Lamp_a(product_prod(heap_ext(product_unit),set(nat)),$o)) )
      & ( y != aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),assn,abs_assn,aTP_Lamp_aa(product_prod(heap_ext(product_unit),set(nat)),$o)) ) ) ) ).

%------------------------------------------------------------------------------