%------------------------------------------------------------------------------
% File     : ITP219_4 : TPTP v9.0.0. Released v8.0.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem VEBT_List_Assn 00252_007651
% 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    : 0062_VEBT_List_Assn_00252_007651 [Des22]

% Status   : Theorem
% Rating   : 1.00 v8.1.0
% Syntax   : Number of formulae    : 11381 (4146 unt;1724 typ;   0 def)
%            Number of atoms       : 17629 (7848 equ)
%            Maximal formula atoms :   22 (   1 avg)
%            Number of connectives : 18368 (2381   ~; 304   |;1595   &)
%                                         (2073 <=>;12015  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   27 (   6 avg)
%            Maximal term depth    :   37 (   2 avg)
%            Number of FOOLs       :  616 ( 370 fml; 246 var)
%            Number of X terms     :  403 (   0  []; 386 ite;  17 let)
%            Number of types       :   15 (  14 usr)
%            Number of type conns  : 1591 (1196   >; 395   *;   0   +;   0  <<)
%            Number of predicates  :  289 ( 286 usr;   2 prp; 0-6 aty)
%            Number of functors    : 1435 (1435 usr;  65 con; 0-7 aty)
%            Number of variables   : 37578 (34148   !; 805   ?;37578   :)
%                                         (2625  !>;   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 17:06:45.939
%------------------------------------------------------------------------------
% Could-be-implicit typings (29)
tff(ty_t_Heap__Time__Monad_OHeap,type,
    heap_Time_Heap: $tType > $tType ).

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_Multiset_Omultiset,type,
    multiset: $tType > $tType ).

tff(ty_t_Assertions_Oassn,type,
    assn: $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_Heap_Oarray,type,
    array: $tType > $tType ).

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

tff(ty_t_Heap_Oref,type,
    ref: $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_d,type,
    d: $tType ).

tff(ty_tf_c,type,
    c: $tType ).

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

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

% Explicit typings (1695)
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_Nat_Osize,type,
    size: 
      !>[A: $tType] : $o ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_cl_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] : ( A > B ) ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_ATP_058Lamp__adj____,type,
    aTP_Lamp_adj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_ATP_058Lamp__zz____,type,
    aTP_Lamp_zz: 
      !>[A: $tType,B: $tType] : ( 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_Array__Time_Oalloc,type,
    array_alloc: 
      !>[A: $tType] : ( ( list(A) * heap_ext(product_unit) ) > product_prod(array(A),heap_ext(product_unit)) ) ).

tff(sy_c_Array__Time_Ofreeze,type,
    array_freeze: 
      !>[A: $tType] : ( array(A) > heap_Time_Heap(list(A)) ) ).

tff(sy_c_Array__Time_Oget,type,
    array_get: 
      !>[A: $tType] : ( ( heap_ext(product_unit) * array(A) ) > list(A) ) ).

tff(sy_c_Array__Time_Olen,type,
    array_len: 
      !>[A: $tType] : ( array(A) > heap_Time_Heap(nat) ) ).

tff(sy_c_Array__Time_Olength,type,
    array_length: 
      !>[A: $tType] : ( ( heap_ext(product_unit) * array(A) ) > nat ) ).

tff(sy_c_Array__Time_Omake,type,
    array_make: 
      !>[A: $tType] : ( ( nat * fun(nat,A) ) > heap_Time_Heap(array(A)) ) ).

tff(sy_c_Array__Time_Omap__entry,type,
    array_map_entry: 
      !>[A: $tType] : ( ( nat * fun(A,A) * array(A) ) > heap_Time_Heap(array(A)) ) ).

tff(sy_c_Array__Time_Onew,type,
    array_new: 
      !>[A: $tType] : ( ( nat * A ) > heap_Time_Heap(array(A)) ) ).

tff(sy_c_Array__Time_Onoteq,type,
    array_noteq: 
      !>[A: $tType,B: $tType] : ( ( array(A) * array(B) ) > $o ) ).

tff(sy_c_Array__Time_Onth,type,
    array_nth: 
      !>[A: $tType] : ( ( array(A) * nat ) > heap_Time_Heap(A) ) ).

tff(sy_c_Array__Time_Oof__list,type,
    array_of_list: 
      !>[A: $tType] : ( list(A) > heap_Time_Heap(array(A)) ) ).

tff(sy_c_Array__Time_Opresent,type,
    array_present: 
      !>[A: $tType] : ( ( heap_ext(product_unit) * array(A) ) > $o ) ).

tff(sy_c_Array__Time_Oset,type,
    array_set: 
      !>[A: $tType] : ( ( array(A) * list(A) * heap_ext(product_unit) ) > heap_ext(product_unit) ) ).

tff(sy_c_Array__Time_Oswap,type,
    array_swap: 
      !>[A: $tType] : ( ( nat * A * array(A) ) > heap_Time_Heap(A) ) ).

tff(sy_c_Array__Time_Oupd,type,
    array_upd: 
      !>[A: $tType] : ( ( nat * A * array(A) ) > heap_Time_Heap(array(A)) ) ).

tff(sy_c_Array__Time_Oupdate,type,
    array_update: 
      !>[A: $tType] : ( ( array(A) * nat * A * heap_ext(product_unit) ) > heap_ext(product_unit) ) ).

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_Oentails,type,
    entails: fun(assn,fun(assn,$o)) ).

tff(sy_c_Assertions_Oentailst,type,
    entailst: ( assn * assn ) > $o ).

tff(sy_c_Assertions_Oex__assn,type,
    ex_assn: 
      !>[A: $tType] : ( fun(A,assn) > assn ) ).

tff(sy_c_Assertions_Oin__range,type,
    in_range: fun(product_prod(heap_ext(product_unit),set(nat)),$o) ).

tff(sy_c_Assertions_Ois__pure__assn,type,
    is_pure_assn: assn > $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_Oprecise,type,
    precise: 
      !>[A: $tType,B: $tType] : ( fun(A,fun(B,assn)) > $o ) ).

tff(sy_c_Assertions_Oproper,type,
    proper: fun(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o) ).

tff(sy_c_Assertions_Opure__assn,type,
    pure_assn: $o > assn ).

tff(sy_c_Assertions_Opure__assn__raw,type,
    pure_assn_raw: 
      !>[A: $tType,B: $tType] : ( $o > fun(product_prod(A,set(B)),$o) ) ).

tff(sy_c_Assertions_Opure__assn__raw__rel,type,
    pure_assn_raw_rel: 
      !>[A: $tType,B: $tType] : fun(product_prod($o,product_prod(A,set(B))),fun(product_prod($o,product_prod(A,set(B))),$o)) ).

tff(sy_c_Assertions_OrelH,type,
    relH: ( set(nat) * heap_ext(product_unit) * heap_ext(product_unit) ) > $o ).

tff(sy_c_Assertions_Osnga__assn,type,
    snga_assn: 
      !>[A: $tType] : ( ( array(A) * list(A) ) > assn ) ).

tff(sy_c_Assertions_Osnga__assn__raw,type,
    snga_assn_raw: 
      !>[A: $tType] : ( ( array(A) * list(A) ) > fun(product_prod(heap_ext(product_unit),set(nat)),$o) ) ).

tff(sy_c_Assertions_Osnga__assn__raw__rel,type,
    snga_assn_raw_rel: 
      !>[A: $tType] : fun(product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),fun(product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),$o)) ).

tff(sy_c_Assertions_Osngr__assn,type,
    sngr_assn: 
      !>[A: $tType] : ( ( ref(A) * A ) > assn ) ).

tff(sy_c_Assertions_Osngr__assn__raw,type,
    sngr_assn_raw: 
      !>[A: $tType] : ( ( ref(A) * A ) > fun(product_prod(heap_ext(product_unit),set(nat)),$o) ) ).

tff(sy_c_Assertions_Osngr__assn__raw__rel,type,
    sngr_assn_raw_rel: 
      !>[A: $tType] : fun(product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),fun(product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),$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_Automation_OFI,type,
    fi: ( list(product_prod(assn,assn)) * assn * assn * assn * assn * assn ) > $o ).

tff(sy_c_Automation_OFI__QUERY,type,
    fI_QUERY: ( assn * assn * assn ) > $o ).

tff(sy_c_Automation_OFI__RESULT,type,
    fI_RESULT: ( list(product_prod(assn,assn)) * assn * assn * assn ) > $o ).

tff(sy_c_Automation_OSLN,type,
    sln: assn ).

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) * 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)) * 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] : ( 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_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__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_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(A,B) ) > $o ) ).

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_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__Embedding_Oembed,type,
    bNF_Wellorder_embed: 
      !>[A: $tType,A2: $tType] : ( ( set(product_prod(A,A)) * set(product_prod(A2,A2)) * 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) * 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_Opred__fun,type,
    basic_pred_fun: 
      !>[A: $tType,B: $tType] : ( ( fun(A,$o) * fun(B,$o) ) > fun(fun(A,B),$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_Osnds,type,
    basic_snds: 
      !>[A: $tType,B: $tType] : ( product_prod(A,B) > set(B) ) ).

tff(sy_c_Binomial_Obinomial,type,
    binomial: ( nat * 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_Oconcat__bit,type,
    bit_concat_bit: ( nat * int * int ) > int ).

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

tff(sy_c_Bit__Operations_Osemiring__bits__class_Obit,type,
    bit_se5641148757651400278ts_bit: 
      !>[A: $tType] : ( ( A * 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_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_Code__Numeral_Obit__cut__integer,type,
    code_bit_cut_integer: code_integer > product_prod(code_integer,$o) ).

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

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

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

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

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

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

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

tff(sy_c_Complete__Partial__Order_Occpo_Oadmissible,type,
    comple1908693960933563346ssible: 
      !>[A: $tType] : ( ( fun(set(A),A) * fun(A,fun(A,$o)) * fun(A,$o) ) > $o ) ).

tff(sy_c_Complete__Partial__Order_Occpo_Ofixp,type,
    comple187402453842119260l_fixp: 
      !>[A: $tType] : ( ( fun(set(A),A) * fun(A,fun(A,$o)) * fun(A,A) ) > A ) ).

tff(sy_c_Complete__Partial__Order_Occpo__class_Ofixp,type,
    comple115746919287870866o_fixp: 
      !>[A: $tType] : ( fun(A,A) > A ) ).

tff(sy_c_Complete__Partial__Order_Ochain,type,
    comple1602240252501008431_chain: 
      !>[A: $tType] : ( ( fun(A,fun(A,$o)) * set(A) ) > $o ) ).

tff(sy_c_Complete__Partial__Order_Omonotone,type,
    comple7038119648293358887notone: 
      !>[A: $tType,B: $tType] : fun(fun(A,fun(A,$o)),fun(fun(B,fun(B,$o)),fun(fun(A,B),$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)) * 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] : ( 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)) * set(A) ) > $o ) ).

tff(sy_c_Divides_Oadjust__div,type,
    adjust_div: product_prod(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_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_Oabstract__filter,type,
    abstract_filter: 
      !>[A: $tType] : ( fun(product_unit,filter(A)) > filter(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: $tType] : fun(fun(A,$o),fun(filter(A),$o)) ).

tff(sy_c_Filter_Ofilter_OAbs__filter,type,
    abs_filter: 
      !>[A: $tType] : fun(fun(fun(A,$o),$o),filter(A)) ).

tff(sy_c_Filter_Ofilter_ORep__filter,type,
    rep_filter: 
      !>[A: $tType] : fun(filter(A),fun(fun(A,$o),$o)) ).

tff(sy_c_Filter_Ofiltercomap,type,
    filtercomap: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * filter(B) ) > filter(A) ) ).

tff(sy_c_Filter_Ofilterlim,type,
    filterlim: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * filter(B) * filter(A) ) > $o ) ).

tff(sy_c_Filter_Ofiltermap,type,
    filtermap: 
      !>[A: $tType,B: $tType] : fun(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: $tType] : fun(fun(A,$o),fun(filter(A),$o)) ).

tff(sy_c_Filter_Ois__filter,type,
    is_filter: 
      !>[A: $tType] : fun(fun(fun(A,$o),$o),$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] : fun(set(A),filter(A)) ).

tff(sy_c_Filter_Oprod__filter,type,
    prod_filter: 
      !>[A: $tType,B: $tType] : fun(filter(A),fun(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_Ocard,type,
    finite_card: 
      !>[B: $tType] : ( set(B) > nat ) ).

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

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

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

tff(sy_c_Finite__Set_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) * B ) > $o ) ).

tff(sy_c_Finite__Set_Ofolding__idem__on,type,
    finite1890593828518410140dem_on: 
      !>[A: $tType,B: $tType] : ( ( set(A) * fun(A,fun(B,B)) ) > $o ) ).

tff(sy_c_Finite__Set_Ofolding__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(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) * B ) > 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_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] : ( set(A) > A ) ).

tff(sy_c_GCD_Osemiring__gcd__class_OLcm__fin,type,
    semiring_gcd_Lcm_fin: 
      !>[A: $tType] : ( set(A) > A ) ).

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

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

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

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

tff(sy_c_Groups_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,type,
    groups778175481326437816id_set: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * A ) > $o ) ).

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

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

tff(sy_c_Groups__List_Ocomm__monoid__list,type,
    groups1828464146339083142d_list: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * A ) > $o ) ).

tff(sy_c_Groups__List_Ocomm__monoid__list__set,type,
    groups4802862169904069756st_set: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * A ) > $o ) ).

tff(sy_c_Groups__List_Ocomm__semiring__0__class_Ohorner__sum,type,
    groups4207007520872428315er_sum: 
      !>[B: $tType,A: $tType] : fun(fun(B,A),fun(A,fun(list(B),A))) ).

tff(sy_c_Groups__List_Omonoid__add_Osum__list,type,
    groups4543113879258116180m_list: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * A * list(A) ) > 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_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_Oaddr__of__array,type,
    addr_of_array: 
      !>[A: $tType] : ( array(A) > nat ) ).

tff(sy_c_Heap_Oaddr__of__ref,type,
    addr_of_ref: 
      !>[A: $tType] : ( ref(A) > nat ) ).

tff(sy_c_Heap_Oarray_OArray,type,
    array2: 
      !>[A: $tType] : ( nat > array(A) ) ).

tff(sy_c_Heap_Oheap_Olim,type,
    lim: 
      !>[Z: $tType] : ( heap_ext(Z) > nat ) ).

tff(sy_c_Heap_Oheap_Olim__update,type,
    lim_update: 
      !>[Z: $tType] : ( ( fun(nat,nat) * heap_ext(Z) ) > heap_ext(Z) ) ).

tff(sy_c_Heap_Oref_ORef,type,
    ref2: 
      !>[A: $tType] : ( nat > ref(A) ) ).

tff(sy_c_Heap__Time__Monad_OHeap_OHeap,type,
    heap_Time_Heap2: 
      !>[A: $tType] : ( fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))) > heap_Time_Heap(A) ) ).

tff(sy_c_Heap__Time__Monad_Oassert,type,
    heap_Time_assert: 
      !>[A: $tType] : ( ( fun(A,$o) * A ) > heap_Time_Heap(A) ) ).

tff(sy_c_Heap__Time__Monad_Obind,type,
    heap_Time_bind: 
      !>[A: $tType,B: $tType] : ( ( heap_Time_Heap(A) * fun(A,heap_Time_Heap(B)) ) > heap_Time_Heap(B) ) ).

tff(sy_c_Heap__Time__Monad_Oeffect,type,
    heap_Time_effect: 
      !>[A: $tType] : ( ( heap_Time_Heap(A) * heap_ext(product_unit) * heap_ext(product_unit) * A * nat ) > $o ) ).

tff(sy_c_Heap__Time__Monad_Oexecute,type,
    heap_Time_execute: 
      !>[A: $tType] : ( heap_Time_Heap(A) > fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))) ) ).

tff(sy_c_Heap__Time__Monad_Oguard,type,
    heap_Time_guard: 
      !>[A: $tType] : ( ( fun(heap_ext(product_unit),$o) * fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))) ) > heap_Time_Heap(A) ) ).

tff(sy_c_Heap__Time__Monad_Oheap,type,
    heap_Time_heap: 
      !>[A: $tType] : ( fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))) > heap_Time_Heap(A) ) ).

tff(sy_c_Heap__Time__Monad_Oraise,type,
    heap_Time_raise: 
      !>[A: $tType] : ( list(char) > heap_Time_Heap(A) ) ).

tff(sy_c_Heap__Time__Monad_Oreturn,type,
    heap_Time_return: 
      !>[A: $tType] : ( A > heap_Time_Heap(A) ) ).

tff(sy_c_Heap__Time__Monad_Osuccess,type,
    heap_Time_success: 
      !>[A: $tType] : ( ( heap_Time_Heap(A) * heap_ext(product_unit) ) > $o ) ).

tff(sy_c_Heap__Time__Monad_Otap,type,
    heap_Time_tap: 
      !>[A: $tType] : ( fun(heap_ext(product_unit),A) > heap_Time_Heap(A) ) ).

tff(sy_c_Heap__Time__Monad_OtimeFrame,type,
    heap_Time_timeFrame: 
      !>[A: $tType] : ( ( nat * option(product_prod(A,product_prod(heap_ext(product_unit),nat))) ) > option(product_prod(A,product_prod(heap_ext(product_unit),nat))) ) ).

tff(sy_c_Heap__Time__Monad_OtimeFrame__rel,type,
    heap_T5500966940807335491me_rel: 
      !>[A: $tType] : fun(product_prod(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),fun(product_prod(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),$o)) ).

tff(sy_c_Heap__Time__Monad_Oureturn,type,
    heap_Time_ureturn: 
      !>[A: $tType] : ( A > heap_Time_Heap(A) ) ).

tff(sy_c_Heap__Time__Monad_Owait,type,
    heap_Time_wait: nat > heap_Time_Heap(product_unit) ).

tff(sy_c_Hoare__Triple_Ohoare__triple,type,
    hoare_hoare_triple: 
      !>[A: $tType] : ( ( assn * heap_Time_Heap(A) * fun(A,assn) ) > $o ) ).

tff(sy_c_Hoare__Triple_Onew__addrs,type,
    hoare_new_addrs: ( heap_ext(product_unit) * set(nat) * heap_ext(product_unit) ) > set(nat) ).

tff(sy_c_Inductive_Ocomplete__lattice__class_Olfp,type,
    complete_lattice_lfp: 
      !>[A: $tType] : ( fun(A,A) > A ) ).

tff(sy_c_Infinite__Set_Owellorder__class_Oenumerate,type,
    infini527867602293511546merate: 
      !>[A: $tType] : ( ( set(A) * nat ) > A ) ).

tff(sy_c_Int_OAbs__Integ,type,
    abs_Integ: fun(product_prod(nat,nat),int) ).

tff(sy_c_Int_ORep__Integ,type,
    rep_Integ: fun(int,product_prod(nat,nat)) ).

tff(sy_c_Int_Ointrel,type,
    intrel: fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)) ).

tff(sy_c_Int_Onat,type,
    nat2: int > nat ).

tff(sy_c_Int_Opcr__int,type,
    pcr_int: fun(product_prod(nat,nat),fun(int,$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_Lattices_Oinf__class_Oinf,type,
    inf_inf: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

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

tff(sy_c_Lattices_Osemilattice__neutr__order,type,
    semila1105856199041335345_order: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * A * fun(A,fun(A,$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_Osup__class_Osup,type,
    sup_sup: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_Lattices__Big_Olinorder__class_OMax,type,
    lattic643756798349783984er_Max: 
      !>[A: $tType] : ( set(A) > A ) ).

tff(sy_c_Lattices__Big_Olinorder__class_OMin,type,
    lattic643756798350308766er_Min: 
      !>[A: $tType] : ( set(A) > A ) ).

tff(sy_c_Lattices__Big_Oord__class_Oarg__max,type,
    lattices_ord_arg_max: 
      !>[B: $tType,A: $tType] : ( ( fun(B,A) * fun(B,$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__max,type,
    lattic501386751176901750rg_max: 
      !>[B: $tType,A: $tType] : ( ( fun(B,A) * fun(B,$o) * B ) > $o ) ).

tff(sy_c_Lattices__Big_Oord__class_Ois__arg__min,type,
    lattic501386751177426532rg_min: 
      !>[B: $tType,A: $tType] : ( ( fun(B,A) * fun(B,$o) * B ) > $o ) ).

tff(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin,type,
    lattic7752659483105999362nf_fin: 
      !>[A: $tType] : ( set(A) > A ) ).

tff(sy_c_Lattices__Big_Osemilattice__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)) * set(A) ) > A ) ).

tff(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin,type,
    lattic5882676163264333800up_fin: 
      !>[A: $tType] : ( 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_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_Oall__interval__int,type,
    all_interval_int: ( fun(int,$o) * int * int ) > $o ).

tff(sy_c_List_Oall__interval__nat,type,
    all_interval_nat: ( fun(nat,$o) * nat * nat ) > $o ).

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] : fun(list(A),list(A)) ).

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

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

tff(sy_c_List_Ocoset,type,
    coset: 
      !>[A: $tType] : ( list(A) > set(A) ) ).

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

tff(sy_c_List_Odistinct,type,
    distinct: 
      !>[A: $tType] : fun(list(A),$o) ).

tff(sy_c_List_Odistinct__adj,type,
    distinct_adj: 
      !>[A: $tType] : fun(list(A),$o) ).

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

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

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

tff(sy_c_List_Ofolding__insort__key__axioms,type,
    foldin3648464208017769352axioms: 
      !>[B: $tType,A: $tType] : ( ( set(B) * fun(B,A) ) > $o ) ).

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_List_Olinorder_Osorted__key__list__of__set,type,
    sorted8670434370408473282of_set: 
      !>[A: $tType,B: $tType] : ( fun(A,fun(A,$o)) > fun(fun(B,A),fun(set(B),list(B))) ) ).

tff(sy_c_List_Olinorder__class_Oinsort__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(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] : fun(B,fun(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: $tType] : fun(fun(A,$o),fun(list(A),$o)) ).

tff(sy_c_List_Olist_Olist__all2,type,
    list_all2: 
      !>[A: $tType,B: $tType] : fun(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o))) ).

tff(sy_c_List_Olist_Omap,type,
    map: 
      !>[A: $tType,Aa: $tType] : fun(fun(A,Aa),fun(list(A),list(Aa))) ).

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

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

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

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

tff(sy_c_List_Olist__ex,type,
    list_ex: 
      !>[A: $tType] : fun(fun(A,$o),fun(list(A),$o)) ).

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] : fun(list(A),fun(nat,fun(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] : fun(set(A),set(list(A))) ).

tff(sy_c_List_Olistset,type,
    listset: 
      !>[A: $tType] : fun(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,type,
    map_tailrec: 
      !>[A: $tType,B: $tType] : fun(fun(A,B),fun(list(A),list(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_Omaps,type,
    maps: 
      !>[A: $tType,B: $tType] : ( ( fun(A,list(B)) * list(A) ) > list(B) ) ).

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] : fun(list(A),fun(set(nat),list(A))) ).

tff(sy_c_List_Onull,type,
    null: 
      !>[A: $tType] : fun(list(A),$o) ).

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_Oord__class_Olexordp__eq,type,
    ord_lexordp_eq: 
      !>[A: $tType] : ( ( list(A) * list(A) ) > $o ) ).

tff(sy_c_List_Opartition,type,
    partition: 
      !>[A: $tType] : fun(fun(A,$o),fun(list(A),product_prod(list(A),list(A)))) ).

tff(sy_c_List_Oproduct,type,
    product: 
      !>[A: $tType,B: $tType] : fun(list(A),fun(list(B),list(product_prod(A,B)))) ).

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

tff(sy_c_List_Oremdups__adj__rel,type,
    remdups_adj_rel: 
      !>[A: $tType] : fun(list(A),fun(list(A),$o)) ).

tff(sy_c_List_Oremove1,type,
    remove1: 
      !>[A: $tType] : fun(A,fun(list(A),list(A))) ).

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

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

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

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

tff(sy_c_List_Orotate1,type,
    rotate1: 
      !>[A: $tType] : fun(list(A),list(A)) ).

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

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

tff(sy_c_List_Osuccessively,type,
    successively: 
      !>[A: $tType] : fun(fun(A,fun(A,$o)),fun(list(A),$o)) ).

tff(sy_c_List_Osuccessively__rel,type,
    successively_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_Otake,type,
    take: 
      !>[A: $tType] : fun(nat,fun(list(A),list(A))) ).

tff(sy_c_List_OtakeWhile,type,
    takeWhile: 
      !>[A: $tType] : fun(fun(A,$o),fun(list(A),list(A))) ).

tff(sy_c_List_Otranspose,type,
    transpose: 
      !>[A: $tType] : ( list(list(A)) > list(list(A)) ) ).

tff(sy_c_List_Otranspose__rel,type,
    transpose_rel: 
      !>[A: $tType] : fun(list(list(A)),fun(list(list(A)),$o)) ).

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

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

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

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

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

tff(sy_c_List_Ozip,type,
    zip: 
      !>[A: $tType,B: $tType] : fun(list(A),fun(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__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_OCODE__ABORT,type,
    cODE_ABORT: 
      !>[A: $tType] : ( fun(product_unit,A) > A ) ).

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_Ofilter__rev,type,
    filter_rev: 
      !>[A: $tType] : fun(fun(A,$o),fun(list(A),list(A))) ).

tff(sy_c_Misc_Ofilter__rev__aux,type,
    filter_rev_aux: 
      !>[A: $tType] : ( list(A) > fun(fun(A,$o),fun(list(A),list(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) ) > fun(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 > fun(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] : ( ( A * 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_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: $tType] : ( ( fun(A,$o) * 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) * multiset(A) ) > multiset(B) ) ).

tff(sy_c_Multiset_Ointer__mset,type,
    inter_mset: 
      !>[A: $tType] : ( ( multiset(A) * 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)) * multiset(A) * 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)) * multiset(A) * 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_Orepeat__mset,type,
    repeat_mset: 
      !>[A: $tType] : ( ( nat * 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_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] : ( ( multiset(A) * multiset(A) ) > multiset(A) ) ).

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] : ( ( A * fun(nat,A) * nat ) > A ) ).

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

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

tff(sy_c_Nat__Bijection_Olist__encode,type,
    nat_list_encode: list(nat) > nat ).

tff(sy_c_Nat__Bijection_Olist__encode__rel,type,
    nat_list_encode_rel: fun(list(nat),fun(list(nat),$o)) ).

tff(sy_c_Nat__Bijection_Oprod__decode__aux,type,
    nat_prod_decode_aux: ( nat * nat ) > product_prod(nat,nat) ).

tff(sy_c_Nat__Bijection_Oprod__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: 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_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__dec,type,
    neg_numeral_dbl_dec: 
      !>[A: $tType] : ( A > A ) ).

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

tff(sy_c_Num_Onum_OBit0,type,
    bit0: num > num ).

tff(sy_c_Num_Onum_OBit1,type,
    bit1: num > num ).

tff(sy_c_Num_Onum_OOne,type,
    one2: num ).

tff(sy_c_Num_Onum__of__nat,type,
    num_of_nat: nat > num ).

tff(sy_c_Num_Onumeral__class_Onumeral,type,
    numeral_numeral: 
      !>[A: $tType] : ( num > A ) ).

tff(sy_c_Num_Opred__numeral,type,
    pred_numeral: num > nat ).

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

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

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

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

tff(sy_c_Option_Ooption_Omap__option,type,
    map_option: 
      !>[A: $tType,Aa: $tType] : ( fun(A,Aa) > fun(option(A),option(Aa)) ) ).

tff(sy_c_Option_Ooption_Othe,type,
    the2: 
      !>[A: $tType] : fun(option(A),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)) * 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__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_Omono,type,
    mono: 
      !>[A: $tType,B: $tType] : ( fun(A,fun(A,$o)) > fun(fun(A,B),$o) ) ).

tff(sy_c_Orderings_Oorder_Ostrict__mono,type,
    strict_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_Oordering,type,
    ordering: 
      !>[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_Otop__class_Otop,type,
    top_top: 
      !>[A: $tType] : A ).

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

tff(sy_c_Partial__Function_Oflat__ord,type,
    partial_flat_ord: 
      !>[A: $tType] : ( A > fun(A,fun(A,$o)) ) ).

tff(sy_c_Partial__Function_Ofun__lub,type,
    partial_fun_lub: 
      !>[C: $tType,B: $tType,A: $tType] : fun(fun(set(C),B),fun(set(fun(A,C)),fun(A,B))) ).

tff(sy_c_Partial__Function_Ofun__ord,type,
    partial_fun_ord: 
      !>[A: $tType,B: $tType,C: $tType] : fun(fun(A,fun(B,$o)),fun(fun(C,A),fun(fun(C,B),$o))) ).

tff(sy_c_Partial__Function_Opartial__function__definitions,type,
    partia7178651479351089652itions: 
      !>[A: $tType] : ( ( fun(A,fun(A,$o)) * fun(set(A),A) ) > $o ) ).

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_Obind,type,
    bind2: 
      !>[A: $tType,B: $tType] : ( ( pred(A) * fun(A,pred(B)) ) > pred(B) ) ).

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_Opred_OPred,type,
    pred2: 
      !>[A: $tType] : ( fun(A,$o) > pred(A) ) ).

tff(sy_c_Predicate_Oseq_OEmpty,type,
    empty: 
      !>[A: $tType] : seq(A) ).

tff(sy_c_Predicate_Oseq_OInsert,type,
    insert2: 
      !>[A: $tType] : ( ( A * pred(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_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_Ocurry,type,
    product_curry: 
      !>[A: $tType,B: $tType,C: $tType] : ( fun(product_prod(A,B),C) > fun(A,fun(B,C)) ) ).

tff(sy_c_Product__Type_Ointernal__case__prod,type,
    produc5280177257484947105e_prod: 
      !>[A: $tType,B: $tType,C: $tType] : ( ( fun(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_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_Oprod_Ocase__prod,type,
    product_case_prod: 
      !>[A: $tType,B: $tType,C: $tType] : fun(fun(A,fun(B,C)),fun(product_prod(A,B),C)) ).

tff(sy_c_Product__Type_Oprod_Ofst,type,
    product_fst: 
      !>[A: $tType,B: $tType] : fun(product_prod(A,B),A) ).

tff(sy_c_Product__Type_Oprod_Osnd,type,
    product_snd: 
      !>[A: $tType,B: $tType] : fun(product_prod(A,B),B) ).

tff(sy_c_Product__Type_Oprod_Oswap,type,
    product_swap: 
      !>[A: $tType,B: $tType] : fun(product_prod(A,B),product_prod(B,A)) ).

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

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

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

tff(sy_c_Random_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)) * 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_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_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_Ofield__char__0__class_Oof__rat,type,
    field_char_0_of_rat: 
      !>[A: $tType] : ( rat > A ) ).

tff(sy_c_Rat_Onormalize,type,
    normalize: product_prod(int,int) > product_prod(int,int) ).

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

tff(sy_c_Rat_Oratrel,type,
    ratrel: fun(product_prod(int,int),fun(product_prod(int,int),$o)) ).

tff(sy_c_Ref__Time_Oalloc,type,
    ref_alloc: 
      !>[A: $tType] : ( ( A * heap_ext(product_unit) ) > product_prod(ref(A),heap_ext(product_unit)) ) ).

tff(sy_c_Ref__Time_Ochange,type,
    ref_change: 
      !>[A: $tType] : ( ( fun(A,A) * ref(A) ) > heap_Time_Heap(A) ) ).

tff(sy_c_Ref__Time_Oget,type,
    ref_get: 
      !>[A: $tType] : ( ( heap_ext(product_unit) * ref(A) ) > A ) ).

tff(sy_c_Ref__Time_Olookup,type,
    ref_lookup: 
      !>[A: $tType] : ( ref(A) > heap_Time_Heap(A) ) ).

tff(sy_c_Ref__Time_Opresent,type,
    ref_present: 
      !>[A: $tType] : ( ( heap_ext(product_unit) * ref(A) ) > $o ) ).

tff(sy_c_Ref__Time_Oref,type,
    ref_ref: 
      !>[A: $tType] : ( A > heap_Time_Heap(ref(A)) ) ).

tff(sy_c_Ref__Time_Oset,type,
    ref_set: 
      !>[A: $tType] : ( ( ref(A) * A * heap_ext(product_unit) ) > heap_ext(product_unit) ) ).

tff(sy_c_Ref__Time_Oupdate,type,
    ref_update: 
      !>[A: $tType] : ( ( ref(A) * A ) > heap_Time_Heap(product_unit) ) ).

tff(sy_c_Refine__Imp__Hol_Oassert_H,type,
    refine_Imp_assert: $o > heap_Time_Heap(product_unit) ).

tff(sy_c_Refine__Imp__Hol_Orefines,type,
    refine_Imp_refines: 
      !>[A: $tType] : ( ( heap_Time_Heap(A) * heap_Time_Heap(A) ) > $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: $tType] : ( fun(A,fun(B,$o)) > fun(A,$o) ) ).

tff(sy_c_Relation_OField,type,
    field2: 
      !>[A: $tType] : fun(set(product_prod(A,A)),set(A)) ).

tff(sy_c_Relation_OId,type,
    id2: 
      !>[A: $tType] : set(product_prod(A,A)) ).

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

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

tff(sy_c_Relation_OPowp,type,
    powp: 
      !>[A: $tType] : ( fun(A,$o) > fun(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: $tType] : ( ( fun(A,fun(B,$o)) * 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] : ( ( A * A ) > A ) ).

tff(sy_c_Rings_Odvd__class_Odvd,type,
    dvd_dvd: 
      !>[A: $tType] : ( ( A * A ) > $o ) ).

tff(sy_c_Rings_Omodulo__class_Omodulo,type,
    modulo_modulo: 
      !>[A: $tType] : ( ( A * A ) > A ) ).

tff(sy_c_Rings_Onormalization__semidom__class_Onormalize,type,
    normal6383669964737779283malize: 
      !>[A: $tType] : ( 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: $tType] : ( set(A) > fun(fun(A,$o),$o) ) ).

tff(sy_c_Set_OCollect,type,
    collect: 
      !>[A: $tType] : fun(fun(A,$o),set(A)) ).

tff(sy_c_Set_OPow,type,
    pow: 
      !>[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,
    insert3: 
      !>[A: $tType] : fun(A,fun(set(A),set(A))) ).

tff(sy_c_Set_Ois__empty,type,
    is_empty: 
      !>[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(A,B) * set(B) ) > set(A) ) ).

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

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

tff(sy_c_Set__Interval_Oord__class_OatLeast,type,
    set_ord_atLeast: 
      !>[A: $tType] : ( 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_Sum__Type_OPlus,type,
    sum_Plus: 
      !>[A: $tType,B: $tType] : ( ( set(A) * set(B) ) > set(sum_sum(A,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_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] : fun(set(product_prod(A,A)),set(product_prod(A,A))) ).

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

tff(sy_c_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_VEBT__List__Assn_OlistI__assn,type,
    vEBT_List_listI_assn: 
      !>[A: $tType,B: $tType] : ( ( set(nat) * fun(A,fun(B,assn)) * list(A) * list(B) ) > assn ) ).

tff(sy_c_VEBT__List__Assn_Olist__assn,type,
    vEBT_List_list_assn: 
      !>[A: $tType,C: $tType] : ( ( fun(A,fun(C,assn)) * list(A) * list(C) ) > assn ) ).

tff(sy_c_VEBT__List__Assn_Olist__assn__rel,type,
    vEBT_L4249061453398456502sn_rel: 
      !>[A: $tType,C: $tType] : fun(product_prod(fun(A,fun(C,assn)),product_prod(list(A),list(C))),fun(product_prod(fun(A,fun(C,assn)),product_prod(list(A),list(C))),$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_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__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 * set(A) ) > $o ) ).

tff(sy_v_A,type,
    a2: fun(c,fun(d,assn)) ).

tff(sy_v_F,type,
    f: assn ).

tff(sy_v_I,type,
    i: set(nat) ).

tff(sy_v_P,type,
    p: assn ).

tff(sy_v_Q,type,
    q: assn ).

tff(sy_v_i,type,
    i2: nat ).

tff(sy_v_x,type,
    x: c ).

tff(sy_v_xi,type,
    xi: d ).

tff(sy_v_xs,type,
    xs: list(c) ).

tff(sy_v_xsi,type,
    xsi: list(d) ).

% Relevant facts (8818)
tff(fact_0_entailsI,axiom,
    ! [Pa: assn,Qa: assn] :
      ( ! [H: 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,Pa),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,Qa),H) )
     => aa(assn,$o,aa(assn,fun(assn,$o),entails,Pa),Qa) ) ).

% entailsI
tff(fact_1__092_060open_062P_A_092_060Longrightarrow_062_092_060_094sub_062A_Afalse_092_060close_062,axiom,
    aa(assn,$o,aa(assn,fun(assn,$o),entails,p),bot_bot(assn)) ).

% \<open>P \<Longrightarrow>\<^sub>A false\<close>
tff(fact_2_assms_I3_J,axiom,
    member(nat,i2,i) ).

% assms(3)
tff(fact_3_ent__iffI,axiom,
    ! [Aa2: assn,B2: assn] :
      ( aa(assn,$o,aa(assn,fun(assn,$o),entails,Aa2),B2)
     => ( aa(assn,$o,aa(assn,fun(assn,$o),entails,B2),Aa2)
       => ( Aa2 = B2 ) ) ) ).

% ent_iffI
tff(fact_4_ent__refl,axiom,
    ! [Pa: assn] : aa(assn,$o,aa(assn,fun(assn,$o),entails,Pa),Pa) ).

% ent_refl
tff(fact_5_ent__trans,axiom,
    ! [Pa: assn,Qa: assn,R: assn] :
      ( aa(assn,$o,aa(assn,fun(assn,$o),entails,Pa),Qa)
     => ( aa(assn,$o,aa(assn,fun(assn,$o),entails,Qa),R)
       => aa(assn,$o,aa(assn,fun(assn,$o),entails,Pa),R) ) ) ).

% ent_trans
tff(fact_6_is__entails,axiom,
    ! [Pa: assn,Qa: assn] :
      ( aa(assn,$o,aa(assn,fun(assn,$o),entails,Pa),Qa)
     => aa(assn,$o,aa(assn,fun(assn,$o),entails,Pa),Qa) ) ).

% is_entails
tff(fact_7_ent__false,axiom,
    ! [Pa: assn] : aa(assn,$o,aa(assn,fun(assn,$o),entails,bot_bot(assn)),Pa) ).

% ent_false
tff(fact_8_ent__fwd,axiom,
    ! [Pa: assn,H2: product_prod(heap_ext(product_unit),set(nat)),Qa: assn] :
      ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,Pa),H2)
     => ( aa(assn,$o,aa(assn,fun(assn,$o),entails,Pa),Qa)
       => aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,Qa),H2) ) ) ).

% ent_fwd
tff(fact_9_entailsD,axiom,
    ! [Pa: assn,Qa: assn,H2: product_prod(heap_ext(product_unit),set(nat))] :
      ( aa(assn,$o,aa(assn,fun(assn,$o),entails,Pa),Qa)
     => ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,Pa),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,Qa),H2) ) ) ).

% entailsD
tff(fact_10_entails__def,axiom,
    ! [Pa: assn,Qa: assn] :
      ( aa(assn,$o,aa(assn,fun(assn,$o),entails,Pa),Qa)
    <=> ! [H3: 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,Pa),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,Qa),H3) ) ) ).

% entails_def
tff(fact_11_list__assn__mono,axiom,
    ! [A: $tType,B: $tType,Pa: fun(A,fun(B,assn)),P: fun(A,fun(B,assn)),L: list(A),L2: list(B)] :
      ( ! [X: A,X2: B] : aa(assn,$o,aa(assn,fun(assn,$o),entails,aa(B,assn,aa(A,fun(B,assn),Pa,X),X2)),aa(B,assn,aa(A,fun(B,assn),P,X),X2))
     => aa(assn,$o,aa(assn,fun(assn,$o),entails,vEBT_List_list_assn(A,B,Pa,L,L2)),vEBT_List_list_assn(A,B,P,L,L2)) ) ).

% list_assn_mono
tff(fact_12_ent__false__iff,axiom,
    ! [Pa: assn] :
      ( aa(assn,$o,aa(assn,fun(assn,$o),entails,Pa),bot_bot(assn))
    <=> ! [H3: 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,Pa),H3) ) ).

% ent_false_iff
tff(fact_13_assms_I4_J,axiom,
    aa(assn,$o,aa(assn,fun(assn,$o),entails,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),vEBT_List_listI_assn(c,d,i,a2,aa(c,list(c),aa(nat,fun(c,list(c)),aa(list(c),fun(nat,fun(c,list(c))),list_update(c),xs),i2),x),aa(d,list(d),aa(nat,fun(d,list(d)),aa(list(d),fun(nat,fun(d,list(d))),list_update(d),xsi),i2),xi))),f)),q) ).

% assms(4)
tff(fact_14_ent__imp__entt,axiom,
    ! [Pa: assn,Qa: assn] :
      ( aa(assn,$o,aa(assn,fun(assn,$o),entails,Pa),Qa)
     => entailst(Pa,Qa) ) ).

% ent_imp_entt
tff(fact_15_Rep__assn__inject,axiom,
    ! [Xa: assn,Y: assn] :
      ( ( aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,Xa) = aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,Y) )
    <=> ( Xa = Y ) ) ).

% Rep_assn_inject
tff(fact_16_entt__refl,axiom,
    ! [Aa2: assn] : entailst(Aa2,Aa2) ).

% entt_refl
tff(fact_17_star__false__left,axiom,
    ! [Pa: assn] : ( aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),bot_bot(assn)),Pa) = bot_bot(assn) ) ).

% star_false_left
tff(fact_18_star__false__right,axiom,
    ! [Pa: assn] : ( aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Pa),bot_bot(assn)) = bot_bot(assn) ) ).

% star_false_right
tff(fact_19_assn__aci_I10_J,axiom,
    ! [A3: assn,B3: assn,C2: assn] : ( aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),A3),B3)),C2) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),A3),C2)),B3) ) ).

% assn_aci(10)
tff(fact_20_star__aci_I3_J,axiom,
    ! [A3: assn,B3: assn,C2: assn] : ( aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),A3),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),B3),C2)) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),B3),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),A3),C2)) ) ).

% star_aci(3)
tff(fact_21_star__aci_I2_J,axiom,
    ! [A3: assn,B3: assn] : ( aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),A3),B3) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),B3),A3) ) ).

% star_aci(2)
tff(fact_22_mod__false,axiom,
    ! [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,bot_bot(assn)),H2) ).

% mod_false
tff(fact_23_mod__starD,axiom,
    ! [Aa2: 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),Aa2),B2)),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,Aa2),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,B2),H22) ) ) ).

% mod_starD
tff(fact_24_mod__starE,axiom,
    ! [A3: 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),A3),B3)),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,B3),H_2) ) ) ).

% mod_starE
tff(fact_25_entt__trans,axiom,
    ! [Aa2: assn,B2: assn,C3: assn] :
      ( entailst(Aa2,B2)
     => ( entailst(B2,C3)
       => entailst(Aa2,C3) ) ) ).

% entt_trans
tff(fact_26_star__assoc,axiom,
    ! [A3: assn,B3: assn,C2: assn] : ( aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),A3),B3)),C2) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),A3),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),B3),C2)) ) ).

% star_assoc
tff(fact_27_entt__fr__drop,axiom,
    ! [Fa: assn,F: assn,Aa2: assn] :
      ( entailst(Fa,F)
     => entailst(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Fa),Aa2),F) ) ).

% entt_fr_drop
tff(fact_28_entt__fr__refl,axiom,
    ! [Fa: assn,F: assn,Aa2: assn] :
      ( entailst(Fa,F)
     => entailst(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Fa),Aa2),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),F),Aa2)) ) ).

% entt_fr_refl
tff(fact_29_entt__frame__fwd,axiom,
    ! [Pa: assn,Qa: assn,Aa2: assn,Fa: assn,B2: assn] :
      ( entailst(Pa,Qa)
     => ( entailst(Aa2,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Pa),Fa))
       => ( entailst(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Qa),Fa),B2)
         => entailst(Aa2,B2) ) ) ) ).

% entt_frame_fwd
tff(fact_30_entt__star__mono,axiom,
    ! [Aa2: assn,B2: assn,C3: assn,D2: assn] :
      ( entailst(Aa2,B2)
     => ( entailst(C3,D2)
       => entailst(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Aa2),C3),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),B2),D2)) ) ) ).

% entt_star_mono
tff(fact_31_assn__times__comm,axiom,
    ! [Pa: assn,Qa: assn] : ( aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Pa),Qa) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Qa),Pa) ) ).

% assn_times_comm
tff(fact_32_assn__times__assoc,axiom,
    ! [Pa: assn,Qa: 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),Pa),Qa)),R) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Pa),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Qa),R)) ) ).

% assn_times_assoc
tff(fact_33_mod__frame__fwd,axiom,
    ! [Ps: assn,H2: product_prod(heap_ext(product_unit),set(nat)),Pa: assn,R: assn,Fa: assn] :
      ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,Ps),H2)
     => ( aa(assn,$o,aa(assn,fun(assn,$o),entails,Pa),R)
       => ( aa(assn,$o,aa(assn,fun(assn,$o),entails,Ps),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Pa),Fa))
         => 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),R),Fa)),H2) ) ) ) ).

% mod_frame_fwd
tff(fact_34_ent__frame__fwd,axiom,
    ! [Pa: assn,R: assn,Ps: assn,Fa: assn,Qa: assn] :
      ( aa(assn,$o,aa(assn,fun(assn,$o),entails,Pa),R)
     => ( aa(assn,$o,aa(assn,fun(assn,$o),entails,Ps),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Pa),Fa))
       => ( aa(assn,$o,aa(assn,fun(assn,$o),entails,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),R),Fa)),Qa)
         => aa(assn,$o,aa(assn,fun(assn,$o),entails,Ps),Qa) ) ) ) ).

% ent_frame_fwd
tff(fact_35_ent__star__mono,axiom,
    ! [Pa: assn,P: assn,Qa: assn,Q: assn] :
      ( aa(assn,$o,aa(assn,fun(assn,$o),entails,Pa),P)
     => ( aa(assn,$o,aa(assn,fun(assn,$o),entails,Qa),Q)
       => aa(assn,$o,aa(assn,fun(assn,$o),entails,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Pa),Qa)),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),Q)) ) ) ).

% ent_star_mono
tff(fact_36_fr__rot__rhs,axiom,
    ! [Aa2: assn,B2: assn,C3: assn] :
      ( aa(assn,$o,aa(assn,fun(assn,$o),entails,Aa2),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),B2),C3))
     => aa(assn,$o,aa(assn,fun(assn,$o),entails,Aa2),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),C3),B2)) ) ).

% fr_rot_rhs
tff(fact_37_fr__refl,axiom,
    ! [Aa2: assn,B2: assn,C3: assn] :
      ( aa(assn,$o,aa(assn,fun(assn,$o),entails,Aa2),B2)
     => aa(assn,$o,aa(assn,fun(assn,$o),entails,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Aa2),C3)),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),B2),C3)) ) ).

% fr_refl
tff(fact_38_fr__rot,axiom,
    ! [Aa2: assn,B2: assn,C3: assn] :
      ( aa(assn,$o,aa(assn,fun(assn,$o),entails,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Aa2),B2)),C3)
     => aa(assn,$o,aa(assn,fun(assn,$o),entails,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),B2),Aa2)),C3) ) ).

% fr_rot
tff(fact_39_assms_I1_J,axiom,
    aa(assn,$o,aa(assn,fun(assn,$o),entails,p),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(d,assn,aa(c,fun(d,assn),a2,x),xi)),vEBT_List_listI_assn(c,d,aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),i),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert3(nat),i2),bot_bot(set(nat)))),a2,xs,xsi))),f)) ).

% assms(1)
tff(fact_40_list__update__overwrite,axiom,
    ! [A: $tType,Xsa: list(A),Ia: nat,Xa: A,Y: A] : ( aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xsa),Ia),Xa)),Ia),Y) = aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xsa),Ia),Y) ) ).

% list_update_overwrite
tff(fact_41_bot__apply,axiom,
    ! [B: $tType,A: $tType] :
      ( bot(A)
     => ! [Xa: B] : ( aa(B,A,bot_bot(fun(B,A)),Xa) = bot_bot(A) ) ) ).

% bot_apply
tff(fact_42_False,axiom,
    aa(list(c),nat,size_size(list(c)),xs) != aa(list(d),nat,size_size(list(d)),xsi) ).

% False
tff(fact_43_assms_I2_J,axiom,
    aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),i2),aa(list(c),nat,size_size(list(c)),xs)) ).

% assms(2)
tff(fact_44_frame__inference__init,axiom,
    ! [Pa: assn,Qa: assn,Fa: assn] :
      ( fI_QUERY(Pa,Qa,Fa)
     => aa(assn,$o,aa(assn,fun(assn,$o),entails,Pa),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Qa),Fa)) ) ).

% frame_inference_init
tff(fact_45_mem__Collect__eq,axiom,
    ! [A: $tType,A3: A,Pa: fun(A,$o)] :
      ( member(A,A3,aa(fun(A,$o),set(A),collect(A),Pa))
    <=> aa(A,$o,Pa,A3) ) ).

% mem_Collect_eq
tff(fact_46_Collect__mem__eq,axiom,
    ! [A: $tType,Aa2: set(A)] : ( aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_a(set(A),fun(A,$o)),Aa2)) = Aa2 ) ).

% Collect_mem_eq
tff(fact_47_Collect__cong,axiom,
    ! [A: $tType,Pa: fun(A,$o),Qa: fun(A,$o)] :
      ( ! [X: A] :
          ( aa(A,$o,Pa,X)
        <=> aa(A,$o,Qa,X) )
     => ( aa(fun(A,$o),set(A),collect(A),Pa) = aa(fun(A,$o),set(A),collect(A),Qa) ) ) ).

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

% ext
tff(fact_49_FI__QUERY__def,axiom,
    ! [Pa: assn,Qa: assn,Fa: assn] :
      ( fI_QUERY(Pa,Qa,Fa)
    <=> aa(assn,$o,aa(assn,fun(assn,$o),entails,Pa),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Qa),Fa)) ) ).

% FI_QUERY_def
tff(fact_50_sngr__same__false,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [P2: ref(A),Xa: A,Y: A] : ( aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),sngr_assn(A,P2,Xa)),sngr_assn(A,P2,Y)) = bot_bot(assn) ) ) ).

% sngr_same_false
tff(fact_51_ent__wand__frameI,axiom,
    ! [Qa: assn,R: assn,Fa: assn,S: assn,Pa: assn,X3: assn] :
      ( aa(assn,$o,aa(assn,fun(assn,$o),entails,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),wand_assn(Qa,R)),Fa)),S)
     => ( aa(assn,$o,aa(assn,fun(assn,$o),entails,Pa),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Fa),X3))
       => ( aa(assn,$o,aa(assn,fun(assn,$o),entails,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Qa),X3)),R)
         => aa(assn,$o,aa(assn,fun(assn,$o),entails,Pa),S) ) ) ) ).

% ent_wand_frameI
tff(fact_52_ent__wandI,axiom,
    ! [Qa: assn,Pa: assn,R: assn] :
      ( aa(assn,$o,aa(assn,fun(assn,$o),entails,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Qa),Pa)),R)
     => aa(assn,$o,aa(assn,fun(assn,$o),entails,Pa),wand_assn(Qa,R)) ) ).

% ent_wandI
tff(fact_53_ent__mp,axiom,
    ! [Pa: assn,Qa: assn] : aa(assn,$o,aa(assn,fun(assn,$o),entails,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Pa),wand_assn(Pa,Qa))),Qa) ).

% ent_mp
tff(fact_54_snga__same__false,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [P2: array(A),Xa: list(A),Y: list(A)] : ( aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),snga_assn(A,P2,Xa)),snga_assn(A,P2,Y)) = bot_bot(assn) ) ) ).

% snga_same_false
tff(fact_55_length__list__update,axiom,
    ! [A: $tType,Xsa: list(A),Ia: nat,Xa: A] : ( aa(list(A),nat,size_size(list(A)),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xsa),Ia),Xa)) = aa(list(A),nat,size_size(list(A)),Xsa) ) ).

% length_list_update
tff(fact_56_order__less__imp__not__less,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xa) ) ) ).

% order_less_imp_not_less
tff(fact_57_order__less__imp__not__eq2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
         => ( Y != Xa ) ) ) ).

% order_less_imp_not_eq2
tff(fact_58_order__less__imp__not__eq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
         => ( Xa != Y ) ) ) ).

% order_less_imp_not_eq
tff(fact_59_linorder__less__linear,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
          | ( Xa = Y )
          | aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xa) ) ) ).

% linorder_less_linear
tff(fact_60_order__less__imp__triv,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xa: A,Y: A,Pa: $o] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xa)
           => (Pa) ) ) ) ).

% order_less_imp_triv
tff(fact_61_order__less__not__sym,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xa) ) ) ).

% order_less_not_sym
tff(fact_62_order__less__subst2,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A3: A,B3: A,F2: fun(A,B),C2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,B3)),C2)
           => ( ! [X: A,Y2: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y2)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,X)),aa(A,B,F2,Y2)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,A3)),C2) ) ) ) ) ).

% order_less_subst2
tff(fact_63_order__less__subst1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A3: A,F2: fun(B,A),B3: B,C2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(B,A,F2,B3))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),B3),C2)
           => ( ! [X: B,Y2: B] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X),Y2)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,F2,X)),aa(B,A,F2,Y2)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(B,A,F2,C2)) ) ) ) ) ).

% order_less_subst1
tff(fact_64_order__less__irrefl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xa: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Xa) ) ).

% order_less_irrefl
tff(fact_65_ord__less__eq__subst,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & ord(A) )
     => ! [A3: A,B3: A,F2: fun(A,B),C2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
         => ( ( aa(A,B,F2,B3) = C2 )
           => ( ! [X: A,Y2: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y2)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,X)),aa(A,B,F2,Y2)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,A3)),C2) ) ) ) ) ).

% ord_less_eq_subst
tff(fact_66_ord__eq__less__subst,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & ord(A) )
     => ! [A3: A,F2: fun(B,A),B3: B,C2: B] :
          ( ( A3 = aa(B,A,F2,B3) )
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),B3),C2)
           => ( ! [X: B,Y2: B] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X),Y2)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,F2,X)),aa(B,A,F2,Y2)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(B,A,F2,C2)) ) ) ) ) ).

% ord_eq_less_subst
tff(fact_67_order__less__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xa: A,Y: A,Z2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Z2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Z2) ) ) ) ).

% order_less_trans
tff(fact_68_order__less__asym_H,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A3) ) ) ).

% order_less_asym'
tff(fact_69_linorder__neq__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Y: A] :
          ( ( Xa != Y )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
            | aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xa) ) ) ) ).

% linorder_neq_iff
tff(fact_70_order__less__asym,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xa) ) ) ).

% order_less_asym
tff(fact_71_linorder__neqE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Y: A] :
          ( ( Xa != Y )
         => ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xa) ) ) ) ).

% linorder_neqE
tff(fact_72_neq__if__length__neq,axiom,
    ! [A: $tType,Xsa: list(A),Ys: list(A)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xsa) != aa(list(A),nat,size_size(list(A)),Ys) )
     => ( Xsa != Ys ) ) ).

% neq_if_length_neq
tff(fact_73_Ex__list__of__length,axiom,
    ! [A: $tType,N: nat] :
    ? [Xs: list(A)] : ( aa(list(A),nat,size_size(list(A)),Xs) = N ) ).

% Ex_list_of_length
tff(fact_74_length__induct,axiom,
    ! [A: $tType,Pa: fun(list(A),$o),Xsa: list(A)] :
      ( ! [Xs: list(A)] :
          ( ! [Ys2: list(A)] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),Ys2)),aa(list(A),nat,size_size(list(A)),Xs))
             => aa(list(A),$o,Pa,Ys2) )
         => aa(list(A),$o,Pa,Xs) )
     => aa(list(A),$o,Pa,Xsa) ) ).

% length_induct
tff(fact_75_dual__order_Ostrict__implies__not__eq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A3)
         => ( A3 != B3 ) ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_76_order_Ostrict__implies__not__eq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
         => ( A3 != B3 ) ) ) ).

% order.strict_implies_not_eq
tff(fact_77_dual__order_Ostrict__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B3: A,A3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),B3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),A3) ) ) ) ).

% dual_order.strict_trans
tff(fact_78_not__less__iff__gr__or__eq,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Y: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xa)
            | ( Xa = Y ) ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_79_order_Ostrict__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),C2) ) ) ) ).

% order.strict_trans
tff(fact_80_linorder__less__wlog,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Pa: fun(A,fun(A,$o)),A3: A,B3: A] :
          ( ! [A4: A,B4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),B4)
             => aa(A,$o,aa(A,fun(A,$o),Pa,A4),B4) )
         => ( ! [A4: A] : aa(A,$o,aa(A,fun(A,$o),Pa,A4),A4)
           => ( ! [A4: A,B4: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),Pa,B4),A4)
                 => aa(A,$o,aa(A,fun(A,$o),Pa,A4),B4) )
             => aa(A,$o,aa(A,fun(A,$o),Pa,A3),B3) ) ) ) ) ).

% linorder_less_wlog
tff(fact_81_exists__least__iff,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [Pa: fun(A,$o)] :
          ( ? [X_12: A] : aa(A,$o,Pa,X_12)
        <=> ? [N2: A] :
              ( aa(A,$o,Pa,N2)
              & ! [M: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),M),N2)
                 => ~ aa(A,$o,Pa,M) ) ) ) ) ).

% exists_least_iff
tff(fact_82_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_83_dual__order_Oasym,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A3)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3) ) ) ).

% dual_order.asym
tff(fact_84_linorder__cases,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Y: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
         => ( ( Xa != Y )
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xa) ) ) ) ).

% linorder_cases
tff(fact_85_antisym__conv3,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Y: A,Xa: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xa)
         => ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
          <=> ( Xa = Y ) ) ) ) ).

% antisym_conv3
tff(fact_86_less__induct,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [Pa: fun(A,$o),A3: A] :
          ( ! [X: A] :
              ( ! [Y3: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y3),X)
                 => aa(A,$o,Pa,Y3) )
             => aa(A,$o,Pa,X) )
         => aa(A,$o,Pa,A3) ) ) ).

% less_induct
tff(fact_87_ord__less__eq__trans,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
         => ( ( B3 = C2 )
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),C2) ) ) ) ).

% ord_less_eq_trans
tff(fact_88_ord__eq__less__trans,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A3: A,B3: A,C2: A] :
          ( ( A3 = B3 )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),C2) ) ) ) ).

% ord_eq_less_trans
tff(fact_89_order_Oasym,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A3) ) ) ).

% order.asym
tff(fact_90_less__imp__neq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
         => ( Xa != Y ) ) ) ).

% less_imp_neq
tff(fact_91_dense,axiom,
    ! [A: $tType] :
      ( dense_order(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
         => ? [Z3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Z3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z3),Y) ) ) ) ).

% dense
tff(fact_92_gt__ex,axiom,
    ! [A: $tType] :
      ( no_top(A)
     => ! [Xa: A] :
        ? [X_1: A] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),X_1) ) ).

% gt_ex
tff(fact_93_lt__ex,axiom,
    ! [A: $tType] :
      ( no_bot(A)
     => ! [Xa: A] :
        ? [Y2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y2),Xa) ) ).

% lt_ex
tff(fact_94_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_95_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_96_list__assn__aux__ineq__len,axiom,
    ! [B: $tType,A: $tType,L: list(A),Li: list(B),Aa2: fun(A,fun(B,assn))] :
      ( ( aa(list(A),nat,size_size(list(A)),L) != aa(list(B),nat,size_size(list(B)),Li) )
     => ( vEBT_List_list_assn(A,B,Aa2,L,Li) = bot_bot(assn) ) ) ).

% list_assn_aux_ineq_len
tff(fact_97_bot__fun__def,axiom,
    ! [A: $tType,B: $tType] :
      ( bot(B)
     => ! [X4: A] : ( aa(A,B,bot_bot(fun(A,B)),X4) = bot_bot(B) ) ) ).

% bot_fun_def
tff(fact_98_list__update__swap,axiom,
    ! [A: $tType,Ia: nat,I: nat,Xsa: list(A),Xa: A,X5: A] :
      ( ( Ia != I )
     => ( aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xsa),Ia),Xa)),I),X5) = aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xsa),I),X5)),Ia),Xa) ) ) ).

% list_update_swap
tff(fact_99_insert__Diff__single,axiom,
    ! [A: $tType,A3: A,Aa2: set(A)] : ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),bot_bot(set(A))))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),Aa2) ) ).

% insert_Diff_single
tff(fact_100_insert__Diff1,axiom,
    ! [A: $tType,Xa: A,B2: set(A),Aa2: set(A)] :
      ( member(A,Xa,B2)
     => ( 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)),insert3(A),Xa),Aa2)),B2) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),B2) ) ) ).

% insert_Diff1
tff(fact_101_Diff__insert0,axiom,
    ! [A: $tType,Xa: A,Aa2: set(A),B2: set(A)] :
      ( ~ member(A,Xa,Aa2)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),B2)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),B2) ) ) ).

% Diff_insert0
tff(fact_102_Diff__empty,axiom,
    ! [A: $tType,Aa2: set(A)] : ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),bot_bot(set(A))) = Aa2 ) ).

% Diff_empty
tff(fact_103_empty__Diff,axiom,
    ! [A: $tType,Aa2: set(A)] : ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),bot_bot(set(A))),Aa2) = bot_bot(set(A)) ) ).

% empty_Diff
tff(fact_104_Diff__cancel,axiom,
    ! [A: $tType,Aa2: set(A)] : ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),Aa2) = bot_bot(set(A)) ) ).

% Diff_cancel
tff(fact_105_singletonI,axiom,
    ! [A: $tType,A3: A] : member(A,A3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),bot_bot(set(A)))) ).

% singletonI
tff(fact_106_Diff__insert,axiom,
    ! [A: $tType,Aa2: set(A),A3: A,B2: set(A)] : ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),B2)) = 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)),Aa2),B2)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),bot_bot(set(A)))) ) ).

% Diff_insert
tff(fact_107_insert__Diff,axiom,
    ! [A: $tType,A3: A,Aa2: set(A)] :
      ( member(A,A3,Aa2)
     => ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),bot_bot(set(A))))) = Aa2 ) ) ).

% insert_Diff
tff(fact_108_Diff__insert2,axiom,
    ! [A: $tType,Aa2: set(A),A3: A,B2: set(A)] : ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),B2)) = 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)),Aa2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),bot_bot(set(A))))),B2) ) ).

% Diff_insert2
tff(fact_109_insert__minus__eq,axiom,
    ! [A: $tType,Xa: A,Y: A,Aa2: set(A)] :
      ( ( Xa != 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)),insert3(A),Xa),Aa2)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Y),bot_bot(set(A)))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Y),bot_bot(set(A))))) ) ) ).

% insert_minus_eq
tff(fact_110_Diff__insert__absorb,axiom,
    ! [A: $tType,Xa: A,Aa2: set(A)] :
      ( ~ member(A,Xa,Aa2)
     => ( 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)),insert3(A),Xa),Aa2)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A)))) = Aa2 ) ) ).

% Diff_insert_absorb
tff(fact_111_empty__Collect__eq,axiom,
    ! [A: $tType,Pa: fun(A,$o)] :
      ( ( bot_bot(set(A)) = aa(fun(A,$o),set(A),collect(A),Pa) )
    <=> ! [X6: A] : ~ aa(A,$o,Pa,X6) ) ).

% empty_Collect_eq
tff(fact_112_Collect__empty__eq,axiom,
    ! [A: $tType,Pa: fun(A,$o)] :
      ( ( aa(fun(A,$o),set(A),collect(A),Pa) = bot_bot(set(A)) )
    <=> ! [X6: A] : ~ aa(A,$o,Pa,X6) ) ).

% Collect_empty_eq
tff(fact_113_all__not__in__conv,axiom,
    ! [A: $tType,Aa2: set(A)] :
      ( ! [X6: A] : ~ member(A,X6,Aa2)
    <=> ( Aa2 = bot_bot(set(A)) ) ) ).

% all_not_in_conv
tff(fact_114_empty__iff,axiom,
    ! [A: $tType,C2: A] : ~ member(A,C2,bot_bot(set(A))) ).

% empty_iff
tff(fact_115_insertCI,axiom,
    ! [A: $tType,A3: A,B2: set(A),B3: A] :
      ( ( ~ member(A,A3,B2)
       => ( A3 = B3 ) )
     => member(A,A3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),B3),B2)) ) ).

% insertCI
tff(fact_116_insert__iff,axiom,
    ! [A: $tType,A3: A,B3: A,Aa2: set(A)] :
      ( member(A,A3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),B3),Aa2))
    <=> ( ( A3 = B3 )
        | member(A,A3,Aa2) ) ) ).

% insert_iff
tff(fact_117_insert__absorb2,axiom,
    ! [A: $tType,Xa: A,Aa2: set(A)] : ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),Aa2)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),Aa2) ) ).

% insert_absorb2
tff(fact_118_DiffI,axiom,
    ! [A: $tType,C2: A,Aa2: set(A),B2: set(A)] :
      ( member(A,C2,Aa2)
     => ( ~ member(A,C2,B2)
       => member(A,C2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),B2)) ) ) ).

% DiffI
tff(fact_119_Diff__iff,axiom,
    ! [A: $tType,C2: A,Aa2: set(A),B2: set(A)] :
      ( member(A,C2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),B2))
    <=> ( member(A,C2,Aa2)
        & ~ member(A,C2,B2) ) ) ).

% Diff_iff
tff(fact_120_Diff__idemp,axiom,
    ! [A: $tType,Aa2: set(A),B2: 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)),Aa2),B2)),B2) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),B2) ) ).

% Diff_idemp
tff(fact_121_not__psubset__empty,axiom,
    ! [A: $tType,Aa2: set(A)] : ~ aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),Aa2),bot_bot(set(A))) ).

% not_psubset_empty
tff(fact_122_psubset__imp__ex__mem,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),Aa2),B2)
     => ? [B4: A] : member(A,B4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B2),Aa2)) ) ).

% psubset_imp_ex_mem
tff(fact_123_remove__subset,axiom,
    ! [A: $tType,Xa: A,S: set(A)] :
      ( member(A,Xa,S)
     => aa(set(A),$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)),insert3(A),Xa),bot_bot(set(A))))),S) ) ).

% remove_subset
tff(fact_124_memb__imp__not__empty,axiom,
    ! [A: $tType,Xa: A,S: set(A)] :
      ( member(A,Xa,S)
     => ( S != bot_bot(set(A)) ) ) ).

% memb_imp_not_empty
tff(fact_125_set__notEmptyE,axiom,
    ! [A: $tType,S: set(A)] :
      ( ( S != bot_bot(set(A)) )
     => ~ ! [X: A] : ~ member(A,X,S) ) ).

% set_notEmptyE
tff(fact_126_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_127_ex__in__conv,axiom,
    ! [A: $tType,Aa2: set(A)] :
      ( ? [X6: A] : member(A,X6,Aa2)
    <=> ( Aa2 != bot_bot(set(A)) ) ) ).

% ex_in_conv
tff(fact_128_equals0I,axiom,
    ! [A: $tType,Aa2: set(A)] :
      ( ! [Y2: A] : ~ member(A,Y2,Aa2)
     => ( Aa2 = bot_bot(set(A)) ) ) ).

% equals0I
tff(fact_129_equals0D,axiom,
    ! [A: $tType,Aa2: set(A),A3: A] :
      ( ( Aa2 = bot_bot(set(A)) )
     => ~ member(A,A3,Aa2) ) ).

% equals0D
tff(fact_130_emptyE,axiom,
    ! [A: $tType,A3: A] : ~ member(A,A3,bot_bot(set(A))) ).

% emptyE
tff(fact_131_insertE,axiom,
    ! [A: $tType,A3: A,B3: A,Aa2: set(A)] :
      ( member(A,A3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),B3),Aa2))
     => ( ( A3 != B3 )
       => member(A,A3,Aa2) ) ) ).

% insertE
tff(fact_132_insertI1,axiom,
    ! [A: $tType,A3: A,B2: set(A)] : member(A,A3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),B2)) ).

% insertI1
tff(fact_133_insertI2,axiom,
    ! [A: $tType,A3: A,B2: set(A),B3: A] :
      ( member(A,A3,B2)
     => member(A,A3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),B3),B2)) ) ).

% insertI2
tff(fact_134_Set_Oset__insert,axiom,
    ! [A: $tType,Xa: A,Aa2: set(A)] :
      ( member(A,Xa,Aa2)
     => ~ ! [B5: set(A)] :
            ( ( Aa2 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),B5) )
           => member(A,Xa,B5) ) ) ).

% Set.set_insert
tff(fact_135_insert__ident,axiom,
    ! [A: $tType,Xa: A,Aa2: set(A),B2: set(A)] :
      ( ~ member(A,Xa,Aa2)
     => ( ~ member(A,Xa,B2)
       => ( ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),Aa2) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),B2) )
        <=> ( Aa2 = B2 ) ) ) ) ).

% insert_ident
tff(fact_136_insert__absorb,axiom,
    ! [A: $tType,A3: A,Aa2: set(A)] :
      ( member(A,A3,Aa2)
     => ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),Aa2) = Aa2 ) ) ).

% insert_absorb
tff(fact_137_insert__eq__iff,axiom,
    ! [A: $tType,A3: A,Aa2: set(A),B3: A,B2: set(A)] :
      ( ~ member(A,A3,Aa2)
     => ( ~ member(A,B3,B2)
       => ( ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),Aa2) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),B3),B2) )
        <=> $ite(
              A3 = B3,
              Aa2 = B2,
              ? [C4: set(A)] :
                ( ( Aa2 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),B3),C4) )
                & ~ member(A,B3,C4)
                & ( B2 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),C4) )
                & ~ member(A,A3,C4) ) ) ) ) ) ).

% insert_eq_iff
tff(fact_138_insert__commute,axiom,
    ! [A: $tType,Xa: A,Y: A,Aa2: set(A)] : ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Y),Aa2)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Y),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),Aa2)) ) ).

% insert_commute
tff(fact_139_mk__disjoint__insert,axiom,
    ! [A: $tType,A3: A,Aa2: set(A)] :
      ( member(A,A3,Aa2)
     => ? [B5: set(A)] :
          ( ( Aa2 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),B5) )
          & ~ member(A,A3,B5) ) ) ).

% mk_disjoint_insert
tff(fact_140_DiffE,axiom,
    ! [A: $tType,C2: A,Aa2: set(A),B2: set(A)] :
      ( member(A,C2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),B2))
     => ~ ( member(A,C2,Aa2)
         => member(A,C2,B2) ) ) ).

% DiffE
tff(fact_141_DiffD1,axiom,
    ! [A: $tType,C2: A,Aa2: set(A),B2: set(A)] :
      ( member(A,C2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),B2))
     => member(A,C2,Aa2) ) ).

% DiffD1
tff(fact_142_DiffD2,axiom,
    ! [A: $tType,C2: A,Aa2: set(A),B2: set(A)] :
      ( member(A,C2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),B2))
     => ~ member(A,C2,B2) ) ).

% DiffD2
tff(fact_143_singleton__inject,axiom,
    ! [A: $tType,A3: A,B3: A] :
      ( ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),bot_bot(set(A))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),B3),bot_bot(set(A))) )
     => ( A3 = B3 ) ) ).

% singleton_inject
tff(fact_144_insert__not__empty,axiom,
    ! [A: $tType,A3: A,Aa2: set(A)] : ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),Aa2) != bot_bot(set(A)) ) ).

% insert_not_empty
tff(fact_145_doubleton__eq__iff,axiom,
    ! [A: $tType,A3: A,B3: A,C2: A,D3: A] :
      ( ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),B3),bot_bot(set(A)))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),C2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),D3),bot_bot(set(A)))) )
    <=> ( ( ( A3 = C2 )
          & ( B3 = D3 ) )
        | ( ( A3 = D3 )
          & ( B3 = C2 ) ) ) ) ).

% doubleton_eq_iff
tff(fact_146_singleton__iff,axiom,
    ! [A: $tType,B3: A,A3: A] :
      ( member(A,B3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),bot_bot(set(A))))
    <=> ( B3 = A3 ) ) ).

% singleton_iff
tff(fact_147_singletonD,axiom,
    ! [A: $tType,B3: A,A3: A] :
      ( member(A,B3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),bot_bot(set(A))))
     => ( B3 = A3 ) ) ).

% singletonD
tff(fact_148_insert__Diff__if,axiom,
    ! [A: $tType,Xa: A,Aa2: set(A),B2: set(A)] :
      ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),Aa2)),B2) = $ite(member(A,Xa,B2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),B2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),B2))) ) ).

% insert_Diff_if
tff(fact_149_set__minus__singleton__eq,axiom,
    ! [A: $tType,Xa: A,X3: set(A)] :
      ( ~ member(A,Xa,X3)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),X3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A)))) = X3 ) ) ).

% set_minus_singleton_eq
tff(fact_150_listI__assn__reinsert,axiom,
    ! [B: $tType,A: $tType,Pa: assn,Aa2: fun(A,fun(B,assn)),Xsa: list(A),Ia: nat,Xsia: list(B),Ia2: set(nat),Fa: assn,Qa: assn] :
      ( aa(assn,$o,aa(assn,fun(assn,$o),entails,Pa),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),Aa2,aa(nat,A,nth(A,Xsa),Ia)),aa(nat,B,nth(B,Xsia),Ia))),vEBT_List_listI_assn(A,B,aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),Ia2),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert3(nat),Ia),bot_bot(set(nat)))),Aa2,Xsa,Xsia))),Fa))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),aa(list(A),nat,size_size(list(A)),Xsa))
       => ( member(nat,Ia,Ia2)
         => ( aa(assn,$o,aa(assn,fun(assn,$o),entails,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),vEBT_List_listI_assn(A,B,Ia2,Aa2,Xsa,Xsia)),Fa)),Qa)
           => aa(assn,$o,aa(assn,fun(assn,$o),entails,Pa),Qa) ) ) ) ) ).

% listI_assn_reinsert
tff(fact_151_the__elem__eq,axiom,
    ! [A: $tType,Xa: A] : ( the_elem(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A)))) = Xa ) ).

% the_elem_eq
tff(fact_152_minus__apply,axiom,
    ! [A: $tType,B: $tType] :
      ( minus(A)
     => ! [Aa2: fun(B,A),B2: fun(B,A),Xa: 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)),Aa2),B2),Xa) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(B,A,Aa2,Xa)),aa(B,A,B2,Xa)) ) ) ).

% minus_apply
tff(fact_153_listI__assn__extract,axiom,
    ! [A: $tType,B: $tType,Ia: nat,Ia2: set(nat),Xsa: list(A),Aa2: fun(A,fun(B,assn)),Xsia: list(B)] :
      ( member(nat,Ia,Ia2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),aa(list(A),nat,size_size(list(A)),Xsa))
       => ( vEBT_List_listI_assn(A,B,Ia2,Aa2,Xsa,Xsia) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),Aa2,aa(nat,A,nth(A,Xsa),Ia)),aa(nat,B,nth(B,Xsia),Ia))),vEBT_List_listI_assn(A,B,aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),Ia2),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert3(nat),Ia),bot_bot(set(nat)))),Aa2,Xsa,Xsia)) ) ) ) ).

% listI_assn_extract
tff(fact_154_is__singletonI,axiom,
    ! [A: $tType,Xa: A] : is_singleton(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A)))) ).

% is_singletonI
tff(fact_155_inf__period_I2_J,axiom,
    ! [A: $tType] :
      ( ( comm_ring(A)
        & dvd(A) )
     => ! [Pa: fun(A,$o),D2: A,Qa: fun(A,$o)] :
          ( ! [X: A,K: A] :
              ( aa(A,$o,Pa,X)
            <=> aa(A,$o,Pa,aa(A,A,aa(A,fun(A,A),minus_minus(A),X),aa(A,A,aa(A,fun(A,A),times_times(A),K),D2))) )
         => ( ! [X: A,K: A] :
                ( aa(A,$o,Qa,X)
              <=> aa(A,$o,Qa,aa(A,A,aa(A,fun(A,A),minus_minus(A),X),aa(A,A,aa(A,fun(A,A),times_times(A),K),D2))) )
           => ! [X4: A,K2: A] :
                ( ( aa(A,$o,Pa,X4)
                  | aa(A,$o,Qa,X4) )
              <=> ( aa(A,$o,Pa,aa(A,A,aa(A,fun(A,A),minus_minus(A),X4),aa(A,A,aa(A,fun(A,A),times_times(A),K2),D2)))
                  | aa(A,$o,Qa,aa(A,A,aa(A,fun(A,A),minus_minus(A),X4),aa(A,A,aa(A,fun(A,A),times_times(A),K2),D2))) ) ) ) ) ) ).

% inf_period(2)
tff(fact_156_inf__period_I1_J,axiom,
    ! [A: $tType] :
      ( ( comm_ring(A)
        & dvd(A) )
     => ! [Pa: fun(A,$o),D2: A,Qa: fun(A,$o)] :
          ( ! [X: A,K: A] :
              ( aa(A,$o,Pa,X)
            <=> aa(A,$o,Pa,aa(A,A,aa(A,fun(A,A),minus_minus(A),X),aa(A,A,aa(A,fun(A,A),times_times(A),K),D2))) )
         => ( ! [X: A,K: A] :
                ( aa(A,$o,Qa,X)
              <=> aa(A,$o,Qa,aa(A,A,aa(A,fun(A,A),minus_minus(A),X),aa(A,A,aa(A,fun(A,A),times_times(A),K),D2))) )
           => ! [X4: A,K2: A] :
                ( ( aa(A,$o,Pa,X4)
                  & aa(A,$o,Qa,X4) )
              <=> ( aa(A,$o,Pa,aa(A,A,aa(A,fun(A,A),minus_minus(A),X4),aa(A,A,aa(A,fun(A,A),times_times(A),K2),D2)))
                  & aa(A,$o,Qa,aa(A,A,aa(A,fun(A,A),minus_minus(A),X4),aa(A,A,aa(A,fun(A,A),times_times(A),K2),D2))) ) ) ) ) ) ).

% inf_period(1)
tff(fact_157_left__diff__distrib,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A3: A,B3: 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),B3)),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),B3),C2)) ) ) ).

% left_diff_distrib
tff(fact_158_right__diff__distrib,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A3: A,B3: 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),B3),C2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)) ) ) ).

% right_diff_distrib
tff(fact_159_left__diff__distrib_H,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [B3: 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),B3),C2)),A3) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B3),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)) ) ) ).

% left_diff_distrib'
tff(fact_160_right__diff__distrib_H,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A3: A,B3: 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),B3),C2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)) ) ) ).

% right_diff_distrib'
tff(fact_161_diff__strict__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B3: A,D3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
         => ( aa(A,$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),B3),D3)) ) ) ) ).

% diff_strict_mono
tff(fact_162_nth__list__update__neq,axiom,
    ! [A: $tType,Ia: nat,J: nat,Xsa: list(A),Xa: A] :
      ( ( Ia != J )
     => ( aa(nat,A,nth(A,aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xsa),Ia),Xa)),J) = aa(nat,A,nth(A,Xsa),J) ) ) ).

% nth_list_update_neq
tff(fact_163_list__update__id,axiom,
    ! [A: $tType,Xsa: list(A),Ia: nat] : ( aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xsa),Ia),aa(nat,A,nth(A,Xsa),Ia)) = Xsa ) ).

% list_update_id
tff(fact_164_nth__list__update__eq,axiom,
    ! [A: $tType,Ia: nat,Xsa: list(A),Xa: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),aa(list(A),nat,size_size(list(A)),Xsa))
     => ( aa(nat,A,nth(A,aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xsa),Ia),Xa)),Ia) = Xa ) ) ).

% nth_list_update_eq
tff(fact_165_nth__update__invalid,axiom,
    ! [A: $tType,Ia: nat,L: list(A),J: nat,Xa: A] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),aa(list(A),nat,size_size(list(A)),L))
     => ( aa(nat,A,nth(A,aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),L),J),Xa)),Ia) = aa(nat,A,nth(A,L),Ia) ) ) ).

% nth_update_invalid
tff(fact_166_psubset__trans,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A),C3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),Aa2),B2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),B2),C3)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),Aa2),C3) ) ) ).

% psubset_trans
tff(fact_167_psubsetD,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A),C2: A] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),Aa2),B2)
     => ( member(A,C2,Aa2)
       => member(A,C2,B2) ) ) ).

% psubsetD
tff(fact_168_list__eq__iff__nth__eq,axiom,
    ! [A: $tType,Xsa: list(A),Ys: list(A)] :
      ( ( Xsa = Ys )
    <=> ( ( aa(list(A),nat,size_size(list(A)),Xsa) = aa(list(A),nat,size_size(list(A)),Ys) )
        & ! [I2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xsa))
           => ( aa(nat,A,nth(A,Xsa),I2) = aa(nat,A,nth(A,Ys),I2) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_169_Skolem__list__nth,axiom,
    ! [A: $tType,K3: nat,Pa: fun(nat,fun(A,$o))] :
      ( ! [I2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),K3)
         => ? [X_12: A] : aa(A,$o,aa(nat,fun(A,$o),Pa,I2),X_12) )
    <=> ? [Xs2: list(A)] :
          ( ( aa(list(A),nat,size_size(list(A)),Xs2) = K3 )
          & ! [I2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),K3)
             => aa(A,$o,aa(nat,fun(A,$o),Pa,I2),aa(nat,A,nth(A,Xs2),I2)) ) ) ) ).

% Skolem_list_nth
tff(fact_170_nth__equalityI,axiom,
    ! [A: $tType,Xsa: list(A),Ys: list(A)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xsa) = aa(list(A),nat,size_size(list(A)),Ys) )
     => ( ! [I3: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),Xsa))
           => ( aa(nat,A,nth(A,Xsa),I3) = aa(nat,A,nth(A,Ys),I3) ) )
       => ( Xsa = Ys ) ) ) ).

% nth_equalityI
tff(fact_171_obtain__list__from__elements,axiom,
    ! [A: $tType,N: nat,Pa: fun(A,fun(nat,$o))] :
      ( ! [I3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),N)
         => ? [Li2: A] : aa(nat,$o,aa(A,fun(nat,$o),Pa,Li2),I3) )
     => ~ ! [L3: list(A)] :
            ( ( aa(list(A),nat,size_size(list(A)),L3) = N )
           => ~ ! [I4: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),N)
                 => aa(nat,$o,aa(A,fun(nat,$o),Pa,aa(nat,A,nth(A,L3),I4)),I4) ) ) ) ).

% obtain_list_from_elements
tff(fact_172_is__singleton__the__elem,axiom,
    ! [A: $tType,Aa2: set(A)] :
      ( is_singleton(A,Aa2)
    <=> ( Aa2 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),the_elem(A,Aa2)),bot_bot(set(A))) ) ) ).

% is_singleton_the_elem
tff(fact_173_is__singletonI_H,axiom,
    ! [A: $tType,Aa2: set(A)] :
      ( ( Aa2 != bot_bot(set(A)) )
     => ( ! [X: A,Y2: A] :
            ( member(A,X,Aa2)
           => ( member(A,Y2,Aa2)
             => ( X = Y2 ) ) )
       => is_singleton(A,Aa2) ) ) ).

% is_singletonI'
tff(fact_174_nth__list__update,axiom,
    ! [A: $tType,Ia: nat,Xsa: list(A),Xa: A,J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),aa(list(A),nat,size_size(list(A)),Xsa))
     => ( aa(nat,A,nth(A,aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xsa),Ia),Xa)),J) = $ite(Ia = J,Xa,aa(nat,A,nth(A,Xsa),J)) ) ) ).

% nth_list_update
tff(fact_175_list__update__same__conv,axiom,
    ! [A: $tType,Ia: nat,Xsa: list(A),Xa: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),aa(list(A),nat,size_size(list(A)),Xsa))
     => ( ( aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xsa),Ia),Xa) = Xsa )
      <=> ( aa(nat,A,nth(A,Xsa),Ia) = Xa ) ) ) ).

% list_update_same_conv
tff(fact_176_nth__list__update_H,axiom,
    ! [A: $tType,L: list(A),Ia: nat,Xa: A,J: nat] :
      ( aa(nat,A,nth(A,aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),L),Ia),Xa)),J) = $ite(
          ( ( Ia = J )
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),aa(list(A),nat,size_size(list(A)),L)) ),
          Xa,
          aa(nat,A,nth(A,L),J) ) ) ).

% nth_list_update'
tff(fact_177_linorder__neqE__linordered__idom,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A,Y: A] :
          ( ( Xa != Y )
         => ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xa) ) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_178_pinf_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Pa: fun(A,$o),P: fun(A,$o),Qa: fun(A,$o),Q: fun(A,$o)] :
          ( ? [Z4: A] :
            ! [X: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z4),X)
             => ( aa(A,$o,Pa,X)
              <=> aa(A,$o,P,X) ) )
         => ( ? [Z4: A] :
              ! [X: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z4),X)
               => ( aa(A,$o,Qa,X)
                <=> aa(A,$o,Q,X) ) )
           => ? [Z3: A] :
              ! [X4: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z3),X4)
               => ( ( aa(A,$o,Pa,X4)
                    & aa(A,$o,Qa,X4) )
                <=> ( aa(A,$o,P,X4)
                    & aa(A,$o,Q,X4) ) ) ) ) ) ) ).

% pinf(1)
tff(fact_179_pinf_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Pa: fun(A,$o),P: fun(A,$o),Qa: fun(A,$o),Q: fun(A,$o)] :
          ( ? [Z4: A] :
            ! [X: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z4),X)
             => ( aa(A,$o,Pa,X)
              <=> aa(A,$o,P,X) ) )
         => ( ? [Z4: A] :
              ! [X: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z4),X)
               => ( aa(A,$o,Qa,X)
                <=> aa(A,$o,Q,X) ) )
           => ? [Z3: A] :
              ! [X4: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z3),X4)
               => ( ( aa(A,$o,Pa,X4)
                    | aa(A,$o,Qa,X4) )
                <=> ( aa(A,$o,P,X4)
                    | aa(A,$o,Q,X4) ) ) ) ) ) ) ).

% pinf(2)
tff(fact_180_pinf_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T2: A] :
        ? [Z3: A] :
        ! [X4: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z3),X4)
         => ( X4 != T2 ) ) ) ).

% pinf(3)
tff(fact_181_pinf_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T2: A] :
        ? [Z3: A] :
        ! [X4: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z3),X4)
         => ( X4 != T2 ) ) ) ).

% pinf(4)
tff(fact_182_pinf_I5_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T2: A] :
        ? [Z3: A] :
        ! [X4: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z3),X4)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),T2) ) ) ).

% pinf(5)
tff(fact_183_pinf_I7_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T2: A] :
        ? [Z3: A] :
        ! [X4: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z3),X4)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),T2),X4) ) ) ).

% pinf(7)
tff(fact_184_pinf_I11_J,axiom,
    ! [A: $tType,B: $tType] :
      ( ord(A)
     => ! [Fa: B] :
        ? [Z3: A] :
        ! [X4: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z3),X4)
         => ( Fa = Fa ) ) ) ).

% pinf(11)
tff(fact_185_minf_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Pa: fun(A,$o),P: fun(A,$o),Qa: fun(A,$o),Q: fun(A,$o)] :
          ( ? [Z4: A] :
            ! [X: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Z4)
             => ( aa(A,$o,Pa,X)
              <=> aa(A,$o,P,X) ) )
         => ( ? [Z4: A] :
              ! [X: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Z4)
               => ( aa(A,$o,Qa,X)
                <=> aa(A,$o,Q,X) ) )
           => ? [Z3: A] :
              ! [X4: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Z3)
               => ( ( aa(A,$o,Pa,X4)
                    & aa(A,$o,Qa,X4) )
                <=> ( aa(A,$o,P,X4)
                    & aa(A,$o,Q,X4) ) ) ) ) ) ) ).

% minf(1)
tff(fact_186_minf_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Pa: fun(A,$o),P: fun(A,$o),Qa: fun(A,$o),Q: fun(A,$o)] :
          ( ? [Z4: A] :
            ! [X: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Z4)
             => ( aa(A,$o,Pa,X)
              <=> aa(A,$o,P,X) ) )
         => ( ? [Z4: A] :
              ! [X: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Z4)
               => ( aa(A,$o,Qa,X)
                <=> aa(A,$o,Q,X) ) )
           => ? [Z3: A] :
              ! [X4: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Z3)
               => ( ( aa(A,$o,Pa,X4)
                    | aa(A,$o,Qa,X4) )
                <=> ( aa(A,$o,P,X4)
                    | aa(A,$o,Q,X4) ) ) ) ) ) ) ).

% minf(2)
tff(fact_187_minf_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T2: A] :
        ? [Z3: A] :
        ! [X4: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Z3)
         => ( X4 != T2 ) ) ) ).

% minf(3)
tff(fact_188_minf_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T2: A] :
        ? [Z3: A] :
        ! [X4: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Z3)
         => ( X4 != T2 ) ) ) ).

% minf(4)
tff(fact_189_minf_I5_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T2: A] :
        ? [Z3: A] :
        ! [X4: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Z3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),T2) ) ) ).

% minf(5)
tff(fact_190_minf_I7_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T2: A] :
        ? [Z3: A] :
        ! [X4: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Z3)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),T2),X4) ) ) ).

% minf(7)
tff(fact_191_minf_I11_J,axiom,
    ! [A: $tType,B: $tType] :
      ( ord(A)
     => ! [Fa: B] :
        ? [Z3: A] :
        ! [X4: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Z3)
         => ( Fa = Fa ) ) ) ).

% minf(11)
tff(fact_192_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [A: $tType] :
      ( ab_semigroup_mult(A)
     => ! [B3: A,A3: A,C2: A] : ( aa(A,A,aa(A,fun(A,A),times_times(A),B3),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),B3),C2)) ) ) ).

% ab_semigroup_mult_class.mult.left_commute
tff(fact_193_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ! [A: $tType] :
      ( ab_semigroup_mult(A)
     => ! [A3: A,B3: A] : ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3) = aa(A,A,aa(A,fun(A,A),times_times(A),B3),A3) ) ) ).

% ab_semigroup_mult_class.mult.commute
tff(fact_194_mult_Oassoc,axiom,
    ! [A: $tType] :
      ( semigroup_mult(A)
     => ! [A3: A,B3: 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),B3)),C2) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2)) ) ) ).

% mult.assoc
tff(fact_195_listI__assn__cong,axiom,
    ! [A: $tType,B: $tType,Ia2: set(nat),I5: set(nat),Xsa: list(A),Xs3: list(A),Xsia: list(B),Xsi: list(B),Aa2: fun(A,fun(B,assn)),A5: fun(A,fun(B,assn))] :
      ( ( Ia2 = I5 )
     => ( ( aa(list(A),nat,size_size(list(A)),Xsa) = aa(list(A),nat,size_size(list(A)),Xs3) )
       => ( ( aa(list(B),nat,size_size(list(B)),Xsia) = aa(list(B),nat,size_size(list(B)),Xsi) )
         => ( ! [I3: nat] :
                ( member(nat,I3,Ia2)
               => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),Xsa))
                 => ( ( aa(list(A),nat,size_size(list(A)),Xsa) = aa(list(B),nat,size_size(list(B)),Xsia) )
                   => ( ( aa(nat,A,nth(A,Xsa),I3) = aa(nat,A,nth(A,Xs3),I3) )
                      & ( aa(nat,B,nth(B,Xsia),I3) = aa(nat,B,nth(B,Xsi),I3) )
                      & ( aa(B,assn,aa(A,fun(B,assn),Aa2,aa(nat,A,nth(A,Xsa),I3)),aa(nat,B,nth(B,Xsia),I3)) = aa(B,assn,aa(A,fun(B,assn),A5,aa(nat,A,nth(A,Xs3),I3)),aa(nat,B,nth(B,Xsi),I3)) ) ) ) ) )
           => ( vEBT_List_listI_assn(A,B,Ia2,Aa2,Xsa,Xsia) = vEBT_List_listI_assn(A,B,I5,A5,Xs3,Xsi) ) ) ) ) ) ).

% listI_assn_cong
tff(fact_196_listI__assn__weak__cong,axiom,
    ! [A: $tType,B: $tType,Ia2: set(nat),I5: set(nat),Aa2: fun(A,fun(B,assn)),A5: fun(A,fun(B,assn)),Xsa: list(A),Xs3: list(A),Xsia: list(B),Xsi: list(B)] :
      ( ( Ia2 = I5 )
     => ( ( Aa2 = A5 )
       => ( ( aa(list(A),nat,size_size(list(A)),Xsa) = aa(list(A),nat,size_size(list(A)),Xs3) )
         => ( ( aa(list(B),nat,size_size(list(B)),Xsia) = aa(list(B),nat,size_size(list(B)),Xsi) )
           => ( ! [I3: nat] :
                  ( member(nat,I3,Ia2)
                 => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),Xsa))
                   => ( ( aa(list(A),nat,size_size(list(A)),Xsa) = aa(list(B),nat,size_size(list(B)),Xsia) )
                     => ( ( aa(nat,A,nth(A,Xsa),I3) = aa(nat,A,nth(A,Xs3),I3) )
                        & ( aa(nat,B,nth(B,Xsia),I3) = aa(nat,B,nth(B,Xsi),I3) ) ) ) ) )
             => ( vEBT_List_listI_assn(A,B,Ia2,Aa2,Xsa,Xsia) = vEBT_List_listI_assn(A,B,I5,A5,Xs3,Xsi) ) ) ) ) ) ) ).

% listI_assn_weak_cong
tff(fact_197_fun__diff__def,axiom,
    ! [B: $tType,A: $tType] :
      ( minus(B)
     => ! [Aa2: fun(A,B),B2: fun(A,B),X4: 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)),Aa2),B2),X4) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,Aa2,X4)),aa(A,B,B2,X4)) ) ) ).

% fun_diff_def
tff(fact_198_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [A3: A,C2: A,B3: 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)),B3) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B3)),C2) ) ) ).

% cancel_ab_semigroup_add_class.diff_right_commute
tff(fact_199_diff__eq__diff__eq,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B3: A,C2: A,D3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B3) = aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),D3) )
         => ( ( A3 = B3 )
          <=> ( C2 = D3 ) ) ) ) ).

% diff_eq_diff_eq
tff(fact_200_is__singletonE,axiom,
    ! [A: $tType,Aa2: set(A)] :
      ( is_singleton(A,Aa2)
     => ~ ! [X: A] : ( Aa2 != aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),X),bot_bot(set(A))) ) ) ).

% is_singletonE
tff(fact_201_is__singleton__def,axiom,
    ! [A: $tType,Aa2: set(A)] :
      ( is_singleton(A,Aa2)
    <=> ? [X6: A] : ( Aa2 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),X6),bot_bot(set(A))) ) ) ).

% is_singleton_def
tff(fact_202_listI__assn__insert,axiom,
    ! [A: $tType,B: $tType,Ia: nat,Ia2: set(nat),Xsa: list(A),Aa2: fun(A,fun(B,assn)),Xsia: list(B)] :
      ( ~ member(nat,Ia,Ia2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),aa(list(A),nat,size_size(list(A)),Xsa))
       => ( vEBT_List_listI_assn(A,B,aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert3(nat),Ia),Ia2),Aa2,Xsa,Xsia) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),Aa2,aa(nat,A,nth(A,Xsa),Ia)),aa(nat,B,nth(B,Xsia),Ia))),vEBT_List_listI_assn(A,B,Ia2,Aa2,Xsa,Xsia)) ) ) ) ).

% listI_assn_insert
tff(fact_203_diff__strict__right__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
         => aa(A,$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),B3),C2)) ) ) ).

% diff_strict_right_mono
tff(fact_204_diff__strict__left__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [B3: A,A3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A3)
         => aa(A,$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),B3)) ) ) ).

% diff_strict_left_mono
tff(fact_205_diff__eq__diff__less,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B3: A,C2: A,D3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B3) = aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),D3) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),D3) ) ) ) ).

% diff_eq_diff_less
tff(fact_206_bot__empty__eq,axiom,
    ! [A: $tType,X4: A] :
      ( aa(A,$o,bot_bot(fun(A,$o)),X4)
    <=> member(A,X4,bot_bot(set(A))) ) ).

% bot_empty_eq
tff(fact_207_Collect__empty__eq__bot,axiom,
    ! [A: $tType,Pa: fun(A,$o)] :
      ( ( aa(fun(A,$o),set(A),collect(A),Pa) = bot_bot(set(A)) )
    <=> ( Pa = bot_bot(fun(A,$o)) ) ) ).

% Collect_empty_eq_bot
tff(fact_208_remove__def,axiom,
    ! [A: $tType,Xa: A,Aa2: set(A)] : ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),remove(A),Xa),Aa2) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A)))) ) ).

% remove_def
tff(fact_209_insert__swap__set__eq,axiom,
    ! [A: $tType,Ia: nat,L: list(A),Xa: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),aa(list(A),nat,size_size(list(A)),L))
     => ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),aa(nat,A,nth(A,L),Ia)),aa(list(A),set(A),set2(A),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),L),Ia),Xa))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),aa(list(A),set(A),set2(A),L)) ) ) ).

% insert_swap_set_eq
tff(fact_210_set__swap,axiom,
    ! [A: $tType,Ia: nat,Xsa: list(A),J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),aa(list(A),nat,size_size(list(A)),Xsa))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),Xsa))
       => ( aa(list(A),set(A),set2(A),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xsa),Ia),aa(nat,A,nth(A,Xsa),J))),J),aa(nat,A,nth(A,Xsa),Ia))) = aa(list(A),set(A),set2(A),Xsa) ) ) ) ).

% set_swap
tff(fact_211_Set_Ois__empty__def,axiom,
    ! [A: $tType,Aa2: set(A)] :
      ( is_empty(A,Aa2)
    <=> ( Aa2 = bot_bot(set(A)) ) ) ).

% Set.is_empty_def
tff(fact_212_distinct__swap,axiom,
    ! [A: $tType,Ia: nat,Xsa: list(A),J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),aa(list(A),nat,size_size(list(A)),Xsa))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),Xsa))
       => ( aa(list(A),$o,distinct(A),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xsa),Ia),aa(nat,A,nth(A,Xsa),J))),J),aa(nat,A,nth(A,Xsa),Ia)))
        <=> aa(list(A),$o,distinct(A),Xsa) ) ) ) ).

% distinct_swap
tff(fact_213_list__ex__length,axiom,
    ! [A: $tType,Pa: fun(A,$o),Xsa: list(A)] :
      ( aa(list(A),$o,aa(fun(A,$o),fun(list(A),$o),list_ex(A),Pa),Xsa)
    <=> ? [N2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N2),aa(list(A),nat,size_size(list(A)),Xsa))
          & aa(A,$o,Pa,aa(nat,A,nth(A,Xsa),N2)) ) ) ).

% list_ex_length
tff(fact_214_mod__h__bot__iff_I3_J,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [P2: ref(A),Xa: A,H2: heap_ext(product_unit)] : ~ aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,sngr_assn(A,P2,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)),H2),bot_bot(set(nat)))) ) ).

% mod_h_bot_iff(3)
tff(fact_215_mod__h__bot__iff_I4_J,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Q2: array(A),Y: list(A),H2: heap_ext(product_unit)] : ~ aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,snga_assn(A,Q2,Y)),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)))) ) ).

% mod_h_bot_iff(4)
tff(fact_216_ent__pure__post__iff,axiom,
    ! [Pa: assn,Qa: assn,B3: $o] :
      ( aa(assn,$o,aa(assn,fun(assn,$o),entails,Pa),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Qa),pure_assn((B3))))
    <=> ( ! [H3: product_prod(heap_ext(product_unit),set(nat))] :
            ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,Pa),H3)
           => (B3) )
        & aa(assn,$o,aa(assn,fun(assn,$o),entails,Pa),Qa) ) ) ).

% ent_pure_post_iff
tff(fact_217_pure__assn__eq__conv,axiom,
    ! [Pa: $o,Qa: $o] :
      ( ( pure_assn((Pa)) = pure_assn((Qa)) )
    <=> ( (Pa)
      <=> (Qa) ) ) ).

% pure_assn_eq_conv
tff(fact_218_member__remove,axiom,
    ! [A: $tType,Xa: A,Y: A,Aa2: set(A)] :
      ( member(A,Xa,aa(set(A),set(A),aa(A,fun(set(A),set(A)),remove(A),Y),Aa2))
    <=> ( member(A,Xa,Aa2)
        & ( Xa != Y ) ) ) ).

% member_remove
tff(fact_219_merge__pure__star,axiom,
    ! [A3: $o,B3: $o] :
      ( aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),pure_assn((A3))),pure_assn((B3))) = pure_assn(( (A3)
          & (B3) )) ) ).

% merge_pure_star
tff(fact_220_pure__false,axiom,
    pure_assn($false) = bot_bot(assn) ).

% pure_false
tff(fact_221_pure__assn__eq__false__iff,axiom,
    ! [Pa: $o] :
      ( ( pure_assn((Pa)) = bot_bot(assn) )
    <=> ~ (Pa) ) ).

% pure_assn_eq_false_iff
tff(fact_222_mod__pure__star__dist,axiom,
    ! [Pa: assn,B3: $o,H2: product_prod(heap_ext(product_unit),set(nat))] :
      ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Pa),pure_assn((B3)))),H2)
    <=> ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,Pa),H2)
        & (B3) ) ) ).

% mod_pure_star_dist
tff(fact_223_ent__pure__pre__iff,axiom,
    ! [Pa: assn,B3: $o,Qa: assn] :
      ( aa(assn,$o,aa(assn,fun(assn,$o),entails,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Pa),pure_assn((B3)))),Qa)
    <=> ( (B3)
       => aa(assn,$o,aa(assn,fun(assn,$o),entails,Pa),Qa) ) ) ).

% ent_pure_pre_iff
tff(fact_224_mod__h__bot__iff_I5_J,axiom,
    ! [Pa: assn,Qa: assn,H2: 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,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Pa),Qa)),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,Pa),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,Qa),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)))) ) ) ).

% mod_h_bot_iff(5)
tff(fact_225_mod__h__bot__iff_I1_J,axiom,
    ! [B3: $o,H2: heap_ext(product_unit)] :
      ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,pure_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)),H2),bot_bot(set(nat))))
    <=> (B3) ) ).

% mod_h_bot_iff(1)
tff(fact_226_list__ex__cong,axiom,
    ! [A: $tType,Xsa: list(A),Ys: list(A),F2: fun(A,$o),G: fun(A,$o)] :
      ( ( Xsa = Ys )
     => ( ! [X: A] :
            ( member(A,X,aa(list(A),set(A),set2(A),Ys))
           => ( aa(A,$o,F2,X)
            <=> aa(A,$o,G,X) ) )
       => ( aa(list(A),$o,aa(fun(A,$o),fun(list(A),$o),list_ex(A),F2),Xsa)
        <=> aa(list(A),$o,aa(fun(A,$o),fun(list(A),$o),list_ex(A),G),Ys) ) ) ) ).

% list_ex_cong
tff(fact_227_sorted__list__of__set_Odistinct__if__distinct__map,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xsa: list(A)] :
          ( aa(list(A),$o,distinct(A),Xsa)
         => aa(list(A),$o,distinct(A),Xsa) ) ) ).

% sorted_list_of_set.distinct_if_distinct_map
tff(fact_228_one__assn__raw_Ocases,axiom,
    ! [Xa: product_prod(heap_ext(product_unit),set(nat))] :
      ~ ! [H: heap_ext(product_unit),As: 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),As) ) ).

% one_assn_raw.cases
tff(fact_229_bex2I,axiom,
    ! [A: $tType,B: $tType,A3: A,B3: B,S: set(product_prod(A,B)),Pa: fun(A,fun(B,$o))] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B3),S)
     => ( ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B3),S)
         => aa(B,$o,aa(A,fun(B,$o),Pa,A3),B3) )
       => ? [A4: A,B4: B] :
            ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B4),S)
            & aa(B,$o,aa(A,fun(B,$o),Pa,A4),B4) ) ) ) ).

% bex2I
tff(fact_230_pairself_Ocases,axiom,
    ! [B: $tType,A: $tType,Xa: product_prod(fun(A,B),product_prod(A,A))] :
      ~ ! [F3: fun(A,B),A4: A,B4: A] : ( Xa != 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)),F3),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),B4)) ) ).

% pairself.cases
tff(fact_231_list__assn__cong,axiom,
    ! [A: $tType,B: $tType,Xsa: list(A),Xs3: list(A),Xsia: list(B),Xsi: list(B),Aa2: fun(A,fun(B,assn)),A5: fun(A,fun(B,assn))] :
      ( ( Xsa = Xs3 )
     => ( ( Xsia = Xsi )
       => ( ! [X: A,Xi: B] :
              ( member(A,X,aa(list(A),set(A),set2(A),Xs3))
             => ( member(B,Xi,aa(list(B),set(B),set2(B),Xsi))
               => ( aa(B,assn,aa(A,fun(B,assn),Aa2,X),Xi) = aa(B,assn,aa(A,fun(B,assn),A5,X),Xi) ) ) )
         => ( vEBT_List_list_assn(A,B,Aa2,Xsa,Xsia) = vEBT_List_list_assn(A,B,A5,Xs3,Xsi) ) ) ) ) ).

% list_assn_cong
tff(fact_232_distinct__Ex1,axiom,
    ! [A: $tType,Xsa: list(A),Xa: A] :
      ( aa(list(A),$o,distinct(A),Xsa)
     => ( member(A,Xa,aa(list(A),set(A),set2(A),Xsa))
       => ? [X: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),aa(list(A),nat,size_size(list(A)),Xsa))
            & ( aa(nat,A,nth(A,Xsa),X) = Xa )
            & ! [Y3: nat] :
                ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y3),aa(list(A),nat,size_size(list(A)),Xsa))
                  & ( aa(nat,A,nth(A,Xsa),Y3) = Xa ) )
               => ( Y3 = X ) ) ) ) ) ).

% distinct_Ex1
tff(fact_233_mod__h__bot__indep,axiom,
    ! [Pa: assn,H2: heap_ext(product_unit),H4: 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,Pa),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,Pa),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),bot_bot(set(nat)))) ) ).

% mod_h_bot_indep
tff(fact_234_distinct__list__update,axiom,
    ! [A: $tType,Xsa: list(A),A3: A,Ia: nat] :
      ( aa(list(A),$o,distinct(A),Xsa)
     => ( ~ 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),Xsa)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),aa(nat,A,nth(A,Xsa),Ia)),bot_bot(set(A)))))
       => aa(list(A),$o,distinct(A),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xsa),Ia),A3)) ) ) ).

% distinct_list_update
tff(fact_235_distinct__conv__nth,axiom,
    ! [A: $tType,Xsa: list(A)] :
      ( aa(list(A),$o,distinct(A),Xsa)
    <=> ! [I2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xsa))
         => ! [J2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J2),aa(list(A),nat,size_size(list(A)),Xsa))
             => ( ( I2 != J2 )
               => ( aa(nat,A,nth(A,Xsa),I2) != aa(nat,A,nth(A,Xsa),J2) ) ) ) ) ) ).

% distinct_conv_nth
tff(fact_236_nth__eq__iff__index__eq,axiom,
    ! [A: $tType,Xsa: list(A),Ia: nat,J: nat] :
      ( aa(list(A),$o,distinct(A),Xsa)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),aa(list(A),nat,size_size(list(A)),Xsa))
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),Xsa))
         => ( ( aa(nat,A,nth(A,Xsa),Ia) = aa(nat,A,nth(A,Xsa),J) )
          <=> ( Ia = J ) ) ) ) ) ).

% nth_eq_iff_index_eq
tff(fact_237_all__set__conv__nth,axiom,
    ! [A: $tType,L: list(A),Pa: fun(A,$o)] :
      ( ! [X6: A] :
          ( member(A,X6,aa(list(A),set(A),set2(A),L))
         => aa(A,$o,Pa,X6) )
    <=> ! [I2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),L))
         => aa(A,$o,Pa,aa(nat,A,nth(A,L),I2)) ) ) ).

% all_set_conv_nth
tff(fact_238_nth__mem,axiom,
    ! [A: $tType,N: nat,Xsa: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xsa))
     => member(A,aa(nat,A,nth(A,Xsa),N),aa(list(A),set(A),set2(A),Xsa)) ) ).

% nth_mem
tff(fact_239_list__ball__nth,axiom,
    ! [A: $tType,N: nat,Xsa: list(A),Pa: fun(A,$o)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xsa))
     => ( ! [X: A] :
            ( member(A,X,aa(list(A),set(A),set2(A),Xsa))
           => aa(A,$o,Pa,X) )
       => aa(A,$o,Pa,aa(nat,A,nth(A,Xsa),N)) ) ) ).

% list_ball_nth
tff(fact_240_in__set__conv__nth,axiom,
    ! [A: $tType,Xa: A,Xsa: list(A)] :
      ( member(A,Xa,aa(list(A),set(A),set2(A),Xsa))
    <=> ? [I2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xsa))
          & ( aa(nat,A,nth(A,Xsa),I2) = Xa ) ) ) ).

% in_set_conv_nth
tff(fact_241_all__nth__imp__all__set,axiom,
    ! [A: $tType,Xsa: list(A),Pa: fun(A,$o),Xa: A] :
      ( ! [I3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),Xsa))
         => aa(A,$o,Pa,aa(nat,A,nth(A,Xsa),I3)) )
     => ( member(A,Xa,aa(list(A),set(A),set2(A),Xsa))
       => aa(A,$o,Pa,Xa) ) ) ).

% all_nth_imp_all_set
tff(fact_242_all__set__conv__all__nth,axiom,
    ! [A: $tType,Xsa: list(A),Pa: fun(A,$o)] :
      ( ! [X6: A] :
          ( member(A,X6,aa(list(A),set(A),set2(A),Xsa))
         => aa(A,$o,Pa,X6) )
    <=> ! [I2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xsa))
         => aa(A,$o,Pa,aa(nat,A,nth(A,Xsa),I2)) ) ) ).

% all_set_conv_all_nth
tff(fact_243_in__set__upd__eq__aux,axiom,
    ! [A: $tType,Ia: nat,L: list(A),Xa: A,Y: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),aa(list(A),nat,size_size(list(A)),L))
     => ( member(A,Xa,aa(list(A),set(A),set2(A),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),L),Ia),Y)))
      <=> ( ( Xa = Y )
          | ! [Y4: A] : member(A,Xa,aa(list(A),set(A),set2(A),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),L),Ia),Y4))) ) ) ) ).

% in_set_upd_eq_aux
tff(fact_244_in__set__upd__cases,axiom,
    ! [A: $tType,Xa: A,L: list(A),Ia: nat,Y: A] :
      ( member(A,Xa,aa(list(A),set(A),set2(A),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),L),Ia),Y)))
     => ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),aa(list(A),nat,size_size(list(A)),L))
         => ( Xa != Y ) )
       => member(A,Xa,aa(list(A),set(A),set2(A),L)) ) ) ).

% in_set_upd_cases
tff(fact_245_in__set__upd__eq,axiom,
    ! [A: $tType,Ia: nat,L: list(A),Xa: A,Y: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),aa(list(A),nat,size_size(list(A)),L))
     => ( member(A,Xa,aa(list(A),set(A),set2(A),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),L),Ia),Y)))
      <=> ( ( Xa = Y )
          | ( member(A,Xa,aa(list(A),set(A),set2(A),L))
            & ! [Y4: A] : member(A,Xa,aa(list(A),set(A),set2(A),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),L),Ia),Y4))) ) ) ) ) ).

% in_set_upd_eq
tff(fact_246_set__update__memI,axiom,
    ! [A: $tType,N: nat,Xsa: list(A),Xa: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xsa))
     => member(A,Xa,aa(list(A),set(A),set2(A),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xsa),N),Xa))) ) ).

% set_update_memI
tff(fact_247_set__update__distinct,axiom,
    ! [A: $tType,Xsa: list(A),N: nat,Xa: A] :
      ( aa(list(A),$o,distinct(A),Xsa)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xsa))
       => ( aa(list(A),set(A),set2(A),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xsa),N),Xa)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(list(A),set(A),set2(A),Xsa)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),aa(nat,A,nth(A,Xsa),N)),bot_bot(set(A))))) ) ) ) ).

% set_update_distinct
tff(fact_248_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_249_old_Oprod_Oinject,axiom,
    ! [A: $tType,B: $tType,A3: A,B3: B,A6: A,B6: B] :
      ( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B3) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A6),B6) )
    <=> ( ( A3 = A6 )
        & ( B3 = B6 ) ) ) ).

% old.prod.inject
tff(fact_250_distinct__union,axiom,
    ! [A: $tType,Xsa: list(A),Ys: list(A)] :
      ( aa(list(A),$o,distinct(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),union(A),Xsa),Ys))
    <=> aa(list(A),$o,distinct(A),Ys) ) ).

% distinct_union
tff(fact_251_set__remove1__eq,axiom,
    ! [A: $tType,Xsa: list(A),Xa: A] :
      ( aa(list(A),$o,distinct(A),Xsa)
     => ( aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),remove1(A),Xa),Xsa)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(list(A),set(A),set2(A),Xsa)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A)))) ) ) ).

% set_remove1_eq
tff(fact_252_is__empty__set,axiom,
    ! [A: $tType,Xsa: list(A)] :
      ( is_empty(A,aa(list(A),set(A),set2(A),Xsa))
    <=> aa(list(A),$o,null(A),Xsa) ) ).

% is_empty_set
tff(fact_253_set__removeAll,axiom,
    ! [A: $tType,Xa: A,Xsa: list(A)] : ( aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),removeAll(A),Xa),Xsa)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(list(A),set(A),set2(A),Xsa)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A)))) ) ).

% set_removeAll
tff(fact_254_listrel1__iff__update,axiom,
    ! [A: $tType,Xsa: list(A),Ys: list(A),R2: set(product_prod(A,A))] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xsa),Ys),listrel1(A,R2))
    <=> ? [Y4: A,N2: nat] :
          ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(nat,A,nth(A,Xsa),N2)),Y4),R2)
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N2),aa(list(A),nat,size_size(list(A)),Xsa))
          & ( Ys = aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xsa),N2),Y4) ) ) ) ).

% listrel1_iff_update
tff(fact_255_uncurry__apply,axiom,
    ! [B: $tType,A: $tType,E: $tType,F2: fun(B,fun(E,A)),A3: B,B3: E] : ( aa(product_prod(B,E),A,uncurry(B,E,A,F2),aa(E,product_prod(B,E),aa(B,fun(E,product_prod(B,E)),product_Pair(B,E),A3),B3)) = aa(E,A,aa(B,fun(E,A),F2,A3),B3) ) ).

% uncurry_apply
tff(fact_256_List_Oset__insert,axiom,
    ! [A: $tType,Xa: A,Xsa: list(A)] : ( aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),insert(A),Xa),Xsa)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),aa(list(A),set(A),set2(A),Xsa)) ) ).

% List.set_insert
tff(fact_257_listrel__iff__nth,axiom,
    ! [B: $tType,A: $tType,Xsa: list(A),Ys: list(B),R2: set(product_prod(A,B))] :
      ( member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xsa),Ys),listrel(A,B,R2))
    <=> ( ( aa(list(A),nat,size_size(list(A)),Xsa) = aa(list(B),nat,size_size(list(B)),Ys) )
        & ! [N2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N2),aa(list(A),nat,size_size(list(A)),Xsa))
           => 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,Xsa),N2)),aa(nat,B,nth(B,Ys),N2)),R2) ) ) ) ).

% listrel_iff_nth
tff(fact_258_ent__pure__post__iff__sng,axiom,
    ! [Pa: assn,B3: $o] :
      ( aa(assn,$o,aa(assn,fun(assn,$o),entails,Pa),pure_assn((B3)))
    <=> ( ! [H3: product_prod(heap_ext(product_unit),set(nat))] :
            ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,Pa),H3)
           => (B3) )
        & aa(assn,$o,aa(assn,fun(assn,$o),entails,Pa),one_one(assn)) ) ) ).

% ent_pure_post_iff_sng
tff(fact_259_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_260_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_261_in__set__remove1,axiom,
    ! [A: $tType,A3: A,B3: A,Xsa: list(A)] :
      ( ( A3 != B3 )
     => ( member(A,A3,aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),remove1(A),B3),Xsa)))
      <=> member(A,A3,aa(list(A),set(A),set2(A),Xsa)) ) ) ).

% in_set_remove1
tff(fact_262_removeAll__id,axiom,
    ! [A: $tType,Xa: A,Xsa: list(A)] :
      ( ~ member(A,Xa,aa(list(A),set(A),set2(A),Xsa))
     => ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),removeAll(A),Xa),Xsa) = Xsa ) ) ).

% removeAll_id
tff(fact_263_pure__true,axiom,
    pure_assn($true) = one_one(assn) ).

% pure_true
tff(fact_264_pure__assn__eq__emp__iff,axiom,
    ! [Pa: $o] :
      ( ( pure_assn((Pa)) = one_one(assn) )
    <=> (Pa) ) ).

% pure_assn_eq_emp_iff
tff(fact_265_entt__emp,axiom,
    ! [Aa2: assn] : entailst(Aa2,one_one(assn)) ).

% entt_emp
tff(fact_266_in__set__insert,axiom,
    ! [A: $tType,Xa: A,Xsa: list(A)] :
      ( member(A,Xa,aa(list(A),set(A),set2(A),Xsa))
     => ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),insert(A),Xa),Xsa) = Xsa ) ) ).

% in_set_insert
tff(fact_267_assn__basic__inequalities_I3_J,axiom,
    bot_bot(assn) != one_one(assn) ).

% assn_basic_inequalities(3)
tff(fact_268_distinct__insert,axiom,
    ! [A: $tType,Xa: A,Xsa: list(A)] :
      ( aa(list(A),$o,distinct(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),insert(A),Xa),Xsa))
    <=> aa(list(A),$o,distinct(A),Xsa) ) ).

% distinct_insert
tff(fact_269_ent__pure__pre__iff__sng,axiom,
    ! [B3: $o,Qa: assn] :
      ( aa(assn,$o,aa(assn,fun(assn,$o),entails,pure_assn((B3))),Qa)
    <=> ( (B3)
       => aa(assn,$o,aa(assn,fun(assn,$o),entails,one_one(assn)),Qa) ) ) ).

% ent_pure_pre_iff_sng
tff(fact_270_remove1__commute,axiom,
    ! [A: $tType,Xa: A,Y: A,Zs: list(A)] : ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),remove1(A),Xa),aa(list(A),list(A),aa(A,fun(list(A),list(A)),remove1(A),Y),Zs)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),remove1(A),Y),aa(list(A),list(A),aa(A,fun(list(A),list(A)),remove1(A),Xa),Zs)) ) ).

% remove1_commute
tff(fact_271_one__reorient,axiom,
    ! [A: $tType] :
      ( one(A)
     => ! [Xa: A] :
          ( ( one_one(A) = Xa )
        <=> ( Xa = one_one(A) ) ) ) ).

% one_reorient
tff(fact_272_times__assn__raw_Ocases,axiom,
    ! [Xa: 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))))] :
      ~ ! [P3: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Q3: fun(product_prod(heap_ext(product_unit),set(nat)),$o),H: heap_ext(product_unit),As: set(nat)] : ( Xa != 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)),$o),product_prod(heap_ext(product_unit),set(nat)))),P3),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$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))),Q3),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),As))) ) ).

% times_assn_raw.cases
tff(fact_273_listrel1__eq__len,axiom,
    ! [A: $tType,Xsa: list(A),Ys: list(A),R2: set(product_prod(A,A))] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xsa),Ys),listrel1(A,R2))
     => ( aa(list(A),nat,size_size(list(A)),Xsa) = aa(list(A),nat,size_size(list(A)),Ys) ) ) ).

% listrel1_eq_len
tff(fact_274_sngr__assn__raw_Ocases,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Xa: product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat))))] :
          ~ ! [R3: ref(A),X: A,H: heap_ext(product_unit),As: set(nat)] : ( Xa != aa(product_prod(A,product_prod(heap_ext(product_unit),set(nat))),product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),aa(ref(A),fun(product_prod(A,product_prod(heap_ext(product_unit),set(nat))),product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat))))),product_Pair(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),R3),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(A,product_prod(heap_ext(product_unit),set(nat))),aa(A,fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),product_Pair(A,product_prod(heap_ext(product_unit),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),As))) ) ) ).

% sngr_assn_raw.cases
tff(fact_275_snga__assn__raw_Ocases,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Xa: product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))))] :
          ~ ! [R3: array(A),X: list(A),H: heap_ext(product_unit),As: set(nat)] : ( Xa != aa(product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),aa(array(A),fun(product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),R3),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),aa(list(A),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(list(A),product_prod(heap_ext(product_unit),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),As))) ) ) ).

% snga_assn_raw.cases
tff(fact_276_listrel__eq__len,axiom,
    ! [A: $tType,B: $tType,Xsa: list(A),Ys: list(B),R2: set(product_prod(A,B))] :
      ( member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xsa),Ys),listrel(A,B,R2))
     => ( aa(list(A),nat,size_size(list(A)),Xsa) = aa(list(B),nat,size_size(list(B)),Ys) ) ) ).

% listrel_eq_len
tff(fact_277_distinct__remove1__removeAll,axiom,
    ! [A: $tType,Xsa: list(A),Xa: A] :
      ( aa(list(A),$o,distinct(A),Xsa)
     => ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),remove1(A),Xa),Xsa) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),removeAll(A),Xa),Xsa) ) ) ).

% distinct_remove1_removeAll
tff(fact_278_pure__assn__raw_Ocases,axiom,
    ! [A: $tType,B: $tType,Xa: product_prod($o,product_prod(A,set(B)))] :
      ~ ! [B4: $o,H: A,As: set(B)] : ( Xa != aa(product_prod(A,set(B)),product_prod($o,product_prod(A,set(B))),aa($o,fun(product_prod(A,set(B)),product_prod($o,product_prod(A,set(B)))),product_Pair($o,product_prod(A,set(B))),(B4)),aa(set(B),product_prod(A,set(B)),aa(A,fun(set(B),product_prod(A,set(B))),product_Pair(A,set(B)),H),As)) ) ).

% pure_assn_raw.cases
tff(fact_279_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_280_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_281_assn__one__left,axiom,
    ! [Pa: assn] : ( aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),one_one(assn)),Pa) = Pa ) ).

% assn_one_left
tff(fact_282_norm__assertion__simps_I1_J,axiom,
    ! [A3: assn] : ( aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),one_one(assn)),A3) = A3 ) ).

% norm_assertion_simps(1)
tff(fact_283_norm__assertion__simps_I2_J,axiom,
    ! [A3: assn] : ( aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),A3),one_one(assn)) = A3 ) ).

% norm_assertion_simps(2)
tff(fact_284_remove1__idem,axiom,
    ! [A: $tType,Xa: A,Xsa: list(A)] :
      ( ~ member(A,Xa,aa(list(A),set(A),set2(A),Xsa))
     => ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),remove1(A),Xa),Xsa) = Xsa ) ) ).

% remove1_idem
tff(fact_285_notin__set__remove1,axiom,
    ! [A: $tType,Xa: A,Xsa: list(A),Y: A] :
      ( ~ member(A,Xa,aa(list(A),set(A),set2(A),Xsa))
     => ~ member(A,Xa,aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),remove1(A),Y),Xsa))) ) ).

% notin_set_remove1
tff(fact_286_distinct__remove1,axiom,
    ! [A: $tType,Xsa: list(A),Xa: A] :
      ( aa(list(A),$o,distinct(A),Xsa)
     => aa(list(A),$o,distinct(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),remove1(A),Xa),Xsa)) ) ).

% distinct_remove1
tff(fact_287_distinct__removeAll,axiom,
    ! [A: $tType,Xsa: list(A),Xa: A] :
      ( aa(list(A),$o,distinct(A),Xsa)
     => aa(list(A),$o,distinct(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),removeAll(A),Xa),Xsa)) ) ).

% distinct_removeAll
tff(fact_288_less__1__mult,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [M2: A,N: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),M2)
         => ( aa(A,$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),M2),N)) ) ) ) ).

% less_1_mult
tff(fact_289_entails__solve__init_I2_J,axiom,
    ! [Pa: assn,Qa: assn] :
      ( fI_QUERY(Pa,Qa,one_one(assn))
     => aa(assn,$o,aa(assn,fun(assn,$o),entails,Pa),Qa) ) ).

% entails_solve_init(2)
tff(fact_290_prod__induct7,axiom,
    ! [I6: $tType,H5: $tType,G2: $tType,F4: $tType,E: $tType,B: $tType,A: $tType,Pa: fun(product_prod(A,product_prod(B,product_prod(E,product_prod(F4,product_prod(G2,product_prod(H5,I6)))))),$o),Xa: product_prod(A,product_prod(B,product_prod(E,product_prod(F4,product_prod(G2,product_prod(H5,I6))))))] :
      ( ! [A4: A,B4: B,C5: E,D4: F4,E2: G2,F3: H5,G3: I6] : aa(product_prod(A,product_prod(B,product_prod(E,product_prod(F4,product_prod(G2,product_prod(H5,I6)))))),$o,Pa,aa(product_prod(B,product_prod(E,product_prod(F4,product_prod(G2,product_prod(H5,I6))))),product_prod(A,product_prod(B,product_prod(E,product_prod(F4,product_prod(G2,product_prod(H5,I6)))))),aa(A,fun(product_prod(B,product_prod(E,product_prod(F4,product_prod(G2,product_prod(H5,I6))))),product_prod(A,product_prod(B,product_prod(E,product_prod(F4,product_prod(G2,product_prod(H5,I6))))))),product_Pair(A,product_prod(B,product_prod(E,product_prod(F4,product_prod(G2,product_prod(H5,I6)))))),A4),aa(product_prod(E,product_prod(F4,product_prod(G2,product_prod(H5,I6)))),product_prod(B,product_prod(E,product_prod(F4,product_prod(G2,product_prod(H5,I6))))),aa(B,fun(product_prod(E,product_prod(F4,product_prod(G2,product_prod(H5,I6)))),product_prod(B,product_prod(E,product_prod(F4,product_prod(G2,product_prod(H5,I6)))))),product_Pair(B,product_prod(E,product_prod(F4,product_prod(G2,product_prod(H5,I6))))),B4),aa(product_prod(F4,product_prod(G2,product_prod(H5,I6))),product_prod(E,product_prod(F4,product_prod(G2,product_prod(H5,I6)))),aa(E,fun(product_prod(F4,product_prod(G2,product_prod(H5,I6))),product_prod(E,product_prod(F4,product_prod(G2,product_prod(H5,I6))))),product_Pair(E,product_prod(F4,product_prod(G2,product_prod(H5,I6)))),C5),aa(product_prod(G2,product_prod(H5,I6)),product_prod(F4,product_prod(G2,product_prod(H5,I6))),aa(F4,fun(product_prod(G2,product_prod(H5,I6)),product_prod(F4,product_prod(G2,product_prod(H5,I6)))),product_Pair(F4,product_prod(G2,product_prod(H5,I6))),D4),aa(product_prod(H5,I6),product_prod(G2,product_prod(H5,I6)),aa(G2,fun(product_prod(H5,I6),product_prod(G2,product_prod(H5,I6))),product_Pair(G2,product_prod(H5,I6)),E2),aa(I6,product_prod(H5,I6),aa(H5,fun(I6,product_prod(H5,I6)),product_Pair(H5,I6),F3),G3)))))))
     => aa(product_prod(A,product_prod(B,product_prod(E,product_prod(F4,product_prod(G2,product_prod(H5,I6)))))),$o,Pa,Xa) ) ).

% prod_induct7
tff(fact_291_prod__induct6,axiom,
    ! [H5: $tType,G2: $tType,F4: $tType,E: $tType,B: $tType,A: $tType,Pa: fun(product_prod(A,product_prod(B,product_prod(E,product_prod(F4,product_prod(G2,H5))))),$o),Xa: product_prod(A,product_prod(B,product_prod(E,product_prod(F4,product_prod(G2,H5)))))] :
      ( ! [A4: A,B4: B,C5: E,D4: F4,E2: G2,F3: H5] : aa(product_prod(A,product_prod(B,product_prod(E,product_prod(F4,product_prod(G2,H5))))),$o,Pa,aa(product_prod(B,product_prod(E,product_prod(F4,product_prod(G2,H5)))),product_prod(A,product_prod(B,product_prod(E,product_prod(F4,product_prod(G2,H5))))),aa(A,fun(product_prod(B,product_prod(E,product_prod(F4,product_prod(G2,H5)))),product_prod(A,product_prod(B,product_prod(E,product_prod(F4,product_prod(G2,H5)))))),product_Pair(A,product_prod(B,product_prod(E,product_prod(F4,product_prod(G2,H5))))),A4),aa(product_prod(E,product_prod(F4,product_prod(G2,H5))),product_prod(B,product_prod(E,product_prod(F4,product_prod(G2,H5)))),aa(B,fun(product_prod(E,product_prod(F4,product_prod(G2,H5))),product_prod(B,product_prod(E,product_prod(F4,product_prod(G2,H5))))),product_Pair(B,product_prod(E,product_prod(F4,product_prod(G2,H5)))),B4),aa(product_prod(F4,product_prod(G2,H5)),product_prod(E,product_prod(F4,product_prod(G2,H5))),aa(E,fun(product_prod(F4,product_prod(G2,H5)),product_prod(E,product_prod(F4,product_prod(G2,H5)))),product_Pair(E,product_prod(F4,product_prod(G2,H5))),C5),aa(product_prod(G2,H5),product_prod(F4,product_prod(G2,H5)),aa(F4,fun(product_prod(G2,H5),product_prod(F4,product_prod(G2,H5))),product_Pair(F4,product_prod(G2,H5)),D4),aa(H5,product_prod(G2,H5),aa(G2,fun(H5,product_prod(G2,H5)),product_Pair(G2,H5),E2),F3))))))
     => aa(product_prod(A,product_prod(B,product_prod(E,product_prod(F4,product_prod(G2,H5))))),$o,Pa,Xa) ) ).

% prod_induct6
tff(fact_292_prod__induct5,axiom,
    ! [G2: $tType,F4: $tType,E: $tType,B: $tType,A: $tType,Pa: fun(product_prod(A,product_prod(B,product_prod(E,product_prod(F4,G2)))),$o),Xa: product_prod(A,product_prod(B,product_prod(E,product_prod(F4,G2))))] :
      ( ! [A4: A,B4: B,C5: E,D4: F4,E2: G2] : aa(product_prod(A,product_prod(B,product_prod(E,product_prod(F4,G2)))),$o,Pa,aa(product_prod(B,product_prod(E,product_prod(F4,G2))),product_prod(A,product_prod(B,product_prod(E,product_prod(F4,G2)))),aa(A,fun(product_prod(B,product_prod(E,product_prod(F4,G2))),product_prod(A,product_prod(B,product_prod(E,product_prod(F4,G2))))),product_Pair(A,product_prod(B,product_prod(E,product_prod(F4,G2)))),A4),aa(product_prod(E,product_prod(F4,G2)),product_prod(B,product_prod(E,product_prod(F4,G2))),aa(B,fun(product_prod(E,product_prod(F4,G2)),product_prod(B,product_prod(E,product_prod(F4,G2)))),product_Pair(B,product_prod(E,product_prod(F4,G2))),B4),aa(product_prod(F4,G2),product_prod(E,product_prod(F4,G2)),aa(E,fun(product_prod(F4,G2),product_prod(E,product_prod(F4,G2))),product_Pair(E,product_prod(F4,G2)),C5),aa(G2,product_prod(F4,G2),aa(F4,fun(G2,product_prod(F4,G2)),product_Pair(F4,G2),D4),E2)))))
     => aa(product_prod(A,product_prod(B,product_prod(E,product_prod(F4,G2)))),$o,Pa,Xa) ) ).

% prod_induct5
tff(fact_293_prod__induct4,axiom,
    ! [F4: $tType,E: $tType,B: $tType,A: $tType,Pa: fun(product_prod(A,product_prod(B,product_prod(E,F4))),$o),Xa: product_prod(A,product_prod(B,product_prod(E,F4)))] :
      ( ! [A4: A,B4: B,C5: E,D4: F4] : aa(product_prod(A,product_prod(B,product_prod(E,F4))),$o,Pa,aa(product_prod(B,product_prod(E,F4)),product_prod(A,product_prod(B,product_prod(E,F4))),aa(A,fun(product_prod(B,product_prod(E,F4)),product_prod(A,product_prod(B,product_prod(E,F4)))),product_Pair(A,product_prod(B,product_prod(E,F4))),A4),aa(product_prod(E,F4),product_prod(B,product_prod(E,F4)),aa(B,fun(product_prod(E,F4),product_prod(B,product_prod(E,F4))),product_Pair(B,product_prod(E,F4)),B4),aa(F4,product_prod(E,F4),aa(E,fun(F4,product_prod(E,F4)),product_Pair(E,F4),C5),D4))))
     => aa(product_prod(A,product_prod(B,product_prod(E,F4))),$o,Pa,Xa) ) ).

% prod_induct4
tff(fact_294_prod__induct3,axiom,
    ! [E: $tType,B: $tType,A: $tType,Pa: fun(product_prod(A,product_prod(B,E)),$o),Xa: product_prod(A,product_prod(B,E))] :
      ( ! [A4: A,B4: B,C5: E] : aa(product_prod(A,product_prod(B,E)),$o,Pa,aa(product_prod(B,E),product_prod(A,product_prod(B,E)),aa(A,fun(product_prod(B,E),product_prod(A,product_prod(B,E))),product_Pair(A,product_prod(B,E)),A4),aa(E,product_prod(B,E),aa(B,fun(E,product_prod(B,E)),product_Pair(B,E),B4),C5)))
     => aa(product_prod(A,product_prod(B,E)),$o,Pa,Xa) ) ).

% prod_induct3
tff(fact_295_prod__cases7,axiom,
    ! [A: $tType,B: $tType,E: $tType,F4: $tType,G2: $tType,H5: $tType,I6: $tType,Y: product_prod(A,product_prod(B,product_prod(E,product_prod(F4,product_prod(G2,product_prod(H5,I6))))))] :
      ~ ! [A4: A,B4: B,C5: E,D4: F4,E2: G2,F3: H5,G3: I6] : ( Y != aa(product_prod(B,product_prod(E,product_prod(F4,product_prod(G2,product_prod(H5,I6))))),product_prod(A,product_prod(B,product_prod(E,product_prod(F4,product_prod(G2,product_prod(H5,I6)))))),aa(A,fun(product_prod(B,product_prod(E,product_prod(F4,product_prod(G2,product_prod(H5,I6))))),product_prod(A,product_prod(B,product_prod(E,product_prod(F4,product_prod(G2,product_prod(H5,I6))))))),product_Pair(A,product_prod(B,product_prod(E,product_prod(F4,product_prod(G2,product_prod(H5,I6)))))),A4),aa(product_prod(E,product_prod(F4,product_prod(G2,product_prod(H5,I6)))),product_prod(B,product_prod(E,product_prod(F4,product_prod(G2,product_prod(H5,I6))))),aa(B,fun(product_prod(E,product_prod(F4,product_prod(G2,product_prod(H5,I6)))),product_prod(B,product_prod(E,product_prod(F4,product_prod(G2,product_prod(H5,I6)))))),product_Pair(B,product_prod(E,product_prod(F4,product_prod(G2,product_prod(H5,I6))))),B4),aa(product_prod(F4,product_prod(G2,product_prod(H5,I6))),product_prod(E,product_prod(F4,product_prod(G2,product_prod(H5,I6)))),aa(E,fun(product_prod(F4,product_prod(G2,product_prod(H5,I6))),product_prod(E,product_prod(F4,product_prod(G2,product_prod(H5,I6))))),product_Pair(E,product_prod(F4,product_prod(G2,product_prod(H5,I6)))),C5),aa(product_prod(G2,product_prod(H5,I6)),product_prod(F4,product_prod(G2,product_prod(H5,I6))),aa(F4,fun(product_prod(G2,product_prod(H5,I6)),product_prod(F4,product_prod(G2,product_prod(H5,I6)))),product_Pair(F4,product_prod(G2,product_prod(H5,I6))),D4),aa(product_prod(H5,I6),product_prod(G2,product_prod(H5,I6)),aa(G2,fun(product_prod(H5,I6),product_prod(G2,product_prod(H5,I6))),product_Pair(G2,product_prod(H5,I6)),E2),aa(I6,product_prod(H5,I6),aa(H5,fun(I6,product_prod(H5,I6)),product_Pair(H5,I6),F3),G3)))))) ) ).

% prod_cases7
tff(fact_296_prod__cases6,axiom,
    ! [A: $tType,B: $tType,E: $tType,F4: $tType,G2: $tType,H5: $tType,Y: product_prod(A,product_prod(B,product_prod(E,product_prod(F4,product_prod(G2,H5)))))] :
      ~ ! [A4: A,B4: B,C5: E,D4: F4,E2: G2,F3: H5] : ( Y != aa(product_prod(B,product_prod(E,product_prod(F4,product_prod(G2,H5)))),product_prod(A,product_prod(B,product_prod(E,product_prod(F4,product_prod(G2,H5))))),aa(A,fun(product_prod(B,product_prod(E,product_prod(F4,product_prod(G2,H5)))),product_prod(A,product_prod(B,product_prod(E,product_prod(F4,product_prod(G2,H5)))))),product_Pair(A,product_prod(B,product_prod(E,product_prod(F4,product_prod(G2,H5))))),A4),aa(product_prod(E,product_prod(F4,product_prod(G2,H5))),product_prod(B,product_prod(E,product_prod(F4,product_prod(G2,H5)))),aa(B,fun(product_prod(E,product_prod(F4,product_prod(G2,H5))),product_prod(B,product_prod(E,product_prod(F4,product_prod(G2,H5))))),product_Pair(B,product_prod(E,product_prod(F4,product_prod(G2,H5)))),B4),aa(product_prod(F4,product_prod(G2,H5)),product_prod(E,product_prod(F4,product_prod(G2,H5))),aa(E,fun(product_prod(F4,product_prod(G2,H5)),product_prod(E,product_prod(F4,product_prod(G2,H5)))),product_Pair(E,product_prod(F4,product_prod(G2,H5))),C5),aa(product_prod(G2,H5),product_prod(F4,product_prod(G2,H5)),aa(F4,fun(product_prod(G2,H5),product_prod(F4,product_prod(G2,H5))),product_Pair(F4,product_prod(G2,H5)),D4),aa(H5,product_prod(G2,H5),aa(G2,fun(H5,product_prod(G2,H5)),product_Pair(G2,H5),E2),F3))))) ) ).

% prod_cases6
tff(fact_297_prod__cases5,axiom,
    ! [A: $tType,B: $tType,E: $tType,F4: $tType,G2: $tType,Y: product_prod(A,product_prod(B,product_prod(E,product_prod(F4,G2))))] :
      ~ ! [A4: A,B4: B,C5: E,D4: F4,E2: G2] : ( Y != aa(product_prod(B,product_prod(E,product_prod(F4,G2))),product_prod(A,product_prod(B,product_prod(E,product_prod(F4,G2)))),aa(A,fun(product_prod(B,product_prod(E,product_prod(F4,G2))),product_prod(A,product_prod(B,product_prod(E,product_prod(F4,G2))))),product_Pair(A,product_prod(B,product_prod(E,product_prod(F4,G2)))),A4),aa(product_prod(E,product_prod(F4,G2)),product_prod(B,product_prod(E,product_prod(F4,G2))),aa(B,fun(product_prod(E,product_prod(F4,G2)),product_prod(B,product_prod(E,product_prod(F4,G2)))),product_Pair(B,product_prod(E,product_prod(F4,G2))),B4),aa(product_prod(F4,G2),product_prod(E,product_prod(F4,G2)),aa(E,fun(product_prod(F4,G2),product_prod(E,product_prod(F4,G2))),product_Pair(E,product_prod(F4,G2)),C5),aa(G2,product_prod(F4,G2),aa(F4,fun(G2,product_prod(F4,G2)),product_Pair(F4,G2),D4),E2)))) ) ).

% prod_cases5
tff(fact_298_prod__cases4,axiom,
    ! [A: $tType,B: $tType,E: $tType,F4: $tType,Y: product_prod(A,product_prod(B,product_prod(E,F4)))] :
      ~ ! [A4: A,B4: B,C5: E,D4: F4] : ( Y != aa(product_prod(B,product_prod(E,F4)),product_prod(A,product_prod(B,product_prod(E,F4))),aa(A,fun(product_prod(B,product_prod(E,F4)),product_prod(A,product_prod(B,product_prod(E,F4)))),product_Pair(A,product_prod(B,product_prod(E,F4))),A4),aa(product_prod(E,F4),product_prod(B,product_prod(E,F4)),aa(B,fun(product_prod(E,F4),product_prod(B,product_prod(E,F4))),product_Pair(B,product_prod(E,F4)),B4),aa(F4,product_prod(E,F4),aa(E,fun(F4,product_prod(E,F4)),product_Pair(E,F4),C5),D4))) ) ).

% prod_cases4
tff(fact_299_prod__cases3,axiom,
    ! [A: $tType,B: $tType,E: $tType,Y: product_prod(A,product_prod(B,E))] :
      ~ ! [A4: A,B4: B,C5: E] : ( Y != aa(product_prod(B,E),product_prod(A,product_prod(B,E)),aa(A,fun(product_prod(B,E),product_prod(A,product_prod(B,E))),product_Pair(A,product_prod(B,E)),A4),aa(E,product_prod(B,E),aa(B,fun(E,product_prod(B,E)),product_Pair(B,E),B4),C5)) ) ).

% prod_cases3
tff(fact_300_Pair__inject,axiom,
    ! [A: $tType,B: $tType,A3: A,B3: B,A6: A,B6: B] :
      ( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B3) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A6),B6) )
     => ~ ( ( A3 = A6 )
         => ( B3 != B6 ) ) ) ).

% Pair_inject
tff(fact_301_prod__cases,axiom,
    ! [B: $tType,A: $tType,Pa: fun(product_prod(A,B),$o),P2: product_prod(A,B)] :
      ( ! [A4: A,B4: B] : aa(product_prod(A,B),$o,Pa,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B4))
     => aa(product_prod(A,B),$o,Pa,P2) ) ).

% prod_cases
tff(fact_302_surj__pair,axiom,
    ! [A: $tType,B: $tType,P2: product_prod(A,B)] :
    ? [X: A,Y2: B] : ( P2 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y2) ) ).

% surj_pair
tff(fact_303_old_Oprod_Oexhaust,axiom,
    ! [A: $tType,B: $tType,Y: product_prod(A,B)] :
      ~ ! [A4: A,B4: B] : ( Y != aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B4) ) ).

% old.prod.exhaust
tff(fact_304_remove__code_I1_J,axiom,
    ! [A: $tType,Xa: A,Xsa: list(A)] : ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),remove(A),Xa),aa(list(A),set(A),set2(A),Xsa)) = aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),removeAll(A),Xa),Xsa)) ) ).

% remove_code(1)
tff(fact_305_mod__emp__simp,axiom,
    ! [H2: 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,one_one(assn)),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)))) ).

% mod_emp_simp
tff(fact_306_length__removeAll__less,axiom,
    ! [A: $tType,Xa: A,Xsa: list(A)] :
      ( member(A,Xa,aa(list(A),set(A),set2(A),Xsa))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),removeAll(A),Xa),Xsa))),aa(list(A),nat,size_size(list(A)),Xsa)) ) ).

% length_removeAll_less
tff(fact_307_old_Oprod_Orec,axiom,
    ! [B: $tType,A: $tType,E: $tType,F1: fun(B,fun(E,A)),A3: B,B3: E] : ( product_rec_prod(B,E,A,F1,aa(E,product_prod(B,E),aa(B,fun(E,product_prod(B,E)),product_Pair(B,E),A3),B3)) = aa(E,A,aa(B,fun(E,A),F1,A3),B3) ) ).

% old.prod.rec
tff(fact_308_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_309_remove__code_I2_J,axiom,
    ! [A: $tType,Xa: A,Xsa: list(A)] : ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),remove(A),Xa),coset(A,Xsa)) = coset(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),insert(A),Xa),Xsa)) ) ).

% remove_code(2)
tff(fact_310_internal__case__prod__conv,axiom,
    ! [B: $tType,A: $tType,E: $tType,C2: fun(B,fun(E,A)),A3: B,B3: E] : ( produc5280177257484947105e_prod(B,E,A,C2,aa(E,product_prod(B,E),aa(B,fun(E,product_prod(B,E)),product_Pair(B,E),A3),B3)) = aa(E,A,aa(B,fun(E,A),C2,A3),B3) ) ).

% internal_case_prod_conv
tff(fact_311_curry__uncurry__id,axiom,
    ! [E: $tType,B: $tType,A: $tType,F2: fun(A,fun(B,E))] : ( product_curry(A,B,E,uncurry(A,B,E,F2)) = F2 ) ).

% curry_uncurry_id
tff(fact_312_uncurry__curry__id,axiom,
    ! [E: $tType,B: $tType,A: $tType,F2: fun(product_prod(A,B),E)] : ( uncurry(A,B,E,product_curry(A,B,E,F2)) = F2 ) ).

% uncurry_curry_id
tff(fact_313_pure__assn__raw_Oelims_I3_J,axiom,
    ! [B: $tType,A: $tType,Xa: $o,Xaa: product_prod(A,set(B))] :
      ( ~ aa(product_prod(A,set(B)),$o,pure_assn_raw(A,B,(Xa)),Xaa)
     => ~ ! [H: A,As: set(B)] :
            ( ( Xaa = aa(set(B),product_prod(A,set(B)),aa(A,fun(set(B),product_prod(A,set(B))),product_Pair(A,set(B)),H),As) )
           => ( ( As = bot_bot(set(B)) )
              & (Xa) ) ) ) ).

% pure_assn_raw.elims(3)
tff(fact_314_pure__assn__raw_Oelims_I2_J,axiom,
    ! [B: $tType,A: $tType,Xa: $o,Xaa: product_prod(A,set(B))] :
      ( aa(product_prod(A,set(B)),$o,pure_assn_raw(A,B,(Xa)),Xaa)
     => ~ ! [H: A,As: set(B)] :
            ( ( Xaa = aa(set(B),product_prod(A,set(B)),aa(A,fun(set(B),product_prod(A,set(B))),product_Pair(A,set(B)),H),As) )
           => ~ ( ( As = bot_bot(set(B)) )
                & (Xa) ) ) ) ).

% pure_assn_raw.elims(2)
tff(fact_315_pure__assn__raw_Oelims_I1_J,axiom,
    ! [B: $tType,A: $tType,Xa: $o,Xaa: product_prod(A,set(B)),Y: $o] :
      ( ( aa(product_prod(A,set(B)),$o,pure_assn_raw(A,B,(Xa)),Xaa)
      <=> (Y) )
     => ~ ! [H: A,As: set(B)] :
            ( ( Xaa = aa(set(B),product_prod(A,set(B)),aa(A,fun(set(B),product_prod(A,set(B))),product_Pair(A,set(B)),H),As) )
           => ( (Y)
            <=> ~ ( ( As = bot_bot(set(B)) )
                  & (Xa) ) ) ) ) ).

% pure_assn_raw.elims(1)
tff(fact_316_pure__assn__raw_Osimps,axiom,
    ! [B: $tType,A: $tType,B3: $o,H2: A,As2: set(B)] :
      ( aa(product_prod(A,set(B)),$o,pure_assn_raw(A,B,(B3)),aa(set(B),product_prod(A,set(B)),aa(A,fun(set(B),product_prod(A,set(B))),product_Pair(A,set(B)),H2),As2))
    <=> ( ( As2 = bot_bot(set(B)) )
        & (B3) ) ) ).

% pure_assn_raw.simps
tff(fact_317_nth__zip,axiom,
    ! [A: $tType,B: $tType,Ia: nat,Xsa: list(A),Ys: list(B)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),aa(list(A),nat,size_size(list(A)),Xsa))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),aa(list(B),nat,size_size(list(B)),Ys))
       => ( aa(nat,product_prod(A,B),nth(product_prod(A,B),aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),Xsa),Ys)),Ia) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(nat,A,nth(A,Xsa),Ia)),aa(nat,B,nth(B,Ys),Ia)) ) ) ) ).

% nth_zip
tff(fact_318_curryI,axiom,
    ! [A: $tType,B: $tType,F2: fun(product_prod(A,B),$o),A3: A,B3: B] :
      ( aa(product_prod(A,B),$o,F2,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B3))
     => aa(B,$o,aa(A,fun(B,$o),product_curry(A,B,$o,F2),A3),B3) ) ).

% curryI
tff(fact_319_zip__eq__zip__same__len,axiom,
    ! [A: $tType,B: $tType,A3: list(A),B3: list(B),A6: list(A),B6: list(B)] :
      ( ( aa(list(A),nat,size_size(list(A)),A3) = aa(list(B),nat,size_size(list(B)),B3) )
     => ( ( aa(list(A),nat,size_size(list(A)),A6) = aa(list(B),nat,size_size(list(B)),B6) )
       => ( ( aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),A3),B3) = aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),A6),B6) )
        <=> ( ( A3 = A6 )
            & ( B3 = B6 ) ) ) ) ) ).

% zip_eq_zip_same_len
tff(fact_320_curry__conv,axiom,
    ! [A: $tType,B: $tType,E: $tType,F2: fun(product_prod(B,E),A),A3: B,B3: E] : ( aa(E,A,aa(B,fun(E,A),product_curry(B,E,A,F2),A3),B3) = aa(product_prod(B,E),A,F2,aa(E,product_prod(B,E),aa(B,fun(E,product_prod(B,E)),product_Pair(B,E),A3),B3)) ) ).

% curry_conv
tff(fact_321_zip__inj,axiom,
    ! [A: $tType,B: $tType,A3: list(A),B3: list(B),A6: list(A),B6: list(B)] :
      ( ( aa(list(A),nat,size_size(list(A)),A3) = aa(list(B),nat,size_size(list(B)),B3) )
     => ( ( aa(list(A),nat,size_size(list(A)),A6) = aa(list(B),nat,size_size(list(B)),B6) )
       => ( ( aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),A3),B3) = aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),A6),B6) )
         => ( ( A3 = A6 )
            & ( B3 = B6 ) ) ) ) ) ).

% zip_inj
tff(fact_322_pair__list__split,axiom,
    ! [A: $tType,B: $tType,L: list(product_prod(A,B))] :
      ~ ! [L1: list(A),L22: list(B)] :
          ( ( L = aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),L1),L22) )
         => ( ( aa(list(A),nat,size_size(list(A)),L1) = aa(list(B),nat,size_size(list(B)),L22) )
           => ( aa(list(product_prod(A,B)),nat,size_size(list(product_prod(A,B))),L) != aa(list(B),nat,size_size(list(B)),L22) ) ) ) ).

% pair_list_split
tff(fact_323_distinct__zipI2,axiom,
    ! [B: $tType,A: $tType,Ys: list(A),Xsa: list(B)] :
      ( aa(list(A),$o,distinct(A),Ys)
     => aa(list(product_prod(B,A)),$o,distinct(product_prod(B,A)),aa(list(A),list(product_prod(B,A)),aa(list(B),fun(list(A),list(product_prod(B,A))),zip(B,A),Xsa),Ys)) ) ).

% distinct_zipI2
tff(fact_324_distinct__zipI1,axiom,
    ! [A: $tType,B: $tType,Xsa: list(A),Ys: list(B)] :
      ( aa(list(A),$o,distinct(A),Xsa)
     => aa(list(product_prod(A,B)),$o,distinct(product_prod(A,B)),aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),Xsa),Ys)) ) ).

% distinct_zipI1
tff(fact_325_curryD,axiom,
    ! [A: $tType,B: $tType,F2: fun(product_prod(A,B),$o),A3: A,B3: B] :
      ( aa(B,$o,aa(A,fun(B,$o),product_curry(A,B,$o,F2),A3),B3)
     => aa(product_prod(A,B),$o,F2,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B3)) ) ).

% curryD
tff(fact_326_curryE,axiom,
    ! [A: $tType,B: $tType,F2: fun(product_prod(A,B),$o),A3: A,B3: B] :
      ( aa(B,$o,aa(A,fun(B,$o),product_curry(A,B,$o,F2),A3),B3)
     => aa(product_prod(A,B),$o,F2,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B3)) ) ).

% curryE
tff(fact_327_set__zip__rightD,axiom,
    ! [A: $tType,B: $tType,Xa: A,Y: B,Xsa: list(A),Ys: list(B)] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa),Y),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),Xsa),Ys)))
     => member(B,Y,aa(list(B),set(B),set2(B),Ys)) ) ).

% set_zip_rightD
tff(fact_328_set__zip__leftD,axiom,
    ! [B: $tType,A: $tType,Xa: A,Y: B,Xsa: list(A),Ys: list(B)] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa),Y),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),Xsa),Ys)))
     => member(A,Xa,aa(list(A),set(A),set2(A),Xsa)) ) ).

% set_zip_leftD
tff(fact_329_in__set__zipE,axiom,
    ! [A: $tType,B: $tType,Xa: A,Y: B,Xsa: list(A),Ys: list(B)] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa),Y),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),Xsa),Ys)))
     => ~ ( member(A,Xa,aa(list(A),set(A),set2(A),Xsa))
         => ~ member(B,Y,aa(list(B),set(B),set2(B),Ys)) ) ) ).

% in_set_zipE
tff(fact_330_zip__same,axiom,
    ! [A: $tType,A3: A,B3: A,Xsa: list(A)] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B3),aa(list(product_prod(A,A)),set(product_prod(A,A)),set2(product_prod(A,A)),aa(list(A),list(product_prod(A,A)),aa(list(A),fun(list(A),list(product_prod(A,A))),zip(A,A),Xsa),Xsa)))
    <=> ( member(A,A3,aa(list(A),set(A),set2(A),Xsa))
        & ( A3 = B3 ) ) ) ).

% zip_same
tff(fact_331_zip__update,axiom,
    ! [A: $tType,B: $tType,Xsa: list(A),Ia: nat,Xa: A,Ys: list(B),Y: B] : ( aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xsa),Ia),Xa)),aa(B,list(B),aa(nat,fun(B,list(B)),aa(list(B),fun(nat,fun(B,list(B))),list_update(B),Ys),Ia),Y)) = aa(product_prod(A,B),list(product_prod(A,B)),aa(nat,fun(product_prod(A,B),list(product_prod(A,B))),aa(list(product_prod(A,B)),fun(nat,fun(product_prod(A,B),list(product_prod(A,B)))),list_update(product_prod(A,B)),aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),Xsa),Ys)),Ia),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa),Y)) ) ).

% zip_update
tff(fact_332_in__set__impl__in__set__zip1,axiom,
    ! [A: $tType,B: $tType,Xsa: list(A),Ys: list(B),Xa: A] :
      ( ( aa(list(A),nat,size_size(list(A)),Xsa) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( member(A,Xa,aa(list(A),set(A),set2(A),Xsa))
       => ~ ! [Y2: B] : ~ member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa),Y2),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),Xsa),Ys))) ) ) ).

% in_set_impl_in_set_zip1
tff(fact_333_in__set__impl__in__set__zip2,axiom,
    ! [A: $tType,B: $tType,Xsa: list(A),Ys: list(B),Y: B] :
      ( ( aa(list(A),nat,size_size(list(A)),Xsa) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( member(B,Y,aa(list(B),set(B),set2(B),Ys))
       => ~ ! [X: A] : ~ 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)),aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),Xsa),Ys))) ) ) ).

% in_set_impl_in_set_zip2
tff(fact_334_length__remove1,axiom,
    ! [A: $tType,Xa: A,Xsa: list(A)] :
      ( aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),remove1(A),Xa),Xsa)) = $ite(member(A,Xa,aa(list(A),set(A),set2(A),Xsa)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xsa)),one_one(nat)),aa(list(A),nat,size_size(list(A)),Xsa)) ) ).

% length_remove1
tff(fact_335_insert__code_I2_J,axiom,
    ! [A: $tType,Xa: A,Xsa: list(A)] : ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),coset(A,Xsa)) = coset(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),removeAll(A),Xa),Xsa)) ) ).

% insert_code(2)
tff(fact_336_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) = aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),L12),L23) ) ) ).

% zipf_zip
tff(fact_337_one__assn__raw_Osimps,axiom,
    ! [H2: heap_ext(product_unit),As2: 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),As2))
    <=> ( As2 = bot_bot(set(nat)) ) ) ).

% one_assn_raw.simps
tff(fact_338_one__assn__raw_Oelims_I1_J,axiom,
    ! [Xa: product_prod(heap_ext(product_unit),set(nat)),Y: $o] :
      ( ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,one_assn_raw,Xa)
      <=> (Y) )
     => ~ ! [H: heap_ext(product_unit),As: 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),As) )
           => ( (Y)
            <=> ( As != bot_bot(set(nat)) ) ) ) ) ).

% one_assn_raw.elims(1)
tff(fact_339_one__assn__raw_Oelims_I2_J,axiom,
    ! [Xa: product_prod(heap_ext(product_unit),set(nat))] :
      ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,one_assn_raw,Xa)
     => ~ ! [H: heap_ext(product_unit),As: 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),As) )
           => ( As != bot_bot(set(nat)) ) ) ) ).

% one_assn_raw.elims(2)
tff(fact_340_one__assn__raw_Oelims_I3_J,axiom,
    ! [Xa: product_prod(heap_ext(product_unit),set(nat))] :
      ( ~ aa(product_prod(heap_ext(product_unit),set(nat)),$o,one_assn_raw,Xa)
     => ~ ! [H: heap_ext(product_unit),As: 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),As) )
           => ( As = bot_bot(set(nat)) ) ) ) ).

% one_assn_raw.elims(3)
tff(fact_341_psubset__insert__iff,axiom,
    ! [A: $tType,Aa2: set(A),Xa: A,B2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),Aa2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),B2))
    <=> $ite(
          member(A,Xa,B2),
          aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),Aa2),B2),
          $ite(member(A,Xa,Aa2),aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A))))),B2),aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),B2)) ) ) ).

% psubset_insert_iff
tff(fact_342_lexn__length,axiom,
    ! [A: $tType,Xsa: list(A),Ys: list(A),R2: set(product_prod(A,A)),N: nat] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xsa),Ys),aa(nat,set(product_prod(list(A),list(A))),lexn(A,R2),N))
     => ( ( aa(list(A),nat,size_size(list(A)),Xsa) = N )
        & ( aa(list(A),nat,size_size(list(A)),Ys) = N ) ) ) ).

% lexn_length
tff(fact_343_pairwise__alt,axiom,
    ! [A: $tType,R: fun(A,fun(A,$o)),S: set(A)] :
      ( pairwise(A,R,S)
    <=> ! [X6: A] :
          ( member(A,X6,S)
         => ! [Xa2: A] :
              ( 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)),insert3(A),X6),bot_bot(set(A)))))
             => aa(A,$o,aa(A,fun(A,$o),R,X6),Xa2) ) ) ) ).

% pairwise_alt
tff(fact_344_preciseD_H,axiom,
    ! [B: $tType,A: $tType,R: fun(A,fun(B,assn)),A3: A,P2: B,Fa: assn,H2: product_prod(heap_ext(product_unit),set(nat)),A6: A,F: assn] :
      ( precise(A,B,R)
     => ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),R,A3),P2)),Fa)),H2)
       => ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),R,A6),P2)),F)),H2)
         => ( A3 = A6 ) ) ) ) ).

% preciseD'
tff(fact_345_and__extract__pure__left__iff,axiom,
    ! [B3: $o,Qa: assn] : ( aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),pure_assn((B3))),Qa) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),one_one(assn)),Qa)),pure_assn((B3))) ) ).

% and_extract_pure_left_iff
tff(fact_346_and__extract__pure__right__iff,axiom,
    ! [Pa: assn,B3: $o] : ( aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),Pa),pure_assn((B3))) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),one_one(assn)),Pa)),pure_assn((B3))) ) ).

% and_extract_pure_right_iff
tff(fact_347_order__refl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xa: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Xa) ) ).

% order_refl
tff(fact_348_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_349_subsetI,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A)] :
      ( ! [X: A] :
          ( member(A,X,Aa2)
         => member(A,X,B2) )
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),B2) ) ).

% subsetI
tff(fact_350_subset__antisym,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),B2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),Aa2)
       => ( Aa2 = B2 ) ) ) ).

% subset_antisym
tff(fact_351_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_352_inf__idem,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xa: A] : ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Xa) = Xa ) ) ).

% inf_idem
tff(fact_353_inf_Oleft__idem,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B3: 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),B3)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B3) ) ) ).

% inf.left_idem
tff(fact_354_inf__left__idem,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xa: A,Y: A] : ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Y)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Y) ) ) ).

% inf_left_idem
tff(fact_355_inf_Oright__idem,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B3: A] : ( aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B3)),B3) = aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B3) ) ) ).

% inf.right_idem
tff(fact_356_inf__right__idem,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xa: 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),Xa),Y)),Y) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Y) ) ) ).

% inf_right_idem
tff(fact_357_inf__apply,axiom,
    ! [A: $tType,B: $tType] :
      ( semilattice_inf(A)
     => ! [F2: fun(B,A),G: fun(B,A),Xa: 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)),F2),G),Xa) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(B,A,F2,Xa)),aa(B,A,G,Xa)) ) ) ).

% inf_apply
tff(fact_358_le__inf__iff,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xa: A,Y: A,Z2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Z2))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Z2) ) ) ) ).

% le_inf_iff
tff(fact_359_inf_Obounded__iff,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),inf_inf(A),B3),C2))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),C2) ) ) ) ).

% inf.bounded_iff
tff(fact_360_inf__bot__right,axiom,
    ! [A: $tType] :
      ( bounded_lattice_bot(A)
     => ! [Xa: A] : ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),bot_bot(A)) = bot_bot(A) ) ) ).

% inf_bot_right
tff(fact_361_inf__bot__left,axiom,
    ! [A: $tType] :
      ( bounded_lattice_bot(A)
     => ! [Xa: A] : ( aa(A,A,aa(A,fun(A,A),inf_inf(A),bot_bot(A)),Xa) = bot_bot(A) ) ) ).

% inf_bot_left
tff(fact_362_subset__empty,axiom,
    ! [A: $tType,Aa2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),bot_bot(set(A)))
    <=> ( Aa2 = bot_bot(set(A)) ) ) ).

% subset_empty
tff(fact_363_empty__subsetI,axiom,
    ! [A: $tType,Aa2: set(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),bot_bot(set(A))),Aa2) ).

% empty_subsetI
tff(fact_364_insert__subset,axiom,
    ! [A: $tType,Xa: A,Aa2: set(A),B2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),Aa2)),B2)
    <=> ( member(A,Xa,B2)
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),B2) ) ) ).

% insert_subset
tff(fact_365_psubsetI,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),B2)
     => ( ( Aa2 != B2 )
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),Aa2),B2) ) ) ).

% psubsetI
tff(fact_366_merge__pure__and,axiom,
    ! [A3: $o,B3: $o] :
      ( aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),pure_assn((A3))),pure_assn((B3))) = pure_assn(( (A3)
          & (B3) )) ) ).

% merge_pure_and
tff(fact_367_singleton__insert__inj__eq,axiom,
    ! [A: $tType,B3: A,A3: A,Aa2: set(A)] :
      ( ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),B3),bot_bot(set(A))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),Aa2) )
    <=> ( ( A3 = B3 )
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),B3),bot_bot(set(A)))) ) ) ).

% singleton_insert_inj_eq
tff(fact_368_singleton__insert__inj__eq_H,axiom,
    ! [A: $tType,A3: A,Aa2: set(A),B3: A] :
      ( ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),Aa2) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),B3),bot_bot(set(A))) )
    <=> ( ( A3 = B3 )
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),B3),bot_bot(set(A)))) ) ) ).

% singleton_insert_inj_eq'
tff(fact_369_Diff__eq__empty__iff,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),B2) = bot_bot(set(A)) )
    <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),B2) ) ).

% Diff_eq_empty_iff
tff(fact_370_and__extract__pure__right__ctx__iff,axiom,
    ! [Pa: assn,Qa: assn,B3: $o] : ( aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),Pa),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Qa),pure_assn((B3)))) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),Pa),Qa)),pure_assn((B3))) ) ).

% and_extract_pure_right_ctx_iff
tff(fact_371_and__extract__pure__left__ctx__iff,axiom,
    ! [Pa: assn,B3: $o,Qa: assn] : ( aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Pa),pure_assn((B3)))),Qa) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),Pa),Qa)),pure_assn((B3))) ) ).

% and_extract_pure_left_ctx_iff
tff(fact_372_mod__h__bot__iff_I6_J,axiom,
    ! [Pa: assn,Qa: assn,H2: 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,aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),Pa),Qa)),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,Pa),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,Qa),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)))) ) ) ).

% mod_h_bot_iff(6)
tff(fact_373_in__mono,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A),Xa: A] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),B2)
     => ( member(A,Xa,Aa2)
       => member(A,Xa,B2) ) ) ).

% in_mono
tff(fact_374_subsetD,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A),C2: A] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),B2)
     => ( member(A,C2,Aa2)
       => member(A,C2,B2) ) ) ).

% subsetD
tff(fact_375_equalityE,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A)] :
      ( ( Aa2 = B2 )
     => ~ ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),B2)
         => ~ aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),Aa2) ) ) ).

% equalityE
tff(fact_376_pairwiseD,axiom,
    ! [A: $tType,R: fun(A,fun(A,$o)),S: set(A),Xa: A,Y: A] :
      ( pairwise(A,R,S)
     => ( member(A,Xa,S)
       => ( member(A,Y,S)
         => ( ( Xa != Y )
           => aa(A,$o,aa(A,fun(A,$o),R,Xa),Y) ) ) ) ) ).

% pairwiseD
tff(fact_377_pairwiseI,axiom,
    ! [A: $tType,S: set(A),R: fun(A,fun(A,$o))] :
      ( ! [X: A,Y2: A] :
          ( member(A,X,S)
         => ( member(A,Y2,S)
           => ( ( X != Y2 )
             => aa(A,$o,aa(A,fun(A,$o),R,X),Y2) ) ) )
     => pairwise(A,R,S) ) ).

% pairwiseI
tff(fact_378_subset__eq,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),B2)
    <=> ! [X6: A] :
          ( member(A,X6,Aa2)
         => member(A,X6,B2) ) ) ).

% subset_eq
tff(fact_379_equalityD1,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A)] :
      ( ( Aa2 = B2 )
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),B2) ) ).

% equalityD1
tff(fact_380_equalityD2,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A)] :
      ( ( Aa2 = B2 )
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),Aa2) ) ).

% equalityD2
tff(fact_381_subset__iff,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),B2)
    <=> ! [T3: A] :
          ( member(A,T3,Aa2)
         => member(A,T3,B2) ) ) ).

% subset_iff
tff(fact_382_subset__refl,axiom,
    ! [A: $tType,Aa2: set(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),Aa2) ).

% subset_refl
tff(fact_383_Collect__mono,axiom,
    ! [A: $tType,Pa: fun(A,$o),Qa: fun(A,$o)] :
      ( ! [X: A] :
          ( aa(A,$o,Pa,X)
         => aa(A,$o,Qa,X) )
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(fun(A,$o),set(A),collect(A),Pa)),aa(fun(A,$o),set(A),collect(A),Qa)) ) ).

% Collect_mono
tff(fact_384_pairwise__def,axiom,
    ! [A: $tType,R: fun(A,fun(A,$o)),S: set(A)] :
      ( pairwise(A,R,S)
    <=> ! [X6: A] :
          ( member(A,X6,S)
         => ! [Xa2: A] :
              ( member(A,Xa2,S)
             => ( ( X6 != Xa2 )
               => aa(A,$o,aa(A,fun(A,$o),R,X6),Xa2) ) ) ) ) ).

% pairwise_def
tff(fact_385_subset__trans,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A),C3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),B2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),C3)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),C3) ) ) ).

% subset_trans
tff(fact_386_pairwise__mono,axiom,
    ! [A: $tType,Pa: fun(A,fun(A,$o)),Aa2: set(A),Qa: fun(A,fun(A,$o)),B2: set(A)] :
      ( pairwise(A,Pa,Aa2)
     => ( ! [X: A,Y2: A] :
            ( aa(A,$o,aa(A,fun(A,$o),Pa,X),Y2)
           => aa(A,$o,aa(A,fun(A,$o),Qa,X),Y2) )
       => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),Aa2)
         => pairwise(A,Qa,B2) ) ) ) ).

% pairwise_mono
tff(fact_387_set__eq__subset,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A)] :
      ( ( Aa2 = B2 )
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),B2)
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),Aa2) ) ) ).

% set_eq_subset
tff(fact_388_pairwise__subset,axiom,
    ! [A: $tType,Pa: fun(A,fun(A,$o)),S: set(A),T4: set(A)] :
      ( pairwise(A,Pa,S)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),T4),S)
       => pairwise(A,Pa,T4) ) ) ).

% pairwise_subset
tff(fact_389_Collect__mono__iff,axiom,
    ! [A: $tType,Pa: fun(A,$o),Qa: 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),Pa)),aa(fun(A,$o),set(A),collect(A),Qa))
    <=> ! [X6: A] :
          ( aa(A,$o,Pa,X6)
         => aa(A,$o,Qa,X6) ) ) ).

% Collect_mono_iff
tff(fact_390_ord__eq__le__eq__trans,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A3: A,B3: A,C2: A,D3: A] :
          ( ( A3 = B3 )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),C2)
           => ( ( C2 = D3 )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),D3) ) ) ) ) ).

% ord_eq_le_eq_trans
tff(fact_391_subset__Collect__conv,axiom,
    ! [A: $tType,S: set(A),Pa: fun(A,$o)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),aa(fun(A,$o),set(A),collect(A),Pa))
    <=> ! [X6: A] :
          ( member(A,X6,S)
         => aa(A,$o,Pa,X6) ) ) ).

% subset_Collect_conv
tff(fact_392_assn__aci_I4_J,axiom,
    ! [Xa: assn,Y: assn] : ( aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),Xa),aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),Xa),Y)) = aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),Xa),Y) ) ).

% assn_aci(4)
tff(fact_393_assn__aci_I3_J,axiom,
    ! [Xa: assn,Y: assn,Z2: assn] : ( aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),Xa),aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),Y),Z2)) = aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),Y),aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),Xa),Z2)) ) ).

% assn_aci(3)
tff(fact_394_assn__aci_I1_J,axiom,
    ! [Xa: assn,Y: assn] : ( aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),Xa),Y) = aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),Y),Xa) ) ).

% assn_aci(1)
tff(fact_395_norm__assertion__simps_I31_J,axiom,
    ! [Xa: assn] : ( aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),Xa),Xa) = Xa ) ).

% norm_assertion_simps(31)
tff(fact_396_norm__assertion__simps_I14_J,axiom,
    ! [Xa: assn,Y: assn,Z2: assn] : ( aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),Xa),Y)),Z2) = aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),Xa),aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),Y),Z2)) ) ).

% norm_assertion_simps(14)
tff(fact_397_nle__le,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3)
            & ( B3 != A3 ) ) ) ) ).

% nle_le
tff(fact_398_le__cases3,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Y: A,Z2: A] :
          ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
           => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z2) )
         => ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xa)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Z2) )
           => ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Z2)
               => ~ aa(A,$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,$o),ord_less_eq(A),Y),Xa) )
               => ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z2)
                   => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z2),Xa) )
                 => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z2),Xa)
                     => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y) ) ) ) ) ) ) ) ).

% le_cases3
tff(fact_399_order__class_Oorder__eq__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xa: A,Y: A] :
          ( ( Xa = Y )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xa) ) ) ) ).

% order_class.order_eq_iff
tff(fact_400_ord__eq__le__trans,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A3: A,B3: A,C2: A] :
          ( ( A3 = B3 )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),C2) ) ) ) ).

% ord_eq_le_trans
tff(fact_401_ord__le__eq__trans,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
         => ( ( B3 = C2 )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),C2) ) ) ) ).

% ord_le_eq_trans
tff(fact_402_order__antisym,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xa)
           => ( Xa = Y ) ) ) ) ).

% order_antisym
tff(fact_403_order_Otrans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),C2) ) ) ) ).

% order.trans
tff(fact_404_order__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xa: A,Y: A,Z2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Z2) ) ) ) ).

% order_trans
tff(fact_405_linorder__wlog,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Pa: fun(A,fun(A,$o)),A3: A,B3: A] :
          ( ! [A4: A,B4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),B4)
             => aa(A,$o,aa(A,fun(A,$o),Pa,A4),B4) )
         => ( ! [A4: A,B4: A] :
                ( aa(A,$o,aa(A,fun(A,$o),Pa,B4),A4)
               => aa(A,$o,aa(A,fun(A,$o),Pa,A4),B4) )
           => aa(A,$o,aa(A,fun(A,$o),Pa,A3),B3) ) ) ) ).

% linorder_wlog
tff(fact_406_dual__order_Oeq__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B3: A] :
          ( ( A3 = B3 )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3) ) ) ) ).

% dual_order.eq_iff
tff(fact_407_dual__order_Oantisym,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
           => ( A3 = B3 ) ) ) ) ).

% dual_order.antisym
tff(fact_408_dual__order_Otrans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B3: A,A3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),B3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A3) ) ) ) ).

% dual_order.trans
tff(fact_409_antisym,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3)
           => ( A3 = B3 ) ) ) ) ).

% antisym
tff(fact_410_le__funD,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F2: fun(A,B),G: fun(A,B),Xa: A] :
          ( aa(fun(A,B),$o,aa(fun(A,B),fun(fun(A,B),$o),ord_less_eq(fun(A,B)),F2),G)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,Xa)),aa(A,B,G,Xa)) ) ) ).

% le_funD
tff(fact_411_le__funE,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F2: fun(A,B),G: fun(A,B),Xa: A] :
          ( aa(fun(A,B),$o,aa(fun(A,B),fun(fun(A,B),$o),ord_less_eq(fun(A,B)),F2),G)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,Xa)),aa(A,B,G,Xa)) ) ) ).

% le_funE
tff(fact_412_le__funI,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F2: fun(A,B),G: fun(A,B)] :
          ( ! [X: A] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X)),aa(A,B,G,X))
         => aa(fun(A,B),$o,aa(fun(A,B),fun(fun(A,B),$o),ord_less_eq(fun(A,B)),F2),G) ) ) ).

% le_funI
tff(fact_413_le__fun__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F2: fun(A,B),G: fun(A,B)] :
          ( aa(fun(A,B),$o,aa(fun(A,B),fun(fun(A,B),$o),ord_less_eq(fun(A,B)),F2),G)
        <=> ! [X6: A] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X6)),aa(A,B,G,X6)) ) ) ).

% le_fun_def
tff(fact_414_Orderings_Oorder__eq__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B3: A] :
          ( ( A3 = B3 )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3) ) ) ) ).

% Orderings.order_eq_iff
tff(fact_415_order__subst1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A3: A,F2: fun(B,A),B3: B,C2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(B,A,F2,B3))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B3),C2)
           => ( ! [X: B,Y2: B] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),Y2)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,F2,X)),aa(B,A,F2,Y2)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(B,A,F2,C2)) ) ) ) ) ).

% order_subst1
tff(fact_416_order__subst2,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A3: A,B3: A,F2: fun(A,B),C2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,B3)),C2)
           => ( ! [X: A,Y2: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y2)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X)),aa(A,B,F2,Y2)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,A3)),C2) ) ) ) ) ).

% order_subst2
tff(fact_417_order__eq__refl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xa: A,Y: A] :
          ( ( Xa = Y )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y) ) ) ).

% order_eq_refl
tff(fact_418_linorder__linear,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
          | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xa) ) ) ).

% linorder_linear
tff(fact_419_ord__eq__le__subst,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & ord(A) )
     => ! [A3: A,F2: fun(B,A),B3: B,C2: B] :
          ( ( A3 = aa(B,A,F2,B3) )
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B3),C2)
           => ( ! [X: B,Y2: B] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),Y2)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,F2,X)),aa(B,A,F2,Y2)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(B,A,F2,C2)) ) ) ) ) ).

% ord_eq_le_subst
tff(fact_420_ord__le__eq__subst,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & ord(A) )
     => ! [A3: A,B3: A,F2: fun(A,B),C2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
         => ( ( aa(A,B,F2,B3) = C2 )
           => ( ! [X: A,Y2: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y2)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X)),aa(A,B,F2,Y2)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,A3)),C2) ) ) ) ) ).

% ord_le_eq_subst
tff(fact_421_linorder__le__cases,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Y: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xa) ) ) ).

% linorder_le_cases
tff(fact_422_order__antisym__conv,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Y: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xa)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
          <=> ( Xa = Y ) ) ) ) ).

% order_antisym_conv
tff(fact_423_inf__sup__aci_I4_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xa: A,Y: A] : ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Y)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Y) ) ) ).

% inf_sup_aci(4)
tff(fact_424_inf__sup__aci_I3_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xa: A,Y: A,Z2: A] : ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),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),Xa),Z2)) ) ) ).

% inf_sup_aci(3)
tff(fact_425_inf__sup__aci_I2_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xa: 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),Xa),Y)),Z2) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Z2)) ) ) ).

% inf_sup_aci(2)
tff(fact_426_inf__sup__aci_I1_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xa: A,Y: A] : ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Y) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Xa) ) ) ).

% inf_sup_aci(1)
tff(fact_427_inf__sup__ord_I2_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xa: A,Y: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Y)),Y) ) ).

% inf_sup_ord(2)
tff(fact_428_inf__sup__ord_I1_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xa: A,Y: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Y)),Xa) ) ).

% inf_sup_ord(1)
tff(fact_429_inf__le1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xa: A,Y: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Y)),Xa) ) ).

% inf_le1
tff(fact_430_inf__le2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xa: A,Y: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Y)),Y) ) ).

% inf_le2
tff(fact_431_le__infE,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xa: A,A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B3))
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),A3)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),B3) ) ) ) ).

% le_infE
tff(fact_432_le__infI,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xa: A,A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),B3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B3)) ) ) ) ).

% le_infI
tff(fact_433_inf__mono,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,C2: A,B3: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),D3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B3)),aa(A,A,aa(A,fun(A,A),inf_inf(A),C2),D3)) ) ) ) ).

% inf_mono
tff(fact_434_le__infI1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,Xa: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),Xa)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B3)),Xa) ) ) ).

% le_infI1
tff(fact_435_le__infI2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B3: A,Xa: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),Xa)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B3)),Xa) ) ) ).

% le_infI2
tff(fact_436_inf_Oassoc,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B3: 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),B3)),C2) = aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),aa(A,A,aa(A,fun(A,A),inf_inf(A),B3),C2)) ) ) ).

% inf.assoc
tff(fact_437_inf__assoc,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xa: 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),Xa),Y)),Z2) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Z2)) ) ) ).

% inf_assoc
tff(fact_438_inf_OorderE,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
         => ( A3 = aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B3) ) ) ) ).

% inf.orderE
tff(fact_439_inf_OorderI,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B3: A] :
          ( ( A3 = aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B3) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3) ) ) ).

% inf.orderI
tff(fact_440_inf__unique,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [F2: fun(A,fun(A,A)),Xa: A,Y: A] :
          ( ! [X: A,Y2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),F2,X),Y2)),X)
         => ( ! [X: A,Y2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),F2,X),Y2)),Y2)
           => ( ! [X: A,Y2: A,Z3: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y2)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Z3)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),F2,Y2),Z3)) ) )
             => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Y) = aa(A,A,aa(A,fun(A,A),F2,Xa),Y) ) ) ) ) ) ).

% inf_unique
tff(fact_441_le__iff__inf,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
        <=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Y) = Xa ) ) ) ).

% le_iff_inf
tff(fact_442_inf_Oabsorb1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B3) = A3 ) ) ) ).

% inf.absorb1
tff(fact_443_inf_Oabsorb2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3)
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B3) = B3 ) ) ) ).

% inf.absorb2
tff(fact_444_inf_Ocommute,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B3: A] : ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B3) = aa(A,A,aa(A,fun(A,A),inf_inf(A),B3),A3) ) ) ).

% inf.commute
tff(fact_445_inf__absorb1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Y) = Xa ) ) ) ).

% inf_absorb1
tff(fact_446_inf__absorb2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Y: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xa)
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Y) = Y ) ) ) ).

% inf_absorb2
tff(fact_447_inf__commute,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xa: A,Y: A] : ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Y) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Xa) ) ) ).

% inf_commute
tff(fact_448_inf_OboundedE,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),inf_inf(A),B3),C2))
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),C2) ) ) ) ).

% inf.boundedE
tff(fact_449_inf_OboundedI,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),inf_inf(A),B3),C2)) ) ) ) ).

% inf.boundedI
tff(fact_450_inf__greatest,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xa: A,Y: A,Z2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Z2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Z2)) ) ) ) ).

% inf_greatest
tff(fact_451_inf_Oorder__iff,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
        <=> ( A3 = aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B3) ) ) ) ).

% inf.order_iff
tff(fact_452_inf_Ocobounded1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B3)),A3) ) ).

% inf.cobounded1
tff(fact_453_inf_Ocobounded2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B3)),B3) ) ).

% inf.cobounded2
tff(fact_454_inf_Oabsorb__iff1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
        <=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B3) = A3 ) ) ) ).

% inf.absorb_iff1
tff(fact_455_inf_Oabsorb__iff2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3)
        <=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B3) = B3 ) ) ) ).

% inf.absorb_iff2
tff(fact_456_inf_OcoboundedI1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,C2: A,B3: A] :
          ( aa(A,$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),B3)),C2) ) ) ).

% inf.coboundedI1
tff(fact_457_inf_OcoboundedI2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B3: A,C2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),C2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B3)),C2) ) ) ).

% inf.coboundedI2
tff(fact_458_inf_Oleft__commute,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B3: A,A3: A,C2: A] : ( aa(A,A,aa(A,fun(A,A),inf_inf(A),B3),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),B3),C2)) ) ) ).

% inf.left_commute
tff(fact_459_inf__left__commute,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xa: A,Y: A,Z2: A] : ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),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),Xa),Z2)) ) ) ).

% inf_left_commute
tff(fact_460_inf__fun__def,axiom,
    ! [B: $tType,A: $tType] :
      ( semilattice_inf(B)
     => ! [F2: fun(A,B),G: fun(A,B),X4: 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)),F2),G),X4) = aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(A,B,F2,X4)),aa(A,B,G,X4)) ) ) ).

% inf_fun_def
tff(fact_461_listrel1__mono,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)),$o),ord_less_eq(set(product_prod(A,A))),R2),S2)
     => aa(set(product_prod(list(A),list(A))),$o,aa(set(product_prod(list(A),list(A))),fun(set(product_prod(list(A),list(A))),$o),ord_less_eq(set(product_prod(list(A),list(A)))),listrel1(A,R2)),listrel1(A,S2)) ) ).

% listrel1_mono
tff(fact_462_listrel__mono,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,B)),S2: set(product_prod(A,B))] :
      ( aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),R2),S2)
     => aa(set(product_prod(list(A),list(B))),$o,aa(set(product_prod(list(A),list(B))),fun(set(product_prod(list(A),list(B))),$o),ord_less_eq(set(product_prod(list(A),list(B)))),listrel(A,B,R2)),listrel(A,B,S2)) ) ).

% listrel_mono
tff(fact_463_inf_Ostrict__coboundedI2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B3: A,C2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),C2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B3)),C2) ) ) ).

% inf.strict_coboundedI2
tff(fact_464_inf_Ostrict__coboundedI1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,C2: A,B3: A] :
          ( aa(A,$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),B3)),C2) ) ) ).

% inf.strict_coboundedI1
tff(fact_465_inf_Ostrict__order__iff,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
        <=> ( ( A3 = aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B3) )
            & ( A3 != B3 ) ) ) ) ).

% inf.strict_order_iff
tff(fact_466_inf_Ostrict__boundedE,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),inf_inf(A),B3),C2))
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),C2) ) ) ) ).

% inf.strict_boundedE
tff(fact_467_inf_Oabsorb4,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A3)
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B3) = B3 ) ) ) ).

% inf.absorb4
tff(fact_468_inf_Oabsorb3,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B3) = A3 ) ) ) ).

% inf.absorb3
tff(fact_469_less__infI2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B3: A,Xa: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),Xa)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B3)),Xa) ) ) ).

% less_infI2
tff(fact_470_less__infI1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,Xa: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),Xa)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B3)),Xa) ) ) ).

% less_infI1
tff(fact_471_pinf_I6_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T2: A] :
        ? [Z3: A] :
        ! [X4: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z3),X4)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),T2) ) ) ).

% pinf(6)
tff(fact_472_pinf_I8_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T2: A] :
        ? [Z3: A] :
        ! [X4: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z3),X4)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),T2),X4) ) ) ).

% pinf(8)
tff(fact_473_minf_I6_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T2: A] :
        ? [Z3: A] :
        ! [X4: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Z3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),T2) ) ) ).

% minf(6)
tff(fact_474_minf_I8_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T2: A] :
        ? [Z3: A] :
        ! [X4: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Z3)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),T2),X4) ) ) ).

% minf(8)
tff(fact_475_order__le__imp__less__or__eq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
            | ( Xa = Y ) ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_476_linorder__le__less__linear,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
          | aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xa) ) ) ).

% linorder_le_less_linear
tff(fact_477_order__less__le__subst2,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A3: A,B3: A,F2: fun(A,B),C2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,B3)),C2)
           => ( ! [X: A,Y2: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y2)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,X)),aa(A,B,F2,Y2)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,A3)),C2) ) ) ) ) ).

% order_less_le_subst2
tff(fact_478_order__less__le__subst1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A3: A,F2: fun(B,A),B3: B,C2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(B,A,F2,B3))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B3),C2)
           => ( ! [X: B,Y2: B] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),Y2)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,F2,X)),aa(B,A,F2,Y2)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(B,A,F2,C2)) ) ) ) ) ).

% order_less_le_subst1
tff(fact_479_order__le__less__subst2,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A3: A,B3: A,F2: fun(A,B),C2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,B3)),C2)
           => ( ! [X: A,Y2: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y2)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X)),aa(A,B,F2,Y2)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,A3)),C2) ) ) ) ) ).

% order_le_less_subst2
tff(fact_480_order__le__less__subst1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A3: A,F2: fun(B,A),B3: B,C2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(B,A,F2,B3))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),B3),C2)
           => ( ! [X: B,Y2: B] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X),Y2)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,F2,X)),aa(B,A,F2,Y2)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(B,A,F2,C2)) ) ) ) ) ).

% order_le_less_subst1
tff(fact_481_order__less__le__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xa: A,Y: A,Z2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Z2) ) ) ) ).

% order_less_le_trans
tff(fact_482_order__le__less__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xa: A,Y: A,Z2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Z2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Z2) ) ) ) ).

% order_le_less_trans
tff(fact_483_order__neq__le__trans,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B3: A] :
          ( ( A3 != B3 )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3) ) ) ) ).

% order_neq_le_trans
tff(fact_484_order__le__neq__trans,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
         => ( ( A3 != B3 )
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3) ) ) ) ).

% order_le_neq_trans
tff(fact_485_order__less__imp__le,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y) ) ) ).

% order_less_imp_le
tff(fact_486_linorder__not__less,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Y: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xa) ) ) ).

% linorder_not_less
tff(fact_487_linorder__not__le,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Y: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xa) ) ) ).

% linorder_not_le
tff(fact_488_order__less__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
            & ( Xa != Y ) ) ) ) ).

% order_less_le
tff(fact_489_order__le__less,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
            | ( Xa = Y ) ) ) ) ).

% order_le_less
tff(fact_490_dual__order_Ostrict__implies__order,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3) ) ) ).

% dual_order.strict_implies_order
tff(fact_491_order_Ostrict__implies__order,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3) ) ) ).

% order.strict_implies_order
tff(fact_492_dual__order_Ostrict__iff__not,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A3)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3)
            & ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3) ) ) ) ).

% dual_order.strict_iff_not
tff(fact_493_dual__order_Ostrict__trans2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B3: A,A3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),B3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),A3) ) ) ) ).

% dual_order.strict_trans2
tff(fact_494_dual__order_Ostrict__trans1,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B3: A,A3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),B3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),A3) ) ) ) ).

% dual_order.strict_trans1
tff(fact_495_dual__order_Ostrict__iff__order,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A3)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3)
            & ( A3 != B3 ) ) ) ) ).

% dual_order.strict_iff_order
tff(fact_496_dual__order_Oorder__iff__strict,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A3)
            | ( A3 = B3 ) ) ) ) ).

% dual_order.order_iff_strict
tff(fact_497_dense__le__bounded,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [Xa: A,Y: A,Z2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
         => ( ! [W: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),W)
               => ( aa(A,$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_498_dense__ge__bounded,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [Z2: A,Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z2),Xa)
         => ( ! [W: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z2),W)
               => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),W),Xa)
                 => aa(A,$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_499_order_Ostrict__iff__not,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
            & ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3) ) ) ) ).

% order.strict_iff_not
tff(fact_500_order_Ostrict__trans2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),C2) ) ) ) ).

% order.strict_trans2
tff(fact_501_order_Ostrict__trans1,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),C2) ) ) ) ).

% order.strict_trans1
tff(fact_502_order_Ostrict__iff__order,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
            & ( A3 != B3 ) ) ) ) ).

% order.strict_iff_order
tff(fact_503_order_Oorder__iff__strict,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
            | ( A3 = B3 ) ) ) ) ).

% order.order_iff_strict
tff(fact_504_not__le__imp__less,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Y: A,Xa: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xa)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y) ) ) ).

% not_le_imp_less
tff(fact_505_less__le__not__le,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
            & ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xa) ) ) ) ).

% less_le_not_le
tff(fact_506_dense__le,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [Y: A,Z2: A] :
          ( ! [X: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Z2) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z2) ) ) ).

% dense_le
tff(fact_507_dense__ge,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [Z2: A,Y: A] :
          ( ! [X: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z2),X)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z2) ) ) ).

% dense_ge
tff(fact_508_antisym__conv2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
         => ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
          <=> ( Xa = Y ) ) ) ) ).

% antisym_conv2
tff(fact_509_antisym__conv1,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xa: A,Y: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
          <=> ( Xa = Y ) ) ) ) ).

% antisym_conv1
tff(fact_510_nless__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B3: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
        <=> ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
            | ( A3 = B3 ) ) ) ) ).

% nless_le
tff(fact_511_leI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Y: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xa) ) ) ).

% leI
tff(fact_512_leD,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Y: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xa)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y) ) ) ).

% leD
tff(fact_513_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_514_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_515_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_516_subrelI,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,B)),S2: set(product_prod(A,B))] :
      ( ! [X: A,Y2: B] :
          ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y2),R2)
         => member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y2),S2) )
     => aa(set(product_prod(A,B)),$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_517_diff__eq__diff__less__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B3: A,C2: A,D3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B3) = aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),D3) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),D3) ) ) ) ).

% diff_eq_diff_less_eq
tff(fact_518_diff__right__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),C2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),B3),C2)) ) ) ).

% diff_right_mono
tff(fact_519_diff__left__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [B3: A,A3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_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),B3)) ) ) ).

% diff_left_mono
tff(fact_520_diff__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B3: A,D3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),D3),C2)
           => aa(A,$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),B3),D3)) ) ) ) ).

% diff_mono
tff(fact_521_mod__and__dist,axiom,
    ! [Pa: assn,Qa: 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),inf_inf(assn),Pa),Qa)),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,Pa),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,Qa),H2) ) ) ).

% mod_and_dist
tff(fact_522_ent__conjI,axiom,
    ! [Aa2: assn,B2: assn,C3: assn] :
      ( aa(assn,$o,aa(assn,fun(assn,$o),entails,Aa2),B2)
     => ( aa(assn,$o,aa(assn,fun(assn,$o),entails,Aa2),C3)
       => aa(assn,$o,aa(assn,fun(assn,$o),entails,Aa2),aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),B2),C3)) ) ) ).

% ent_conjI
tff(fact_523_ent__conjE1,axiom,
    ! [Aa2: assn,C3: assn,B2: assn] :
      ( aa(assn,$o,aa(assn,fun(assn,$o),entails,Aa2),C3)
     => aa(assn,$o,aa(assn,fun(assn,$o),entails,aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),Aa2),B2)),C3) ) ).

% ent_conjE1
tff(fact_524_ent__conjE2,axiom,
    ! [B2: assn,C3: assn,Aa2: assn] :
      ( aa(assn,$o,aa(assn,fun(assn,$o),entails,B2),C3)
     => aa(assn,$o,aa(assn,fun(assn,$o),entails,aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),Aa2),B2)),C3) ) ).

% ent_conjE2
tff(fact_525_subset__insertI2,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A),B3: A] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),B2)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),B3),B2)) ) ).

% subset_insertI2
tff(fact_526_subset__insertI,axiom,
    ! [A: $tType,B2: set(A),A3: A] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),B2)) ).

% subset_insertI
tff(fact_527_subset__insert,axiom,
    ! [A: $tType,Xa: A,Aa2: set(A),B2: set(A)] :
      ( ~ member(A,Xa,Aa2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),B2))
      <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),B2) ) ) ).

% subset_insert
tff(fact_528_insert__mono,axiom,
    ! [A: $tType,C3: set(A),D2: set(A),A3: A] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C3),D2)
     => aa(set(A),$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)),insert3(A),A3),C3)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),D2)) ) ).

% insert_mono
tff(fact_529_subset__code_I1_J,axiom,
    ! [A: $tType,Xsa: list(A),B2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xsa)),B2)
    <=> ! [X6: A] :
          ( member(A,X6,aa(list(A),set(A),set2(A),Xsa))
         => member(A,X6,B2) ) ) ).

% subset_code(1)
tff(fact_530_double__diff,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A),C3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),B2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),C3)
       => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),C3),Aa2)) = Aa2 ) ) ) ).

% double_diff
tff(fact_531_Diff__subset,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A)] : aa(set(A),$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)),Aa2),B2)),Aa2) ).

% Diff_subset
tff(fact_532_Diff__mono,axiom,
    ! [A: $tType,Aa2: set(A),C3: set(A),D2: set(A),B2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),C3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),D2),B2)
       => aa(set(A),$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)),Aa2),B2)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),C3),D2)) ) ) ).

% Diff_mono
tff(fact_533_subset__iff__psubset__eq,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),B2)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),Aa2),B2)
        | ( Aa2 = B2 ) ) ) ).

% subset_iff_psubset_eq
tff(fact_534_subset__psubset__trans,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A),C3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),B2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),B2),C3)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),Aa2),C3) ) ) ).

% subset_psubset_trans
tff(fact_535_subset__not__subset__eq,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),Aa2),B2)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),B2)
        & ~ aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),Aa2) ) ) ).

% subset_not_subset_eq
tff(fact_536_psubset__subset__trans,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A),C3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),Aa2),B2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),C3)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),Aa2),C3) ) ) ).

% psubset_subset_trans
tff(fact_537_psubset__imp__subset,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),Aa2),B2)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),B2) ) ).

% psubset_imp_subset
tff(fact_538_psubset__eq,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),Aa2),B2)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),B2)
        & ( Aa2 != B2 ) ) ) ).

% psubset_eq
tff(fact_539_psubsetE,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),Aa2),B2)
     => ~ ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),B2)
         => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),Aa2) ) ) ).

% psubsetE
tff(fact_540_norm__assertion__simps_I9_J,axiom,
    ! [Xa: assn] : ( aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),bot_bot(assn)),Xa) = bot_bot(assn) ) ).

% norm_assertion_simps(9)
tff(fact_541_norm__assertion__simps_I10_J,axiom,
    ! [Xa: assn] : ( aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),Xa),bot_bot(assn)) = bot_bot(assn) ) ).

% norm_assertion_simps(10)
tff(fact_542_precise__def,axiom,
    ! [B: $tType,A: $tType,R: fun(A,fun(B,assn))] :
      ( precise(A,B,R)
    <=> ! [A7: A,A8: A,H3: product_prod(heap_ext(product_unit),set(nat)),P4: B,F5: assn,F6: assn] :
          ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),R,A7),P4)),F5)),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),R,A8),P4)),F6))),H3)
         => ( A7 = A8 ) ) ) ).

% precise_def
tff(fact_543_preciseI,axiom,
    ! [B: $tType,A: $tType,R: fun(A,fun(B,assn))] :
      ( ! [A4: A,A9: A,H: product_prod(heap_ext(product_unit),set(nat)),P5: B,F7: assn,F8: assn] :
          ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),R,A4),P5)),F7)),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),R,A9),P5)),F8))),H)
         => ( A4 = A9 ) )
     => precise(A,B,R) ) ).

% preciseI
tff(fact_544_preciseD,axiom,
    ! [B: $tType,A: $tType,R: fun(A,fun(B,assn)),A3: A,P2: B,Fa: assn,A6: A,F: assn,H2: product_prod(heap_ext(product_unit),set(nat))] :
      ( precise(A,B,R)
     => ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),R,A3),P2)),Fa)),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),R,A6),P2)),F))),H2)
       => ( A3 = A6 ) ) ) ).

% preciseD
tff(fact_545_pairwise__empty,axiom,
    ! [A: $tType,Pa: fun(A,fun(A,$o))] : pairwise(A,Pa,bot_bot(set(A))) ).

% pairwise_empty
tff(fact_546_pairwise__insert,axiom,
    ! [A: $tType,R2: fun(A,fun(A,$o)),Xa: A,S2: set(A)] :
      ( pairwise(A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),S2))
    <=> ( ! [Y4: A] :
            ( ( member(A,Y4,S2)
              & ( Y4 != Xa ) )
           => ( aa(A,$o,aa(A,fun(A,$o),R2,Xa),Y4)
              & aa(A,$o,aa(A,fun(A,$o),R2,Y4),Xa) ) )
        & pairwise(A,R2,S2) ) ) ).

% pairwise_insert
tff(fact_547_prec__frame,axiom,
    ! [B: $tType,A: $tType,Pa: fun(A,fun(B,assn)),R1: assn,R22: assn,H2: product_prod(heap_ext(product_unit),set(nat)),Xa: A,P2: B,F12: assn,Y: A,F22: assn] :
      ( precise(A,B,Pa)
     => ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),R1),R22)),H2)
       => ( aa(assn,$o,aa(assn,fun(assn,$o),entails,R1),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),Pa,Xa),P2)),F12))
         => ( aa(assn,$o,aa(assn,fun(assn,$o),entails,R22),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),Pa,Y),P2)),F22))
           => ( Xa = Y ) ) ) ) ) ).

% prec_frame
tff(fact_548_subset__singletonD,axiom,
    ! [A: $tType,Aa2: set(A),Xa: A] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A))))
     => ( ( Aa2 = bot_bot(set(A)) )
        | ( Aa2 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A))) ) ) ) ).

% subset_singletonD
tff(fact_549_subset__singleton__iff,axiom,
    ! [A: $tType,X3: set(A),A3: A] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),bot_bot(set(A))))
    <=> ( ( X3 = bot_bot(set(A)) )
        | ( X3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),bot_bot(set(A))) ) ) ) ).

% subset_singleton_iff
tff(fact_550_subset__minus__empty,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),B2)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),B2) = bot_bot(set(A)) ) ) ).

% subset_minus_empty
tff(fact_551_subset__Diff__insert,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A),Xa: A,C3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),C3)))
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B2),C3))
        & ~ member(A,Xa,Aa2) ) ) ).

% subset_Diff_insert
tff(fact_552_set__update__subsetI,axiom,
    ! [A: $tType,Xsa: list(A),Aa2: set(A),Xa: A,Ia: nat] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xsa)),Aa2)
     => ( member(A,Xa,Aa2)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xsa),Ia),Xa))),Aa2) ) ) ).

% set_update_subsetI
tff(fact_553_set__remove1__subset,axiom,
    ! [A: $tType,Xa: A,Xsa: list(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),remove1(A),Xa),Xsa))),aa(list(A),set(A),set2(A),Xsa)) ).

% set_remove1_subset
tff(fact_554_subset__code_I2_J,axiom,
    ! [A: $tType,Aa2: set(A),Ys: list(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),coset(A,Ys))
    <=> ! [X6: A] :
          ( member(A,X6,aa(list(A),set(A),set2(A),Ys))
         => ~ member(A,X6,Aa2) ) ) ).

% subset_code(2)
tff(fact_555_pairwise__singleton,axiom,
    ! [A: $tType,Pa: fun(A,fun(A,$o)),Aa2: A] : pairwise(A,Pa,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Aa2),bot_bot(set(A)))) ).

% pairwise_singleton
tff(fact_556_prec__frame__expl,axiom,
    ! [A: $tType,Pa: fun(A,assn),F12: assn,F22: assn,H2: product_prod(heap_ext(product_unit),set(nat)),R1: assn,R22: assn,Xa: A,Y: A] :
      ( ! [X: A,Y2: A] :
          ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(A,assn,Pa,X)),F12)),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(A,assn,Pa,Y2)),F22))),H2)
         => ( X = Y2 ) )
     => ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),R1),R22)),H2)
       => ( aa(assn,$o,aa(assn,fun(assn,$o),entails,R1),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(A,assn,Pa,Xa)),F12))
         => ( aa(assn,$o,aa(assn,fun(assn,$o),entails,R22),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(A,assn,Pa,Y)),F22))
           => ( Xa = Y ) ) ) ) ) ).

% prec_frame_expl
tff(fact_557_prec__frame_H,axiom,
    ! [A: $tType,Pa: fun(A,assn),Xa: A,F12: assn,Y: A,F22: assn,H2: product_prod(heap_ext(product_unit),set(nat)),R1: assn,R22: assn] :
      ( ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(A,assn,Pa,Xa)),F12)),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(A,assn,Pa,Y)),F22))),H2)
       => ( Xa = Y ) )
     => ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),R1),R22)),H2)
       => ( aa(assn,$o,aa(assn,fun(assn,$o),entails,R1),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(A,assn,Pa,Xa)),F12))
         => ( aa(assn,$o,aa(assn,fun(assn,$o),entails,R22),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(A,assn,Pa,Y)),F22))
           => ( Xa = Y ) ) ) ) ) ).

% prec_frame'
tff(fact_558_subset__insert__iff,axiom,
    ! [A: $tType,Aa2: set(A),Xa: A,B2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),B2))
    <=> $ite(member(A,Xa,Aa2),aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A))))),B2),aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),B2)) ) ).

% subset_insert_iff
tff(fact_559_Diff__single__insert,axiom,
    ! [A: $tType,Aa2: set(A),Xa: A,B2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A))))),B2)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),B2)) ) ).

% Diff_single_insert
tff(fact_560_set__update__subset__insert,axiom,
    ! [A: $tType,Xsa: list(A),Ia: nat,Xa: A] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xsa),Ia),Xa))),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),aa(list(A),set(A),set2(A),Xsa))) ).

% set_update_subset_insert
tff(fact_561_boolean__algebra_Oconj__zero__right,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A] : ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),bot_bot(A)) = bot_bot(A) ) ) ).

% boolean_algebra.conj_zero_right
tff(fact_562_boolean__algebra_Oconj__zero__left,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A] : ( aa(A,A,aa(A,fun(A,A),inf_inf(A),bot_bot(A)),Xa) = bot_bot(A) ) ) ).

% boolean_algebra.conj_zero_left
tff(fact_563_diff__shunt__var,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),minus_minus(A),Xa),Y) = bot_bot(A) )
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y) ) ) ).

% diff_shunt_var
tff(fact_564_length__product,axiom,
    ! [A: $tType,B: $tType,Xsa: list(A),Ys: list(B)] : ( aa(list(product_prod(A,B)),nat,size_size(list(product_prod(A,B))),aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),product(A,B),Xsa),Ys)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(list(A),nat,size_size(list(A)),Xsa)),aa(list(B),nat,size_size(list(B)),Ys)) ) ).

% length_product
tff(fact_565_less__imp__diff__less,axiom,
    ! [J: nat,K3: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),K3)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),N)),K3) ) ).

% less_imp_diff_less
tff(fact_566_diff__less__mono2,axiom,
    ! [M2: nat,N: nat,L: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),N)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),L)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),L),N)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),L),M2)) ) ) ).

% diff_less_mono2
tff(fact_567_insert__subsetI,axiom,
    ! [A: $tType,Xa: A,Aa2: set(A),X3: set(A)] :
      ( member(A,Xa,Aa2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X3),Aa2)
       => aa(set(A),$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)),insert3(A),Xa),X3)),Aa2) ) ) ).

% insert_subsetI
tff(fact_568_distinct__foldl__invar,axiom,
    ! [B: $tType,A: $tType,S: list(A),Ia2: fun(set(A),fun(B,$o)),Sigma_0: B,F2: fun(B,fun(A,B))] :
      ( aa(list(A),$o,distinct(A),S)
     => ( aa(B,$o,aa(set(A),fun(B,$o),Ia2,aa(list(A),set(A),set2(A),S)),Sigma_0)
       => ( ! [X: A,It: set(A),Sigma: B] :
              ( member(A,X,It)
             => ( aa(set(A),$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,$o),Ia2,It),Sigma)
                 => aa(B,$o,aa(set(A),fun(B,$o),Ia2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),It),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),X),bot_bot(set(A))))),aa(A,B,aa(B,fun(A,B),F2,Sigma),X)) ) ) )
         => aa(B,$o,aa(set(A),fun(B,$o),Ia2,bot_bot(set(A))),aa(list(A),B,aa(B,fun(list(A),B),aa(fun(B,fun(A,B)),fun(B,fun(list(A),B)),foldl(B,A),F2),Sigma_0),S)) ) ) ) ).

% distinct_foldl_invar
tff(fact_569_subset__emptyI,axiom,
    ! [A: $tType,Aa2: set(A)] :
      ( ! [X: A] : ~ member(A,X,Aa2)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),bot_bot(set(A))) ) ).

% subset_emptyI
tff(fact_570_complete__interval,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [A3: A,B3: A,Pa: fun(A,$o)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
         => ( aa(A,$o,Pa,A3)
           => ( ~ aa(A,$o,Pa,B3)
             => ? [C5: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),C5)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C5),B3)
                  & ! [X4: A] :
                      ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),X4)
                        & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),C5) )
                     => aa(A,$o,Pa,X4) )
                  & ! [D5: A] :
                      ( ! [X: A] :
                          ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),X)
                            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),D5) )
                         => aa(A,$o,Pa,X) )
                     => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),D5),C5) ) ) ) ) ) ) ).

% complete_interval
tff(fact_571_verit__comp__simplify1_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B6: A,A6: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B6),A6)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A6),B6) ) ) ).

% verit_comp_simplify1(3)
tff(fact_572_Int__subset__iff,axiom,
    ! [A: $tType,C3: set(A),Aa2: set(A),B2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),B2))
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C3),Aa2)
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C3),B2) ) ) ).

% Int_subset_iff
tff(fact_573_Int__insert__left__if0,axiom,
    ! [A: $tType,A3: A,C3: set(A),B2: set(A)] :
      ( ~ 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)),insert3(A),A3),B2)),C3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B2),C3) ) ) ).

% Int_insert_left_if0
tff(fact_574_Int__insert__left__if1,axiom,
    ! [A: $tType,A3: A,C3: set(A),B2: set(A)] :
      ( 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)),insert3(A),A3),B2)),C3) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B2),C3)) ) ) ).

% Int_insert_left_if1
tff(fact_575_insert__inter__insert,axiom,
    ! [A: $tType,A3: A,Aa2: set(A),B2: 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)),insert3(A),A3),Aa2)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),B2)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),B2)) ) ).

% insert_inter_insert
tff(fact_576_Int__insert__right__if0,axiom,
    ! [A: $tType,A3: A,Aa2: set(A),B2: set(A)] :
      ( ~ member(A,A3,Aa2)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),B2)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),B2) ) ) ).

% Int_insert_right_if0
tff(fact_577_Int__insert__right__if1,axiom,
    ! [A: $tType,A3: A,Aa2: set(A),B2: set(A)] :
      ( member(A,A3,Aa2)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),B2)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),B2)) ) ) ).

% Int_insert_right_if1
tff(fact_578_insert__disjoint_I1_J,axiom,
    ! [A: $tType,A3: A,Aa2: set(A),B2: 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)),insert3(A),A3),Aa2)),B2) = bot_bot(set(A)) )
    <=> ( ~ member(A,A3,B2)
        & ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),B2) = bot_bot(set(A)) ) ) ) ).

% insert_disjoint(1)
tff(fact_579_insert__disjoint_I2_J,axiom,
    ! [A: $tType,A3: A,Aa2: set(A),B2: 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)),insert3(A),A3),Aa2)),B2) )
    <=> ( ~ member(A,A3,B2)
        & ( bot_bot(set(A)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),B2) ) ) ) ).

% insert_disjoint(2)
tff(fact_580_disjoint__insert_I1_J,axiom,
    ! [A: $tType,B2: set(A),A3: A,Aa2: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),Aa2)) = bot_bot(set(A)) )
    <=> ( ~ member(A,A3,B2)
        & ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B2),Aa2) = bot_bot(set(A)) ) ) ) ).

% disjoint_insert(1)
tff(fact_581_disjoint__insert_I2_J,axiom,
    ! [A: $tType,Aa2: set(A),B3: A,B2: set(A)] :
      ( ( bot_bot(set(A)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),B3),B2)) )
    <=> ( ~ member(A,B3,Aa2)
        & ( bot_bot(set(A)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),B2) ) ) ) ).

% disjoint_insert(2)
tff(fact_582_Diff__disjoint,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A)] : ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B2),Aa2)) = bot_bot(set(A)) ) ).

% Diff_disjoint
tff(fact_583_list__update__beyond,axiom,
    ! [A: $tType,Xsa: list(A),Ia: nat,Xa: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xsa)),Ia)
     => ( aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xsa),Ia),Xa) = Xsa ) ) ).

% list_update_beyond
tff(fact_584_less__fun__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F2: fun(A,B),G: fun(A,B)] :
          ( aa(fun(A,B),$o,aa(fun(A,B),fun(fun(A,B),$o),ord_less(fun(A,B)),F2),G)
        <=> ( aa(fun(A,B),$o,aa(fun(A,B),fun(fun(A,B),$o),ord_less_eq(fun(A,B)),F2),G)
            & ~ aa(fun(A,B),$o,aa(fun(A,B),fun(fun(A,B),$o),ord_less_eq(fun(A,B)),G),F2) ) ) ) ).

% less_fun_def
tff(fact_585_foldl__A1__eq,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),N: A,Ia: A,Ww: list(A)] :
      ( ! [E2: A] : ( aa(A,A,aa(A,fun(A,A),F2,N),E2) = E2 )
     => ( ! [E2: A] : ( aa(A,A,aa(A,fun(A,A),F2,E2),N) = E2 )
       => ( ! [A4: A,B4: A,C5: A] : ( aa(A,A,aa(A,fun(A,A),F2,A4),aa(A,A,aa(A,fun(A,A),F2,B4),C5)) = aa(A,A,aa(A,fun(A,A),F2,aa(A,A,aa(A,fun(A,A),F2,A4),B4)),C5) )
         => ( aa(list(A),A,aa(A,fun(list(A),A),aa(fun(A,fun(A,A)),fun(A,fun(list(A),A)),foldl(A,A),F2),Ia),Ww) = aa(A,A,aa(A,fun(A,A),F2,Ia),aa(list(A),A,aa(A,fun(list(A),A),aa(fun(A,fun(A,A)),fun(A,fun(list(A),A)),foldl(A,A),F2),N),Ww)) ) ) ) ) ).

% foldl_A1_eq
tff(fact_586_nat__less__le,axiom,
    ! [M2: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),N)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),N)
        & ( M2 != N ) ) ) ).

% nat_less_le
tff(fact_587_less__imp__le__nat,axiom,
    ! [M2: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),N)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),N) ) ).

% less_imp_le_nat
tff(fact_588_le__eq__less__or__eq,axiom,
    ! [M2: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),N)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),N)
        | ( M2 = N ) ) ) ).

% le_eq_less_or_eq
tff(fact_589_less__or__eq__imp__le,axiom,
    ! [M2: nat,N: nat] :
      ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),N)
        | ( M2 = N ) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),N) ) ).

% less_or_eq_imp_le
tff(fact_590_le__neq__implies__less,axiom,
    ! [M2: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),N)
     => ( ( M2 != N )
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),N) ) ) ).

% le_neq_implies_less
tff(fact_591_less__mono__imp__le__mono,axiom,
    ! [F2: fun(nat,nat),Ia: nat,J: nat] :
      ( ! [I3: nat,J3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),J3)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,F2,I3)),aa(nat,nat,F2,J3)) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ia),J)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,F2,Ia)),aa(nat,nat,F2,J)) ) ) ).

% less_mono_imp_le_mono
tff(fact_592_disjoint__iff__not__equal,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),B2) = bot_bot(set(A)) )
    <=> ! [X6: A] :
          ( member(A,X6,Aa2)
         => ! [Xa2: A] :
              ( member(A,Xa2,B2)
             => ( X6 != Xa2 ) ) ) ) ).

% disjoint_iff_not_equal
tff(fact_593_Int__empty__right,axiom,
    ! [A: $tType,Aa2: set(A)] : ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),bot_bot(set(A))) = bot_bot(set(A)) ) ).

% Int_empty_right
tff(fact_594_Int__empty__left,axiom,
    ! [A: $tType,B2: set(A)] : ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),bot_bot(set(A))),B2) = bot_bot(set(A)) ) ).

% Int_empty_left
tff(fact_595_disjoint__iff,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),B2) = bot_bot(set(A)) )
    <=> ! [X6: A] :
          ( member(A,X6,Aa2)
         => ~ member(A,X6,B2) ) ) ).

% disjoint_iff
tff(fact_596_Int__emptyI,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A)] :
      ( ! [X: A] :
          ( member(A,X,Aa2)
         => ~ member(A,X,B2) )
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),B2) = bot_bot(set(A)) ) ) ).

% Int_emptyI
tff(fact_597_disjointI,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( ! [X: A] :
          ( member(A,X,A3)
         => ~ member(A,X,B3) )
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3) = bot_bot(set(A)) ) ) ).

% disjointI
tff(fact_598_inter__eq__subsetI,axiom,
    ! [A: $tType,S: set(A),S3: set(A),Aa2: set(A),B2: set(A)] :
      ( aa(set(A),$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)),Aa2),S3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B2),S3) )
       => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),S) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B2),S) ) ) ) ).

% inter_eq_subsetI
tff(fact_599_Int__Collect__mono,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A),Pa: fun(A,$o),Qa: fun(A,$o)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),B2)
     => ( ! [X: A] :
            ( member(A,X,Aa2)
           => ( aa(A,$o,Pa,X)
             => aa(A,$o,Qa,X) ) )
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),aa(fun(A,$o),set(A),collect(A),Pa))),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B2),aa(fun(A,$o),set(A),collect(A),Qa))) ) ) ).

% Int_Collect_mono
tff(fact_600_Int__greatest,axiom,
    ! [A: $tType,C3: set(A),Aa2: set(A),B2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C3),Aa2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C3),B2)
       => aa(set(A),$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)),Aa2),B2)) ) ) ).

% Int_greatest
tff(fact_601_Int__absorb2,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),B2)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),B2) = Aa2 ) ) ).

% Int_absorb2
tff(fact_602_Int__absorb1,axiom,
    ! [A: $tType,B2: set(A),Aa2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),Aa2)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),B2) = B2 ) ) ).

% Int_absorb1
tff(fact_603_Int__lower2,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A)] : aa(set(A),$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)),Aa2),B2)),B2) ).

% Int_lower2
tff(fact_604_Int__lower1,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A)] : aa(set(A),$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)),Aa2),B2)),Aa2) ).

% Int_lower1
tff(fact_605_Int__mono,axiom,
    ! [A: $tType,Aa2: set(A),C3: set(A),B2: set(A),D2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),C3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),D2)
       => aa(set(A),$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)),Aa2),B2)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),C3),D2)) ) ) ).

% Int_mono
tff(fact_606_Int__insert__left,axiom,
    ! [A: $tType,A3: A,B2: 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)),insert3(A),A3),B2)),C3) = $ite(member(A,A3,C3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B2),C3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B2),C3)) ) ).

% Int_insert_left
tff(fact_607_Int__insert__right,axiom,
    ! [A: $tType,Aa2: set(A),A3: A,B2: set(A)] :
      ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),B2)) = $ite(member(A,A3,Aa2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),B2)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),B2)) ) ).

% Int_insert_right
tff(fact_608_Int__Diff,axiom,
    ! [A: $tType,Aa2: set(A),B2: 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)),Aa2),B2)),C3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B2),C3)) ) ).

% Int_Diff
tff(fact_609_Diff__Int2,axiom,
    ! [A: $tType,Aa2: set(A),C3: set(A),B2: 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)),Aa2),C3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B2),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)),Aa2),C3)),B2) ) ).

% Diff_Int2
tff(fact_610_Diff__Diff__Int,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A)] : ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),B2)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),B2) ) ).

% Diff_Diff_Int
tff(fact_611_Diff__Int__distrib,axiom,
    ! [A: $tType,C3: set(A),Aa2: set(A),B2: 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)),Aa2),B2)) = 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),Aa2)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),C3),B2)) ) ).

% Diff_Int_distrib
tff(fact_612_Diff__Int__distrib2,axiom,
    ! [A: $tType,Aa2: set(A),B2: 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)),Aa2),B2)),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)),Aa2),C3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B2),C3)) ) ).

% Diff_Int_distrib2
tff(fact_613_exists__leI,axiom,
    ! [N: nat,Pa: fun(nat,$o)] :
      ( ( ! [N3: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N3),N)
           => ~ aa(nat,$o,Pa,N3) )
       => aa(nat,$o,Pa,N) )
     => ? [N4: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N4),N)
          & aa(nat,$o,Pa,N4) ) ) ).

% exists_leI
tff(fact_614_foldl__cong,axiom,
    ! [A: $tType,B: $tType,A3: A,B3: A,L: list(B),K3: list(B),F2: fun(A,fun(B,A)),G: fun(A,fun(B,A))] :
      ( ( A3 = B3 )
     => ( ( L = K3 )
       => ( ! [A4: A,X: B] :
              ( member(B,X,aa(list(B),set(B),set2(B),L))
             => ( aa(B,A,aa(A,fun(B,A),F2,A4),X) = aa(B,A,aa(A,fun(B,A),G,A4),X) ) )
         => ( aa(list(B),A,aa(A,fun(list(B),A),aa(fun(A,fun(B,A)),fun(A,fun(list(B),A)),foldl(A,B),F2),A3),L) = aa(list(B),A,aa(A,fun(list(B),A),aa(fun(A,fun(B,A)),fun(A,fun(list(B),A)),foldl(A,B),G),B3),K3) ) ) ) ) ).

% foldl_cong
tff(fact_615_less__eq__assn__def,axiom,
    ! [A3: assn,B3: assn] :
      ( aa(assn,$o,aa(assn,fun(assn,$o),ord_less_eq(assn),A3),B3)
    <=> ( A3 = aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),A3),B3) ) ) ).

% less_eq_assn_def
tff(fact_616_less__diff__iff,axiom,
    ! [K3: nat,M2: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K3),M2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K3),N)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),K3)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K3))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),N) ) ) ) ).

% less_diff_iff
tff(fact_617_diff__less__mono,axiom,
    ! [A3: nat,B3: nat,C2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A3),B3)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),C2),A3)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),A3),C2)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),B3),C2)) ) ) ).

% diff_less_mono
tff(fact_618_disjoint__mono,axiom,
    ! [A: $tType,A3: set(A),A6: set(A),B3: set(A),B6: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),A6)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),B6)
       => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),B6) = bot_bot(set(A)) )
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3) = bot_bot(set(A)) ) ) ) ) ).

% disjoint_mono
tff(fact_619_disjoint__alt__simp2,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),B2) != Aa2 )
    <=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),B2) != bot_bot(set(A)) ) ) ).

% disjoint_alt_simp2
tff(fact_620_disjoint__alt__simp1,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),B2) = Aa2 )
    <=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),B2) = bot_bot(set(A)) ) ) ).

% disjoint_alt_simp1
tff(fact_621_Int__Diff__disjoint,axiom,
    ! [A: $tType,Aa2: set(A),B2: 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)),Aa2),B2)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),B2)) = bot_bot(set(A)) ) ).

% Int_Diff_disjoint
tff(fact_622_Diff__triv,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),B2) = bot_bot(set(A)) )
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),B2) = Aa2 ) ) ).

% Diff_triv
tff(fact_623_foldl__absorb1,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Xa: A,Zs: list(A)] : ( aa(A,A,aa(A,fun(A,A),times_times(A),Xa),aa(list(A),A,aa(A,fun(list(A),A),aa(fun(A,fun(A,A)),fun(A,fun(list(A),A)),foldl(A,A),times_times(A)),one_one(A)),Zs)) = aa(list(A),A,aa(A,fun(list(A),A),aa(fun(A,fun(A,A)),fun(A,fun(list(A),A)),foldl(A,A),times_times(A)),Xa),Zs) ) ) ).

% foldl_absorb1
tff(fact_624_length__removeAll__less__eq,axiom,
    ! [A: $tType,Xa: A,Xsa: list(A)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),removeAll(A),Xa),Xsa))),aa(list(A),nat,size_size(list(A)),Xsa)) ).

% length_removeAll_less_eq
tff(fact_625_disjoint__alt__simp3,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A)] :
      ( aa(set(A),$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)),Aa2),B2)),Aa2)
    <=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),B2) != bot_bot(set(A)) ) ) ).

% disjoint_alt_simp3
tff(fact_626_verit__comp__simplify1_I1_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),A3) ) ).

% verit_comp_simplify1(1)
tff(fact_627_ex__gt__or__lt,axiom,
    ! [A: $tType] :
      ( condit5016429287641298734tinuum(A)
     => ! [A3: A] :
        ? [B4: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B4)
          | aa(A,$o,aa(A,fun(A,$o),ord_less(A),B4),A3) ) ) ).

% ex_gt_or_lt
tff(fact_628_measure__induct,axiom,
    ! [B: $tType,A: $tType] :
      ( wellorder(B)
     => ! [F2: fun(A,B),Pa: fun(A,$o),A3: A] :
          ( ! [X: A] :
              ( ! [Y3: A] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,Y3)),aa(A,B,F2,X))
                 => aa(A,$o,Pa,Y3) )
             => aa(A,$o,Pa,X) )
         => aa(A,$o,Pa,A3) ) ) ).

% measure_induct
tff(fact_629_measure__induct__rule,axiom,
    ! [B: $tType,A: $tType] :
      ( wellorder(B)
     => ! [F2: fun(A,B),Pa: fun(A,$o),A3: A] :
          ( ! [X: A] :
              ( ! [Y3: A] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,Y3)),aa(A,B,F2,X))
                 => aa(A,$o,Pa,Y3) )
             => aa(A,$o,Pa,X) )
         => aa(A,$o,Pa,A3) ) ) ).

% measure_induct_rule
tff(fact_630_ssubst__Pair__rhs,axiom,
    ! [B: $tType,A: $tType,R2: A,S2: B,R: set(product_prod(A,B)),S4: B] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),R2),S2),R)
     => ( ( S4 = S2 )
       => 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_631_distinct__length__le,axiom,
    ! [A: $tType,Ys: list(A),Xsa: list(A)] :
      ( aa(list(A),$o,distinct(A),Ys)
     => ( ( aa(list(A),set(A),set2(A),Ys) = aa(list(A),set(A),set2(A),Xsa) )
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Ys)),aa(list(A),nat,size_size(list(A)),Xsa)) ) ) ).

% distinct_length_le
tff(fact_632_nat__neq__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( M2 != N )
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),N)
        | aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),M2) ) ) ).

% nat_neq_iff
tff(fact_633_less__not__refl,axiom,
    ! [N: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),N) ).

% less_not_refl
tff(fact_634_less__not__refl2,axiom,
    ! [N: nat,M2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),M2)
     => ( M2 != N ) ) ).

% less_not_refl2
tff(fact_635_less__not__refl3,axiom,
    ! [S2: nat,T2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),S2),T2)
     => ( S2 != T2 ) ) ).

% less_not_refl3
tff(fact_636_less__irrefl__nat,axiom,
    ! [N: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),N) ).

% less_irrefl_nat
tff(fact_637_nat__less__induct,axiom,
    ! [Pa: fun(nat,$o),N: nat] :
      ( ! [N5: nat] :
          ( ! [M3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M3),N5)
             => aa(nat,$o,Pa,M3) )
         => aa(nat,$o,Pa,N5) )
     => aa(nat,$o,Pa,N) ) ).

% nat_less_induct
tff(fact_638_infinite__descent,axiom,
    ! [Pa: fun(nat,$o),N: nat] :
      ( ! [N5: nat] :
          ( ~ aa(nat,$o,Pa,N5)
         => ? [M3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M3),N5)
              & ~ aa(nat,$o,Pa,M3) ) )
     => aa(nat,$o,Pa,N) ) ).

% infinite_descent
tff(fact_639_linorder__neqE__nat,axiom,
    ! [Xa: nat,Y: nat] :
      ( ( Xa != Y )
     => ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Y)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y),Xa) ) ) ).

% linorder_neqE_nat
tff(fact_640_infinite__descent__measure,axiom,
    ! [A: $tType,Pa: fun(A,$o),V: fun(A,nat),Xa: A] :
      ( ! [X: A] :
          ( ~ aa(A,$o,Pa,X)
         => ? [Y3: A] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,V,Y3)),aa(A,nat,V,X))
              & ~ aa(A,$o,Pa,Y3) ) )
     => aa(A,$o,Pa,Xa) ) ).

% infinite_descent_measure
tff(fact_641_size__neq__size__imp__neq,axiom,
    ! [A: $tType] :
      ( size(A)
     => ! [Xa: A,Y: A] :
          ( ( aa(A,nat,size_size(A),Xa) != aa(A,nat,size_size(A),Y) )
         => ( Xa != Y ) ) ) ).

% size_neq_size_imp_neq
tff(fact_642_distinct__product,axiom,
    ! [A: $tType,B: $tType,Xsa: list(A),Ys: list(B)] :
      ( aa(list(A),$o,distinct(A),Xsa)
     => ( aa(list(B),$o,distinct(B),Ys)
       => aa(list(product_prod(A,B)),$o,distinct(product_prod(A,B)),aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),product(A,B),Xsa),Ys)) ) ) ).

% distinct_product
tff(fact_643_distinct__sorted__mono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: list(A),Ia: nat,J: nat] :
          ( sorted_wrt(A,ord_less_eq(A),L)
         => ( aa(list(A),$o,distinct(A),L)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),J)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),L))
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,nth(A,L),Ia)),aa(nat,A,nth(A,L),J)) ) ) ) ) ) ).

% distinct_sorted_mono
tff(fact_644_distinct__sorted__strict__mono__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: list(A),Ia: nat,J: nat] :
          ( aa(list(A),$o,distinct(A),L)
         => ( sorted_wrt(A,ord_less_eq(A),L)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),aa(list(A),nat,size_size(list(A)),L))
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),L))
               => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,nth(A,L),Ia)),aa(nat,A,nth(A,L),J))
                <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),J) ) ) ) ) ) ) ).

% distinct_sorted_strict_mono_iff
tff(fact_645_slice__len,axiom,
    ! [A: $tType,From: nat,To: nat,Xsa: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),From),To)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),To),aa(list(A),nat,size_size(list(A)),Xsa))
       => ( aa(list(A),nat,size_size(list(A)),slice(A,From,To,Xsa)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),To),From) ) ) ) ).

% slice_len
tff(fact_646_rev__update,axiom,
    ! [A: $tType,K3: nat,Xsa: list(A),Y: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K3),aa(list(A),nat,size_size(list(A)),Xsa))
     => ( aa(list(A),list(A),rev(A),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xsa),K3),Y)) = aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),aa(list(A),list(A),rev(A),Xsa)),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)),Xsa)),K3)),one_one(nat))),Y) ) ) ).

% rev_update
tff(fact_647_butlast__list__update,axiom,
    ! [A: $tType,Xsa: list(A),K3: nat,Xa: A] :
      ( aa(list(A),list(A),butlast(A),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xsa),K3),Xa)) = $ite(K3 = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xsa)),one_one(nat)),aa(list(A),list(A),butlast(A),Xsa),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),aa(list(A),list(A),butlast(A),Xsa)),K3),Xa)) ) ).

% butlast_list_update
tff(fact_648_distinct__sorted__mono__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: list(A),Ia: nat,J: nat] :
          ( aa(list(A),$o,distinct(A),L)
         => ( sorted_wrt(A,ord_less_eq(A),L)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),aa(list(A),nat,size_size(list(A)),L))
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),L))
               => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,nth(A,L),Ia)),aa(nat,A,nth(A,L),J))
                <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ia),J) ) ) ) ) ) ) ).

% distinct_sorted_mono_iff
tff(fact_649_length__zip,axiom,
    ! [A: $tType,B: $tType,Xsa: list(A),Ys: list(B)] : ( aa(list(product_prod(A,B)),nat,size_size(list(product_prod(A,B))),aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),Xsa),Ys)) = aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(list(A),nat,size_size(list(A)),Xsa)),aa(list(B),nat,size_size(list(B)),Ys)) ) ).

% length_zip
tff(fact_650_star__assnI,axiom,
    ! [Pa: assn,H2: heap_ext(product_unit),As2: set(nat),Qa: 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,Pa),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))
     => ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,Qa),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)),As2),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),Pa),Qa)),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)),As2),As3))) ) ) ) ).

% star_assnI
tff(fact_651_mod__star__conv,axiom,
    ! [Aa2: 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),Aa2),B2)),H2)
    <=> ? [Hr: heap_ext(product_unit),As1: set(nat),As22: 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),As22)) )
          & ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As1),As22) = 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,Aa2),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,B2),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),As22)) ) ) ).

% mod_star_conv
tff(fact_652_rotate1__length01,axiom,
    ! [A: $tType,Xsa: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xsa)),one_one(nat))
     => ( aa(list(A),list(A),rotate1(A),Xsa) = Xsa ) ) ).

% rotate1_length01
tff(fact_653_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_654_sup__idem,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xa: A] : ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Xa) = Xa ) ) ).

% sup_idem
tff(fact_655_sup_Oleft__idem,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: A,B3: 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),B3)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3) ) ) ).

% sup.left_idem
tff(fact_656_sup__left__idem,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xa: A,Y: A] : ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Y)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Y) ) ) ).

% sup_left_idem
tff(fact_657_sup_Oright__idem,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: A,B3: A] : ( aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3)),B3) = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3) ) ) ).

% sup.right_idem
tff(fact_658_sup__apply,axiom,
    ! [A: $tType,B: $tType] :
      ( semilattice_sup(A)
     => ! [F2: fun(B,A),G: fun(B,A),Xa: 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)),F2),G),Xa) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(B,A,F2,Xa)),aa(B,A,G,Xa)) ) ) ).

% sup_apply
tff(fact_659_IntI,axiom,
    ! [A: $tType,C2: A,Aa2: set(A),B2: set(A)] :
      ( member(A,C2,Aa2)
     => ( member(A,C2,B2)
       => member(A,C2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),B2)) ) ) ).

% IntI
tff(fact_660_Int__iff,axiom,
    ! [A: $tType,C2: A,Aa2: set(A),B2: set(A)] :
      ( member(A,C2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),B2))
    <=> ( member(A,C2,Aa2)
        & member(A,C2,B2) ) ) ).

% Int_iff
tff(fact_661_UnCI,axiom,
    ! [A: $tType,C2: A,B2: set(A),Aa2: set(A)] :
      ( ( ~ member(A,C2,B2)
       => member(A,C2,Aa2) )
     => member(A,C2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),B2)) ) ).

% UnCI
tff(fact_662_Un__iff,axiom,
    ! [A: $tType,C2: A,Aa2: set(A),B2: set(A)] :
      ( member(A,C2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),B2))
    <=> ( member(A,C2,Aa2)
        | member(A,C2,B2) ) ) ).

% Un_iff
tff(fact_663_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_664_min_Oleft__idem,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: 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),B3)) = aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B3) ) ) ).

% min.left_idem
tff(fact_665_min_Oright__idem,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A] : ( aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B3)),B3) = aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B3) ) ) ).

% min.right_idem
tff(fact_666_rev__rev__ident,axiom,
    ! [A: $tType,Xsa: list(A)] : ( aa(list(A),list(A),rev(A),aa(list(A),list(A),rev(A),Xsa)) = Xsa ) ).

% rev_rev_ident
tff(fact_667_rev__is__rev__conv,axiom,
    ! [A: $tType,Xsa: list(A),Ys: list(A)] :
      ( ( aa(list(A),list(A),rev(A),Xsa) = aa(list(A),list(A),rev(A),Ys) )
    <=> ( Xsa = Ys ) ) ).

% rev_is_rev_conv
tff(fact_668_le__sup__iff,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xa: A,Y: A,Z2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Y)),Z2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Z2)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z2) ) ) ) ).

% le_sup_iff
tff(fact_669_sup_Obounded__iff,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B3: A,C2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),B3),C2)),A3)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A3) ) ) ) ).

% sup.bounded_iff
tff(fact_670_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_671_sup__bot_Oneutr__eq__iff,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [A3: A,B3: A] :
          ( ( bot_bot(A) = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3) )
        <=> ( ( A3 = bot_bot(A) )
            & ( B3 = bot_bot(A) ) ) ) ) ).

% sup_bot.neutr_eq_iff
tff(fact_672_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_673_sup__bot_Oeq__neutr__iff,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [A3: A,B3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3) = bot_bot(A) )
        <=> ( ( A3 = bot_bot(A) )
            & ( B3 = bot_bot(A) ) ) ) ) ).

% sup_bot.eq_neutr_iff
tff(fact_674_sup__eq__bot__iff,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [Xa: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Y) = bot_bot(A) )
        <=> ( ( Xa = bot_bot(A) )
            & ( Y = bot_bot(A) ) ) ) ) ).

% sup_eq_bot_iff
tff(fact_675_bot__eq__sup__iff,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [Xa: A,Y: A] :
          ( ( bot_bot(A) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Y) )
        <=> ( ( Xa = bot_bot(A) )
            & ( Y = bot_bot(A) ) ) ) ) ).

% bot_eq_sup_iff
tff(fact_676_sup__bot__right,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [Xa: A] : ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),bot_bot(A)) = Xa ) ) ).

% sup_bot_right
tff(fact_677_sup__bot__left,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [Xa: A] : ( aa(A,A,aa(A,fun(A,A),sup_sup(A),bot_bot(A)),Xa) = Xa ) ) ).

% sup_bot_left
tff(fact_678_inf__sup__absorb,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xa: A,Y: A] : ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Y)) = Xa ) ) ).

% inf_sup_absorb
tff(fact_679_sup__inf__absorb,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xa: A,Y: A] : ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Y)) = Xa ) ) ).

% sup_inf_absorb
tff(fact_680_Un__empty,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),B2) = bot_bot(set(A)) )
    <=> ( ( Aa2 = bot_bot(set(A)) )
        & ( B2 = bot_bot(set(A)) ) ) ) ).

% Un_empty
tff(fact_681_min__arg__le_I2_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [M2: A,N: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),M2),aa(A,A,aa(A,fun(A,A),ord_min(A),M2),N))
        <=> ( aa(A,A,aa(A,fun(A,A),ord_min(A),M2),N) = M2 ) ) ) ).

% min_arg_le(2)
tff(fact_682_min__arg__le_I1_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [N: A,M2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),N),aa(A,A,aa(A,fun(A,A),ord_min(A),M2),N))
        <=> ( aa(A,A,aa(A,fun(A,A),ord_min(A),M2),N) = N ) ) ) ).

% min_arg_le(1)
tff(fact_683_min__eq__arg_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [M2: A,N: A] :
          ( ( aa(A,A,aa(A,fun(A,A),ord_min(A),M2),N) = N )
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),N),M2) ) ) ).

% min_eq_arg(2)
tff(fact_684_min__eq__arg_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [M2: A,N: A] :
          ( ( aa(A,A,aa(A,fun(A,A),ord_min(A),M2),N) = M2 )
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),M2),N) ) ) ).

% min_eq_arg(1)
tff(fact_685_min_Oabsorb1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
         => ( aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B3) = A3 ) ) ) ).

% min.absorb1
tff(fact_686_min_Oabsorb2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3)
         => ( aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B3) = B3 ) ) ) ).

% min.absorb2
tff(fact_687_min_Obounded__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),ord_min(A),B3),C2))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),C2) ) ) ) ).

% min.bounded_iff
tff(fact_688_Un__subset__iff,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A),C3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),B2)),C3)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),C3)
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),C3) ) ) ).

% Un_subset_iff
tff(fact_689_Un__insert__right,axiom,
    ! [A: $tType,Aa2: set(A),A3: A,B2: set(A)] : ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),B2)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),B2)) ) ).

% Un_insert_right
tff(fact_690_Un__insert__left,axiom,
    ! [A: $tType,A3: A,B2: 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)),insert3(A),A3),B2)),C3) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B2),C3)) ) ).

% Un_insert_left
tff(fact_691_min__less__iff__conj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Z2: A,Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z2),aa(A,A,aa(A,fun(A,A),ord_min(A),Xa),Y))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z2),Xa)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z2),Y) ) ) ) ).

% min_less_iff_conj
tff(fact_692_min_Oabsorb4,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A3)
         => ( aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B3) = B3 ) ) ) ).

% min.absorb4
tff(fact_693_min_Oabsorb3,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
         => ( aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B3) = A3 ) ) ) ).

% min.absorb3
tff(fact_694_min__arg__not__ge_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [M2: A,N: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_min(A),M2),N)),M2)
        <=> ( aa(A,A,aa(A,fun(A,A),ord_min(A),M2),N) = M2 ) ) ) ).

% min_arg_not_ge(1)
tff(fact_695_min__arg__not__ge_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [M2: A,N: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_min(A),M2),N)),N)
        <=> ( aa(A,A,aa(A,fun(A,A),ord_min(A),M2),N) = N ) ) ) ).

% min_arg_not_ge(2)
tff(fact_696_min__less__self__conv_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B3)),A3)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A3) ) ) ).

% min_less_self_conv(1)
tff(fact_697_min__less__self__conv_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B3)),B3)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3) ) ) ).

% min_less_self_conv(2)
tff(fact_698_min__simps_I1_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
         => ( aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B3) = A3 ) ) ) ).

% min_simps(1)
tff(fact_699_min__simps_I2_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A3)
         => ( aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B3) = B3 ) ) ) ).

% min_simps(2)
tff(fact_700_min__bot2,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [Xa: A] : ( aa(A,A,aa(A,fun(A,A),ord_min(A),Xa),bot_bot(A)) = bot_bot(A) ) ) ).

% min_bot2
tff(fact_701_min__bot,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [Xa: A] : ( aa(A,A,aa(A,fun(A,A),ord_min(A),bot_bot(A)),Xa) = bot_bot(A) ) ) ).

% min_bot
tff(fact_702_Int__Un__eq_I4_J,axiom,
    ! [A: $tType,T4: set(A),S: set(A)] : ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),T4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S),T4)) = T4 ) ).

% Int_Un_eq(4)
tff(fact_703_Int__Un__eq_I3_J,axiom,
    ! [A: $tType,S: set(A),T4: 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),T4)) = S ) ).

% Int_Un_eq(3)
tff(fact_704_Int__Un__eq_I2_J,axiom,
    ! [A: $tType,S: set(A),T4: 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),T4)),T4) = T4 ) ).

% Int_Un_eq(2)
tff(fact_705_Int__Un__eq_I1_J,axiom,
    ! [A: $tType,S: set(A),T4: 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),T4)),S) = S ) ).

% Int_Un_eq(1)
tff(fact_706_Un__Int__eq_I4_J,axiom,
    ! [A: $tType,T4: set(A),S: set(A)] : ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),T4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),S),T4)) = T4 ) ).

% Un_Int_eq(4)
tff(fact_707_Un__Int__eq_I3_J,axiom,
    ! [A: $tType,S: set(A),T4: 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),T4)) = S ) ).

% Un_Int_eq(3)
tff(fact_708_Un__Int__eq_I2_J,axiom,
    ! [A: $tType,S: set(A),T4: 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),T4)),T4) = T4 ) ).

% Un_Int_eq(2)
tff(fact_709_Un__Int__eq_I1_J,axiom,
    ! [A: $tType,S: set(A),T4: 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),T4)),S) = S ) ).

% Un_Int_eq(1)
tff(fact_710_Un__Diff__cancel2,axiom,
    ! [A: $tType,B2: set(A),Aa2: 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)),B2),Aa2)),Aa2) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B2),Aa2) ) ).

% Un_Diff_cancel2
tff(fact_711_Un__Diff__cancel,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A)] : ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B2),Aa2)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),B2) ) ).

% Un_Diff_cancel
tff(fact_712_set__rev,axiom,
    ! [A: $tType,Xsa: list(A)] : ( aa(list(A),set(A),set2(A),aa(list(A),list(A),rev(A),Xsa)) = aa(list(A),set(A),set2(A),Xsa) ) ).

% set_rev
tff(fact_713_length__rev,axiom,
    ! [A: $tType,Xsa: list(A)] : ( aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),rev(A),Xsa)) = aa(list(A),nat,size_size(list(A)),Xsa) ) ).

% length_rev
tff(fact_714_distinct__rev,axiom,
    ! [A: $tType,Xsa: list(A)] :
      ( aa(list(A),$o,distinct(A),aa(list(A),list(A),rev(A),Xsa))
    <=> aa(list(A),$o,distinct(A),Xsa) ) ).

% distinct_rev
tff(fact_715_set__rotate1,axiom,
    ! [A: $tType,Xsa: list(A)] : ( aa(list(A),set(A),set2(A),aa(list(A),list(A),rotate1(A),Xsa)) = aa(list(A),set(A),set2(A),Xsa) ) ).

% set_rotate1
tff(fact_716_length__rotate1,axiom,
    ! [A: $tType,Xsa: list(A)] : ( aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),rotate1(A),Xsa)) = aa(list(A),nat,size_size(list(A)),Xsa) ) ).

% length_rotate1
tff(fact_717_distinct1__rotate,axiom,
    ! [A: $tType,Xsa: list(A)] :
      ( aa(list(A),$o,distinct(A),aa(list(A),list(A),rotate1(A),Xsa))
    <=> aa(list(A),$o,distinct(A),Xsa) ) ).

% distinct1_rotate
tff(fact_718_list__ex__rev,axiom,
    ! [A: $tType,Pa: fun(A,$o),Xsa: list(A)] :
      ( aa(list(A),$o,aa(fun(A,$o),fun(list(A),$o),list_ex(A),Pa),aa(list(A),list(A),rev(A),Xsa))
    <=> aa(list(A),$o,aa(fun(A,$o),fun(list(A),$o),list_ex(A),Pa),Xsa) ) ).

% list_ex_rev
tff(fact_719_set__union,axiom,
    ! [A: $tType,Xsa: list(A),Ys: list(A)] : ( aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),union(A),Xsa),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),Xsa)),aa(list(A),set(A),set2(A),Ys)) ) ).

% set_union
tff(fact_720_length__butlast,axiom,
    ! [A: $tType,Xsa: list(A)] : ( aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),butlast(A),Xsa)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xsa)),one_one(nat)) ) ).

% length_butlast
tff(fact_721_IntE,axiom,
    ! [A: $tType,C2: A,Aa2: set(A),B2: set(A)] :
      ( member(A,C2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),B2))
     => ~ ( member(A,C2,Aa2)
         => ~ member(A,C2,B2) ) ) ).

% IntE
tff(fact_722_IntD1,axiom,
    ! [A: $tType,C2: A,Aa2: set(A),B2: set(A)] :
      ( member(A,C2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),B2))
     => member(A,C2,Aa2) ) ).

% IntD1
tff(fact_723_IntD2,axiom,
    ! [A: $tType,C2: A,Aa2: set(A),B2: set(A)] :
      ( member(A,C2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),B2))
     => member(A,C2,B2) ) ).

% IntD2
tff(fact_724_Int__assoc,axiom,
    ! [A: $tType,Aa2: set(A),B2: 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)),Aa2),B2)),C3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B2),C3)) ) ).

% Int_assoc
tff(fact_725_Int__absorb,axiom,
    ! [A: $tType,Aa2: set(A)] : ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),Aa2) = Aa2 ) ).

% Int_absorb
tff(fact_726_Int__commute,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A)] : ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),B2) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B2),Aa2) ) ).

% Int_commute
tff(fact_727_Un__Int__crazy,axiom,
    ! [A: $tType,Aa2: set(A),B2: 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)),Aa2),B2)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B2),C3))),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),C3),Aa2)) = 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)),Aa2),B2)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B2),C3))),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),C3),Aa2)) ) ).

% Un_Int_crazy
tff(fact_728_Int__Un__distrib,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A),C3: set(A)] : ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B2),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)),Aa2),B2)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),C3)) ) ).

% Int_Un_distrib
tff(fact_729_Un__Int__distrib,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A),C3: set(A)] : ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B2),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)),Aa2),B2)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),C3)) ) ).

% Un_Int_distrib
tff(fact_730_Int__Un__distrib2,axiom,
    ! [A: $tType,B2: set(A),C3: set(A),Aa2: 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)),B2),C3)),Aa2) = 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)),B2),Aa2)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),C3),Aa2)) ) ).

% Int_Un_distrib2
tff(fact_731_Int__left__absorb,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A)] : ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),B2)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),B2) ) ).

% Int_left_absorb
tff(fact_732_Un__Int__distrib2,axiom,
    ! [A: $tType,B2: set(A),C3: set(A),Aa2: 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)),B2),C3)),Aa2) = 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)),B2),Aa2)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),C3),Aa2)) ) ).

% Un_Int_distrib2
tff(fact_733_Int__left__commute,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A),C3: set(A)] : ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B2),C3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),C3)) ) ).

% Int_left_commute
tff(fact_734_less__assn__def,axiom,
    ! [A3: assn,B3: assn] :
      ( aa(assn,$o,aa(assn,fun(assn,$o),ord_less(assn),A3),B3)
    <=> ( aa(assn,$o,aa(assn,fun(assn,$o),ord_less_eq(assn),A3),B3)
        & ( A3 != B3 ) ) ) ).

% less_assn_def
tff(fact_735_UnE,axiom,
    ! [A: $tType,C2: A,Aa2: set(A),B2: set(A)] :
      ( member(A,C2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),B2))
     => ( ~ member(A,C2,Aa2)
       => member(A,C2,B2) ) ) ).

% UnE
tff(fact_736_UnI1,axiom,
    ! [A: $tType,C2: A,Aa2: set(A),B2: set(A)] :
      ( member(A,C2,Aa2)
     => member(A,C2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),B2)) ) ).

% UnI1
tff(fact_737_UnI2,axiom,
    ! [A: $tType,C2: A,B2: set(A),Aa2: set(A)] :
      ( member(A,C2,B2)
     => member(A,C2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),B2)) ) ).

% UnI2
tff(fact_738_bex__Un,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A),Pa: fun(A,$o)] :
      ( ? [X6: A] :
          ( member(A,X6,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),B2))
          & aa(A,$o,Pa,X6) )
    <=> ( ? [X6: A] :
            ( member(A,X6,Aa2)
            & aa(A,$o,Pa,X6) )
        | ? [X6: A] :
            ( member(A,X6,B2)
            & aa(A,$o,Pa,X6) ) ) ) ).

% bex_Un
tff(fact_739_ball__Un,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A),Pa: fun(A,$o)] :
      ( ! [X6: A] :
          ( member(A,X6,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),B2))
         => aa(A,$o,Pa,X6) )
    <=> ( ! [X6: A] :
            ( member(A,X6,Aa2)
           => aa(A,$o,Pa,X6) )
        & ! [X6: A] :
            ( member(A,X6,B2)
           => aa(A,$o,Pa,X6) ) ) ) ).

% ball_Un
tff(fact_740_Un__assoc,axiom,
    ! [A: $tType,Aa2: set(A),B2: 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)),Aa2),B2)),C3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B2),C3)) ) ).

% Un_assoc
tff(fact_741_Un__absorb,axiom,
    ! [A: $tType,Aa2: set(A)] : ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),Aa2) = Aa2 ) ).

% Un_absorb
tff(fact_742_Un__commute,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A)] : ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),B2) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B2),Aa2) ) ).

% Un_commute
tff(fact_743_Un__left__absorb,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A)] : ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),B2)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),B2) ) ).

% Un_left_absorb
tff(fact_744_Un__left__commute,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A),C3: set(A)] : ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B2),C3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),C3)) ) ).

% Un_left_commute
tff(fact_745_rev__swap,axiom,
    ! [A: $tType,Xsa: list(A),Ys: list(A)] :
      ( ( aa(list(A),list(A),rev(A),Xsa) = Ys )
    <=> ( Xsa = aa(list(A),list(A),rev(A),Ys) ) ) ).

% rev_swap
tff(fact_746_inf__sup__aci_I8_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xa: A,Y: A] : ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Y)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Y) ) ) ).

% inf_sup_aci(8)
tff(fact_747_inf__sup__aci_I7_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xa: A,Y: A,Z2: A] : ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),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),Xa),Z2)) ) ) ).

% inf_sup_aci(7)
tff(fact_748_inf__sup__aci_I6_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xa: 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),Xa),Y)),Z2) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z2)) ) ) ).

% inf_sup_aci(6)
tff(fact_749_inf__sup__aci_I5_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xa: A,Y: A] : ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Y) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Xa) ) ) ).

% inf_sup_aci(5)
tff(fact_750_min_Oassoc,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: 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),B3)),C2) = aa(A,A,aa(A,fun(A,A),ord_min(A),A3),aa(A,A,aa(A,fun(A,A),ord_min(A),B3),C2)) ) ) ).

% min.assoc
tff(fact_751_min_Ocommute,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A] : ( aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B3) = aa(A,A,aa(A,fun(A,A),ord_min(A),B3),A3) ) ) ).

% min.commute
tff(fact_752_min_Oleft__commute,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B3: A,A3: A,C2: A] : ( aa(A,A,aa(A,fun(A,A),ord_min(A),B3),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),B3),C2)) ) ) ).

% min.left_commute
tff(fact_753_sup_Oassoc,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: A,B3: 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),B3)),C2) = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),aa(A,A,aa(A,fun(A,A),sup_sup(A),B3),C2)) ) ) ).

% sup.assoc
tff(fact_754_sup__assoc,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xa: 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),Xa),Y)),Z2) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z2)) ) ) ).

% sup_assoc
tff(fact_755_sup_Ocommute,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: A,B3: A] : ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3) = aa(A,A,aa(A,fun(A,A),sup_sup(A),B3),A3) ) ) ).

% sup.commute
tff(fact_756_sup__commute,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xa: A,Y: A] : ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Y) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Xa) ) ) ).

% sup_commute
tff(fact_757_sup_Oleft__commute,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B3: A,A3: A,C2: A] : ( aa(A,A,aa(A,fun(A,A),sup_sup(A),B3),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),B3),C2)) ) ) ).

% sup.left_commute
tff(fact_758_sup__left__commute,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xa: A,Y: A,Z2: A] : ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),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),Xa),Z2)) ) ) ).

% sup_left_commute
tff(fact_759_sup__fun__def,axiom,
    ! [B: $tType,A: $tType] :
      ( semilattice_sup(B)
     => ! [F2: fun(A,B),G: fun(A,B),X4: 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)),F2),G),X4) = aa(B,B,aa(B,fun(B,B),sup_sup(B),aa(A,B,F2,X4)),aa(A,B,G,X4)) ) ) ).

% sup_fun_def
tff(fact_760_zip__rev,axiom,
    ! [A: $tType,B: $tType,Xsa: list(A),Ys: list(B)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xsa) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),aa(list(A),list(A),rev(A),Xsa)),aa(list(B),list(B),rev(B),Ys)) = aa(list(product_prod(A,B)),list(product_prod(A,B)),rev(product_prod(A,B)),aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),Xsa),Ys)) ) ) ).

% zip_rev
tff(fact_761_sorted__wrt__less__idx,axiom,
    ! [Ns: list(nat),Ia: nat] :
      ( sorted_wrt(nat,ord_less(nat),Ns)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),aa(list(nat),nat,size_size(list(nat)),Ns))
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ia),aa(nat,nat,nth(nat,Ns),Ia)) ) ) ).

% sorted_wrt_less_idx
tff(fact_762_sorted__wrt__mono__rel,axiom,
    ! [A: $tType,Xsa: list(A),Pa: fun(A,fun(A,$o)),Qa: fun(A,fun(A,$o))] :
      ( ! [X: A,Y2: A] :
          ( member(A,X,aa(list(A),set(A),set2(A),Xsa))
         => ( member(A,Y2,aa(list(A),set(A),set2(A),Xsa))
           => ( aa(A,$o,aa(A,fun(A,$o),Pa,X),Y2)
             => aa(A,$o,aa(A,fun(A,$o),Qa,X),Y2) ) ) )
     => ( sorted_wrt(A,Pa,Xsa)
       => sorted_wrt(A,Qa,Xsa) ) ) ).

% sorted_wrt_mono_rel
tff(fact_763_min__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A3: A,B3: A] :
          ( aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B3) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3),A3,B3) ) ) ).

% min_def
tff(fact_764_min__absorb1,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
         => ( aa(A,A,aa(A,fun(A,A),ord_min(A),Xa),Y) = Xa ) ) ) ).

% min_absorb1
tff(fact_765_min__absorb2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Y: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xa)
         => ( aa(A,A,aa(A,fun(A,A),ord_min(A),Xa),Y) = Y ) ) ) ).

% min_absorb2
tff(fact_766_min_Omono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,C2: A,B3: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),D3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B3)),aa(A,A,aa(A,fun(A,A),ord_min(A),C2),D3)) ) ) ) ).

% min.mono
tff(fact_767_min_OorderE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
         => ( A3 = aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B3) ) ) ) ).

% min.orderE
tff(fact_768_min_OorderI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A] :
          ( ( A3 = aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B3) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3) ) ) ).

% min.orderI
tff(fact_769_min_OboundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),ord_min(A),B3),C2))
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),C2) ) ) ) ).

% min.boundedE
tff(fact_770_min_OboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),ord_min(A),B3),C2)) ) ) ) ).

% min.boundedI
tff(fact_771_min_Oorder__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
        <=> ( A3 = aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B3) ) ) ) ).

% min.order_iff
tff(fact_772_min_Ocobounded1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B3)),A3) ) ).

% min.cobounded1
tff(fact_773_min_Ocobounded2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B3)),B3) ) ).

% min.cobounded2
tff(fact_774_min_Oabsorb__iff1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
        <=> ( aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B3) = A3 ) ) ) ).

% min.absorb_iff1
tff(fact_775_min_Oabsorb__iff2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3)
        <=> ( aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B3) = B3 ) ) ) ).

% min.absorb_iff2
tff(fact_776_min_OcoboundedI1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,C2: A,B3: A] :
          ( aa(A,$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),B3)),C2) ) ) ).

% min.coboundedI1
tff(fact_777_min_OcoboundedI2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B3: A,C2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),C2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B3)),C2) ) ) ).

% min.coboundedI2
tff(fact_778_min__le__iff__disj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Y: A,Z2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),Xa),Y)),Z2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Z2)
            | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z2) ) ) ) ).

% min_le_iff_disj
tff(fact_779_min_Ostrict__coboundedI2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B3: A,C2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),C2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B3)),C2) ) ) ).

% min.strict_coboundedI2
tff(fact_780_min_Ostrict__coboundedI1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,C2: A,B3: A] :
          ( aa(A,$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),B3)),C2) ) ) ).

% min.strict_coboundedI1
tff(fact_781_min_Ostrict__order__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
        <=> ( ( A3 = aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B3) )
            & ( A3 != B3 ) ) ) ) ).

% min.strict_order_iff
tff(fact_782_min_Ostrict__boundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),ord_min(A),B3),C2))
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),C2) ) ) ) ).

% min.strict_boundedE
tff(fact_783_min__less__iff__disj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Y: A,Z2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_min(A),Xa),Y)),Z2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Z2)
            | aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Z2) ) ) ) ).

% min_less_iff_disj
tff(fact_784_min__diff__distrib__left,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [Xa: 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),Xa),Y)),Z2) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xa),Z2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),Y),Z2)) ) ) ).

% min_diff_distrib_left
tff(fact_785_inf__sup__ord_I4_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Y: A,Xa: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Y)) ) ).

% inf_sup_ord(4)
tff(fact_786_inf__sup__ord_I3_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xa: A,Y: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Y)) ) ).

% inf_sup_ord(3)
tff(fact_787_le__supE,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: A,B3: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3)),Xa)
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),Xa)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),Xa) ) ) ) ).

% le_supE
tff(fact_788_le__supI,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: A,Xa: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),Xa)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),Xa)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3)),Xa) ) ) ) ).

% le_supI
tff(fact_789_sup__ge1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xa: A,Y: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Y)) ) ).

% sup_ge1
tff(fact_790_sup__ge2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Y: A,Xa: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Y)) ) ).

% sup_ge2
tff(fact_791_le__supI1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xa: A,A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),A3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3)) ) ) ).

% le_supI1
tff(fact_792_le__supI2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xa: A,B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),B3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3)) ) ) ).

% le_supI2
tff(fact_793_sup_Omono,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [C2: A,A3: A,D3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),D3),B3)
           => aa(A,$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),B3)) ) ) ) ).

% sup.mono
tff(fact_794_sup__mono,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: A,C2: A,B3: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),D3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3)),aa(A,A,aa(A,fun(A,A),sup_sup(A),C2),D3)) ) ) ) ).

% sup_mono
tff(fact_795_sup__least,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Y: A,Xa: A,Z2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xa)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z2),Xa)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z2)),Xa) ) ) ) ).

% sup_least
tff(fact_796_le__iff__sup,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
        <=> ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Y) = Y ) ) ) ).

% le_iff_sup
tff(fact_797_sup_OorderE,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3)
         => ( A3 = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3) ) ) ) ).

% sup.orderE
tff(fact_798_sup_OorderI,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: A,B3: A] :
          ( ( A3 = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3) ) ) ).

% sup.orderI
tff(fact_799_sup__unique,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [F2: fun(A,fun(A,A)),Xa: A,Y: A] :
          ( ! [X: A,Y2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),F2,X),Y2))
         => ( ! [X: A,Y2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y2),aa(A,A,aa(A,fun(A,A),F2,X),Y2))
           => ( ! [X: A,Y2: A,Z3: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y2),X)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z3),X)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),F2,Y2),Z3)),X) ) )
             => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Y) = aa(A,A,aa(A,fun(A,A),F2,Xa),Y) ) ) ) ) ) ).

% sup_unique
tff(fact_800_sup_Oabsorb1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3)
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3) = A3 ) ) ) ).

% sup.absorb1
tff(fact_801_sup_Oabsorb2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3) = B3 ) ) ) ).

% sup.absorb2
tff(fact_802_sup__absorb1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Y: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xa)
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Y) = Xa ) ) ) ).

% sup_absorb1
tff(fact_803_sup__absorb2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Y) = Y ) ) ) ).

% sup_absorb2
tff(fact_804_sup_OboundedE,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B3: A,C2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),B3),C2)),A3)
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A3) ) ) ) ).

% sup.boundedE
tff(fact_805_sup_OboundedI,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B3: A,A3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),B3),C2)),A3) ) ) ) ).

% sup.boundedI
tff(fact_806_sup_Oorder__iff,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3)
        <=> ( A3 = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3) ) ) ) ).

% sup.order_iff
tff(fact_807_sup_Ocobounded1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: A,B3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3)) ) ).

% sup.cobounded1
tff(fact_808_sup_Ocobounded2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B3: A,A3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3)) ) ).

% sup.cobounded2
tff(fact_809_sup_Oabsorb__iff1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3)
        <=> ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3) = A3 ) ) ) ).

% sup.absorb_iff1
tff(fact_810_sup_Oabsorb__iff2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
        <=> ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3) = B3 ) ) ) ).

% sup.absorb_iff2
tff(fact_811_sup_OcoboundedI1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [C2: A,A3: A,B3: A] :
          ( aa(A,$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),B3)) ) ) ).

% sup.coboundedI1
tff(fact_812_sup_OcoboundedI2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [C2: A,B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),B3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3)) ) ) ).

% sup.coboundedI2
tff(fact_813_inf__min,axiom,
    ! [A: $tType] :
      ( ( semilattice_inf(A)
        & linorder(A) )
     => ( inf_inf(A) = ord_min(A) ) ) ).

% inf_min
tff(fact_814_sup_Ostrict__coboundedI2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [C2: A,B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),B3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3)) ) ) ).

% sup.strict_coboundedI2
tff(fact_815_sup_Ostrict__coboundedI1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [C2: A,A3: A,B3: A] :
          ( aa(A,$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),B3)) ) ) ).

% sup.strict_coboundedI1
tff(fact_816_sup_Ostrict__order__iff,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A3)
        <=> ( ( A3 = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3) )
            & ( A3 != B3 ) ) ) ) ).

% sup.strict_order_iff
tff(fact_817_sup_Ostrict__boundedE,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B3: A,C2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),B3),C2)),A3)
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A3)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),A3) ) ) ) ).

% sup.strict_boundedE
tff(fact_818_sup_Oabsorb4,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3) = B3 ) ) ) ).

% sup.absorb4
tff(fact_819_sup_Oabsorb3,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A3)
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3) = A3 ) ) ) ).

% sup.absorb3
tff(fact_820_less__supI2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xa: A,B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),B3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3)) ) ) ).

% less_supI2
tff(fact_821_less__supI1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xa: A,A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),A3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3)) ) ) ).

% less_supI1
tff(fact_822_boolean__algebra_Odisj__zero__right,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A] : ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),bot_bot(A)) = Xa ) ) ).

% boolean_algebra.disj_zero_right
tff(fact_823_distrib__imp1,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xa: A,Y: A,Z2: A] :
          ( ! [X: A,Y2: A,Z3: A] : ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X),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),X),Y2)),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Z3)) )
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),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),Xa),Y)),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Z2)) ) ) ) ).

% distrib_imp1
tff(fact_824_distrib__imp2,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xa: A,Y: A,Z2: A] :
          ( ! [X: A,Y2: A,Z3: A] : ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),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),X),Y2)),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Z3)) )
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),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),Xa),Y)),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Z2)) ) ) ) ).

% distrib_imp2
tff(fact_825_inf__sup__distrib1,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [Xa: A,Y: A,Z2: A] : ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),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),Xa),Y)),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Z2)) ) ) ).

% inf_sup_distrib1
tff(fact_826_inf__sup__distrib2,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [Y: A,Z2: A,Xa: 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)),Xa) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Xa)),aa(A,A,aa(A,fun(A,A),inf_inf(A),Z2),Xa)) ) ) ).

% inf_sup_distrib2
tff(fact_827_sup__inf__distrib1,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [Xa: A,Y: A,Z2: A] : ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),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),Xa),Y)),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Z2)) ) ) ).

% sup_inf_distrib1
tff(fact_828_sup__inf__distrib2,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [Y: A,Z2: A,Xa: 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)),Xa) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Xa)),aa(A,A,aa(A,fun(A,A),sup_sup(A),Z2),Xa)) ) ) ).

% sup_inf_distrib2
tff(fact_829_Un__empty__left,axiom,
    ! [A: $tType,B2: set(A)] : ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),bot_bot(set(A))),B2) = B2 ) ).

% Un_empty_left
tff(fact_830_Un__empty__right,axiom,
    ! [A: $tType,Aa2: set(A)] : ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),bot_bot(set(A))) = Aa2 ) ).

% Un_empty_right
tff(fact_831_foldl__un__empty__eq,axiom,
    ! [A: $tType,Ia: set(A),Ww: list(set(A))] : ( aa(list(set(A)),set(A),aa(set(A),fun(list(set(A)),set(A)),aa(fun(set(A),fun(set(A),set(A))),fun(set(A),fun(list(set(A)),set(A))),foldl(set(A),set(A)),sup_sup(set(A))),Ia),Ww) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Ia),aa(list(set(A)),set(A),aa(set(A),fun(list(set(A)),set(A)),aa(fun(set(A),fun(set(A),set(A))),fun(set(A),fun(list(set(A)),set(A))),foldl(set(A),set(A)),sup_sup(set(A))),bot_bot(set(A))),Ww)) ) ).

% foldl_un_empty_eq
tff(fact_832_Un__mono,axiom,
    ! [A: $tType,Aa2: set(A),C3: set(A),B2: set(A),D2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),C3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),D2)
       => aa(set(A),$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)),Aa2),B2)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),C3),D2)) ) ) ).

% Un_mono
tff(fact_833_Un__least,axiom,
    ! [A: $tType,Aa2: set(A),C3: set(A),B2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),C3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),C3)
       => aa(set(A),$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)),Aa2),B2)),C3) ) ) ).

% Un_least
tff(fact_834_Un__upper1,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),B2)) ).

% Un_upper1
tff(fact_835_Un__upper2,axiom,
    ! [A: $tType,B2: set(A),Aa2: set(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),B2)) ).

% Un_upper2
tff(fact_836_Un__absorb1,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),B2)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),B2) = B2 ) ) ).

% Un_absorb1
tff(fact_837_Un__absorb2,axiom,
    ! [A: $tType,B2: set(A),Aa2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),Aa2)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),B2) = Aa2 ) ) ).

% Un_absorb2
tff(fact_838_subset__UnE,axiom,
    ! [A: $tType,C3: set(A),Aa2: set(A),B2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),B2))
     => ~ ! [A10: set(A)] :
            ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A10),Aa2)
           => ! [B7: set(A)] :
                ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B7),B2)
               => ( C3 != aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A10),B7) ) ) ) ) ).

% subset_UnE
tff(fact_839_subset__Un__eq,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),B2)
    <=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),B2) = B2 ) ) ).

% subset_Un_eq
tff(fact_840_Un__Int__assoc__eq,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A),C3: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),B2)),C3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B2),C3)) )
    <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C3),Aa2) ) ).

% Un_Int_assoc_eq
tff(fact_841_set__diff__diff__left,axiom,
    ! [A: $tType,Aa2: set(A),B2: 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)),Aa2),B2)),C3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B2),C3)) ) ).

% set_diff_diff_left
tff(fact_842_Un__Diff,axiom,
    ! [A: $tType,Aa2: set(A),B2: 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)),Aa2),B2)),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)),Aa2),C3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B2),C3)) ) ).

% Un_Diff
tff(fact_843_Diff__Un,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A),C3: set(A)] : ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B2),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)),Aa2),B2)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),C3)) ) ).

% Diff_Un
tff(fact_844_Diff__Int,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A),C3: set(A)] : ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B2),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)),Aa2),B2)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),C3)) ) ).

% Diff_Int
tff(fact_845_Int__Diff__Un,axiom,
    ! [A: $tType,Aa2: set(A),B2: 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)),Aa2),B2)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),B2)) = Aa2 ) ).

% Int_Diff_Un
tff(fact_846_Un__Diff__Int,axiom,
    ! [A: $tType,Aa2: set(A),B2: 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)),Aa2),B2)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),B2)) = Aa2 ) ).

% Un_Diff_Int
tff(fact_847_in__set__butlastD,axiom,
    ! [A: $tType,Xa: A,Xsa: list(A)] :
      ( member(A,Xa,aa(list(A),set(A),set2(A),aa(list(A),list(A),butlast(A),Xsa)))
     => member(A,Xa,aa(list(A),set(A),set2(A),Xsa)) ) ).

% in_set_butlastD
tff(fact_848_distinct__butlast,axiom,
    ! [A: $tType,Xsa: list(A)] :
      ( aa(list(A),$o,distinct(A),Xsa)
     => aa(list(A),$o,distinct(A),aa(list(A),list(A),butlast(A),Xsa)) ) ).

% distinct_butlast
tff(fact_849_butlast__update_H,axiom,
    ! [A: $tType,L: list(A),Ia: nat,Xa: A] : ( aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),aa(list(A),list(A),butlast(A),L)),Ia),Xa) = aa(list(A),list(A),butlast(A),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),L),Ia),Xa)) ) ).

% butlast_update'
tff(fact_850_sorted__rev__iff__nth__mono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xsa: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),rev(A),Xsa))
        <=> ! [I2: nat,J2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),J2)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J2),aa(list(A),nat,size_size(list(A)),Xsa))
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,nth(A,Xsa),J2)),aa(nat,A,nth(A,Xsa),I2)) ) ) ) ) ).

% sorted_rev_iff_nth_mono
tff(fact_851_sorted__rev__nth__mono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xsa: list(A),Ia: nat,J: nat] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),rev(A),Xsa))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ia),J)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),Xsa))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,nth(A,Xsa),J)),aa(nat,A,nth(A,Xsa),Ia)) ) ) ) ) ).

% sorted_rev_nth_mono
tff(fact_852_strict__sorted__imp__sorted,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xsa: list(A)] :
          ( sorted_wrt(A,ord_less(A),Xsa)
         => sorted_wrt(A,ord_less_eq(A),Xsa) ) ) ).

% strict_sorted_imp_sorted
tff(fact_853_strict__sorted__equal,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xsa: list(A),Ys: list(A)] :
          ( sorted_wrt(A,ord_less(A),Xsa)
         => ( sorted_wrt(A,ord_less(A),Ys)
           => ( ( aa(list(A),set(A),set2(A),Ys) = aa(list(A),set(A),set2(A),Xsa) )
             => ( Ys = Xsa ) ) ) ) ) ).

% strict_sorted_equal
tff(fact_854_sorted__wrt01,axiom,
    ! [A: $tType,Xsa: list(A),Pa: fun(A,fun(A,$o))] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xsa)),one_one(nat))
     => sorted_wrt(A,Pa,Xsa) ) ).

% sorted_wrt01
tff(fact_855_distrib__inf__le,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xa: A,Y: A,Z2: A] : aa(A,$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),Xa),Y)),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Z2))),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z2))) ) ).

% distrib_inf_le
tff(fact_856_distrib__sup__le,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xa: A,Y: A,Z2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),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),Xa),Y)),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Z2))) ) ).

% distrib_sup_le
tff(fact_857_sorted__remove1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xsa: list(A),A3: A] :
          ( sorted_wrt(A,ord_less_eq(A),Xsa)
         => sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),remove1(A),A3),Xsa)) ) ) ).

% sorted_remove1
tff(fact_858_insert__is__Un,axiom,
    ! [A: $tType,A3: A,Aa2: set(A)] : ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),Aa2) = 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)),insert3(A),A3),bot_bot(set(A)))),Aa2) ) ).

% insert_is_Un
tff(fact_859_Un__singleton__iff,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A),Xa: A] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),B2) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A))) )
    <=> ( ( ( Aa2 = bot_bot(set(A)) )
          & ( B2 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A))) ) )
        | ( ( Aa2 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A))) )
          & ( B2 = bot_bot(set(A)) ) )
        | ( ( Aa2 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A))) )
          & ( B2 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A))) ) ) ) ) ).

% Un_singleton_iff
tff(fact_860_singleton__Un__iff,axiom,
    ! [A: $tType,Xa: A,Aa2: set(A),B2: set(A)] :
      ( ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),B2) )
    <=> ( ( ( Aa2 = bot_bot(set(A)) )
          & ( B2 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A))) ) )
        | ( ( Aa2 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A))) )
          & ( B2 = bot_bot(set(A)) ) )
        | ( ( Aa2 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A))) )
          & ( B2 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A))) ) ) ) ) ).

% singleton_Un_iff
tff(fact_861_Diff__partition,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),B2)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B2),Aa2)) = B2 ) ) ).

% Diff_partition
tff(fact_862_Diff__subset__conv,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A),C3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),B2)),C3)
    <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B2),C3)) ) ).

% Diff_subset_conv
tff(fact_863_strict__sorted__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: list(A)] :
          ( sorted_wrt(A,ord_less(A),L)
        <=> ( sorted_wrt(A,ord_less_eq(A),L)
            & aa(list(A),$o,distinct(A),L) ) ) ) ).

% strict_sorted_iff
tff(fact_864_sorted01,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xsa: list(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xsa)),one_one(nat))
         => sorted_wrt(A,ord_less_eq(A),Xsa) ) ) ).

% sorted01
tff(fact_865_sorted__distinct__set__unique,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xsa: list(A),Ys: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),Xsa)
         => ( aa(list(A),$o,distinct(A),Xsa)
           => ( sorted_wrt(A,ord_less_eq(A),Ys)
             => ( aa(list(A),$o,distinct(A),Ys)
               => ( ( aa(list(A),set(A),set2(A),Xsa) = aa(list(A),set(A),set2(A),Ys) )
                 => ( Xsa = Ys ) ) ) ) ) ) ) ).

% sorted_distinct_set_unique
tff(fact_866_sorted__wrt__iff__nth__less,axiom,
    ! [A: $tType,Pa: fun(A,fun(A,$o)),Xsa: list(A)] :
      ( sorted_wrt(A,Pa,Xsa)
    <=> ! [I2: nat,J2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),J2)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J2),aa(list(A),nat,size_size(list(A)),Xsa))
           => aa(A,$o,aa(A,fun(A,$o),Pa,aa(nat,A,nth(A,Xsa),I2)),aa(nat,A,nth(A,Xsa),J2)) ) ) ) ).

% sorted_wrt_iff_nth_less
tff(fact_867_sorted__wrt__nth__less,axiom,
    ! [A: $tType,Pa: fun(A,fun(A,$o)),Xsa: list(A),Ia: nat,J: nat] :
      ( sorted_wrt(A,Pa,Xsa)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),J)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),Xsa))
         => aa(A,$o,aa(A,fun(A,$o),Pa,aa(nat,A,nth(A,Xsa),Ia)),aa(nat,A,nth(A,Xsa),J)) ) ) ) ).

% sorted_wrt_nth_less
tff(fact_868_nth__butlast,axiom,
    ! [A: $tType,N: nat,Xsa: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),butlast(A),Xsa)))
     => ( aa(nat,A,nth(A,aa(list(A),list(A),butlast(A),Xsa)),N) = aa(nat,A,nth(A,Xsa),N) ) ) ).

% nth_butlast
tff(fact_869_sorted__iff__nth__mono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xsa: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),Xsa)
        <=> ! [I2: nat,J2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),J2)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J2),aa(list(A),nat,size_size(list(A)),Xsa))
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,nth(A,Xsa),I2)),aa(nat,A,nth(A,Xsa),J2)) ) ) ) ) ).

% sorted_iff_nth_mono
tff(fact_870_sorted__nth__mono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xsa: list(A),Ia: nat,J: nat] :
          ( sorted_wrt(A,ord_less_eq(A),Xsa)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ia),J)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),Xsa))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,nth(A,Xsa),Ia)),aa(nat,A,nth(A,Xsa),J)) ) ) ) ) ).

% sorted_nth_mono
tff(fact_871_sorted__iff__nth__mono__less,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xsa: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),Xsa)
        <=> ! [I2: nat,J2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),J2)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J2),aa(list(A),nat,size_size(list(A)),Xsa))
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,nth(A,Xsa),I2)),aa(nat,A,nth(A,Xsa),J2)) ) ) ) ) ).

% sorted_iff_nth_mono_less
tff(fact_872_mergesort__remdups__correct,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: list(A)] :
          ( aa(list(A),$o,distinct(A),aa(list(A),list(A),mergesort_remdups(A),L))
          & sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),mergesort_remdups(A),L))
          & ( aa(list(A),set(A),set2(A),aa(list(A),list(A),mergesort_remdups(A),L)) = aa(list(A),set(A),set2(A),L) ) ) ) ).

% mergesort_remdups_correct
tff(fact_873_times__assn__raw_Osimps,axiom,
    ! [Pa: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Qa: fun(product_prod(heap_ext(product_unit),set(nat)),$o),H2: heap_ext(product_unit),As2: set(nat)] :
      ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,times_assn_raw(Pa,Qa),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As2))
    <=> ? [As1: set(nat),As22: set(nat)] :
          ( ( As2 = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As1),As22) )
          & ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As1),As22) = bot_bot(set(nat)) )
          & aa(product_prod(heap_ext(product_unit),set(nat)),$o,Pa,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,Qa,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),As22)) ) ) ).

% times_assn_raw.simps
tff(fact_874_times__assn__raw_Oelims_I1_J,axiom,
    ! [Xa: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xaa: fun(product_prod(heap_ext(product_unit),set(nat)),$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(Xa,Xaa),Xb)
      <=> (Y) )
     => ~ ! [H: heap_ext(product_unit),As: 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),As) )
           => ( (Y)
            <=> ~ ? [As1: set(nat),As22: set(nat)] :
                    ( ( As = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As1),As22) )
                    & ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As1),As22) = bot_bot(set(nat)) )
                    & aa(product_prod(heap_ext(product_unit),set(nat)),$o,Xa,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As1))
                    & aa(product_prod(heap_ext(product_unit),set(nat)),$o,Xaa,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(1)
tff(fact_875_times__assn__raw_Oelims_I2_J,axiom,
    ! [Xa: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xaa: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xb: product_prod(heap_ext(product_unit),set(nat))] :
      ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,times_assn_raw(Xa,Xaa),Xb)
     => ~ ! [H: heap_ext(product_unit),As: set(nat)] :
            ( ( Xb = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As) )
           => ~ ? [As12: set(nat),As23: set(nat)] :
                  ( ( As = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As12),As23) )
                  & ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As12),As23) = bot_bot(set(nat)) )
                  & aa(product_prod(heap_ext(product_unit),set(nat)),$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),As12))
                  & aa(product_prod(heap_ext(product_unit),set(nat)),$o,Xaa,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(2)
tff(fact_876_times__assn__raw_Oelims_I3_J,axiom,
    ! [Xa: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xaa: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xb: product_prod(heap_ext(product_unit),set(nat))] :
      ( ~ aa(product_prod(heap_ext(product_unit),set(nat)),$o,times_assn_raw(Xa,Xaa),Xb)
     => ~ ! [H: heap_ext(product_unit),As: set(nat)] :
            ( ( Xb = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As) )
           => ? [As13: set(nat),As24: set(nat)] :
                ( ( As = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As13),As24) )
                & ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As13),As24) = bot_bot(set(nat)) )
                & aa(product_prod(heap_ext(product_unit),set(nat)),$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),As13))
                & aa(product_prod(heap_ext(product_unit),set(nat)),$o,Xaa,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),As24)) ) ) ) ).

% times_assn_raw.elims(3)
tff(fact_877_Lattices__Big_Oex__has__greatest__nat,axiom,
    ! [A: $tType,Pa: fun(A,$o),K3: A,F2: fun(A,nat),B3: nat] :
      ( aa(A,$o,Pa,K3)
     => ( ! [Y2: A] :
            ( aa(A,$o,Pa,Y2)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,F2,Y2)),B3) )
       => ? [X: A] :
            ( aa(A,$o,Pa,X)
            & ! [Y3: A] :
                ( aa(A,$o,Pa,Y3)
               => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,F2,Y3)),aa(A,nat,F2,X)) ) ) ) ) ).

% Lattices_Big.ex_has_greatest_nat
tff(fact_878_nat__descend__induct,axiom,
    ! [N: nat,Pa: fun(nat,$o),M2: nat] :
      ( ! [K: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),K)
         => aa(nat,$o,Pa,K) )
     => ( ! [K: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),N)
           => ( ! [I4: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),I4)
                 => aa(nat,$o,Pa,I4) )
             => aa(nat,$o,Pa,K) ) )
       => aa(nat,$o,Pa,M2) ) ) ).

% nat_descend_induct
tff(fact_879_merge__correct,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L12: list(A),L23: list(A)] :
          ( ( aa(list(A),$o,distinct(A),L12)
            & sorted_wrt(A,ord_less_eq(A),L12) )
         => ( ( aa(list(A),$o,distinct(A),L23)
              & sorted_wrt(A,ord_less_eq(A),L23) )
           => ( aa(list(A),$o,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_880_slice__nth,axiom,
    ! [A: $tType,From: nat,To: nat,Xsa: list(A),Ia: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),From),To)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),To),aa(list(A),nat,size_size(list(A)),Xsa))
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),To),From))
         => ( aa(nat,A,nth(A,slice(A,From,To,Xsa)),Ia) = aa(nat,A,nth(A,Xsa),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),From),Ia)) ) ) ) ) ).

% slice_nth
tff(fact_881_sorted__rev__iff__nth__Suc,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xsa: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),rev(A),Xsa))
        <=> ! [I2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,I2)),aa(list(A),nat,size_size(list(A)),Xsa))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,nth(A,Xsa),aa(nat,nat,suc,I2))),aa(nat,A,nth(A,Xsa),I2)) ) ) ) ).

% sorted_rev_iff_nth_Suc
tff(fact_882_add__right__cancel,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [B3: A,A3: A,C2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A3) )
        <=> ( B3 = C2 ) ) ) ).

% add_right_cancel
tff(fact_883_add__left__cancel,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [A3: A,B3: A,C2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2) )
        <=> ( B3 = C2 ) ) ) ).

% add_left_cancel
tff(fact_884_mod__or__dist,axiom,
    ! [Pa: assn,Qa: 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),sup_sup(assn),Pa),Qa)),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,Pa),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,Qa),H2) ) ) ).

% mod_or_dist
tff(fact_885_merge__pure__or,axiom,
    ! [A3: $o,B3: $o] :
      ( aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),pure_assn((A3))),pure_assn((B3))) = pure_assn(( (A3)
          | (B3) )) ) ).

% merge_pure_or
tff(fact_886_add__le__cancel__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [A3: A,C2: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),C2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3) ) ) ).

% add_le_cancel_right
tff(fact_887_add__le__cancel__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [C2: A,A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3) ) ) ).

% add_le_cancel_left
tff(fact_888_add__less__cancel__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [C2: A,A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3) ) ) ).

% add_less_cancel_left
tff(fact_889_add__less__cancel__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [A3: A,C2: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),C2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3) ) ) ).

% add_less_cancel_right
tff(fact_890_add__diff__cancel,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B3: A] : ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3)),B3) = A3 ) ) ).

% add_diff_cancel
tff(fact_891_diff__add__cancel,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B3: A] : ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B3)),B3) = A3 ) ) ).

% diff_add_cancel
tff(fact_892_add__diff__cancel__left,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [C2: A,A3: A,B3: 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),B3)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B3) ) ) ).

% add_diff_cancel_left
tff(fact_893_add__diff__cancel__left_H,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [A3: A,B3: A] : ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3)),A3) = B3 ) ) ).

% add_diff_cancel_left'
tff(fact_894_add__diff__cancel__right,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [A3: A,C2: A,B3: 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),B3),C2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B3) ) ) ).

% add_diff_cancel_right
tff(fact_895_add__diff__cancel__right_H,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [A3: A,B3: A] : ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3)),B3) = A3 ) ) ).

% add_diff_cancel_right'
tff(fact_896_lessI,axiom,
    ! [N: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),aa(nat,nat,suc,N)) ).

% lessI
tff(fact_897_Suc__mono,axiom,
    ! [M2: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),N)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,M2)),aa(nat,nat,suc,N)) ) ).

% Suc_mono
tff(fact_898_Suc__less__eq,axiom,
    ! [M2: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,M2)),aa(nat,nat,suc,N))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),N) ) ).

% Suc_less_eq
tff(fact_899_nat__add__left__cancel__less,axiom,
    ! [K3: nat,M2: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K3),M2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K3),N))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),N) ) ).

% nat_add_left_cancel_less
tff(fact_900_le__add__diff__inverse,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3)
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B3)) = A3 ) ) ) ).

% le_add_diff_inverse
tff(fact_901_le__add__diff__inverse2,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3)
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B3)),B3) = A3 ) ) ) ).

% le_add_diff_inverse2
tff(fact_902_min__Suc__gt_I1_J,axiom,
    ! [A3: nat,B3: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A3),B3)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(nat,nat,suc,A3)),B3) = aa(nat,nat,suc,A3) ) ) ).

% min_Suc_gt(1)
tff(fact_903_min__Suc__gt_I2_J,axiom,
    ! [A3: nat,B3: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A3),B3)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),B3),aa(nat,nat,suc,A3)) = aa(nat,nat,suc,A3) ) ) ).

% min_Suc_gt(2)
tff(fact_904_mod__h__bot__iff_I7_J,axiom,
    ! [Pa: assn,Qa: assn,H2: 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,aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),Pa),Qa)),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,Pa),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,Qa),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)))) ) ) ).

% mod_h_bot_iff(7)
tff(fact_905_Suc__diff,axiom,
    ! [M2: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),N)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),M2)
       => ( aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M2)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),one_one(nat))) ) ) ) ).

% Suc_diff
tff(fact_906_less__natE,axiom,
    ! [M2: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),N)
     => ~ ! [Q4: nat] : ( N != aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),Q4)) ) ) ).

% less_natE
tff(fact_907_less__add__Suc1,axiom,
    ! [Ia: nat,M2: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ia),M2))) ).

% less_add_Suc1
tff(fact_908_less__add__Suc2,axiom,
    ! [Ia: nat,M2: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),Ia))) ).

% less_add_Suc2
tff(fact_909_less__iff__Suc__add,axiom,
    ! [M2: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),N)
    <=> ? [K4: nat] : ( N = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),K4)) ) ) ).

% less_iff_Suc_add
tff(fact_910_less__imp__Suc__add,axiom,
    ! [M2: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),N)
     => ? [K: nat] : ( N = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),K)) ) ) ).

% less_imp_Suc_add
tff(fact_911_add__right__imp__eq,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [B3: A,A3: A,C2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A3) )
         => ( B3 = C2 ) ) ) ).

% add_right_imp_eq
tff(fact_912_add__left__imp__eq,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [A3: A,B3: A,C2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2) )
         => ( B3 = C2 ) ) ) ).

% add_left_imp_eq
tff(fact_913_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [A: $tType] :
      ( ab_semigroup_add(A)
     => ! [B3: A,A3: A,C2: A] : ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),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),B3),C2)) ) ) ).

% ab_semigroup_add_class.add.left_commute
tff(fact_914_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ! [A: $tType] :
      ( ab_semigroup_add(A)
     => ! [A3: A,B3: A] : ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),A3) ) ) ).

% ab_semigroup_add_class.add.commute
tff(fact_915_add_Oright__cancel,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [B3: A,A3: A,C2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A3) )
        <=> ( B3 = C2 ) ) ) ).

% add.right_cancel
tff(fact_916_add_Oleft__cancel,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B3: A,C2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2) )
        <=> ( B3 = C2 ) ) ) ).

% add.left_cancel
tff(fact_917_add_Oassoc,axiom,
    ! [A: $tType] :
      ( semigroup_add(A)
     => ! [A3: A,B3: 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),B3)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),C2)) ) ) ).

% add.assoc
tff(fact_918_group__cancel_Oadd2,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [B2: A,K3: A,B3: A,A3: A] :
          ( ( B2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K3),B3) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),K3),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3)) ) ) ) ).

% group_cancel.add2
tff(fact_919_group__cancel_Oadd1,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Aa2: A,K3: A,A3: A,B3: A] :
          ( ( Aa2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K3),A3) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),B3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),K3),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3)) ) ) ) ).

% group_cancel.add1
tff(fact_920_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [Ia: A,J: A,K3: A,L: A] :
          ( ( ( Ia = J )
            & ( K3 = L ) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),Ia),K3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L) ) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_921_add__le__imp__le__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [A3: A,C2: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),C2))
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3) ) ) ).

% add_le_imp_le_right
tff(fact_922_add__le__imp__le__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [C2: A,A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B3))
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3) ) ) ).

% add_le_imp_le_left
tff(fact_923_le__iff__add,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
        <=> ? [C6: A] : ( B3 = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C6) ) ) ) ).

% le_iff_add
tff(fact_924_add__right__mono,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),C2)) ) ) ).

% add_right_mono
tff(fact_925_less__eqE,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
         => ~ ! [C5: A] : ( B3 != aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C5) ) ) ) ).

% less_eqE
tff(fact_926_add__left__mono,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B3)) ) ) ).

% add_left_mono
tff(fact_927_add__mono,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [A3: A,B3: A,C2: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),D3)
           => aa(A,$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),B3),D3)) ) ) ) ).

% add_mono
tff(fact_928_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [Ia: A,J: A,K3: A,L: A] :
          ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ia),J)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),K3),L) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Ia),K3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_929_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [Ia: A,J: A,K3: A,L: A] :
          ( ( ( Ia = J )
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),K3),L) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Ia),K3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_930_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [Ia: A,J: A,K3: A,L: A] :
          ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ia),J)
            & ( K3 = L ) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Ia),K3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_931_add__mono__thms__linordered__field_I5_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [Ia: A,J: A,K3: A,L: A] :
          ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ia),J)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),K3),L) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Ia),K3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_932_add__mono__thms__linordered__field_I2_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [Ia: A,J: A,K3: A,L: A] :
          ( ( ( Ia = J )
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),K3),L) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Ia),K3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_933_add__mono__thms__linordered__field_I1_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [Ia: A,J: A,K3: A,L: A] :
          ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ia),J)
            & ( K3 = L ) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Ia),K3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_934_add__strict__mono,axiom,
    ! [A: $tType] :
      ( strict9044650504122735259up_add(A)
     => ! [A3: A,B3: A,C2: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
         => ( aa(A,$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),B3),D3)) ) ) ) ).

% add_strict_mono
tff(fact_935_add__strict__left__mono,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
         => aa(A,$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),B3)) ) ) ).

% add_strict_left_mono
tff(fact_936_add__strict__right__mono,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
         => aa(A,$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),B3),C2)) ) ) ).

% add_strict_right_mono
tff(fact_937_add__less__imp__less__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [C2: A,A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B3))
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3) ) ) ).

% add_less_imp_less_left
tff(fact_938_add__less__imp__less__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [A3: A,C2: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),C2))
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3) ) ) ).

% add_less_imp_less_right
tff(fact_939_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A3: A,B3: 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),B3)),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),B3),C2)) ) ) ).

% ring_class.ring_distribs(2)
tff(fact_940_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A3: A,B3: 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),B3),C2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)) ) ) ).

% ring_class.ring_distribs(1)
tff(fact_941_comm__semiring__class_Odistrib,axiom,
    ! [A: $tType] :
      ( comm_semiring(A)
     => ! [A3: A,B3: 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),B3)),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),B3),C2)) ) ) ).

% comm_semiring_class.distrib
tff(fact_942_distrib__left,axiom,
    ! [A: $tType] :
      ( semiring(A)
     => ! [A3: A,B3: 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),B3),C2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)) ) ) ).

% distrib_left
tff(fact_943_distrib__right,axiom,
    ! [A: $tType] :
      ( semiring(A)
     => ! [A3: A,B3: 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),B3)),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),B3),C2)) ) ) ).

% distrib_right
tff(fact_944_combine__common__factor,axiom,
    ! [A: $tType] :
      ( semiring(A)
     => ! [A3: A,E3: A,B3: 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),E3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B3),E3)),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),B3)),E3)),C2) ) ) ).

% combine_common_factor
tff(fact_945_group__cancel_Osub1,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Aa2: A,K3: A,A3: A,B3: A] :
          ( ( Aa2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K3),A3) )
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),Aa2),B3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),K3),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B3)) ) ) ) ).

% group_cancel.sub1
tff(fact_946_diff__eq__eq,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B3: A,C2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B3) = C2 )
        <=> ( A3 = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B3) ) ) ) ).

% diff_eq_eq
tff(fact_947_eq__diff__eq,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,C2: A,B3: A] :
          ( ( A3 = aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),B3) )
        <=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3) = C2 ) ) ) ).

% eq_diff_eq
tff(fact_948_add__diff__eq,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B3: 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),B3),C2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3)),C2) ) ) ).

% add_diff_eq
tff(fact_949_diff__diff__eq2,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B3: 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),B3),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)),B3) ) ) ).

% diff_diff_eq2
tff(fact_950_diff__add__eq,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A,B3: 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),B3)),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)),B3) ) ) ).

% diff_add_eq
tff(fact_951_diff__add__eq__diff__diff__swap,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B3: 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),B3),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)),B3) ) ) ).

% diff_add_eq_diff_diff_swap
tff(fact_952_add__implies__diff,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [C2: A,B3: A,A3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B3) = A3 )
         => ( C2 = aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B3) ) ) ) ).

% add_implies_diff
tff(fact_953_diff__diff__eq,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [A3: A,B3: 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),B3)),C2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),C2)) ) ) ).

% diff_diff_eq
tff(fact_954_star__or__dist1,axiom,
    ! [Aa2: assn,B2: assn,C3: assn] : ( aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),Aa2),B2)),C3) = aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Aa2),C3)),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),B2),C3)) ) ).

% star_or_dist1
tff(fact_955_star__or__dist2,axiom,
    ! [C3: assn,Aa2: assn,B2: assn] : ( aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),C3),aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),Aa2),B2)) = aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),C3),Aa2)),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),C3),B2)) ) ).

% star_or_dist2
tff(fact_956_Nat_OlessE,axiom,
    ! [Ia: nat,K3: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),K3)
     => ( ( K3 != aa(nat,nat,suc,Ia) )
       => ~ ! [J3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),J3)
             => ( K3 != aa(nat,nat,suc,J3) ) ) ) ) ).

% Nat.lessE
tff(fact_957_Suc__lessD,axiom,
    ! [M2: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,M2)),N)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),N) ) ).

% Suc_lessD
tff(fact_958_Suc__lessE,axiom,
    ! [Ia: nat,K3: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,Ia)),K3)
     => ~ ! [J3: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),J3)
           => ( K3 != aa(nat,nat,suc,J3) ) ) ) ).

% Suc_lessE
tff(fact_959_Suc__lessI,axiom,
    ! [M2: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),N)
     => ( ( aa(nat,nat,suc,M2) != N )
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,M2)),N) ) ) ).

% Suc_lessI
tff(fact_960_less__SucE,axiom,
    ! [M2: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),aa(nat,nat,suc,N))
     => ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),N)
       => ( M2 = N ) ) ) ).

% less_SucE
tff(fact_961_less__SucI,axiom,
    ! [M2: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),N)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),aa(nat,nat,suc,N)) ) ).

% less_SucI
tff(fact_962_Ex__less__Suc,axiom,
    ! [N: nat,Pa: fun(nat,$o)] :
      ( ? [I2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(nat,nat,suc,N))
          & aa(nat,$o,Pa,I2) )
    <=> ( aa(nat,$o,Pa,N)
        | ? [I2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),N)
            & aa(nat,$o,Pa,I2) ) ) ) ).

% Ex_less_Suc
tff(fact_963_less__Suc__eq,axiom,
    ! [M2: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),aa(nat,nat,suc,N))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),N)
        | ( M2 = N ) ) ) ).

% less_Suc_eq
tff(fact_964_not__less__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),N)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),aa(nat,nat,suc,M2)) ) ).

% not_less_eq
tff(fact_965_All__less__Suc,axiom,
    ! [N: nat,Pa: fun(nat,$o)] :
      ( ! [I2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(nat,nat,suc,N))
         => aa(nat,$o,Pa,I2) )
    <=> ( aa(nat,$o,Pa,N)
        & ! [I2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),N)
           => aa(nat,$o,Pa,I2) ) ) ) ).

% All_less_Suc
tff(fact_966_Suc__less__eq2,axiom,
    ! [N: nat,M2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,N)),M2)
    <=> ? [M4: nat] :
          ( ( M2 = aa(nat,nat,suc,M4) )
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),M4) ) ) ).

% Suc_less_eq2
tff(fact_967_less__antisym,axiom,
    ! [N: nat,M2: nat] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),M2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),aa(nat,nat,suc,M2))
       => ( M2 = N ) ) ) ).

% less_antisym
tff(fact_968_Suc__less__SucD,axiom,
    ! [M2: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,M2)),aa(nat,nat,suc,N))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),N) ) ).

% Suc_less_SucD
tff(fact_969_less__trans__Suc,axiom,
    ! [Ia: nat,J: nat,K3: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),K3)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,Ia)),K3) ) ) ).

% less_trans_Suc
tff(fact_970_less__Suc__induct,axiom,
    ! [Ia: nat,J: nat,Pa: fun(nat,fun(nat,$o))] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),J)
     => ( ! [I3: nat] : aa(nat,$o,aa(nat,fun(nat,$o),Pa,I3),aa(nat,nat,suc,I3))
       => ( ! [I3: nat,J3: nat,K: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),J3)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),K)
               => ( aa(nat,$o,aa(nat,fun(nat,$o),Pa,I3),J3)
                 => ( aa(nat,$o,aa(nat,fun(nat,$o),Pa,J3),K)
                   => aa(nat,$o,aa(nat,fun(nat,$o),Pa,I3),K) ) ) ) )
         => aa(nat,$o,aa(nat,fun(nat,$o),Pa,Ia),J) ) ) ) ).

% less_Suc_induct
tff(fact_971_strict__inc__induct,axiom,
    ! [Ia: nat,J: nat,Pa: fun(nat,$o)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),J)
     => ( ! [I3: nat] :
            ( ( J = aa(nat,nat,suc,I3) )
           => aa(nat,$o,Pa,I3) )
       => ( ! [I3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),J)
             => ( aa(nat,$o,Pa,aa(nat,nat,suc,I3))
               => aa(nat,$o,Pa,I3) ) )
         => aa(nat,$o,Pa,Ia) ) ) ) ).

% strict_inc_induct
tff(fact_972_not__less__less__Suc__eq,axiom,
    ! [N: nat,M2: nat] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),M2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),aa(nat,nat,suc,M2))
      <=> ( N = M2 ) ) ) ).

% not_less_less_Suc_eq
tff(fact_973_add__lessD1,axiom,
    ! [Ia: nat,J: nat,K3: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ia),J)),K3)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),K3) ) ).

% add_lessD1
tff(fact_974_add__less__mono,axiom,
    ! [Ia: nat,J: nat,K3: nat,L: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K3),L)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ia),K3)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),L)) ) ) ).

% add_less_mono
tff(fact_975_not__add__less1,axiom,
    ! [Ia: nat,J: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ia),J)),Ia) ).

% not_add_less1
tff(fact_976_not__add__less2,axiom,
    ! [J: nat,Ia: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),Ia)),Ia) ).

% not_add_less2
tff(fact_977_add__less__mono1,axiom,
    ! [Ia: nat,J: nat,K3: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),J)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ia),K3)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K3)) ) ).

% add_less_mono1
tff(fact_978_trans__less__add1,axiom,
    ! [Ia: nat,J: nat,M2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),J)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),M2)) ) ).

% trans_less_add1
tff(fact_979_trans__less__add2,axiom,
    ! [Ia: nat,J: nat,M2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),J)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),J)) ) ).

% trans_less_add2
tff(fact_980_less__add__eq__less,axiom,
    ! [K3: nat,L: nat,M2: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K3),L)
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),L) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K3),N) )
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),N) ) ) ).

% less_add_eq_less
tff(fact_981_ent__disjI2__direct,axiom,
    ! [B2: assn,Aa2: assn] : aa(assn,$o,aa(assn,fun(assn,$o),entails,B2),aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),Aa2),B2)) ).

% ent_disjI2_direct
tff(fact_982_ent__disjI1__direct,axiom,
    ! [Aa2: assn,B2: assn] : aa(assn,$o,aa(assn,fun(assn,$o),entails,Aa2),aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),Aa2),B2)) ).

% ent_disjI1_direct
tff(fact_983_ent__disjI2_H,axiom,
    ! [Aa2: assn,C3: assn,B2: assn] :
      ( aa(assn,$o,aa(assn,fun(assn,$o),entails,Aa2),C3)
     => aa(assn,$o,aa(assn,fun(assn,$o),entails,Aa2),aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),B2),C3)) ) ).

% ent_disjI2'
tff(fact_984_ent__disjI1_H,axiom,
    ! [Aa2: assn,B2: assn,C3: assn] :
      ( aa(assn,$o,aa(assn,fun(assn,$o),entails,Aa2),B2)
     => aa(assn,$o,aa(assn,fun(assn,$o),entails,Aa2),aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),B2),C3)) ) ).

% ent_disjI1'
tff(fact_985_ent__disjI2,axiom,
    ! [Pa: assn,Qa: assn,R: assn] :
      ( aa(assn,$o,aa(assn,fun(assn,$o),entails,aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),Pa),Qa)),R)
     => aa(assn,$o,aa(assn,fun(assn,$o),entails,Qa),R) ) ).

% ent_disjI2
tff(fact_986_ent__disjI1,axiom,
    ! [Pa: assn,Qa: assn,R: assn] :
      ( aa(assn,$o,aa(assn,fun(assn,$o),entails,aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),Pa),Qa)),R)
     => aa(assn,$o,aa(assn,fun(assn,$o),entails,Pa),R) ) ).

% ent_disjI1
tff(fact_987_ent__disjE,axiom,
    ! [Aa2: assn,C3: assn,B2: assn] :
      ( aa(assn,$o,aa(assn,fun(assn,$o),entails,Aa2),C3)
     => ( aa(assn,$o,aa(assn,fun(assn,$o),entails,B2),C3)
       => aa(assn,$o,aa(assn,fun(assn,$o),entails,aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),Aa2),B2)),C3) ) ) ).

% ent_disjE
tff(fact_988_min__add__distrib__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [Xa: A,Y: A,Z2: A] : ( aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),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),Xa),Y)),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Z2)) ) ) ).

% min_add_distrib_right
tff(fact_989_min__add__distrib__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [Xa: 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),Xa),Y)),Z2) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Z2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),Y),Z2)) ) ) ).

% min_add_distrib_left
tff(fact_990_entt__disjE,axiom,
    ! [Aa2: assn,M5: assn,B2: assn] :
      ( entailst(Aa2,M5)
     => ( entailst(B2,M5)
       => entailst(aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),Aa2),B2),M5) ) ) ).

% entt_disjE
tff(fact_991_entt__disjD1,axiom,
    ! [Aa2: assn,B2: assn,C3: assn] :
      ( entailst(aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),Aa2),B2),C3)
     => entailst(Aa2,C3) ) ).

% entt_disjD1
tff(fact_992_entt__disjD2,axiom,
    ! [Aa2: assn,B2: assn,C3: assn] :
      ( entailst(aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),Aa2),B2),C3)
     => entailst(B2,C3) ) ).

% entt_disjD2
tff(fact_993_entt__disjI1_H,axiom,
    ! [Aa2: assn,B2: assn,C3: assn] :
      ( entailst(Aa2,B2)
     => entailst(Aa2,aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),B2),C3)) ) ).

% entt_disjI1'
tff(fact_994_entt__disjI2_H,axiom,
    ! [Aa2: assn,C3: assn,B2: assn] :
      ( entailst(Aa2,C3)
     => entailst(Aa2,aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),B2),C3)) ) ).

% entt_disjI2'
tff(fact_995_entt__disjI1__direct,axiom,
    ! [Aa2: assn,B2: assn] : entailst(Aa2,aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),Aa2),B2)) ).

% entt_disjI1_direct
tff(fact_996_entt__disjI2__direct,axiom,
    ! [B2: assn,Aa2: assn] : entailst(B2,aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),Aa2),B2)) ).

% entt_disjI2_direct
tff(fact_997_norm__assertion__simps_I6_J,axiom,
    ! [Xa: assn] : ( aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),Xa),bot_bot(assn)) = Xa ) ).

% norm_assertion_simps(6)
tff(fact_998_norm__assertion__simps_I5_J,axiom,
    ! [Xa: assn] : ( aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),bot_bot(assn)),Xa) = Xa ) ).

% norm_assertion_simps(5)
tff(fact_999_ex__has__greatest__nat__lemma,axiom,
    ! [A: $tType,Pa: fun(A,$o),K3: A,F2: fun(A,nat),N: nat] :
      ( aa(A,$o,Pa,K3)
     => ( ! [X: A] :
            ( aa(A,$o,Pa,X)
           => ? [Y3: A] :
                ( aa(A,$o,Pa,Y3)
                & ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,F2,Y3)),aa(A,nat,F2,X)) ) )
       => ? [Y2: A] :
            ( aa(A,$o,Pa,Y2)
            & ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,F2,Y2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,F2,K3)),N)) ) ) ) ).

% ex_has_greatest_nat_lemma
tff(fact_1000_add__mono__thms__linordered__field_I4_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [Ia: A,J: A,K3: A,L: A] :
          ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ia),J)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),K3),L) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Ia),K3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_1001_add__mono__thms__linordered__field_I3_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [Ia: A,J: A,K3: A,L: A] :
          ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ia),J)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),K3),L) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Ia),K3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_1002_add__le__less__mono,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [A3: A,B3: A,C2: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
         => ( aa(A,$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),B3),D3)) ) ) ) ).

% add_le_less_mono
tff(fact_1003_add__less__le__mono,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [A3: A,B3: A,C2: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
         => ( aa(A,$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),B3),D3)) ) ) ) ).

% add_less_le_mono
tff(fact_1004_diff__le__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B3)),C2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B3)) ) ) ).

% diff_le_eq
tff(fact_1005_le__diff__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,C2: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),B3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3)),C2) ) ) ).

% le_diff_eq
tff(fact_1006_add__le__imp__le__diff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Ia: A,K3: A,N: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Ia),K3)),N)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ia),aa(A,A,aa(A,fun(A,A),minus_minus(A),N),K3)) ) ) ).

% add_le_imp_le_diff
tff(fact_1007_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B3),A3)),A3) = B3 ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add
tff(fact_1008_add__le__add__imp__diff__le,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Ia: A,K3: A,N: A,J: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Ia),K3)),N)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),N),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),K3))
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Ia),K3)),N)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),N),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),K3))
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),N),K3)),J) ) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_1009_le__add__diff,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),C2)),A3)) ) ) ).

% le_add_diff
tff(fact_1010_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),B3),A3))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A3)),B3) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
tff(fact_1011_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),B3),A3)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B3)),A3) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
tff(fact_1012_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B3)),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),B3),A3)) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
tff(fact_1013_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B3),A3)),C2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),C2)),A3) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
tff(fact_1014_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),C2)),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B3),A3)),C2) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
tff(fact_1015_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),B3),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)),B3) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_diff_right
tff(fact_1016_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),minus_minus(A),B3),A3)) = B3 ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
tff(fact_1017_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
           => ( ( aa(A,A,aa(A,fun(A,A),minus_minus(A),B3),A3) = C2 )
            <=> ( B3 = 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_1018_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_1019_add__mono1,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
         => aa(A,$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),B3),one_one(A))) ) ) ).

% add_mono1
tff(fact_1020_diff__less__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B3)),C2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B3)) ) ) ).

% diff_less_eq
tff(fact_1021_less__diff__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,C2: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),B3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3)),C2) ) ) ).

% less_diff_eq
tff(fact_1022_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,B3: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B3)) = A3 ) ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_1023_square__diff__square__factored,axiom,
    ! [A: $tType] :
      ( comm_ring(A)
     => ! [Xa: 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),Xa),Xa)),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),Xa),Y)),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xa),Y)) ) ) ).

% square_diff_square_factored
tff(fact_1024_eq__add__iff2,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A3: A,E3: A,C2: A,B3: A,D3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),E3)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B3),E3)),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),B3),A3)),E3)),D3) ) ) ) ).

% eq_add_iff2
tff(fact_1025_eq__add__iff1,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A3: A,E3: A,C2: A,B3: A,D3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),E3)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B3),E3)),D3) )
        <=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B3)),E3)),C2) = D3 ) ) ) ).

% eq_add_iff1
tff(fact_1026_lift__Suc__mono__less,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(nat,A),N: nat,N6: nat] :
          ( ! [N5: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,F2,N5)),aa(nat,A,F2,aa(nat,nat,suc,N5)))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),N6)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,F2,N)),aa(nat,A,F2,N6)) ) ) ) ).

% lift_Suc_mono_less
tff(fact_1027_lift__Suc__mono__less__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(nat,A),N: nat,M2: nat] :
          ( ! [N5: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,F2,N5)),aa(nat,A,F2,aa(nat,nat,suc,N5)))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,F2,N)),aa(nat,A,F2,M2))
          <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),M2) ) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_1028_Suc__leI,axiom,
    ! [M2: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),N)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,M2)),N) ) ).

% Suc_leI
tff(fact_1029_Suc__le__eq,axiom,
    ! [M2: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,M2)),N)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),N) ) ).

% Suc_le_eq
tff(fact_1030_dec__induct,axiom,
    ! [Ia: nat,J: nat,Pa: fun(nat,$o)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ia),J)
     => ( aa(nat,$o,Pa,Ia)
       => ( ! [N5: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ia),N5)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N5),J)
               => ( aa(nat,$o,Pa,N5)
                 => aa(nat,$o,Pa,aa(nat,nat,suc,N5)) ) ) )
         => aa(nat,$o,Pa,J) ) ) ) ).

% dec_induct
tff(fact_1031_inc__induct,axiom,
    ! [Ia: nat,J: nat,Pa: fun(nat,$o)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ia),J)
     => ( aa(nat,$o,Pa,J)
       => ( ! [N5: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ia),N5)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N5),J)
               => ( aa(nat,$o,Pa,aa(nat,nat,suc,N5))
                 => aa(nat,$o,Pa,N5) ) ) )
         => aa(nat,$o,Pa,Ia) ) ) ) ).

% inc_induct
tff(fact_1032_Suc__le__lessD,axiom,
    ! [M2: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,M2)),N)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),N) ) ).

% Suc_le_lessD
tff(fact_1033_le__less__Suc__eq,axiom,
    ! [M2: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),N)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),aa(nat,nat,suc,M2))
      <=> ( N = M2 ) ) ) ).

% le_less_Suc_eq
tff(fact_1034_less__Suc__eq__le,axiom,
    ! [M2: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),aa(nat,nat,suc,N))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),N) ) ).

% less_Suc_eq_le
tff(fact_1035_less__eq__Suc__le,axiom,
    ! [N: nat,M2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),M2)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,N)),M2) ) ).

% less_eq_Suc_le
tff(fact_1036_le__imp__less__Suc,axiom,
    ! [M2: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),N)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),aa(nat,nat,suc,N)) ) ).

% le_imp_less_Suc
tff(fact_1037_nat__in__between__eq_I1_J,axiom,
    ! [A3: nat,B3: nat] :
      ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A3),B3)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),B3),aa(nat,nat,suc,A3)) )
    <=> ( B3 = aa(nat,nat,suc,A3) ) ) ).

% nat_in_between_eq(1)
tff(fact_1038_nat__in__between__eq_I2_J,axiom,
    ! [A3: nat,B3: nat] :
      ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A3),B3)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),B3),aa(nat,nat,suc,A3)) )
    <=> ( B3 = A3 ) ) ).

% nat_in_between_eq(2)
tff(fact_1039_Suc__diff__Suc,axiom,
    ! [N: nat,M2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),M2)
     => ( aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),aa(nat,nat,suc,N))) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),N) ) ) ).

% Suc_diff_Suc
tff(fact_1040_diff__less__Suc,axiom,
    ! [M2: nat,N: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),N)),aa(nat,nat,suc,M2)) ).

% diff_less_Suc
tff(fact_1041_mono__nat__linear__lb,axiom,
    ! [F2: fun(nat,nat),M2: nat,K3: nat] :
      ( ! [M6: nat,N5: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M6),N5)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,F2,M6)),aa(nat,nat,F2,N5)) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,F2,M2)),K3)),aa(nat,nat,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),K3))) ) ).

% mono_nat_linear_lb
tff(fact_1042_Suc__mult__less__cancel1,axiom,
    ! [K3: nat,M2: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K3)),M2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K3)),N))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),N) ) ).

% Suc_mult_less_cancel1
tff(fact_1043_less__diff__conv,axiom,
    ! [Ia: nat,J: nat,K3: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K3))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ia),K3)),J) ) ).

% less_diff_conv
tff(fact_1044_add__diff__inverse__nat,axiom,
    ! [M2: nat,N: nat] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),N)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),N)) = M2 ) ) ).

% add_diff_inverse_nat
tff(fact_1045_mlex__bound,axiom,
    ! [A3: nat,Aa2: nat,B3: nat,N7: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A3),Aa2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),B3),N7)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),N7)),B3)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Aa2),N7)) ) ) ).

% mlex_bound
tff(fact_1046_mlex__fst__decrI,axiom,
    ! [A3: nat,A6: nat,B3: nat,N7: nat,B6: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A3),A6)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),B3),N7)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),B6),N7)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),N7)),B3)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A6),N7)),B6)) ) ) ) ).

% mlex_fst_decrI
tff(fact_1047_mlex__snd__decrI,axiom,
    ! [A3: nat,A6: nat,B3: nat,B6: nat,N7: nat] :
      ( ( A3 = A6 )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),B3),B6)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),N7)),B3)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A6),N7)),B6)) ) ) ).

% mlex_snd_decrI
tff(fact_1048_mlex__leI,axiom,
    ! [A3: nat,A6: nat,B3: nat,B6: nat,N7: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A3),A6)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),B3),B6)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),N7)),B3)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A6),N7)),B6)) ) ) ).

% mlex_leI
tff(fact_1049_le__add__iff2,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A3: A,E3: A,C2: A,B3: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),E3)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B3),E3)),D3))
        <=> aa(A,$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),B3),A3)),E3)),D3)) ) ) ).

% le_add_iff2
tff(fact_1050_le__add__iff1,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A3: A,E3: A,C2: A,B3: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),E3)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B3),E3)),D3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B3)),E3)),C2)),D3) ) ) ).

% le_add_iff1
tff(fact_1051_less__add__iff2,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A3: A,E3: A,C2: A,B3: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),E3)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B3),E3)),D3))
        <=> aa(A,$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),B3),A3)),E3)),D3)) ) ) ).

% less_add_iff2
tff(fact_1052_less__add__iff1,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A3: A,E3: A,C2: A,B3: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),E3)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B3),E3)),D3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B3)),E3)),C2)),D3) ) ) ).

% less_add_iff1
tff(fact_1053_square__diff__one__factored,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Xa: A] : ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xa),Xa)),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),Xa),one_one(A))),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xa),one_one(A))) ) ) ).

% square_diff_one_factored
tff(fact_1054_less__diff__conv2,axiom,
    ! [K3: nat,J: nat,Ia: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K3),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K3)),Ia)
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ia),K3)) ) ) ).

% less_diff_conv2
tff(fact_1055_Suc__n__minus__m__eq,axiom,
    ! [M2: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),N)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),one_one(nat)),M2)
       => ( aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M2)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),one_one(nat))) ) ) ) ).

% Suc_n_minus_m_eq
tff(fact_1056_sorted__iff__nth__Suc,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xsa: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),Xsa)
        <=> ! [I2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,I2)),aa(list(A),nat,size_size(list(A)),Xsa))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,nth(A,Xsa),I2)),aa(nat,A,nth(A,Xsa),aa(nat,nat,suc,I2))) ) ) ) ).

% sorted_iff_nth_Suc
tff(fact_1057_rev__nth,axiom,
    ! [A: $tType,N: nat,Xsa: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xsa))
     => ( aa(nat,A,nth(A,aa(list(A),list(A),rev(A),Xsa)),N) = aa(nat,A,nth(A,Xsa),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xsa)),aa(nat,nat,suc,N))) ) ) ).

% rev_nth
tff(fact_1058_nat__less__add__iff1,axiom,
    ! [J: nat,Ia: nat,U: nat,M2: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),Ia)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Ia),U)),M2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),N))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ia),J)),U)),M2)),N) ) ) ).

% nat_less_add_iff1
tff(fact_1059_nat__less__add__iff2,axiom,
    ! [Ia: nat,J: nat,U: nat,M2: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ia),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Ia),U)),M2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),N))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),Ia)),U)),N)) ) ) ).

% nat_less_add_iff2
tff(fact_1060_discrete,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
        <=> aa(A,$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))),B3) ) ) ).

% discrete
tff(fact_1061_sorted__in__between,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ia: nat,J: nat,L: list(A),Xa: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),zero_zero(nat)),Ia)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),J)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),L))
             => ( sorted_wrt(A,ord_less_eq(A),L)
               => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,nth(A,L),Ia)),Xa)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),aa(nat,A,nth(A,L),J))
                   => ~ ! [K: nat] :
                          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ia),K)
                         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),J)
                           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,nth(A,L),K)),Xa)
                             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),aa(nat,A,nth(A,L),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),one_one(nat)))) ) ) ) ) ) ) ) ) ) ) ).

% sorted_in_between
tff(fact_1062_crossproduct__eq,axiom,
    ! [A: $tType] :
      ( semiri1453513574482234551roduct(A)
     => ! [W2: A,Y: A,Xa: 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),Xa),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),Xa),Y)) )
        <=> ( ( W2 = Xa )
            | ( Y = Z2 ) ) ) ) ).

% crossproduct_eq
tff(fact_1063_crossproduct__noteq,axiom,
    ! [A: $tType] :
      ( semiri1453513574482234551roduct(A)
     => ! [A3: A,B3: A,C2: A,D3: A] :
          ( ( ( A3 != B3 )
            & ( 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),B3),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),B3),C2)) ) ) ) ).

% crossproduct_noteq
tff(fact_1064_nth__step__trancl,axiom,
    ! [A: $tType,Xsa: list(A),R: set(product_prod(A,A)),N: nat,M2: nat] :
      ( ! [N5: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N5),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xsa)),one_one(nat)))
         => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(nat,A,nth(A,Xsa),aa(nat,nat,suc,N5))),aa(nat,A,nth(A,Xsa),N5)),R) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xsa))
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),N)
         => 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,Xsa),N)),aa(nat,A,nth(A,Xsa),M2)),transitive_trancl(A,R)) ) ) ) ).

% nth_step_trancl
tff(fact_1065_dbl__dec__def,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Xa: A] : ( neg_numeral_dbl_dec(A,Xa) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Xa)),one_one(A)) ) ) ).

% dbl_dec_def
tff(fact_1066_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_1067_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_1068_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_1069_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_1070_add__cancel__left__left,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [B3: A,A3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),A3) = A3 )
        <=> ( B3 = zero_zero(A) ) ) ) ).

% add_cancel_left_left
tff(fact_1071_add__cancel__left__right,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [A3: A,B3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3) = A3 )
        <=> ( B3 = zero_zero(A) ) ) ) ).

% add_cancel_left_right
tff(fact_1072_add__cancel__right__left,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [A3: A,B3: A] :
          ( ( A3 = aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),A3) )
        <=> ( B3 = zero_zero(A) ) ) ) ).

% add_cancel_right_left
tff(fact_1073_add__cancel__right__right,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [A3: A,B3: A] :
          ( ( A3 = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3) )
        <=> ( B3 = zero_zero(A) ) ) ) ).

% add_cancel_right_right
tff(fact_1074_add__eq__0__iff__both__eq__0,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Xa: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Y) = zero_zero(A) )
        <=> ( ( Xa = zero_zero(A) )
            & ( Y = zero_zero(A) ) ) ) ) ).

% add_eq_0_iff_both_eq_0
tff(fact_1075_zero__eq__add__iff__both__eq__0,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Xa: A,Y: A] :
          ( ( zero_zero(A) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Y) )
        <=> ( ( Xa = zero_zero(A) )
            & ( Y = zero_zero(A) ) ) ) ) ).

% zero_eq_add_iff_both_eq_0
tff(fact_1076_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_1077_mult__cancel__right,axiom,
    ! [A: $tType] :
      ( semiri6575147826004484403cancel(A)
     => ! [A3: A,C2: A,B3: 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),B3),C2) )
        <=> ( ( C2 = zero_zero(A) )
            | ( A3 = B3 ) ) ) ) ).

% mult_cancel_right
tff(fact_1078_mult__cancel__left,axiom,
    ! [A: $tType] :
      ( semiri6575147826004484403cancel(A)
     => ! [C2: A,A3: A,B3: 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),B3) )
        <=> ( ( C2 = zero_zero(A) )
            | ( A3 = B3 ) ) ) ) ).

% mult_cancel_left
tff(fact_1079_mult__eq__0__iff,axiom,
    ! [A: $tType] :
      ( semiri3467727345109120633visors(A)
     => ! [A3: A,B3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3) = zero_zero(A) )
        <=> ( ( A3 = zero_zero(A) )
            | ( B3 = zero_zero(A) ) ) ) ) ).

% mult_eq_0_iff
tff(fact_1080_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_1081_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_1082_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_1083_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_1084_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_1085_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_1086_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_1087_less__nat__zero__code,axiom,
    ! [N: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),zero_zero(nat)) ).

% less_nat_zero_code
tff(fact_1088_neq0__conv,axiom,
    ! [N: nat] :
      ( ( N != zero_zero(nat) )
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N) ) ).

% neq0_conv
tff(fact_1089_bot__nat__0_Onot__eq__extremum,axiom,
    ! [A3: nat] :
      ( ( A3 != zero_zero(nat) )
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),A3) ) ).

% bot_nat_0.not_eq_extremum
tff(fact_1090_trancl__single,axiom,
    ! [A: $tType,A3: A,B3: 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))),insert3(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B3)),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))),insert3(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B3)),bot_bot(set(product_prod(A,A)))) ) ).

% trancl_single
tff(fact_1091_add__le__same__cancel1,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),A3)),B3)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),zero_zero(A)) ) ) ).

% add_le_same_cancel1
tff(fact_1092_add__le__same__cancel2,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3)),B3)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),zero_zero(A)) ) ) ).

% add_le_same_cancel2
tff(fact_1093_le__add__same__cancel1,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B3) ) ) ).

% le_add_same_cancel1
tff(fact_1094_le__add__same__cancel2,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),A3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B3) ) ) ).

% le_add_same_cancel2
tff(fact_1095_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_1096_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_1097_diff__ge__0__iff__ge,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3) ) ) ).

% diff_ge_0_iff_ge
tff(fact_1098_zero__comp__diff__simps_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3) ) ) ).

% zero_comp_diff_simps(1)
tff(fact_1099_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_1100_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_1101_less__add__same__cancel2,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),A3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B3) ) ) ).

% less_add_same_cancel2
tff(fact_1102_less__add__same__cancel1,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B3) ) ) ).

% less_add_same_cancel1
tff(fact_1103_add__less__same__cancel2,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3)),B3)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A)) ) ) ).

% add_less_same_cancel2
tff(fact_1104_add__less__same__cancel1,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),A3)),B3)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A)) ) ) ).

% add_less_same_cancel1
tff(fact_1105_diff__gt__0__iff__gt,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A3) ) ) ).

% diff_gt_0_iff_gt
tff(fact_1106_zero__comp__diff__simps_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A3) ) ) ).

% zero_comp_diff_simps(2)
tff(fact_1107_mult__cancel__left1,axiom,
    ! [A: $tType] :
      ( ring_15535105094025558882visors(A)
     => ! [C2: A,B3: A] :
          ( ( C2 = aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3) )
        <=> ( ( C2 = zero_zero(A) )
            | ( B3 = one_one(A) ) ) ) ) ).

% mult_cancel_left1
tff(fact_1108_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_1109_mult__cancel__right1,axiom,
    ! [A: $tType] :
      ( ring_15535105094025558882visors(A)
     => ! [C2: A,B3: A] :
          ( ( C2 = aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2) )
        <=> ( ( C2 = zero_zero(A) )
            | ( B3 = one_one(A) ) ) ) ) ).

% mult_cancel_right1
tff(fact_1110_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_1111_diff__add__zero,axiom,
    ! [A: $tType] :
      ( comm_monoid_diff(A)
     => ! [A3: A,B3: 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),B3)) = zero_zero(A) ) ) ).

% diff_add_zero
tff(fact_1112_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_1113_less__Suc0,axiom,
    ! [N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),aa(nat,nat,suc,zero_zero(nat)))
    <=> ( N = zero_zero(nat) ) ) ).

% less_Suc0
tff(fact_1114_zero__less__Suc,axiom,
    ! [N: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,suc,N)) ).

% zero_less_Suc
tff(fact_1115_add__gr__0,axiom,
    ! [M2: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),N))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M2)
        | aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N) ) ) ).

% add_gr_0
tff(fact_1116_zero__less__diff,axiom,
    ! [N: nat,M2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M2))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),N) ) ).

% zero_less_diff
tff(fact_1117_nat__mult__less__cancel__disj,axiom,
    ! [K3: nat,M2: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K3),M2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K3),N))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K3)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),N) ) ) ).

% nat_mult_less_cancel_disj
tff(fact_1118_mult__less__cancel2,axiom,
    ! [M2: nat,K3: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M2),K3)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),K3))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K3)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),N) ) ) ).

% mult_less_cancel2
tff(fact_1119_nat__0__less__mult__iff,axiom,
    ! [M2: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M2),N))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M2)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N) ) ) ).

% nat_0_less_mult_iff
tff(fact_1120_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_1121_slice__complete,axiom,
    ! [A: $tType,Xsa: list(A)] : ( slice(A,zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xsa),Xsa) = Xsa ) ).

% slice_complete
tff(fact_1122_Suc__pred,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),aa(nat,nat,suc,zero_zero(nat)))) = N ) ) ).

% Suc_pred
tff(fact_1123_nat__mult__le__cancel__disj,axiom,
    ! [K3: nat,M2: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K3),M2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K3),N))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K3)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),N) ) ) ).

% nat_mult_le_cancel_disj
tff(fact_1124_mult__le__cancel2,axiom,
    ! [M2: nat,K3: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M2),K3)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),K3))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K3)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),N) ) ) ).

% mult_le_cancel2
tff(fact_1125_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_1126_assn__aci_I8_J,axiom,
    ! [Xa: assn,Y: assn] : ( aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),Xa),aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),Xa),Y)) = aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),Xa),Y) ) ).

% assn_aci(8)
tff(fact_1127_assn__aci_I7_J,axiom,
    ! [Xa: assn,Y: assn,Z2: assn] : ( aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),Xa),aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),Y),Z2)) = aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),Y),aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),Xa),Z2)) ) ).

% assn_aci(7)
tff(fact_1128_assn__aci_I5_J,axiom,
    ! [Xa: assn,Y: assn] : ( aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),Xa),Y) = aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),Y),Xa) ) ).

% assn_aci(5)
tff(fact_1129_norm__assertion__simps_I32_J,axiom,
    ! [Xa: assn] : ( aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),Xa),Xa) = Xa ) ).

% norm_assertion_simps(32)
tff(fact_1130_norm__assertion__simps_I15_J,axiom,
    ! [Xa: assn,Y: assn,Z2: assn] : ( aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),Xa),Y)),Z2) = aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),Xa),aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),Y),Z2)) ) ).

% norm_assertion_simps(15)
tff(fact_1131_zero__reorient,axiom,
    ! [A: $tType] :
      ( zero(A)
     => ! [Xa: A] :
          ( ( zero_zero(A) = Xa )
        <=> ( Xa = zero_zero(A) ) ) ) ).

% zero_reorient
tff(fact_1132_trancl__mono__mp,axiom,
    ! [A: $tType,U2: set(product_prod(A,A)),V: set(product_prod(A,A)),Xa: product_prod(A,A)] :
      ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),U2),V)
     => ( member(product_prod(A,A),Xa,transitive_trancl(A,U2))
       => member(product_prod(A,A),Xa,transitive_trancl(A,V)) ) ) ).

% trancl_mono_mp
tff(fact_1133_trancl__sub,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),transitive_trancl(A,R)) ).

% trancl_sub
tff(fact_1134_trancl__sub__insert__trancl,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),Xa: product_prod(A,A)] : aa(set(product_prod(A,A)),$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,R)),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))),insert3(product_prod(A,A)),Xa),R))) ).

% trancl_sub_insert_trancl
tff(fact_1135_zero__le,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Xa: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xa) ) ).

% zero_le
tff(fact_1136_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_1137_gr__implies__not__zero,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [M2: A,N: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),M2),N)
         => ( N != zero_zero(A) ) ) ) ).

% gr_implies_not_zero
tff(fact_1138_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_1139_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_1140_less__numeral__extra_I3_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),zero_zero(A)) ) ).

% less_numeral_extra(3)
tff(fact_1141_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_1142_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_1143_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_1144_mult__right__cancel,axiom,
    ! [A: $tType] :
      ( semiri6575147826004484403cancel(A)
     => ! [C2: A,A3: A,B3: 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),B3),C2) )
          <=> ( A3 = B3 ) ) ) ) ).

% mult_right_cancel
tff(fact_1145_mult__left__cancel,axiom,
    ! [A: $tType] :
      ( semiri6575147826004484403cancel(A)
     => ! [C2: A,A3: A,B3: 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),B3) )
          <=> ( A3 = B3 ) ) ) ) ).

% mult_left_cancel
tff(fact_1146_no__zero__divisors,axiom,
    ! [A: $tType] :
      ( semiri3467727345109120633visors(A)
     => ! [A3: A,B3: A] :
          ( ( A3 != zero_zero(A) )
         => ( ( B3 != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3) != zero_zero(A) ) ) ) ) ).

% no_zero_divisors
tff(fact_1147_divisors__zero,axiom,
    ! [A: $tType] :
      ( semiri3467727345109120633visors(A)
     => ! [A3: A,B3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3) = zero_zero(A) )
         => ( ( A3 = zero_zero(A) )
            | ( B3 = zero_zero(A) ) ) ) ) ).

% divisors_zero
tff(fact_1148_mult__not__zero,axiom,
    ! [A: $tType] :
      ( mult_zero(A)
     => ! [A3: A,B3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3) != zero_zero(A) )
         => ( ( A3 != zero_zero(A) )
            & ( B3 != zero_zero(A) ) ) ) ) ).

% mult_not_zero
tff(fact_1149_zero__neq__one,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ( zero_zero(A) != one_one(A) ) ) ).

% zero_neq_one
tff(fact_1150_eq__iff__diff__eq__0,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B3: A] :
          ( ( A3 = B3 )
        <=> ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B3) = zero_zero(A) ) ) ) ).

% eq_iff_diff_eq_0
tff(fact_1151_infinite__descent0__measure,axiom,
    ! [A: $tType,V: fun(A,nat),Pa: fun(A,$o),Xa: A] :
      ( ! [X: A] :
          ( ( aa(A,nat,V,X) = zero_zero(nat) )
         => aa(A,$o,Pa,X) )
     => ( ! [X: A] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(A,nat,V,X))
           => ( ~ aa(A,$o,Pa,X)
             => ? [Y3: A] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,V,Y3)),aa(A,nat,V,X))
                  & ~ aa(A,$o,Pa,Y3) ) ) )
       => aa(A,$o,Pa,Xa) ) ) ).

% infinite_descent0_measure
tff(fact_1152_infinite__descent0,axiom,
    ! [Pa: fun(nat,$o),N: nat] :
      ( aa(nat,$o,Pa,zero_zero(nat))
     => ( ! [N5: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N5)
           => ( ~ aa(nat,$o,Pa,N5)
             => ? [M3: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M3),N5)
                  & ~ aa(nat,$o,Pa,M3) ) ) )
       => aa(nat,$o,Pa,N) ) ) ).

% infinite_descent0
tff(fact_1153_gr__implies__not0,axiom,
    ! [M2: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),N)
     => ( N != zero_zero(nat) ) ) ).

% gr_implies_not0
tff(fact_1154_less__zeroE,axiom,
    ! [N: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),zero_zero(nat)) ).

% less_zeroE
tff(fact_1155_not__less0,axiom,
    ! [N: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),zero_zero(nat)) ).

% not_less0
tff(fact_1156_not__gr0,axiom,
    ! [N: nat] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
    <=> ( N = zero_zero(nat) ) ) ).

% not_gr0
tff(fact_1157_gr0I,axiom,
    ! [N: nat] :
      ( ( N != zero_zero(nat) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N) ) ).

% gr0I
tff(fact_1158_bot__nat__0_Oextremum__strict,axiom,
    ! [A3: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A3),zero_zero(nat)) ).

% bot_nat_0.extremum_strict
tff(fact_1159_add__scale__eq__noteq,axiom,
    ! [A: $tType] :
      ( semiri1453513574482234551roduct(A)
     => ! [R2: A,A3: A,B3: A,C2: A,D3: A] :
          ( ( R2 != zero_zero(A) )
         => ( ( ( A3 = B3 )
              & ( 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),B3),aa(A,A,aa(A,fun(A,A),times_times(A),R2),D3)) ) ) ) ) ).

% add_scale_eq_noteq
tff(fact_1160_nat__mult__eq__cancel1,axiom,
    ! [K3: nat,M2: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K3)
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K3),M2) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K3),N) )
      <=> ( M2 = N ) ) ) ).

% nat_mult_eq_cancel1
tff(fact_1161_nat__mult__less__cancel1,axiom,
    ! [K3: nat,M2: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K3)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K3),M2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K3),N))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),N) ) ) ).

% nat_mult_less_cancel1
tff(fact_1162_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_1163_add__nonpos__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),zero_zero(A))
         => ( aa(A,$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),Xa),Y) = zero_zero(A) )
            <=> ( ( Xa = zero_zero(A) )
                & ( Y = zero_zero(A) ) ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_1164_add__nonneg__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xa)
         => ( aa(A,$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),Xa),Y) = zero_zero(A) )
            <=> ( ( Xa = zero_zero(A) )
                & ( Y = zero_zero(A) ) ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_1165_add__nonpos__nonpos,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3)),zero_zero(A)) ) ) ) ).

% add_nonpos_nonpos
tff(fact_1166_add__nonneg__nonneg,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B3)
           => aa(A,$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),B3)) ) ) ) ).

% add_nonneg_nonneg
tff(fact_1167_add__increasing2,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [C2: A,B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)) ) ) ) ).

% add_increasing2
tff(fact_1168_add__decreasing2,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [C2: A,A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)),B3) ) ) ) ).

% add_decreasing2
tff(fact_1169_add__increasing,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)) ) ) ) ).

% add_increasing
tff(fact_1170_add__decreasing,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: A,C2: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),B3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)),B3) ) ) ) ).

% add_decreasing
tff(fact_1171_mult__mono,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A3: A,B3: A,C2: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),D3)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B3)
             => ( aa(A,$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),B3),D3)) ) ) ) ) ) ).

% mult_mono
tff(fact_1172_mult__mono_H,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A3: A,B3: A,C2: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),D3)
           => ( aa(A,$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),B3),D3)) ) ) ) ) ) ).

% mult_mono'
tff(fact_1173_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_1174_split__mult__pos__le,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A3: A,B3: A] :
          ( ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B3) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),zero_zero(A)) ) )
         => aa(A,$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),B3)) ) ) ).

% split_mult_pos_le
tff(fact_1175_mult__left__mono__neg,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [B3: A,A3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
           => aa(A,$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),B3)) ) ) ) ).

% mult_left_mono_neg
tff(fact_1176_mult__nonpos__nonpos,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),zero_zero(A))
           => aa(A,$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),B3)) ) ) ) ).

% mult_nonpos_nonpos
tff(fact_1177_mult__left__mono,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
           => aa(A,$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),B3)) ) ) ) ).

% mult_left_mono
tff(fact_1178_mult__right__mono__neg,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [B3: A,A3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
           => aa(A,$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),B3),C2)) ) ) ) ).

% mult_right_mono_neg
tff(fact_1179_mult__right__mono,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
           => aa(A,$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),B3),C2)) ) ) ) ).

% mult_right_mono
tff(fact_1180_mult__le__0__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)),zero_zero(A))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),zero_zero(A)) )
            | ( aa(A,$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)),B3) ) ) ) ) ).

% mult_le_0_iff
tff(fact_1181_split__mult__neg__le,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A3: A,B3: A] :
          ( ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),zero_zero(A)) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B3) ) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)),zero_zero(A)) ) ) ).

% split_mult_neg_le
tff(fact_1182_mult__nonneg__nonneg,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B3)
           => aa(A,$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),B3)) ) ) ) ).

% mult_nonneg_nonneg
tff(fact_1183_mult__nonneg__nonpos,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)),zero_zero(A)) ) ) ) ).

% mult_nonneg_nonpos
tff(fact_1184_mult__nonpos__nonneg,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)),zero_zero(A)) ) ) ) ).

% mult_nonpos_nonneg
tff(fact_1185_mult__nonneg__nonpos2,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),B3),A3)),zero_zero(A)) ) ) ) ).

% mult_nonneg_nonpos2
tff(fact_1186_zero__le__mult__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B3) )
            | ( aa(A,$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),B3),zero_zero(A)) ) ) ) ) ).

% zero_le_mult_iff
tff(fact_1187_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: $tType] :
      ( ordere2520102378445227354miring(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
           => aa(A,$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),B3)) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_1188_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_1189_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_1190_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_1191_le__iff__diff__le__0,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B3)),zero_zero(A)) ) ) ).

% le_iff_diff_le_0
tff(fact_1192_add__less__zeroD,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Y)),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),zero_zero(A))
            | aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),zero_zero(A)) ) ) ) ).

% add_less_zeroD
tff(fact_1193_add__neg__neg,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3)),zero_zero(A)) ) ) ) ).

% add_neg_neg
tff(fact_1194_add__pos__pos,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3)) ) ) ) ).

% add_pos_pos
tff(fact_1195_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
         => ~ ! [C5: A] :
                ( ( B3 = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C5) )
               => ( C5 = zero_zero(A) ) ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
tff(fact_1196_pos__add__strict,axiom,
    ! [A: $tType] :
      ( strict7427464778891057005id_add(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)) ) ) ) ).

% pos_add_strict
tff(fact_1197_mult__neg__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)) ) ) ) ).

% mult_neg_neg
tff(fact_1198_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_1199_mult__less__0__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)),zero_zero(A))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),zero_zero(A)) )
            | ( aa(A,$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)),B3) ) ) ) ) ).

% mult_less_0_iff
tff(fact_1200_mult__neg__pos,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)),zero_zero(A)) ) ) ) ).

% mult_neg_pos
tff(fact_1201_mult__pos__neg,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)),zero_zero(A)) ) ) ) ).

% mult_pos_neg
tff(fact_1202_mult__pos__pos,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)) ) ) ) ).

% mult_pos_pos
tff(fact_1203_mult__pos__neg2,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),B3),A3)),zero_zero(A)) ) ) ) ).

% mult_pos_neg2
tff(fact_1204_zero__less__mult__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B3) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),zero_zero(A)) ) ) ) ) ).

% zero_less_mult_iff
tff(fact_1205_zero__less__mult__pos,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3))
         => ( aa(A,$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)),B3) ) ) ) ).

% zero_less_mult_pos
tff(fact_1206_zero__less__mult__pos2,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),A3))
         => ( aa(A,$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)),B3) ) ) ) ).

% zero_less_mult_pos2
tff(fact_1207_mult__less__cancel__left__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A3) ) ) ) ).

% mult_less_cancel_left_neg
tff(fact_1208_mult__less__cancel__left__pos,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3) ) ) ) ).

% mult_less_cancel_left_pos
tff(fact_1209_mult__strict__left__mono__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [B3: A,A3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A3)
         => ( aa(A,$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),B3)) ) ) ) ).

% mult_strict_left_mono_neg
tff(fact_1210_mult__strict__left__mono,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
         => ( aa(A,$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),B3)) ) ) ) ).

% mult_strict_left_mono
tff(fact_1211_mult__less__cancel__left__disj,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A3) ) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_1212_mult__strict__right__mono__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [B3: A,A3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A3)
         => ( aa(A,$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),B3),C2)) ) ) ) ).

% mult_strict_right_mono_neg
tff(fact_1213_mult__strict__right__mono,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
         => ( aa(A,$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),B3),C2)) ) ) ) ).

% mult_strict_right_mono
tff(fact_1214_mult__less__cancel__right__disj,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A3: A,C2: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A3) ) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_1215_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: $tType] :
      ( linord2810124833399127020strict(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
         => ( aa(A,$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),B3)) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_1216_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_1217_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_1218_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_1219_less__iff__diff__less__0,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B3)),zero_zero(A)) ) ) ).

% less_iff_diff_less_0
tff(fact_1220_nat__mult__le__cancel1,axiom,
    ! [K3: nat,M2: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K3)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K3),M2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K3),N))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),N) ) ) ).

% nat_mult_le_cancel1
tff(fact_1221_Ex__less__Suc2,axiom,
    ! [N: nat,Pa: fun(nat,$o)] :
      ( ? [I2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(nat,nat,suc,N))
          & aa(nat,$o,Pa,I2) )
    <=> ( aa(nat,$o,Pa,zero_zero(nat))
        | ? [I2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),N)
            & aa(nat,$o,Pa,aa(nat,nat,suc,I2)) ) ) ) ).

% Ex_less_Suc2
tff(fact_1222_gr0__conv__Suc,axiom,
    ! [N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
    <=> ? [M: nat] : ( N = aa(nat,nat,suc,M) ) ) ).

% gr0_conv_Suc
tff(fact_1223_All__less__Suc2,axiom,
    ! [N: nat,Pa: fun(nat,$o)] :
      ( ! [I2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(nat,nat,suc,N))
         => aa(nat,$o,Pa,I2) )
    <=> ( aa(nat,$o,Pa,zero_zero(nat))
        & ! [I2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),N)
           => aa(nat,$o,Pa,aa(nat,nat,suc,I2)) ) ) ) ).

% All_less_Suc2
tff(fact_1224_gr0__implies__Suc,axiom,
    ! [N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
     => ? [M6: nat] : ( N = aa(nat,nat,suc,M6) ) ) ).

% gr0_implies_Suc
tff(fact_1225_less__Suc__eq__0__disj,axiom,
    ! [M2: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),aa(nat,nat,suc,N))
    <=> ( ( M2 = zero_zero(nat) )
        | ? [J2: nat] :
            ( ( M2 = aa(nat,nat,suc,J2) )
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J2),N) ) ) ) ).

% less_Suc_eq_0_disj
tff(fact_1226_nat__compl__induct,axiom,
    ! [Pa: fun(nat,$o),N: nat] :
      ( aa(nat,$o,Pa,zero_zero(nat))
     => ( ! [N5: nat] :
            ( ! [Nn: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nn),N5)
               => aa(nat,$o,Pa,Nn) )
           => aa(nat,$o,Pa,aa(nat,nat,suc,N5)) )
       => aa(nat,$o,Pa,N) ) ) ).

% nat_compl_induct
tff(fact_1227_nat__compl__induct_H,axiom,
    ! [Pa: fun(nat,$o),N: nat] :
      ( aa(nat,$o,Pa,zero_zero(nat))
     => ( ! [N5: nat] :
            ( ! [Nn: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nn),N5)
               => aa(nat,$o,Pa,Nn) )
           => aa(nat,$o,Pa,aa(nat,nat,suc,N5)) )
       => aa(nat,$o,Pa,N) ) ) ).

% nat_compl_induct'
tff(fact_1228_ex__least__nat__le,axiom,
    ! [Pa: fun(nat,$o),N: nat] :
      ( aa(nat,$o,Pa,N)
     => ( ~ aa(nat,$o,Pa,zero_zero(nat))
       => ? [K: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),N)
            & ! [I4: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),K)
               => ~ aa(nat,$o,Pa,I4) )
            & aa(nat,$o,Pa,K) ) ) ) ).

% ex_least_nat_le
tff(fact_1229_less__imp__add__positive,axiom,
    ! [Ia: nat,J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),J)
     => ? [K: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
          & ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ia),K) = J ) ) ) ).

% less_imp_add_positive
tff(fact_1230_Suc__to__right,axiom,
    ! [N: nat,M2: nat] :
      ( ( aa(nat,nat,suc,N) = M2 )
     => ( N = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),aa(nat,nat,suc,zero_zero(nat))) ) ) ).

% Suc_to_right
tff(fact_1231_diff__less,axiom,
    ! [N: nat,M2: 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)),M2)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),N)),M2) ) ) ).

% diff_less
tff(fact_1232_mult__less__mono1,axiom,
    ! [Ia: nat,J: nat,K3: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K3)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Ia),K3)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),K3)) ) ) ).

% mult_less_mono1
tff(fact_1233_mult__less__mono2,axiom,
    ! [Ia: nat,J: nat,K3: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K3)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K3),Ia)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K3),J)) ) ) ).

% mult_less_mono2
tff(fact_1234_nat__geq__1__eq__neqz,axiom,
    ! [Xa: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),Xa)
    <=> ( Xa != zero_zero(nat) ) ) ).

% nat_geq_1_eq_neqz
tff(fact_1235_add__neg__nonpos,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3)),zero_zero(A)) ) ) ) ).

% add_neg_nonpos
tff(fact_1236_add__nonneg__pos,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3)) ) ) ) ).

% add_nonneg_pos
tff(fact_1237_add__nonpos__neg,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3)),zero_zero(A)) ) ) ) ).

% add_nonpos_neg
tff(fact_1238_add__pos__nonneg,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3)) ) ) ) ).

% add_pos_nonneg
tff(fact_1239_add__strict__increasing,axiom,
    ! [A: $tType] :
      ( ordere8940638589300402666id_add(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)) ) ) ) ).

% add_strict_increasing
tff(fact_1240_add__strict__increasing2,axiom,
    ! [A: $tType] :
      ( ordere8940638589300402666id_add(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)) ) ) ) ).

% add_strict_increasing2
tff(fact_1241_mult__le__cancel__left,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3) )
            & ( aa(A,$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),B3),A3) ) ) ) ) ).

% mult_le_cancel_left
tff(fact_1242_mult__le__cancel__right,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A3: A,C2: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3) )
            & ( aa(A,$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),B3),A3) ) ) ) ) ).

% mult_le_cancel_right
tff(fact_1243_mult__left__less__imp__less,axiom,
    ! [A: $tType] :
      ( linordered_semiring(A)
     => ! [C2: A,A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3))
         => ( aa(A,$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),B3) ) ) ) ).

% mult_left_less_imp_less
tff(fact_1244_mult__strict__mono,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B3: A,C2: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),D3)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B3)
             => ( aa(A,$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),B3),D3)) ) ) ) ) ) ).

% mult_strict_mono
tff(fact_1245_mult__less__cancel__left,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3) )
            & ( aa(A,$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),B3),A3) ) ) ) ) ).

% mult_less_cancel_left
tff(fact_1246_mult__right__less__imp__less,axiom,
    ! [A: $tType] :
      ( linordered_semiring(A)
     => ! [A3: A,C2: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2))
         => ( aa(A,$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),B3) ) ) ) ).

% mult_right_less_imp_less
tff(fact_1247_mult__strict__mono_H,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B3: A,C2: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
         => ( aa(A,$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),B3),D3)) ) ) ) ) ) ).

% mult_strict_mono'
tff(fact_1248_mult__less__cancel__right,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A3: A,C2: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3) )
            & ( aa(A,$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),B3),A3) ) ) ) ) ).

% mult_less_cancel_right
tff(fact_1249_mult__le__cancel__left__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3) ) ) ) ).

% mult_le_cancel_left_neg
tff(fact_1250_mult__le__cancel__left__pos,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3) ) ) ) ).

% mult_le_cancel_left_pos
tff(fact_1251_mult__left__le__imp__le,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [C2: A,A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3))
         => ( aa(A,$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),B3) ) ) ) ).

% mult_left_le_imp_le
tff(fact_1252_mult__right__le__imp__le,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,C2: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2))
         => ( aa(A,$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),B3) ) ) ) ).

% mult_right_le_imp_le
tff(fact_1253_mult__le__less__imp__less,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B3: A,C2: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
         => ( aa(A,$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),B3),D3)) ) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_1254_mult__less__le__imp__less,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B3: A,C2: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
         => ( aa(A,$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),B3),D3)) ) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_1255_sum__squares__ge__zero,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [Xa: A,Y: A] : aa(A,$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),Xa),Xa)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y))) ) ).

% sum_squares_ge_zero
tff(fact_1256_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_1257_mult__le__one,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),one_one(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B3)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),one_one(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)),one_one(A)) ) ) ) ) ).

% mult_le_one
tff(fact_1258_mult__right__le__one__le,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xa)
         => ( aa(A,$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),Xa),Y)),Xa) ) ) ) ) ).

% mult_right_le_one_le
tff(fact_1259_mult__left__le__one__le,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xa)
         => ( aa(A,$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),Xa)),Xa) ) ) ) ) ).

% mult_left_le_one_le
tff(fact_1260_not__sum__squares__lt__zero,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [Xa: A,Y: A] : ~ aa(A,$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),Xa),Xa)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y))),zero_zero(A)) ) ).

% not_sum_squares_lt_zero
tff(fact_1261_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_1262_ex__least__nat__less,axiom,
    ! [Pa: fun(nat,$o),N: nat] :
      ( aa(nat,$o,Pa,N)
     => ( ~ aa(nat,$o,Pa,zero_zero(nat))
       => ? [K: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),N)
            & ! [I4: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I4),K)
               => ~ aa(nat,$o,Pa,I4) )
            & aa(nat,$o,Pa,aa(nat,nat,suc,K)) ) ) ) ).

% ex_least_nat_less
tff(fact_1263_length__pos__if__in__set,axiom,
    ! [A: $tType,Xa: A,Xsa: list(A)] :
      ( member(A,Xa,aa(list(A),set(A),set2(A),Xsa))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(list(A),nat,size_size(list(A)),Xsa)) ) ).

% length_pos_if_in_set
tff(fact_1264_diff__Suc__less,axiom,
    ! [N: nat,Ia: 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),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,suc,Ia))),N) ) ).

% diff_Suc_less
tff(fact_1265_n__less__n__mult__m,axiom,
    ! [N: nat,M2: 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),aa(nat,nat,suc,zero_zero(nat))),M2)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),M2)) ) ) ).

% n_less_n_mult_m
tff(fact_1266_n__less__m__mult__n,axiom,
    ! [N: nat,M2: 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),aa(nat,nat,suc,zero_zero(nat))),M2)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M2),N)) ) ) ).

% n_less_m_mult_n
tff(fact_1267_one__less__mult,axiom,
    ! [N: nat,M2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),N)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),M2)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M2),N)) ) ) ).

% one_less_mult
tff(fact_1268_nat__induct__non__zero,axiom,
    ! [N: nat,Pa: fun(nat,$o)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
     => ( aa(nat,$o,Pa,one_one(nat))
       => ( ! [N5: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N5)
             => ( aa(nat,$o,Pa,N5)
               => aa(nat,$o,Pa,aa(nat,nat,suc,N5)) ) )
         => aa(nat,$o,Pa,N) ) ) ) ).

% nat_induct_non_zero
tff(fact_1269_nat__diff__split,axiom,
    ! [Pa: fun(nat,$o),A3: nat,B3: nat] :
      ( aa(nat,$o,Pa,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),A3),B3))
    <=> ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A3),B3)
         => aa(nat,$o,Pa,zero_zero(nat)) )
        & ! [D6: nat] :
            ( ( A3 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),B3),D6) )
           => aa(nat,$o,Pa,D6) ) ) ) ).

% nat_diff_split
tff(fact_1270_nat__diff__split__asm,axiom,
    ! [Pa: fun(nat,$o),A3: nat,B3: nat] :
      ( aa(nat,$o,Pa,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),A3),B3))
    <=> ~ ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A3),B3)
            & ~ aa(nat,$o,Pa,zero_zero(nat)) )
          | ? [D6: nat] :
              ( ( A3 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),B3),D6) )
              & ~ aa(nat,$o,Pa,D6) ) ) ) ).

% nat_diff_split_asm
tff(fact_1271_mult__le__cancel__left1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),B3) )
            & ( aa(A,$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),B3),one_one(A)) ) ) ) ) ).

% mult_le_cancel_left1
tff(fact_1272_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_1273_mult__le__cancel__right1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),B3) )
            & ( aa(A,$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),B3),one_one(A)) ) ) ) ) ).

% mult_le_cancel_right1
tff(fact_1274_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_1275_mult__less__cancel__left1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),B3) )
            & ( aa(A,$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),B3),one_one(A)) ) ) ) ) ).

% mult_less_cancel_left1
tff(fact_1276_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_1277_mult__less__cancel__right1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),B3) )
            & ( aa(A,$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),B3),one_one(A)) ) ) ) ) ).

% mult_less_cancel_right1
tff(fact_1278_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_1279_convex__bound__le,axiom,
    ! [A: $tType] :
      ( linord6961819062388156250ring_1(A)
     => ! [Xa: A,A3: A,Y: A,U: A,V2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),A3)
         => ( aa(A,$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,$o),ord_less_eq(A),zero_zero(A)),V2)
               => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),U),V2) = one_one(A) )
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),U),Xa)),aa(A,A,aa(A,fun(A,A),times_times(A),V2),Y))),A3) ) ) ) ) ) ) ).

% convex_bound_le
tff(fact_1280_nz__le__conv__less,axiom,
    ! [K3: nat,M2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K3)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K3),M2)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K3),aa(nat,nat,suc,zero_zero(nat)))),M2) ) ) ).

% nz_le_conv_less
tff(fact_1281_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_1282_Suc__diff__eq__diff__pred,axiom,
    ! [N: nat,M2: 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,M2)),N) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))) ) ) ).

% Suc_diff_eq_diff_pred
tff(fact_1283_convex__bound__lt,axiom,
    ! [A: $tType] :
      ( linord715952674999750819strict(A)
     => ! [Xa: A,A3: A,Y: A,U: A,V2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),A3)
         => ( aa(A,$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,$o),ord_less_eq(A),zero_zero(A)),V2)
               => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),U),V2) = one_one(A) )
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),U),Xa)),aa(A,A,aa(A,fun(A,A),times_times(A),V2),Y))),A3) ) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_1284_field__le__mult__one__interval,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A,Y: A] :
          ( ! [Z3: A] :
              ( aa(A,$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,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z3),Xa)),Y) ) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y) ) ) ).

% field_le_mult_one_interval
tff(fact_1285_sum__squares__gt__zero__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$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),Xa),Xa)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y)))
        <=> ( ( Xa != zero_zero(A) )
            | ( Y != zero_zero(A) ) ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_1286_sum__squares__le__zero__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$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),Xa),Xa)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y))),zero_zero(A))
        <=> ( ( Xa = zero_zero(A) )
            & ( Y = zero_zero(A) ) ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_1287_mult__le__cancel__iff1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z2: A,Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Z2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xa),Z2)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y) ) ) ) ).

% mult_le_cancel_iff1
tff(fact_1288_mult__le__cancel__iff2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z2: A,Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Z2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z2),Xa)),aa(A,A,aa(A,fun(A,A),times_times(A),Z2),Y))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y) ) ) ) ).

% mult_le_cancel_iff2
tff(fact_1289_field__le__epsilon,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A,Y: A] :
          ( ! [E2: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),E2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(A,A,aa(A,fun(A,A),plus_plus(A),Y),E2)) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y) ) ) ).

% field_le_epsilon
tff(fact_1290_divides__aux__eq,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Q2: A,R2: A] :
          ( unique5940410009612947441es_aux(A,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q2),R2))
        <=> ( R2 = zero_zero(A) ) ) ) ).

% divides_aux_eq
tff(fact_1291_sum__squares__eq__zero__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [Xa: 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),Xa),Xa)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y)) = zero_zero(A) )
        <=> ( ( Xa = zero_zero(A) )
            & ( Y = zero_zero(A) ) ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_1292_mult__less__iff1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z2: A,Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Z2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xa),Z2)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y) ) ) ) ).

% mult_less_iff1
tff(fact_1293_linordered__field__no__lb,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X4: A] :
        ? [Y2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y2),X4) ) ).

% linordered_field_no_lb
tff(fact_1294_linordered__field__no__ub,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X4: A] :
        ? [X_1: A] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),X_1) ) ).

% linordered_field_no_ub
tff(fact_1295_converse__trancl__induct,axiom,
    ! [A: $tType,A3: A,B3: A,R2: set(product_prod(A,A)),Pa: fun(A,$o)] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B3),transitive_trancl(A,R2))
     => ( ! [Y2: A] :
            ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),B3),R2)
           => aa(A,$o,Pa,Y2) )
       => ( ! [Y2: A,Z3: A] :
              ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),Z3),R2)
             => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z3),B3),transitive_trancl(A,R2))
               => ( aa(A,$o,Pa,Z3)
                 => aa(A,$o,Pa,Y2) ) ) )
         => aa(A,$o,Pa,A3) ) ) ) ).

% converse_trancl_induct
tff(fact_1296_trancl__trans__induct,axiom,
    ! [A: $tType,Xa: A,Y: A,R2: set(product_prod(A,A)),Pa: fun(A,fun(A,$o))] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Y),transitive_trancl(A,R2))
     => ( ! [X: A,Y2: A] :
            ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y2),R2)
           => aa(A,$o,aa(A,fun(A,$o),Pa,X),Y2) )
       => ( ! [X: A,Y2: A,Z3: A] :
              ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y2),transitive_trancl(A,R2))
             => ( aa(A,$o,aa(A,fun(A,$o),Pa,X),Y2)
               => ( 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),Pa,Y2),Z3)
                   => aa(A,$o,aa(A,fun(A,$o),Pa,X),Z3) ) ) ) )
         => aa(A,$o,aa(A,fun(A,$o),Pa,Xa),Y) ) ) ) ).

% trancl_trans_induct
tff(fact_1297_trancl__into__trancl2,axiom,
    ! [A: $tType,A3: A,B3: A,R2: set(product_prod(A,A)),C2: A] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B3),R2)
     => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B3),C2),transitive_trancl(A,R2))
       => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),C2),transitive_trancl(A,R2)) ) ) ).

% trancl_into_trancl2
tff(fact_1298_Transitive__Closure_Otrancl__into__trancl,axiom,
    ! [A: $tType,A3: A,B3: A,R2: set(product_prod(A,A)),C2: A] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B3),transitive_trancl(A,R2))
     => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B3),C2),R2)
       => 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_1299_irrefl__trancl__rD,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),Xa: A,Y: A] :
      ( ! [X: A] : ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),X),transitive_trancl(A,R2))
     => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Y),R2)
       => ( Xa != Y ) ) ) ).

% irrefl_trancl_rD
tff(fact_1300_converse__tranclE,axiom,
    ! [A: $tType,Xa: A,Z2: A,R2: set(product_prod(A,A))] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Z2),transitive_trancl(A,R2))
     => ( ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Z2),R2)
       => ~ ! [Y2: A] :
              ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Y2),R2)
             => ~ 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_1301_r__r__into__trancl,axiom,
    ! [A: $tType,A3: A,B3: A,R: set(product_prod(A,A)),C2: A] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B3),R)
     => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B3),C2),R)
       => 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_1302_trancl__induct,axiom,
    ! [A: $tType,A3: A,B3: A,R2: set(product_prod(A,A)),Pa: fun(A,$o)] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B3),transitive_trancl(A,R2))
     => ( ! [Y2: A] :
            ( 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,$o,Pa,Y2) )
       => ( ! [Y2: A,Z3: A] :
              ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),Y2),transitive_trancl(A,R2))
             => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),Z3),R2)
               => ( aa(A,$o,Pa,Y2)
                 => aa(A,$o,Pa,Z3) ) ) )
         => aa(A,$o,Pa,B3) ) ) ) ).

% trancl_induct
tff(fact_1303_trancl__induct2,axiom,
    ! [A: $tType,B: $tType,Ax: A,Ay: B,Bx: A,By: B,R2: set(product_prod(product_prod(A,B),product_prod(A,B))),Pa: fun(A,fun(B,$o))] :
      ( member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Bx),By)),transitive_trancl(product_prod(A,B),R2))
     => ( ! [A4: A,B4: B] :
            ( member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B4)),R2)
           => aa(B,$o,aa(A,fun(B,$o),Pa,A4),B4) )
       => ( ! [A4: A,B4: B,Aa3: A,Ba: B] :
              ( member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B4)),transitive_trancl(product_prod(A,B),R2))
             => ( member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B4)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa3),Ba)),R2)
               => ( aa(B,$o,aa(A,fun(B,$o),Pa,A4),B4)
                 => aa(B,$o,aa(A,fun(B,$o),Pa,Aa3),Ba) ) ) )
         => aa(B,$o,aa(A,fun(B,$o),Pa,Bx),By) ) ) ) ).

% trancl_induct2
tff(fact_1304_trancl_Ocases,axiom,
    ! [A: $tType,A1: A,A22: A,R2: set(product_prod(A,A))] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A1),A22),transitive_trancl(A,R2))
     => ( ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A1),A22),R2)
       => ~ ! [B4: A] :
              ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A1),B4),transitive_trancl(A,R2))
             => ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B4),A22),R2) ) ) ) ).

% trancl.cases
tff(fact_1305_trancl_Osimps,axiom,
    ! [A: $tType,A1: A,A22: A,R2: set(product_prod(A,A))] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A1),A22),transitive_trancl(A,R2))
    <=> ( ? [A7: A,B8: A] :
            ( ( A1 = A7 )
            & ( A22 = B8 )
            & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A7),B8),R2) )
        | ? [A7: A,B8: A,C6: A] :
            ( ( A1 = A7 )
            & ( A22 = C6 )
            & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A7),B8),transitive_trancl(A,R2))
            & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B8),C6),R2) ) ) ) ).

% trancl.simps
tff(fact_1306_trancl_Or__into__trancl,axiom,
    ! [A: $tType,A3: A,B3: A,R2: set(product_prod(A,A))] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B3),R2)
     => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B3),transitive_trancl(A,R2)) ) ).

% trancl.r_into_trancl
tff(fact_1307_tranclE,axiom,
    ! [A: $tType,A3: A,B3: A,R2: set(product_prod(A,A))] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B3),transitive_trancl(A,R2))
     => ( ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B3),R2)
       => ~ ! [C5: A] :
              ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),C5),transitive_trancl(A,R2))
             => ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),C5),B3),R2) ) ) ) ).

% tranclE
tff(fact_1308_trancl__trans,axiom,
    ! [A: $tType,Xa: A,Y: A,R2: set(product_prod(A,A)),Z2: A] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Y),transitive_trancl(A,R2))
     => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z2),transitive_trancl(A,R2))
       => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Z2),transitive_trancl(A,R2)) ) ) ).

% trancl_trans
tff(fact_1309_nonempty__has__size,axiom,
    ! [A: $tType,S: multiset(A)] :
      ( ( S != zero_zero(multiset(A)) )
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(multiset(A),nat,size_size(multiset(A)),S)) ) ).

% nonempty_has_size
tff(fact_1310_bijective__Empty,axiom,
    ! [B: $tType,A: $tType] : bijective(A,B,bot_bot(set(product_prod(A,B)))) ).

% bijective_Empty
tff(fact_1311_total__on__singleton,axiom,
    ! [A: $tType,Xa: A] : total_on(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),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))),insert3(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Xa)),bot_bot(set(product_prod(A,A))))) ).

% total_on_singleton
tff(fact_1312_nth__Cons__pos,axiom,
    ! [A: $tType,N: nat,Xa: A,Xsa: list(A)] :
      ( aa(nat,$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),Xa),Xsa)),N) = aa(nat,A,nth(A,Xsa),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))) ) ) ).

% nth_Cons_pos
tff(fact_1313_Field__insert,axiom,
    ! [A: $tType,A3: A,B3: A,R2: set(product_prod(A,A))] : ( aa(set(product_prod(A,A)),set(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))),insert3(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B3)),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)),insert3(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),B3),bot_bot(set(A))))),aa(set(product_prod(A,A)),set(A),field2(A),R2)) ) ).

% Field_insert
tff(fact_1314_nth__enumerate__eq,axiom,
    ! [A: $tType,M2: nat,Xsa: list(A),N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),aa(list(A),nat,size_size(list(A)),Xsa))
     => ( aa(nat,product_prod(nat,A),nth(product_prod(nat,A),enumerate(A,N,Xsa)),M2) = aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M2)),aa(nat,A,nth(A,Xsa),M2)) ) ) ).

% nth_enumerate_eq
tff(fact_1315_times__assn__raw_Opelims_I3_J,axiom,
    ! [Xa: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xaa: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xb: product_prod(heap_ext(product_unit),set(nat))] :
      ( ~ aa(product_prod(heap_ext(product_unit),set(nat)),$o,times_assn_raw(Xa,Xaa),Xb)
     => ( accp(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$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)))),Xa),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))),Xaa),Xb)))
       => ~ ! [H: heap_ext(product_unit),As: 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),As) )
             => ( 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)),$o),product_prod(heap_ext(product_unit),set(nat)))),Xa),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$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)),$o),product_prod(heap_ext(product_unit),set(nat))),Xaa),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As))))
               => ? [As13: set(nat),As24: set(nat)] :
                    ( ( As = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As13),As24) )
                    & ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As13),As24) = bot_bot(set(nat)) )
                    & aa(product_prod(heap_ext(product_unit),set(nat)),$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),As13))
                    & aa(product_prod(heap_ext(product_unit),set(nat)),$o,Xaa,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),As24)) ) ) ) ) ) ).

% times_assn_raw.pelims(3)
tff(fact_1316_times__assn__raw_Opelims_I2_J,axiom,
    ! [Xa: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xaa: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xb: product_prod(heap_ext(product_unit),set(nat))] :
      ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,times_assn_raw(Xa,Xaa),Xb)
     => ( accp(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$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)))),Xa),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))),Xaa),Xb)))
       => ~ ! [H: heap_ext(product_unit),As: 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),As) )
             => ( 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)),$o),product_prod(heap_ext(product_unit),set(nat)))),Xa),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$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)),$o),product_prod(heap_ext(product_unit),set(nat))),Xaa),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As))))
               => ~ ? [As12: set(nat),As23: set(nat)] :
                      ( ( As = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As12),As23) )
                      & ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As12),As23) = bot_bot(set(nat)) )
                      & aa(product_prod(heap_ext(product_unit),set(nat)),$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),As12))
                      & aa(product_prod(heap_ext(product_unit),set(nat)),$o,Xaa,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(2)
tff(fact_1317_list_Oinject,axiom,
    ! [A: $tType,X21: A,X222: list(A),Y21: A,Y222: list(A)] :
      ( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X21),X222) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y21),Y222) )
    <=> ( ( X21 = Y21 )
        & ( X222 = Y222 ) ) ) ).

% list.inject
tff(fact_1318_Field__Un,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),S2: set(product_prod(A,A))] : ( aa(set(product_prod(A,A)),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))),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)),aa(set(product_prod(A,A)),set(A),field2(A),R2)),aa(set(product_prod(A,A)),set(A),field2(A),S2)) ) ).

% Field_Un
tff(fact_1319_list__ex__simps_I1_J,axiom,
    ! [A: $tType,Pa: fun(A,$o),Xa: A,Xsa: list(A)] :
      ( aa(list(A),$o,aa(fun(A,$o),fun(list(A),$o),list_ex(A),Pa),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa))
    <=> ( aa(A,$o,Pa,Xa)
        | aa(list(A),$o,aa(fun(A,$o),fun(list(A),$o),list_ex(A),Pa),Xsa) ) ) ).

% list_ex_simps(1)
tff(fact_1320_length__enumerate,axiom,
    ! [A: $tType,N: nat,Xsa: list(A)] : ( aa(list(product_prod(nat,A)),nat,size_size(list(product_prod(nat,A))),enumerate(A,N,Xsa)) = aa(list(A),nat,size_size(list(A)),Xsa) ) ).

% length_enumerate
tff(fact_1321_list_Osimps_I15_J,axiom,
    ! [A: $tType,X21: A,X222: list(A)] : ( aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X21),X222)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),X21),aa(list(A),set(A),set2(A),X222)) ) ).

% list.simps(15)
tff(fact_1322_nth__Cons__0,axiom,
    ! [A: $tType,Xa: A,Xsa: list(A)] : ( aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa)),zero_zero(nat)) = Xa ) ).

% nth_Cons_0
tff(fact_1323_nth__Cons__Suc,axiom,
    ! [A: $tType,Xa: A,Xsa: list(A),N: nat] : ( aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa)),aa(nat,nat,suc,N)) = aa(nat,A,nth(A,Xsa),N) ) ).

% nth_Cons_Suc
tff(fact_1324_zip__Cons__Cons,axiom,
    ! [A: $tType,B: $tType,Xa: A,Xsa: list(A),Y: B,Ys: list(B)] : ( aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa)),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),Xa),Y)),aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),Xsa),Ys)) ) ).

% zip_Cons_Cons
tff(fact_1325_Field__empty,axiom,
    ! [A: $tType] : ( aa(set(product_prod(A,A)),set(A),field2(A),bot_bot(set(product_prod(A,A)))) = bot_bot(set(A)) ) ).

% Field_empty
tff(fact_1326_not__in__set__insert,axiom,
    ! [A: $tType,Xa: A,Xsa: list(A)] :
      ( ~ member(A,Xa,aa(list(A),set(A),set2(A),Xsa))
     => ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),insert(A),Xa),Xsa) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa) ) ) ).

% not_in_set_insert
tff(fact_1327_Cons__listrel1__Cons,axiom,
    ! [A: $tType,Xa: A,Xsa: list(A),Y: A,Ys: list(A),R2: set(product_prod(A,A))] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),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),Xa),Xsa)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)),listrel1(A,R2))
    <=> ( ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Y),R2)
          & ( Xsa = Ys ) )
        | ( ( Xa = Y )
          & 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)),Xsa),Ys),listrel1(A,R2)) ) ) ) ).

% Cons_listrel1_Cons
tff(fact_1328_enumerate__simps_I2_J,axiom,
    ! [A: $tType,N: nat,Xa: A,Xsa: list(A)] : ( enumerate(A,N,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa)) = 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),Xa)),enumerate(A,aa(nat,nat,suc,N),Xsa)) ) ).

% enumerate_simps(2)
tff(fact_1329_fold__atLeastAtMost__nat_Ocases,axiom,
    ! [A: $tType,Xa: product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A)))] :
      ~ ! [F3: fun(nat,fun(A,A)),A4: nat,B4: nat,Acc: A] : ( Xa != aa(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),aa(fun(nat,fun(A,A)),fun(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A)))),product_Pair(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),F3),aa(product_prod(nat,A),product_prod(nat,product_prod(nat,A)),aa(nat,fun(product_prod(nat,A),product_prod(nat,product_prod(nat,A))),product_Pair(nat,product_prod(nat,A)),A4),aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),B4),Acc))) ) ).

% fold_atLeastAtMost_nat.cases
tff(fact_1330_distinct__enumerate,axiom,
    ! [A: $tType,N: nat,Xsa: list(A)] : aa(list(product_prod(nat,A)),$o,distinct(product_prod(nat,A)),enumerate(A,N,Xsa)) ).

% distinct_enumerate
tff(fact_1331_not__Cons__self2,axiom,
    ! [A: $tType,Xa: A,Xsa: list(A)] : ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa) != Xsa ) ).

% not_Cons_self2
tff(fact_1332_list__tail__coinc,axiom,
    ! [A: $tType,N1: A,R12: list(A),N22: A,R23: list(A)] :
      ( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),N1),R12) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),N22),R23) )
     => ( ( N1 = N22 )
        & ( R12 = R23 ) ) ) ).

% list_tail_coinc
tff(fact_1333_union__diff__assoc,axiom,
    ! [A: $tType,C3: multiset(A),B2: multiset(A),Aa2: multiset(A)] :
      ( ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),C3),B2) = zero_zero(multiset(A)) )
     => ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),Aa2),B2)),C3) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),Aa2),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),B2),C3)) ) ) ).

% union_diff_assoc
tff(fact_1334_zip__eq__ConsE,axiom,
    ! [A: $tType,B: $tType,Xsa: list(A),Ys: list(B),Xy: product_prod(A,B),Xys: list(product_prod(A,B))] :
      ( ( aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),Xsa),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) )
     => ~ ! [X: A,Xs4: list(A)] :
            ( ( Xsa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs4) )
           => ! [Y2: B,Ys3: list(B)] :
                ( ( Ys = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y2),Ys3) )
               => ( ( Xy = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y2) )
                 => ( Xys != aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),Xs4),Ys3) ) ) ) ) ) ).

% zip_eq_ConsE
tff(fact_1335_FieldI1,axiom,
    ! [A: $tType,Ia: A,J: A,R: set(product_prod(A,A))] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Ia),J),R)
     => member(A,Ia,aa(set(product_prod(A,A)),set(A),field2(A),R)) ) ).

% FieldI1
tff(fact_1336_FieldI2,axiom,
    ! [A: $tType,Ia: A,J: A,R: set(product_prod(A,A))] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Ia),J),R)
     => member(A,J,aa(set(product_prod(A,A)),set(A),field2(A),R)) ) ).

% FieldI2
tff(fact_1337_set__ConsD,axiom,
    ! [A: $tType,Y: A,Xa: A,Xsa: list(A)] :
      ( member(A,Y,aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa)))
     => ( ( Y = Xa )
        | member(A,Y,aa(list(A),set(A),set2(A),Xsa)) ) ) ).

% set_ConsD
tff(fact_1338_list_Oset__cases,axiom,
    ! [A: $tType,E3: A,A3: list(A)] :
      ( member(A,E3,aa(list(A),set(A),set2(A),A3))
     => ( ! [Z22: list(A)] : ( A3 != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E3),Z22) )
       => ~ ! [Z1: A,Z22: list(A)] :
              ( ( A3 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Z1),Z22) )
             => ~ member(A,E3,aa(list(A),set(A),set2(A),Z22)) ) ) ) ).

% list.set_cases
tff(fact_1339_list_Oset__intros_I1_J,axiom,
    ! [A: $tType,X21: A,X222: list(A)] : member(A,X21,aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X21),X222))) ).

% list.set_intros(1)
tff(fact_1340_list_Oset__intros_I2_J,axiom,
    ! [A: $tType,Y: A,X222: list(A),X21: A] :
      ( member(A,Y,aa(list(A),set(A),set2(A),X222))
     => member(A,Y,aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X21),X222))) ) ).

% list.set_intros(2)
tff(fact_1341_distinct__length__2__or__more,axiom,
    ! [A: $tType,A3: A,B3: A,Xsa: list(A)] :
      ( aa(list(A),$o,distinct(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),B3),Xsa)))
    <=> ( ( A3 != B3 )
        & aa(list(A),$o,distinct(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),Xsa))
        & aa(list(A),$o,distinct(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),B3),Xsa)) ) ) ).

% distinct_length_2_or_more
tff(fact_1342_foldl__Cons,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,fun(B,A)),A3: A,Xa: B,Xsa: list(B)] : ( aa(list(B),A,aa(A,fun(list(B),A),aa(fun(A,fun(B,A)),fun(A,fun(list(B),A)),foldl(A,B),F2),A3),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Xa),Xsa)) = aa(list(B),A,aa(A,fun(list(B),A),aa(fun(A,fun(B,A)),fun(A,fun(list(B),A)),foldl(A,B),F2),aa(B,A,aa(A,fun(B,A),F2,A3),Xa)),Xsa) ) ).

% foldl_Cons
tff(fact_1343_remove1_Osimps_I2_J,axiom,
    ! [A: $tType,Xa: A,Y: A,Xsa: list(A)] :
      ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),remove1(A),Xa),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Xsa)) = $ite(Xa = Y,Xsa,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),aa(list(A),list(A),aa(A,fun(list(A),list(A)),remove1(A),Xa),Xsa))) ) ).

% remove1.simps(2)
tff(fact_1344_total__onI,axiom,
    ! [A: $tType,Aa2: set(A),R2: set(product_prod(A,A))] :
      ( ! [X: A,Y2: A] :
          ( member(A,X,Aa2)
         => ( member(A,Y2,Aa2)
           => ( ( X != Y2 )
             => ( 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)
                | member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),X),R2) ) ) ) )
     => total_on(A,Aa2,R2) ) ).

% total_onI
tff(fact_1345_total__on__def,axiom,
    ! [A: $tType,Aa2: set(A),R2: set(product_prod(A,A))] :
      ( total_on(A,Aa2,R2)
    <=> ! [X6: A] :
          ( member(A,X6,Aa2)
         => ! [Xa2: A] :
              ( member(A,Xa2,Aa2)
             => ( ( X6 != Xa2 )
               => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X6),Xa2),R2)
                  | member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa2),X6),R2) ) ) ) ) ) ).

% total_on_def
tff(fact_1346_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_1347_removeAll_Osimps_I2_J,axiom,
    ! [A: $tType,Xa: A,Y: A,Xsa: list(A)] :
      ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),removeAll(A),Xa),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Xsa)) = $ite(Xa = Y,aa(list(A),list(A),aa(A,fun(list(A),list(A)),removeAll(A),Xa),Xsa),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),aa(list(A),list(A),aa(A,fun(list(A),list(A)),removeAll(A),Xa),Xsa))) ) ).

% removeAll.simps(2)
tff(fact_1348_zipf_Osimps_I2_J,axiom,
    ! [B: $tType,A: $tType,E: $tType,F2: fun(B,fun(E,A)),A3: B,As2: list(B),B3: E,Bs: list(E)] : ( zipf(B,E,A,F2,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),A3),As2),aa(list(E),list(E),aa(E,fun(list(E),list(E)),cons(E),B3),Bs)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(E,A,aa(B,fun(E,A),F2,A3),B3)),zipf(B,E,A,F2,As2,Bs)) ) ).

% zipf.simps(2)
tff(fact_1349_null__rec_I1_J,axiom,
    ! [A: $tType,Xa: A,Xsa: list(A)] : ~ aa(list(A),$o,null(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa)) ).

% null_rec(1)
tff(fact_1350_length__Suc__conv,axiom,
    ! [A: $tType,Xsa: list(A),N: nat] :
      ( ( aa(list(A),nat,size_size(list(A)),Xsa) = aa(nat,nat,suc,N) )
    <=> ? [Y4: A,Ys4: list(A)] :
          ( ( Xsa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys4) )
          & ( aa(list(A),nat,size_size(list(A)),Ys4) = N ) ) ) ).

% length_Suc_conv
tff(fact_1351_Suc__length__conv,axiom,
    ! [A: $tType,N: nat,Xsa: list(A)] :
      ( ( aa(nat,nat,suc,N) = aa(list(A),nat,size_size(list(A)),Xsa) )
    <=> ? [Y4: A,Ys4: list(A)] :
          ( ( Xsa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys4) )
          & ( aa(list(A),nat,size_size(list(A)),Ys4) = N ) ) ) ).

% Suc_length_conv
tff(fact_1352_set__subset__Cons,axiom,
    ! [A: $tType,Xsa: list(A),Xa: A] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xsa)),aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa))) ).

% set_subset_Cons
tff(fact_1353_mono__Field,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)),$o),ord_less_eq(set(product_prod(A,A))),R2),S2)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(product_prod(A,A)),set(A),field2(A),R2)),aa(set(product_prod(A,A)),set(A),field2(A),S2)) ) ).

% mono_Field
tff(fact_1354_impossible__Cons,axiom,
    ! [A: $tType,Xsa: list(A),Ys: list(A),Xa: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xsa)),aa(list(A),nat,size_size(list(A)),Ys))
     => ( Xsa != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Ys) ) ) ).

% impossible_Cons
tff(fact_1355_sorted2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Y: A,Zs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Zs)))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
            & sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Zs)) ) ) ) ).

% sorted2
tff(fact_1356_distinct_Osimps_I2_J,axiom,
    ! [A: $tType,Xa: A,Xsa: list(A)] :
      ( aa(list(A),$o,distinct(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa))
    <=> ( ~ member(A,Xa,aa(list(A),set(A),set2(A),Xsa))
        & aa(list(A),$o,distinct(A),Xsa) ) ) ).

% distinct.simps(2)
tff(fact_1357_list__update__code_I2_J,axiom,
    ! [A: $tType,Xa: A,Xsa: list(A),Y: A] : ( aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa)),zero_zero(nat)),Y) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Xsa) ) ).

% list_update_code(2)
tff(fact_1358_list__update__code_I3_J,axiom,
    ! [A: $tType,Xa: A,Xsa: list(A),Ia: nat,Y: A] : ( aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa)),aa(nat,nat,suc,Ia)),Y) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xsa),Ia),Y)) ) ).

% list_update_code(3)
tff(fact_1359_listrel1I2,axiom,
    ! [A: $tType,Xsa: list(A),Ys: list(A),R2: set(product_prod(A,A)),Xa: A] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xsa),Ys),listrel1(A,R2))
     => member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Ys)),listrel1(A,R2)) ) ).

% listrel1I2
tff(fact_1360_merge_Osimps_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X1: A,L12: list(A),X22: A,L23: list(A)] :
          ( merge(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X1),L12),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X22),L23)) = $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),X1),X22),
              aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X1),merge(A,L12,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X22),L23))),
              $ite(X1 = X22,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X1),merge(A,L12,L23)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X22),merge(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X1),L12),L23))) ) ) ) ).

% merge.simps(3)
tff(fact_1361_list__assn_Osimps_I2_J,axiom,
    ! [A: $tType,B: $tType,Pa: fun(A,fun(B,assn)),A3: A,As2: list(A),C2: B,Cs: list(B)] : ( vEBT_List_list_assn(A,B,Pa,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),As2),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),C2),Cs)) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),Pa,A3),C2)),vEBT_List_list_assn(A,B,Pa,As2,Cs)) ) ).

% list_assn.simps(2)
tff(fact_1362_list__assn__simps_I2_J,axiom,
    ! [A: $tType,B: $tType,Pa: fun(A,fun(B,assn)),A3: A,As2: list(A),C2: B,Cs: list(B)] : ( vEBT_List_list_assn(A,B,Pa,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),As2),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),C2),Cs)) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),Pa,A3),C2)),vEBT_List_list_assn(A,B,Pa,As2,Cs)) ) ).

% list_assn_simps(2)
tff(fact_1363_List_Oinsert__def,axiom,
    ! [A: $tType,Xa: A,Xsa: list(A)] :
      ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),insert(A),Xa),Xsa) = $ite(member(A,Xa,aa(list(A),set(A),set2(A),Xsa)),Xsa,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa)) ) ).

% List.insert_def
tff(fact_1364_Suc__le__length__iff,axiom,
    ! [A: $tType,N: nat,Xsa: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,N)),aa(list(A),nat,size_size(list(A)),Xsa))
    <=> ? [X6: A,Ys4: list(A)] :
          ( ( Xsa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X6),Ys4) )
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),aa(list(A),nat,size_size(list(A)),Ys4)) ) ) ).

% Suc_le_length_iff
tff(fact_1365_sorted__simps_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Ys: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Ys))
        <=> ( ! [X6: A] :
                ( member(A,X6,aa(list(A),set(A),set2(A),Ys))
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),X6) )
            & sorted_wrt(A,ord_less_eq(A),Ys) ) ) ) ).

% sorted_simps(2)
tff(fact_1366_strict__sorted__simps_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Ys: list(A)] :
          ( sorted_wrt(A,ord_less(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Ys))
        <=> ( ! [X6: A] :
                ( member(A,X6,aa(list(A),set(A),set2(A),Ys))
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),X6) )
            & sorted_wrt(A,ord_less(A),Ys) ) ) ) ).

% strict_sorted_simps(2)
tff(fact_1367_bijective__def,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,B))] :
      ( bijective(A,B,R)
    <=> ( ! [X6: A,Y4: B,Z5: B] :
            ( ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X6),Y4),R)
              & member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X6),Z5),R) )
           => ( Y4 = Z5 ) )
        & ! [X6: A,Y4: A,Z5: B] :
            ( ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X6),Z5),R)
              & member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Y4),Z5),R) )
           => ( X6 = Y4 ) ) ) ) ).

% bijective_def
tff(fact_1368_Cons__listrel1E2,axiom,
    ! [A: $tType,Xsa: list(A),Y: A,Ys: list(A),R2: set(product_prod(A,A))] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xsa),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)),listrel1(A,R2))
     => ( ! [X: A] :
            ( ( Xsa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Ys) )
           => ~ 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) )
       => ~ ! [Zs2: list(A)] :
              ( ( Xsa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Zs2) )
             => ~ member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),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_1369_Cons__listrel1E1,axiom,
    ! [A: $tType,Xa: A,Xsa: list(A),Ys: list(A),R2: set(product_prod(A,A))] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa)),Ys),listrel1(A,R2))
     => ( ! [Y2: A] :
            ( ( Ys = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Xsa) )
           => ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Y2),R2) )
       => ~ ! [Zs2: list(A)] :
              ( ( Ys = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Zs2) )
             => ~ member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xsa),Zs2),listrel1(A,R2)) ) ) ) ).

% Cons_listrel1E1
tff(fact_1370_listrel1I1,axiom,
    ! [A: $tType,Xa: A,Y: A,R2: set(product_prod(A,A)),Xsa: list(A)] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Y),R2)
     => member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Xsa)),listrel1(A,R2)) ) ).

% listrel1I1
tff(fact_1371_listrel__Cons2,axiom,
    ! [B: $tType,A: $tType,Xsa: list(A),Y: B,Ys: list(B),R2: set(product_prod(A,B))] :
      ( member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xsa),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),Ys)),listrel(A,B,R2))
     => ~ ! [X: A,Xs: list(A)] :
            ( ( Xsa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) )
           => ( 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)
             => ~ 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)) ) ) ) ).

% listrel_Cons2
tff(fact_1372_listrel__Cons1,axiom,
    ! [B: $tType,A: $tType,Y: A,Ys: list(A),Xsa: list(B),R2: set(product_prod(A,B))] :
      ( member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)),Xsa),listrel(A,B,R2))
     => ~ ! [Y2: B,Ys5: list(B)] :
            ( ( Xsa = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y2),Ys5) )
           => ( 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)
             => ~ member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Ys),Ys5),listrel(A,B,R2)) ) ) ) ).

% listrel_Cons1
tff(fact_1373_listrel_OCons,axiom,
    ! [B: $tType,A: $tType,Xa: A,Y: B,R2: set(product_prod(A,B)),Xsa: list(A),Ys: list(B)] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa),Y),R2)
     => ( member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xsa),Ys),listrel(A,B,R2))
       => member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),Ys)),listrel(A,B,R2)) ) ) ).

% listrel.Cons
tff(fact_1374_list_Osize_I4_J,axiom,
    ! [A: $tType,X21: A,X222: list(A)] : ( aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X21),X222)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),X222)),aa(nat,nat,suc,zero_zero(nat))) ) ).

% list.size(4)
tff(fact_1375_nth__Cons_H,axiom,
    ! [A: $tType,Xa: A,Xsa: list(A),N: nat] :
      ( aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa)),N) = $ite(N = zero_zero(nat),Xa,aa(nat,A,nth(A,Xsa),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)))) ) ).

% nth_Cons'
tff(fact_1376_nth__equal__first__eq,axiom,
    ! [A: $tType,Xa: A,Xsa: list(A),N: nat] :
      ( ~ member(A,Xa,aa(list(A),set(A),set2(A),Xsa))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),aa(list(A),nat,size_size(list(A)),Xsa))
       => ( ( aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa)),N) = Xa )
        <=> ( N = zero_zero(nat) ) ) ) ) ).

% nth_equal_first_eq
tff(fact_1377_nth__non__equal__first__eq,axiom,
    ! [A: $tType,Xa: A,Y: A,Xsa: list(A),N: nat] :
      ( ( Xa != Y )
     => ( ( aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa)),N) = Y )
      <=> ( ( aa(nat,A,nth(A,Xsa),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))) = Y )
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N) ) ) ) ).

% nth_non_equal_first_eq
tff(fact_1378_slice__Cons,axiom,
    ! [A: $tType,Begin: nat,End: nat,Xa: A,Xsa: list(A)] :
      ( slice(A,Begin,End,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa)) = $ite(
          ( ( Begin = zero_zero(nat) )
          & aa(nat,$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),Xa),slice(A,Begin,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),End),one_one(nat)),Xsa)),
          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)),Xsa) ) ) ).

% slice_Cons
tff(fact_1379_times__assn__raw_Opelims_I1_J,axiom,
    ! [Xa: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xaa: fun(product_prod(heap_ext(product_unit),set(nat)),$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(Xa,Xaa),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)),$o),product_prod(heap_ext(product_unit),set(nat)))),Xa),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$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)),$o),product_prod(heap_ext(product_unit),set(nat))),Xaa),Xb)))
       => ~ ! [H: heap_ext(product_unit),As: set(nat)] :
              ( ( Xb = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As) )
             => ( ( (Y)
                <=> ? [As1: set(nat),As22: set(nat)] :
                      ( ( As = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As1),As22) )
                      & ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As1),As22) = bot_bot(set(nat)) )
                      & aa(product_prod(heap_ext(product_unit),set(nat)),$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),As1))
                      & aa(product_prod(heap_ext(product_unit),set(nat)),$o,Xaa,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As22)) ) )
               => ~ accp(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$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)))),Xa),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))),Xaa),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),As)))) ) ) ) ) ).

% times_assn_raw.pelims(1)
tff(fact_1380_length__Cons,axiom,
    ! [A: $tType,Xa: A,Xsa: list(A)] : ( aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa)) = aa(nat,nat,suc,aa(list(A),nat,size_size(list(A)),Xsa)) ) ).

% length_Cons
tff(fact_1381_Cons__lenlex__iff,axiom,
    ! [A: $tType,M2: A,Ms: list(A),N: A,Ns: list(A),R2: set(product_prod(A,A))] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),M2),Ms)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),N),Ns)),lenlex(A,R2))
    <=> ( aa(nat,$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) )
          & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),M2),N),R2) )
        | ( ( M2 = N )
          & 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_1382_Cons__in__lex,axiom,
    ! [A: $tType,Xa: A,Xsa: list(A),Y: A,Ys: list(A),R2: set(product_prod(A,A))] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),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),Xa),Xsa)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)),lex(A,R2))
    <=> ( ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Y),R2)
          & ( aa(list(A),nat,size_size(list(A)),Xsa) = aa(list(A),nat,size_size(list(A)),Ys) ) )
        | ( ( Xa = Y )
          & 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)),Xsa),Ys),lex(A,R2)) ) ) ) ).

% Cons_in_lex
tff(fact_1383_list__assn_Oelims,axiom,
    ! [B: $tType,A: $tType,Xa: fun(A,fun(B,assn)),Xaa: list(A),Xb: list(B),Y: assn] :
      ( ( vEBT_List_list_assn(A,B,Xa,Xaa,Xb) = Y )
     => ( ( ( Xaa = nil(A) )
         => ( ( Xb = nil(B) )
           => ( Y != one_one(assn) ) ) )
       => ( ! [A4: A,As: list(A)] :
              ( ( Xaa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A4),As) )
             => ! [C5: B,Cs2: list(B)] :
                  ( ( Xb = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),C5),Cs2) )
                 => ( Y != aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),Xa,A4),C5)),vEBT_List_list_assn(A,B,Xa,As,Cs2)) ) ) )
         => ( ( ? [V3: A,Va: list(A)] : ( Xaa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V3),Va) )
             => ( ( Xb = nil(B) )
               => ( Y != bot_bot(assn) ) ) )
           => ~ ( ( Xaa = nil(A) )
               => ( ? [V3: B,Va: list(B)] : ( Xb = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),V3),Va) )
                 => ( Y != bot_bot(assn) ) ) ) ) ) ) ) ).

% list_assn.elims
tff(fact_1384_count__list_Osimps_I2_J,axiom,
    ! [A: $tType,Xa: A,Xsa: list(A),Y: A] :
      ( aa(A,nat,count_list(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa)),Y) = $ite(Xa = Y,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,count_list(A,Xsa),Y)),one_one(nat)),aa(A,nat,count_list(A,Xsa),Y)) ) ).

% count_list.simps(2)
tff(fact_1385_count__notin,axiom,
    ! [A: $tType,Xa: A,Xsa: list(A)] :
      ( ~ member(A,Xa,aa(list(A),set(A),set2(A),Xsa))
     => ( aa(A,nat,count_list(A,Xsa),Xa) = zero_zero(nat) ) ) ).

% count_notin
tff(fact_1386_power__decreasing__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B3: A,M2: nat,N: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),one_one(A))
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),B3),M2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B3),N))
            <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),M2) ) ) ) ) ).

% power_decreasing_iff
tff(fact_1387_in__measures_I2_J,axiom,
    ! [A: $tType,Xa: A,Y: A,F2: fun(A,nat),Fs: list(fun(A,nat))] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Y),measures(A,aa(list(fun(A,nat)),list(fun(A,nat)),aa(fun(A,nat),fun(list(fun(A,nat)),list(fun(A,nat))),cons(fun(A,nat)),F2),Fs)))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,F2,Xa)),aa(A,nat,F2,Y))
        | ( ( aa(A,nat,F2,Xa) = aa(A,nat,F2,Y) )
          & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Y),measures(A,Fs)) ) ) ) ).

% in_measures(2)
tff(fact_1388_nat__zero__less__power__iff,axiom,
    ! [Xa: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Xa),N))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Xa)
        | ( N = zero_zero(nat) ) ) ) ).

% nat_zero_less_power_iff
tff(fact_1389_zip__eq__Nil__iff,axiom,
    ! [A: $tType,B: $tType,Xsa: list(A),Ys: list(B)] :
      ( ( aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),Xsa),Ys) = nil(product_prod(A,B)) )
    <=> ( ( Xsa = nil(A) )
        | ( Ys = nil(B) ) ) ) ).

% zip_eq_Nil_iff
tff(fact_1390_Nil__eq__zip__iff,axiom,
    ! [A: $tType,B: $tType,Xsa: list(A),Ys: list(B)] :
      ( ( nil(product_prod(A,B)) = aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),Xsa),Ys) )
    <=> ( ( Xsa = nil(A) )
        | ( Ys = nil(B) ) ) ) ).

% Nil_eq_zip_iff
tff(fact_1391_zip__Nil,axiom,
    ! [B: $tType,A: $tType,Ys: list(B)] : ( aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),nil(A)),Ys) = nil(product_prod(A,B)) ) ).

% zip_Nil
tff(fact_1392_Nil__is__rev__conv,axiom,
    ! [A: $tType,Xsa: list(A)] :
      ( ( nil(A) = aa(list(A),list(A),rev(A),Xsa) )
    <=> ( Xsa = nil(A) ) ) ).

% Nil_is_rev_conv
tff(fact_1393_rev__is__Nil__conv,axiom,
    ! [A: $tType,Xsa: list(A)] :
      ( ( aa(list(A),list(A),rev(A),Xsa) = nil(A) )
    <=> ( Xsa = nil(A) ) ) ).

% rev_is_Nil_conv
tff(fact_1394_list__update__nonempty,axiom,
    ! [A: $tType,Xsa: list(A),K3: nat,Xa: A] :
      ( ( aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xsa),K3),Xa) = nil(A) )
    <=> ( Xsa = nil(A) ) ) ).

% list_update_nonempty
tff(fact_1395_enumerate__simps_I1_J,axiom,
    ! [A: $tType,N: nat] : ( enumerate(A,N,nil(A)) = nil(product_prod(nat,A)) ) ).

% enumerate_simps(1)
tff(fact_1396_rotate1__is__Nil__conv,axiom,
    ! [A: $tType,Xsa: list(A)] :
      ( ( aa(list(A),list(A),rotate1(A),Xsa) = nil(A) )
    <=> ( Xsa = nil(A) ) ) ).

% rotate1_is_Nil_conv
tff(fact_1397_list__ex__simps_I2_J,axiom,
    ! [A: $tType,Pa: fun(A,$o)] : ~ aa(list(A),$o,aa(fun(A,$o),fun(list(A),$o),list_ex(A),Pa),nil(A)) ).

% list_ex_simps(2)
tff(fact_1398_slice__Nil,axiom,
    ! [A: $tType,Begin: nat,End: nat] : ( slice(A,Begin,End,nil(A)) = nil(A) ) ).

% slice_Nil
tff(fact_1399_slice__eq__bounds__empty,axiom,
    ! [A: $tType,Ia: nat,Xsa: list(A)] : ( slice(A,Ia,Ia,Xsa) = nil(A) ) ).

% slice_eq_bounds_empty
tff(fact_1400_in__measures_I1_J,axiom,
    ! [A: $tType,Xa: A,Y: A] : ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Y),measures(A,nil(fun(A,nat)))) ).

% in_measures(1)
tff(fact_1401_power__inject__exp,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,M2: nat,N: nat] :
          ( aa(A,$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),M2) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N) )
          <=> ( M2 = N ) ) ) ) ).

% power_inject_exp
tff(fact_1402_set__empty,axiom,
    ! [A: $tType,Xsa: list(A)] :
      ( ( aa(list(A),set(A),set2(A),Xsa) = bot_bot(set(A)) )
    <=> ( Xsa = nil(A) ) ) ).

% set_empty
tff(fact_1403_set__empty2,axiom,
    ! [A: $tType,Xsa: list(A)] :
      ( ( bot_bot(set(A)) = aa(list(A),set(A),set2(A),Xsa) )
    <=> ( Xsa = nil(A) ) ) ).

% set_empty2
tff(fact_1404_length__0__conv,axiom,
    ! [A: $tType,Xsa: list(A)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xsa) = zero_zero(nat) )
    <=> ( Xsa = nil(A) ) ) ).

% length_0_conv
tff(fact_1405_singleton__rev__conv,axiom,
    ! [A: $tType,Xa: A,Xsa: list(A)] :
      ( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),nil(A)) = aa(list(A),list(A),rev(A),Xsa) )
    <=> ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),nil(A)) = Xsa ) ) ).

% singleton_rev_conv
tff(fact_1406_rev__singleton__conv,axiom,
    ! [A: $tType,Xsa: list(A),Xa: A] :
      ( ( aa(list(A),list(A),rev(A),Xsa) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),nil(A)) )
    <=> ( Xsa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),nil(A)) ) ) ).

% rev_singleton_conv
tff(fact_1407_Nil__lenlex__iff1,axiom,
    ! [A: $tType,Ns: list(A),R2: set(product_prod(A,A))] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),Ns),lenlex(A,R2))
    <=> ( Ns != nil(A) ) ) ).

% Nil_lenlex_iff1
tff(fact_1408_insert__Nil,axiom,
    ! [A: $tType,Xa: A] : ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),insert(A),Xa),nil(A)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),nil(A)) ) ).

% insert_Nil
tff(fact_1409_list__assn__aux__simps_I1_J,axiom,
    ! [A: $tType,B: $tType,Pa: fun(A,fun(B,assn)),L2: list(B)] : ( vEBT_List_list_assn(A,B,Pa,nil(A),L2) = pure_assn(L2 = nil(B)) ) ).

% list_assn_aux_simps(1)
tff(fact_1410_list__assn__aux__simps_I2_J,axiom,
    ! [B: $tType,A: $tType,Pa: fun(A,fun(B,assn)),L: list(A)] : ( vEBT_List_list_assn(A,B,Pa,L,nil(B)) = pure_assn(L = nil(A)) ) ).

% list_assn_aux_simps(2)
tff(fact_1411_power__strict__increasing__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B3: A,Xa: nat,Y: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),B3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),B3),Xa)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B3),Y))
          <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Y) ) ) ) ).

% power_strict_increasing_iff
tff(fact_1412_power__eq__0__iff,axiom,
    ! [A: $tType] :
      ( semiri2026040879449505780visors(A)
     => ! [A3: A,N: nat] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N) = zero_zero(A) )
        <=> ( ( A3 = zero_zero(A) )
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N) ) ) ) ).

% power_eq_0_iff
tff(fact_1413_length__greater__0__conv,axiom,
    ! [A: $tType,Xsa: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(list(A),nat,size_size(list(A)),Xsa))
    <=> ( Xsa != nil(A) ) ) ).

% length_greater_0_conv
tff(fact_1414_power__strict__decreasing__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B3: A,M2: nat,N: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),one_one(A))
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),B3),M2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B3),N))
            <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),M2) ) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_1415_power__mono__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,B3: A,N: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B3)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B3),N))
              <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3) ) ) ) ) ) ).

% power_mono_iff
tff(fact_1416_power__increasing__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B3: A,Xa: nat,Y: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),B3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),B3),Xa)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B3),Y))
          <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xa),Y) ) ) ) ).

% power_increasing_iff
tff(fact_1417_length__ge__1__conv,axiom,
    ! [A: $tType,L: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),aa(list(A),nat,size_size(list(A)),L))
    <=> ( L != nil(A) ) ) ).

% length_ge_1_conv
tff(fact_1418_transpose_Ocases,axiom,
    ! [A: $tType,Xa: list(list(A))] :
      ( ( Xa != nil(list(A)) )
     => ( ! [Xss: list(list(A))] : ( Xa != aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),nil(A)),Xss) )
       => ~ ! [X: A,Xs: list(A),Xss: list(list(A))] : ( Xa != 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.cases
tff(fact_1419_Nil__lenlex__iff2,axiom,
    ! [A: $tType,Ns: list(A),R2: set(product_prod(A,A))] : ~ member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Ns),nil(A)),lenlex(A,R2)) ).

% Nil_lenlex_iff2
tff(fact_1420_Nil2__notin__lex,axiom,
    ! [A: $tType,Xsa: list(A),R2: set(product_prod(A,A))] : ~ member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xsa),nil(A)),lex(A,R2)) ).

% Nil2_notin_lex
tff(fact_1421_Nil__notin__lex,axiom,
    ! [A: $tType,Ys: list(A),R2: set(product_prod(A,A))] : ~ member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),Ys),lex(A,R2)) ).

% Nil_notin_lex
tff(fact_1422_zip_Osimps_I1_J,axiom,
    ! [B: $tType,A: $tType,Xsa: list(A)] : ( aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),Xsa),nil(B)) = nil(product_prod(A,B)) ) ).

% zip.simps(1)
tff(fact_1423_mset__distrib,axiom,
    ! [A: $tType,Aa2: multiset(A),B2: multiset(A),M5: multiset(A),N7: multiset(A)] :
      ( ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),Aa2),B2) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M5),N7) )
     => ~ ! [Am: multiset(A),An: multiset(A)] :
            ( ( Aa2 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),Am),An) )
           => ! [Bm: multiset(A),Bn: multiset(A)] :
                ( ( B2 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),Bm),Bn) )
               => ( ( M5 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),Am),Bm) )
                 => ( N7 != aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),An),Bn) ) ) ) ) ) ).

% mset_distrib
tff(fact_1424_count__list_Osimps_I1_J,axiom,
    ! [A: $tType,Y: A] : ( aa(A,nat,count_list(A,nil(A)),Y) = zero_zero(nat) ) ).

% count_list.simps(1)
tff(fact_1425_product_Osimps_I1_J,axiom,
    ! [B: $tType,A: $tType,Uu: list(B)] : ( aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),product(A,B),nil(A)),Uu) = nil(product_prod(A,B)) ) ).

% product.simps(1)
tff(fact_1426_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_1427_power__mult__distrib,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A,B3: 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),B3)),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),B3),N)) ) ) ).

% power_mult_distrib
tff(fact_1428_power__commuting__commutes,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Xa: A,Y: A,N: nat] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),Xa),Y) = aa(A,A,aa(A,fun(A,A),times_times(A),Y),Xa) )
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),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),Xa),N)) ) ) ) ).

% power_commuting_commutes
tff(fact_1429_successively_Ocases,axiom,
    ! [A: $tType,Xa: product_prod(fun(A,fun(A,$o)),list(A))] :
      ( ! [P3: fun(A,fun(A,$o))] : ( Xa != aa(list(A),product_prod(fun(A,fun(A,$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,$o)),list(A)),P3),nil(A)) )
     => ( ! [P3: fun(A,fun(A,$o)),X: A] : ( Xa != 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)),P3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A))) )
       => ~ ! [P3: fun(A,fun(A,$o)),X: A,Y2: A,Xs: list(A)] : ( Xa != aa(list(A),product_prod(fun(A,fun(A,$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)),P3),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),Y2),Xs))) ) ) ) ).

% successively.cases
tff(fact_1430_arg__min__list_Ocases,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Xa: product_prod(fun(A,B),list(A))] :
          ( ! [F3: fun(A,B),X: A] : ( Xa != 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)),F3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A))) )
         => ( ! [F3: fun(A,B),X: A,Y2: A,Zs2: list(A)] : ( Xa != 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)),F3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Zs2))) )
           => ~ ! [A4: fun(A,B)] : ( Xa != aa(list(A),product_prod(fun(A,B),list(A)),aa(fun(A,B),fun(list(A),product_prod(fun(A,B),list(A))),product_Pair(fun(A,B),list(A)),A4),nil(A)) ) ) ) ) ).

% arg_min_list.cases
tff(fact_1431_sorted__wrt_Ocases,axiom,
    ! [A: $tType,Xa: product_prod(fun(A,fun(A,$o)),list(A))] :
      ( ! [P3: fun(A,fun(A,$o))] : ( Xa != aa(list(A),product_prod(fun(A,fun(A,$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,$o)),list(A)),P3),nil(A)) )
     => ~ ! [P3: fun(A,fun(A,$o)),X: A,Ys5: list(A)] : ( Xa != 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)),P3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Ys5)) ) ) ).

% sorted_wrt.cases
tff(fact_1432_neq__NilE,axiom,
    ! [A: $tType,L: list(A)] :
      ( ( L != nil(A) )
     => ~ ! [X: A,Xs: list(A)] : ( L != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) ) ) ).

% neq_NilE
tff(fact_1433_list__2pre__induct,axiom,
    ! [A: $tType,B: $tType,Pa: fun(list(A),fun(list(B),$o)),W1: list(A),W22: list(B)] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),Pa,nil(A)),nil(B))
     => ( ! [E2: A,W12: list(A),W23: list(B)] :
            ( aa(list(B),$o,aa(list(A),fun(list(B),$o),Pa,W12),W23)
           => aa(list(B),$o,aa(list(A),fun(list(B),$o),Pa,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E2),W12)),W23) )
       => ( ! [E2: B,W13: list(A),W24: list(B)] :
              ( aa(list(B),$o,aa(list(A),fun(list(B),$o),Pa,W13),W24)
             => aa(list(B),$o,aa(list(A),fun(list(B),$o),Pa,W13),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),E2),W24)) )
         => aa(list(B),$o,aa(list(A),fun(list(B),$o),Pa,W1),W22) ) ) ) ).

% list_2pre_induct
tff(fact_1434_list__induct__first2,axiom,
    ! [A: $tType,Pa: fun(list(A),$o),Xsa: list(A)] :
      ( aa(list(A),$o,Pa,nil(A))
     => ( ! [X: A] : aa(list(A),$o,Pa,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)))
       => ( ! [X12: A,X23: A,Xs: list(A)] :
              ( aa(list(A),$o,Pa,Xs)
             => aa(list(A),$o,Pa,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),Xs))) )
         => aa(list(A),$o,Pa,Xsa) ) ) ) ).

% list_induct_first2
tff(fact_1435_mergesort__by__rel__merge__induct,axiom,
    ! [A: $tType,B: $tType,Pa: fun(list(A),fun(list(B),$o)),R: fun(A,fun(B,$o)),Xsa: list(A),Ys: list(B)] :
      ( ! [Xs: list(A)] : aa(list(B),$o,aa(list(A),fun(list(B),$o),Pa,Xs),nil(B))
     => ( ! [Ys5: list(B)] : aa(list(B),$o,aa(list(A),fun(list(B),$o),Pa,nil(A)),Ys5)
       => ( ! [X: A,Xs: list(A),Y2: B,Ys5: list(B)] :
              ( aa(B,$o,aa(A,fun(B,$o),R,X),Y2)
             => ( aa(list(B),$o,aa(list(A),fun(list(B),$o),Pa,Xs),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y2),Ys5))
               => aa(list(B),$o,aa(list(A),fun(list(B),$o),Pa,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y2),Ys5)) ) )
         => ( ! [X: A,Xs: list(A),Y2: B,Ys5: list(B)] :
                ( ~ aa(B,$o,aa(A,fun(B,$o),R,X),Y2)
               => ( aa(list(B),$o,aa(list(A),fun(list(B),$o),Pa,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),Ys5)
                 => aa(list(B),$o,aa(list(A),fun(list(B),$o),Pa,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y2),Ys5)) ) )
           => aa(list(B),$o,aa(list(A),fun(list(B),$o),Pa,Xsa),Ys) ) ) ) ) ).

% mergesort_by_rel_merge_induct
tff(fact_1436_list_Odistinct_I1_J,axiom,
    ! [A: $tType,X21: A,X222: list(A)] : ( nil(A) != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X21),X222) ) ).

% list.distinct(1)
tff(fact_1437_list_OdiscI,axiom,
    ! [A: $tType,List: list(A),X21: A,X222: list(A)] :
      ( ( List = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X21),X222) )
     => ( List != nil(A) ) ) ).

% list.discI
tff(fact_1438_list_Oexhaust,axiom,
    ! [A: $tType,Y: list(A)] :
      ( ( Y != nil(A) )
     => ~ ! [X212: A,X223: list(A)] : ( Y != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X212),X223) ) ) ).

% list.exhaust
tff(fact_1439_min__list_Ocases,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Xa: list(A)] :
          ( ! [X: A,Xs: list(A)] : ( Xa != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) )
         => ( Xa = nil(A) ) ) ) ).

% min_list.cases
tff(fact_1440_remdups__adj_Ocases,axiom,
    ! [A: $tType,Xa: list(A)] :
      ( ( Xa != nil(A) )
     => ( ! [X: A] : ( Xa != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)) )
       => ~ ! [X: A,Y2: A,Xs: list(A)] : ( Xa != 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),Y2),Xs)) ) ) ) ).

% remdups_adj.cases
tff(fact_1441_neq__Nil__conv,axiom,
    ! [A: $tType,Xsa: list(A)] :
      ( ( Xsa != nil(A) )
    <=> ? [Y4: A,Ys4: list(A)] : ( Xsa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys4) ) ) ).

% neq_Nil_conv
tff(fact_1442_list__induct2_H,axiom,
    ! [A: $tType,B: $tType,Pa: fun(list(A),fun(list(B),$o)),Xsa: list(A),Ys: list(B)] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),Pa,nil(A)),nil(B))
     => ( ! [X: A,Xs: list(A)] : aa(list(B),$o,aa(list(A),fun(list(B),$o),Pa,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),nil(B))
       => ( ! [Y2: B,Ys5: list(B)] : aa(list(B),$o,aa(list(A),fun(list(B),$o),Pa,nil(A)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y2),Ys5))
         => ( ! [X: A,Xs: list(A),Y2: B,Ys5: list(B)] :
                ( aa(list(B),$o,aa(list(A),fun(list(B),$o),Pa,Xs),Ys5)
               => aa(list(B),$o,aa(list(A),fun(list(B),$o),Pa,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),Y2),Ys5)) )
           => aa(list(B),$o,aa(list(A),fun(list(B),$o),Pa,Xsa),Ys) ) ) ) ) ).

% list_induct2'
tff(fact_1443_list__nonempty__induct,axiom,
    ! [A: $tType,Xsa: list(A),Pa: fun(list(A),$o)] :
      ( ( Xsa != nil(A) )
     => ( ! [X: A] : aa(list(A),$o,Pa,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)))
       => ( ! [X: A,Xs: list(A)] :
              ( ( Xs != nil(A) )
             => ( aa(list(A),$o,Pa,Xs)
               => aa(list(A),$o,Pa,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) ) )
         => aa(list(A),$o,Pa,Xsa) ) ) ) ).

% list_nonempty_induct
tff(fact_1444_distinct_Osimps_I1_J,axiom,
    ! [A: $tType] : aa(list(A),$o,distinct(A),nil(A)) ).

% distinct.simps(1)
tff(fact_1445_sorted__wrt_Osimps_I1_J,axiom,
    ! [A: $tType,Pa: fun(A,fun(A,$o))] : sorted_wrt(A,Pa,nil(A)) ).

% sorted_wrt.simps(1)
tff(fact_1446_rev_Osimps_I1_J,axiom,
    ! [A: $tType] : ( aa(list(A),list(A),rev(A),nil(A)) = nil(A) ) ).

% rev.simps(1)
tff(fact_1447_list__update_Osimps_I1_J,axiom,
    ! [A: $tType,Ia: nat,V2: A] : ( aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),nil(A)),Ia),V2) = nil(A) ) ).

% list_update.simps(1)
tff(fact_1448_list__update__code_I1_J,axiom,
    ! [A: $tType,Ia: nat,Y: A] : ( aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),nil(A)),Ia),Y) = nil(A) ) ).

% list_update_code(1)
tff(fact_1449_butlast_Osimps_I1_J,axiom,
    ! [A: $tType] : ( aa(list(A),list(A),butlast(A),nil(A)) = nil(A) ) ).

% butlast.simps(1)
tff(fact_1450_foldl__Nil,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,fun(B,A)),A3: A] : ( aa(list(B),A,aa(A,fun(list(B),A),aa(fun(A,fun(B,A)),fun(A,fun(list(B),A)),foldl(A,B),F2),A3),nil(B)) = A3 ) ).

% foldl_Nil
tff(fact_1451_remove1_Osimps_I1_J,axiom,
    ! [A: $tType,Xa: A] : ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),remove1(A),Xa),nil(A)) = nil(A) ) ).

% remove1.simps(1)
tff(fact_1452_removeAll_Osimps_I1_J,axiom,
    ! [A: $tType,Xa: A] : ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),removeAll(A),Xa),nil(A)) = nil(A) ) ).

% removeAll.simps(1)
tff(fact_1453_merge_Osimps_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L23: list(A)] : ( merge(A,nil(A),L23) = L23 ) ) ).

% merge.simps(1)
tff(fact_1454_rotate1_Osimps_I1_J,axiom,
    ! [A: $tType] : ( aa(list(A),list(A),rotate1(A),nil(A)) = nil(A) ) ).

% rotate1.simps(1)
tff(fact_1455_zipf_Osimps_I1_J,axiom,
    ! [E: $tType,B: $tType,A: $tType,F2: fun(B,fun(E,A))] : ( zipf(B,E,A,F2,nil(B),nil(E)) = nil(A) ) ).

% zipf.simps(1)
tff(fact_1456_null__rec_I2_J,axiom,
    ! [A: $tType] : aa(list(A),$o,null(A),nil(A)) ).

% null_rec(2)
tff(fact_1457_eq__Nil__null,axiom,
    ! [A: $tType,Xsa: list(A)] :
      ( ( Xsa = nil(A) )
    <=> aa(list(A),$o,null(A),Xsa) ) ).

% eq_Nil_null
tff(fact_1458_zero__less__power,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),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)) ) ) ).

% zero_less_power
tff(fact_1459_left__right__inverse__power,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Xa: A,Y: A,N: nat] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),Xa),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),Xa),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),N)) = one_one(A) ) ) ) ).

% left_right_inverse_power
tff(fact_1460_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_1461_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_1462_power__add,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A3: A,M2: nat,N: nat] : ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),M2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)) ) ) ).

% power_add
tff(fact_1463_nat__power__less__imp__less,axiom,
    ! [Ia: nat,M2: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Ia)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Ia),M2)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Ia),N))
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),N) ) ) ).

% nat_power_less_imp_less
tff(fact_1464_list__induct2,axiom,
    ! [A: $tType,B: $tType,Xsa: list(A),Ys: list(B),Pa: fun(list(A),fun(list(B),$o))] :
      ( ( aa(list(A),nat,size_size(list(A)),Xsa) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( aa(list(B),$o,aa(list(A),fun(list(B),$o),Pa,nil(A)),nil(B))
       => ( ! [X: A,Xs: list(A),Y2: B,Ys5: list(B)] :
              ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys5) )
             => ( aa(list(B),$o,aa(list(A),fun(list(B),$o),Pa,Xs),Ys5)
               => aa(list(B),$o,aa(list(A),fun(list(B),$o),Pa,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),Y2),Ys5)) ) )
         => aa(list(B),$o,aa(list(A),fun(list(B),$o),Pa,Xsa),Ys) ) ) ) ).

% list_induct2
tff(fact_1465_list__induct3,axiom,
    ! [B: $tType,A: $tType,E: $tType,Xsa: list(A),Ys: list(B),Zs: list(E),Pa: fun(list(A),fun(list(B),fun(list(E),$o)))] :
      ( ( aa(list(A),nat,size_size(list(A)),Xsa) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( ( aa(list(B),nat,size_size(list(B)),Ys) = aa(list(E),nat,size_size(list(E)),Zs) )
       => ( aa(list(E),$o,aa(list(B),fun(list(E),$o),aa(list(A),fun(list(B),fun(list(E),$o)),Pa,nil(A)),nil(B)),nil(E))
         => ( ! [X: A,Xs: list(A),Y2: B,Ys5: list(B),Z3: E,Zs2: list(E)] :
                ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys5) )
               => ( ( aa(list(B),nat,size_size(list(B)),Ys5) = aa(list(E),nat,size_size(list(E)),Zs2) )
                 => ( aa(list(E),$o,aa(list(B),fun(list(E),$o),aa(list(A),fun(list(B),fun(list(E),$o)),Pa,Xs),Ys5),Zs2)
                   => aa(list(E),$o,aa(list(B),fun(list(E),$o),aa(list(A),fun(list(B),fun(list(E),$o)),Pa,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y2),Ys5)),aa(list(E),list(E),aa(E,fun(list(E),list(E)),cons(E),Z3),Zs2)) ) ) )
           => aa(list(E),$o,aa(list(B),fun(list(E),$o),aa(list(A),fun(list(B),fun(list(E),$o)),Pa,Xsa),Ys),Zs) ) ) ) ) ).

% list_induct3
tff(fact_1466_list__induct4,axiom,
    ! [E: $tType,A: $tType,B: $tType,F4: $tType,Xsa: list(A),Ys: list(B),Zs: list(E),Ws: list(F4),Pa: fun(list(A),fun(list(B),fun(list(E),fun(list(F4),$o))))] :
      ( ( aa(list(A),nat,size_size(list(A)),Xsa) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( ( aa(list(B),nat,size_size(list(B)),Ys) = aa(list(E),nat,size_size(list(E)),Zs) )
       => ( ( aa(list(E),nat,size_size(list(E)),Zs) = aa(list(F4),nat,size_size(list(F4)),Ws) )
         => ( aa(list(F4),$o,aa(list(E),fun(list(F4),$o),aa(list(B),fun(list(E),fun(list(F4),$o)),aa(list(A),fun(list(B),fun(list(E),fun(list(F4),$o))),Pa,nil(A)),nil(B)),nil(E)),nil(F4))
           => ( ! [X: A,Xs: list(A),Y2: B,Ys5: list(B),Z3: E,Zs2: list(E),W: F4,Ws2: list(F4)] :
                  ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys5) )
                 => ( ( aa(list(B),nat,size_size(list(B)),Ys5) = aa(list(E),nat,size_size(list(E)),Zs2) )
                   => ( ( aa(list(E),nat,size_size(list(E)),Zs2) = aa(list(F4),nat,size_size(list(F4)),Ws2) )
                     => ( aa(list(F4),$o,aa(list(E),fun(list(F4),$o),aa(list(B),fun(list(E),fun(list(F4),$o)),aa(list(A),fun(list(B),fun(list(E),fun(list(F4),$o))),Pa,Xs),Ys5),Zs2),Ws2)
                       => aa(list(F4),$o,aa(list(E),fun(list(F4),$o),aa(list(B),fun(list(E),fun(list(F4),$o)),aa(list(A),fun(list(B),fun(list(E),fun(list(F4),$o))),Pa,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),Y2),Ys5)),aa(list(E),list(E),aa(E,fun(list(E),list(E)),cons(E),Z3),Zs2)),aa(list(F4),list(F4),aa(F4,fun(list(F4),list(F4)),cons(F4),W),Ws2)) ) ) ) )
             => aa(list(F4),$o,aa(list(E),fun(list(F4),$o),aa(list(B),fun(list(E),fun(list(F4),$o)),aa(list(A),fun(list(B),fun(list(E),fun(list(F4),$o))),Pa,Xsa),Ys),Zs),Ws) ) ) ) ) ) ).

% list_induct4
tff(fact_1467_empty__set,axiom,
    ! [A: $tType] : ( bot_bot(set(A)) = aa(list(A),set(A),set2(A),nil(A)) ) ).

% empty_set
tff(fact_1468_list_Osize_I3_J,axiom,
    ! [A: $tType] : ( aa(list(A),nat,size_size(list(A)),nil(A)) = zero_zero(nat) ) ).

% list.size(3)
tff(fact_1469_len__greater__imp__nonempty,axiom,
    ! [A: $tType,Xa: nat,L: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),aa(list(A),nat,size_size(list(A)),L))
     => ( L != nil(A) ) ) ).

% len_greater_imp_nonempty
tff(fact_1470_zipf_Ocases,axiom,
    ! [E: $tType,A: $tType,B: $tType,Xa: product_prod(fun(A,fun(B,E)),product_prod(list(A),list(B)))] :
      ( ! [F3: fun(A,fun(B,E))] : ( Xa != aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,E)),product_prod(list(A),list(B))),aa(fun(A,fun(B,E)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,E)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,E)),product_prod(list(A),list(B))),F3),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))) )
     => ( ! [F3: fun(A,fun(B,E)),A4: A,As: list(A),B4: B,Bs2: list(B)] : ( Xa != aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,E)),product_prod(list(A),list(B))),aa(fun(A,fun(B,E)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,E)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,E)),product_prod(list(A),list(B))),F3),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A4),As)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B4),Bs2))) )
       => ( ! [A4: fun(A,fun(B,E)),V3: A,Va: list(A)] : ( Xa != aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,E)),product_prod(list(A),list(B))),aa(fun(A,fun(B,E)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,E)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,E)),product_prod(list(A),list(B))),A4),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),V3),Va)),nil(B))) )
         => ~ ! [A4: fun(A,fun(B,E)),V3: B,Va: list(B)] : ( Xa != aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,E)),product_prod(list(A),list(B))),aa(fun(A,fun(B,E)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,E)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,E)),product_prod(list(A),list(B))),A4),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),V3),Va))) ) ) ) ) ).

% zipf.cases
tff(fact_1471_merge_Ocases,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: product_prod(list(A),list(A))] :
          ( ! [L22: 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)),nil(A)),L22) )
         => ( ! [V3: A,Va: 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)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V3),Va)),nil(A)) )
           => ~ ! [X12: A,L1: list(A),X23: A,L22: 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)),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_1472_list__all__zip_Ocases,axiom,
    ! [A: $tType,B: $tType,Xa: product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B)))] :
      ( ! [P3: fun(A,fun(B,$o))] : ( Xa != aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$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,$o)),product_prod(list(A),list(B))),P3),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),nil(A)),nil(B))) )
     => ( ! [P3: fun(A,fun(B,$o)),A4: A,As: list(A),B4: B,Bs2: list(B)] : ( Xa != 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))),P3),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A4),As)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B4),Bs2))) )
       => ( ! [P3: fun(A,fun(B,$o)),V3: A,Va: list(A)] : ( Xa != aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$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,$o)),product_prod(list(A),list(B))),P3),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V3),Va)),nil(B))) )
         => ~ ! [P3: fun(A,fun(B,$o)),V3: B,Va: list(B)] : ( Xa != 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))),P3),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),V3),Va))) ) ) ) ) ).

% list_all_zip.cases
tff(fact_1473_partition__rev_Ocases,axiom,
    ! [A: $tType,Xa: product_prod(fun(A,$o),product_prod(product_prod(list(A),list(A)),list(A)))] :
      ( ! [P3: fun(A,$o),Yes: list(A),No: list(A)] : ( Xa != aa(product_prod(product_prod(list(A),list(A)),list(A)),product_prod(fun(A,$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,$o),product_prod(product_prod(list(A),list(A)),list(A))),P3),aa(list(A),product_prod(product_prod(list(A),list(A)),list(A)),aa(product_prod(list(A),list(A)),fun(list(A),product_prod(product_prod(list(A),list(A)),list(A))),product_Pair(product_prod(list(A),list(A)),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes),No)),nil(A))) )
     => ~ ! [P3: fun(A,$o),Yes: list(A),No: list(A),X: A,Xs: list(A)] : ( Xa != 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))),P3),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),X),Xs))) ) ) ).

% partition_rev.cases
tff(fact_1474_quicksort__by__rel_Ocases,axiom,
    ! [A: $tType,Xa: product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A)))] :
      ( ! [R4: fun(A,fun(A,$o)),Sl: list(A)] : ( Xa != aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$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,$o)),product_prod(list(A),list(A))),R4),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Sl),nil(A))) )
     => ~ ! [R4: fun(A,fun(A,$o)),Sl: list(A),X: A,Xs: list(A)] : ( Xa != 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))),R4),aa(list(A),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),X),Xs))) ) ) ).

% quicksort_by_rel.cases
tff(fact_1475_mergesort__by__rel__merge_Ocases,axiom,
    ! [A: $tType,Xa: product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A)))] :
      ( ! [R4: fun(A,fun(A,$o)),X: A,Xs: list(A),Y2: A,Ys5: list(A)] : ( Xa != aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$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,$o)),product_prod(list(A),list(A))),R4),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys5))) )
     => ( ! [R4: fun(A,fun(A,$o)),Xs: list(A)] : ( Xa != 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))),R4),aa(list(A),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))) )
       => ~ ! [R4: fun(A,fun(A,$o)),V3: A,Va: list(A)] : ( Xa != aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$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))),R4),aa(list(A),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),V3),Va))) ) ) ) ).

% mergesort_by_rel_merge.cases
tff(fact_1476_mergesort__by__rel__split_Ocases,axiom,
    ! [A: $tType,Xa: product_prod(product_prod(list(A),list(A)),list(A))] :
      ( ! [Xs1: list(A),Xs22: list(A)] : ( Xa != 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)),Xs1),Xs22)),nil(A)) )
     => ( ! [Xs1: list(A),Xs22: list(A),X: A] : ( Xa != 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)),Xs1),Xs22)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A))) )
       => ~ ! [Xs1: list(A),Xs22: list(A),X12: A,X23: A,Xs: list(A)] : ( Xa != 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)),Xs1),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),Xs))) ) ) ) ).

% mergesort_by_rel_split.cases
tff(fact_1477_list__assn_Ocases,axiom,
    ! [A: $tType,B: $tType,Xa: product_prod(fun(A,fun(B,assn)),product_prod(list(A),list(B)))] :
      ( ! [P3: fun(A,fun(B,assn))] : ( Xa != aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,assn)),product_prod(list(A),list(B))),aa(fun(A,fun(B,assn)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,assn)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,assn)),product_prod(list(A),list(B))),P3),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))) )
     => ( ! [P3: fun(A,fun(B,assn)),A4: A,As: list(A),C5: B,Cs2: list(B)] : ( Xa != aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,assn)),product_prod(list(A),list(B))),aa(fun(A,fun(B,assn)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,assn)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,assn)),product_prod(list(A),list(B))),P3),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A4),As)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),C5),Cs2))) )
       => ( ! [Uu2: fun(A,fun(B,assn)),V3: A,Va: list(A)] : ( Xa != aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,assn)),product_prod(list(A),list(B))),aa(fun(A,fun(B,assn)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,assn)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,assn)),product_prod(list(A),list(B))),Uu2),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),V3),Va)),nil(B))) )
         => ~ ! [Uu2: fun(A,fun(B,assn)),V3: B,Va: list(B)] : ( Xa != aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,assn)),product_prod(list(A),list(B))),aa(fun(A,fun(B,assn)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,assn)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,assn)),product_prod(list(A),list(B))),Uu2),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),V3),Va))) ) ) ) ) ).

% list_assn.cases
tff(fact_1478_shuffles_Ocases,axiom,
    ! [A: $tType,Xa: product_prod(list(A),list(A))] :
      ( ! [Ys5: 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)),nil(A)),Ys5) )
     => ( ! [Xs: 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)),Xs),nil(A)) )
       => ~ ! [X: A,Xs: list(A),Y2: A,Ys5: 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)),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),Y2),Ys5)) ) ) ) ).

% shuffles.cases
tff(fact_1479_map__tailrec__rev_Ocases,axiom,
    ! [A: $tType,B: $tType,Xa: product_prod(fun(A,B),product_prod(list(A),list(B)))] :
      ( ! [F3: fun(A,B),Bs2: list(B)] : ( Xa != 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))),F3),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)) )
     => ~ ! [F3: fun(A,B),A4: A,As: list(A),Bs2: list(B)] : ( Xa != 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))),F3),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A4),As)),Bs2)) ) ) ).

% map_tailrec_rev.cases
tff(fact_1480_subset__eq__mset__impl_Ocases,axiom,
    ! [A: $tType,Xa: product_prod(list(A),list(A))] :
      ( ! [Ys5: 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)),nil(A)),Ys5) )
     => ~ ! [X: A,Xs: list(A),Ys5: 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)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),Ys5) ) ) ).

% subset_eq_mset_impl.cases
tff(fact_1481_sorted0,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => sorted_wrt(A,ord_less_eq(A),nil(A)) ) ).

% sorted0
tff(fact_1482_strict__sorted__simps_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => sorted_wrt(A,ord_less(A),nil(A)) ) ).

% strict_sorted_simps(1)
tff(fact_1483_distinct__singleton,axiom,
    ! [A: $tType,Xa: A] : aa(list(A),$o,distinct(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),nil(A))) ).

% distinct_singleton
tff(fact_1484_sorted__wrt1,axiom,
    ! [A: $tType,Pa: fun(A,fun(A,$o)),Xa: A] : sorted_wrt(A,Pa,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),nil(A))) ).

% sorted_wrt1
tff(fact_1485_butlast_Osimps_I2_J,axiom,
    ! [A: $tType,Xa: A,Xsa: list(A)] :
      ( aa(list(A),list(A),butlast(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa)) = $ite(Xsa = nil(A),nil(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),aa(list(A),list(A),butlast(A),Xsa))) ) ).

% butlast.simps(2)
tff(fact_1486_not__listrel1__Nil,axiom,
    ! [A: $tType,Xsa: list(A),R2: set(product_prod(A,A))] : ~ member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xsa),nil(A)),listrel1(A,R2)) ).

% not_listrel1_Nil
tff(fact_1487_not__Nil__listrel1,axiom,
    ! [A: $tType,Xsa: list(A),R2: set(product_prod(A,A))] : ~ member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),Xsa),listrel1(A,R2)) ).

% not_Nil_listrel1
tff(fact_1488_listrel__Nil2,axiom,
    ! [B: $tType,A: $tType,Xsa: list(A),R2: set(product_prod(A,B))] :
      ( member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xsa),nil(B)),listrel(A,B,R2))
     => ( Xsa = nil(A) ) ) ).

% listrel_Nil2
tff(fact_1489_listrel__Nil1,axiom,
    ! [A: $tType,B: $tType,Xsa: list(B),R2: set(product_prod(A,B))] :
      ( member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),nil(A)),Xsa),listrel(A,B,R2))
     => ( Xsa = nil(B) ) ) ).

% listrel_Nil1
tff(fact_1490_listrel_ONil,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,B))] : member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),nil(A)),nil(B)),listrel(A,B,R2)) ).

% listrel.Nil
tff(fact_1491_merge_Osimps_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [V2: A,Va2: list(A)] : ( merge(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V2),Va2),nil(A)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V2),Va2) ) ) ).

% merge.simps(2)
tff(fact_1492_list__assn__simps_I1_J,axiom,
    ! [B: $tType,A: $tType,Pa: fun(A,fun(B,assn))] : ( vEBT_List_list_assn(A,B,Pa,nil(A),nil(B)) = one_one(assn) ) ).

% list_assn_simps(1)
tff(fact_1493_list__assn_Osimps_I1_J,axiom,
    ! [B: $tType,A: $tType,Pa: fun(A,fun(B,assn))] : ( vEBT_List_list_assn(A,B,Pa,nil(A),nil(B)) = one_one(assn) ) ).

% list_assn.simps(1)
tff(fact_1494_power__less__imp__less__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N: nat,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B3),N))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3) ) ) ) ).

% power_less_imp_less_base
tff(fact_1495_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_1496_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_1497_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_1498_power__strict__increasing,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat,N7: nat,A3: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),N7)
         => ( aa(A,$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),N7)) ) ) ) ).

% power_strict_increasing
tff(fact_1499_power__less__imp__less__exp,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,M2: nat,N: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),M2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N))
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),N) ) ) ) ).

% power_less_imp_less_exp
tff(fact_1500_zero__power,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [N: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),N) = zero_zero(A) ) ) ) ).

% zero_power
tff(fact_1501_power__gt__expt,axiom,
    ! [N: nat,K3: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),N)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K3),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),N),K3)) ) ).

% power_gt_expt
tff(fact_1502_lenlex__irreflexive,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),Xsa: list(A)] :
      ( ! [X: A] : ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),X),R2)
     => ~ member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xsa),Xsa),lenlex(A,R2)) ) ).

% lenlex_irreflexive
tff(fact_1503_length__compl__induct,axiom,
    ! [A: $tType,Pa: fun(list(A),$o),L: list(A)] :
      ( aa(list(A),$o,Pa,nil(A))
     => ( ! [E2: A,L3: list(A)] :
            ( ! [Ll: list(A)] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Ll)),aa(list(A),nat,size_size(list(A)),L3))
               => aa(list(A),$o,Pa,Ll) )
           => aa(list(A),$o,Pa,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E2),L3)) )
       => aa(list(A),$o,Pa,L) ) ) ).

% length_compl_induct
tff(fact_1504_sorted1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A] : sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),nil(A))) ) ).

% sorted1
tff(fact_1505_list__decomp__1,axiom,
    ! [A: $tType,L: list(A)] :
      ( ( aa(list(A),nat,size_size(list(A)),L) = one_one(nat) )
     => ? [A4: A] : ( L = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A4),nil(A)) ) ) ).

% list_decomp_1
tff(fact_1506_butlast__subset,axiom,
    ! [A: $tType,Xsa: list(A),Aa2: set(A)] :
      ( ( Xsa != nil(A) )
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xsa)),Aa2)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),aa(list(A),list(A),butlast(A),Xsa))),Aa2) ) ) ).

% butlast_subset
tff(fact_1507_sorted__butlast,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xsa: list(A)] :
          ( ( Xsa != nil(A) )
         => ( sorted_wrt(A,ord_less_eq(A),Xsa)
           => sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),butlast(A),Xsa)) ) ) ) ).

% sorted_butlast
tff(fact_1508_merge_Oelims,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: list(A),Xaa: list(A),Y: list(A)] :
          ( ( merge(A,Xa,Xaa) = Y )
         => ( ( ( Xa = nil(A) )
             => ( Y != Xaa ) )
           => ( ! [V3: A,Va: list(A)] :
                  ( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V3),Va) )
                 => ( ( Xaa = nil(A) )
                   => ( Y != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V3),Va) ) ) )
             => ~ ! [X12: A,L1: list(A)] :
                    ( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X12),L1) )
                   => ! [X23: A,L22: list(A)] :
                        ( ( Xaa = 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))) ) ) ) ) ) ) ) ) ).

% merge.elims
tff(fact_1509_subset__code_I3_J,axiom,
    ! [A: $tType] : ~ aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),coset(A,nil(A))),aa(list(A),set(A),set2(A),nil(A))) ).

% subset_code(3)
tff(fact_1510_list__assn__simps_I3_J,axiom,
    ! [B: $tType,A: $tType,Pa: fun(A,fun(B,assn)),A3: A,As2: list(A)] : ( vEBT_List_list_assn(A,B,Pa,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),As2),nil(B)) = bot_bot(assn) ) ).

% list_assn_simps(3)
tff(fact_1511_list__assn__simps_I4_J,axiom,
    ! [B: $tType,A: $tType,Pa: fun(A,fun(B,assn)),C2: B,Cs: list(B)] : ( vEBT_List_list_assn(A,B,Pa,nil(A),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),C2),Cs)) = bot_bot(assn) ) ).

% list_assn_simps(4)
tff(fact_1512_list__assn_Osimps_I3_J,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,fun(B,assn)),V2: A,Va2: list(A)] : ( vEBT_List_list_assn(A,B,Uu,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V2),Va2),nil(B)) = bot_bot(assn) ) ).

% list_assn.simps(3)
tff(fact_1513_list__assn_Osimps_I4_J,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,fun(B,assn)),V2: B,Va2: list(B)] : ( vEBT_List_list_assn(A,B,Uu,nil(A),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),V2),Va2)) = bot_bot(assn) ) ).

% list_assn.simps(4)
tff(fact_1514_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_1515_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_1516_power__strict__decreasing,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat,N7: nat,A3: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),N7)
         => ( aa(A,$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),N7)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)) ) ) ) ) ).

% power_strict_decreasing
tff(fact_1517_power__eq__iff__eq__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat,A3: A,B3: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B3)
             => ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N) = aa(nat,A,aa(A,fun(nat,A),power_power(A),B3),N) )
              <=> ( A3 = B3 ) ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_1518_power__eq__imp__eq__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N: nat,B3: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N) = aa(nat,A,aa(A,fun(nat,A),power_power(A),B3),N) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B3)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
               => ( A3 = B3 ) ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_1519_power__le__imp__le__exp,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,M2: nat,N: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),M2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N))
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),N) ) ) ) ).

% power_le_imp_le_exp
tff(fact_1520_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_1521_the__elem__set,axiom,
    ! [A: $tType,Xa: A] : ( the_elem(A,aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),nil(A)))) = Xa ) ).

% the_elem_set
tff(fact_1522_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_1523_measures__less,axiom,
    ! [A: $tType,F2: fun(A,nat),Xa: A,Y: A,Fs: list(fun(A,nat))] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,F2,Xa)),aa(A,nat,F2,Y))
     => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),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)),F2),Fs))) ) ).

% measures_less
tff(fact_1524_measures__lesseq,axiom,
    ! [A: $tType,F2: fun(A,nat),Xa: A,Y: A,Fs: list(fun(A,nat))] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,F2,Xa)),aa(A,nat,F2,Y))
     => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Y),measures(A,Fs))
       => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),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)),F2),Fs))) ) ) ).

% measures_lesseq
tff(fact_1525_count__le__length,axiom,
    ! [A: $tType,Xsa: list(A),Xa: A] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,count_list(A,Xsa),Xa)),aa(list(A),nat,size_size(list(A)),Xsa)) ).

% count_le_length
tff(fact_1526_lenlex__length,axiom,
    ! [A: $tType,Ms: list(A),Ns: list(A),R2: set(product_prod(A,A))] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Ms),Ns),lenlex(A,R2))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Ms)),aa(list(A),nat,size_size(list(A)),Ns)) ) ).

% lenlex_length
tff(fact_1527_listrel_Osimps,axiom,
    ! [B: $tType,A: $tType,A1: list(A),A22: list(B),R2: set(product_prod(A,B))] :
      ( member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),A1),A22),listrel(A,B,R2))
    <=> ( ( ( A1 = nil(A) )
          & ( A22 = nil(B) ) )
        | ? [X6: A,Y4: B,Xs2: list(A),Ys4: list(B)] :
            ( ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X6),Xs2) )
            & ( A22 = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y4),Ys4) )
            & member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X6),Y4),R2)
            & member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xs2),Ys4),listrel(A,B,R2)) ) ) ) ).

% listrel.simps
tff(fact_1528_listrel_Ocases,axiom,
    ! [B: $tType,A: $tType,A1: list(A),A22: list(B),R2: set(product_prod(A,B))] :
      ( member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),A1),A22),listrel(A,B,R2))
     => ( ( ( A1 = nil(A) )
         => ( A22 != nil(B) ) )
       => ~ ! [X: A,Y2: B,Xs: list(A)] :
              ( ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) )
             => ! [Ys5: list(B)] :
                  ( ( A22 = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y2),Ys5) )
                 => ( 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),R2)
                   => ~ member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xs),Ys5),listrel(A,B,R2)) ) ) ) ) ) ).

% listrel.cases
tff(fact_1529_power__strict__mono,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,B3: A,N: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B3),N)) ) ) ) ) ).

% power_strict_mono
tff(fact_1530_power__eq__if,axiom,
    ! [A: $tType] :
      ( power(A)
     => ! [P2: A,M2: nat] :
          ( aa(nat,A,aa(A,fun(nat,A),power_power(A),P2),M2) = $ite(M2 = zero_zero(nat),one_one(A),aa(A,A,aa(A,fun(A,A),times_times(A),P2),aa(nat,A,aa(A,fun(nat,A),power_power(A),P2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),one_one(nat))))) ) ) ).

% power_eq_if
tff(fact_1531_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_1532_subset__mset_Osum__list__update,axiom,
    ! [A: $tType,K3: nat,Xsa: list(multiset(A)),Xa: multiset(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K3),aa(list(multiset(A)),nat,size_size(list(multiset(A))),Xsa))
     => ( groups4543113879258116180m_list(multiset(A),plus_plus(multiset(A)),zero_zero(multiset(A)),aa(multiset(A),list(multiset(A)),aa(nat,fun(multiset(A),list(multiset(A))),aa(list(multiset(A)),fun(nat,fun(multiset(A),list(multiset(A)))),list_update(multiset(A)),Xsa),K3),Xa)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),groups4543113879258116180m_list(multiset(A),plus_plus(multiset(A)),zero_zero(multiset(A)),Xsa)),Xa)),aa(nat,multiset(A),nth(multiset(A),Xsa),K3)) ) ) ).

% subset_mset.sum_list_update
tff(fact_1533_n__lists__Nil,axiom,
    ! [A: $tType,N: nat] :
      ( n_lists(A,N,nil(A)) = $ite(N = zero_zero(nat),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),nil(A)),nil(list(A))),nil(list(A))) ) ).

% n_lists_Nil
tff(fact_1534_lex__take__index,axiom,
    ! [A: $tType,Xsa: list(A),Ys: list(A),R2: set(product_prod(A,A))] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xsa),Ys),lex(A,R2))
     => ~ ! [I3: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),Xsa))
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),Ys))
             => ( ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),I3),Xsa) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),I3),Ys) )
               => ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(nat,A,nth(A,Xsa),I3)),aa(nat,A,nth(A,Ys),I3)),R2) ) ) ) ) ).

% lex_take_index
tff(fact_1535_of__nat__zero__less__power__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xa: nat,N: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,semiring_1_of_nat(A),Xa)),N))
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Xa)
            | ( N = zero_zero(nat) ) ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_1536_take__minus__one__conv__butlast,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))
     => ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,suc,zero_zero(nat)))),L) = aa(list(A),list(A),butlast(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),L)) ) ) ).

% take_minus_one_conv_butlast
tff(fact_1537_take__butlast__conv,axiom,
    ! [A: $tType,L: list(A)] : ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),L)),aa(nat,nat,suc,zero_zero(nat)))),L) = aa(list(A),list(A),butlast(A),L) ) ).

% take_butlast_conv
tff(fact_1538_refl__on__singleton,axiom,
    ! [A: $tType,Xa: A] : refl_on(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),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))),insert3(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Xa)),bot_bot(set(product_prod(A,A))))) ).

% refl_on_singleton
tff(fact_1539_greaterThan__Suc,axiom,
    ! [K3: nat] : ( aa(nat,set(nat),set_ord_greaterThan(nat),aa(nat,nat,suc,K3)) = 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),K3)),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert3(nat),aa(nat,nat,suc,K3)),bot_bot(set(nat)))) ) ).

% greaterThan_Suc
tff(fact_1540_greaterThan__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Ia: A,K3: A] :
          ( member(A,Ia,aa(A,set(A),set_ord_greaterThan(A),K3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),K3),Ia) ) ) ).

% greaterThan_iff
tff(fact_1541_take__update,axiom,
    ! [A: $tType,N: nat,L: list(A),Ia: nat,Xa: A] : ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),L),Ia),Xa)) = aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),L)),Ia),Xa) ) ).

% take_update
tff(fact_1542_take__update__swap,axiom,
    ! [A: $tType,M2: nat,Xsa: list(A),N: nat,Xa: A] : ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),M2),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xsa),N),Xa)) = aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),M2),Xsa)),N),Xa) ) ).

% take_update_swap
tff(fact_1543_take__take,axiom,
    ! [A: $tType,N: nat,M2: nat,Xsa: list(A)] : ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),M2),Xsa)) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),N),M2)),Xsa) ) ).

% take_take
tff(fact_1544_of__nat__less__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M2: nat,N: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,semiring_1_of_nat(A),M2)),aa(nat,A,semiring_1_of_nat(A),N))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),N) ) ) ).

% of_nat_less_iff
tff(fact_1545_of__nat__mult,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [M2: nat,N: nat] : ( aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M2),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),M2)),aa(nat,A,semiring_1_of_nat(A),N)) ) ) ).

% of_nat_mult
tff(fact_1546_take__Suc__Cons,axiom,
    ! [A: $tType,N: nat,Xa: A,Xsa: list(A)] : ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),aa(nat,nat,suc,N)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),Xsa)) ) ).

% take_Suc_Cons
tff(fact_1547_take0,axiom,
    ! [A: $tType,X4: list(A)] : ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),zero_zero(nat)),X4) = nil(A) ) ).

% take0
tff(fact_1548_take__eq__Nil,axiom,
    ! [A: $tType,N: nat,Xsa: list(A)] :
      ( ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),Xsa) = nil(A) )
    <=> ( ( N = zero_zero(nat) )
        | ( Xsa = nil(A) ) ) ) ).

% take_eq_Nil
tff(fact_1549_take__eq__Nil2,axiom,
    ! [A: $tType,N: nat,Xsa: list(A)] :
      ( ( nil(A) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),Xsa) )
    <=> ( ( N = zero_zero(nat) )
        | ( Xsa = nil(A) ) ) ) ).

% take_eq_Nil2
tff(fact_1550_take__all,axiom,
    ! [A: $tType,Xsa: list(A),N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xsa)),N)
     => ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),Xsa) = Xsa ) ) ).

% take_all
tff(fact_1551_take__all__iff,axiom,
    ! [A: $tType,N: nat,Xsa: list(A)] :
      ( ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),Xsa) = Xsa )
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xsa)),N) ) ).

% take_all_iff
tff(fact_1552_nth__take,axiom,
    ! [A: $tType,Ia: nat,N: nat,Xsa: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),N)
     => ( aa(nat,A,nth(A,aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),Xsa)),Ia) = aa(nat,A,nth(A,Xsa),Ia) ) ) ).

% nth_take
tff(fact_1553_take__update__cancel,axiom,
    ! [A: $tType,N: nat,M2: nat,Xsa: list(A),Y: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),M2)
     => ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xsa),M2),Y)) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),Xsa) ) ) ).

% take_update_cancel
tff(fact_1554_length__take,axiom,
    ! [A: $tType,N: nat,Xsa: list(A)] : ( aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),Xsa)) = aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(list(A),nat,size_size(list(A)),Xsa)),N) ) ).

% length_take
tff(fact_1555_subset__mset_Osum__list__eq__0__iff,axiom,
    ! [A: $tType,Ns: list(multiset(A))] :
      ( ( groups4543113879258116180m_list(multiset(A),plus_plus(multiset(A)),zero_zero(multiset(A)),Ns) = zero_zero(multiset(A)) )
    <=> ! [X6: multiset(A)] :
          ( member(multiset(A),X6,aa(list(multiset(A)),set(multiset(A)),set2(multiset(A)),Ns))
         => ( X6 = zero_zero(multiset(A)) ) ) ) ).

% subset_mset.sum_list_eq_0_iff
tff(fact_1556_of__nat__0__less__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [N: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,semiring_1_of_nat(A),N))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N) ) ) ).

% of_nat_0_less_iff
tff(fact_1557_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B3: nat,W2: nat,Xa: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,semiring_1_of_nat(A),B3)),W2)),aa(nat,A,semiring_1_of_nat(A),Xa))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B3),W2)),Xa) ) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_1558_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xa: nat,B3: nat,W2: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,semiring_1_of_nat(A),Xa)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,semiring_1_of_nat(A),B3)),W2))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B3),W2)) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_1559_merge__list_Ocases,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: product_prod(list(list(A)),list(list(A)))] :
          ( ( Xa != 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))) )
         => ( ! [L3: list(A)] : ( Xa != 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)),L3),nil(list(A)))) )
           => ( ! [La: list(A),Acc2: list(list(A))] : ( Xa != 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),Acc2)),nil(list(A))) )
             => ( ! [La: list(A),Acc2: list(list(A)),L3: list(A)] : ( Xa != 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),Acc2)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L3),nil(list(A)))) )
               => ~ ! [Acc2: list(list(A)),L1: list(A),L22: list(A),Ls: list(list(A))] : ( Xa != 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))),Acc2),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_1560_mergesort__by__rel_Ocases,axiom,
    ! [A: $tType,Xa: product_prod(fun(A,fun(A,$o)),list(A))] :
      ~ ! [R4: fun(A,fun(A,$o)),Xs: list(A)] : ( Xa != aa(list(A),product_prod(fun(A,fun(A,$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)),R4),Xs) ) ).

% mergesort_by_rel.cases
tff(fact_1561_take__equalityI,axiom,
    ! [A: $tType,Xsa: list(A),Ys: list(A)] :
      ( ! [I3: nat] : ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),I3),Xsa) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),I3),Ys) )
     => ( Xsa = Ys ) ) ).

% take_equalityI
tff(fact_1562_take__zip,axiom,
    ! [A: $tType,B: $tType,N: nat,Xsa: list(A),Ys: list(B)] : ( aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(nat,fun(list(product_prod(A,B)),list(product_prod(A,B))),take(product_prod(A,B)),N),aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),Xsa),Ys)) = aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),Xsa)),aa(list(B),list(B),aa(nat,fun(list(B),list(B)),take(B),N),Ys)) ) ).

% take_zip
tff(fact_1563_take__Nil,axiom,
    ! [A: $tType,N: nat] : ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),nil(A)) = nil(A) ) ).

% take_Nil
tff(fact_1564_in__set__takeD,axiom,
    ! [A: $tType,Xa: A,N: nat,Xsa: list(A)] :
      ( member(A,Xa,aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),Xsa)))
     => member(A,Xa,aa(list(A),set(A),set2(A),Xsa)) ) ).

% in_set_takeD
tff(fact_1565_mult__of__nat__commute,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Xa: nat,Y: A] : ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Xa)),Y) = aa(A,A,aa(A,fun(A,A),times_times(A),Y),aa(nat,A,semiring_1_of_nat(A),Xa)) ) ) ).

% mult_of_nat_commute
tff(fact_1566_distinct__take,axiom,
    ! [A: $tType,Xsa: list(A),Ia: nat] :
      ( aa(list(A),$o,distinct(A),Xsa)
     => aa(list(A),$o,distinct(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),Ia),Xsa)) ) ).

% distinct_take
tff(fact_1567_sorted__wrt__take,axiom,
    ! [A: $tType,F2: fun(A,fun(A,$o)),Xsa: list(A),N: nat] :
      ( sorted_wrt(A,F2,Xsa)
     => sorted_wrt(A,F2,aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),Xsa)) ) ).

% sorted_wrt_take
tff(fact_1568_nat__int__comparison_I2_J,axiom,
    ! [A3: nat,B3: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A3),B3)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(nat,int,semiring_1_of_nat(int),A3)),aa(nat,int,semiring_1_of_nat(int),B3)) ) ).

% nat_int_comparison(2)
tff(fact_1569_refl__onD,axiom,
    ! [A: $tType,Aa2: set(A),R2: set(product_prod(A,A)),A3: A] :
      ( refl_on(A,Aa2,R2)
     => ( member(A,A3,Aa2)
       => 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_1570_refl__onD1,axiom,
    ! [A: $tType,Aa2: set(A),R2: set(product_prod(A,A)),Xa: A,Y: A] :
      ( refl_on(A,Aa2,R2)
     => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Y),R2)
       => member(A,Xa,Aa2) ) ) ).

% refl_onD1
tff(fact_1571_refl__onD2,axiom,
    ! [A: $tType,Aa2: set(A),R2: set(product_prod(A,A)),Xa: A,Y: A] :
      ( refl_on(A,Aa2,R2)
     => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Y),R2)
       => member(A,Y,Aa2) ) ) ).

% refl_onD2
tff(fact_1572_refl__on__Int,axiom,
    ! [A: $tType,Aa2: set(A),R2: set(product_prod(A,A)),B2: set(A),S2: set(product_prod(A,A))] :
      ( refl_on(A,Aa2,R2)
     => ( refl_on(A,B2,S2)
       => refl_on(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),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))),inf_inf(set(product_prod(A,A))),R2),S2)) ) ) ).

% refl_on_Int
tff(fact_1573_greaterThan__non__empty,axiom,
    ! [A: $tType] :
      ( no_top(A)
     => ! [Xa: A] : ( aa(A,set(A),set_ord_greaterThan(A),Xa) != bot_bot(set(A)) ) ) ).

% greaterThan_non_empty
tff(fact_1574_take__0,axiom,
    ! [A: $tType,Xsa: list(A)] : ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),zero_zero(nat)),Xsa) = nil(A) ) ).

% take_0
tff(fact_1575_set__take__subset,axiom,
    ! [A: $tType,N: nat,Xsa: list(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),Xsa))),aa(list(A),set(A),set2(A),Xsa)) ).

% set_take_subset
tff(fact_1576_sorted__take,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xsa: list(A),N: nat] :
          ( sorted_wrt(A,ord_less_eq(A),Xsa)
         => sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),Xsa)) ) ) ).

% sorted_take
tff(fact_1577_of__nat__less__0__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M2: nat] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,semiring_1_of_nat(A),M2)),zero_zero(A)) ) ).

% of_nat_less_0_iff
tff(fact_1578_less__imp__of__nat__less,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M2: nat,N: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),N)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,semiring_1_of_nat(A),M2)),aa(nat,A,semiring_1_of_nat(A),N)) ) ) ).

% less_imp_of_nat_less
tff(fact_1579_of__nat__less__imp__less,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M2: nat,N: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,semiring_1_of_nat(A),M2)),aa(nat,A,semiring_1_of_nat(A),N))
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),N) ) ) ).

% of_nat_less_imp_less
tff(fact_1580_length__n__lists__elem,axiom,
    ! [A: $tType,Ys: list(A),N: nat,Xsa: list(A)] :
      ( member(list(A),Ys,aa(list(list(A)),set(list(A)),set2(list(A)),n_lists(A,N,Xsa)))
     => ( aa(list(A),nat,size_size(list(A)),Ys) = N ) ) ).

% length_n_lists_elem
tff(fact_1581_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_1582_refl__on__Un,axiom,
    ! [A: $tType,Aa2: set(A),R2: set(product_prod(A,A)),B2: set(A),S2: set(product_prod(A,A))] :
      ( refl_on(A,Aa2,R2)
     => ( refl_on(A,B2,S2)
       => refl_on(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),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))),sup_sup(set(product_prod(A,A))),R2),S2)) ) ) ).

% refl_on_Un
tff(fact_1583_set__take__subset__set__take,axiom,
    ! [A: $tType,M2: nat,N: nat,Xsa: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),N)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),M2),Xsa))),aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),Xsa))) ) ).

% set_take_subset_set_take
tff(fact_1584_take__butlast,axiom,
    ! [A: $tType,N: nat,Xsa: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xsa))
     => ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),aa(list(A),list(A),butlast(A),Xsa)) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),Xsa) ) ) ).

% take_butlast
tff(fact_1585_of__nat__diff,axiom,
    ! [A: $tType] :
      ( semiring_1_cancel(A)
     => ! [N: nat,M2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),M2)
         => ( aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),N)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,semiring_1_of_nat(A),M2)),aa(nat,A,semiring_1_of_nat(A),N)) ) ) ) ).

% of_nat_diff
tff(fact_1586_zip__obtain__same__length,axiom,
    ! [A: $tType,B: $tType,Xsa: list(A),Ys: list(B),Pa: fun(list(product_prod(A,B)),$o)] :
      ( ! [Zs2: list(A),Ws2: list(B),N5: nat] :
          ( ( aa(list(A),nat,size_size(list(A)),Zs2) = aa(list(B),nat,size_size(list(B)),Ws2) )
         => ( ( N5 = aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(list(A),nat,size_size(list(A)),Xsa)),aa(list(B),nat,size_size(list(B)),Ys)) )
           => ( ( Zs2 = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N5),Xsa) )
             => ( ( Ws2 = aa(list(B),list(B),aa(nat,fun(list(B),list(B)),take(B),N5),Ys) )
               => aa(list(product_prod(A,B)),$o,Pa,aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),Zs2),Ws2)) ) ) ) )
     => aa(list(product_prod(A,B)),$o,Pa,aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),Xsa),Ys)) ) ).

% zip_obtain_same_length
tff(fact_1587_zdiff__int__split,axiom,
    ! [Pa: fun(int,$o),Xa: nat,Y: nat] :
      ( aa(int,$o,Pa,aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Xa),Y)))
    <=> ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y),Xa)
         => aa(int,$o,Pa,aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,semiring_1_of_nat(int),Xa)),aa(nat,int,semiring_1_of_nat(int),Y))) )
        & ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Y)
         => aa(int,$o,Pa,zero_zero(int)) ) ) ) ).

% zdiff_int_split
tff(fact_1588_nth__take__lemma,axiom,
    ! [A: $tType,K3: nat,Xsa: list(A),Ys: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K3),aa(list(A),nat,size_size(list(A)),Xsa))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K3),aa(list(A),nat,size_size(list(A)),Ys))
       => ( ! [I3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),K3)
             => ( aa(nat,A,nth(A,Xsa),I3) = aa(nat,A,nth(A,Ys),I3) ) )
         => ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),K3),Xsa) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),K3),Ys) ) ) ) ) ).

% nth_take_lemma
tff(fact_1589_butlast__conv__take,axiom,
    ! [A: $tType,Xsa: list(A)] : ( aa(list(A),list(A),butlast(A),Xsa) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xsa)),one_one(nat))),Xsa) ) ).

% butlast_conv_take
tff(fact_1590_n__lists_Osimps_I1_J,axiom,
    ! [A: $tType,Xsa: list(A)] : ( n_lists(A,zero_zero(nat),Xsa) = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),nil(A)),nil(list(A))) ) ).

% n_lists.simps(1)
tff(fact_1591_take__Cons_H,axiom,
    ! [A: $tType,N: nat,Xa: A,Xsa: list(A)] :
      ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa)) = $ite(N = zero_zero(nat),nil(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))),Xsa))) ) ).

% take_Cons'
tff(fact_1592_butlast__take,axiom,
    ! [A: $tType,N: nat,Xsa: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),aa(list(A),nat,size_size(list(A)),Xsa))
     => ( aa(list(A),list(A),butlast(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),Xsa)) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))),Xsa) ) ) ).

% butlast_take
tff(fact_1593_ex__less__of__nat__mult,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xa)
         => ? [N5: nat] : aa(A,$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),N5)),Xa)) ) ) ).

% ex_less_of_nat_mult
tff(fact_1594_length__n__lists,axiom,
    ! [A: $tType,N: nat,Xsa: list(A)] : ( aa(list(list(A)),nat,size_size(list(list(A))),n_lists(A,N,Xsa)) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(list(A),nat,size_size(list(A)),Xsa)),N) ) ).

% length_n_lists
tff(fact_1595_subset__mset_Oelem__le__sum__list,axiom,
    ! [A: $tType,K3: nat,Ns: list(multiset(A))] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K3),aa(list(multiset(A)),nat,size_size(list(multiset(A))),Ns))
     => aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),aa(nat,multiset(A),nth(multiset(A),Ns),K3)),groups4543113879258116180m_list(multiset(A),plus_plus(multiset(A)),zero_zero(multiset(A)),Ns)) ) ).

% subset_mset.elem_le_sum_list
tff(fact_1596_distinct__n__lists,axiom,
    ! [A: $tType,Xsa: list(A),N: nat] :
      ( aa(list(A),$o,distinct(A),Xsa)
     => aa(list(list(A)),$o,distinct(list(A)),n_lists(A,N,Xsa)) ) ).

% distinct_n_lists
tff(fact_1597_lexord__take__index__conv,axiom,
    ! [A: $tType,Xa: list(A),Y: list(A),R2: set(product_prod(A,A))] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xa),Y),lexord(A,R2))
    <=> ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),Xa)),aa(list(A),nat,size_size(list(A)),Y))
          & ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),aa(list(A),nat,size_size(list(A)),Xa)),Y) = Xa ) )
        | ? [I2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(list(A),nat,size_size(list(A)),Xa)),aa(list(A),nat,size_size(list(A)),Y)))
            & ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),I2),Xa) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),I2),Y) )
            & 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,Xa),I2)),aa(nat,A,nth(A,Y),I2)),R2) ) ) ) ).

% lexord_take_index_conv
tff(fact_1598_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,aa(list(A),list(A),aa(nat,fun(list(A),list(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_1599_refl__on__domain,axiom,
    ! [A: $tType,Aa2: set(A),R2: set(product_prod(A,A)),A3: A,B3: A] :
      ( refl_on(A,Aa2,R2)
     => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B3),R2)
       => ( member(A,A3,Aa2)
          & member(A,B3,Aa2) ) ) ) ).

% refl_on_domain
tff(fact_1600_take__update__last,axiom,
    ! [A: $tType,N: nat,List: list(A),Xa: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),List))
     => ( aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),aa(nat,nat,suc,N)),List)),N),Xa) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),List)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),nil(A))) ) ) ).

% take_update_last
tff(fact_1601_append_Oassoc,axiom,
    ! [A: $tType,A3: list(A),B3: list(A),C2: list(A)] : ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),A3),B3)),C2) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),A3),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),B3),C2)) ) ).

% append.assoc
tff(fact_1602_append__assoc,axiom,
    ! [A: $tType,Xsa: list(A),Ys: list(A),Zs: list(A)] : ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),Ys)),Zs) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Zs)) ) ).

% append_assoc
tff(fact_1603_append__same__eq,axiom,
    ! [A: $tType,Ys: list(A),Xsa: list(A),Zs: list(A)] :
      ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Xsa) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Zs),Xsa) )
    <=> ( Ys = Zs ) ) ).

% append_same_eq
tff(fact_1604_same__append__eq,axiom,
    ! [A: $tType,Xsa: list(A),Ys: list(A),Zs: list(A)] :
      ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),Ys) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),Zs) )
    <=> ( Ys = Zs ) ) ).

% same_append_eq
tff(fact_1605_empty__append__eq__id,axiom,
    ! [A: $tType,X4: list(A)] : ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),nil(A)),X4) = X4 ) ).

% empty_append_eq_id
tff(fact_1606_append_Oright__neutral,axiom,
    ! [A: $tType,A3: list(A)] : ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),A3),nil(A)) = A3 ) ).

% append.right_neutral
tff(fact_1607_append__Nil2,axiom,
    ! [A: $tType,Xsa: list(A)] : ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),nil(A)) = Xsa ) ).

% append_Nil2
tff(fact_1608_append__self__conv,axiom,
    ! [A: $tType,Xsa: list(A),Ys: list(A)] :
      ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),Ys) = Xsa )
    <=> ( Ys = nil(A) ) ) ).

% append_self_conv
tff(fact_1609_self__append__conv,axiom,
    ! [A: $tType,Y: list(A),Ys: list(A)] :
      ( ( Y = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Y),Ys) )
    <=> ( Ys = nil(A) ) ) ).

% self_append_conv
tff(fact_1610_append__self__conv2,axiom,
    ! [A: $tType,Xsa: list(A),Ys: list(A)] :
      ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),Ys) = Ys )
    <=> ( Xsa = nil(A) ) ) ).

% append_self_conv2
tff(fact_1611_self__append__conv2,axiom,
    ! [A: $tType,Y: list(A),Xsa: list(A)] :
      ( ( Y = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),Y) )
    <=> ( Xsa = nil(A) ) ) ).

% self_append_conv2
tff(fact_1612_Nil__is__append__conv,axiom,
    ! [A: $tType,Xsa: list(A),Ys: list(A)] :
      ( ( nil(A) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),Ys) )
    <=> ( ( Xsa = nil(A) )
        & ( Ys = nil(A) ) ) ) ).

% Nil_is_append_conv
tff(fact_1613_append__is__Nil__conv,axiom,
    ! [A: $tType,Xsa: list(A),Ys: list(A)] :
      ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),Ys) = nil(A) )
    <=> ( ( Xsa = nil(A) )
        & ( Ys = nil(A) ) ) ) ).

% append_is_Nil_conv
tff(fact_1614_append__eq__append__conv,axiom,
    ! [A: $tType,Xsa: list(A),Ys: list(A),Us: list(A),Vs: list(A)] :
      ( ( ( aa(list(A),nat,size_size(list(A)),Xsa) = aa(list(A),nat,size_size(list(A)),Ys) )
        | ( aa(list(A),nat,size_size(list(A)),Us) = aa(list(A),nat,size_size(list(A)),Vs) ) )
     => ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),Us) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Vs) )
      <=> ( ( Xsa = Ys )
          & ( Us = Vs ) ) ) ) ).

% append_eq_append_conv
tff(fact_1615_rev__append,axiom,
    ! [A: $tType,Xsa: list(A),Ys: list(A)] : ( aa(list(A),list(A),rev(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),Ys)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),rev(A),Ys)),aa(list(A),list(A),rev(A),Xsa)) ) ).

% rev_append
tff(fact_1616_foldl__append,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,fun(B,A)),A3: A,Xsa: list(B),Ys: list(B)] : ( aa(list(B),A,aa(A,fun(list(B),A),aa(fun(A,fun(B,A)),fun(A,fun(list(B),A)),foldl(A,B),F2),A3),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Xsa),Ys)) = aa(list(B),A,aa(A,fun(list(B),A),aa(fun(A,fun(B,A)),fun(A,fun(list(B),A)),foldl(A,B),F2),aa(list(B),A,aa(A,fun(list(B),A),aa(fun(A,fun(B,A)),fun(A,fun(list(B),A)),foldl(A,B),F2),A3),Xsa)),Ys) ) ).

% foldl_append
tff(fact_1617_removeAll__append,axiom,
    ! [A: $tType,Xa: A,Xsa: list(A),Ys: list(A)] : ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),removeAll(A),Xa),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),Ys)) = 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)),removeAll(A),Xa),Xsa)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),removeAll(A),Xa),Ys)) ) ).

% removeAll_append
tff(fact_1618_list__ex__append,axiom,
    ! [A: $tType,Pa: fun(A,$o),Xsa: list(A),Ys: list(A)] :
      ( aa(list(A),$o,aa(fun(A,$o),fun(list(A),$o),list_ex(A),Pa),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),Ys))
    <=> ( aa(list(A),$o,aa(fun(A,$o),fun(list(A),$o),list_ex(A),Pa),Xsa)
        | aa(list(A),$o,aa(fun(A,$o),fun(list(A),$o),list_ex(A),Pa),Ys) ) ) ).

% list_ex_append
tff(fact_1619_append1__eq__conv,axiom,
    ! [A: $tType,Xsa: list(A),Xa: A,Ys: list(A),Y: A] :
      ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),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))) )
    <=> ( ( Xsa = Ys )
        & ( Xa = Y ) ) ) ).

% append1_eq_conv
tff(fact_1620_list__ee__eq__leel_I1_J,axiom,
    ! [A: $tType,E1: A,E22: A,L12: list(A),E12: A,E23: A,L23: list(A)] :
      ( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E22),nil(A))) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L12),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E12),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E23),L23))) )
    <=> ( ( L12 = nil(A) )
        & ( E1 = E12 )
        & ( E22 = E23 )
        & ( L23 = nil(A) ) ) ) ).

% list_ee_eq_leel(1)
tff(fact_1621_list__ee__eq__leel_I2_J,axiom,
    ! [A: $tType,L12: list(A),E12: A,E23: A,L23: list(A),E1: A,E22: A] :
      ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L12),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E12),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E23),L23))) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E22),nil(A))) )
    <=> ( ( L12 = nil(A) )
        & ( E1 = E12 )
        & ( E22 = E23 )
        & ( L23 = nil(A) ) ) ) ).

% list_ee_eq_leel(2)
tff(fact_1622_list__se__match_I1_J,axiom,
    ! [A: $tType,L12: list(A),L23: list(A),A3: A] :
      ( ( L12 != nil(A) )
     => ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L12),L23) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),nil(A)) )
      <=> ( ( L12 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),nil(A)) )
          & ( L23 = nil(A) ) ) ) ) ).

% list_se_match(1)
tff(fact_1623_list__se__match_I2_J,axiom,
    ! [A: $tType,L23: list(A),L12: list(A),A3: A] :
      ( ( L23 != nil(A) )
     => ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L12),L23) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),nil(A)) )
      <=> ( ( L12 = nil(A) )
          & ( L23 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),nil(A)) ) ) ) ) ).

% list_se_match(2)
tff(fact_1624_list__se__match_I3_J,axiom,
    ! [A: $tType,L12: list(A),A3: A,L23: list(A)] :
      ( ( L12 != nil(A) )
     => ( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),nil(A)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L12),L23) )
      <=> ( ( L12 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),nil(A)) )
          & ( L23 = nil(A) ) ) ) ) ).

% list_se_match(3)
tff(fact_1625_list__se__match_I4_J,axiom,
    ! [A: $tType,L23: list(A),A3: A,L12: list(A)] :
      ( ( L23 != nil(A) )
     => ( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),nil(A)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L12),L23) )
      <=> ( ( L12 = nil(A) )
          & ( L23 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),nil(A)) ) ) ) ) ).

% list_se_match(4)
tff(fact_1626_list__e__eq__lel_I1_J,axiom,
    ! [A: $tType,E3: A,L12: list(A),E4: A,L23: list(A)] :
      ( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E3),nil(A)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L12),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E4),L23)) )
    <=> ( ( L12 = nil(A) )
        & ( E4 = E3 )
        & ( L23 = nil(A) ) ) ) ).

% list_e_eq_lel(1)
tff(fact_1627_list__e__eq__lel_I2_J,axiom,
    ! [A: $tType,L12: list(A),E4: A,L23: list(A),E3: A] :
      ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L12),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E4),L23)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E3),nil(A)) )
    <=> ( ( L12 = nil(A) )
        & ( E4 = E3 )
        & ( L23 = nil(A) ) ) ) ).

% list_e_eq_lel(2)
tff(fact_1628_length__append,axiom,
    ! [A: $tType,Xsa: list(A),Ys: list(A)] : ( aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),Ys)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),Xsa)),aa(list(A),nat,size_size(list(A)),Ys)) ) ).

% length_append
tff(fact_1629_zip__append,axiom,
    ! [A: $tType,B: $tType,Xsa: list(A),Us: list(B),Ys: list(A),Vs: list(B)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xsa) = aa(list(B),nat,size_size(list(B)),Us) )
     => ( aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),Ys)),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Us),Vs)) = aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(list(product_prod(A,B)),fun(list(product_prod(A,B)),list(product_prod(A,B))),append(product_prod(A,B)),aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),Xsa),Us)),aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),Ys),Vs)) ) ) ).

% zip_append
tff(fact_1630_List_Olast__in__set,axiom,
    ! [A: $tType,As2: list(A)] :
      ( ( As2 != nil(A) )
     => member(A,last(A,As2),aa(list(A),set(A),set2(A),As2)) ) ).

% List.last_in_set
tff(fact_1631_Misc_Olast__in__set,axiom,
    ! [A: $tType,L: list(A)] :
      ( ( L != nil(A) )
     => member(A,last(A,L),aa(list(A),set(A),set2(A),L)) ) ).

% Misc.last_in_set
tff(fact_1632_last__appendL,axiom,
    ! [A: $tType,Ys: list(A),Xsa: list(A)] :
      ( ( Ys = nil(A) )
     => ( last(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),Ys)) = last(A,Xsa) ) ) ).

% last_appendL
tff(fact_1633_last__appendR,axiom,
    ! [A: $tType,Ys: list(A),Xsa: list(A)] :
      ( ( Ys != nil(A) )
     => ( last(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),Ys)) = last(A,Ys) ) ) ).

% last_appendR
tff(fact_1634_nth__append__length,axiom,
    ! [A: $tType,Xsa: list(A),Xa: A,Ys: list(A)] : ( aa(nat,A,nth(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Ys))),aa(list(A),nat,size_size(list(A)),Xsa)) = Xa ) ).

% nth_append_length
tff(fact_1635_nth__append__first,axiom,
    ! [A: $tType,Ia: nat,L: list(A),L2: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),aa(list(A),nat,size_size(list(A)),L))
     => ( aa(nat,A,nth(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L),L2)),Ia) = aa(nat,A,nth(A,L),Ia) ) ) ).

% nth_append_first
tff(fact_1636_nth__append__length__plus,axiom,
    ! [A: $tType,Xsa: list(A),Ys: list(A),N: nat] : ( aa(nat,A,nth(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),Ys)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),Xsa)),N)) = aa(nat,A,nth(A,Ys),N) ) ).

% nth_append_length_plus
tff(fact_1637_rev__eq__Cons__iff,axiom,
    ! [A: $tType,Xsa: list(A),Y: A,Ys: list(A)] :
      ( ( aa(list(A),list(A),rev(A),Xsa) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys) )
    <=> ( Xsa = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),rev(A),Ys)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),nil(A))) ) ) ).

% rev_eq_Cons_iff
tff(fact_1638_list__update__length,axiom,
    ! [A: $tType,Xsa: list(A),Xa: A,Ys: list(A),Y: A] : ( aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Ys))),aa(list(A),nat,size_size(list(A)),Xsa)),Y) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)) ) ).

% list_update_length
tff(fact_1639_take__append,axiom,
    ! [A: $tType,N: nat,Xsa: list(A),Ys: list(A)] : ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),Ys)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),Xsa)),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(list(A),nat,size_size(list(A)),Xsa))),Ys)) ) ).

% take_append
tff(fact_1640_last__snoc,axiom,
    ! [A: $tType,Xsa: list(A),Xa: A] : ( last(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),nil(A)))) = Xa ) ).

% last_snoc
tff(fact_1641_butlast__snoc,axiom,
    ! [A: $tType,Xsa: list(A),Xa: A] : ( aa(list(A),list(A),butlast(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),nil(A)))) = Xsa ) ).

% butlast_snoc
tff(fact_1642_lexord__cons__cons,axiom,
    ! [A: $tType,A3: A,Xa: list(A),B3: A,Y: list(A),R2: set(product_prod(A,A))] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),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),Xa)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),B3),Y)),lexord(A,R2))
    <=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B3),R2)
        | ( ( A3 = B3 )
          & 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)),Xa),Y),lexord(A,R2)) ) ) ) ).

% lexord_cons_cons
tff(fact_1643_lexord__Nil__left,axiom,
    ! [A: $tType,Y: list(A),R2: set(product_prod(A,A))] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),Y),lexord(A,R2))
    <=> ? [A7: A,X6: list(A)] : ( Y = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A7),X6) ) ) ).

% lexord_Nil_left
tff(fact_1644_list__assn__aux__append2,axiom,
    ! [A: $tType,B: $tType,L23: list(A),L24: list(B),Pa: fun(A,fun(B,assn)),L12: list(A),L13: list(B)] :
      ( ( aa(list(A),nat,size_size(list(A)),L23) = aa(list(B),nat,size_size(list(B)),L24) )
     => ( vEBT_List_list_assn(A,B,Pa,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L12),L23),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),L13),L24)) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),vEBT_List_list_assn(A,B,Pa,L12,L13)),vEBT_List_list_assn(A,B,Pa,L23,L24)) ) ) ).

% list_assn_aux_append2
tff(fact_1645_list__assn__aux__append,axiom,
    ! [A: $tType,B: $tType,L12: list(A),L13: list(B),Pa: fun(A,fun(B,assn)),L23: list(A),L24: list(B)] :
      ( ( aa(list(A),nat,size_size(list(A)),L12) = aa(list(B),nat,size_size(list(B)),L13) )
     => ( vEBT_List_list_assn(A,B,Pa,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L12),L23),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),L13),L24)) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),vEBT_List_list_assn(A,B,Pa,L12,L13)),vEBT_List_list_assn(A,B,Pa,L23,L24)) ) ) ).

% list_assn_aux_append
tff(fact_1646_distinct__append,axiom,
    ! [A: $tType,Xsa: list(A),Ys: list(A)] :
      ( aa(list(A),$o,distinct(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),Ys))
    <=> ( aa(list(A),$o,distinct(A),Xsa)
        & aa(list(A),$o,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),Xsa)),aa(list(A),set(A),set2(A),Ys)) = bot_bot(set(A)) ) ) ) ).

% distinct_append
tff(fact_1647_append__butlast__last__id,axiom,
    ! [A: $tType,Xsa: list(A)] :
      ( ( Xsa != nil(A) )
     => ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),butlast(A),Xsa)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),last(A,Xsa)),nil(A))) = Xsa ) ) ).

% append_butlast_last_id
tff(fact_1648_lexord__append__leftI,axiom,
    ! [A: $tType,U: list(A),V2: list(A),R2: set(product_prod(A,A)),Xa: list(A)] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),U),V2),lexord(A,R2))
     => member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xa),U)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xa),V2)),lexord(A,R2)) ) ).

% lexord_append_leftI
tff(fact_1649_last__append,axiom,
    ! [A: $tType,Xsa: list(A),Ys: list(A)] :
      ( last(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),Ys)) = $ite(Ys = nil(A),last(A,Xsa),last(A,Ys)) ) ).

% last_append
tff(fact_1650_longest__common__suffix,axiom,
    ! [A: $tType,Xsa: list(A),Ys: list(A)] :
    ? [Ss: list(A),Xs4: list(A),Ys3: list(A)] :
      ( ( Xsa = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs4),Ss) )
      & ( Ys = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys3),Ss) )
      & ( ( Xs4 = nil(A) )
        | ( Ys3 = nil(A) )
        | ( last(A,Xs4) != last(A,Ys3) ) ) ) ).

% longest_common_suffix
tff(fact_1651_imp__le__cong,axiom,
    ! [Xa: int,X5: int,Pa: $o,P: $o] :
      ( ( Xa = X5 )
     => ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),X5)
         => ( (Pa)
          <=> (P) ) )
       => ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Xa)
           => (Pa) )
        <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),X5)
           => (P) ) ) ) ) ).

% imp_le_cong
tff(fact_1652_conj__le__cong,axiom,
    ! [Xa: int,X5: int,Pa: $o,P: $o] :
      ( ( Xa = X5 )
     => ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),X5)
         => ( (Pa)
          <=> (P) ) )
       => ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Xa)
            & (Pa) )
        <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),X5)
            & (P) ) ) ) ) ).

% conj_le_cong
tff(fact_1653_append__eq__appendI,axiom,
    ! [A: $tType,Xsa: list(A),Xs12: list(A),Zs: list(A),Ys: list(A),Us: list(A)] :
      ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),Xs12) = Zs )
     => ( ( Ys = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs12),Us) )
       => ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),Ys) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Zs),Us) ) ) ) ).

% append_eq_appendI
tff(fact_1654_append__eq__append__conv2,axiom,
    ! [A: $tType,Xsa: list(A),Ys: list(A),Zs: list(A),Ts: list(A)] :
      ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),Ys) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Zs),Ts) )
    <=> ? [Us2: list(A)] :
          ( ( ( Xsa = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Zs),Us2) )
            & ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us2),Ys) = Ts ) )
          | ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),Us2) = Zs )
            & ( Ys = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us2),Ts) ) ) ) ) ).

% append_eq_append_conv2
tff(fact_1655_incr__mult__lemma,axiom,
    ! [D3: int,Pa: fun(int,$o),K3: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D3)
     => ( ! [X: int] :
            ( aa(int,$o,Pa,X)
           => aa(int,$o,Pa,aa(int,int,aa(int,fun(int,int),plus_plus(int),X),D3)) )
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K3)
         => ! [X4: int] :
              ( aa(int,$o,Pa,X4)
             => aa(int,$o,Pa,aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),aa(int,int,aa(int,fun(int,int),times_times(int),K3),D3))) ) ) ) ) ).

% incr_mult_lemma
tff(fact_1656_decr__mult__lemma,axiom,
    ! [D3: int,Pa: fun(int,$o),K3: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D3)
     => ( ! [X: int] :
            ( aa(int,$o,Pa,X)
           => aa(int,$o,Pa,aa(int,int,aa(int,fun(int,int),minus_minus(int),X),D3)) )
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K3)
         => ! [X4: int] :
              ( aa(int,$o,Pa,X4)
             => aa(int,$o,Pa,aa(int,int,aa(int,fun(int,int),minus_minus(int),X4),aa(int,int,aa(int,fun(int,int),times_times(int),K3),D3))) ) ) ) ) ).

% decr_mult_lemma
tff(fact_1657_minusinfinity,axiom,
    ! [D3: int,P1: fun(int,$o),Pa: fun(int,$o)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D3)
     => ( ! [X: int,K: int] :
            ( aa(int,$o,P1,X)
          <=> aa(int,$o,P1,aa(int,int,aa(int,fun(int,int),minus_minus(int),X),aa(int,int,aa(int,fun(int,int),times_times(int),K),D3))) )
       => ( ? [Z4: int] :
            ! [X: int] :
              ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),X),Z4)
             => ( aa(int,$o,Pa,X)
              <=> aa(int,$o,P1,X) ) )
         => ( ? [X_13: int] : aa(int,$o,P1,X_13)
           => ? [X_1: int] : aa(int,$o,Pa,X_1) ) ) ) ) ).

% minusinfinity
tff(fact_1658_plusinfinity,axiom,
    ! [D3: int,P: fun(int,$o),Pa: fun(int,$o)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D3)
     => ( ! [X: int,K: int] :
            ( aa(int,$o,P,X)
          <=> aa(int,$o,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),X),aa(int,int,aa(int,fun(int,int),times_times(int),K),D3))) )
       => ( ? [Z4: int] :
            ! [X: int] :
              ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z4),X)
             => ( aa(int,$o,Pa,X)
              <=> aa(int,$o,P,X) ) )
         => ( ? [X_13: int] : aa(int,$o,P,X_13)
           => ? [X_1: int] : aa(int,$o,Pa,X_1) ) ) ) ) ).

% plusinfinity
tff(fact_1659_lexord__append__leftD,axiom,
    ! [A: $tType,Xa: list(A),U: list(A),V2: list(A),R2: set(product_prod(A,A))] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),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),Xa),U)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xa),V2)),lexord(A,R2))
     => ( ! [A4: A] : ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),A4),R2)
       => member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),U),V2),lexord(A,R2)) ) ) ).

% lexord_append_leftD
tff(fact_1660_lexord__append__rightI,axiom,
    ! [A: $tType,Y: list(A),Xa: list(A),R2: set(product_prod(A,A))] :
      ( ? [B9: A,Z4: list(A)] : ( Y = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),B9),Z4) )
     => 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)),Xa),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xa),Y)),lexord(A,R2)) ) ).

% lexord_append_rightI
tff(fact_1661_lexord__sufE,axiom,
    ! [A: $tType,Xsa: list(A),Zs: list(A),Ys: list(A),Qs: list(A),R2: set(product_prod(A,A))] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),Zs)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Qs)),lexord(A,R2))
     => ( ( Xsa != Ys )
       => ( ( aa(list(A),nat,size_size(list(A)),Xsa) = aa(list(A),nat,size_size(list(A)),Ys) )
         => ( ( aa(list(A),nat,size_size(list(A)),Zs) = aa(list(A),nat,size_size(list(A)),Qs) )
           => member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xsa),Ys),lexord(A,R2)) ) ) ) ) ).

% lexord_sufE
tff(fact_1662_list__match__lel__lel,axiom,
    ! [A: $tType,C1: list(A),Qs: A,C22: list(A),C12: list(A),Qs2: A,C23: list(A)] :
      ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),C1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Qs),C22)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),C12),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Qs2),C23)) )
     => ( ! [C21: list(A)] :
            ( ( C1 = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),C12),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Qs2),C21)) )
           => ( C23 != aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),C21),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Qs),C22)) ) )
       => ( ( ( C12 = C1 )
           => ( ( Qs2 = Qs )
             => ( C23 != C22 ) ) )
         => ~ ! [C212: list(A)] :
                ( ( C12 = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),C1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Qs),C212)) )
               => ( C22 != aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),C212),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Qs2),C23)) ) ) ) ) ) ).

% list_match_lel_lel
tff(fact_1663_append__Cons,axiom,
    ! [A: $tType,Xa: A,Xsa: list(A),Ys: list(A)] : ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa)),Ys) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),Ys)) ) ).

% append_Cons
tff(fact_1664_Cons__eq__appendI,axiom,
    ! [A: $tType,Xa: A,Xs12: list(A),Ys: list(A),Xsa: list(A),Zs: list(A)] :
      ( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xs12) = Ys )
     => ( ( Xsa = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs12),Zs) )
       => ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Zs) ) ) ) ).

% Cons_eq_appendI
tff(fact_1665_foldl__conc__empty__eq,axiom,
    ! [A: $tType,Ia: list(A),Ww: list(list(A))] : ( aa(list(list(A)),list(A),aa(list(A),fun(list(list(A)),list(A)),aa(fun(list(A),fun(list(A),list(A))),fun(list(A),fun(list(list(A)),list(A))),foldl(list(A),list(A)),append(A)),Ia),Ww) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ia),aa(list(list(A)),list(A),aa(list(A),fun(list(list(A)),list(A)),aa(fun(list(A),fun(list(A),list(A))),fun(list(A),fun(list(list(A)),list(A))),foldl(list(A),list(A)),append(A)),nil(A)),Ww)) ) ).

% foldl_conc_empty_eq
tff(fact_1666_append__Nil,axiom,
    ! [A: $tType,Ys: list(A)] : ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),nil(A)),Ys) = Ys ) ).

% append_Nil
tff(fact_1667_append_Oleft__neutral,axiom,
    ! [A: $tType,A3: list(A)] : ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),nil(A)),A3) = A3 ) ).

% append.left_neutral
tff(fact_1668_eq__Nil__appendI,axiom,
    ! [A: $tType,Xsa: list(A),Ys: list(A)] :
      ( ( Xsa = Ys )
     => ( Xsa = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),nil(A)),Ys) ) ) ).

% eq_Nil_appendI
tff(fact_1669_mset__le__addE,axiom,
    ! [A: $tType,Xsa: multiset(A),Ys: multiset(A)] :
      ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),Xsa),Ys)
     => ~ ! [Zs2: multiset(A)] : ( Ys != aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),Xsa),Zs2) ) ) ).

% mset_le_addE
tff(fact_1670_mset__le__distrib,axiom,
    ! [A: $tType,X3: multiset(A),Aa2: multiset(A),B2: multiset(A)] :
      ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),X3),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),Aa2),B2))
     => ~ ! [Xa3: multiset(A),Xb2: multiset(A)] :
            ( ( X3 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),Xa3),Xb2) )
           => ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),Xa3),Aa2)
             => ~ aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),Xb2),B2) ) ) ) ).

% mset_le_distrib
tff(fact_1671_mset__union__subset,axiom,
    ! [A: $tType,Aa2: multiset(A),B2: multiset(A),C3: multiset(A)] :
      ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),Aa2),B2)),C3)
     => ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),Aa2),C3)
        & aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),B2),C3) ) ) ).

% mset_union_subset
tff(fact_1672_mset__le__decr__left1,axiom,
    ! [A: $tType,A3: multiset(A),C2: multiset(A),B3: multiset(A)] :
      ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A3),C2)),B3)
     => aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),A3),B3) ) ).

% mset_le_decr_left1
tff(fact_1673_mset__le__decr__left2,axiom,
    ! [A: $tType,C2: multiset(A),A3: multiset(A),B3: multiset(A)] :
      ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),C2),A3)),B3)
     => aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),A3),B3) ) ).

% mset_le_decr_left2
tff(fact_1674_mset__le__incr__right1,axiom,
    ! [A: $tType,A3: multiset(A),B3: multiset(A),C2: multiset(A)] :
      ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),A3),B3)
     => aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),A3),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),B3),C2)) ) ).

% mset_le_incr_right1
tff(fact_1675_mset__le__incr__right2,axiom,
    ! [A: $tType,A3: multiset(A),B3: multiset(A),C2: multiset(A)] :
      ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),A3),B3)
     => aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),A3),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),C2),B3)) ) ).

% mset_le_incr_right2
tff(fact_1676_mset__le__subtract,axiom,
    ! [A: $tType,Aa2: multiset(A),B2: multiset(A),C3: multiset(A)] :
      ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),Aa2),B2)
     => aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),Aa2),C3)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),B2),C3)) ) ).

% mset_le_subtract
tff(fact_1677_lexord__append__left__rightI,axiom,
    ! [A: $tType,A3: A,B3: A,R2: set(product_prod(A,A)),U: list(A),Xa: list(A),Y: list(A)] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B3),R2)
     => member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),U),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),Xa))),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),B3),Y))),lexord(A,R2)) ) ).

% lexord_append_left_rightI
tff(fact_1678_snoc__eq__iff__butlast,axiom,
    ! [A: $tType,Xsa: list(A),Xa: A,Ys: list(A)] :
      ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),nil(A))) = Ys )
    <=> ( ( Ys != nil(A) )
        & ( aa(list(A),list(A),butlast(A),Ys) = Xsa )
        & ( last(A,Ys) = Xa ) ) ) ).

% snoc_eq_iff_butlast
tff(fact_1679_snoc__eq__iff__butlast_H,axiom,
    ! [A: $tType,Ys: list(A),Xsa: list(A),Xa: A] :
      ( ( Ys = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),nil(A))) )
    <=> ( ( Ys != nil(A) )
        & ( aa(list(A),list(A),butlast(A),Ys) = Xsa )
        & ( last(A,Ys) = Xa ) ) ) ).

% snoc_eq_iff_butlast'
tff(fact_1680_lexord__same__pref__iff,axiom,
    ! [A: $tType,Xsa: list(A),Ys: list(A),Zs: list(A),R2: set(product_prod(A,A))] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),Ys)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),Zs)),lexord(A,R2))
    <=> ( ? [X6: A] :
            ( member(A,X6,aa(list(A),set(A),set2(A),Xsa))
            & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X6),X6),R2) )
        | member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Ys),Zs),lexord(A,R2)) ) ) ).

% lexord_same_pref_iff
tff(fact_1681_lexord__sufI,axiom,
    ! [A: $tType,U: list(A),W2: list(A),R2: set(product_prod(A,A)),V2: list(A),Z2: list(A)] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),U),W2),lexord(A,R2))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),W2)),aa(list(A),nat,size_size(list(A)),U))
       => member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),U),V2)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),W2),Z2)),lexord(A,R2)) ) ) ).

% lexord_sufI
tff(fact_1682_enumerate__append__eq,axiom,
    ! [A: $tType,N: nat,Xsa: list(A),Ys: list(A)] : ( enumerate(A,N,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),Ys)) = aa(list(product_prod(nat,A)),list(product_prod(nat,A)),aa(list(product_prod(nat,A)),fun(list(product_prod(nat,A)),list(product_prod(nat,A))),append(product_prod(nat,A)),enumerate(A,N,Xsa)),enumerate(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(list(A),nat,size_size(list(A)),Xsa)),Ys)) ) ).

% enumerate_append_eq
tff(fact_1683_rev__nonempty__induct,axiom,
    ! [A: $tType,Xsa: list(A),Pa: fun(list(A),$o)] :
      ( ( Xsa != nil(A) )
     => ( ! [X: A] : aa(list(A),$o,Pa,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)))
       => ( ! [X: A,Xs: list(A)] :
              ( ( Xs != nil(A) )
             => ( aa(list(A),$o,Pa,Xs)
               => aa(list(A),$o,Pa,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)))) ) )
         => aa(list(A),$o,Pa,Xsa) ) ) ) ).

% rev_nonempty_induct
tff(fact_1684_append__eq__Cons__conv,axiom,
    ! [A: $tType,Ys: list(A),Zs: list(A),Xa: A,Xsa: list(A)] :
      ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Zs) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa) )
    <=> ( ( ( Ys = nil(A) )
          & ( Zs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa) ) )
        | ? [Ys6: list(A)] :
            ( ( Ys = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Ys6) )
            & ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys6),Zs) = Xsa ) ) ) ) ).

% append_eq_Cons_conv
tff(fact_1685_Cons__eq__append__conv,axiom,
    ! [A: $tType,Xa: A,Xsa: list(A),Ys: list(A),Zs: list(A)] :
      ( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Zs) )
    <=> ( ( ( Ys = nil(A) )
          & ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa) = Zs ) )
        | ? [Ys6: list(A)] :
            ( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Ys6) = Ys )
            & ( Xsa = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys6),Zs) ) ) ) ) ).

% Cons_eq_append_conv
tff(fact_1686_rev__exhaust,axiom,
    ! [A: $tType,Xsa: list(A)] :
      ( ( Xsa != nil(A) )
     => ~ ! [Ys5: list(A),Y2: A] : ( Xsa != aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys5),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),nil(A))) ) ) ).

% rev_exhaust
tff(fact_1687_rev__induct,axiom,
    ! [A: $tType,Pa: fun(list(A),$o),Xsa: list(A)] :
      ( aa(list(A),$o,Pa,nil(A))
     => ( ! [X: A,Xs: list(A)] :
            ( aa(list(A),$o,Pa,Xs)
           => aa(list(A),$o,Pa,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)))) )
       => aa(list(A),$o,Pa,Xsa) ) ) ).

% rev_induct
tff(fact_1688_list__append__eq__Cons__cases,axiom,
    ! [A: $tType,Ys: list(A),Zs: list(A),Xa: A,Xsa: list(A)] :
      ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Zs) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa) )
     => ( ( ( Ys = nil(A) )
         => ( Zs != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa) ) )
       => ~ ! [Ys3: list(A)] :
              ( ( Ys = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Ys3) )
             => ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys3),Zs) != Xsa ) ) ) ) ).

% list_append_eq_Cons_cases
tff(fact_1689_list__Cons__eq__append__cases,axiom,
    ! [A: $tType,Xa: A,Xsa: list(A),Ys: list(A),Zs: list(A)] :
      ( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Zs) )
     => ( ( ( Ys = nil(A) )
         => ( Zs != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa) ) )
       => ~ ! [Ys3: list(A)] :
              ( ( Ys = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Ys3) )
             => ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys3),Zs) != Xsa ) ) ) ) ).

% list_Cons_eq_append_cases
tff(fact_1690_rev__nonempty__induct2_H,axiom,
    ! [A: $tType,B: $tType,Xsa: list(A),Ys: list(B),Pa: fun(list(A),fun(list(B),$o))] :
      ( ( Xsa != nil(A) )
     => ( ( Ys != nil(B) )
       => ( ! [X: A,Y2: B] : aa(list(B),$o,aa(list(A),fun(list(B),$o),Pa,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A))),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y2),nil(B)))
         => ( ! [X: A,Xs: list(A),Y2: B] :
                ( ( Xs != nil(A) )
               => aa(list(B),$o,aa(list(A),fun(list(B),$o),Pa,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(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y2),nil(B))) )
           => ( ! [X: A,Y2: B,Ys5: list(B)] :
                  ( ( Ys5 != nil(B) )
                 => aa(list(B),$o,aa(list(A),fun(list(B),$o),Pa,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A))),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Ys5),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y2),nil(B)))) )
             => ( ! [X: A,Xs: list(A),Y2: B,Ys5: list(B)] :
                    ( aa(list(B),$o,aa(list(A),fun(list(B),$o),Pa,Xs),Ys5)
                   => ( ( Xs != nil(A) )
                     => ( ( Ys5 != nil(B) )
                       => aa(list(B),$o,aa(list(A),fun(list(B),$o),Pa,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(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Ys5),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y2),nil(B)))) ) ) )
               => aa(list(B),$o,aa(list(A),fun(list(B),$o),Pa,Xsa),Ys) ) ) ) ) ) ) ).

% rev_nonempty_induct2'
tff(fact_1691_neq__Nil__rev__conv,axiom,
    ! [A: $tType,L: list(A)] :
      ( ( L != nil(A) )
    <=> ? [Xs2: list(A),X6: A] : ( L = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X6),nil(A))) ) ) ).

% neq_Nil_rev_conv
tff(fact_1692_rev__induct2_H,axiom,
    ! [A: $tType,B: $tType,Pa: fun(list(A),fun(list(B),$o)),Xsa: list(A),Ys: list(B)] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),Pa,nil(A)),nil(B))
     => ( ! [X: A,Xs: list(A)] : aa(list(B),$o,aa(list(A),fun(list(B),$o),Pa,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)))),nil(B))
       => ( ! [Y2: B,Ys5: list(B)] : aa(list(B),$o,aa(list(A),fun(list(B),$o),Pa,nil(A)),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Ys5),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y2),nil(B))))
         => ( ! [X: A,Xs: list(A),Y2: B,Ys5: list(B)] :
                ( aa(list(B),$o,aa(list(A),fun(list(B),$o),Pa,Xs),Ys5)
               => aa(list(B),$o,aa(list(A),fun(list(B),$o),Pa,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(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Ys5),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y2),nil(B)))) )
           => aa(list(B),$o,aa(list(A),fun(list(B),$o),Pa,Xsa),Ys) ) ) ) ) ).

% rev_induct2'
tff(fact_1693_neq__Nil__revE,axiom,
    ! [A: $tType,L: list(A)] :
      ( ( L != nil(A) )
     => ~ ! [Ll2: list(A),E2: A] : ( L != aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ll2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E2),nil(A))) ) ) ).

% neq_Nil_revE
tff(fact_1694_last__zip,axiom,
    ! [A: $tType,B: $tType,Xsa: list(A),Ys: list(B)] :
      ( ( Xsa != nil(A) )
     => ( ( Ys != nil(B) )
       => ( ( aa(list(A),nat,size_size(list(A)),Xsa) = aa(list(B),nat,size_size(list(B)),Ys) )
         => ( last(product_prod(A,B),aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),Xsa),Ys)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),last(A,Xsa)),last(B,Ys)) ) ) ) ) ).

% last_zip
tff(fact_1695_split__list__first__prop__iff,axiom,
    ! [A: $tType,Xsa: list(A),Pa: fun(A,$o)] :
      ( ? [X6: A] :
          ( member(A,X6,aa(list(A),set(A),set2(A),Xsa))
          & aa(A,$o,Pa,X6) )
    <=> ? [Ys4: list(A),X6: A] :
          ( ? [Zs3: list(A)] : ( Xsa = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys4),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X6),Zs3)) )
          & aa(A,$o,Pa,X6)
          & ! [Xa2: A] :
              ( member(A,Xa2,aa(list(A),set(A),set2(A),Ys4))
             => ~ aa(A,$o,Pa,Xa2) ) ) ) ).

% split_list_first_prop_iff
tff(fact_1696_split__list__last__prop__iff,axiom,
    ! [A: $tType,Xsa: list(A),Pa: fun(A,$o)] :
      ( ? [X6: A] :
          ( member(A,X6,aa(list(A),set(A),set2(A),Xsa))
          & aa(A,$o,Pa,X6) )
    <=> ? [Ys4: list(A),X6: A,Zs3: list(A)] :
          ( ( Xsa = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys4),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X6),Zs3)) )
          & aa(A,$o,Pa,X6)
          & ! [Xa2: A] :
              ( member(A,Xa2,aa(list(A),set(A),set2(A),Zs3))
             => ~ aa(A,$o,Pa,Xa2) ) ) ) ).

% split_list_last_prop_iff
tff(fact_1697_in__set__conv__decomp__first,axiom,
    ! [A: $tType,Xa: A,Xsa: list(A)] :
      ( member(A,Xa,aa(list(A),set(A),set2(A),Xsa))
    <=> ? [Ys4: list(A),Zs3: list(A)] :
          ( ( Xsa = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys4),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Zs3)) )
          & ~ member(A,Xa,aa(list(A),set(A),set2(A),Ys4)) ) ) ).

% in_set_conv_decomp_first
tff(fact_1698_in__set__conv__decomp__last,axiom,
    ! [A: $tType,Xa: A,Xsa: list(A)] :
      ( member(A,Xa,aa(list(A),set(A),set2(A),Xsa))
    <=> ? [Ys4: list(A),Zs3: list(A)] :
          ( ( Xsa = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys4),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Zs3)) )
          & ~ member(A,Xa,aa(list(A),set(A),set2(A),Zs3)) ) ) ).

% in_set_conv_decomp_last
tff(fact_1699_split__list__first__propE,axiom,
    ! [A: $tType,Xsa: list(A),Pa: fun(A,$o)] :
      ( ? [X4: A] :
          ( member(A,X4,aa(list(A),set(A),set2(A),Xsa))
          & aa(A,$o,Pa,X4) )
     => ~ ! [Ys5: list(A),X: A] :
            ( ? [Zs2: list(A)] : ( Xsa = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys5),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Zs2)) )
           => ( aa(A,$o,Pa,X)
             => ~ ! [Xa4: A] :
                    ( member(A,Xa4,aa(list(A),set(A),set2(A),Ys5))
                   => ~ aa(A,$o,Pa,Xa4) ) ) ) ) ).

% split_list_first_propE
tff(fact_1700_split__list__last__propE,axiom,
    ! [A: $tType,Xsa: list(A),Pa: fun(A,$o)] :
      ( ? [X4: A] :
          ( member(A,X4,aa(list(A),set(A),set2(A),Xsa))
          & aa(A,$o,Pa,X4) )
     => ~ ! [Ys5: list(A),X: A,Zs2: list(A)] :
            ( ( Xsa = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys5),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Zs2)) )
           => ( aa(A,$o,Pa,X)
             => ~ ! [Xa4: A] :
                    ( member(A,Xa4,aa(list(A),set(A),set2(A),Zs2))
                   => ~ aa(A,$o,Pa,Xa4) ) ) ) ) ).

% split_list_last_propE
tff(fact_1701_split__list__first__prop,axiom,
    ! [A: $tType,Xsa: list(A),Pa: fun(A,$o)] :
      ( ? [X4: A] :
          ( member(A,X4,aa(list(A),set(A),set2(A),Xsa))
          & aa(A,$o,Pa,X4) )
     => ? [Ys5: list(A),X: A] :
          ( ? [Zs2: list(A)] : ( Xsa = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys5),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Zs2)) )
          & aa(A,$o,Pa,X)
          & ! [Xa4: A] :
              ( member(A,Xa4,aa(list(A),set(A),set2(A),Ys5))
             => ~ aa(A,$o,Pa,Xa4) ) ) ) ).

% split_list_first_prop
tff(fact_1702_split__list__last__prop,axiom,
    ! [A: $tType,Xsa: list(A),Pa: fun(A,$o)] :
      ( ? [X4: A] :
          ( member(A,X4,aa(list(A),set(A),set2(A),Xsa))
          & aa(A,$o,Pa,X4) )
     => ? [Ys5: list(A),X: A,Zs2: list(A)] :
          ( ( Xsa = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys5),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Zs2)) )
          & aa(A,$o,Pa,X)
          & ! [Xa4: A] :
              ( member(A,Xa4,aa(list(A),set(A),set2(A),Zs2))
             => ~ aa(A,$o,Pa,Xa4) ) ) ) ).

% split_list_last_prop
tff(fact_1703_in__set__conv__decomp,axiom,
    ! [A: $tType,Xa: A,Xsa: list(A)] :
      ( member(A,Xa,aa(list(A),set(A),set2(A),Xsa))
    <=> ? [Ys4: list(A),Zs3: list(A)] : ( Xsa = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys4),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Zs3)) ) ) ).

% in_set_conv_decomp
tff(fact_1704_append__Cons__eq__iff,axiom,
    ! [A: $tType,Xa: A,Xsa: list(A),Ys: list(A),Xs3: list(A),Ys7: list(A)] :
      ( ~ member(A,Xa,aa(list(A),set(A),set2(A),Xsa))
     => ( ~ member(A,Xa,aa(list(A),set(A),set2(A),Ys))
       => ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Ys)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Ys7)) )
        <=> ( ( Xsa = Xs3 )
            & ( Ys = Ys7 ) ) ) ) ) ).

% append_Cons_eq_iff
tff(fact_1705_split__list__propE,axiom,
    ! [A: $tType,Xsa: list(A),Pa: fun(A,$o)] :
      ( ? [X4: A] :
          ( member(A,X4,aa(list(A),set(A),set2(A),Xsa))
          & aa(A,$o,Pa,X4) )
     => ~ ! [Ys5: list(A),X: A] :
            ( ? [Zs2: list(A)] : ( Xsa = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys5),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Zs2)) )
           => ~ aa(A,$o,Pa,X) ) ) ).

% split_list_propE
tff(fact_1706_split__list__first,axiom,
    ! [A: $tType,Xa: A,Xsa: list(A)] :
      ( member(A,Xa,aa(list(A),set(A),set2(A),Xsa))
     => ? [Ys5: list(A),Zs2: list(A)] :
          ( ( Xsa = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys5),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Zs2)) )
          & ~ member(A,Xa,aa(list(A),set(A),set2(A),Ys5)) ) ) ).

% split_list_first
tff(fact_1707_split__list__prop,axiom,
    ! [A: $tType,Xsa: list(A),Pa: fun(A,$o)] :
      ( ? [X4: A] :
          ( member(A,X4,aa(list(A),set(A),set2(A),Xsa))
          & aa(A,$o,Pa,X4) )
     => ? [Ys5: list(A),X: A] :
          ( ? [Zs2: list(A)] : ( Xsa = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys5),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Zs2)) )
          & aa(A,$o,Pa,X) ) ) ).

% split_list_prop
tff(fact_1708_split__list__last,axiom,
    ! [A: $tType,Xa: A,Xsa: list(A)] :
      ( member(A,Xa,aa(list(A),set(A),set2(A),Xsa))
     => ? [Ys5: list(A),Zs2: list(A)] :
          ( ( Xsa = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys5),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Zs2)) )
          & ~ member(A,Xa,aa(list(A),set(A),set2(A),Zs2)) ) ) ).

% split_list_last
tff(fact_1709_split__list,axiom,
    ! [A: $tType,Xa: A,Xsa: list(A)] :
      ( member(A,Xa,aa(list(A),set(A),set2(A),Xsa))
     => ? [Ys5: list(A),Zs2: list(A)] : ( Xsa = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys5),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Zs2)) ) ) ).

% split_list
tff(fact_1710_in__set__list__format,axiom,
    ! [A: $tType,E3: A,L: list(A)] :
      ( member(A,E3,aa(list(A),set(A),set2(A),L))
     => ~ ! [L1: list(A),L22: list(A)] : ( L != aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E3),L22)) ) ) ).

% in_set_list_format
tff(fact_1711_xy__in__set__cases,axiom,
    ! [A: $tType,Xa: A,L: list(A),Y: A] :
      ( member(A,Xa,aa(list(A),set(A),set2(A),L))
     => ( member(A,Y,aa(list(A),set(A),set2(A),L))
       => ( ( ( Xa = Y )
           => ! [L1: list(A),L22: list(A)] : ( L != aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),L22)) ) )
         => ( ( ( Xa != Y )
             => ! [L1: list(A),L22: list(A),L32: list(A)] : ( L != aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L22),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),L32)))) ) )
           => ~ ( ( Xa != Y )
               => ! [L1: list(A),L22: list(A),L32: list(A)] : ( L != aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L22),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),L32)))) ) ) ) ) ) ) ).

% xy_in_set_cases
tff(fact_1712_list__rest__coinc,axiom,
    ! [A: $tType,S22: list(A),S1: list(A),R12: list(A),R23: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),S22)),aa(list(A),nat,size_size(list(A)),S1))
     => ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),S1),R12) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),S22),R23) )
       => ? [R1p: list(A)] : ( R23 = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),R1p),R12) ) ) ) ).

% list_rest_coinc
tff(fact_1713_set__union__code,axiom,
    ! [A: $tType,Xsa: 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),Xsa)),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),Xsa),Ys)) ) ).

% set_union_code
tff(fact_1714_distinct__match,axiom,
    ! [A: $tType,Al: list(A),E3: A,Bl: list(A),Al2: list(A),Bl2: list(A)] :
      ( aa(list(A),$o,distinct(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Al),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E3),Bl)))
     => ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Al),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E3),Bl)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Al2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E3),Bl2)) )
      <=> ( ( Al = Al2 )
          & ( Bl = Bl2 ) ) ) ) ).

% distinct_match
tff(fact_1715_sorted__wrt__append,axiom,
    ! [A: $tType,Pa: fun(A,fun(A,$o)),Xsa: list(A),Ys: list(A)] :
      ( sorted_wrt(A,Pa,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),Ys))
    <=> ( sorted_wrt(A,Pa,Xsa)
        & sorted_wrt(A,Pa,Ys)
        & ! [X6: A] :
            ( member(A,X6,aa(list(A),set(A),set2(A),Xsa))
           => ! [Xa2: A] :
                ( member(A,Xa2,aa(list(A),set(A),set2(A),Ys))
               => aa(A,$o,aa(A,fun(A,$o),Pa,X6),Xa2) ) ) ) ) ).

% sorted_wrt_append
tff(fact_1716_last__ConsR,axiom,
    ! [A: $tType,Xsa: list(A),Xa: A] :
      ( ( Xsa != nil(A) )
     => ( last(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa)) = last(A,Xsa) ) ) ).

% last_ConsR
tff(fact_1717_last__ConsL,axiom,
    ! [A: $tType,Xsa: list(A),Xa: A] :
      ( ( Xsa = nil(A) )
     => ( last(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa)) = Xa ) ) ).

% last_ConsL
tff(fact_1718_last_Osimps,axiom,
    ! [A: $tType,Xa: A,Xsa: list(A)] :
      ( last(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa)) = $ite(Xsa = nil(A),Xa,last(A,Xsa)) ) ).

% last.simps
tff(fact_1719_butlast__append,axiom,
    ! [A: $tType,Xsa: list(A),Ys: list(A)] :
      ( aa(list(A),list(A),butlast(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),Ys)) = $ite(Ys = nil(A),aa(list(A),list(A),butlast(A),Xsa),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),aa(list(A),list(A),butlast(A),Ys))) ) ).

% butlast_append
tff(fact_1720_mset__le__subtract__right,axiom,
    ! [A: $tType,Aa2: multiset(A),B2: multiset(A),X3: multiset(A)] :
      ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),Aa2),B2)),X3)
     => ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),Aa2),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),X3),B2))
        & aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),B2),X3) ) ) ).

% mset_le_subtract_right
tff(fact_1721_mset__le__subtract__left,axiom,
    ! [A: $tType,Aa2: multiset(A),B2: multiset(A),X3: multiset(A)] :
      ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),Aa2),B2)),X3)
     => ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),B2),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),X3),Aa2))
        & aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),Aa2),X3) ) ) ).

% mset_le_subtract_left
tff(fact_1722_in__set__butlast__appendI,axiom,
    ! [A: $tType,Xa: A,Xsa: list(A),Ys: list(A)] :
      ( ( member(A,Xa,aa(list(A),set(A),set2(A),aa(list(A),list(A),butlast(A),Xsa)))
        | member(A,Xa,aa(list(A),set(A),set2(A),aa(list(A),list(A),butlast(A),Ys))) )
     => member(A,Xa,aa(list(A),set(A),set2(A),aa(list(A),list(A),butlast(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),Ys)))) ) ).

% in_set_butlast_appendI
tff(fact_1723_remove1__append,axiom,
    ! [A: $tType,Xa: A,Xsa: list(A),Ys: list(A)] :
      ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),remove1(A),Xa),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),Ys)) = $ite(member(A,Xa,aa(list(A),set(A),set2(A),Xsa)),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)),remove1(A),Xa),Xsa)),Ys),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),aa(list(A),list(A),aa(A,fun(list(A),list(A)),remove1(A),Xa),Ys))) ) ).

% remove1_append
tff(fact_1724_append__listrel1I,axiom,
    ! [A: $tType,Xsa: list(A),Ys: list(A),R2: set(product_prod(A,A)),Us: list(A),Vs: list(A)] :
      ( ( ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xsa),Ys),listrel1(A,R2))
          & ( Us = Vs ) )
        | ( ( Xsa = Ys )
          & member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Us),Vs),listrel1(A,R2)) ) )
     => member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),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_1725_lex__append__leftI,axiom,
    ! [A: $tType,Ys: list(A),Zs: list(A),R2: set(product_prod(A,A)),Xsa: list(A)] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Ys),Zs),lex(A,R2))
     => member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),Ys)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),Zs)),lex(A,R2)) ) ).

% lex_append_leftI
tff(fact_1726_lexord__linear,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),Xa: list(A),Y: list(A)] :
      ( ! [A4: A,B4: A] :
          ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),B4),R2)
          | ( A4 = B4 )
          | member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B4),A4),R2) )
     => ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xa),Y),lexord(A,R2))
        | ( Xa = Y )
        | 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),Xa),lexord(A,R2)) ) ) ).

% lexord_linear
tff(fact_1727_lexord__irreflexive,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),Xsa: list(A)] :
      ( ! [X: A] : ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),X),R2)
     => ~ member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xsa),Xsa),lexord(A,R2)) ) ).

% lexord_irreflexive
tff(fact_1728_lexord__Nil__right,axiom,
    ! [A: $tType,Xa: list(A),R2: set(product_prod(A,A))] : ~ member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xa),nil(A)),lexord(A,R2)) ).

% lexord_Nil_right
tff(fact_1729_same__length__different,axiom,
    ! [A: $tType,Xsa: list(A),Ys: list(A)] :
      ( ( Xsa != Ys )
     => ( ( aa(list(A),nat,size_size(list(A)),Xsa) = aa(list(A),nat,size_size(list(A)),Ys) )
       => ? [Pre: list(A),X: A,Xs4: list(A),Y2: A,Ys3: list(A)] :
            ( ( X != Y2 )
            & ( Xsa = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Pre),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A))),Xs4)) )
            & ( Ys = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Pre),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),nil(A))),Ys3)) ) ) ) ) ).

% same_length_different
tff(fact_1730_sorted__append,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xsa: list(A),Ys: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),Ys))
        <=> ( sorted_wrt(A,ord_less_eq(A),Xsa)
            & sorted_wrt(A,ord_less_eq(A),Ys)
            & ! [X6: A] :
                ( member(A,X6,aa(list(A),set(A),set2(A),Xsa))
               => ! [Xa2: A] :
                    ( member(A,Xa2,aa(list(A),set(A),set2(A),Ys))
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X6),Xa2) ) ) ) ) ) ).

% sorted_append
tff(fact_1731_not__distinct__decomp,axiom,
    ! [A: $tType,Ws: list(A)] :
      ( ~ aa(list(A),$o,distinct(A),Ws)
     => ? [Xs: list(A),Ys5: list(A),Zs2: list(A),Y2: A] : ( Ws = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),nil(A))),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys5),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),Y2),nil(A))),Zs2)))) ) ) ).

% not_distinct_decomp
tff(fact_1732_not__distinct__conv__prefix,axiom,
    ! [A: $tType,As2: list(A)] :
      ( ~ aa(list(A),$o,distinct(A),As2)
    <=> ? [Xs2: list(A),Y4: A,Ys4: list(A)] :
          ( member(A,Y4,aa(list(A),set(A),set2(A),Xs2))
          & aa(list(A),$o,distinct(A),Xs2)
          & ( As2 = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys4)) ) ) ) ).

% not_distinct_conv_prefix
tff(fact_1733_rev_Osimps_I2_J,axiom,
    ! [A: $tType,Xa: A,Xsa: list(A)] : ( aa(list(A),list(A),rev(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),rev(A),Xsa)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),nil(A))) ) ).

% rev.simps(2)
tff(fact_1734_list__update__append1,axiom,
    ! [A: $tType,Ia: nat,Xsa: list(A),Ys: list(A),Xa: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),aa(list(A),nat,size_size(list(A)),Xsa))
     => ( aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),Ys)),Ia),Xa) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xsa),Ia),Xa)),Ys) ) ) ).

% list_update_append1
tff(fact_1735_butlast__eq__cons__conv,axiom,
    ! [A: $tType,L: list(A),Xa: A,Xsa: list(A)] :
      ( ( aa(list(A),list(A),butlast(A),L) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa) )
    <=> ? [Xl: A] : ( L = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xl),nil(A)))) ) ) ).

% butlast_eq_cons_conv
tff(fact_1736_butlast__eq__consE,axiom,
    ! [A: $tType,L: list(A),Xa: A,Xsa: list(A)] :
      ( ( aa(list(A),list(A),butlast(A),L) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa) )
     => ~ ! [Xl2: A] : ( L != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xl2),nil(A)))) ) ) ).

% butlast_eq_consE
tff(fact_1737_foldl__rule,axiom,
    ! [A: $tType,B: $tType,Ia2: fun(A,fun(list(B),fun(list(B),$o))),Sigma_0: A,L0: list(B),F2: fun(A,fun(B,A))] :
      ( aa(list(B),$o,aa(list(B),fun(list(B),$o),aa(A,fun(list(B),fun(list(B),$o)),Ia2,Sigma_0),nil(B)),L0)
     => ( ! [L1: list(B),L22: list(B),X: B,Sigma: A] :
            ( ( L0 = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),L1),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),L22)) )
           => ( aa(list(B),$o,aa(list(B),fun(list(B),$o),aa(A,fun(list(B),fun(list(B),$o)),Ia2,Sigma),L1),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),L22))
             => aa(list(B),$o,aa(list(B),fun(list(B),$o),aa(A,fun(list(B),fun(list(B),$o)),Ia2,aa(B,A,aa(A,fun(B,A),F2,Sigma),X)),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),L1),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),nil(B)))),L22) ) )
       => aa(list(B),$o,aa(list(B),fun(list(B),$o),aa(A,fun(list(B),fun(list(B),$o)),Ia2,aa(list(B),A,aa(A,fun(list(B),A),aa(fun(A,fun(B,A)),fun(A,fun(list(B),A)),foldl(A,B),F2),Sigma_0),L0)),L0),nil(B)) ) ) ).

% foldl_rule
tff(fact_1738_foldl__rule__P,axiom,
    ! [A: $tType,B: $tType,Ia2: fun(A,fun(list(B),fun(list(B),$o))),Sigma_0: A,L0: list(B),F2: fun(A,fun(B,A)),Pa: fun(A,$o)] :
      ( aa(list(B),$o,aa(list(B),fun(list(B),$o),aa(A,fun(list(B),fun(list(B),$o)),Ia2,Sigma_0),nil(B)),L0)
     => ( ! [L1: list(B),L22: list(B),X: B,Sigma: A] :
            ( ( L0 = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),L1),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),L22)) )
           => ( aa(list(B),$o,aa(list(B),fun(list(B),$o),aa(A,fun(list(B),fun(list(B),$o)),Ia2,Sigma),L1),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),L22))
             => aa(list(B),$o,aa(list(B),fun(list(B),$o),aa(A,fun(list(B),fun(list(B),$o)),Ia2,aa(B,A,aa(A,fun(B,A),F2,Sigma),X)),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),L1),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),nil(B)))),L22) ) )
       => ( ! [Sigma: A] :
              ( aa(list(B),$o,aa(list(B),fun(list(B),$o),aa(A,fun(list(B),fun(list(B),$o)),Ia2,Sigma),L0),nil(B))
             => aa(A,$o,Pa,Sigma) )
         => aa(A,$o,Pa,aa(list(B),A,aa(A,fun(list(B),A),aa(fun(A,fun(B,A)),fun(A,fun(list(B),A)),foldl(A,B),F2),Sigma_0),L0)) ) ) ) ).

% foldl_rule_P
tff(fact_1739_foldl__rule__aux,axiom,
    ! [A: $tType,B: $tType,Ia2: fun(A,fun(list(B),$o)),Sigma_0: A,L0: list(B),F2: fun(A,fun(B,A))] :
      ( aa(list(B),$o,aa(A,fun(list(B),$o),Ia2,Sigma_0),L0)
     => ( ! [L1: list(B),L22: list(B),X: B,Sigma: A] :
            ( ( L0 = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),L1),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),L22)) )
           => ( aa(list(B),$o,aa(A,fun(list(B),$o),Ia2,Sigma),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),L22))
             => aa(list(B),$o,aa(A,fun(list(B),$o),Ia2,aa(B,A,aa(A,fun(B,A),F2,Sigma),X)),L22) ) )
       => aa(list(B),$o,aa(A,fun(list(B),$o),Ia2,aa(list(B),A,aa(A,fun(list(B),A),aa(fun(A,fun(B,A)),fun(A,fun(list(B),A)),foldl(A,B),F2),Sigma_0),L0)),nil(B)) ) ) ).

% foldl_rule_aux
tff(fact_1740_foldl__rule__aux__P,axiom,
    ! [A: $tType,B: $tType,Ia2: fun(A,fun(list(B),$o)),Sigma_0: A,L0: list(B),F2: fun(A,fun(B,A)),Pa: fun(A,$o)] :
      ( aa(list(B),$o,aa(A,fun(list(B),$o),Ia2,Sigma_0),L0)
     => ( ! [L1: list(B),L22: list(B),X: B,Sigma: A] :
            ( ( L0 = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),L1),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),L22)) )
           => ( aa(list(B),$o,aa(A,fun(list(B),$o),Ia2,Sigma),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),L22))
             => aa(list(B),$o,aa(A,fun(list(B),$o),Ia2,aa(B,A,aa(A,fun(B,A),F2,Sigma),X)),L22) ) )
       => ( ! [Sigma: A] :
              ( aa(list(B),$o,aa(A,fun(list(B),$o),Ia2,Sigma),nil(B))
             => aa(A,$o,Pa,Sigma) )
         => aa(A,$o,Pa,aa(list(B),A,aa(A,fun(list(B),A),aa(fun(A,fun(B,A)),fun(A,fun(list(B),A)),foldl(A,B),F2),Sigma_0),L0)) ) ) ) ).

% foldl_rule_aux_P
tff(fact_1741_remove1__split,axiom,
    ! [A: $tType,A3: A,Xsa: list(A),Ys: list(A)] :
      ( member(A,A3,aa(list(A),set(A),set2(A),Xsa))
     => ( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),remove1(A),A3),Xsa) = Ys )
      <=> ? [Ls2: list(A),Rs: list(A)] :
            ( ( Xsa = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ls2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),Rs)) )
            & ~ member(A,A3,aa(list(A),set(A),set2(A),Ls2))
            & ( Ys = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ls2),Rs) ) ) ) ) ).

% remove1_split
tff(fact_1742_lex__append__leftD,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),Xsa: list(A),Ys: list(A),Zs: list(A)] :
      ( ! [X: A] : ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),X),R2)
     => ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),Ys)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),Zs)),lex(A,R2))
       => member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Ys),Zs),lex(A,R2)) ) ) ).

% lex_append_leftD
tff(fact_1743_lex__append__left__iff,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),Xsa: list(A),Ys: list(A),Zs: list(A)] :
      ( ! [X: A] : ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),X),R2)
     => ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),Ys)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),Zs)),lex(A,R2))
      <=> member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Ys),Zs),lex(A,R2)) ) ) ).

% lex_append_left_iff
tff(fact_1744_rotate1_Osimps_I2_J,axiom,
    ! [A: $tType,Xa: A,Xsa: list(A)] : ( aa(list(A),list(A),rotate1(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),nil(A))) ) ).

% rotate1.simps(2)
tff(fact_1745_lex__append__rightI,axiom,
    ! [A: $tType,Xsa: list(A),Ys: list(A),R2: set(product_prod(A,A)),Vs: list(A),Us: list(A)] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xsa),Ys),lex(A,R2))
     => ( ( aa(list(A),nat,size_size(list(A)),Vs) = aa(list(A),nat,size_size(list(A)),Us) )
       => member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),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_1746_lenlex__append1,axiom,
    ! [A: $tType,Us: list(A),Xsa: list(A),R: set(product_prod(A,A)),Vs: list(A),Ys: list(A)] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Us),Xsa),lenlex(A,R))
     => ( ( aa(list(A),nat,size_size(list(A)),Vs) = aa(list(A),nat,size_size(list(A)),Ys) )
       => member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us),Vs)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),Ys)),lenlex(A,R)) ) ) ).

% lenlex_append1
tff(fact_1747_lexord__partial__trans,axiom,
    ! [A: $tType,Xsa: list(A),R2: set(product_prod(A,A)),Ys: list(A),Zs: list(A)] :
      ( ! [X: A,Y2: A,Z3: A] :
          ( member(A,X,aa(list(A),set(A),set2(A),Xsa))
         => ( 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)
           => ( 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)
             => 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),R2) ) ) )
     => ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xsa),Ys),lexord(A,R2))
       => ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Ys),Zs),lexord(A,R2))
         => member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xsa),Zs),lexord(A,R2)) ) ) ) ).

% lexord_partial_trans
tff(fact_1748_distinct__butlast__swap,axiom,
    ! [A: $tType,Pq: list(A),Ia: nat] :
      ( aa(list(A),$o,distinct(A),Pq)
     => aa(list(A),$o,distinct(A),aa(list(A),list(A),butlast(A),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Pq),Ia),last(A,Pq)))) ) ).

% distinct_butlast_swap
tff(fact_1749_subset__mset_Osum__list__nonneg,axiom,
    ! [A: $tType,Xsa: list(multiset(A))] :
      ( ! [X: multiset(A)] :
          ( member(multiset(A),X,aa(list(multiset(A)),set(multiset(A)),set2(multiset(A)),Xsa))
         => aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),zero_zero(multiset(A))),X) )
     => aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),zero_zero(multiset(A))),groups4543113879258116180m_list(multiset(A),plus_plus(multiset(A)),zero_zero(multiset(A)),Xsa)) ) ).

% subset_mset.sum_list_nonneg
tff(fact_1750_subset__mset_Osum__list__nonpos,axiom,
    ! [A: $tType,Xsa: list(multiset(A))] :
      ( ! [X: multiset(A)] :
          ( member(multiset(A),X,aa(list(multiset(A)),set(multiset(A)),set2(multiset(A)),Xsa))
         => aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),X),zero_zero(multiset(A))) )
     => aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),groups4543113879258116180m_list(multiset(A),plus_plus(multiset(A)),zero_zero(multiset(A)),Xsa)),zero_zero(multiset(A))) ) ).

% subset_mset.sum_list_nonpos
tff(fact_1751_subset__mset_Omember__le__sum__list,axiom,
    ! [A: $tType,Xa: multiset(A),Xsa: list(multiset(A))] :
      ( member(multiset(A),Xa,aa(list(multiset(A)),set(multiset(A)),set2(multiset(A)),Xsa))
     => aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),Xa),groups4543113879258116180m_list(multiset(A),plus_plus(multiset(A)),zero_zero(multiset(A)),Xsa)) ) ).

% subset_mset.member_le_sum_list
tff(fact_1752_subset__mset_Osum__list__nonneg__eq__0__iff,axiom,
    ! [A: $tType,Xsa: list(multiset(A))] :
      ( ! [X: multiset(A)] :
          ( member(multiset(A),X,aa(list(multiset(A)),set(multiset(A)),set2(multiset(A)),Xsa))
         => aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),zero_zero(multiset(A))),X) )
     => ( ( groups4543113879258116180m_list(multiset(A),plus_plus(multiset(A)),zero_zero(multiset(A)),Xsa) = zero_zero(multiset(A)) )
      <=> ! [X6: multiset(A)] :
            ( member(multiset(A),X6,aa(list(multiset(A)),set(multiset(A)),set2(multiset(A)),Xsa))
           => ( X6 = zero_zero(multiset(A)) ) ) ) ) ).

% subset_mset.sum_list_nonneg_eq_0_iff
tff(fact_1753_lexord__lex,axiom,
    ! [A: $tType,Xa: list(A),Y: list(A),R2: set(product_prod(A,A))] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xa),Y),lex(A,R2))
    <=> ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xa),Y),lexord(A,R2))
        & ( aa(list(A),nat,size_size(list(A)),Xa) = aa(list(A),nat,size_size(list(A)),Y) ) ) ) ).

% lexord_lex
tff(fact_1754_length__append__singleton,axiom,
    ! [A: $tType,Xsa: list(A),Xa: A] : ( aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),nil(A)))) = aa(nat,nat,suc,aa(list(A),nat,size_size(list(A)),Xsa)) ) ).

% length_append_singleton
tff(fact_1755_length__Suc__conv__rev,axiom,
    ! [A: $tType,Xsa: list(A),N: nat] :
      ( ( aa(list(A),nat,size_size(list(A)),Xsa) = aa(nat,nat,suc,N) )
    <=> ? [Y4: A,Ys4: list(A)] :
          ( ( Xsa = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys4),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),nil(A))) )
          & ( aa(list(A),nat,size_size(list(A)),Ys4) = N ) ) ) ).

% length_Suc_conv_rev
tff(fact_1756_length__Suc__rev__conv,axiom,
    ! [A: $tType,Xsa: list(A),N: nat] :
      ( ( aa(list(A),nat,size_size(list(A)),Xsa) = aa(nat,nat,suc,N) )
    <=> ? [Ys4: list(A),Y4: A] :
          ( ( Xsa = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys4),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),nil(A))) )
          & ( aa(list(A),nat,size_size(list(A)),Ys4) = N ) ) ) ).

% length_Suc_rev_conv
tff(fact_1757_length__compl__rev__induct,axiom,
    ! [A: $tType,Pa: fun(list(A),$o),L: list(A)] :
      ( aa(list(A),$o,Pa,nil(A))
     => ( ! [L3: list(A),E2: A] :
            ( ! [Ll: list(A)] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Ll)),aa(list(A),nat,size_size(list(A)),L3))
               => aa(list(A),$o,Pa,Ll) )
           => aa(list(A),$o,Pa,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E2),nil(A)))) )
       => aa(list(A),$o,Pa,L) ) ) ).

% length_compl_rev_induct
tff(fact_1758_not__distinct__split__distinct,axiom,
    ! [A: $tType,Xsa: list(A)] :
      ( ~ aa(list(A),$o,distinct(A),Xsa)
     => ~ ! [Y2: A,Ys5: list(A)] :
            ( aa(list(A),$o,distinct(A),Ys5)
           => ( member(A,Y2,aa(list(A),set(A),set2(A),Ys5))
             => ! [Zs2: list(A)] : ( Xsa != aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys5),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),Y2),nil(A))),Zs2)) ) ) ) ) ).

% not_distinct_split_distinct
tff(fact_1759_nth__append,axiom,
    ! [A: $tType,Xsa: list(A),Ys: list(A),N: nat] :
      ( aa(nat,A,nth(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),Ys)),N) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xsa)),aa(nat,A,nth(A,Xsa),N),aa(nat,A,nth(A,Ys),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(list(A),nat,size_size(list(A)),Xsa)))) ) ).

% nth_append
tff(fact_1760_list__update__append,axiom,
    ! [A: $tType,Xsa: list(A),Ys: list(A),N: nat,Xa: A] :
      ( aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),Ys)),N),Xa) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xsa)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xsa),N),Xa)),Ys),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Ys),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(list(A),nat,size_size(list(A)),Xsa))),Xa))) ) ).

% list_update_append
tff(fact_1761_listrel1I,axiom,
    ! [A: $tType,Xa: A,Y: A,R2: set(product_prod(A,A)),Xsa: list(A),Us: list(A),Vs: list(A),Ys: list(A)] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Y),R2)
     => ( ( Xsa = 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),Xa),Vs)) )
       => ( ( Ys = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Vs)) )
         => member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xsa),Ys),listrel1(A,R2)) ) ) ) ).

% listrel1I
tff(fact_1762_listrel1E,axiom,
    ! [A: $tType,Xsa: list(A),Ys: list(A),R2: set(product_prod(A,A))] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xsa),Ys),listrel1(A,R2))
     => ~ ! [X: A,Y2: A] :
            ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y2),R2)
           => ! [Us3: list(A),Vs2: list(A)] :
                ( ( Xsa = 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),X),Vs2)) )
               => ( Ys != aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Vs2)) ) ) ) ) ).

% listrel1E
tff(fact_1763_sorted__append__bigger,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xsa: list(A),Y: A] :
          ( sorted_wrt(A,ord_less_eq(A),Xsa)
         => ( ! [X: A] :
                ( member(A,X,aa(list(A),set(A),set2(A),Xsa))
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y) )
           => sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),nil(A)))) ) ) ) ).

% sorted_append_bigger
tff(fact_1764_snoc__listrel1__snoc__iff,axiom,
    ! [A: $tType,Xsa: list(A),Xa: A,Ys: list(A),Y: A,R2: set(product_prod(A,A))] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),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),Xsa),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),nil(A)))),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),nil(A)))),listrel1(A,R2))
    <=> ( ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xsa),Ys),listrel1(A,R2))
          & ( Xa = Y ) )
        | ( ( Xsa = Ys )
          & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Y),R2) ) ) ) ).

% snoc_listrel1_snoc_iff
tff(fact_1765_last__conv__nth,axiom,
    ! [A: $tType,Xsa: list(A)] :
      ( ( Xsa != nil(A) )
     => ( last(A,Xsa) = aa(nat,A,nth(A,Xsa),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xsa)),one_one(nat))) ) ) ).

% last_conv_nth
tff(fact_1766_last__list__update,axiom,
    ! [A: $tType,Xsa: list(A),K3: nat,Xa: A] :
      ( ( Xsa != nil(A) )
     => ( last(A,aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xsa),K3),Xa)) = $ite(K3 = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xsa)),one_one(nat)),Xa,last(A,Xsa)) ) ) ).

% last_list_update
tff(fact_1767_reals__Archimedean2,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xa: A] :
        ? [N5: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),aa(nat,A,semiring_1_of_nat(A),N5)) ) ).

% reals_Archimedean2
tff(fact_1768_take__Suc__conv__app__nth,axiom,
    ! [A: $tType,Ia: nat,Xsa: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),aa(list(A),nat,size_size(list(A)),Xsa))
     => ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),aa(nat,nat,suc,Ia)),Xsa) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),Ia),Xsa)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(nat,A,nth(A,Xsa),Ia)),nil(A))) ) ) ).

% take_Suc_conv_app_nth
tff(fact_1769_pos__int__cases,axiom,
    ! [K3: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K3)
     => ~ ! [N5: nat] :
            ( ( K3 = aa(nat,int,semiring_1_of_nat(int),N5) )
           => ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N5) ) ) ).

% pos_int_cases
tff(fact_1770_zero__less__imp__eq__int,axiom,
    ! [K3: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K3)
     => ? [N5: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N5)
          & ( K3 = aa(nat,int,semiring_1_of_nat(int),N5) ) ) ) ).

% zero_less_imp_eq_int
tff(fact_1771_zmult__zless__mono2__lemma,axiom,
    ! [Ia: int,J: int,K3: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Ia),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K3)
       => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,semiring_1_of_nat(int),K3)),Ia)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,semiring_1_of_nat(int),K3)),J)) ) ) ).

% zmult_zless_mono2_lemma
tff(fact_1772_distinct__concat__iff,axiom,
    ! [A: $tType,Xsa: list(list(A))] :
      ( aa(list(A),$o,distinct(A),aa(list(list(A)),list(A),concat(A),Xsa))
    <=> ( aa(list(list(A)),$o,distinct(list(A)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),removeAll(list(A)),nil(A)),Xsa))
        & ! [Ys4: list(A)] :
            ( member(list(A),Ys4,aa(list(list(A)),set(list(A)),set2(list(A)),Xsa))
           => aa(list(A),$o,distinct(A),Ys4) )
        & ! [Ys4: list(A),Zs3: list(A)] :
            ( ( member(list(A),Ys4,aa(list(list(A)),set(list(A)),set2(list(A)),Xsa))
              & member(list(A),Zs3,aa(list(list(A)),set(list(A)),set2(list(A)),Xsa))
              & ( Ys4 != Zs3 ) )
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Ys4)),aa(list(A),set(A),set2(A),Zs3)) = bot_bot(set(A)) ) ) ) ) ).

% distinct_concat_iff
tff(fact_1773_horner__sum__append,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(A)
     => ! [F2: fun(B,A),A3: A,Xsa: 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),F2),A3),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Xsa),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),F2),A3),Xsa)),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)),Xsa))),aa(list(B),A,aa(A,fun(list(B),A),aa(fun(B,A),fun(A,fun(list(B),A)),groups4207007520872428315er_sum(B,A),F2),A3),Ys))) ) ) ).

% horner_sum_append
tff(fact_1774_drop__last__conv,axiom,
    ! [A: $tType,L: list(A)] :
      ( ( L != nil(A) )
     => ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),L)),aa(nat,nat,suc,zero_zero(nat)))),L) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),last(A,L)),nil(A)) ) ) ).

% drop_last_conv
tff(fact_1775_distinct__product__lists,axiom,
    ! [A: $tType,Xss2: list(list(A))] :
      ( ! [X: list(A)] :
          ( member(list(A),X,aa(list(list(A)),set(list(A)),set2(list(A)),Xss2))
         => aa(list(A),$o,distinct(A),X) )
     => aa(list(list(A)),$o,distinct(list(A)),aa(list(list(A)),list(list(A)),product_lists(A),Xss2)) ) ).

% distinct_product_lists
tff(fact_1776_distinct__concat,axiom,
    ! [A: $tType,Xsa: list(list(A))] :
      ( aa(list(list(A)),$o,distinct(list(A)),Xsa)
     => ( ! [Ys5: list(A)] :
            ( member(list(A),Ys5,aa(list(list(A)),set(list(A)),set2(list(A)),Xsa))
           => aa(list(A),$o,distinct(A),Ys5) )
       => ( ! [Ys5: list(A),Zs2: list(A)] :
              ( member(list(A),Ys5,aa(list(list(A)),set(list(A)),set2(list(A)),Xsa))
             => ( member(list(A),Zs2,aa(list(list(A)),set(list(A)),set2(list(A)),Xsa))
               => ( ( Ys5 != Zs2 )
                 => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Ys5)),aa(list(A),set(A),set2(A),Zs2)) = bot_bot(set(A)) ) ) ) )
         => aa(list(A),$o,distinct(A),aa(list(list(A)),list(A),concat(A),Xsa)) ) ) ) ).

% distinct_concat
tff(fact_1777_drop0,axiom,
    ! [A: $tType,X4: list(A)] : ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),zero_zero(nat)),X4) = X4 ) ).

% drop0
tff(fact_1778_drop__drop,axiom,
    ! [A: $tType,N: nat,M2: nat,Xsa: list(A)] : ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),M2),Xsa)) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M2)),Xsa) ) ).

% drop_drop
tff(fact_1779_length__concat__rev,axiom,
    ! [A: $tType,Xsa: list(list(A))] : ( aa(list(A),nat,size_size(list(A)),aa(list(list(A)),list(A),concat(A),aa(list(list(A)),list(list(A)),rev(list(A)),Xsa))) = aa(list(A),nat,size_size(list(A)),aa(list(list(A)),list(A),concat(A),Xsa)) ) ).

% length_concat_rev
tff(fact_1780_concat__append,axiom,
    ! [A: $tType,Xsa: list(list(A)),Ys: list(list(A))] : ( aa(list(list(A)),list(A),concat(A),aa(list(list(A)),list(list(A)),aa(list(list(A)),fun(list(list(A)),list(list(A))),append(list(A)),Xsa),Ys)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(list(A)),list(A),concat(A),Xsa)),aa(list(list(A)),list(A),concat(A),Ys)) ) ).

% concat_append
tff(fact_1781_drop__Suc__Cons,axiom,
    ! [A: $tType,N: nat,Xa: A,Xsa: list(A)] : ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),aa(nat,nat,suc,N)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa)) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),Xsa) ) ).

% drop_Suc_Cons
tff(fact_1782_length__drop,axiom,
    ! [A: $tType,N: nat,Xsa: list(A)] : ( aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),Xsa)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xsa)),N) ) ).

% length_drop
tff(fact_1783_drop__upd__irrelevant,axiom,
    ! [A: $tType,M2: nat,N: nat,L: list(A),Xa: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),N)
     => ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),L),M2),Xa)) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),L) ) ) ).

% drop_upd_irrelevant
tff(fact_1784_drop__update__cancel,axiom,
    ! [A: $tType,N: nat,M2: nat,Xsa: list(A),Xa: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),M2)
     => ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),M2),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xsa),N),Xa)) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),M2),Xsa) ) ) ).

% drop_update_cancel
tff(fact_1785_append__take__drop__id,axiom,
    ! [A: $tType,N: nat,Xsa: list(A)] : ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),Xsa)),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),Xsa)) = Xsa ) ).

% append_take_drop_id
tff(fact_1786_concat__eq__Nil__conv,axiom,
    ! [A: $tType,Xss2: list(list(A))] :
      ( ( aa(list(list(A)),list(A),concat(A),Xss2) = nil(A) )
    <=> ! [X6: list(A)] :
          ( member(list(A),X6,aa(list(list(A)),set(list(A)),set2(list(A)),Xss2))
         => ( X6 = nil(A) ) ) ) ).

% concat_eq_Nil_conv
tff(fact_1787_Nil__eq__concat__conv,axiom,
    ! [A: $tType,Xss2: list(list(A))] :
      ( ( nil(A) = aa(list(list(A)),list(A),concat(A),Xss2) )
    <=> ! [X6: list(A)] :
          ( member(list(A),X6,aa(list(list(A)),set(list(A)),set2(list(A)),Xss2))
         => ( X6 = nil(A) ) ) ) ).

% Nil_eq_concat_conv
tff(fact_1788_drop__eq__Nil2,axiom,
    ! [A: $tType,N: nat,Xsa: list(A)] :
      ( ( nil(A) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),Xsa) )
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xsa)),N) ) ).

% drop_eq_Nil2
tff(fact_1789_drop__eq__Nil,axiom,
    ! [A: $tType,N: nat,Xsa: list(A)] :
      ( ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),Xsa) = nil(A) )
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xsa)),N) ) ).

% drop_eq_Nil
tff(fact_1790_drop__all,axiom,
    ! [A: $tType,Xsa: list(A),N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xsa)),N)
     => ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),Xsa) = nil(A) ) ) ).

% drop_all
tff(fact_1791_drop__append,axiom,
    ! [A: $tType,N: nat,Xsa: list(A),Ys: list(A)] : ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),Ys)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),Xsa)),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(list(A),nat,size_size(list(A)),Xsa))),Ys)) ) ).

% drop_append
tff(fact_1792_horner__sum__simps_I2_J,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_0(A)
     => ! [F2: fun(B,A),A3: A,Xa: B,Xsa: 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),F2),A3),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Xa),Xsa)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,F2,Xa)),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),F2),A3),Xsa))) ) ) ).

% horner_sum_simps(2)
tff(fact_1793_last__drop,axiom,
    ! [A: $tType,N: nat,Xsa: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xsa))
     => ( last(A,aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),Xsa)) = last(A,Xsa) ) ) ).

% last_drop
tff(fact_1794_nth__drop,axiom,
    ! [A: $tType,N: nat,Xsa: list(A),Ia: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),aa(list(A),nat,size_size(list(A)),Xsa))
     => ( aa(nat,A,nth(A,aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),Xsa)),Ia) = aa(nat,A,nth(A,Xsa),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),Ia)) ) ) ).

% nth_drop
tff(fact_1795_drop__zip,axiom,
    ! [A: $tType,B: $tType,N: nat,Xsa: list(A),Ys: list(B)] : ( aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(nat,fun(list(product_prod(A,B)),list(product_prod(A,B))),drop(product_prod(A,B)),N),aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),Xsa),Ys)) = aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),Xsa)),aa(list(B),list(B),aa(nat,fun(list(B),list(B)),drop(B),N),Ys)) ) ).

% drop_zip
tff(fact_1796_drop__Nil,axiom,
    ! [A: $tType,N: nat] : ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),nil(A)) = nil(A) ) ).

% drop_Nil
tff(fact_1797_drop__0,axiom,
    ! [A: $tType,Xsa: list(A)] : ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),zero_zero(nat)),Xsa) = Xsa ) ).

% drop_0
tff(fact_1798_in__set__dropD,axiom,
    ! [A: $tType,Xa: A,N: nat,Xsa: list(A)] :
      ( member(A,Xa,aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),Xsa)))
     => member(A,Xa,aa(list(A),set(A),set2(A),Xsa)) ) ).

% in_set_dropD
tff(fact_1799_distinct__drop,axiom,
    ! [A: $tType,Xsa: list(A),Ia: nat] :
      ( aa(list(A),$o,distinct(A),Xsa)
     => aa(list(A),$o,distinct(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),Ia),Xsa)) ) ).

% distinct_drop
tff(fact_1800_sorted__wrt__drop,axiom,
    ! [A: $tType,F2: fun(A,fun(A,$o)),Xsa: list(A),N: nat] :
      ( sorted_wrt(A,F2,Xsa)
     => sorted_wrt(A,F2,aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),Xsa)) ) ).

% sorted_wrt_drop
tff(fact_1801_drop__butlast,axiom,
    ! [A: $tType,N: nat,Xsa: list(A)] : ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),aa(list(A),list(A),butlast(A),Xsa)) = aa(list(A),list(A),butlast(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),Xsa)) ) ).

% drop_butlast
tff(fact_1802_foldl__foldl__conv__concat,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,fun(B,A)),A3: A,Xsa: list(list(B))] : ( aa(list(list(B)),A,aa(A,fun(list(list(B)),A),aa(fun(A,fun(list(B),A)),fun(A,fun(list(list(B)),A)),foldl(A,list(B)),aa(fun(A,fun(B,A)),fun(A,fun(list(B),A)),foldl(A,B),F2)),A3),Xsa) = aa(list(B),A,aa(A,fun(list(B),A),aa(fun(A,fun(B,A)),fun(A,fun(list(B),A)),foldl(A,B),F2),A3),aa(list(list(B)),list(B),concat(B),Xsa)) ) ).

% foldl_foldl_conv_concat
tff(fact_1803_drop__eq__ConsD,axiom,
    ! [A: $tType,N: nat,Xsa: list(A),Xa: A,Xs3: list(A)] :
      ( ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),Xsa) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xs3) )
     => ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),aa(nat,nat,suc,N)),Xsa) = Xs3 ) ) ).

% drop_eq_ConsD
tff(fact_1804_set__drop__subset,axiom,
    ! [A: $tType,N: nat,Xsa: list(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),Xsa))),aa(list(A),set(A),set2(A),Xsa)) ).

% set_drop_subset
tff(fact_1805_nth__via__drop,axiom,
    ! [A: $tType,N: nat,Xsa: list(A),Y: A,Ys: list(A)] :
      ( ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),Xsa) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys) )
     => ( aa(nat,A,nth(A,Xsa),N) = Y ) ) ).

% nth_via_drop
tff(fact_1806_sorted__drop,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xsa: list(A),N: nat] :
          ( sorted_wrt(A,ord_less_eq(A),Xsa)
         => sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),Xsa)) ) ) ).

% sorted_drop
tff(fact_1807_take__drop,axiom,
    ! [A: $tType,N: nat,M2: nat,Xsa: list(A)] : ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),M2),Xsa)) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),M2),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M2)),Xsa)) ) ).

% take_drop
tff(fact_1808_drop__take,axiom,
    ! [A: $tType,N: nat,M2: nat,Xsa: list(A)] : ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),M2),Xsa)) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),N)),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),Xsa)) ) ).

% drop_take
tff(fact_1809_concat_Osimps_I1_J,axiom,
    ! [A: $tType] : ( aa(list(list(A)),list(A),concat(A),nil(list(A))) = nil(A) ) ).

% concat.simps(1)
tff(fact_1810_concat_Osimps_I2_J,axiom,
    ! [A: $tType,Xa: list(A),Xsa: list(list(A))] : ( aa(list(list(A)),list(A),concat(A),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),Xa),Xsa)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xa),aa(list(list(A)),list(A),concat(A),Xsa)) ) ).

% concat.simps(2)
tff(fact_1811_set__drop__subset__set__drop,axiom,
    ! [A: $tType,N: nat,M2: nat,Xsa: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),M2)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),M2),Xsa))),aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),Xsa))) ) ).

% set_drop_subset_set_drop
tff(fact_1812_append__eq__conv__conj,axiom,
    ! [A: $tType,Xsa: list(A),Ys: list(A),Zs: list(A)] :
      ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),Ys) = Zs )
    <=> ( ( Xsa = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),aa(list(A),nat,size_size(list(A)),Xsa)),Zs) )
        & ( Ys = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),aa(list(A),nat,size_size(list(A)),Xsa)),Zs) ) ) ) ).

% append_eq_conv_conj
tff(fact_1813_drop__take__drop__unsplit,axiom,
    ! [A: $tType,Ia: nat,J: nat,L: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ia),J)
     => ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),Ia),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),J),L))),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),J),L)) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),Ia),L) ) ) ).

% drop_take_drop_unsplit
tff(fact_1814_take__add,axiom,
    ! [A: $tType,Ia: nat,J: nat,Xsa: list(A)] : ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ia),J)),Xsa) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),Ia),Xsa)),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),J),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),Ia),Xsa))) ) ).

% take_add
tff(fact_1815_drop__update__swap,axiom,
    ! [A: $tType,M2: nat,N: nat,Xsa: list(A),Xa: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),N)
     => ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),M2),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xsa),N),Xa)) = aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),M2),Xsa)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M2)),Xa) ) ) ).

% drop_update_swap
tff(fact_1816_slice__def,axiom,
    ! [A: $tType,From: nat,To: nat,List: list(A)] : ( slice(A,From,To,List) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),To),From)),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),From),List)) ) ).

% slice_def
tff(fact_1817_concat__eq__appendD,axiom,
    ! [A: $tType,Xss2: list(list(A)),Ys: list(A),Zs: list(A)] :
      ( ( aa(list(list(A)),list(A),concat(A),Xss2) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Zs) )
     => ( ( Xss2 != nil(list(A)) )
       => ? [Xss1: list(list(A)),Xs: list(A),Xs4: list(A),Xss22: list(list(A))] :
            ( ( Xss2 = aa(list(list(A)),list(list(A)),aa(list(list(A)),fun(list(list(A)),list(list(A))),append(list(A)),Xss1),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Xs4)),Xss22)) )
            & ( Ys = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(list(A)),list(A),concat(A),Xss1)),Xs) )
            & ( Zs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs4),aa(list(list(A)),list(A),concat(A),Xss22)) ) ) ) ) ).

% concat_eq_appendD
tff(fact_1818_drop__Cons_H,axiom,
    ! [A: $tType,N: nat,Xa: A,Xsa: list(A)] :
      ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa)) = $ite(N = zero_zero(nat),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))),Xsa)) ) ).

% drop_Cons'
tff(fact_1819_append__eq__append__conv__if,axiom,
    ! [A: $tType,Xs_1: list(A),Xs_2: list(A),Ys_1: list(A),Ys_2: list(A)] :
      ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs_1),Xs_2) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys_1),Ys_2) )
    <=> $ite(
          aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs_1)),aa(list(A),nat,size_size(list(A)),Ys_1)),
          ( ( Xs_1 = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),aa(list(A),nat,size_size(list(A)),Xs_1)),Ys_1) )
          & ( Xs_2 = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),aa(list(A),nat,size_size(list(A)),Xs_1)),Ys_1)),Ys_2) ) ),
          ( ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),aa(list(A),nat,size_size(list(A)),Ys_1)),Xs_1) = Ys_1 )
          & ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),aa(list(A),nat,size_size(list(A)),Ys_1)),Xs_1)),Xs_2) = Ys_2 ) ) ) ) ).

% append_eq_append_conv_if
tff(fact_1820_zip__append1,axiom,
    ! [A: $tType,B: $tType,Xsa: list(A),Ys: list(A),Zs: list(B)] : ( aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),Ys)),Zs) = aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(list(product_prod(A,B)),fun(list(product_prod(A,B)),list(product_prod(A,B))),append(product_prod(A,B)),aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),Xsa),aa(list(B),list(B),aa(nat,fun(list(B),list(B)),take(B),aa(list(A),nat,size_size(list(A)),Xsa)),Zs))),aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),Ys),aa(list(B),list(B),aa(nat,fun(list(B),list(B)),drop(B),aa(list(A),nat,size_size(list(A)),Xsa)),Zs))) ) ).

% zip_append1
tff(fact_1821_zip__append2,axiom,
    ! [A: $tType,B: $tType,Xsa: list(A),Ys: list(B),Zs: list(B)] : ( aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),Xsa),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Ys),Zs)) = aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(list(product_prod(A,B)),fun(list(product_prod(A,B)),list(product_prod(A,B))),append(product_prod(A,B)),aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),aa(list(B),nat,size_size(list(B)),Ys)),Xsa)),Ys)),aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),aa(list(B),nat,size_size(list(B)),Ys)),Xsa)),Zs)) ) ).

% zip_append2
tff(fact_1822_take__rev,axiom,
    ! [A: $tType,N: nat,Xsa: list(A)] : ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),aa(list(A),list(A),rev(A),Xsa)) = aa(list(A),list(A),rev(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xsa)),N)),Xsa)) ) ).

% take_rev
tff(fact_1823_rev__take,axiom,
    ! [A: $tType,Ia: nat,Xsa: list(A)] : ( aa(list(A),list(A),rev(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),Ia),Xsa)) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xsa)),Ia)),aa(list(A),list(A),rev(A),Xsa)) ) ).

% rev_take
tff(fact_1824_rev__drop,axiom,
    ! [A: $tType,Ia: nat,Xsa: list(A)] : ( aa(list(A),list(A),rev(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),Ia),Xsa)) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xsa)),Ia)),aa(list(A),list(A),rev(A),Xsa)) ) ).

% rev_drop
tff(fact_1825_drop__rev,axiom,
    ! [A: $tType,N: nat,Xsa: list(A)] : ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),aa(list(A),list(A),rev(A),Xsa)) = aa(list(A),list(A),rev(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xsa)),N)),Xsa)) ) ).

% drop_rev
tff(fact_1826_concat__eq__append__conv,axiom,
    ! [A: $tType,Xss2: list(list(A)),Ys: list(A),Zs: list(A)] :
      ( ( aa(list(list(A)),list(A),concat(A),Xss2) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Zs) )
    <=> $ite(
          Xss2 = nil(list(A)),
          ( ( Ys = nil(A) )
          & ( Zs = nil(A) ) ),
          ? [Xss12: list(list(A)),Xs2: list(A),Xs5: list(A),Xss23: list(list(A))] :
            ( ( Xss2 = aa(list(list(A)),list(list(A)),aa(list(list(A)),fun(list(list(A)),list(list(A))),append(list(A)),Xss12),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs2),Xs5)),Xss23)) )
            & ( Ys = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(list(A)),list(A),concat(A),Xss12)),Xs2) )
            & ( Zs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs5),aa(list(list(A)),list(A),concat(A),Xss23)) ) ) ) ) ).

% concat_eq_append_conv
tff(fact_1827_Cons__nth__drop__Suc,axiom,
    ! [A: $tType,Ia: nat,Xsa: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),aa(list(A),nat,size_size(list(A)),Xsa))
     => ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(nat,A,nth(A,Xsa),Ia)),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),aa(nat,nat,suc,Ia)),Xsa)) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),Ia),Xsa) ) ) ).

% Cons_nth_drop_Suc
tff(fact_1828_in__set__drop__conv__nth,axiom,
    ! [A: $tType,Xa: A,N: nat,L: list(A)] :
      ( member(A,Xa,aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),L)))
    <=> ? [I2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),I2)
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),L))
          & ( Xa = aa(nat,A,nth(A,L),I2) ) ) ) ).

% in_set_drop_conv_nth
tff(fact_1829_set__take__disj__set__drop__if__distinct,axiom,
    ! [A: $tType,Vs: list(A),Ia: nat,J: nat] :
      ( aa(list(A),$o,distinct(A),Vs)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ia),J)
       => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),Ia),Vs))),aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),J),Vs))) = bot_bot(set(A)) ) ) ) ).

% set_take_disj_set_drop_if_distinct
tff(fact_1830_foldl__list__update,axiom,
    ! [B: $tType,A: $tType,N: nat,Xsa: list(A),F2: fun(B,fun(A,B)),A3: B,Xa: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xsa))
     => ( aa(list(A),B,aa(B,fun(list(A),B),aa(fun(B,fun(A,B)),fun(B,fun(list(A),B)),foldl(B,A),F2),A3),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xsa),N),Xa)) = aa(list(A),B,aa(B,fun(list(A),B),aa(fun(B,fun(A,B)),fun(B,fun(list(A),B)),foldl(B,A),F2),aa(A,B,aa(B,fun(A,B),F2,aa(list(A),B,aa(B,fun(list(A),B),aa(fun(B,fun(A,B)),fun(B,fun(list(A),B)),foldl(B,A),F2),A3),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),Xsa))),Xa)),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),aa(nat,nat,suc,N)),Xsa)) ) ) ).

% foldl_list_update
tff(fact_1831_product__lists_Osimps_I1_J,axiom,
    ! [A: $tType] : ( aa(list(list(A)),list(list(A)),product_lists(A),nil(list(A))) = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),nil(A)),nil(list(A))) ) ).

% product_lists.simps(1)
tff(fact_1832_id__take__nth__drop,axiom,
    ! [A: $tType,Ia: nat,Xsa: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),aa(list(A),nat,size_size(list(A)),Xsa))
     => ( Xsa = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),Ia),Xsa)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(nat,A,nth(A,Xsa),Ia)),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),aa(nat,nat,suc,Ia)),Xsa))) ) ) ).

% id_take_nth_drop
tff(fact_1833_upd__conv__take__nth__drop,axiom,
    ! [A: $tType,Ia: nat,Xsa: list(A),A3: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),aa(list(A),nat,size_size(list(A)),Xsa))
     => ( aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xsa),Ia),A3) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),Ia),Xsa)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),aa(nat,nat,suc,Ia)),Xsa))) ) ) ).

% upd_conv_take_nth_drop
tff(fact_1834_in__set__product__lists__length,axiom,
    ! [A: $tType,Xsa: list(A),Xss2: list(list(A))] :
      ( member(list(A),Xsa,aa(list(list(A)),set(list(A)),set2(list(A)),aa(list(list(A)),list(list(A)),product_lists(A),Xss2)))
     => ( aa(list(A),nat,size_size(list(A)),Xsa) = aa(list(list(A)),nat,size_size(list(list(A))),Xss2) ) ) ).

% in_set_product_lists_length
tff(fact_1835_merge__list__correct,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ls3: list(list(A)),As2: list(list(A))] :
          ( ! [L3: list(A)] :
              ( member(list(A),L3,aa(list(list(A)),set(list(A)),set2(list(A)),Ls3))
             => ( aa(list(A),$o,distinct(A),L3)
                & sorted_wrt(A,ord_less_eq(A),L3) ) )
         => ( ! [L3: list(A)] :
                ( member(list(A),L3,aa(list(list(A)),set(list(A)),set2(list(A)),As2))
               => ( aa(list(A),$o,distinct(A),L3)
                  & sorted_wrt(A,ord_less_eq(A),L3) ) )
           => ( aa(list(A),$o,distinct(A),merge_list(A,As2,Ls3))
              & sorted_wrt(A,ord_less_eq(A),merge_list(A,As2,Ls3))
              & ( aa(list(A),set(A),set2(A),merge_list(A,As2,Ls3)) = aa(list(A),set(A),set2(A),aa(list(list(A)),list(A),concat(A),aa(list(list(A)),list(list(A)),aa(list(list(A)),fun(list(list(A)),list(list(A))),append(list(A)),As2),Ls3))) ) ) ) ) ) ).

% merge_list_correct
tff(fact_1836_take__hd__drop,axiom,
    ! [A: $tType,N: nat,Xsa: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xsa))
     => ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),Xsa)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(list(A),A,hd(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),Xsa))),nil(A))) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),aa(nat,nat,suc,N)),Xsa) ) ) ).

% take_hd_drop
tff(fact_1837_eucl__rel__int__iff,axiom,
    ! [K3: int,L: int,Q2: int,R2: int] :
      ( eucl_rel_int(K3,L,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),R2))
    <=> ( ( K3 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),L),Q2)),R2) )
        & $ite(
            aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),L),
            ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),R2)
            & aa(int,$o,aa(int,fun(int,$o),ord_less(int),R2),L) ),
            $ite(
              aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),zero_zero(int)),
              ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),R2)
              & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),R2),zero_zero(int)) ),
              Q2 = zero_zero(int) ) ) ) ) ).

% eucl_rel_int_iff
tff(fact_1838_linear__order__on__singleton,axiom,
    ! [A: $tType,Xa: A] : order_679001287576687338der_on(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),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))),insert3(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Xa)),bot_bot(set(product_prod(A,A))))) ).

% linear_order_on_singleton
tff(fact_1839_Total__subset__Id,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( total_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R2),id2(A))
       => ( ( R2 = bot_bot(set(product_prod(A,A))) )
          | ? [A4: A] : ( R2 = aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(product_prod(A,A),fun(set(product_prod(A,A)),set(product_prod(A,A))),insert3(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),A4)),bot_bot(set(product_prod(A,A)))) ) ) ) ) ).

% Total_subset_Id
tff(fact_1840_nat0__intermed__int__val,axiom,
    ! [N: nat,F2: fun(nat,int),K3: int] :
      ( ! [I3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),N)
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),abs_abs(int,aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I3),one_one(nat)))),aa(nat,int,F2,I3)))),one_one(int)) )
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,F2,zero_zero(nat))),K3)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K3),aa(nat,int,F2,N))
         => ? [I3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I3),N)
              & ( aa(nat,int,F2,I3) = K3 ) ) ) ) ) ).

% nat0_intermed_int_val
tff(fact_1841_revg__fun,axiom,
    ! [A: $tType,A3: list(A),B3: list(A)] : ( revg(A,A3,B3) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),rev(A),A3)),B3) ) ).

% revg_fun
tff(fact_1842_abs__abs,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A3: A] : ( abs_abs(A,abs_abs(A,A3)) = abs_abs(A,A3) ) ) ).

% abs_abs
tff(fact_1843_abs__idempotent,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] : ( abs_abs(A,abs_abs(A,A3)) = abs_abs(A,A3) ) ) ).

% abs_idempotent
tff(fact_1844_abs__0,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ( abs_abs(A,zero_zero(A)) = zero_zero(A) ) ) ).

% abs_0
tff(fact_1845_abs__0__eq,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] :
          ( ( zero_zero(A) = abs_abs(A,A3) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% abs_0_eq
tff(fact_1846_abs__eq__0,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] :
          ( ( abs_abs(A,A3) = zero_zero(A) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% abs_eq_0
tff(fact_1847_abs__zero,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ( abs_abs(A,zero_zero(A)) = zero_zero(A) ) ) ).

% abs_zero
tff(fact_1848_abs__add__abs,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B3: A] : ( abs_abs(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),abs_abs(A,A3)),abs_abs(A,B3))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),abs_abs(A,A3)),abs_abs(A,B3)) ) ) ).

% abs_add_abs
tff(fact_1849_abs__mult__self__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] : ( aa(A,A,aa(A,fun(A,A),times_times(A),abs_abs(A,A3)),abs_abs(A,A3)) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),A3) ) ) ).

% abs_mult_self_eq
tff(fact_1850_abs__1,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ( abs_abs(A,one_one(A)) = one_one(A) ) ) ).

% abs_1
tff(fact_1851_pair__in__Id__conv,axiom,
    ! [A: $tType,A3: A,B3: A] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B3),id2(A))
    <=> ( A3 = B3 ) ) ).

% pair_in_Id_conv
tff(fact_1852_IdI,axiom,
    ! [A: $tType,A3: A] : 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_1853_total__on__diff__Id,axiom,
    ! [A: $tType,Aa2: set(A),R2: set(product_prod(A,A))] :
      ( total_on(A,Aa2,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)))
    <=> total_on(A,Aa2,R2) ) ).

% total_on_diff_Id
tff(fact_1854_bijective__Id,axiom,
    ! [A: $tType] : bijective(A,A,id2(A)) ).

% bijective_Id
tff(fact_1855_abs__le__zero__iff,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),abs_abs(A,A3)),zero_zero(A))
        <=> ( A3 = zero_zero(A) ) ) ) ).

% abs_le_zero_iff
tff(fact_1856_abs__le__self__iff,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),abs_abs(A,A3)),A3)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3) ) ) ).

% abs_le_self_iff
tff(fact_1857_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)
         => ( abs_abs(A,A3) = A3 ) ) ) ).

% abs_of_nonneg
tff(fact_1858_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)),abs_abs(A,A3))
        <=> ( A3 != zero_zero(A) ) ) ) ).

% zero_less_abs_iff
tff(fact_1859_hd__append2,axiom,
    ! [A: $tType,Xsa: list(A),Ys: list(A)] :
      ( ( Xsa != nil(A) )
     => ( aa(list(A),A,hd(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),Ys)) = aa(list(A),A,hd(A),Xsa) ) ) ).

% hd_append2
tff(fact_1860_hd__take,axiom,
    ! [A: $tType,J: nat,Xsa: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),J)
     => ( aa(list(A),A,hd(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),J),Xsa)) = aa(list(A),A,hd(A),Xsa) ) ) ).

% hd_take
tff(fact_1861_zero__less__power__abs__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,N: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),abs_abs(A,A3)),N))
        <=> ( ( A3 != zero_zero(A) )
            | ( N = zero_zero(nat) ) ) ) ) ).

% zero_less_power_abs_iff
tff(fact_1862_unique__remainder,axiom,
    ! [A3: int,B3: int,Q2: int,R2: int,Q5: int,R5: int] :
      ( eucl_rel_int(A3,B3,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),R2))
     => ( eucl_rel_int(A3,B3,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q5),R5))
       => ( R2 = R5 ) ) ) ).

% unique_remainder
tff(fact_1863_unique__quotient,axiom,
    ! [A3: int,B3: int,Q2: int,R2: int,Q5: int,R5: int] :
      ( eucl_rel_int(A3,B3,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),R2))
     => ( eucl_rel_int(A3,B3,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q5),R5))
       => ( Q2 = Q5 ) ) ) ).

% unique_quotient
tff(fact_1864_abs__le__D1,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),abs_abs(A,A3)),B3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3) ) ) ).

% abs_le_D1
tff(fact_1865_abs__ge__self,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),abs_abs(A,A3)) ) ).

% abs_ge_self
tff(fact_1866_abs__eq__0__iff,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A3: A] :
          ( ( abs_abs(A,A3) = zero_zero(A) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% abs_eq_0_iff
tff(fact_1867_abs__mult,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A3: A,B3: A] : ( abs_abs(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)) = aa(A,A,aa(A,fun(A,A),times_times(A),abs_abs(A,A3)),abs_abs(A,B3)) ) ) ).

% abs_mult
tff(fact_1868_abs__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ( abs_abs(A,one_one(A)) = one_one(A) ) ) ).

% abs_one
tff(fact_1869_abs__minus__commute,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B3: A] : ( abs_abs(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B3)) = abs_abs(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),B3),A3)) ) ) ).

% abs_minus_commute
tff(fact_1870_list_Osel_I1_J,axiom,
    ! [A: $tType,X21: A,X222: list(A)] : ( aa(list(A),A,hd(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X21),X222)) = X21 ) ).

% list.sel(1)
tff(fact_1871_Linear__order__in__diff__Id,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A3: A,B3: A] :
      ( order_679001287576687338der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => ( member(A,A3,aa(set(product_prod(A,A)),set(A),field2(A),R2))
       => ( member(A,B3,aa(set(product_prod(A,A)),set(A),field2(A),R2))
         => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B3),R2)
          <=> ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B3),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_1872_BNF__Greatest__Fixpoint_OIdD,axiom,
    ! [A: $tType,A3: A,B3: A] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B3),id2(A))
     => ( A3 = B3 ) ) ).

% BNF_Greatest_Fixpoint.IdD
tff(fact_1873_IdE,axiom,
    ! [A: $tType,P2: product_prod(A,A)] :
      ( member(product_prod(A,A),P2,id2(A))
     => ~ ! [X: A] : ( P2 != aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),X) ) ) ).

% IdE
tff(fact_1874_hd__zip,axiom,
    ! [A: $tType,B: $tType,Xsa: list(A),Ys: list(B)] :
      ( ( Xsa != nil(A) )
     => ( ( Ys != nil(B) )
       => ( aa(list(product_prod(A,B)),product_prod(A,B),hd(product_prod(A,B)),aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),Xsa),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),Xsa)),aa(list(B),B,hd(B),Ys)) ) ) ) ).

% hd_zip
tff(fact_1875_hd__concat,axiom,
    ! [A: $tType,Xsa: list(list(A))] :
      ( ( Xsa != nil(list(A)) )
     => ( ( aa(list(list(A)),list(A),hd(list(A)),Xsa) != nil(A) )
       => ( aa(list(A),A,hd(A),aa(list(list(A)),list(A),concat(A),Xsa)) = aa(list(A),A,hd(A),aa(list(list(A)),list(A),hd(list(A)),Xsa)) ) ) ) ).

% hd_concat
tff(fact_1876_eucl__rel__int__by0,axiom,
    ! [K3: int] : eucl_rel_int(K3,zero_zero(int),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),K3)) ).

% eucl_rel_int_by0
tff(fact_1877_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)),abs_abs(A,A3)) ) ).

% abs_ge_zero
tff(fact_1878_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)
         => ( abs_abs(A,A3) = A3 ) ) ) ).

% abs_of_pos
tff(fact_1879_abs__not__less__zero,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),abs_abs(A,A3)),zero_zero(A)) ) ).

% abs_not_less_zero
tff(fact_1880_abs__triangle__ineq,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),abs_abs(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3))),aa(A,A,aa(A,fun(A,A),plus_plus(A),abs_abs(A,A3)),abs_abs(A,B3))) ) ).

% abs_triangle_ineq
tff(fact_1881_abs__triangle__ineq2__sym,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),abs_abs(A,A3)),abs_abs(A,B3))),abs_abs(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),B3),A3))) ) ).

% abs_triangle_ineq2_sym
tff(fact_1882_abs__triangle__ineq3,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),abs_abs(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),abs_abs(A,A3)),abs_abs(A,B3)))),abs_abs(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B3))) ) ).

% abs_triangle_ineq3
tff(fact_1883_abs__triangle__ineq2,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),abs_abs(A,A3)),abs_abs(A,B3))),abs_abs(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B3))) ) ).

% abs_triangle_ineq2
tff(fact_1884_abs__mult__less,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,C2: A,B3: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),abs_abs(A,A3)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),abs_abs(A,B3)),D3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),abs_abs(A,A3)),abs_abs(A,B3))),aa(A,A,aa(A,fun(A,A),times_times(A),C2),D3)) ) ) ) ).

% abs_mult_less
tff(fact_1885_hd__in__set,axiom,
    ! [A: $tType,Xsa: list(A)] :
      ( ( Xsa != nil(A) )
     => member(A,aa(list(A),A,hd(A),Xsa),aa(list(A),set(A),set2(A),Xsa)) ) ).

% hd_in_set
tff(fact_1886_list_Oset__sel_I1_J,axiom,
    ! [A: $tType,A3: list(A)] :
      ( ( A3 != nil(A) )
     => member(A,aa(list(A),A,hd(A),A3),aa(list(A),set(A),set2(A),A3)) ) ).

% list.set_sel(1)
tff(fact_1887_hd__append,axiom,
    ! [A: $tType,Xsa: list(A),Ys: list(A)] :
      ( aa(list(A),A,hd(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),Ys)) = $ite(Xsa = nil(A),aa(list(A),A,hd(A),Ys),aa(list(A),A,hd(A),Xsa)) ) ).

% hd_append
tff(fact_1888_longest__common__prefix,axiom,
    ! [A: $tType,Xsa: list(A),Ys: list(A)] :
    ? [Ps2: list(A),Xs4: list(A),Ys3: list(A)] :
      ( ( Xsa = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ps2),Xs4) )
      & ( Ys = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ps2),Ys3) )
      & ( ( Xs4 = nil(A) )
        | ( Ys3 = nil(A) )
        | ( aa(list(A),A,hd(A),Xs4) != aa(list(A),A,hd(A),Ys3) ) ) ) ).

% longest_common_prefix
tff(fact_1889_hd__Nil__eq__last,axiom,
    ! [A: $tType] : ( aa(list(A),A,hd(A),nil(A)) = last(A,nil(A)) ) ).

% hd_Nil_eq_last
tff(fact_1890_last__rev,axiom,
    ! [A: $tType,Xsa: list(A)] : ( last(A,aa(list(A),list(A),rev(A),Xsa)) = aa(list(A),A,hd(A),Xsa) ) ).

% last_rev
tff(fact_1891_hd__rev,axiom,
    ! [A: $tType,Xsa: list(A)] : ( aa(list(A),A,hd(A),aa(list(A),list(A),rev(A),Xsa)) = last(A,Xsa) ) ).

% hd_rev
tff(fact_1892_revg_Osimps_I2_J,axiom,
    ! [A: $tType,A3: A,As2: list(A),B3: list(A)] : ( revg(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),As2),B3) = revg(A,As2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),B3)) ) ).

% revg.simps(2)
tff(fact_1893_revg_Osimps_I1_J,axiom,
    ! [A: $tType,B3: list(A)] : ( revg(A,nil(A),B3) = B3 ) ).

% revg.simps(1)
tff(fact_1894_dense__eq0__I,axiom,
    ! [A: $tType] :
      ( ( ordere166539214618696060dd_abs(A)
        & dense_linorder(A) )
     => ! [Xa: A] :
          ( ! [E2: A] :
              ( aa(A,$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),abs_abs(A,Xa)),E2) )
         => ( Xa = zero_zero(A) ) ) ) ).

% dense_eq0_I
tff(fact_1895_merge__list_Osimps_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( merge_list(A,nil(list(A)),nil(list(A))) = nil(A) ) ) ).

% merge_list.simps(1)
tff(fact_1896_abs__eq__mult,axiom,
    ! [A: $tType] :
      ( ordered_ring_abs(A)
     => ! [A3: A,B3: A] :
          ( ( ( aa(A,$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,$o),ord_less_eq(A),zero_zero(A)),B3)
              | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),zero_zero(A)) ) )
         => ( abs_abs(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)) = aa(A,A,aa(A,fun(A,A),times_times(A),abs_abs(A,A3)),abs_abs(A,B3)) ) ) ) ).

% abs_eq_mult
tff(fact_1897_abs__mult__pos,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xa)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),abs_abs(A,Y)),Xa) = abs_abs(A,aa(A,A,aa(A,fun(A,A),times_times(A),Y),Xa)) ) ) ) ).

% abs_mult_pos
tff(fact_1898_abs__diff__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A,A3: A,R2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),abs_abs(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Xa),A3))),R2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),R2)),Xa)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),R2)) ) ) ) ).

% abs_diff_le_iff
tff(fact_1899_abs__triangle__ineq4,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),abs_abs(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B3))),aa(A,A,aa(A,fun(A,A),plus_plus(A),abs_abs(A,A3)),abs_abs(A,B3))) ) ).

% abs_triangle_ineq4
tff(fact_1900_abs__diff__triangle__ineq,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B3: A,C2: A,D3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(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),B3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),D3)))),aa(A,A,aa(A,fun(A,A),plus_plus(A),abs_abs(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),C2))),abs_abs(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),B3),D3)))) ) ).

% abs_diff_triangle_ineq
tff(fact_1901_abs__diff__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A,A3: A,R2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),abs_abs(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Xa),A3))),R2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),R2)),Xa)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),R2)) ) ) ) ).

% abs_diff_less_iff
tff(fact_1902_merge__list_Osimps_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [La2: list(A),Acc22: list(list(A)),L: 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),Acc22),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),Acc22))) ) ) ).

% merge_list.simps(4)
tff(fact_1903_merge__list_Osimps_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [La2: list(A),Acc22: list(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),Acc22),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),Acc22)) ) ) ).

% merge_list.simps(3)
tff(fact_1904_merge__list_Osimps_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: 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.simps(2)
tff(fact_1905_hd__conv__nth,axiom,
    ! [A: $tType,Xsa: list(A)] :
      ( ( Xsa != nil(A) )
     => ( aa(list(A),A,hd(A),Xsa) = aa(nat,A,nth(A,Xsa),zero_zero(nat)) ) ) ).

% hd_conv_nth
tff(fact_1906_eucl__rel__int__dividesI,axiom,
    ! [L: int,K3: int,Q2: int] :
      ( ( L != zero_zero(int) )
     => ( ( K3 = aa(int,int,aa(int,fun(int,int),times_times(int),Q2),L) )
       => eucl_rel_int(K3,L,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),zero_zero(int))) ) ) ).

% eucl_rel_int_dividesI
tff(fact_1907_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_1908_merge__list_Osimps_I5_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Acc22: list(list(A)),L12: list(A),L23: list(A),Ls3: list(list(A))] : ( merge_list(A,Acc22,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),Ls3))) = 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)),Acc22),Ls3) ) ) ).

% merge_list.simps(5)
tff(fact_1909_abs__add__one__gt__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A] : aa(A,$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)),abs_abs(A,Xa))) ) ).

% abs_add_one_gt_zero
tff(fact_1910_sorted__hd__min,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xsa: list(A)] :
          ( ( Xsa != nil(A) )
         => ( sorted_wrt(A,ord_less_eq(A),Xsa)
           => ! [X4: A] :
                ( member(A,X4,aa(list(A),set(A),set2(A),Xsa))
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(list(A),A,hd(A),Xsa)),X4) ) ) ) ) ).

% sorted_hd_min
tff(fact_1911_hd__drop__conv__nth,axiom,
    ! [A: $tType,N: nat,Xsa: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xsa))
     => ( aa(list(A),A,hd(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),Xsa)) = aa(nat,A,nth(A,Xsa),N) ) ) ).

% hd_drop_conv_nth
tff(fact_1912_sorted__hd__last,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),L)
         => ( ( L != nil(A) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(list(A),A,hd(A),L)),last(A,L)) ) ) ) ).

% sorted_hd_last
tff(fact_1913_hd__last__singletonI,axiom,
    ! [A: $tType,Xsa: list(A)] :
      ( ( Xsa != nil(A) )
     => ( ( aa(list(A),A,hd(A),Xsa) = last(A,Xsa) )
       => ( aa(list(A),$o,distinct(A),Xsa)
         => ( Xsa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(list(A),A,hd(A),Xsa)),nil(A)) ) ) ) ) ).

% hd_last_singletonI
tff(fact_1914_hd__butlast,axiom,
    ! [A: $tType,Xsa: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),one_one(nat)),aa(list(A),nat,size_size(list(A)),Xsa))
     => ( aa(list(A),A,hd(A),aa(list(A),list(A),butlast(A),Xsa)) = aa(list(A),A,hd(A),Xsa) ) ) ).

% hd_butlast
tff(fact_1915_revg_Oelims,axiom,
    ! [A: $tType,Xa: list(A),Xaa: list(A),Y: list(A)] :
      ( ( revg(A,Xa,Xaa) = Y )
     => ( ( ( Xa = nil(A) )
         => ( Y != Xaa ) )
       => ~ ! [A4: A,As: list(A)] :
              ( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A4),As) )
             => ( Y != revg(A,As,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A4),Xaa)) ) ) ) ) ).

% revg.elims
tff(fact_1916_nat__intermed__int__val,axiom,
    ! [M2: nat,N: nat,F2: fun(nat,int),K3: int] :
      ( ! [I3: nat] :
          ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),I3)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),N) )
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),abs_abs(int,aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,F2,aa(nat,nat,suc,I3))),aa(nat,int,F2,I3)))),one_one(int)) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),N)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,F2,M2)),K3)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K3),aa(nat,int,F2,N))
           => ? [I3: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),I3)
                & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I3),N)
                & ( aa(nat,int,F2,I3) = K3 ) ) ) ) ) ) ).

% nat_intermed_int_val
tff(fact_1917_incr__lemma,axiom,
    ! [D3: int,Z2: int,Xa: 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),Xa),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),abs_abs(int,aa(int,int,aa(int,fun(int,int),minus_minus(int),Xa),Z2))),one_one(int))),D3))) ) ).

% incr_lemma
tff(fact_1918_decr__lemma,axiom,
    ! [D3: int,Xa: 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),Xa),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),abs_abs(int,aa(int,int,aa(int,fun(int,int),minus_minus(int),Xa),Z2))),one_one(int))),D3))),Z2) ) ).

% decr_lemma
tff(fact_1919_slice__head,axiom,
    ! [A: $tType,From: nat,To: nat,Xsa: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),From),To)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),To),aa(list(A),nat,size_size(list(A)),Xsa))
       => ( aa(list(A),A,hd(A),slice(A,From,To,Xsa)) = aa(nat,A,nth(A,Xsa),From) ) ) ) ).

% slice_head
tff(fact_1920_merge__list_Oelims,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: list(list(A)),Xaa: list(list(A)),Y: list(A)] :
          ( ( merge_list(A,Xa,Xaa) = Y )
         => ( ( ( Xa = nil(list(A)) )
             => ( ( Xaa = nil(list(A)) )
               => ( Y != nil(A) ) ) )
           => ( ( ( Xa = nil(list(A)) )
               => ! [L3: list(A)] :
                    ( ( Xaa = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L3),nil(list(A))) )
                   => ( Y != L3 ) ) )
             => ( ! [La: list(A),Acc2: list(list(A))] :
                    ( ( Xa = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc2) )
                   => ( ( Xaa = 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),Acc2)) ) ) )
               => ( ! [La: list(A),Acc2: list(list(A))] :
                      ( ( Xa = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc2) )
                     => ! [L3: list(A)] :
                          ( ( Xaa = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L3),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)),L3),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc2))) ) ) )
                 => ~ ! [L1: list(A),L22: list(A),Ls: list(list(A))] :
                        ( ( Xaa = 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)),Xa),Ls) ) ) ) ) ) ) ) ) ).

% merge_list.elims
tff(fact_1921_nat__ivt__aux,axiom,
    ! [N: nat,F2: fun(nat,int),K3: int] :
      ( ! [I3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),N)
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),abs_abs(int,aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,F2,aa(nat,nat,suc,I3))),aa(nat,int,F2,I3)))),one_one(int)) )
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,F2,zero_zero(nat))),K3)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K3),aa(nat,int,F2,N))
         => ? [I3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I3),N)
              & ( aa(nat,int,F2,I3) = K3 ) ) ) ) ) ).

% nat_ivt_aux
tff(fact_1922_upto__aux__rec,axiom,
    ! [Ia: int,J: int,Js: list(int)] :
      ( upto_aux(Ia,J,Js) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less(int),J),Ia),Js,upto_aux(Ia,aa(int,int,aa(int,fun(int,int),minus_minus(int),J),one_one(int)),aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),J),Js))) ) ).

% upto_aux_rec
tff(fact_1923_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) ) )
        | ? [R6: 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),R6) )
            & ( aa(int,int,sgn_sgn(int),R6) = aa(int,int,sgn_sgn(int),L4) )
            & aa(int,$o,aa(int,fun(int,$o),ord_less(int),abs_abs(int,R6)),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)),R6) ) ) ) ) ).

% eucl_rel_int.simps
tff(fact_1924_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) ) )
       => ( ! [Q4: int] :
              ( ( A32 = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q4),zero_zero(int)) )
             => ( ( A22 != zero_zero(int) )
               => ( A1 != aa(int,int,aa(int,fun(int,int),times_times(int),Q4),A22) ) ) )
         => ~ ! [R3: int,Q4: int] :
                ( ( A32 = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q4),R3) )
               => ( ( aa(int,int,sgn_sgn(int),R3) = aa(int,int,sgn_sgn(int),A22) )
                 => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),abs_abs(int,R3)),abs_abs(int,A22))
                   => ( A1 != aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Q4),A22)),R3) ) ) ) ) ) ) ) ).

% eucl_rel_int.cases
tff(fact_1925_listrel1__subset__listrel,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),R5: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R2),R5)
     => ( refl_on(A,top_top(set(A)),R5)
       => aa(set(product_prod(list(A),list(A))),$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,R5)) ) ) ).

% listrel1_subset_listrel
tff(fact_1926_bind__simps_I2_J,axiom,
    ! [A: $tType,B: $tType,Xa: B,Xsa: list(B),F2: fun(B,list(A))] : ( bind(B,A,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Xa),Xsa),F2) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(B,list(A),F2,Xa)),bind(B,A,Xsa,F2)) ) ).

% bind_simps(2)
tff(fact_1927_sum__list__update,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [K3: nat,Xsa: list(A),Xa: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K3),aa(list(A),nat,size_size(list(A)),Xsa))
         => ( groups8242544230860333062m_list(A,aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xsa),K3),Xa)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),groups8242544230860333062m_list(A,Xsa)),Xa)),aa(nat,A,nth(A,Xsa),K3)) ) ) ) ).

% sum_list_update
tff(fact_1928_top__apply,axiom,
    ! [B: $tType,A: $tType] :
      ( top(A)
     => ! [Xa: B] : ( aa(B,A,top_top(fun(B,A)),Xa) = top_top(A) ) ) ).

% top_apply
tff(fact_1929_UNIV__I,axiom,
    ! [A: $tType,Xa: A] : member(A,Xa,top_top(set(A))) ).

% UNIV_I
tff(fact_1930_sgn__sgn,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A3: A] : ( aa(A,A,sgn_sgn(A),aa(A,A,sgn_sgn(A),A3)) = aa(A,A,sgn_sgn(A),A3) ) ) ).

% sgn_sgn
tff(fact_1931_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_1932_inf__top_Oneutr__eq__iff,axiom,
    ! [A: $tType] :
      ( bounde4346867609351753570nf_top(A)
     => ! [A3: A,B3: A] :
          ( ( top_top(A) = aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B3) )
        <=> ( ( A3 = top_top(A) )
            & ( B3 = top_top(A) ) ) ) ) ).

% inf_top.neutr_eq_iff
tff(fact_1933_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_1934_inf__top_Oeq__neutr__iff,axiom,
    ! [A: $tType] :
      ( bounde4346867609351753570nf_top(A)
     => ! [A3: A,B3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B3) = top_top(A) )
        <=> ( ( A3 = top_top(A) )
            & ( B3 = top_top(A) ) ) ) ) ).

% inf_top.eq_neutr_iff
tff(fact_1935_top__eq__inf__iff,axiom,
    ! [A: $tType] :
      ( bounde4346867609351753570nf_top(A)
     => ! [Xa: A,Y: A] :
          ( ( top_top(A) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Y) )
        <=> ( ( Xa = top_top(A) )
            & ( Y = top_top(A) ) ) ) ) ).

% top_eq_inf_iff
tff(fact_1936_inf__eq__top__iff,axiom,
    ! [A: $tType] :
      ( bounde4346867609351753570nf_top(A)
     => ! [Xa: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Y) = top_top(A) )
        <=> ( ( Xa = top_top(A) )
            & ( Y = top_top(A) ) ) ) ) ).

% inf_eq_top_iff
tff(fact_1937_inf__top__right,axiom,
    ! [A: $tType] :
      ( bounde4346867609351753570nf_top(A)
     => ! [Xa: A] : ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),top_top(A)) = Xa ) ) ).

% inf_top_right
tff(fact_1938_inf__top__left,axiom,
    ! [A: $tType] :
      ( bounde4346867609351753570nf_top(A)
     => ! [Xa: A] : ( aa(A,A,aa(A,fun(A,A),inf_inf(A),top_top(A)),Xa) = Xa ) ) ).

% inf_top_left
tff(fact_1939_sup__top__right,axiom,
    ! [A: $tType] :
      ( bounded_lattice_top(A)
     => ! [Xa: A] : ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),top_top(A)) = top_top(A) ) ) ).

% sup_top_right
tff(fact_1940_sup__top__left,axiom,
    ! [A: $tType] :
      ( bounded_lattice_top(A)
     => ! [Xa: A] : ( aa(A,A,aa(A,fun(A,A),sup_sup(A),top_top(A)),Xa) = top_top(A) ) ) ).

% sup_top_left
tff(fact_1941_Int__UNIV,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),B2) = top_top(set(A)) )
    <=> ( ( Aa2 = top_top(set(A)) )
        & ( B2 = top_top(set(A)) ) ) ) ).

% Int_UNIV
tff(fact_1942_sgn__0,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ( aa(A,A,sgn_sgn(A),zero_zero(A)) = zero_zero(A) ) ) ).

% sgn_0
tff(fact_1943_min__top2,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [Xa: A] : ( aa(A,A,aa(A,fun(A,A),ord_min(A),Xa),top_top(A)) = Xa ) ) ).

% min_top2
tff(fact_1944_min__top,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [Xa: A] : ( aa(A,A,aa(A,fun(A,A),ord_min(A),top_top(A)),Xa) = Xa ) ) ).

% min_top
tff(fact_1945_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_1946_bind__simps_I1_J,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,list(A))] : ( bind(B,A,nil(B),F2) = nil(A) ) ).

% bind_simps(1)
tff(fact_1947_sgn__less,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,sgn_sgn(A),A3)),zero_zero(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A)) ) ) ).

% sgn_less
tff(fact_1948_sgn__greater,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,sgn_sgn(A),A3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3) ) ) ).

% sgn_greater
tff(fact_1949_Diff__UNIV,axiom,
    ! [A: $tType,Aa2: set(A)] : ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),top_top(set(A))) = bot_bot(set(A)) ) ).

% Diff_UNIV
tff(fact_1950_sum__list__eq__0__iff,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Ns: list(A)] :
          ( ( groups8242544230860333062m_list(A,Ns) = zero_zero(A) )
        <=> ! [X6: A] :
              ( member(A,X6,aa(list(A),set(A),set2(A),Ns))
             => ( X6 = zero_zero(A) ) ) ) ) ).

% sum_list_eq_0_iff
tff(fact_1951_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_1952_abs__sgn__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( ( A3 != zero_zero(A) )
         => ( abs_abs(A,aa(A,A,sgn_sgn(A),A3)) = one_one(A) ) ) ) ).

% abs_sgn_eq_1
tff(fact_1953_UNIV__witness,axiom,
    ! [A: $tType] :
    ? [X: A] : member(A,X,top_top(set(A))) ).

% UNIV_witness
tff(fact_1954_UNIV__eq__I,axiom,
    ! [A: $tType,Aa2: set(A)] :
      ( ! [X: A] : member(A,X,Aa2)
     => ( top_top(set(A)) = Aa2 ) ) ).

% UNIV_eq_I
tff(fact_1955_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_1956_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_1957_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_1958_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_1959_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_1960_sgn__eq__0__iff,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A3: A] :
          ( ( aa(A,A,sgn_sgn(A),A3) = zero_zero(A) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% sgn_eq_0_iff
tff(fact_1961_sgn__0__0,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( ( aa(A,A,sgn_sgn(A),A3) = zero_zero(A) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% sgn_0_0
tff(fact_1962_same__sgn__sgn__add,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [B3: A,A3: A] :
          ( ( aa(A,A,sgn_sgn(A),B3) = aa(A,A,sgn_sgn(A),A3) )
         => ( aa(A,A,sgn_sgn(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3)) = aa(A,A,sgn_sgn(A),A3) ) ) ) ).

% same_sgn_sgn_add
tff(fact_1963_sgn__mult,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A3: A,B3: A] : ( aa(A,A,sgn_sgn(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,sgn_sgn(A),A3)),aa(A,A,sgn_sgn(A),B3)) ) ) ).

% sgn_mult
tff(fact_1964_empty__not__UNIV,axiom,
    ! [A: $tType] : ( bot_bot(set(A)) != top_top(set(A)) ) ).

% empty_not_UNIV
tff(fact_1965_subset__UNIV,axiom,
    ! [A: $tType,Aa2: set(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),top_top(set(A))) ).

% subset_UNIV
tff(fact_1966_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_1967_insert__UNIV,axiom,
    ! [A: $tType,Xa: A] : ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),top_top(set(A))) = top_top(set(A)) ) ).

% insert_UNIV
tff(fact_1968_Int__UNIV__right,axiom,
    ! [A: $tType,Aa2: set(A)] : ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),top_top(set(A))) = Aa2 ) ).

% Int_UNIV_right
tff(fact_1969_Int__UNIV__left,axiom,
    ! [A: $tType,B2: set(A)] : ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),top_top(set(A))),B2) = B2 ) ).

% Int_UNIV_left
tff(fact_1970_Un__UNIV__right,axiom,
    ! [A: $tType,Aa2: set(A)] : ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),top_top(set(A))) = top_top(set(A)) ) ).

% Un_UNIV_right
tff(fact_1971_Un__UNIV__left,axiom,
    ! [A: $tType,B2: set(A)] : ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),top_top(set(A))),B2) = top_top(set(A)) ) ).

% Un_UNIV_left
tff(fact_1972_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_1973_member__le__sum__list,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Xa: A,Xsa: list(A)] :
          ( member(A,Xa,aa(list(A),set(A),set2(A),Xsa))
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),groups8242544230860333062m_list(A,Xsa)) ) ) ).

% member_le_sum_list
tff(fact_1974_same__sgn__abs__add,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [B3: A,A3: A] :
          ( ( aa(A,A,sgn_sgn(A),B3) = aa(A,A,sgn_sgn(A),A3) )
         => ( abs_abs(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),abs_abs(A,A3)),abs_abs(A,B3)) ) ) ) ).

% same_sgn_abs_add
tff(fact_1975_mult__sgn__abs,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A] : ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,sgn_sgn(A),Xa)),abs_abs(A,Xa)) = Xa ) ) ).

% mult_sgn_abs
tff(fact_1976_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)),abs_abs(A,A3)) = A3 ) ) ).

% sgn_mult_abs
tff(fact_1977_abs__mult__sgn,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A3: A] : ( aa(A,A,aa(A,fun(A,A),times_times(A),abs_abs(A,A3)),aa(A,A,sgn_sgn(A),A3)) = A3 ) ) ).

% abs_mult_sgn
tff(fact_1978_linordered__idom__class_Oabs__sgn,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [K3: A] : ( abs_abs(A,K3) = aa(A,A,aa(A,fun(A,A),times_times(A),K3),aa(A,A,sgn_sgn(A),K3)) ) ) ).

% linordered_idom_class.abs_sgn
tff(fact_1979_list__bind__cong,axiom,
    ! [B: $tType,A: $tType,Xsa: list(A),Ys: list(A),F2: fun(A,list(B)),G: fun(A,list(B))] :
      ( ( Xsa = Ys )
     => ( ! [X: A] :
            ( member(A,X,aa(list(A),set(A),set2(A),Xsa))
           => ( aa(A,list(B),F2,X) = aa(A,list(B),G,X) ) )
       => ( bind(A,B,Xsa,F2) = bind(A,B,Ys,G) ) ) ) ).

% list_bind_cong
tff(fact_1980_refl__Id,axiom,
    ! [A: $tType] : refl_on(A,top_top(set(A)),id2(A)) ).

% refl_Id
tff(fact_1981_UNIV__coset,axiom,
    ! [A: $tType] : ( top_top(set(A)) = coset(A,nil(A)) ) ).

% UNIV_coset
tff(fact_1982_boolean__algebra_Ocomplement__unique,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [A3: A,Xa: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),Xa) = bot_bot(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),Xa) = 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) )
               => ( Xa = Y ) ) ) ) ) ) ).

% boolean_algebra.complement_unique
tff(fact_1983_Groups__List_Osum__list__nonneg,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [Xsa: list(A)] :
          ( ! [X: A] :
              ( member(A,X,aa(list(A),set(A),set2(A),Xsa))
             => aa(A,$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)),groups8242544230860333062m_list(A,Xsa)) ) ) ).

% Groups_List.sum_list_nonneg
tff(fact_1984_sum__list__nonneg__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [Xsa: list(A)] :
          ( ! [X: A] :
              ( member(A,X,aa(list(A),set(A),set2(A),Xsa))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),X) )
         => ( ( groups8242544230860333062m_list(A,Xsa) = zero_zero(A) )
          <=> ! [X6: A] :
                ( member(A,X6,aa(list(A),set(A),set2(A),Xsa))
               => ( X6 = zero_zero(A) ) ) ) ) ) ).

% sum_list_nonneg_eq_0_iff
tff(fact_1985_sum__list__nonpos,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [Xsa: list(A)] :
          ( ! [X: A] :
              ( member(A,X,aa(list(A),set(A),set2(A),Xsa))
             => aa(A,$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),groups8242544230860333062m_list(A,Xsa)),zero_zero(A)) ) ) ).

% sum_list_nonpos
tff(fact_1986_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_1987_abs__sgn__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: 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_1988_elem__le__sum__list,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [K3: nat,Ns: list(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K3),aa(list(A),nat,size_size(list(A)),Ns))
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,nth(A,Ns),K3)),groups8242544230860333062m_list(A,Ns)) ) ) ).

% elem_le_sum_list
tff(fact_1989_eucl__rel__int__remainderI,axiom,
    ! [R2: int,L: int,K3: int,Q2: 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),abs_abs(int,R2)),abs_abs(int,L))
       => ( ( K3 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Q2),L)),R2) )
         => eucl_rel_int(K3,L,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),R2)) ) ) ) ).

% eucl_rel_int_remainderI
tff(fact_1990_Linear__order__wf__diff__Id,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( order_679001287576687338der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => ( wf(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),minus_minus(set(product_prod(A,A))),R2),id2(A)))
      <=> ! [A11: set(A)] :
            ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A11),aa(set(product_prod(A,A)),set(A),field2(A),R2))
           => ( ( A11 != bot_bot(set(A)) )
             => ? [X6: A] :
                  ( member(A,X6,A11)
                  & ! [Xa2: A] :
                      ( member(A,Xa2,A11)
                     => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X6),Xa2),R2) ) ) ) ) ) ) ).

% Linear_order_wf_diff_Id
tff(fact_1991_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_1992_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_1993_fact__num__eq__if,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [M2: nat] :
          ( semiring_char_0_fact(A,M2) = $ite(M2 = 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),M2)),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),one_one(nat))))) ) ) ).

% fact_num_eq_if
tff(fact_1994_list__assn_Opelims,axiom,
    ! [A: $tType,B: $tType,Xa: fun(A,fun(B,assn)),Xaa: list(A),Xb: list(B),Y: assn] :
      ( ( vEBT_List_list_assn(A,B,Xa,Xaa,Xb) = Y )
     => ( accp(product_prod(fun(A,fun(B,assn)),product_prod(list(A),list(B))),vEBT_L4249061453398456502sn_rel(A,B),aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,assn)),product_prod(list(A),list(B))),aa(fun(A,fun(B,assn)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,assn)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,assn)),product_prod(list(A),list(B))),Xa),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)),Xaa),Xb)))
       => ( ( ( Xaa = nil(A) )
           => ( ( Xb = nil(B) )
             => ( ( Y = one_one(assn) )
               => ~ accp(product_prod(fun(A,fun(B,assn)),product_prod(list(A),list(B))),vEBT_L4249061453398456502sn_rel(A,B),aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,assn)),product_prod(list(A),list(B))),aa(fun(A,fun(B,assn)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,assn)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,assn)),product_prod(list(A),list(B))),Xa),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)))) ) ) )
         => ( ! [A4: A,As: list(A)] :
                ( ( Xaa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A4),As) )
               => ! [C5: B,Cs2: list(B)] :
                    ( ( Xb = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),C5),Cs2) )
                   => ( ( Y = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),Xa,A4),C5)),vEBT_List_list_assn(A,B,Xa,As,Cs2)) )
                     => ~ accp(product_prod(fun(A,fun(B,assn)),product_prod(list(A),list(B))),vEBT_L4249061453398456502sn_rel(A,B),aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,assn)),product_prod(list(A),list(B))),aa(fun(A,fun(B,assn)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,assn)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,assn)),product_prod(list(A),list(B))),Xa),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A4),As)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),C5),Cs2)))) ) ) )
           => ( ! [V3: A,Va: list(A)] :
                  ( ( Xaa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V3),Va) )
                 => ( ( Xb = nil(B) )
                   => ( ( Y = bot_bot(assn) )
                     => ~ accp(product_prod(fun(A,fun(B,assn)),product_prod(list(A),list(B))),vEBT_L4249061453398456502sn_rel(A,B),aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,assn)),product_prod(list(A),list(B))),aa(fun(A,fun(B,assn)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,assn)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,assn)),product_prod(list(A),list(B))),Xa),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),V3),Va)),nil(B)))) ) ) )
             => ~ ( ( Xaa = nil(A) )
                 => ! [V3: B,Va: list(B)] :
                      ( ( Xb = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),V3),Va) )
                     => ( ( Y = bot_bot(assn) )
                       => ~ accp(product_prod(fun(A,fun(B,assn)),product_prod(list(A),list(B))),vEBT_L4249061453398456502sn_rel(A,B),aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,assn)),product_prod(list(A),list(B))),aa(fun(A,fun(B,assn)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,assn)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,assn)),product_prod(list(A),list(B))),Xa),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),V3),Va)))) ) ) ) ) ) ) ) ) ).

% list_assn.pelims
tff(fact_1995_rev__split__conv,axiom,
    ! [A: $tType,L: list(A)] :
      ( ( L != nil(A) )
     => ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),rev(A),aa(list(A),list(A),tl(A),L))),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(list(A),A,hd(A),L)),nil(A))) = aa(list(A),list(A),rev(A),L) ) ) ).

% rev_split_conv
tff(fact_1996_int__power__div__base,axiom,
    ! [M2: nat,K3: int] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M2)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K3)
       => ( divide_divide(int,aa(nat,int,aa(int,fun(nat,int),power_power(int),K3),M2),K3) = aa(nat,int,aa(int,fun(nat,int),power_power(int),K3),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),aa(nat,nat,suc,zero_zero(nat)))) ) ) ) ).

% int_power_div_base
tff(fact_1997_merge__true__star,axiom,
    aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),top_top(assn)),top_top(assn)) = top_top(assn) ).

% merge_true_star
tff(fact_1998_assn__basic__inequalities_I1_J,axiom,
    top_top(assn) != one_one(assn) ).

% assn_basic_inequalities(1)
tff(fact_1999_entt__true,axiom,
    ! [Aa2: assn] : entailst(Aa2,top_top(assn)) ).

% entt_true
tff(fact_2000_assn__basic__inequalities_I5_J,axiom,
    top_top(assn) != bot_bot(assn) ).

% assn_basic_inequalities(5)
tff(fact_2001_div__0,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A3: A] : ( divide_divide(A,zero_zero(A),A3) = zero_zero(A) ) ) ).

% div_0
tff(fact_2002_div__by__0,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A3: A] : ( divide_divide(A,A3,zero_zero(A)) = zero_zero(A) ) ) ).

% div_by_0
tff(fact_2003_times__divide__eq__right,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B3: A,C2: A] : ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),divide_divide(A,B3,C2)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3),C2) ) ) ).

% times_divide_eq_right
tff(fact_2004_divide__divide__eq__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A,B3: A,C2: A] : ( divide_divide(A,A3,divide_divide(A,B3,C2)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2),B3) ) ) ).

% divide_divide_eq_right
tff(fact_2005_divide__divide__eq__left,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A,B3: A,C2: A] : ( divide_divide(A,divide_divide(A,A3,B3),C2) = divide_divide(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2)) ) ) ).

% divide_divide_eq_left
tff(fact_2006_times__divide__eq__left,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [B3: A,C2: A,A3: A] : ( aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,B3,C2)),A3) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),B3),A3),C2) ) ) ).

% times_divide_eq_left
tff(fact_2007_div__by__1,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A3: A] : ( divide_divide(A,A3,one_one(A)) = A3 ) ) ).

% div_by_1
tff(fact_2008_entt__star__true__simp_I1_J,axiom,
    ! [Aa2: assn,B2: assn] :
      ( entailst(Aa2,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),B2),top_top(assn)))
    <=> entailst(Aa2,B2) ) ).

% entt_star_true_simp(1)
tff(fact_2009_entt__star__true__simp_I2_J,axiom,
    ! [Aa2: assn,B2: assn] :
      ( entailst(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Aa2),top_top(assn)),B2)
    <=> entailst(Aa2,B2) ) ).

% entt_star_true_simp(2)
tff(fact_2010_wf__listrel1__iff,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( wf(list(A),listrel1(A,R2))
    <=> wf(A,R2) ) ).

% wf_listrel1_iff
tff(fact_2011_wf__lex,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( wf(A,R2)
     => wf(list(A),lex(A,R2)) ) ).

% wf_lex
tff(fact_2012_wf__lenlex,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( wf(A,R2)
     => wf(list(A),lenlex(A,R2)) ) ).

% wf_lenlex
tff(fact_2013_nonzero__mult__div__cancel__left,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A3: A,B3: A] :
          ( ( A3 != zero_zero(A) )
         => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3),A3) = B3 ) ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_2014_nonzero__mult__div__cancel__right,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [B3: A,A3: A] :
          ( ( B3 != zero_zero(A) )
         => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3),B3) = A3 ) ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_2015_mult__divide__mult__cancel__left__if,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: A,A3: A,B3: 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),B3)) = $ite(C2 = zero_zero(A),zero_zero(A),divide_divide(A,A3,B3)) ) ) ).

% mult_divide_mult_cancel_left_if
tff(fact_2016_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: A,A3: A,B3: A] :
          ( ( C2 != zero_zero(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),B3)) = divide_divide(A,A3,B3) ) ) ) ).

% nonzero_mult_divide_mult_cancel_left
tff(fact_2017_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: A,A3: A,B3: A] :
          ( ( C2 != zero_zero(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),B3),C2)) = divide_divide(A,A3,B3) ) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
tff(fact_2018_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: A,A3: A,B3: A] :
          ( ( C2 != zero_zero(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),B3),C2)) = divide_divide(A,A3,B3) ) ) ) ).

% nonzero_mult_divide_mult_cancel_right
tff(fact_2019_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: A,A3: A,B3: A] :
          ( ( C2 != zero_zero(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),B3)) = divide_divide(A,A3,B3) ) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
tff(fact_2020_div__self,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A3: A] :
          ( ( A3 != zero_zero(A) )
         => ( divide_divide(A,A3,A3) = one_one(A) ) ) ) ).

% div_self
tff(fact_2021_divide__sgn,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] : ( divide_divide(A,A3,aa(A,A,sgn_sgn(A),B3)) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,sgn_sgn(A),B3)) ) ) ).

% divide_sgn
tff(fact_2022_tl__append2,axiom,
    ! [A: $tType,Xsa: list(A),Ys: list(A)] :
      ( ( Xsa != nil(A) )
     => ( aa(list(A),list(A),tl(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),Ys)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),tl(A),Xsa)),Ys) ) ) ).

% tl_append2
tff(fact_2023_butlast__rev,axiom,
    ! [A: $tType,Xsa: list(A)] : ( aa(list(A),list(A),butlast(A),aa(list(A),list(A),rev(A),Xsa)) = aa(list(A),list(A),rev(A),aa(list(A),list(A),tl(A),Xsa)) ) ).

% butlast_rev
tff(fact_2024_divide__less__0__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,one_one(A),A3)),zero_zero(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A)) ) ) ).

% divide_less_0_1_iff
tff(fact_2025_divide__less__eq__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,B3,A3)),one_one(A))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3) ) ) ) ).

% divide_less_eq_1_neg
tff(fact_2026_divide__less__eq__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,B3,A3)),one_one(A))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A3) ) ) ) ).

% divide_less_eq_1_pos
tff(fact_2027_less__divide__eq__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),divide_divide(A,B3,A3))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A3) ) ) ) ).

% less_divide_eq_1_neg
tff(fact_2028_less__divide__eq__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),divide_divide(A,B3,A3))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3) ) ) ) ).

% less_divide_eq_1_pos
tff(fact_2029_zero__less__divide__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),divide_divide(A,one_one(A),A3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3) ) ) ).

% zero_less_divide_1_iff
tff(fact_2030_nonzero__divide__mult__cancel__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [B3: A,A3: A] :
          ( ( B3 != zero_zero(A) )
         => ( divide_divide(A,B3,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)) = divide_divide(A,one_one(A),A3) ) ) ) ).

% nonzero_divide_mult_cancel_right
tff(fact_2031_nonzero__divide__mult__cancel__left,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A,B3: A] :
          ( ( A3 != zero_zero(A) )
         => ( divide_divide(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)) = divide_divide(A,one_one(A),B3) ) ) ) ).

% nonzero_divide_mult_cancel_left
tff(fact_2032_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_2033_length__tl,axiom,
    ! [A: $tType,Xsa: list(A)] : ( aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),tl(A),Xsa)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xsa)),one_one(nat)) ) ).

% length_tl
tff(fact_2034_hd__Cons__tl,axiom,
    ! [A: $tType,Xsa: list(A)] :
      ( ( Xsa != nil(A) )
     => ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(list(A),A,hd(A),Xsa)),aa(list(A),list(A),tl(A),Xsa)) = Xsa ) ) ).

% hd_Cons_tl
tff(fact_2035_list_Ocollapse,axiom,
    ! [A: $tType,List: list(A)] :
      ( ( List != nil(A) )
     => ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(list(A),A,hd(A),List)),aa(list(A),list(A),tl(A),List)) = List ) ) ).

% list.collapse
tff(fact_2036_in__hd__or__tl__conv,axiom,
    ! [A: $tType,L: list(A),Xa: A] :
      ( ( L != nil(A) )
     => ( ( ( Xa = aa(list(A),A,hd(A),L) )
          | member(A,Xa,aa(list(A),set(A),set2(A),aa(list(A),list(A),tl(A),L))) )
      <=> member(A,Xa,aa(list(A),set(A),set2(A),L)) ) ) ).

% in_hd_or_tl_conv
tff(fact_2037_divide__le__eq__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,B3,A3)),one_one(A))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3) ) ) ) ).

% divide_le_eq_1_neg
tff(fact_2038_divide__le__eq__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,B3,A3)),one_one(A))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3) ) ) ) ).

% divide_le_eq_1_pos
tff(fact_2039_le__divide__eq__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),divide_divide(A,B3,A3))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3) ) ) ) ).

% le_divide_eq_1_neg
tff(fact_2040_le__divide__eq__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),divide_divide(A,B3,A3))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3) ) ) ) ).

% le_divide_eq_1_pos
tff(fact_2041_wf__lexn,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),N: nat] :
      ( wf(A,R2)
     => wf(list(A),aa(nat,set(product_prod(list(A),list(A))),lexn(A,R2),N)) ) ).

% wf_lexn
tff(fact_2042_semilattice__neutr__order_Oeq__neutr__iff,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z2: A,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A3: A,B3: A] :
      ( semila1105856199041335345_order(A,F2,Z2,Less_eq,Less)
     => ( ( aa(A,A,aa(A,fun(A,A),F2,A3),B3) = Z2 )
      <=> ( ( A3 = Z2 )
          & ( B3 = Z2 ) ) ) ) ).

% semilattice_neutr_order.eq_neutr_iff
tff(fact_2043_semilattice__neutr__order_Oneutr__eq__iff,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z2: A,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A3: A,B3: A] :
      ( semila1105856199041335345_order(A,F2,Z2,Less_eq,Less)
     => ( ( Z2 = aa(A,A,aa(A,fun(A,A),F2,A3),B3) )
      <=> ( ( A3 = Z2 )
          & ( B3 = Z2 ) ) ) ) ).

% semilattice_neutr_order.neutr_eq_iff
tff(fact_2044_top__empty__eq,axiom,
    ! [A: $tType,X4: A] :
      ( aa(A,$o,top_top(fun(A,$o)),X4)
    <=> member(A,X4,top_top(set(A))) ) ).

% top_empty_eq
tff(fact_2045_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_2046_list_Osel_I3_J,axiom,
    ! [A: $tType,X21: A,X222: list(A)] : ( aa(list(A),list(A),tl(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X21),X222)) = X222 ) ).

% list.sel(3)
tff(fact_2047_list_Osel_I2_J,axiom,
    ! [A: $tType] : ( aa(list(A),list(A),tl(A),nil(A)) = nil(A) ) ).

% list.sel(2)
tff(fact_2048_in__set__tlD,axiom,
    ! [A: $tType,Xa: A,Xsa: list(A)] :
      ( member(A,Xa,aa(list(A),set(A),set2(A),aa(list(A),list(A),tl(A),Xsa)))
     => member(A,Xa,aa(list(A),set(A),set2(A),Xsa)) ) ).

% in_set_tlD
tff(fact_2049_times__divide__times__eq,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Xa: A,Y: A,Z2: A,W2: A] : ( aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,Xa,Y)),divide_divide(A,Z2,W2)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),Xa),Z2),aa(A,A,aa(A,fun(A,A),times_times(A),Y),W2)) ) ) ).

% times_divide_times_eq
tff(fact_2050_divide__divide__times__eq,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Xa: A,Y: A,Z2: A,W2: A] : ( divide_divide(A,divide_divide(A,Xa,Y),divide_divide(A,Z2,W2)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),Xa),W2),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z2)) ) ) ).

% divide_divide_times_eq
tff(fact_2051_divide__divide__eq__left_H,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B3: A,C2: A] : ( divide_divide(A,divide_divide(A,A3,B3),C2) = divide_divide(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3)) ) ) ).

% divide_divide_eq_left'
tff(fact_2052_diff__divide__distrib,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B3: A,C2: A] : ( divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B3),C2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),divide_divide(A,A3,C2)),divide_divide(A,B3,C2)) ) ) ).

% diff_divide_distrib
tff(fact_2053_distinct__tl,axiom,
    ! [A: $tType,Xsa: list(A)] :
      ( aa(list(A),$o,distinct(A),Xsa)
     => aa(list(A),$o,distinct(A),aa(list(A),list(A),tl(A),Xsa)) ) ).

% distinct_tl
tff(fact_2054_merge__true__star__ctx,axiom,
    ! [Pa: assn] : ( aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),top_top(assn)),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),top_top(assn)),Pa)) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),top_top(assn)),Pa) ) ).

% merge_true_star_ctx
tff(fact_2055_ent__true,axiom,
    ! [Pa: assn] : aa(assn,$o,aa(assn,fun(assn,$o),entails,Pa),top_top(assn)) ).

% ent_true
tff(fact_2056_tl__drop,axiom,
    ! [A: $tType,N: nat,Xsa: list(A)] : ( aa(list(A),list(A),tl(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),Xsa)) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),aa(list(A),list(A),tl(A),Xsa)) ) ).

% tl_drop
tff(fact_2057_butlast__tl,axiom,
    ! [A: $tType,Xsa: list(A)] : ( aa(list(A),list(A),butlast(A),aa(list(A),list(A),tl(A),Xsa)) = aa(list(A),list(A),tl(A),aa(list(A),list(A),butlast(A),Xsa)) ) ).

% butlast_tl
tff(fact_2058_norm__assertion__simps_I12_J,axiom,
    ! [Xa: assn] : ( aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),Xa),top_top(assn)) = top_top(assn) ) ).

% norm_assertion_simps(12)
tff(fact_2059_norm__assertion__simps_I11_J,axiom,
    ! [Xa: assn] : ( aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),top_top(assn)),Xa) = top_top(assn) ) ).

% norm_assertion_simps(11)
tff(fact_2060_norm__assertion__simps_I4_J,axiom,
    ! [Xa: assn] : ( aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),Xa),top_top(assn)) = Xa ) ).

% norm_assertion_simps(4)
tff(fact_2061_norm__assertion__simps_I3_J,axiom,
    ! [Xa: assn] : ( aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),top_top(assn)),Xa) = Xa ) ).

% norm_assertion_simps(3)
tff(fact_2062_wf__measures,axiom,
    ! [A: $tType,Fs: list(fun(A,nat))] : wf(A,measures(A,Fs)) ).

% wf_measures
tff(fact_2063_fact__less__mono__nat,axiom,
    ! [M2: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),N)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),semiring_char_0_fact(nat,M2)),semiring_char_0_fact(nat,N)) ) ) ).

% fact_less_mono_nat
tff(fact_2064_tl__Nil,axiom,
    ! [A: $tType,Xsa: list(A)] :
      ( ( aa(list(A),list(A),tl(A),Xsa) = nil(A) )
    <=> ( ( Xsa = nil(A) )
        | ? [X6: A] : ( Xsa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X6),nil(A)) ) ) ) ).

% tl_Nil
tff(fact_2065_Nil__tl,axiom,
    ! [A: $tType,Xsa: list(A)] :
      ( ( nil(A) = aa(list(A),list(A),tl(A),Xsa) )
    <=> ( ( Xsa = nil(A) )
        | ? [X6: A] : ( Xsa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X6),nil(A)) ) ) ) ).

% Nil_tl
tff(fact_2066_tl__obtain__elem,axiom,
    ! [A: $tType,Xsa: list(A)] :
      ( ( Xsa != nil(A) )
     => ( ( aa(list(A),list(A),tl(A),Xsa) = nil(A) )
       => ~ ! [E2: A] : ( Xsa != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E2),nil(A)) ) ) ) ).

% tl_obtain_elem
tff(fact_2067_divide__neg__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),divide_divide(A,Xa,Y)) ) ) ) ).

% divide_neg_neg
tff(fact_2068_divide__neg__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Y)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,Xa,Y)),zero_zero(A)) ) ) ) ).

% divide_neg_pos
tff(fact_2069_divide__pos__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xa)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,Xa,Y)),zero_zero(A)) ) ) ) ).

% divide_pos_neg
tff(fact_2070_divide__pos__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xa)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Y)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),divide_divide(A,Xa,Y)) ) ) ) ).

% divide_pos_pos
tff(fact_2071_divide__less__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,A3,B3)),zero_zero(A))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),zero_zero(A)) )
            | ( aa(A,$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)),B3) ) ) ) ) ).

% divide_less_0_iff
tff(fact_2072_divide__less__cancel,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,C2: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,A3,C2)),divide_divide(A,B3,C2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A3) )
            & ( C2 != zero_zero(A) ) ) ) ) ).

% divide_less_cancel
tff(fact_2073_zero__less__divide__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),divide_divide(A,A3,B3))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B3) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),zero_zero(A)) ) ) ) ) ).

% zero_less_divide_iff
tff(fact_2074_divide__strict__right__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,A3,C2)),divide_divide(A,B3,C2)) ) ) ) ).

% divide_strict_right_mono
tff(fact_2075_divide__strict__right__mono__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B3: A,A3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,A3,C2)),divide_divide(A,B3,C2)) ) ) ) ).

% divide_strict_right_mono_neg
tff(fact_2076_frac__eq__eq,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y: A,Z2: A,Xa: A,W2: A] :
          ( ( Y != zero_zero(A) )
         => ( ( Z2 != zero_zero(A) )
           => ( ( divide_divide(A,Xa,Y) = divide_divide(A,W2,Z2) )
            <=> ( aa(A,A,aa(A,fun(A,A),times_times(A),Xa),Z2) = aa(A,A,aa(A,fun(A,A),times_times(A),W2),Y) ) ) ) ) ) ).

% frac_eq_eq
tff(fact_2077_divide__eq__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B3: A,C2: A,A3: A] :
          ( ( divide_divide(A,B3,C2) = A3 )
        <=> $ite(C2 != zero_zero(A),B3 = aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2),A3 = zero_zero(A)) ) ) ).

% divide_eq_eq
tff(fact_2078_eq__divide__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B3: A,C2: A] :
          ( ( A3 = divide_divide(A,B3,C2) )
        <=> $ite(C2 != zero_zero(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2) = B3,A3 = zero_zero(A)) ) ) ).

% eq_divide_eq
tff(fact_2079_divide__eq__imp,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [C2: A,B3: A,A3: A] :
          ( ( C2 != zero_zero(A) )
         => ( ( B3 = aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2) )
           => ( divide_divide(A,B3,C2) = A3 ) ) ) ) ).

% divide_eq_imp
tff(fact_2080_eq__divide__imp,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [C2: A,A3: A,B3: A] :
          ( ( C2 != zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2) = B3 )
           => ( A3 = divide_divide(A,B3,C2) ) ) ) ) ).

% eq_divide_imp
tff(fact_2081_nonzero__divide__eq__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [C2: A,B3: A,A3: A] :
          ( ( C2 != zero_zero(A) )
         => ( ( divide_divide(A,B3,C2) = A3 )
          <=> ( B3 = aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2) ) ) ) ) ).

% nonzero_divide_eq_eq
tff(fact_2082_nonzero__eq__divide__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [C2: A,A3: A,B3: A] :
          ( ( C2 != zero_zero(A) )
         => ( ( A3 = divide_divide(A,B3,C2) )
          <=> ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2) = B3 ) ) ) ) ).

% nonzero_eq_divide_eq
tff(fact_2083_list_Oset__sel_I2_J,axiom,
    ! [A: $tType,A3: list(A),Xa: A] :
      ( ( A3 != nil(A) )
     => ( member(A,Xa,aa(list(A),set(A),set2(A),aa(list(A),list(A),tl(A),A3)))
       => member(A,Xa,aa(list(A),set(A),set2(A),A3)) ) ) ).

% list.set_sel(2)
tff(fact_2084_sorted__tl,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xsa: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),Xsa)
         => sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),tl(A),Xsa)) ) ) ).

% sorted_tl
tff(fact_2085_fact__gt__zero,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),semiring_char_0_fact(A,N)) ) ).

% fact_gt_zero
tff(fact_2086_fact__not__neg,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),semiring_char_0_fact(A,N)),zero_zero(A)) ) ).

% fact_not_neg
tff(fact_2087_take__tl,axiom,
    ! [A: $tType,N: nat,Xsa: list(A)] : ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),aa(list(A),list(A),tl(A),Xsa)) = aa(list(A),list(A),tl(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),aa(nat,nat,suc,N)),Xsa)) ) ).

% take_tl
tff(fact_2088_drop__Suc,axiom,
    ! [A: $tType,N: nat,Xsa: list(A)] : ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),aa(nat,nat,suc,N)),Xsa) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),aa(list(A),list(A),tl(A),Xsa)) ) ).

% drop_Suc
tff(fact_2089_list_Oexpand,axiom,
    ! [A: $tType,List: list(A),List2: list(A)] :
      ( ( ( List = nil(A) )
      <=> ( List2 = nil(A) ) )
     => ( ( ( List != nil(A) )
         => ( ( List2 != nil(A) )
           => ( ( aa(list(A),A,hd(A),List) = aa(list(A),A,hd(A),List2) )
              & ( aa(list(A),list(A),tl(A),List) = aa(list(A),list(A),tl(A),List2) ) ) ) )
       => ( List = List2 ) ) ) ).

% list.expand
tff(fact_2090_not__hd__in__tl,axiom,
    ! [A: $tType,Xa: A,Xsa: list(A)] :
      ( ( Xa != aa(list(A),A,hd(A),Xsa) )
     => ( member(A,Xa,aa(list(A),set(A),set2(A),Xsa))
       => member(A,Xa,aa(list(A),set(A),set2(A),aa(list(A),list(A),tl(A),Xsa))) ) ) ).

% not_hd_in_tl
tff(fact_2091_tl__last,axiom,
    ! [A: $tType,Xsa: list(A)] :
      ( ( aa(list(A),list(A),tl(A),Xsa) != nil(A) )
     => ( last(A,Xsa) = last(A,aa(list(A),list(A),tl(A),Xsa)) ) ) ).

% tl_last
tff(fact_2092_last__tl,axiom,
    ! [A: $tType,Xsa: list(A)] :
      ( ( ( Xsa = nil(A) )
        | ( aa(list(A),list(A),tl(A),Xsa) != nil(A) ) )
     => ( last(A,aa(list(A),list(A),tl(A),Xsa)) = last(A,Xsa) ) ) ).

% last_tl
tff(fact_2093_mod__star__trueI,axiom,
    ! [Pa: 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,Pa),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,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Pa),top_top(assn))),H2) ) ).

% mod_star_trueI
tff(fact_2094_mod__star__trueE,axiom,
    ! [Pa: 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),Pa),top_top(assn))),H2)
     => ~ ! [H6: 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,Pa),H6) ) ).

% mod_star_trueE
tff(fact_2095_ent__star__mono__true,axiom,
    ! [Aa2: assn,A5: assn,B2: assn,B10: assn] :
      ( aa(assn,$o,aa(assn,fun(assn,$o),entails,Aa2),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),A5),top_top(assn)))
     => ( aa(assn,$o,aa(assn,fun(assn,$o),entails,B2),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),B10),top_top(assn)))
       => aa(assn,$o,aa(assn,fun(assn,$o),entails,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Aa2),B2)),top_top(assn))),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),A5),B10)),top_top(assn))) ) ) ).

% ent_star_mono_true
tff(fact_2096_ent__refl__true,axiom,
    ! [Aa2: assn] : aa(assn,$o,aa(assn,fun(assn,$o),entails,Aa2),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Aa2),top_top(assn))) ).

% ent_refl_true
tff(fact_2097_ent__true__drop_I1_J,axiom,
    ! [Pa: assn,Qa: assn,R: assn] :
      ( aa(assn,$o,aa(assn,fun(assn,$o),entails,Pa),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Qa),top_top(assn)))
     => aa(assn,$o,aa(assn,fun(assn,$o),entails,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Pa),R)),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Qa),top_top(assn))) ) ).

% ent_true_drop(1)
tff(fact_2098_ent__true__drop_I2_J,axiom,
    ! [Pa: assn,Qa: assn] :
      ( aa(assn,$o,aa(assn,fun(assn,$o),entails,Pa),Qa)
     => aa(assn,$o,aa(assn,fun(assn,$o),entails,Pa),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Qa),top_top(assn))) ) ).

% ent_true_drop(2)
tff(fact_2099_rev__butlast__is__tl__rev,axiom,
    ! [A: $tType,L: list(A)] : ( aa(list(A),list(A),rev(A),aa(list(A),list(A),butlast(A),L)) = aa(list(A),list(A),tl(A),aa(list(A),list(A),rev(A),L)) ) ).

% rev_butlast_is_tl_rev
tff(fact_2100_div__int__unique,axiom,
    ! [K3: int,L: int,Q2: int,R2: int] :
      ( eucl_rel_int(K3,L,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),R2))
     => ( divide_divide(int,K3,L) = Q2 ) ) ).

% div_int_unique
tff(fact_2101_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$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),B3)
           => ( divide_divide(A,A3,B3) = zero_zero(A) ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
tff(fact_2102_div__positive,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),divide_divide(A,A3,B3)) ) ) ) ).

% div_positive
tff(fact_2103_frac__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Xa: A,W2: A,Z2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),W2)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),W2),Z2)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,Xa,Z2)),divide_divide(A,Y,W2)) ) ) ) ) ) ).

% frac_le
tff(fact_2104_frac__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A,Y: A,W2: A,Z2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xa)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),W2)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),W2),Z2)
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,Xa,Z2)),divide_divide(A,Y,W2)) ) ) ) ) ) ).

% frac_less
tff(fact_2105_frac__less2,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A,Y: A,W2: A,Z2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xa)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),W2)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),W2),Z2)
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,Xa,Z2)),divide_divide(A,Y,W2)) ) ) ) ) ) ).

% frac_less2
tff(fact_2106_divide__le__cancel,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,C2: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,A3,C2)),divide_divide(A,B3,C2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3) )
            & ( aa(A,$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),B3),A3) ) ) ) ) ).

% divide_le_cancel
tff(fact_2107_divide__nonneg__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xa)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,Xa,Y)),zero_zero(A)) ) ) ) ).

% divide_nonneg_neg
tff(fact_2108_divide__nonneg__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xa)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Y)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),divide_divide(A,Xa,Y)) ) ) ) ).

% divide_nonneg_pos
tff(fact_2109_divide__nonpos__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),divide_divide(A,Xa,Y)) ) ) ) ).

% divide_nonpos_neg
tff(fact_2110_divide__nonpos__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Y)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,Xa,Y)),zero_zero(A)) ) ) ) ).

% divide_nonpos_pos
tff(fact_2111_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [C2: A,A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
         => ( divide_divide(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2)) = divide_divide(A,divide_divide(A,A3,B3),C2) ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_2112_divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B3: A,C2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,B3,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),B3),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,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),B3),aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)) ) ) ) ).

% divide_less_eq
tff(fact_2113_less__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),divide_divide(A,B3,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)),B3),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A))) ) ) ) ).

% less_divide_eq
tff(fact_2114_neg__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,B3,C2)),A3)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),B3) ) ) ) ).

% neg_divide_less_eq
tff(fact_2115_neg__less__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),divide_divide(A,B3,C2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)) ) ) ) ).

% neg_less_divide_eq
tff(fact_2116_pos__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,B3,C2)),A3)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)) ) ) ) ).

% pos_divide_less_eq
tff(fact_2117_pos__less__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),divide_divide(A,B3,C2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),B3) ) ) ) ).

% pos_less_divide_eq
tff(fact_2118_mult__imp__div__pos__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Xa: A,Z2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),aa(A,A,aa(A,fun(A,A),times_times(A),Z2),Y))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,Xa,Y)),Z2) ) ) ) ).

% mult_imp_div_pos_less
tff(fact_2119_mult__imp__less__div__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Z2: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z2),Y)),Xa)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z2),divide_divide(A,Xa,Y)) ) ) ) ).

% mult_imp_less_div_pos
tff(fact_2120_divide__strict__left__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B3: A,A3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,C2,A3)),divide_divide(A,C2,B3)) ) ) ) ) ).

% divide_strict_left_mono
tff(fact_2121_divide__strict__left__mono__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,C2,A3)),divide_divide(A,C2,B3)) ) ) ) ) ).

% divide_strict_left_mono_neg
tff(fact_2122_divide__less__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,B3,A3)),one_one(A))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A3) )
            | ( aa(A,$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),B3) )
            | ( A3 = zero_zero(A) ) ) ) ) ).

% divide_less_eq_1
tff(fact_2123_less__divide__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),divide_divide(A,B3,A3))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A3) ) ) ) ) ).

% less_divide_eq_1
tff(fact_2124_add__divide__eq__if__simps_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,Z2: A,B3: A] :
          ( aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,A3,Z2)),B3) = $ite(Z2 = zero_zero(A),B3,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),B3),Z2)),Z2)) ) ) ).

% add_divide_eq_if_simps(2)
tff(fact_2125_add__divide__eq__if__simps_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B3: A,Z2: A] :
          ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),divide_divide(A,B3,Z2)) = $ite(Z2 = zero_zero(A),A3,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)),B3),Z2)) ) ) ).

% add_divide_eq_if_simps(1)
tff(fact_2126_add__frac__eq,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y: A,Z2: A,Xa: A,W2: A] :
          ( ( Y != zero_zero(A) )
         => ( ( Z2 != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,Xa,Y)),divide_divide(A,W2,Z2)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xa),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_2127_add__frac__num,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y: A,Xa: A,Z2: A] :
          ( ( Y != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,Xa,Y)),Z2) = divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),aa(A,A,aa(A,fun(A,A),times_times(A),Z2),Y)),Y) ) ) ) ).

% add_frac_num
tff(fact_2128_add__num__frac,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y: A,Z2: A,Xa: A] :
          ( ( Y != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),Z2),divide_divide(A,Xa,Y)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),aa(A,A,aa(A,fun(A,A),times_times(A),Z2),Y)),Y) ) ) ) ).

% add_num_frac
tff(fact_2129_add__divide__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z2: A,Xa: A,Y: A] :
          ( ( Z2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),divide_divide(A,Y,Z2)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xa),Z2)),Y),Z2) ) ) ) ).

% add_divide_eq_iff
tff(fact_2130_divide__add__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z2: A,Xa: A,Y: A] :
          ( ( Z2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,Xa,Z2)),Y) = divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z2)),Z2) ) ) ) ).

% divide_add_eq_iff
tff(fact_2131_tl__subset,axiom,
    ! [A: $tType,Xsa: list(A),Aa2: set(A)] :
      ( ( Xsa != nil(A) )
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xsa)),Aa2)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),aa(list(A),list(A),tl(A),Xsa))),Aa2) ) ) ).

% tl_subset
tff(fact_2132_add__divide__eq__if__simps_I4_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B3: A,Z2: A] :
          ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),divide_divide(A,B3,Z2)) = $ite(Z2 = zero_zero(A),A3,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)),B3),Z2)) ) ) ).

% add_divide_eq_if_simps(4)
tff(fact_2133_diff__frac__eq,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y: A,Z2: A,Xa: A,W2: A] :
          ( ( Y != zero_zero(A) )
         => ( ( Z2 != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),divide_divide(A,Xa,Y)),divide_divide(A,W2,Z2)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xa),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_2134_diff__divide__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z2: A,Xa: A,Y: A] :
          ( ( Z2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),Xa),divide_divide(A,Y,Z2)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xa),Z2)),Y),Z2) ) ) ) ).

% diff_divide_eq_iff
tff(fact_2135_divide__diff__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z2: A,Xa: A,Y: A] :
          ( ( Z2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),divide_divide(A,Xa,Z2)),Y) = divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Xa),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z2)),Z2) ) ) ) ).

% divide_diff_eq_iff
tff(fact_2136_gt__half__sum,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),one_one(A)))),B3) ) ) ).

% gt_half_sum
tff(fact_2137_less__half__sum,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),one_one(A)))) ) ) ).

% less_half_sum
tff(fact_2138_Misc_Onth__tl,axiom,
    ! [A: $tType,Xsa: list(A),N: nat] :
      ( ( Xsa != nil(A) )
     => ( aa(nat,A,nth(A,aa(list(A),list(A),tl(A),Xsa)),N) = aa(nat,A,nth(A,Xsa),aa(nat,nat,suc,N)) ) ) ).

% Misc.nth_tl
tff(fact_2139_abs__div__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Y)
         => ( divide_divide(A,abs_abs(A,Xa),Y) = abs_abs(A,divide_divide(A,Xa,Y)) ) ) ) ).

% abs_div_pos
tff(fact_2140_list__take__induct__tl2,axiom,
    ! [B: $tType,A: $tType,Xsa: list(A),Ys: list(B),Pa: fun(B,fun(A,$o))] :
      ( ( aa(list(A),nat,size_size(list(A)),Xsa) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( ! [N5: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N5),aa(list(A),nat,size_size(list(A)),Xsa))
           => aa(A,$o,aa(B,fun(A,$o),Pa,aa(nat,B,nth(B,Ys),N5)),aa(nat,A,nth(A,Xsa),N5)) )
       => ! [N8: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N8),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),tl(A),Xsa)))
           => aa(A,$o,aa(B,fun(A,$o),Pa,aa(nat,B,nth(B,aa(list(B),list(B),tl(B),Ys)),N8)),aa(nat,A,nth(A,aa(list(A),list(A),tl(A),Xsa)),N8)) ) ) ) ).

% list_take_induct_tl2
tff(fact_2141_list_Oexhaust__sel,axiom,
    ! [A: $tType,List: list(A)] :
      ( ( List != nil(A) )
     => ( List = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(list(A),A,hd(A),List)),aa(list(A),list(A),tl(A),List)) ) ) ).

% list.exhaust_sel
tff(fact_2142_fact__less__mono,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [M2: nat,N: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M2)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),N)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),semiring_char_0_fact(A,M2)),semiring_char_0_fact(A,N)) ) ) ) ).

% fact_less_mono
tff(fact_2143_tl__take,axiom,
    ! [A: $tType,N: nat,Xsa: list(A)] : ( aa(list(A),list(A),tl(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),Xsa)) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))),aa(list(A),list(A),tl(A),Xsa)) ) ).

% tl_take
tff(fact_2144_distinct__hd__tl,axiom,
    ! [A: $tType,Xsa: list(A),Xa: A] :
      ( aa(list(A),$o,distinct(A),Xsa)
     => ( ( Xa = aa(list(A),A,hd(A),Xsa) )
       => ~ member(A,Xa,aa(list(A),set(A),set2(A),aa(list(A),list(A),tl(A),Xsa))) ) ) ).

% distinct_hd_tl
tff(fact_2145_butlast__rev__tl,axiom,
    ! [A: $tType,Xsa: list(A)] :
      ( ( Xsa != nil(A) )
     => ( aa(list(A),list(A),butlast(A),aa(list(A),list(A),rev(A),Xsa)) = aa(list(A),list(A),rev(A),aa(list(A),list(A),tl(A),Xsa)) ) ) ).

% butlast_rev_tl
tff(fact_2146_mod__h__bot__iff_I2_J,axiom,
    ! [H2: 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,top_top(assn)),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)))) ).

% mod_h_bot_iff(2)
tff(fact_2147_remove1__tl,axiom,
    ! [A: $tType,Xsa: list(A)] :
      ( ( Xsa != nil(A) )
     => ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),remove1(A),aa(list(A),A,hd(A),Xsa)),Xsa) = aa(list(A),list(A),tl(A),Xsa) ) ) ).

% remove1_tl
tff(fact_2148_entt__def__true,axiom,
    ! [Pa: assn,Qa: assn] :
      ( entailst(Pa,Qa)
    <=> aa(assn,$o,aa(assn,fun(assn,$o),entails,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Pa),top_top(assn))),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Qa),top_top(assn))) ) ).

% entt_def_true
tff(fact_2149_entailst__def,axiom,
    ! [Aa2: assn,B2: assn] :
      ( entailst(Aa2,B2)
    <=> aa(assn,$o,aa(assn,fun(assn,$o),entails,Aa2),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),B2),top_top(assn))) ) ).

% entailst_def
tff(fact_2150_enttI__true,axiom,
    ! [Pa: assn,Qa: assn] :
      ( aa(assn,$o,aa(assn,fun(assn,$o),entails,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Pa),top_top(assn))),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Qa),top_top(assn)))
     => entailst(Pa,Qa) ) ).

% enttI_true
tff(fact_2151_enttI,axiom,
    ! [Aa2: assn,B2: assn] :
      ( aa(assn,$o,aa(assn,fun(assn,$o),entails,Aa2),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),B2),top_top(assn)))
     => entailst(Aa2,B2) ) ).

% enttI
tff(fact_2152_enttD,axiom,
    ! [Aa2: assn,B2: assn] :
      ( entailst(Aa2,B2)
     => aa(assn,$o,aa(assn,fun(assn,$o),entails,Aa2),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),B2),top_top(assn))) ) ).

% enttD
tff(fact_2153_wf__eq__minimal2,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( wf(A,R2)
    <=> ! [A11: set(A)] :
          ( ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A11),aa(set(product_prod(A,A)),set(A),field2(A),R2))
            & ( A11 != bot_bot(set(A)) ) )
         => ? [X6: A] :
              ( member(A,X6,A11)
              & ! [Xa2: A] :
                  ( member(A,Xa2,A11)
                 => ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa2),X6),R2) ) ) ) ) ).

% wf_eq_minimal2
tff(fact_2154_entails__solve__init_I1_J,axiom,
    ! [Pa: assn,Qa: assn] :
      ( fI_QUERY(Pa,Qa,top_top(assn))
     => aa(assn,$o,aa(assn,fun(assn,$o),entails,Pa),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Qa),top_top(assn))) ) ).

% entails_solve_init(1)
tff(fact_2155_divide__left__mono__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,C2,A3)),divide_divide(A,C2,B3)) ) ) ) ) ).

% divide_left_mono_neg
tff(fact_2156_mult__imp__le__div__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Z2: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z2),Y)),Xa)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z2),divide_divide(A,Xa,Y)) ) ) ) ).

% mult_imp_le_div_pos
tff(fact_2157_mult__imp__div__pos__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Xa: A,Z2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(A,A,aa(A,fun(A,A),times_times(A),Z2),Y))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,Xa,Y)),Z2) ) ) ) ).

% mult_imp_div_pos_le
tff(fact_2158_pos__le__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),divide_divide(A,B3,C2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),B3) ) ) ) ).

% pos_le_divide_eq
tff(fact_2159_pos__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,B3,C2)),A3)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)) ) ) ) ).

% pos_divide_le_eq
tff(fact_2160_neg__le__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),divide_divide(A,B3,C2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)) ) ) ) ).

% neg_le_divide_eq
tff(fact_2161_neg__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,B3,C2)),A3)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),B3) ) ) ) ).

% neg_divide_le_eq
tff(fact_2162_divide__left__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B3: A,A3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,C2,A3)),divide_divide(A,C2,B3)) ) ) ) ) ).

% divide_left_mono
tff(fact_2163_le__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),divide_divide(A,B3,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)),B3),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),zero_zero(A))) ) ) ) ).

% le_divide_eq
tff(fact_2164_divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B3: A,C2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,B3,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),B3),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,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),B3),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)) ) ) ) ).

% divide_le_eq
tff(fact_2165_divide__le__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,B3,A3)),one_one(A))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3) )
            | ( aa(A,$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),B3) )
            | ( A3 = zero_zero(A) ) ) ) ) ).

% divide_le_eq_1
tff(fact_2166_le__divide__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),divide_divide(A,B3,A3))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3) )
            | ( aa(A,$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),B3),A3) ) ) ) ) ).

% le_divide_eq_1
tff(fact_2167_frac__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Z2: A,Xa: A,W2: A] :
          ( ( Y != zero_zero(A) )
         => ( ( Z2 != zero_zero(A) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,Xa,Y)),divide_divide(A,W2,Z2))
            <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xa),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_2168_frac__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Z2: A,Xa: A,W2: A] :
          ( ( Y != zero_zero(A) )
         => ( ( Z2 != zero_zero(A) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,Xa,Y)),divide_divide(A,W2,Z2))
            <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xa),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_2169_List_Onth__tl,axiom,
    ! [A: $tType,N: nat,Xsa: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),tl(A),Xsa)))
     => ( aa(nat,A,nth(A,aa(list(A),list(A),tl(A),Xsa)),N) = aa(nat,A,nth(A,Xsa),aa(nat,nat,suc,N)) ) ) ).

% List.nth_tl
tff(fact_2170_scaling__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [U: A,V2: A,R2: A,S2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),U),V2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),R2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),R2),S2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),U),divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),R2),aa(A,A,aa(A,fun(A,A),minus_minus(A),V2),U)),S2))),V2) ) ) ) ) ).

% scaling_mono
tff(fact_2171_nat__approx__posE,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [E3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),E3)
         => ~ ! [N5: nat] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,one_one(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,N5)))),E3) ) ) ).

% nat_approx_posE
tff(fact_2172_take__Suc,axiom,
    ! [A: $tType,Xsa: list(A),N: nat] :
      ( ( Xsa != nil(A) )
     => ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),aa(nat,nat,suc,N)),Xsa) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(list(A),A,hd(A),Xsa)),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),aa(list(A),list(A),tl(A),Xsa))) ) ) ).

% take_Suc
tff(fact_2173_rotate1__hd__tl,axiom,
    ! [A: $tType,Xsa: list(A)] :
      ( ( Xsa != nil(A) )
     => ( aa(list(A),list(A),rotate1(A),Xsa) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),tl(A),Xsa)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(list(A),A,hd(A),Xsa)),nil(A))) ) ) ).

% rotate1_hd_tl
tff(fact_2174_div__mult__self1,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B3: A,A3: A,C2: A] :
          ( ( B3 != zero_zero(A) )
         => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3)),B3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),divide_divide(A,A3,B3)) ) ) ) ).

% div_mult_self1
tff(fact_2175_div__mult__self2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B3: A,A3: A,C2: A] :
          ( ( B3 != zero_zero(A) )
         => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2)),B3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),divide_divide(A,A3,B3)) ) ) ) ).

% div_mult_self2
tff(fact_2176_div__mult__self3,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B3: A,C2: A,A3: A] :
          ( ( B3 != zero_zero(A) )
         => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3)),A3),B3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),divide_divide(A,A3,B3)) ) ) ) ).

% div_mult_self3
tff(fact_2177_div__mult__self4,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B3: A,C2: A,A3: A] :
          ( ( B3 != zero_zero(A) )
         => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2)),A3),B3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),divide_divide(A,A3,B3)) ) ) ) ).

% div_mult_self4
tff(fact_2178_div__mult__mult1,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,A3: A,B3: A] :
          ( ( C2 != zero_zero(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),B3)) = divide_divide(A,A3,B3) ) ) ) ).

% div_mult_mult1
tff(fact_2179_div__mult__mult2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,A3: A,B3: A] :
          ( ( C2 != zero_zero(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),B3),C2)) = divide_divide(A,A3,B3) ) ) ) ).

% div_mult_mult2
tff(fact_2180_div__mult__mult1__if,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,A3: A,B3: 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),B3)) = $ite(C2 = zero_zero(A),zero_zero(A),divide_divide(A,A3,B3)) ) ) ).

% div_mult_mult1_if
tff(fact_2181_top1I,axiom,
    ! [A: $tType,Xa: A] : aa(A,$o,top_top(fun(A,$o)),Xa) ).

% top1I
tff(fact_2182_div__less,axiom,
    ! [M2: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),N)
     => ( divide_divide(nat,M2,N) = zero_zero(nat) ) ) ).

% div_less
tff(fact_2183_div__mult__self__is__m,axiom,
    ! [N: nat,M2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
     => ( divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M2),N),N) = M2 ) ) ).

% div_mult_self_is_m
tff(fact_2184_div__mult__self1__is__m,axiom,
    ! [N: nat,M2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
     => ( divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),M2),N) = M2 ) ) ).

% div_mult_self1_is_m
tff(fact_2185_Euclidean__Division_Odiv__eq__0__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( divide_divide(nat,M2,N) = zero_zero(nat) )
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),N)
        | ( N = zero_zero(nat) ) ) ) ).

% Euclidean_Division.div_eq_0_iff
tff(fact_2186_less__mult__imp__div__less,axiom,
    ! [M2: nat,Ia: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Ia),N))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),divide_divide(nat,M2,N)),Ia) ) ).

% less_mult_imp_div_less
tff(fact_2187_div__greater__zero__iff,axiom,
    ! [M2: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),divide_divide(nat,M2,N))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),M2)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N) ) ) ).

% div_greater_zero_iff
tff(fact_2188_div__le__mono2,axiom,
    ! [M2: nat,N: nat,K3: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),N)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),divide_divide(nat,K3,N)),divide_divide(nat,K3,M2)) ) ) ).

% div_le_mono2
tff(fact_2189_div__less__iff__less__mult,axiom,
    ! [Q2: nat,M2: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Q2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),divide_divide(nat,M2,Q2)),N)
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),Q2)) ) ) ).

% div_less_iff_less_mult
tff(fact_2190_div__eq__dividend__iff,axiom,
    ! [M2: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M2)
     => ( ( divide_divide(nat,M2,N) = M2 )
      <=> ( N = one_one(nat) ) ) ) ).

% div_eq_dividend_iff
tff(fact_2191_div__less__dividend,axiom,
    ! [N: nat,M2: 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)),M2)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),divide_divide(nat,M2,N)),M2) ) ) ).

% div_less_dividend
tff(fact_2192_div__if,axiom,
    ! [M2: nat,N: nat] :
      ( divide_divide(nat,M2,N) = $ite(
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),N)
          | ( N = zero_zero(nat) ) ),
          zero_zero(nat),
          aa(nat,nat,suc,divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),N),N)) ) ) ).

% div_if
tff(fact_2193_less__eq__div__iff__mult__less__eq,axiom,
    ! [Q2: nat,M2: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Q2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),divide_divide(nat,N,Q2))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M2),Q2)),N) ) ) ).

% less_eq_div_iff_mult_less_eq
tff(fact_2194_div__nat__eqI,axiom,
    ! [N: nat,Q2: nat,M2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),Q2)),M2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,suc,Q2)))
       => ( divide_divide(nat,M2,N) = Q2 ) ) ) ).

% div_nat_eqI
tff(fact_2195_split__div,axiom,
    ! [Pa: fun(nat,$o),M2: nat,N: nat] :
      ( aa(nat,$o,Pa,divide_divide(nat,M2,N))
    <=> ( ( ( N = zero_zero(nat) )
         => aa(nat,$o,Pa,zero_zero(nat)) )
        & ( ( N != zero_zero(nat) )
         => ! [I2: nat,J2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J2),N)
             => ( ( M2 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),I2)),J2) )
               => aa(nat,$o,Pa,I2) ) ) ) ) ) ).

% split_div
tff(fact_2196_dividend__less__div__times,axiom,
    ! [N: nat,M2: 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),M2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),divide_divide(nat,M2,N)),N))) ) ).

% dividend_less_div_times
tff(fact_2197_dividend__less__times__div,axiom,
    ! [N: nat,M2: 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),M2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),divide_divide(nat,M2,N)))) ) ).

% dividend_less_times_div
tff(fact_2198_nat__mult__div__cancel1,axiom,
    ! [K3: nat,M2: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K3)
     => ( divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K3),M2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K3),N)) = divide_divide(nat,M2,N) ) ) ).

% nat_mult_div_cancel1
tff(fact_2199_le__div__geq,axiom,
    ! [N: nat,M2: 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_eq(nat),N),M2)
       => ( divide_divide(nat,M2,N) = aa(nat,nat,suc,divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),N),N)) ) ) ) ).

% le_div_geq
tff(fact_2200_split__div_H,axiom,
    ! [Pa: fun(nat,$o),M2: nat,N: nat] :
      ( aa(nat,$o,Pa,divide_divide(nat,M2,N))
    <=> ( ( ( N = zero_zero(nat) )
          & aa(nat,$o,Pa,zero_zero(nat)) )
        | ? [Q6: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),Q6)),M2)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,suc,Q6)))
            & aa(nat,$o,Pa,Q6) ) ) ) ).

% split_div'
tff(fact_2201_div__geq,axiom,
    ! [N: nat,M2: 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),M2),N)
       => ( divide_divide(nat,M2,N) = aa(nat,nat,suc,divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),N),N)) ) ) ) ).

% div_geq
tff(fact_2202_div__mult2__eq_H,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A3: A,M2: nat,N: nat] : ( divide_divide(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),M2)),aa(nat,A,semiring_1_of_nat(A),N))) = divide_divide(A,divide_divide(A,A3,aa(nat,A,semiring_1_of_nat(A),M2)),aa(nat,A,semiring_1_of_nat(A),N)) ) ) ).

% div_mult2_eq'
tff(fact_2203_Nitpick_Osize__list__simp_I2_J,axiom,
    ! [A: $tType,Xsa: list(A)] :
      ( aa(list(A),nat,size_size(list(A)),Xsa) = $ite(Xsa = nil(A),zero_zero(nat),aa(nat,nat,suc,aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),tl(A),Xsa)))) ) ).

% Nitpick.size_list_simp(2)
tff(fact_2204_brk__rel__wf,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( wf(A,R)
     => wf(product_prod($o,A),brk_rel(A,A,R)) ) ).

% brk_rel_wf
tff(fact_2205_wf__bounded__measure,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),Ub: fun(A,nat),F2: fun(A,nat)] :
      ( ! [A4: A,B4: A] :
          ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B4),A4),R2)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,Ub,B4)),aa(A,nat,Ub,A4))
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,F2,B4)),aa(A,nat,Ub,A4))
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,F2,A4)),aa(A,nat,F2,B4)) ) )
     => wf(A,R2) ) ).

% wf_bounded_measure
tff(fact_2206_wf__iff__no__infinite__down__chain,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( wf(A,R2)
    <=> ~ ? [F9: fun(nat,A)] :
          ! [I2: nat] : member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(nat,A,F9,aa(nat,nat,suc,I2))),aa(nat,A,F9,I2)),R2) ) ).

% wf_iff_no_infinite_down_chain
tff(fact_2207_wf__no__infinite__down__chainE,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),F2: fun(nat,A)] :
      ( wf(A,R2)
     => ~ ! [K: nat] : member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(nat,A,F2,aa(nat,nat,suc,K))),aa(nat,A,F2,K)),R2) ) ).

% wf_no_infinite_down_chainE
tff(fact_2208_wfE__min_H,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),Qa: set(A)] :
      ( wf(A,R)
     => ( ( Qa != bot_bot(set(A)) )
       => ~ ! [Z3: A] :
              ( member(A,Z3,Qa)
             => ~ ! [Y3: A] :
                    ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),Z3),R)
                   => ~ member(A,Y3,Qa) ) ) ) ) ).

% wfE_min'
tff(fact_2209_product__nth,axiom,
    ! [A: $tType,B: $tType,N: nat,Xsa: list(A),Ys: list(B)] :
      ( aa(nat,$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)),Xsa)),aa(list(B),nat,size_size(list(B)),Ys)))
     => ( aa(nat,product_prod(A,B),nth(product_prod(A,B),aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),product(A,B),Xsa),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,Xsa),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_2210_mod__0,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A3: A] : ( modulo_modulo(A,zero_zero(A),A3) = zero_zero(A) ) ) ).

% mod_0
tff(fact_2211_mod__by__0,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A3: A] : ( modulo_modulo(A,A3,zero_zero(A)) = A3 ) ) ).

% mod_by_0
tff(fact_2212_mod__self,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A3: A] : ( modulo_modulo(A,A3,A3) = zero_zero(A) ) ) ).

% mod_self
tff(fact_2213_minus__mod__self2,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A3: A,B3: A] : ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B3),B3) = modulo_modulo(A,A3,B3) ) ) ).

% minus_mod_self2
tff(fact_2214_mod__less,axiom,
    ! [M2: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),N)
     => ( modulo_modulo(nat,M2,N) = M2 ) ) ).

% mod_less
tff(fact_2215_mod__mult__self1__is__0,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B3: A,A3: A] : ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),B3),A3),B3) = zero_zero(A) ) ) ).

% mod_mult_self1_is_0
tff(fact_2216_mod__mult__self2__is__0,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A3: A,B3: A] : ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3),B3) = zero_zero(A) ) ) ).

% mod_mult_self2_is_0
tff(fact_2217_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_2218_mod__mult__self1,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A3: A,C2: A,B3: 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),B3)),B3) = modulo_modulo(A,A3,B3) ) ) ).

% mod_mult_self1
tff(fact_2219_mod__mult__self2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A3: A,B3: 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),B3),C2)),B3) = modulo_modulo(A,A3,B3) ) ) ).

% mod_mult_self2
tff(fact_2220_mod__mult__self3,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,B3: 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),B3)),A3),B3) = modulo_modulo(A,A3,B3) ) ) ).

% mod_mult_self3
tff(fact_2221_mod__mult__self4,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B3: 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),B3),C2)),A3),B3) = modulo_modulo(A,A3,B3) ) ) ).

% mod_mult_self4
tff(fact_2222_mod__mult__right__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A3: A,B3: A,C2: A] : ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),modulo_modulo(A,B3,C2)),C2) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3),C2) ) ) ).

% mod_mult_right_eq
tff(fact_2223_mod__mult__left__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A3: A,C2: A,B3: A] : ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),modulo_modulo(A,A3,C2)),B3),C2) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3),C2) ) ) ).

% mod_mult_left_eq
tff(fact_2224_mult__mod__right,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,A3: A,B3: A] : ( aa(A,A,aa(A,fun(A,A),times_times(A),C2),modulo_modulo(A,A3,B3)) = 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),B3)) ) ) ).

% mult_mod_right
tff(fact_2225_mod__mult__mult2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A3: A,C2: A,B3: 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),B3),C2)) = aa(A,A,aa(A,fun(A,A),times_times(A),modulo_modulo(A,A3,B3)),C2) ) ) ).

% mod_mult_mult2
tff(fact_2226_mod__mult__cong,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A3: A,C2: A,A6: A,B3: A,B6: A] :
          ( ( modulo_modulo(A,A3,C2) = modulo_modulo(A,A6,C2) )
         => ( ( modulo_modulo(A,B3,C2) = modulo_modulo(A,B6,C2) )
           => ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3),C2) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A6),B6),C2) ) ) ) ) ).

% mod_mult_cong
tff(fact_2227_mod__mult__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A3: A,C2: A,B3: A] : ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),modulo_modulo(A,A3,C2)),modulo_modulo(A,B3,C2)),C2) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3),C2) ) ) ).

% mod_mult_eq
tff(fact_2228_mod__diff__right__eq,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A3: A,B3: A,C2: A] : ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),modulo_modulo(A,B3,C2)),C2) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B3),C2) ) ) ).

% mod_diff_right_eq
tff(fact_2229_mod__diff__left__eq,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A3: A,C2: A,B3: A] : ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),modulo_modulo(A,A3,C2)),B3),C2) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B3),C2) ) ) ).

% mod_diff_left_eq
tff(fact_2230_mod__diff__cong,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A3: A,C2: A,A6: A,B3: A,B6: A] :
          ( ( modulo_modulo(A,A3,C2) = modulo_modulo(A,A6,C2) )
         => ( ( modulo_modulo(A,B3,C2) = modulo_modulo(A,B6,C2) )
           => ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B3),C2) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A6),B6),C2) ) ) ) ) ).

% mod_diff_cong
tff(fact_2231_mod__diff__eq,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A3: A,C2: A,B3: A] : ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),modulo_modulo(A,A3,C2)),modulo_modulo(A,B3,C2)),C2) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B3),C2) ) ) ).

% mod_diff_eq
tff(fact_2232_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),modulo_modulo(A,A3,B3)),B3) ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
tff(fact_2233_mod__eqE,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A3: A,C2: A,B3: A] :
          ( ( modulo_modulo(A,A3,C2) = modulo_modulo(A,B3,C2) )
         => ~ ! [D4: A] : ( B3 != aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),C2),D4)) ) ) ) ).

% mod_eqE
tff(fact_2234_mod__less__divisor,axiom,
    ! [N: nat,M2: 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),modulo_modulo(nat,M2,N)),N) ) ).

% mod_less_divisor
tff(fact_2235_div__less__mono,axiom,
    ! [Aa2: nat,B2: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Aa2),B2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
       => ( ( modulo_modulo(nat,Aa2,N) = zero_zero(nat) )
         => ( ( modulo_modulo(nat,B2,N) = zero_zero(nat) )
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),divide_divide(nat,Aa2,N)),divide_divide(nat,B2,N)) ) ) ) ) ).

% div_less_mono
tff(fact_2236_mod__induct,axiom,
    ! [Pa: fun(nat,$o),N: nat,P2: nat,M2: nat] :
      ( aa(nat,$o,Pa,N)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),P2)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),P2)
         => ( ! [N5: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N5),P2)
               => ( aa(nat,$o,Pa,N5)
                 => aa(nat,$o,Pa,modulo_modulo(nat,aa(nat,nat,suc,N5),P2)) ) )
           => aa(nat,$o,Pa,M2) ) ) ) ) ).

% mod_induct
tff(fact_2237_mod__geq,axiom,
    ! [M2: nat,N: nat] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),N)
     => ( modulo_modulo(nat,M2,N) = modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),N),N) ) ) ).

% mod_geq
tff(fact_2238_mod__if,axiom,
    ! [M2: nat,N: nat] :
      ( modulo_modulo(nat,M2,N) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),N),M2,modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),N),N)) ) ).

% mod_if
tff(fact_2239_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$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),B3)
           => ( modulo_modulo(A,A3,B3) = A3 ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
tff(fact_2240_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),modulo_modulo(A,A3,B3)) ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
tff(fact_2241_cancel__div__mod__rules_I2_J,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [B3: 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),B3),divide_divide(A,A3,B3))),modulo_modulo(A,A3,B3))),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2) ) ) ).

% cancel_div_mod_rules(2)
tff(fact_2242_cancel__div__mod__rules_I1_J,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A3: A,B3: A,C2: A] : ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,A3,B3)),B3)),modulo_modulo(A,A3,B3))),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2) ) ) ).

% cancel_div_mod_rules(1)
tff(fact_2243_mod__div__decomp,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A3: A,B3: A] : ( A3 = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,A3,B3)),B3)),modulo_modulo(A,A3,B3)) ) ) ).

% mod_div_decomp
tff(fact_2244_div__mult__mod__eq,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A3: A,B3: A] : ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,A3,B3)),B3)),modulo_modulo(A,A3,B3)) = A3 ) ) ).

% div_mult_mod_eq
tff(fact_2245_mod__div__mult__eq,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A3: A,B3: A] : ( aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A3,B3)),aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,A3,B3)),B3)) = A3 ) ) ).

% mod_div_mult_eq
tff(fact_2246_mod__mult__div__eq,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A3: A,B3: A] : ( aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A3,B3)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),divide_divide(A,A3,B3))) = A3 ) ) ).

% mod_mult_div_eq
tff(fact_2247_mult__div__mod__eq,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [B3: 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),B3),divide_divide(A,A3,B3))),modulo_modulo(A,A3,B3)) = A3 ) ) ).

% mult_div_mod_eq
tff(fact_2248_div__mult1__eq,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [A3: A,B3: A,C2: A] : ( divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),divide_divide(A,B3,C2))),divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),modulo_modulo(A,B3,C2)),C2)) ) ) ).

% div_mult1_eq
tff(fact_2249_minus__div__mult__eq__mod,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A3: A,B3: A] : ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,A3,B3)),B3)) = modulo_modulo(A,A3,B3) ) ) ).

% minus_div_mult_eq_mod
tff(fact_2250_minus__mod__eq__div__mult,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A3: A,B3: A] : ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),modulo_modulo(A,A3,B3)) = aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,A3,B3)),B3) ) ) ).

% minus_mod_eq_div_mult
tff(fact_2251_minus__mod__eq__mult__div,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A3: A,B3: A] : ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),modulo_modulo(A,A3,B3)) = aa(A,A,aa(A,fun(A,A),times_times(A),B3),divide_divide(A,A3,B3)) ) ) ).

% minus_mod_eq_mult_div
tff(fact_2252_minus__mult__div__eq__mod,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A3: A,B3: A] : ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B3),divide_divide(A,A3,B3))) = modulo_modulo(A,A3,B3) ) ) ).

% minus_mult_div_eq_mod
tff(fact_2253_field__char__0__class_Oof__nat__div,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [M2: nat,N: nat] : ( aa(nat,A,semiring_1_of_nat(A),divide_divide(nat,M2,N)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,semiring_1_of_nat(A),M2)),aa(nat,A,semiring_1_of_nat(A),modulo_modulo(nat,M2,N))),aa(nat,A,semiring_1_of_nat(A),N)) ) ) ).

% field_char_0_class.of_nat_div
tff(fact_2254_mod__le__divisor,axiom,
    ! [N: nat,M2: 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_eq(nat),modulo_modulo(nat,M2,N)),N) ) ).

% mod_le_divisor
tff(fact_2255_mod__mult2__eq_H,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A3: A,M2: nat,N: nat] : ( modulo_modulo(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),M2)),aa(nat,A,semiring_1_of_nat(A),N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),M2)),modulo_modulo(A,divide_divide(A,A3,aa(nat,A,semiring_1_of_nat(A),M2)),aa(nat,A,semiring_1_of_nat(A),N)))),modulo_modulo(A,A3,aa(nat,A,semiring_1_of_nat(A),M2))) ) ) ).

% mod_mult2_eq'
tff(fact_2256_split__mod,axiom,
    ! [Pa: fun(nat,$o),M2: nat,N: nat] :
      ( aa(nat,$o,Pa,modulo_modulo(nat,M2,N))
    <=> ( ( ( N = zero_zero(nat) )
         => aa(nat,$o,Pa,M2) )
        & ( ( N != zero_zero(nat) )
         => ! [I2: nat,J2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J2),N)
             => ( ( M2 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),I2)),J2) )
               => aa(nat,$o,Pa,J2) ) ) ) ) ) ).

% split_mod
tff(fact_2257_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [C2: A,A3: A,B3: A] :
          ( aa(A,$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),B3),C2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B3),modulo_modulo(A,divide_divide(A,A3,B3),C2))),modulo_modulo(A,A3,B3)) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_2258_verit__le__mono__div,axiom,
    ! [Aa2: nat,B2: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Aa2),B2)
     => ( 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),divide_divide(nat,Aa2,N)),
                $ite(modulo_modulo(nat,B2,N) = zero_zero(nat),one_one(nat),zero_zero(nat)))),
            divide_divide(nat,B2,N)) ) ) ).

% verit_le_mono_div
tff(fact_2259_Suc__times__mod__eq,axiom,
    ! [M2: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),M2)
     => ( modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M2),N)),M2) = one_one(nat) ) ) ).

% Suc_times_mod_eq
tff(fact_2260_nth__rotate1,axiom,
    ! [A: $tType,N: nat,Xsa: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xsa))
     => ( aa(nat,A,nth(A,aa(list(A),list(A),rotate1(A),Xsa)),N) = aa(nat,A,nth(A,Xsa),modulo_modulo(nat,aa(nat,nat,suc,N),aa(list(A),nat,size_size(list(A)),Xsa))) ) ) ).

% nth_rotate1
tff(fact_2261_wf__induct__rule,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),Pa: fun(A,$o),A3: A] :
      ( wf(A,R2)
     => ( ! [X: A] :
            ( ! [Y3: A] :
                ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),X),R2)
               => aa(A,$o,Pa,Y3) )
           => aa(A,$o,Pa,X) )
       => aa(A,$o,Pa,A3) ) ) ).

% wf_induct_rule
tff(fact_2262_wf__eq__minimal,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( wf(A,R2)
    <=> ! [Q7: set(A)] :
          ( ? [X6: A] : member(A,X6,Q7)
         => ? [X6: A] :
              ( member(A,X6,Q7)
              & ! [Y4: A] :
                  ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),X6),R2)
                 => ~ member(A,Y4,Q7) ) ) ) ) ).

% wf_eq_minimal
tff(fact_2263_wf__not__refl,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A3: A] :
      ( wf(A,R2)
     => ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),A3),R2) ) ).

% wf_not_refl
tff(fact_2264_wf__not__sym,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A3: A,Xa: A] :
      ( wf(A,R2)
     => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),Xa),R2)
       => ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),A3),R2) ) ) ).

% wf_not_sym
tff(fact_2265_wf__irrefl,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A3: A] :
      ( wf(A,R2)
     => ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),A3),R2) ) ).

% wf_irrefl
tff(fact_2266_wf__induct,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),Pa: fun(A,$o),A3: A] :
      ( wf(A,R2)
     => ( ! [X: A] :
            ( ! [Y3: A] :
                ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),X),R2)
               => aa(A,$o,Pa,Y3) )
           => aa(A,$o,Pa,X) )
       => aa(A,$o,Pa,A3) ) ) ).

% wf_induct
tff(fact_2267_wf__asym,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A3: A,Xa: A] :
      ( wf(A,R2)
     => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),Xa),R2)
       => ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),A3),R2) ) ) ).

% wf_asym
tff(fact_2268_wfUNIVI,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( ! [P3: fun(A,$o),X: A] :
          ( ! [Xa4: A] :
              ( ! [Y2: A] :
                  ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),Xa4),R2)
                 => aa(A,$o,P3,Y2) )
             => aa(A,$o,P3,Xa4) )
         => aa(A,$o,P3,X) )
     => wf(A,R2) ) ).

% wfUNIVI
tff(fact_2269_wfI__min,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( ! [X: A,Q3: set(A)] :
          ( member(A,X,Q3)
         => ? [Xa4: A] :
              ( member(A,Xa4,Q3)
              & ! [Y2: A] :
                  ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),Xa4),R)
                 => ~ member(A,Y2,Q3) ) ) )
     => wf(A,R) ) ).

% wfI_min
tff(fact_2270_wfE__min,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),Xa: A,Qa: set(A)] :
      ( wf(A,R)
     => ( member(A,Xa,Qa)
       => ~ ! [Z3: A] :
              ( member(A,Z3,Qa)
             => ~ ! [Y3: A] :
                    ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),Z3),R)
                   => ~ member(A,Y3,Qa) ) ) ) ) ).

% wfE_min
tff(fact_2271_wf__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( wf(A,R2)
    <=> ! [P6: fun(A,$o)] :
          ( ! [X6: A] :
              ( ! [Y4: A] :
                  ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),X6),R2)
                 => aa(A,$o,P6,Y4) )
             => aa(A,$o,P6,X6) )
         => ! [X_12: A] : aa(A,$o,P6,X_12) ) ) ).

% wf_def
tff(fact_2272_in__measure,axiom,
    ! [A: $tType,Xa: A,Y: A,F2: fun(A,nat)] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Y),measure(A,F2))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,F2,Xa)),aa(A,nat,F2,Y)) ) ).

% in_measure
tff(fact_2273_gcd__nat__induct,axiom,
    ! [Pa: fun(nat,fun(nat,$o)),M2: nat,N: nat] :
      ( ! [M6: nat] : aa(nat,$o,aa(nat,fun(nat,$o),Pa,M6),zero_zero(nat))
     => ( ! [M6: nat,N5: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N5)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),Pa,N5),modulo_modulo(nat,M6,N5))
             => aa(nat,$o,aa(nat,fun(nat,$o),Pa,M6),N5) ) )
       => aa(nat,$o,aa(nat,fun(nat,$o),Pa,M2),N) ) ) ).

% gcd_nat_induct
tff(fact_2274_in__lex__prod,axiom,
    ! [A: $tType,B: $tType,A3: A,B3: B,A6: A,B6: B,R2: set(product_prod(A,A)),S2: set(product_prod(B,B))] :
      ( member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B3)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A6),B6)),lex_prod(A,B,R2,S2))
    <=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),A6),R2)
        | ( ( A3 = A6 )
          & member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),B3),B6),S2) ) ) ) ).

% in_lex_prod
tff(fact_2275_dependent__wf__choice,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,A)),Pa: fun(fun(A,B),fun(A,fun(B,$o)))] :
      ( wf(A,R)
     => ( ! [F3: fun(A,B),G3: fun(A,B),X: A,R3: B] :
            ( ! [Z4: A] :
                ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z4),X),R)
               => ( aa(A,B,F3,Z4) = aa(A,B,G3,Z4) ) )
           => ( aa(B,$o,aa(A,fun(B,$o),aa(fun(A,B),fun(A,fun(B,$o)),Pa,F3),X),R3)
            <=> aa(B,$o,aa(A,fun(B,$o),aa(fun(A,B),fun(A,fun(B,$o)),Pa,G3),X),R3) ) )
       => ( ! [X: A,F3: fun(A,B)] :
              ( ! [Y3: A] :
                  ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),X),R)
                 => aa(B,$o,aa(A,fun(B,$o),aa(fun(A,B),fun(A,fun(B,$o)),Pa,F3),Y3),aa(A,B,F3,Y3)) )
             => ? [X_13: B] : aa(B,$o,aa(A,fun(B,$o),aa(fun(A,B),fun(A,fun(B,$o)),Pa,F3),X),X_13) )
         => ? [F3: fun(A,B)] :
            ! [X4: A] : aa(B,$o,aa(A,fun(B,$o),aa(fun(A,B),fun(A,fun(B,$o)),Pa,F3),X4),aa(A,B,F3,X4)) ) ) ) ).

% dependent_wf_choice
tff(fact_2276_gbinomial__absorption_H,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K3: nat,A3: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K3)
         => ( aa(nat,A,gbinomial(A,A3),K3) = aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,A3,aa(nat,A,semiring_1_of_nat(A),K3))),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),K3),one_one(nat)))) ) ) ) ).

% gbinomial_absorption'
tff(fact_2277_rotate__drop__take,axiom,
    ! [A: $tType,N: nat,Xsa: list(A)] : ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),rotate(A),N),Xsa) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),modulo_modulo(nat,N,aa(list(A),nat,size_size(list(A)),Xsa))),Xsa)),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),modulo_modulo(nat,N,aa(list(A),nat,size_size(list(A)),Xsa))),Xsa)) ) ).

% rotate_drop_take
tff(fact_2278_rotate__is__Nil__conv,axiom,
    ! [A: $tType,N: nat,Xsa: list(A)] :
      ( ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),rotate(A),N),Xsa) = nil(A) )
    <=> ( Xsa = nil(A) ) ) ).

% rotate_is_Nil_conv
tff(fact_2279_set__rotate,axiom,
    ! [A: $tType,N: nat,Xsa: list(A)] : ( aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),rotate(A),N),Xsa)) = aa(list(A),set(A),set2(A),Xsa) ) ).

% set_rotate
tff(fact_2280_length__rotate,axiom,
    ! [A: $tType,N: nat,Xsa: list(A)] : ( aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),rotate(A),N),Xsa)) = aa(list(A),nat,size_size(list(A)),Xsa) ) ).

% length_rotate
tff(fact_2281_distinct__rotate,axiom,
    ! [A: $tType,N: nat,Xsa: list(A)] :
      ( aa(list(A),$o,distinct(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),rotate(A),N),Xsa))
    <=> aa(list(A),$o,distinct(A),Xsa) ) ).

% distinct_rotate
tff(fact_2282_rotate__Suc,axiom,
    ! [A: $tType,N: nat,Xsa: list(A)] : ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),rotate(A),aa(nat,nat,suc,N)),Xsa) = aa(list(A),list(A),rotate1(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),rotate(A),N),Xsa)) ) ).

% rotate_Suc
tff(fact_2283_rotate__length01,axiom,
    ! [A: $tType,Xsa: list(A),N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xsa)),one_one(nat))
     => ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),rotate(A),N),Xsa) = Xsa ) ) ).

% rotate_length01
tff(fact_2284_rotate__id,axiom,
    ! [A: $tType,N: nat,Xsa: list(A)] :
      ( ( modulo_modulo(nat,N,aa(list(A),nat,size_size(list(A)),Xsa)) = zero_zero(nat) )
     => ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),rotate(A),N),Xsa) = Xsa ) ) ).

% rotate_id
tff(fact_2285_rotate__rotate,axiom,
    ! [A: $tType,M2: nat,N: nat,Xsa: list(A)] : ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),rotate(A),M2),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),rotate(A),N),Xsa)) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),rotate(A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),N)),Xsa) ) ).

% rotate_rotate
tff(fact_2286_rotate1__rotate__swap,axiom,
    ! [A: $tType,N: nat,Xsa: list(A)] : ( aa(list(A),list(A),rotate1(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),rotate(A),N),Xsa)) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),rotate(A),N),aa(list(A),list(A),rotate1(A),Xsa)) ) ).

% rotate1_rotate_swap
tff(fact_2287_mod__int__unique,axiom,
    ! [K3: int,L: int,Q2: int,R2: int] :
      ( eucl_rel_int(K3,L,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),R2))
     => ( modulo_modulo(int,K3,L) = R2 ) ) ).

% mod_int_unique
tff(fact_2288_rotate__append,axiom,
    ! [A: $tType,L: list(A),Q2: list(A)] : ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),rotate(A),aa(list(A),nat,size_size(list(A)),L)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L),Q2)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Q2),L) ) ).

% rotate_append
tff(fact_2289_rotate__conv__mod,axiom,
    ! [A: $tType,N: nat,Xsa: list(A)] : ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),rotate(A),N),Xsa) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),rotate(A),modulo_modulo(nat,N,aa(list(A),nat,size_size(list(A)),Xsa))),Xsa) ) ).

% rotate_conv_mod
tff(fact_2290_gbinomial__addition__formula,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K3: nat] : ( aa(nat,A,gbinomial(A,A3),aa(nat,nat,suc,K3)) = 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,K3))),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),one_one(A))),K3)) ) ) ).

% gbinomial_addition_formula
tff(fact_2291_gbinomial__absorb__comp,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K3: 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),K3))),aa(nat,A,gbinomial(A,A3),K3)) = 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))),K3)) ) ) ).

% gbinomial_absorb_comp
tff(fact_2292_gbinomial__mult__1_H,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K3: nat] : ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,A3),K3)),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),K3)),aa(nat,A,gbinomial(A,A3),K3))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,K3))),aa(nat,A,gbinomial(A,A3),aa(nat,nat,suc,K3)))) ) ) ).

% gbinomial_mult_1'
tff(fact_2293_gbinomial__mult__1,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K3: nat] : ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(nat,A,gbinomial(A,A3),K3)) = 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),K3)),aa(nat,A,gbinomial(A,A3),K3))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,K3))),aa(nat,A,gbinomial(A,A3),aa(nat,nat,suc,K3)))) ) ) ).

% gbinomial_mult_1
tff(fact_2294_dependent__wellorder__choice,axiom,
    ! [A: $tType,B: $tType] :
      ( wellorder(B)
     => ! [Pa: fun(fun(B,A),fun(B,fun(A,$o)))] :
          ( ! [R3: A,F3: fun(B,A),G3: fun(B,A),X: B] :
              ( ! [Y3: B] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),Y3),X)
                 => ( aa(B,A,F3,Y3) = aa(B,A,G3,Y3) ) )
             => ( aa(A,$o,aa(B,fun(A,$o),aa(fun(B,A),fun(B,fun(A,$o)),Pa,F3),X),R3)
              <=> aa(A,$o,aa(B,fun(A,$o),aa(fun(B,A),fun(B,fun(A,$o)),Pa,G3),X),R3) ) )
         => ( ! [X: B,F3: fun(B,A)] :
                ( ! [Y3: B] :
                    ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),Y3),X)
                   => aa(A,$o,aa(B,fun(A,$o),aa(fun(B,A),fun(B,fun(A,$o)),Pa,F3),Y3),aa(B,A,F3,Y3)) )
               => ? [X_13: A] : aa(A,$o,aa(B,fun(A,$o),aa(fun(B,A),fun(B,fun(A,$o)),Pa,F3),X),X_13) )
           => ? [F3: fun(B,A)] :
              ! [X4: B] : aa(A,$o,aa(B,fun(A,$o),aa(fun(B,A),fun(B,fun(A,$o)),Pa,F3),X4),aa(B,A,F3,X4)) ) ) ) ).

% dependent_wellorder_choice
tff(fact_2295_eucl__rel__int,axiom,
    ! [K3: int,L: int] : eucl_rel_int(K3,L,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),divide_divide(int,K3,L)),modulo_modulo(int,K3,L))) ).

% eucl_rel_int
tff(fact_2296_Suc__times__gbinomial,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K3: 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,K3))),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A))),aa(nat,nat,suc,K3))) = 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),K3)) ) ) ).

% Suc_times_gbinomial
tff(fact_2297_gbinomial__absorption,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K3: 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,K3))),aa(nat,A,gbinomial(A,A3),aa(nat,nat,suc,K3))) = 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))),K3)) ) ) ).

% gbinomial_absorption
tff(fact_2298_gbinomial__trinomial__revision,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K3: nat,M2: nat,A3: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K3),M2)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,A3),M2)),aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),M2)),K3)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,A3),K3)),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),K3))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),K3))) ) ) ) ).

% gbinomial_trinomial_revision
tff(fact_2299_gbinomial__rec,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K3: 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,K3)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,A3),K3)),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,K3)))) ) ) ).

% gbinomial_rec
tff(fact_2300_gbinomial__factors,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K3: 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,K3)) = aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,K3)))),aa(nat,A,gbinomial(A,A3),K3)) ) ) ).

% gbinomial_factors
tff(fact_2301_gbinomial__reduce__nat,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K3: nat,A3: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K3)
         => ( aa(nat,A,gbinomial(A,A3),K3) = 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),K3),one_one(nat)))),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),one_one(A))),K3)) ) ) ) ).

% gbinomial_reduce_nat
tff(fact_2302_rotate__rev,axiom,
    ! [A: $tType,N: nat,Xsa: list(A)] : ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),rotate(A),N),aa(list(A),list(A),rev(A),Xsa)) = aa(list(A),list(A),rev(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),rotate(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xsa)),modulo_modulo(nat,N,aa(list(A),nat,size_size(list(A)),Xsa)))),Xsa)) ) ).

% rotate_rev
tff(fact_2303_nth__rotate,axiom,
    ! [A: $tType,N: nat,Xsa: list(A),M2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xsa))
     => ( aa(nat,A,nth(A,aa(list(A),list(A),aa(nat,fun(list(A),list(A)),rotate(A),M2),Xsa)),N) = aa(nat,A,nth(A,Xsa),modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),N),aa(list(A),nat,size_size(list(A)),Xsa))) ) ) ).

% nth_rotate
tff(fact_2304_hd__rotate__conv__nth,axiom,
    ! [A: $tType,Xsa: list(A),N: nat] :
      ( ( Xsa != nil(A) )
     => ( aa(list(A),A,hd(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),rotate(A),N),Xsa)) = aa(nat,A,nth(A,Xsa),modulo_modulo(nat,N,aa(list(A),nat,size_size(list(A)),Xsa))) ) ) ).

% hd_rotate_conv_nth
tff(fact_2305_bezw__0,axiom,
    ! [Xa: nat] : ( bezw(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)) ) ).

% bezw_0
tff(fact_2306_gbinomial__pochhammer_H,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K3: nat] : ( aa(nat,A,gbinomial(A,A3),K3) = 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),K3))),one_one(A)),K3),semiring_char_0_fact(A,K3)) ) ) ).

% gbinomial_pochhammer'
tff(fact_2307_gbinomial__minus,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K3: nat] : ( aa(nat,A,gbinomial(A,aa(A,A,uminus_uminus(A),A3)),K3) = 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))),K3)),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),K3))),one_one(A))),K3)) ) ) ).

% gbinomial_minus
tff(fact_2308_list_Osplit__sel,axiom,
    ! [A: $tType,B: $tType,Pa: fun(A,$o),F1: A,F23: fun(B,fun(list(B),A)),List: list(B)] :
      ( aa(A,$o,Pa,aa(list(B),A,aa(fun(B,fun(list(B),A)),fun(list(B),A),aa(A,fun(fun(B,fun(list(B),A)),fun(list(B),A)),case_list(A,B),F1),F23),List))
    <=> ( ( ( List = nil(B) )
         => aa(A,$o,Pa,F1) )
        & ( ( List = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),aa(list(B),B,hd(B),List)),aa(list(B),list(B),tl(B),List)) )
         => aa(A,$o,Pa,aa(list(B),A,aa(B,fun(list(B),A),F23,aa(list(B),B,hd(B),List)),aa(list(B),list(B),tl(B),List))) ) ) ) ).

% list.split_sel
tff(fact_2309_list_Osplit__sel__asm,axiom,
    ! [A: $tType,B: $tType,Pa: fun(A,$o),F1: A,F23: fun(B,fun(list(B),A)),List: list(B)] :
      ( aa(A,$o,Pa,aa(list(B),A,aa(fun(B,fun(list(B),A)),fun(list(B),A),aa(A,fun(fun(B,fun(list(B),A)),fun(list(B),A)),case_list(A,B),F1),F23),List))
    <=> ~ ( ( ( List = nil(B) )
            & ~ aa(A,$o,Pa,F1) )
          | ( ( List = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),aa(list(B),B,hd(B),List)),aa(list(B),list(B),tl(B),List)) )
            & ~ aa(A,$o,Pa,aa(list(B),A,aa(B,fun(list(B),A),F23,aa(list(B),B,hd(B),List)),aa(list(B),list(B),tl(B),List))) ) ) ) ).

% list.split_sel_asm
tff(fact_2310_Gcd__0__iff,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [Aa2: set(A)] :
          ( ( gcd_Gcd(A,Aa2) = zero_zero(A) )
        <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),zero_zero(A)),bot_bot(set(A)))) ) ) ).

% Gcd_0_iff
tff(fact_2311_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_2312_neg__equal__iff__equal,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B3: A] :
          ( ( aa(A,A,uminus_uminus(A),A3) = aa(A,A,uminus_uminus(A),B3) )
        <=> ( A3 = B3 ) ) ) ).

% neg_equal_iff_equal
tff(fact_2313_uminus__apply,axiom,
    ! [A: $tType,B: $tType] :
      ( uminus(A)
     => ! [Aa2: fun(B,A),Xa: B] : ( aa(B,A,aa(fun(B,A),fun(B,A),uminus_uminus(fun(B,A)),Aa2),Xa) = aa(A,A,uminus_uminus(A),aa(B,A,Aa2,Xa)) ) ) ).

% uminus_apply
tff(fact_2314_Compl__anti__mono,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),B2)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),B2)),aa(set(A),set(A),uminus_uminus(set(A)),Aa2)) ) ).

% Compl_anti_mono
tff(fact_2315_Compl__subset__Compl__iff,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),Aa2)),aa(set(A),set(A),uminus_uminus(set(A)),B2))
    <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),Aa2) ) ).

% Compl_subset_Compl_iff
tff(fact_2316_neg__le__iff__le,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),B3)),aa(A,A,uminus_uminus(A),A3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3) ) ) ).

% neg_le_iff_le
tff(fact_2317_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_2318_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_2319_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_2320_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_2321_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_2322_neg__less__iff__less,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),B3)),aa(A,A,uminus_uminus(A),A3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3) ) ) ).

% neg_less_iff_less
tff(fact_2323_compl__less__compl__iff,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),Xa)),aa(A,A,uminus_uminus(A),Y))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xa) ) ) ).

% compl_less_compl_iff
tff(fact_2324_add__minus__cancel,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B3: 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)),B3)) = B3 ) ) ).

% add_minus_cancel
tff(fact_2325_minus__add__cancel,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B3: 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),B3)) = B3 ) ) ).

% minus_add_cancel
tff(fact_2326_minus__add__distrib,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A,B3: A] : ( aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A3)),aa(A,A,uminus_uminus(A),B3)) ) ) ).

% minus_add_distrib
tff(fact_2327_mult__minus__right,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A3: A,B3: A] : ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,uminus_uminus(A),B3)) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)) ) ) ).

% mult_minus_right
tff(fact_2328_minus__mult__minus,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A3: A,B3: 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),B3)) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3) ) ) ).

% minus_mult_minus
tff(fact_2329_mult__minus__left,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A3: A,B3: A] : ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),A3)),B3) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)) ) ) ).

% mult_minus_left
tff(fact_2330_minus__diff__eq,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B3: A] : ( aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B3)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),B3),A3) ) ) ).

% minus_diff_eq
tff(fact_2331_Compl__disjoint2,axiom,
    ! [A: $tType,Aa2: 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)),Aa2)),Aa2) = bot_bot(set(A)) ) ).

% Compl_disjoint2
tff(fact_2332_Compl__disjoint,axiom,
    ! [A: $tType,Aa2: set(A)] : ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),aa(set(A),set(A),uminus_uminus(set(A)),Aa2)) = bot_bot(set(A)) ) ).

% Compl_disjoint
tff(fact_2333_abs__minus,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A3: A] : ( abs_abs(A,aa(A,A,uminus_uminus(A),A3)) = abs_abs(A,A3) ) ) ).

% abs_minus
tff(fact_2334_abs__minus__cancel,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] : ( abs_abs(A,aa(A,A,uminus_uminus(A),A3)) = abs_abs(A,A3) ) ) ).

% abs_minus_cancel
tff(fact_2335_Diff__Compl,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A)] : ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),uminus_uminus(set(A)),B2)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),B2) ) ).

% Diff_Compl
tff(fact_2336_inter__compl__diff__conv,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A)] : ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),aa(set(A),set(A),uminus_uminus(set(A)),B2)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),B2) ) ).

% inter_compl_diff_conv
tff(fact_2337_Compl__Diff__eq,axiom,
    ! [A: $tType,Aa2: set(A),B2: 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)),Aa2),B2)) = 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)),Aa2)),B2) ) ).

% Compl_Diff_eq
tff(fact_2338_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_2339_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_2340_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_2341_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_2342_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_2343_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_2344_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_2345_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_2346_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_2347_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_2348_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_2349_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_2350_verit__minus__simplify_I3_J,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [B3: A] : ( aa(A,A,aa(A,fun(A,A),minus_minus(A),zero_zero(A)),B3) = aa(A,A,uminus_uminus(A),B3) ) ) ).

% verit_minus_simplify(3)
tff(fact_2351_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_2352_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_2353_uminus__add__conv__diff,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A,B3: A] : ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A3)),B3) = aa(A,A,aa(A,fun(A,A),minus_minus(A),B3),A3) ) ) ).

% uminus_add_conv_diff
tff(fact_2354_diff__minus__eq__add,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B3: A] : ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),aa(A,A,uminus_uminus(A),B3)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3) ) ) ).

% diff_minus_eq_add
tff(fact_2355_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_2356_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_2357_boolean__algebra_Oconj__cancel__right,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A] : ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),aa(A,A,uminus_uminus(A),Xa)) = bot_bot(A) ) ) ).

% boolean_algebra.conj_cancel_right
tff(fact_2358_boolean__algebra_Oconj__cancel__left,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A] : ( aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,uminus_uminus(A),Xa)),Xa) = bot_bot(A) ) ) ).

% boolean_algebra.conj_cancel_left
tff(fact_2359_inf__compl__bot__right,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A,Y: A] : ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),aa(A,A,uminus_uminus(A),Xa))) = bot_bot(A) ) ) ).

% inf_compl_bot_right
tff(fact_2360_inf__compl__bot__left2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A,Y: A] : ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,uminus_uminus(A),Xa)),Y)) = bot_bot(A) ) ) ).

% inf_compl_bot_left2
tff(fact_2361_inf__compl__bot__left1,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A,Y: A] : ( aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,uminus_uminus(A),Xa)),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Y)) = bot_bot(A) ) ) ).

% inf_compl_bot_left1
tff(fact_2362_subset__Compl__singleton,axiom,
    ! [A: $tType,Aa2: set(A),B3: A] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),B3),bot_bot(set(A)))))
    <=> ~ member(A,B3,Aa2) ) ).

% subset_Compl_singleton
tff(fact_2363_minus__mod__self1,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [B3: A,A3: A] : ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),B3),A3),B3) = modulo_modulo(A,aa(A,A,uminus_uminus(A),A3),B3) ) ) ).

% minus_mod_self1
tff(fact_2364_Gcd__empty,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ( gcd_Gcd(A,bot_bot(set(A))) = zero_zero(A) ) ) ).

% Gcd_empty
tff(fact_2365_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_2366_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))
         => ( abs_abs(A,A3) = aa(A,A,uminus_uminus(A),A3) ) ) ) ).

% abs_of_nonpos
tff(fact_2367_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_2368_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_2369_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_2370_equation__minus__iff,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B3: A] :
          ( ( A3 = aa(A,A,uminus_uminus(A),B3) )
        <=> ( B3 = aa(A,A,uminus_uminus(A),A3) ) ) ) ).

% equation_minus_iff
tff(fact_2371_minus__equation__iff,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B3: A] :
          ( ( aa(A,A,uminus_uminus(A),A3) = B3 )
        <=> ( aa(A,A,uminus_uminus(A),B3) = A3 ) ) ) ).

% minus_equation_iff
tff(fact_2372_fun__Compl__def,axiom,
    ! [B: $tType,A: $tType] :
      ( uminus(B)
     => ! [Aa2: fun(A,B),X4: A] : ( aa(A,B,aa(fun(A,B),fun(A,B),uminus_uminus(fun(A,B)),Aa2),X4) = aa(B,B,uminus_uminus(B),aa(A,B,Aa2,X4)) ) ) ).

% fun_Compl_def
tff(fact_2373_le__minus__iff,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,uminus_uminus(A),B3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),aa(A,A,uminus_uminus(A),A3)) ) ) ).

% le_minus_iff
tff(fact_2374_minus__le__iff,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),A3)),B3)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),B3)),A3) ) ) ).

% minus_le_iff
tff(fact_2375_le__imp__neg__le,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),B3)),aa(A,A,uminus_uminus(A),A3)) ) ) ).

% le_imp_neg_le
tff(fact_2376_minus__less__iff,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),A3)),B3)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),B3)),A3) ) ) ).

% minus_less_iff
tff(fact_2377_less__minus__iff,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(A,A,uminus_uminus(A),B3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),aa(A,A,uminus_uminus(A),A3)) ) ) ).

% less_minus_iff
tff(fact_2378_compl__less__swap2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Y: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),Y)),Xa)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),Xa)),Y) ) ) ).

% compl_less_swap2
tff(fact_2379_compl__less__swap1,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Y: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),aa(A,A,uminus_uminus(A),Xa))
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),aa(A,A,uminus_uminus(A),Y)) ) ) ).

% compl_less_swap1
tff(fact_2380_verit__negate__coefficient_I2_J,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),B3)),aa(A,A,uminus_uminus(A),A3)) ) ) ).

% verit_negate_coefficient(2)
tff(fact_2381_group__cancel_Oneg1,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Aa2: A,K3: A,A3: A] :
          ( ( Aa2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K3),A3) )
         => ( aa(A,A,uminus_uminus(A),Aa2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),K3)),aa(A,A,uminus_uminus(A),A3)) ) ) ) ).

% group_cancel.neg1
tff(fact_2382_add_Oinverse__distrib__swap,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B3: A] : ( aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),B3)),aa(A,A,uminus_uminus(A),A3)) ) ) ).

% add.inverse_distrib_swap
tff(fact_2383_minus__mult__commute,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A3: A,B3: A] : ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),A3)),B3) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,uminus_uminus(A),B3)) ) ) ).

% minus_mult_commute
tff(fact_2384_square__eq__iff,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [A3: A,B3: 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),B3),B3) )
        <=> ( ( A3 = B3 )
            | ( A3 = aa(A,A,uminus_uminus(A),B3) ) ) ) ) ).

% square_eq_iff
tff(fact_2385_minus__diff__commute,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [B3: A,A3: A] : ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),B3)),A3) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),A3)),B3) ) ) ).

% minus_diff_commute
tff(fact_2386_abs__eq__iff,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [Xa: A,Y: A] :
          ( ( abs_abs(A,Xa) = abs_abs(A,Y) )
        <=> ( ( Xa = Y )
            | ( Xa = aa(A,A,uminus_uminus(A),Y) ) ) ) ) ).

% abs_eq_iff
tff(fact_2387_minus__assn__def,axiom,
    ! [A3: assn,B3: assn] : ( aa(assn,assn,aa(assn,fun(assn,assn),minus_minus(assn),A3),B3) = aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),A3),aa(assn,assn,uminus_uminus(assn),B3)) ) ).

% minus_assn_def
tff(fact_2388_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [N: nat,K3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),K3)
         => ( comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),N)),K3) = zero_zero(A) ) ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_2389_pochhammer__of__nat__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ( ring_char_0(A)
        & idom(A) )
     => ! [N: nat,K3: nat] :
          ( ( comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),N)),K3) = zero_zero(A) )
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),K3) ) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_2390_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_2391_list_Osimps_I5_J,axiom,
    ! [A: $tType,B: $tType,F1: A,F23: fun(B,fun(list(B),A)),X21: B,X222: list(B)] : ( aa(list(B),A,aa(fun(B,fun(list(B),A)),fun(list(B),A),aa(A,fun(fun(B,fun(list(B),A)),fun(list(B),A)),case_list(A,B),F1),F23),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X21),X222)) = aa(list(B),A,aa(B,fun(list(B),A),F23,X21),X222) ) ).

% list.simps(5)
tff(fact_2392_list_Osimps_I4_J,axiom,
    ! [B: $tType,A: $tType,F1: A,F23: fun(B,fun(list(B),A))] : ( aa(list(B),A,aa(fun(B,fun(list(B),A)),fun(list(B),A),aa(A,fun(fun(B,fun(list(B),A)),fun(list(B),A)),case_list(A,B),F1),F23),nil(B)) = F1 ) ).

% list.simps(4)
tff(fact_2393_neg__eq__iff__add__eq__0,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B3: A] :
          ( ( aa(A,A,uminus_uminus(A),A3) = B3 )
        <=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3) = zero_zero(A) ) ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_2394_eq__neg__iff__add__eq__0,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B3: A] :
          ( ( A3 = aa(A,A,uminus_uminus(A),B3) )
        <=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3) = zero_zero(A) ) ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_2395_add_Oinverse__unique,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3) = zero_zero(A) )
         => ( aa(A,A,uminus_uminus(A),A3) = B3 ) ) ) ).

% add.inverse_unique
tff(fact_2396_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_2397_add__eq__0__iff,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3) = zero_zero(A) )
        <=> ( B3 = aa(A,A,uminus_uminus(A),A3) ) ) ) ).

% add_eq_0_iff
tff(fact_2398_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_2399_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_2400_square__eq__1__iff,axiom,
    ! [A: $tType] :
      ( ring_15535105094025558882visors(A)
     => ! [Xa: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),Xa),Xa) = one_one(A) )
        <=> ( ( Xa = one_one(A) )
            | ( Xa = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ) ).

% square_eq_1_iff
tff(fact_2401_group__cancel_Osub2,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [B2: A,K3: A,B3: A,A3: A] :
          ( ( B2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K3),B3) )
         => ( 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,uminus_uminus(A),K3)),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B3)) ) ) ) ).

% group_cancel.sub2
tff(fact_2402_diff__conv__add__uminus,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B3: A] : ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,uminus_uminus(A),B3)) ) ) ).

% diff_conv_add_uminus
tff(fact_2403_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A,B3: A] : ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,uminus_uminus(A),B3)) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_2404_inf__cancel__left2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A,A3: A,B3: 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),Xa)),A3)),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),B3)) = bot_bot(A) ) ) ).

% inf_cancel_left2
tff(fact_2405_inf__cancel__left1,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A,A3: A,B3: A] : ( aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),A3)),aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,uminus_uminus(A),Xa)),B3)) = bot_bot(A) ) ) ).

% inf_cancel_left1
tff(fact_2406_diff__eq,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A,Y: A] : ( aa(A,A,aa(A,fun(A,A),minus_minus(A),Xa),Y) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),aa(A,A,uminus_uminus(A),Y)) ) ) ).

% diff_eq
tff(fact_2407_subset__Compl__self__eq,axiom,
    ! [A: $tType,Aa2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),aa(set(A),set(A),uminus_uminus(set(A)),Aa2))
    <=> ( Aa2 = bot_bot(set(A)) ) ) ).

% subset_Compl_self_eq
tff(fact_2408_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)),abs_abs(A,A3)) ) ).

% abs_ge_minus_self
tff(fact_2409_abs__le__iff,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),abs_abs(A,A3)),B3)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),A3)),B3) ) ) ) ).

% abs_le_iff
tff(fact_2410_abs__le__D2,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),abs_abs(A,A3)),B3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),A3)),B3) ) ) ).

% abs_le_D2
tff(fact_2411_abs__leI,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),A3)),B3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),abs_abs(A,A3)),B3) ) ) ) ).

% abs_leI
tff(fact_2412_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_2413_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_2414_abs__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),abs_abs(A,A3)),B3)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),A3)),B3) ) ) ) ).

% abs_less_iff
tff(fact_2415_Compl__partition2,axiom,
    ! [A: $tType,Aa2: 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)),Aa2)),Aa2) = top_top(set(A)) ) ).

% Compl_partition2
tff(fact_2416_Compl__partition,axiom,
    ! [A: $tType,Aa2: set(A)] : ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),aa(set(A),set(A),uminus_uminus(set(A)),Aa2)) = top_top(set(A)) ) ).

% Compl_partition
tff(fact_2417_Compl__eq__Diff__UNIV,axiom,
    ! [A: $tType,Aa2: set(A)] : ( aa(set(A),set(A),uminus_uminus(set(A)),Aa2) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),top_top(set(A))),Aa2) ) ).

% Compl_eq_Diff_UNIV
tff(fact_2418_sgn__not__eq__imp,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [B3: A,A3: A] :
          ( ( aa(A,A,sgn_sgn(A),B3) != aa(A,A,sgn_sgn(A),A3) )
         => ( ( aa(A,A,sgn_sgn(A),A3) != zero_zero(A) )
           => ( ( aa(A,A,sgn_sgn(A),B3) != zero_zero(A) )
             => ( aa(A,A,sgn_sgn(A),A3) = aa(A,A,uminus_uminus(A),aa(A,A,sgn_sgn(A),B3)) ) ) ) ) ) ).

% sgn_not_eq_imp
tff(fact_2419_pochhammer__pos,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xa: A,N: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xa)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),comm_s3205402744901411588hammer(A,Xa,N)) ) ) ).

% pochhammer_pos
tff(fact_2420_Compl__Un,axiom,
    ! [A: $tType,Aa2: set(A),B2: 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)),Aa2),B2)) = 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)),Aa2)),aa(set(A),set(A),uminus_uminus(set(A)),B2)) ) ).

% Compl_Un
tff(fact_2421_Compl__Int,axiom,
    ! [A: $tType,Aa2: set(A),B2: 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)),Aa2),B2)) = 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)),Aa2)),aa(set(A),set(A),uminus_uminus(set(A)),B2)) ) ).

% Compl_Int
tff(fact_2422_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_2423_pochhammer__absorb__comp,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [R2: A,K3: 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),K3))),comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),R2),K3)) = 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)),K3)) ) ) ).

% pochhammer_absorb_comp
tff(fact_2424_coset__def,axiom,
    ! [A: $tType,Xsa: list(A)] : ( coset(A,Xsa) = aa(set(A),set(A),uminus_uminus(set(A)),aa(list(A),set(A),set2(A),Xsa)) ) ).

% coset_def
tff(fact_2425_compl__coset,axiom,
    ! [A: $tType,Xsa: list(A)] : ( aa(set(A),set(A),uminus_uminus(set(A)),coset(A,Xsa)) = aa(list(A),set(A),set2(A),Xsa) ) ).

% compl_coset
tff(fact_2426_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_2427_pochhammer__minus_H,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [B3: A,K3: 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),B3),aa(nat,A,semiring_1_of_nat(A),K3))),one_one(A)),K3) = 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))),K3)),comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),B3),K3)) ) ) ).

% pochhammer_minus'
tff(fact_2428_pochhammer__minus,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [B3: A,K3: nat] : ( comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),B3),K3) = 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))),K3)),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),B3),aa(nat,A,semiring_1_of_nat(A),K3))),one_one(A)),K3)) ) ) ).

% pochhammer_minus
tff(fact_2429_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_2430_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_2431_eq__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B3: A,C2: A] :
          ( ( A3 = aa(A,A,uminus_uminus(A),divide_divide(A,B3,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),B3),A3 = zero_zero(A)) ) ) ).

% eq_minus_divide_eq
tff(fact_2432_minus__divide__eq__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B3: A,C2: A,A3: A] :
          ( ( aa(A,A,uminus_uminus(A),divide_divide(A,B3,C2)) = A3 )
        <=> $ite(C2 != zero_zero(A),aa(A,A,uminus_uminus(A),B3) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2),A3 = zero_zero(A)) ) ) ).

% minus_divide_eq_eq
tff(fact_2433_nonzero__neg__divide__eq__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B3: A,A3: A,C2: A] :
          ( ( B3 != zero_zero(A) )
         => ( ( aa(A,A,uminus_uminus(A),divide_divide(A,A3,B3)) = C2 )
          <=> ( aa(A,A,uminus_uminus(A),A3) = aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3) ) ) ) ) ).

% nonzero_neg_divide_eq_eq
tff(fact_2434_nonzero__neg__divide__eq__eq2,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B3: A,C2: A,A3: A] :
          ( ( B3 != zero_zero(A) )
         => ( ( C2 = aa(A,A,uminus_uminus(A),divide_divide(A,A3,B3)) )
          <=> ( aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3) = aa(A,A,uminus_uminus(A),A3) ) ) ) ) ).

% nonzero_neg_divide_eq_eq2
tff(fact_2435_inf__shunt,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Y) = bot_bot(A) )
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(A,A,uminus_uminus(A),Y)) ) ) ).

% inf_shunt
tff(fact_2436_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),abs_abs(A,A3))),zero_zero(A)) ) ).

% abs_minus_le_zero
tff(fact_2437_eq__abs__iff_H,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [A3: A,B3: A] :
          ( ( A3 = abs_abs(A,B3) )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
            & ( ( B3 = A3 )
              | ( B3 = aa(A,A,uminus_uminus(A),A3) ) ) ) ) ) ).

% eq_abs_iff'
tff(fact_2438_abs__eq__iff_H,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [A3: A,B3: A] :
          ( ( abs_abs(A,A3) = B3 )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B3)
            & ( ( A3 = B3 )
              | ( A3 = aa(A,A,uminus_uminus(A),B3) ) ) ) ) ) ).

% abs_eq_iff'
tff(fact_2439_abs__if__raw,axiom,
    ! [A: $tType] :
      ( abs_if(A)
     => ! [X4: A] :
          ( abs_abs(A,X4) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),zero_zero(A)),aa(A,A,uminus_uminus(A),X4),X4) ) ) ).

% abs_if_raw
tff(fact_2440_abs__if,axiom,
    ! [A: $tType] :
      ( abs_if(A)
     => ! [A3: 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_2441_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))
         => ( abs_abs(A,A3) = aa(A,A,uminus_uminus(A),A3) ) ) ) ).

% abs_of_neg
tff(fact_2442_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_2443_gbinomial__pochhammer,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K3: nat] : ( aa(nat,A,gbinomial(A,A3),K3) = 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))),K3)),comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),A3),K3)),semiring_char_0_fact(A,K3)) ) ) ).

% gbinomial_pochhammer
tff(fact_2444_disjoint__eq__subset__Compl,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),B2) = bot_bot(set(A)) )
    <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),aa(set(A),set(A),uminus_uminus(set(A)),B2)) ) ).

% disjoint_eq_subset_Compl
tff(fact_2445_Compl__insert,axiom,
    ! [A: $tType,Xa: A,Aa2: set(A)] : ( aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),Aa2)) = 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)),Aa2)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A)))) ) ).

% Compl_insert
tff(fact_2446_pochhammer__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xa: A,N: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xa)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),comm_s3205402744901411588hammer(A,Xa,N)) ) ) ).

% pochhammer_nonneg
tff(fact_2447_int__cases4,axiom,
    ! [M2: int] :
      ( ! [N5: nat] : ( M2 != aa(nat,int,semiring_1_of_nat(int),N5) )
     => ~ ! [N5: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N5)
           => ( M2 != aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N5)) ) ) ) ).

% int_cases4
tff(fact_2448_less__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(A,A,uminus_uminus(A),divide_divide(A,B3,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),B3)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),B3)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A))) ) ) ) ).

% less_minus_divide_eq
tff(fact_2449_minus__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B3: A,C2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),divide_divide(A,B3,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),B3)),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,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,uminus_uminus(A),B3)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)) ) ) ) ).

% minus_divide_less_eq
tff(fact_2450_neg__less__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(A,A,uminus_uminus(A),divide_divide(A,B3,C2)))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),B3)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)) ) ) ) ).

% neg_less_minus_divide_eq
tff(fact_2451_neg__minus__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),divide_divide(A,B3,C2))),A3)
          <=> aa(A,$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),B3)) ) ) ) ).

% neg_minus_divide_less_eq
tff(fact_2452_pos__less__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(A,A,uminus_uminus(A),divide_divide(A,B3,C2)))
          <=> aa(A,$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),B3)) ) ) ) ).

% pos_less_minus_divide_eq
tff(fact_2453_pos__minus__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),divide_divide(A,B3,C2))),A3)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),B3)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)) ) ) ) ).

% pos_minus_divide_less_eq
tff(fact_2454_minus__divide__add__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z2: A,Xa: A,Y: A] :
          ( ( Z2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),divide_divide(A,Xa,Z2))),Y) = divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),Xa)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z2)),Z2) ) ) ) ).

% minus_divide_add_eq_iff
tff(fact_2455_add__divide__eq__if__simps_I3_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,Z2: A,B3: A] :
          ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),divide_divide(A,A3,Z2))),B3) = $ite(Z2 = zero_zero(A),B3,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),B3),Z2)),Z2)) ) ) ).

% add_divide_eq_if_simps(3)
tff(fact_2456_minus__divide__diff__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z2: A,Xa: A,Y: A] :
          ( ( Z2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),divide_divide(A,Xa,Z2))),Y) = divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),Xa)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z2)),Z2) ) ) ) ).

% minus_divide_diff_eq_iff
tff(fact_2457_add__divide__eq__if__simps_I5_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,Z2: A,B3: A] :
          ( aa(A,A,aa(A,fun(A,A),minus_minus(A),divide_divide(A,A3,Z2)),B3) = $ite(Z2 = zero_zero(A),aa(A,A,uminus_uminus(A),B3),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),B3),Z2)),Z2)) ) ) ).

% add_divide_eq_if_simps(5)
tff(fact_2458_add__divide__eq__if__simps_I6_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,Z2: A,B3: A] :
          ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),divide_divide(A,A3,Z2))),B3) = $ite(Z2 = zero_zero(A),aa(A,A,uminus_uminus(A),B3),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),B3),Z2)),Z2)) ) ) ).

% add_divide_eq_if_simps(6)
tff(fact_2459_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Y) = bot_bot(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Y) = top_top(A) )
           => ( aa(A,A,uminus_uminus(A),Xa) = Y ) ) ) ) ).

% boolean_algebra_class.boolean_algebra.compl_unique
tff(fact_2460_sgn__if,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A] :
          ( aa(A,A,sgn_sgn(A),Xa) = $ite(
              Xa = zero_zero(A),
              zero_zero(A),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xa),one_one(A),aa(A,A,uminus_uminus(A),one_one(A))) ) ) ) ).

% sgn_if
tff(fact_2461_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_2462_int__cases3,axiom,
    ! [K3: int] :
      ( ( K3 != zero_zero(int) )
     => ( ! [N5: nat] :
            ( ( K3 = aa(nat,int,semiring_1_of_nat(int),N5) )
           => ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N5) )
       => ~ ! [N5: nat] :
              ( ( K3 = aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N5)) )
             => ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N5) ) ) ) ).

% int_cases3
tff(fact_2463_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_2464_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_2465_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_2466_pochhammer__product_H,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Z2: A,N: nat,M2: nat] : ( comm_s3205402744901411588hammer(A,Z2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M2)) = aa(A,A,aa(A,fun(A,A),times_times(A),comm_s3205402744901411588hammer(A,Z2,N)),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z2),aa(nat,A,semiring_1_of_nat(A),N)),M2)) ) ) ).

% pochhammer_product'
tff(fact_2467_pos__minus__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),divide_divide(A,B3,C2))),A3)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),B3)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)) ) ) ) ).

% pos_minus_divide_le_eq
tff(fact_2468_pos__le__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,uminus_uminus(A),divide_divide(A,B3,C2)))
          <=> aa(A,$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),B3)) ) ) ) ).

% pos_le_minus_divide_eq
tff(fact_2469_neg__minus__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),divide_divide(A,B3,C2))),A3)
          <=> aa(A,$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),B3)) ) ) ) ).

% neg_minus_divide_le_eq
tff(fact_2470_neg__le__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,uminus_uminus(A),divide_divide(A,B3,C2)))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),B3)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)) ) ) ) ).

% neg_le_minus_divide_eq
tff(fact_2471_minus__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B3: A,C2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),divide_divide(A,B3,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),B3)),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,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,uminus_uminus(A),B3)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)) ) ) ) ).

% minus_divide_le_eq
tff(fact_2472_le__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,uminus_uminus(A),divide_divide(A,B3,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),B3)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),B3)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),zero_zero(A))) ) ) ) ).

% le_minus_divide_eq
tff(fact_2473_list_Ocase__eq__if,axiom,
    ! [A: $tType,B: $tType,F1: A,F23: fun(B,fun(list(B),A)),List: list(B)] :
      ( aa(list(B),A,aa(fun(B,fun(list(B),A)),fun(list(B),A),aa(A,fun(fun(B,fun(list(B),A)),fun(list(B),A)),case_list(A,B),F1),F23),List) = $ite(List = nil(B),F1,aa(list(B),A,aa(B,fun(list(B),A),F23,aa(list(B),B,hd(B),List)),aa(list(B),list(B),tl(B),List))) ) ).

% list.case_eq_if
tff(fact_2474_neg__int__cases,axiom,
    ! [K3: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K3),zero_zero(int))
     => ~ ! [N5: nat] :
            ( ( K3 = aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N5)) )
           => ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N5) ) ) ).

% neg_int_cases
tff(fact_2475_pochhammer__product,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [M2: nat,N: nat,Z2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),N)
         => ( comm_s3205402744901411588hammer(A,Z2,N) = aa(A,A,aa(A,fun(A,A),times_times(A),comm_s3205402744901411588hammer(A,Z2,M2)),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z2),aa(nat,A,semiring_1_of_nat(A),M2)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M2))) ) ) ) ).

% pochhammer_product
tff(fact_2476_zminus1__lemma,axiom,
    ! [A3: int,B3: int,Q2: int,R2: int] :
      ( eucl_rel_int(A3,B3,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),R2))
     => ( ( B3 != zero_zero(int) )
       => eucl_rel_int(aa(int,int,uminus_uminus(int),A3),B3,
            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),Q2),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,uminus_uminus(int),Q2)),one_one(int)))),
              $ite(R2 = zero_zero(int),zero_zero(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),B3),R2)))) ) ) ).

% zminus1_lemma
tff(fact_2477_gbinomial__index__swap,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K3: 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))),K3)),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))),K3)) = 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),K3))),one_one(A))),N)) ) ) ).

% gbinomial_index_swap
tff(fact_2478_gbinomial__negated__upper,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K3: nat] : ( aa(nat,A,gbinomial(A,A3),K3) = 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))),K3)),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),K3)),A3)),one_one(A))),K3)) ) ) ).

% gbinomial_negated_upper
tff(fact_2479_power__minus_H,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Xa: A,N: nat] :
          ( nO_MATCH(A,A,one_one(A),Xa)
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),Xa)),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),Xa),N)) ) ) ) ).

% power_minus'
tff(fact_2480_lenlex__append2,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),Us: list(A),Xsa: list(A),Ys: list(A)] :
      ( irrefl(A,R)
     => ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us),Xsa)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us),Ys)),lenlex(A,R))
      <=> member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xsa),Ys),lenlex(A,R)) ) ) ).

% lenlex_append2
tff(fact_2481_entails__solve__finalize_I1_J,axiom,
    ! [M5: list(product_prod(assn,assn)),Pa: assn] : fI_RESULT(M5,Pa,one_one(assn),top_top(assn)) ).

% entails_solve_finalize(1)
tff(fact_2482_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)),aa(list(set(A)),list(set(A)),aa(nat,fun(list(set(A)),list(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)),aa(list(set(A)),list(set(A)),aa(nat,fun(list(set(A)),list(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_2483_zero__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),archimedean_ceiling(A,Xa))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),one_one(A))),Xa) ) ) ).

% zero_le_ceiling
tff(fact_2484_divide__le__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B3: A,C2: A,W2: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,B3,C2)),aa(A,A,uminus_uminus(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),B3),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(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,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),numeral_numeral(A,W2))),C2)),B3),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,uminus_uminus(A),numeral_numeral(A,W2)))) ) ) ) ).

% divide_le_eq_numeral(2)
tff(fact_2485_ComplI,axiom,
    ! [A: $tType,C2: A,Aa2: set(A)] :
      ( ~ member(A,C2,Aa2)
     => member(A,C2,aa(set(A),set(A),uminus_uminus(set(A)),Aa2)) ) ).

% ComplI
tff(fact_2486_Compl__iff,axiom,
    ! [A: $tType,C2: A,Aa2: set(A)] :
      ( member(A,C2,aa(set(A),set(A),uminus_uminus(set(A)),Aa2))
    <=> ~ member(A,C2,Aa2) ) ).

% Compl_iff
tff(fact_2487_Compl__eq__Compl__iff,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A)] :
      ( ( aa(set(A),set(A),uminus_uminus(set(A)),Aa2) = aa(set(A),set(A),uminus_uminus(set(A)),B2) )
    <=> ( Aa2 = B2 ) ) ).

% Compl_eq_Compl_iff
tff(fact_2488_numeral__less__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M2: num,N: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),numeral_numeral(A,M2)),numeral_numeral(A,N))
        <=> aa(num,$o,aa(num,fun(num,$o),ord_less(num),M2),N) ) ) ).

% numeral_less_iff
tff(fact_2489_numeral__times__numeral,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [M2: num,N: num] : ( aa(A,A,aa(A,fun(A,A),times_times(A),numeral_numeral(A,M2)),numeral_numeral(A,N)) = numeral_numeral(A,aa(num,num,aa(num,fun(num,num),times_times(num),M2),N)) ) ) ).

% numeral_times_numeral
tff(fact_2490_mult__numeral__left__semiring__numeral,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [V2: num,W2: num,Z2: A] : ( aa(A,A,aa(A,fun(A,A),times_times(A),numeral_numeral(A,V2)),aa(A,A,aa(A,fun(A,A),times_times(A),numeral_numeral(A,W2)),Z2)) = aa(A,A,aa(A,fun(A,A),times_times(A),numeral_numeral(A,aa(num,num,aa(num,fun(num,num),times_times(num),V2),W2))),Z2) ) ) ).

% mult_numeral_left_semiring_numeral
tff(fact_2491_power__add__numeral,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A3: A,M2: num,N: num] : ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),numeral_numeral(nat,M2))),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),numeral_numeral(nat,N))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),numeral_numeral(nat,aa(num,num,aa(num,fun(num,num),plus_plus(num),M2),N))) ) ) ).

% power_add_numeral
tff(fact_2492_power__add__numeral2,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A3: A,M2: num,N: num,B3: A] : ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),numeral_numeral(nat,M2))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),numeral_numeral(nat,N))),B3)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),numeral_numeral(nat,aa(num,num,aa(num,fun(num,num),plus_plus(num),M2),N)))),B3) ) ) ).

% power_add_numeral2
tff(fact_2493_distrib__left__numeral,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & semiring(A) )
     => ! [V2: num,B3: A,C2: A] : ( aa(A,A,aa(A,fun(A,A),times_times(A),numeral_numeral(A,V2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),C2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),numeral_numeral(A,V2)),B3)),aa(A,A,aa(A,fun(A,A),times_times(A),numeral_numeral(A,V2)),C2)) ) ) ).

% distrib_left_numeral
tff(fact_2494_distrib__right__numeral,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & semiring(A) )
     => ! [A3: A,B3: A,V2: num] : ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3)),numeral_numeral(A,V2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),numeral_numeral(A,V2))),aa(A,A,aa(A,fun(A,A),times_times(A),B3),numeral_numeral(A,V2))) ) ) ).

% distrib_right_numeral
tff(fact_2495_right__diff__distrib__numeral,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & ring(A) )
     => ! [V2: num,B3: A,C2: A] : ( aa(A,A,aa(A,fun(A,A),times_times(A),numeral_numeral(A,V2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),B3),C2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),numeral_numeral(A,V2)),B3)),aa(A,A,aa(A,fun(A,A),times_times(A),numeral_numeral(A,V2)),C2)) ) ) ).

% right_diff_distrib_numeral
tff(fact_2496_left__diff__distrib__numeral,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & ring(A) )
     => ! [A3: A,B3: A,V2: num] : ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B3)),numeral_numeral(A,V2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),numeral_numeral(A,V2))),aa(A,A,aa(A,fun(A,A),times_times(A),B3),numeral_numeral(A,V2))) ) ) ).

% left_diff_distrib_numeral
tff(fact_2497_neg__numeral__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M2: num,N: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),numeral_numeral(A,M2))),aa(A,A,uminus_uminus(A),numeral_numeral(A,N)))
        <=> aa(num,$o,aa(num,fun(num,$o),ord_less(num),N),M2) ) ) ).

% neg_numeral_less_iff
tff(fact_2498_mult__neg__numeral__simps_I3_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [M2: num,N: num] : ( aa(A,A,aa(A,fun(A,A),times_times(A),numeral_numeral(A,M2)),aa(A,A,uminus_uminus(A),numeral_numeral(A,N))) = aa(A,A,uminus_uminus(A),numeral_numeral(A,aa(num,num,aa(num,fun(num,num),times_times(num),M2),N))) ) ) ).

% mult_neg_numeral_simps(3)
tff(fact_2499_mult__neg__numeral__simps_I2_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [M2: num,N: num] : ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),numeral_numeral(A,M2))),numeral_numeral(A,N)) = aa(A,A,uminus_uminus(A),numeral_numeral(A,aa(num,num,aa(num,fun(num,num),times_times(num),M2),N))) ) ) ).

% mult_neg_numeral_simps(2)
tff(fact_2500_mult__neg__numeral__simps_I1_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [M2: num,N: num] : ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),numeral_numeral(A,M2))),aa(A,A,uminus_uminus(A),numeral_numeral(A,N))) = numeral_numeral(A,aa(num,num,aa(num,fun(num,num),times_times(num),M2),N)) ) ) ).

% mult_neg_numeral_simps(1)
tff(fact_2501_semiring__norm_I170_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [V2: num,W2: num,Y: A] : ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),numeral_numeral(A,V2))),aa(A,A,aa(A,fun(A,A),times_times(A),numeral_numeral(A,W2)),Y)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),numeral_numeral(A,aa(num,num,aa(num,fun(num,num),times_times(num),V2),W2)))),Y) ) ) ).

% semiring_norm(170)
tff(fact_2502_semiring__norm_I171_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [V2: num,W2: num,Y: A] : ( aa(A,A,aa(A,fun(A,A),times_times(A),numeral_numeral(A,V2)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),numeral_numeral(A,W2))),Y)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),numeral_numeral(A,aa(num,num,aa(num,fun(num,num),times_times(num),V2),W2)))),Y) ) ) ).

% semiring_norm(171)
tff(fact_2503_semiring__norm_I172_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [V2: num,W2: num,Y: A] : ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),numeral_numeral(A,V2))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),numeral_numeral(A,W2))),Y)) = aa(A,A,aa(A,fun(A,A),times_times(A),numeral_numeral(A,aa(num,num,aa(num,fun(num,num),times_times(num),V2),W2))),Y) ) ) ).

% semiring_norm(172)
tff(fact_2504_diff__numeral__simps_I2_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M2: num,N: num] : ( aa(A,A,aa(A,fun(A,A),minus_minus(A),numeral_numeral(A,M2)),aa(A,A,uminus_uminus(A),numeral_numeral(A,N))) = numeral_numeral(A,aa(num,num,aa(num,fun(num,num),plus_plus(num),M2),N)) ) ) ).

% diff_numeral_simps(2)
tff(fact_2505_diff__numeral__simps_I3_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M2: num,N: num] : ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),numeral_numeral(A,M2))),numeral_numeral(A,N)) = aa(A,A,uminus_uminus(A),numeral_numeral(A,aa(num,num,aa(num,fun(num,num),plus_plus(num),M2),N))) ) ) ).

% diff_numeral_simps(3)
tff(fact_2506_cSup__singleton,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Xa: A] : ( aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A)))) = Xa ) ) ).

% cSup_singleton
tff(fact_2507_le__divide__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A,W2: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),divide_divide(A,B3,numeral_numeral(A,W2)))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),numeral_numeral(A,W2))),B3) ) ) ).

% le_divide_eq_numeral1(1)
tff(fact_2508_divide__le__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B3: A,W2: num,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,B3,numeral_numeral(A,W2))),A3)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),aa(A,A,aa(A,fun(A,A),times_times(A),A3),numeral_numeral(A,W2))) ) ) ).

% divide_le_eq_numeral1(1)
tff(fact_2509_divide__eq__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B3: A,W2: num,A3: A] :
          ( ( divide_divide(A,B3,numeral_numeral(A,W2)) = A3 )
        <=> $ite(numeral_numeral(A,W2) != zero_zero(A),B3 = aa(A,A,aa(A,fun(A,A),times_times(A),A3),numeral_numeral(A,W2)),A3 = zero_zero(A)) ) ) ).

% divide_eq_eq_numeral1(1)
tff(fact_2510_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B3: A,W2: num] :
          ( ( A3 = divide_divide(A,B3,numeral_numeral(A,W2)) )
        <=> $ite(numeral_numeral(A,W2) != zero_zero(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),numeral_numeral(A,W2)) = B3,A3 = zero_zero(A)) ) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_2511_less__divide__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A,W2: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),divide_divide(A,B3,numeral_numeral(A,W2)))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),numeral_numeral(A,W2))),B3) ) ) ).

% less_divide_eq_numeral1(1)
tff(fact_2512_divide__less__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B3: A,W2: num,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,B3,numeral_numeral(A,W2))),A3)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),aa(A,A,aa(A,fun(A,A),times_times(A),A3),numeral_numeral(A,W2))) ) ) ).

% divide_less_eq_numeral1(1)
tff(fact_2513_nth__Cons__numeral,axiom,
    ! [A: $tType,Xa: A,Xsa: list(A),V2: num] : ( aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa)),numeral_numeral(nat,V2)) = aa(nat,A,nth(A,Xsa),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),numeral_numeral(nat,V2)),one_one(nat))) ) ).

% nth_Cons_numeral
tff(fact_2514_numeral__less__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V2: num,Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),numeral_numeral(int,V2)),archimedean_ceiling(A,Xa))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),numeral_numeral(A,V2)),Xa) ) ) ).

% numeral_less_ceiling
tff(fact_2515_take__Cons__numeral,axiom,
    ! [A: $tType,V2: num,Xa: A,Xsa: list(A)] : ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),numeral_numeral(nat,V2)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),numeral_numeral(nat,V2)),one_one(nat))),Xsa)) ) ).

% take_Cons_numeral
tff(fact_2516_drop__Cons__numeral,axiom,
    ! [A: $tType,V2: num,Xa: A,Xsa: list(A)] : ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),numeral_numeral(nat,V2)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa)) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),numeral_numeral(nat,V2)),one_one(nat))),Xsa) ) ).

% drop_Cons_numeral
tff(fact_2517_ceiling__diff__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,V2: num] : ( archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Xa),numeral_numeral(A,V2))) = aa(int,int,aa(int,fun(int,int),minus_minus(int),archimedean_ceiling(A,Xa)),numeral_numeral(int,V2)) ) ) ).

% ceiling_diff_numeral
tff(fact_2518_Sup__greaterThanAtLeast,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),top_top(A))
         => ( aa(set(A),A,complete_Sup_Sup(A),aa(A,set(A),set_ord_greaterThan(A),Xa)) = top_top(A) ) ) ) ).

% Sup_greaterThanAtLeast
tff(fact_2519_lexord__same__pref__if__irrefl,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),Xsa: list(A),Ys: list(A),Zs: list(A)] :
      ( irrefl(A,R2)
     => ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),Ys)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),Zs)),lexord(A,R2))
      <=> member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Ys),Zs),lexord(A,R2)) ) ) ).

% lexord_same_pref_if_irrefl
tff(fact_2520_le__divide__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A,W2: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),divide_divide(A,B3,aa(A,A,uminus_uminus(A),numeral_numeral(A,W2))))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,uminus_uminus(A),numeral_numeral(A,W2)))) ) ) ).

% le_divide_eq_numeral1(2)
tff(fact_2521_divide__le__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B3: A,W2: num,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,B3,aa(A,A,uminus_uminus(A),numeral_numeral(A,W2)))),A3)
        <=> aa(A,$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),numeral_numeral(A,W2)))),B3) ) ) ).

% divide_le_eq_numeral1(2)
tff(fact_2522_divide__eq__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B3: A,W2: num,A3: A] :
          ( ( divide_divide(A,B3,aa(A,A,uminus_uminus(A),numeral_numeral(A,W2))) = A3 )
        <=> $ite(aa(A,A,uminus_uminus(A),numeral_numeral(A,W2)) != zero_zero(A),B3 = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,uminus_uminus(A),numeral_numeral(A,W2))),A3 = zero_zero(A)) ) ) ).

% divide_eq_eq_numeral1(2)
tff(fact_2523_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B3: A,W2: num] :
          ( ( A3 = divide_divide(A,B3,aa(A,A,uminus_uminus(A),numeral_numeral(A,W2))) )
        <=> $ite(aa(A,A,uminus_uminus(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),numeral_numeral(A,W2))) = B3,A3 = zero_zero(A)) ) ) ).

% eq_divide_eq_numeral1(2)
tff(fact_2524_less__divide__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A,W2: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),divide_divide(A,B3,aa(A,A,uminus_uminus(A),numeral_numeral(A,W2))))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,uminus_uminus(A),numeral_numeral(A,W2)))) ) ) ).

% less_divide_eq_numeral1(2)
tff(fact_2525_divide__less__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B3: A,W2: num,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,B3,aa(A,A,uminus_uminus(A),numeral_numeral(A,W2)))),A3)
        <=> aa(A,$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),numeral_numeral(A,W2)))),B3) ) ) ).

% divide_less_eq_numeral1(2)
tff(fact_2526_zero__less__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),archimedean_ceiling(A,Xa))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xa) ) ) ).

% zero_less_ceiling
tff(fact_2527_one__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),one_one(int)),archimedean_ceiling(A,Xa))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xa) ) ) ).

% one_le_ceiling
tff(fact_2528_one__less__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),one_one(int)),archimedean_ceiling(A,Xa))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),Xa) ) ) ).

% one_less_ceiling
tff(fact_2529_ceiling__diff__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] : ( archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Xa),one_one(A))) = aa(int,int,aa(int,fun(int,int),minus_minus(int),archimedean_ceiling(A,Xa)),one_one(int)) ) ) ).

% ceiling_diff_one
tff(fact_2530_ceiling__less__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,V2: num] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archimedean_ceiling(A,Xa)),numeral_numeral(int,V2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(A,A,aa(A,fun(A,A),minus_minus(A),numeral_numeral(A,V2)),one_one(A))) ) ) ).

% ceiling_less_numeral
tff(fact_2531_numeral__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V2: num,Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),numeral_numeral(int,V2)),archimedean_ceiling(A,Xa))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),numeral_numeral(A,V2)),one_one(A))),Xa) ) ) ).

% numeral_le_ceiling
tff(fact_2532_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Ia: num,N: nat,Xa: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),numeral_numeral(A,Ia)),N)),aa(nat,A,semiring_1_of_nat(A),Xa))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),numeral_numeral(nat,Ia)),N)),Xa) ) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_2533_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xa: nat,Ia: num,N: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,semiring_1_of_nat(A),Xa)),aa(nat,A,aa(A,fun(nat,A),power_power(A),numeral_numeral(A,Ia)),N))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),numeral_numeral(nat,Ia)),N)) ) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_2534_neg__numeral__less__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V2: num,Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,uminus_uminus(int),numeral_numeral(int,V2))),archimedean_ceiling(A,Xa))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),numeral_numeral(A,V2))),Xa) ) ) ).

% neg_numeral_less_ceiling
tff(fact_2535_ceiling__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,V2: num] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archimedean_ceiling(A,Xa)),aa(int,int,uminus_uminus(int),numeral_numeral(int,V2)))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),numeral_numeral(A,V2))),one_one(A))) ) ) ).

% ceiling_less_neg_numeral
tff(fact_2536_neg__numeral__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V2: num,Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),numeral_numeral(int,V2))),archimedean_ceiling(A,Xa))
        <=> aa(A,$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),numeral_numeral(A,V2))),one_one(A))),Xa) ) ) ).

% neg_numeral_le_ceiling
tff(fact_2537_ComplD,axiom,
    ! [A: $tType,C2: A,Aa2: set(A)] :
      ( member(A,C2,aa(set(A),set(A),uminus_uminus(set(A)),Aa2))
     => ~ member(A,C2,Aa2) ) ).

% ComplD
tff(fact_2538_double__complement,axiom,
    ! [A: $tType,Aa2: set(A)] : ( aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),Aa2)) = Aa2 ) ).

% double_complement
tff(fact_2539_lexord__irrefl,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( irrefl(A,R)
     => irrefl(list(A),lexord(A,R)) ) ).

% lexord_irrefl
tff(fact_2540_irrefl__lex,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( irrefl(A,R2)
     => irrefl(list(A),lex(A,R2)) ) ).

% irrefl_lex
tff(fact_2541_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)),aa(list(set(A)),list(set(A)),aa(nat,fun(list(set(A)),list(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)),aa(list(set(A)),list(set(A)),aa(nat,fun(list(set(A)),list(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_2542_irreflI,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( ! [A4: A] : ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),A4),R)
     => irrefl(A,R) ) ).

% irreflI
tff(fact_2543_irrefl__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( irrefl(A,R2)
    <=> ! [A7: A] : ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A7),A7),R2) ) ).

% irrefl_def
tff(fact_2544_cSup__least,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X3: set(A),Z2: A] :
          ( ( X3 != bot_bot(set(A)) )
         => ( ! [X: A] :
                ( member(A,X,X3)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Z2) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),X3)),Z2) ) ) ) ).

% cSup_least
tff(fact_2545_cSup__eq__non__empty,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X3: set(A),A3: A] :
          ( ( X3 != bot_bot(set(A)) )
         => ( ! [X: A] :
                ( member(A,X,X3)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),A3) )
           => ( ! [Y2: A] :
                  ( ! [X4: A] :
                      ( member(A,X4,X3)
                     => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),Y2) )
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),Y2) )
             => ( aa(set(A),A,complete_Sup_Sup(A),X3) = A3 ) ) ) ) ) ).

% cSup_eq_non_empty
tff(fact_2546_less__cSupE,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [Y: A,X3: set(A)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),aa(set(A),A,complete_Sup_Sup(A),X3))
         => ( ( X3 != bot_bot(set(A)) )
           => ~ ! [X: A] :
                  ( member(A,X,X3)
                 => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),X) ) ) ) ) ).

% less_cSupE
tff(fact_2547_less__cSupD,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X3: set(A),Z2: A] :
          ( ( X3 != bot_bot(set(A)) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z2),aa(set(A),A,complete_Sup_Sup(A),X3))
           => ? [X: A] :
                ( member(A,X,X3)
                & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z2),X) ) ) ) ) ).

% less_cSupD
tff(fact_2548_Sup__inf__eq__bot__iff,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [B2: set(A),A3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Sup_Sup(A),B2)),A3) = bot_bot(A) )
        <=> ! [X6: A] :
              ( member(A,X6,B2)
             => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X6),A3) = bot_bot(A) ) ) ) ) ).

% Sup_inf_eq_bot_iff
tff(fact_2549_not__numeral__less__zero,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [N: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),numeral_numeral(A,N)),zero_zero(A)) ) ).

% not_numeral_less_zero
tff(fact_2550_zero__less__numeral,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [N: num] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),numeral_numeral(A,N)) ) ).

% zero_less_numeral
tff(fact_2551_ceiling__less__cancel,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,Y: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archimedean_ceiling(A,Xa)),archimedean_ceiling(A,Y))
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y) ) ) ).

% ceiling_less_cancel
tff(fact_2552_not__numeral__less__one,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [N: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),numeral_numeral(A,N)),one_one(A)) ) ).

% not_numeral_less_one
tff(fact_2553_not__numeral__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M2: num,N: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),numeral_numeral(A,M2)),aa(A,A,uminus_uminus(A),numeral_numeral(A,N))) ) ).

% not_numeral_less_neg_numeral
tff(fact_2554_neg__numeral__less__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M2: num,N: num] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),numeral_numeral(A,M2))),numeral_numeral(A,N)) ) ).

% neg_numeral_less_numeral
tff(fact_2555_numeral__times__minus__swap,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [W2: num,Xa: A] : ( aa(A,A,aa(A,fun(A,A),times_times(A),numeral_numeral(A,W2)),aa(A,A,uminus_uminus(A),Xa)) = aa(A,A,aa(A,fun(A,A),times_times(A),Xa),aa(A,A,uminus_uminus(A),numeral_numeral(A,W2))) ) ) ).

% numeral_times_minus_swap
tff(fact_2556_cSup__abs__le,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & linordered_idom(A) )
     => ! [S: set(A),A3: A] :
          ( ( S != bot_bot(set(A)) )
         => ( ! [X: A] :
                ( member(A,X,S)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),abs_abs(A,X)),A3) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),abs_abs(A,aa(set(A),A,complete_Sup_Sup(A),S))),A3) ) ) ) ).

% cSup_abs_le
tff(fact_2557_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),numeral_numeral(A,N))),zero_zero(A)) ) ).

% neg_numeral_less_zero
tff(fact_2558_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),numeral_numeral(A,N))) ) ).

% not_zero_less_neg_numeral
tff(fact_2559_divide__eq__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B3: A,C2: A,W2: num] :
          ( ( divide_divide(A,B3,C2) = numeral_numeral(A,W2) )
        <=> $ite(C2 != zero_zero(A),B3 = aa(A,A,aa(A,fun(A,A),times_times(A),numeral_numeral(A,W2)),C2),numeral_numeral(A,W2) = zero_zero(A)) ) ) ).

% divide_eq_eq_numeral(1)
tff(fact_2560_eq__divide__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [W2: num,B3: A,C2: A] :
          ( ( numeral_numeral(A,W2) = divide_divide(A,B3,C2) )
        <=> $ite(C2 != zero_zero(A),aa(A,A,aa(A,fun(A,A),times_times(A),numeral_numeral(A,W2)),C2) = B3,numeral_numeral(A,W2) = zero_zero(A)) ) ) ).

% eq_divide_eq_numeral(1)
tff(fact_2561_not__neg__one__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M2: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(A,A,uminus_uminus(A),numeral_numeral(A,M2))) ) ).

% not_neg_one_less_neg_numeral
tff(fact_2562_not__one__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M2: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(A,A,uminus_uminus(A),numeral_numeral(A,M2))) ) ).

% not_one_less_neg_numeral
tff(fact_2563_not__numeral__less__neg__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M2: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),numeral_numeral(A,M2)),aa(A,A,uminus_uminus(A),one_one(A))) ) ).

% not_numeral_less_neg_one
tff(fact_2564_neg__one__less__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M2: num] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),one_one(A))),numeral_numeral(A,M2)) ) ).

% neg_one_less_numeral
tff(fact_2565_neg__numeral__less__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M2: num] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),numeral_numeral(A,M2))),one_one(A)) ) ).

% neg_numeral_less_one
tff(fact_2566_lexl__not__refl,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),Xa: list(A)] :
      ( irrefl(A,R2)
     => ~ member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xa),Xa),lex(A,R2)) ) ).

% lexl_not_refl
tff(fact_2567_irrefl__diff__Id,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : irrefl(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))) ).

% irrefl_diff_Id
tff(fact_2568_frame__inference__finalize,axiom,
    ! [M5: list(product_prod(assn,assn)),Fa: assn] : fI_RESULT(M5,Fa,one_one(assn),Fa) ).

% frame_inference_finalize
tff(fact_2569_entails__solve__finalize_I2_J,axiom,
    ! [M5: list(product_prod(assn,assn))] : fI_RESULT(M5,one_one(assn),one_one(assn),one_one(assn)) ).

% entails_solve_finalize(2)
tff(fact_2570_cSup__asclose,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & linordered_idom(A) )
     => ! [S: set(A),L: A,E3: A] :
          ( ( S != bot_bot(set(A)) )
         => ( ! [X: A] :
                ( member(A,X,S)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),abs_abs(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X),L))),E3) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(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))),E3) ) ) ) ).

% cSup_asclose
tff(fact_2571_divide__less__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B3: A,C2: A,W2: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,B3,C2)),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),B3),aa(A,A,aa(A,fun(A,A),times_times(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,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),numeral_numeral(A,W2)),C2)),B3),aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),numeral_numeral(A,W2))) ) ) ) ).

% divide_less_eq_numeral(1)
tff(fact_2572_less__divide__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [W2: num,B3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),numeral_numeral(A,W2)),divide_divide(A,B3,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),numeral_numeral(A,W2)),C2)),B3),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),aa(A,A,aa(A,fun(A,A),times_times(A),numeral_numeral(A,W2)),C2)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),numeral_numeral(A,W2)),zero_zero(A))) ) ) ) ).

% less_divide_eq_numeral(1)
tff(fact_2573_eq__divide__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [W2: num,B3: A,C2: A] :
          ( ( aa(A,A,uminus_uminus(A),numeral_numeral(A,W2)) = divide_divide(A,B3,C2) )
        <=> $ite(C2 != zero_zero(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),numeral_numeral(A,W2))),C2) = B3,aa(A,A,uminus_uminus(A),numeral_numeral(A,W2)) = zero_zero(A)) ) ) ).

% eq_divide_eq_numeral(2)
tff(fact_2574_divide__eq__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B3: A,C2: A,W2: num] :
          ( ( divide_divide(A,B3,C2) = aa(A,A,uminus_uminus(A),numeral_numeral(A,W2)) )
        <=> $ite(C2 != zero_zero(A),B3 = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),numeral_numeral(A,W2))),C2),aa(A,A,uminus_uminus(A),numeral_numeral(A,W2)) = zero_zero(A)) ) ) ).

% divide_eq_eq_numeral(2)
tff(fact_2575_mult__ceiling__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B3)
           => aa(int,$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),B3))),aa(int,int,aa(int,fun(int,int),times_times(int),archimedean_ceiling(A,A3)),archimedean_ceiling(A,B3))) ) ) ) ).

% mult_ceiling_le
tff(fact_2576_le__divide__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [W2: num,B3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),numeral_numeral(A,W2)),divide_divide(A,B3,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),numeral_numeral(A,W2)),C2)),B3),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),aa(A,A,aa(A,fun(A,A),times_times(A),numeral_numeral(A,W2)),C2)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),numeral_numeral(A,W2)),zero_zero(A))) ) ) ) ).

% le_divide_eq_numeral(1)
tff(fact_2577_divide__le__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B3: A,C2: A,W2: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,B3,C2)),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),B3),aa(A,A,aa(A,fun(A,A),times_times(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,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),numeral_numeral(A,W2)),C2)),B3),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),numeral_numeral(A,W2))) ) ) ) ).

% divide_le_eq_numeral(1)
tff(fact_2578_divide__less__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B3: A,C2: A,W2: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,B3,C2)),aa(A,A,uminus_uminus(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),B3),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(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,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),numeral_numeral(A,W2))),C2)),B3),aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,uminus_uminus(A),numeral_numeral(A,W2)))) ) ) ) ).

% divide_less_eq_numeral(2)
tff(fact_2579_less__divide__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [W2: num,B3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),numeral_numeral(A,W2))),divide_divide(A,B3,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),numeral_numeral(A,W2))),C2)),B3),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),numeral_numeral(A,W2))),C2)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),numeral_numeral(A,W2))),zero_zero(A))) ) ) ) ).

% less_divide_eq_numeral(2)
tff(fact_2580_distrib__right__NO__MATCH,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring(B)
     => ! [Xa: A,Y: A,C2: B,A3: B,B3: B] :
          ( nO_MATCH(A,B,divide_divide(A,Xa,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),B3)),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),B3),C2)) ) ) ) ).

% distrib_right_NO_MATCH
tff(fact_2581_distrib__left__NO__MATCH,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring(B)
     => ! [Xa: A,Y: A,A3: B,B3: B,C2: B] :
          ( nO_MATCH(A,B,divide_divide(A,Xa,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),B3),C2)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),A3),B3)),aa(B,B,aa(B,fun(B,B),times_times(B),A3),C2)) ) ) ) ).

% distrib_left_NO_MATCH
tff(fact_2582_right__diff__distrib__NO__MATCH,axiom,
    ! [A: $tType,B: $tType] :
      ( ring(B)
     => ! [Xa: A,Y: A,A3: B,B3: B,C2: B] :
          ( nO_MATCH(A,B,divide_divide(A,Xa,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),B3),C2)) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),times_times(B),A3),B3)),aa(B,B,aa(B,fun(B,B),times_times(B),A3),C2)) ) ) ) ).

% right_diff_distrib_NO_MATCH
tff(fact_2583_left__diff__distrib__NO__MATCH,axiom,
    ! [A: $tType,B: $tType] :
      ( ring(B)
     => ! [Xa: A,Y: A,C2: B,A3: B,B3: B] :
          ( nO_MATCH(A,B,divide_divide(A,Xa,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),B3)),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),B3),C2)) ) ) ) ).

% left_diff_distrib_NO_MATCH
tff(fact_2584_le__divide__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [W2: num,B3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),numeral_numeral(A,W2))),divide_divide(A,B3,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),numeral_numeral(A,W2))),C2)),B3),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),numeral_numeral(A,W2))),C2)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),numeral_numeral(A,W2))),zero_zero(A))) ) ) ) ).

% le_divide_eq_numeral(2)
tff(fact_2585_Sup__insert,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: A,Aa2: set(A)] : ( aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),Aa2)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),aa(set(A),A,complete_Sup_Sup(A),Aa2)) ) ) ).

% Sup_insert
tff(fact_2586_ccpo__Sup__singleton,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [Xa: A] : ( aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A)))) = Xa ) ) ).

% ccpo_Sup_singleton
tff(fact_2587_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_2588_Sup__eq__top__iff,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [Aa2: set(A)] :
          ( ( aa(set(A),A,complete_Sup_Sup(A),Aa2) = top_top(A) )
        <=> ! [X6: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X6),top_top(A))
             => ? [Xa2: A] :
                  ( member(A,Xa2,Aa2)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X6),Xa2) ) ) ) ) ).

% Sup_eq_top_iff
tff(fact_2589_Sup__bot__conv_I1_J,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Aa2: set(A)] :
          ( ( aa(set(A),A,complete_Sup_Sup(A),Aa2) = bot_bot(A) )
        <=> ! [X6: A] :
              ( member(A,X6,Aa2)
             => ( X6 = bot_bot(A) ) ) ) ) ).

% Sup_bot_conv(1)
tff(fact_2590_Sup__bot__conv_I2_J,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Aa2: set(A)] :
          ( ( bot_bot(A) = aa(set(A),A,complete_Sup_Sup(A),Aa2) )
        <=> ! [X6: A] :
              ( member(A,X6,Aa2)
             => ( X6 = bot_bot(A) ) ) ) ) ).

% Sup_bot_conv(2)
tff(fact_2591_Sup__nat__empty,axiom,
    aa(set(nat),nat,complete_Sup_Sup(nat),bot_bot(set(nat))) = zero_zero(nat) ).

% Sup_nat_empty
tff(fact_2592_less__Sup__iff,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [A3: A,S: set(A)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(set(A),A,complete_Sup_Sup(A),S))
        <=> ? [X6: A] :
              ( member(A,X6,S)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),X6) ) ) ) ).

% less_Sup_iff
tff(fact_2593_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_2594_Union__empty__conv,axiom,
    ! [A: $tType,Aa2: set(set(A))] :
      ( ( aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),Aa2) = bot_bot(set(A)) )
    <=> ! [X6: set(A)] :
          ( member(set(A),X6,Aa2)
         => ( X6 = bot_bot(set(A)) ) ) ) ).

% Union_empty_conv
tff(fact_2595_empty__Union__conv,axiom,
    ! [A: $tType,Aa2: set(set(A))] :
      ( ( bot_bot(set(A)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),Aa2) )
    <=> ! [X6: set(A)] :
          ( member(set(A),X6,Aa2)
         => ( X6 = bot_bot(set(A)) ) ) ) ).

% empty_Union_conv
tff(fact_2596_le__Sup__iff,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [Xa: A,Aa2: set(A)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(set(A),A,complete_Sup_Sup(A),Aa2))
        <=> ! [Y4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y4),Xa)
             => ? [X6: A] :
                  ( member(A,X6,Aa2)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y4),X6) ) ) ) ) ).

% le_Sup_iff
tff(fact_2597_less__eq__Sup,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Aa2: set(A),U: A] :
          ( ! [V3: A] :
              ( member(A,V3,Aa2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),U),V3) )
         => ( ( Aa2 != bot_bot(set(A)) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),U),aa(set(A),A,complete_Sup_Sup(A),Aa2)) ) ) ) ).

% less_eq_Sup
tff(fact_2598_Union__disjoint,axiom,
    ! [A: $tType,C3: set(set(A)),Aa2: 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)),Aa2) = bot_bot(set(A)) )
    <=> ! [X6: set(A)] :
          ( member(set(A),X6,C3)
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),X6),Aa2) = bot_bot(set(A)) ) ) ) ).

% Union_disjoint
tff(fact_2599_insert__partition,axiom,
    ! [A: $tType,Xa: set(A),Fa: set(set(A))] :
      ( ~ member(set(A),Xa,Fa)
     => ( ! [X: set(A)] :
            ( member(set(A),X,aa(set(set(A)),set(set(A)),aa(set(A),fun(set(set(A)),set(set(A))),insert3(set(A)),Xa),Fa))
           => ! [Xa5: set(A)] :
                ( member(set(A),Xa5,aa(set(set(A)),set(set(A)),aa(set(A),fun(set(set(A)),set(set(A))),insert3(set(A)),Xa),Fa))
               => ( ( X != Xa5 )
                 => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),X),Xa5) = bot_bot(set(A)) ) ) ) )
       => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Xa),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),Fa)) = bot_bot(set(A)) ) ) ) ).

% insert_partition
tff(fact_2600_Sup__finite__insert,axiom,
    ! [A: $tType] :
      ( finite_lattice(A)
     => ! [A3: A,Aa2: set(A)] : ( aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),Aa2)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),aa(set(A),A,complete_Sup_Sup(A),Aa2)) ) ) ).

% Sup_finite_insert
tff(fact_2601_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: int,Xa: num,N: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(int,A,ring_1_of_int(A),A3)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),numeral_numeral(A,Xa))),N))
        <=> aa(int,$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),numeral_numeral(int,Xa))),N)) ) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_2602_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: num,N: nat,A3: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),numeral_numeral(A,Xa))),N)),aa(int,A,ring_1_of_int(A),A3))
        <=> aa(int,$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),numeral_numeral(int,Xa))),N)),A3) ) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_2603_floor__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,V2: num] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archim6421214686448440834_floor(A,Xa)),aa(int,int,uminus_uminus(int),numeral_numeral(int,V2)))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),numeral_numeral(A,V2))),one_one(A))) ) ) ).

% floor_le_neg_numeral
tff(fact_2604_upto__rec__numeral_I2_J,axiom,
    ! [M2: num,N: num] :
      ( upto(numeral_numeral(int,M2),aa(int,int,uminus_uminus(int),numeral_numeral(int,N))) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),numeral_numeral(int,M2)),aa(int,int,uminus_uminus(int),numeral_numeral(int,N))),aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),numeral_numeral(int,M2)),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),numeral_numeral(int,M2)),one_one(int)),aa(int,int,uminus_uminus(int),numeral_numeral(int,N)))),nil(int)) ) ).

% upto_rec_numeral(2)
tff(fact_2605_of__int__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [W2: int,Z2: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(int,A,ring_1_of_int(A),W2)),aa(int,A,ring_1_of_int(A),Z2))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),W2),Z2) ) ) ).

% of_int_less_iff
tff(fact_2606_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_2607_of__int__diff,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),minus_minus(int),W2),Z2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),W2)),aa(int,A,ring_1_of_int(A),Z2)) ) ) ).

% of_int_diff
tff(fact_2608_upto__empty,axiom,
    ! [J: int,Ia: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),J),Ia)
     => ( upto(Ia,J) = nil(int) ) ) ).

% upto_empty
tff(fact_2609_upto__Nil2,axiom,
    ! [Ia: int,J: int] :
      ( ( nil(int) = upto(Ia,J) )
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),J),Ia) ) ).

% upto_Nil2
tff(fact_2610_upto__Nil,axiom,
    ! [Ia: int,J: int] :
      ( ( upto(Ia,J) = nil(int) )
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),J),Ia) ) ).

% upto_Nil
tff(fact_2611_upto__single,axiom,
    ! [Ia: int] : ( upto(Ia,Ia) = aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),Ia),nil(int)) ) ).

% upto_single
tff(fact_2612_floor__diff__of__int,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,Z2: int] : ( archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Xa),aa(int,A,ring_1_of_int(A),Z2))) = aa(int,int,aa(int,fun(int,int),minus_minus(int),archim6421214686448440834_floor(A,Xa)),Z2) ) ) ).

% floor_diff_of_int
tff(fact_2613_ceiling__diff__of__int,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,Z2: int] : ( archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Xa),aa(int,A,ring_1_of_int(A),Z2))) = aa(int,int,aa(int,fun(int,int),minus_minus(int),archimedean_ceiling(A,Xa)),Z2) ) ) ).

% ceiling_diff_of_int
tff(fact_2614_nth__upto,axiom,
    ! [Ia: int,K3: nat,J: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),Ia),aa(nat,int,semiring_1_of_nat(int),K3))),J)
     => ( aa(nat,int,nth(int,upto(Ia,J)),K3) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Ia),aa(nat,int,semiring_1_of_nat(int),K3)) ) ) ).

% nth_upto
tff(fact_2615_of__int__less__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z2: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(int,A,ring_1_of_int(A),Z2)),zero_zero(A))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z2),zero_zero(int)) ) ) ).

% of_int_less_0_iff
tff(fact_2616_of__int__0__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z2: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(int,A,ring_1_of_int(A),Z2))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Z2) ) ) ).

% of_int_0_less_iff
tff(fact_2617_of__int__numeral__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: num,Z2: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),numeral_numeral(A,N)),aa(int,A,ring_1_of_int(A),Z2))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),numeral_numeral(int,N)),Z2) ) ) ).

% of_int_numeral_less_iff
tff(fact_2618_of__int__less__numeral__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z2: int,N: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(int,A,ring_1_of_int(A),Z2)),numeral_numeral(A,N))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z2),numeral_numeral(int,N)) ) ) ).

% of_int_less_numeral_iff
tff(fact_2619_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_2620_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_2621_floor__less__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archim6421214686448440834_floor(A,Xa)),zero_zero(int))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),zero_zero(A)) ) ) ).

% floor_less_zero
tff(fact_2622_floor__le__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archim6421214686448440834_floor(A,Xa)),zero_zero(int))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),one_one(A)) ) ) ).

% floor_le_zero
tff(fact_2623_floor__less__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,V2: num] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archim6421214686448440834_floor(A,Xa)),numeral_numeral(int,V2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),numeral_numeral(A,V2)) ) ) ).

% floor_less_numeral
tff(fact_2624_floor__less__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archim6421214686448440834_floor(A,Xa)),one_one(int))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),one_one(A)) ) ) ).

% floor_less_one
tff(fact_2625_floor__diff__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,V2: num] : ( archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Xa),numeral_numeral(A,V2))) = aa(int,int,aa(int,fun(int,int),minus_minus(int),archim6421214686448440834_floor(A,Xa)),numeral_numeral(int,V2)) ) ) ).

% floor_diff_numeral
tff(fact_2626_floor__diff__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] : ( archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Xa),one_one(A))) = aa(int,int,aa(int,fun(int,int),minus_minus(int),archim6421214686448440834_floor(A,Xa)),one_one(int)) ) ) ).

% floor_diff_one
tff(fact_2627_of__int__power__less__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: int,B3: int,W2: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(int,A,ring_1_of_int(A),Xa)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(int,A,ring_1_of_int(A),B3)),W2))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),Xa),aa(nat,int,aa(int,fun(nat,int),power_power(int),B3),W2)) ) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_2628_of__int__less__of__int__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [B3: int,W2: nat,Xa: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(int,A,ring_1_of_int(A),B3)),W2)),aa(int,A,ring_1_of_int(A),Xa))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),B3),W2)),Xa) ) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_2629_floor__le__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,V2: num] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archim6421214686448440834_floor(A,Xa)),numeral_numeral(int,V2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),aa(A,A,aa(A,fun(A,A),plus_plus(A),numeral_numeral(A,V2)),one_one(A))) ) ) ).

% floor_le_numeral
tff(fact_2630_floor__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,V2: num] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archim6421214686448440834_floor(A,Xa)),aa(int,int,uminus_uminus(int),numeral_numeral(int,V2)))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),aa(A,A,uminus_uminus(A),numeral_numeral(A,V2))) ) ) ).

% floor_less_neg_numeral
tff(fact_2631_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: int,Xa: num,N: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(int,A,ring_1_of_int(A),A3)),aa(nat,A,aa(A,fun(nat,A),power_power(A),numeral_numeral(A,Xa)),N))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),A3),aa(nat,int,aa(int,fun(nat,int),power_power(int),numeral_numeral(int,Xa)),N)) ) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_2632_numeral__power__less__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: num,N: nat,A3: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),numeral_numeral(A,Xa)),N)),aa(int,A,ring_1_of_int(A),A3))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),numeral_numeral(int,Xa)),N)),A3) ) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_2633_upto__rec__numeral_I1_J,axiom,
    ! [M2: num,N: num] :
      ( upto(numeral_numeral(int,M2),numeral_numeral(int,N)) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),numeral_numeral(int,M2)),numeral_numeral(int,N)),aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),numeral_numeral(int,M2)),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),numeral_numeral(int,M2)),one_one(int)),numeral_numeral(int,N))),nil(int)) ) ).

% upto_rec_numeral(1)
tff(fact_2634_upto__rec__numeral_I4_J,axiom,
    ! [M2: num,N: num] :
      ( upto(aa(int,int,uminus_uminus(int),numeral_numeral(int,M2)),aa(int,int,uminus_uminus(int),numeral_numeral(int,N))) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),numeral_numeral(int,M2))),aa(int,int,uminus_uminus(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),numeral_numeral(int,M2))),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,uminus_uminus(int),numeral_numeral(int,M2))),one_one(int)),aa(int,int,uminus_uminus(int),numeral_numeral(int,N)))),nil(int)) ) ).

% upto_rec_numeral(4)
tff(fact_2635_upto__rec__numeral_I3_J,axiom,
    ! [M2: num,N: num] :
      ( upto(aa(int,int,uminus_uminus(int),numeral_numeral(int,M2)),numeral_numeral(int,N)) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),numeral_numeral(int,M2))),numeral_numeral(int,N)),aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),aa(int,int,uminus_uminus(int),numeral_numeral(int,M2))),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,uminus_uminus(int),numeral_numeral(int,M2))),one_one(int)),numeral_numeral(int,N))),nil(int)) ) ).

% upto_rec_numeral(3)
tff(fact_2636_floor__less__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,Z2: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archim6421214686448440834_floor(A,Xa)),Z2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),aa(int,A,ring_1_of_int(A),Z2)) ) ) ).

% floor_less_iff
tff(fact_2637_distinct__upto,axiom,
    ! [Ia: int,J: int] : aa(list(int),$o,distinct(int),upto(Ia,J)) ).

% distinct_upto
tff(fact_2638_ex__of__int__less,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xa: A] :
        ? [Z3: int] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(int,A,ring_1_of_int(A),Z3)),Xa) ) ).

% ex_of_int_less
tff(fact_2639_ex__less__of__int,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xa: A] :
        ? [Z3: int] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),aa(int,A,ring_1_of_int(A),Z3)) ) ).

% ex_less_of_int
tff(fact_2640_mult__of__int__commute,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Xa: int,Y: A] : ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(int,A,ring_1_of_int(A),Xa)),Y) = aa(A,A,aa(A,fun(A,A),times_times(A),Y),aa(int,A,ring_1_of_int(A),Xa)) ) ) ).

% mult_of_int_commute
tff(fact_2641_sorted__upto,axiom,
    ! [M2: int,N: int] : sorted_wrt(int,ord_less_eq(int),upto(M2,N)) ).

% sorted_upto
tff(fact_2642_sorted__wrt__upto,axiom,
    ! [Ia: int,J: int] : sorted_wrt(int,ord_less(int),upto(Ia,J)) ).

% sorted_wrt_upto
tff(fact_2643_floor__unique,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z2: int,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z2)),Xa)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),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,Xa) = Z2 ) ) ) ) ).

% floor_unique
tff(fact_2644_floor__eq__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,A3: int] :
          ( ( archim6421214686448440834_floor(A,Xa) = A3 )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),A3)),Xa)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),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_2645_floor__split,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Pa: fun(int,$o),T2: A] :
          ( aa(int,$o,Pa,archim6421214686448440834_floor(A,T2))
        <=> ! [I2: int] :
              ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),I2)),T2)
                & aa(A,$o,aa(A,fun(A,$o),ord_less(A),T2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),I2)),one_one(A))) )
             => aa(int,$o,Pa,I2) ) ) ) ).

% floor_split
tff(fact_2646_floor__le__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,Z2: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archim6421214686448440834_floor(A,Xa)),Z2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),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_2647_floor__correct,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,Xa))),Xa)
          & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,Xa)),one_one(int)))) ) ) ).

% floor_correct
tff(fact_2648_upto__aux__def,axiom,
    ! [Ia: int,J: int,Js: list(int)] : ( upto_aux(Ia,J,Js) = aa(list(int),list(int),aa(list(int),fun(list(int),list(int)),append(int),upto(Ia,J)),Js) ) ).

% upto_aux_def
tff(fact_2649_upto__code,axiom,
    ! [Ia: int,J: int] : ( upto(Ia,J) = upto_aux(Ia,J,nil(int)) ) ).

% upto_code
tff(fact_2650_floor__less__cancel,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,Y: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archim6421214686448440834_floor(A,Xa)),archim6421214686448440834_floor(A,Y))
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y) ) ) ).

% floor_less_cancel
tff(fact_2651_floor__divide__lower,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q2: A,P2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Q2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,divide_divide(A,P2,Q2)))),Q2)),P2) ) ) ).

% floor_divide_lower
tff(fact_2652_floor__divide__upper,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q2: A,P2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Q2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),P2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,divide_divide(A,P2,Q2)))),one_one(A))),Q2)) ) ) ).

% floor_divide_upper
tff(fact_2653_less__ceiling__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z2: int,Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z2),archimedean_ceiling(A,Xa))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(int,A,ring_1_of_int(A),Z2)),Xa) ) ) ).

% less_ceiling_iff
tff(fact_2654_upto__split2,axiom,
    ! [Ia: int,J: int,K3: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Ia),J)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),J),K3)
       => ( upto(Ia,K3) = aa(list(int),list(int),aa(list(int),fun(list(int),list(int)),append(int),upto(Ia,J)),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),J),one_one(int)),K3)) ) ) ) ).

% upto_split2
tff(fact_2655_upto__split1,axiom,
    ! [Ia: int,J: int,K3: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Ia),J)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),J),K3)
       => ( upto(Ia,K3) = aa(list(int),list(int),aa(list(int),fun(list(int),list(int)),append(int),upto(Ia,aa(int,int,aa(int,fun(int,int),minus_minus(int),J),one_one(int)))),upto(J,K3)) ) ) ) ).

% upto_split1
tff(fact_2656_of__int__pos,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z2: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Z2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(int,A,ring_1_of_int(A),Z2)) ) ) ).

% of_int_pos
tff(fact_2657_of__int__lessD,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: int,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),abs_abs(A,aa(int,A,ring_1_of_int(A),N))),Xa)
         => ( ( N = zero_zero(int) )
            | aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),Xa) ) ) ) ).

% of_int_lessD
tff(fact_2658_floor__exists1,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xa: A] :
        ? [X: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),X)),Xa)
          & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),plus_plus(int),X),one_one(int))))
          & ! [Y3: int] :
              ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Y3)),Xa)
                & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),plus_plus(int),Y3),one_one(int)))) )
             => ( Y3 = X ) ) ) ) ).

% floor_exists1
tff(fact_2659_floor__exists,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xa: A] :
        ? [Z3: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z3)),Xa)
          & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),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_2660_of__nat__less__of__int__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: nat,Xa: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,semiring_1_of_nat(A),N)),aa(int,A,ring_1_of_int(A),Xa))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(nat,int,semiring_1_of_nat(int),N)),Xa) ) ) ).

% of_nat_less_of_int_iff
tff(fact_2661_le__mult__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B3)
           => aa(int,$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,B3))),archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3))) ) ) ) ).

% le_mult_floor
tff(fact_2662_ceiling__correct,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,Xa))),one_one(A))),Xa)
          & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,Xa))) ) ) ).

% ceiling_correct
tff(fact_2663_ceiling__unique,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z2: int,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),Z2)),one_one(A))),Xa)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(int,A,ring_1_of_int(A),Z2))
           => ( archimedean_ceiling(A,Xa) = Z2 ) ) ) ) ).

% ceiling_unique
tff(fact_2664_ceiling__eq__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,A3: int] :
          ( ( archimedean_ceiling(A,Xa) = A3 )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),A3)),one_one(A))),Xa)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(int,A,ring_1_of_int(A),A3)) ) ) ) ).

% ceiling_eq_iff
tff(fact_2665_ceiling__split,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Pa: fun(int,$o),T2: A] :
          ( aa(int,$o,Pa,archimedean_ceiling(A,T2))
        <=> ! [I2: int] :
              ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),I2)),one_one(A))),T2)
                & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),T2),aa(int,A,ring_1_of_int(A),I2)) )
             => aa(int,$o,Pa,I2) ) ) ) ).

% ceiling_split
tff(fact_2666_upto__rec1,axiom,
    ! [Ia: int,J: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Ia),J)
     => ( upto(Ia,J) = aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),Ia),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),Ia),one_one(int)),J)) ) ) ).

% upto_rec1
tff(fact_2667_upto_Osimps,axiom,
    ! [Ia: int,J: int] :
      ( upto(Ia,J) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Ia),J),aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),Ia),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),Ia),one_one(int)),J)),nil(int)) ) ).

% upto.simps
tff(fact_2668_upto_Oelims,axiom,
    ! [Xa: int,Xaa: int,Y: list(int)] :
      ( ( upto(Xa,Xaa) = Y )
     => ( Y = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Xa),Xaa),aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),Xa),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),Xa),one_one(int)),Xaa)),nil(int)) ) ) ).

% upto.elims
tff(fact_2669_upto__rec2,axiom,
    ! [Ia: int,J: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Ia),J)
     => ( upto(Ia,J) = aa(list(int),list(int),aa(list(int),fun(list(int),list(int)),append(int),upto(Ia,aa(int,int,aa(int,fun(int,int),minus_minus(int),J),one_one(int)))),aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),J),nil(int))) ) ) ).

% upto_rec2
tff(fact_2670_ceiling__less__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,Z2: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archimedean_ceiling(A,Xa)),Z2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),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_2671_le__ceiling__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z2: int,Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Z2),archimedean_ceiling(A,Xa))
        <=> aa(A,$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))),Xa) ) ) ).

% le_ceiling_iff
tff(fact_2672_ceiling__divide__upper,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q2: A,P2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Q2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),P2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,divide_divide(A,P2,Q2)))),Q2)) ) ) ).

% ceiling_divide_upper
tff(fact_2673_upto__split3,axiom,
    ! [Ia: int,J: int,K3: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Ia),J)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),J),K3)
       => ( upto(Ia,K3) = aa(list(int),list(int),aa(list(int),fun(list(int),list(int)),append(int),upto(Ia,aa(int,int,aa(int,fun(int,int),minus_minus(int),J),one_one(int)))),aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),J),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),J),one_one(int)),K3))) ) ) ) ).

% upto_split3
tff(fact_2674_ceiling__divide__lower,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q2: A,P2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Q2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,divide_divide(A,P2,Q2)))),one_one(A))),Q2)),P2) ) ) ).

% ceiling_divide_lower
tff(fact_2675_floor__add,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,Y: A] :
          ( archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),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,Xa)),archimedean_frac(A,Y))),one_one(A)),aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,Xa)),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,Xa)),archim6421214686448440834_floor(A,Y))),one_one(int))) ) ) ).

% floor_add
tff(fact_2676_numeral__power__less__nat__cancel__iff,axiom,
    ! [Xa: num,N: nat,A3: int] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),numeral_numeral(nat,Xa)),N)),nat2(A3))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),numeral_numeral(int,Xa)),N)),A3) ) ).

% numeral_power_less_nat_cancel_iff
tff(fact_2677_nat__less__numeral__power__cancel__iff,axiom,
    ! [A3: int,Xa: num,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),nat2(A3)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),numeral_numeral(nat,Xa)),N))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),A3),aa(nat,int,aa(int,fun(nat,int),power_power(int),numeral_numeral(int,Xa)),N)) ) ).

% nat_less_numeral_power_cancel_iff
tff(fact_2678_mult__ceiling__le__Ints,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(A)
        & linordered_idom(B) )
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
         => ( member(A,A3,ring_1_Ints(A))
           => aa(B,$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),B3)))),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,B3)))) ) ) ) ).

% mult_ceiling_le_Ints
tff(fact_2679_le__mult__floor__Ints,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(A)
        & linordered_idom(B) )
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
         => ( member(A,A3,ring_1_Ints(A))
           => aa(B,$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,B3)))),aa(int,B,ring_1_of_int(B),archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)))) ) ) ) ).

% le_mult_floor_Ints
tff(fact_2680_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),numeral_numeral(A,N))) = numeral_numeral(A,aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),N)) ) ) ).

% diff_numeral_special(3)
tff(fact_2681_zless__nat__conj,axiom,
    ! [W2: int,Z2: int] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),nat2(W2)),nat2(Z2))
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Z2)
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),W2),Z2) ) ) ).

% zless_nat_conj
tff(fact_2682_frac__gt__0__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),archimedean_frac(A,Xa))
        <=> ~ member(A,Xa,ring_1_Ints(A)) ) ) ).

% frac_gt_0_iff
tff(fact_2683_neg__numeral__less__neg__one__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M2: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),numeral_numeral(A,M2))),aa(A,A,uminus_uminus(A),one_one(A)))
        <=> ( M2 != one2 ) ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_2684_zero__less__nat__eq,axiom,
    ! [Z2: int] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),nat2(Z2))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Z2) ) ).

% zero_less_nat_eq
tff(fact_2685_one__less__numeral__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [N: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),numeral_numeral(A,N))
        <=> aa(num,$o,aa(num,fun(num,$o),ord_less(num),one2),N) ) ) ).

% one_less_numeral_iff
tff(fact_2686_length__upto,axiom,
    ! [Ia: int,J: int] : ( aa(list(int),nat,size_size(list(int)),upto(Ia,J)) = nat2(aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),J),Ia)),one_one(int))) ) ).

% length_upto
tff(fact_2687_diff__numeral__special_I4_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M2: num] : ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),numeral_numeral(A,M2))),one_one(A)) = aa(A,A,uminus_uminus(A),numeral_numeral(A,aa(num,num,aa(num,fun(num,num),plus_plus(num),M2),one2))) ) ) ).

% diff_numeral_special(4)
tff(fact_2688_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))),nat2(Z2))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),one_one(int)),Z2) ) ).

% one_less_nat_eq
tff(fact_2689_Ints__mult,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A3: A,B3: A] :
          ( member(A,A3,ring_1_Ints(A))
         => ( member(A,B3,ring_1_Ints(A))
           => member(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3),ring_1_Ints(A)) ) ) ) ).

% Ints_mult
tff(fact_2690_Ints__diff,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A3: A,B3: A] :
          ( member(A,A3,ring_1_Ints(A))
         => ( member(A,B3,ring_1_Ints(A))
           => member(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B3),ring_1_Ints(A)) ) ) ) ).

% Ints_diff
tff(fact_2691_mult__numeral__1__right,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [A3: A] : ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),numeral_numeral(A,one2)) = A3 ) ) ).

% mult_numeral_1_right
tff(fact_2692_mult__numeral__1,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [A3: A] : ( aa(A,A,aa(A,fun(A,A),times_times(A),numeral_numeral(A,one2)),A3) = A3 ) ) ).

% mult_numeral_1
tff(fact_2693_frac__neg,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          ( archimedean_frac(A,aa(A,A,uminus_uminus(A),Xa)) = $ite(member(A,Xa,ring_1_Ints(A)),zero_zero(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),archimedean_frac(A,Xa))) ) ) ).

% frac_neg
tff(fact_2694_frac__lt__1,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),archimedean_frac(A,Xa)),one_one(A)) ) ).

% frac_lt_1
tff(fact_2695_frac__unique__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,A3: A] :
          ( ( archimedean_frac(A,Xa) = A3 )
        <=> ( member(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Xa),A3),ring_1_Ints(A))
            & aa(A,$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_2696_nat__mono__iff,axiom,
    ! [Z2: int,W2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Z2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),nat2(W2)),nat2(Z2))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),W2),Z2) ) ) ).

% nat_mono_iff
tff(fact_2697_zless__nat__eq__int__zless,axiom,
    ! [M2: nat,Z2: int] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),nat2(Z2))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(nat,int,semiring_1_of_nat(int),M2)),Z2) ) ).

% zless_nat_eq_int_zless
tff(fact_2698_mult__1s__ring__1_I2_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [B3: A] : ( aa(A,A,aa(A,fun(A,A),times_times(A),B3),aa(A,A,uminus_uminus(A),numeral_numeral(A,one2))) = aa(A,A,uminus_uminus(A),B3) ) ) ).

% mult_1s_ring_1(2)
tff(fact_2699_mult__1s__ring__1_I1_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [B3: A] : ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),numeral_numeral(A,one2))),B3) = aa(A,A,uminus_uminus(A),B3) ) ) ).

% mult_1s_ring_1(1)
tff(fact_2700_nat__less__eq__zless,axiom,
    ! [W2: int,Z2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),W2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),nat2(W2)),nat2(Z2))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),W2),Z2) ) ) ).

% nat_less_eq_zless
tff(fact_2701_frac__def,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] : ( archimedean_frac(A,Xa) = aa(A,A,aa(A,fun(A,A),minus_minus(A),Xa),aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,Xa))) ) ) ).

% frac_def
tff(fact_2702_Ints__odd__less__0,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( member(A,A3,ring_1_Ints(A))
         => ( aa(A,$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_2703_Ints__nonzero__abs__less1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A] :
          ( member(A,Xa,ring_1_Ints(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),abs_abs(A,Xa)),one_one(A))
           => ( Xa = zero_zero(A) ) ) ) ) ).

% Ints_nonzero_abs_less1
tff(fact_2704_Ints__eq__abs__less1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A,Y: A] :
          ( member(A,Xa,ring_1_Ints(A))
         => ( member(A,Y,ring_1_Ints(A))
           => ( ( Xa = Y )
            <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),abs_abs(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Xa),Y))),one_one(A)) ) ) ) ) ).

% Ints_eq_abs_less1
tff(fact_2705_nat__less__iff,axiom,
    ! [W2: int,M2: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),W2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),nat2(W2)),M2)
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),W2),aa(nat,int,semiring_1_of_nat(int),M2)) ) ) ).

% nat_less_iff
tff(fact_2706_frac__eq,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          ( ( archimedean_frac(A,Xa) = Xa )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xa)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),one_one(A)) ) ) ) ).

% frac_eq
tff(fact_2707_frac__add,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,Y: A] :
          ( archimedean_frac(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),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,Xa)),archimedean_frac(A,Y))),one_one(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,Xa)),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,Xa)),archimedean_frac(A,Y))),one_one(A))) ) ) ).

% frac_add
tff(fact_2708_butlast__upd__last__eq,axiom,
    ! [A: $tType,L: list(A),Xa: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),numeral_numeral(nat,bit0(one2))),aa(list(A),nat,size_size(list(A)),L))
     => ( aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),aa(list(A),list(A),butlast(A),L)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),L)),numeral_numeral(nat,bit0(one2)))),Xa) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),L)),numeral_numeral(nat,bit0(one2)))),L)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),nil(A))) ) ) ).

% butlast_upd_last_eq
tff(fact_2709_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),numeral_numeral(A,bit0(one2))),Z2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),numeral_numeral(nat,bit0(one2))),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),numeral_numeral(nat,bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),numeral_numeral(nat,bit0(one2))),N)))),comm_s3205402744901411588hammer(A,Z2,N))),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z2),divide_divide(A,one_one(A),numeral_numeral(A,bit0(one2)))),N)) ) ) ).

% pochhammer_double
tff(fact_2710_neg__eucl__rel__int__mult__2,axiom,
    ! [B3: int,A3: int,Q2: int,R2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),B3),zero_zero(int))
     => ( eucl_rel_int(aa(int,int,aa(int,fun(int,int),plus_plus(int),A3),one_one(int)),B3,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),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),numeral_numeral(int,bit0(one2))),A3)),aa(int,int,aa(int,fun(int,int),times_times(int),numeral_numeral(int,bit0(one2))),B3),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,aa(int,fun(int,int),times_times(int),numeral_numeral(int,bit0(one2))),R2)),one_one(int)))) ) ) ).

% neg_eucl_rel_int_mult_2
tff(fact_2711_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),numeral_numeral(nat,bit0(one2))),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),numeral_numeral(A,bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),numeral_numeral(nat,bit0(one2))),N))),comm_s3205402744901411588hammer(A,divide_divide(A,one_one(A),numeral_numeral(A,bit0(one2))),N))),semiring_char_0_fact(A,N)) ) ) ).

% fact_double
tff(fact_2712_divmod__digit__1_I1_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B3)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),modulo_modulo(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),numeral_numeral(A,bit0(one2))),B3)))
             => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),numeral_numeral(A,bit0(one2))),divide_divide(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),numeral_numeral(A,bit0(one2))),B3)))),one_one(A)) = divide_divide(A,A3,B3) ) ) ) ) ) ).

% divmod_digit_1(1)
tff(fact_2713_pos__eucl__rel__int__mult__2,axiom,
    ! [B3: int,A3: int,Q2: int,R2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),B3)
     => ( eucl_rel_int(A3,B3,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),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),numeral_numeral(int,bit0(one2))),A3)),aa(int,int,aa(int,fun(int,int),times_times(int),numeral_numeral(int,bit0(one2))),B3),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),numeral_numeral(int,bit0(one2))),R2)))) ) ) ).

% pos_eucl_rel_int_mult_2
tff(fact_2714_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))) = numeral_numeral(A,bit0(one2)) ) ) ).

% diff_numeral_special(11)
tff(fact_2715_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),numeral_numeral(A,bit0(one2))) ) ) ).

% diff_numeral_special(10)
tff(fact_2716_zero__less__power2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),numeral_numeral(nat,bit0(one2))))
        <=> ( A3 != zero_zero(A) ) ) ) ).

% zero_less_power2
tff(fact_2717_floor__le__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archim6421214686448440834_floor(A,Xa)),one_one(int))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),numeral_numeral(A,bit0(one2))) ) ) ).

% floor_le_one
tff(fact_2718_mod2__gr__0,axiom,
    ! [M2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),modulo_modulo(nat,M2,numeral_numeral(nat,bit0(one2))))
    <=> ( modulo_modulo(nat,M2,numeral_numeral(nat,bit0(one2))) = one_one(nat) ) ) ).

% mod2_gr_0
tff(fact_2719_left__add__twice,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [A3: A,B3: 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),B3)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),numeral_numeral(A,bit0(one2))),A3)),B3) ) ) ).

% left_add_twice
tff(fact_2720_mult__2__right,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [Z2: A] : ( aa(A,A,aa(A,fun(A,A),times_times(A),Z2),numeral_numeral(A,bit0(one2))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Z2),Z2) ) ) ).

% mult_2_right
tff(fact_2721_mult__2,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [Z2: A] : ( aa(A,A,aa(A,fun(A,A),times_times(A),numeral_numeral(A,bit0(one2))),Z2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Z2),Z2) ) ) ).

% mult_2
tff(fact_2722_power4__eq__xxxx,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Xa: A] : ( aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),numeral_numeral(nat,bit0(bit0(one2)))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xa),Xa)),Xa)),Xa) ) ) ).

% power4_eq_xxxx
tff(fact_2723_power2__eq__square,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A3: A] : ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),numeral_numeral(nat,bit0(one2))) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),A3) ) ) ).

% power2_eq_square
tff(fact_2724_power2__commute,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Xa: 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),Xa),Y)),numeral_numeral(nat,bit0(one2))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Y),Xa)),numeral_numeral(nat,bit0(one2))) ) ) ).

% power2_commute
tff(fact_2725_less__exp,axiom,
    ! [N: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),numeral_numeral(nat,bit0(one2))),N)) ).

% less_exp
tff(fact_2726_half__gt__zero,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),zero_zero(A)),divide_divide(A,A3,numeral_numeral(A,bit0(one2)))) ) ) ).

% half_gt_zero
tff(fact_2727_half__gt__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),divide_divide(A,A3,numeral_numeral(A,bit0(one2))))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3) ) ) ).

% half_gt_zero_iff
tff(fact_2728_power2__less__0,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),numeral_numeral(nat,bit0(one2)))),zero_zero(A)) ) ).

% power2_less_0
tff(fact_2729_list__decomp__2,axiom,
    ! [A: $tType,L: list(A)] :
      ( ( aa(list(A),nat,size_size(list(A)),L) = numeral_numeral(nat,bit0(one2)) )
     => ? [A4: A,B4: A] : ( L = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A4),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),B4),nil(A))) ) ) ).

% list_decomp_2
tff(fact_2730_less__2__cases,axiom,
    ! [N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),numeral_numeral(nat,bit0(one2)))
     => ( ( N = zero_zero(nat) )
        | ( N = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).

% less_2_cases
tff(fact_2731_less__2__cases__iff,axiom,
    ! [N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),numeral_numeral(nat,bit0(one2)))
    <=> ( ( N = zero_zero(nat) )
        | ( N = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).

% less_2_cases_iff
tff(fact_2732_divmod__digit__0_I2_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),modulo_modulo(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),numeral_numeral(A,bit0(one2))),B3))),B3)
           => ( modulo_modulo(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),numeral_numeral(A,bit0(one2))),B3)) = modulo_modulo(A,A3,B3) ) ) ) ) ).

% divmod_digit_0(2)
tff(fact_2733_power2__less__imp__less,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),numeral_numeral(nat,bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),numeral_numeral(nat,bit0(one2))))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Y)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y) ) ) ) ).

% power2_less_imp_less
tff(fact_2734_not__sum__power2__lt__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A,Y: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),numeral_numeral(nat,bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),numeral_numeral(nat,bit0(one2))))),zero_zero(A)) ) ).

% not_sum_power2_lt_zero
tff(fact_2735_sum__power2__gt__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),numeral_numeral(nat,bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),numeral_numeral(nat,bit0(one2)))))
        <=> ( ( Xa != zero_zero(A) )
            | ( Y != zero_zero(A) ) ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_2736_power2__sum,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Xa: 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),Xa),Y)),numeral_numeral(nat,bit0(one2))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),numeral_numeral(nat,bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),numeral_numeral(nat,bit0(one2))))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),numeral_numeral(A,bit0(one2))),Xa)),Y)) ) ) ).

% power2_sum
tff(fact_2737_abs__square__less__1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),numeral_numeral(nat,bit0(one2)))),one_one(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),abs_abs(A,Xa)),one_one(A)) ) ) ).

% abs_square_less_1
tff(fact_2738_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),numeral_numeral(nat,bit0(one2))),N)) ) ) ).

% minus_power_mult_self
tff(fact_2739_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),numeral_numeral(nat,bit0(one2))),N))) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)),numeral_numeral(nat,bit0(one2)))) ) ) ).

% power_odd_eq
tff(fact_2740_div__2__gt__zero,axiom,
    ! [N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),N)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),divide_divide(nat,N,numeral_numeral(nat,bit0(one2)))) ) ).

% div_2_gt_zero
tff(fact_2741_Suc__n__div__2__gt__zero,axiom,
    ! [N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),divide_divide(nat,aa(nat,nat,suc,N),numeral_numeral(nat,bit0(one2)))) ) ).

% Suc_n_div_2_gt_zero
tff(fact_2742_divmod__digit__0_I1_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),modulo_modulo(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),numeral_numeral(A,bit0(one2))),B3))),B3)
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),numeral_numeral(A,bit0(one2))),divide_divide(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),numeral_numeral(A,bit0(one2))),B3))) = divide_divide(A,A3,B3) ) ) ) ) ).

% divmod_digit_0(1)
tff(fact_2743_power2__diff,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Xa: 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),Xa),Y)),numeral_numeral(nat,bit0(one2))) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),numeral_numeral(nat,bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),numeral_numeral(nat,bit0(one2))))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),numeral_numeral(A,bit0(one2))),Xa)),Y)) ) ) ).

% power2_diff
tff(fact_2744_odd__power__less__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,N: nat] :
          ( aa(A,$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(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),numeral_numeral(nat,bit0(one2))),N)))),zero_zero(A)) ) ) ).

% odd_power_less_zero
tff(fact_2745_mask__mod__exp,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [N: nat,M2: 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),numeral_numeral(A,bit0(one2))),N)),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),numeral_numeral(A,bit0(one2))),M2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),numeral_numeral(A,bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),M2),N))),one_one(A)) ) ) ).

% mask_mod_exp
tff(fact_2746_ex__power__ivl1,axiom,
    ! [B3: nat,K3: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),numeral_numeral(nat,bit0(one2))),B3)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),K3)
       => ? [N5: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B3),N5)),K3)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K3),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B3),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N5),one_one(nat)))) ) ) ) ).

% ex_power_ivl1
tff(fact_2747_ex__power__ivl2,axiom,
    ! [B3: nat,K3: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),numeral_numeral(nat,bit0(one2))),B3)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),numeral_numeral(nat,bit0(one2))),K3)
       => ? [N5: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B3),N5)),K3)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K3),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B3),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N5),one_one(nat)))) ) ) ) ).

% ex_power_ivl2
tff(fact_2748_mod__double__modulus,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M2: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),M2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xa)
           => ( ( modulo_modulo(A,Xa,aa(A,A,aa(A,fun(A,A),times_times(A),numeral_numeral(A,bit0(one2))),M2)) = modulo_modulo(A,Xa,M2) )
              | ( modulo_modulo(A,Xa,aa(A,A,aa(A,fun(A,A),times_times(A),numeral_numeral(A,bit0(one2))),M2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,Xa,M2)),M2) ) ) ) ) ) ).

% mod_double_modulus
tff(fact_2749_divmod__digit__1_I2_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B3)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),modulo_modulo(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),numeral_numeral(A,bit0(one2))),B3)))
             => ( 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),numeral_numeral(A,bit0(one2))),B3))),B3) = modulo_modulo(A,A3,B3) ) ) ) ) ) ).

% divmod_digit_1(2)
tff(fact_2750_divmod__step__eq,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [L: num,Q2: A,R2: A] :
          ( unique1321980374590559556d_step(A,L,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q2),R2)) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),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),numeral_numeral(A,bit0(one2))),Q2)),one_one(A))),aa(A,A,aa(A,fun(A,A),minus_minus(A),R2),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),numeral_numeral(A,bit0(one2))),Q2)),R2)) ) ) ).

% divmod_step_eq
tff(fact_2751_mult__exp__mod__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M2: nat,N: nat,A3: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),N)
         => ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),numeral_numeral(A,bit0(one2))),M2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),numeral_numeral(A,bit0(one2))),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),modulo_modulo(A,A3,aa(nat,A,aa(A,fun(nat,A),power_power(A),numeral_numeral(A,bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),numeral_numeral(A,bit0(one2))),M2)) ) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_2752_nat__bit__induct,axiom,
    ! [Pa: fun(nat,$o),N: nat] :
      ( aa(nat,$o,Pa,zero_zero(nat))
     => ( ! [N5: nat] :
            ( aa(nat,$o,Pa,N5)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N5)
             => aa(nat,$o,Pa,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),numeral_numeral(nat,bit0(one2))),N5)) ) )
       => ( ! [N5: nat] :
              ( aa(nat,$o,Pa,N5)
             => aa(nat,$o,Pa,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),numeral_numeral(nat,bit0(one2))),N5))) )
         => aa(nat,$o,Pa,N) ) ) ) ).

% nat_bit_induct
tff(fact_2753_round__unique,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,Y: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xa),divide_divide(A,one_one(A),numeral_numeral(A,bit0(one2))))),aa(int,A,ring_1_of_int(A),Y))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Y)),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),divide_divide(A,one_one(A),numeral_numeral(A,bit0(one2)))))
           => ( archimedean_round(A,Xa) = Y ) ) ) ) ).

% round_unique
tff(fact_2754_round__unique_H,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,N: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),abs_abs(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Xa),aa(int,A,ring_1_of_int(A),N)))),divide_divide(A,one_one(A),numeral_numeral(A,bit0(one2))))
         => ( archimedean_round(A,Xa) = N ) ) ) ).

% round_unique'
tff(fact_2755_of__int__round__abs__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(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,Xa))),Xa))),divide_divide(A,one_one(A),numeral_numeral(A,bit0(one2)))) ) ).

% of_int_round_abs_le
tff(fact_2756_round__diff__minimal,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z2: A,M2: int] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),abs_abs(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Z2),aa(int,A,ring_1_of_int(A),archimedean_round(A,Z2))))),abs_abs(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Z2),aa(int,A,ring_1_of_int(A),M2)))) ) ).

% round_diff_minimal
tff(fact_2757_of__int__round__ge,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xa),divide_divide(A,one_one(A),numeral_numeral(A,bit0(one2))))),aa(int,A,ring_1_of_int(A),archimedean_round(A,Xa))) ) ).

% of_int_round_ge
tff(fact_2758_of__int__round__gt,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xa),divide_divide(A,one_one(A),numeral_numeral(A,bit0(one2))))),aa(int,A,ring_1_of_int(A),archimedean_round(A,Xa))) ) ).

% of_int_round_gt
tff(fact_2759_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),numeral_numeral(A,bit0(one2))),divide_divide(A,A3,numeral_numeral(A,bit0(one2))))) ) ) ).

% set_bit_0
tff(fact_2760_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),numeral_numeral(A,bit0(one2))),divide_divide(A,A3,numeral_numeral(A,bit0(one2)))) ) ) ).

% unset_bit_0
tff(fact_2761_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,numeral_numeral(A,bit0(one2)))),aa(A,A,aa(A,fun(A,A),times_times(A),numeral_numeral(A,bit0(one2))),bit_se5668285175392031749et_bit(A,N,divide_divide(A,A3,numeral_numeral(A,bit0(one2)))))) ) ) ).

% set_bit_Suc
tff(fact_2762_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,numeral_numeral(A,bit0(one2)))),aa(A,A,aa(A,fun(A,A),times_times(A),numeral_numeral(A,bit0(one2))),bit_se8732182000553998342ip_bit(A,N,divide_divide(A,A3,numeral_numeral(A,bit0(one2)))))) ) ) ).

% flip_bit_Suc
tff(fact_2763_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,numeral_numeral(A,bit0(one2)))),aa(A,A,aa(A,fun(A,A),times_times(A),numeral_numeral(A,bit0(one2))),bit_se2638667681897837118et_bit(A,N,divide_divide(A,A3,numeral_numeral(A,bit0(one2)))))) ) ) ).

% unset_bit_Suc
tff(fact_2764_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,numeral_numeral(A,bit0(one2)))),aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A3,numeral_numeral(A,bit0(one2)))),aa(A,A,aa(A,fun(A,A),times_times(A),numeral_numeral(A,bit0(one2))),bit_ri4674362597316999326ke_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)),divide_divide(A,A3,numeral_numeral(A,bit0(one2))))))) ) ) ).

% signed_take_bit_rec
tff(fact_2765_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,numeral_numeral(A,bit0(one2)))),aa(A,A,aa(A,fun(A,A),times_times(A),numeral_numeral(A,bit0(one2))),bit_ri4674362597316999326ke_bit(A,N,divide_divide(A,A3,numeral_numeral(A,bit0(one2)))))) ) ) ).

% signed_take_bit_Suc
tff(fact_2766_even__succ__mod__exp,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,N: nat] :
          ( dvd_dvd(A,numeral_numeral(A,bit0(one2)),A3)
         => ( aa(nat,$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),numeral_numeral(A,bit0(one2))),N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),modulo_modulo(A,A3,aa(nat,A,aa(A,fun(nat,A),power_power(A),numeral_numeral(A,bit0(one2))),N))) ) ) ) ) ).

% even_succ_mod_exp
tff(fact_2767_even__succ__div__exp,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,N: nat] :
          ( dvd_dvd(A,numeral_numeral(A,bit0(one2)),A3)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
           => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),numeral_numeral(A,bit0(one2))),N)) = divide_divide(A,A3,aa(nat,A,aa(A,fun(nat,A),power_power(A),numeral_numeral(A,bit0(one2))),N)) ) ) ) ) ).

% even_succ_div_exp
tff(fact_2768_length__subseqs,axiom,
    ! [A: $tType,Xsa: list(A)] : ( aa(list(list(A)),nat,size_size(list(list(A))),aa(list(A),list(list(A)),subseqs(A),Xsa)) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),numeral_numeral(nat,bit0(one2))),aa(list(A),nat,size_size(list(A)),Xsa)) ) ).

% length_subseqs
tff(fact_2769_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)),divide_divide(A,A3,numeral_numeral(A,bit0(one2))))),numeral_numeral(A,bit0(one2)))),modulo_modulo(A,A3,numeral_numeral(A,bit0(one2))))) ) ) ).

% take_bit_rec
tff(fact_2770_even__mult__exp__div__exp__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,M2: nat,N: nat] :
          ( dvd_dvd(A,numeral_numeral(A,bit0(one2)),divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),numeral_numeral(A,bit0(one2))),M2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),numeral_numeral(A,bit0(one2))),N)))
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),M2)
            | ( aa(nat,A,aa(A,fun(nat,A),power_power(A),numeral_numeral(A,bit0(one2))),N) = zero_zero(A) )
            | ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),N)
              & dvd_dvd(A,numeral_numeral(A,bit0(one2)),divide_divide(A,A3,aa(nat,A,aa(A,fun(nat,A),power_power(A),numeral_numeral(A,bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M2)))) ) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_2771_one__mod__2__pow__eq,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [N: nat] : ( modulo_modulo(A,one_one(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),numeral_numeral(A,bit0(one2))),N)) = aa($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_2772_dvd__0__right,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A] : dvd_dvd(A,A3,zero_zero(A)) ) ).

% dvd_0_right
tff(fact_2773_dvd__0__left__iff,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A] :
          ( dvd_dvd(A,zero_zero(A),A3)
        <=> ( A3 = zero_zero(A) ) ) ) ).

% dvd_0_left_iff
tff(fact_2774_dvd__add__triv__left__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A3: A,B3: A] :
          ( dvd_dvd(A,A3,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3))
        <=> dvd_dvd(A,A3,B3) ) ) ).

% dvd_add_triv_left_iff
tff(fact_2775_dvd__add__triv__right__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A3: A,B3: A] :
          ( dvd_dvd(A,A3,aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),A3))
        <=> dvd_dvd(A,A3,B3) ) ) ).

% dvd_add_triv_right_iff
tff(fact_2776_dvd__minus__iff,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Xa: A,Y: A] :
          ( dvd_dvd(A,Xa,aa(A,A,uminus_uminus(A),Y))
        <=> dvd_dvd(A,Xa,Y) ) ) ).

% dvd_minus_iff
tff(fact_2777_minus__dvd__iff,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Xa: A,Y: A] :
          ( dvd_dvd(A,aa(A,A,uminus_uminus(A),Xa),Y)
        <=> dvd_dvd(A,Xa,Y) ) ) ).

% minus_dvd_iff
tff(fact_2778_div__dvd__div,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B3: A,C2: A] :
          ( dvd_dvd(A,A3,B3)
         => ( dvd_dvd(A,A3,C2)
           => ( dvd_dvd(A,divide_divide(A,B3,A3),divide_divide(A,C2,A3))
            <=> dvd_dvd(A,B3,C2) ) ) ) ) ).

% div_dvd_div
tff(fact_2779_abs__dvd__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M2: A,K3: A] :
          ( dvd_dvd(A,abs_abs(A,M2),K3)
        <=> dvd_dvd(A,M2,K3) ) ) ).

% abs_dvd_iff
tff(fact_2780_dvd__abs__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M2: A,K3: A] :
          ( dvd_dvd(A,M2,abs_abs(A,K3))
        <=> dvd_dvd(A,M2,K3) ) ) ).

% dvd_abs_iff
tff(fact_2781_of__bool__less__eq__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Pa: $o,Qa: $o] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa($o,A,zero_neq_one_of_bool(A),(Pa))),aa($o,A,zero_neq_one_of_bool(A),(Qa)))
        <=> ( (Pa)
           => (Qa) ) ) ) ).

% of_bool_less_eq_iff
tff(fact_2782_of__bool__eq_I1_J,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ( aa($o,A,zero_neq_one_of_bool(A),$false) = zero_zero(A) ) ) ).

% of_bool_eq(1)
tff(fact_2783_of__bool__eq__0__iff,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ! [Pa: $o] :
          ( ( aa($o,A,zero_neq_one_of_bool(A),(Pa)) = zero_zero(A) )
        <=> ~ (Pa) ) ) ).

% of_bool_eq_0_iff
tff(fact_2784_of__bool__less__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Pa: $o,Qa: $o] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa($o,A,zero_neq_one_of_bool(A),(Pa))),aa($o,A,zero_neq_one_of_bool(A),(Qa)))
        <=> ( ~ (Pa)
            & (Qa) ) ) ) ).

% of_bool_less_iff
tff(fact_2785_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_2786_of__bool__eq__1__iff,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ! [Pa: $o] :
          ( ( aa($o,A,zero_neq_one_of_bool(A),(Pa)) = one_one(A) )
        <=> (Pa) ) ) ).

% of_bool_eq_1_iff
tff(fact_2787_abs__bool__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Pa: $o] : ( abs_abs(A,aa($o,A,zero_neq_one_of_bool(A),(Pa))) = aa($o,A,zero_neq_one_of_bool(A),(Pa)) ) ) ).

% abs_bool_eq
tff(fact_2788_dvd__mult__cancel__left,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [C2: A,A3: A,B3: A] :
          ( 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),B3))
        <=> ( ( C2 = zero_zero(A) )
            | dvd_dvd(A,A3,B3) ) ) ) ).

% dvd_mult_cancel_left
tff(fact_2789_dvd__mult__cancel__right,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [A3: A,C2: A,B3: A] :
          ( 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),B3),C2))
        <=> ( ( C2 = zero_zero(A) )
            | dvd_dvd(A,A3,B3) ) ) ) ).

% dvd_mult_cancel_right
tff(fact_2790_dvd__times__left__cancel__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B3: A,C2: A] :
          ( ( A3 != zero_zero(A) )
         => ( dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2))
          <=> dvd_dvd(A,B3,C2) ) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_2791_dvd__times__right__cancel__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B3: A,C2: A] :
          ( ( A3 != zero_zero(A) )
         => ( dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),B3),A3),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3))
          <=> dvd_dvd(A,B3,C2) ) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_2792_dvd__add__times__triv__left__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A3: A,C2: A,B3: A] :
          ( 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)),B3))
        <=> dvd_dvd(A,A3,B3) ) ) ).

% dvd_add_times_triv_left_iff
tff(fact_2793_dvd__add__times__triv__right__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A3: A,B3: A,C2: A] :
          ( dvd_dvd(A,A3,aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)))
        <=> dvd_dvd(A,A3,B3) ) ) ).

% dvd_add_times_triv_right_iff
tff(fact_2794_unit__prod,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B3: A] :
          ( dvd_dvd(A,A3,one_one(A))
         => ( dvd_dvd(A,B3,one_one(A))
           => dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3),one_one(A)) ) ) ) ).

% unit_prod
tff(fact_2795_div__add,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,A3: A,B3: A] :
          ( dvd_dvd(A,C2,A3)
         => ( dvd_dvd(A,C2,B3)
           => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,A3,C2)),divide_divide(A,B3,C2)) ) ) ) ) ).

% div_add
tff(fact_2796_dvd__div__mult__self,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B3: A] :
          ( dvd_dvd(A,A3,B3)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,B3,A3)),A3) = B3 ) ) ) ).

% dvd_div_mult_self
tff(fact_2797_dvd__mult__div__cancel,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B3: A] :
          ( dvd_dvd(A,A3,B3)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),divide_divide(A,B3,A3)) = B3 ) ) ) ).

% dvd_mult_div_cancel
tff(fact_2798_unit__div,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B3: A] :
          ( dvd_dvd(A,A3,one_one(A))
         => ( dvd_dvd(A,B3,one_one(A))
           => dvd_dvd(A,divide_divide(A,A3,B3),one_one(A)) ) ) ) ).

% unit_div
tff(fact_2799_unit__div__1__unit,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A] :
          ( dvd_dvd(A,A3,one_one(A))
         => dvd_dvd(A,divide_divide(A,one_one(A),A3),one_one(A)) ) ) ).

% unit_div_1_unit
tff(fact_2800_unit__div__1__div__1,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A] :
          ( dvd_dvd(A,A3,one_one(A))
         => ( divide_divide(A,one_one(A),divide_divide(A,one_one(A),A3)) = A3 ) ) ) ).

% unit_div_1_div_1
tff(fact_2801_div__diff,axiom,
    ! [A: $tType] :
      ( idom_modulo(A)
     => ! [C2: A,A3: A,B3: A] :
          ( dvd_dvd(A,C2,A3)
         => ( dvd_dvd(A,C2,B3)
           => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B3),C2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),divide_divide(A,A3,C2)),divide_divide(A,B3,C2)) ) ) ) ) ).

% div_diff
tff(fact_2802_dvd__imp__mod__0,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A3: A,B3: A] :
          ( dvd_dvd(A,A3,B3)
         => ( modulo_modulo(A,B3,A3) = zero_zero(A) ) ) ) ).

% dvd_imp_mod_0
tff(fact_2803_take__bit__of__Suc__0,axiom,
    ! [N: nat] : ( bit_se2584673776208193580ke_bit(nat,N,aa(nat,nat,suc,zero_zero(nat))) = aa($o,nat,zero_neq_one_of_bool(nat),aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)) ) ).

% take_bit_of_Suc_0
tff(fact_2804_zero__less__of__bool__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Pa: $o] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa($o,A,zero_neq_one_of_bool(A),(Pa)))
        <=> (Pa) ) ) ).

% zero_less_of_bool_iff
tff(fact_2805_of__bool__less__one__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Pa: $o] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa($o,A,zero_neq_one_of_bool(A),(Pa))),one_one(A))
        <=> ~ (Pa) ) ) ).

% of_bool_less_one_iff
tff(fact_2806_of__bool__not__iff,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Pa: $o] : ( aa($o,A,zero_neq_one_of_bool(A),~ (Pa)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa($o,A,zero_neq_one_of_bool(A),(Pa))) ) ) ).

% of_bool_not_iff
tff(fact_2807_unit__div__mult__self,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B3: A] :
          ( dvd_dvd(A,A3,one_one(A))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,B3,A3)),A3) = B3 ) ) ) ).

% unit_div_mult_self
tff(fact_2808_unit__mult__div__div,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B3: A] :
          ( dvd_dvd(A,A3,one_one(A))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),B3),divide_divide(A,one_one(A),A3)) = divide_divide(A,B3,A3) ) ) ) ).

% unit_mult_div_div
tff(fact_2809_pow__divides__pow__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [N: nat,A3: A,B3: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
         => ( dvd_dvd(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N),aa(nat,A,aa(A,fun(nat,A),power_power(A),B3),N))
          <=> dvd_dvd(A,A3,B3) ) ) ) ).

% pow_divides_pow_iff
tff(fact_2810_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_2811_idom__abs__sgn__class_Oabs__sgn,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A3: A] : ( aa(A,A,sgn_sgn(A),abs_abs(A,A3)) = aa($o,A,zero_neq_one_of_bool(A),A3 != zero_zero(A)) ) ) ).

% idom_abs_sgn_class.abs_sgn
tff(fact_2812_sgn__abs,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A3: A] : ( abs_abs(A,aa(A,A,sgn_sgn(A),A3)) = aa($o,A,zero_neq_one_of_bool(A),A3 != zero_zero(A)) ) ) ).

% sgn_abs
tff(fact_2813_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_2814_sgn__of__nat,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: nat] : ( aa(A,A,sgn_sgn(A),aa(nat,A,semiring_1_of_nat(A),N)) = aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)) ) ) ).

% sgn_of_nat
tff(fact_2815_take__bit__of__exp,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M2: nat,N: nat] : ( bit_se2584673776208193580ke_bit(A,M2,aa(nat,A,aa(A,fun(nat,A),power_power(A),numeral_numeral(A,bit0(one2))),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),M2))),aa(nat,A,aa(A,fun(nat,A),power_power(A),numeral_numeral(A,bit0(one2))),N)) ) ) ).

% take_bit_of_exp
tff(fact_2816_take__bit__of__2,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: nat] : ( bit_se2584673776208193580ke_bit(A,N,numeral_numeral(A,bit0(one2))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),numeral_numeral(nat,bit0(one2))),N))),numeral_numeral(A,bit0(one2))) ) ) ).

% take_bit_of_2
tff(fact_2817_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),dvd_dvd(A,numeral_numeral(A,bit0(one2)),A3))),aa(A,A,aa(A,fun(A,A),times_times(A),numeral_numeral(A,bit0(one2))),divide_divide(A,A3,numeral_numeral(A,bit0(one2))))) ) ) ).

% flip_bit_0
tff(fact_2818_dvd__refl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A] : dvd_dvd(A,A3,A3) ) ).

% dvd_refl
tff(fact_2819_dvd__trans,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A,B3: A,C2: A] :
          ( dvd_dvd(A,A3,B3)
         => ( dvd_dvd(A,B3,C2)
           => dvd_dvd(A,A3,C2) ) ) ) ).

% dvd_trans
tff(fact_2820_of__bool__eq__iff,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ! [P2: $o,Q2: $o] :
          ( ( aa($o,A,zero_neq_one_of_bool(A),(P2)) = aa($o,A,zero_neq_one_of_bool(A),(Q2)) )
        <=> ( (P2)
          <=> (Q2) ) ) ) ).

% of_bool_eq_iff
tff(fact_2821_of__bool__conj,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Pa: $o,Qa: $o] :
          ( aa($o,A,zero_neq_one_of_bool(A),
              ( (Pa)
              & (Qa) )) = aa(A,A,aa(A,fun(A,A),times_times(A),aa($o,A,zero_neq_one_of_bool(A),(Pa))),aa($o,A,zero_neq_one_of_bool(A),(Qa))) ) ) ).

% of_bool_conj
tff(fact_2822_dvd__0__left,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A] :
          ( dvd_dvd(A,zero_zero(A),A3)
         => ( A3 = zero_zero(A) ) ) ) ).

% dvd_0_left
tff(fact_2823_dvd__add,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A,B3: A,C2: A] :
          ( dvd_dvd(A,A3,B3)
         => ( dvd_dvd(A,A3,C2)
           => dvd_dvd(A,A3,aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),C2)) ) ) ) ).

% dvd_add
tff(fact_2824_dvd__add__left__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A3: A,C2: A,B3: A] :
          ( dvd_dvd(A,A3,C2)
         => ( dvd_dvd(A,A3,aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),C2))
          <=> dvd_dvd(A,A3,B3) ) ) ) ).

% dvd_add_left_iff
tff(fact_2825_dvd__add__right__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A3: A,B3: A,C2: A] :
          ( dvd_dvd(A,A3,B3)
         => ( dvd_dvd(A,A3,aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),C2))
          <=> dvd_dvd(A,A3,C2) ) ) ) ).

% dvd_add_right_iff
tff(fact_2826_dvd__triv__right,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A,B3: A] : dvd_dvd(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),B3),A3)) ) ).

% dvd_triv_right
tff(fact_2827_dvd__mult__right,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A,B3: A,C2: A] :
          ( dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3),C2)
         => dvd_dvd(A,B3,C2) ) ) ).

% dvd_mult_right
tff(fact_2828_mult__dvd__mono,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A,B3: A,C2: A,D3: A] :
          ( dvd_dvd(A,A3,B3)
         => ( dvd_dvd(A,C2,D3)
           => 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),B3),D3)) ) ) ) ).

% mult_dvd_mono
tff(fact_2829_dvd__triv__left,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A,B3: A] : dvd_dvd(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)) ) ).

% dvd_triv_left
tff(fact_2830_dvd__mult__left,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A,B3: A,C2: A] :
          ( dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3),C2)
         => dvd_dvd(A,A3,C2) ) ) ).

% dvd_mult_left
tff(fact_2831_dvd__mult2,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A,B3: A,C2: A] :
          ( dvd_dvd(A,A3,B3)
         => dvd_dvd(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2)) ) ) ).

% dvd_mult2
tff(fact_2832_dvd__mult,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A,C2: A,B3: A] :
          ( dvd_dvd(A,A3,C2)
         => dvd_dvd(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2)) ) ) ).

% dvd_mult
tff(fact_2833_dvd__def,axiom,
    ! [A: $tType] :
      ( dvd(A)
     => ! [B3: A,A3: A] :
          ( dvd_dvd(A,B3,A3)
        <=> ? [K4: A] : ( A3 = aa(A,A,aa(A,fun(A,A),times_times(A),B3),K4) ) ) ) ).

% dvd_def
tff(fact_2834_dvdI,axiom,
    ! [A: $tType] :
      ( dvd(A)
     => ! [A3: A,B3: A,K3: A] :
          ( ( A3 = aa(A,A,aa(A,fun(A,A),times_times(A),B3),K3) )
         => dvd_dvd(A,B3,A3) ) ) ).

% dvdI
tff(fact_2835_dvdE,axiom,
    ! [A: $tType] :
      ( dvd(A)
     => ! [B3: A,A3: A] :
          ( dvd_dvd(A,B3,A3)
         => ~ ! [K: A] : ( A3 != aa(A,A,aa(A,fun(A,A),times_times(A),B3),K) ) ) ) ).

% dvdE
tff(fact_2836_division__decomp,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B3: A,C2: A] :
          ( dvd_dvd(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2))
         => ? [B11: A,C7: A] :
              ( ( A3 = aa(A,A,aa(A,fun(A,A),times_times(A),B11),C7) )
              & dvd_dvd(A,B11,B3)
              & dvd_dvd(A,C7,C2) ) ) ) ).

% division_decomp
tff(fact_2837_dvd__productE,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [P2: A,A3: A,B3: A] :
          ( dvd_dvd(A,P2,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3))
         => ~ ! [X: A,Y2: A] :
                ( ( P2 = aa(A,A,aa(A,fun(A,A),times_times(A),X),Y2) )
               => ( dvd_dvd(A,X,A3)
                 => ~ dvd_dvd(A,Y2,B3) ) ) ) ) ).

% dvd_productE
tff(fact_2838_one__dvd,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A] : dvd_dvd(A,one_one(A),A3) ) ).

% one_dvd
tff(fact_2839_unit__imp__dvd,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B3: A,A3: A] :
          ( dvd_dvd(A,B3,one_one(A))
         => dvd_dvd(A,B3,A3) ) ) ).

% unit_imp_dvd
tff(fact_2840_dvd__unit__imp__unit,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B3: A] :
          ( dvd_dvd(A,A3,B3)
         => ( dvd_dvd(A,B3,one_one(A))
           => dvd_dvd(A,A3,one_one(A)) ) ) ) ).

% dvd_unit_imp_unit
tff(fact_2841_dvd__diff,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Xa: A,Y: A,Z2: A] :
          ( dvd_dvd(A,Xa,Y)
         => ( dvd_dvd(A,Xa,Z2)
           => dvd_dvd(A,Xa,aa(A,A,aa(A,fun(A,A),minus_minus(A),Y),Z2)) ) ) ) ).

% dvd_diff
tff(fact_2842_dvd__diff__commute,axiom,
    ! [A: $tType] :
      ( euclid5891614535332579305n_ring(A)
     => ! [A3: A,C2: A,B3: A] :
          ( dvd_dvd(A,A3,aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),B3))
        <=> dvd_dvd(A,A3,aa(A,A,aa(A,fun(A,A),minus_minus(A),B3),C2)) ) ) ).

% dvd_diff_commute
tff(fact_2843_dvd__div__eq__iff,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [C2: A,A3: A,B3: A] :
          ( dvd_dvd(A,C2,A3)
         => ( dvd_dvd(A,C2,B3)
           => ( ( divide_divide(A,A3,C2) = divide_divide(A,B3,C2) )
            <=> ( A3 = B3 ) ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_2844_dvd__div__eq__cancel,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A3: A,C2: A,B3: A] :
          ( ( divide_divide(A,A3,C2) = divide_divide(A,B3,C2) )
         => ( dvd_dvd(A,C2,A3)
           => ( dvd_dvd(A,C2,B3)
             => ( A3 = B3 ) ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_2845_div__div__div__same,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [D3: A,B3: A,A3: A] :
          ( dvd_dvd(A,D3,B3)
         => ( dvd_dvd(A,B3,A3)
           => ( divide_divide(A,divide_divide(A,A3,D3),divide_divide(A,B3,D3)) = divide_divide(A,A3,B3) ) ) ) ) ).

% div_div_div_same
tff(fact_2846_dvd__mod__iff,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [C2: A,B3: A,A3: A] :
          ( dvd_dvd(A,C2,B3)
         => ( dvd_dvd(A,C2,modulo_modulo(A,A3,B3))
          <=> dvd_dvd(A,C2,A3) ) ) ) ).

% dvd_mod_iff
tff(fact_2847_dvd__mod__imp__dvd,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [C2: A,A3: A,B3: A] :
          ( dvd_dvd(A,C2,modulo_modulo(A,A3,B3))
         => ( dvd_dvd(A,C2,B3)
           => dvd_dvd(A,C2,A3) ) ) ) ).

% dvd_mod_imp_dvd
tff(fact_2848_dvd__if__abs__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [L: A,K3: A] :
          ( ( abs_abs(A,L) = abs_abs(A,K3) )
         => dvd_dvd(A,L,K3) ) ) ).

% dvd_if_abs_eq
tff(fact_2849_uminus__dvd__conv_I1_J,axiom,
    ! [D3: int,T2: int] :
      ( dvd_dvd(int,D3,T2)
    <=> dvd_dvd(int,aa(int,int,uminus_uminus(int),D3),T2) ) ).

% uminus_dvd_conv(1)
tff(fact_2850_uminus__dvd__conv_I2_J,axiom,
    ! [D3: int,T2: int] :
      ( dvd_dvd(int,D3,T2)
    <=> dvd_dvd(int,D3,aa(int,int,uminus_uminus(int),T2)) ) ).

% uminus_dvd_conv(2)
tff(fact_2851_zero__less__eq__of__bool,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Pa: $o] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa($o,A,zero_neq_one_of_bool(A),(Pa))) ) ).

% zero_less_eq_of_bool
tff(fact_2852_of__bool__less__eq__one,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Pa: $o] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa($o,A,zero_neq_one_of_bool(A),(Pa))),one_one(A)) ) ).

% of_bool_less_eq_one
tff(fact_2853_split__of__bool__asm,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ! [Pa: fun(A,$o),P2: $o] :
          ( aa(A,$o,Pa,aa($o,A,zero_neq_one_of_bool(A),(P2)))
        <=> ~ ( ( (P2)
                & ~ aa(A,$o,Pa,one_one(A)) )
              | ( ~ (P2)
                & ~ aa(A,$o,Pa,zero_zero(A)) ) ) ) ) ).

% split_of_bool_asm
tff(fact_2854_split__of__bool,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ! [Pa: fun(A,$o),P2: $o] :
          ( aa(A,$o,Pa,aa($o,A,zero_neq_one_of_bool(A),(P2)))
        <=> ( ( (P2)
             => aa(A,$o,Pa,one_one(A)) )
            & ( ~ (P2)
             => aa(A,$o,Pa,zero_zero(A)) ) ) ) ) ).

% split_of_bool
tff(fact_2855_of__bool__def,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ! [P2: $o] :
          ( aa($o,A,zero_neq_one_of_bool(A),(P2)) = $ite((P2),one_one(A),zero_zero(A)) ) ) ).

% of_bool_def
tff(fact_2856_not__is__unit__0,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ~ dvd_dvd(A,zero_zero(A),one_one(A)) ) ).

% not_is_unit_0
tff(fact_2857_pinf_I9_J,axiom,
    ! [A: $tType] :
      ( ( plus(A)
        & linorder(A)
        & dvd(A) )
     => ! [D3: A,S2: A] :
        ? [Z3: A] :
        ! [X4: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z3),X4)
         => ( dvd_dvd(A,D3,aa(A,A,aa(A,fun(A,A),plus_plus(A),X4),S2))
          <=> dvd_dvd(A,D3,aa(A,A,aa(A,fun(A,A),plus_plus(A),X4),S2)) ) ) ) ).

% pinf(9)
tff(fact_2858_pinf_I10_J,axiom,
    ! [A: $tType] :
      ( ( plus(A)
        & linorder(A)
        & dvd(A) )
     => ! [D3: A,S2: A] :
        ? [Z3: A] :
        ! [X4: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z3),X4)
         => ( ~ dvd_dvd(A,D3,aa(A,A,aa(A,fun(A,A),plus_plus(A),X4),S2))
          <=> ~ dvd_dvd(A,D3,aa(A,A,aa(A,fun(A,A),plus_plus(A),X4),S2)) ) ) ) ).

% pinf(10)
tff(fact_2859_minf_I9_J,axiom,
    ! [A: $tType] :
      ( ( plus(A)
        & linorder(A)
        & dvd(A) )
     => ! [D3: A,S2: A] :
        ? [Z3: A] :
        ! [X4: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Z3)
         => ( dvd_dvd(A,D3,aa(A,A,aa(A,fun(A,A),plus_plus(A),X4),S2))
          <=> dvd_dvd(A,D3,aa(A,A,aa(A,fun(A,A),plus_plus(A),X4),S2)) ) ) ) ).

% minf(9)
tff(fact_2860_minf_I10_J,axiom,
    ! [A: $tType] :
      ( ( plus(A)
        & linorder(A)
        & dvd(A) )
     => ! [D3: A,S2: A] :
        ? [Z3: A] :
        ! [X4: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Z3)
         => ( ~ dvd_dvd(A,D3,aa(A,A,aa(A,fun(A,A),plus_plus(A),X4),S2))
          <=> ~ dvd_dvd(A,D3,aa(A,A,aa(A,fun(A,A),plus_plus(A),X4),S2)) ) ) ) ).

% minf(10)
tff(fact_2861_is__unit__mult__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B3: A] :
          ( dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3),one_one(A))
        <=> ( dvd_dvd(A,A3,one_one(A))
            & dvd_dvd(A,B3,one_one(A)) ) ) ) ).

% is_unit_mult_iff
tff(fact_2862_dvd__mult__unit__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B3: A,A3: A,C2: A] :
          ( dvd_dvd(A,B3,one_one(A))
         => ( dvd_dvd(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3))
          <=> dvd_dvd(A,A3,C2) ) ) ) ).

% dvd_mult_unit_iff
tff(fact_2863_mult__unit__dvd__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B3: A,A3: A,C2: A] :
          ( dvd_dvd(A,B3,one_one(A))
         => ( dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3),C2)
          <=> dvd_dvd(A,A3,C2) ) ) ) ).

% mult_unit_dvd_iff
tff(fact_2864_dvd__mult__unit__iff_H,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B3: A,A3: A,C2: A] :
          ( dvd_dvd(A,B3,one_one(A))
         => ( dvd_dvd(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2))
          <=> dvd_dvd(A,A3,C2) ) ) ) ).

% dvd_mult_unit_iff'
tff(fact_2865_mult__unit__dvd__iff_H,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B3: A,C2: A] :
          ( dvd_dvd(A,A3,one_one(A))
         => ( dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3),C2)
          <=> dvd_dvd(A,B3,C2) ) ) ) ).

% mult_unit_dvd_iff'
tff(fact_2866_unit__mult__left__cancel,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B3: A,C2: A] :
          ( dvd_dvd(A,A3,one_one(A))
         => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2) )
          <=> ( B3 = C2 ) ) ) ) ).

% unit_mult_left_cancel
tff(fact_2867_unit__mult__right__cancel,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B3: A,C2: A] :
          ( dvd_dvd(A,A3,one_one(A))
         => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),B3),A3) = aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3) )
          <=> ( B3 = C2 ) ) ) ) ).

% unit_mult_right_cancel
tff(fact_2868_dvd__div__eq__0__iff,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [B3: A,A3: A] :
          ( dvd_dvd(A,B3,A3)
         => ( ( divide_divide(A,A3,B3) = zero_zero(A) )
          <=> ( A3 = zero_zero(A) ) ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_2869_dvd__div__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,B3: A,A3: A] :
          ( dvd_dvd(A,C2,B3)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,B3,C2)),A3) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),B3),A3),C2) ) ) ) ).

% dvd_div_mult
tff(fact_2870_div__mult__swap,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,B3: A,A3: A] :
          ( dvd_dvd(A,C2,B3)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),divide_divide(A,B3,C2)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3),C2) ) ) ) ).

% div_mult_swap
tff(fact_2871_div__div__eq__right,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,B3: A,A3: A] :
          ( dvd_dvd(A,C2,B3)
         => ( dvd_dvd(A,B3,A3)
           => ( divide_divide(A,A3,divide_divide(A,B3,C2)) = aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,A3,B3)),C2) ) ) ) ) ).

% div_div_eq_right
tff(fact_2872_dvd__div__mult2__eq,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B3: A,C2: A,A3: A] :
          ( dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2),A3)
         => ( divide_divide(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2)) = divide_divide(A,divide_divide(A,A3,B3),C2) ) ) ) ).

% dvd_div_mult2_eq
tff(fact_2873_dvd__mult__imp__div,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,C2: A,B3: A] :
          ( dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2),B3)
         => dvd_dvd(A,A3,divide_divide(A,B3,C2)) ) ) ).

% dvd_mult_imp_div
tff(fact_2874_div__mult__div__if__dvd,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B3: A,A3: A,D3: A,C2: A] :
          ( dvd_dvd(A,B3,A3)
         => ( dvd_dvd(A,D3,C2)
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,A3,B3)),divide_divide(A,C2,D3)) = 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),B3),D3)) ) ) ) ) ).

% div_mult_div_if_dvd
tff(fact_2875_unit__div__cancel,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B3: A,C2: A] :
          ( dvd_dvd(A,A3,one_one(A))
         => ( ( divide_divide(A,B3,A3) = divide_divide(A,C2,A3) )
          <=> ( B3 = C2 ) ) ) ) ).

% unit_div_cancel
tff(fact_2876_div__unit__dvd__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B3: A,A3: A,C2: A] :
          ( dvd_dvd(A,B3,one_one(A))
         => ( dvd_dvd(A,divide_divide(A,A3,B3),C2)
          <=> dvd_dvd(A,A3,C2) ) ) ) ).

% div_unit_dvd_iff
tff(fact_2877_dvd__div__unit__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B3: A,A3: A,C2: A] :
          ( dvd_dvd(A,B3,one_one(A))
         => ( dvd_dvd(A,A3,divide_divide(A,C2,B3))
          <=> dvd_dvd(A,A3,C2) ) ) ) ).

% dvd_div_unit_iff
tff(fact_2878_mod__0__imp__dvd,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A3: A,B3: A] :
          ( ( modulo_modulo(A,A3,B3) = zero_zero(A) )
         => dvd_dvd(A,B3,A3) ) ) ).

% mod_0_imp_dvd
tff(fact_2879_dvd__eq__mod__eq__0,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A3: A,B3: A] :
          ( dvd_dvd(A,A3,B3)
        <=> ( modulo_modulo(A,B3,A3) = zero_zero(A) ) ) ) ).

% dvd_eq_mod_eq_0
tff(fact_2880_mod__eq__0__iff__dvd,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A3: A,B3: A] :
          ( ( modulo_modulo(A,A3,B3) = zero_zero(A) )
        <=> dvd_dvd(A,B3,A3) ) ) ).

% mod_eq_0_iff_dvd
tff(fact_2881_dvd__neg__div,axiom,
    ! [A: $tType] :
      ( idom_divide(A)
     => ! [B3: A,A3: A] :
          ( dvd_dvd(A,B3,A3)
         => ( divide_divide(A,aa(A,A,uminus_uminus(A),A3),B3) = aa(A,A,uminus_uminus(A),divide_divide(A,A3,B3)) ) ) ) ).

% dvd_neg_div
tff(fact_2882_dvd__div__neg,axiom,
    ! [A: $tType] :
      ( idom_divide(A)
     => ! [B3: A,A3: A] :
          ( dvd_dvd(A,B3,A3)
         => ( divide_divide(A,A3,aa(A,A,uminus_uminus(A),B3)) = aa(A,A,uminus_uminus(A),divide_divide(A,A3,B3)) ) ) ) ).

% dvd_div_neg
tff(fact_2883_mod__eq__dvd__iff,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A3: A,C2: A,B3: A] :
          ( ( modulo_modulo(A,A3,C2) = modulo_modulo(A,B3,C2) )
        <=> dvd_dvd(A,C2,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B3)) ) ) ).

% mod_eq_dvd_iff
tff(fact_2884_dvd__minus__mod,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [B3: A,A3: A] : dvd_dvd(A,B3,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),modulo_modulo(A,A3,B3))) ) ).

% dvd_minus_mod
tff(fact_2885_nat__dvd__not__less,axiom,
    ! [M2: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),N)
       => ~ dvd_dvd(nat,N,M2) ) ) ).

% nat_dvd_not_less
tff(fact_2886_dvd__pos__nat,axiom,
    ! [N: nat,M2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
     => ( dvd_dvd(nat,M2,N)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M2) ) ) ).

% dvd_pos_nat
tff(fact_2887_dvd__minus__self,axiom,
    ! [M2: nat,N: nat] :
      ( dvd_dvd(nat,M2,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M2))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),M2)
        | dvd_dvd(nat,M2,N) ) ) ).

% dvd_minus_self
tff(fact_2888_zdvd__mono,axiom,
    ! [K3: int,M2: int,T2: int] :
      ( ( K3 != zero_zero(int) )
     => ( dvd_dvd(int,M2,T2)
      <=> dvd_dvd(int,aa(int,int,aa(int,fun(int,int),times_times(int),K3),M2),aa(int,int,aa(int,fun(int,int),times_times(int),K3),T2)) ) ) ).

% zdvd_mono
tff(fact_2889_subseqs__refl,axiom,
    ! [A: $tType,Xsa: list(A)] : member(list(A),Xsa,aa(list(list(A)),set(list(A)),set2(list(A)),aa(list(A),list(list(A)),subseqs(A),Xsa))) ).

% subseqs_refl
tff(fact_2890_unity__coeff__ex,axiom,
    ! [A: $tType] :
      ( ( dvd(A)
        & semiring_0(A) )
     => ! [Pa: fun(A,$o),L: A] :
          ( ? [X6: A] : aa(A,$o,Pa,aa(A,A,aa(A,fun(A,A),times_times(A),L),X6))
        <=> ? [X6: A] :
              ( dvd_dvd(A,L,aa(A,A,aa(A,fun(A,A),plus_plus(A),X6),zero_zero(A)))
              & aa(A,$o,Pa,X6) ) ) ) ).

% unity_coeff_ex
tff(fact_2891_unit__dvdE,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B3: A] :
          ( dvd_dvd(A,A3,one_one(A))
         => ~ ( ( A3 != zero_zero(A) )
             => ! [C5: A] : ( B3 != aa(A,A,aa(A,fun(A,A),times_times(A),A3),C5) ) ) ) ) ).

% unit_dvdE
tff(fact_2892_inf__period_I4_J,axiom,
    ! [A: $tType] :
      ( ( comm_ring(A)
        & dvd(A) )
     => ! [D3: A,D2: A,T2: A] :
          ( dvd_dvd(A,D3,D2)
         => ! [X4: A,K2: A] :
              ( ~ dvd_dvd(A,D3,aa(A,A,aa(A,fun(A,A),plus_plus(A),X4),T2))
            <=> ~ 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),X4),aa(A,A,aa(A,fun(A,A),times_times(A),K2),D2))),T2)) ) ) ) ).

% inf_period(4)
tff(fact_2893_inf__period_I3_J,axiom,
    ! [A: $tType] :
      ( ( comm_ring(A)
        & dvd(A) )
     => ! [D3: A,D2: A,T2: A] :
          ( dvd_dvd(A,D3,D2)
         => ! [X4: A,K2: A] :
              ( dvd_dvd(A,D3,aa(A,A,aa(A,fun(A,A),plus_plus(A),X4),T2))
            <=> 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),X4),aa(A,A,aa(A,fun(A,A),times_times(A),K2),D2))),T2)) ) ) ) ).

% inf_period(3)
tff(fact_2894_dvd__div__eq__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B3: A,C2: A] :
          ( ( A3 != zero_zero(A) )
         => ( dvd_dvd(A,A3,B3)
           => ( ( divide_divide(A,B3,A3) = C2 )
            <=> ( B3 = aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3) ) ) ) ) ) ).

% dvd_div_eq_mult
tff(fact_2895_div__dvd__iff__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B3: A,A3: A,C2: A] :
          ( ( B3 != zero_zero(A) )
         => ( dvd_dvd(A,B3,A3)
           => ( dvd_dvd(A,divide_divide(A,A3,B3),C2)
            <=> dvd_dvd(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3)) ) ) ) ) ).

% div_dvd_iff_mult
tff(fact_2896_dvd__div__iff__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,B3: A,A3: A] :
          ( ( C2 != zero_zero(A) )
         => ( dvd_dvd(A,C2,B3)
           => ( dvd_dvd(A,A3,divide_divide(A,B3,C2))
            <=> dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2),B3) ) ) ) ) ).

% dvd_div_iff_mult
tff(fact_2897_dvd__div__div__eq__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,C2: A,B3: A,D3: A] :
          ( ( A3 != zero_zero(A) )
         => ( ( C2 != zero_zero(A) )
           => ( dvd_dvd(A,A3,B3)
             => ( dvd_dvd(A,C2,D3)
               => ( ( divide_divide(A,B3,A3) = divide_divide(A,D3,C2) )
                <=> ( aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),D3) ) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
tff(fact_2898_unit__div__eq__0__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B3: A,A3: A] :
          ( dvd_dvd(A,B3,one_one(A))
         => ( ( divide_divide(A,A3,B3) = zero_zero(A) )
          <=> ( A3 = zero_zero(A) ) ) ) ) ).

% unit_div_eq_0_iff
tff(fact_2899_is__unit__div__mult2__eq,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B3: A,C2: A,A3: A] :
          ( dvd_dvd(A,B3,one_one(A))
         => ( dvd_dvd(A,C2,one_one(A))
           => ( divide_divide(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2)) = divide_divide(A,divide_divide(A,A3,B3),C2) ) ) ) ) ).

% is_unit_div_mult2_eq
tff(fact_2900_unit__div__mult__swap,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,A3: A,B3: A] :
          ( dvd_dvd(A,C2,one_one(A))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),divide_divide(A,B3,C2)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3),C2) ) ) ) ).

% unit_div_mult_swap
tff(fact_2901_unit__div__commute,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B3: A,A3: A,C2: A] :
          ( dvd_dvd(A,B3,one_one(A))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,A3,B3)),C2) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2),B3) ) ) ) ).

% unit_div_commute
tff(fact_2902_div__mult__unit2,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,B3: A,A3: A] :
          ( dvd_dvd(A,C2,one_one(A))
         => ( dvd_dvd(A,B3,A3)
           => ( divide_divide(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2)) = divide_divide(A,divide_divide(A,A3,B3),C2) ) ) ) ) ).

% div_mult_unit2
tff(fact_2903_unit__eq__div2,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B3: A,A3: A,C2: A] :
          ( dvd_dvd(A,B3,one_one(A))
         => ( ( A3 = divide_divide(A,C2,B3) )
          <=> ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3) = C2 ) ) ) ) ).

% unit_eq_div2
tff(fact_2904_unit__eq__div1,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B3: A,A3: A,C2: A] :
          ( dvd_dvd(A,B3,one_one(A))
         => ( ( divide_divide(A,A3,B3) = C2 )
          <=> ( A3 = aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3) ) ) ) ) ).

% unit_eq_div1
tff(fact_2905_dvd__imp__le,axiom,
    ! [K3: nat,N: nat] :
      ( dvd_dvd(nat,K3,N)
     => ( 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),K3),N) ) ) ).

% dvd_imp_le
tff(fact_2906_nat__mult__dvd__cancel1,axiom,
    ! [K3: nat,M2: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K3)
     => ( dvd_dvd(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K3),M2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K3),N))
      <=> dvd_dvd(nat,M2,N) ) ) ).

% nat_mult_dvd_cancel1
tff(fact_2907_dvd__mult__cancel,axiom,
    ! [K3: nat,M2: nat,N: nat] :
      ( dvd_dvd(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K3),M2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K3),N))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K3)
       => dvd_dvd(nat,M2,N) ) ) ).

% dvd_mult_cancel
tff(fact_2908_fact__fact__dvd__fact,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K3: nat,N: nat] : dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),semiring_char_0_fact(A,K3)),semiring_char_0_fact(A,N)),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K3),N))) ) ).

% fact_fact_dvd_fact
tff(fact_2909_mod__greater__zero__iff__not__dvd,axiom,
    ! [M2: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),modulo_modulo(nat,M2,N))
    <=> ~ dvd_dvd(nat,N,M2) ) ).

% mod_greater_zero_iff_not_dvd
tff(fact_2910_stable__imp__take__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A,N: nat] :
          ( ( divide_divide(A,A3,numeral_numeral(A,bit0(one2))) = A3 )
         => ( bit_se2584673776208193580ke_bit(A,N,A3) = $ite(dvd_dvd(A,numeral_numeral(A,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),numeral_numeral(A,bit0(one2))),N)),one_one(A))) ) ) ) ).

% stable_imp_take_bit_eq
tff(fact_2911_Cons__in__subseqsD,axiom,
    ! [A: $tType,Y: A,Ys: list(A),Xsa: list(A)] :
      ( member(list(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys),aa(list(list(A)),set(list(A)),set2(list(A)),aa(list(A),list(list(A)),subseqs(A),Xsa)))
     => member(list(A),Ys,aa(list(list(A)),set(list(A)),set2(list(A)),aa(list(A),list(list(A)),subseqs(A),Xsa))) ) ).

% Cons_in_subseqsD
tff(fact_2912_is__unit__div__mult__cancel__right,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B3: A] :
          ( ( A3 != zero_zero(A) )
         => ( dvd_dvd(A,B3,one_one(A))
           => ( divide_divide(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),B3),A3)) = divide_divide(A,one_one(A),B3) ) ) ) ) ).

% is_unit_div_mult_cancel_right
tff(fact_2913_is__unit__div__mult__cancel__left,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B3: A] :
          ( ( A3 != zero_zero(A) )
         => ( dvd_dvd(A,B3,one_one(A))
           => ( divide_divide(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)) = divide_divide(A,one_one(A),B3) ) ) ) ) ).

% is_unit_div_mult_cancel_left
tff(fact_2914_is__unitE,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,C2: A] :
          ( dvd_dvd(A,A3,one_one(A))
         => ~ ( ( A3 != zero_zero(A) )
             => ! [B4: A] :
                  ( ( B4 != zero_zero(A) )
                 => ( dvd_dvd(A,B4,one_one(A))
                   => ( ( divide_divide(A,one_one(A),A3) = B4 )
                     => ( ( divide_divide(A,one_one(A),B4) = A3 )
                       => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),B4) = one_one(A) )
                         => ( divide_divide(A,C2,A3) != aa(A,A,aa(A,fun(A,A),times_times(A),C2),B4) ) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_2915_subseqs__distinctD,axiom,
    ! [A: $tType,Ys: list(A),Xsa: list(A)] :
      ( member(list(A),Ys,aa(list(list(A)),set(list(A)),set2(list(A)),aa(list(A),list(list(A)),subseqs(A),Xsa)))
     => ( aa(list(A),$o,distinct(A),Xsa)
       => aa(list(A),$o,distinct(A),Ys) ) ) ).

% subseqs_distinctD
tff(fact_2916_dvd__power,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [N: nat,Xa: A] :
          ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
            | ( Xa = one_one(A) ) )
         => dvd_dvd(A,Xa,aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),N)) ) ) ).

% dvd_power
tff(fact_2917_dvd__mult__cancel2,axiom,
    ! [M2: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M2)
     => ( dvd_dvd(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),M2),M2)
      <=> ( N = one_one(nat) ) ) ) ).

% dvd_mult_cancel2
tff(fact_2918_dvd__mult__cancel1,axiom,
    ! [M2: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M2)
     => ( dvd_dvd(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M2),N),M2)
      <=> ( N = one_one(nat) ) ) ) ).

% dvd_mult_cancel1
tff(fact_2919_choose__dvd,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K3: nat,N: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K3),N)
         => dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),semiring_char_0_fact(A,K3)),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K3))),semiring_char_0_fact(A,N)) ) ) ).

% choose_dvd
tff(fact_2920_even__even__mod__4__iff,axiom,
    ! [N: nat] :
      ( dvd_dvd(nat,numeral_numeral(nat,bit0(one2)),N)
    <=> dvd_dvd(nat,numeral_numeral(nat,bit0(one2)),modulo_modulo(nat,N,numeral_numeral(nat,bit0(bit0(one2))))) ) ).

% even_even_mod_4_iff
tff(fact_2921_mod__nat__eqI,axiom,
    ! [R2: nat,N: nat,M2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),R2),N)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),R2),M2)
       => ( dvd_dvd(nat,N,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),R2))
         => ( modulo_modulo(nat,M2,N) = R2 ) ) ) ) ).

% mod_nat_eqI
tff(fact_2922_power__dvd__imp__le,axiom,
    ! [Ia: nat,M2: nat,N: nat] :
      ( dvd_dvd(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Ia),M2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Ia),N))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),one_one(nat)),Ia)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),N) ) ) ).

% power_dvd_imp_le
tff(fact_2923_take__bit__Suc__bit0,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: nat,K3: num] : ( bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,N),numeral_numeral(A,bit0(K3))) = aa(A,A,aa(A,fun(A,A),times_times(A),bit_se2584673776208193580ke_bit(A,N,numeral_numeral(A,K3))),numeral_numeral(A,bit0(one2))) ) ) ).

% take_bit_Suc_bit0
tff(fact_2924_take__bit__nat__eq__self,axiom,
    ! [M2: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),numeral_numeral(nat,bit0(one2))),N))
     => ( bit_se2584673776208193580ke_bit(nat,N,M2) = M2 ) ) ).

% take_bit_nat_eq_self
tff(fact_2925_take__bit__nat__less__exp,axiom,
    ! [N: nat,M2: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),bit_se2584673776208193580ke_bit(nat,N,M2)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),numeral_numeral(nat,bit0(one2))),N)) ).

% take_bit_nat_less_exp
tff(fact_2926_take__bit__nat__eq__self__iff,axiom,
    ! [N: nat,M2: nat] :
      ( ( bit_se2584673776208193580ke_bit(nat,N,M2) = M2 )
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),numeral_numeral(nat,bit0(one2))),N)) ) ).

% take_bit_nat_eq_self_iff
tff(fact_2927_subseqs_Osimps_I1_J,axiom,
    ! [A: $tType] : ( aa(list(A),list(list(A)),subseqs(A),nil(A)) = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),nil(A)),nil(list(A))) ) ).

% subseqs.simps(1)
tff(fact_2928_take__bit__nat__less__self__iff,axiom,
    ! [N: nat,M2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),bit_se2584673776208193580ke_bit(nat,N,M2)),M2)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),numeral_numeral(nat,bit0(one2))),N)),M2) ) ).

% take_bit_nat_less_self_iff
tff(fact_2929_bits__induct,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Pa: fun(A,$o),A3: A] :
          ( ! [A4: A] :
              ( ( divide_divide(A,A4,numeral_numeral(A,bit0(one2))) = A4 )
             => aa(A,$o,Pa,A4) )
         => ( ! [A4: A,B4: $o] :
                ( aa(A,$o,Pa,A4)
               => ( ( divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa($o,A,zero_neq_one_of_bool(A),(B4))),aa(A,A,aa(A,fun(A,A),times_times(A),numeral_numeral(A,bit0(one2))),A4)),numeral_numeral(A,bit0(one2))) = A4 )
                 => aa(A,$o,Pa,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa($o,A,zero_neq_one_of_bool(A),(B4))),aa(A,A,aa(A,fun(A,A),times_times(A),numeral_numeral(A,bit0(one2))),A4))) ) )
           => aa(A,$o,Pa,A3) ) ) ) ).

% bits_induct
tff(fact_2930_exp__mod__exp,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [M2: nat,N: nat] : ( modulo_modulo(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),numeral_numeral(A,bit0(one2))),M2),aa(nat,A,aa(A,fun(nat,A),power_power(A),numeral_numeral(A,bit0(one2))),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),N))),aa(nat,A,aa(A,fun(nat,A),power_power(A),numeral_numeral(A,bit0(one2))),M2)) ) ) ).

% exp_mod_exp
tff(fact_2931_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),numeral_numeral(A,bit0(one2))),aa(nat,nat,suc,N))),one_one(A)) ) ) ).

% take_bit_Suc_minus_1_eq
tff(fact_2932_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,divide_divide(A,A3,numeral_numeral(A,bit0(one2))))),numeral_numeral(A,bit0(one2)))),modulo_modulo(A,A3,numeral_numeral(A,bit0(one2)))) ) ) ).

% take_bit_Suc
tff(fact_2933_take__bit__numeral__minus__1__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [K3: num] : ( bit_se2584673776208193580ke_bit(A,numeral_numeral(nat,K3),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),numeral_numeral(A,bit0(one2))),numeral_numeral(nat,K3))),one_one(A)) ) ) ).

% take_bit_numeral_minus_1_eq
tff(fact_2934_take__bit__int__less__eq,axiom,
    ! [N: nat,K3: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),numeral_numeral(int,bit0(one2))),N)),K3)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),bit_se2584673776208193580ke_bit(int,N,K3)),aa(int,int,aa(int,fun(int,int),minus_minus(int),K3),aa(nat,int,aa(int,fun(nat,int),power_power(int),numeral_numeral(int,bit0(one2))),N))) ) ) ).

% take_bit_int_less_eq
tff(fact_2935_even__mod__4__div__2,axiom,
    ! [N: nat] :
      ( ( modulo_modulo(nat,N,numeral_numeral(nat,bit0(bit0(one2)))) = aa(nat,nat,suc,zero_zero(nat)) )
     => dvd_dvd(nat,numeral_numeral(nat,bit0(one2)),divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,suc,zero_zero(nat))),numeral_numeral(nat,bit0(one2)))) ) ).

% even_mod_4_div_2
tff(fact_2936_exp__div__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M2: nat,N: nat] :
          ( divide_divide(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),numeral_numeral(A,bit0(one2))),M2),aa(nat,A,aa(A,fun(nat,A),power_power(A),numeral_numeral(A,bit0(one2))),N)) = aa(A,A,
              aa(A,fun(A,A),times_times(A),
                aa($o,A,zero_neq_one_of_bool(A),
                  ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),numeral_numeral(A,bit0(one2))),M2) != zero_zero(A) )
                  & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),M2) ))),
              aa(nat,A,aa(A,fun(nat,A),power_power(A),numeral_numeral(A,bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),N))) ) ) ).

% exp_div_exp_eq
tff(fact_2937_even__mask__div__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M2: nat,N: nat] :
          ( dvd_dvd(A,numeral_numeral(A,bit0(one2)),divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),numeral_numeral(A,bit0(one2))),M2)),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),numeral_numeral(A,bit0(one2))),N)))
        <=> ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),numeral_numeral(A,bit0(one2))),N) = zero_zero(A) )
            | aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),N) ) ) ) ).

% even_mask_div_iff
tff(fact_2938_power__le__zero__eq__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,W2: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),numeral_numeral(nat,W2))),zero_zero(A))
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),numeral_numeral(nat,W2))
            & ( ( ~ dvd_dvd(nat,numeral_numeral(nat,bit0(one2)),numeral_numeral(nat,W2))
                & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),zero_zero(A)) )
              | ( dvd_dvd(nat,numeral_numeral(nat,bit0(one2)),numeral_numeral(nat,W2))
                & ( A3 = zero_zero(A) ) ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_2939_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [N: nat] :
          ( dvd_dvd(A,numeral_numeral(A,bit0(one2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),numeral_numeral(A,bit0(one2))),N)),one_one(A)))
        <=> ( N = zero_zero(nat) ) ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_2940_odd__two__times__div__two__succ,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A3: A] :
          ( ~ dvd_dvd(A,numeral_numeral(A,bit0(one2)),A3)
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),numeral_numeral(A,bit0(one2))),divide_divide(A,A3,numeral_numeral(A,bit0(one2))))),one_one(A)) = A3 ) ) ) ).

% odd_two_times_div_two_succ
tff(fact_2941_even__power,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A3: A,N: nat] :
          ( dvd_dvd(A,numeral_numeral(A,bit0(one2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N))
        <=> ( dvd_dvd(A,numeral_numeral(A,bit0(one2)),A3)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N) ) ) ) ).

% even_power
tff(fact_2942_zero__less__power__eq__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,W2: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),numeral_numeral(nat,W2)))
        <=> ( ( numeral_numeral(nat,W2) = zero_zero(nat) )
            | ( dvd_dvd(nat,numeral_numeral(nat,bit0(one2)),numeral_numeral(nat,W2))
              & ( A3 != zero_zero(A) ) )
            | ( ~ dvd_dvd(nat,numeral_numeral(nat,bit0(one2)),numeral_numeral(nat,W2))
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3) ) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_2943_even__diff__nat,axiom,
    ! [M2: nat,N: nat] :
      ( dvd_dvd(nat,numeral_numeral(nat,bit0(one2)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),N))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),N)
        | dvd_dvd(nat,numeral_numeral(nat,bit0(one2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),N)) ) ) ).

% even_diff_nat
tff(fact_2944_even__mult__iff,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A3: A,B3: A] :
          ( dvd_dvd(A,numeral_numeral(A,bit0(one2)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3))
        <=> ( dvd_dvd(A,numeral_numeral(A,bit0(one2)),A3)
            | dvd_dvd(A,numeral_numeral(A,bit0(one2)),B3) ) ) ) ).

% even_mult_iff
tff(fact_2945_even__diff,axiom,
    ! [A: $tType] :
      ( ring_parity(A)
     => ! [A3: A,B3: A] :
          ( dvd_dvd(A,numeral_numeral(A,bit0(one2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B3))
        <=> dvd_dvd(A,numeral_numeral(A,bit0(one2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3)) ) ) ).

% even_diff
tff(fact_2946_power__less__zero__eq__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,W2: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),numeral_numeral(nat,W2))),zero_zero(A))
        <=> ( ~ dvd_dvd(nat,numeral_numeral(nat,bit0(one2)),numeral_numeral(nat,W2))
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A)) ) ) ) ).

% power_less_zero_eq_numeral
tff(fact_2947_power__less__zero__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,N: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)),zero_zero(A))
        <=> ( ~ dvd_dvd(nat,numeral_numeral(nat,bit0(one2)),N)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A)) ) ) ) ).

% power_less_zero_eq
tff(fact_2948_evenE,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A3: A] :
          ( dvd_dvd(A,numeral_numeral(A,bit0(one2)),A3)
         => ~ ! [B4: A] : ( A3 != aa(A,A,aa(A,fun(A,A),times_times(A),numeral_numeral(A,bit0(one2))),B4) ) ) ) ).

% evenE
tff(fact_2949_odd__pos,axiom,
    ! [N: nat] :
      ( ~ dvd_dvd(nat,numeral_numeral(nat,bit0(one2)),N)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N) ) ).

% odd_pos
tff(fact_2950_even__two__times__div__two,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A3: A] :
          ( dvd_dvd(A,numeral_numeral(A,bit0(one2)),A3)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),numeral_numeral(A,bit0(one2))),divide_divide(A,A3,numeral_numeral(A,bit0(one2)))) = A3 ) ) ) ).

% even_two_times_div_two
tff(fact_2951_oddE,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A3: A] :
          ( ~ dvd_dvd(A,numeral_numeral(A,bit0(one2)),A3)
         => ~ ! [B4: A] : ( A3 != aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),numeral_numeral(A,bit0(one2))),B4)),one_one(A)) ) ) ) ).

% oddE
tff(fact_2952_zero__less__power__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,N: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N))
        <=> ( ( N = zero_zero(nat) )
            | ( dvd_dvd(nat,numeral_numeral(nat,bit0(one2)),N)
              & ( A3 != zero_zero(A) ) )
            | ( ~ dvd_dvd(nat,numeral_numeral(nat,bit0(one2)),N)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3) ) ) ) ) ).

% zero_less_power_eq
tff(fact_2953_power__le__zero__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,N: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)),zero_zero(A))
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
            & ( ( ~ dvd_dvd(nat,numeral_numeral(nat,bit0(one2)),N)
                & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),zero_zero(A)) )
              | ( dvd_dvd(nat,numeral_numeral(nat,bit0(one2)),N)
                & ( A3 = zero_zero(A) ) ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_2954_even__mask__div__iff_H,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [M2: nat,N: nat] :
          ( dvd_dvd(A,numeral_numeral(A,bit0(one2)),divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),numeral_numeral(A,bit0(one2))),M2)),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),numeral_numeral(A,bit0(one2))),N)))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),N) ) ) ).

% even_mask_div_iff'
tff(fact_2955_Divides_Oadjust__div__eq,axiom,
    ! [Q2: int,R2: int] : ( adjust_div(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),R2)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Q2),aa($o,int,zero_neq_one_of_bool(int),R2 != zero_zero(int))) ) ).

% Divides.adjust_div_eq
tff(fact_2956_odd__mod__4__div__2,axiom,
    ! [N: nat] :
      ( ( modulo_modulo(nat,N,numeral_numeral(nat,bit0(bit0(one2)))) = numeral_numeral(nat,bit1(one2)) )
     => ~ dvd_dvd(nat,numeral_numeral(nat,bit0(one2)),divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,suc,zero_zero(nat))),numeral_numeral(nat,bit0(one2)))) ) ).

% odd_mod_4_div_2
tff(fact_2957_divmod__divmod__step,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M2: num,N: num] :
          ( unique8689654367752047608divmod(A,M2,N) = $ite(aa(num,$o,aa(num,fun(num,$o),ord_less(num),M2),N),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),zero_zero(A)),numeral_numeral(A,M2)),unique1321980374590559556d_step(A,N,unique8689654367752047608divmod(A,M2,bit0(N)))) ) ) ).

% divmod_divmod_step
tff(fact_2958_divmod__algorithm__code_I3_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [N: num] : ( unique8689654367752047608divmod(A,one2,bit0(N)) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),zero_zero(A)),numeral_numeral(A,one2)) ) ) ).

% divmod_algorithm_code(3)
tff(fact_2959_one__xor__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : ( bit_se5824344971392196577ns_xor(A,one_one(A),A3) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa($o,A,zero_neq_one_of_bool(A),dvd_dvd(A,numeral_numeral(A,bit0(one2)),A3)))),aa($o,A,zero_neq_one_of_bool(A),~ dvd_dvd(A,numeral_numeral(A,bit0(one2)),A3))) ) ) ).

% one_xor_eq
tff(fact_2960_xor__numerals_I3_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xa: num,Y: num] : ( bit_se5824344971392196577ns_xor(A,numeral_numeral(A,bit0(Xa)),numeral_numeral(A,bit0(Y))) = aa(A,A,aa(A,fun(A,A),times_times(A),numeral_numeral(A,bit0(one2))),bit_se5824344971392196577ns_xor(A,numeral_numeral(A,Xa),numeral_numeral(A,Y))) ) ) ).

% xor_numerals(3)
tff(fact_2961_divmod__algorithm__code_I2_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M2: num] : ( unique8689654367752047608divmod(A,M2,one2) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),numeral_numeral(A,M2)),zero_zero(A)) ) ) ).

% divmod_algorithm_code(2)
tff(fact_2962_xor__numerals_I7_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xa: num,Y: num] : ( bit_se5824344971392196577ns_xor(A,numeral_numeral(A,bit1(Xa)),numeral_numeral(A,bit1(Y))) = aa(A,A,aa(A,fun(A,A),times_times(A),numeral_numeral(A,bit0(one2))),bit_se5824344971392196577ns_xor(A,numeral_numeral(A,Xa),numeral_numeral(A,Y))) ) ) ).

% xor_numerals(7)
tff(fact_2963_divmod__algorithm__code_I4_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [N: num] : ( unique8689654367752047608divmod(A,one2,bit1(N)) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),zero_zero(A)),numeral_numeral(A,one2)) ) ) ).

% divmod_algorithm_code(4)
tff(fact_2964_divmod__algorithm__code_I7_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M2: num,N: num] :
          ( unique8689654367752047608divmod(A,bit0(M2),bit1(N)) = $ite(aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),M2),N),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),zero_zero(A)),numeral_numeral(A,bit0(M2))),unique1321980374590559556d_step(A,bit1(N),unique8689654367752047608divmod(A,bit0(M2),bit0(bit1(N))))) ) ) ).

% divmod_algorithm_code(7)
tff(fact_2965_xor__numerals_I6_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xa: num,Y: num] : ( bit_se5824344971392196577ns_xor(A,numeral_numeral(A,bit1(Xa)),numeral_numeral(A,bit0(Y))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),numeral_numeral(A,bit0(one2))),bit_se5824344971392196577ns_xor(A,numeral_numeral(A,Xa),numeral_numeral(A,Y)))) ) ) ).

% xor_numerals(6)
tff(fact_2966_xor__numerals_I4_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xa: num,Y: num] : ( bit_se5824344971392196577ns_xor(A,numeral_numeral(A,bit0(Xa)),numeral_numeral(A,bit1(Y))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),numeral_numeral(A,bit0(one2))),bit_se5824344971392196577ns_xor(A,numeral_numeral(A,Xa),numeral_numeral(A,Y)))) ) ) ).

% xor_numerals(4)
tff(fact_2967_divmod__algorithm__code_I8_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M2: num,N: num] :
          ( unique8689654367752047608divmod(A,bit1(M2),bit1(N)) = $ite(aa(num,$o,aa(num,fun(num,$o),ord_less(num),M2),N),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),zero_zero(A)),numeral_numeral(A,bit1(M2))),unique1321980374590559556d_step(A,bit1(N),unique8689654367752047608divmod(A,bit1(M2),bit0(bit1(N))))) ) ) ).

% divmod_algorithm_code(8)
tff(fact_2968_xor__num_Ocases,axiom,
    ! [Xa: product_prod(num,num)] :
      ( ( Xa != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),one2) )
     => ( ! [N5: num] : ( Xa != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),bit0(N5)) )
       => ( ! [N5: num] : ( Xa != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),bit1(N5)) )
         => ( ! [M6: num] : ( Xa != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),bit0(M6)),one2) )
           => ( ! [M6: num,N5: num] : ( Xa != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),bit0(M6)),bit0(N5)) )
             => ( ! [M6: num,N5: num] : ( Xa != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),bit0(M6)),bit1(N5)) )
               => ( ! [M6: num] : ( Xa != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),bit1(M6)),one2) )
                 => ( ! [M6: num,N5: num] : ( Xa != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),bit1(M6)),bit0(N5)) )
                   => ~ ! [M6: num,N5: num] : ( Xa != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),bit1(M6)),bit1(N5)) ) ) ) ) ) ) ) ) ) ).

% xor_num.cases
tff(fact_2969_power3__eq__cube,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A3: A] : ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),numeral_numeral(nat,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_2970_divmod_H__nat__def,axiom,
    ! [M2: num,N: num] : ( unique8689654367752047608divmod(nat,M2,N) = aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),divide_divide(nat,numeral_numeral(nat,M2),numeral_numeral(nat,N))),modulo_modulo(nat,numeral_numeral(nat,M2),numeral_numeral(nat,N))) ) ).

% divmod'_nat_def
tff(fact_2971_divmod__int__def,axiom,
    ! [M2: num,N: num] : ( unique8689654367752047608divmod(int,M2,N) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),divide_divide(int,numeral_numeral(int,M2),numeral_numeral(int,N))),modulo_modulo(int,numeral_numeral(int,M2),numeral_numeral(int,N))) ) ).

% divmod_int_def
tff(fact_2972_divmod__def,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M2: num,N: num] : ( unique8689654367752047608divmod(A,M2,N) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),divide_divide(A,numeral_numeral(A,M2),numeral_numeral(A,N))),modulo_modulo(A,numeral_numeral(A,M2),numeral_numeral(A,N))) ) ) ).

% divmod_def
tff(fact_2973_take__bit__Suc__bit1,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: nat,K3: num] : ( bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,N),numeral_numeral(A,bit1(K3))) = 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,numeral_numeral(A,K3))),numeral_numeral(A,bit0(one2)))),one_one(A)) ) ) ).

% take_bit_Suc_bit1
tff(fact_2974_xor__one__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : ( bit_se5824344971392196577ns_xor(A,A3,one_one(A)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa($o,A,zero_neq_one_of_bool(A),dvd_dvd(A,numeral_numeral(A,bit0(one2)),A3)))),aa($o,A,zero_neq_one_of_bool(A),~ dvd_dvd(A,numeral_numeral(A,bit0(one2)),A3))) ) ) ).

% xor_one_eq
tff(fact_2975_take__bit__numeral__bit1,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [L: num,K3: num] : ( bit_se2584673776208193580ke_bit(A,numeral_numeral(nat,L),numeral_numeral(A,bit1(K3))) = 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),numeral_numeral(A,K3))),numeral_numeral(A,bit0(one2)))),one_one(A)) ) ) ).

% take_bit_numeral_bit1
tff(fact_2976_divmod__BitM__2__eq,axiom,
    ! [M2: num] : ( unique8689654367752047608divmod(int,bitM(M2),bit0(one2)) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),minus_minus(int),numeral_numeral(int,M2)),one_one(int))),one_one(int)) ) ).

% divmod_BitM_2_eq
tff(fact_2977_mset__size2elem,axiom,
    ! [A: $tType,Pa: multiset(A),Q2: A,Q5: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(multiset(A),nat,size_size(multiset(A)),Pa)),numeral_numeral(nat,bit0(one2)))
     => ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),add_mset(A,Q2,zero_zero(multiset(A)))),add_mset(A,Q5,zero_zero(multiset(A))))),Pa)
       => ( Pa = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),add_mset(A,Q2,zero_zero(multiset(A)))),add_mset(A,Q5,zero_zero(multiset(A)))) ) ) ) ).

% mset_size2elem
tff(fact_2978_or__numerals_I7_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xa: num,Y: num] : ( bit_se1065995026697491101ons_or(A,numeral_numeral(A,bit1(Xa)),numeral_numeral(A,bit1(Y))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),numeral_numeral(A,bit0(one2))),bit_se1065995026697491101ons_or(A,numeral_numeral(A,Xa),numeral_numeral(A,Y)))) ) ) ).

% or_numerals(7)
tff(fact_2979_or__numerals_I6_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xa: num,Y: num] : ( bit_se1065995026697491101ons_or(A,numeral_numeral(A,bit1(Xa)),numeral_numeral(A,bit0(Y))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),numeral_numeral(A,bit0(one2))),bit_se1065995026697491101ons_or(A,numeral_numeral(A,Xa),numeral_numeral(A,Y)))) ) ) ).

% or_numerals(6)
tff(fact_2980_less__numeral__Suc,axiom,
    ! [K3: num,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),numeral_numeral(nat,K3)),aa(nat,nat,suc,N))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pred_numeral(K3)),N) ) ).

% less_numeral_Suc
tff(fact_2981_less__Suc__numeral,axiom,
    ! [N: nat,K3: num] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,N)),numeral_numeral(nat,K3))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),pred_numeral(K3)) ) ).

% less_Suc_numeral
tff(fact_2982_or__numerals_I3_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xa: num,Y: num] : ( bit_se1065995026697491101ons_or(A,numeral_numeral(A,bit0(Xa)),numeral_numeral(A,bit0(Y))) = aa(A,A,aa(A,fun(A,A),times_times(A),numeral_numeral(A,bit0(one2))),bit_se1065995026697491101ons_or(A,numeral_numeral(A,Xa),numeral_numeral(A,Y))) ) ) ).

% or_numerals(3)
tff(fact_2983_or__numerals_I4_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xa: num,Y: num] : ( bit_se1065995026697491101ons_or(A,numeral_numeral(A,bit0(Xa)),numeral_numeral(A,bit1(Y))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),numeral_numeral(A,bit0(one2))),bit_se1065995026697491101ons_or(A,numeral_numeral(A,Xa),numeral_numeral(A,Y)))) ) ) ).

% or_numerals(4)
tff(fact_2984_mset__le__add__mset__decr__left1,axiom,
    ! [A: $tType,C2: A,A3: multiset(A),B3: multiset(A)] :
      ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),add_mset(A,C2,A3)),B3)
     => aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),A3),B3) ) ).

% mset_le_add_mset_decr_left1
tff(fact_2985_mset__union__2__elem,axiom,
    ! [A: $tType,A3: A,B3: A,C2: A,M5: multiset(A)] :
      ( ( add_mset(A,A3,add_mset(A,B3,zero_zero(multiset(A)))) = add_mset(A,C2,M5) )
     => ( ( ( add_mset(A,A3,zero_zero(multiset(A))) = M5 )
          & ( B3 = C2 ) )
        | ( ( A3 = C2 )
          & ( add_mset(A,B3,zero_zero(multiset(A))) = M5 ) ) ) ) ).

% mset_union_2_elem
tff(fact_2986_mset__le__add__mset,axiom,
    ! [A: $tType,Xa: A,B2: multiset(A),C3: multiset(A)] :
      ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),add_mset(A,Xa,B2)),C3)
     => ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),add_mset(A,Xa,zero_zero(multiset(A)))),C3)
        & aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),B2),C3) ) ) ).

% mset_le_add_mset
tff(fact_2987_mset__le__single__cases,axiom,
    ! [A: $tType,M5: multiset(A),A3: A] :
      ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),M5),add_mset(A,A3,zero_zero(multiset(A))))
     => ( ( M5 != zero_zero(multiset(A)) )
       => ( M5 = add_mset(A,A3,zero_zero(multiset(A))) ) ) ) ).

% mset_le_single_cases
tff(fact_2988_mset__le__add__mset__decr__left2,axiom,
    ! [A: $tType,C2: A,A3: multiset(A),B3: multiset(A)] :
      ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),add_mset(A,C2,A3)),B3)
     => aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),add_mset(A,C2,zero_zero(multiset(A)))),B3) ) ).

% mset_le_add_mset_decr_left2
tff(fact_2989_mset__le__subtract__add__mset__right,axiom,
    ! [A: $tType,Xa: A,B2: multiset(A),X3: multiset(A)] :
      ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),add_mset(A,Xa,B2)),X3)
     => ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),add_mset(A,Xa,zero_zero(multiset(A)))),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),X3),B2))
        & aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),B2),X3) ) ) ).

% mset_le_subtract_add_mset_right
tff(fact_2990_mset__le__subtract__add__mset__left,axiom,
    ! [A: $tType,Xa: A,B2: multiset(A),X3: multiset(A)] :
      ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),add_mset(A,Xa,B2)),X3)
     => ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),B2),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),X3),add_mset(A,Xa,zero_zero(multiset(A)))))
        & aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),add_mset(A,Xa,zero_zero(multiset(A)))),X3) ) ) ).

% mset_le_subtract_add_mset_left
tff(fact_2991_mset__unplusm__dist__cases2,axiom,
    ! [A: $tType,B2: multiset(A),C3: multiset(A),S2: A,Aa2: multiset(A)] :
      ( ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),B2),C3) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),add_mset(A,S2,zero_zero(multiset(A)))),Aa2) )
     => ( ( ( B2 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),add_mset(A,S2,zero_zero(multiset(A)))),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),B2),add_mset(A,S2,zero_zero(multiset(A))))) )
         => ( Aa2 != 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)),minus_minus(multiset(A)),B2),add_mset(A,S2,zero_zero(multiset(A))))),C3) ) )
       => ~ ( ( C3 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),add_mset(A,S2,zero_zero(multiset(A)))),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),C3),add_mset(A,S2,zero_zero(multiset(A))))) )
           => ( Aa2 != aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),B2),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),C3),add_mset(A,S2,zero_zero(multiset(A))))) ) ) ) ) ).

% mset_unplusm_dist_cases2
tff(fact_2992_mset__unplusm__dist__cases,axiom,
    ! [A: $tType,S2: A,Aa2: multiset(A),B2: multiset(A),C3: multiset(A)] :
      ( ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),add_mset(A,S2,zero_zero(multiset(A)))),Aa2) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),B2),C3) )
     => ( ( ( B2 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),add_mset(A,S2,zero_zero(multiset(A)))),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),B2),add_mset(A,S2,zero_zero(multiset(A))))) )
         => ( Aa2 != 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)),minus_minus(multiset(A)),B2),add_mset(A,S2,zero_zero(multiset(A))))),C3) ) )
       => ~ ( ( C3 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),add_mset(A,S2,zero_zero(multiset(A)))),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),C3),add_mset(A,S2,zero_zero(multiset(A))))) )
           => ( Aa2 != aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),B2),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),C3),add_mset(A,S2,zero_zero(multiset(A))))) ) ) ) ) ).

% mset_unplusm_dist_cases
tff(fact_2993_mset__single__cases2_H,axiom,
    ! [A: $tType,S2: A,C2: multiset(A),R5: A,C8: multiset(A)] :
      ( ( add_mset(A,S2,C2) = add_mset(A,R5,C8) )
     => ( ( ( S2 = R5 )
         => ( C2 != C8 ) )
       => ~ ! [Cc: multiset(A)] :
              ( ( C8 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),Cc),add_mset(A,S2,zero_zero(multiset(A)))) )
             => ( ( C2 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),Cc),add_mset(A,R5,zero_zero(multiset(A)))) )
               => ( ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),C8),add_mset(A,S2,zero_zero(multiset(A)))) = Cc )
                 => ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),C2),add_mset(A,R5,zero_zero(multiset(A)))) != Cc ) ) ) ) ) ) ).

% mset_single_cases2'
tff(fact_2994_mset__single__cases2,axiom,
    ! [A: $tType,S2: A,C2: multiset(A),R5: A,C8: multiset(A)] :
      ( ( add_mset(A,S2,C2) = add_mset(A,R5,C8) )
     => ( ( ( S2 = R5 )
         => ( C2 != C8 ) )
       => ~ ( ( C8 = 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)),minus_minus(multiset(A)),C8),add_mset(A,S2,zero_zero(multiset(A))))),add_mset(A,S2,zero_zero(multiset(A)))) )
           => ( ( C2 = 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)),minus_minus(multiset(A)),C2),add_mset(A,R5,zero_zero(multiset(A))))),add_mset(A,R5,zero_zero(multiset(A)))) )
             => ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),C2),add_mset(A,R5,zero_zero(multiset(A)))) != aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),C8),add_mset(A,S2,zero_zero(multiset(A)))) ) ) ) ) ) ).

% mset_single_cases2
tff(fact_2995_mset__single__cases_H,axiom,
    ! [A: $tType,S2: A,C2: multiset(A),R5: A,C8: multiset(A)] :
      ( ( add_mset(A,S2,C2) = add_mset(A,R5,C8) )
     => ( ( ( S2 = R5 )
         => ( C2 != C8 ) )
       => ~ ! [Cc: multiset(A)] :
              ( ( C8 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),add_mset(A,S2,zero_zero(multiset(A)))),Cc) )
             => ( ( C2 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),add_mset(A,R5,zero_zero(multiset(A)))),Cc) )
               => ( ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),C8),add_mset(A,S2,zero_zero(multiset(A)))) = Cc )
                 => ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),C2),add_mset(A,R5,zero_zero(multiset(A)))) != Cc ) ) ) ) ) ) ).

% mset_single_cases'
tff(fact_2996_mset__single__cases,axiom,
    ! [A: $tType,S2: A,C2: multiset(A),R5: A,C8: multiset(A)] :
      ( ( add_mset(A,S2,C2) = add_mset(A,R5,C8) )
     => ( ( ( S2 = R5 )
         => ( C2 != C8 ) )
       => ~ ( ( C8 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),add_mset(A,S2,zero_zero(multiset(A)))),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),C8),add_mset(A,S2,zero_zero(multiset(A))))) )
           => ( ( C2 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),add_mset(A,R5,zero_zero(multiset(A)))),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),C2),add_mset(A,R5,zero_zero(multiset(A))))) )
             => ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),C2),add_mset(A,R5,zero_zero(multiset(A)))) != aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),C8),add_mset(A,S2,zero_zero(multiset(A)))) ) ) ) ) ) ).

% mset_single_cases
tff(fact_2997_fact__numeral,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [K3: num] : ( semiring_char_0_fact(A,numeral_numeral(nat,K3)) = aa(A,A,aa(A,fun(A,A),times_times(A),numeral_numeral(A,K3)),semiring_char_0_fact(A,pred_numeral(K3))) ) ) ).

% fact_numeral
tff(fact_2998_numeral__BitM,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [N: num] : ( numeral_numeral(A,bitM(N)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),numeral_numeral(A,bit0(N))),one_one(A)) ) ) ).

% numeral_BitM
tff(fact_2999_mset__size__le1__cases,axiom,
    ! [A: $tType,M5: multiset(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(multiset(A),nat,size_size(multiset(A)),M5)),aa(nat,nat,suc,zero_zero(nat)))
     => ( ( M5 != zero_zero(multiset(A)) )
       => ~ ! [M6: A] : ( M5 != add_mset(A,M6,zero_zero(multiset(A))) ) ) ) ).

% mset_size_le1_cases
tff(fact_3000_take__bit__numeral__bit0,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [L: num,K3: num] : ( bit_se2584673776208193580ke_bit(A,numeral_numeral(nat,L),numeral_numeral(A,bit0(K3))) = aa(A,A,aa(A,fun(A,A),times_times(A),bit_se2584673776208193580ke_bit(A,pred_numeral(L),numeral_numeral(A,K3))),numeral_numeral(A,bit0(one2))) ) ) ).

% take_bit_numeral_bit0
tff(fact_3001_mask__numeral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: num] : ( bit_se2239418461657761734s_mask(A,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),numeral_numeral(A,bit0(one2))),bit_se2239418461657761734s_mask(A,pred_numeral(N)))) ) ) ).

% mask_numeral
tff(fact_3002_mask__Suc__double,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat] : ( bit_se2239418461657761734s_mask(A,aa(nat,nat,suc,N)) = bit_se1065995026697491101ons_or(A,one_one(A),aa(A,A,aa(A,fun(A,A),times_times(A),numeral_numeral(A,bit0(one2))),bit_se2239418461657761734s_mask(A,N))) ) ) ).

% mask_Suc_double
tff(fact_3003_divmod__nat__def,axiom,
    ! [M2: nat,N: nat] : ( divmod_nat(M2,N) = aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),divide_divide(nat,M2,N)),modulo_modulo(nat,M2,N)) ) ).

% divmod_nat_def
tff(fact_3004_mset__zip__take__Cons__drop__twice,axiom,
    ! [A: $tType,B: $tType,Xsa: list(A),Ys: list(B),J: nat,Xa: A,Y: B] :
      ( ( aa(list(A),nat,size_size(list(A)),Xsa) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),aa(list(A),nat,size_size(list(A)),Xsa))
       => ( aa(list(product_prod(A,B)),multiset(product_prod(A,B)),mset(product_prod(A,B)),aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),J),Xsa)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),J),Xsa)))),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),aa(list(B),list(B),aa(nat,fun(list(B),list(B)),take(B),J),Ys)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),aa(list(B),list(B),aa(nat,fun(list(B),list(B)),drop(B),J),Ys))))) = add_mset(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa),Y),aa(list(product_prod(A,B)),multiset(product_prod(A,B)),mset(product_prod(A,B)),aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),Xsa),Ys))) ) ) ) ).

% mset_zip_take_Cons_drop_twice
tff(fact_3005_and__numerals_I7_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xa: num,Y: num] : ( bit_se5824344872417868541ns_and(A,numeral_numeral(A,bit1(Xa)),numeral_numeral(A,bit1(Y))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),numeral_numeral(A,bit0(one2))),bit_se5824344872417868541ns_and(A,numeral_numeral(A,Xa),numeral_numeral(A,Y)))) ) ) ).

% and_numerals(7)
tff(fact_3006_mask__nat__positive__iff,axiom,
    ! [N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),bit_se2239418461657761734s_mask(nat,N))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N) ) ).

% mask_nat_positive_iff
tff(fact_3007_size__mset,axiom,
    ! [A: $tType,Xsa: list(A)] : ( aa(multiset(A),nat,size_size(multiset(A)),aa(list(A),multiset(A),mset(A),Xsa)) = aa(list(A),nat,size_size(list(A)),Xsa) ) ).

% size_mset
tff(fact_3008_and__numerals_I3_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xa: num,Y: num] : ( bit_se5824344872417868541ns_and(A,numeral_numeral(A,bit0(Xa)),numeral_numeral(A,bit0(Y))) = aa(A,A,aa(A,fun(A,A),times_times(A),numeral_numeral(A,bit0(one2))),bit_se5824344872417868541ns_and(A,numeral_numeral(A,Xa),numeral_numeral(A,Y))) ) ) ).

% and_numerals(3)
tff(fact_3009_and__numerals_I6_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xa: num,Y: num] : ( bit_se5824344872417868541ns_and(A,numeral_numeral(A,bit1(Xa)),numeral_numeral(A,bit0(Y))) = aa(A,A,aa(A,fun(A,A),times_times(A),numeral_numeral(A,bit0(one2))),bit_se5824344872417868541ns_and(A,numeral_numeral(A,Xa),numeral_numeral(A,Y))) ) ) ).

% and_numerals(6)
tff(fact_3010_and__numerals_I4_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xa: num,Y: num] : ( bit_se5824344872417868541ns_and(A,numeral_numeral(A,bit0(Xa)),numeral_numeral(A,bit1(Y))) = aa(A,A,aa(A,fun(A,A),times_times(A),numeral_numeral(A,bit0(one2))),bit_se5824344872417868541ns_and(A,numeral_numeral(A,Xa),numeral_numeral(A,Y))) ) ) ).

% and_numerals(4)
tff(fact_3011_ex__mset__zip__left,axiom,
    ! [A: $tType,B: $tType,Xsa: list(A),Ys: list(B),Xs3: list(A)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xsa) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( ( aa(list(A),multiset(A),mset(A),Xs3) = aa(list(A),multiset(A),mset(A),Xsa) )
       => ? [Ys3: list(B)] :
            ( ( aa(list(B),nat,size_size(list(B)),Ys3) = aa(list(A),nat,size_size(list(A)),Xs3) )
            & ( aa(list(product_prod(A,B)),multiset(product_prod(A,B)),mset(product_prod(A,B)),aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),Xs3),Ys3)) = aa(list(product_prod(A,B)),multiset(product_prod(A,B)),mset(product_prod(A,B)),aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),Xsa),Ys)) ) ) ) ) ).

% ex_mset_zip_left
tff(fact_3012_mset__eq__length,axiom,
    ! [A: $tType,Xsa: list(A),Ys: list(A)] :
      ( ( aa(list(A),multiset(A),mset(A),Xsa) = aa(list(A),multiset(A),mset(A),Ys) )
     => ( aa(list(A),nat,size_size(list(A)),Xsa) = aa(list(A),nat,size_size(list(A)),Ys) ) ) ).

% mset_eq_length
tff(fact_3013_mset__eq__setD,axiom,
    ! [A: $tType,Xsa: list(A),Ys: list(A)] :
      ( ( aa(list(A),multiset(A),mset(A),Xsa) = aa(list(A),multiset(A),mset(A),Ys) )
     => ( aa(list(A),set(A),set2(A),Xsa) = aa(list(A),set(A),set2(A),Ys) ) ) ).

% mset_eq_setD
tff(fact_3014_mset__eq__imp__distinct__iff,axiom,
    ! [A: $tType,Xsa: list(A),Ys: list(A)] :
      ( ( aa(list(A),multiset(A),mset(A),Xsa) = aa(list(A),multiset(A),mset(A),Ys) )
     => ( aa(list(A),$o,distinct(A),Xsa)
      <=> aa(list(A),$o,distinct(A),Ys) ) ) ).

% mset_eq_imp_distinct_iff
tff(fact_3015_set__eq__iff__mset__eq__distinct,axiom,
    ! [A: $tType,Xa: list(A),Y: list(A)] :
      ( aa(list(A),$o,distinct(A),Xa)
     => ( aa(list(A),$o,distinct(A),Y)
       => ( ( aa(list(A),set(A),set2(A),Xa) = aa(list(A),set(A),set2(A),Y) )
        <=> ( aa(list(A),multiset(A),mset(A),Xa) = aa(list(A),multiset(A),mset(A),Y) ) ) ) ) ).

% set_eq_iff_mset_eq_distinct
tff(fact_3016_less__mask,axiom,
    ! [N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),N)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),bit_se2239418461657761734s_mask(nat,N)) ) ).

% less_mask
tff(fact_3017_mset__swap,axiom,
    ! [A: $tType,Ia: nat,Ls3: list(A),J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),aa(list(A),nat,size_size(list(A)),Ls3))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),Ls3))
       => ( aa(list(A),multiset(A),mset(A),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Ls3),J),aa(nat,A,nth(A,Ls3),Ia))),Ia),aa(nat,A,nth(A,Ls3),J))) = aa(list(A),multiset(A),mset(A),Ls3) ) ) ) ).

% mset_swap
tff(fact_3018_mask__nat__less__exp,axiom,
    ! [N: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),bit_se2239418461657761734s_mask(nat,N)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),numeral_numeral(nat,bit0(one2))),N)) ).

% mask_nat_less_exp
tff(fact_3019_mset__update,axiom,
    ! [A: $tType,Ia: nat,Ls3: list(A),V2: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),aa(list(A),nat,size_size(list(A)),Ls3))
     => ( aa(list(A),multiset(A),mset(A),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Ls3),Ia),V2)) = add_mset(A,V2,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),aa(list(A),multiset(A),mset(A),Ls3)),add_mset(A,aa(nat,A,nth(A,Ls3),Ia),zero_zero(multiset(A))))) ) ) ).

% mset_update
tff(fact_3020_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),numeral_numeral(A,bit0(one2))),N)),one_one(A)) ) ) ).

% mask_eq_exp_minus_1
tff(fact_3021_and__int_Opelims,axiom,
    ! [Xa: int,Xaa: int,Y: int] :
      ( ( bit_se5824344872417868541ns_and(int,Xa,Xaa) = 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),Xa),Xaa))
       => ~ ( ( Y = $ite(
                  ( member(int,Xa,aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert3(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert3(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))
                  & member(int,Xaa,aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert3(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert3(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))) ),
                  aa(int,int,uminus_uminus(int),
                    aa($o,int,zero_neq_one_of_bool(int),
                      ( ~ dvd_dvd(int,numeral_numeral(int,bit0(one2)),Xa)
                      & ~ dvd_dvd(int,numeral_numeral(int,bit0(one2)),Xaa) ))),
                  aa(int,int,
                    aa(int,fun(int,int),plus_plus(int),
                      aa($o,int,zero_neq_one_of_bool(int),
                        ( ~ dvd_dvd(int,numeral_numeral(int,bit0(one2)),Xa)
                        & ~ dvd_dvd(int,numeral_numeral(int,bit0(one2)),Xaa) ))),
                    aa(int,int,aa(int,fun(int,int),times_times(int),numeral_numeral(int,bit0(one2))),bit_se5824344872417868541ns_and(int,divide_divide(int,Xa,numeral_numeral(int,bit0(one2))),divide_divide(int,Xaa,numeral_numeral(int,bit0(one2)))))) ) )
           => ~ accp(product_prod(int,int),bit_and_int_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Xa),Xaa)) ) ) ) ).

% and_int.pelims
tff(fact_3022_and__int_Opsimps,axiom,
    ! [K3: 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),K3),L))
     => ( bit_se5824344872417868541ns_and(int,K3,L) = $ite(
            ( member(int,K3,aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert3(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert3(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))
            & member(int,L,aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert3(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert3(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))) ),
            aa(int,int,uminus_uminus(int),
              aa($o,int,zero_neq_one_of_bool(int),
                ( ~ dvd_dvd(int,numeral_numeral(int,bit0(one2)),K3)
                & ~ dvd_dvd(int,numeral_numeral(int,bit0(one2)),L) ))),
            aa(int,int,
              aa(int,fun(int,int),plus_plus(int),
                aa($o,int,zero_neq_one_of_bool(int),
                  ( ~ dvd_dvd(int,numeral_numeral(int,bit0(one2)),K3)
                  & ~ dvd_dvd(int,numeral_numeral(int,bit0(one2)),L) ))),
              aa(int,int,aa(int,fun(int,int),times_times(int),numeral_numeral(int,bit0(one2))),bit_se5824344872417868541ns_and(int,divide_divide(int,K3,numeral_numeral(int,bit0(one2))),divide_divide(int,L,numeral_numeral(int,bit0(one2)))))) ) ) ) ).

% and_int.psimps
tff(fact_3023_bit__sum__mult__2__cases,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A,B3: A,N: nat] :
          ( ! [J3: nat] : ~ bit_se5641148757651400278ts_bit(A,A3,aa(nat,nat,suc,J3))
         => ( 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),numeral_numeral(A,bit0(one2))),B3)),N)
          <=> $ite(N = zero_zero(nat),~ dvd_dvd(A,numeral_numeral(A,bit0(one2)),A3),bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),times_times(A),numeral_numeral(A,bit0(one2))),B3),N)) ) ) ) ).

% bit_sum_mult_2_cases
tff(fact_3024_diff__numeral__special_I8_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M2: num] : ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),numeral_numeral(A,M2))),aa(A,A,uminus_uminus(A),one_one(A))) = neg_numeral_sub(A,one2,M2) ) ) ).

% diff_numeral_special(8)
tff(fact_3025_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),numeral_numeral(A,N))) = neg_numeral_sub(A,N,one2) ) ) ).

% diff_numeral_special(7)
tff(fact_3026_diff__numeral__simps_I1_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M2: num,N: num] : ( aa(A,A,aa(A,fun(A,A),minus_minus(A),numeral_numeral(A,M2)),numeral_numeral(A,N)) = neg_numeral_sub(A,M2,N) ) ) ).

% diff_numeral_simps(1)
tff(fact_3027_diff__numeral__simps_I4_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M2: num,N: num] : ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),numeral_numeral(A,M2))),aa(A,A,uminus_uminus(A),numeral_numeral(A,N))) = neg_numeral_sub(A,N,M2) ) ) ).

% diff_numeral_simps(4)
tff(fact_3028_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)),numeral_numeral(A,N)) = neg_numeral_sub(A,one2,N) ) ) ).

% diff_numeral_special(1)
tff(fact_3029_diff__numeral__special_I2_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M2: num] : ( aa(A,A,aa(A,fun(A,A),minus_minus(A),numeral_numeral(A,M2)),one_one(A)) = neg_numeral_sub(A,M2,one2) ) ) ).

% diff_numeral_special(2)
tff(fact_3030_minus__sub__one__diff__one,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [M2: num] : ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),neg_numeral_sub(A,M2,one2))),one_one(A)) = aa(A,A,uminus_uminus(A),numeral_numeral(A,M2)) ) ) ).

% minus_sub_one_diff_one
tff(fact_3031_bit__take__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M2: nat,A3: A,N: nat] :
          ( bit_se5641148757651400278ts_bit(A,bit_se2584673776208193580ke_bit(A,M2,A3),N)
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),M2)
            & bit_se5641148757651400278ts_bit(A,A3,N) ) ) ) ).

% bit_take_bit_iff
tff(fact_3032_neg__numeral__class_Osub__def,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [K3: num,L: num] : ( neg_numeral_sub(A,K3,L) = aa(A,A,aa(A,fun(A,A),minus_minus(A),numeral_numeral(A,K3)),numeral_numeral(A,L)) ) ) ).

% neg_numeral_class.sub_def
tff(fact_3033_sub__negative,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: num,M2: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),neg_numeral_sub(A,N,M2)),zero_zero(A))
        <=> aa(num,$o,aa(num,fun(num,$o),ord_less(num),N),M2) ) ) ).

% sub_negative
tff(fact_3034_sub__positive,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: num,M2: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),neg_numeral_sub(A,N,M2))
        <=> aa(num,$o,aa(num,fun(num,$o),ord_less(num),M2),N) ) ) ).

% sub_positive
tff(fact_3035_bit__imp__take__bit__positive,axiom,
    ! [N: nat,M2: nat,K3: int] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),M2)
     => ( bit_se5641148757651400278ts_bit(int,K3,N)
       => aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),bit_se2584673776208193580ke_bit(int,M2,K3)) ) ) ).

% bit_imp_take_bit_positive
tff(fact_3036_bit__horner__sum__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Bs: list($o),N: nat] :
          ( bit_se5641148757651400278ts_bit(A,aa(list($o),A,aa(A,fun(list($o),A),aa(fun($o,A),fun(A,fun(list($o),A)),groups4207007520872428315er_sum($o,A),zero_neq_one_of_bool(A)),numeral_numeral(A,bit0(one2))),Bs),N)
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),aa(list($o),nat,size_size(list($o)),Bs))
            & aa(nat,$o,nth($o,Bs),N) ) ) ) ).

% bit_horner_sum_bit_iff
tff(fact_3037_int__bit__bound,axiom,
    ! [K3: int] :
      ~ ! [N5: nat] :
          ( ! [M3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N5),M3)
             => ( bit_se5641148757651400278ts_bit(int,K3,M3)
              <=> bit_se5641148757651400278ts_bit(int,K3,N5) ) )
         => ~ ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N5)
             => ( bit_se5641148757651400278ts_bit(int,K3,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N5),one_one(nat)))
              <=> ~ bit_se5641148757651400278ts_bit(int,K3,N5) ) ) ) ).

% int_bit_bound
tff(fact_3038_and__int_Opinduct,axiom,
    ! [A0: int,A1: int,Pa: fun(int,fun(int,$o))] :
      ( accp(product_prod(int,int),bit_and_int_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),A0),A1))
     => ( ! [K: int,L3: int] :
            ( accp(product_prod(int,int),bit_and_int_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),K),L3))
           => ( ( ~ ( member(int,K,aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert3(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert3(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))
                    & member(int,L3,aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert3(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert3(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))) )
               => aa(int,$o,aa(int,fun(int,$o),Pa,divide_divide(int,K,numeral_numeral(int,bit0(one2)))),divide_divide(int,L3,numeral_numeral(int,bit0(one2)))) )
             => aa(int,$o,aa(int,fun(int,$o),Pa,K),L3) ) )
       => aa(int,$o,aa(int,fun(int,$o),Pa,A0),A1) ) ) ).

% and_int.pinduct
tff(fact_3039_odd__bit__iff__bit__pred,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,N: nat] :
          ( ~ dvd_dvd(A,numeral_numeral(A,bit0(one2)),A3)
         => ( bit_se5641148757651400278ts_bit(A,A3,N)
          <=> ( 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_3040_upto_Opsimps,axiom,
    ! [Ia: 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),Ia),J))
     => ( upto(Ia,J) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Ia),J),aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),Ia),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),Ia),one_one(int)),J)),nil(int)) ) ) ).

% upto.psimps
tff(fact_3041_upto_Opelims,axiom,
    ! [Xa: int,Xaa: int,Y: list(int)] :
      ( ( upto(Xa,Xaa) = 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),Xa),Xaa))
       => ~ ( ( Y = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Xa),Xaa),aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),Xa),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),Xa),one_one(int)),Xaa)),nil(int)) )
           => ~ accp(product_prod(int,int),upto_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Xa),Xaa)) ) ) ) ).

% upto.pelims
tff(fact_3042_upto_Opinduct,axiom,
    ! [A0: int,A1: int,Pa: fun(int,fun(int,$o))] :
      ( accp(product_prod(int,int),upto_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),A0),A1))
     => ( ! [I3: int,J3: int] :
            ( accp(product_prod(int,int),upto_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),I3),J3))
           => ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I3),J3)
               => aa(int,$o,aa(int,fun(int,$o),Pa,aa(int,int,aa(int,fun(int,int),plus_plus(int),I3),one_one(int))),J3) )
             => aa(int,$o,aa(int,fun(int,$o),Pa,I3),J3) ) )
       => aa(int,$o,aa(int,fun(int,$o),Pa,A0),A1) ) ) ).

% upto.pinduct
tff(fact_3043_signed__take__bit__def,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat,A3: A] : ( bit_ri4674362597316999326ke_bit(A,N,A3) = bit_se1065995026697491101ons_or(A,bit_se2584673776208193580ke_bit(A,N,A3),aa(A,A,aa(A,fun(A,A),times_times(A),aa($o,A,zero_neq_one_of_bool(A),bit_se5641148757651400278ts_bit(A,A3,N))),bit_ri4277139882892585799ns_not(A,bit_se2239418461657761734s_mask(A,N)))) ) ) ).

% signed_take_bit_def
tff(fact_3044_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))),numeral_numeral(A,N)) = aa(A,A,uminus_uminus(A),numeral_numeral(A,inc(N))) ) ) ).

% diff_numeral_special(5)
tff(fact_3045_diff__numeral__special_I6_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M2: num] : ( aa(A,A,aa(A,fun(A,A),minus_minus(A),numeral_numeral(A,M2)),aa(A,A,uminus_uminus(A),one_one(A))) = numeral_numeral(A,inc(M2)) ) ) ).

% diff_numeral_special(6)
tff(fact_3046_not__diff__distrib,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A3: A,B3: A] : ( bit_ri4277139882892585799ns_not(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B3)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),bit_ri4277139882892585799ns_not(A,A3)),B3) ) ) ).

% not_diff_distrib
tff(fact_3047_not__add__distrib,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A3: A,B3: A] : ( bit_ri4277139882892585799ns_not(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),bit_ri4277139882892585799ns_not(A,A3)),B3) ) ) ).

% not_add_distrib
tff(fact_3048_minus__eq__not__minus__1,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A3: A] : ( aa(A,A,uminus_uminus(A),A3) = 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_3049_not__eq__complement,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A3: 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_3050_disjunctive__diff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [B3: A,A3: A] :
          ( ! [N5: nat] :
              ( bit_se5641148757651400278ts_bit(A,B3,N5)
             => bit_se5641148757651400278ts_bit(A,A3,N5) )
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B3) = bit_se5824344872417868541ns_and(A,A3,bit_ri4277139882892585799ns_not(A,B3)) ) ) ) ).

% disjunctive_diff
tff(fact_3051_take__bit__not__eq__mask__diff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat,A3: A] : ( bit_se2584673776208193580ke_bit(A,N,bit_ri4277139882892585799ns_not(A,A3)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),bit_se2239418461657761734s_mask(A,N)),bit_se2584673776208193580ke_bit(A,N,A3)) ) ) ).

% take_bit_not_eq_mask_diff
tff(fact_3052_num__of__nat_Osimps_I2_J,axiom,
    ! [N: nat] :
      ( num_of_nat(aa(nat,nat,suc,N)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N),inc(num_of_nat(N)),one2) ) ).

% num_of_nat.simps(2)
tff(fact_3053_binomial__code,axiom,
    ! [N: nat,K3: nat] :
      ( binomial(N,K3) = $ite(
          aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),K3),
          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),numeral_numeral(nat,bit0(one2))),K3)),binomial(N,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K3)),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),K3)),one_one(nat)),N,one_one(nat)),semiring_char_0_fact(nat,K3))) ) ) ).

% binomial_code
tff(fact_3054_set__decode__plus__power__2,axiom,
    ! [N: nat,Z2: nat] :
      ( ~ member(nat,N,nat_set_decode(Z2))
     => ( nat_set_decode(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),numeral_numeral(nat,bit0(one2))),N)),Z2)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert3(nat),N),nat_set_decode(Z2)) ) ) ).

% set_decode_plus_power_2
tff(fact_3055_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),numeral_numeral(A,bit0(one2)))) ) ) ).

% push_bit_Suc
tff(fact_3056_size__diff__se,axiom,
    ! [A: $tType,T2: A,S: multiset(A)] :
      ( member(A,T2,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),add_mset(A,T2,zero_zero(multiset(A)))))),one_one(nat)) ) ) ).

% size_diff_se
tff(fact_3057_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_3058_set__mset__eq__empty__iff,axiom,
    ! [A: $tType,M5: multiset(A)] :
      ( ( aa(multiset(A),set(A),set_mset(A),M5) = bot_bot(set(A)) )
    <=> ( M5 = zero_zero(multiset(A)) ) ) ).

% set_mset_eq_empty_iff
tff(fact_3059_set__mset__add__mset__insert,axiom,
    ! [A: $tType,A3: A,Aa2: multiset(A)] : ( aa(multiset(A),set(A),set_mset(A),add_mset(A,A3,Aa2)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),aa(multiset(A),set(A),set_mset(A),Aa2)) ) ).

% set_mset_add_mset_insert
tff(fact_3060_binomial__eq__0__iff,axiom,
    ! [N: nat,K3: nat] :
      ( ( binomial(N,K3) = zero_zero(nat) )
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),K3) ) ).

% binomial_eq_0_iff
tff(fact_3061_set__mset__mset,axiom,
    ! [A: $tType,Xsa: list(A)] : ( aa(multiset(A),set(A),set_mset(A),aa(list(A),multiset(A),mset(A),Xsa)) = aa(list(A),set(A),set2(A),Xsa) ) ).

% set_mset_mset
tff(fact_3062_set__decode__zero,axiom,
    nat_set_decode(zero_zero(nat)) = bot_bot(set(nat)) ).

% set_decode_zero
tff(fact_3063_zero__less__binomial__iff,axiom,
    ! [N: nat,K3: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),binomial(N,K3))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K3),N) ) ).

% zero_less_binomial_iff
tff(fact_3064_mset__diff__cancel1elem,axiom,
    ! [A: $tType,A3: A,B2: multiset(A)] :
      ( ~ member(A,A3,aa(multiset(A),set(A),set_mset(A),B2))
     => ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),add_mset(A,A3,zero_zero(multiset(A)))),B2) = add_mset(A,A3,zero_zero(multiset(A))) ) ) ).

% mset_diff_cancel1elem
tff(fact_3065_mset__un__cases,axiom,
    ! [A: $tType,A3: A,Aa2: multiset(A),B2: multiset(A)] :
      ( member(A,A3,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)),Aa2),B2)))
     => ( ~ member(A,A3,aa(multiset(A),set(A),set_mset(A),Aa2))
       => member(A,A3,aa(multiset(A),set(A),set_mset(A),B2)) ) ) ).

% mset_un_cases
tff(fact_3066_mset__left__cancel__union,axiom,
    ! [A: $tType,A3: A,Aa2: multiset(A),B2: multiset(A)] :
      ( member(A,A3,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)),Aa2),B2)))
     => ( ~ member(A,A3,aa(multiset(A),set(A),set_mset(A),Aa2))
       => member(A,A3,aa(multiset(A),set(A),set_mset(A),B2)) ) ) ).

% mset_left_cancel_union
tff(fact_3067_mset__right__cancel__union,axiom,
    ! [A: $tType,A3: A,Aa2: multiset(A),B2: multiset(A)] :
      ( member(A,A3,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)),Aa2),B2)))
     => ( ~ member(A,A3,aa(multiset(A),set(A),set_mset(A),B2))
       => member(A,A3,aa(multiset(A),set(A),set_mset(A),Aa2)) ) ) ).

% mset_right_cancel_union
tff(fact_3068_ex__Melem__conv,axiom,
    ! [A: $tType,Aa2: multiset(A)] :
      ( ? [X6: A] : member(A,X6,aa(multiset(A),set(A),set_mset(A),Aa2))
    <=> ( Aa2 != zero_zero(multiset(A)) ) ) ).

% ex_Melem_conv
tff(fact_3069_binomial__eq__0,axiom,
    ! [N: nat,K3: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),K3)
     => ( binomial(N,K3) = zero_zero(nat) ) ) ).

% binomial_eq_0
tff(fact_3070_in__multiset__in__set,axiom,
    ! [A: $tType,Xa: A,Xsa: list(A)] :
      ( member(A,Xa,aa(multiset(A),set(A),set_mset(A),aa(list(A),multiset(A),mset(A),Xsa)))
    <=> member(A,Xa,aa(list(A),set(A),set2(A),Xsa)) ) ).

% in_multiset_in_set
tff(fact_3071_zero__less__binomial,axiom,
    ! [K3: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K3),N)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),binomial(N,K3)) ) ).

% zero_less_binomial
tff(fact_3072_mset__right__cancel__elem,axiom,
    ! [A: $tType,A3: A,Aa2: multiset(A),B3: A] :
      ( member(A,A3,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)),Aa2),add_mset(A,B3,zero_zero(multiset(A))))))
     => ( ( A3 != B3 )
       => member(A,A3,aa(multiset(A),set(A),set_mset(A),Aa2)) ) ) ).

% mset_right_cancel_elem
tff(fact_3073_mset__left__cancel__elem,axiom,
    ! [A: $tType,A3: A,B3: A,Aa2: multiset(A)] :
      ( member(A,A3,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)),add_mset(A,B3,zero_zero(multiset(A)))),Aa2)))
     => ( ( A3 != B3 )
       => member(A,A3,aa(multiset(A),set(A),set_mset(A),Aa2)) ) ) ).

% mset_left_cancel_elem
tff(fact_3074_prod__decode__aux_Ocases,axiom,
    ! [Xa: product_prod(nat,nat)] :
      ~ ! [K: nat,M6: nat] : ( Xa != aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),K),M6) ) ).

% prod_decode_aux.cases
tff(fact_3075_set__mset__single,axiom,
    ! [A: $tType,B3: A] : ( aa(multiset(A),set(A),set_mset(A),add_mset(A,B3,zero_zero(multiset(A)))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),B3),bot_bot(set(A))) ) ).

% set_mset_single
tff(fact_3076_mset__2dist2__cases,axiom,
    ! [A: $tType,A3: A,B3: A,Aa2: multiset(A),B2: multiset(A)] :
      ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),add_mset(A,A3,zero_zero(multiset(A)))),add_mset(A,B3,zero_zero(multiset(A))))),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),Aa2),B2))
     => ( ~ aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),add_mset(A,A3,zero_zero(multiset(A)))),add_mset(A,B3,zero_zero(multiset(A))))),Aa2)
       => ( ~ aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),add_mset(A,A3,zero_zero(multiset(A)))),add_mset(A,B3,zero_zero(multiset(A))))),B2)
         => ( ( member(A,A3,aa(multiset(A),set(A),set_mset(A),Aa2))
             => ~ member(A,B3,aa(multiset(A),set(A),set_mset(A),B2)) )
           => ~ ( member(A,A3,aa(multiset(A),set(A),set_mset(A),B2))
               => ~ member(A,B3,aa(multiset(A),set(A),set_mset(A),Aa2)) ) ) ) ) ) ).

% mset_2dist2_cases
tff(fact_3077_mset__union__subset__s,axiom,
    ! [A: $tType,A3: A,B2: multiset(A),C3: multiset(A)] :
      ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),add_mset(A,A3,zero_zero(multiset(A)))),B2)),C3)
     => ( member(A,A3,aa(multiset(A),set(A),set_mset(A),C3))
        & aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),B2),C3) ) ) ).

% mset_union_subset_s
tff(fact_3078_mset__le__mono__add__single,axiom,
    ! [A: $tType,A3: A,Ys: multiset(A),B3: A,Ws: multiset(A)] :
      ( member(A,A3,aa(multiset(A),set(A),set_mset(A),Ys))
     => ( member(A,B3,aa(multiset(A),set(A),set_mset(A),Ws))
       => aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),add_mset(A,A3,zero_zero(multiset(A)))),add_mset(A,B3,zero_zero(multiset(A))))),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),Ys),Ws)) ) ) ).

% mset_le_mono_add_single
tff(fact_3079_nth__mem__mset,axiom,
    ! [A: $tType,Ia: nat,Ls3: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),aa(list(A),nat,size_size(list(A)),Ls3))
     => member(A,aa(nat,A,nth(A,Ls3),Ia),aa(multiset(A),set(A),set_mset(A),aa(list(A),multiset(A),mset(A),Ls3))) ) ).

% nth_mem_mset
tff(fact_3080_mset__contains__eq,axiom,
    ! [A: $tType,M2: A,M5: multiset(A)] :
      ( member(A,M2,aa(multiset(A),set(A),set_mset(A),M5))
    <=> ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),add_mset(A,M2,zero_zero(multiset(A)))),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),M5),add_mset(A,M2,zero_zero(multiset(A))))) = M5 ) ) ).

% mset_contains_eq
tff(fact_3081_mset__union__diff__comm,axiom,
    ! [A: $tType,T2: A,S: multiset(A),T4: multiset(A)] :
      ( member(A,T2,aa(multiset(A),set(A),set_mset(A),S))
     => ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),T4),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),S),add_mset(A,T2,zero_zero(multiset(A))))) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),T4),S)),add_mset(A,T2,zero_zero(multiset(A)))) ) ) ).

% mset_union_diff_comm
tff(fact_3082_diff__union__single__conv2,axiom,
    ! [A: $tType,A3: A,J4: multiset(A),Ia2: multiset(A)] :
      ( member(A,A3,aa(multiset(A),set(A),set_mset(A),J4))
     => ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),J4),Ia2)),add_mset(A,A3,zero_zero(multiset(A)))) = 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)),minus_minus(multiset(A)),J4),add_mset(A,A3,zero_zero(multiset(A))))),Ia2) ) ) ).

% diff_union_single_conv2
tff(fact_3083_mset__un__single__un__cases,axiom,
    ! [A: $tType,A3: A,Aa2: multiset(A),B2: multiset(A),C3: multiset(A)] :
      ( ( add_mset(A,A3,Aa2) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),B2),C3) )
     => ( ( member(A,A3,aa(multiset(A),set(A),set_mset(A),B2))
         => ( Aa2 != 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)),minus_minus(multiset(A)),B2),add_mset(A,A3,zero_zero(multiset(A))))),C3) ) )
       => ~ ( member(A,A3,aa(multiset(A),set(A),set_mset(A),C3))
           => ( Aa2 != aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),B2),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),C3),add_mset(A,A3,zero_zero(multiset(A))))) ) ) ) ) ).

% mset_un_single_un_cases
tff(fact_3084_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),numeral_numeral(A,bit0(one2)))) = aa(A,A,aa(A,fun(A,A),times_times(A),bit_se4730199178511100633sh_bit(A,N,A3)),numeral_numeral(A,bit0(one2))) ) ) ).

% push_bit_double
tff(fact_3085_numeral__num__of__nat,axiom,
    ! [N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
     => ( numeral_numeral(nat,num_of_nat(N)) = N ) ) ).

% numeral_num_of_nat
tff(fact_3086_choose__reduce__nat,axiom,
    ! [N: nat,K3: 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)),K3)
       => ( binomial(N,K3) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(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),K3),one_one(nat)))),binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)),K3)) ) ) ) ).

% choose_reduce_nat
tff(fact_3087_times__binomial__minus1__eq,axiom,
    ! [K3: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K3)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K3),binomial(N,K3)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),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),K3),one_one(nat)))) ) ) ).

% times_binomial_minus1_eq
tff(fact_3088_fold__atLeastAtMost__nat_Osimps,axiom,
    ! [A: $tType,F2: fun(nat,fun(A,A)),A3: nat,B3: nat,Acc3: A] :
      ( set_fo6178422350223883121st_nat(A,F2,A3,B3,Acc3) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),B3),A3),Acc3,set_fo6178422350223883121st_nat(A,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A3),one_one(nat)),B3,aa(A,A,aa(nat,fun(A,A),F2,A3),Acc3))) ) ).

% fold_atLeastAtMost_nat.simps
tff(fact_3089_fold__atLeastAtMost__nat_Oelims,axiom,
    ! [A: $tType,Xa: fun(nat,fun(A,A)),Xaa: nat,Xb: nat,Xc: A,Y: A] :
      ( ( set_fo6178422350223883121st_nat(A,Xa,Xaa,Xb,Xc) = Y )
     => ( Y = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),Xaa),Xc,set_fo6178422350223883121st_nat(A,Xa,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Xaa),one_one(nat)),Xb,aa(A,A,aa(nat,fun(A,A),Xa,Xaa),Xc))) ) ) ).

% fold_atLeastAtMost_nat.elims
tff(fact_3090_mset__size1elem,axiom,
    ! [A: $tType,Pa: multiset(A),Q2: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(multiset(A),nat,size_size(multiset(A)),Pa)),one_one(nat))
     => ( member(A,Q2,aa(multiset(A),set(A),set_mset(A),Pa))
       => ( Pa = add_mset(A,Q2,zero_zero(multiset(A))) ) ) ) ).

% mset_size1elem
tff(fact_3091_size__Diff1__less,axiom,
    ! [A: $tType,Xa: A,M5: multiset(A)] :
      ( member(A,Xa,aa(multiset(A),set(A),set_mset(A),M5))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(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)),M5),add_mset(A,Xa,zero_zero(multiset(A)))))),aa(multiset(A),nat,size_size(multiset(A)),M5)) ) ).

% size_Diff1_less
tff(fact_3092_size__Diff2__less,axiom,
    ! [A: $tType,Xa: A,M5: multiset(A),Y: A] :
      ( member(A,Xa,aa(multiset(A),set(A),set_mset(A),M5))
     => ( member(A,Y,aa(multiset(A),set(A),set_mset(A),M5))
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(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)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),M5),add_mset(A,Xa,zero_zero(multiset(A))))),add_mset(A,Y,zero_zero(multiset(A)))))),aa(multiset(A),nat,size_size(multiset(A)),M5)) ) ) ).

% size_Diff2_less
tff(fact_3093_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),numeral_numeral(A,bit0(one2))),N)) ) ) ).

% push_bit_eq_mult
tff(fact_3094_num__of__nat__double,axiom,
    ! [N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
     => ( num_of_nat(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),N)) = bit0(num_of_nat(N)) ) ) ).

% num_of_nat_double
tff(fact_3095_num__of__nat__plus__distrib,axiom,
    ! [M2: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
       => ( num_of_nat(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),N)) = aa(num,num,aa(num,fun(num,num),plus_plus(num),num_of_nat(M2)),num_of_nat(N)) ) ) ) ).

% num_of_nat_plus_distrib
tff(fact_3096_binomial__addition__formula,axiom,
    ! [N: nat,K3: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
     => ( binomial(N,aa(nat,nat,suc,K3)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)),aa(nat,nat,suc,K3))),binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)),K3)) ) ) ).

% binomial_addition_formula
tff(fact_3097_binomial__fact,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K3: nat,N: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K3),N)
         => ( aa(nat,A,semiring_1_of_nat(A),binomial(N,K3)) = 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,K3)),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K3)))) ) ) ) ).

% binomial_fact
tff(fact_3098_fact__binomial,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K3: nat,N: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K3),N)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),semiring_char_0_fact(A,K3)),aa(nat,A,semiring_1_of_nat(A),binomial(N,K3))) = 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),K3))) ) ) ) ).

% fact_binomial
tff(fact_3099_at__most__one__mset__mset__diff,axiom,
    ! [A: $tType,A3: A,M5: multiset(A)] :
      ( ~ 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)),M5),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)),M5),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),M5)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),bot_bot(set(A)))) ) ) ).

% at_most_one_mset_mset_diff
tff(fact_3100_prod__decode__aux_Oelims,axiom,
    ! [Xa: nat,Xaa: nat,Y: product_prod(nat,nat)] :
      ( ( nat_prod_decode_aux(Xa,Xaa) = Y )
     => ( Y = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xaa),Xa),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xaa),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Xa),Xaa)),nat_prod_decode_aux(aa(nat,nat,suc,Xa),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Xaa),aa(nat,nat,suc,Xa)))) ) ) ).

% prod_decode_aux.elims
tff(fact_3101_prod__decode__aux_Osimps,axiom,
    ! [K3: nat,M2: nat] :
      ( nat_prod_decode_aux(K3,M2) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),K3),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),M2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K3),M2)),nat_prod_decode_aux(aa(nat,nat,suc,K3),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),aa(nat,nat,suc,K3)))) ) ).

% prod_decode_aux.simps
tff(fact_3102_set__mset__replicate__mset__subset,axiom,
    ! [A: $tType,N: nat,Xa: A] :
      ( aa(multiset(A),set(A),set_mset(A),replicate_mset(A,N,Xa)) = $ite(N = zero_zero(nat),bot_bot(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A)))) ) ).

% set_mset_replicate_mset_subset
tff(fact_3103_mult1E,axiom,
    ! [A: $tType,N7: multiset(A),M5: multiset(A),R2: set(product_prod(A,A))] :
      ( 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)),N7),M5),mult1(A,R2))
     => ~ ! [A4: A,M0: multiset(A)] :
            ( ( M5 = add_mset(A,A4,M0) )
           => ! [K5: multiset(A)] :
                ( ( N7 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M0),K5) )
               => ~ ! [B9: A] :
                      ( member(A,B9,aa(multiset(A),set(A),set_mset(A),K5))
                     => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B9),A4),R2) ) ) ) ) ).

% mult1E
tff(fact_3104_mult1I,axiom,
    ! [A: $tType,M5: multiset(A),A3: A,M02: multiset(A),N7: multiset(A),K6: multiset(A),R2: set(product_prod(A,A))] :
      ( ( M5 = add_mset(A,A3,M02) )
     => ( ( N7 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M02),K6) )
       => ( ! [B4: A] :
              ( member(A,B4,aa(multiset(A),set(A),set_mset(A),K6))
             => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B4),A3),R2) )
         => 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)),N7),M5),mult1(A,R2)) ) ) ) ).

% mult1I
tff(fact_3105_in__replicate__mset,axiom,
    ! [A: $tType,Xa: A,N: nat,Y: A] :
      ( member(A,Xa,aa(multiset(A),set(A),set_mset(A),replicate_mset(A,N,Y)))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
        & ( Xa = Y ) ) ) ).

% in_replicate_mset
tff(fact_3106_not__less__empty,axiom,
    ! [A: $tType,M5: multiset(A),R2: set(product_prod(A,A))] : ~ 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)),M5),zero_zero(multiset(A))),mult1(A,R2)) ).

% not_less_empty
tff(fact_3107_mult1__union,axiom,
    ! [A: $tType,B2: multiset(A),D2: multiset(A),R2: set(product_prod(A,A)),C3: multiset(A)] :
      ( 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)),B2),D2),mult1(A,R2))
     => 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),B2)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),C3),D2)),mult1(A,R2)) ) ).

% mult1_union
tff(fact_3108_less__add,axiom,
    ! [A: $tType,N7: multiset(A),A3: A,M02: multiset(A),R2: set(product_prod(A,A))] :
      ( 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)),N7),add_mset(A,A3,M02)),mult1(A,R2))
     => ( ? [M7: multiset(A)] :
            ( 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),M02),mult1(A,R2))
            & ( N7 = add_mset(A,A3,M7) ) )
        | ? [K5: multiset(A)] :
            ( ! [B9: A] :
                ( member(A,B9,aa(multiset(A),set(A),set_mset(A),K5))
               => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B9),A3),R2) )
            & ( N7 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M02),K5) ) ) ) ) ).

% less_add
tff(fact_3109_one__step__implies__mult,axiom,
    ! [A: $tType,J4: multiset(A),K6: multiset(A),R2: set(product_prod(A,A)),Ia2: multiset(A)] :
      ( ( J4 != zero_zero(multiset(A)) )
     => ( ! [X: A] :
            ( member(A,X,aa(multiset(A),set(A),set_mset(A),K6))
           => ? [Xa4: A] :
                ( member(A,Xa4,aa(multiset(A),set(A),set_mset(A),J4))
                & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Xa4),R2) ) )
       => 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)),Ia2),K6)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),Ia2),J4)),mult(A,R2)) ) ) ).

% one_step_implies_mult
tff(fact_3110_fold__atLeastAtMost__nat_Opsimps,axiom,
    ! [A: $tType,F2: fun(nat,fun(A,A)),A3: nat,B3: nat,Acc3: 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)),A3),aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),B3),Acc3))))
     => ( set_fo6178422350223883121st_nat(A,F2,A3,B3,Acc3) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),B3),A3),Acc3,set_fo6178422350223883121st_nat(A,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A3),one_one(nat)),B3,aa(A,A,aa(nat,fun(A,A),F2,A3),Acc3))) ) ) ).

% fold_atLeastAtMost_nat.psimps
tff(fact_3111_fold__atLeastAtMost__nat_Opelims,axiom,
    ! [A: $tType,Xa: fun(nat,fun(A,A)),Xaa: nat,Xb: nat,Xc: A,Y: A] :
      ( ( set_fo6178422350223883121st_nat(A,Xa,Xaa,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))),Xa),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)),Xaa),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),Xaa),Xc,set_fo6178422350223883121st_nat(A,Xa,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Xaa),one_one(nat)),Xb,aa(A,A,aa(nat,fun(A,A),Xa,Xaa),Xc))) )
           => ~ accp(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),set_fo1817059534552279752at_rel(A),aa(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),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))),Xa),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)),Xaa),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_3112_set__sorted__list__of__multiset,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [M5: multiset(A)] : ( aa(list(A),set(A),set2(A),linord6283353356039996273ltiset(A,M5)) = aa(multiset(A),set(A),set_mset(A),M5) ) ) ).

% set_sorted_list_of_multiset
tff(fact_3113_cppi,axiom,
    ! [D2: int,Pa: fun(int,$o),P: fun(int,$o),Aa2: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D2)
     => ( ? [Z4: int] :
          ! [X: int] :
            ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z4),X)
           => ( aa(int,$o,Pa,X)
            <=> aa(int,$o,P,X) ) )
       => ( ! [X: int] :
              ( ! [Xa4: int] :
                  ( member(int,Xa4,set_or1337092689740270186AtMost(int,one_one(int),D2))
                 => ! [Xb3: int] :
                      ( member(int,Xb3,Aa2)
                     => ( X != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa4) ) ) )
             => ( aa(int,$o,Pa,X)
               => aa(int,$o,Pa,aa(int,int,aa(int,fun(int,int),plus_plus(int),X),D2)) ) )
         => ( ! [X: int,K: int] :
                ( aa(int,$o,P,X)
              <=> aa(int,$o,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),X),aa(int,int,aa(int,fun(int,int),times_times(int),K),D2))) )
           => ( ? [X_12: int] : aa(int,$o,Pa,X_12)
            <=> ( ? [X6: int] :
                    ( member(int,X6,set_or1337092689740270186AtMost(int,one_one(int),D2))
                    & aa(int,$o,P,X6) )
                | ? [X6: int] :
                    ( member(int,X6,set_or1337092689740270186AtMost(int,one_one(int),D2))
                    & ? [Xa2: int] :
                        ( member(int,Xa2,Aa2)
                        & aa(int,$o,Pa,aa(int,int,aa(int,fun(int,int),minus_minus(int),Xa2),X6)) ) ) ) ) ) ) ) ) ).

% cppi
tff(fact_3114_atLeastatMost__empty__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B3: A] :
          ( ( set_or1337092689740270186AtMost(A,A3,B3) = bot_bot(set(A)) )
        <=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3) ) ) ).

% atLeastatMost_empty_iff
tff(fact_3115_atLeastatMost__empty__iff2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B3: A] :
          ( ( bot_bot(set(A)) = set_or1337092689740270186AtMost(A,A3,B3) )
        <=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3) ) ) ).

% atLeastatMost_empty_iff2
tff(fact_3116_atLeastatMost__empty,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A3)
         => ( set_or1337092689740270186AtMost(A,A3,B3) = bot_bot(set(A)) ) ) ) ).

% atLeastatMost_empty
tff(fact_3117_atLeastAtMost__singleton__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B3: A,C2: A] :
          ( ( set_or1337092689740270186AtMost(A,A3,B3) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),C2),bot_bot(set(A))) )
        <=> ( ( A3 = B3 )
            & ( B3 = C2 ) ) ) ) ).

% atLeastAtMost_singleton_iff
tff(fact_3118_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)),insert3(A),A3),bot_bot(set(A))) ) ) ).

% atLeastAtMost_singleton
tff(fact_3119_atLeastAtMost__upto,axiom,
    ! [Ia: int,J: int] : ( set_or1337092689740270186AtMost(int,Ia,J) = aa(list(int),set(int),set2(int),upto(Ia,J)) ) ).

% atLeastAtMost_upto
tff(fact_3120_atLeastAtMost__singleton_H,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B3: A] :
          ( ( A3 = B3 )
         => ( set_or1337092689740270186AtMost(A,A3,B3) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),bot_bot(set(A))) ) ) ) ).

% atLeastAtMost_singleton'
tff(fact_3121_atLeastAtMost__eq__UNIV__iff,axiom,
    ! [A: $tType] :
      ( bounded_lattice(A)
     => ! [Xa: A,Y: A] :
          ( ( set_or1337092689740270186AtMost(A,Xa,Y) = top_top(set(A)) )
        <=> ( ( Xa = bot_bot(A) )
            & ( Y = top_top(A) ) ) ) ) ).

% atLeastAtMost_eq_UNIV_iff
tff(fact_3122_atLeastatMost__psubset__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B3: A,C2: A,D3: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),set_or1337092689740270186AtMost(A,A3,B3)),set_or1337092689740270186AtMost(A,C2,D3))
        <=> ( ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
              | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A3)
                & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),D3)
                & ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),A3)
                  | aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),D3) ) ) )
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),D3) ) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_3123_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_3124_bset_I1_J,axiom,
    ! [D2: int,B2: set(int),Pa: fun(int,$o),Qa: fun(int,$o)] :
      ( ! [X: int] :
          ( ! [Xa4: int] :
              ( member(int,Xa4,set_or1337092689740270186AtMost(int,one_one(int),D2))
             => ! [Xb3: int] :
                  ( member(int,Xb3,B2)
                 => ( X != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa4) ) ) )
         => ( aa(int,$o,Pa,X)
           => aa(int,$o,Pa,aa(int,int,aa(int,fun(int,int),minus_minus(int),X),D2)) ) )
     => ( ! [X: int] :
            ( ! [Xa4: int] :
                ( member(int,Xa4,set_or1337092689740270186AtMost(int,one_one(int),D2))
               => ! [Xb3: int] :
                    ( member(int,Xb3,B2)
                   => ( X != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa4) ) ) )
           => ( aa(int,$o,Qa,X)
             => aa(int,$o,Qa,aa(int,int,aa(int,fun(int,int),minus_minus(int),X),D2)) ) )
       => ! [X4: int] :
            ( ! [Xa5: int] :
                ( member(int,Xa5,set_or1337092689740270186AtMost(int,one_one(int),D2))
               => ! [Xb4: int] :
                    ( member(int,Xb4,B2)
                   => ( X4 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb4),Xa5) ) ) )
           => ( ( aa(int,$o,Pa,X4)
                & aa(int,$o,Qa,X4) )
             => ( aa(int,$o,Pa,aa(int,int,aa(int,fun(int,int),minus_minus(int),X4),D2))
                & aa(int,$o,Qa,aa(int,int,aa(int,fun(int,int),minus_minus(int),X4),D2)) ) ) ) ) ) ).

% bset(1)
tff(fact_3125_bset_I2_J,axiom,
    ! [D2: int,B2: set(int),Pa: fun(int,$o),Qa: fun(int,$o)] :
      ( ! [X: int] :
          ( ! [Xa4: int] :
              ( member(int,Xa4,set_or1337092689740270186AtMost(int,one_one(int),D2))
             => ! [Xb3: int] :
                  ( member(int,Xb3,B2)
                 => ( X != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa4) ) ) )
         => ( aa(int,$o,Pa,X)
           => aa(int,$o,Pa,aa(int,int,aa(int,fun(int,int),minus_minus(int),X),D2)) ) )
     => ( ! [X: int] :
            ( ! [Xa4: int] :
                ( member(int,Xa4,set_or1337092689740270186AtMost(int,one_one(int),D2))
               => ! [Xb3: int] :
                    ( member(int,Xb3,B2)
                   => ( X != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa4) ) ) )
           => ( aa(int,$o,Qa,X)
             => aa(int,$o,Qa,aa(int,int,aa(int,fun(int,int),minus_minus(int),X),D2)) ) )
       => ! [X4: int] :
            ( ! [Xa5: int] :
                ( member(int,Xa5,set_or1337092689740270186AtMost(int,one_one(int),D2))
               => ! [Xb4: int] :
                    ( member(int,Xb4,B2)
                   => ( X4 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb4),Xa5) ) ) )
           => ( ( aa(int,$o,Pa,X4)
                | aa(int,$o,Qa,X4) )
             => ( aa(int,$o,Pa,aa(int,int,aa(int,fun(int,int),minus_minus(int),X4),D2))
                | aa(int,$o,Qa,aa(int,int,aa(int,fun(int,int),minus_minus(int),X4),D2)) ) ) ) ) ) ).

% bset(2)
tff(fact_3126_aset_I1_J,axiom,
    ! [D2: int,Aa2: set(int),Pa: fun(int,$o),Qa: fun(int,$o)] :
      ( ! [X: int] :
          ( ! [Xa4: int] :
              ( member(int,Xa4,set_or1337092689740270186AtMost(int,one_one(int),D2))
             => ! [Xb3: int] :
                  ( member(int,Xb3,Aa2)
                 => ( X != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa4) ) ) )
         => ( aa(int,$o,Pa,X)
           => aa(int,$o,Pa,aa(int,int,aa(int,fun(int,int),plus_plus(int),X),D2)) ) )
     => ( ! [X: int] :
            ( ! [Xa4: int] :
                ( member(int,Xa4,set_or1337092689740270186AtMost(int,one_one(int),D2))
               => ! [Xb3: int] :
                    ( member(int,Xb3,Aa2)
                   => ( X != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa4) ) ) )
           => ( aa(int,$o,Qa,X)
             => aa(int,$o,Qa,aa(int,int,aa(int,fun(int,int),plus_plus(int),X),D2)) ) )
       => ! [X4: int] :
            ( ! [Xa5: int] :
                ( member(int,Xa5,set_or1337092689740270186AtMost(int,one_one(int),D2))
               => ! [Xb4: int] :
                    ( member(int,Xb4,Aa2)
                   => ( X4 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb4),Xa5) ) ) )
           => ( ( aa(int,$o,Pa,X4)
                & aa(int,$o,Qa,X4) )
             => ( aa(int,$o,Pa,aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),D2))
                & aa(int,$o,Qa,aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),D2)) ) ) ) ) ) ).

% aset(1)
tff(fact_3127_aset_I2_J,axiom,
    ! [D2: int,Aa2: set(int),Pa: fun(int,$o),Qa: fun(int,$o)] :
      ( ! [X: int] :
          ( ! [Xa4: int] :
              ( member(int,Xa4,set_or1337092689740270186AtMost(int,one_one(int),D2))
             => ! [Xb3: int] :
                  ( member(int,Xb3,Aa2)
                 => ( X != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa4) ) ) )
         => ( aa(int,$o,Pa,X)
           => aa(int,$o,Pa,aa(int,int,aa(int,fun(int,int),plus_plus(int),X),D2)) ) )
     => ( ! [X: int] :
            ( ! [Xa4: int] :
                ( member(int,Xa4,set_or1337092689740270186AtMost(int,one_one(int),D2))
               => ! [Xb3: int] :
                    ( member(int,Xb3,Aa2)
                   => ( X != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa4) ) ) )
           => ( aa(int,$o,Qa,X)
             => aa(int,$o,Qa,aa(int,int,aa(int,fun(int,int),plus_plus(int),X),D2)) ) )
       => ! [X4: int] :
            ( ! [Xa5: int] :
                ( member(int,Xa5,set_or1337092689740270186AtMost(int,one_one(int),D2))
               => ! [Xb4: int] :
                    ( member(int,Xb4,Aa2)
                   => ( X4 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb4),Xa5) ) ) )
           => ( ( aa(int,$o,Pa,X4)
                | aa(int,$o,Qa,X4) )
             => ( aa(int,$o,Pa,aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),D2))
                | aa(int,$o,Qa,aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),D2)) ) ) ) ) ) ).

% aset(2)
tff(fact_3128_aset_I10_J,axiom,
    ! [D3: int,D2: int,Aa2: set(int),T2: int] :
      ( dvd_dvd(int,D3,D2)
     => ! [X4: int] :
          ( ! [Xa5: int] :
              ( member(int,Xa5,set_or1337092689740270186AtMost(int,one_one(int),D2))
             => ! [Xb4: int] :
                  ( member(int,Xb4,Aa2)
                 => ( X4 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb4),Xa5) ) ) )
         => ( ~ dvd_dvd(int,D3,aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),T2))
           => ~ dvd_dvd(int,D3,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),D2)),T2)) ) ) ) ).

% aset(10)
tff(fact_3129_aset_I9_J,axiom,
    ! [D3: int,D2: int,Aa2: set(int),T2: int] :
      ( dvd_dvd(int,D3,D2)
     => ! [X4: int] :
          ( ! [Xa5: int] :
              ( member(int,Xa5,set_or1337092689740270186AtMost(int,one_one(int),D2))
             => ! [Xb4: int] :
                  ( member(int,Xb4,Aa2)
                 => ( X4 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb4),Xa5) ) ) )
         => ( dvd_dvd(int,D3,aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),T2))
           => dvd_dvd(int,D3,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),D2)),T2)) ) ) ) ).

% aset(9)
tff(fact_3130_bset_I10_J,axiom,
    ! [D3: int,D2: int,B2: set(int),T2: int] :
      ( dvd_dvd(int,D3,D2)
     => ! [X4: int] :
          ( ! [Xa5: int] :
              ( member(int,Xa5,set_or1337092689740270186AtMost(int,one_one(int),D2))
             => ! [Xb4: int] :
                  ( member(int,Xb4,B2)
                 => ( X4 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb4),Xa5) ) ) )
         => ( ~ dvd_dvd(int,D3,aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),T2))
           => ~ 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),X4),D2)),T2)) ) ) ) ).

% bset(10)
tff(fact_3131_bset_I9_J,axiom,
    ! [D3: int,D2: int,B2: set(int),T2: int] :
      ( dvd_dvd(int,D3,D2)
     => ! [X4: int] :
          ( ! [Xa5: int] :
              ( member(int,Xa5,set_or1337092689740270186AtMost(int,one_one(int),D2))
             => ! [Xb4: int] :
                  ( member(int,Xb4,B2)
                 => ( X4 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb4),Xa5) ) ) )
         => ( dvd_dvd(int,D3,aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),T2))
           => 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),X4),D2)),T2)) ) ) ) ).

% bset(9)
tff(fact_3132_periodic__finite__ex,axiom,
    ! [D3: int,Pa: fun(int,$o)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D3)
     => ( ! [X: int,K: int] :
            ( aa(int,$o,Pa,X)
          <=> aa(int,$o,Pa,aa(int,int,aa(int,fun(int,int),minus_minus(int),X),aa(int,int,aa(int,fun(int,int),times_times(int),K),D3))) )
       => ( ? [X_12: int] : aa(int,$o,Pa,X_12)
        <=> ? [X6: int] :
              ( member(int,X6,set_or1337092689740270186AtMost(int,one_one(int),D3))
              & aa(int,$o,Pa,X6) ) ) ) ) ).

% periodic_finite_ex
tff(fact_3133_aset_I7_J,axiom,
    ! [D2: int,Aa2: set(int),T2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D2)
     => ! [X4: int] :
          ( ! [Xa5: int] :
              ( member(int,Xa5,set_or1337092689740270186AtMost(int,one_one(int),D2))
             => ! [Xb4: int] :
                  ( member(int,Xb4,Aa2)
                 => ( X4 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb4),Xa5) ) ) )
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),T2),X4)
           => aa(int,$o,aa(int,fun(int,$o),ord_less(int),T2),aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),D2)) ) ) ) ).

% aset(7)
tff(fact_3134_aset_I5_J,axiom,
    ! [D2: int,T2: int,Aa2: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D2)
     => ( member(int,T2,Aa2)
       => ! [X4: int] :
            ( ! [Xa5: int] :
                ( member(int,Xa5,set_or1337092689740270186AtMost(int,one_one(int),D2))
               => ! [Xb4: int] :
                    ( member(int,Xb4,Aa2)
                   => ( X4 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb4),Xa5) ) ) )
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),X4),T2)
             => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),D2)),T2) ) ) ) ) ).

% aset(5)
tff(fact_3135_aset_I4_J,axiom,
    ! [D2: int,T2: int,Aa2: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D2)
     => ( member(int,T2,Aa2)
       => ! [X4: int] :
            ( ! [Xa5: int] :
                ( member(int,Xa5,set_or1337092689740270186AtMost(int,one_one(int),D2))
               => ! [Xb4: int] :
                    ( member(int,Xb4,Aa2)
                   => ( X4 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb4),Xa5) ) ) )
           => ( ( X4 != T2 )
             => ( aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),D2) != T2 ) ) ) ) ) ).

% aset(4)
tff(fact_3136_aset_I3_J,axiom,
    ! [D2: int,T2: int,Aa2: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D2)
     => ( member(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),T2),one_one(int)),Aa2)
       => ! [X4: int] :
            ( ! [Xa5: int] :
                ( member(int,Xa5,set_or1337092689740270186AtMost(int,one_one(int),D2))
               => ! [Xb4: int] :
                    ( member(int,Xb4,Aa2)
                   => ( X4 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb4),Xa5) ) ) )
           => ( ( X4 = T2 )
             => ( aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),D2) = T2 ) ) ) ) ) ).

% aset(3)
tff(fact_3137_bset_I7_J,axiom,
    ! [D2: int,T2: int,B2: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D2)
     => ( member(int,T2,B2)
       => ! [X4: int] :
            ( ! [Xa5: int] :
                ( member(int,Xa5,set_or1337092689740270186AtMost(int,one_one(int),D2))
               => ! [Xb4: int] :
                    ( member(int,Xb4,B2)
                   => ( X4 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb4),Xa5) ) ) )
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),T2),X4)
             => aa(int,$o,aa(int,fun(int,$o),ord_less(int),T2),aa(int,int,aa(int,fun(int,int),minus_minus(int),X4),D2)) ) ) ) ) ).

% bset(7)
tff(fact_3138_bset_I5_J,axiom,
    ! [D2: int,B2: set(int),T2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D2)
     => ! [X4: int] :
          ( ! [Xa5: int] :
              ( member(int,Xa5,set_or1337092689740270186AtMost(int,one_one(int),D2))
             => ! [Xb4: int] :
                  ( member(int,Xb4,B2)
                 => ( X4 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb4),Xa5) ) ) )
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),X4),T2)
           => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),X4),D2)),T2) ) ) ) ).

% bset(5)
tff(fact_3139_bset_I4_J,axiom,
    ! [D2: int,T2: int,B2: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D2)
     => ( member(int,T2,B2)
       => ! [X4: int] :
            ( ! [Xa5: int] :
                ( member(int,Xa5,set_or1337092689740270186AtMost(int,one_one(int),D2))
               => ! [Xb4: int] :
                    ( member(int,Xb4,B2)
                   => ( X4 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb4),Xa5) ) ) )
           => ( ( X4 != T2 )
             => ( aa(int,int,aa(int,fun(int,int),minus_minus(int),X4),D2) != T2 ) ) ) ) ) ).

% bset(4)
tff(fact_3140_bset_I3_J,axiom,
    ! [D2: int,T2: int,B2: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D2)
     => ( member(int,aa(int,int,aa(int,fun(int,int),minus_minus(int),T2),one_one(int)),B2)
       => ! [X4: int] :
            ( ! [Xa5: int] :
                ( member(int,Xa5,set_or1337092689740270186AtMost(int,one_one(int),D2))
               => ! [Xb4: int] :
                    ( member(int,Xb4,B2)
                   => ( X4 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb4),Xa5) ) ) )
           => ( ( X4 = T2 )
             => ( aa(int,int,aa(int,fun(int,int),minus_minus(int),X4),D2) = T2 ) ) ) ) ) ).

% bset(3)
tff(fact_3141_fold__atLeastAtMost__nat_Opinduct,axiom,
    ! [A: $tType,A0: fun(nat,fun(A,A)),A1: nat,A22: nat,A32: A,Pa: fun(fun(nat,fun(A,A)),fun(nat,fun(nat,fun(A,$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))))
     => ( ! [F3: fun(nat,fun(A,A)),A4: nat,B4: nat,Acc: A] :
            ( accp(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),set_fo1817059534552279752at_rel(A),aa(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),aa(fun(nat,fun(A,A)),fun(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A)))),product_Pair(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),F3),aa(product_prod(nat,A),product_prod(nat,product_prod(nat,A)),aa(nat,fun(product_prod(nat,A),product_prod(nat,product_prod(nat,A))),product_Pair(nat,product_prod(nat,A)),A4),aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),B4),Acc))))
           => ( ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),B4),A4)
               => aa(A,$o,aa(nat,fun(A,$o),aa(nat,fun(nat,fun(A,$o)),aa(fun(nat,fun(A,A)),fun(nat,fun(nat,fun(A,$o))),Pa,F3),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A4),one_one(nat))),B4),aa(A,A,aa(nat,fun(A,A),F3,A4),Acc)) )
             => aa(A,$o,aa(nat,fun(A,$o),aa(nat,fun(nat,fun(A,$o)),aa(fun(nat,fun(A,A)),fun(nat,fun(nat,fun(A,$o))),Pa,F3),A4),B4),Acc) ) )
       => aa(A,$o,aa(nat,fun(A,$o),aa(nat,fun(nat,fun(A,$o)),aa(fun(nat,fun(A,A)),fun(nat,fun(nat,fun(A,$o))),Pa,A0),A1),A22),A32) ) ) ).

% fold_atLeastAtMost_nat.pinduct
tff(fact_3142_simp__from__to,axiom,
    ! [Ia: int,J: int] :
      ( set_or1337092689740270186AtMost(int,Ia,J) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less(int),J),Ia),bot_bot(set(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert3(int),Ia),set_or1337092689740270186AtMost(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),Ia),one_one(int)),J))) ) ).

% simp_from_to
tff(fact_3143_bset_I6_J,axiom,
    ! [D2: int,B2: set(int),T2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D2)
     => ! [X4: int] :
          ( ! [Xa5: int] :
              ( member(int,Xa5,set_or1337092689740270186AtMost(int,one_one(int),D2))
             => ! [Xb4: int] :
                  ( member(int,Xb4,B2)
                 => ( X4 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb4),Xa5) ) ) )
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),X4),T2)
           => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),X4),D2)),T2) ) ) ) ).

% bset(6)
tff(fact_3144_bset_I8_J,axiom,
    ! [D2: int,T2: int,B2: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D2)
     => ( member(int,aa(int,int,aa(int,fun(int,int),minus_minus(int),T2),one_one(int)),B2)
       => ! [X4: int] :
            ( ! [Xa5: int] :
                ( member(int,Xa5,set_or1337092689740270186AtMost(int,one_one(int),D2))
               => ! [Xb4: int] :
                    ( member(int,Xb4,B2)
                   => ( X4 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb4),Xa5) ) ) )
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),T2),X4)
             => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),T2),aa(int,int,aa(int,fun(int,int),minus_minus(int),X4),D2)) ) ) ) ) ).

% bset(8)
tff(fact_3145_aset_I6_J,axiom,
    ! [D2: int,T2: int,Aa2: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D2)
     => ( member(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),T2),one_one(int)),Aa2)
       => ! [X4: int] :
            ( ! [Xa5: int] :
                ( member(int,Xa5,set_or1337092689740270186AtMost(int,one_one(int),D2))
               => ! [Xb4: int] :
                    ( member(int,Xb4,Aa2)
                   => ( X4 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb4),Xa5) ) ) )
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),X4),T2)
             => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),D2)),T2) ) ) ) ) ).

% aset(6)
tff(fact_3146_aset_I8_J,axiom,
    ! [D2: int,Aa2: set(int),T2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D2)
     => ! [X4: int] :
          ( ! [Xa5: int] :
              ( member(int,Xa5,set_or1337092689740270186AtMost(int,one_one(int),D2))
             => ! [Xb4: int] :
                  ( member(int,Xb4,Aa2)
                 => ( X4 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb4),Xa5) ) ) )
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),T2),X4)
           => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),T2),aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),D2)) ) ) ) ).

% aset(8)
tff(fact_3147_cpmi,axiom,
    ! [D2: int,Pa: fun(int,$o),P: fun(int,$o),B2: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D2)
     => ( ? [Z4: int] :
          ! [X: int] :
            ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),X),Z4)
           => ( aa(int,$o,Pa,X)
            <=> aa(int,$o,P,X) ) )
       => ( ! [X: int] :
              ( ! [Xa4: int] :
                  ( member(int,Xa4,set_or1337092689740270186AtMost(int,one_one(int),D2))
                 => ! [Xb3: int] :
                      ( member(int,Xb3,B2)
                     => ( X != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa4) ) ) )
             => ( aa(int,$o,Pa,X)
               => aa(int,$o,Pa,aa(int,int,aa(int,fun(int,int),minus_minus(int),X),D2)) ) )
         => ( ! [X: int,K: int] :
                ( aa(int,$o,P,X)
              <=> aa(int,$o,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),X),aa(int,int,aa(int,fun(int,int),times_times(int),K),D2))) )
           => ( ? [X_12: int] : aa(int,$o,Pa,X_12)
            <=> ( ? [X6: int] :
                    ( member(int,X6,set_or1337092689740270186AtMost(int,one_one(int),D2))
                    & aa(int,$o,P,X6) )
                | ? [X6: int] :
                    ( member(int,X6,set_or1337092689740270186AtMost(int,one_one(int),D2))
                    & ? [Xa2: int] :
                        ( member(int,Xa2,B2)
                        & aa(int,$o,Pa,aa(int,int,aa(int,fun(int,int),plus_plus(int),Xa2),X6)) ) ) ) ) ) ) ) ) ).

% cpmi
tff(fact_3148_mult__implies__one__step,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),M5: multiset(A),N7: multiset(A)] :
      ( trans(A,R2)
     => ( 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)),M5),N7),mult(A,R2))
       => ? [I7: multiset(A),J5: multiset(A)] :
            ( ( N7 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),I7),J5) )
            & ? [K5: multiset(A)] :
                ( ( M5 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),I7),K5) )
                & ( J5 != zero_zero(multiset(A)) )
                & ! [X4: A] :
                    ( member(A,X4,aa(multiset(A),set(A),set_mset(A),K5))
                   => ? [Xa5: A] :
                        ( member(A,Xa5,aa(multiset(A),set(A),set_mset(A),J5))
                        & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Xa5),R2) ) ) ) ) ) ) ).

% mult_implies_one_step
tff(fact_3149_map__tailrec__rev_Opelims,axiom,
    ! [B: $tType,A: $tType,Xa: fun(B,A),Xaa: list(B),Xb: list(A),Y: list(A)] :
      ( ( map_tailrec_rev(B,A,Xa,Xaa,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))),Xa),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)),Xaa),Xb)))
       => ( ( ( Xaa = 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))),Xa),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))) ) )
         => ~ ! [A4: B,As: list(B)] :
                ( ( Xaa = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),A4),As) )
               => ( ( Y = map_tailrec_rev(B,A,Xa,As,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(B,A,Xa,A4)),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))),Xa),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),A4),As)),Xb))) ) ) ) ) ) ).

% map_tailrec_rev.pelims
tff(fact_3150_Pow__set_I1_J,axiom,
    ! [A: $tType] : ( pow(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))),insert3(set(A)),bot_bot(set(A))),bot_bot(set(set(A)))) ) ).

% Pow_set(1)
tff(fact_3151_card__insert__le__m1,axiom,
    ! [A: $tType,N: nat,Y: set(A),Xa: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),finite_card(A,Y)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)))
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),finite_card(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),Y))),N) ) ) ).

% card_insert_le_m1
tff(fact_3152_less__eq,axiom,
    ! [M2: nat,N: nat] :
      ( member(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),M2),N),transitive_trancl(nat,pred_nat))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),N) ) ).

% less_eq
tff(fact_3153_card__eq__UNIV,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [S: set(A)] :
          ( ( finite_card(A,S) = finite_card(A,top_top(set(A))) )
        <=> ( S = top_top(set(A)) ) ) ) ).

% card_eq_UNIV
tff(fact_3154_card__eq__UNIV2,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [S: set(A)] :
          ( ( finite_card(A,top_top(set(A))) = finite_card(A,S) )
        <=> ( S = top_top(set(A)) ) ) ) ).

% card_eq_UNIV2
tff(fact_3155_PowI,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),B2)
     => member(set(A),Aa2,pow(A,B2)) ) ).

% PowI
tff(fact_3156_Pow__iff,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A)] :
      ( member(set(A),Aa2,pow(A,B2))
    <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),B2) ) ).

% Pow_iff
tff(fact_3157_Pow__UNIV,axiom,
    ! [A: $tType] : ( pow(A,top_top(set(A))) = top_top(set(set(A))) ) ).

% Pow_UNIV
tff(fact_3158_Pow__Int__eq,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A)] : ( pow(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),B2)) = aa(set(set(A)),set(set(A)),aa(set(set(A)),fun(set(set(A)),set(set(A))),inf_inf(set(set(A))),pow(A,Aa2)),pow(A,B2)) ) ).

% Pow_Int_eq
tff(fact_3159_lenlex__transI,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( trans(A,R2)
     => trans(list(A),lenlex(A,R2)) ) ).

% lenlex_transI
tff(fact_3160_card_Oempty,axiom,
    ! [A: $tType] : ( finite_card(A,bot_bot(set(A))) = zero_zero(nat) ) ).

% card.empty
tff(fact_3161_card__ge__UNIV,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [S: set(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),finite_card(A,top_top(set(A)))),finite_card(A,S))
        <=> ( S = top_top(set(A)) ) ) ) ).

% card_ge_UNIV
tff(fact_3162_Pow__singleton__iff,axiom,
    ! [A: $tType,X3: set(A),Y5: set(A)] :
      ( ( pow(A,X3) = aa(set(set(A)),set(set(A)),aa(set(A),fun(set(set(A)),set(set(A))),insert3(set(A)),Y5),bot_bot(set(set(A)))) )
    <=> ( ( X3 = bot_bot(set(A)) )
        & ( Y5 = bot_bot(set(A)) ) ) ) ).

% Pow_singleton_iff
tff(fact_3163_Pow__empty,axiom,
    ! [A: $tType] : ( pow(A,bot_bot(set(A))) = aa(set(set(A)),set(set(A)),aa(set(A),fun(set(set(A)),set(set(A))),insert3(set(A)),bot_bot(set(A))),bot_bot(set(set(A)))) ) ).

% Pow_empty
tff(fact_3164_card__Diff__insert,axiom,
    ! [A: $tType,A3: A,Aa2: set(A),B2: set(A)] :
      ( member(A,A3,Aa2)
     => ( ~ member(A,A3,B2)
       => ( finite_card(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),B2))) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),finite_card(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),B2))),one_one(nat)) ) ) ) ).

% card_Diff_insert
tff(fact_3165_card__doubleton__eq__2__iff,axiom,
    ! [A: $tType,A3: A,B3: A] :
      ( ( finite_card(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),B3),bot_bot(set(A))))) = numeral_numeral(nat,bit0(one2)) )
    <=> ( A3 != B3 ) ) ).

% card_doubleton_eq_2_iff
tff(fact_3166_lexn__transI,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),N: nat] :
      ( trans(A,R2)
     => trans(list(A),aa(nat,set(product_prod(list(A),list(A))),lexn(A,R2),N)) ) ).

% lexn_transI
tff(fact_3167_Pow__top,axiom,
    ! [A: $tType,Aa2: set(A)] : member(set(A),Aa2,pow(A,Aa2)) ).

% Pow_top
tff(fact_3168_listrel__trans,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( trans(A,R2)
     => trans(list(A),listrel(A,A,R2)) ) ).

% listrel_trans
tff(fact_3169_lex__transI,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( trans(A,R2)
     => trans(list(A),lex(A,R2)) ) ).

% lex_transI
tff(fact_3170_lexord__transI,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( trans(A,R2)
     => trans(list(A),lexord(A,R2)) ) ).

% lexord_transI
tff(fact_3171_trans__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( trans(A,R2)
    <=> ! [X6: A,Y4: A,Z5: A] :
          ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X6),Y4),R2)
         => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),Z5),R2)
           => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X6),Z5),R2) ) ) ) ).

% trans_def
tff(fact_3172_transI,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( ! [X: A,Y2: A,Z3: A] :
          ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y2),R2)
         => ( 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)
           => 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),R2) ) )
     => trans(A,R2) ) ).

% transI
tff(fact_3173_transE,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),Xa: A,Y: A,Z2: A] :
      ( trans(A,R2)
     => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Y),R2)
       => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z2),R2)
         => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Z2),R2) ) ) ) ).

% transE
tff(fact_3174_transD,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),Xa: A,Y: A,Z2: A] :
      ( trans(A,R2)
     => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Y),R2)
       => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z2),R2)
         => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Z2),R2) ) ) ) ).

% transD
tff(fact_3175_Pow__bottom,axiom,
    ! [A: $tType,B2: set(A)] : member(set(A),bot_bot(set(A)),pow(A,B2)) ).

% Pow_bottom
tff(fact_3176_PowD,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A)] :
      ( member(set(A),Aa2,pow(A,B2))
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),B2) ) ).

% PowD
tff(fact_3177_Pow__mono,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),B2)
     => aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),pow(A,Aa2)),pow(A,B2)) ) ).

% Pow_mono
tff(fact_3178_trans__empty,axiom,
    ! [A: $tType] : trans(A,bot_bot(set(product_prod(A,A)))) ).

% trans_empty
tff(fact_3179_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_3180_Un__Pow__subset,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A)] : aa(set(set(A)),$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))),pow(A,Aa2)),pow(A,B2))),pow(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),B2))) ).

% Un_Pow_subset
tff(fact_3181_trans__Id,axiom,
    ! [A: $tType] : trans(A,id2(A)) ).

% trans_Id
tff(fact_3182_Pow__not__empty,axiom,
    ! [A: $tType,Aa2: set(A)] : ( pow(A,Aa2) != bot_bot(set(set(A))) ) ).

% Pow_not_empty
tff(fact_3183_card__insert__le,axiom,
    ! [A: $tType,Aa2: set(A),Xa: A] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),finite_card(A,Aa2)),finite_card(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),Aa2))) ).

% card_insert_le
tff(fact_3184_all__nat__less,axiom,
    ! [N: nat,Pa: fun(nat,$o)] :
      ( ! [M: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),N)
         => aa(nat,$o,Pa,M) )
    <=> ! [X6: nat] :
          ( member(nat,X6,set_or1337092689740270186AtMost(nat,zero_zero(nat),N))
         => aa(nat,$o,Pa,X6) ) ) ).

% all_nat_less
tff(fact_3185_ex__nat__less,axiom,
    ! [N: nat,Pa: fun(nat,$o)] :
      ( ? [M: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),N)
          & aa(nat,$o,Pa,M) )
    <=> ? [X6: nat] :
          ( member(nat,X6,set_or1337092689740270186AtMost(nat,zero_zero(nat),N))
          & aa(nat,$o,Pa,X6) ) ) ).

% ex_nat_less
tff(fact_3186_map__tailrec__rev_Osimps_I2_J,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),A3: B,As2: list(B),Bs: list(A)] : ( map_tailrec_rev(B,A,F2,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),A3),As2),Bs) = map_tailrec_rev(B,A,F2,As2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(B,A,F2,A3)),Bs)) ) ).

% map_tailrec_rev.simps(2)
tff(fact_3187_map__tailrec__rev_Osimps_I1_J,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),Bs: list(A)] : ( map_tailrec_rev(B,A,F2,nil(B),Bs) = Bs ) ).

% map_tailrec_rev.simps(1)
tff(fact_3188_lexord__trans,axiom,
    ! [A: $tType,Xa: list(A),Y: list(A),R2: set(product_prod(A,A)),Z2: list(A)] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xa),Y),lexord(A,R2))
     => ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Y),Z2),lexord(A,R2))
       => ( trans(A,R2)
         => member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xa),Z2),lexord(A,R2)) ) ) ) ).

% lexord_trans
tff(fact_3189_lenlex__trans,axiom,
    ! [A: $tType,Xa: list(A),Y: list(A),R2: set(product_prod(A,A)),Z2: list(A)] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xa),Y),lenlex(A,R2))
     => ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Y),Z2),lenlex(A,R2))
       => ( trans(A,R2)
         => member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xa),Z2),lenlex(A,R2)) ) ) ) ).

% lenlex_trans
tff(fact_3190_is__singleton__altdef,axiom,
    ! [A: $tType,Aa2: set(A)] :
      ( is_singleton(A,Aa2)
    <=> ( finite_card(A,Aa2) = one_one(nat) ) ) ).

% is_singleton_altdef
tff(fact_3191_card__length,axiom,
    ! [A: $tType,Xsa: list(A)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),finite_card(A,aa(list(A),set(A),set2(A),Xsa))),aa(list(A),nat,size_size(list(A)),Xsa)) ).

% card_length
tff(fact_3192_card__1__singletonE,axiom,
    ! [A: $tType,Aa2: set(A)] :
      ( ( finite_card(A,Aa2) = one_one(nat) )
     => ~ ! [X: A] : ( Aa2 != aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),X),bot_bot(set(A))) ) ) ).

% card_1_singletonE
tff(fact_3193_distinct__card,axiom,
    ! [A: $tType,Xsa: list(A)] :
      ( aa(list(A),$o,distinct(A),Xsa)
     => ( finite_card(A,aa(list(A),set(A),set2(A),Xsa)) = aa(list(A),nat,size_size(list(A)),Xsa) ) ) ).

% distinct_card
tff(fact_3194_card__distinct,axiom,
    ! [A: $tType,Xsa: list(A)] :
      ( ( finite_card(A,aa(list(A),set(A),set2(A),Xsa)) = aa(list(A),nat,size_size(list(A)),Xsa) )
     => aa(list(A),$o,distinct(A),Xsa) ) ).

% card_distinct
tff(fact_3195_atLeast0__atMost__Suc,axiom,
    ! [N: nat] : ( set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,N)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert3(nat),aa(nat,nat,suc,N)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) ) ).

% atLeast0_atMost_Suc
tff(fact_3196_atLeastAtMost__insertL,axiom,
    ! [M2: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),N)
     => ( aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert3(nat),M2),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M2),N)) = set_or1337092689740270186AtMost(nat,M2,N) ) ) ).

% atLeastAtMost_insertL
tff(fact_3197_atLeastAtMostSuc__conv,axiom,
    ! [M2: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),aa(nat,nat,suc,N))
     => ( set_or1337092689740270186AtMost(nat,M2,aa(nat,nat,suc,N)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert3(nat),aa(nat,nat,suc,N)),set_or1337092689740270186AtMost(nat,M2,N)) ) ) ).

% atLeastAtMostSuc_conv
tff(fact_3198_Icc__eq__insert__lb__nat,axiom,
    ! [M2: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),N)
     => ( set_or1337092689740270186AtMost(nat,M2,N) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert3(nat),M2),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M2),N)) ) ) ).

% Icc_eq_insert_lb_nat
tff(fact_3199_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))),insert3(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_3200_map__tailrec__rev_Oelims,axiom,
    ! [B: $tType,A: $tType,Xa: fun(B,A),Xaa: list(B),Xb: list(A),Y: list(A)] :
      ( ( map_tailrec_rev(B,A,Xa,Xaa,Xb) = Y )
     => ( ( ( Xaa = nil(B) )
         => ( Y != Xb ) )
       => ~ ! [A4: B,As: list(B)] :
              ( ( Xaa = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),A4),As) )
             => ( Y != map_tailrec_rev(B,A,Xa,As,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(B,A,Xa,A4)),Xb)) ) ) ) ) ).

% map_tailrec_rev.elims
tff(fact_3201_card__1__singleton__iff,axiom,
    ! [A: $tType,Aa2: set(A)] :
      ( ( finite_card(A,Aa2) = aa(nat,nat,suc,zero_zero(nat)) )
    <=> ? [X6: A] : ( Aa2 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),X6),bot_bot(set(A))) ) ) ).

% card_1_singleton_iff
tff(fact_3202_card__eq__SucD,axiom,
    ! [A: $tType,Aa2: set(A),K3: nat] :
      ( ( finite_card(A,Aa2) = aa(nat,nat,suc,K3) )
     => ? [B4: A,B5: set(A)] :
          ( ( Aa2 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),B4),B5) )
          & ~ member(A,B4,B5)
          & ( finite_card(A,B5) = K3 )
          & ( ( K3 = zero_zero(nat) )
           => ( B5 = bot_bot(set(A)) ) ) ) ) ).

% card_eq_SucD
tff(fact_3203_card__Suc__eq,axiom,
    ! [A: $tType,Aa2: set(A),K3: nat] :
      ( ( finite_card(A,Aa2) = aa(nat,nat,suc,K3) )
    <=> ? [B8: A,B12: set(A)] :
          ( ( Aa2 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),B8),B12) )
          & ~ member(A,B8,B12)
          & ( finite_card(A,B12) = K3 )
          & ( ( K3 = zero_zero(nat) )
           => ( B12 = bot_bot(set(A)) ) ) ) ) ).

% card_Suc_eq
tff(fact_3204_card__Diff1__le,axiom,
    ! [A: $tType,Aa2: set(A),Xa: A] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),finite_card(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A)))))),finite_card(A,Aa2)) ).

% card_Diff1_le
tff(fact_3205_mult__cancel__add__mset,axiom,
    ! [A: $tType,S2: set(product_prod(A,A)),Uu: A,X3: multiset(A),Y5: multiset(A)] :
      ( trans(A,S2)
     => ( irrefl(A,S2)
       => ( 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)),add_mset(A,Uu,X3)),add_mset(A,Uu,Y5)),mult(A,S2))
        <=> 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)),X3),Y5),mult(A,S2)) ) ) ) ).

% mult_cancel_add_mset
tff(fact_3206_mult__cancel,axiom,
    ! [A: $tType,S2: set(product_prod(A,A)),X3: multiset(A),Z6: multiset(A),Y5: multiset(A)] :
      ( trans(A,S2)
     => ( irrefl(A,S2)
       => ( 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)),X3),Z6)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),Y5),Z6)),mult(A,S2))
        <=> 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)),X3),Y5),mult(A,S2)) ) ) ) ).

% mult_cancel
tff(fact_3207_card__2__iff,axiom,
    ! [A: $tType,S: set(A)] :
      ( ( finite_card(A,S) = numeral_numeral(nat,bit0(one2)) )
    <=> ? [X6: A,Y4: A] :
          ( ( S = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),X6),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Y4),bot_bot(set(A)))) )
          & ( X6 != Y4 ) ) ) ).

% card_2_iff
tff(fact_3208_mult__cancel__max,axiom,
    ! [A: $tType,S2: set(product_prod(A,A)),X3: multiset(A),Y5: multiset(A)] :
      ( trans(A,S2)
     => ( irrefl(A,S2)
       => ( 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)),X3),Y5),mult(A,S2))
        <=> 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)),X3),Y5)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),Y5),X3)),mult(A,S2)) ) ) ) ).

% mult_cancel_max
tff(fact_3209_card__3__iff,axiom,
    ! [A: $tType,S: set(A)] :
      ( ( finite_card(A,S) = numeral_numeral(nat,bit1(one2)) )
    <=> ? [X6: A,Y4: A,Z5: A] :
          ( ( S = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),X6),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Y4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Z5),bot_bot(set(A))))) )
          & ( X6 != Y4 )
          & ( Y4 != Z5 )
          & ( X6 != Z5 ) ) ) ).

% card_3_iff
tff(fact_3210_odd__card__imp__not__empty,axiom,
    ! [A: $tType,Aa2: set(A)] :
      ( ~ dvd_dvd(nat,numeral_numeral(nat,bit0(one2)),finite_card(A,Aa2))
     => ( Aa2 != bot_bot(set(A)) ) ) ).

% odd_card_imp_not_empty
tff(fact_3211_card__Diff__singleton,axiom,
    ! [A: $tType,Xa: A,Aa2: set(A)] :
      ( member(A,Xa,Aa2)
     => ( finite_card(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A))))) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),finite_card(A,Aa2)),one_one(nat)) ) ) ).

% card_Diff_singleton
tff(fact_3212_card__Diff__singleton__if,axiom,
    ! [A: $tType,Aa2: set(A),Xa: A] :
      ( finite_card(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A))))) = $ite(member(A,Xa,Aa2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),finite_card(A,Aa2)),one_one(nat)),finite_card(A,Aa2)) ) ).

% card_Diff_singleton_if
tff(fact_3213_multeqp__code__iff__reflcl__mult,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),Pa: fun(A,fun(A,$o)),N7: multiset(A),M5: multiset(A)] :
      ( irrefl(A,R)
     => ( trans(A,R)
       => ( ! [X: A,Y2: A] :
              ( aa(A,$o,aa(A,fun(A,$o),Pa,X),Y2)
            <=> 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) )
         => ( multeqp_code(A,Pa,N7,M5)
          <=> 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)),N7),M5),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_3214_multp__code__iff__mult,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),Pa: fun(A,fun(A,$o)),N7: multiset(A),M5: multiset(A)] :
      ( irrefl(A,R)
     => ( trans(A,R)
       => ( ! [X: A,Y2: A] :
              ( aa(A,$o,aa(A,fun(A,$o),Pa,X),Y2)
            <=> 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) )
         => ( multp_code(A,Pa,N7,M5)
          <=> 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)),N7),M5),mult(A,R)) ) ) ) ) ).

% multp_code_iff_mult
tff(fact_3215_map__tailrec__def,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),As2: list(B)] : ( aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map_tailrec(B,A),F2),As2) = aa(list(A),list(A),rev(A),map_tailrec_rev(B,A,F2,As2,nil(A))) ) ).

% map_tailrec_def
tff(fact_3216_trancl__set__ntrancl,axiom,
    ! [A: $tType,Xsa: list(product_prod(A,A))] : ( transitive_trancl(A,aa(list(product_prod(A,A)),set(product_prod(A,A)),set2(product_prod(A,A)),Xsa)) = transitive_ntrancl(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),finite_card(product_prod(A,A),aa(list(product_prod(A,A)),set(product_prod(A,A)),set2(product_prod(A,A)),Xsa))),one_one(nat)),aa(list(product_prod(A,A)),set(product_prod(A,A)),set2(product_prod(A,A)),Xsa)) ) ).

% trancl_set_ntrancl
tff(fact_3217_card__disjoint__shuffles,axiom,
    ! [A: $tType,Xsa: 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),Xsa)),aa(list(A),set(A),set2(A),Ys)) = bot_bot(set(A)) )
     => ( finite_card(list(A),aa(list(A),set(list(A)),aa(list(A),fun(list(A),set(list(A))),shuffles(A),Xsa),Ys)) = binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),Xsa)),aa(list(A),nat,size_size(list(A)),Ys)),aa(list(A),nat,size_size(list(A)),Xsa)) ) ) ).

% card_disjoint_shuffles
tff(fact_3218_Nil__in__shuffles,axiom,
    ! [A: $tType,Xsa: list(A),Ys: list(A)] :
      ( member(list(A),nil(A),aa(list(A),set(list(A)),aa(list(A),fun(list(A),set(list(A))),shuffles(A),Xsa),Ys))
    <=> ( ( Xsa = nil(A) )
        & ( Ys = nil(A) ) ) ) ).

% Nil_in_shuffles
tff(fact_3219_shuffles__commutes,axiom,
    ! [A: $tType,Xsa: list(A),Ys: list(A)] : ( aa(list(A),set(list(A)),aa(list(A),fun(list(A),set(list(A))),shuffles(A),Xsa),Ys) = aa(list(A),set(list(A)),aa(list(A),fun(list(A),set(list(A))),shuffles(A),Ys),Xsa) ) ).

% shuffles_commutes
tff(fact_3220_Cons__in__shuffles__rightI,axiom,
    ! [A: $tType,Zs: list(A),Xsa: list(A),Ys: list(A),Z2: A] :
      ( member(list(A),Zs,aa(list(A),set(list(A)),aa(list(A),fun(list(A),set(list(A))),shuffles(A),Xsa),Ys))
     => member(list(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Z2),Zs),aa(list(A),set(list(A)),aa(list(A),fun(list(A),set(list(A))),shuffles(A),Xsa),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Z2),Ys))) ) ).

% Cons_in_shuffles_rightI
tff(fact_3221_Cons__in__shuffles__leftI,axiom,
    ! [A: $tType,Zs: list(A),Xsa: list(A),Ys: list(A),Z2: A] :
      ( member(list(A),Zs,aa(list(A),set(list(A)),aa(list(A),fun(list(A),set(list(A))),shuffles(A),Xsa),Ys))
     => member(list(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Z2),Zs),aa(list(A),set(list(A)),aa(list(A),fun(list(A),set(list(A))),shuffles(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Z2),Xsa)),Ys)) ) ).

% Cons_in_shuffles_leftI
tff(fact_3222_Nil__in__shufflesI,axiom,
    ! [A: $tType,Xsa: list(A),Ys: list(A)] :
      ( ( Xsa = nil(A) )
     => ( ( Ys = nil(A) )
       => member(list(A),nil(A),aa(list(A),set(list(A)),aa(list(A),fun(list(A),set(list(A))),shuffles(A),Xsa),Ys)) ) ) ).

% Nil_in_shufflesI
tff(fact_3223_shuffles_Osimps_I1_J,axiom,
    ! [A: $tType,Ys: list(A)] : ( aa(list(A),set(list(A)),aa(list(A),fun(list(A),set(list(A))),shuffles(A),nil(A)),Ys) = aa(set(list(A)),set(list(A)),aa(list(A),fun(set(list(A)),set(list(A))),insert3(list(A)),Ys),bot_bot(set(list(A)))) ) ).

% shuffles.simps(1)
tff(fact_3224_shuffles_Osimps_I2_J,axiom,
    ! [A: $tType,Xsa: list(A)] : ( aa(list(A),set(list(A)),aa(list(A),fun(list(A),set(list(A))),shuffles(A),Xsa),nil(A)) = aa(set(list(A)),set(list(A)),aa(list(A),fun(set(list(A)),set(list(A))),insert3(list(A)),Xsa),bot_bot(set(list(A)))) ) ).

% shuffles.simps(2)
tff(fact_3225_shufflesE,axiom,
    ! [A: $tType,Zs: list(A),Xsa: list(A),Ys: list(A)] :
      ( member(list(A),Zs,aa(list(A),set(list(A)),aa(list(A),fun(list(A),set(list(A))),shuffles(A),Xsa),Ys))
     => ( ( ( Zs = Xsa )
         => ( Ys != nil(A) ) )
       => ( ( ( Zs = Ys )
           => ( Xsa != nil(A) ) )
         => ( ! [X: A,Xs4: list(A)] :
                ( ( Xsa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs4) )
               => ! [Z3: A,Zs4: list(A)] :
                    ( ( Zs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Z3),Zs4) )
                   => ( ( X = Z3 )
                     => ~ member(list(A),Zs4,aa(list(A),set(list(A)),aa(list(A),fun(list(A),set(list(A))),shuffles(A),Xs4),Ys)) ) ) )
           => ~ ! [Y2: A,Ys3: list(A)] :
                  ( ( Ys = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys3) )
                 => ! [Z3: A,Zs4: list(A)] :
                      ( ( Zs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Z3),Zs4) )
                     => ( ( Y2 = Z3 )
                       => ~ member(list(A),Zs4,aa(list(A),set(list(A)),aa(list(A),fun(list(A),set(list(A))),shuffles(A),Xsa),Ys3)) ) ) ) ) ) ) ) ).

% shufflesE
tff(fact_3226_length__shuffles,axiom,
    ! [A: $tType,Zs: list(A),Xsa: list(A),Ys: list(A)] :
      ( member(list(A),Zs,aa(list(A),set(list(A)),aa(list(A),fun(list(A),set(list(A))),shuffles(A),Xsa),Ys))
     => ( aa(list(A),nat,size_size(list(A)),Zs) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),Xsa)),aa(list(A),nat,size_size(list(A)),Ys)) ) ) ).

% length_shuffles
tff(fact_3227_set__shuffles,axiom,
    ! [A: $tType,Zs: list(A),Xsa: list(A),Ys: list(A)] :
      ( member(list(A),Zs,aa(list(A),set(list(A)),aa(list(A),fun(list(A),set(list(A))),shuffles(A),Xsa),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),Xsa)),aa(list(A),set(A),set2(A),Ys)) ) ) ).

% set_shuffles
tff(fact_3228_distinct__disjoint__shuffles,axiom,
    ! [A: $tType,Xsa: list(A),Ys: list(A),Zs: list(A)] :
      ( aa(list(A),$o,distinct(A),Xsa)
     => ( aa(list(A),$o,distinct(A),Ys)
       => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Xsa)),aa(list(A),set(A),set2(A),Ys)) = bot_bot(set(A)) )
         => ( member(list(A),Zs,aa(list(A),set(list(A)),aa(list(A),fun(list(A),set(list(A))),shuffles(A),Xsa),Ys))
           => aa(list(A),$o,distinct(A),Zs) ) ) ) ) ).

% distinct_disjoint_shuffles
tff(fact_3229_Cons__in__shuffles__iff,axiom,
    ! [A: $tType,Z2: A,Zs: list(A),Xsa: list(A),Ys: list(A)] :
      ( member(list(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Z2),Zs),aa(list(A),set(list(A)),aa(list(A),fun(list(A),set(list(A))),shuffles(A),Xsa),Ys))
    <=> ( ( ( Xsa != nil(A) )
          & ( aa(list(A),A,hd(A),Xsa) = Z2 )
          & member(list(A),Zs,aa(list(A),set(list(A)),aa(list(A),fun(list(A),set(list(A))),shuffles(A),aa(list(A),list(A),tl(A),Xsa)),Ys)) )
        | ( ( Ys != nil(A) )
          & ( aa(list(A),A,hd(A),Ys) = Z2 )
          & member(list(A),Zs,aa(list(A),set(list(A)),aa(list(A),fun(list(A),set(list(A))),shuffles(A),Xsa),aa(list(A),list(A),tl(A),Ys))) ) ) ) ).

% Cons_in_shuffles_iff
tff(fact_3230_greaterThanLessThan__upto,axiom,
    ! [Ia: int,J: int] : ( set_or5935395276787703475ssThan(int,Ia,J) = aa(list(int),set(int),set2(int),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),Ia),one_one(int)),aa(int,int,aa(int,fun(int,int),minus_minus(int),J),one_one(int)))) ) ).

% greaterThanLessThan_upto
tff(fact_3231_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)),insert3(nat),zero_zero(nat)),bot_bot(set(nat)))) ) ).

% atLeast1_atMost_eq_remove0
tff(fact_3232_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_3233_mult__cancel__max0,axiom,
    ! [A: $tType,S2: set(product_prod(A,A)),X3: multiset(A),Y5: multiset(A)] :
      ( trans(A,S2)
     => ( irrefl(A,S2)
       => ( 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)),X3),Y5),mult(A,S2))
        <=> 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)),X3),inter_mset(A,X3,Y5))),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),Y5),inter_mset(A,X3,Y5))),mult(A,S2)) ) ) ) ).

% mult_cancel_max0
tff(fact_3234_wo__rel_Ocases__Total3,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A3: A,B3: A,Phi: fun(A,fun(A,$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)),insert3(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),B3),bot_bot(set(A))))),aa(set(product_prod(A,A)),set(A),field2(A),R2))
       => ( ( ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B3),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),minus_minus(set(product_prod(A,A))),R2),id2(A)))
              | member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B3),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),B3) )
         => ( ( ( A3 = B3 )
             => aa(A,$o,aa(A,fun(A,$o),Phi,A3),B3) )
           => aa(A,$o,aa(A,fun(A,$o),Phi,A3),B3) ) ) ) ) ).

% wo_rel.cases_Total3
tff(fact_3235_greaterThanLessThan__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Ia: A,L: A,U: A] :
          ( member(A,Ia,set_or5935395276787703475ssThan(A,L,U))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),L),Ia)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ia),U) ) ) ) ).

% greaterThanLessThan_iff
tff(fact_3236_greaterThanLessThan__empty__iff2,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B3: A] :
          ( ( bot_bot(set(A)) = set_or5935395276787703475ssThan(A,A3,B3) )
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3) ) ) ).

% greaterThanLessThan_empty_iff2
tff(fact_3237_greaterThanLessThan__empty__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B3: A] :
          ( ( set_or5935395276787703475ssThan(A,A3,B3) = bot_bot(set(A)) )
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3) ) ) ).

% greaterThanLessThan_empty_iff
tff(fact_3238_greaterThanLessThan__empty,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,K3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),K3)
         => ( set_or5935395276787703475ssThan(A,K3,L) = bot_bot(set(A)) ) ) ) ).

% greaterThanLessThan_empty
tff(fact_3239_Sup__greaterThanLessThan,axiom,
    ! [A: $tType] :
      ( ( comple6319245703460814977attice(A)
        & dense_linorder(A) )
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or5935395276787703475ssThan(A,Xa,Y)) = Y ) ) ) ).

% Sup_greaterThanLessThan
tff(fact_3240_cSup__greaterThanLessThan,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & dense_linorder(A) )
     => ! [Y: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xa)
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or5935395276787703475ssThan(A,Y,Xa)) = Xa ) ) ) ).

% cSup_greaterThanLessThan
tff(fact_3241_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)),insert3(nat),zero_zero(nat)),bot_bot(set(nat))) ).

% atMost_0
tff(fact_3242_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_3243_wo__rel_Owell__order__induct,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),Pa: fun(A,$o),A3: A] :
      ( bNF_Wellorder_wo_rel(A,R2)
     => ( ! [X: A] :
            ( ! [Y3: A] :
                ( ( ( Y3 != X )
                  & 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),R2) )
               => aa(A,$o,Pa,Y3) )
           => aa(A,$o,Pa,X) )
       => aa(A,$o,Pa,A3) ) ) ).

% wo_rel.well_order_induct
tff(fact_3244_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_3245_wo__rel_OTOTALS,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( bNF_Wellorder_wo_rel(A,R2)
     => ! [X4: A] :
          ( member(A,X4,aa(set(product_prod(A,A)),set(A),field2(A),R2))
         => ! [Xa4: A] :
              ( member(A,Xa4,aa(set(product_prod(A,A)),set(A),field2(A),R2))
             => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Xa4),R2)
                | member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa4),X4),R2) ) ) ) ) ).

% wo_rel.TOTALS
tff(fact_3246_atMost__Suc,axiom,
    ! [K3: nat] : ( aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,suc,K3)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert3(nat),aa(nat,nat,suc,K3)),aa(nat,set(nat),set_ord_atMost(nat),K3)) ) ).

% atMost_Suc
tff(fact_3247_well__order__induct__imp,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),Pa: fun(A,$o),A3: A] :
      ( bNF_Wellorder_wo_rel(A,R2)
     => ( ! [X: A] :
            ( ! [Y3: A] :
                ( ( ( Y3 != X )
                  & 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),R2) )
               => ( member(A,Y3,aa(set(product_prod(A,A)),set(A),field2(A),R2))
                 => aa(A,$o,Pa,Y3) ) )
           => ( member(A,X,aa(set(product_prod(A,A)),set(A),field2(A),R2))
             => aa(A,$o,Pa,X) ) )
       => ( member(A,A3,aa(set(product_prod(A,A)),set(A),field2(A),R2))
         => aa(A,$o,Pa,A3) ) ) ) ).

% well_order_induct_imp
tff(fact_3248_atMost__nat__numeral,axiom,
    ! [K3: num] : ( aa(nat,set(nat),set_ord_atMost(nat),numeral_numeral(nat,K3)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert3(nat),numeral_numeral(nat,K3)),aa(nat,set(nat),set_ord_atMost(nat),pred_numeral(K3))) ) ).

% atMost_nat_numeral
tff(fact_3249_greaterThanLessThan__subseteq__greaterThanLessThan,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B3: A,C2: A,D3: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or5935395276787703475ssThan(A,A3,B3)),set_or5935395276787703475ssThan(A,C2,D3))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),D3) ) ) ) ) ).

% greaterThanLessThan_subseteq_greaterThanLessThan
tff(fact_3250_ivl__disj__int__two_I5_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,M2: A,U: A] : ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or5935395276787703475ssThan(A,L,M2)),set_or1337092689740270186AtMost(A,M2,U)) = bot_bot(set(A)) ) ) ).

% ivl_disj_int_two(5)
tff(fact_3251_ivl__disj__int__two_I4_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,M2: A,U: A] : ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or1337092689740270186AtMost(A,L,M2)),set_or5935395276787703475ssThan(A,M2,U)) = bot_bot(set(A)) ) ) ).

% ivl_disj_int_two(4)
tff(fact_3252_wo__rel_Ocases__Total,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A3: A,B3: A,Phi: fun(A,fun(A,$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)),insert3(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),B3),bot_bot(set(A))))),aa(set(product_prod(A,A)),set(A),field2(A),R2))
       => ( ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B3),R2)
           => aa(A,$o,aa(A,fun(A,$o),Phi,A3),B3) )
         => ( ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B3),A3),R2)
             => aa(A,$o,aa(A,fun(A,$o),Phi,A3),B3) )
           => aa(A,$o,aa(A,fun(A,$o),Phi,A3),B3) ) ) ) ) ).

% wo_rel.cases_Total
tff(fact_3253_greaterThanLessThan__subseteq__atLeastAtMost__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B3: A,C2: A,D3: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or5935395276787703475ssThan(A,A3,B3)),set_or1337092689740270186AtMost(A,C2,D3))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),D3) ) ) ) ) ).

% greaterThanLessThan_subseteq_atLeastAtMost_iff
tff(fact_3254_atLeastAtMost__diff__ends,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A] : ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),set_or1337092689740270186AtMost(A,A3,B3)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),B3),bot_bot(set(A))))) = set_or5935395276787703475ssThan(A,A3,B3) ) ) ).

% atLeastAtMost_diff_ends
tff(fact_3255_listset_Osimps_I1_J,axiom,
    ! [A: $tType] : ( aa(list(set(A)),set(list(A)),listset(A),nil(set(A))) = aa(set(list(A)),set(list(A)),aa(list(A),fun(set(list(A)),set(list(A))),insert3(list(A)),nil(A)),bot_bot(set(list(A)))) ) ).

% listset.simps(1)
tff(fact_3256_wo__rel_OWell__order__isMinim__exists,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),B2: set(A)] :
      ( bNF_Wellorder_wo_rel(A,R2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),aa(set(product_prod(A,A)),set(A),field2(A),R2))
       => ( ( B2 != bot_bot(set(A)) )
         => ? [X_1: A] : bNF_We4791949203932849705sMinim(A,R2,B2,X_1) ) ) ) ).

% wo_rel.Well_order_isMinim_exists
tff(fact_3257_wo__rel_Ominim__inField,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),B2: set(A)] :
      ( bNF_Wellorder_wo_rel(A,R2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),aa(set(product_prod(A,A)),set(A),field2(A),R2))
       => ( ( B2 != bot_bot(set(A)) )
         => member(A,bNF_We6954850376910717587_minim(A,R2,B2),aa(set(product_prod(A,A)),set(A),field2(A),R2)) ) ) ) ).

% wo_rel.minim_inField
tff(fact_3258_wo__rel_Ominim__in,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),B2: set(A)] :
      ( bNF_Wellorder_wo_rel(A,R2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),aa(set(product_prod(A,A)),set(A),field2(A),R2))
       => ( ( B2 != bot_bot(set(A)) )
         => member(A,bNF_We6954850376910717587_minim(A,R2,B2),B2) ) ) ) ).

% wo_rel.minim_in
tff(fact_3259_wo__rel_Oequals__minim,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),B2: set(A),A3: A] :
      ( bNF_Wellorder_wo_rel(A,R2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),aa(set(product_prod(A,A)),set(A),field2(A),R2))
       => ( member(A,A3,B2)
         => ( ! [B4: A] :
                ( member(A,B4,B2)
               => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B4),R2) )
           => ( A3 = bNF_We6954850376910717587_minim(A,R2,B2) ) ) ) ) ) ).

% wo_rel.equals_minim
tff(fact_3260_wo__rel_Ominim__isMinim,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),B2: set(A)] :
      ( bNF_Wellorder_wo_rel(A,R2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),aa(set(product_prod(A,A)),set(A),field2(A),R2))
       => ( ( B2 != bot_bot(set(A)) )
         => bNF_We4791949203932849705sMinim(A,R2,B2,bNF_We6954850376910717587_minim(A,R2,B2)) ) ) ) ).

% wo_rel.minim_isMinim
tff(fact_3261_wo__rel_OisMinim__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),Aa2: set(A),B3: A] :
      ( bNF_Wellorder_wo_rel(A,R2)
     => ( bNF_We4791949203932849705sMinim(A,R2,Aa2,B3)
      <=> ( member(A,B3,Aa2)
          & ! [X6: A] :
              ( member(A,X6,Aa2)
             => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B3),X6),R2) ) ) ) ) ).

% wo_rel.isMinim_def
tff(fact_3262_wo__rel_Ominim__least,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),B2: set(A),B3: A] :
      ( bNF_Wellorder_wo_rel(A,R2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),aa(set(product_prod(A,A)),set(A),field2(A),R2))
       => ( member(A,B3,B2)
         => 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,B2)),B3),R2) ) ) ) ).

% wo_rel.minim_least
tff(fact_3263_wo__rel_Omax2__greater__among,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A3: A,B3: A] :
      ( bNF_Wellorder_wo_rel(A,R2)
     => ( member(A,A3,aa(set(product_prod(A,A)),set(A),field2(A),R2))
       => ( member(A,B3,aa(set(product_prod(A,A)),set(A),field2(A),R2))
         => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),bNF_We1388413361240627857o_max2(A,R2,A3,B3)),R2)
            & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B3),bNF_We1388413361240627857o_max2(A,R2,A3,B3)),R2)
            & member(A,bNF_We1388413361240627857o_max2(A,R2,A3,B3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),B3),bot_bot(set(A))))) ) ) ) ) ).

% wo_rel.max2_greater_among
tff(fact_3264_nth__sorted__list__of__set__greaterThanLessThan,axiom,
    ! [N: nat,J: nat,Ia: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),aa(nat,nat,suc,Ia)))
     => ( aa(nat,nat,nth(nat,aa(set(nat),list(nat),linord4507533701916653071of_set(nat),set_or5935395276787703475ssThan(nat,Ia,J))),N) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ia),N)) ) ) ).

% nth_sorted_list_of_set_greaterThanLessThan
tff(fact_3265_lists__empty,axiom,
    ! [A: $tType] : ( aa(set(A),set(list(A)),lists(A),bot_bot(set(A))) = aa(set(list(A)),set(list(A)),aa(list(A),fun(set(list(A)),set(list(A))),insert3(list(A)),nil(A)),bot_bot(set(list(A)))) ) ).

% lists_empty
tff(fact_3266_wo__rel_Omax2__among,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A3: A,B3: A] :
      ( bNF_Wellorder_wo_rel(A,R2)
     => ( member(A,A3,aa(set(product_prod(A,A)),set(A),field2(A),R2))
       => ( member(A,B3,aa(set(product_prod(A,A)),set(A),field2(A),R2))
         => member(A,bNF_We1388413361240627857o_max2(A,R2,A3,B3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),B3),bot_bot(set(A))))) ) ) ) ).

% wo_rel.max2_among
tff(fact_3267_image__mult__atLeastAtMost__if,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,Xa: A,Y: A] :
          ( aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),times_times(A),C2)),set_or1337092689740270186AtMost(A,Xa,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),Xa),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),Xa),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),Xa)),bot_bot(set(A))) ) ) ) ).

% image_mult_atLeastAtMost_if
tff(fact_3268_image__eqI,axiom,
    ! [A: $tType,B: $tType,B3: A,F2: fun(B,A),Xa: B,Aa2: set(B)] :
      ( ( B3 = aa(B,A,F2,Xa) )
     => ( member(B,Xa,Aa2)
       => member(A,B3,aa(set(B),set(A),image2(B,A,F2),Aa2)) ) ) ).

% image_eqI
tff(fact_3269_image__empty,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A)] : ( aa(set(B),set(A),image2(B,A,F2),bot_bot(set(B))) = bot_bot(set(A)) ) ).

% image_empty
tff(fact_3270_empty__is__image,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Aa2: set(B)] :
      ( ( bot_bot(set(A)) = aa(set(B),set(A),image2(B,A,F2),Aa2) )
    <=> ( Aa2 = bot_bot(set(B)) ) ) ).

% empty_is_image
tff(fact_3271_image__is__empty,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Aa2: set(B)] :
      ( ( aa(set(B),set(A),image2(B,A,F2),Aa2) = bot_bot(set(A)) )
    <=> ( Aa2 = bot_bot(set(B)) ) ) ).

% image_is_empty
tff(fact_3272_insert__image,axiom,
    ! [B: $tType,A: $tType,Xa: A,Aa2: set(A),F2: fun(A,B)] :
      ( member(A,Xa,Aa2)
     => ( aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert3(B),aa(A,B,F2,Xa)),aa(set(A),set(B),image2(A,B,F2),Aa2)) = aa(set(A),set(B),image2(A,B,F2),Aa2) ) ) ).

% insert_image
tff(fact_3273_image__insert,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A3: B,B2: set(B)] : ( aa(set(B),set(A),image2(B,A,F2),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert3(B),A3),B2)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),aa(B,A,F2,A3)),aa(set(B),set(A),image2(B,A,F2),B2)) ) ).

% image_insert
tff(fact_3274_op__conc__empty__img__id,axiom,
    ! [A: $tType,L5: set(list(A))] : ( aa(set(list(A)),set(list(A)),image2(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),nil(A))),L5) = L5 ) ).

% op_conc_empty_img_id
tff(fact_3275_Field__Union,axiom,
    ! [A: $tType,R: set(set(product_prod(A,A)))] : ( aa(set(product_prod(A,A)),set(A),field2(A),aa(set(set(product_prod(A,A))),set(product_prod(A,A)),complete_Sup_Sup(set(product_prod(A,A))),R)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(set(product_prod(A,A))),set(set(A)),image2(set(product_prod(A,A)),set(A),field2(A)),R)) ) ).

% Field_Union
tff(fact_3276_Cons__in__lists__iff,axiom,
    ! [A: $tType,Xa: A,Xsa: list(A),Aa2: set(A)] :
      ( member(list(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa),aa(set(A),set(list(A)),lists(A),Aa2))
    <=> ( member(A,Xa,Aa2)
        & member(list(A),Xsa,aa(set(A),set(list(A)),lists(A),Aa2)) ) ) ).

% Cons_in_lists_iff
tff(fact_3277_in__listsI,axiom,
    ! [A: $tType,Xsa: list(A),Aa2: set(A)] :
      ( ! [X: A] :
          ( member(A,X,aa(list(A),set(A),set2(A),Xsa))
         => member(A,X,Aa2) )
     => member(list(A),Xsa,aa(set(A),set(list(A)),lists(A),Aa2)) ) ).

% in_listsI
tff(fact_3278_lists__Int__eq,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A)] : ( aa(set(A),set(list(A)),lists(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),B2)) = aa(set(list(A)),set(list(A)),aa(set(list(A)),fun(set(list(A)),set(list(A))),inf_inf(set(list(A))),aa(set(A),set(list(A)),lists(A),Aa2)),aa(set(A),set(list(A)),lists(A),B2)) ) ).

% lists_Int_eq
tff(fact_3279_append__in__lists__conv,axiom,
    ! [A: $tType,Xsa: list(A),Ys: list(A),Aa2: set(A)] :
      ( member(list(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),Ys),aa(set(A),set(list(A)),lists(A),Aa2))
    <=> ( member(list(A),Xsa,aa(set(A),set(list(A)),lists(A),Aa2))
        & member(list(A),Ys,aa(set(A),set(list(A)),lists(A),Aa2)) ) ) ).

% append_in_lists_conv
tff(fact_3280_image__add__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [S: set(A)] : ( aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),plus_plus(A),zero_zero(A))),S) = S ) ) ).

% image_add_0
tff(fact_3281_image__diff__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [D3: A,A3: A,B3: A] : ( aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),minus_minus(A),D3)),set_or1337092689740270186AtMost(A,A3,B3)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),D3),B3),aa(A,A,aa(A,fun(A,A),minus_minus(A),D3),A3)) ) ) ).

% image_diff_atLeastAtMost
tff(fact_3282_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_3283_sorted__list__of__set_Olength__sorted__key__list__of__set,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Aa2: set(A)] : ( aa(list(A),nat,size_size(list(A)),aa(set(A),list(A),linord4507533701916653071of_set(A),Aa2)) = finite_card(A,Aa2) ) ) ).

% sorted_list_of_set.length_sorted_key_list_of_set
tff(fact_3284_sorted__list__of__set__atMost__Suc,axiom,
    ! [K3: nat] : ( aa(set(nat),list(nat),linord4507533701916653071of_set(nat),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,suc,K3))) = aa(list(nat),list(nat),aa(list(nat),fun(list(nat),list(nat)),append(nat),aa(set(nat),list(nat),linord4507533701916653071of_set(nat),aa(nat,set(nat),set_ord_atMost(nat),K3))),aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),aa(nat,nat,suc,K3)),nil(nat))) ) ).

% sorted_list_of_set_atMost_Suc
tff(fact_3285_lists__UNIV,axiom,
    ! [A: $tType] : ( aa(set(A),set(list(A)),lists(A),top_top(set(A))) = top_top(set(list(A))) ) ).

% lists_UNIV
tff(fact_3286_image__mult__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [D3: A,A3: A,B3: A] :
          ( aa(A,$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,B3)) = 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),B3)) ) ) ) ).

% image_mult_atLeastAtMost
tff(fact_3287_rangeI,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xa: B] : member(A,aa(B,A,F2,Xa),aa(set(B),set(A),image2(B,A,F2),top_top(set(B)))) ).

% rangeI
tff(fact_3288_range__eqI,axiom,
    ! [A: $tType,B: $tType,B3: A,F2: fun(B,A),Xa: B] :
      ( ( B3 = aa(B,A,F2,Xa) )
     => member(A,B3,aa(set(B),set(A),image2(B,A,F2),top_top(set(B)))) ) ).

% range_eqI
tff(fact_3289_subset__image__iff,axiom,
    ! [A: $tType,B: $tType,B2: set(A),F2: fun(B,A),Aa2: set(B)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),aa(set(B),set(A),image2(B,A,F2),Aa2))
    <=> ? [AA: set(B)] :
          ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),AA),Aa2)
          & ( B2 = aa(set(B),set(A),image2(B,A,F2),AA) ) ) ) ).

% subset_image_iff
tff(fact_3290_image__subset__iff,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),Aa2: set(B),B2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(B),set(A),image2(B,A,F2),Aa2)),B2)
    <=> ! [X6: B] :
          ( member(B,X6,Aa2)
         => member(A,aa(B,A,F2,X6),B2) ) ) ).

% image_subset_iff
tff(fact_3291_subset__imageE,axiom,
    ! [A: $tType,B: $tType,B2: set(A),F2: fun(B,A),Aa2: set(B)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),aa(set(B),set(A),image2(B,A,F2),Aa2))
     => ~ ! [C9: set(B)] :
            ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),C9),Aa2)
           => ( B2 != aa(set(B),set(A),image2(B,A,F2),C9) ) ) ) ).

% subset_imageE
tff(fact_3292_image__subsetI,axiom,
    ! [A: $tType,B: $tType,Aa2: set(A),F2: fun(A,B),B2: set(B)] :
      ( ! [X: A] :
          ( member(A,X,Aa2)
         => member(B,aa(A,B,F2,X),B2) )
     => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F2),Aa2)),B2) ) ).

% image_subsetI
tff(fact_3293_image__mono,axiom,
    ! [B: $tType,A: $tType,Aa2: set(A),B2: set(A),F2: fun(A,B)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),B2)
     => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F2),Aa2)),aa(set(A),set(B),image2(A,B,F2),B2)) ) ).

% image_mono
tff(fact_3294_image__Un,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Aa2: set(B),B2: set(B)] : ( aa(set(B),set(A),image2(B,A,F2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),Aa2),B2)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(B),set(A),image2(B,A,F2),Aa2)),aa(set(B),set(A),image2(B,A,F2),B2)) ) ).

% image_Un
tff(fact_3295_Sup__SUP__eq,axiom,
    ! [A: $tType,S: set(fun(A,$o)),X4: A] :
      ( aa(A,$o,aa(set(fun(A,$o)),fun(A,$o),complete_Sup_Sup(fun(A,$o)),S),X4)
    <=> member(A,X4,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(fun(A,$o)),set(set(A)),image2(fun(A,$o),set(A),collect(A)),S))) ) ).

% Sup_SUP_eq
tff(fact_3296_imageI,axiom,
    ! [B: $tType,A: $tType,Xa: A,Aa2: set(A),F2: fun(A,B)] :
      ( member(A,Xa,Aa2)
     => member(B,aa(A,B,F2,Xa),aa(set(A),set(B),image2(A,B,F2),Aa2)) ) ).

% imageI
tff(fact_3297_image__iff,axiom,
    ! [A: $tType,B: $tType,Z2: A,F2: fun(B,A),Aa2: set(B)] :
      ( member(A,Z2,aa(set(B),set(A),image2(B,A,F2),Aa2))
    <=> ? [X6: B] :
          ( member(B,X6,Aa2)
          & ( Z2 = aa(B,A,F2,X6) ) ) ) ).

% image_iff
tff(fact_3298_bex__imageD,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Aa2: set(B),Pa: fun(A,$o)] :
      ( ? [X4: A] :
          ( member(A,X4,aa(set(B),set(A),image2(B,A,F2),Aa2))
          & aa(A,$o,Pa,X4) )
     => ? [X: B] :
          ( member(B,X,Aa2)
          & aa(A,$o,Pa,aa(B,A,F2,X)) ) ) ).

% bex_imageD
tff(fact_3299_image__cong,axiom,
    ! [B: $tType,A: $tType,M5: set(A),N7: set(A),F2: fun(A,B),G: fun(A,B)] :
      ( ( M5 = N7 )
     => ( ! [X: A] :
            ( member(A,X,N7)
           => ( aa(A,B,F2,X) = aa(A,B,G,X) ) )
       => ( aa(set(A),set(B),image2(A,B,F2),M5) = aa(set(A),set(B),image2(A,B,G),N7) ) ) ) ).

% image_cong
tff(fact_3300_ball__imageD,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Aa2: set(B),Pa: fun(A,$o)] :
      ( ! [X: A] :
          ( member(A,X,aa(set(B),set(A),image2(B,A,F2),Aa2))
         => aa(A,$o,Pa,X) )
     => ! [X4: B] :
          ( member(B,X4,Aa2)
         => aa(A,$o,Pa,aa(B,A,F2,X4)) ) ) ).

% ball_imageD
tff(fact_3301_rev__image__eqI,axiom,
    ! [B: $tType,A: $tType,Xa: A,Aa2: set(A),B3: B,F2: fun(A,B)] :
      ( member(A,Xa,Aa2)
     => ( ( B3 = aa(A,B,F2,Xa) )
       => member(B,B3,aa(set(A),set(B),image2(A,B,F2),Aa2)) ) ) ).

% rev_image_eqI
tff(fact_3302_refl__on__UNION,axiom,
    ! [B: $tType,A: $tType,S: set(A),Aa2: fun(A,set(B)),R2: fun(A,set(product_prod(B,B)))] :
      ( ! [X: A] :
          ( member(A,X,S)
         => refl_on(B,aa(A,set(B),Aa2,X),aa(A,set(product_prod(B,B)),R2,X)) )
     => refl_on(B,aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),Aa2),S)),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),S))) ) ).

% refl_on_UNION
tff(fact_3303_image__Pow__surj,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),Aa2: set(B),B2: set(A)] :
      ( ( aa(set(B),set(A),image2(B,A,F2),Aa2) = B2 )
     => ( aa(set(set(B)),set(set(A)),image2(set(B),set(A),image2(B,A,F2)),pow(B,Aa2)) = pow(A,B2) ) ) ).

% image_Pow_surj
tff(fact_3304_lists_OCons,axiom,
    ! [A: $tType,A3: A,Aa2: set(A),L: list(A)] :
      ( member(A,A3,Aa2)
     => ( member(list(A),L,aa(set(A),set(list(A)),lists(A),Aa2))
       => member(list(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),L),aa(set(A),set(list(A)),lists(A),Aa2)) ) ) ).

% lists.Cons
tff(fact_3305_listsE,axiom,
    ! [A: $tType,Xa: A,L: list(A),Aa2: set(A)] :
      ( member(list(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),L),aa(set(A),set(list(A)),lists(A),Aa2))
     => ~ ( member(A,Xa,Aa2)
         => ~ member(list(A),L,aa(set(A),set(list(A)),lists(A),Aa2)) ) ) ).

% listsE
tff(fact_3306_lists_ONil,axiom,
    ! [A: $tType,Aa2: set(A)] : member(list(A),nil(A),aa(set(A),set(list(A)),lists(A),Aa2)) ).

% lists.Nil
tff(fact_3307_lists__mono,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),B2)
     => aa(set(list(A)),$o,aa(set(list(A)),fun(set(list(A)),$o),ord_less_eq(set(list(A))),aa(set(A),set(list(A)),lists(A),Aa2)),aa(set(A),set(list(A)),lists(A),B2)) ) ).

% lists_mono
tff(fact_3308_in__listsD,axiom,
    ! [A: $tType,Xsa: list(A),Aa2: set(A)] :
      ( member(list(A),Xsa,aa(set(A),set(list(A)),lists(A),Aa2))
     => ! [X4: A] :
          ( member(A,X4,aa(list(A),set(A),set2(A),Xsa))
         => member(A,X4,Aa2) ) ) ).

% in_listsD
tff(fact_3309_in__lists__conv__set,axiom,
    ! [A: $tType,Xsa: list(A),Aa2: set(A)] :
      ( member(list(A),Xsa,aa(set(A),set(list(A)),lists(A),Aa2))
    <=> ! [X6: A] :
          ( member(A,X6,aa(list(A),set(A),set2(A),Xsa))
         => member(A,X6,Aa2) ) ) ).

% in_lists_conv_set
tff(fact_3310_lists__IntI,axiom,
    ! [A: $tType,L: list(A),Aa2: set(A),B2: set(A)] :
      ( member(list(A),L,aa(set(A),set(list(A)),lists(A),Aa2))
     => ( member(list(A),L,aa(set(A),set(list(A)),lists(A),B2))
       => member(list(A),L,aa(set(A),set(list(A)),lists(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),B2))) ) ) ).

% lists_IntI
tff(fact_3311_pairwise__imageI,axiom,
    ! [B: $tType,A: $tType,Aa2: set(A),F2: fun(A,B),Pa: fun(B,fun(B,$o))] :
      ( ! [X: A,Y2: A] :
          ( member(A,X,Aa2)
         => ( member(A,Y2,Aa2)
           => ( ( X != Y2 )
             => ( ( aa(A,B,F2,X) != aa(A,B,F2,Y2) )
               => aa(B,$o,aa(B,fun(B,$o),Pa,aa(A,B,F2,X)),aa(A,B,F2,Y2)) ) ) ) )
     => pairwise(B,Pa,aa(set(A),set(B),image2(A,B,F2),Aa2)) ) ).

% pairwise_imageI
tff(fact_3312_image__Pow__mono,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),Aa2: set(B),B2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(B),set(A),image2(B,A,F2),Aa2)),B2)
     => aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),aa(set(set(B)),set(set(A)),image2(set(B),set(A),image2(B,A,F2)),pow(B,Aa2))),pow(A,B2)) ) ).

% image_Pow_mono
tff(fact_3313_sorted__list__of__set_Odistinct__sorted__key__list__of__set,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Aa2: set(A)] : aa(list(A),$o,distinct(A),aa(set(A),list(A),linord4507533701916653071of_set(A),Aa2)) ) ).

% sorted_list_of_set.distinct_sorted_key_list_of_set
tff(fact_3314_SUP__eq__const,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [Ia2: set(A),F2: fun(A,B),Xa: B] :
          ( ( Ia2 != bot_bot(set(A)) )
         => ( ! [I3: A] :
                ( member(A,I3,Ia2)
               => ( aa(A,B,F2,I3) = Xa ) )
           => ( aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),Ia2)) = Xa ) ) ) ) ).

% SUP_eq_const
tff(fact_3315_subset__mset_OcSUP__least,axiom,
    ! [A: $tType,B: $tType,Aa2: set(A),F2: fun(A,multiset(B)),M5: multiset(B)] :
      ( ( Aa2 != bot_bot(set(A)) )
     => ( ! [X: A] :
            ( member(A,X,Aa2)
           => aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),aa(A,multiset(B),F2,X)),M5) )
       => aa(multiset(B),$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),F2),Aa2))),M5) ) ) ).

% subset_mset.cSUP_least
tff(fact_3316_range__subsetD,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),B2: set(A),Ia: B] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(B),set(A),image2(B,A,F2),top_top(set(B)))),B2)
     => member(A,aa(B,A,F2,Ia),B2) ) ).

% range_subsetD
tff(fact_3317_image__Int__subset,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Aa2: set(B),B2: set(B)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(B),set(A),image2(B,A,F2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),Aa2),B2))),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(B),set(A),image2(B,A,F2),Aa2)),aa(set(B),set(A),image2(B,A,F2),B2))) ).

% image_Int_subset
tff(fact_3318_image__diff__subset,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Aa2: set(B),B2: set(B)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(B),set(A),image2(B,A,F2),Aa2)),aa(set(B),set(A),image2(B,A,F2),B2))),aa(set(B),set(A),image2(B,A,F2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),Aa2),B2))) ).

% image_diff_subset
tff(fact_3319_Cons__shuffles__subset2,axiom,
    ! [A: $tType,Y: A,Xsa: list(A),Ys: list(A)] : aa(set(list(A)),$o,aa(set(list(A)),fun(set(list(A)),$o),ord_less_eq(set(list(A))),aa(set(list(A)),set(list(A)),image2(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y)),aa(list(A),set(list(A)),aa(list(A),fun(list(A),set(list(A))),shuffles(A),Xsa),Ys))),aa(list(A),set(list(A)),aa(list(A),fun(list(A),set(list(A))),shuffles(A),Xsa),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys))) ).

% Cons_shuffles_subset2
tff(fact_3320_Cons__shuffles__subset1,axiom,
    ! [A: $tType,Xa: A,Xsa: list(A),Ys: list(A)] : aa(set(list(A)),$o,aa(set(list(A)),fun(set(list(A)),$o),ord_less_eq(set(list(A))),aa(set(list(A)),set(list(A)),image2(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa)),aa(list(A),set(list(A)),aa(list(A),fun(list(A),set(list(A))),shuffles(A),Xsa),Ys))),aa(list(A),set(list(A)),aa(list(A),fun(list(A),set(list(A))),shuffles(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa)),Ys)) ).

% Cons_shuffles_subset1
tff(fact_3321_shuffles_Osimps_I3_J,axiom,
    ! [A: $tType,Xa: A,Xsa: list(A),Y: A,Ys: list(A)] : ( aa(list(A),set(list(A)),aa(list(A),fun(list(A),set(list(A))),shuffles(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa)),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),Xa)),aa(list(A),set(list(A)),aa(list(A),fun(list(A),set(list(A))),shuffles(A),Xsa),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)),aa(list(A),set(list(A)),aa(list(A),fun(list(A),set(list(A))),shuffles(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa)),Ys))) ) ).

% shuffles.simps(3)
tff(fact_3322_lists_Osimps,axiom,
    ! [A: $tType,A3: list(A),Aa2: set(A)] :
      ( member(list(A),A3,aa(set(A),set(list(A)),lists(A),Aa2))
    <=> ( ( A3 = nil(A) )
        | ? [A7: A,L4: list(A)] :
            ( ( A3 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A7),L4) )
            & member(A,A7,Aa2)
            & member(list(A),L4,aa(set(A),set(list(A)),lists(A),Aa2)) ) ) ) ).

% lists.simps
tff(fact_3323_lists_Ocases,axiom,
    ! [A: $tType,A3: list(A),Aa2: set(A)] :
      ( member(list(A),A3,aa(set(A),set(list(A)),lists(A),Aa2))
     => ( ( A3 != nil(A) )
       => ~ ! [A4: A,L3: list(A)] :
              ( ( A3 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A4),L3) )
             => ( member(A,A4,Aa2)
               => ~ member(list(A),L3,aa(set(A),set(list(A)),lists(A),Aa2)) ) ) ) ) ).

% lists.cases
tff(fact_3324_sorted__list__of__set__greaterThanLessThan,axiom,
    ! [Ia: nat,J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,Ia)),J)
     => ( aa(set(nat),list(nat),linord4507533701916653071of_set(nat),set_or5935395276787703475ssThan(nat,Ia,J)) = aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),aa(nat,nat,suc,Ia)),aa(set(nat),list(nat),linord4507533701916653071of_set(nat),set_or5935395276787703475ssThan(nat,aa(nat,nat,suc,Ia),J))) ) ) ).

% sorted_list_of_set_greaterThanLessThan
tff(fact_3325_the__elem__image__unique,axiom,
    ! [B: $tType,A: $tType,Aa2: set(A),F2: fun(A,B),Xa: A] :
      ( ( Aa2 != bot_bot(set(A)) )
     => ( ! [Y2: A] :
            ( member(A,Y2,Aa2)
           => ( aa(A,B,F2,Y2) = aa(A,B,F2,Xa) ) )
       => ( the_elem(B,aa(set(A),set(B),image2(A,B,F2),Aa2)) = aa(A,B,F2,Xa) ) ) ) ).

% the_elem_image_unique
tff(fact_3326_listrel__refl__on,axiom,
    ! [A: $tType,Aa2: set(A),R2: set(product_prod(A,A))] :
      ( refl_on(A,Aa2,R2)
     => refl_on(list(A),aa(set(A),set(list(A)),lists(A),Aa2),listrel(A,A,R2)) ) ).

% listrel_refl_on
tff(fact_3327_sorted__list__of__set_Osorted__sorted__key__list__of__set,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Aa2: set(A)] : sorted_wrt(A,ord_less_eq(A),aa(set(A),list(A),linord4507533701916653071of_set(A),Aa2)) ) ).

% sorted_list_of_set.sorted_sorted_key_list_of_set
tff(fact_3328_sorted__list__of__set_Ostrict__sorted__key__list__of__set,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Aa2: set(A)] : sorted_wrt(A,ord_less(A),aa(set(A),list(A),linord4507533701916653071of_set(A),Aa2)) ) ).

% sorted_list_of_set.strict_sorted_key_list_of_set
tff(fact_3329_le__SUP__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [Xa: A,F2: fun(B,A),Aa2: set(B)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F2),Aa2)))
        <=> ! [Y4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y4),Xa)
             => ? [X6: B] :
                  ( member(B,X6,Aa2)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y4),aa(B,A,F2,X6)) ) ) ) ) ).

% le_SUP_iff
tff(fact_3330_SUP__eq__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [Ia2: set(A),C2: B,F2: fun(A,B)] :
          ( ( Ia2 != bot_bot(set(A)) )
         => ( ! [I3: A] :
                ( member(A,I3,Ia2)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),C2),aa(A,B,F2,I3)) )
           => ( ( aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),Ia2)) = C2 )
            <=> ! [X6: A] :
                  ( member(A,X6,Ia2)
                 => ( aa(A,B,F2,X6) = C2 ) ) ) ) ) ) ).

% SUP_eq_iff
tff(fact_3331_cSUP__least,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [Aa2: set(A),F2: fun(A,B),M5: B] :
          ( ( Aa2 != bot_bot(set(A)) )
         => ( ! [X: A] :
                ( member(A,X,Aa2)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X)),M5) )
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),Aa2))),M5) ) ) ) ).

% cSUP_least
tff(fact_3332_range__eq__singletonD,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),A3: A,Xa: B] :
      ( ( aa(set(B),set(A),image2(B,A,F2),top_top(set(B))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),bot_bot(set(A))) )
     => ( aa(B,A,F2,Xa) = A3 ) ) ).

% range_eq_singletonD
tff(fact_3333_in__image__insert__iff,axiom,
    ! [A: $tType,B2: set(set(A)),Xa: A,Aa2: set(A)] :
      ( ! [C9: set(A)] :
          ( member(set(A),C9,B2)
         => ~ member(A,Xa,C9) )
     => ( member(set(A),Aa2,aa(set(set(A)),set(set(A)),image2(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa)),B2))
      <=> ( member(A,Xa,Aa2)
          & member(set(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A)))),B2) ) ) ) ).

% in_image_insert_iff
tff(fact_3334_wo__rel_Omax2__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A3: A,B3: A] :
      ( bNF_Wellorder_wo_rel(A,R2)
     => ( bNF_We1388413361240627857o_max2(A,R2,A3,B3) = $ite(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B3),R2),B3,A3) ) ) ).

% wo_rel.max2_def
tff(fact_3335_Pow__insert,axiom,
    ! [A: $tType,A3: A,Aa2: set(A)] : ( pow(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),Aa2)) = aa(set(set(A)),set(set(A)),aa(set(set(A)),fun(set(set(A)),set(set(A))),sup_sup(set(set(A))),pow(A,Aa2)),aa(set(set(A)),set(set(A)),image2(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3)),pow(A,Aa2))) ) ).

% Pow_insert
tff(fact_3336_set__image__eq__pointwiseI,axiom,
    ! [B: $tType,A: $tType,L: list(A),L2: list(A),F2: fun(A,B)] :
      ( ( aa(list(A),nat,size_size(list(A)),L) = aa(list(A),nat,size_size(list(A)),L2) )
     => ( ! [I3: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),L))
           => ( aa(A,B,F2,aa(nat,A,nth(A,L),I3)) = aa(A,B,F2,aa(nat,A,nth(A,L2),I3)) ) )
       => ( aa(set(A),set(B),image2(A,B,F2),aa(list(A),set(A),set2(A),L)) = aa(set(A),set(B),image2(A,B,F2),aa(list(A),set(A),set2(A),L2)) ) ) ) ).

% set_image_eq_pointwiseI
tff(fact_3337_in__set__image__conv__nth,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xa: B,L: list(B)] :
      ( member(A,aa(B,A,F2,Xa),aa(set(B),set(A),image2(B,A,F2),aa(list(B),set(B),set2(B),L)))
    <=> ? [I2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(B),nat,size_size(list(B)),L))
          & ( aa(B,A,F2,aa(nat,B,nth(B,L),I2)) = aa(B,A,F2,Xa) ) ) ) ).

% in_set_image_conv_nth
tff(fact_3338_atLeast0__atMost__Suc__eq__insert__0,axiom,
    ! [N: nat] : ( set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,N)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert3(nat),zero_zero(nat)),aa(set(nat),set(nat),image2(nat,nat,suc),set_or1337092689740270186AtMost(nat,zero_zero(nat),N))) ) ).

% atLeast0_atMost_Suc_eq_insert_0
tff(fact_3339_atMost__Suc__eq__insert__0,axiom,
    ! [N: nat] : ( aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,suc,N)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert3(nat),zero_zero(nat)),aa(set(nat),set(nat),image2(nat,nat,suc),aa(nat,set(nat),set_ord_atMost(nat),N))) ) ).

% atMost_Suc_eq_insert_0
tff(fact_3340_subset__subseqs,axiom,
    ! [A: $tType,X3: set(A),Xsa: list(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X3),aa(list(A),set(A),set2(A),Xsa))
     => member(set(A),X3,aa(set(list(A)),set(set(A)),image2(list(A),set(A),set2(A)),aa(list(list(A)),set(list(A)),set2(list(A)),aa(list(A),list(list(A)),subseqs(A),Xsa)))) ) ).

% subset_subseqs
tff(fact_3341_wo__rel_Omax2__greater,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A3: A,B3: A] :
      ( bNF_Wellorder_wo_rel(A,R2)
     => ( member(A,A3,aa(set(product_prod(A,A)),set(A),field2(A),R2))
       => ( member(A,B3,aa(set(product_prod(A,A)),set(A),field2(A),R2))
         => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),bNF_We1388413361240627857o_max2(A,R2,A3,B3)),R2)
            & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B3),bNF_We1388413361240627857o_max2(A,R2,A3,B3)),R2) ) ) ) ) ).

% wo_rel.max2_greater
tff(fact_3342_wo__rel_Omax2__equals2,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A3: A,B3: A] :
      ( bNF_Wellorder_wo_rel(A,R2)
     => ( member(A,A3,aa(set(product_prod(A,A)),set(A),field2(A),R2))
       => ( member(A,B3,aa(set(product_prod(A,A)),set(A),field2(A),R2))
         => ( ( bNF_We1388413361240627857o_max2(A,R2,A3,B3) = B3 )
          <=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B3),R2) ) ) ) ) ).

% wo_rel.max2_equals2
tff(fact_3343_wo__rel_Omax2__equals1,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A3: A,B3: A] :
      ( bNF_Wellorder_wo_rel(A,R2)
     => ( member(A,A3,aa(set(product_prod(A,A)),set(A),field2(A),R2))
       => ( member(A,B3,aa(set(product_prod(A,A)),set(A),field2(A),R2))
         => ( ( bNF_We1388413361240627857o_max2(A,R2,A3,B3) = A3 )
          <=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B3),A3),R2) ) ) ) ) ).

% wo_rel.max2_equals1
tff(fact_3344_subseqs__powset,axiom,
    ! [A: $tType,Xsa: list(A)] : ( aa(set(list(A)),set(set(A)),image2(list(A),set(A),set2(A)),aa(list(list(A)),set(list(A)),set2(list(A)),aa(list(A),list(list(A)),subseqs(A),Xsa))) = pow(A,aa(list(A),set(A),set2(A),Xsa)) ) ).

% subseqs_powset
tff(fact_3345_sorted__list__of__set_Oidem__if__sorted__distinct,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xsa: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),Xsa)
         => ( aa(list(A),$o,distinct(A),Xsa)
           => ( aa(set(A),list(A),linord4507533701916653071of_set(A),aa(list(A),set(A),set2(A),Xsa)) = Xsa ) ) ) ) ).

% sorted_list_of_set.idem_if_sorted_distinct
tff(fact_3346_shuffles_Oelims,axiom,
    ! [A: $tType,Xa: list(A),Xaa: list(A),Y: set(list(A))] :
      ( ( aa(list(A),set(list(A)),aa(list(A),fun(list(A),set(list(A))),shuffles(A),Xa),Xaa) = Y )
     => ( ( ( Xa = nil(A) )
         => ( Y != aa(set(list(A)),set(list(A)),aa(list(A),fun(set(list(A)),set(list(A))),insert3(list(A)),Xaa),bot_bot(set(list(A)))) ) )
       => ( ( ( Xaa = nil(A) )
           => ( Y != aa(set(list(A)),set(list(A)),aa(list(A),fun(set(list(A)),set(list(A))),insert3(list(A)),Xa),bot_bot(set(list(A)))) ) )
         => ~ ! [X: A,Xs: list(A)] :
                ( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) )
               => ! [Y2: A,Ys5: list(A)] :
                    ( ( Xaa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys5) )
                   => ( 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),X)),aa(list(A),set(list(A)),aa(list(A),fun(list(A),set(list(A))),shuffles(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys5)))),aa(set(list(A)),set(list(A)),image2(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2)),aa(list(A),set(list(A)),aa(list(A),fun(list(A),set(list(A))),shuffles(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),Ys5))) ) ) ) ) ) ) ).

% shuffles.elims
tff(fact_3347_Union__image__insert,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,set(A)),A3: B,B2: set(B)] : ( aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),F2),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert3(B),A3),B2))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(B,set(A),F2,A3)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),F2),B2))) ) ).

% Union_image_insert
tff(fact_3348_Union__image__empty,axiom,
    ! [B: $tType,A: $tType,Aa2: set(A),F2: fun(B,set(A))] : ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),F2),bot_bot(set(B))))) = Aa2 ) ).

% Union_image_empty
tff(fact_3349_translation__diff,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A,S2: set(A),T2: 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)),minus_minus(set(A)),S2),T2)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),plus_plus(A),A3)),S2)),aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),plus_plus(A),A3)),T2)) ) ) ).

% translation_diff
tff(fact_3350_nth__sorted__list__of__set__greaterThanAtMost,axiom,
    ! [N: nat,J: nat,Ia: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),Ia))
     => ( aa(nat,nat,nth(nat,aa(set(nat),list(nat),linord4507533701916653071of_set(nat),set_or3652927894154168847AtMost(nat,Ia,J))),N) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ia),N)) ) ) ).

% nth_sorted_list_of_set_greaterThanAtMost
tff(fact_3351_sorted__list__of__set_Osorted__key__list__of__set__unique,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Aa2: set(A),L: list(A)] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( ( sorted_wrt(A,ord_less(A),L)
              & ( aa(list(A),set(A),set2(A),L) = Aa2 )
              & ( aa(list(A),nat,size_size(list(A)),L) = finite_card(A,Aa2) ) )
          <=> ( aa(set(A),list(A),linord4507533701916653071of_set(A),Aa2) = L ) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_unique
tff(fact_3352_finite__shuffles,axiom,
    ! [A: $tType,Xsa: list(A),Ys: list(A)] : aa(set(list(A)),$o,finite_finite2(list(A)),aa(list(A),set(list(A)),aa(list(A),fun(list(A),set(list(A))),shuffles(A),Xsa),Ys)) ).

% finite_shuffles
tff(fact_3353_finite__insert,axiom,
    ! [A: $tType,A3: A,Aa2: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),Aa2))
    <=> aa(set(A),$o,finite_finite2(A),Aa2) ) ).

% finite_insert
tff(fact_3354_List_Ofinite__set,axiom,
    ! [A: $tType,Xsa: list(A)] : aa(set(A),$o,finite_finite2(A),aa(list(A),set(A),set2(A),Xsa)) ).

% List.finite_set
tff(fact_3355_finite__Field__eq__finite,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( aa(set(A),$o,finite_finite2(A),aa(set(product_prod(A,A)),set(A),field2(A),R))
    <=> aa(set(product_prod(A,A)),$o,finite_finite2(product_prod(A,A)),R) ) ).

% finite_Field_eq_finite
tff(fact_3356_finite__Diff2,axiom,
    ! [A: $tType,B2: set(A),Aa2: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),B2)
     => ( aa(set(A),$o,finite_finite2(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),B2))
      <=> aa(set(A),$o,finite_finite2(A),Aa2) ) ) ).

% finite_Diff2
tff(fact_3357_finite__Diff,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),Aa2)
     => aa(set(A),$o,finite_finite2(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),B2)) ) ).

% finite_Diff
tff(fact_3358_listI__assn__finite,axiom,
    ! [B: $tType,A: $tType,Ia2: set(nat),Aa2: fun(A,fun(B,assn)),Xsa: list(A),Xsia: list(B)] :
      ( ~ aa(set(nat),$o,finite_finite2(nat),Ia2)
     => ( vEBT_List_listI_assn(A,B,Ia2,Aa2,Xsa,Xsia) = bot_bot(assn) ) ) ).

% listI_assn_finite
tff(fact_3359_infinite__Icc__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B3: A] :
          ( ~ aa(set(A),$o,finite_finite2(A),set_or1337092689740270186AtMost(A,A3,B3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3) ) ) ).

% infinite_Icc_iff
tff(fact_3360_finite__Diff__insert,axiom,
    ! [A: $tType,Aa2: set(A),A3: A,B2: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),B2)))
    <=> aa(set(A),$o,finite_finite2(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),B2)) ) ).

% finite_Diff_insert
tff(fact_3361_greaterThanAtMost__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Ia: A,L: A,U: A] :
          ( member(A,Ia,set_or3652927894154168847AtMost(A,L,U))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),L),Ia)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ia),U) ) ) ) ).

% greaterThanAtMost_iff
tff(fact_3362_greaterThanAtMost__empty,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,K3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),K3)
         => ( set_or3652927894154168847AtMost(A,K3,L) = bot_bot(set(A)) ) ) ) ).

% greaterThanAtMost_empty
tff(fact_3363_greaterThanAtMost__empty__iff2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [K3: A,L: A] :
          ( ( bot_bot(set(A)) = set_or3652927894154168847AtMost(A,K3,L) )
        <=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),K3),L) ) ) ).

% greaterThanAtMost_empty_iff2
tff(fact_3364_greaterThanAtMost__empty__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [K3: A,L: A] :
          ( ( set_or3652927894154168847AtMost(A,K3,L) = bot_bot(set(A)) )
        <=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),K3),L) ) ) ).

% greaterThanAtMost_empty_iff
tff(fact_3365_infinite__Ioc__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B3: A] :
          ( ~ aa(set(A),$o,finite_finite2(A),set_or3652927894154168847AtMost(A,A3,B3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3) ) ) ).

% infinite_Ioc_iff
tff(fact_3366_infinite__Ioo__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B3: A] :
          ( ~ aa(set(A),$o,finite_finite2(A),set_or5935395276787703475ssThan(A,A3,B3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3) ) ) ).

% infinite_Ioo_iff
tff(fact_3367_cSup__greaterThanAtMost,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Y: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xa)
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or3652927894154168847AtMost(A,Y,Xa)) = Xa ) ) ) ).

% cSup_greaterThanAtMost
tff(fact_3368_Sup__greaterThanAtMost,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or3652927894154168847AtMost(A,Xa,Y)) = Y ) ) ) ).

% Sup_greaterThanAtMost
tff(fact_3369_sorted__list__of__set_Ofold__insort__key_Oinfinite,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Aa2: set(A)] :
          ( ~ aa(set(A),$o,finite_finite2(A),Aa2)
         => ( aa(set(A),list(A),linord4507533701916653071of_set(A),Aa2) = nil(A) ) ) ) ).

% sorted_list_of_set.fold_insort_key.infinite
tff(fact_3370_sorted__list__of__set_Oset__sorted__key__list__of__set,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Aa2: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( aa(list(A),set(A),set2(A),aa(set(A),list(A),linord4507533701916653071of_set(A),Aa2)) = Aa2 ) ) ) ).

% sorted_list_of_set.set_sorted_key_list_of_set
tff(fact_3371_card__0__eq,axiom,
    ! [A: $tType,Aa2: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),Aa2)
     => ( ( finite_card(A,Aa2) = zero_zero(nat) )
      <=> ( Aa2 = bot_bot(set(A)) ) ) ) ).

% card_0_eq
tff(fact_3372_card__insert__disjoint,axiom,
    ! [A: $tType,Aa2: set(A),Xa: A] :
      ( aa(set(A),$o,finite_finite2(A),Aa2)
     => ( ~ member(A,Xa,Aa2)
       => ( finite_card(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),Aa2)) = aa(nat,nat,suc,finite_card(A,Aa2)) ) ) ) ).

% card_insert_disjoint
tff(fact_3373_sorted__list__of__set_Osorted__key__list__of__set__eq__Nil__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Aa2: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( ( aa(set(A),list(A),linord4507533701916653071of_set(A),Aa2) = nil(A) )
          <=> ( Aa2 = bot_bot(set(A)) ) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_eq_Nil_iff
tff(fact_3374_finite__set__image,axiom,
    ! [A: $tType,Aa2: set(list(A))] :
      ( aa(set(set(A)),$o,finite_finite2(set(A)),aa(set(list(A)),set(set(A)),image2(list(A),set(A),set2(A)),Aa2))
     => ( ! [Xs: list(A)] :
            ( member(list(A),Xs,Aa2)
           => aa(list(A),$o,distinct(A),Xs) )
       => aa(set(list(A)),$o,finite_finite2(list(A)),Aa2) ) ) ).

% finite_set_image
tff(fact_3375_finite_OinsertI,axiom,
    ! [A: $tType,Aa2: set(A),A3: A] :
      ( aa(set(A),$o,finite_finite2(A),Aa2)
     => aa(set(A),$o,finite_finite2(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),Aa2)) ) ).

% finite.insertI
tff(fact_3376_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_3377_finite_OemptyI,axiom,
    ! [A: $tType] : aa(set(A),$o,finite_finite2(A),bot_bot(set(A))) ).

% finite.emptyI
tff(fact_3378_infinite__Ioc,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
         => ~ aa(set(A),$o,finite_finite2(A),set_or3652927894154168847AtMost(A,A3,B3)) ) ) ).

% infinite_Ioc
tff(fact_3379_finite__Field,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,finite_finite2(product_prod(A,A)),R2)
     => aa(set(A),$o,finite_finite2(A),aa(set(product_prod(A,A)),set(A),field2(A),R2)) ) ).

% finite_Field
tff(fact_3380_finite__list,axiom,
    ! [A: $tType,Aa2: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),Aa2)
     => ? [Xs: list(A)] : ( aa(list(A),set(A),set2(A),Xs) = Aa2 ) ) ).

% finite_list
tff(fact_3381_finite__nat__set__iff__bounded,axiom,
    ! [N7: set(nat)] :
      ( aa(set(nat),$o,finite_finite2(nat),N7)
    <=> ? [M: nat] :
        ! [X6: nat] :
          ( member(nat,X6,N7)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X6),M) ) ) ).

% finite_nat_set_iff_bounded
tff(fact_3382_bounded__nat__set__is__finite,axiom,
    ! [N7: set(nat),N: nat] :
      ( ! [X: nat] :
          ( member(nat,X,N7)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),N) )
     => aa(set(nat),$o,finite_finite2(nat),N7) ) ).

% bounded_nat_set_is_finite
tff(fact_3383_Diff__infinite__finite,axiom,
    ! [A: $tType,T4: set(A),S: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),T4)
     => ( ~ 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),T4)) ) ) ).

% Diff_infinite_finite
tff(fact_3384_finite__UNION__then__finite,axiom,
    ! [A: $tType,B: $tType,B2: fun(B,set(A)),Aa2: set(B),A3: B] :
      ( aa(set(A),$o,finite_finite2(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B2),Aa2)))
     => ( member(B,A3,Aa2)
       => aa(set(A),$o,finite_finite2(A),aa(B,set(A),B2,A3)) ) ) ).

% finite_UNION_then_finite
tff(fact_3385_finite__psubset__induct,axiom,
    ! [A: $tType,Aa2: set(A),Pa: fun(set(A),$o)] :
      ( aa(set(A),$o,finite_finite2(A),Aa2)
     => ( ! [A12: set(A)] :
            ( aa(set(A),$o,finite_finite2(A),A12)
           => ( ! [B13: set(A)] :
                  ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),B13),A12)
                 => aa(set(A),$o,Pa,B13) )
             => aa(set(A),$o,Pa,A12) ) )
       => aa(set(A),$o,Pa,Aa2) ) ) ).

% finite_psubset_induct
tff(fact_3386_sorted__list__of__set_Osorted__key__list__of__set__inject,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Aa2: set(A),B2: set(A)] :
          ( ( aa(set(A),list(A),linord4507533701916653071of_set(A),Aa2) = aa(set(A),list(A),linord4507533701916653071of_set(A),B2) )
         => ( aa(set(A),$o,finite_finite2(A),Aa2)
           => ( aa(set(A),$o,finite_finite2(A),B2)
             => ( Aa2 = B2 ) ) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_inject
tff(fact_3387_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_3388_finite__has__maximal,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Aa2: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( ( Aa2 != bot_bot(set(A)) )
           => ? [X: A] :
                ( member(A,X,Aa2)
                & ! [Xa4: A] :
                    ( member(A,Xa4,Aa2)
                   => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Xa4)
                     => ( X = Xa4 ) ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_3389_finite__has__minimal,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Aa2: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( ( Aa2 != bot_bot(set(A)) )
           => ? [X: A] :
                ( member(A,X,Aa2)
                & ! [Xa4: A] :
                    ( member(A,Xa4,Aa2)
                   => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa4),X)
                     => ( X = Xa4 ) ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_3390_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)) )
           => ? [X: A] :
                ( member(A,X,S)
                & ~ ? [Xa4: A] :
                      ( member(A,Xa4,S)
                      & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa4),X) ) ) ) ) ) ).

% ex_min_if_finite
tff(fact_3391_infinite__growing,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X3: set(A)] :
          ( ( X3 != bot_bot(set(A)) )
         => ( ! [X: A] :
                ( member(A,X,X3)
               => ? [Xa4: A] :
                    ( member(A,Xa4,X3)
                    & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Xa4) ) )
           => ~ aa(set(A),$o,finite_finite2(A),X3) ) ) ) ).

% infinite_growing
tff(fact_3392_finite__imp__Sup__less,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X3: set(A),Xa: A,A3: A] :
          ( aa(set(A),$o,finite_finite2(A),X3)
         => ( member(A,Xa,X3)
           => ( ! [X: A] :
                  ( member(A,X,X3)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),A3) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(A),A,complete_Sup_Sup(A),X3)),A3) ) ) ) ) ).

% finite_imp_Sup_less
tff(fact_3393_infinite__finite__induct,axiom,
    ! [A: $tType,Pa: fun(set(A),$o),Aa2: set(A)] :
      ( ! [A12: set(A)] :
          ( ~ aa(set(A),$o,finite_finite2(A),A12)
         => aa(set(A),$o,Pa,A12) )
     => ( aa(set(A),$o,Pa,bot_bot(set(A)))
       => ( ! [X: A,F7: set(A)] :
              ( aa(set(A),$o,finite_finite2(A),F7)
             => ( ~ member(A,X,F7)
               => ( aa(set(A),$o,Pa,F7)
                 => aa(set(A),$o,Pa,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),X),F7)) ) ) )
         => aa(set(A),$o,Pa,Aa2) ) ) ) ).

% infinite_finite_induct
tff(fact_3394_finite__ne__induct,axiom,
    ! [A: $tType,Fa: set(A),Pa: fun(set(A),$o)] :
      ( aa(set(A),$o,finite_finite2(A),Fa)
     => ( ( Fa != bot_bot(set(A)) )
       => ( ! [X: A] : aa(set(A),$o,Pa,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),X),bot_bot(set(A))))
         => ( ! [X: A,F7: set(A)] :
                ( aa(set(A),$o,finite_finite2(A),F7)
               => ( ( F7 != bot_bot(set(A)) )
                 => ( ~ member(A,X,F7)
                   => ( aa(set(A),$o,Pa,F7)
                     => aa(set(A),$o,Pa,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),X),F7)) ) ) ) )
           => aa(set(A),$o,Pa,Fa) ) ) ) ) ).

% finite_ne_induct
tff(fact_3395_finite__induct,axiom,
    ! [A: $tType,Fa: set(A),Pa: fun(set(A),$o)] :
      ( aa(set(A),$o,finite_finite2(A),Fa)
     => ( aa(set(A),$o,Pa,bot_bot(set(A)))
       => ( ! [X: A,F7: set(A)] :
              ( aa(set(A),$o,finite_finite2(A),F7)
             => ( ~ member(A,X,F7)
               => ( aa(set(A),$o,Pa,F7)
                 => aa(set(A),$o,Pa,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),X),F7)) ) ) )
         => aa(set(A),$o,Pa,Fa) ) ) ) ).

% finite_induct
tff(fact_3396_finite_Osimps,axiom,
    ! [A: $tType,A3: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A3)
    <=> ( ( A3 = bot_bot(set(A)) )
        | ? [A11: set(A),A7: A] :
            ( ( A3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A7),A11) )
            & aa(set(A),$o,finite_finite2(A),A11) ) ) ) ).

% finite.simps
tff(fact_3397_finite_Ocases,axiom,
    ! [A: $tType,A3: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( ( A3 != bot_bot(set(A)) )
       => ~ ! [A12: set(A)] :
              ( ? [A4: A] : ( A3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A4),A12) )
             => ~ aa(set(A),$o,finite_finite2(A),A12) ) ) ) ).

% finite.cases
tff(fact_3398_infinite__Icc,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
         => ~ aa(set(A),$o,finite_finite2(A),set_or1337092689740270186AtMost(A,A3,B3)) ) ) ).

% infinite_Icc
tff(fact_3399_finite__maxlen,axiom,
    ! [A: $tType,M5: set(list(A))] :
      ( aa(set(list(A)),$o,finite_finite2(list(A)),M5)
     => ? [N5: nat] :
        ! [X4: list(A)] :
          ( member(list(A),X4,M5)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),X4)),N5) ) ) ).

% finite_maxlen
tff(fact_3400_finite__distinct__list,axiom,
    ! [A: $tType,Aa2: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),Aa2)
     => ? [Xs: list(A)] :
          ( ( aa(list(A),set(A),set2(A),Xs) = Aa2 )
          & aa(list(A),$o,distinct(A),Xs) ) ) ).

% finite_distinct_list
tff(fact_3401_ivl__disj__int__two_I6_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,M2: A,U: A] : ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or3652927894154168847AtMost(A,L,M2)),set_or3652927894154168847AtMost(A,M2,U)) = bot_bot(set(A)) ) ) ).

% ivl_disj_int_two(6)
tff(fact_3402_infinite__Ioo,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
         => ~ aa(set(A),$o,finite_finite2(A),set_or5935395276787703475ssThan(A,A3,B3)) ) ) ).

% infinite_Ioo
tff(fact_3403_card__partition,axiom,
    ! [A: $tType,C3: set(set(A)),K3: nat] :
      ( aa(set(set(A)),$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))
       => ( ! [C5: set(A)] :
              ( member(set(A),C5,C3)
             => ( finite_card(A,C5) = K3 ) )
         => ( ! [C13: set(A),C24: set(A)] :
                ( member(set(A),C13,C3)
               => ( member(set(A),C24,C3)
                 => ( ( C13 != C24 )
                   => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),C13),C24) = bot_bot(set(A)) ) ) ) )
           => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K3),finite_card(set(A),C3)) = finite_card(A,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C3)) ) ) ) ) ) ).

% card_partition
tff(fact_3404_finite__ranking__induct,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [S: set(A),Pa: fun(set(A),$o),F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),S)
         => ( aa(set(A),$o,Pa,bot_bot(set(A)))
           => ( ! [X: A,S5: set(A)] :
                  ( aa(set(A),$o,finite_finite2(A),S5)
                 => ( ! [Y3: A] :
                        ( member(A,Y3,S5)
                       => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,Y3)),aa(A,B,F2,X)) )
                   => ( aa(set(A),$o,Pa,S5)
                     => aa(set(A),$o,Pa,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),X),S5)) ) ) )
             => aa(set(A),$o,Pa,S) ) ) ) ) ).

% finite_ranking_induct
tff(fact_3405_finite__linorder__min__induct,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Aa2: set(A),Pa: fun(set(A),$o)] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( aa(set(A),$o,Pa,bot_bot(set(A)))
           => ( ! [B4: A,A12: set(A)] :
                  ( aa(set(A),$o,finite_finite2(A),A12)
                 => ( ! [X4: A] :
                        ( member(A,X4,A12)
                       => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B4),X4) )
                   => ( aa(set(A),$o,Pa,A12)
                     => aa(set(A),$o,Pa,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),B4),A12)) ) ) )
             => aa(set(A),$o,Pa,Aa2) ) ) ) ) ).

% finite_linorder_min_induct
tff(fact_3406_finite__linorder__max__induct,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Aa2: set(A),Pa: fun(set(A),$o)] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( aa(set(A),$o,Pa,bot_bot(set(A)))
           => ( ! [B4: A,A12: set(A)] :
                  ( aa(set(A),$o,finite_finite2(A),A12)
                 => ( ! [X4: A] :
                        ( member(A,X4,A12)
                       => aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),B4) )
                   => ( aa(set(A),$o,Pa,A12)
                     => aa(set(A),$o,Pa,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),B4),A12)) ) ) )
             => aa(set(A),$o,Pa,Aa2) ) ) ) ) ).

% finite_linorder_max_induct
tff(fact_3407_finite__Sup__less__iff,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X3: set(A),A3: A] :
          ( aa(set(A),$o,finite_finite2(A),X3)
         => ( ( X3 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(A),A,complete_Sup_Sup(A),X3)),A3)
            <=> ! [X6: A] :
                  ( member(A,X6,X3)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),X6),A3) ) ) ) ) ) ).

% finite_Sup_less_iff
tff(fact_3408_finite__subset__induct_H,axiom,
    ! [A: $tType,Fa: set(A),Aa2: set(A),Pa: fun(set(A),$o)] :
      ( aa(set(A),$o,finite_finite2(A),Fa)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Fa),Aa2)
       => ( aa(set(A),$o,Pa,bot_bot(set(A)))
         => ( ! [A4: A,F7: set(A)] :
                ( aa(set(A),$o,finite_finite2(A),F7)
               => ( member(A,A4,Aa2)
                 => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),F7),Aa2)
                   => ( ~ member(A,A4,F7)
                     => ( aa(set(A),$o,Pa,F7)
                       => aa(set(A),$o,Pa,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A4),F7)) ) ) ) ) )
           => aa(set(A),$o,Pa,Fa) ) ) ) ) ).

% finite_subset_induct'
tff(fact_3409_finite__subset__induct,axiom,
    ! [A: $tType,Fa: set(A),Aa2: set(A),Pa: fun(set(A),$o)] :
      ( aa(set(A),$o,finite_finite2(A),Fa)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Fa),Aa2)
       => ( aa(set(A),$o,Pa,bot_bot(set(A)))
         => ( ! [A4: A,F7: set(A)] :
                ( aa(set(A),$o,finite_finite2(A),F7)
               => ( member(A,A4,Aa2)
                 => ( ~ member(A,A4,F7)
                   => ( aa(set(A),$o,Pa,F7)
                     => aa(set(A),$o,Pa,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A4),F7)) ) ) ) )
           => aa(set(A),$o,Pa,Fa) ) ) ) ) ).

% finite_subset_induct
tff(fact_3410_finite__Sup__in,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Aa2: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( ( Aa2 != bot_bot(set(A)) )
           => ( ! [X: A,Y2: A] :
                  ( member(A,X,Aa2)
                 => ( member(A,Y2,Aa2)
                   => member(A,aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y2),Aa2) ) )
             => member(A,aa(set(A),A,complete_Sup_Sup(A),Aa2),Aa2) ) ) ) ) ).

% finite_Sup_in
tff(fact_3411_finite__empty__induct,axiom,
    ! [A: $tType,Aa2: set(A),Pa: fun(set(A),$o)] :
      ( aa(set(A),$o,finite_finite2(A),Aa2)
     => ( aa(set(A),$o,Pa,Aa2)
       => ( ! [A4: A,A12: set(A)] :
              ( aa(set(A),$o,finite_finite2(A),A12)
             => ( member(A,A4,A12)
               => ( aa(set(A),$o,Pa,A12)
                 => aa(set(A),$o,Pa,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A12),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A4),bot_bot(set(A))))) ) ) )
         => aa(set(A),$o,Pa,bot_bot(set(A))) ) ) ) ).

% finite_empty_induct
tff(fact_3412_infinite__coinduct,axiom,
    ! [A: $tType,X3: fun(set(A),$o),Aa2: set(A)] :
      ( aa(set(A),$o,X3,Aa2)
     => ( ! [A12: set(A)] :
            ( aa(set(A),$o,X3,A12)
           => ? [X4: A] :
                ( member(A,X4,A12)
                & ( aa(set(A),$o,X3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A12),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),X4),bot_bot(set(A)))))
                  | ~ aa(set(A),$o,finite_finite2(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A12),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),X4),bot_bot(set(A))))) ) ) )
       => ~ aa(set(A),$o,finite_finite2(A),Aa2) ) ) ).

% infinite_coinduct
tff(fact_3413_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)),insert3(A),A3),bot_bot(set(A))))) ) ).

% infinite_remove
tff(fact_3414_card__eq__0__iff,axiom,
    ! [A: $tType,Aa2: set(A)] :
      ( ( finite_card(A,Aa2) = zero_zero(nat) )
    <=> ( ( Aa2 = bot_bot(set(A)) )
        | ~ aa(set(A),$o,finite_finite2(A),Aa2) ) ) ).

% card_eq_0_iff
tff(fact_3415_card__ge__0__finite,axiom,
    ! [A: $tType,Aa2: set(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),finite_card(A,Aa2))
     => aa(set(A),$o,finite_finite2(A),Aa2) ) ).

% card_ge_0_finite
tff(fact_3416_card__Suc__eq__finite,axiom,
    ! [A: $tType,Aa2: set(A),K3: nat] :
      ( ( finite_card(A,Aa2) = aa(nat,nat,suc,K3) )
    <=> ? [B8: A,B12: set(A)] :
          ( ( Aa2 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),B8),B12) )
          & ~ member(A,B8,B12)
          & ( finite_card(A,B12) = K3 )
          & aa(set(A),$o,finite_finite2(A),B12) ) ) ).

% card_Suc_eq_finite
tff(fact_3417_card__insert__if,axiom,
    ! [A: $tType,Aa2: set(A),Xa: A] :
      ( aa(set(A),$o,finite_finite2(A),Aa2)
     => ( finite_card(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),Aa2)) = $ite(member(A,Xa,Aa2),finite_card(A,Aa2),aa(nat,nat,suc,finite_card(A,Aa2))) ) ) ).

% card_insert_if
tff(fact_3418_card__less__sym__Diff,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),Aa2)
     => ( aa(set(A),$o,finite_finite2(A),B2)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),finite_card(A,Aa2)),finite_card(A,B2))
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),finite_card(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),B2))),finite_card(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B2),Aa2))) ) ) ) ).

% card_less_sym_Diff
tff(fact_3419_card__le__sym__Diff,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),Aa2)
     => ( aa(set(A),$o,finite_finite2(A),B2)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),finite_card(A,Aa2)),finite_card(A,B2))
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),finite_card(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),B2))),finite_card(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B2),Aa2))) ) ) ) ).

% card_le_sym_Diff
tff(fact_3420_Ioc__disjoint,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A,C2: A,D3: A] :
          ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or3652927894154168847AtMost(A,A3,B3)),set_or3652927894154168847AtMost(A,C2,D3)) = bot_bot(set(A)) )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3)
            | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),D3),C2)
            | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),C2)
            | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),D3),A3) ) ) ) ).

% Ioc_disjoint
tff(fact_3421_psubset__card__mono,axiom,
    ! [A: $tType,B2: set(A),Aa2: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),B2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),Aa2),B2)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),finite_card(A,Aa2)),finite_card(A,B2)) ) ) ).

% psubset_card_mono
tff(fact_3422_ivl__disj__int__two_I8_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,M2: A,U: A] : ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or1337092689740270186AtMost(A,L,M2)),set_or3652927894154168847AtMost(A,M2,U)) = bot_bot(set(A)) ) ) ).

% ivl_disj_int_two(8)
tff(fact_3423_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_3424_ivl__disj__int__two_I2_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,M2: A,U: A] : ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or3652927894154168847AtMost(A,L,M2)),set_or5935395276787703475ssThan(A,M2,U)) = bot_bot(set(A)) ) ) ).

% ivl_disj_int_two(2)
tff(fact_3425_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_3426_sorted__list__of__set__greaterThanAtMost,axiom,
    ! [Ia: nat,J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,Ia)),J)
     => ( aa(set(nat),list(nat),linord4507533701916653071of_set(nat),set_or3652927894154168847AtMost(nat,Ia,J)) = aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),aa(nat,nat,suc,Ia)),aa(set(nat),list(nat),linord4507533701916653071of_set(nat),set_or3652927894154168847AtMost(nat,aa(nat,nat,suc,Ia),J))) ) ) ).

% sorted_list_of_set_greaterThanAtMost
tff(fact_3427_finite__remove__induct,axiom,
    ! [A: $tType,B2: set(A),Pa: fun(set(A),$o)] :
      ( aa(set(A),$o,finite_finite2(A),B2)
     => ( aa(set(A),$o,Pa,bot_bot(set(A)))
       => ( ! [A12: set(A)] :
              ( aa(set(A),$o,finite_finite2(A),A12)
             => ( ( A12 != bot_bot(set(A)) )
               => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A12),B2)
                 => ( ! [X4: A] :
                        ( member(A,X4,A12)
                       => aa(set(A),$o,Pa,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A12),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),X4),bot_bot(set(A))))) )
                   => aa(set(A),$o,Pa,A12) ) ) ) )
         => aa(set(A),$o,Pa,B2) ) ) ) ).

% finite_remove_induct
tff(fact_3428_remove__induct,axiom,
    ! [A: $tType,Pa: fun(set(A),$o),B2: set(A)] :
      ( aa(set(A),$o,Pa,bot_bot(set(A)))
     => ( ( ~ aa(set(A),$o,finite_finite2(A),B2)
         => aa(set(A),$o,Pa,B2) )
       => ( ! [A12: set(A)] :
              ( aa(set(A),$o,finite_finite2(A),A12)
             => ( ( A12 != bot_bot(set(A)) )
               => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A12),B2)
                 => ( ! [X4: A] :
                        ( member(A,X4,A12)
                       => aa(set(A),$o,Pa,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A12),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),X4),bot_bot(set(A))))) )
                   => aa(set(A),$o,Pa,A12) ) ) ) )
         => aa(set(A),$o,Pa,B2) ) ) ) ).

% remove_induct
tff(fact_3429_card__gt__0__iff,axiom,
    ! [A: $tType,Aa2: set(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),finite_card(A,Aa2))
    <=> ( ( Aa2 != bot_bot(set(A)) )
        & aa(set(A),$o,finite_finite2(A),Aa2) ) ) ).

% card_gt_0_iff
tff(fact_3430_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)),finite_card(A,top_top(set(A)))) ) ).

% finite_UNIV_card_ge_0
tff(fact_3431_card__le__Suc__iff,axiom,
    ! [A: $tType,N: nat,Aa2: set(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,N)),finite_card(A,Aa2))
    <=> ? [A7: A,B12: set(A)] :
          ( ( Aa2 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A7),B12) )
          & ~ member(A,A7,B12)
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),finite_card(A,B12))
          & aa(set(A),$o,finite_finite2(A),B12) ) ) ).

% card_le_Suc_iff
tff(fact_3432_card__1__singletonI,axiom,
    ! [A: $tType,S: set(A),Xa: A] :
      ( aa(set(A),$o,finite_finite2(A),S)
     => ( ( finite_card(A,S) = one_one(nat) )
       => ( member(A,Xa,S)
         => ( S = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A))) ) ) ) ) ).

% card_1_singletonI
tff(fact_3433_finite__induct__select,axiom,
    ! [A: $tType,S: set(A),Pa: fun(set(A),$o)] :
      ( aa(set(A),$o,finite_finite2(A),S)
     => ( aa(set(A),$o,Pa,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),$o,Pa,T5)
               => ? [X4: A] :
                    ( member(A,X4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S),T5))
                    & aa(set(A),$o,Pa,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),X4),T5)) ) ) )
         => aa(set(A),$o,Pa,S) ) ) ) ).

% finite_induct_select
tff(fact_3434_card__Diff__subset,axiom,
    ! [A: $tType,B2: set(A),Aa2: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),B2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),Aa2)
       => ( finite_card(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),B2)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),finite_card(A,Aa2)),finite_card(A,B2)) ) ) ) ).

% card_Diff_subset
tff(fact_3435_finite__sorted__distinct__unique,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Aa2: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ? [X: list(A)] :
              ( ( aa(list(A),set(A),set2(A),X) = Aa2 )
              & sorted_wrt(A,ord_less_eq(A),X)
              & aa(list(A),$o,distinct(A),X)
              & ! [Y3: list(A)] :
                  ( ( ( aa(list(A),set(A),set2(A),Y3) = Aa2 )
                    & sorted_wrt(A,ord_less_eq(A),Y3)
                    & aa(list(A),$o,distinct(A),Y3) )
                 => ( Y3 = X ) ) ) ) ) ).

% finite_sorted_distinct_unique
tff(fact_3436_card__psubset,axiom,
    ! [A: $tType,B2: set(A),Aa2: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),B2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),B2)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),finite_card(A,Aa2)),finite_card(A,B2))
         => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),Aa2),B2) ) ) ) ).

% card_psubset
tff(fact_3437_diff__card__le__card__Diff,axiom,
    ! [A: $tType,B2: set(A),Aa2: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),B2)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),finite_card(A,Aa2)),finite_card(A,B2))),finite_card(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),B2))) ) ).

% diff_card_le_card_Diff
tff(fact_3438_wf__bounded__set,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,A)),Ub: fun(A,set(B)),F2: fun(A,set(B))] :
      ( ! [A4: A,B4: A] :
          ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B4),A4),R2)
         => ( aa(set(B),$o,finite_finite2(B),aa(A,set(B),Ub,A4))
            & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(A,set(B),Ub,B4)),aa(A,set(B),Ub,A4))
            & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(A,set(B),F2,B4)),aa(A,set(B),Ub,A4))
            & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less(set(B)),aa(A,set(B),F2,A4)),aa(A,set(B),F2,B4)) ) )
     => wf(A,R2) ) ).

% wf_bounded_set
tff(fact_3439_card__Diff__subset__Int,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),B2))
     => ( finite_card(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),B2)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),finite_card(A,Aa2)),finite_card(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),B2))) ) ) ).

% card_Diff_subset_Int
tff(fact_3440_greaterThanAtMost__subseteq__atLeastAtMost__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B3: A,C2: A,D3: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or3652927894154168847AtMost(A,A3,B3)),set_or1337092689740270186AtMost(A,C2,D3))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),D3) ) ) ) ) ).

% greaterThanAtMost_subseteq_atLeastAtMost_iff
tff(fact_3441_ivl__disj__un__two__touch_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M2: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),L),M2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),M2),U)
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or3652927894154168847AtMost(A,L,M2)),set_or1337092689740270186AtMost(A,M2,U)) = set_or3652927894154168847AtMost(A,L,U) ) ) ) ) ).

% ivl_disj_un_two_touch(3)
tff(fact_3442_greaterThanAtMost__eq__atLeastAtMost__diff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B3: A] : ( set_or3652927894154168847AtMost(A,A3,B3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),set_or1337092689740270186AtMost(A,A3,B3)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),bot_bot(set(A)))) ) ) ).

% greaterThanAtMost_eq_atLeastAtMost_diff
tff(fact_3443_greaterThanLessThan__subseteq__greaterThanAtMost__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B3: A,C2: A,D3: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or5935395276787703475ssThan(A,A3,B3)),set_or3652927894154168847AtMost(A,C2,D3))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),D3) ) ) ) ) ).

% greaterThanLessThan_subseteq_greaterThanAtMost_iff
tff(fact_3444_ivl__disj__un__two_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M2: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),M2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),M2),U)
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or3652927894154168847AtMost(A,L,M2)),set_or5935395276787703475ssThan(A,M2,U)) = set_or5935395276787703475ssThan(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(2)
tff(fact_3445_Gcd__remove0__nat,axiom,
    ! [M5: set(nat)] :
      ( aa(set(nat),$o,finite_finite2(nat),M5)
     => ( gcd_Gcd(nat,M5) = gcd_Gcd(nat,aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),M5),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert3(nat),zero_zero(nat)),bot_bot(set(nat))))) ) ) ).

% Gcd_remove0_nat
tff(fact_3446_card__range__greater__zero,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A)] :
      ( aa(set(A),$o,finite_finite2(A),aa(set(B),set(A),image2(B,A,F2),top_top(set(B))))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),finite_card(A,aa(set(B),set(A),image2(B,A,F2),top_top(set(B))))) ) ).

% card_range_greater_zero
tff(fact_3447_card__Suc__Diff1,axiom,
    ! [A: $tType,Aa2: set(A),Xa: A] :
      ( aa(set(A),$o,finite_finite2(A),Aa2)
     => ( member(A,Xa,Aa2)
       => ( aa(nat,nat,suc,finite_card(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A)))))) = finite_card(A,Aa2) ) ) ) ).

% card_Suc_Diff1
tff(fact_3448_card_Oinsert__remove,axiom,
    ! [A: $tType,Aa2: set(A),Xa: A] :
      ( aa(set(A),$o,finite_finite2(A),Aa2)
     => ( finite_card(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),Aa2)) = aa(nat,nat,suc,finite_card(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A)))))) ) ) ).

% card.insert_remove
tff(fact_3449_card_Oremove,axiom,
    ! [A: $tType,Aa2: set(A),Xa: A] :
      ( aa(set(A),$o,finite_finite2(A),Aa2)
     => ( member(A,Xa,Aa2)
       => ( finite_card(A,Aa2) = aa(nat,nat,suc,finite_card(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A)))))) ) ) ) ).

% card.remove
tff(fact_3450_card__insert__disjoint_H,axiom,
    ! [A: $tType,Aa2: set(A),Xa: A] :
      ( aa(set(A),$o,finite_finite2(A),Aa2)
     => ( ~ member(A,Xa,Aa2)
       => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),finite_card(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),Aa2))),aa(nat,nat,suc,zero_zero(nat))) = finite_card(A,Aa2) ) ) ) ).

% card_insert_disjoint'
tff(fact_3451_card__Diff1__less__iff,axiom,
    ! [A: $tType,Aa2: set(A),Xa: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),finite_card(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A)))))),finite_card(A,Aa2))
    <=> ( aa(set(A),$o,finite_finite2(A),Aa2)
        & member(A,Xa,Aa2) ) ) ).

% card_Diff1_less_iff
tff(fact_3452_card__Diff2__less,axiom,
    ! [A: $tType,Aa2: set(A),Xa: A,Y: A] :
      ( aa(set(A),$o,finite_finite2(A),Aa2)
     => ( member(A,Xa,Aa2)
       => ( member(A,Y,Aa2)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(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)),Aa2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A))))),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Y),bot_bot(set(A)))))),finite_card(A,Aa2)) ) ) ) ).

% card_Diff2_less
tff(fact_3453_card__Diff1__less,axiom,
    ! [A: $tType,Aa2: set(A),Xa: A] :
      ( aa(set(A),$o,finite_finite2(A),Aa2)
     => ( member(A,Xa,Aa2)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),finite_card(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A)))))),finite_card(A,Aa2)) ) ) ).

% card_Diff1_less
tff(fact_3454_sorted__list__of__set_Ofinite__set__strict__sorted,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Aa2: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ~ ! [L3: list(A)] :
                ( sorted_wrt(A,ord_less(A),L3)
               => ( ( aa(list(A),set(A),set2(A),L3) = Aa2 )
                 => ( aa(list(A),nat,size_size(list(A)),L3) != finite_card(A,Aa2) ) ) ) ) ) ).

% sorted_list_of_set.finite_set_strict_sorted
tff(fact_3455_card__Un__disjoint,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),Aa2)
     => ( aa(set(A),$o,finite_finite2(A),B2)
       => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),B2) = bot_bot(set(A)) )
         => ( finite_card(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),B2)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),finite_card(A,Aa2)),finite_card(A,B2)) ) ) ) ) ).

% card_Un_disjoint
tff(fact_3456_dvd__partition,axiom,
    ! [A: $tType,C3: set(set(A)),K3: nat] :
      ( aa(set(A),$o,finite_finite2(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C3))
     => ( ! [X: set(A)] :
            ( member(set(A),X,C3)
           => dvd_dvd(nat,K3,finite_card(A,X)) )
       => ( ! [X: set(A)] :
              ( member(set(A),X,C3)
             => ! [Xa5: set(A)] :
                  ( member(set(A),Xa5,C3)
                 => ( ( X != Xa5 )
                   => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),X),Xa5) = bot_bot(set(A)) ) ) ) )
         => dvd_dvd(nat,K3,finite_card(A,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C3))) ) ) ) ).

% dvd_partition
tff(fact_3457_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)),insert3(A),L),bot_bot(set(A)))),set_or3652927894154168847AtMost(A,L,U)) = set_or1337092689740270186AtMost(A,L,U) ) ) ) ).

% ivl_disj_un_singleton(5)
tff(fact_3458_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)),insert3(A),U),bot_bot(set(A)))) = set_or3652927894154168847AtMost(A,L,U) ) ) ) ).

% ivl_disj_un_singleton(4)
tff(fact_3459_ivl__disj__un__two_I5_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M2: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),L),M2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),M2),U)
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or5935395276787703475ssThan(A,L,M2)),set_or1337092689740270186AtMost(A,M2,U)) = set_or3652927894154168847AtMost(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(5)
tff(fact_3460_sorted__list__of__set_Osorted__key__list__of__set__remove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Aa2: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( 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)),Aa2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A))))) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),remove1(A),Xa),aa(set(A),list(A),linord4507533701916653071of_set(A),Aa2)) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_remove
tff(fact_3461_set__encode__insert,axiom,
    ! [Aa2: set(nat),N: nat] :
      ( aa(set(nat),$o,finite_finite2(nat),Aa2)
     => ( ~ member(nat,N,Aa2)
       => ( nat_set_encode(aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert3(nat),N),Aa2)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),numeral_numeral(nat,bit0(one2))),N)),nat_set_encode(Aa2)) ) ) ) ).

% set_encode_insert
tff(fact_3462_in__finite__psubset,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A)] :
      ( member(product_prod(set(A),set(A)),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),Aa2),B2),finite_psubset(A))
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),Aa2),B2)
        & aa(set(A),$o,finite_finite2(A),B2) ) ) ).

% in_finite_psubset
tff(fact_3463_subset__mset_Osum__mono2,axiom,
    ! [B: $tType,A: $tType,B2: set(A),Aa2: set(A),F2: fun(A,multiset(B))] :
      ( aa(set(A),$o,finite_finite2(A),B2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),B2)
       => ( ! [B4: A] :
              ( member(A,B4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B2),Aa2))
             => aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),zero_zero(multiset(B))),aa(A,multiset(B),F2,B4)) )
         => 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)),F2,Aa2)),groups3894954378712506084id_sum(multiset(B),A,plus_plus(multiset(B)),zero_zero(multiset(B)),F2,B2)) ) ) ) ).

% subset_mset.sum_mono2
tff(fact_3464_UNION__fun__upd,axiom,
    ! [B: $tType,A: $tType,Aa2: fun(B,set(A)),Ia: B,B2: set(A),J4: set(B)] :
      ( aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),fun_upd(B,set(A),Aa2,Ia,B2)),J4)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),Aa2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),J4),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert3(B),Ia),bot_bot(set(B))))))),
          $ite(member(B,Ia,J4),B2,bot_bot(set(A)))) ) ).

% UNION_fun_upd
tff(fact_3465_arg__min__if__finite_I2_J,axiom,
    ! [B: $tType,A: $tType] :
      ( order(B)
     => ! [S: set(A),F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),S)
         => ( ( S != bot_bot(set(A)) )
           => ~ ? [X4: A] :
                  ( member(A,X4,S)
                  & aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,X4)),aa(A,B,F2,lattic7623131987881927897min_on(A,B,F2,S))) ) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_3466_image__update,axiom,
    ! [B: $tType,A: $tType,Xa: A,Aa2: set(A),F2: fun(A,B),N: B] :
      ( ~ member(A,Xa,Aa2)
     => ( aa(set(A),set(B),image2(A,B,fun_upd(A,B,F2,Xa,N)),Aa2) = aa(set(A),set(B),image2(A,B,F2),Aa2) ) ) ).

% image_update
tff(fact_3467_set__encode__empty,axiom,
    nat_set_encode(bot_bot(set(nat))) = zero_zero(nat) ).

% set_encode_empty
tff(fact_3468_fun__upd__image,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xa: B,Y: A,Aa2: set(B)] :
      ( aa(set(B),set(A),image2(B,A,fun_upd(B,A,F2,Xa,Y)),Aa2) = $ite(member(B,Xa,Aa2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Y),aa(set(B),set(A),image2(B,A,F2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),Aa2),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert3(B),Xa),bot_bot(set(B)))))),aa(set(B),set(A),image2(B,A,F2),Aa2)) ) ).

% fun_upd_image
tff(fact_3469_arg__min__if__finite_I1_J,axiom,
    ! [B: $tType,A: $tType] :
      ( order(B)
     => ! [S: set(A),F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),S)
         => ( ( S != bot_bot(set(A)) )
           => member(A,lattic7623131987881927897min_on(A,B,F2,S),S) ) ) ) ).

% arg_min_if_finite(1)
tff(fact_3470_greaterThanAtMost__upto,axiom,
    ! [Ia: int,J: int] : ( set_or3652927894154168847AtMost(int,Ia,J) = aa(list(int),set(int),set2(int),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),Ia),one_one(int)),J)) ) ).

% greaterThanAtMost_upto
tff(fact_3471_arg__min__least,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [S: set(A),Y: A,F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),S)
         => ( ( S != bot_bot(set(A)) )
           => ( member(A,Y,S)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,lattic7623131987881927897min_on(A,B,F2,S))),aa(A,B,F2,Y)) ) ) ) ) ).

% arg_min_least
tff(fact_3472_unbounded__k__infinite,axiom,
    ! [K3: nat,S: set(nat)] :
      ( ! [M6: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K3),M6)
         => ? [N8: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M6),N8)
              & member(nat,N8,S) ) )
     => ~ aa(set(nat),$o,finite_finite2(nat),S) ) ).

% unbounded_k_infinite
tff(fact_3473_infinite__nat__iff__unbounded,axiom,
    ! [S: set(nat)] :
      ( ~ aa(set(nat),$o,finite_finite2(nat),S)
    <=> ! [M: nat] :
        ? [N2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),N2)
          & member(nat,N2,S) ) ) ).

% infinite_nat_iff_unbounded
tff(fact_3474_nth__image,axiom,
    ! [A: $tType,L: nat,Xsa: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),L),aa(list(A),nat,size_size(list(A)),Xsa))
     => ( aa(set(nat),set(A),image2(nat,A,nth(A,Xsa)),set_or7035219750837199246ssThan(nat,zero_zero(nat),L)) = aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),L),Xsa)) ) ) ).

% nth_image
tff(fact_3475_sorted__list__of__set__lessThan__Suc,axiom,
    ! [K3: nat] : ( aa(set(nat),list(nat),linord4507533701916653071of_set(nat),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,K3))) = aa(list(nat),list(nat),aa(list(nat),fun(list(nat),list(nat)),append(nat),aa(set(nat),list(nat),linord4507533701916653071of_set(nat),aa(nat,set(nat),set_ord_lessThan(nat),K3))),aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),K3),nil(nat))) ) ).

% sorted_list_of_set_lessThan_Suc
tff(fact_3476_finite__transitivity__chain,axiom,
    ! [A: $tType,Aa2: set(A),R: fun(A,fun(A,$o))] :
      ( aa(set(A),$o,finite_finite2(A),Aa2)
     => ( ! [X: A] : ~ aa(A,$o,aa(A,fun(A,$o),R,X),X)
       => ( ! [X: A,Y2: A,Z3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),R,X),Y2)
             => ( aa(A,$o,aa(A,fun(A,$o),R,Y2),Z3)
               => aa(A,$o,aa(A,fun(A,$o),R,X),Z3) ) )
         => ( ! [X: A] :
                ( member(A,X,Aa2)
               => ? [Y3: A] :
                    ( member(A,Y3,Aa2)
                    & aa(A,$o,aa(A,fun(A,$o),R,X),Y3) ) )
           => ( Aa2 = bot_bot(set(A)) ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_3477_lessThan__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Ia: A,K3: A] :
          ( member(A,Ia,aa(A,set(A),set_ord_lessThan(A),K3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ia),K3) ) ) ).

% lessThan_iff
tff(fact_3478_atLeastLessThan__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Ia: A,L: A,U: A] :
          ( member(A,Ia,set_or7035219750837199246ssThan(A,L,U))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),Ia)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ia),U) ) ) ) ).

% atLeastLessThan_iff
tff(fact_3479_atLeastLessThan__empty,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3)
         => ( set_or7035219750837199246ssThan(A,A3,B3) = bot_bot(set(A)) ) ) ) ).

% atLeastLessThan_empty
tff(fact_3480_atLeastLessThan__empty__iff2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B3: A] :
          ( ( bot_bot(set(A)) = set_or7035219750837199246ssThan(A,A3,B3) )
        <=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3) ) ) ).

% atLeastLessThan_empty_iff2
tff(fact_3481_atLeastLessThan__empty__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B3: A] :
          ( ( set_or7035219750837199246ssThan(A,A3,B3) = bot_bot(set(A)) )
        <=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3) ) ) ).

% atLeastLessThan_empty_iff
tff(fact_3482_infinite__Ico__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B3: A] :
          ( ~ aa(set(A),$o,finite_finite2(A),set_or7035219750837199246ssThan(A,A3,B3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3) ) ) ).

% infinite_Ico_iff
tff(fact_3483_cSup__atLeastLessThan,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & dense_linorder(A) )
     => ! [Y: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xa)
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or7035219750837199246ssThan(A,Y,Xa)) = Xa ) ) ) ).

% cSup_atLeastLessThan
tff(fact_3484_Sup__atLeastLessThan,axiom,
    ! [A: $tType] :
      ( ( comple6319245703460814977attice(A)
        & dense_linorder(A) )
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or7035219750837199246ssThan(A,Xa,Y)) = Y ) ) ) ).

% Sup_atLeastLessThan
tff(fact_3485_ivl__diff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ia: A,N: A,M2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ia),N)
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),set_or7035219750837199246ssThan(A,Ia,M2)),set_or7035219750837199246ssThan(A,Ia,N)) = set_or7035219750837199246ssThan(A,N,M2) ) ) ) ).

% ivl_diff
tff(fact_3486_lessThan__minus__lessThan,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [N: A,M2: A] : ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(A,set(A),set_ord_lessThan(A),N)),aa(A,set(A),set_ord_lessThan(A),M2)) = set_or7035219750837199246ssThan(A,M2,N) ) ) ).

% lessThan_minus_lessThan
tff(fact_3487_lessThan__0,axiom,
    aa(nat,set(nat),set_ord_lessThan(nat),zero_zero(nat)) = bot_bot(set(nat)) ).

% lessThan_0
tff(fact_3488_single__Diff__lessThan,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [K3: 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)),insert3(A),K3),bot_bot(set(A)))),aa(A,set(A),set_ord_lessThan(A),K3)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),K3),bot_bot(set(A))) ) ) ).

% single_Diff_lessThan
tff(fact_3489_image__diff__atLeastLessThan,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,A3: A,B3: A] : ( aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),minus_minus(A),C2)),set_or7035219750837199246ssThan(A,A3,B3)) = set_or3652927894154168847AtMost(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),B3),aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),A3)) ) ) ).

% image_diff_atLeastLessThan
tff(fact_3490_image__minus__const__greaterThanAtMost,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,A3: A,B3: A] : ( aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),minus_minus(A),C2)),set_or3652927894154168847AtMost(A,A3,B3)) = set_or7035219750837199246ssThan(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),B3),aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),A3)) ) ) ).

% image_minus_const_greaterThanAtMost
tff(fact_3491_atLeastLessThan__singleton,axiom,
    ! [M2: nat] : ( set_or7035219750837199246ssThan(nat,M2,aa(nat,nat,suc,M2)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert3(nat),M2),bot_bot(set(nat))) ) ).

% atLeastLessThan_singleton
tff(fact_3492_nth__image__indices,axiom,
    ! [A: $tType,L: list(A)] : ( aa(set(nat),set(A),image2(nat,A,nth(A,L)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(A),nat,size_size(list(A)),L))) = aa(list(A),set(A),set2(A),L) ) ).

% nth_image_indices
tff(fact_3493_atLeastLessThan__inj_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A,C2: A,D3: A] :
          ( ( set_or7035219750837199246ssThan(A,A3,B3) = set_or7035219750837199246ssThan(A,C2,D3) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),D3)
             => ( B3 = D3 ) ) ) ) ) ).

% atLeastLessThan_inj(2)
tff(fact_3494_atLeastLessThan__inj_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A,C2: A,D3: A] :
          ( ( set_or7035219750837199246ssThan(A,A3,B3) = set_or7035219750837199246ssThan(A,C2,D3) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),D3)
             => ( A3 = C2 ) ) ) ) ) ).

% atLeastLessThan_inj(1)
tff(fact_3495_atLeastLessThan__eq__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A,C2: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),D3)
           => ( ( set_or7035219750837199246ssThan(A,A3,B3) = set_or7035219750837199246ssThan(A,C2,D3) )
            <=> ( ( A3 = C2 )
                & ( B3 = D3 ) ) ) ) ) ) ).

% atLeastLessThan_eq_iff
tff(fact_3496_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_3497_lessThan__non__empty,axiom,
    ! [A: $tType] :
      ( no_bot(A)
     => ! [Xa: A] : ( aa(A,set(A),set_ord_lessThan(A),Xa) != bot_bot(set(A)) ) ) ).

% lessThan_non_empty
tff(fact_3498_infinite__Ico,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
         => ~ aa(set(A),$o,finite_finite2(A),set_or7035219750837199246ssThan(A,A3,B3)) ) ) ).

% infinite_Ico
tff(fact_3499_ex__nat__less__eq,axiom,
    ! [N: nat,Pa: fun(nat,$o)] :
      ( ? [M: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),N)
          & aa(nat,$o,Pa,M) )
    <=> ? [X6: nat] :
          ( member(nat,X6,set_or7035219750837199246ssThan(nat,zero_zero(nat),N))
          & aa(nat,$o,Pa,X6) ) ) ).

% ex_nat_less_eq
tff(fact_3500_all__nat__less__eq,axiom,
    ! [N: nat,Pa: fun(nat,$o)] :
      ( ! [M: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),N)
         => aa(nat,$o,Pa,M) )
    <=> ! [X6: nat] :
          ( member(nat,X6,set_or7035219750837199246ssThan(nat,zero_zero(nat),N))
         => aa(nat,$o,Pa,X6) ) ) ).

% all_nat_less_eq
tff(fact_3501_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_3502_ivl__disj__int__two_I3_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,M2: A,U: A] : ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or7035219750837199246ssThan(A,L,M2)),set_or7035219750837199246ssThan(A,M2,U)) = bot_bot(set(A)) ) ) ).

% ivl_disj_int_two(3)
tff(fact_3503_lessThan__Suc,axiom,
    ! [K3: nat] : ( aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,K3)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert3(nat),K3),aa(nat,set(nat),set_ord_lessThan(nat),K3)) ) ).

% lessThan_Suc
tff(fact_3504_atLeastLessThan0,axiom,
    ! [M2: nat] : ( set_or7035219750837199246ssThan(nat,M2,zero_zero(nat)) = bot_bot(set(nat)) ) ).

% atLeastLessThan0
tff(fact_3505_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_3506_lessThan__strict__subset__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [M2: A,N: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),aa(A,set(A),set_ord_lessThan(A),M2)),aa(A,set(A),set_ord_lessThan(A),N))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),M2),N) ) ) ).

% lessThan_strict_subset_iff
tff(fact_3507_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)),insert3(nat),zero_zero(nat)),bot_bot(set(nat)))) ) ).

% atLeast1_lessThan_eq_remove0
tff(fact_3508_ivl__disj__int__two_I7_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,M2: A,U: A] : ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or7035219750837199246ssThan(A,L,M2)),set_or1337092689740270186AtMost(A,M2,U)) = bot_bot(set(A)) ) ) ).

% ivl_disj_int_two(7)
tff(fact_3509_atLeast0__lessThan__Suc,axiom,
    ! [N: nat] : ( set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,N)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert3(nat),N),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)) ) ).

% atLeast0_lessThan_Suc
tff(fact_3510_ivl__disj__int__two_I1_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,M2: A,U: A] : ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or5935395276787703475ssThan(A,L,M2)),set_or7035219750837199246ssThan(A,M2,U)) = bot_bot(set(A)) ) ) ).

% ivl_disj_int_two(1)
tff(fact_3511_lessThan__nat__numeral,axiom,
    ! [K3: num] : ( aa(nat,set(nat),set_ord_lessThan(nat),numeral_numeral(nat,K3)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert3(nat),pred_numeral(K3)),aa(nat,set(nat),set_ord_lessThan(nat),pred_numeral(K3))) ) ).

% lessThan_nat_numeral
tff(fact_3512_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_3513_Iic__subset__Iio__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(A,set(A),set_ord_atMost(A),A3)),aa(A,set(A),set_ord_lessThan(A),B3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3) ) ) ).

% Iic_subset_Iio_iff
tff(fact_3514_atLeastLessThan__subseteq__atLeastAtMost__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B3: A,C2: A,D3: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or7035219750837199246ssThan(A,A3,B3)),set_or1337092689740270186AtMost(A,C2,D3))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),D3) ) ) ) ) ).

% atLeastLessThan_subseteq_atLeastAtMost_iff
tff(fact_3515_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B3: A,C2: A,D3: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or1337092689740270186AtMost(A,A3,B3)),set_or7035219750837199246ssThan(A,C2,D3))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),D3) ) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
tff(fact_3516_ivl__disj__un__two__touch_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M2: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),M2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),M2),U)
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or1337092689740270186AtMost(A,L,M2)),set_or7035219750837199246ssThan(A,M2,U)) = set_or7035219750837199246ssThan(A,L,U) ) ) ) ) ).

% ivl_disj_un_two_touch(2)
tff(fact_3517_atLeastLessThan__eq__atLeastAtMost__diff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B3: A] : ( set_or7035219750837199246ssThan(A,A3,B3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),set_or1337092689740270186AtMost(A,A3,B3)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),B3),bot_bot(set(A)))) ) ) ).

% atLeastLessThan_eq_atLeastAtMost_diff
tff(fact_3518_greaterThanAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B3: A,C2: A,D3: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or3652927894154168847AtMost(A,A3,B3)),set_or7035219750837199246ssThan(A,C2,D3))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),D3) ) ) ) ) ).

% greaterThanAtMost_subseteq_atLeastLessThan_iff
tff(fact_3519_greaterThanLessThan__subseteq__atLeastLessThan__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B3: A,C2: A,D3: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or5935395276787703475ssThan(A,A3,B3)),set_or7035219750837199246ssThan(A,C2,D3))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),D3) ) ) ) ) ).

% greaterThanLessThan_subseteq_atLeastLessThan_iff
tff(fact_3520_ivl__disj__un__two_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M2: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),L),M2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),M2),U)
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or5935395276787703475ssThan(A,L,M2)),set_or7035219750837199246ssThan(A,M2,U)) = set_or5935395276787703475ssThan(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(1)
tff(fact_3521_lessThan__Suc__eq__insert__0,axiom,
    ! [N: nat] : ( aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,N)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert3(nat),zero_zero(nat)),aa(set(nat),set(nat),image2(nat,nat,suc),aa(nat,set(nat),set_ord_lessThan(nat),N))) ) ).

% lessThan_Suc_eq_insert_0
tff(fact_3522_ivl__disj__un__two__touch_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M2: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),L),M2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),M2),U)
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or3652927894154168847AtMost(A,L,M2)),set_or7035219750837199246ssThan(A,M2,U)) = set_or5935395276787703475ssThan(A,L,U) ) ) ) ) ).

% ivl_disj_un_two_touch(1)
tff(fact_3523_atLeastLessThanSuc,axiom,
    ! [M2: nat,N: nat] :
      ( set_or7035219750837199246ssThan(nat,M2,aa(nat,nat,suc,N)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),N),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert3(nat),N),set_or7035219750837199246ssThan(nat,M2,N)),bot_bot(set(nat))) ) ).

% atLeastLessThanSuc
tff(fact_3524_atLeast0__lessThan__Suc__eq__insert__0,axiom,
    ! [N: nat] : ( set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,N)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert3(nat),zero_zero(nat)),aa(set(nat),set(nat),image2(nat,nat,suc),set_or7035219750837199246ssThan(nat,zero_zero(nat),N))) ) ).

% atLeast0_lessThan_Suc_eq_insert_0
tff(fact_3525_listI__assn__conv,axiom,
    ! [A: $tType,B: $tType,N: nat,Xsa: list(A),Aa2: fun(A,fun(B,assn)),Xsia: list(B)] :
      ( ( N = aa(list(A),nat,size_size(list(A)),Xsa) )
     => ( vEBT_List_listI_assn(A,B,set_or7035219750837199246ssThan(nat,zero_zero(nat),N),Aa2,Xsa,Xsia) = vEBT_List_list_assn(A,B,Aa2,Xsa,Xsia) ) ) ).

% listI_assn_conv
tff(fact_3526_list__assn__conv__idx,axiom,
    ! [A: $tType,B: $tType,Aa2: fun(A,fun(B,assn)),Xsa: list(A),Xsia: list(B)] : ( vEBT_List_list_assn(A,B,Aa2,Xsa,Xsia) = vEBT_List_listI_assn(A,B,set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xsa)),Aa2,Xsa,Xsia) ) ).

% list_assn_conv_idx
tff(fact_3527_Iio__Int__singleton,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [K3: A,Xa: 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),K3)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A)))) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),K3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A))),bot_bot(set(A))) ) ) ).

% Iio_Int_singleton
tff(fact_3528_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)),insert3(A),U),bot_bot(set(A)))) = aa(A,set(A),set_ord_atMost(A),U) ) ) ).

% ivl_disj_un_singleton(2)
tff(fact_3529_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_3530_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)),insert3(A),U),bot_bot(set(A)))) = set_or1337092689740270186AtMost(A,L,U) ) ) ) ).

% ivl_disj_un_singleton(6)
tff(fact_3531_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)),insert3(A),L),bot_bot(set(A)))),set_or5935395276787703475ssThan(A,L,U)) = set_or7035219750837199246ssThan(A,L,U) ) ) ) ).

% ivl_disj_un_singleton(3)
tff(fact_3532_ivl__disj__un__two_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M2: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),M2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),M2),U)
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or1337092689740270186AtMost(A,L,M2)),set_or5935395276787703475ssThan(A,M2,U)) = set_or7035219750837199246ssThan(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(4)
tff(fact_3533_atLeastLessThan__nat__numeral,axiom,
    ! [M2: nat,K3: num] :
      ( set_or7035219750837199246ssThan(nat,M2,numeral_numeral(nat,K3)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),pred_numeral(K3)),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert3(nat),pred_numeral(K3)),set_or7035219750837199246ssThan(nat,M2,pred_numeral(K3))),bot_bot(set(nat))) ) ).

% atLeastLessThan_nat_numeral
tff(fact_3534_listI__assn__conv_H,axiom,
    ! [B: $tType,A: $tType,N: nat,Xsa: list(A),Aa2: fun(A,fun(B,assn)),Xsia: list(B),Fa: assn] :
      ( ( N = aa(list(A),nat,size_size(list(A)),Xsa) )
     => ( aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),vEBT_List_listI_assn(A,B,set_or7035219750837199246ssThan(nat,zero_zero(nat),N),Aa2,Xsa,Xsia)),Fa) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),vEBT_List_list_assn(A,B,Aa2,Xsa,Xsia)),Fa) ) ) ).

% listI_assn_conv'
tff(fact_3535_interval__cases,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [S: set(A)] :
          ( ! [A4: A,B4: A,X: A] :
              ( member(A,A4,S)
             => ( member(A,B4,S)
               => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),X)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),B4)
                   => member(A,X,S) ) ) ) )
         => ? [A4: A,B4: A] :
              ( ( S = bot_bot(set(A)) )
              | ( S = top_top(set(A)) )
              | ( S = aa(A,set(A),set_ord_lessThan(A),B4) )
              | ( S = aa(A,set(A),set_ord_atMost(A),B4) )
              | ( S = aa(A,set(A),set_ord_greaterThan(A),A4) )
              | ( S = set_ord_atLeast(A,A4) )
              | ( S = set_or5935395276787703475ssThan(A,A4,B4) )
              | ( S = set_or3652927894154168847AtMost(A,A4,B4) )
              | ( S = set_or7035219750837199246ssThan(A,A4,B4) )
              | ( S = set_or1337092689740270186AtMost(A,A4,B4) ) ) ) ) ).

% interval_cases
tff(fact_3536_fact__split,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [K3: nat,N: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K3),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),K3),N)))),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K3))) ) ) ) ).

% fact_split
tff(fact_3537_sorted__list__of__set__nonempty,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Aa2: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( ( Aa2 != bot_bot(set(A)) )
           => ( aa(set(A),list(A),linord4507533701916653071of_set(A),Aa2) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),lattic643756798350308766er_Min(A,Aa2)),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)),Aa2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),lattic643756798350308766er_Min(A,Aa2)),bot_bot(set(A)))))) ) ) ) ) ).

% sorted_list_of_set_nonempty
tff(fact_3538_max__ext_Ocases,axiom,
    ! [A: $tType,A1: set(A),A22: set(A),R: set(product_prod(A,A))] :
      ( member(product_prod(set(A),set(A)),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),A1),A22),max_ext(A,R))
     => ~ ( aa(set(A),$o,finite_finite2(A),A1)
         => ( aa(set(A),$o,finite_finite2(A),A22)
           => ( ( A22 != bot_bot(set(A)) )
             => ~ ! [X4: A] :
                    ( member(A,X4,A1)
                   => ? [Xa5: A] :
                        ( member(A,Xa5,A22)
                        & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Xa5),R) ) ) ) ) ) ) ).

% max_ext.cases
tff(fact_3539_max__ext_Osimps,axiom,
    ! [A: $tType,A1: set(A),A22: set(A),R: set(product_prod(A,A))] :
      ( member(product_prod(set(A),set(A)),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),A1),A22),max_ext(A,R))
    <=> ( aa(set(A),$o,finite_finite2(A),A1)
        & aa(set(A),$o,finite_finite2(A),A22)
        & ( A22 != bot_bot(set(A)) )
        & ! [X6: A] :
            ( member(A,X6,A1)
           => ? [Xa2: A] :
                ( member(A,Xa2,A22)
                & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X6),Xa2),R) ) ) ) ) ).

% max_ext.simps
tff(fact_3540_atLeast__empty__triv,axiom,
    ! [A: $tType] : ( set_ord_atLeast(set(A),bot_bot(set(A))) = top_top(set(set(A))) ) ).

% atLeast_empty_triv
tff(fact_3541_Min__singleton,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A] : ( lattic643756798350308766er_Min(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A)))) = Xa ) ) ).

% Min_singleton
tff(fact_3542_Min_Obounded__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Aa2: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( ( Aa2 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),lattic643756798350308766er_Min(A,Aa2))
            <=> ! [X6: A] :
                  ( member(A,X6,Aa2)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),X6) ) ) ) ) ) ).

% Min.bounded_iff
tff(fact_3543_Min__gr__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Aa2: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( ( Aa2 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),lattic643756798350308766er_Min(A,Aa2))
            <=> ! [X6: A] :
                  ( member(A,X6,Aa2)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),X6) ) ) ) ) ) ).

% Min_gr_iff
tff(fact_3544_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_3545_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_3546_image__minus__const__AtMost,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,B3: A] : ( aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),minus_minus(A),C2)),aa(A,set(A),set_ord_atMost(A),B3)) = set_ord_atLeast(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),B3)) ) ) ).

% image_minus_const_AtMost
tff(fact_3547_image__minus__const__atLeast,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,A3: A] : ( aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),minus_minus(A),C2)),set_ord_atLeast(A,A3)) = aa(A,set(A),set_ord_atMost(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),A3)) ) ) ).

% image_minus_const_atLeast
tff(fact_3548_Min__insert,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Aa2: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( ( Aa2 != bot_bot(set(A)) )
           => ( lattic643756798350308766er_Min(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),Aa2)) = aa(A,A,aa(A,fun(A,A),ord_min(A),Xa),lattic643756798350308766er_Min(A,Aa2)) ) ) ) ) ).

% Min_insert
tff(fact_3549_prod_Oop__ivl__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M2: nat,N: nat] :
          ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,M2,aa(nat,nat,suc,N))) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),M2),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,M2,N))),aa(nat,A,G,N))) ) ) ).

% prod.op_ivl_Suc
tff(fact_3550_prod_Ocl__ivl__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M2: nat,N: nat] :
          ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M2,aa(nat,nat,suc,N))) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,N)),M2),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,M2,N))),aa(nat,A,G,aa(nat,nat,suc,N)))) ) ) ).

% prod.cl_ivl_Suc
tff(fact_3551_not__empty__eq__Ici__eq__empty,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [L: A] : ( bot_bot(set(A)) != set_ord_atLeast(A,L) ) ) ).

% not_empty_eq_Ici_eq_empty
tff(fact_3552_Min__in,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Aa2: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( ( Aa2 != bot_bot(set(A)) )
           => member(A,lattic643756798350308766er_Min(A,Aa2),Aa2) ) ) ) ).

% Min_in
tff(fact_3553_atLeast__eq__UNIV__iff,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [Xa: A] :
          ( ( set_ord_atLeast(A,Xa) = top_top(set(A)) )
        <=> ( Xa = bot_bot(A) ) ) ) ).

% atLeast_eq_UNIV_iff
tff(fact_3554_prod_Oivl__cong,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ord(A)
        & comm_monoid_mult(B) )
     => ! [A3: A,C2: A,B3: A,D3: A,G: fun(A,B),H2: fun(A,B)] :
          ( ( A3 = C2 )
         => ( ( B3 = D3 )
           => ( ! [X: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),X)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),D3)
                   => ( aa(A,B,G,X) = aa(A,B,H2,X) ) ) )
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),set_or7035219750837199246ssThan(A,A3,B3)) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),H2),set_or7035219750837199246ssThan(A,C2,D3)) ) ) ) ) ) ).

% prod.ivl_cong
tff(fact_3555_prod_OatLeastLessThan__concat,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M2: nat,N: nat,P2: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),N)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),P2)
           => ( 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,M2,N))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,N,P2))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,M2,P2)) ) ) ) ) ).

% prod.atLeastLessThan_concat
tff(fact_3556_Min_OboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Aa2: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( ( Aa2 != bot_bot(set(A)) )
           => ( ! [A4: A] :
                  ( member(A,A4,Aa2)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),A4) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),lattic643756798350308766er_Min(A,Aa2)) ) ) ) ) ).

% Min.boundedI
tff(fact_3557_Min_OboundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Aa2: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( ( Aa2 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),lattic643756798350308766er_Min(A,Aa2))
             => ! [A13: A] :
                  ( member(A,A13,Aa2)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),A13) ) ) ) ) ) ).

% Min.boundedE
tff(fact_3558_eq__Min__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Aa2: set(A),M2: A] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( ( Aa2 != bot_bot(set(A)) )
           => ( ( M2 = lattic643756798350308766er_Min(A,Aa2) )
            <=> ( member(A,M2,Aa2)
                & ! [X6: A] :
                    ( member(A,X6,Aa2)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),M2),X6) ) ) ) ) ) ) ).

% eq_Min_iff
tff(fact_3559_Min__le__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Aa2: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( ( Aa2 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),lattic643756798350308766er_Min(A,Aa2)),Xa)
            <=> ? [X6: A] :
                  ( member(A,X6,Aa2)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X6),Xa) ) ) ) ) ) ).

% Min_le_iff
tff(fact_3560_Min__eq__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Aa2: set(A),M2: A] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( ( Aa2 != bot_bot(set(A)) )
           => ( ( lattic643756798350308766er_Min(A,Aa2) = M2 )
            <=> ( member(A,M2,Aa2)
                & ! [X6: A] :
                    ( member(A,X6,Aa2)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),M2),X6) ) ) ) ) ) ) ).

% Min_eq_iff
tff(fact_3561_Min__less__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Aa2: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( ( Aa2 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),lattic643756798350308766er_Min(A,Aa2)),Xa)
            <=> ? [X6: A] :
                  ( member(A,X6,Aa2)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X6),Xa) ) ) ) ) ) ).

% Min_less_iff
tff(fact_3562_Min__insert2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Aa2: set(A),A3: A] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( ! [B4: A] :
                ( member(A,B4,Aa2)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B4) )
           => ( lattic643756798350308766er_Min(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),Aa2)) = A3 ) ) ) ) ).

% Min_insert2
tff(fact_3563_max__ext__additive,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A),R: set(product_prod(A,A)),C3: set(A),D2: set(A)] :
      ( member(product_prod(set(A),set(A)),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),Aa2),B2),max_ext(A,R))
     => ( member(product_prod(set(A),set(A)),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),C3),D2),max_ext(A,R))
       => member(product_prod(set(A),set(A)),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),C3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B2),D2)),max_ext(A,R)) ) ) ).

% max_ext_additive
tff(fact_3564_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)),set_ord_atLeast(A,U)) = bot_bot(set(A)) ) ) ).

% ivl_disj_int_one(8)
tff(fact_3565_Ici__subset__Ioi__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_ord_atLeast(A,A3)),aa(A,set(A),set_ord_greaterThan(A),B3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A3) ) ) ).

% Ici_subset_Ioi_iff
tff(fact_3566_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)),set_ord_atLeast(A,U)) = bot_bot(set(A)) ) ) ).

% ivl_disj_int_one(6)
tff(fact_3567_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_3568_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_3569_prod_OatLeast__Suc__lessThan,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M2: nat,N: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),N)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,M2,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,M2)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M2),N))) ) ) ) ).

% prod.atLeast_Suc_lessThan
tff(fact_3570_prod_OatLeastLessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: nat,B3: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A3),B3)
         => ( 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,B3))) = 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,B3))),aa(nat,A,G,B3)) ) ) ) ).

% prod.atLeastLessThan_Suc
tff(fact_3571_prod_Onat__ivl__Suc_H,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M2: nat,N: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),aa(nat,nat,suc,N))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M2,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,M2,N))) ) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_3572_prod_OatLeast__Suc__atMost,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M2: nat,N: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),N)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M2,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,M2)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M2),N))) ) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_3573_prod_Olast__plus,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M2: nat,N: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),N)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M2,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,M2,N))) ) ) ) ).

% prod.last_plus
tff(fact_3574_Min_Osubset__imp,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Aa2: set(A),B2: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),B2)
         => ( ( Aa2 != bot_bot(set(A)) )
           => ( aa(set(A),$o,finite_finite2(A),B2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),lattic643756798350308766er_Min(A,B2)),lattic643756798350308766er_Min(A,Aa2)) ) ) ) ) ).

% Min.subset_imp
tff(fact_3575_Min__antimono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [M5: set(A),N7: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),M5),N7)
         => ( ( M5 != bot_bot(set(A)) )
           => ( aa(set(A),$o,finite_finite2(A),N7)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),lattic643756798350308766er_Min(A,N7)),lattic643756798350308766er_Min(A,M5)) ) ) ) ) ).

% Min_antimono
tff(fact_3576_prod_Ohead,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M2: nat,N: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),N)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M2,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,M2)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or3652927894154168847AtMost(nat,M2,N))) ) ) ) ).

% prod.head
tff(fact_3577_hom__Min__commute,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [H2: fun(A,A),N7: set(A)] :
          ( ! [X: A,Y2: A] : ( aa(A,A,H2,aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y2)) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,H2,X)),aa(A,A,H2,Y2)) )
         => ( aa(set(A),$o,finite_finite2(A),N7)
           => ( ( N7 != bot_bot(set(A)) )
             => ( aa(A,A,H2,lattic643756798350308766er_Min(A,N7)) = lattic643756798350308766er_Min(A,aa(set(A),set(A),image2(A,A,H2),N7)) ) ) ) ) ) ).

% hom_Min_commute
tff(fact_3578_Min_Osubset,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Aa2: set(A),B2: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( ( B2 != bot_bot(set(A)) )
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),Aa2)
             => ( aa(A,A,aa(A,fun(A,A),ord_min(A),lattic643756798350308766er_Min(A,B2)),lattic643756798350308766er_Min(A,Aa2)) = lattic643756798350308766er_Min(A,Aa2) ) ) ) ) ) ).

% Min.subset
tff(fact_3579_Min_Oinsert__not__elem,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Aa2: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( ~ member(A,Xa,Aa2)
           => ( ( Aa2 != bot_bot(set(A)) )
             => ( lattic643756798350308766er_Min(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),Aa2)) = aa(A,A,aa(A,fun(A,A),ord_min(A),Xa),lattic643756798350308766er_Min(A,Aa2)) ) ) ) ) ) ).

% Min.insert_not_elem
tff(fact_3580_Min_Oclosed,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Aa2: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( ( Aa2 != bot_bot(set(A)) )
           => ( ! [X: A,Y2: A] : member(A,aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),X),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Y2),bot_bot(set(A)))))
             => member(A,lattic643756798350308766er_Min(A,Aa2),Aa2) ) ) ) ) ).

% Min.closed
tff(fact_3581_Min_Ounion,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Aa2: set(A),B2: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( ( Aa2 != bot_bot(set(A)) )
           => ( aa(set(A),$o,finite_finite2(A),B2)
             => ( ( B2 != bot_bot(set(A)) )
               => ( lattic643756798350308766er_Min(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),B2)) = aa(A,A,aa(A,fun(A,A),ord_min(A),lattic643756798350308766er_Min(A,Aa2)),lattic643756798350308766er_Min(A,B2)) ) ) ) ) ) ) ).

% Min.union
tff(fact_3582_atLeastLessThan__upto,axiom,
    ! [Ia: int,J: int] : ( set_or7035219750837199246ssThan(int,Ia,J) = aa(list(int),set(int),set2(int),upto(Ia,aa(int,int,aa(int,fun(int,int),minus_minus(int),J),one_one(int)))) ) ).

% atLeastLessThan_upto
tff(fact_3583_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)),set_ord_atLeast(A,N)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),N),bot_bot(set(A))) ) ) ).

% atMost_Int_atLeast
tff(fact_3584_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)),insert3(A),L),bot_bot(set(A)))),aa(A,set(A),set_ord_greaterThan(A),L)) = set_ord_atLeast(A,L) ) ) ).

% ivl_disj_un_singleton(1)
tff(fact_3585_prod_Ohead__if,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M2: nat,N: nat] :
          ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M2,N)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),M2),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,M2,N))),aa(nat,A,G,N))) ) ) ).

% prod.head_if
tff(fact_3586_prod_Oub__add__nat,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M2: nat,N: nat,G: fun(nat,A),P2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),P2))) = 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,M2,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),P2)))) ) ) ) ).

% prod.ub_add_nat
tff(fact_3587_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)),set_ord_atLeast(A,U)) = aa(A,set(A),set_ord_greaterThan(A),L) ) ) ) ).

% ivl_disj_un_one(6)
tff(fact_3588_atLeast__Suc,axiom,
    ! [K3: nat] : ( set_ord_atLeast(nat,aa(nat,nat,suc,K3)) = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),set_ord_atLeast(nat,K3)),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert3(nat),K3),bot_bot(set(nat)))) ) ).

% atLeast_Suc
tff(fact_3589_Min_Oremove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Aa2: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( member(A,Xa,Aa2)
           => ( lattic643756798350308766er_Min(A,Aa2) = $ite(aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A)))) = bot_bot(set(A)),Xa,aa(A,A,aa(A,fun(A,A),ord_min(A),Xa),lattic643756798350308766er_Min(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A))))))) ) ) ) ) ).

% Min.remove
tff(fact_3590_Min_Oinsert__remove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Aa2: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( lattic643756798350308766er_Min(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),Aa2)) = $ite(aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A)))) = bot_bot(set(A)),Xa,aa(A,A,aa(A,fun(A,A),ord_min(A),Xa),lattic643756798350308766er_Min(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A))))))) ) ) ) ).

% Min.insert_remove
tff(fact_3591_max__ext_Omax__extI,axiom,
    ! [A: $tType,X3: set(A),Y5: set(A),R: set(product_prod(A,A))] :
      ( aa(set(A),$o,finite_finite2(A),X3)
     => ( aa(set(A),$o,finite_finite2(A),Y5)
       => ( ( Y5 != bot_bot(set(A)) )
         => ( ! [X: A] :
                ( member(A,X,X3)
               => ? [Xa4: A] :
                    ( member(A,Xa4,Y5)
                    & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Xa4),R) ) )
           => member(product_prod(set(A),set(A)),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),X3),Y5),max_ext(A,R)) ) ) ) ) ).

% max_ext.max_extI
tff(fact_3592_prod__pos__nat__iff,axiom,
    ! [A: $tType,Aa2: set(A),F2: fun(A,nat)] :
      ( aa(set(A),$o,finite_finite2(A),Aa2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7121269368397514597t_prod(A,nat),F2),Aa2))
      <=> ! [X6: A] :
            ( member(A,X6,Aa2)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(A,nat,F2,X6)) ) ) ) ).

% prod_pos_nat_iff
tff(fact_3593_prod_Oinsert,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [Aa2: set(A),Xa: A,G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( ~ member(A,Xa,Aa2)
           => ( 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)),insert3(A),Xa),Aa2)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,G,Xa)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),Aa2)) ) ) ) ) ).

% prod.insert
tff(fact_3594_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_3595_prod__Un,axiom,
    ! [B: $tType,A: $tType] :
      ( field(B)
     => ! [Aa2: set(A),B2: set(A),F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( aa(set(A),$o,finite_finite2(A),B2)
           => ( ! [X: A] :
                  ( member(A,X,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),B2))
                 => ( aa(A,B,F2,X) != zero_zero(B) ) )
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),B2)) = divide_divide(B,aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),Aa2)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),B2)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),B2))) ) ) ) ) ) ).

% prod_Un
tff(fact_3596_prod__diff1,axiom,
    ! [B: $tType,A: $tType] :
      ( semidom_divide(B)
     => ! [Aa2: set(A),F2: fun(A,B),A3: A] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( ( aa(A,B,F2,A3) != zero_zero(B) )
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),bot_bot(set(A))))) = $ite(member(A,A3,Aa2),divide_divide(B,aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),Aa2),aa(A,B,F2,A3)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),Aa2)) ) ) ) ) ).

% prod_diff1
tff(fact_3597_prod__pos,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_semidom(B)
     => ! [Aa2: set(A),F2: fun(A,B)] :
          ( ! [X: A] :
              ( member(A,X,Aa2)
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(A,B,F2,X)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),Aa2)) ) ) ).

% prod_pos
tff(fact_3598_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),$o,finite_finite2(B),S)
             => ( ! [X: B] :
                    ( member(B,X,S)
                   => aa(A,$o,aa(A,fun(A,$o),R,aa(B,A,H2,X)),aa(B,A,G,X)) )
               => aa(A,$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_3599_prod_Oinsert__if,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [Aa2: set(A),G: fun(A,B),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( 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)),insert3(A),Xa),Aa2)) = $ite(member(A,Xa,Aa2),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),Aa2),aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,G,Xa)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),Aa2))) ) ) ) ).

% prod.insert_if
tff(fact_3600_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [A: $tType,E: $tType,B: $tType] :
      ( comm_monoid_mult(E)
     => ! [S3: set(A),T6: set(B),S: set(A),Ia: fun(B,A),J: fun(A,B),T4: set(B),G: fun(A,E),H2: fun(B,E)] :
          ( aa(set(A),$o,finite_finite2(A),S3)
         => ( aa(set(B),$o,finite_finite2(B),T6)
           => ( ! [A4: A] :
                  ( member(A,A4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S),S3))
                 => ( aa(B,A,Ia,aa(A,B,J,A4)) = A4 ) )
             => ( ! [A4: A] :
                    ( member(A,A4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S),S3))
                   => member(B,aa(A,B,J,A4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T4),T6)) )
               => ( ! [B4: B] :
                      ( member(B,B4,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T4),T6))
                     => ( aa(A,B,J,aa(B,A,Ia,B4)) = B4 ) )
                 => ( ! [B4: B] :
                        ( member(B,B4,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T4),T6))
                       => member(A,aa(B,A,Ia,B4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S),S3)) )
                   => ( ! [A4: A] :
                          ( member(A,A4,S3)
                         => ( aa(A,E,G,A4) = one_one(E) ) )
                     => ( ! [B4: B] :
                            ( member(B,B4,T6)
                           => ( aa(B,E,H2,B4) = one_one(E) ) )
                       => ( ! [A4: A] :
                              ( member(A,A4,S)
                             => ( aa(B,E,H2,aa(A,B,J,A4)) = aa(A,E,G,A4) ) )
                         => ( aa(set(A),E,aa(fun(A,E),fun(set(A),E),groups7121269368397514597t_prod(A,E),G),S) = aa(set(B),E,aa(fun(B,E),fun(set(B),E),groups7121269368397514597t_prod(B,E),H2),T4) ) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_3601_less__1__prod2,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_idom(B)
     => ! [Ia2: set(A),Ia: A,F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),Ia2)
         => ( member(A,Ia,Ia2)
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),one_one(B)),aa(A,B,F2,Ia))
             => ( ! [I3: A] :
                    ( member(A,I3,Ia2)
                   => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),one_one(B)),aa(A,B,F2,I3)) )
               => aa(B,$o,aa(B,fun(B,$o),ord_less(B),one_one(B)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),Ia2)) ) ) ) ) ) ).

% less_1_prod2
tff(fact_3602_less__1__prod,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_idom(B)
     => ! [Ia2: set(A),F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),Ia2)
         => ( ( Ia2 != bot_bot(set(A)) )
           => ( ! [I3: A] :
                  ( member(A,I3,Ia2)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less(B),one_one(B)),aa(A,B,F2,I3)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),one_one(B)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),Ia2)) ) ) ) ) ).

% less_1_prod
tff(fact_3603_prod_Osubset__diff,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [B2: set(A),Aa2: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),Aa2)
         => ( aa(set(A),$o,finite_finite2(A),Aa2)
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),Aa2) = 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)),Aa2),B2))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),B2)) ) ) ) ) ).

% prod.subset_diff
tff(fact_3604_prod_Osame__carrier,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [C3: set(A),Aa2: set(A),B2: set(A),G: fun(A,B),H2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),C3)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),C3)
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),C3)
             => ( ! [A4: A] :
                    ( member(A,A4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),C3),Aa2))
                   => ( aa(A,B,G,A4) = one_one(B) ) )
               => ( ! [B4: A] :
                      ( member(A,B4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),C3),B2))
                     => ( aa(A,B,H2,B4) = one_one(B) ) )
                 => ( ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),Aa2) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),H2),B2) )
                  <=> ( 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_3605_prod_Osame__carrierI,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [C3: set(A),Aa2: set(A),B2: set(A),G: fun(A,B),H2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),C3)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),C3)
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),C3)
             => ( ! [A4: A] :
                    ( member(A,A4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),C3),Aa2))
                   => ( aa(A,B,G,A4) = one_one(B) ) )
               => ( ! [B4: A] :
                      ( member(A,B4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),C3),B2))
                     => ( aa(A,B,H2,B4) = 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),Aa2) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),H2),B2) ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_3606_prod_Omono__neutral__left,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [T4: set(A),S: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),T4)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),T4)
           => ( ! [X: A] :
                  ( member(A,X,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T4),S))
                 => ( aa(A,B,G,X) = 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),T4) ) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_3607_prod_Omono__neutral__right,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [T4: set(A),S: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),T4)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),T4)
           => ( ! [X: A] :
                  ( member(A,X,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T4),S))
                 => ( aa(A,B,G,X) = one_one(B) ) )
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),T4) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),S) ) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_3608_prod_Omono__neutral__cong__left,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [T4: set(A),S: set(A),H2: fun(A,B),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),T4)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),T4)
           => ( ! [X: A] :
                  ( member(A,X,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T4),S))
                 => ( aa(A,B,H2,X) = one_one(B) ) )
             => ( ! [X: A] :
                    ( member(A,X,S)
                   => ( aa(A,B,G,X) = aa(A,B,H2,X) ) )
               => ( 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),T4) ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_3609_prod_Omono__neutral__cong__right,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [T4: set(A),S: set(A),G: fun(A,B),H2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),T4)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),T4)
           => ( ! [X: A] :
                  ( member(A,X,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T4),S))
                 => ( aa(A,B,G,X) = one_one(B) ) )
             => ( ! [X: A] :
                    ( member(A,X,S)
                   => ( aa(A,B,G,X) = aa(A,B,H2,X) ) )
               => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),T4) = 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_3610_prod_Ounion__inter,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [Aa2: set(A),B2: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( aa(set(A),$o,finite_finite2(A),B2)
           => ( 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)),Aa2),B2))),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)),Aa2),B2))) = 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),Aa2)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),B2)) ) ) ) ) ).

% prod.union_inter
tff(fact_3611_prod_OInt__Diff,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [Aa2: set(A),G: fun(A,B),B2: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),Aa2) = 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)),Aa2),B2))),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)),Aa2),B2))) ) ) ) ).

% prod.Int_Diff
tff(fact_3612_prod_Omono__neutral__cong,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [T4: set(A),S: set(A),H2: fun(A,B),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),T4)
         => ( aa(set(A),$o,finite_finite2(A),S)
           => ( ! [I3: A] :
                  ( member(A,I3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T4),S))
                 => ( aa(A,B,H2,I3) = one_one(B) ) )
             => ( ! [I3: A] :
                    ( member(A,I3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S),T4))
                   => ( aa(A,B,G,I3) = one_one(B) ) )
               => ( ! [X: A] :
                      ( member(A,X,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S),T4))
                     => ( aa(A,B,G,X) = aa(A,B,H2,X) ) )
                 => ( 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),T4) ) ) ) ) ) ) ) ).

% prod.mono_neutral_cong
tff(fact_3613_prod__mono__strict,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_semidom(B)
     => ! [Aa2: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( ! [I3: A] :
                ( member(A,I3,Aa2)
               => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,I3))
                  & aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,I3)),aa(A,B,G,I3)) ) )
           => ( ( Aa2 != bot_bot(set(A)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),Aa2)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),Aa2)) ) ) ) ) ).

% prod_mono_strict
tff(fact_3614_prod_Oinsert__remove,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [Aa2: set(A),G: fun(A,B),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( 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)),insert3(A),Xa),Aa2)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,G,Xa)),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)),Aa2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A)))))) ) ) ) ).

% prod.insert_remove
tff(fact_3615_prod_Oremove,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [Aa2: set(A),Xa: A,G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( member(A,Xa,Aa2)
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),Aa2) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,G,Xa)),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)),Aa2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A)))))) ) ) ) ) ).

% prod.remove
tff(fact_3616_prod_Ounion__inter__neutral,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [Aa2: set(A),B2: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( aa(set(A),$o,finite_finite2(A),B2)
           => ( ! [X: A] :
                  ( member(A,X,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),B2))
                 => ( aa(A,B,G,X) = 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)),Aa2),B2)) = 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),Aa2)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),B2)) ) ) ) ) ) ).

% prod.union_inter_neutral
tff(fact_3617_prod_Ounion__disjoint,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [Aa2: set(A),B2: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( aa(set(A),$o,finite_finite2(A),B2)
           => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),B2) = 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)),Aa2),B2)) = 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),Aa2)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),B2)) ) ) ) ) ) ).

% prod.union_disjoint
tff(fact_3618_prod_Ounion__diff2,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [Aa2: set(A),B2: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( aa(set(A),$o,finite_finite2(A),B2)
           => ( 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)),Aa2),B2)) = 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)),Aa2),B2))),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)),B2),Aa2)))),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)),Aa2),B2))) ) ) ) ) ).

% prod.union_diff2
tff(fact_3619_prod__mono2,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_idom(B)
     => ! [B2: set(A),Aa2: set(A),F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),B2)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),B2)
           => ( ! [B4: A] :
                  ( member(A,B4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B2),Aa2))
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),one_one(B)),aa(A,B,F2,B4)) )
             => ( ! [A4: A] :
                    ( member(A,A4,Aa2)
                   => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,A4)) )
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),Aa2)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),B2)) ) ) ) ) ) ).

% prod_mono2
tff(fact_3620_pochhammer__times__pochhammer__half,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Z2: A,N: nat] : ( aa(A,A,aa(A,fun(A,A),times_times(A),comm_s3205402744901411588hammer(A,Z2,aa(nat,nat,suc,N))),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z2),divide_divide(A,one_one(A),numeral_numeral(A,bit0(one2)))),aa(nat,nat,suc,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_aa(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),numeral_numeral(nat,bit0(one2))),N)),one_one(nat)))) ) ) ).

% pochhammer_times_pochhammer_half
tff(fact_3621_f__arg__min__list__f,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Xsa: list(A),F2: fun(A,B)] :
          ( ( Xsa != nil(A) )
         => ( aa(A,B,F2,arg_min_list(A,B,F2,Xsa)) = lattic643756798350308766er_Min(B,aa(set(A),set(B),image2(A,B,F2),aa(list(A),set(A),set2(A),Xsa))) ) ) ) ).

% f_arg_min_list_f
tff(fact_3622_min__list__Min,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xsa: list(A)] :
          ( ( Xsa != nil(A) )
         => ( min_list(A,Xsa) = lattic643756798350308766er_Min(A,aa(list(A),set(A),set2(A),Xsa)) ) ) ) ).

% min_list_Min
tff(fact_3623_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),lattic643756798349783984er_Max(A,S)) = lattic643756798350308766er_Min(A,aa(set(A),set(A),image2(A,A,uminus_uminus(A)),S)) ) ) ) ) ).

% minus_Max_eq_Min
tff(fact_3624_image__ident,axiom,
    ! [A: $tType,Y5: set(A)] : ( aa(set(A),set(A),image2(A,A,aTP_Lamp_ab(A,A)),Y5) = Y5 ) ).

% image_ident
tff(fact_3625_card__Collect__less__nat,axiom,
    ! [N: nat] : ( finite_card(nat,aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_ac(nat,fun(nat,$o)),N))) = N ) ).

% card_Collect_less_nat
tff(fact_3626_singleton__conv,axiom,
    ! [A: $tType,A3: A] : ( aa(fun(A,$o),set(A),collect(A),aTP_Lamp_ad(A,fun(A,$o),A3)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),bot_bot(set(A))) ) ).

% singleton_conv
tff(fact_3627_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)),insert3(A),A3),bot_bot(set(A))) ) ).

% singleton_conv2
tff(fact_3628_Collect__const,axiom,
    ! [A: $tType,Pa: $o] :
      ( aa(fun(A,$o),set(A),collect(A),aTP_Lamp_ae($o,fun(A,$o),(Pa))) = $ite((Pa),top_top(set(A)),bot_bot(set(A))) ) ).

% Collect_const
tff(fact_3629_finite__Collect__less__nat,axiom,
    ! [K3: nat] : aa(set(nat),$o,finite_finite2(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_ac(nat,fun(nat,$o)),K3))) ).

% finite_Collect_less_nat
tff(fact_3630_Max__singleton,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A] : ( lattic643756798349783984er_Max(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A)))) = Xa ) ) ).

% Max_singleton
tff(fact_3631_surj__diff__right,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A] : ( aa(set(A),set(A),image2(A,A,aTP_Lamp_af(A,fun(A,A),A3)),top_top(set(A))) = top_top(set(A)) ) ) ).

% surj_diff_right
tff(fact_3632_SUP__bot,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Aa2: set(B)] : ( aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,aTP_Lamp_ag(B,A)),Aa2)) = bot_bot(A) ) ) ).

% SUP_bot
tff(fact_3633_SUP__bot__conv_I1_J,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [B2: fun(B,A),Aa2: set(B)] :
          ( ( aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,B2),Aa2)) = bot_bot(A) )
        <=> ! [X6: B] :
              ( member(B,X6,Aa2)
             => ( aa(B,A,B2,X6) = bot_bot(A) ) ) ) ) ).

% SUP_bot_conv(1)
tff(fact_3634_SUP__bot__conv_I2_J,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [B2: fun(B,A),Aa2: set(B)] :
          ( ( bot_bot(A) = aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,B2),Aa2)) )
        <=> ! [X6: B] :
              ( member(B,X6,Aa2)
             => ( aa(B,A,B2,X6) = bot_bot(A) ) ) ) ) ).

% SUP_bot_conv(2)
tff(fact_3635_cSUP__const,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [Aa2: set(A),C2: B] :
          ( ( Aa2 != bot_bot(set(A)) )
         => ( aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,aTP_Lamp_ah(B,fun(A,B),C2)),Aa2)) = C2 ) ) ) ).

% cSUP_const
tff(fact_3636_SUP__const,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [Aa2: set(A),F2: B] :
          ( ( Aa2 != bot_bot(set(A)) )
         => ( aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,aTP_Lamp_ai(B,fun(A,B),F2)),Aa2)) = F2 ) ) ) ).

% SUP_const
tff(fact_3637_foldl__length,axiom,
    ! [A: $tType,L: list(A)] : ( aa(list(A),nat,aa(nat,fun(list(A),nat),aa(fun(nat,fun(A,nat)),fun(nat,fun(list(A),nat)),foldl(nat,A),aTP_Lamp_aj(nat,fun(A,nat))),zero_zero(nat)),L) = aa(list(A),nat,size_size(list(A)),L) ) ).

% foldl_length
tff(fact_3638_image__minus__const__atLeastAtMost_H,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [D3: A,A3: A,B3: A] : ( aa(set(A),set(A),image2(A,A,aTP_Lamp_ak(A,fun(A,A),D3)),set_or1337092689740270186AtMost(A,A3,B3)) = 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),B3),D3)) ) ) ).

% image_minus_const_atLeastAtMost'
tff(fact_3639_if__image__distrib,axiom,
    ! [A: $tType,B: $tType,Pa: fun(B,$o),F2: 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_al(fun(B,$o),fun(fun(B,A),fun(fun(B,A),fun(B,A))),Pa),F2),G)),S) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(B),set(A),image2(B,A,F2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),S),aa(fun(B,$o),set(B),collect(B),Pa)))),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,$o),set(B),collect(B),aTP_Lamp_am(fun(B,$o),fun(B,$o),Pa))))) ) ).

% if_image_distrib
tff(fact_3640_UN__constant,axiom,
    ! [B: $tType,A: $tType,C2: set(A),Aa2: 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_an(set(A),fun(B,set(A)),C2)),Aa2)) = $ite(Aa2 = bot_bot(set(B)),bot_bot(set(A)),C2) ) ).

% UN_constant
tff(fact_3641_sorted__wrt__rev__linord,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: list(A)] :
          ( sorted_wrt(A,aTP_Lamp_ao(A,fun(A,$o)),L)
        <=> sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),rev(A),L)) ) ) ).

% sorted_wrt_rev_linord
tff(fact_3642_subset__mset_OcSUP__const,axiom,
    ! [A: $tType,B: $tType,Aa2: set(A),C2: multiset(B)] :
      ( ( Aa2 != 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_ap(multiset(B),fun(A,multiset(B)),C2)),Aa2)) = C2 ) ) ).

% subset_mset.cSUP_const
tff(fact_3643_precise__extr__pure_I1_J,axiom,
    ! [B: $tType,A: $tType,Pa: $o,R: fun(A,fun(B,assn))] :
      ( precise(A,B,aa(fun(A,fun(B,assn)),fun(A,fun(B,assn)),aTP_Lamp_aq($o,fun(fun(A,fun(B,assn)),fun(A,fun(B,assn))),(Pa)),R))
    <=> ( (Pa)
       => precise(A,B,R) ) ) ).

% precise_extr_pure(1)
tff(fact_3644_precise__extr__pure_I2_J,axiom,
    ! [B: $tType,A: $tType,R: fun(A,fun(B,assn)),Pa: $o] :
      ( precise(A,B,aa($o,fun(A,fun(B,assn)),aTP_Lamp_ar(fun(A,fun(B,assn)),fun($o,fun(A,fun(B,assn))),R),(Pa)))
    <=> ( (Pa)
       => precise(A,B,R) ) ) ).

% precise_extr_pure(2)
tff(fact_3645_Max_Obounded__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Aa2: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( ( Aa2 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),lattic643756798349783984er_Max(A,Aa2)),Xa)
            <=> ! [X6: A] :
                  ( member(A,X6,Aa2)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X6),Xa) ) ) ) ) ) ).

% Max.bounded_iff
tff(fact_3646_Max__less__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Aa2: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( ( Aa2 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),lattic643756798349783984er_Max(A,Aa2)),Xa)
            <=> ! [X6: A] :
                  ( member(A,X6,Aa2)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),X6),Xa) ) ) ) ) ) ).

% Max_less_iff
tff(fact_3647_SUP__eq__top__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [F2: fun(B,A),Aa2: set(B)] :
          ( ( aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F2),Aa2)) = top_top(A) )
        <=> ! [X6: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X6),top_top(A))
             => ? [Xa2: B] :
                  ( member(B,Xa2,Aa2)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X6),aa(B,A,F2,Xa2)) ) ) ) ) ).

% SUP_eq_top_iff
tff(fact_3648_range__constant,axiom,
    ! [B: $tType,A: $tType,Xa: A] : ( aa(set(B),set(A),image2(B,A,aTP_Lamp_as(A,fun(B,A),Xa)),top_top(set(B))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A))) ) ).

% range_constant
tff(fact_3649_UN__singleton,axiom,
    ! [A: $tType,Aa2: 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_at(A,set(A))),Aa2)) = Aa2 ) ).

% UN_singleton
tff(fact_3650_UN__simps_I1_J,axiom,
    ! [A: $tType,B: $tType,A3: A,B2: 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_au(A,fun(fun(B,set(A)),fun(B,set(A))),A3),B2)),C3)) = $ite(C3 = bot_bot(set(B)),bot_bot(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B2),C3)))) ) ).

% UN_simps(1)
tff(fact_3651_UN__simps_I3_J,axiom,
    ! [A: $tType,B: $tType,Aa2: set(A),B2: 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_av(set(A),fun(fun(B,set(A)),fun(B,set(A))),Aa2),B2)),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)),Aa2),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B2),C3)))) ) ).

% UN_simps(3)
tff(fact_3652_UN__simps_I2_J,axiom,
    ! [B: $tType,A: $tType,Aa2: fun(B,set(A)),B2: 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_aw(fun(B,set(A)),fun(set(A),fun(B,set(A))),Aa2),B2)),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),Aa2),C3))),B2)) ) ).

% UN_simps(2)
tff(fact_3653_UN__insert,axiom,
    ! [A: $tType,B: $tType,B2: fun(B,set(A)),A3: B,Aa2: set(B)] : ( aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B2),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert3(B),A3),Aa2))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(B,set(A),B2,A3)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B2),Aa2))) ) ).

% UN_insert
tff(fact_3654_Max__const,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(B)
     => ! [Aa2: set(A),C2: B] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( ( Aa2 != bot_bot(set(A)) )
           => ( lattic643756798349783984er_Max(B,aa(set(A),set(B),image2(A,B,aTP_Lamp_ax(B,fun(A,B),C2)),Aa2)) = C2 ) ) ) ) ).

% Max_const
tff(fact_3655_Min__const,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(B)
     => ! [Aa2: set(A),C2: B] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( ( Aa2 != bot_bot(set(A)) )
           => ( lattic643756798350308766er_Min(B,aa(set(A),set(B),image2(A,B,aTP_Lamp_ax(B,fun(A,B),C2)),Aa2)) = C2 ) ) ) ) ).

% Min_const
tff(fact_3656_image__divide__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [D3: A,A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),D3)
         => ( aa(set(A),set(A),image2(A,A,aTP_Lamp_ay(A,fun(A,A),D3)),set_or1337092689740270186AtMost(A,A3,B3)) = set_or1337092689740270186AtMost(A,divide_divide(A,A3,D3),divide_divide(A,B3,D3)) ) ) ) ).

% image_divide_atLeastAtMost
tff(fact_3657_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),lattic643756798350308766er_Min(A,S)) = lattic643756798349783984er_Max(A,aa(set(A),set(A),image2(A,A,uminus_uminus(A)),S)) ) ) ) ) ).

% minus_Min_eq_Max
tff(fact_3658_set__concat,axiom,
    ! [A: $tType,Xsa: list(list(A))] : ( aa(list(A),set(A),set2(A),aa(list(list(A)),list(A),concat(A),Xsa)) = 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)),Xsa))) ) ).

% set_concat
tff(fact_3659_finite__lists__distinct__length__eq,axiom,
    ! [A: $tType,Aa2: set(A),N: nat] :
      ( aa(set(A),$o,finite_finite2(A),Aa2)
     => 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),$o),aTP_Lamp_az(set(A),fun(nat,fun(list(A),$o)),Aa2),N))) ) ).

% finite_lists_distinct_length_eq
tff(fact_3660_prod_Odistrib,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(B,A),H2: fun(B,A),Aa2: 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_ba(fun(B,A),fun(fun(B,A),fun(B,A)),G),H2)),Aa2) = 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),Aa2)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),H2),Aa2)) ) ) ).

% prod.distrib
tff(fact_3661_prod_Ofinite__Collect__op,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [Ia2: set(A),Xa: fun(A,B),Y: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_bb(set(A),fun(fun(A,B),fun(A,$o)),Ia2),Xa)))
         => ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_bb(set(A),fun(fun(A,B),fun(A,$o)),Ia2),Y)))
           => aa(set(A),$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,$o)),aTP_Lamp_bc(set(A),fun(fun(A,B),fun(fun(A,B),fun(A,$o))),Ia2),Xa),Y))) ) ) ) ).

% prod.finite_Collect_op
tff(fact_3662_image__constant__conv,axiom,
    ! [B: $tType,A: $tType,C2: A,Aa2: set(B)] :
      ( aa(set(B),set(A),image2(B,A,aTP_Lamp_as(A,fun(B,A),C2)),Aa2) = $ite(Aa2 = bot_bot(set(B)),bot_bot(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),C2),bot_bot(set(A)))) ) ).

% image_constant_conv
tff(fact_3663_image__constant,axiom,
    ! [A: $tType,B: $tType,Xa: A,Aa2: set(A),C2: B] :
      ( member(A,Xa,Aa2)
     => ( aa(set(A),set(B),image2(A,B,aTP_Lamp_bd(B,fun(A,B),C2)),Aa2) = aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert3(B),C2),bot_bot(set(B))) ) ) ).

% image_constant
tff(fact_3664_filter__preserves__multiset,axiom,
    ! [A: $tType,M5: fun(A,nat),Pa: fun(A,$o)] :
      ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_be(fun(A,nat),fun(A,$o),M5)))
     => aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_bf(fun(A,nat),fun(fun(A,$o),fun(A,$o)),M5),Pa))) ) ).

% filter_preserves_multiset
tff(fact_3665_card__less__Suc2,axiom,
    ! [M5: set(nat),Ia: nat] :
      ( ~ member(nat,zero_zero(nat),M5)
     => ( finite_card(nat,aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_bg(set(nat),fun(nat,fun(nat,$o)),M5),Ia))) = finite_card(nat,aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_bh(set(nat),fun(nat,fun(nat,$o)),M5),Ia))) ) ) ).

% card_less_Suc2
tff(fact_3666_card__less__Suc,axiom,
    ! [M5: set(nat),Ia: nat] :
      ( member(nat,zero_zero(nat),M5)
     => ( aa(nat,nat,suc,finite_card(nat,aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_bg(set(nat),fun(nat,fun(nat,$o)),M5),Ia)))) = finite_card(nat,aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_bh(set(nat),fun(nat,fun(nat,$o)),M5),Ia))) ) ) ).

% card_less_Suc
tff(fact_3667_card__less,axiom,
    ! [M5: set(nat),Ia: nat] :
      ( member(nat,zero_zero(nat),M5)
     => ( finite_card(nat,aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_bh(set(nat),fun(nat,fun(nat,$o)),M5),Ia))) != zero_zero(nat) ) ) ).

% card_less
tff(fact_3668_translation__subtract__Int,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A,S2: set(A),T2: set(A)] : ( aa(set(A),set(A),image2(A,A,aTP_Lamp_af(A,fun(A,A),A3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S2),T2)) = 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_af(A,fun(A,A),A3)),S2)),aa(set(A),set(A),image2(A,A,aTP_Lamp_af(A,fun(A,A),A3)),T2)) ) ) ).

% translation_subtract_Int
tff(fact_3669_translation__subtract__diff,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A,S2: set(A),T2: set(A)] : ( aa(set(A),set(A),image2(A,A,aTP_Lamp_af(A,fun(A,A),A3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S2),T2)) = 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_af(A,fun(A,A),A3)),S2)),aa(set(A),set(A),image2(A,A,aTP_Lamp_af(A,fun(A,A),A3)),T2)) ) ) ).

% translation_subtract_diff
tff(fact_3670_SUP__Sup__eq2,axiom,
    ! [B: $tType,A: $tType,S: set(set(product_prod(A,B))),X4: A,Xa4: B] :
      ( aa(B,$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_bi(set(product_prod(A,B)),fun(A,fun(B,$o)))),S)),X4),Xa4)
    <=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Xa4),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_3671_Max__add__commute,axiom,
    ! [A: $tType,B: $tType] :
      ( linord4140545234300271783up_add(B)
     => ! [S: set(A),F2: fun(A,B),K3: B] :
          ( aa(set(A),$o,finite_finite2(A),S)
         => ( ( S != bot_bot(set(A)) )
           => ( lattic643756798349783984er_Max(B,aa(set(A),set(B),image2(A,B,aa(B,fun(A,B),aTP_Lamp_bj(fun(A,B),fun(B,fun(A,B)),F2),K3)),S)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),lattic643756798349783984er_Max(B,aa(set(A),set(B),image2(A,B,F2),S))),K3) ) ) ) ) ).

% Max_add_commute
tff(fact_3672_pairwise__image,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(A,$o)),F2: fun(B,A),S2: set(B)] :
      ( pairwise(A,R2,aa(set(B),set(A),image2(B,A,F2),S2))
    <=> pairwise(B,aa(fun(B,A),fun(B,fun(B,$o)),aTP_Lamp_bk(fun(A,fun(A,$o)),fun(fun(B,A),fun(B,fun(B,$o))),R2),F2),S2) ) ).

% pairwise_image
tff(fact_3673_Compr__image__eq,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),Aa2: set(B),Pa: fun(A,$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,$o)),aTP_Lamp_bl(fun(B,A),fun(set(B),fun(fun(A,$o),fun(A,$o))),F2),Aa2),Pa)) = aa(set(B),set(A),image2(B,A,F2),aa(fun(B,$o),set(B),collect(B),aa(fun(A,$o),fun(B,$o),aa(set(B),fun(fun(A,$o),fun(B,$o)),aTP_Lamp_bm(fun(B,A),fun(set(B),fun(fun(A,$o),fun(B,$o))),F2),Aa2),Pa))) ) ).

% Compr_image_eq
tff(fact_3674_image__image,axiom,
    ! [B: $tType,A: $tType,E: $tType,F2: fun(B,A),G: fun(E,B),Aa2: set(E)] : ( aa(set(B),set(A),image2(B,A,F2),aa(set(E),set(B),image2(E,B,G),Aa2)) = aa(set(E),set(A),image2(E,A,aa(fun(E,B),fun(E,A),aTP_Lamp_bn(fun(B,A),fun(fun(E,B),fun(E,A)),F2),G)),Aa2) ) ).

% image_image
tff(fact_3675_imageE,axiom,
    ! [A: $tType,B: $tType,B3: A,F2: fun(B,A),Aa2: set(B)] :
      ( member(A,B3,aa(set(B),set(A),image2(B,A,F2),Aa2))
     => ~ ! [X: B] :
            ( ( B3 = aa(B,A,F2,X) )
           => ~ member(B,X,Aa2) ) ) ).

% imageE
tff(fact_3676_SUP__Sup__eq,axiom,
    ! [A: $tType,S: set(set(A)),X4: A] :
      ( aa(A,$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,$o),aTP_Lamp_a(set(A),fun(A,$o))),S)),X4)
    <=> member(A,X4,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),S)) ) ).

% SUP_Sup_eq
tff(fact_3677_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_3678_range__composition,axiom,
    ! [A: $tType,E: $tType,B: $tType,F2: fun(E,A),G: fun(B,E)] : ( aa(set(B),set(A),image2(B,A,aa(fun(B,E),fun(B,A),aTP_Lamp_bo(fun(E,A),fun(fun(B,E),fun(B,A)),F2),G)),top_top(set(B))) = aa(set(E),set(A),image2(E,A,F2),aa(set(B),set(E),image2(B,E,G),top_top(set(B)))) ) ).

% range_composition
tff(fact_3679_rangeE,axiom,
    ! [A: $tType,B: $tType,B3: A,F2: fun(B,A)] :
      ( member(A,B3,aa(set(B),set(A),image2(B,A,F2),top_top(set(B))))
     => ~ ! [X: B] : ( B3 != aa(B,A,F2,X) ) ) ).

% rangeE
tff(fact_3680_insert__def,axiom,
    ! [A: $tType,A3: A,B2: set(A)] : ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),B2) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_ad(A,fun(A,$o),A3))),B2) ) ).

% insert_def
tff(fact_3681_min__list_Osimps,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Xa: A,Xsa: list(A)] : ( min_list(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa)) = aa(list(A),A,aa(fun(A,fun(list(A),A)),fun(list(A),A),aa(A,fun(fun(A,fun(list(A),A)),fun(list(A),A)),case_list(A,A),Xa),aa(list(A),fun(A,fun(list(A),A)),aTP_Lamp_bp(A,fun(list(A),fun(A,fun(list(A),A))),Xa),Xsa)),Xsa) ) ) ).

% min_list.simps
tff(fact_3682_tl__append,axiom,
    ! [A: $tType,Xsa: 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),Xsa),Ys)) = aa(list(A),list(A),aa(fun(A,fun(list(A),list(A))),fun(list(A),list(A)),aa(list(A),fun(fun(A,fun(list(A),list(A))),fun(list(A),list(A))),case_list(list(A),A),aa(list(A),list(A),tl(A),Ys)),aTP_Lamp_bq(list(A),fun(A,fun(list(A),list(A))),Ys)),Xsa) ) ).

% tl_append
tff(fact_3683_less__eq__set__def,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),B2)
    <=> aa(fun(A,$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_a(set(A),fun(A,$o)),Aa2)),aa(set(A),fun(A,$o),aTP_Lamp_a(set(A),fun(A,$o)),B2)) ) ).

% less_eq_set_def
tff(fact_3684_Collect__subset,axiom,
    ! [A: $tType,Aa2: set(A),Pa: fun(A,$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,$o),aTP_Lamp_br(set(A),fun(fun(A,$o),fun(A,$o)),Aa2),Pa))),Aa2) ).

% Collect_subset
tff(fact_3685_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_a(set(A),fun(A,$o)),R)),aa(set(A),fun(A,$o),aTP_Lamp_a(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_3686_arg__min__list_Osimps_I2_J,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),Xa: A,Y: A,Zs: list(A)] :
          ( arg_min_list(A,B,F2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),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,F2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Zs)),
              $ite(aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,Xa)),aa(A,B,F2,m)),Xa,m) ) ) ) ).

% arg_min_list.simps(2)
tff(fact_3687_subset__divisors__dvd,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A,B3: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_bs(A,fun(A,$o),A3))),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_bs(A,fun(A,$o),B3)))
        <=> dvd_dvd(A,A3,B3) ) ) ).

% subset_divisors_dvd
tff(fact_3688_Collect__imp__eq,axiom,
    ! [A: $tType,Pa: fun(A,$o),Qa: fun(A,$o)] : ( aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_bt(fun(A,$o),fun(fun(A,$o),fun(A,$o)),Pa),Qa)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),aa(fun(A,$o),set(A),collect(A),Pa))),aa(fun(A,$o),set(A),collect(A),Qa)) ) ).

% Collect_imp_eq
tff(fact_3689_sup__Un__eq,axiom,
    ! [A: $tType,R: set(A),S: set(A),X4: A] :
      ( aa(A,$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,$o),aTP_Lamp_a(set(A),fun(A,$o)),R)),aa(set(A),fun(A,$o),aTP_Lamp_a(set(A),fun(A,$o)),S)),X4)
    <=> member(A,X4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),R),S)) ) ).

% sup_Un_eq
tff(fact_3690_Un__def,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A)] : ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),B2) = aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_bu(set(A),fun(set(A),fun(A,$o)),Aa2),B2)) ) ).

% Un_def
tff(fact_3691_sup__set__def,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A)] : ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),B2) = aa(fun(A,$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,$o),aTP_Lamp_a(set(A),fun(A,$o)),Aa2)),aa(set(A),fun(A,$o),aTP_Lamp_a(set(A),fun(A,$o)),B2))) ) ).

% sup_set_def
tff(fact_3692_Collect__disj__eq,axiom,
    ! [A: $tType,Pa: fun(A,$o),Qa: fun(A,$o)] : ( aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_bv(fun(A,$o),fun(fun(A,$o),fun(A,$o)),Pa),Qa)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(fun(A,$o),set(A),collect(A),Pa)),aa(fun(A,$o),set(A),collect(A),Qa)) ) ).

% Collect_disj_eq
tff(fact_3693_inf__Int__eq,axiom,
    ! [A: $tType,R: set(A),S: set(A),X4: A] :
      ( aa(A,$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,$o),aTP_Lamp_a(set(A),fun(A,$o)),R)),aa(set(A),fun(A,$o),aTP_Lamp_a(set(A),fun(A,$o)),S)),X4)
    <=> member(A,X4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),R),S)) ) ).

% inf_Int_eq
tff(fact_3694_Int__def,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A)] : ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),B2) = aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_bw(set(A),fun(set(A),fun(A,$o)),Aa2),B2)) ) ).

% Int_def
tff(fact_3695_Int__Collect,axiom,
    ! [A: $tType,Xa: A,Aa2: set(A),Pa: fun(A,$o)] :
      ( member(A,Xa,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),aa(fun(A,$o),set(A),collect(A),Pa)))
    <=> ( member(A,Xa,Aa2)
        & aa(A,$o,Pa,Xa) ) ) ).

% Int_Collect
tff(fact_3696_inf__set__def,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A)] : ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),B2) = aa(fun(A,$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,$o),aTP_Lamp_a(set(A),fun(A,$o)),Aa2)),aa(set(A),fun(A,$o),aTP_Lamp_a(set(A),fun(A,$o)),B2))) ) ).

% inf_set_def
tff(fact_3697_Collect__conj__eq,axiom,
    ! [A: $tType,Pa: fun(A,$o),Qa: fun(A,$o)] : ( aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_bx(fun(A,$o),fun(fun(A,$o),fun(A,$o)),Pa),Qa)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(fun(A,$o),set(A),collect(A),Pa)),aa(fun(A,$o),set(A),collect(A),Qa)) ) ).

% Collect_conj_eq
tff(fact_3698_Pow__def,axiom,
    ! [A: $tType,Aa2: set(A)] : ( pow(A,Aa2) = aa(fun(set(A),$o),set(set(A)),collect(set(A)),aTP_Lamp_by(set(A),fun(set(A),$o),Aa2)) ) ).

% Pow_def
tff(fact_3699_list_Odisc__eq__case_I2_J,axiom,
    ! [A: $tType,List: list(A)] :
      ( ( List != nil(A) )
    <=> aa(list(A),$o,aa(fun(A,fun(list(A),$o)),fun(list(A),$o),aa($o,fun(fun(A,fun(list(A),$o)),fun(list(A),$o)),case_list($o,A),$false),aTP_Lamp_bz(A,fun(list(A),$o))),List) ) ).

% list.disc_eq_case(2)
tff(fact_3700_list_Odisc__eq__case_I1_J,axiom,
    ! [A: $tType,List: list(A)] :
      ( ( List = nil(A) )
    <=> aa(list(A),$o,aa(fun(A,fun(list(A),$o)),fun(list(A),$o),aa($o,fun(fun(A,fun(list(A),$o)),fun(list(A),$o)),case_list($o,A),$true),aTP_Lamp_ca(A,fun(list(A),$o))),List) ) ).

% list.disc_eq_case(1)
tff(fact_3701_tl__def,axiom,
    ! [A: $tType,List: list(A)] : ( aa(list(A),list(A),tl(A),List) = aa(list(A),list(A),aa(fun(A,fun(list(A),list(A))),fun(list(A),list(A)),aa(list(A),fun(fun(A,fun(list(A),list(A))),fun(list(A),list(A))),case_list(list(A),A),nil(A)),aTP_Lamp_cb(A,fun(list(A),list(A)))),List) ) ).

% tl_def
tff(fact_3702_UNIV__def,axiom,
    ! [A: $tType] : ( top_top(set(A)) = aa(fun(A,$o),set(A),collect(A),aTP_Lamp_cc(A,$o)) ) ).

% UNIV_def
tff(fact_3703_strict__subset__divisors__dvd,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A,B3: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_bs(A,fun(A,$o),A3))),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_bs(A,fun(A,$o),B3)))
        <=> ( dvd_dvd(A,A3,B3)
            & ~ dvd_dvd(A,B3,A3) ) ) ) ).

% strict_subset_divisors_dvd
tff(fact_3704_Compl__eq,axiom,
    ! [A: $tType,Aa2: set(A)] : ( aa(set(A),set(A),uminus_uminus(set(A)),Aa2) = aa(fun(A,$o),set(A),collect(A),aTP_Lamp_cd(set(A),fun(A,$o),Aa2)) ) ).

% Compl_eq
tff(fact_3705_Collect__neg__eq,axiom,
    ! [A: $tType,Pa: fun(A,$o)] : ( aa(fun(A,$o),set(A),collect(A),aTP_Lamp_ce(fun(A,$o),fun(A,$o),Pa)) = aa(set(A),set(A),uminus_uminus(set(A)),aa(fun(A,$o),set(A),collect(A),Pa)) ) ).

% Collect_neg_eq
tff(fact_3706_uminus__set__def,axiom,
    ! [A: $tType,Aa2: set(A)] : ( aa(set(A),set(A),uminus_uminus(set(A)),Aa2) = aa(fun(A,$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_a(set(A),fun(A,$o)),Aa2))) ) ).

% uminus_set_def
tff(fact_3707_set__diff__eq,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A)] : ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),B2) = aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_cf(set(A),fun(set(A),fun(A,$o)),Aa2),B2)) ) ).

% set_diff_eq
tff(fact_3708_minus__set__def,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A)] : ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),B2) = aa(fun(A,$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,$o),aTP_Lamp_a(set(A),fun(A,$o)),Aa2)),aa(set(A),fun(A,$o),aTP_Lamp_a(set(A),fun(A,$o)),B2))) ) ).

% minus_set_def
tff(fact_3709_pairwise__trivial,axiom,
    ! [A: $tType,Ia2: set(A)] : pairwise(A,aTP_Lamp_cg(A,fun(A,$o)),Ia2) ).

% pairwise_trivial
tff(fact_3710_sngr__prec,axiom,
    ! [A: $tType] :
      ( heap(A)
     => precise(A,ref(A),aTP_Lamp_ch(A,fun(ref(A),assn))) ) ).

% sngr_prec
tff(fact_3711_snga__prec,axiom,
    ! [A: $tType] :
      ( heap(A)
     => precise(list(A),array(A),aTP_Lamp_ci(list(A),fun(array(A),assn))) ) ).

% snga_prec
tff(fact_3712_list_Ocase__distrib,axiom,
    ! [B: $tType,A: $tType,E: $tType,H2: fun(B,A),F1: B,F23: fun(E,fun(list(E),B)),List: list(E)] : ( aa(B,A,H2,aa(list(E),B,aa(fun(E,fun(list(E),B)),fun(list(E),B),aa(B,fun(fun(E,fun(list(E),B)),fun(list(E),B)),case_list(B,E),F1),F23),List)) = aa(list(E),A,aa(fun(E,fun(list(E),A)),fun(list(E),A),aa(A,fun(fun(E,fun(list(E),A)),fun(list(E),A)),case_list(A,E),aa(B,A,H2,F1)),aa(fun(E,fun(list(E),B)),fun(E,fun(list(E),A)),aTP_Lamp_cj(fun(B,A),fun(fun(E,fun(list(E),B)),fun(E,fun(list(E),A))),H2),F23)),List) ) ).

% list.case_distrib
tff(fact_3713_less__set__def,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),Aa2),B2)
    <=> aa(fun(A,$o),$o,aa(fun(A,$o),fun(fun(A,$o),$o),ord_less(fun(A,$o)),aa(set(A),fun(A,$o),aTP_Lamp_a(set(A),fun(A,$o)),Aa2)),aa(set(A),fun(A,$o),aTP_Lamp_a(set(A),fun(A,$o)),B2)) ) ).

% less_set_def
tff(fact_3714_lambda__one,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ( aTP_Lamp_ck(A,A) = aa(A,fun(A,A),times_times(A),one_one(A)) ) ) ).

% lambda_one
tff(fact_3715_lambda__zero,axiom,
    ! [A: $tType] :
      ( mult_zero(A)
     => ( aTP_Lamp_cl(A,A) = aa(A,fun(A,A),times_times(A),zero_zero(A)) ) ) ).

% lambda_zero
tff(fact_3716_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_3717_sorted__wrt__true,axiom,
    ! [A: $tType,Xsa: list(A)] : sorted_wrt(A,aTP_Lamp_cm(A,fun(A,$o)),Xsa) ).

% sorted_wrt_true
tff(fact_3718_sorted__wrt__rev,axiom,
    ! [A: $tType,Pa: fun(A,fun(A,$o)),Xsa: list(A)] :
      ( sorted_wrt(A,Pa,aa(list(A),list(A),rev(A),Xsa))
    <=> sorted_wrt(A,aTP_Lamp_cn(fun(A,fun(A,$o)),fun(A,fun(A,$o)),Pa),Xsa) ) ).

% sorted_wrt_rev
tff(fact_3719_insert__Collect,axiom,
    ! [A: $tType,A3: A,Pa: fun(A,$o)] : ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),aa(fun(A,$o),set(A),collect(A),Pa)) = aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_co(A,fun(fun(A,$o),fun(A,$o)),A3),Pa)) ) ).

% insert_Collect
tff(fact_3720_insert__compr,axiom,
    ! [A: $tType,A3: A,B2: set(A)] : ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),B2) = aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_cp(A,fun(set(A),fun(A,$o)),A3),B2)) ) ).

% insert_compr
tff(fact_3721_Set_Oempty__def,axiom,
    ! [A: $tType] : ( bot_bot(set(A)) = aa(fun(A,$o),set(A),collect(A),aTP_Lamp_cq(A,$o)) ) ).

% Set.empty_def
tff(fact_3722_Collect__conv__if,axiom,
    ! [A: $tType,A3: A,Pa: fun(A,$o)] :
      ( aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_cr(A,fun(fun(A,$o),fun(A,$o)),A3),Pa)) = $ite(aa(A,$o,Pa,A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),bot_bot(set(A))),bot_bot(set(A))) ) ).

% Collect_conv_if
tff(fact_3723_Collect__conv__if2,axiom,
    ! [A: $tType,A3: A,Pa: fun(A,$o)] :
      ( aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_cs(A,fun(fun(A,$o),fun(A,$o)),A3),Pa)) = $ite(aa(A,$o,Pa,A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),bot_bot(set(A))),bot_bot(set(A))) ) ).

% Collect_conv_if2
tff(fact_3724_bot__empty__eq2,axiom,
    ! [B: $tType,A: $tType,X4: A,Xa4: B] :
      ( aa(B,$o,aa(A,fun(B,$o),bot_bot(fun(A,fun(B,$o))),X4),Xa4)
    <=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Xa4),bot_bot(set(product_prod(A,B)))) ) ).

% bot_empty_eq2
tff(fact_3725_curry__def,axiom,
    ! [E: $tType,A: $tType,B: $tType,X4: fun(product_prod(A,B),E),Xa4: A,Xb3: B] : ( aa(B,E,aa(A,fun(B,E),product_curry(A,B,E,X4),Xa4),Xb3) = aa(product_prod(A,B),E,X4,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa4),Xb3)) ) ).

% curry_def
tff(fact_3726_pred__equals__eq2,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,B)),S: set(product_prod(A,B))] :
      ( ! [X6: A,Xa2: B] :
          ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X6),Xa2),R)
        <=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X6),Xa2),S) )
    <=> ( R = S ) ) ).

% pred_equals_eq2
tff(fact_3727_top__empty__eq2,axiom,
    ! [B: $tType,A: $tType,X4: A,Xa4: B] :
      ( aa(B,$o,aa(A,fun(B,$o),top_top(fun(A,fun(B,$o))),X4),Xa4)
    <=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Xa4),top_top(set(product_prod(A,B)))) ) ).

% top_empty_eq2
tff(fact_3728_sup__Un__eq2,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,B)),S: set(product_prod(A,B)),X4: A,Xa4: B] :
      ( aa(B,$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_bi(set(product_prod(A,B)),fun(A,fun(B,$o))),R)),aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_bi(set(product_prod(A,B)),fun(A,fun(B,$o))),S)),X4),Xa4)
    <=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Xa4),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_3729_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_bi(set(product_prod(A,B)),fun(A,fun(B,$o))),R)),aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_bi(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_3730_inf__Int__eq2,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,B)),S: set(product_prod(A,B)),X4: A,Xa4: B] :
      ( aa(B,$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_bi(set(product_prod(A,B)),fun(A,fun(B,$o))),R)),aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_bi(set(product_prod(A,B)),fun(A,fun(B,$o))),S)),X4),Xa4)
    <=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Xa4),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_3731_finite__M__bounded__by__nat,axiom,
    ! [Pa: fun(nat,$o),Ia: nat] : aa(set(nat),$o,finite_finite2(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_ct(fun(nat,$o),fun(nat,fun(nat,$o)),Pa),Ia))) ).

% finite_M_bounded_by_nat
tff(fact_3732_less__SUP__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [A3: A,F2: fun(B,A),Aa2: set(B)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F2),Aa2)))
        <=> ? [X6: B] :
              ( member(B,X6,Aa2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(B,A,F2,X6)) ) ) ) ).

% less_SUP_iff
tff(fact_3733_SUP__lessD,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(B,A),Aa2: set(B),Y: A,Ia: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F2),Aa2))),Y)
         => ( member(B,Ia,Aa2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,F2,Ia)),Y) ) ) ) ).

% SUP_lessD
tff(fact_3734_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_cu(A,fun(A,$o),U)) ) ) ).

% lessThan_def
tff(fact_3735_foldl__length__aux,axiom,
    ! [A: $tType,A3: nat,L: list(A)] : ( aa(list(A),nat,aa(nat,fun(list(A),nat),aa(fun(nat,fun(A,nat)),fun(nat,fun(list(A),nat)),foldl(nat,A),aTP_Lamp_aj(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_3736_UNION__empty__conv_I2_J,axiom,
    ! [B: $tType,A: $tType,B2: fun(B,set(A)),Aa2: set(B)] :
      ( ( aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B2),Aa2)) = bot_bot(set(A)) )
    <=> ! [X6: B] :
          ( member(B,X6,Aa2)
         => ( aa(B,set(A),B2,X6) = bot_bot(set(A)) ) ) ) ).

% UNION_empty_conv(2)
tff(fact_3737_UNION__empty__conv_I1_J,axiom,
    ! [B: $tType,A: $tType,B2: fun(B,set(A)),Aa2: 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),B2),Aa2)) )
    <=> ! [X6: B] :
          ( member(B,X6,Aa2)
         => ( aa(B,set(A),B2,X6) = bot_bot(set(A)) ) ) ) ).

% UNION_empty_conv(1)
tff(fact_3738_UN__empty,axiom,
    ! [B: $tType,A: $tType,B2: 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),B2),bot_bot(set(B)))) = bot_bot(set(A)) ) ).

% UN_empty
tff(fact_3739_UN__empty2,axiom,
    ! [B: $tType,A: $tType,Aa2: 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_cv(B,set(A))),Aa2)) = bot_bot(set(A)) ) ).

% UN_empty2
tff(fact_3740_UN__insert__distrib,axiom,
    ! [B: $tType,A: $tType,U: A,Aa2: set(A),A3: B,B2: fun(A,set(B))] :
      ( member(A,U,Aa2)
     => ( 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_cw(B,fun(fun(A,set(B)),fun(A,set(B))),A3),B2)),Aa2)) = aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert3(B),A3),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),B2),Aa2))) ) ) ).

% UN_insert_distrib
tff(fact_3741_translation__subtract__Compl,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A,T2: set(A)] : ( aa(set(A),set(A),image2(A,A,aTP_Lamp_af(A,fun(A,A),A3)),aa(set(A),set(A),uminus_uminus(set(A)),T2)) = aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),image2(A,A,aTP_Lamp_af(A,fun(A,A),A3)),T2)) ) ) ).

% translation_subtract_Compl
tff(fact_3742_nat__less__as__int,axiom,
    ! [X4: nat,Xa4: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X4),Xa4)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(nat,int,semiring_1_of_nat(int),X4)),aa(nat,int,semiring_1_of_nat(int),Xa4)) ) ).

% nat_less_as_int
tff(fact_3743_UN__extend__simps_I6_J,axiom,
    ! [A: $tType,B: $tType,Aa2: fun(B,set(A)),C3: set(B),B2: 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),Aa2),C3))),B2) = 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_cx(fun(B,set(A)),fun(set(A),fun(B,set(A))),Aa2),B2)),C3)) ) ).

% UN_extend_simps(6)
tff(fact_3744_nat__seg__image__imp__finite,axiom,
    ! [A: $tType,Aa2: set(A),F2: fun(nat,A),N: nat] :
      ( ( Aa2 = aa(set(nat),set(A),image2(nat,A,F2),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_ac(nat,fun(nat,$o)),N))) )
     => aa(set(A),$o,finite_finite2(A),Aa2) ) ).

% nat_seg_image_imp_finite
tff(fact_3745_finite__conv__nat__seg__image,axiom,
    ! [A: $tType,Aa2: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),Aa2)
    <=> ? [N2: nat,F9: fun(nat,A)] : ( Aa2 = aa(set(nat),set(A),image2(nat,A,F9),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_ac(nat,fun(nat,$o)),N2))) ) ) ).

% finite_conv_nat_seg_image
tff(fact_3746_lists__eq__set,axiom,
    ! [A: $tType,Aa2: set(A)] : ( aa(set(A),set(list(A)),lists(A),Aa2) = aa(fun(list(A),$o),set(list(A)),collect(list(A)),aTP_Lamp_cy(set(A),fun(list(A),$o),Aa2)) ) ).

% lists_eq_set
tff(fact_3747_reflcl__set__eq,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),X4: A,Xa4: A] :
      ( aa(A,$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,$o))),aTP_Lamp_cz(set(product_prod(A,A)),fun(A,fun(A,$o)),R2)),fequal(A)),X4),Xa4)
    <=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Xa4),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_3748_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),$o),aTP_Lamp_da(set(A),fun(set(A),fun(set(A),$o)),S),S4))) ) ).

% finite_if_eq_beyond_finite
tff(fact_3749_SUP__UN__eq2,axiom,
    ! [A: $tType,B: $tType,E: $tType,R2: fun(E,set(product_prod(A,B))),S: set(E),X4: A,Xa4: B] :
      ( aa(B,$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(E),set(fun(A,fun(B,$o))),image2(E,fun(A,fun(B,$o)),aTP_Lamp_db(fun(E,set(product_prod(A,B))),fun(E,fun(A,fun(B,$o))),R2)),S)),X4),Xa4)
    <=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Xa4),aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),aa(set(E),set(set(product_prod(A,B))),image2(E,set(product_prod(A,B)),R2),S))) ) ).

% SUP_UN_eq2
tff(fact_3750_distinct__finite__set,axiom,
    ! [A: $tType,Xa: set(A)] : aa(set(list(A)),$o,finite_finite2(list(A)),aa(fun(list(A),$o),set(list(A)),collect(list(A)),aTP_Lamp_dc(set(A),fun(list(A),$o),Xa))) ).

% distinct_finite_set
tff(fact_3751_SUP__UN__eq,axiom,
    ! [A: $tType,B: $tType,R2: fun(B,set(A)),S: set(B),X4: A] :
      ( aa(A,$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,$o),aTP_Lamp_dd(fun(B,set(A)),fun(B,fun(A,$o)),R2)),S)),X4)
    <=> member(A,X4,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_3752_SUP__constant,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [C2: A,Aa2: set(B)] :
          ( aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,aTP_Lamp_de(A,fun(B,A),C2)),Aa2)) = $ite(Aa2 = bot_bot(set(B)),bot_bot(A),C2) ) ) ).

% SUP_constant
tff(fact_3753_SUP__empty,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(B,A)] : ( aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F2),bot_bot(set(B)))) = bot_bot(A) ) ) ).

% SUP_empty
tff(fact_3754_add__mset__in__multiset,axiom,
    ! [A: $tType,M5: fun(A,nat),A3: A] :
      ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_be(fun(A,nat),fun(A,$o),M5)))
     => aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(A,fun(A,$o),aTP_Lamp_df(fun(A,nat),fun(A,fun(A,$o)),M5),A3))) ) ).

% add_mset_in_multiset
tff(fact_3755_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),numeral_numeral(nat,bit0(W2))) = $let(
              w: A,
              w:= aa(nat,A,aa(A,fun(nat,A),power_power(A),Z2),numeral_numeral(nat,W2)),
              aa(A,A,aa(A,fun(A,A),times_times(A),w),w) ) ) ) ).

% power_numeral_even
tff(fact_3756_SUP__insert,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(B,A),A3: B,Aa2: set(B)] : ( aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F2),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert3(B),A3),Aa2))) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(B,A,F2,A3)),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F2),Aa2))) ) ) ).

% SUP_insert
tff(fact_3757_diff__preserves__multiset,axiom,
    ! [A: $tType,M5: fun(A,nat),N7: fun(A,nat)] :
      ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_be(fun(A,nat),fun(A,$o),M5)))
     => aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,nat),fun(A,$o),aTP_Lamp_dg(fun(A,nat),fun(fun(A,nat),fun(A,$o)),M5),N7))) ) ).

% diff_preserves_multiset
tff(fact_3758_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),numeral_numeral(nat,bit1(W2))) = $let(
              w: A,
              w:= aa(nat,A,aa(A,fun(nat,A),power_power(A),Z2),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_3759_prod_Osetdiff__irrelevant,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [Aa2: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( 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)),Aa2),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_dh(fun(A,B),fun(A,$o),G)))) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),Aa2) ) ) ) ).

% prod.setdiff_irrelevant
tff(fact_3760_UN__extend__simps_I1_J,axiom,
    ! [A: $tType,B: $tType,A3: A,B2: fun(B,set(A)),C3: set(B)] :
      ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B2),C3))) = $ite(C3 = bot_bot(set(B)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(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_au(A,fun(fun(B,set(A)),fun(B,set(A))),A3),B2)),C3))) ) ).

% UN_extend_simps(1)
tff(fact_3761_UN__extend__simps_I3_J,axiom,
    ! [A: $tType,B: $tType,Aa2: set(A),B2: fun(B,set(A)),C3: set(B)] :
      ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B2),C3))) = $ite(C3 = bot_bot(set(B)),Aa2,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_av(set(A),fun(fun(B,set(A)),fun(B,set(A))),Aa2),B2)),C3))) ) ).

% UN_extend_simps(3)
tff(fact_3762_UN__extend__simps_I2_J,axiom,
    ! [A: $tType,B: $tType,Aa2: fun(B,set(A)),C3: set(B),B2: 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),Aa2),C3))),B2) = $ite(C3 = bot_bot(set(B)),B2,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_aw(fun(B,set(A)),fun(set(A),fun(B,set(A))),Aa2),B2)),C3))) ) ).

% UN_extend_simps(2)
tff(fact_3763_finite__divisors__nat,axiom,
    ! [M2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M2)
     => aa(set(nat),$o,finite_finite2(nat),aa(fun(nat,$o),set(nat),collect(nat),aTP_Lamp_di(nat,fun(nat,$o),M2))) ) ).

% finite_divisors_nat
tff(fact_3764_set__list__bind,axiom,
    ! [A: $tType,B: $tType,Xsa: list(B),F2: fun(B,list(A))] : ( aa(list(A),set(A),set2(A),bind(B,A,Xsa,F2)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aTP_Lamp_dj(fun(B,list(A)),fun(B,set(A)),F2)),aa(list(B),set(B),set2(B),Xsa))) ) ).

% set_list_bind
tff(fact_3765_Max__in,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Aa2: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( ( Aa2 != bot_bot(set(A)) )
           => member(A,lattic643756798349783984er_Max(A,Aa2),Aa2) ) ) ) ).

% Max_in
tff(fact_3766_UNION__singleton__eq__range,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Aa2: 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_dk(fun(B,A),fun(B,set(A)),F2)),Aa2)) = aa(set(B),set(A),image2(B,A,F2),Aa2) ) ).

% UNION_singleton_eq_range
tff(fact_3767_Min__add__commute,axiom,
    ! [A: $tType,B: $tType] :
      ( linord4140545234300271783up_add(B)
     => ! [S: set(A),F2: fun(A,B),K3: B] :
          ( aa(set(A),$o,finite_finite2(A),S)
         => ( ( S != bot_bot(set(A)) )
           => ( lattic643756798350308766er_Min(B,aa(set(A),set(B),image2(A,B,aa(B,fun(A,B),aTP_Lamp_bj(fun(A,B),fun(B,fun(A,B)),F2),K3)),S)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),lattic643756798350308766er_Min(B,aa(set(A),set(B),image2(A,B,F2),S))),K3) ) ) ) ) ).

% Min_add_commute
tff(fact_3768_prod_OSuc__reindex__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M2: nat,N: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),N)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M2,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,M2)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_dl(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,M2,N))) ) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_3769_prod_OIf__cases,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [Aa2: set(A),Pa: fun(A,$o),H2: fun(A,B),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_dm(fun(A,$o),fun(fun(A,B),fun(fun(A,B),fun(A,B))),Pa),H2),G)),Aa2) = 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)),Aa2),aa(fun(A,$o),set(A),collect(A),Pa)))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),aa(set(A),set(A),uminus_uminus(set(A)),aa(fun(A,$o),set(A),collect(A),Pa))))) ) ) ) ).

% prod.If_cases
tff(fact_3770_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_dl(fun(nat,A),fun(nat,A),G)),aa(nat,set(nat),set_ord_atMost(nat),N))) ) ) ).

% prod.atMost_Suc_shift
tff(fact_3771_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_dl(fun(nat,A),fun(nat,A),G)),aa(nat,set(nat),set_ord_lessThan(nat),N))) ) ) ).

% prod.lessThan_Suc_shift
tff(fact_3772_sup__bot_Osemilattice__neutr__order__axioms,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => semila1105856199041335345_order(A,sup_sup(A),bot_bot(A),aTP_Lamp_dn(A,fun(A,$o)),aTP_Lamp_do(A,fun(A,$o))) ) ).

% sup_bot.semilattice_neutr_order_axioms
tff(fact_3773_finite__lists__length__eq,axiom,
    ! [A: $tType,Aa2: set(A),N: nat] :
      ( aa(set(A),$o,finite_finite2(A),Aa2)
     => 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),$o),aTP_Lamp_dp(set(A),fun(nat,fun(list(A),$o)),Aa2),N))) ) ).

% finite_lists_length_eq
tff(fact_3774_wf__finite__segments,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( irrefl(A,R2)
     => ( trans(A,R2)
       => ( ! [X: A] : aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(A,fun(A,$o),aTP_Lamp_dq(set(product_prod(A,A)),fun(A,fun(A,$o)),R2),X)))
         => wf(A,R2) ) ) ) ).

% wf_finite_segments
tff(fact_3775_slice__prepend,axiom,
    ! [A: $tType,Ia: nat,K3: nat,Xsa: list(A),Ys: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ia),K3)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K3),aa(list(A),nat,size_size(list(A)),Xsa))
       => $let(
            p2: nat,
            p2:= aa(list(A),nat,size_size(list(A)),Ys),
            slice(A,Ia,K3,Xsa) = slice(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ia),p2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K3),p2),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Xsa)) ) ) ) ).

% slice_prepend
tff(fact_3776_distinct__finite__subset,axiom,
    ! [A: $tType,Xa: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),Xa)
     => aa(set(list(A)),$o,finite_finite2(list(A)),aa(fun(list(A),$o),set(list(A)),collect(list(A)),aTP_Lamp_dr(set(A),fun(list(A),$o),Xa))) ) ).

% distinct_finite_subset
tff(fact_3777_prod__atLeastAtMost__code,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [F2: fun(nat,A),A3: nat,B3: nat] : ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),F2),set_or1337092689740270186AtMost(nat,A3,B3)) = set_fo6178422350223883121st_nat(A,aTP_Lamp_ds(fun(nat,A),fun(nat,fun(A,A)),F2),A3,B3,one_one(A)) ) ) ).

% prod_atLeastAtMost_code
tff(fact_3778_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_dt(nat,fun(nat,nat),N)),top_top(set(nat))) = set_or7035219750837199246ssThan(nat,zero_zero(nat),N) ) ) ).

% range_mod
tff(fact_3779_zip__Cons,axiom,
    ! [B: $tType,A: $tType,Xsa: list(A),Y: B,Ys: list(B)] : ( aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),Xsa),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),Ys)) = aa(list(A),list(product_prod(A,B)),aa(fun(A,fun(list(A),list(product_prod(A,B)))),fun(list(A),list(product_prod(A,B))),aa(list(product_prod(A,B)),fun(fun(A,fun(list(A),list(product_prod(A,B)))),fun(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_du(B,fun(list(B),fun(A,fun(list(A),list(product_prod(A,B))))),Y),Ys)),Xsa) ) ).

% zip_Cons
tff(fact_3780_zip__Cons1,axiom,
    ! [A: $tType,B: $tType,Xa: A,Xsa: list(A),Ys: list(B)] : ( aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa)),Ys) = aa(list(B),list(product_prod(A,B)),aa(fun(B,fun(list(B),list(product_prod(A,B)))),fun(list(B),list(product_prod(A,B))),aa(list(product_prod(A,B)),fun(fun(B,fun(list(B),list(product_prod(A,B)))),fun(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_dv(A,fun(list(A),fun(B,fun(list(B),list(product_prod(A,B))))),Xa),Xsa)),Ys) ) ).

% zip_Cons1
tff(fact_3781_set__n__lists,axiom,
    ! [A: $tType,N: nat,Xsa: list(A)] : ( aa(list(list(A)),set(list(A)),set2(list(A)),n_lists(A,N,Xsa)) = aa(fun(list(A),$o),set(list(A)),collect(list(A)),aa(list(A),fun(list(A),$o),aTP_Lamp_dw(nat,fun(list(A),fun(list(A),$o)),N),Xsa)) ) ).

% set_n_lists
tff(fact_3782_prod_Odelta__remove,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [S: set(A),A3: A,B3: fun(A,B),C2: fun(A,B)] :
          ( aa(set(A),$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_dx(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),A3),B3),C2)),S) = $ite(member(A,A3,S),aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,B3,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)),insert3(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)),insert3(A),A3),bot_bot(set(A)))))) ) ) ) ).

% prod.delta_remove
tff(fact_3783_prod_OUNION__disjoint,axiom,
    ! [B: $tType,E: $tType,A: $tType] :
      ( comm_monoid_mult(E)
     => ! [Ia2: set(A),Aa2: fun(A,set(B)),G: fun(B,E)] :
          ( aa(set(A),$o,finite_finite2(A),Ia2)
         => ( ! [X: A] :
                ( member(A,X,Ia2)
               => aa(set(B),$o,finite_finite2(B),aa(A,set(B),Aa2,X)) )
           => ( ! [X: A] :
                  ( member(A,X,Ia2)
                 => ! [Xa5: A] :
                      ( member(A,Xa5,Ia2)
                     => ( ( X != Xa5 )
                       => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(A,set(B),Aa2,X)),aa(A,set(B),Aa2,Xa5)) = bot_bot(set(B)) ) ) ) )
             => ( aa(set(B),E,aa(fun(B,E),fun(set(B),E),groups7121269368397514597t_prod(B,E),G),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),Aa2),Ia2))) = aa(set(A),E,aa(fun(A,E),fun(set(A),E),groups7121269368397514597t_prod(A,E),aa(fun(B,E),fun(A,E),aTP_Lamp_dy(fun(A,set(B)),fun(fun(B,E),fun(A,E)),Aa2),G)),Ia2) ) ) ) ) ) ).

% prod.UNION_disjoint
tff(fact_3784_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_dl(fun(nat,A),fun(nat,A),G)),aa(nat,set(nat),set_ord_lessThan(nat),N))) ) ) ).

% prod.atMost_shift
tff(fact_3785_finite__lists__length__le,axiom,
    ! [A: $tType,Aa2: set(A),N: nat] :
      ( aa(set(A),$o,finite_finite2(A),Aa2)
     => 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),$o),aTP_Lamp_dz(set(A),fun(nat,fun(list(A),$o)),Aa2),N))) ) ).

% finite_lists_length_le
tff(fact_3786_lists__of__len__fin1,axiom,
    ! [A: $tType,Pa: set(A),N: nat] :
      ( aa(set(A),$o,finite_finite2(A),Pa)
     => aa(set(list(A)),$o,finite_finite2(list(A)),aa(set(list(A)),set(list(A)),aa(set(list(A)),fun(set(list(A)),set(list(A))),inf_inf(set(list(A))),aa(set(A),set(list(A)),lists(A),Pa)),aa(fun(list(A),$o),set(list(A)),collect(list(A)),aTP_Lamp_ea(nat,fun(list(A),$o),N)))) ) ).

% lists_of_len_fin1
tff(fact_3787_lists__of__len__fin2,axiom,
    ! [A: $tType,Pa: set(A),N: nat] :
      ( aa(set(A),$o,finite_finite2(A),Pa)
     => aa(set(list(A)),$o,finite_finite2(list(A)),aa(set(list(A)),set(list(A)),aa(set(list(A)),fun(set(list(A)),set(list(A))),inf_inf(set(list(A))),aa(set(A),set(list(A)),lists(A),Pa)),aa(fun(list(A),$o),set(list(A)),collect(list(A)),aTP_Lamp_eb(nat,fun(list(A),$o),N)))) ) ).

% lists_of_len_fin2
tff(fact_3788_foldl__set,axiom,
    ! [A: $tType,L: list(set(A))] : ( aa(list(set(A)),set(A),aa(set(A),fun(list(set(A)),set(A)),aa(fun(set(A),fun(set(A),set(A))),fun(set(A),fun(list(set(A)),set(A))),foldl(set(A),set(A)),sup_sup(set(A))),bot_bot(set(A))),L) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(fun(set(A),$o),set(set(A)),collect(set(A)),aTP_Lamp_ec(list(set(A)),fun(set(A),$o),L))) ) ).

% foldl_set
tff(fact_3789_Pow__set_I2_J,axiom,
    ! [A: $tType,Xa: A,Xsa: list(A)] :
      ( pow(A,aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa))) = $let(
          a3: set(set(A)),
          a3:= pow(A,aa(list(A),set(A),set2(A),Xsa)),
          aa(set(set(A)),set(set(A)),aa(set(set(A)),fun(set(set(A)),set(set(A))),sup_sup(set(set(A))),a3),aa(set(set(A)),set(set(A)),image2(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa)),a3)) ) ) ).

% Pow_set(2)
tff(fact_3790_image__mult__atLeastAtMost__if_H,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,Xa: A,Y: A] :
          ( aa(set(A),set(A),image2(A,A,aTP_Lamp_ed(A,fun(A,A),C2)),set_or1337092689740270186AtMost(A,Xa,Y)) = $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),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),Xa),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),Xa),C2))),
              bot_bot(set(A)) ) ) ) ).

% image_mult_atLeastAtMost_if'
tff(fact_3791_image__affinity__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [M2: A,C2: A,A3: A,B3: A] :
          ( aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),aTP_Lamp_ee(A,fun(A,fun(A,A)),M2),C2)),set_or1337092689740270186AtMost(A,A3,B3)) = $ite(
              set_or1337092689740270186AtMost(A,A3,B3) = bot_bot(set(A)),
              bot_bot(set(A)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),M2),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M2),A3)),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M2),B3)),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),M2),B3)),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M2),A3)),C2))) ) ) ) ).

% image_affinity_atLeastAtMost
tff(fact_3792_image__affinity__atLeastAtMost__diff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [M2: A,C2: A,A3: A,B3: A] :
          ( aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),aTP_Lamp_ef(A,fun(A,fun(A,A)),M2),C2)),set_or1337092689740270186AtMost(A,A3,B3)) = $ite(
              set_or1337092689740270186AtMost(A,A3,B3) = bot_bot(set(A)),
              bot_bot(set(A)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),M2),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M2),A3)),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M2),B3)),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),M2),B3)),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M2),A3)),C2))) ) ) ) ).

% image_affinity_atLeastAtMost_diff
tff(fact_3793_image__affinity__atLeastAtMost__div,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [M2: A,C2: A,A3: A,B3: A] :
          ( aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),aTP_Lamp_eg(A,fun(A,fun(A,A)),M2),C2)),set_or1337092689740270186AtMost(A,A3,B3)) = $ite(
              set_or1337092689740270186AtMost(A,A3,B3) = bot_bot(set(A)),
              bot_bot(set(A)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),M2),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,A3,M2)),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,B3,M2)),C2)),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,B3,M2)),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,A3,M2)),C2))) ) ) ) ).

% image_affinity_atLeastAtMost_div
tff(fact_3794_image__affinity__atLeastAtMost__div__diff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [M2: A,C2: A,A3: A,B3: A] :
          ( aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),aTP_Lamp_eh(A,fun(A,fun(A,A)),M2),C2)),set_or1337092689740270186AtMost(A,A3,B3)) = $ite(
              set_or1337092689740270186AtMost(A,A3,B3) = bot_bot(set(A)),
              bot_bot(set(A)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),M2),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),divide_divide(A,A3,M2)),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),divide_divide(A,B3,M2)),C2)),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),divide_divide(A,B3,M2)),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),divide_divide(A,A3,M2)),C2))) ) ) ) ).

% image_affinity_atLeastAtMost_div_diff
tff(fact_3795_prod__gen__delta,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [S: set(A),A3: A,B3: fun(A,B),C2: B] :
          ( aa(set(A),$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_ei(A,fun(fun(A,B),fun(B,fun(A,B))),A3),B3),C2)),S) = $ite(member(A,A3,S),aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,B3,A3)),aa(nat,B,aa(B,fun(nat,B),power_power(B),C2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),finite_card(A,S)),one_one(nat)))),aa(nat,B,aa(B,fun(nat,B),power_power(B),C2),finite_card(A,S))) ) ) ) ).

% prod_gen_delta
tff(fact_3796_finite__mono__strict__prefix__implies__finite__fixpoint,axiom,
    ! [A: $tType,F2: fun(nat,set(A)),S: set(A)] :
      ( ! [I3: nat] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(nat,set(A),F2,I3)),S)
     => ( aa(set(A),$o,finite_finite2(A),S)
       => ( ? [N9: nat] :
              ( ! [N5: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N5),N9)
                 => ! [M6: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M6),N9)
                     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M6),N5)
                       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),aa(nat,set(A),F2,M6)),aa(nat,set(A),F2,N5)) ) ) )
              & ! [N5: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N9),N5)
                 => ( aa(nat,set(A),F2,N9) = aa(nat,set(A),F2,N5) ) ) )
         => ( aa(nat,set(A),F2,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),F2),top_top(set(nat)))) ) ) ) ) ).

% finite_mono_strict_prefix_implies_finite_fixpoint
tff(fact_3797_Max_OboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Aa2: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( ( Aa2 != bot_bot(set(A)) )
           => ( ! [A4: A] :
                  ( member(A,A4,Aa2)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),Xa) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),lattic643756798349783984er_Max(A,Aa2)),Xa) ) ) ) ) ).

% Max.boundedI
tff(fact_3798_Max_OboundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Aa2: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( ( Aa2 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),lattic643756798349783984er_Max(A,Aa2)),Xa)
             => ! [A13: A] :
                  ( member(A,A13,Aa2)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A13),Xa) ) ) ) ) ) ).

% Max.boundedE
tff(fact_3799_eq__Max__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Aa2: set(A),M2: A] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( ( Aa2 != bot_bot(set(A)) )
           => ( ( M2 = lattic643756798349783984er_Max(A,Aa2) )
            <=> ( member(A,M2,Aa2)
                & ! [X6: A] :
                    ( member(A,X6,Aa2)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X6),M2) ) ) ) ) ) ) ).

% eq_Max_iff
tff(fact_3800_Max__ge__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Aa2: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( ( Aa2 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),lattic643756798349783984er_Max(A,Aa2))
            <=> ? [X6: A] :
                  ( member(A,X6,Aa2)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),X6) ) ) ) ) ) ).

% Max_ge_iff
tff(fact_3801_Max__eq__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Aa2: set(A),M2: A] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( ( Aa2 != bot_bot(set(A)) )
           => ( ( lattic643756798349783984er_Max(A,Aa2) = M2 )
            <=> ( member(A,M2,Aa2)
                & ! [X6: A] :
                    ( member(A,X6,Aa2)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X6),M2) ) ) ) ) ) ) ).

% Max_eq_iff
tff(fact_3802_Max__gr__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Aa2: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( ( Aa2 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),lattic643756798349783984er_Max(A,Aa2))
            <=> ? [X6: A] :
                  ( member(A,X6,Aa2)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),X6) ) ) ) ) ) ).

% Max_gr_iff
tff(fact_3803_Max__insert2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Aa2: set(A),A3: A] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( ! [B4: A] :
                ( member(A,B4,Aa2)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B4),A3) )
           => ( lattic643756798349783984er_Max(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),Aa2)) = A3 ) ) ) ) ).

% Max_insert2
tff(fact_3804_cSup__eq__Max,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X3: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),X3)
         => ( ( X3 != bot_bot(set(A)) )
           => ( aa(set(A),A,complete_Sup_Sup(A),X3) = lattic643756798349783984er_Max(A,X3) ) ) ) ) ).

% cSup_eq_Max
tff(fact_3805_Max__Sup,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [Aa2: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( ( Aa2 != bot_bot(set(A)) )
           => ( lattic643756798349783984er_Max(A,Aa2) = aa(set(A),A,complete_Sup_Sup(A),Aa2) ) ) ) ) ).

% Max_Sup
tff(fact_3806_card__lists__length__eq,axiom,
    ! [A: $tType,Aa2: set(A),N: nat] :
      ( aa(set(A),$o,finite_finite2(A),Aa2)
     => ( finite_card(list(A),aa(fun(list(A),$o),set(list(A)),collect(list(A)),aa(nat,fun(list(A),$o),aTP_Lamp_dp(set(A),fun(nat,fun(list(A),$o)),Aa2),N))) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),finite_card(A,Aa2)),N) ) ) ).

% card_lists_length_eq
tff(fact_3807_image__minus__const__atLeastLessThan__nat,axiom,
    ! [C2: nat,Xa: nat,Y: nat] :
      ( aa(set(nat),set(nat),image2(nat,nat,aTP_Lamp_ej(nat,fun(nat,nat),C2)),set_or7035219750837199246ssThan(nat,Xa,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),Xa),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),Xa),Y),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert3(nat),zero_zero(nat)),bot_bot(set(nat))),bot_bot(set(nat))) ) ) ).

% image_minus_const_atLeastLessThan_nat
tff(fact_3808_prod_Oin__pairs,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M2: 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),numeral_numeral(nat,bit0(one2))),M2),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),numeral_numeral(nat,bit0(one2))),N)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_ek(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,M2,N)) ) ) ).

% prod.in_pairs
tff(fact_3809_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),numeral_numeral(nat,bit0(one2))),N)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_ek(fun(nat,A),fun(nat,A),G)),aa(nat,set(nat),set_ord_atMost(nat),N)) ) ) ).

% prod.in_pairs_0
tff(fact_3810_gbinomial__altdef__of__nat,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K3: nat] : ( aa(nat,A,gbinomial(A,A3),K3) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_el(A,fun(nat,fun(nat,A)),A3),K3)),set_or7035219750837199246ssThan(nat,zero_zero(nat),K3)) ) ) ).

% gbinomial_altdef_of_nat
tff(fact_3811_gbinomial__mult__fact_H,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K3: nat] : ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,A3),K3)),semiring_char_0_fact(A,K3)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_em(A,fun(nat,A),A3)),set_or7035219750837199246ssThan(nat,zero_zero(nat),K3)) ) ) ).

% gbinomial_mult_fact'
tff(fact_3812_gbinomial__mult__fact,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K3: nat,A3: A] : ( aa(A,A,aa(A,fun(A,A),times_times(A),semiring_char_0_fact(A,K3)),aa(nat,A,gbinomial(A,A3),K3)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_em(A,fun(nat,A),A3)),set_or7035219750837199246ssThan(nat,zero_zero(nat),K3)) ) ) ).

% gbinomial_mult_fact
tff(fact_3813_gbinomial__prod__rev,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [A3: A,K3: nat] : ( aa(nat,A,gbinomial(A,A3),K3) = divide_divide(A,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_en(A,fun(nat,A),A3)),set_or7035219750837199246ssThan(nat,zero_zero(nat),K3)),semiring_char_0_fact(A,K3)) ) ) ).

% gbinomial_prod_rev
tff(fact_3814_arg__min__list_Osimps_I1_J,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),Xa: A] : ( arg_min_list(A,B,F2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),nil(A))) = Xa ) ) ).

% arg_min_list.simps(1)
tff(fact_3815_arg__min__list__in,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Xsa: list(A),F2: fun(A,B)] :
          ( ( Xsa != nil(A) )
         => member(A,arg_min_list(A,B,F2,Xsa),aa(list(A),set(A),set2(A),Xsa)) ) ) ).

% arg_min_list_in
tff(fact_3816_Sup__nat__def,axiom,
    ! [X3: set(nat)] :
      ( aa(set(nat),nat,complete_Sup_Sup(nat),X3) = $ite(X3 = bot_bot(set(nat)),zero_zero(nat),lattic643756798349783984er_Max(nat,X3)) ) ).

% Sup_nat_def
tff(fact_3817_prod_Ozero__middle,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [P2: nat,K3: nat,G: fun(nat,A),H2: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),P2)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K3),P2)
           => ( 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_eo(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K3),G),H2)),aa(nat,set(nat),set_ord_atMost(nat),P2)) = 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_ep(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K3),G),H2)),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),P2),aa(nat,nat,suc,zero_zero(nat))))) ) ) ) ) ).

% prod.zero_middle
tff(fact_3818_gbinomial__Suc,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [A3: A,K3: nat] : ( aa(nat,A,gbinomial(A,A3),aa(nat,nat,suc,K3)) = divide_divide(A,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_en(A,fun(nat,A),A3)),set_or1337092689740270186AtMost(nat,zero_zero(nat),K3)),semiring_char_0_fact(A,aa(nat,nat,suc,K3))) ) ) ).

% gbinomial_Suc
tff(fact_3819_Max_Osubset__imp,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Aa2: set(A),B2: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),B2)
         => ( ( Aa2 != bot_bot(set(A)) )
           => ( aa(set(A),$o,finite_finite2(A),B2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),lattic643756798349783984er_Max(A,Aa2)),lattic643756798349783984er_Max(A,B2)) ) ) ) ) ).

% Max.subset_imp
tff(fact_3820_Max__mono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [M5: set(A),N7: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),M5),N7)
         => ( ( M5 != bot_bot(set(A)) )
           => ( aa(set(A),$o,finite_finite2(A),N7)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),lattic643756798349783984er_Max(A,M5)),lattic643756798349783984er_Max(A,N7)) ) ) ) ) ).

% Max_mono
tff(fact_3821_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_eq(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_3822_gbinomial__code,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K3: nat] :
          ( aa(nat,A,gbinomial(A,A3),K3) = $ite(K3 = zero_zero(nat),one_one(A),divide_divide(A,set_fo6178422350223883121st_nat(A,aTP_Lamp_er(A,fun(nat,fun(A,A)),A3),zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K3),one_one(nat)),one_one(A)),semiring_char_0_fact(A,K3))) ) ) ).

% gbinomial_code
tff(fact_3823_card__lists__distinct__length__eq_H,axiom,
    ! [A: $tType,K3: nat,Aa2: set(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K3),finite_card(A,Aa2))
     => ( finite_card(list(A),aa(fun(list(A),$o),set(list(A)),collect(list(A)),aa(set(A),fun(list(A),$o),aTP_Lamp_es(nat,fun(set(A),fun(list(A),$o)),K3),Aa2))) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),aTP_Lamp_et(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),finite_card(A,Aa2)),K3)),one_one(nat)),finite_card(A,Aa2))) ) ) ).

% card_lists_distinct_length_eq'
tff(fact_3824_card__lists__distinct__length__eq,axiom,
    ! [A: $tType,Aa2: set(A),K3: nat] :
      ( aa(set(A),$o,finite_finite2(A),Aa2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K3),finite_card(A,Aa2))
       => ( finite_card(list(A),aa(fun(list(A),$o),set(list(A)),collect(list(A)),aa(nat,fun(list(A),$o),aTP_Lamp_az(set(A),fun(nat,fun(list(A),$o)),Aa2),K3))) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),aTP_Lamp_et(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),finite_card(A,Aa2)),K3)),one_one(nat)),finite_card(A,Aa2))) ) ) ) ).

% card_lists_distinct_length_eq
tff(fact_3825_set__Cons__sing__Nil,axiom,
    ! [A: $tType,Aa2: set(A)] : ( aa(set(list(A)),set(list(A)),aa(set(A),fun(set(list(A)),set(list(A))),set_Cons(A),Aa2),aa(set(list(A)),set(list(A)),aa(list(A),fun(set(list(A)),set(list(A))),insert3(list(A)),nil(A)),bot_bot(set(list(A))))) = aa(set(A),set(list(A)),image2(A,list(A),aTP_Lamp_eu(A,list(A))),Aa2) ) ).

% set_Cons_sing_Nil
tff(fact_3826_same__fst__trancl,axiom,
    ! [B: $tType,A: $tType,Pa: fun(A,$o),R: fun(A,set(product_prod(B,B)))] : ( transitive_trancl(product_prod(A,B),same_fst(A,B,Pa,R)) = same_fst(A,B,Pa,aTP_Lamp_ev(fun(A,set(product_prod(B,B))),fun(A,set(product_prod(B,B))),R)) ) ).

% same_fst_trancl
tff(fact_3827_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),numeral_numeral(A,bit0(one2))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_ew(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),numeral_numeral(A,bit0(one2))),N) ) ) ) ).

% choose_even_sum
tff(fact_3828_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),numeral_numeral(A,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),numeral_numeral(A,bit0(one2))),N) ) ) ) ).

% choose_odd_sum
tff(fact_3829_predicate2I,axiom,
    ! [B: $tType,A: $tType,Pa: fun(A,fun(B,$o)),Qa: fun(A,fun(B,$o))] :
      ( ! [X: A,Y2: B] :
          ( aa(B,$o,aa(A,fun(B,$o),Pa,X),Y2)
         => aa(B,$o,aa(A,fun(B,$o),Qa,X),Y2) )
     => aa(fun(A,fun(B,$o)),$o,aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),$o),ord_less_eq(fun(A,fun(B,$o))),Pa),Qa) ) ).

% predicate2I
tff(fact_3830_predicate1I,axiom,
    ! [A: $tType,Pa: fun(A,$o),Qa: fun(A,$o)] :
      ( ! [X: A] :
          ( aa(A,$o,Pa,X)
         => aa(A,$o,Qa,X) )
     => aa(fun(A,$o),$o,aa(fun(A,$o),fun(fun(A,$o),$o),ord_less_eq(fun(A,$o)),Pa),Qa) ) ).

% predicate1I
tff(fact_3831_top2I,axiom,
    ! [A: $tType,B: $tType,Xa: A,Y: B] : aa(B,$o,aa(A,fun(B,$o),top_top(fun(A,fun(B,$o))),Xa),Y) ).

% top2I
tff(fact_3832_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_3833_sum_Oinsert,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [Aa2: set(A),Xa: A,G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( ~ member(A,Xa,Aa2)
           => ( 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)),insert3(A),Xa),Aa2)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,G,Xa)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),Aa2)) ) ) ) ) ).

% sum.insert
tff(fact_3834_sum__constant,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(A)
     => ! [Y: A,Aa2: set(B)] : ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aTP_Lamp_ey(A,fun(B,A),Y)),Aa2) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),finite_card(B,Aa2))),Y) ) ) ).

% sum_constant
tff(fact_3835_sum__of__bool__mult__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(B)
     => ! [Aa2: set(A),Pa: fun(A,$o),F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aa(fun(A,B),fun(A,B),aTP_Lamp_ez(fun(A,$o),fun(fun(A,B),fun(A,B)),Pa),F2)),Aa2) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),aa(fun(A,$o),set(A),collect(A),Pa))) ) ) ) ).

% sum_of_bool_mult_eq
tff(fact_3836_sum__mult__of__bool__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(B)
     => ! [Aa2: set(A),F2: fun(A,B),Pa: fun(A,$o)] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aa(fun(A,$o),fun(A,B),aTP_Lamp_fa(fun(A,B),fun(fun(A,$o),fun(A,B)),F2),Pa)),Aa2) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),aa(fun(A,$o),set(A),collect(A),Pa))) ) ) ) ).

% sum_mult_of_bool_eq
tff(fact_3837_sum_Oop__ivl__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M2: nat,N: nat] :
          ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,M2,aa(nat,nat,suc,N))) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),M2),zero_zero(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,M2,N))),aa(nat,A,G,N))) ) ) ).

% sum.op_ivl_Suc
tff(fact_3838_sum_Ocl__ivl__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M2: nat,N: nat] :
          ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M2,aa(nat,nat,suc,N))) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,N)),M2),zero_zero(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M2,N))),aa(nat,A,G,aa(nat,nat,suc,N)))) ) ) ).

% sum.cl_ivl_Suc
tff(fact_3839_sum__zero__power,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [C2: fun(nat,A),Aa2: set(nat)] :
          ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_fb(fun(nat,A),fun(nat,A),C2)),Aa2) = $ite(
              ( aa(set(nat),$o,finite_finite2(nat),Aa2)
              & member(nat,zero_zero(nat),Aa2) ),
              aa(nat,A,C2,zero_zero(nat)),
              zero_zero(A) ) ) ) ).

% sum_zero_power
tff(fact_3840_sum__zero__power_H,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: fun(nat,A),D3: fun(nat,A),Aa2: 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_fc(fun(nat,A),fun(fun(nat,A),fun(nat,A)),C2),D3)),Aa2) = $ite(
              ( aa(set(nat),$o,finite_finite2(nat),Aa2)
              & member(nat,zero_zero(nat),Aa2) ),
              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_3841_predicate1D,axiom,
    ! [A: $tType,Pa: fun(A,$o),Qa: fun(A,$o),Xa: A] :
      ( aa(fun(A,$o),$o,aa(fun(A,$o),fun(fun(A,$o),$o),ord_less_eq(fun(A,$o)),Pa),Qa)
     => ( aa(A,$o,Pa,Xa)
       => aa(A,$o,Qa,Xa) ) ) ).

% predicate1D
tff(fact_3842_predicate2D,axiom,
    ! [A: $tType,B: $tType,Pa: fun(A,fun(B,$o)),Qa: fun(A,fun(B,$o)),Xa: A,Y: B] :
      ( aa(fun(A,fun(B,$o)),$o,aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),$o),ord_less_eq(fun(A,fun(B,$o))),Pa),Qa)
     => ( aa(B,$o,aa(A,fun(B,$o),Pa,Xa),Y)
       => aa(B,$o,aa(A,fun(B,$o),Qa,Xa),Y) ) ) ).

% predicate2D
tff(fact_3843_rev__predicate1D,axiom,
    ! [A: $tType,Pa: fun(A,$o),Xa: A,Qa: fun(A,$o)] :
      ( aa(A,$o,Pa,Xa)
     => ( aa(fun(A,$o),$o,aa(fun(A,$o),fun(fun(A,$o),$o),ord_less_eq(fun(A,$o)),Pa),Qa)
       => aa(A,$o,Qa,Xa) ) ) ).

% rev_predicate1D
tff(fact_3844_rev__predicate2D,axiom,
    ! [A: $tType,B: $tType,Pa: fun(A,fun(B,$o)),Xa: A,Y: B,Qa: fun(A,fun(B,$o))] :
      ( aa(B,$o,aa(A,fun(B,$o),Pa,Xa),Y)
     => ( aa(fun(A,fun(B,$o)),$o,aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),$o),ord_less_eq(fun(A,fun(B,$o))),Pa),Qa)
       => aa(B,$o,aa(A,fun(B,$o),Qa,Xa),Y) ) ) ).

% rev_predicate2D
tff(fact_3845_sum__distrib__left,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_0(A)
     => ! [R2: A,F2: fun(B,A),Aa2: 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),F2),Aa2)) = 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_fd(A,fun(fun(B,A),fun(B,A)),R2),F2)),Aa2) ) ) ).

% sum_distrib_left
tff(fact_3846_sum__distrib__right,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_0(A)
     => ! [F2: fun(B,A),Aa2: 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),F2),Aa2)),R2) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(A,fun(B,A),aTP_Lamp_fe(fun(B,A),fun(A,fun(B,A)),F2),R2)),Aa2) ) ) ).

% sum_distrib_right
tff(fact_3847_sum__product,axiom,
    ! [E: $tType,A: $tType,B: $tType] :
      ( semiring_0(A)
     => ! [F2: fun(B,A),Aa2: set(B),G: fun(E,A),B2: set(E)] : ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),Aa2)),aa(set(E),A,aa(fun(E,A),fun(set(E),A),groups7311177749621191930dd_sum(E,A),G),B2)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(set(E),fun(B,A),aa(fun(E,A),fun(set(E),fun(B,A)),aTP_Lamp_fg(fun(B,A),fun(fun(E,A),fun(set(E),fun(B,A))),F2),G),B2)),Aa2) ) ) ).

% sum_product
tff(fact_3848_sum__subtractf,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_add(A)
     => ! [F2: fun(B,A),G: fun(B,A),Aa2: 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_fh(fun(B,A),fun(fun(B,A),fun(B,A)),F2),G)),Aa2) = 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),F2),Aa2)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),Aa2)) ) ) ).

% sum_subtractf
tff(fact_3849_sum__strict__mono__ex1,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere8940638589300402666id_add(B)
     => ! [Aa2: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( ! [X: A] :
                ( member(A,X,Aa2)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X)),aa(A,B,G,X)) )
           => ( ? [X4: A] :
                  ( member(A,X4,Aa2)
                  & aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,X4)),aa(A,B,G,X4)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),Aa2)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),Aa2)) ) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_3850_sum__strict__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( strict7427464778891057005id_add(B)
     => ! [Aa2: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( ( Aa2 != bot_bot(set(A)) )
           => ( ! [X: A] :
                  ( member(A,X,Aa2)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,X)),aa(A,B,G,X)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),Aa2)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),Aa2)) ) ) ) ) ).

% sum_strict_mono
tff(fact_3851_sum_Oinsert__if,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [Aa2: set(A),G: fun(A,B),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( 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)),insert3(A),Xa),Aa2)) = $ite(member(A,Xa,Aa2),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),Aa2),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,G,Xa)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),Aa2))) ) ) ) ).

% sum.insert_if
tff(fact_3852_sum_Oivl__cong,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ord(A)
        & comm_monoid_add(B) )
     => ! [A3: A,C2: A,B3: A,D3: A,G: fun(A,B),H2: fun(A,B)] :
          ( ( A3 = C2 )
         => ( ( B3 = D3 )
           => ( ! [X: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),X)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),D3)
                   => ( aa(A,B,G,X) = aa(A,B,H2,X) ) ) )
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),set_or7035219750837199246ssThan(A,A3,B3)) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),H2),set_or7035219750837199246ssThan(A,C2,D3)) ) ) ) ) ) ).

% sum.ivl_cong
tff(fact_3853_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [A: $tType,E: $tType,B: $tType] :
      ( comm_monoid_add(E)
     => ! [S3: set(A),T6: set(B),S: set(A),Ia: fun(B,A),J: fun(A,B),T4: set(B),G: fun(A,E),H2: fun(B,E)] :
          ( aa(set(A),$o,finite_finite2(A),S3)
         => ( aa(set(B),$o,finite_finite2(B),T6)
           => ( ! [A4: A] :
                  ( member(A,A4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S),S3))
                 => ( aa(B,A,Ia,aa(A,B,J,A4)) = A4 ) )
             => ( ! [A4: A] :
                    ( member(A,A4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S),S3))
                   => member(B,aa(A,B,J,A4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T4),T6)) )
               => ( ! [B4: B] :
                      ( member(B,B4,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T4),T6))
                     => ( aa(A,B,J,aa(B,A,Ia,B4)) = B4 ) )
                 => ( ! [B4: B] :
                        ( member(B,B4,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T4),T6))
                       => member(A,aa(B,A,Ia,B4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S),S3)) )
                   => ( ! [A4: A] :
                          ( member(A,A4,S3)
                         => ( aa(A,E,G,A4) = zero_zero(E) ) )
                     => ( ! [B4: B] :
                            ( member(B,B4,T6)
                           => ( aa(B,E,H2,B4) = zero_zero(E) ) )
                       => ( ! [A4: A] :
                              ( member(A,A4,S)
                             => ( aa(B,E,H2,aa(A,B,J,A4)) = aa(A,E,G,A4) ) )
                         => ( aa(set(A),E,aa(fun(A,E),fun(set(A),E),groups7311177749621191930dd_sum(A,E),G),S) = aa(set(B),E,aa(fun(B,E),fun(set(B),E),groups7311177749621191930dd_sum(B,E),H2),T4) ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_3854_sum__diff__nat__ivl,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [M2: nat,N: nat,P2: nat,F2: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),N)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),P2)
           => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),set_or7035219750837199246ssThan(nat,M2,P2))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),set_or7035219750837199246ssThan(nat,M2,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),set_or7035219750837199246ssThan(nat,N,P2)) ) ) ) ) ).

% sum_diff_nat_ivl
tff(fact_3855_sum_Osetdiff__irrelevant,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [Aa2: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( 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)),Aa2),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_fi(fun(A,B),fun(A,$o),G)))) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),Aa2) ) ) ) ).

% sum.setdiff_irrelevant
tff(fact_3856_sum__power__add,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Xa: A,M2: nat,Ia2: 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_fj(A,fun(nat,fun(nat,A)),Xa),M2)),Ia2) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),M2)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),Xa)),Ia2)) ) ) ).

% sum_power_add
tff(fact_3857_distinct__sum__list__conv__Sum,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Xsa: list(A)] :
          ( aa(list(A),$o,distinct(A),Xsa)
         => ( groups8242544230860333062m_list(A,Xsa) = aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_fk(A,A)),aa(list(A),set(A),set2(A),Xsa)) ) ) ) ).

% distinct_sum_list_conv_Sum
tff(fact_3858_sum__pos2,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [Ia2: set(A),Ia: A,F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),Ia2)
         => ( member(A,Ia,Ia2)
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(A,B,F2,Ia))
             => ( ! [I3: A] :
                    ( member(A,I3,Ia2)
                   => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,I3)) )
               => aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),Ia2)) ) ) ) ) ) ).

% sum_pos2
tff(fact_3859_sum__pos,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [Ia2: set(A),F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),Ia2)
         => ( ( Ia2 != bot_bot(set(A)) )
           => ( ! [I3: A] :
                  ( member(A,I3,Ia2)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(A,B,F2,I3)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),Ia2)) ) ) ) ) ).

% sum_pos
tff(fact_3860_sum_Omono__neutral__cong__right,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [T4: set(A),S: set(A),G: fun(A,B),H2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),T4)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),T4)
           => ( ! [X: A] :
                  ( member(A,X,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T4),S))
                 => ( aa(A,B,G,X) = zero_zero(B) ) )
             => ( ! [X: A] :
                    ( member(A,X,S)
                   => ( aa(A,B,G,X) = aa(A,B,H2,X) ) )
               => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),T4) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),H2),S) ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_3861_sum_Omono__neutral__cong__left,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [T4: set(A),S: set(A),H2: fun(A,B),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),T4)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),T4)
           => ( ! [X: A] :
                  ( member(A,X,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T4),S))
                 => ( aa(A,B,H2,X) = zero_zero(B) ) )
             => ( ! [X: A] :
                    ( member(A,X,S)
                   => ( aa(A,B,G,X) = aa(A,B,H2,X) ) )
               => ( 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),T4) ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_3862_sum_Omono__neutral__right,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [T4: set(A),S: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),T4)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),T4)
           => ( ! [X: A] :
                  ( member(A,X,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T4),S))
                 => ( aa(A,B,G,X) = zero_zero(B) ) )
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),T4) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),S) ) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_3863_sum_Omono__neutral__left,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [T4: set(A),S: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),T4)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),T4)
           => ( ! [X: A] :
                  ( member(A,X,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T4),S))
                 => ( aa(A,B,G,X) = zero_zero(B) ) )
             => ( 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),G),T4) ) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_3864_sum_Osame__carrierI,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [C3: set(A),Aa2: set(A),B2: set(A),G: fun(A,B),H2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),C3)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),C3)
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),C3)
             => ( ! [A4: A] :
                    ( member(A,A4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),C3),Aa2))
                   => ( aa(A,B,G,A4) = zero_zero(B) ) )
               => ( ! [B4: A] :
                      ( member(A,B4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),C3),B2))
                     => ( aa(A,B,H2,B4) = zero_zero(B) ) )
                 => ( ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),C3) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),H2),C3) )
                   => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),Aa2) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),H2),B2) ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_3865_sum_Osame__carrier,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [C3: set(A),Aa2: set(A),B2: set(A),G: fun(A,B),H2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),C3)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),C3)
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),C3)
             => ( ! [A4: A] :
                    ( member(A,A4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),C3),Aa2))
                   => ( aa(A,B,G,A4) = zero_zero(B) ) )
               => ( ! [B4: A] :
                      ( member(A,B4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),C3),B2))
                     => ( aa(A,B,H2,B4) = zero_zero(B) ) )
                 => ( ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),Aa2) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),H2),B2) )
                  <=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),C3) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),H2),C3) ) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_3866_sum_Osubset__diff,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [B2: set(A),Aa2: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),Aa2)
         => ( aa(set(A),$o,finite_finite2(A),Aa2)
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),Aa2) = 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)),Aa2),B2))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),B2)) ) ) ) ) ).

% sum.subset_diff
tff(fact_3867_sum__bounded__below,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ordere6911136660526730532id_add(B)
        & semiring_1(B) )
     => ! [Aa2: set(A),K6: B,F2: fun(A,B)] :
          ( ! [I3: A] :
              ( member(A,I3,Aa2)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),K6),aa(A,B,F2,I3)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,semiring_1_of_nat(B),finite_card(A,Aa2))),K6)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),Aa2)) ) ) ).

% sum_bounded_below
tff(fact_3868_sum__bounded__above,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ordere6911136660526730532id_add(B)
        & semiring_1(B) )
     => ! [Aa2: set(A),F2: fun(A,B),K6: B] :
          ( ! [I3: A] :
              ( member(A,I3,Aa2)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,I3)),K6) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),Aa2)),aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,semiring_1_of_nat(B),finite_card(A,Aa2))),K6)) ) ) ).

% sum_bounded_above
tff(fact_3869_sum__diff,axiom,
    ! [B: $tType,A: $tType] :
      ( ab_group_add(B)
     => ! [Aa2: set(A),B2: set(A),F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),Aa2)
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),B2)) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),Aa2)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),B2)) ) ) ) ) ).

% sum_diff
tff(fact_3870_sum_Omono__neutral__cong,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [T4: set(A),S: set(A),H2: fun(A,B),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),T4)
         => ( aa(set(A),$o,finite_finite2(A),S)
           => ( ! [I3: A] :
                  ( member(A,I3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T4),S))
                 => ( aa(A,B,H2,I3) = zero_zero(B) ) )
             => ( ! [I3: A] :
                    ( member(A,I3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S),T4))
                   => ( aa(A,B,G,I3) = zero_zero(B) ) )
               => ( ! [X: A] :
                      ( member(A,X,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S),T4))
                     => ( aa(A,B,G,X) = aa(A,B,H2,X) ) )
                 => ( 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),T4) ) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
tff(fact_3871_sum_OInt__Diff,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [Aa2: set(A),G: fun(A,B),B2: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),Aa2) = 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)),Aa2),B2))),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)),Aa2),B2))) ) ) ) ).

% sum.Int_Diff
tff(fact_3872_sum_OatLeast__Suc__lessThan,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M2: nat,N: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),N)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,M2,N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,M2)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M2),N))) ) ) ) ).

% sum.atLeast_Suc_lessThan
tff(fact_3873_listset_Osimps_I2_J,axiom,
    ! [A: $tType,Aa2: set(A),As4: list(set(A))] : ( aa(list(set(A)),set(list(A)),listset(A),aa(list(set(A)),list(set(A)),aa(set(A),fun(list(set(A)),list(set(A))),cons(set(A)),Aa2),As4)) = aa(set(list(A)),set(list(A)),aa(set(A),fun(set(list(A)),set(list(A))),set_Cons(A),Aa2),aa(list(set(A)),set(list(A)),listset(A),As4)) ) ).

% listset.simps(2)
tff(fact_3874_sum__Suc__diff_H,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [M2: nat,N: nat,F2: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),N)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_fl(fun(nat,A),fun(nat,A),F2)),set_or7035219750837199246ssThan(nat,M2,N)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F2,N)),aa(nat,A,F2,M2)) ) ) ) ).

% sum_Suc_diff'
tff(fact_3875_sum__Suc__diff,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [M2: nat,N: nat,F2: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),aa(nat,nat,suc,N))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_fl(fun(nat,A),fun(nat,A),F2)),set_or1337092689740270186AtMost(nat,M2,N)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F2,aa(nat,nat,suc,N))),aa(nat,A,F2,M2)) ) ) ) ).

% sum_Suc_diff
tff(fact_3876_sum__telescope,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [F2: fun(nat,A),Ia: nat] : ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_fm(fun(nat,A),fun(nat,A),F2)),aa(nat,set(nat),set_ord_atMost(nat),Ia)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F2,zero_zero(nat))),aa(nat,A,F2,aa(nat,nat,suc,Ia))) ) ) ).

% sum_telescope
tff(fact_3877_sum__lessThan__telescope,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [F2: fun(nat,A),M2: nat] : ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_fl(fun(nat,A),fun(nat,A),F2)),aa(nat,set(nat),set_ord_lessThan(nat),M2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F2,M2)),aa(nat,A,F2,zero_zero(nat))) ) ) ).

% sum_lessThan_telescope
tff(fact_3878_sum__lessThan__telescope_H,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [F2: fun(nat,A),M2: nat] : ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_fm(fun(nat,A),fun(nat,A),F2)),aa(nat,set(nat),set_ord_lessThan(nat),M2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F2,zero_zero(nat))),aa(nat,A,F2,M2)) ) ) ).

% sum_lessThan_telescope'
tff(fact_3879_sum__mono2,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [B2: set(A),Aa2: set(A),F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),B2)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),B2)
           => ( ! [B4: A] :
                  ( member(A,B4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B2),Aa2))
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,B4)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),Aa2)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),B2)) ) ) ) ) ).

% sum_mono2
tff(fact_3880_sum_Oinsert__remove,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [Aa2: set(A),G: fun(A,B),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( 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)),insert3(A),Xa),Aa2)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,G,Xa)),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)),Aa2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A)))))) ) ) ) ).

% sum.insert_remove
tff(fact_3881_sum_Oremove,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [Aa2: set(A),Xa: A,G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( member(A,Xa,Aa2)
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),Aa2) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,G,Xa)),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)),Aa2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A)))))) ) ) ) ) ).

% sum.remove
tff(fact_3882_sum__diff1,axiom,
    ! [B: $tType,A: $tType] :
      ( ab_group_add(B)
     => ! [Aa2: set(A),F2: fun(A,B),A3: A] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),bot_bot(set(A))))) = $ite(member(A,A3,Aa2),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),Aa2)),aa(A,B,F2,A3)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),Aa2)) ) ) ) ).

% sum_diff1
tff(fact_3883_sum__Un,axiom,
    ! [B: $tType,A: $tType] :
      ( ab_group_add(B)
     => ! [Aa2: set(A),B2: set(A),F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( aa(set(A),$o,finite_finite2(A),B2)
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),B2)) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),Aa2)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),B2))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),B2))) ) ) ) ) ).

% sum_Un
tff(fact_3884_sum_Ounion__disjoint,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [Aa2: set(A),B2: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( aa(set(A),$o,finite_finite2(A),B2)
           => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),B2) = 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)),Aa2),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),G),Aa2)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),B2)) ) ) ) ) ) ).

% sum.union_disjoint
tff(fact_3885_sum__Un2,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [Aa2: set(A),B2: set(A),F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),B2))
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),B2)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),B2))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B2),Aa2)))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),B2))) ) ) ) ).

% sum_Un2
tff(fact_3886_sum_Ounion__diff2,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [Aa2: set(A),B2: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( aa(set(A),$o,finite_finite2(A),B2)
           => ( 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)),Aa2),B2)) = 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)),Aa2),B2))),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)),B2),Aa2)))),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)),Aa2),B2))) ) ) ) ) ).

% sum.union_diff2
tff(fact_3887_sum_Ohead__if,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M2: nat,N: nat] :
          ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M2,N)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),M2),zero_zero(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,M2,N))),aa(nat,A,G,N))) ) ) ).

% sum.head_if
tff(fact_3888_sum_Odelta__remove,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [S: set(A),A3: A,B3: fun(A,B),C2: fun(A,B)] :
          ( aa(set(A),$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_fn(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),A3),B3),C2)),S) = $ite(member(A,A3,S),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,B3,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)),insert3(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)),insert3(A),A3),bot_bot(set(A)))))) ) ) ) ).

% sum.delta_remove
tff(fact_3889_sum_OUNION__disjoint,axiom,
    ! [B: $tType,E: $tType,A: $tType] :
      ( comm_monoid_add(E)
     => ! [Ia2: set(A),Aa2: fun(A,set(B)),G: fun(B,E)] :
          ( aa(set(A),$o,finite_finite2(A),Ia2)
         => ( ! [X: A] :
                ( member(A,X,Ia2)
               => aa(set(B),$o,finite_finite2(B),aa(A,set(B),Aa2,X)) )
           => ( ! [X: A] :
                  ( member(A,X,Ia2)
                 => ! [Xa5: A] :
                      ( member(A,Xa5,Ia2)
                     => ( ( X != Xa5 )
                       => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(A,set(B),Aa2,X)),aa(A,set(B),Aa2,Xa5)) = bot_bot(set(B)) ) ) ) )
             => ( aa(set(B),E,aa(fun(B,E),fun(set(B),E),groups7311177749621191930dd_sum(B,E),G),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),Aa2),Ia2))) = aa(set(A),E,aa(fun(A,E),fun(set(A),E),groups7311177749621191930dd_sum(A,E),aa(fun(B,E),fun(A,E),aTP_Lamp_fo(fun(A,set(B)),fun(fun(B,E),fun(A,E)),Aa2),G)),Ia2) ) ) ) ) ) ).

% sum.UNION_disjoint
tff(fact_3890_power__diff__1__eq,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Xa: 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),Xa),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),Xa),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),Xa)),aa(nat,set(nat),set_ord_lessThan(nat),N))) ) ) ).

% power_diff_1_eq
tff(fact_3891_one__diff__power__eq,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Xa: 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),Xa),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)),Xa)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),Xa)),aa(nat,set(nat),set_ord_lessThan(nat),N))) ) ) ).

% one_diff_power_eq
tff(fact_3892_geometric__sum,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Xa: A,N: nat] :
          ( ( Xa != 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),Xa)),aa(nat,set(nat),set_ord_lessThan(nat),N)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),N)),one_one(A)),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xa),one_one(A))) ) ) ) ).

% geometric_sum
tff(fact_3893_sum__list__sum__nth,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Xsa: list(A)] : ( groups8242544230860333062m_list(A,Xsa) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),nth(A,Xsa)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xsa))) ) ) ).

% sum_list_sum_nth
tff(fact_3894_sum__strict__mono2,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere8940638589300402666id_add(B)
     => ! [B2: set(A),Aa2: set(A),B3: A,F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),B2)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),B2)
           => ( member(A,B3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B2),Aa2))
             => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(A,B,F2,B3))
               => ( ! [X: A] :
                      ( member(A,X,B2)
                     => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,X)) )
                 => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),Aa2)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),B2)) ) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_3895_member__le__sum,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ordere6911136660526730532id_add(B)
        & semiring_1(B) )
     => ! [Ia: A,Aa2: set(A),F2: fun(A,B)] :
          ( member(A,Ia,Aa2)
         => ( ! [X: A] :
                ( member(A,X,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Ia),bot_bot(set(A)))))
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,X)) )
           => ( aa(set(A),$o,finite_finite2(A),Aa2)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,Ia)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),Aa2)) ) ) ) ) ).

% member_le_sum
tff(fact_3896_sum__bounded__above__divide,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_field(B)
     => ! [Aa2: set(A),F2: fun(A,B),K6: B] :
          ( ! [I3: A] :
              ( member(A,I3,Aa2)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,I3)),divide_divide(B,K6,aa(nat,B,semiring_1_of_nat(B),finite_card(A,Aa2)))) )
         => ( aa(set(A),$o,finite_finite2(A),Aa2)
           => ( ( Aa2 != bot_bot(set(A)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),Aa2)),K6) ) ) ) ) ).

% sum_bounded_above_divide
tff(fact_3897_sum__bounded__above__strict,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ordere8940638589300402666id_add(B)
        & semiring_1(B) )
     => ! [Aa2: set(A),F2: fun(A,B),K6: B] :
          ( ! [I3: A] :
              ( member(A,I3,Aa2)
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,I3)),K6) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),finite_card(A,Aa2))
           => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),Aa2)),aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,semiring_1_of_nat(B),finite_card(A,Aa2))),K6)) ) ) ) ).

% sum_bounded_above_strict
tff(fact_3898_convex__sum__bound__le,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_idom(B)
     => ! [Ia2: set(A),Xa: fun(A,B),A3: fun(A,B),B3: B,Delta: B] :
          ( ! [I3: A] :
              ( member(A,I3,Ia2)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,Xa,I3)) )
         => ( ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),Xa),Ia2) = one_one(B) )
           => ( ! [I3: A] :
                  ( member(A,I3,Ia2)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),abs_abs(B,aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,A3,I3)),B3))),Delta) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(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_fp(fun(A,B),fun(fun(A,B),fun(A,B)),Xa),A3)),Ia2)),B3))),Delta) ) ) ) ) ).

% convex_sum_bound_le
tff(fact_3899_sum__fun__comp,axiom,
    ! [E: $tType,A: $tType,B: $tType] :
      ( semiring_1(E)
     => ! [S: set(A),R: set(B),G: fun(A,B),F2: fun(B,E)] :
          ( aa(set(A),$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),E,aa(fun(A,E),fun(set(A),E),groups7311177749621191930dd_sum(A,E),aa(fun(B,E),fun(A,E),aTP_Lamp_fq(fun(A,B),fun(fun(B,E),fun(A,E)),G),F2)),S) = aa(set(B),E,aa(fun(B,E),fun(set(B),E),groups7311177749621191930dd_sum(B,E),aa(fun(B,E),fun(B,E),aa(fun(A,B),fun(fun(B,E),fun(B,E)),aTP_Lamp_fs(set(A),fun(fun(A,B),fun(fun(B,E),fun(B,E))),S),G),F2)),R) ) ) ) ) ) ).

% sum_fun_comp
tff(fact_3900_sum__gp__basic,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Xa: 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)),Xa)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),Xa)),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),Xa),aa(nat,nat,suc,N))) ) ) ).

% sum_gp_basic
tff(fact_3901_sum__natinterval__diff,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [F2: fun(nat,A),M2: nat,N: nat] :
          ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_ft(fun(nat,A),fun(nat,A),F2)),set_or1337092689740270186AtMost(nat,M2,N)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),N),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F2,M2)),aa(nat,A,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)))),zero_zero(A)) ) ) ).

% sum_natinterval_diff
tff(fact_3902_diff__power__eq__sum,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Xa: 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),Xa),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),Xa),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_fu(A,fun(nat,fun(A,fun(nat,A))),Xa),N),Y)),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,N)))) ) ) ).

% diff_power_eq_sum
tff(fact_3903_power__diff__sumr2,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Xa: 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),Xa),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),Xa),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_fv(A,fun(nat,fun(A,fun(nat,A))),Xa),N),Y)),aa(nat,set(nat),set_ord_lessThan(nat),N))) ) ) ).

% power_diff_sumr2
tff(fact_3904_sum__gp__strict,axiom,
    ! [A: $tType] :
      ( ( division_ring(A)
        & comm_ring(A) )
     => ! [Xa: 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),Xa)),aa(nat,set(nat),set_ord_lessThan(nat),N)) = $ite(Xa = one_one(A),aa(nat,A,semiring_1_of_nat(A),N),divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),N)),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),Xa))) ) ) ).

% sum_gp_strict
tff(fact_3905_sum__telescope_H_H,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [M2: nat,N: nat,F2: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),N)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_fw(fun(nat,A),fun(nat,A),F2)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M2),N)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F2,N)),aa(nat,A,F2,M2)) ) ) ) ).

% sum_telescope''
tff(fact_3906_sum__power__shift,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [M2: nat,N: nat,Xa: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),N)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),Xa)),set_or1337092689740270186AtMost(nat,M2,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),M2)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),Xa)),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M2)))) ) ) ) ).

% sum_power_shift
tff(fact_3907_mask__eq__sum__exp,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [N: nat] : ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),numeral_numeral(A,bit0(one2))),N)),one_one(A)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),numeral_numeral(A,bit0(one2)))),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_ac(nat,fun(nat,$o)),N))) ) ) ).

% mask_eq_sum_exp
tff(fact_3908_sum__gp__multiplied,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [M2: nat,N: nat,Xa: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),N)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),Xa)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),Xa)),set_or1337092689740270186AtMost(nat,M2,N))) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),M2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),aa(nat,nat,suc,N))) ) ) ) ).

% sum_gp_multiplied
tff(fact_3909_one__diff__power__eq_H,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Xa: 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),Xa),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)),Xa)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aTP_Lamp_fx(A,fun(nat,fun(nat,A)),Xa),N)),aa(nat,set(nat),set_ord_lessThan(nat),N))) ) ) ).

% one_diff_power_eq'
tff(fact_3910_gbinomial__sum__lower__neg,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,M2: nat] : ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_fy(A,fun(nat,A),A3)),aa(nat,set(nat),set_ord_atMost(nat),M2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),M2)),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),one_one(A))),M2)) ) ) ).

% gbinomial_sum_lower_neg
tff(fact_3911_binomial__ring,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A,B3: 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),B3)),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_fz(A,fun(A,fun(nat,fun(nat,A))),A3),B3),N)),aa(nat,set(nat),set_ord_atMost(nat),N)) ) ) ).

% binomial_ring
tff(fact_3912_pochhammer__binomial__sum,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [A3: A,B3: A,N: nat] : ( comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3),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_ga(A,fun(A,fun(nat,fun(nat,A))),A3),B3),N)),aa(nat,set(nat),set_ord_atMost(nat),N)) ) ) ).

% pochhammer_binomial_sum
tff(fact_3913_horner__sum__eq__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(A)
     => ! [F2: fun(B,A),A3: A,Xsa: 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),F2),A3),Xsa) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(list(B),fun(nat,A),aa(A,fun(list(B),fun(nat,A)),aTP_Lamp_gb(fun(B,A),fun(A,fun(list(B),fun(nat,A))),F2),A3),Xsa)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(B),nat,size_size(list(B)),Xsa))) ) ) ).

% horner_sum_eq_sum
tff(fact_3914_sum_Ozero__middle,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [P2: nat,K3: nat,G: fun(nat,A),H2: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),P2)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K3),P2)
           => ( 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_gc(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K3),G),H2)),aa(nat,set(nat),set_ord_atMost(nat),P2)) = 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_gd(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K3),G),H2)),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),P2),aa(nat,nat,suc,zero_zero(nat))))) ) ) ) ) ).

% sum.zero_middle
tff(fact_3915_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),numeral_numeral(nat,bit0(one2))),N)),aa(nat,nat,suc,zero_zero(nat))) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(nat,fun(nat,nat),power_power(nat),numeral_numeral(nat,bit0(one2)))),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_ac(nat,fun(nat,$o)),N))) ) ).

% mask_eq_sum_exp_nat
tff(fact_3916_gbinomial__partial__sum__poly,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [M2: nat,A3: A,Xa: 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_ge(nat,fun(A,fun(A,fun(A,fun(nat,A)))),M2),A3),Xa),Y)),aa(nat,set(nat),set_ord_atMost(nat),M2)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aa(A,fun(A,fun(A,fun(nat,A))),aTP_Lamp_gf(nat,fun(A,fun(A,fun(A,fun(nat,A)))),M2),A3),Xa),Y)),aa(nat,set(nat),set_ord_atMost(nat),M2)) ) ) ).

% gbinomial_partial_sum_poly
tff(fact_3917_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),numeral_numeral(A,bit0(one2))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_gg(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),numeral_numeral(A,bit0(one2))),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),D3))) ) ) ).

% double_arith_series
tff(fact_3918_double__gauss__sum,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [N: nat] : ( aa(A,A,aa(A,fun(A,A),times_times(A),numeral_numeral(A,bit0(one2))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),semiring_1_of_nat(A)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N)),one_one(A))) ) ) ).

% double_gauss_sum
tff(fact_3919_sum__gp0,axiom,
    ! [A: $tType] :
      ( ( division_ring(A)
        & comm_ring(A) )
     => ! [Xa: 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),Xa)),aa(nat,set(nat),set_ord_atMost(nat),N)) = $ite(Xa = 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))),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),Xa),aa(nat,nat,suc,N))),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),Xa))) ) ) ).

% sum_gp0
tff(fact_3920_gbinomial__partial__sum__poly__xpos,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [M2: nat,A3: A,Xa: 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_ge(nat,fun(A,fun(A,fun(A,fun(nat,A)))),M2),A3),Xa),Y)),aa(nat,set(nat),set_ord_atMost(nat),M2)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aa(A,fun(A,fun(A,fun(nat,A))),aTP_Lamp_gh(nat,fun(A,fun(A,fun(A,fun(nat,A)))),M2),A3),Xa),Y)),aa(nat,set(nat),set_ord_atMost(nat),M2)) ) ) ).

% gbinomial_partial_sum_poly_xpos
tff(fact_3921_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_gi(nat,fun(nat,A),N)),aa(nat,set(nat),set_ord_atMost(nat),N)) = zero_zero(A) ) ) ) ).

% choose_alternating_linear_sum
tff(fact_3922_card__lists__length__le,axiom,
    ! [A: $tType,Aa2: set(A),N: nat] :
      ( aa(set(A),$o,finite_finite2(A),Aa2)
     => ( finite_card(list(A),aa(fun(list(A),$o),set(list(A)),collect(list(A)),aa(nat,fun(list(A),$o),aTP_Lamp_dz(set(A),fun(nat,fun(list(A),$o)),Aa2),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),finite_card(A,Aa2))),aa(nat,set(nat),set_ord_atMost(nat),N)) ) ) ).

% card_lists_length_le
tff(fact_3923_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),numeral_numeral(A,bit0(one2))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),semiring_1_of_nat(A)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N)),one_one(A))) ) ) ).

% double_gauss_sum_from_Suc_0
tff(fact_3924_gauss__sum,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [N: nat] : ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),semiring_1_of_nat(A)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N)),one_one(A))),numeral_numeral(A,bit0(one2))) ) ) ).

% gauss_sum
tff(fact_3925_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_gj(A,fun(A,fun(nat,A)),A3),D3)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N)),one_one(A))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),numeral_numeral(A,bit0(one2))),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),D3))),numeral_numeral(A,bit0(one2))) ) ) ).

% arith_series
tff(fact_3926_sum__gp__offset,axiom,
    ! [A: $tType] :
      ( ( division_ring(A)
        & comm_ring(A) )
     => ! [Xa: A,M2: 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),Xa)),set_or1337092689740270186AtMost(nat,M2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),N))) = $ite(Xa = 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)),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),Xa),M2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),aa(nat,nat,suc,N)))),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),Xa))) ) ) ).

% sum_gp_offset
tff(fact_3927_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_gk(nat,fun(nat,A),N)),aa(nat,set(nat),set_ord_atMost(nat),N)) = zero_zero(A) ) ) ) ).

% choose_alternating_sum
tff(fact_3928_gauss__sum__from__Suc__0,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [N: nat] : ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),semiring_1_of_nat(A)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),N)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N)),one_one(A))),numeral_numeral(A,bit0(one2))) ) ) ).

% gauss_sum_from_Suc_0
tff(fact_3929_gchoose__row__sum__weighted,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [R2: A,M2: nat] : ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_gl(A,fun(nat,A),R2)),set_or1337092689740270186AtMost(nat,zero_zero(nat),M2)) = aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,M2)),numeral_numeral(A,bit0(one2)))),aa(nat,A,gbinomial(A,R2),aa(nat,nat,suc,M2))) ) ) ).

% gchoose_row_sum_weighted
tff(fact_3930_sum__gp,axiom,
    ! [A: $tType] :
      ( ( division_ring(A)
        & comm_ring(A) )
     => ! [Xa: A,M2: 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),Xa)),set_or1337092689740270186AtMost(nat,M2,N)) = $ite(
              aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),M2),
              zero_zero(A),
              $ite(Xa = 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))),M2)),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),Xa),M2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),aa(nat,nat,suc,N))),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),Xa))) ) ) ) ).

% sum_gp
tff(fact_3931_gbinomial__r__part__sum,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [M2: nat] : ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),numeral_numeral(A,bit0(one2))),aa(nat,A,semiring_1_of_nat(A),M2))),one_one(A)))),aa(nat,set(nat),set_ord_atMost(nat),M2)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),numeral_numeral(A,bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),numeral_numeral(nat,bit0(one2))),M2)) ) ) ).

% gbinomial_r_part_sum
tff(fact_3932_gbinomial__partial__row__sum,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,M2: nat] : ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_gl(A,fun(nat,A),A3)),aa(nat,set(nat),set_ord_atMost(nat),M2)) = aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),M2)),one_one(A)),numeral_numeral(A,bit0(one2)))),aa(nat,A,gbinomial(A,A3),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),one_one(nat)))) ) ) ).

% gbinomial_partial_row_sum
tff(fact_3933_same__fstI,axiom,
    ! [B: $tType,A: $tType,Pa: fun(A,$o),Xa: A,Y6: B,Y: B,R: fun(A,set(product_prod(B,B)))] :
      ( aa(A,$o,Pa,Xa)
     => ( 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,Xa))
       => 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),Y6)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa),Y)),same_fst(A,B,Pa,R)) ) ) ).

% same_fstI
tff(fact_3934_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_gm(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),R)),aa(fun(nat,$o),set(nat),collect(nat),aTP_Lamp_gn(set(product_prod(A,A)),fun(nat,$o),R)))) ) ) ).

% trancl_finite_eq_relpow
tff(fact_3935_relpow__finite__bounded1,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),K3: nat] :
      ( aa(set(product_prod(A,A)),$o,finite_finite2(product_prod(A,A)),R)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K3)
       => 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)),K3,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_gm(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),R)),aa(fun(nat,$o),set(nat),collect(nat),aTP_Lamp_gn(set(product_prod(A,A)),fun(nat,$o),R))))) ) ) ).

% relpow_finite_bounded1
tff(fact_3936_divmod__algorithm__code_I6_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M2: num,N: num] : ( unique8689654367752047608divmod(A,bit1(M2),bit0(N)) = aa(product_prod(A,A),product_prod(A,A),aa(fun(A,fun(A,product_prod(A,A))),fun(product_prod(A,A),product_prod(A,A)),product_case_prod(A,A,product_prod(A,A)),aTP_Lamp_go(A,fun(A,product_prod(A,A)))),unique8689654367752047608divmod(A,M2,N)) ) ) ).

% divmod_algorithm_code(6)
tff(fact_3937_pair__imageI,axiom,
    ! [E: $tType,B: $tType,A: $tType,A3: A,B3: B,Aa2: set(product_prod(A,B)),F2: fun(A,fun(B,E))] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B3),Aa2)
     => member(E,aa(B,E,aa(A,fun(B,E),F2,A3),B3),aa(set(product_prod(A,B)),set(E),image2(product_prod(A,B),E,aa(fun(A,fun(B,E)),fun(product_prod(A,B),E),product_case_prod(A,B,E),F2)),Aa2)) ) ).

% pair_imageI
tff(fact_3938_case__prod__conv,axiom,
    ! [B: $tType,A: $tType,E: $tType,F2: fun(B,fun(E,A)),A3: B,B3: E] : ( aa(product_prod(B,E),A,aa(fun(B,fun(E,A)),fun(product_prod(B,E),A),product_case_prod(B,E,A),F2),aa(E,product_prod(B,E),aa(B,fun(E,product_prod(B,E)),product_Pair(B,E),A3),B3)) = aa(E,A,aa(B,fun(E,A),F2,A3),B3) ) ).

% case_prod_conv
tff(fact_3939_finite__relpow,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),N: nat] :
      ( aa(set(product_prod(A,A)),$o,finite_finite2(product_prod(A,A)),R)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
       => aa(set(product_prod(A,A)),$o,finite_finite2(product_prod(A,A)),compow(set(product_prod(A,A)),N,R)) ) ) ).

% finite_relpow
tff(fact_3940_divmod__algorithm__code_I5_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M2: num,N: num] : ( unique8689654367752047608divmod(A,bit0(M2),bit0(N)) = aa(product_prod(A,A),product_prod(A,A),aa(fun(A,fun(A,product_prod(A,A))),fun(product_prod(A,A),product_prod(A,A)),product_case_prod(A,A,product_prod(A,A)),aTP_Lamp_gp(A,fun(A,product_prod(A,A)))),unique8689654367752047608divmod(A,M2,N)) ) ) ).

% divmod_algorithm_code(5)
tff(fact_3941_relpow__Suc__D2_H,axiom,
    ! [A: $tType,N: nat,R: set(product_prod(A,A)),X4: A,Y3: A,Z4: A] :
      ( ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Y3),compow(set(product_prod(A,A)),N,R))
        & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),Z4),R) )
     => ? [W: A] :
          ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),W),R)
          & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),W),Z4),compow(set(product_prod(A,A)),N,R)) ) ) ).

% relpow_Suc_D2'
tff(fact_3942_split__cong,axiom,
    ! [E: $tType,B: $tType,A: $tType,Q2: product_prod(A,B),F2: fun(A,fun(B,E)),G: fun(A,fun(B,E)),P2: product_prod(A,B)] :
      ( ! [X: A,Y2: B] :
          ( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y2) = Q2 )
         => ( aa(B,E,aa(A,fun(B,E),F2,X),Y2) = aa(B,E,aa(A,fun(B,E),G,X),Y2) ) )
     => ( ( P2 = Q2 )
       => ( aa(product_prod(A,B),E,aa(fun(A,fun(B,E)),fun(product_prod(A,B),E),product_case_prod(A,B,E),F2),P2) = aa(product_prod(A,B),E,aa(fun(A,fun(B,E)),fun(product_prod(A,B),E),product_case_prod(A,B,E),G),Q2) ) ) ) ).

% split_cong
tff(fact_3943_old_Oprod_Ocase,axiom,
    ! [B: $tType,A: $tType,E: $tType,F2: fun(B,fun(E,A)),X1: B,X22: E] : ( aa(product_prod(B,E),A,aa(fun(B,fun(E,A)),fun(product_prod(B,E),A),product_case_prod(B,E,A),F2),aa(E,product_prod(B,E),aa(B,fun(E,product_prod(B,E)),product_Pair(B,E),X1),X22)) = aa(E,A,aa(B,fun(E,A),F2,X1),X22) ) ).

% old.prod.case
tff(fact_3944_nested__case__prod__simp,axiom,
    ! [F4: $tType,A: $tType,E: $tType,B: $tType,F2: fun(B,fun(E,fun(F4,A))),Xa: product_prod(B,E),Y: F4] : ( aa(F4,A,aa(product_prod(B,E),fun(F4,A),aa(fun(B,fun(E,fun(F4,A))),fun(product_prod(B,E),fun(F4,A)),product_case_prod(B,E,fun(F4,A)),F2),Xa),Y) = aa(product_prod(B,E),A,aa(fun(B,fun(E,A)),fun(product_prod(B,E),A),product_case_prod(B,E,A),aa(F4,fun(B,fun(E,A)),aTP_Lamp_gq(fun(B,fun(E,fun(F4,A))),fun(F4,fun(B,fun(E,A))),F2),Y)),Xa) ) ).

% nested_case_prod_simp
tff(fact_3945_case__prodE2,axiom,
    ! [B: $tType,A: $tType,E: $tType,Qa: fun(A,$o),Pa: fun(B,fun(E,A)),Z2: product_prod(B,E)] :
      ( aa(A,$o,Qa,aa(product_prod(B,E),A,aa(fun(B,fun(E,A)),fun(product_prod(B,E),A),product_case_prod(B,E,A),Pa),Z2))
     => ~ ! [X: B,Y2: E] :
            ( ( Z2 = aa(E,product_prod(B,E),aa(B,fun(E,product_prod(B,E)),product_Pair(B,E),X),Y2) )
           => ~ aa(A,$o,Qa,aa(E,A,aa(B,fun(E,A),Pa,X),Y2)) ) ) ).

% case_prodE2
tff(fact_3946_case__prod__eta,axiom,
    ! [E: $tType,B: $tType,A: $tType,F2: fun(product_prod(A,B),E)] : ( aa(fun(A,fun(B,E)),fun(product_prod(A,B),E),product_case_prod(A,B,E),aTP_Lamp_gr(fun(product_prod(A,B),E),fun(A,fun(B,E)),F2)) = F2 ) ).

% case_prod_eta
tff(fact_3947_cond__case__prod__eta,axiom,
    ! [E: $tType,B: $tType,A: $tType,F2: fun(A,fun(B,E)),G: fun(product_prod(A,B),E)] :
      ( ! [X: A,Y2: B] : ( aa(B,E,aa(A,fun(B,E),F2,X),Y2) = aa(product_prod(A,B),E,G,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y2)) )
     => ( aa(fun(A,fun(B,E)),fun(product_prod(A,B),E),product_case_prod(A,B,E),F2) = G ) ) ).

% cond_case_prod_eta
tff(fact_3948_uncurry__def,axiom,
    ! [E: $tType,B: $tType,A: $tType,F2: fun(A,fun(B,E))] : ( uncurry(A,B,E,F2) = aa(fun(A,fun(B,E)),fun(product_prod(A,B),E),product_case_prod(A,B,E),F2) ) ).

% uncurry_def
tff(fact_3949_relpow__0__E,axiom,
    ! [A: $tType,Xa: A,Y: A,R: set(product_prod(A,A))] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Y),compow(set(product_prod(A,A)),zero_zero(nat),R))
     => ( Xa = Y ) ) ).

% relpow_0_E
tff(fact_3950_relpow__0__I,axiom,
    ! [A: $tType,Xa: A,R: set(product_prod(A,A))] : member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Xa),compow(set(product_prod(A,A)),zero_zero(nat),R)) ).

% relpow_0_I
tff(fact_3951_relpow__Suc__E,axiom,
    ! [A: $tType,Xa: A,Z2: A,N: nat,R: set(product_prod(A,A))] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Z2),compow(set(product_prod(A,A)),aa(nat,nat,suc,N),R))
     => ~ ! [Y2: A] :
            ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Y2),compow(set(product_prod(A,A)),N,R))
           => ~ 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_3952_relpow__Suc__I,axiom,
    ! [A: $tType,Xa: A,Y: A,N: nat,R: set(product_prod(A,A)),Z2: A] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Y),compow(set(product_prod(A,A)),N,R))
     => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z2),R)
       => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Z2),compow(set(product_prod(A,A)),aa(nat,nat,suc,N),R)) ) ) ).

% relpow_Suc_I
tff(fact_3953_relpow__Suc__D2,axiom,
    ! [A: $tType,Xa: A,Z2: A,N: nat,R: set(product_prod(A,A))] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Z2),compow(set(product_prod(A,A)),aa(nat,nat,suc,N),R))
     => ? [Y2: A] :
          ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Y2),R)
          & 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_3954_relpow__Suc__E2,axiom,
    ! [A: $tType,Xa: A,Z2: A,N: nat,R: set(product_prod(A,A))] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Z2),compow(set(product_prod(A,A)),aa(nat,nat,suc,N),R))
     => ~ ! [Y2: A] :
            ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Y2),R)
           => ~ 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_3955_relpow__Suc__I2,axiom,
    ! [A: $tType,Xa: A,Y: A,R: set(product_prod(A,A)),Z2: A,N: nat] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Y),R)
     => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z2),compow(set(product_prod(A,A)),N,R))
       => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Z2),compow(set(product_prod(A,A)),aa(nat,nat,suc,N),R)) ) ) ).

% relpow_Suc_I2
tff(fact_3956_sum__SucD,axiom,
    ! [A: $tType,F2: fun(A,nat),Aa2: set(A),N: nat] :
      ( ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F2),Aa2) = aa(nat,nat,suc,N) )
     => ? [X: A] :
          ( member(A,X,Aa2)
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(A,nat,F2,X)) ) ) ).

% sum_SucD
tff(fact_3957_relpow__E,axiom,
    ! [A: $tType,Xa: A,Z2: A,N: nat,R: set(product_prod(A,A))] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Z2),compow(set(product_prod(A,A)),N,R))
     => ( ( ( N = zero_zero(nat) )
         => ( Xa != Z2 ) )
       => ~ ! [Y2: A,M6: nat] :
              ( ( N = aa(nat,nat,suc,M6) )
             => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Y2),compow(set(product_prod(A,A)),M6,R))
               => ~ 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_3958_relpow__E2,axiom,
    ! [A: $tType,Xa: A,Z2: A,N: nat,R: set(product_prod(A,A))] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Z2),compow(set(product_prod(A,A)),N,R))
     => ( ( ( N = zero_zero(nat) )
         => ( Xa != Z2 ) )
       => ~ ! [Y2: A,M6: nat] :
              ( ( N = aa(nat,nat,suc,M6) )
             => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Y2),R)
               => ~ 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)),M6,R)) ) ) ) ) ).

% relpow_E2
tff(fact_3959_trancl__power,axiom,
    ! [A: $tType,P2: product_prod(A,A),R: set(product_prod(A,A))] :
      ( member(product_prod(A,A),P2,transitive_trancl(A,R))
    <=> ? [N2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N2)
          & member(product_prod(A,A),P2,compow(set(product_prod(A,A)),N2,R)) ) ) ).

% trancl_power
tff(fact_3960_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_3961_sum__diff__nat,axiom,
    ! [A: $tType,B2: set(A),Aa2: set(A),F2: fun(A,nat)] :
      ( aa(set(A),$o,finite_finite2(A),B2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),Aa2)
       => ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),B2)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F2),Aa2)),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F2),B2)) ) ) ) ).

% sum_diff_nat
tff(fact_3962_sum__diff1__nat,axiom,
    ! [A: $tType,F2: fun(A,nat),Aa2: set(A),A3: A] :
      ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),bot_bot(set(A))))) = $ite(member(A,A3,Aa2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F2),Aa2)),aa(A,nat,F2,A3)),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F2),Aa2)) ) ).

% sum_diff1_nat
tff(fact_3963_relpow__fun__conv,axiom,
    ! [A: $tType,A3: A,B3: A,N: nat,R: set(product_prod(A,A))] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B3),compow(set(product_prod(A,A)),N,R))
    <=> ? [F9: fun(nat,A)] :
          ( ( aa(nat,A,F9,zero_zero(nat)) = A3 )
          & ( aa(nat,A,F9,N) = B3 )
          & ! [I2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),N)
             => 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,F9,I2)),aa(nat,A,F9,aa(nat,nat,suc,I2))),R) ) ) ) ).

% relpow_fun_conv
tff(fact_3964_foldl__snd__zip,axiom,
    ! [B: $tType,E: $tType,A: $tType,Ys: list(A),Xsa: list(B),F2: fun(E,fun(A,E)),B3: E] :
      ( aa(nat,$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)),Xsa))
     => ( aa(list(product_prod(B,A)),E,aa(E,fun(list(product_prod(B,A)),E),aa(fun(E,fun(product_prod(B,A),E)),fun(E,fun(list(product_prod(B,A)),E)),foldl(E,product_prod(B,A)),aTP_Lamp_gt(fun(E,fun(A,E)),fun(E,fun(product_prod(B,A),E)),F2)),B3),aa(list(A),list(product_prod(B,A)),aa(list(B),fun(list(A),list(product_prod(B,A))),zip(B,A),Xsa),Ys)) = aa(list(A),E,aa(E,fun(list(A),E),aa(fun(E,fun(A,E)),fun(E,fun(list(A),E)),foldl(E,A),F2),B3),Ys) ) ) ).

% foldl_snd_zip
tff(fact_3965_sum__count__set,axiom,
    ! [A: $tType,Xsa: list(A),X3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xsa)),X3)
     => ( aa(set(A),$o,finite_finite2(A),X3)
       => ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),count_list(A,Xsa)),X3) = aa(list(A),nat,size_size(list(A)),Xsa) ) ) ) ).

% sum_count_set
tff(fact_3966_card__UN__disjoint,axiom,
    ! [B: $tType,A: $tType,Ia2: set(A),Aa2: fun(A,set(B))] :
      ( aa(set(A),$o,finite_finite2(A),Ia2)
     => ( ! [X: A] :
            ( member(A,X,Ia2)
           => aa(set(B),$o,finite_finite2(B),aa(A,set(B),Aa2,X)) )
       => ( ! [X: A] :
              ( member(A,X,Ia2)
             => ! [Xa5: A] :
                  ( member(A,Xa5,Ia2)
                 => ( ( X != Xa5 )
                   => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(A,set(B),Aa2,X)),aa(A,set(B),Aa2,Xa5)) = bot_bot(set(B)) ) ) ) )
         => ( finite_card(B,aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),Aa2),Ia2))) = aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aTP_Lamp_gu(fun(A,set(B)),fun(A,nat),Aa2)),Ia2) ) ) ) ) ).

% card_UN_disjoint
tff(fact_3967_divmod__nat__if,axiom,
    ! [M2: nat,N: nat] :
      ( divmod_nat(M2,N) = $ite(
          ( ( N = zero_zero(nat) )
          | aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),N) ),
          aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),zero_zero(nat)),M2),
          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_gv(nat,fun(nat,product_prod(nat,nat)))),divmod_nat(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),N),N)) ) ) ).

% divmod_nat_if
tff(fact_3968_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_gw(num,fun(nat,fun(nat,product_prod(nat,nat))),L)),Qr) ) ).

% divmod_step_nat_def
tff(fact_3969_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_gx(num,fun(int,fun(int,product_prod(int,int))),L)),Qr) ) ).

% divmod_step_int_def
tff(fact_3970_ntrancl__def,axiom,
    ! [A: $tType,N: nat,R: set(product_prod(A,A))] : ( transitive_ntrancl(A,N,R) = aa(set(set(product_prod(A,A))),set(product_prod(A,A)),complete_Sup_Sup(set(product_prod(A,A))),aa(set(nat),set(set(product_prod(A,A))),image2(nat,set(product_prod(A,A)),aTP_Lamp_gm(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),R)),aa(fun(nat,$o),set(nat),collect(nat),aTP_Lamp_gy(nat,fun(nat,$o),N)))) ) ).

% ntrancl_def
tff(fact_3971_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_gz(num,fun(A,fun(A,product_prod(A,A))),L)),Qr) ) ) ).

% divmod_step_def
tff(fact_3972_sorted__wrt__less__sum__mono__lowerbound,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [F2: fun(nat,A),Ns: list(nat)] :
          ( ! [X: nat,Y2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X),Y2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F2,X)),aa(nat,A,F2,Y2)) )
         => ( sorted_wrt(nat,ord_less(nat),Ns)
           => aa(A,$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),F2),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(nat),nat,size_size(list(nat)),Ns)))),groups8242544230860333062m_list(A,aa(list(nat),list(A),aa(fun(nat,A),fun(list(nat),list(A)),map(nat,A),F2),Ns))) ) ) ) ).

% sorted_wrt_less_sum_mono_lowerbound
tff(fact_3973_length__remdups__concat,axiom,
    ! [A: $tType,Xss2: list(list(A))] : ( aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),remdups(A),aa(list(list(A)),list(A),concat(A),Xss2))) = 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)),Xss2)))) ) ).

% length_remdups_concat
tff(fact_3974_Pow__fold,axiom,
    ! [A: $tType,Aa2: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),Aa2)
     => ( pow(A,Aa2) = finite_fold(A,set(set(A)),aTP_Lamp_ha(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))),insert3(set(A)),bot_bot(set(A))),bot_bot(set(set(A)))),Aa2) ) ) ).

% Pow_fold
tff(fact_3975_listI__assn__def,axiom,
    ! [A: $tType,B: $tType,Ia2: set(nat),Aa2: fun(A,fun(B,assn)),Xsa: list(A),Xsia: list(B)] :
      ( vEBT_List_listI_assn(A,B,Ia2,Aa2,Xsa,Xsia) = aa(assn,assn,
          aa(assn,fun(assn,assn),times_times(assn),
            pure_assn(( ( aa(list(B),nat,size_size(list(B)),Xsia) = aa(list(A),nat,size_size(list(A)),Xsa) )
              & aa(set(nat),$o,aa(set(nat),fun(set(nat),$o),ord_less_eq(set(nat)),Ia2),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xsa))) ))),
          finite_fold(nat,assn,aa(list(B),fun(nat,fun(assn,assn)),aa(list(A),fun(list(B),fun(nat,fun(assn,assn))),aTP_Lamp_hb(fun(A,fun(B,assn)),fun(list(A),fun(list(B),fun(nat,fun(assn,assn)))),Aa2),Xsa),Xsia),one_one(assn),Ia2)) ) ).

% listI_assn_def
tff(fact_3976_case__prodI2,axiom,
    ! [B: $tType,A: $tType,P2: product_prod(A,B),C2: fun(A,fun(B,$o))] :
      ( ! [A4: A,B4: B] :
          ( ( P2 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B4) )
         => aa(B,$o,aa(A,fun(B,$o),C2,A4),B4) )
     => aa(product_prod(A,B),$o,aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),C2),P2) ) ).

% case_prodI2
tff(fact_3977_case__prodI,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,fun(B,$o)),A3: A,B3: B] :
      ( aa(B,$o,aa(A,fun(B,$o),F2,A3),B3)
     => aa(product_prod(A,B),$o,aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),F2),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B3)) ) ).

% case_prodI
tff(fact_3978_mem__case__prodI2,axiom,
    ! [E: $tType,B: $tType,A: $tType,P2: product_prod(A,B),Z2: E,C2: fun(A,fun(B,set(E)))] :
      ( ! [A4: A,B4: B] :
          ( ( P2 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B4) )
         => member(E,Z2,aa(B,set(E),aa(A,fun(B,set(E)),C2,A4),B4)) )
     => member(E,Z2,aa(product_prod(A,B),set(E),aa(fun(A,fun(B,set(E))),fun(product_prod(A,B),set(E)),product_case_prod(A,B,set(E)),C2),P2)) ) ).

% mem_case_prodI2
tff(fact_3979_mem__case__prodI,axiom,
    ! [A: $tType,B: $tType,E: $tType,Z2: A,C2: fun(B,fun(E,set(A))),A3: B,B3: E] :
      ( member(A,Z2,aa(E,set(A),aa(B,fun(E,set(A)),C2,A3),B3))
     => member(A,Z2,aa(product_prod(B,E),set(A),aa(fun(B,fun(E,set(A))),fun(product_prod(B,E),set(A)),product_case_prod(B,E,set(A)),C2),aa(E,product_prod(B,E),aa(B,fun(E,product_prod(B,E)),product_Pair(B,E),A3),B3))) ) ).

% mem_case_prodI
tff(fact_3980_case__prodI2_H,axiom,
    ! [A: $tType,B: $tType,E: $tType,P2: product_prod(A,B),C2: fun(A,fun(B,fun(E,$o))),Xa: E] :
      ( ! [A4: A,B4: B] :
          ( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B4) = P2 )
         => aa(E,$o,aa(B,fun(E,$o),aa(A,fun(B,fun(E,$o)),C2,A4),B4),Xa) )
     => aa(E,$o,aa(product_prod(A,B),fun(E,$o),aa(fun(A,fun(B,fun(E,$o))),fun(product_prod(A,B),fun(E,$o)),product_case_prod(A,B,fun(E,$o)),C2),P2),Xa) ) ).

% case_prodI2'
tff(fact_3981_map__ident,axiom,
    ! [A: $tType,X4: list(A)] : ( aa(list(A),list(A),aa(fun(A,A),fun(list(A),list(A)),map(A,A),aTP_Lamp_ab(A,A)),X4) = X4 ) ).

% map_ident
tff(fact_3982_list_Omap__disc__iff,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A3: list(B)] :
      ( ( aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),A3) = nil(A) )
    <=> ( A3 = nil(B) ) ) ).

% list.map_disc_iff
tff(fact_3983_Nil__is__map__conv,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xsa: list(B)] :
      ( ( nil(A) = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Xsa) )
    <=> ( Xsa = nil(B) ) ) ).

% Nil_is_map_conv
tff(fact_3984_map__is__Nil__conv,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xsa: list(B)] :
      ( ( aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Xsa) = nil(A) )
    <=> ( Xsa = nil(B) ) ) ).

% map_is_Nil_conv
tff(fact_3985_map__eq__conv,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xsa: list(B),G: fun(B,A)] :
      ( ( aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Xsa) = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),G),Xsa) )
    <=> ! [X6: B] :
          ( member(B,X6,aa(list(B),set(B),set2(B),Xsa))
         => ( aa(B,A,F2,X6) = aa(B,A,G,X6) ) ) ) ).

% map_eq_conv
tff(fact_3986_length__map,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xsa: list(B)] : ( aa(list(A),nat,size_size(list(A)),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Xsa)) = aa(list(B),nat,size_size(list(B)),Xsa) ) ).

% length_map
tff(fact_3987_map__append,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xsa: list(B),Ys: list(B)] : ( aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Xsa),Ys)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Xsa)),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Ys)) ) ).

% map_append
tff(fact_3988_fold__empty,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,fun(A,A)),Z2: A] : ( finite_fold(B,A,F2,Z2,bot_bot(set(B))) = Z2 ) ).

% fold_empty
tff(fact_3989_remdups__eq__nil__right__iff,axiom,
    ! [A: $tType,Xa: list(A)] :
      ( ( nil(A) = aa(list(A),list(A),remdups(A),Xa) )
    <=> ( Xa = nil(A) ) ) ).

% remdups_eq_nil_right_iff
tff(fact_3990_remdups__eq__nil__iff,axiom,
    ! [A: $tType,Xa: list(A)] :
      ( ( aa(list(A),list(A),remdups(A),Xa) = nil(A) )
    <=> ( Xa = nil(A) ) ) ).

% remdups_eq_nil_iff
tff(fact_3991_set__remdups,axiom,
    ! [A: $tType,Xsa: list(A)] : ( aa(list(A),set(A),set2(A),aa(list(A),list(A),remdups(A),Xsa)) = aa(list(A),set(A),set2(A),Xsa) ) ).

% set_remdups
tff(fact_3992_length__remdups__eq,axiom,
    ! [A: $tType,Xsa: list(A)] :
      ( ( aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),remdups(A),Xsa)) = aa(list(A),nat,size_size(list(A)),Xsa) )
    <=> ( aa(list(A),list(A),remdups(A),Xsa) = Xsa ) ) ).

% length_remdups_eq
tff(fact_3993_remdups__id__iff__distinct,axiom,
    ! [A: $tType,Xsa: list(A)] :
      ( ( aa(list(A),list(A),remdups(A),Xsa) = Xsa )
    <=> aa(list(A),$o,distinct(A),Xsa) ) ).

% remdups_id_iff_distinct
tff(fact_3994_distinct__remdups,axiom,
    ! [A: $tType,Xsa: list(A)] : aa(list(A),$o,distinct(A),aa(list(A),list(A),remdups(A),Xsa)) ).

% distinct_remdups
tff(fact_3995_list_Oset__map,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),V2: list(B)] : ( aa(list(A),set(A),set2(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),V2)) = aa(set(B),set(A),image2(B,A,F2),aa(list(B),set(B),set2(B),V2)) ) ).

% list.set_map
tff(fact_3996_map__fun__upd,axiom,
    ! [B: $tType,A: $tType,Y: A,Xsa: list(A),F2: fun(A,B),V2: B] :
      ( ~ member(A,Y,aa(list(A),set(A),set2(A),Xsa))
     => ( aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),fun_upd(A,B,F2,Y,V2)),Xsa) = aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),Xsa) ) ) ).

% map_fun_upd
tff(fact_3997_length__remdups__leq,axiom,
    ! [A: $tType,Xsa: list(A)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),remdups(A),Xsa))),aa(list(A),nat,size_size(list(A)),Xsa)) ).

% length_remdups_leq
tff(fact_3998_nth__map,axiom,
    ! [B: $tType,A: $tType,N: nat,Xsa: list(A),F2: fun(A,B)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xsa))
     => ( aa(nat,B,nth(B,aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),Xsa)),N) = aa(A,B,F2,aa(nat,A,nth(A,Xsa),N)) ) ) ).

% nth_map
tff(fact_3999_concat__map__singleton,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xsa: list(B)] : ( aa(list(list(A)),list(A),concat(A),aa(list(B),list(list(A)),aa(fun(B,list(A)),fun(list(B),list(list(A))),map(B,list(A)),aTP_Lamp_hc(fun(B,A),fun(B,list(A)),F2)),Xsa)) = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Xsa) ) ).

% concat_map_singleton
tff(fact_4000_same__fst__def,axiom,
    ! [B: $tType,A: $tType,Pa: fun(A,$o),R: fun(A,set(product_prod(B,B)))] : ( same_fst(A,B,Pa,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_he(fun(A,$o),fun(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(product_prod(A,B),$o)))),Pa),R)))) ) ).

% same_fst_def
tff(fact_4001_case__prodE,axiom,
    ! [A: $tType,B: $tType,C2: fun(A,fun(B,$o)),P2: 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),P2)
     => ~ ! [X: A,Y2: B] :
            ( ( P2 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y2) )
           => ~ aa(B,$o,aa(A,fun(B,$o),C2,X),Y2) ) ) ).

% case_prodE
tff(fact_4002_case__prodD,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,fun(B,$o)),A3: A,B3: B] :
      ( aa(product_prod(A,B),$o,aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),F2),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B3))
     => aa(B,$o,aa(A,fun(B,$o),F2,A3),B3) ) ).

% case_prodD
tff(fact_4003_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_4004_map__consI_I1_J,axiom,
    ! [A: $tType,B: $tType,W2: list(A),F2: fun(B,A),Ww: list(B),A3: B] :
      ( ( W2 = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Ww) )
     => ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(B,A,F2,A3)),W2) = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),A3),Ww)) ) ) ).

% map_consI(1)
tff(fact_4005_map__eq__consE,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),Ls3: list(B),Fa2: A,Fl: list(A)] :
      ( ( aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Ls3) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Fa2),Fl) )
     => ~ ! [A4: B,L3: list(B)] :
            ( ( Ls3 = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),A4),L3) )
           => ( ( aa(B,A,F2,A4) = Fa2 )
             => ( aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),L3) != Fl ) ) ) ) ).

% map_eq_consE
tff(fact_4006_list_Osimps_I9_J,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),X21: B,X222: list(B)] : ( aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X21),X222)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(B,A,F2,X21)),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),X222)) ) ).

% list.simps(9)
tff(fact_4007_Cons__eq__map__D,axiom,
    ! [A: $tType,B: $tType,Xa: A,Xsa: list(A),F2: fun(B,A),Ys: list(B)] :
      ( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa) = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Ys) )
     => ? [Z3: B,Zs2: list(B)] :
          ( ( Ys = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Z3),Zs2) )
          & ( Xa = aa(B,A,F2,Z3) )
          & ( Xsa = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Zs2) ) ) ) ).

% Cons_eq_map_D
tff(fact_4008_map__eq__Cons__D,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),Xsa: list(B),Y: A,Ys: list(A)] :
      ( ( aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Xsa) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys) )
     => ? [Z3: B,Zs2: list(B)] :
          ( ( Xsa = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Z3),Zs2) )
          & ( aa(B,A,F2,Z3) = Y )
          & ( aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Zs2) = Ys ) ) ) ).

% map_eq_Cons_D
tff(fact_4009_Cons__eq__map__conv,axiom,
    ! [A: $tType,B: $tType,Xa: A,Xsa: list(A),F2: fun(B,A),Ys: list(B)] :
      ( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa) = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Ys) )
    <=> ? [Z5: B,Zs3: list(B)] :
          ( ( Ys = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Z5),Zs3) )
          & ( Xa = aa(B,A,F2,Z5) )
          & ( Xsa = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Zs3) ) ) ) ).

% Cons_eq_map_conv
tff(fact_4010_map__eq__Cons__conv,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),Xsa: list(B),Y: A,Ys: list(A)] :
      ( ( aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Xsa) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys) )
    <=> ? [Z5: B,Zs3: list(B)] :
          ( ( Xsa = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Z5),Zs3) )
          & ( aa(B,A,F2,Z5) = Y )
          & ( aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Zs3) = Ys ) ) ) ).

% map_eq_Cons_conv
tff(fact_4011_mem__case__prodE,axiom,
    ! [B: $tType,A: $tType,E: $tType,Z2: A,C2: fun(B,fun(E,set(A))),P2: product_prod(B,E)] :
      ( member(A,Z2,aa(product_prod(B,E),set(A),aa(fun(B,fun(E,set(A))),fun(product_prod(B,E),set(A)),product_case_prod(B,E,set(A)),C2),P2))
     => ~ ! [X: B,Y2: E] :
            ( ( P2 = aa(E,product_prod(B,E),aa(B,fun(E,product_prod(B,E)),product_Pair(B,E),X),Y2) )
           => ~ member(A,Z2,aa(E,set(A),aa(B,fun(E,set(A)),C2,X),Y2)) ) ) ).

% mem_case_prodE
tff(fact_4012_lex__prod__def,axiom,
    ! [A: $tType,B: $tType,Ra: set(product_prod(A,A)),Rb: set(product_prod(B,B))] : ( lex_prod(A,B,Ra,Rb) = aa(fun(product_prod(product_prod(A,B),product_prod(A,B)),$o),set(product_prod(product_prod(A,B),product_prod(A,B))),collect(product_prod(product_prod(A,B),product_prod(A,B))),aa(fun(product_prod(A,B),fun(product_prod(A,B),$o)),fun(product_prod(product_prod(A,B),product_prod(A,B)),$o),product_case_prod(product_prod(A,B),product_prod(A,B),$o),aa(fun(A,fun(B,fun(product_prod(A,B),$o))),fun(product_prod(A,B),fun(product_prod(A,B),$o)),product_case_prod(A,B,fun(product_prod(A,B),$o)),aa(set(product_prod(B,B)),fun(A,fun(B,fun(product_prod(A,B),$o))),aTP_Lamp_hg(set(product_prod(A,A)),fun(set(product_prod(B,B)),fun(A,fun(B,fun(product_prod(A,B),$o)))),Ra),Rb)))) ) ).

% lex_prod_def
tff(fact_4013_map2__map__map,axiom,
    ! [E: $tType,A: $tType,B: $tType,F4: $tType,H2: fun(B,fun(E,A)),F2: fun(F4,B),Xsa: list(F4),G: fun(F4,E)] : ( aa(list(product_prod(B,E)),list(A),aa(fun(product_prod(B,E),A),fun(list(product_prod(B,E)),list(A)),map(product_prod(B,E),A),aa(fun(B,fun(E,A)),fun(product_prod(B,E),A),product_case_prod(B,E,A),H2)),aa(list(E),list(product_prod(B,E)),aa(list(B),fun(list(E),list(product_prod(B,E))),zip(B,E),aa(list(F4),list(B),aa(fun(F4,B),fun(list(F4),list(B)),map(F4,B),F2),Xsa)),aa(list(F4),list(E),aa(fun(F4,E),fun(list(F4),list(E)),map(F4,E),G),Xsa))) = aa(list(F4),list(A),aa(fun(F4,A),fun(list(F4),list(A)),map(F4,A),aa(fun(F4,E),fun(F4,A),aa(fun(F4,B),fun(fun(F4,E),fun(F4,A)),aTP_Lamp_hh(fun(B,fun(E,A)),fun(fun(F4,B),fun(fun(F4,E),fun(F4,A))),H2),F2),G)),Xsa) ) ).

% map2_map_map
tff(fact_4014_list_Omap__ident,axiom,
    ! [A: $tType,T2: list(A)] : ( aa(list(A),list(A),aa(fun(A,A),fun(list(A),list(A)),map(A,A),aTP_Lamp_ab(A,A)),T2) = T2 ) ).

% list.map_ident
tff(fact_4015_rotate1__map,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xsa: list(B)] : ( aa(list(A),list(A),rotate1(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Xsa)) = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),aa(list(B),list(B),rotate1(B),Xsa)) ) ).

% rotate1_map
tff(fact_4016_remdups_Osimps_I1_J,axiom,
    ! [A: $tType] : ( aa(list(A),list(A),remdups(A),nil(A)) = nil(A) ) ).

% remdups.simps(1)
tff(fact_4017_rotate__map,axiom,
    ! [A: $tType,B: $tType,N: nat,F2: fun(B,A),Xsa: list(B)] : ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),rotate(A),N),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Xsa)) = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),aa(list(B),list(B),aa(nat,fun(list(B),list(B)),rotate(B),N),Xsa)) ) ).

% rotate_map
tff(fact_4018_ex__map__conv,axiom,
    ! [B: $tType,A: $tType,Ys: list(B),F2: fun(A,B)] :
      ( ? [Xs2: list(A)] : ( Ys = aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),Xs2) )
    <=> ! [X6: B] :
          ( member(B,X6,aa(list(B),set(B),set2(B),Ys))
         => ? [Xa2: A] : ( X6 = aa(A,B,F2,Xa2) ) ) ) ).

% ex_map_conv
tff(fact_4019_map__cong,axiom,
    ! [B: $tType,A: $tType,Xsa: list(A),Ys: list(A),F2: fun(A,B),G: fun(A,B)] :
      ( ( Xsa = Ys )
     => ( ! [X: A] :
            ( member(A,X,aa(list(A),set(A),set2(A),Ys))
           => ( aa(A,B,F2,X) = aa(A,B,G,X) ) )
       => ( aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),Xsa) = aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),G),Ys) ) ) ) ).

% map_cong
tff(fact_4020_map__idI,axiom,
    ! [A: $tType,Xsa: list(A),F2: fun(A,A)] :
      ( ! [X: A] :
          ( member(A,X,aa(list(A),set(A),set2(A),Xsa))
         => ( aa(A,A,F2,X) = X ) )
     => ( aa(list(A),list(A),aa(fun(A,A),fun(list(A),list(A)),map(A,A),F2),Xsa) = Xsa ) ) ).

% map_idI
tff(fact_4021_map__ext,axiom,
    ! [B: $tType,A: $tType,Xsa: list(A),F2: fun(A,B),G: fun(A,B)] :
      ( ! [X: A] :
          ( member(A,X,aa(list(A),set(A),set2(A),Xsa))
         => ( aa(A,B,F2,X) = aa(A,B,G,X) ) )
     => ( aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),Xsa) = aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),G),Xsa) ) ) ).

% map_ext
tff(fact_4022_list_Oinj__map__strong,axiom,
    ! [B: $tType,A: $tType,Xa: list(A),Xaa: list(A),F2: fun(A,B),Fa2: fun(A,B)] :
      ( ! [Z3: A,Za: A] :
          ( member(A,Z3,aa(list(A),set(A),set2(A),Xa))
         => ( member(A,Za,aa(list(A),set(A),set2(A),Xaa))
           => ( ( aa(A,B,F2,Z3) = aa(A,B,Fa2,Za) )
             => ( Z3 = Za ) ) ) )
     => ( ( aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),Xa) = aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),Fa2),Xaa) )
       => ( Xa = Xaa ) ) ) ).

% list.inj_map_strong
tff(fact_4023_list_Omap__cong0,axiom,
    ! [B: $tType,A: $tType,Xa: list(A),F2: fun(A,B),G: fun(A,B)] :
      ( ! [Z3: A] :
          ( member(A,Z3,aa(list(A),set(A),set2(A),Xa))
         => ( aa(A,B,F2,Z3) = aa(A,B,G,Z3) ) )
     => ( aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),Xa) = aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),G),Xa) ) ) ).

% list.map_cong0
tff(fact_4024_list_Omap__cong,axiom,
    ! [B: $tType,A: $tType,Xa: list(A),Ya: list(A),F2: fun(A,B),G: fun(A,B)] :
      ( ( Xa = Ya )
     => ( ! [Z3: A] :
            ( member(A,Z3,aa(list(A),set(A),set2(A),Ya))
           => ( aa(A,B,F2,Z3) = aa(A,B,G,Z3) ) )
       => ( aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),Xa) = aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),G),Ya) ) ) ) ).

% list.map_cong
tff(fact_4025_map__tl,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xsa: list(B)] : ( aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),aa(list(B),list(B),tl(B),Xsa)) = aa(list(A),list(A),tl(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Xsa)) ) ).

% map_tl
tff(fact_4026_drop__map,axiom,
    ! [A: $tType,B: $tType,N: nat,F2: fun(B,A),Xsa: list(B)] : ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Xsa)) = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),aa(list(B),list(B),aa(nat,fun(list(B),list(B)),drop(B),N),Xsa)) ) ).

% drop_map
tff(fact_4027_map__update,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xsa: list(B),K3: nat,Y: B] : ( aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),aa(B,list(B),aa(nat,fun(B,list(B)),aa(list(B),fun(nat,fun(B,list(B))),list_update(B),Xsa),K3),Y)) = aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Xsa)),K3),aa(B,A,F2,Y)) ) ).

% map_update
tff(fact_4028_map__eq__imp__length__eq,axiom,
    ! [A: $tType,B: $tType,E: $tType,F2: fun(B,A),Xsa: list(B),G: fun(E,A),Ys: list(E)] :
      ( ( aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Xsa) = aa(list(E),list(A),aa(fun(E,A),fun(list(E),list(A)),map(E,A),G),Ys) )
     => ( aa(list(B),nat,size_size(list(B)),Xsa) = aa(list(E),nat,size_size(list(E)),Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_4029_map__eq__nth__eq,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),L: list(B),L2: list(B),Ia: nat] :
      ( ( aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),L) = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),L2) )
     => ( aa(B,A,F2,aa(nat,B,nth(B,L),Ia)) = aa(B,A,F2,aa(nat,B,nth(B,L2),Ia)) ) ) ).

% map_eq_nth_eq
tff(fact_4030_take__map,axiom,
    ! [A: $tType,B: $tType,N: nat,F2: fun(B,A),Xsa: list(B)] : ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Xsa)) = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),aa(list(B),list(B),aa(nat,fun(list(B),list(B)),take(B),N),Xsa)) ) ).

% take_map
tff(fact_4031_distinct__mapI,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),L: list(B)] :
      ( aa(list(A),$o,distinct(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),L))
     => aa(list(B),$o,distinct(B),L) ) ).

% distinct_mapI
tff(fact_4032_remdups__map__remdups,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xsa: list(B)] : ( aa(list(A),list(A),remdups(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),aa(list(B),list(B),remdups(B),Xsa))) = aa(list(A),list(A),remdups(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Xsa)) ) ).

% remdups_map_remdups
tff(fact_4033_remdups__remdups,axiom,
    ! [A: $tType,Xsa: list(A)] : ( aa(list(A),list(A),remdups(A),aa(list(A),list(A),remdups(A),Xsa)) = aa(list(A),list(A),remdups(A),Xsa) ) ).

% remdups_remdups
tff(fact_4034_map__concat,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xsa: list(list(B))] : ( aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),aa(list(list(B)),list(B),concat(B),Xsa)) = aa(list(list(A)),list(A),concat(A),aa(list(list(B)),list(list(A)),aa(fun(list(B),list(A)),fun(list(list(B)),list(list(A))),map(list(B),list(A)),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2)),Xsa)) ) ).

% map_concat
tff(fact_4035_rev__map,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xsa: list(B)] : ( aa(list(A),list(A),rev(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Xsa)) = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),aa(list(B),list(B),rev(B),Xsa)) ) ).

% rev_map
tff(fact_4036_map__butlast,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xsa: list(B)] : ( aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),aa(list(B),list(B),butlast(B),Xsa)) = aa(list(A),list(A),butlast(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Xsa)) ) ).

% map_butlast
tff(fact_4037_list_Osimps_I8_J,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A)] : ( aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),nil(B)) = nil(A) ) ).

% list.simps(8)
tff(fact_4038_List_Omap__eq__append__conv,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xsa: list(B),Ys: list(A),Zs: list(A)] :
      ( ( aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Xsa) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Zs) )
    <=> ? [Us2: list(B),Vs3: list(B)] :
          ( ( Xsa = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Us2),Vs3) )
          & ( Ys = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Us2) )
          & ( Zs = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Vs3) ) ) ) ).

% List.map_eq_append_conv
tff(fact_4039_List_Oappend__eq__map__conv,axiom,
    ! [A: $tType,B: $tType,Ys: list(A),Zs: list(A),F2: fun(B,A),Xsa: list(B)] :
      ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Zs) = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Xsa) )
    <=> ? [Us2: list(B),Vs3: list(B)] :
          ( ( Xsa = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Us2),Vs3) )
          & ( Ys = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Us2) )
          & ( Zs = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Vs3) ) ) ) ).

% List.append_eq_map_conv
tff(fact_4040_Misc_Omap__eq__append__conv,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),Ls3: list(B),Fl: list(A),Fl2: list(A)] :
      ( ( aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Ls3) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Fl),Fl2) )
    <=> ? [L4: list(B),L6: list(B)] :
          ( ( Ls3 = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),L4),L6) )
          & ( aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),L4) = Fl )
          & ( aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),L6) = Fl2 ) ) ) ).

% Misc.map_eq_append_conv
tff(fact_4041_Misc_Oappend__eq__map__conv,axiom,
    ! [B: $tType,A: $tType,Fl: list(A),Fl2: list(A),F2: fun(B,A),Ls3: list(B)] :
      ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Fl),Fl2) = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Ls3) )
    <=> ? [L4: list(B),L6: list(B)] :
          ( ( Ls3 = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),L4),L6) )
          & ( aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),L4) = Fl )
          & ( aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),L6) = Fl2 ) ) ) ).

% Misc.append_eq_map_conv
tff(fact_4042_map__eq__appendE,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),Ls3: list(B),Fl: list(A),Fl2: list(A)] :
      ( ( aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Ls3) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Fl),Fl2) )
     => ~ ! [L3: list(B),L7: list(B)] :
            ( ( Ls3 = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),L3),L7) )
           => ( ( aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),L3) = Fl )
             => ( aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),L7) != Fl2 ) ) ) ) ).

% map_eq_appendE
tff(fact_4043_append__eq__mapE,axiom,
    ! [B: $tType,A: $tType,Fl: list(A),Fl2: list(A),F2: fun(B,A),Ls3: list(B)] :
      ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Fl),Fl2) = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Ls3) )
     => ~ ! [L3: list(B),L7: list(B)] :
            ( ( Ls3 = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),L3),L7) )
           => ( ( aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),L3) = Fl )
             => ( aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),L7) != Fl2 ) ) ) ) ).

% append_eq_mapE
tff(fact_4044_distinct__remdups__id,axiom,
    ! [A: $tType,Xsa: list(A)] :
      ( aa(list(A),$o,distinct(A),Xsa)
     => ( aa(list(A),list(A),remdups(A),Xsa) = Xsa ) ) ).

% distinct_remdups_id
tff(fact_4045_remdups__append2,axiom,
    ! [A: $tType,Xsa: list(A),Ys: list(A)] : ( aa(list(A),list(A),remdups(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),aa(list(A),list(A),remdups(A),Ys))) = aa(list(A),list(A),remdups(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),Ys)) ) ).

% remdups_append2
tff(fact_4046_sorted__wrt__map,axiom,
    ! [A: $tType,B: $tType,R: fun(A,fun(A,$o)),F2: fun(B,A),Xsa: list(B)] :
      ( sorted_wrt(A,R,aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Xsa))
    <=> sorted_wrt(B,aa(fun(B,A),fun(B,fun(B,$o)),aTP_Lamp_hi(fun(A,fun(A,$o)),fun(fun(B,A),fun(B,fun(B,$o))),R),F2),Xsa) ) ).

% sorted_wrt_map
tff(fact_4047_case__prodE_H,axiom,
    ! [B: $tType,A: $tType,E: $tType,C2: fun(A,fun(B,fun(E,$o))),P2: product_prod(A,B),Z2: E] :
      ( aa(E,$o,aa(product_prod(A,B),fun(E,$o),aa(fun(A,fun(B,fun(E,$o))),fun(product_prod(A,B),fun(E,$o)),product_case_prod(A,B,fun(E,$o)),C2),P2),Z2)
     => ~ ! [X: A,Y2: B] :
            ( ( P2 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y2) )
           => ~ aa(E,$o,aa(B,fun(E,$o),aa(A,fun(B,fun(E,$o)),C2,X),Y2),Z2) ) ) ).

% case_prodE'
tff(fact_4048_case__prodD_H,axiom,
    ! [B: $tType,A: $tType,E: $tType,R: fun(A,fun(B,fun(E,$o))),A3: A,B3: B,C2: E] :
      ( aa(E,$o,aa(product_prod(A,B),fun(E,$o),aa(fun(A,fun(B,fun(E,$o))),fun(product_prod(A,B),fun(E,$o)),product_case_prod(A,B,fun(E,$o)),R),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B3)),C2)
     => aa(E,$o,aa(B,fun(E,$o),aa(A,fun(B,fun(E,$o)),R,A3),B3),C2) ) ).

% case_prodD'
tff(fact_4049_map__zip__map2,axiom,
    ! [E: $tType,A: $tType,B: $tType,F4: $tType,F2: fun(product_prod(B,E),A),Xsa: list(B),G: fun(F4,E),Ys: list(F4)] : ( aa(list(product_prod(B,E)),list(A),aa(fun(product_prod(B,E),A),fun(list(product_prod(B,E)),list(A)),map(product_prod(B,E),A),F2),aa(list(E),list(product_prod(B,E)),aa(list(B),fun(list(E),list(product_prod(B,E))),zip(B,E),Xsa),aa(list(F4),list(E),aa(fun(F4,E),fun(list(F4),list(E)),map(F4,E),G),Ys))) = aa(list(product_prod(B,F4)),list(A),aa(fun(product_prod(B,F4),A),fun(list(product_prod(B,F4)),list(A)),map(product_prod(B,F4),A),aa(fun(B,fun(F4,A)),fun(product_prod(B,F4),A),product_case_prod(B,F4,A),aa(fun(F4,E),fun(B,fun(F4,A)),aTP_Lamp_hj(fun(product_prod(B,E),A),fun(fun(F4,E),fun(B,fun(F4,A))),F2),G))),aa(list(F4),list(product_prod(B,F4)),aa(list(B),fun(list(F4),list(product_prod(B,F4))),zip(B,F4),Xsa),Ys)) ) ).

% map_zip_map2
tff(fact_4050_map__zip__map,axiom,
    ! [B: $tType,A: $tType,F4: $tType,E: $tType,F2: fun(product_prod(B,E),A),G: fun(F4,B),Xsa: list(F4),Ys: list(E)] : ( aa(list(product_prod(B,E)),list(A),aa(fun(product_prod(B,E),A),fun(list(product_prod(B,E)),list(A)),map(product_prod(B,E),A),F2),aa(list(E),list(product_prod(B,E)),aa(list(B),fun(list(E),list(product_prod(B,E))),zip(B,E),aa(list(F4),list(B),aa(fun(F4,B),fun(list(F4),list(B)),map(F4,B),G),Xsa)),Ys)) = aa(list(product_prod(F4,E)),list(A),aa(fun(product_prod(F4,E),A),fun(list(product_prod(F4,E)),list(A)),map(product_prod(F4,E),A),aa(fun(F4,fun(E,A)),fun(product_prod(F4,E),A),product_case_prod(F4,E,A),aa(fun(F4,B),fun(F4,fun(E,A)),aTP_Lamp_hk(fun(product_prod(B,E),A),fun(fun(F4,B),fun(F4,fun(E,A))),F2),G))),aa(list(E),list(product_prod(F4,E)),aa(list(F4),fun(list(E),list(product_prod(F4,E))),zip(F4,E),Xsa),Ys)) ) ).

% map_zip_map
tff(fact_4051_sum__code,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(B,A),Xsa: list(B)] : ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),aa(list(B),set(B),set2(B),Xsa)) = groups8242544230860333062m_list(A,aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),G),aa(list(B),list(B),remdups(B),Xsa))) ) ) ).

% sum_code
tff(fact_4052_foldl__map,axiom,
    ! [A: $tType,B: $tType,E: $tType,G: fun(A,fun(B,A)),A3: A,F2: fun(E,B),Xsa: list(E)] : ( aa(list(B),A,aa(A,fun(list(B),A),aa(fun(A,fun(B,A)),fun(A,fun(list(B),A)),foldl(A,B),G),A3),aa(list(E),list(B),aa(fun(E,B),fun(list(E),list(B)),map(E,B),F2),Xsa)) = aa(list(E),A,aa(A,fun(list(E),A),aa(fun(A,fun(E,A)),fun(A,fun(list(E),A)),foldl(A,E),aa(fun(E,B),fun(A,fun(E,A)),aTP_Lamp_hl(fun(A,fun(B,A)),fun(fun(E,B),fun(A,fun(E,A))),G),F2)),A3),Xsa) ) ).

% foldl_map
tff(fact_4053_zip__map2,axiom,
    ! [B: $tType,A: $tType,E: $tType,Xsa: list(A),F2: fun(E,B),Ys: list(E)] : ( aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),Xsa),aa(list(E),list(B),aa(fun(E,B),fun(list(E),list(B)),map(E,B),F2),Ys)) = aa(list(product_prod(A,E)),list(product_prod(A,B)),aa(fun(product_prod(A,E),product_prod(A,B)),fun(list(product_prod(A,E)),list(product_prod(A,B))),map(product_prod(A,E),product_prod(A,B)),aa(fun(A,fun(E,product_prod(A,B))),fun(product_prod(A,E),product_prod(A,B)),product_case_prod(A,E,product_prod(A,B)),aTP_Lamp_hm(fun(E,B),fun(A,fun(E,product_prod(A,B))),F2))),aa(list(E),list(product_prod(A,E)),aa(list(A),fun(list(E),list(product_prod(A,E))),zip(A,E),Xsa),Ys)) ) ).

% zip_map2
tff(fact_4054_zip__map1,axiom,
    ! [A: $tType,E: $tType,B: $tType,F2: fun(E,A),Xsa: list(E),Ys: list(B)] : ( aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),aa(list(E),list(A),aa(fun(E,A),fun(list(E),list(A)),map(E,A),F2),Xsa)),Ys) = aa(list(product_prod(E,B)),list(product_prod(A,B)),aa(fun(product_prod(E,B),product_prod(A,B)),fun(list(product_prod(E,B)),list(product_prod(A,B))),map(product_prod(E,B),product_prod(A,B)),aa(fun(E,fun(B,product_prod(A,B))),fun(product_prod(E,B),product_prod(A,B)),product_case_prod(E,B,product_prod(A,B)),aTP_Lamp_hn(fun(E,A),fun(E,fun(B,product_prod(A,B))),F2))),aa(list(B),list(product_prod(E,B)),aa(list(E),fun(list(B),list(product_prod(E,B))),zip(E,B),Xsa),Ys)) ) ).

% zip_map1
tff(fact_4055_map__prod__fun__zip,axiom,
    ! [E: $tType,A: $tType,B: $tType,F4: $tType,F2: fun(E,A),G: fun(F4,B),Xsa: list(E),Ys: list(F4)] : ( aa(list(product_prod(E,F4)),list(product_prod(A,B)),aa(fun(product_prod(E,F4),product_prod(A,B)),fun(list(product_prod(E,F4)),list(product_prod(A,B))),map(product_prod(E,F4),product_prod(A,B)),aa(fun(E,fun(F4,product_prod(A,B))),fun(product_prod(E,F4),product_prod(A,B)),product_case_prod(E,F4,product_prod(A,B)),aa(fun(F4,B),fun(E,fun(F4,product_prod(A,B))),aTP_Lamp_ho(fun(E,A),fun(fun(F4,B),fun(E,fun(F4,product_prod(A,B)))),F2),G))),aa(list(F4),list(product_prod(E,F4)),aa(list(E),fun(list(F4),list(product_prod(E,F4))),zip(E,F4),Xsa),Ys)) = aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),aa(list(E),list(A),aa(fun(E,A),fun(list(E),list(A)),map(E,A),F2),Xsa)),aa(list(F4),list(B),aa(fun(F4,B),fun(list(F4),list(B)),map(F4,B),G),Ys)) ) ).

% map_prod_fun_zip
tff(fact_4056_insert__remdups,axiom,
    ! [A: $tType,Xa: A,Xsa: list(A)] : ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),insert(A),Xa),aa(list(A),list(A),remdups(A),Xsa)) = aa(list(A),list(A),remdups(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),insert(A),Xa),Xsa)) ) ).

% insert_remdups
tff(fact_4057_image__set,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xsa: list(B)] : ( aa(set(B),set(A),image2(B,A,F2),aa(list(B),set(B),set2(B),Xsa)) = aa(list(A),set(A),set2(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Xsa)) ) ).

% image_set
tff(fact_4058_map__consI_I2_J,axiom,
    ! [B: $tType,A: $tType,W2: list(A),L: list(A),F2: fun(B,A),Ww: list(B),A3: B] :
      ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),W2),L) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Ww)),L) )
     => ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(B,A,F2,A3)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),W2),L)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),A3),Ww))),L) ) ) ).

% map_consI(2)
tff(fact_4059_distinct__map__eq,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),L: list(B),Xa: B,Y: B] :
      ( aa(list(A),$o,distinct(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),L))
     => ( ( aa(B,A,F2,Xa) = aa(B,A,F2,Y) )
       => ( member(B,Xa,aa(list(B),set(B),set2(B),L))
         => ( member(B,Y,aa(list(B),set(B),set2(B),L))
           => ( Xa = Y ) ) ) ) ) ).

% distinct_map_eq
tff(fact_4060_hd__map,axiom,
    ! [B: $tType,A: $tType,Xsa: list(A),F2: fun(A,B)] :
      ( ( Xsa != nil(A) )
     => ( aa(list(B),B,hd(B),aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),Xsa)) = aa(A,B,F2,aa(list(A),A,hd(A),Xsa)) ) ) ).

% hd_map
tff(fact_4061_list_Omap__sel_I1_J,axiom,
    ! [B: $tType,A: $tType,A3: list(A),F2: fun(A,B)] :
      ( ( A3 != nil(A) )
     => ( aa(list(B),B,hd(B),aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),A3)) = aa(A,B,F2,aa(list(A),A,hd(A),A3)) ) ) ).

% list.map_sel(1)
tff(fact_4062_list_Omap__sel_I2_J,axiom,
    ! [B: $tType,A: $tType,A3: list(A),F2: fun(A,B)] :
      ( ( A3 != nil(A) )
     => ( aa(list(B),list(B),tl(B),aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),A3)) = aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),aa(list(A),list(A),tl(A),A3)) ) ) ).

% list.map_sel(2)
tff(fact_4063_last__map,axiom,
    ! [B: $tType,A: $tType,Xsa: list(A),F2: fun(A,B)] :
      ( ( Xsa != nil(A) )
     => ( last(B,aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),Xsa)) = aa(A,B,F2,last(A,Xsa)) ) ) ).

% last_map
tff(fact_4064_lists__image__witness,axiom,
    ! [A: $tType,B: $tType,Xa: list(A),F2: fun(B,A),Qa: set(B)] :
      ( member(list(A),Xa,aa(set(A),set(list(A)),lists(A),aa(set(B),set(A),image2(B,A,F2),Qa)))
     => ~ ! [Xo: list(B)] :
            ( member(list(B),Xo,aa(set(B),set(list(B)),lists(B),Qa))
           => ( Xa != aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Xo) ) ) ) ).

% lists_image_witness
tff(fact_4065_map__eq__map__tailrec,axiom,
    ! [B: $tType,A: $tType] : ( map(A,B) = map_tailrec(A,B) ) ).

% map_eq_map_tailrec
tff(fact_4066_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_4067_sorted__map,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F2: fun(B,A),Xsa: list(B)] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Xsa))
        <=> sorted_wrt(B,aTP_Lamp_hp(fun(B,A),fun(B,fun(B,$o)),F2),Xsa) ) ) ).

% sorted_map
tff(fact_4068_wf__if__measure,axiom,
    ! [A: $tType,Pa: fun(A,$o),F2: fun(A,nat),G: fun(A,A)] :
      ( ! [X: A] :
          ( aa(A,$o,Pa,X)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,F2,aa(A,A,G,X))),aa(A,nat,F2,X)) )
     => wf(A,aa(fun(product_prod(A,A),$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_hq(fun(A,$o),fun(fun(A,A),fun(A,fun(A,$o))),Pa),G)))) ) ).

% wf_if_measure
tff(fact_4069_remdups_Osimps_I2_J,axiom,
    ! [A: $tType,Xa: A,Xsa: list(A)] :
      ( aa(list(A),list(A),remdups(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa)) = $ite(member(A,Xa,aa(list(A),set(A),set2(A),Xsa)),aa(list(A),list(A),remdups(A),Xsa),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),aa(list(A),list(A),remdups(A),Xsa))) ) ).

% remdups.simps(2)
tff(fact_4070_sum__list__mult__const,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_0(A)
     => ! [F2: fun(B,A),C2: A,Xsa: list(B)] : ( groups8242544230860333062m_list(A,aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),aa(A,fun(B,A),aTP_Lamp_fe(fun(B,A),fun(A,fun(B,A)),F2),C2)),Xsa)) = aa(A,A,aa(A,fun(A,A),times_times(A),groups8242544230860333062m_list(A,aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Xsa))),C2) ) ) ).

% sum_list_mult_const
tff(fact_4071_sum__list__const__mult,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_0(A)
     => ! [C2: A,F2: fun(B,A),Xsa: list(B)] : ( groups8242544230860333062m_list(A,aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),aa(fun(B,A),fun(B,A),aTP_Lamp_fd(A,fun(fun(B,A),fun(B,A)),C2),F2)),Xsa)) = aa(A,A,aa(A,fun(A,A),times_times(A),C2),groups8242544230860333062m_list(A,aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Xsa))) ) ) ).

% sum_list_const_mult
tff(fact_4072_sum__list__subtractf,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_add(A)
     => ! [F2: fun(B,A),G: fun(B,A),Xsa: list(B)] : ( groups8242544230860333062m_list(A,aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),aa(fun(B,A),fun(B,A),aTP_Lamp_fh(fun(B,A),fun(fun(B,A),fun(B,A)),F2),G)),Xsa)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),groups8242544230860333062m_list(A,aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Xsa))),groups8242544230860333062m_list(A,aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),G),Xsa))) ) ) ).

% sum_list_subtractf
tff(fact_4073_sorted__remdups,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xsa: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),Xsa)
         => sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),remdups(A),Xsa)) ) ) ).

% sorted_remdups
tff(fact_4074_set__eq__iff__mset__remdups__eq,axiom,
    ! [A: $tType,Xa: list(A),Y: list(A)] :
      ( ( aa(list(A),set(A),set2(A),Xa) = aa(list(A),set(A),set2(A),Y) )
    <=> ( aa(list(A),multiset(A),mset(A),aa(list(A),list(A),remdups(A),Xa)) = aa(list(A),multiset(A),mset(A),aa(list(A),list(A),remdups(A),Y)) ) ) ).

% set_eq_iff_mset_remdups_eq
tff(fact_4075_remove1__remdups,axiom,
    ! [A: $tType,Xsa: list(A),Xa: A] :
      ( aa(list(A),$o,distinct(A),Xsa)
     => ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),remove1(A),Xa),aa(list(A),list(A),remdups(A),Xsa)) = aa(list(A),list(A),remdups(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),remove1(A),Xa),Xsa)) ) ) ).

% remove1_remdups
tff(fact_4076_union__fold__insert,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),Aa2)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),B2) = finite_fold(A,set(A),insert3(A),B2,Aa2) ) ) ).

% union_fold_insert
tff(fact_4077_sorted__map__remove1,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F2: fun(B,A),Xsa: list(B),Xa: B] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Xsa))
         => sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),aa(list(B),list(B),aa(B,fun(list(B),list(B)),remove1(B),Xa),Xsa))) ) ) ).

% sorted_map_remove1
tff(fact_4078_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,$o),aTP_Lamp_hr(nat,fun(nat,fun(nat,$o)),N)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_ht(fun(nat,fun(nat,A)),fun(nat,A),G)),aa(nat,set(nat),set_ord_lessThan(nat),N)) ) ) ).

% sum.triangle_reindex
tff(fact_4079_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,$o),aTP_Lamp_hr(nat,fun(nat,fun(nat,$o)),N)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_hv(fun(nat,fun(nat,A)),fun(nat,A),G)),aa(nat,set(nat),set_ord_lessThan(nat),N)) ) ) ).

% prod.triangle_reindex
tff(fact_4080_finite__subset__wf,axiom,
    ! [A: $tType,Aa2: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),Aa2)
     => 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_hw(set(A),fun(set(A),fun(set(A),$o)),Aa2)))) ) ).

% finite_subset_wf
tff(fact_4081_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_hx(set(A),fun(set(A),fun(set(A),$o)),S)))) ) ).

% wf_bounded_supset
tff(fact_4082_sum__list__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( monoid_add(B)
        & ordere6658533253407199908up_add(B) )
     => ! [Xsa: list(A),F2: fun(A,B),G: fun(A,B)] :
          ( ! [X: A] :
              ( member(A,X,aa(list(A),set(A),set2(A),Xsa))
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X)),aa(A,B,G,X)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),groups8242544230860333062m_list(B,aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),Xsa))),groups8242544230860333062m_list(B,aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),G),Xsa))) ) ) ).

% sum_list_mono
tff(fact_4083_lists__image,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Aa2: set(B)] : ( aa(set(A),set(list(A)),lists(A),aa(set(B),set(A),image2(B,A,F2),Aa2)) = aa(set(list(B)),set(list(A)),image2(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2)),aa(set(B),set(list(B)),lists(B),Aa2)) ) ).

% lists_image
tff(fact_4084_map__tailrec__rev,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),As2: list(B),Bs: list(A)] : ( map_tailrec_rev(B,A,F2,As2,Bs) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),rev(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),As2))),Bs) ) ).

% map_tailrec_rev
tff(fact_4085_minus__fold__remove,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),Aa2)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B2),Aa2) = finite_fold(A,set(A),remove(A),B2,Aa2) ) ) ).

% minus_fold_remove
tff(fact_4086_length__remdups__card__conv,axiom,
    ! [A: $tType,Xsa: list(A)] : ( aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),remdups(A),Xsa)) = finite_card(A,aa(list(A),set(A),set2(A),Xsa)) ) ).

% length_remdups_card_conv
tff(fact_4087_length__remdups__card,axiom,
    ! [A: $tType,L: list(A)] : ( aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),remdups(A),L)) = finite_card(A,aa(list(A),set(A),set2(A),L)) ) ).

% length_remdups_card
tff(fact_4088_Sup__SUP__eq2,axiom,
    ! [B: $tType,A: $tType,S: set(fun(A,fun(B,$o))),X4: A,Xa4: 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,$o))),S),X4),Xa4)
    <=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Xa4),aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),aa(set(fun(product_prod(A,B),$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_4089_sum__list__strict__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( monoid_add(B)
        & strict9044650504122735259up_add(B) )
     => ! [Xsa: list(A),F2: fun(A,B),G: fun(A,B)] :
          ( ( Xsa != nil(A) )
         => ( ! [X: A] :
                ( member(A,X,aa(list(A),set(A),set2(A),Xsa))
               => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,X)),aa(A,B,G,X)) )
           => aa(B,$o,aa(B,fun(B,$o),ord_less(B),groups8242544230860333062m_list(B,aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),Xsa))),groups8242544230860333062m_list(B,aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),G),Xsa))) ) ) ) ).

% sum_list_strict_mono
tff(fact_4090_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),$o),aTP_Lamp_hy(set(A),fun(set(A),$o)))) ) ).

% finite_psubset_def
tff(fact_4091_distinct__idx,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),L: list(B),Ia: nat,J: nat] :
      ( aa(list(A),$o,distinct(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),L))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),aa(list(B),nat,size_size(list(B)),L))
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(B),nat,size_size(list(B)),L))
         => ( ( aa(B,A,F2,aa(nat,B,nth(B,L),Ia)) = aa(B,A,F2,aa(nat,B,nth(B,L),J)) )
           => ( Ia = J ) ) ) ) ) ).

% distinct_idx
tff(fact_4092_map__upd__eq,axiom,
    ! [B: $tType,A: $tType,Ia: nat,L: list(A),F2: fun(A,B),Xa: A] :
      ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),aa(list(A),nat,size_size(list(A)),L))
       => ( aa(A,B,F2,aa(nat,A,nth(A,L),Ia)) = aa(A,B,F2,Xa) ) )
     => ( aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),L),Ia),Xa)) = aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),L) ) ) ).

% map_upd_eq
tff(fact_4093_sum__list__distinct__conv__sum__set,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [Xsa: list(A),F2: fun(A,B)] :
          ( aa(list(A),$o,distinct(A),Xsa)
         => ( groups8242544230860333062m_list(B,aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),Xsa)) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),aa(list(A),set(A),set2(A),Xsa)) ) ) ) ).

% sum_list_distinct_conv_sum_set
tff(fact_4094_sum_Odistinct__set__conv__list,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [Xsa: list(A),G: fun(A,B)] :
          ( aa(list(A),$o,distinct(A),Xsa)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),aa(list(A),set(A),set2(A),Xsa)) = groups8242544230860333062m_list(B,aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),G),Xsa)) ) ) ) ).

% sum.distinct_set_conv_list
tff(fact_4095_sum__list__map__remove1,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [Xa: A,Xsa: list(A),F2: fun(A,B)] :
          ( member(A,Xa,aa(list(A),set(A),set2(A),Xsa))
         => ( groups8242544230860333062m_list(B,aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),Xsa)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,F2,Xa)),groups8242544230860333062m_list(B,aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),remove1(A),Xa),Xsa)))) ) ) ) ).

% sum_list_map_remove1
tff(fact_4096_Sup__fold__sup,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Aa2: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( aa(set(A),A,complete_Sup_Sup(A),Aa2) = finite_fold(A,A,sup_sup(A),bot_bot(A),Aa2) ) ) ) ).

% Sup_fold_sup
tff(fact_4097_sum__list__triv,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(A)
     => ! [R2: A,Xsa: list(B)] : ( groups8242544230860333062m_list(A,aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),aTP_Lamp_ey(A,fun(B,A),R2)),Xsa)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(list(B),nat,size_size(list(B)),Xsa))),R2) ) ) ).

% sum_list_triv
tff(fact_4098_Min_Oeq__fold,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Aa2: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( lattic643756798350308766er_Min(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),Aa2)) = finite_fold(A,A,ord_min(A),Xa,Aa2) ) ) ) ).

% Min.eq_fold
tff(fact_4099_image__fold__insert,axiom,
    ! [B: $tType,A: $tType,Aa2: set(A),F2: fun(A,B)] :
      ( aa(set(A),$o,finite_finite2(A),Aa2)
     => ( aa(set(A),set(B),image2(A,B,F2),Aa2) = finite_fold(A,set(B),aTP_Lamp_hz(fun(A,B),fun(A,fun(set(B),set(B))),F2),bot_bot(set(B)),Aa2) ) ) ).

% image_fold_insert
tff(fact_4100_trancl__insert2,axiom,
    ! [A: $tType,A3: A,B3: A,R2: set(product_prod(A,A))] : ( transitive_trancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(product_prod(A,A),fun(set(product_prod(A,A)),set(product_prod(A,A))),insert3(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B3)),R2)) = aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),transitive_trancl(A,R2)),aa(fun(product_prod(A,A),$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,$o))),aTP_Lamp_ia(A,fun(A,fun(set(product_prod(A,A)),fun(A,fun(A,$o)))),A3),B3),R2)))) ) ).

% trancl_insert2
tff(fact_4101_map__by__foldl,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),L: list(B)] : ( aa(list(B),list(A),aa(list(A),fun(list(B),list(A)),aa(fun(list(A),fun(B,list(A))),fun(list(A),fun(list(B),list(A))),foldl(list(A),B),aTP_Lamp_ib(fun(B,A),fun(list(A),fun(B,list(A))),F2)),nil(A)),L) = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),L) ) ).

% map_by_foldl
tff(fact_4102_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),$o),aTP_Lamp_ic(set(product_prod(A,A)),fun(list(A),fun(list(A),$o)),R2))) ) ).

% lenlex_conv
tff(fact_4103_map__distinct__upd__conv,axiom,
    ! [B: $tType,A: $tType,Ia: nat,L: list(A),F2: fun(A,B),Xa: B] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),aa(list(A),nat,size_size(list(A)),L))
     => ( aa(list(A),$o,distinct(A),L)
       => ( aa(B,list(B),aa(nat,fun(B,list(B)),aa(list(B),fun(nat,fun(B,list(B))),list_update(B),aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),L)),Ia),Xa) = aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),fun_upd(A,B,F2,aa(nat,A,nth(A,L),Ia),Xa)),L) ) ) ) ).

% map_distinct_upd_conv
tff(fact_4104_aux,axiom,
    ! [B: $tType,A: $tType,Pa: fun(A,fun(B,assn)),A3: A,As2: list(A),C2: B,Cs: list(B)] : ( finite_fold(nat,assn,aa(list(B),fun(nat,fun(assn,assn)),aa(B,fun(list(B),fun(nat,fun(assn,assn))),aa(list(A),fun(B,fun(list(B),fun(nat,fun(assn,assn)))),aa(A,fun(list(A),fun(B,fun(list(B),fun(nat,fun(assn,assn))))),aTP_Lamp_id(fun(A,fun(B,assn)),fun(A,fun(list(A),fun(B,fun(list(B),fun(nat,fun(assn,assn)))))),Pa),A3),As2),C2),Cs),one_one(assn),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,aa(list(A),nat,size_size(list(A)),As2)))) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),Pa,A3),C2)),finite_fold(nat,assn,aa(list(B),fun(nat,fun(assn,assn)),aa(list(A),fun(list(B),fun(nat,fun(assn,assn))),aTP_Lamp_hb(fun(A,fun(B,assn)),fun(list(A),fun(list(B),fun(nat,fun(assn,assn)))),Pa),As2),Cs),one_one(assn),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(A),nat,size_size(list(A)),As2)))) ) ).

% aux
tff(fact_4105_List_Olexordp__def,axiom,
    ! [A: $tType,R2: fun(A,fun(A,$o)),Xsa: list(A),Ys: list(A)] :
      ( lexordp(A,R2,Xsa,Ys)
    <=> member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xsa),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_4106_listrel1p__def,axiom,
    ! [A: $tType,R2: fun(A,fun(A,$o)),Xsa: list(A),Ys: list(A)] :
      ( listrel1p(A,R2,Xsa,Ys)
    <=> member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xsa),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_4107_mult1__lessE,axiom,
    ! [A: $tType,N7: multiset(A),M5: multiset(A),R2: fun(A,fun(A,$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)),N7),M5),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)
       => ~ ! [A4: A,M0: multiset(A)] :
              ( ( M5 = add_mset(A,A4,M0) )
             => ! [K5: multiset(A)] :
                  ( ( N7 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M0),K5) )
                 => ( ~ member(A,A4,aa(multiset(A),set(A),set_mset(A),K5))
                   => ~ ! [B9: A] :
                          ( member(A,B9,aa(multiset(A),set(A),set_mset(A),K5))
                         => aa(A,$o,aa(A,fun(A,$o),R2,B9),A4) ) ) ) ) ) ) ).

% mult1_lessE
tff(fact_4108_folding__insort__key_Ofinite__set__strict__sorted,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),S: set(B),F2: fun(B,A),Aa2: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S,F2)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),Aa2),S)
       => ( aa(set(B),$o,finite_finite2(B),Aa2)
         => ~ ! [L3: list(B)] :
                ( sorted_wrt(A,Less,aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),L3))
               => ( ( aa(list(B),set(B),set2(B),L3) = Aa2 )
                 => ( aa(list(B),nat,size_size(list(B)),L3) != finite_card(B,Aa2) ) ) ) ) ) ) ).

% folding_insort_key.finite_set_strict_sorted
tff(fact_4109_product__concat__map,axiom,
    ! [B: $tType,A: $tType,Xsa: list(A),Ys: list(B)] : ( aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),product(A,B),Xsa),Ys) = aa(list(list(product_prod(A,B))),list(product_prod(A,B)),concat(product_prod(A,B)),aa(list(A),list(list(product_prod(A,B))),aa(fun(A,list(product_prod(A,B))),fun(list(A),list(list(product_prod(A,B)))),map(A,list(product_prod(A,B))),aTP_Lamp_ie(list(B),fun(A,list(product_prod(A,B))),Ys)),Xsa)) ) ).

% product_concat_map
tff(fact_4110_asymp_Ocases,axiom,
    ! [A: $tType,A3: fun(A,fun(A,$o))] :
      ( asymp(A,A3)
     => ! [A13: A,B9: A] :
          ( aa(A,$o,aa(A,fun(A,$o),A3,A13),B9)
         => ~ aa(A,$o,aa(A,fun(A,$o),A3,B9),A13) ) ) ).

% asymp.cases
tff(fact_4111_asymp_Osimps,axiom,
    ! [A: $tType,A3: fun(A,fun(A,$o))] :
      ( asymp(A,A3)
    <=> ? [R7: fun(A,fun(A,$o))] :
          ( ( A3 = R7 )
          & ! [X6: A,Xa2: A] :
              ( aa(A,$o,aa(A,fun(A,$o),R7,X6),Xa2)
             => ~ aa(A,$o,aa(A,fun(A,$o),R7,Xa2),X6) ) ) ) ).

% asymp.simps
tff(fact_4112_asymp_Ointros,axiom,
    ! [A: $tType,R: fun(A,fun(A,$o))] :
      ( ! [A4: A,B4: A] :
          ( aa(A,$o,aa(A,fun(A,$o),R,A4),B4)
         => ~ aa(A,$o,aa(A,fun(A,$o),R,B4),A4) )
     => asymp(A,R) ) ).

% asymp.intros
tff(fact_4113_asympD,axiom,
    ! [A: $tType,R: fun(A,fun(A,$o)),Xa: A,Y: A] :
      ( asymp(A,R)
     => ( aa(A,$o,aa(A,fun(A,$o),R,Xa),Y)
       => ~ aa(A,$o,aa(A,fun(A,$o),R,Y),Xa) ) ) ).

% asympD
tff(fact_4114_n__lists_Osimps_I2_J,axiom,
    ! [A: $tType,N: nat,Xsa: list(A)] : ( n_lists(A,aa(nat,nat,suc,N),Xsa) = aa(list(list(list(A))),list(list(A)),concat(list(A)),aa(list(list(A)),list(list(list(A))),aa(fun(list(A),list(list(A))),fun(list(list(A)),list(list(list(A)))),map(list(A),list(list(A))),aTP_Lamp_ig(list(A),fun(list(A),list(list(A))),Xsa)),n_lists(A,N,Xsa))) ) ).

% n_lists.simps(2)
tff(fact_4115_product__lists_Osimps_I2_J,axiom,
    ! [A: $tType,Xsa: list(A),Xss2: list(list(A))] : ( aa(list(list(A)),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)),Xsa),Xss2)) = aa(list(list(list(A))),list(list(A)),concat(list(A)),aa(list(A),list(list(list(A))),aa(fun(A,list(list(A))),fun(list(A),list(list(list(A)))),map(A,list(list(A))),aTP_Lamp_ih(list(list(A)),fun(A,list(list(A))),Xss2)),Xsa)) ) ).

% product_lists.simps(2)
tff(fact_4116_asymp__less,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => asymp(A,ord_less(A)) ) ).

% asymp_less
tff(fact_4117_length__concat,axiom,
    ! [A: $tType,Xss2: list(list(A))] : ( aa(list(A),nat,size_size(list(A)),aa(list(list(A)),list(A),concat(A),Xss2)) = groups8242544230860333062m_list(nat,aa(list(list(A)),list(nat),aa(fun(list(A),nat),fun(list(list(A)),list(nat)),map(list(A),nat),size_size(list(A))),Xss2)) ) ).

% length_concat
tff(fact_4118_zip__same__conv__map,axiom,
    ! [A: $tType,Xsa: list(A)] : ( aa(list(A),list(product_prod(A,A)),aa(list(A),fun(list(A),list(product_prod(A,A))),zip(A,A),Xsa),Xsa) = aa(list(A),list(product_prod(A,A)),aa(fun(A,product_prod(A,A)),fun(list(A),list(product_prod(A,A))),map(A,product_prod(A,A)),aTP_Lamp_ii(A,product_prod(A,A))),Xsa) ) ).

% zip_same_conv_map
tff(fact_4119_List_Obind__def,axiom,
    ! [A: $tType,B: $tType,Xsa: list(B),F2: fun(B,list(A))] : ( bind(B,A,Xsa,F2) = aa(list(list(A)),list(A),concat(A),aa(list(B),list(list(A)),aa(fun(B,list(A)),fun(list(B),list(list(A))),map(B,list(A)),F2),Xsa)) ) ).

% List.bind_def
tff(fact_4120_asymp__greater,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => asymp(A,aTP_Lamp_ij(A,fun(A,$o))) ) ).

% asymp_greater
tff(fact_4121_folding__insort__key_Odistinct__if__distinct__map,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),S: set(B),F2: fun(B,A),Xsa: list(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S,F2)
     => ( aa(list(A),$o,distinct(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Xsa))
       => aa(list(B),$o,distinct(B),Xsa) ) ) ).

% folding_insort_key.distinct_if_distinct_map
tff(fact_4122_sum__list__Suc,axiom,
    ! [A: $tType,F2: fun(A,nat),Xsa: list(A)] : ( groups8242544230860333062m_list(nat,aa(list(A),list(nat),aa(fun(A,nat),fun(list(A),list(nat)),map(A,nat),aTP_Lamp_ik(fun(A,nat),fun(A,nat),F2)),Xsa)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),groups8242544230860333062m_list(nat,aa(list(A),list(nat),aa(fun(A,nat),fun(list(A),list(nat)),map(A,nat),F2),Xsa))),aa(list(A),nat,size_size(list(A)),Xsa)) ) ).

% sum_list_Suc
tff(fact_4123_distinct__set__subseqs,axiom,
    ! [A: $tType,Xsa: list(A)] :
      ( aa(list(A),$o,distinct(A),Xsa)
     => aa(list(set(A)),$o,distinct(set(A)),aa(list(list(A)),list(set(A)),aa(fun(list(A),set(A)),fun(list(list(A)),list(set(A))),map(list(A),set(A)),set2(A)),aa(list(A),list(list(A)),subseqs(A),Xsa))) ) ).

% distinct_set_subseqs
tff(fact_4124_zip__left__commute,axiom,
    ! [B: $tType,A: $tType,E: $tType,Xsa: list(A),Ys: list(B),Zs: list(E)] : ( aa(list(product_prod(B,E)),list(product_prod(A,product_prod(B,E))),aa(list(A),fun(list(product_prod(B,E)),list(product_prod(A,product_prod(B,E)))),zip(A,product_prod(B,E)),Xsa),aa(list(E),list(product_prod(B,E)),aa(list(B),fun(list(E),list(product_prod(B,E))),zip(B,E),Ys),Zs)) = aa(list(product_prod(B,product_prod(A,E))),list(product_prod(A,product_prod(B,E))),aa(fun(product_prod(B,product_prod(A,E)),product_prod(A,product_prod(B,E))),fun(list(product_prod(B,product_prod(A,E))),list(product_prod(A,product_prod(B,E)))),map(product_prod(B,product_prod(A,E)),product_prod(A,product_prod(B,E))),aa(fun(B,fun(product_prod(A,E),product_prod(A,product_prod(B,E)))),fun(product_prod(B,product_prod(A,E)),product_prod(A,product_prod(B,E))),product_case_prod(B,product_prod(A,E),product_prod(A,product_prod(B,E))),aTP_Lamp_im(B,fun(product_prod(A,E),product_prod(A,product_prod(B,E)))))),aa(list(product_prod(A,E)),list(product_prod(B,product_prod(A,E))),aa(list(B),fun(list(product_prod(A,E)),list(product_prod(B,product_prod(A,E)))),zip(B,product_prod(A,E)),Ys),aa(list(E),list(product_prod(A,E)),aa(list(A),fun(list(E),list(product_prod(A,E))),zip(A,E),Xsa),Zs))) ) ).

% zip_left_commute
tff(fact_4125_zip__assoc,axiom,
    ! [B: $tType,A: $tType,E: $tType,Xsa: list(A),Ys: list(B),Zs: list(E)] : ( aa(list(product_prod(B,E)),list(product_prod(A,product_prod(B,E))),aa(list(A),fun(list(product_prod(B,E)),list(product_prod(A,product_prod(B,E)))),zip(A,product_prod(B,E)),Xsa),aa(list(E),list(product_prod(B,E)),aa(list(B),fun(list(E),list(product_prod(B,E))),zip(B,E),Ys),Zs)) = aa(list(product_prod(product_prod(A,B),E)),list(product_prod(A,product_prod(B,E))),aa(fun(product_prod(product_prod(A,B),E),product_prod(A,product_prod(B,E))),fun(list(product_prod(product_prod(A,B),E)),list(product_prod(A,product_prod(B,E)))),map(product_prod(product_prod(A,B),E),product_prod(A,product_prod(B,E))),aa(fun(product_prod(A,B),fun(E,product_prod(A,product_prod(B,E)))),fun(product_prod(product_prod(A,B),E),product_prod(A,product_prod(B,E))),product_case_prod(product_prod(A,B),E,product_prod(A,product_prod(B,E))),aa(fun(A,fun(B,fun(E,product_prod(A,product_prod(B,E))))),fun(product_prod(A,B),fun(E,product_prod(A,product_prod(B,E)))),product_case_prod(A,B,fun(E,product_prod(A,product_prod(B,E)))),aTP_Lamp_in(A,fun(B,fun(E,product_prod(A,product_prod(B,E)))))))),aa(list(E),list(product_prod(product_prod(A,B),E)),aa(list(product_prod(A,B)),fun(list(E),list(product_prod(product_prod(A,B),E))),zip(product_prod(A,B),E),aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),Xsa),Ys)),Zs)) ) ).

% zip_assoc
tff(fact_4126_zip__commute,axiom,
    ! [B: $tType,A: $tType,Xsa: list(A),Ys: list(B)] : ( aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),Xsa),Ys) = aa(list(product_prod(B,A)),list(product_prod(A,B)),aa(fun(product_prod(B,A),product_prod(A,B)),fun(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_io(B,fun(A,product_prod(A,B))))),aa(list(A),list(product_prod(B,A)),aa(list(B),fun(list(A),list(product_prod(B,A))),zip(B,A),Ys),Xsa)) ) ).

% zip_commute
tff(fact_4127_sum__set__upto__conv__sum__list__int,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [F2: fun(int,A),Ia: int,J: int] : ( aa(set(int),A,aa(fun(int,A),fun(set(int),A),groups7311177749621191930dd_sum(int,A),F2),aa(list(int),set(int),set2(int),upto(Ia,J))) = groups8242544230860333062m_list(A,aa(list(int),list(A),aa(fun(int,A),fun(list(int),list(A)),map(int,A),F2),upto(Ia,J))) ) ) ).

% sum_set_upto_conv_sum_list_int
tff(fact_4128_interv__sum__list__conv__sum__set__int,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [F2: fun(int,A),K3: int,L: int] : ( groups8242544230860333062m_list(A,aa(list(int),list(A),aa(fun(int,A),fun(list(int),list(A)),map(int,A),F2),upto(K3,L))) = aa(set(int),A,aa(fun(int,A),fun(set(int),A),groups7311177749621191930dd_sum(int,A),F2),aa(list(int),set(int),set2(int),upto(K3,L))) ) ) ).

% interv_sum_list_conv_sum_set_int
tff(fact_4129_rev__concat,axiom,
    ! [A: $tType,Xsa: list(list(A))] : ( aa(list(A),list(A),rev(A),aa(list(list(A)),list(A),concat(A),Xsa)) = aa(list(list(A)),list(A),concat(A),aa(list(list(A)),list(list(A)),aa(fun(list(A),list(A)),fun(list(list(A)),list(list(A))),map(list(A),list(A)),rev(A)),aa(list(list(A)),list(list(A)),rev(list(A)),Xsa))) ) ).

% rev_concat
tff(fact_4130_subseqs_Osimps_I2_J,axiom,
    ! [A: $tType,Xa: A,Xsa: list(A)] :
      ( aa(list(A),list(list(A)),subseqs(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa)) = $let(
          xss: list(list(A)),
          xss:= aa(list(A),list(list(A)),subseqs(A),Xsa),
          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)),aa(fun(list(A),list(A)),fun(list(list(A)),list(list(A))),map(list(A),list(A)),aa(A,fun(list(A),list(A)),cons(A),Xa)),xss)),xss) ) ) ).

% subseqs.simps(2)
tff(fact_4131_product_Osimps_I2_J,axiom,
    ! [A: $tType,B: $tType,Xa: A,Xsa: list(A),Ys: list(B)] : ( aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),product(A,B),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa)),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)),aa(fun(B,product_prod(A,B)),fun(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),Xa)),Ys)),aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),product(A,B),Xsa),Ys)) ) ).

% product.simps(2)
tff(fact_4132_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_ip(A,A)) ) ).

% sorted_list_of_set.folding_insort_key_axioms
tff(fact_4133_sum__list__map__eq__sum__count,axiom,
    ! [A: $tType,F2: fun(A,nat),Xsa: list(A)] : ( groups8242544230860333062m_list(nat,aa(list(A),list(nat),aa(fun(A,nat),fun(list(A),list(nat)),map(A,nat),F2),Xsa)) = aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aa(list(A),fun(A,nat),aTP_Lamp_iq(fun(A,nat),fun(list(A),fun(A,nat)),F2),Xsa)),aa(list(A),set(A),set2(A),Xsa)) ) ).

% sum_list_map_eq_sum_count
tff(fact_4134_mergesort__remdups__def,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xsa: list(A)] : ( aa(list(A),list(A),mergesort_remdups(A),Xsa) = merge_list(A,nil(list(A)),aa(list(A),list(list(A)),aa(fun(A,list(A)),fun(list(A),list(list(A))),map(A,list(A)),aTP_Lamp_ir(A,list(A))),Xsa)) ) ) ).

% mergesort_remdups_def
tff(fact_4135_sum__list__map__eq__sum__count2,axiom,
    ! [A: $tType,Xsa: list(A),X3: set(A),F2: fun(A,nat)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xsa)),X3)
     => ( aa(set(A),$o,finite_finite2(A),X3)
       => ( groups8242544230860333062m_list(nat,aa(list(A),list(nat),aa(fun(A,nat),fun(list(A),list(nat)),map(A,nat),F2),Xsa)) = aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aa(fun(A,nat),fun(A,nat),aTP_Lamp_is(list(A),fun(fun(A,nat),fun(A,nat)),Xsa),F2)),X3) ) ) ) ).

% sum_list_map_eq_sum_count2
tff(fact_4136_fold__union__pair,axiom,
    ! [B: $tType,A: $tType,B2: set(A),Xa: B,Aa2: set(product_prod(B,A))] :
      ( aa(set(A),$o,finite_finite2(A),B2)
     => ( 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_it(B,fun(A,set(product_prod(B,A))),Xa)),B2))),Aa2) = finite_fold(A,set(product_prod(B,A)),aTP_Lamp_iu(B,fun(A,fun(set(product_prod(B,A)),set(product_prod(B,A)))),Xa),Aa2,B2) ) ) ).

% fold_union_pair
tff(fact_4137_product__code,axiom,
    ! [B: $tType,A: $tType,Xsa: list(A),Ys: list(B)] : ( product_product(A,B,aa(list(A),set(A),set2(A),Xsa),aa(list(B),set(B),set2(B),Ys)) = aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),aa(list(list(product_prod(A,B))),list(product_prod(A,B)),concat(product_prod(A,B)),aa(list(A),list(list(product_prod(A,B))),aa(fun(A,list(product_prod(A,B))),fun(list(A),list(list(product_prod(A,B)))),map(A,list(product_prod(A,B))),aTP_Lamp_ie(list(B),fun(A,list(product_prod(A,B))),Ys)),Xsa))) ) ).

% product_code
tff(fact_4138_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_iw(set(product_prod(A,A)),fun(A,fun(A,fun(product_prod(A,A),$o))),R2)))) ) ).

% bsqr_def
tff(fact_4139_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_ix(nat,fun(nat,fun(nat,$o)),N)))) ).

% natLeq_on_wo_rel
tff(fact_4140_Field__natLeq__on,axiom,
    ! [N: nat] : ( aa(set(product_prod(nat,nat)),set(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_ix(nat,fun(nat,fun(nat,$o)),N)))) = aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_ac(nat,fun(nat,$o)),N)) ) ).

% Field_natLeq_on
tff(fact_4141_transpose_Osimps_I3_J,axiom,
    ! [A: $tType,Xa: A,Xsa: list(A),Xss2: 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),Xa),Xsa)),Xss2)) = 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),Xa),aa(list(list(A)),list(A),concat(A),aa(list(list(A)),list(list(A)),aa(fun(list(A),list(A)),fun(list(list(A)),list(list(A))),map(list(A),list(A)),aa(fun(A,fun(list(A),list(A))),fun(list(A),list(A)),aa(list(A),fun(fun(A,fun(list(A),list(A))),fun(list(A),list(A))),case_list(list(A),A),nil(A)),aTP_Lamp_iy(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)),Xsa),aa(list(list(list(A))),list(list(A)),concat(list(A)),aa(list(list(A)),list(list(list(A))),aa(fun(list(A),list(list(A))),fun(list(list(A)),list(list(list(A)))),map(list(A),list(list(A))),aa(fun(A,fun(list(A),list(list(A)))),fun(list(A),list(list(A))),aa(list(list(A)),fun(fun(A,fun(list(A),list(list(A)))),fun(list(A),list(list(A)))),case_list(list(list(A)),A),nil(list(A))),aTP_Lamp_iz(A,fun(list(A),list(list(A)))))),Xss2))))) ) ).

% transpose.simps(3)
tff(fact_4142_transpose_Oelims,axiom,
    ! [A: $tType,Xa: list(list(A)),Y: list(list(A))] :
      ( ( transpose(A,Xa) = Y )
     => ( ( ( Xa = nil(list(A)) )
         => ( Y != nil(list(A)) ) )
       => ( ! [Xss: list(list(A))] :
              ( ( Xa = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),nil(A)),Xss) )
             => ( Y != transpose(A,Xss) ) )
         => ~ ! [X: A,Xs: list(A),Xss: list(list(A))] :
                ( ( Xa = 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) )
               => ( 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),X),aa(list(list(A)),list(A),concat(A),aa(list(list(A)),list(list(A)),aa(fun(list(A),list(A)),fun(list(list(A)),list(list(A))),map(list(A),list(A)),aa(fun(A,fun(list(A),list(A))),fun(list(A),list(A)),aa(list(A),fun(fun(A,fun(list(A),list(A))),fun(list(A),list(A))),case_list(list(A),A),nil(A)),aTP_Lamp_iy(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),aa(list(list(list(A))),list(list(A)),concat(list(A)),aa(list(list(A)),list(list(list(A))),aa(fun(list(A),list(list(A))),fun(list(list(A)),list(list(list(A)))),map(list(A),list(list(A))),aa(fun(A,fun(list(A),list(list(A)))),fun(list(A),list(list(A))),aa(list(list(A)),fun(fun(A,fun(list(A),list(list(A)))),fun(list(A),list(list(A)))),case_list(list(list(A)),A),nil(list(A))),aTP_Lamp_iz(A,fun(list(A),list(list(A)))))),Xss))))) ) ) ) ) ) ).

% transpose.elims
tff(fact_4143_relChain__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [R2: set(product_prod(A,A)),As4: fun(A,B)] :
          ( bNF_Ca3754400796208372196lChain(A,B,R2,As4)
        <=> ! [I2: A,J2: A] :
              ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),I2),J2),R2)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,As4,I2)),aa(A,B,As4,J2)) ) ) ) ).

% relChain_def
tff(fact_4144_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_4145_transpose_Osimps_I1_J,axiom,
    ! [A: $tType] : ( transpose(A,nil(list(A))) = nil(list(A)) ) ).

% transpose.simps(1)
tff(fact_4146_transpose_Osimps_I2_J,axiom,
    ! [A: $tType,Xss2: list(list(A))] : ( transpose(A,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),nil(A)),Xss2)) = transpose(A,Xss2) ) ).

% transpose.simps(2)
tff(fact_4147_transpose__map__map,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xsa: list(list(B))] : ( transpose(A,aa(list(list(B)),list(list(A)),aa(fun(list(B),list(A)),fun(list(list(B)),list(list(A))),map(list(B),list(A)),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2)),Xsa)) = aa(list(list(B)),list(list(A)),aa(fun(list(B),list(A)),fun(list(list(B)),list(list(A))),map(list(B),list(A)),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2)),transpose(B,Xsa)) ) ).

% transpose_map_map
tff(fact_4148_transpose__empty,axiom,
    ! [A: $tType,Xsa: list(list(A))] :
      ( ( transpose(A,Xsa) = nil(list(A)) )
    <=> ! [X6: list(A)] :
          ( member(list(A),X6,aa(list(list(A)),set(list(A)),set2(list(A)),Xsa))
         => ( X6 = nil(A) ) ) ) ).

% transpose_empty
tff(fact_4149_sorted__transpose,axiom,
    ! [A: $tType,Xsa: list(list(A))] : sorted_wrt(nat,ord_less_eq(nat),aa(list(nat),list(nat),rev(nat),aa(list(list(A)),list(nat),aa(fun(list(A),nat),fun(list(list(A)),list(nat)),map(list(A),nat),size_size(list(A))),transpose(A,Xsa)))) ).

% sorted_transpose
tff(fact_4150_length__transpose__sorted,axiom,
    ! [A: $tType,Xsa: list(list(A))] :
      ( sorted_wrt(nat,ord_less_eq(nat),aa(list(nat),list(nat),rev(nat),aa(list(list(A)),list(nat),aa(fun(list(A),nat),fun(list(list(A)),list(nat)),map(list(A),nat),size_size(list(A))),Xsa)))
     => ( aa(list(list(A)),nat,size_size(list(list(A))),transpose(A,Xsa)) = $ite(Xsa = nil(list(A)),zero_zero(nat),aa(list(A),nat,size_size(list(A)),aa(nat,list(A),nth(list(A),Xsa),zero_zero(nat)))) ) ) ).

% length_transpose_sorted
tff(fact_4151_folding__insort__key_Osorted__key__list__of__set__unique,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),S: set(B),F2: fun(B,A),Aa2: set(B),L: list(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S,F2)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),Aa2),S)
       => ( aa(set(B),$o,finite_finite2(B),Aa2)
         => ( ( sorted_wrt(A,Less,aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),L))
              & ( aa(list(B),set(B),set2(B),L) = Aa2 )
              & ( aa(list(B),nat,size_size(list(B)),L) = finite_card(B,Aa2) ) )
          <=> ( aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F2),Aa2) = L ) ) ) ) ) ).

% folding_insort_key.sorted_key_list_of_set_unique
tff(fact_4152_nth__nth__transpose__sorted,axiom,
    ! [A: $tType,Xsa: list(list(A)),Ia: nat,J: nat] :
      ( sorted_wrt(nat,ord_less_eq(nat),aa(list(nat),list(nat),rev(nat),aa(list(list(A)),list(nat),aa(fun(list(A),nat),fun(list(list(A)),list(nat)),map(list(A),nat),size_size(list(A))),Xsa)))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),aa(list(list(A)),nat,size_size(list(list(A))),transpose(A,Xsa)))
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(list(A)),nat,size_size(list(list(A))),aa(list(list(A)),list(list(A)),aa(fun(list(A),$o),fun(list(list(A)),list(list(A))),filter2(list(A)),aTP_Lamp_ja(nat,fun(list(A),$o),Ia)),Xsa)))
         => ( aa(nat,A,nth(A,aa(nat,list(A),nth(list(A),transpose(A,Xsa)),Ia)),J) = aa(nat,A,nth(A,aa(nat,list(A),nth(list(A),Xsa),J)),Ia) ) ) ) ) ).

% nth_nth_transpose_sorted
tff(fact_4153_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,$o)),S: set(B),F2: fun(B,A),Xa: B,Aa2: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S,F2)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert3(B),Xa),Aa2)),S)
       => ( aa(set(B),$o,finite_finite2(B),Aa2)
         => ( aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),Aa2),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert3(B),Xa),bot_bot(set(B))))) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),remove1(B),Xa),aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F2),Aa2)) ) ) ) ) ).

% folding_insort_key.sorted_key_list_of_set_remove
tff(fact_4154_transpose__column,axiom,
    ! [A: $tType,Xsa: list(list(A)),Ia: nat] :
      ( sorted_wrt(nat,ord_less_eq(nat),aa(list(nat),list(nat),rev(nat),aa(list(list(A)),list(nat),aa(fun(list(A),nat),fun(list(list(A)),list(nat)),map(list(A),nat),size_size(list(A))),Xsa)))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),aa(list(list(A)),nat,size_size(list(list(A))),Xsa))
       => ( aa(list(list(A)),list(A),aa(fun(list(A),A),fun(list(list(A)),list(A)),map(list(A),A),aTP_Lamp_jb(nat,fun(list(A),A),Ia)),aa(list(list(A)),list(list(A)),aa(fun(list(A),$o),fun(list(list(A)),list(list(A))),filter2(list(A)),aTP_Lamp_ja(nat,fun(list(A),$o),Ia)),transpose(A,Xsa))) = aa(nat,list(A),nth(list(A),Xsa),Ia) ) ) ) ).

% transpose_column
tff(fact_4155_filter__filter,axiom,
    ! [A: $tType,Pa: fun(A,$o),Qa: fun(A,$o),Xsa: list(A)] : ( aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),Pa),aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),Qa),Xsa)) = aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_jc(fun(A,$o),fun(fun(A,$o),fun(A,$o)),Pa),Qa)),Xsa) ) ).

% filter_filter
tff(fact_4156_filter__True,axiom,
    ! [A: $tType,Xsa: list(A),Pa: fun(A,$o)] :
      ( ! [X: A] :
          ( member(A,X,aa(list(A),set(A),set2(A),Xsa))
         => aa(A,$o,Pa,X) )
     => ( aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),Pa),Xsa) = Xsa ) ) ).

% filter_True
tff(fact_4157_filter__append,axiom,
    ! [A: $tType,Pa: fun(A,$o),Xsa: list(A),Ys: list(A)] : ( aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),Pa),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),Ys)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),Pa),Xsa)),aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),Pa),Ys)) ) ).

% filter_append
tff(fact_4158_remove1__filter__not,axiom,
    ! [A: $tType,Pa: fun(A,$o),Xa: A,Xsa: list(A)] :
      ( ~ aa(A,$o,Pa,Xa)
     => ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),remove1(A),Xa),aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),Pa),Xsa)) = aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),Pa),Xsa) ) ) ).

% remove1_filter_not
tff(fact_4159_removeAll__filter__not,axiom,
    ! [A: $tType,Pa: fun(A,$o),Xa: A,Xsa: list(A)] :
      ( ~ aa(A,$o,Pa,Xa)
     => ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),removeAll(A),Xa),aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),Pa),Xsa)) = aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),Pa),Xsa) ) ) ).

% removeAll_filter_not
tff(fact_4160_set__filter,axiom,
    ! [A: $tType,Pa: fun(A,$o),Xsa: list(A)] : ( aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),Pa),Xsa)) = aa(fun(A,$o),set(A),collect(A),aa(list(A),fun(A,$o),aTP_Lamp_jd(fun(A,$o),fun(list(A),fun(A,$o)),Pa),Xsa)) ) ).

% set_filter
tff(fact_4161_filter__False,axiom,
    ! [A: $tType,Xsa: list(A),Pa: fun(A,$o)] :
      ( ! [X: A] :
          ( member(A,X,aa(list(A),set(A),set2(A),Xsa))
         => ~ aa(A,$o,Pa,X) )
     => ( aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),Pa),Xsa) = nil(A) ) ) ).

% filter_False
tff(fact_4162_partition__in__shuffles,axiom,
    ! [A: $tType,Xsa: list(A),Pa: fun(A,$o)] : member(list(A),Xsa,aa(list(A),set(list(A)),aa(list(A),fun(list(A),set(list(A))),shuffles(A),aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),Pa),Xsa)),aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),aTP_Lamp_ce(fun(A,$o),fun(A,$o),Pa)),Xsa))) ).

% partition_in_shuffles
tff(fact_4163_filter_Osimps_I2_J,axiom,
    ! [A: $tType,Pa: fun(A,$o),Xa: A,Xsa: list(A)] :
      ( aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),Pa),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa)) = $ite(aa(A,$o,Pa,Xa),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),Pa),Xsa)),aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),Pa),Xsa)) ) ).

% filter.simps(2)
tff(fact_4164_filter__id__conv,axiom,
    ! [A: $tType,Pa: fun(A,$o),Xsa: list(A)] :
      ( ( aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),Pa),Xsa) = Xsa )
    <=> ! [X6: A] :
          ( member(A,X6,aa(list(A),set(A),set2(A),Xsa))
         => aa(A,$o,Pa,X6) ) ) ).

% filter_id_conv
tff(fact_4165_filter__cong,axiom,
    ! [A: $tType,Xsa: list(A),Ys: list(A),Pa: fun(A,$o),Qa: fun(A,$o)] :
      ( ( Xsa = Ys )
     => ( ! [X: A] :
            ( member(A,X,aa(list(A),set(A),set2(A),Ys))
           => ( aa(A,$o,Pa,X)
            <=> aa(A,$o,Qa,X) ) )
       => ( aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),Pa),Xsa) = aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),Qa),Ys) ) ) ) ).

% filter_cong
tff(fact_4166_linorder_Osorted__key__list__of__set_Ocong,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(B,fun(B,$o))] : ( sorted8670434370408473282of_set(B,A,Less_eq) = sorted8670434370408473282of_set(B,A,Less_eq) ) ).

% linorder.sorted_key_list_of_set.cong
tff(fact_4167_rev__filter,axiom,
    ! [A: $tType,Pa: fun(A,$o),Xsa: list(A)] : ( aa(list(A),list(A),rev(A),aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),Pa),Xsa)) = aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),Pa),aa(list(A),list(A),rev(A),Xsa)) ) ).

% rev_filter
tff(fact_4168_sorted__wrt__filter,axiom,
    ! [A: $tType,F2: fun(A,fun(A,$o)),Xsa: list(A),Pa: fun(A,$o)] :
      ( sorted_wrt(A,F2,Xsa)
     => sorted_wrt(A,F2,aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),Pa),Xsa)) ) ).

% sorted_wrt_filter
tff(fact_4169_filter__remove1,axiom,
    ! [A: $tType,Qa: fun(A,$o),Xa: A,Xsa: list(A)] : ( aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),Qa),aa(list(A),list(A),aa(A,fun(list(A),list(A)),remove1(A),Xa),Xsa)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),remove1(A),Xa),aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),Qa),Xsa)) ) ).

% filter_remove1
tff(fact_4170_filter_Osimps_I1_J,axiom,
    ! [A: $tType,Pa: fun(A,$o)] : ( aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),Pa),nil(A)) = nil(A) ) ).

% filter.simps(1)
tff(fact_4171_remdups__filter,axiom,
    ! [A: $tType,Pa: fun(A,$o),Xsa: list(A)] : ( aa(list(A),list(A),remdups(A),aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),Pa),Xsa)) = aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),Pa),aa(list(A),list(A),remdups(A),Xsa)) ) ).

% remdups_filter
tff(fact_4172_distinct__filter,axiom,
    ! [A: $tType,Xsa: list(A),Pa: fun(A,$o)] :
      ( aa(list(A),$o,distinct(A),Xsa)
     => aa(list(A),$o,distinct(A),aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),Pa),Xsa)) ) ).

% distinct_filter
tff(fact_4173_removeAll__filter__not__eq,axiom,
    ! [A: $tType,Xa: A] : ( aa(A,fun(list(A),list(A)),removeAll(A),Xa) = aa(fun(A,$o),fun(list(A),list(A)),filter2(A),aa(A,fun(A,$o),aTP_Lamp_je(A,fun(A,$o)),Xa)) ) ).

% removeAll_filter_not_eq
tff(fact_4174_empty__filter__conv,axiom,
    ! [A: $tType,Pa: fun(A,$o),Xsa: list(A)] :
      ( ( nil(A) = aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),Pa),Xsa) )
    <=> ! [X6: A] :
          ( member(A,X6,aa(list(A),set(A),set2(A),Xsa))
         => ~ aa(A,$o,Pa,X6) ) ) ).

% empty_filter_conv
tff(fact_4175_filter__empty__conv,axiom,
    ! [A: $tType,Pa: fun(A,$o),Xsa: list(A)] :
      ( ( aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),Pa),Xsa) = nil(A) )
    <=> ! [X6: A] :
          ( member(A,X6,aa(list(A),set(A),set2(A),Xsa))
         => ~ aa(A,$o,Pa,X6) ) ) ).

% filter_empty_conv
tff(fact_4176_filter__is__subset,axiom,
    ! [A: $tType,Pa: fun(A,$o),Xsa: list(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),Pa),Xsa))),aa(list(A),set(A),set2(A),Xsa)) ).

% filter_is_subset
tff(fact_4177_length__filter__le,axiom,
    ! [A: $tType,Pa: fun(A,$o),Xsa: list(A)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),Pa),Xsa))),aa(list(A),nat,size_size(list(A)),Xsa)) ).

% length_filter_le
tff(fact_4178_sorted__filter_H,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: list(A),Pa: fun(A,$o)] :
          ( sorted_wrt(A,ord_less_eq(A),L)
         => sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),Pa),L)) ) ) ).

% sorted_filter'
tff(fact_4179_distinct__map__filter,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xsa: list(B),Pa: fun(B,$o)] :
      ( aa(list(A),$o,distinct(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Xsa))
     => aa(list(A),$o,distinct(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),aa(list(B),list(B),aa(fun(B,$o),fun(list(B),list(B)),filter2(B),Pa),Xsa))) ) ).

% distinct_map_filter
tff(fact_4180_last__filter,axiom,
    ! [A: $tType,Xsa: list(A),Pa: fun(A,$o)] :
      ( ( Xsa != nil(A) )
     => ( aa(A,$o,Pa,last(A,Xsa))
       => ( last(A,aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),Pa),Xsa)) = last(A,Xsa) ) ) ) ).

% last_filter
tff(fact_4181_sum__length__filter__compl,axiom,
    ! [A: $tType,Pa: fun(A,$o),Xsa: list(A)] : ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),Pa),Xsa))),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),aTP_Lamp_ce(fun(A,$o),fun(A,$o),Pa)),Xsa))) = aa(list(A),nat,size_size(list(A)),Xsa) ) ).

% sum_length_filter_compl
tff(fact_4182_filter__concat,axiom,
    ! [A: $tType,P2: fun(A,$o),Xsa: list(list(A))] : ( aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),P2),aa(list(list(A)),list(A),concat(A),Xsa)) = aa(list(list(A)),list(A),concat(A),aa(list(list(A)),list(list(A)),aa(fun(list(A),list(A)),fun(list(list(A)),list(list(A))),map(list(A),list(A)),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),P2)),Xsa)) ) ).

% filter_concat
tff(fact_4183_inter__set__filter,axiom,
    ! [A: $tType,Aa2: set(A),Xsa: list(A)] : ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),aa(list(A),set(A),set2(A),Xsa)) = aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),aa(set(A),fun(A,$o),aTP_Lamp_a(set(A),fun(A,$o)),Aa2)),Xsa)) ) ).

% inter_set_filter
tff(fact_4184_sorted__same,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [G: fun(list(A),A),Xsa: list(A)] : sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),aa(list(A),fun(A,$o),aTP_Lamp_jf(fun(list(A),A),fun(list(A),fun(A,$o)),G),Xsa)),Xsa)) ) ).

% sorted_same
tff(fact_4185_mset__eq__length__filter,axiom,
    ! [A: $tType,Xsa: list(A),Ys: list(A),Z2: A] :
      ( ( aa(list(A),multiset(A),mset(A),Xsa) = aa(list(A),multiset(A),mset(A),Ys) )
     => ( aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),aa(A,fun(A,$o),fequal(A),Z2)),Xsa)) = aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),aa(A,fun(A,$o),fequal(A),Z2)),Ys)) ) ) ).

% mset_eq_length_filter
tff(fact_4186_filter__shuffles,axiom,
    ! [A: $tType,Pa: fun(A,$o),Xsa: list(A),Ys: list(A)] : ( aa(set(list(A)),set(list(A)),image2(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),Pa)),aa(list(A),set(list(A)),aa(list(A),fun(list(A),set(list(A))),shuffles(A),Xsa),Ys)) = aa(list(A),set(list(A)),aa(list(A),fun(list(A),set(list(A))),shuffles(A),aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),Pa),Xsa)),aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),Pa),Ys)) ) ).

% filter_shuffles
tff(fact_4187_concat__filter__neq__Nil,axiom,
    ! [A: $tType,Xsa: list(list(A))] : ( aa(list(list(A)),list(A),concat(A),aa(list(list(A)),list(list(A)),aa(fun(list(A),$o),fun(list(list(A)),list(list(A))),filter2(list(A)),aTP_Lamp_jg(list(A),$o)),Xsa)) = aa(list(list(A)),list(A),concat(A),Xsa) ) ).

% concat_filter_neq_Nil
tff(fact_4188_union__coset__filter,axiom,
    ! [A: $tType,Xsa: list(A),Aa2: set(A)] : ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),coset(A,Xsa)),Aa2) = coset(A,aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),aTP_Lamp_cd(set(A),fun(A,$o),Aa2)),Xsa)) ) ).

% union_coset_filter
tff(fact_4189_length__filter__less,axiom,
    ! [A: $tType,Xa: A,Xsa: list(A),Pa: fun(A,$o)] :
      ( member(A,Xa,aa(list(A),set(A),set2(A),Xsa))
     => ( ~ aa(A,$o,Pa,Xa)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),Pa),Xsa))),aa(list(A),nat,size_size(list(A)),Xsa)) ) ) ).

% length_filter_less
tff(fact_4190_filter__eq__Cons__iff,axiom,
    ! [A: $tType,Pa: fun(A,$o),Ys: list(A),Xa: A,Xsa: list(A)] :
      ( ( aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),Pa),Ys) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa) )
    <=> ? [Us2: list(A),Vs3: list(A)] :
          ( ( Ys = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Vs3)) )
          & ! [X6: A] :
              ( member(A,X6,aa(list(A),set(A),set2(A),Us2))
             => ~ aa(A,$o,Pa,X6) )
          & aa(A,$o,Pa,Xa)
          & ( Xsa = aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),Pa),Vs3) ) ) ) ).

% filter_eq_Cons_iff
tff(fact_4191_Cons__eq__filter__iff,axiom,
    ! [A: $tType,Xa: A,Xsa: list(A),Pa: fun(A,$o),Ys: list(A)] :
      ( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa) = aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),Pa),Ys) )
    <=> ? [Us2: list(A),Vs3: list(A)] :
          ( ( Ys = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Vs3)) )
          & ! [X6: A] :
              ( member(A,X6,aa(list(A),set(A),set2(A),Us2))
             => ~ aa(A,$o,Pa,X6) )
          & aa(A,$o,Pa,Xa)
          & ( Xsa = aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),Pa),Vs3) ) ) ) ).

% Cons_eq_filter_iff
tff(fact_4192_filter__eq__ConsD,axiom,
    ! [A: $tType,Pa: fun(A,$o),Ys: list(A),Xa: A,Xsa: list(A)] :
      ( ( aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),Pa),Ys) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa) )
     => ? [Us3: list(A),Vs2: list(A)] :
          ( ( Ys = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Vs2)) )
          & ! [X4: A] :
              ( member(A,X4,aa(list(A),set(A),set2(A),Us3))
             => ~ aa(A,$o,Pa,X4) )
          & aa(A,$o,Pa,Xa)
          & ( Xsa = aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),Pa),Vs2) ) ) ) ).

% filter_eq_ConsD
tff(fact_4193_Cons__eq__filterD,axiom,
    ! [A: $tType,Xa: A,Xsa: list(A),Pa: fun(A,$o),Ys: list(A)] :
      ( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa) = aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),Pa),Ys) )
     => ? [Us3: list(A),Vs2: list(A)] :
          ( ( Ys = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Vs2)) )
          & ! [X4: A] :
              ( member(A,X4,aa(list(A),set(A),set2(A),Us3))
             => ~ aa(A,$o,Pa,X4) )
          & aa(A,$o,Pa,Xa)
          & ( Xsa = aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),Pa),Vs2) ) ) ) ).

% Cons_eq_filterD
tff(fact_4194_sorted__filter,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F2: fun(B,A),Xsa: list(B),Pa: fun(B,$o)] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Xsa))
         => sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),aa(list(B),list(B),aa(fun(B,$o),fun(list(B),list(B)),filter2(B),Pa),Xsa))) ) ) ).

% sorted_filter
tff(fact_4195_sorted__map__same,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F2: fun(B,A),G: fun(list(B),A),Xsa: list(B)] : sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),aa(list(B),list(B),aa(fun(B,$o),fun(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_jh(fun(B,A),fun(fun(list(B),A),fun(list(B),fun(B,$o))),F2),G),Xsa)),Xsa))) ) ).

% sorted_map_same
tff(fact_4196_minus__coset__filter,axiom,
    ! [A: $tType,Aa2: set(A),Xsa: list(A)] : ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),coset(A,Xsa)) = aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),aa(set(A),fun(A,$o),aTP_Lamp_a(set(A),fun(A,$o)),Aa2)),Xsa)) ) ).

% minus_coset_filter
tff(fact_4197_filter__eq__snocD,axiom,
    ! [A: $tType,Pa: fun(A,$o),L: list(A),L2: list(A),Xa: A] :
      ( ( aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),Pa),L) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),nil(A))) )
     => ( member(A,Xa,aa(list(A),set(A),set2(A),L))
        & aa(A,$o,Pa,Xa) ) ) ).

% filter_eq_snocD
tff(fact_4198_sum__list__map__filter,axiom,
    ! [B: $tType,A: $tType] :
      ( monoid_add(B)
     => ! [Xsa: list(A),Pa: fun(A,$o),F2: fun(A,B)] :
          ( ! [X: A] :
              ( member(A,X,aa(list(A),set(A),set2(A),Xsa))
             => ( ~ aa(A,$o,Pa,X)
               => ( aa(A,B,F2,X) = zero_zero(B) ) ) )
         => ( groups8242544230860333062m_list(B,aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),Pa),Xsa))) = groups8242544230860333062m_list(B,aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),Xsa)) ) ) ) ).

% sum_list_map_filter
tff(fact_4199_set__minus__filter__out,axiom,
    ! [A: $tType,Xsa: list(A),Y: A] : ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(list(A),set(A),set2(A),Xsa)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Y),bot_bot(set(A)))) = aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),aa(A,fun(A,$o),aTP_Lamp_cg(A,fun(A,$o)),Y)),Xsa)) ) ).

% set_minus_filter_out
tff(fact_4200_filter__shuffles__disjoint2_I1_J,axiom,
    ! [A: $tType,Xsa: list(A),Ys: list(A),Zs: list(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Xsa)),aa(list(A),set(A),set2(A),Ys)) = bot_bot(set(A)) )
     => ( member(list(A),Zs,aa(list(A),set(list(A)),aa(list(A),fun(list(A),set(list(A))),shuffles(A),Xsa),Ys))
       => ( aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),aTP_Lamp_ji(list(A),fun(A,$o),Ys)),Zs) = Ys ) ) ) ).

% filter_shuffles_disjoint2(1)
tff(fact_4201_filter__shuffles__disjoint2_I2_J,axiom,
    ! [A: $tType,Xsa: list(A),Ys: list(A),Zs: list(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Xsa)),aa(list(A),set(A),set2(A),Ys)) = bot_bot(set(A)) )
     => ( member(list(A),Zs,aa(list(A),set(list(A)),aa(list(A),fun(list(A),set(list(A))),shuffles(A),Xsa),Ys))
       => ( aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),aTP_Lamp_jj(list(A),fun(A,$o),Ys)),Zs) = Xsa ) ) ) ).

% filter_shuffles_disjoint2(2)
tff(fact_4202_filter__shuffles__disjoint1_I1_J,axiom,
    ! [A: $tType,Xsa: list(A),Ys: list(A),Zs: list(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Xsa)),aa(list(A),set(A),set2(A),Ys)) = bot_bot(set(A)) )
     => ( member(list(A),Zs,aa(list(A),set(list(A)),aa(list(A),fun(list(A),set(list(A))),shuffles(A),Xsa),Ys))
       => ( aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),aTP_Lamp_ji(list(A),fun(A,$o),Xsa)),Zs) = Xsa ) ) ) ).

% filter_shuffles_disjoint1(1)
tff(fact_4203_filter__shuffles__disjoint1_I2_J,axiom,
    ! [A: $tType,Xsa: list(A),Ys: list(A),Zs: list(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Xsa)),aa(list(A),set(A),set2(A),Ys)) = bot_bot(set(A)) )
     => ( member(list(A),Zs,aa(list(A),set(list(A)),aa(list(A),fun(list(A),set(list(A))),shuffles(A),Xsa),Ys))
       => ( aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),aTP_Lamp_jj(list(A),fun(A,$o),Xsa)),Zs) = Ys ) ) ) ).

% filter_shuffles_disjoint1(2)
tff(fact_4204_length__filter__conv__card,axiom,
    ! [A: $tType,P2: fun(A,$o),Xsa: list(A)] : ( aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),P2),Xsa)) = finite_card(nat,aa(fun(nat,$o),set(nat),collect(nat),aa(list(A),fun(nat,$o),aTP_Lamp_jk(fun(A,$o),fun(list(A),fun(nat,$o)),P2),Xsa))) ) ).

% length_filter_conv_card
tff(fact_4205_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),F2: fun(B,A)] :
      ( folding_insort_key(A,B,Less_eq,Less,S,F2)
     => ( aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F2),bot_bot(set(B))) = nil(B) ) ) ).

% folding_insort_key.sorted_key_list_of_set_empty
tff(fact_4206_folding__insort__key_Osorted__key__list__of__set__inject,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),S: set(B),F2: fun(B,A),Aa2: set(B),B2: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S,F2)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),Aa2),S)
       => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),B2),S)
         => ( ( aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F2),Aa2) = aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F2),B2) )
           => ( aa(set(B),$o,finite_finite2(B),Aa2)
             => ( aa(set(B),$o,finite_finite2(B),B2)
               => ( Aa2 = B2 ) ) ) ) ) ) ) ).

% folding_insort_key.sorted_key_list_of_set_inject
tff(fact_4207_filter__nth__ex__nth,axiom,
    ! [A: $tType,N: nat,Pa: fun(A,$o),Xsa: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),Pa),Xsa)))
     => ? [M6: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),M6)
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M6),aa(list(A),nat,size_size(list(A)),Xsa))
          & ( aa(nat,A,nth(A,aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),Pa),Xsa)),N) = aa(nat,A,nth(A,Xsa),M6) )
          & ( aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),Pa),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),M6),Xsa)) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),Pa),Xsa)) ) ) ) ).

% filter_nth_ex_nth
tff(fact_4208_distinct__length__filter,axiom,
    ! [A: $tType,Xsa: list(A),Pa: fun(A,$o)] :
      ( aa(list(A),$o,distinct(A),Xsa)
     => ( aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),Pa),Xsa)) = 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),Pa)),aa(list(A),set(A),set2(A),Xsa))) ) ) ).

% distinct_length_filter
tff(fact_4209_folding__insort__key_Oset__sorted__key__list__of__set,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),S: set(B),F2: fun(B,A),Aa2: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S,F2)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),Aa2),S)
       => ( aa(set(B),$o,finite_finite2(B),Aa2)
         => ( aa(list(B),set(B),set2(B),aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F2),Aa2)) = Aa2 ) ) ) ) ).

% folding_insort_key.set_sorted_key_list_of_set
tff(fact_4210_folding__insort__key_Olength__sorted__key__list__of__set,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),S: set(B),F2: fun(B,A),Aa2: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S,F2)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),Aa2),S)
       => ( aa(list(B),nat,size_size(list(B)),aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F2),Aa2)) = finite_card(B,Aa2) ) ) ) ).

% folding_insort_key.length_sorted_key_list_of_set
tff(fact_4211_folding__insort__key_Odistinct__sorted__key__list__of__set,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),S: set(B),F2: fun(B,A),Aa2: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S,F2)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),Aa2),S)
       => aa(list(A),$o,distinct(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F2),Aa2))) ) ) ).

% folding_insort_key.distinct_sorted_key_list_of_set
tff(fact_4212_folding__insort__key_Ostrict__sorted__key__list__of__set,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),S: set(B),F2: fun(B,A),Aa2: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S,F2)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),Aa2),S)
       => sorted_wrt(A,Less,aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F2),Aa2))) ) ) ).

% folding_insort_key.strict_sorted_key_list_of_set
tff(fact_4213_folding__insort__key_Osorted__sorted__key__list__of__set,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),S: set(B),F2: fun(B,A),Aa2: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S,F2)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),Aa2),S)
       => sorted_wrt(A,Less_eq,aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F2),Aa2))) ) ) ).

% folding_insort_key.sorted_sorted_key_list_of_set
tff(fact_4214_nth__transpose,axiom,
    ! [A: $tType,Ia: nat,Xsa: list(list(A))] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),aa(list(list(A)),nat,size_size(list(list(A))),transpose(A,Xsa)))
     => ( aa(nat,list(A),nth(list(A),transpose(A,Xsa)),Ia) = aa(list(list(A)),list(A),aa(fun(list(A),A),fun(list(list(A)),list(A)),map(list(A),A),aTP_Lamp_jb(nat,fun(list(A),A),Ia)),aa(list(list(A)),list(list(A)),aa(fun(list(A),$o),fun(list(list(A)),list(list(A))),filter2(list(A)),aTP_Lamp_ja(nat,fun(list(A),$o),Ia)),Xsa)) ) ) ).

% nth_transpose
tff(fact_4215_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,$o)),S: set(B),F2: fun(B,A),Aa2: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S,F2)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),Aa2),S)
       => ( aa(set(B),$o,finite_finite2(B),Aa2)
         => ( ( aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F2),Aa2) = nil(B) )
          <=> ( Aa2 = bot_bot(set(B)) ) ) ) ) ) ).

% folding_insort_key.sorted_key_list_of_set_eq_Nil_iff
tff(fact_4216_transpose__aux__filter__head,axiom,
    ! [A: $tType,Xss2: list(list(A))] : ( aa(list(list(A)),list(A),concat(A),aa(list(list(A)),list(list(A)),aa(fun(list(A),list(A)),fun(list(list(A)),list(list(A))),map(list(A),list(A)),aa(fun(A,fun(list(A),list(A))),fun(list(A),list(A)),aa(list(A),fun(fun(A,fun(list(A),list(A))),fun(list(A),list(A))),case_list(list(A),A),nil(A)),aTP_Lamp_iy(A,fun(list(A),list(A))))),Xss2)) = aa(list(list(A)),list(A),aa(fun(list(A),A),fun(list(list(A)),list(A)),map(list(A),A),hd(A)),aa(list(list(A)),list(list(A)),aa(fun(list(A),$o),fun(list(list(A)),list(list(A))),filter2(list(A)),aTP_Lamp_jg(list(A),$o)),Xss2)) ) ).

% transpose_aux_filter_head
tff(fact_4217_transpose__column__length,axiom,
    ! [A: $tType,Xsa: list(list(A)),Ia: nat] :
      ( sorted_wrt(nat,ord_less_eq(nat),aa(list(nat),list(nat),rev(nat),aa(list(list(A)),list(nat),aa(fun(list(A),nat),fun(list(list(A)),list(nat)),map(list(A),nat),size_size(list(A))),Xsa)))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),aa(list(list(A)),nat,size_size(list(list(A))),Xsa))
       => ( aa(list(list(A)),nat,size_size(list(list(A))),aa(list(list(A)),list(list(A)),aa(fun(list(A),$o),fun(list(list(A)),list(list(A))),filter2(list(A)),aTP_Lamp_ja(nat,fun(list(A),$o),Ia)),transpose(A,Xsa))) = aa(list(A),nat,size_size(list(A)),aa(nat,list(A),nth(list(A),Xsa),Ia)) ) ) ) ).

% transpose_column_length
tff(fact_4218_distinct__concat_H,axiom,
    ! [A: $tType,Xsa: list(list(A))] :
      ( aa(list(list(A)),$o,distinct(list(A)),aa(list(list(A)),list(list(A)),aa(fun(list(A),$o),fun(list(list(A)),list(list(A))),filter2(list(A)),aTP_Lamp_jg(list(A),$o)),Xsa))
     => ( ! [Ys5: list(A)] :
            ( member(list(A),Ys5,aa(list(list(A)),set(list(A)),set2(list(A)),Xsa))
           => aa(list(A),$o,distinct(A),Ys5) )
       => ( ! [Ys5: list(A),Zs2: list(A)] :
              ( member(list(A),Ys5,aa(list(list(A)),set(list(A)),set2(list(A)),Xsa))
             => ( member(list(A),Zs2,aa(list(list(A)),set(list(A)),set2(list(A)),Xsa))
               => ( ( Ys5 != Zs2 )
                 => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Ys5)),aa(list(A),set(A),set2(A),Zs2)) = bot_bot(set(A)) ) ) ) )
         => aa(list(A),$o,distinct(A),aa(list(list(A)),list(A),concat(A),Xsa)) ) ) ) ).

% distinct_concat'
tff(fact_4219_folding__insort__key_Oidem__if__sorted__distinct,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),S: set(B),F2: fun(B,A),Xsa: list(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S,F2)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(list(B),set(B),set2(B),Xsa)),S)
       => ( sorted_wrt(A,Less_eq,aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Xsa))
         => ( aa(list(B),$o,distinct(B),Xsa)
           => ( aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F2),aa(list(B),set(B),set2(B),Xsa)) = Xsa ) ) ) ) ) ).

% folding_insort_key.idem_if_sorted_distinct
tff(fact_4220_transpose__aux__filter__tail,axiom,
    ! [A: $tType,Xss2: list(list(A))] : ( aa(list(list(list(A))),list(list(A)),concat(list(A)),aa(list(list(A)),list(list(list(A))),aa(fun(list(A),list(list(A))),fun(list(list(A)),list(list(list(A)))),map(list(A),list(list(A))),aa(fun(A,fun(list(A),list(list(A)))),fun(list(A),list(list(A))),aa(list(list(A)),fun(fun(A,fun(list(A),list(list(A)))),fun(list(A),list(list(A)))),case_list(list(list(A)),A),nil(list(A))),aTP_Lamp_iz(A,fun(list(A),list(list(A)))))),Xss2)) = aa(list(list(A)),list(list(A)),aa(fun(list(A),list(A)),fun(list(list(A)),list(list(A))),map(list(A),list(A)),tl(A)),aa(list(list(A)),list(list(A)),aa(fun(list(A),$o),fun(list(list(A)),list(list(A))),filter2(list(A)),aTP_Lamp_jg(list(A),$o)),Xss2)) ) ).

% transpose_aux_filter_tail
tff(fact_4221_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,$o)),S: set(B),F2: fun(B,A),Xa: B,Aa2: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S,F2)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert3(B),Xa),Aa2)),S)
       => ( aa(set(B),$o,finite_finite2(B),Aa2)
         => ( aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F2),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert3(B),Xa),Aa2)) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),aa(fun(B,A),fun(B,fun(list(B),list(B))),insort_key(A,B,Less_eq),F2),Xa),aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),Aa2),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert3(B),Xa),bot_bot(set(B)))))) ) ) ) ) ).

% folding_insort_key.sorted_key_list_of_set_insert_remove
tff(fact_4222_remove__rev__alt__def,axiom,
    ! [A: $tType,Xa: A,Xsa: list(A)] : ( aa(list(A),list(A),remove_rev(A,Xa),Xsa) = aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),aa(A,fun(A,$o),aTP_Lamp_cg(A,fun(A,$o)),Xa)),aa(list(A),list(A),rev(A),Xsa)) ) ).

% remove_rev_alt_def
tff(fact_4223_folding__insort__key_Osorted__key__list__of__set__insert,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),S: set(B),F2: fun(B,A),Xa: B,Aa2: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S,F2)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert3(B),Xa),Aa2)),S)
       => ( aa(set(B),$o,finite_finite2(B),Aa2)
         => ( ~ member(B,Xa,Aa2)
           => ( aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F2),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert3(B),Xa),Aa2)) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),aa(fun(B,A),fun(B,fun(list(B),list(B))),insort_key(A,B,Less_eq),F2),Xa),aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F2),Aa2)) ) ) ) ) ) ).

% folding_insort_key.sorted_key_list_of_set_insert
tff(fact_4224_sort__by__quicksort,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xsa: list(A)] : ( aa(list(A),list(A),aa(fun(A,A),fun(list(A),list(A)),linorder_sort_key(A,A),aTP_Lamp_ip(A,A)),Xsa) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(fun(A,A),fun(list(A),list(A)),linorder_sort_key(A,A),aTP_Lamp_ip(A,A)),aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),aTP_Lamp_jl(list(A),fun(A,$o),Xsa)),Xsa))),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),aTP_Lamp_jm(list(A),fun(A,$o),Xsa)),Xsa)),aa(list(A),list(A),aa(fun(A,A),fun(list(A),list(A)),linorder_sort_key(A,A),aTP_Lamp_ip(A,A)),aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),aa(A,fun(A,$o),ord_less(A),aa(nat,A,nth(A,Xsa),divide_divide(nat,aa(list(A),nat,size_size(list(A)),Xsa),numeral_numeral(nat,bit0(one2)))))),Xsa)))) ) ) ).

% sort_by_quicksort
tff(fact_4225_sort__upto,axiom,
    ! [Ia: int,J: int] : ( aa(list(int),list(int),aa(fun(int,int),fun(list(int),list(int)),linorder_sort_key(int,int),aTP_Lamp_jn(int,int)),upto(Ia,J)) = upto(Ia,J) ) ).

% sort_upto
tff(fact_4226_sort__key__simps_I1_J,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B)] : ( aa(list(A),list(A),aa(fun(A,B),fun(list(A),list(A)),linorder_sort_key(A,B),F2),nil(A)) = nil(A) ) ) ).

% sort_key_simps(1)
tff(fact_4227_set__sort,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),Xsa: list(A)] : ( aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(fun(A,B),fun(list(A),list(A)),linorder_sort_key(A,B),F2),Xsa)) = aa(list(A),set(A),set2(A),Xsa) ) ) ).

% set_sort
tff(fact_4228_length__sort,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),Xsa: list(A)] : ( aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(fun(A,B),fun(list(A),list(A)),linorder_sort_key(A,B),F2),Xsa)) = aa(list(A),nat,size_size(list(A)),Xsa) ) ) ).

% length_sort
tff(fact_4229_distinct__sort,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),Xsa: list(A)] :
          ( aa(list(A),$o,distinct(A),aa(list(A),list(A),aa(fun(A,B),fun(list(A),list(A)),linorder_sort_key(A,B),F2),Xsa))
        <=> aa(list(A),$o,distinct(A),Xsa) ) ) ).

% distinct_sort
tff(fact_4230_filter__sort,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Pa: fun(A,$o),F2: fun(A,B),Xsa: list(A)] : ( aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),Pa),aa(list(A),list(A),aa(fun(A,B),fun(list(A),list(A)),linorder_sort_key(A,B),F2),Xsa)) = aa(list(A),list(A),aa(fun(A,B),fun(list(A),list(A)),linorder_sort_key(A,B),F2),aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),Pa),Xsa)) ) ) ).

% filter_sort
tff(fact_4231_linorder_Oinsort__key_Ocong,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(B,fun(B,$o))] : ( insort_key(B,A,Less_eq) = insort_key(B,A,Less_eq) ) ).

% linorder.insort_key.cong
tff(fact_4232_sort__key__const,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [C2: B,Xsa: list(A)] : ( aa(list(A),list(A),aa(fun(A,B),fun(list(A),list(A)),linorder_sort_key(A,B),aTP_Lamp_ax(B,fun(A,B),C2)),Xsa) = Xsa ) ) ).

% sort_key_const
tff(fact_4233_sort__key__stable,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),K3: B,Xsa: list(A)] : ( aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),aa(B,fun(A,$o),aTP_Lamp_jo(fun(A,B),fun(B,fun(A,$o)),F2),K3)),aa(list(A),list(A),aa(fun(A,B),fun(list(A),list(A)),linorder_sort_key(A,B),F2),Xsa)) = aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),aa(B,fun(A,$o),aTP_Lamp_jo(fun(A,B),fun(B,fun(A,$o)),F2),K3)),Xsa) ) ) ).

% sort_key_stable
tff(fact_4234_sorted__sort,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xsa: list(A)] : sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),aa(fun(A,A),fun(list(A),list(A)),linorder_sort_key(A,A),aTP_Lamp_ip(A,A)),Xsa)) ) ).

% sorted_sort
tff(fact_4235_sorted__sort__id,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xsa: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),Xsa)
         => ( aa(list(A),list(A),aa(fun(A,A),fun(list(A),list(A)),linorder_sort_key(A,A),aTP_Lamp_ip(A,A)),Xsa) = Xsa ) ) ) ).

% sorted_sort_id
tff(fact_4236_sorted__sort__key,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F2: fun(B,A),Xsa: list(B)] : sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),aa(list(B),list(B),aa(fun(B,A),fun(list(B),list(B)),linorder_sort_key(B,A),F2),Xsa))) ) ).

% sorted_sort_key
tff(fact_4237_sorted__list__of__set__sort__remdups,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xsa: list(A)] : ( aa(set(A),list(A),linord4507533701916653071of_set(A),aa(list(A),set(A),set2(A),Xsa)) = aa(list(A),list(A),aa(fun(A,A),fun(list(A),list(A)),linorder_sort_key(A,A),aTP_Lamp_ip(A,A)),aa(list(A),list(A),remdups(A),Xsa)) ) ) ).

% sorted_list_of_set_sort_remdups
tff(fact_4238_properties__for__sort__key,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Ys: list(A),Xsa: list(A),F2: fun(A,B)] :
          ( ( aa(list(A),multiset(A),mset(A),Ys) = aa(list(A),multiset(A),mset(A),Xsa) )
         => ( ! [K: A] :
                ( member(A,K,aa(list(A),set(A),set2(A),Ys))
               => ( aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),aa(A,fun(A,$o),aTP_Lamp_jp(fun(A,B),fun(A,fun(A,$o)),F2),K)),Ys) = aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),aa(A,fun(A,$o),aTP_Lamp_jp(fun(A,B),fun(A,fun(A,$o)),F2),K)),Xsa) ) )
           => ( sorted_wrt(B,ord_less_eq(B),aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),Ys))
             => ( aa(list(A),list(A),aa(fun(A,B),fun(list(A),list(A)),linorder_sort_key(A,B),F2),Xsa) = Ys ) ) ) ) ) ).

% properties_for_sort_key
tff(fact_4239_sort__key__by__quicksort,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),Xsa: list(A)] : ( aa(list(A),list(A),aa(fun(A,B),fun(list(A),list(A)),linorder_sort_key(A,B),F2),Xsa) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(fun(A,B),fun(list(A),list(A)),linorder_sort_key(A,B),F2),aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),aa(list(A),fun(A,$o),aTP_Lamp_jq(fun(A,B),fun(list(A),fun(A,$o)),F2),Xsa)),Xsa))),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),aa(list(A),fun(A,$o),aTP_Lamp_jr(fun(A,B),fun(list(A),fun(A,$o)),F2),Xsa)),Xsa)),aa(list(A),list(A),aa(fun(A,B),fun(list(A),list(A)),linorder_sort_key(A,B),F2),aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),aa(list(A),fun(A,$o),aTP_Lamp_js(fun(A,B),fun(list(A),fun(A,$o)),F2),Xsa)),Xsa)))) ) ) ).

% sort_key_by_quicksort
tff(fact_4240_sort__key__by__quicksort__code,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),Xsa: list(A)] : ( aa(list(A),list(A),aa(fun(A,B),fun(list(A),list(A)),linorder_sort_key(A,B),F2),Xsa) = aa(list(A),list(A),aa(fun(A,fun(list(A),list(A))),fun(list(A),list(A)),aa(list(A),fun(fun(A,fun(list(A),list(A))),fun(list(A),list(A))),case_list(list(A),A),nil(A)),aa(list(A),fun(A,fun(list(A),list(A))),aTP_Lamp_jx(fun(A,B),fun(list(A),fun(A,fun(list(A),list(A)))),F2),Xsa)),Xsa) ) ) ).

% sort_key_by_quicksort_code
tff(fact_4241_transpose__rectangle,axiom,
    ! [A: $tType,Xsa: list(list(A)),N: nat] :
      ( ( ( Xsa = nil(list(A)) )
       => ( N = zero_zero(nat) ) )
     => ( ! [I3: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(list(list(A)),nat,size_size(list(list(A))),Xsa))
           => ( aa(list(A),nat,size_size(list(A)),aa(nat,list(A),nth(list(A),Xsa),I3)) = N ) )
       => ( transpose(A,Xsa) = aa(list(nat),list(list(A)),aa(fun(nat,list(A)),fun(list(nat),list(list(A))),map(nat,list(A)),aTP_Lamp_jz(list(list(A)),fun(nat,list(A)),Xsa)),upt(zero_zero(nat),N)) ) ) ) ).

% transpose_rectangle
tff(fact_4242_mlex__eq,axiom,
    ! [A: $tType,F2: fun(A,nat),R: set(product_prod(A,A))] : ( mlex_prod(A,F2,R) = aa(fun(product_prod(A,A),$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_ka(fun(A,nat),fun(set(product_prod(A,A)),fun(A,fun(A,$o))),F2),R))) ) ).

% mlex_eq
tff(fact_4243_Bleast__code,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xsa: list(A),Pa: fun(A,$o)] : ( bleast(A,aa(list(A),set(A),set2(A),Xsa),Pa) = aa(list(A),A,aa(fun(A,fun(list(A),A)),fun(list(A),A),aa(A,fun(fun(A,fun(list(A),A)),fun(list(A),A)),case_list(A,A),abort_Bleast(A,aa(list(A),set(A),set2(A),Xsa),Pa)),aTP_Lamp_kb(A,fun(list(A),A))),aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),Pa),aa(list(A),list(A),aa(fun(A,A),fun(list(A),list(A)),linorder_sort_key(A,A),aTP_Lamp_ip(A,A)),Xsa))) ) ) ).

% Bleast_code
tff(fact_4244_remdups__upt,axiom,
    ! [M2: nat,N: nat] : ( aa(list(nat),list(nat),remdups(nat),upt(M2,N)) = upt(M2,N) ) ).

% remdups_upt
tff(fact_4245_sort__upt,axiom,
    ! [M2: nat,N: nat] : ( aa(list(nat),list(nat),aa(fun(nat,nat),fun(list(nat),list(nat)),linorder_sort_key(nat,nat),aTP_Lamp_et(nat,nat)),upt(M2,N)) = upt(M2,N) ) ).

% sort_upt
tff(fact_4246_tl__upt,axiom,
    ! [M2: nat,N: nat] : ( aa(list(nat),list(nat),tl(nat),upt(M2,N)) = upt(aa(nat,nat,suc,M2),N) ) ).

% tl_upt
tff(fact_4247_hd__upt,axiom,
    ! [Ia: nat,J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),J)
     => ( aa(list(nat),nat,hd(nat),upt(Ia,J)) = Ia ) ) ).

% hd_upt
tff(fact_4248_drop__upt,axiom,
    ! [M2: nat,Ia: nat,J: nat] : ( aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),drop(nat),M2),upt(Ia,J)) = upt(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ia),M2),J) ) ).

% drop_upt
tff(fact_4249_take__upt,axiom,
    ! [Ia: nat,M2: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ia),M2)),N)
     => ( aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),take(nat),M2),upt(Ia,N)) = upt(Ia,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ia),M2)) ) ) ).

% take_upt
tff(fact_4250_upt__0__eq__Nil__conv,axiom,
    ! [J: nat] :
      ( ( upt(zero_zero(nat),J) = nil(nat) )
    <=> ( J = zero_zero(nat) ) ) ).

% upt_0_eq_Nil_conv
tff(fact_4251_upt__conv__Nil,axiom,
    ! [J: nat,Ia: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),Ia)
     => ( upt(Ia,J) = nil(nat) ) ) ).

% upt_conv_Nil
tff(fact_4252_upt__merge,axiom,
    ! [Ia: nat,J: nat,K3: nat] :
      ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ia),J)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),K3) )
     => ( aa(list(nat),list(nat),aa(list(nat),fun(list(nat),list(nat)),append(nat),upt(Ia,J)),upt(J,K3)) = upt(Ia,K3) ) ) ).

% upt_merge
tff(fact_4253_sorted__list__of__set__range,axiom,
    ! [M2: nat,N: nat] : ( aa(set(nat),list(nat),linord4507533701916653071of_set(nat),set_or7035219750837199246ssThan(nat,M2,N)) = upt(M2,N) ) ).

% sorted_list_of_set_range
tff(fact_4254_length__upt,axiom,
    ! [Ia: nat,J: nat] : ( aa(list(nat),nat,size_size(list(nat)),upt(Ia,J)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),Ia) ) ).

% length_upt
tff(fact_4255_last__upt,axiom,
    ! [Ia: nat,J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),J)
     => ( last(nat,upt(Ia,J)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),one_one(nat)) ) ) ).

% last_upt
tff(fact_4256_upt__eq__Nil__conv,axiom,
    ! [Ia: nat,J: nat] :
      ( ( upt(Ia,J) = nil(nat) )
    <=> ( ( J = zero_zero(nat) )
        | aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),Ia) ) ) ).

% upt_eq_Nil_conv
tff(fact_4257_nth__upt,axiom,
    ! [Ia: nat,K3: nat,J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ia),K3)),J)
     => ( aa(nat,nat,nth(nat,upt(Ia,J)),K3) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ia),K3) ) ) ).

% nth_upt
tff(fact_4258_upt__rec__numeral,axiom,
    ! [M2: num,N: num] :
      ( upt(numeral_numeral(nat,M2),numeral_numeral(nat,N)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),numeral_numeral(nat,M2)),numeral_numeral(nat,N)),aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),numeral_numeral(nat,M2)),upt(aa(nat,nat,suc,numeral_numeral(nat,M2)),numeral_numeral(nat,N))),nil(nat)) ) ).

% upt_rec_numeral
tff(fact_4259_atLeastLessThan__upt,axiom,
    ! [Ia: nat,J: nat] : ( set_or7035219750837199246ssThan(nat,Ia,J) = aa(list(nat),set(nat),set2(nat),upt(Ia,J)) ) ).

% atLeastLessThan_upt
tff(fact_4260_distinct__upt,axiom,
    ! [Ia: nat,J: nat] : aa(list(nat),$o,distinct(nat),upt(Ia,J)) ).

% distinct_upt
tff(fact_4261_map__add__upt_H,axiom,
    ! [Ofs: nat,A3: nat,B3: nat] : ( aa(list(nat),list(nat),aa(fun(nat,nat),fun(list(nat),list(nat)),map(nat,nat),aTP_Lamp_kc(nat,fun(nat,nat),Ofs)),upt(A3,B3)) = 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),B3),Ofs)) ) ).

% map_add_upt'
tff(fact_4262_map__Suc__upt,axiom,
    ! [M2: nat,N: nat] : ( aa(list(nat),list(nat),aa(fun(nat,nat),fun(list(nat),list(nat)),map(nat,nat),suc),upt(M2,N)) = upt(aa(nat,nat,suc,M2),aa(nat,nat,suc,N)) ) ).

% map_Suc_upt
tff(fact_4263_atLeastAtMost__upt,axiom,
    ! [N: nat,M2: nat] : ( set_or1337092689740270186AtMost(nat,N,M2) = aa(list(nat),set(nat),set2(nat),upt(N,aa(nat,nat,suc,M2))) ) ).

% atLeastAtMost_upt
tff(fact_4264_butlast__upt,axiom,
    ! [M2: nat,N: nat] : ( aa(list(nat),list(nat),butlast(nat),upt(M2,N)) = upt(M2,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))) ) ).

% butlast_upt
tff(fact_4265_atLeast__upt,axiom,
    ! [N: nat] : ( aa(nat,set(nat),set_ord_lessThan(nat),N) = aa(list(nat),set(nat),set2(nat),upt(zero_zero(nat),N)) ) ).

% atLeast_upt
tff(fact_4266_upt__conv__Cons__Cons,axiom,
    ! [M2: nat,N: nat,Ns: list(nat),Q2: nat] :
      ( ( aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),M2),aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),N),Ns)) = upt(M2,Q2) )
    <=> ( aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),N),Ns) = upt(aa(nat,nat,suc,M2),Q2) ) ) ).

% upt_conv_Cons_Cons
tff(fact_4267_upt__0,axiom,
    ! [Ia: nat] : ( upt(Ia,zero_zero(nat)) = nil(nat) ) ).

% upt_0
tff(fact_4268_sorted__wrt__upt,axiom,
    ! [M2: nat,N: nat] : sorted_wrt(nat,ord_less(nat),upt(M2,N)) ).

% sorted_wrt_upt
tff(fact_4269_map__add__upt,axiom,
    ! [N: nat,M2: nat] : ( aa(list(nat),list(nat),aa(fun(nat,nat),fun(list(nat),list(nat)),map(nat,nat),aTP_Lamp_kc(nat,fun(nat,nat),N)),upt(zero_zero(nat),M2)) = upt(N,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),N)) ) ).

% map_add_upt
tff(fact_4270_sorted__upt,axiom,
    ! [M2: nat,N: nat] : sorted_wrt(nat,ord_less_eq(nat),upt(M2,N)) ).

% sorted_upt
tff(fact_4271_greaterThanAtMost__upt,axiom,
    ! [N: nat,M2: nat] : ( set_or3652927894154168847AtMost(nat,N,M2) = aa(list(nat),set(nat),set2(nat),upt(aa(nat,nat,suc,N),aa(nat,nat,suc,M2))) ) ).

% greaterThanAtMost_upt
tff(fact_4272_upt__eq__append__conv,axiom,
    ! [Ia: nat,J: nat,Xsa: list(nat),Ys: list(nat)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ia),J)
     => ( ( upt(Ia,J) = aa(list(nat),list(nat),aa(list(nat),fun(list(nat),list(nat)),append(nat),Xsa),Ys) )
      <=> ? [K4: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ia),K4)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K4),J)
            & ( upt(Ia,K4) = Xsa )
            & ( upt(K4,J) = Ys ) ) ) ) ).

% upt_eq_append_conv
tff(fact_4273_greaterThanLessThan__upt,axiom,
    ! [N: nat,M2: nat] : ( set_or5935395276787703475ssThan(nat,N,M2) = aa(list(nat),set(nat),set2(nat),upt(aa(nat,nat,suc,N),M2)) ) ).

% greaterThanLessThan_upt
tff(fact_4274_filter__upt__take__conv,axiom,
    ! [A: $tType,Pa: fun(A,$o),M2: nat,L: list(A),N: nat] : ( aa(list(nat),list(nat),aa(fun(nat,$o),fun(list(nat),list(nat)),filter2(nat),aa(list(A),fun(nat,$o),aa(nat,fun(list(A),fun(nat,$o)),aTP_Lamp_kd(fun(A,$o),fun(nat,fun(list(A),fun(nat,$o))),Pa),M2),L)),upt(N,M2)) = aa(list(nat),list(nat),aa(fun(nat,$o),fun(list(nat),list(nat)),filter2(nat),aa(list(A),fun(nat,$o),aTP_Lamp_ke(fun(A,$o),fun(list(A),fun(nat,$o)),Pa),L)),upt(N,M2)) ) ).

% filter_upt_take_conv
tff(fact_4275_atMost__upto,axiom,
    ! [N: nat] : ( aa(nat,set(nat),set_ord_atMost(nat),N) = aa(list(nat),set(nat),set2(nat),upt(zero_zero(nat),aa(nat,nat,suc,N))) ) ).

% atMost_upto
tff(fact_4276_upt__conv__Cons,axiom,
    ! [Ia: nat,J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),J)
     => ( upt(Ia,J) = aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),Ia),upt(aa(nat,nat,suc,Ia),J)) ) ) ).

% upt_conv_Cons
tff(fact_4277_enumerate__eq__zip,axiom,
    ! [A: $tType,N: nat,Xsa: list(A)] : ( enumerate(A,N,Xsa) = aa(list(A),list(product_prod(nat,A)),aa(list(nat),fun(list(A),list(product_prod(nat,A))),zip(nat,A),upt(N,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(list(A),nat,size_size(list(A)),Xsa)))),Xsa) ) ).

% enumerate_eq_zip
tff(fact_4278_upt__filter__extend,axiom,
    ! [U: nat,U3: nat,Pa: fun(nat,$o)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),U),U3)
     => ( ! [I3: nat] :
            ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),U),I3)
              & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),U3) )
           => ~ aa(nat,$o,Pa,I3) )
       => ( aa(list(nat),list(nat),aa(fun(nat,$o),fun(list(nat),list(nat)),filter2(nat),Pa),upt(zero_zero(nat),U)) = aa(list(nat),list(nat),aa(fun(nat,$o),fun(list(nat),list(nat)),filter2(nat),Pa),upt(zero_zero(nat),U3)) ) ) ) ).

% upt_filter_extend
tff(fact_4279_map__decr__upt,axiom,
    ! [M2: nat,N: nat] : ( aa(list(nat),list(nat),aa(fun(nat,nat),fun(list(nat),list(nat)),map(nat,nat),aTP_Lamp_kf(nat,nat)),upt(aa(nat,nat,suc,M2),aa(nat,nat,suc,N))) = upt(M2,N) ) ).

% map_decr_upt
tff(fact_4280_upt__append,axiom,
    ! [Ia: nat,J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),J)
     => ( aa(list(nat),list(nat),aa(list(nat),fun(list(nat),list(nat)),append(nat),upt(zero_zero(nat),Ia)),upt(Ia,J)) = upt(zero_zero(nat),J) ) ) ).

% upt_append
tff(fact_4281_interv__sum__list__conv__sum__set__nat,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [F2: fun(nat,A),M2: nat,N: nat] : ( groups8242544230860333062m_list(A,aa(list(nat),list(A),aa(fun(nat,A),fun(list(nat),list(A)),map(nat,A),F2),upt(M2,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),aa(list(nat),set(nat),set2(nat),upt(M2,N))) ) ) ).

% interv_sum_list_conv_sum_set_nat
tff(fact_4282_sum__set__upt__conv__sum__list__nat,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [F2: fun(nat,A),M2: nat,N: nat] : ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),aa(list(nat),set(nat),set2(nat),upt(M2,N))) = groups8242544230860333062m_list(A,aa(list(nat),list(A),aa(fun(nat,A),fun(list(nat),list(A)),map(nat,A),F2),upt(M2,N))) ) ) ).

% sum_set_upt_conv_sum_list_nat
tff(fact_4283_upt__add__eq__append,axiom,
    ! [Ia: nat,J: nat,K3: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ia),J)
     => ( upt(Ia,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K3)) = aa(list(nat),list(nat),aa(list(nat),fun(list(nat),list(nat)),append(nat),upt(Ia,J)),upt(J,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K3))) ) ) ).

% upt_add_eq_append
tff(fact_4284_enumerate__map__upt,axiom,
    ! [A: $tType,N: nat,F2: fun(nat,A),M2: nat] : ( enumerate(A,N,aa(list(nat),list(A),aa(fun(nat,A),fun(list(nat),list(A)),map(nat,A),F2),upt(N,M2))) = aa(list(nat),list(product_prod(nat,A)),aa(fun(nat,product_prod(nat,A)),fun(list(nat),list(product_prod(nat,A))),map(nat,product_prod(nat,A)),aTP_Lamp_kg(fun(nat,A),fun(nat,product_prod(nat,A)),F2)),upt(N,M2)) ) ).

% enumerate_map_upt
tff(fact_4285_upt__eq__Cons__conv,axiom,
    ! [Ia: nat,J: nat,Xa: nat,Xsa: list(nat)] :
      ( ( upt(Ia,J) = aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),Xa),Xsa) )
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),J)
        & ( Ia = Xa )
        & ( upt(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ia),one_one(nat)),J) = Xsa ) ) ) ).

% upt_eq_Cons_conv
tff(fact_4286_upt__rec,axiom,
    ! [Ia: nat,J: nat] :
      ( upt(Ia,J) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),J),aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),Ia),upt(aa(nat,nat,suc,Ia),J)),nil(nat)) ) ).

% upt_rec
tff(fact_4287_map__upt__Suc,axiom,
    ! [A: $tType,F2: fun(nat,A),N: nat] : ( aa(list(nat),list(A),aa(fun(nat,A),fun(list(nat),list(A)),map(nat,A),F2),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,F2,zero_zero(nat))),aa(list(nat),list(A),aa(fun(nat,A),fun(list(nat),list(A)),map(nat,A),aTP_Lamp_kh(fun(nat,A),fun(nat,A),F2)),upt(zero_zero(nat),N))) ) ).

% map_upt_Suc
tff(fact_4288_map__nth,axiom,
    ! [A: $tType,Xsa: list(A)] : ( aa(list(nat),list(A),aa(fun(nat,A),fun(list(nat),list(A)),map(nat,A),nth(A,Xsa)),upt(zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xsa))) = Xsa ) ).

% map_nth
tff(fact_4289_mlex__less,axiom,
    ! [A: $tType,F2: fun(A,nat),Xa: A,Y: A,R: set(product_prod(A,A))] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,F2,Xa)),aa(A,nat,F2,Y))
     => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Y),mlex_prod(A,F2,R)) ) ).

% mlex_less
tff(fact_4290_mlex__iff,axiom,
    ! [A: $tType,Xa: A,Y: A,F2: fun(A,nat),R: set(product_prod(A,A))] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Y),mlex_prod(A,F2,R))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,F2,Xa)),aa(A,nat,F2,Y))
        | ( ( aa(A,nat,F2,Xa) = aa(A,nat,F2,Y) )
          & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Y),R) ) ) ) ).

% mlex_iff
tff(fact_4291_mlex__leq,axiom,
    ! [A: $tType,F2: fun(A,nat),Xa: A,Y: A,R: set(product_prod(A,A))] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,F2,Xa)),aa(A,nat,F2,Y))
     => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Y),R)
       => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Y),mlex_prod(A,F2,R)) ) ) ).

% mlex_leq
tff(fact_4292_nth__map__upt,axiom,
    ! [A: $tType,Ia: nat,N: nat,M2: nat,F2: fun(nat,A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M2))
     => ( aa(nat,A,nth(A,aa(list(nat),list(A),aa(fun(nat,A),fun(list(nat),list(A)),map(nat,A),F2),upt(M2,N))),Ia) = aa(nat,A,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),Ia)) ) ) ).

% nth_map_upt
tff(fact_4293_part__code_I1_J,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),Pivot: B] : ( linorder_part(A,B,F2,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_4294_upt__eq__lel__conv,axiom,
    ! [L: nat,H2: nat,Is1: list(nat),Ia: nat,Is2: list(nat)] :
      ( ( upt(L,H2) = aa(list(nat),list(nat),aa(list(nat),fun(list(nat),list(nat)),append(nat),Is1),aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),Ia),Is2)) )
    <=> ( ( Is1 = upt(L,Ia) )
        & ( Is2 = upt(aa(nat,nat,suc,Ia),H2) )
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),L),Ia)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),H2) ) ) ).

% upt_eq_lel_conv
tff(fact_4295_upt__Suc,axiom,
    ! [Ia: nat,J: nat] :
      ( upt(Ia,aa(nat,nat,suc,J)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ia),J),aa(list(nat),list(nat),aa(list(nat),fun(list(nat),list(nat)),append(nat),upt(Ia,J)),aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),J),nil(nat))),nil(nat)) ) ).

% upt_Suc
tff(fact_4296_upt__Suc__append,axiom,
    ! [Ia: nat,J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ia),J)
     => ( upt(Ia,aa(nat,nat,suc,J)) = aa(list(nat),list(nat),aa(list(nat),fun(list(nat),list(nat)),append(nat),upt(Ia,J)),aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),J),nil(nat))) ) ) ).

% upt_Suc_append
tff(fact_4297_map__upt__eqI,axiom,
    ! [A: $tType,Xsa: list(A),N: nat,M2: nat,F2: fun(nat,A)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xsa) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M2) )
     => ( ! [I3: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),Xsa))
           => ( aa(nat,A,nth(A,Xsa),I3) = aa(nat,A,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),I3)) ) )
       => ( aa(list(nat),list(A),aa(fun(nat,A),fun(list(nat),list(A)),map(nat,A),F2),upt(M2,N)) = Xsa ) ) ) ).

% map_upt_eqI
tff(fact_4298_map__nth__upt__drop__take__conv,axiom,
    ! [A: $tType,N7: nat,L: list(A),M5: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N7),aa(list(A),nat,size_size(list(A)),L))
     => ( aa(list(nat),list(A),aa(fun(nat,A),fun(list(nat),list(A)),map(nat,A),nth(A,L)),upt(M5,N7)) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),M5),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N7),L)) ) ) ).

% map_nth_upt_drop_take_conv
tff(fact_4299_filter__upt__last,axiom,
    ! [A: $tType,Pa: fun(A,$o),L: list(A),Js: list(nat),J: nat,Ia: nat] :
      ( ( aa(list(nat),list(nat),aa(fun(nat,$o),fun(list(nat),list(nat)),filter2(nat),aa(list(A),fun(nat,$o),aTP_Lamp_ke(fun(A,$o),fun(list(A),fun(nat,$o)),Pa),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),Ia)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),aa(list(A),nat,size_size(list(A)),L))
         => ~ aa(A,$o,Pa,aa(nat,A,nth(A,L),Ia)) ) ) ) ).

% filter_upt_last
tff(fact_4300_part__code_I2_J,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),Pivot: B,Xa: A,Xsa: list(A)] : ( linorder_part(A,B,F2,Pivot,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa)) = 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_kj(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))))))),F2),Pivot),Xa)),linorder_part(A,B,F2,Pivot,Xsa)) ) ) ).

% part_code(2)
tff(fact_4301_part__def,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),Pivot: B,Xsa: list(A)] : ( linorder_part(A,B,F2,Pivot,Xsa) = aa(product_prod(list(A),list(A)),product_prod(list(A),product_prod(list(A),list(A))),aa(list(A),fun(product_prod(list(A),list(A)),product_prod(list(A),product_prod(list(A),list(A)))),product_Pair(list(A),product_prod(list(A),list(A))),aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),aa(B,fun(A,$o),aTP_Lamp_kk(fun(A,B),fun(B,fun(A,$o)),F2),Pivot)),Xsa)),aa(list(A),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(fun(A,$o),fun(list(A),list(A)),filter2(A),aa(B,fun(A,$o),aTP_Lamp_jo(fun(A,B),fun(B,fun(A,$o)),F2),Pivot)),Xsa)),aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),aa(B,fun(A,$o),aTP_Lamp_kl(fun(A,B),fun(B,fun(A,$o)),F2),Pivot)),Xsa))) ) ) ).

% part_def
tff(fact_4302_transpose__transpose,axiom,
    ! [A: $tType,Xsa: list(list(A))] :
      ( sorted_wrt(nat,ord_less_eq(nat),aa(list(nat),list(nat),rev(nat),aa(list(list(A)),list(nat),aa(fun(list(A),nat),fun(list(list(A)),list(nat)),map(list(A),nat),size_size(list(A))),Xsa)))
     => ( transpose(A,transpose(A,Xsa)) = aa(list(list(A)),list(list(A)),aa(fun(list(A),$o),fun(list(list(A)),list(list(A))),takeWhile(list(A)),aTP_Lamp_jg(list(A),$o)),Xsa) ) ) ).

% transpose_transpose
tff(fact_4303_Set__filter__fold,axiom,
    ! [A: $tType,Aa2: set(A),Pa: fun(A,$o)] :
      ( aa(set(A),$o,finite_finite2(A),Aa2)
     => ( filter3(A,Pa,Aa2) = finite_fold(A,set(A),aTP_Lamp_km(fun(A,$o),fun(A,fun(set(A),set(A))),Pa),bot_bot(set(A)),Aa2) ) ) ).

% Set_filter_fold
tff(fact_4304_transpose_Opsimps_I3_J,axiom,
    ! [A: $tType,Xa: A,Xsa: 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),Xa),Xsa)),Xss2))
     => ( 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),Xa),Xsa)),Xss2)) = 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),Xa),aa(list(list(A)),list(A),concat(A),aa(list(list(A)),list(list(A)),aa(fun(list(A),list(A)),fun(list(list(A)),list(list(A))),map(list(A),list(A)),aa(fun(A,fun(list(A),list(A))),fun(list(A),list(A)),aa(list(A),fun(fun(A,fun(list(A),list(A))),fun(list(A),list(A))),case_list(list(A),A),nil(A)),aTP_Lamp_iy(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)),Xsa),aa(list(list(list(A))),list(list(A)),concat(list(A)),aa(list(list(A)),list(list(list(A))),aa(fun(list(A),list(list(A))),fun(list(list(A)),list(list(list(A)))),map(list(A),list(list(A))),aa(fun(A,fun(list(A),list(list(A)))),fun(list(A),list(list(A))),aa(list(list(A)),fun(fun(A,fun(list(A),list(list(A)))),fun(list(A),list(list(A)))),case_list(list(list(A)),A),nil(list(A))),aTP_Lamp_iz(A,fun(list(A),list(list(A)))))),Xss2))))) ) ) ).

% transpose.psimps(3)
tff(fact_4305_transpose_Opelims,axiom,
    ! [A: $tType,Xa: list(list(A)),Y: list(list(A))] :
      ( ( transpose(A,Xa) = Y )
     => ( accp(list(list(A)),transpose_rel(A),Xa)
       => ( ( ( Xa = nil(list(A)) )
           => ( ( Y = nil(list(A)) )
             => ~ accp(list(list(A)),transpose_rel(A),nil(list(A))) ) )
         => ( ! [Xss: list(list(A))] :
                ( ( Xa = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),nil(A)),Xss) )
               => ( ( Y = transpose(A,Xss) )
                 => ~ accp(list(list(A)),transpose_rel(A),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),nil(A)),Xss)) ) )
           => ~ ! [X: A,Xs: list(A),Xss: list(list(A))] :
                  ( ( Xa = 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) )
                 => ( ( 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),X),aa(list(list(A)),list(A),concat(A),aa(list(list(A)),list(list(A)),aa(fun(list(A),list(A)),fun(list(list(A)),list(list(A))),map(list(A),list(A)),aa(fun(A,fun(list(A),list(A))),fun(list(A),list(A)),aa(list(A),fun(fun(A,fun(list(A),list(A))),fun(list(A),list(A))),case_list(list(A),A),nil(A)),aTP_Lamp_iy(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),aa(list(list(list(A))),list(list(A)),concat(list(A)),aa(list(list(A)),list(list(list(A))),aa(fun(list(A),list(list(A))),fun(list(list(A)),list(list(list(A)))),map(list(A),list(list(A))),aa(fun(A,fun(list(A),list(list(A)))),fun(list(A),list(list(A))),aa(list(list(A)),fun(fun(A,fun(list(A),list(list(A)))),fun(list(A),list(list(A)))),case_list(list(list(A)),A),nil(list(A))),aTP_Lamp_iz(A,fun(list(A),list(list(A)))))),Xss))))) )
                   => ~ 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.pelims
tff(fact_4306_takeWhile__idem,axiom,
    ! [A: $tType,Pa: fun(A,$o),Xsa: list(A)] : ( aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),takeWhile(A),Pa),aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),takeWhile(A),Pa),Xsa)) = aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),takeWhile(A),Pa),Xsa) ) ).

% takeWhile_idem
tff(fact_4307_member__filter,axiom,
    ! [A: $tType,Xa: A,Pa: fun(A,$o),Aa2: set(A)] :
      ( member(A,Xa,filter3(A,Pa,Aa2))
    <=> ( member(A,Xa,Aa2)
        & aa(A,$o,Pa,Xa) ) ) ).

% member_filter
tff(fact_4308_takeWhile__eq__all__conv,axiom,
    ! [A: $tType,Pa: fun(A,$o),Xsa: list(A)] :
      ( ( aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),takeWhile(A),Pa),Xsa) = Xsa )
    <=> ! [X6: A] :
          ( member(A,X6,aa(list(A),set(A),set2(A),Xsa))
         => aa(A,$o,Pa,X6) ) ) ).

% takeWhile_eq_all_conv
tff(fact_4309_takeWhile__append2,axiom,
    ! [A: $tType,Xsa: list(A),Pa: fun(A,$o),Ys: list(A)] :
      ( ! [X: A] :
          ( member(A,X,aa(list(A),set(A),set2(A),Xsa))
         => aa(A,$o,Pa,X) )
     => ( aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),takeWhile(A),Pa),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),Ys)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),takeWhile(A),Pa),Ys)) ) ) ).

% takeWhile_append2
tff(fact_4310_takeWhile__append1,axiom,
    ! [A: $tType,Xa: A,Xsa: list(A),Pa: fun(A,$o),Ys: list(A)] :
      ( member(A,Xa,aa(list(A),set(A),set2(A),Xsa))
     => ( ~ aa(A,$o,Pa,Xa)
       => ( aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),takeWhile(A),Pa),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),Ys)) = aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),takeWhile(A),Pa),Xsa) ) ) ) ).

% takeWhile_append1
tff(fact_4311_distinct__takeWhile,axiom,
    ! [A: $tType,Xsa: list(A),Pa: fun(A,$o)] :
      ( aa(list(A),$o,distinct(A),Xsa)
     => aa(list(A),$o,distinct(A),aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),takeWhile(A),Pa),Xsa)) ) ).

% distinct_takeWhile
tff(fact_4312_takeWhile_Osimps_I1_J,axiom,
    ! [A: $tType,Pa: fun(A,$o)] : ( aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),takeWhile(A),Pa),nil(A)) = nil(A) ) ).

% takeWhile.simps(1)
tff(fact_4313_takeWhile__cong,axiom,
    ! [A: $tType,L: list(A),K3: list(A),Pa: fun(A,$o),Qa: fun(A,$o)] :
      ( ( L = K3 )
     => ( ! [X: A] :
            ( member(A,X,aa(list(A),set(A),set2(A),L))
           => ( aa(A,$o,Pa,X)
            <=> aa(A,$o,Qa,X) ) )
       => ( aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),takeWhile(A),Pa),L) = aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),takeWhile(A),Qa),K3) ) ) ) ).

% takeWhile_cong
tff(fact_4314_set__takeWhileD,axiom,
    ! [A: $tType,Xa: A,Pa: fun(A,$o),Xsa: list(A)] :
      ( member(A,Xa,aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),takeWhile(A),Pa),Xsa)))
     => ( member(A,Xa,aa(list(A),set(A),set2(A),Xsa))
        & aa(A,$o,Pa,Xa) ) ) ).

% set_takeWhileD
tff(fact_4315_Set_Ofilter__def,axiom,
    ! [A: $tType,Pa: fun(A,$o),Aa2: set(A)] : ( filter3(A,Pa,Aa2) = aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_kn(fun(A,$o),fun(set(A),fun(A,$o)),Pa),Aa2)) ) ).

% Set.filter_def
tff(fact_4316_transpose_Opsimps_I1_J,axiom,
    ! [A: $tType] :
      ( accp(list(list(A)),transpose_rel(A),nil(list(A)))
     => ( transpose(A,nil(list(A))) = nil(list(A)) ) ) ).

% transpose.psimps(1)
tff(fact_4317_transpose_Opsimps_I2_J,axiom,
    ! [A: $tType,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))
     => ( transpose(A,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),nil(A)),Xss2)) = transpose(A,Xss2) ) ) ).

% transpose.psimps(2)
tff(fact_4318_takeWhile_Osimps_I2_J,axiom,
    ! [A: $tType,Pa: fun(A,$o),Xa: A,Xsa: list(A)] :
      ( aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),takeWhile(A),Pa),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa)) = $ite(aa(A,$o,Pa,Xa),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),takeWhile(A),Pa),Xsa)),nil(A)) ) ).

% takeWhile.simps(2)
tff(fact_4319_takeWhile__tail,axiom,
    ! [A: $tType,Pa: fun(A,$o),Xa: A,Xsa: list(A),L: list(A)] :
      ( ~ aa(A,$o,Pa,Xa)
     => ( aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),takeWhile(A),Pa),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),L))) = aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),takeWhile(A),Pa),Xsa) ) ) ).

% takeWhile_tail
tff(fact_4320_length__takeWhile__le,axiom,
    ! [A: $tType,Pa: fun(A,$o),Xsa: list(A)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),takeWhile(A),Pa),Xsa))),aa(list(A),nat,size_size(list(A)),Xsa)) ).

% length_takeWhile_le
tff(fact_4321_sorted__takeWhile,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xsa: list(A),Pa: fun(A,$o)] :
          ( sorted_wrt(A,ord_less_eq(A),Xsa)
         => sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),takeWhile(A),Pa),Xsa)) ) ) ).

% sorted_takeWhile
tff(fact_4322_takeWhile__eq__take,axiom,
    ! [A: $tType,Pa: fun(A,$o),Xsa: list(A)] : ( aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),takeWhile(A),Pa),Xsa) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),takeWhile(A),Pa),Xsa))),Xsa) ) ).

% takeWhile_eq_take
tff(fact_4323_takeWhile__eq__Nil__iff,axiom,
    ! [A: $tType,Pa: fun(A,$o),Xsa: list(A)] :
      ( ( aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),takeWhile(A),Pa),Xsa) = nil(A) )
    <=> ( ( Xsa = nil(A) )
        | ~ aa(A,$o,Pa,aa(list(A),A,hd(A),Xsa)) ) ) ).

% takeWhile_eq_Nil_iff
tff(fact_4324_nth__length__takeWhile,axiom,
    ! [A: $tType,Pa: fun(A,$o),Xsa: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),takeWhile(A),Pa),Xsa))),aa(list(A),nat,size_size(list(A)),Xsa))
     => ~ aa(A,$o,Pa,aa(nat,A,nth(A,Xsa),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),takeWhile(A),Pa),Xsa)))) ) ).

% nth_length_takeWhile
tff(fact_4325_takeWhile__nth,axiom,
    ! [A: $tType,J: nat,Pa: fun(A,$o),Xsa: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),takeWhile(A),Pa),Xsa)))
     => ( aa(nat,A,nth(A,aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),takeWhile(A),Pa),Xsa)),J) = aa(nat,A,nth(A,Xsa),J) ) ) ).

% takeWhile_nth
tff(fact_4326_drop__takeWhile,axiom,
    ! [A: $tType,Ia: nat,Pa: fun(A,$o),L: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ia),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),takeWhile(A),Pa),L)))
     => ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),Ia),aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),takeWhile(A),Pa),L)) = aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),takeWhile(A),Pa),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),Ia),L)) ) ) ).

% drop_takeWhile
tff(fact_4327_takeWhile__not__last,axiom,
    ! [A: $tType,Xsa: list(A)] :
      ( aa(list(A),$o,distinct(A),Xsa)
     => ( aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),takeWhile(A),aTP_Lamp_ko(list(A),fun(A,$o),Xsa)),Xsa) = aa(list(A),list(A),butlast(A),Xsa) ) ) ).

% takeWhile_not_last
tff(fact_4328_length__takeWhile__less__P__nth,axiom,
    ! [A: $tType,J: nat,Pa: fun(A,$o),Xsa: list(A)] :
      ( ! [I3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),J)
         => aa(A,$o,Pa,aa(nat,A,nth(A,Xsa),I3)) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),aa(list(A),nat,size_size(list(A)),Xsa))
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),takeWhile(A),Pa),Xsa))) ) ) ).

% length_takeWhile_less_P_nth
tff(fact_4329_less__length__takeWhile__conv,axiom,
    ! [A: $tType,Ia: nat,Pa: fun(A,$o),L: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),takeWhile(A),Pa),L)))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),aa(list(A),nat,size_size(list(A)),L))
        & ! [J2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J2),Ia)
           => aa(A,$o,Pa,aa(nat,A,nth(A,L),J2)) ) ) ) ).

% less_length_takeWhile_conv
tff(fact_4330_eq__len__takeWhile__conv,axiom,
    ! [A: $tType,Ia: nat,Pa: fun(A,$o),L: list(A)] :
      ( ( Ia = aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),takeWhile(A),Pa),L)) )
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ia),aa(list(A),nat,size_size(list(A)),L))
        & ! [J2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J2),Ia)
           => aa(A,$o,Pa,aa(nat,A,nth(A,L),J2)) )
        & ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),aa(list(A),nat,size_size(list(A)),L))
         => ~ aa(A,$o,Pa,aa(nat,A,nth(A,L),Ia)) ) ) ) ).

% eq_len_takeWhile_conv
tff(fact_4331_filter__set,axiom,
    ! [A: $tType,Pa: fun(A,$o),Xsa: list(A)] : ( filter3(A,Pa,aa(list(A),set(A),set2(A),Xsa)) = aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),Pa),Xsa)) ) ).

% filter_set
tff(fact_4332_takeWhile__eq__take__P__nth,axiom,
    ! [A: $tType,N: nat,Xsa: list(A),Pa: fun(A,$o)] :
      ( ! [I3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),N)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),Xsa))
           => aa(A,$o,Pa,aa(nat,A,nth(A,Xsa),I3)) ) )
     => ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xsa))
         => ~ aa(A,$o,Pa,aa(nat,A,nth(A,Xsa),N)) )
       => ( aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),takeWhile(A),Pa),Xsa) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),Xsa) ) ) ) ).

% takeWhile_eq_take_P_nth
tff(fact_4333_transpose_Opinduct,axiom,
    ! [A: $tType,A0: list(list(A)),Pa: fun(list(list(A)),$o)] :
      ( accp(list(list(A)),transpose_rel(A),A0)
     => ( ( accp(list(list(A)),transpose_rel(A),nil(list(A)))
         => aa(list(list(A)),$o,Pa,nil(list(A))) )
       => ( ! [Xss: list(list(A))] :
              ( accp(list(list(A)),transpose_rel(A),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),nil(A)),Xss))
             => ( aa(list(list(A)),$o,Pa,Xss)
               => aa(list(list(A)),$o,Pa,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),nil(A)),Xss)) ) )
         => ( ! [X: A,Xs: list(A),Xss: list(list(A))] :
                ( accp(list(list(A)),transpose_rel(A),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),Xss))
               => ( aa(list(list(A)),$o,Pa,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),Xs),aa(list(list(list(A))),list(list(A)),concat(list(A)),aa(list(list(A)),list(list(list(A))),aa(fun(list(A),list(list(A))),fun(list(list(A)),list(list(list(A)))),map(list(A),list(list(A))),aa(fun(A,fun(list(A),list(list(A)))),fun(list(A),list(list(A))),aa(list(list(A)),fun(fun(A,fun(list(A),list(list(A)))),fun(list(A),list(list(A)))),case_list(list(list(A)),A),nil(list(A))),aTP_Lamp_iz(A,fun(list(A),list(list(A)))))),Xss))))
                 => aa(list(list(A)),$o,Pa,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),Xss)) ) )
           => aa(list(list(A)),$o,Pa,A0) ) ) ) ) ).

% transpose.pinduct
tff(fact_4334_filter__equals__takeWhile__sorted__rev,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F2: fun(B,A),Xsa: list(B),T2: A] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),rev(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Xsa)))
         => ( aa(list(B),list(B),aa(fun(B,$o),fun(list(B),list(B)),filter2(B),aa(A,fun(B,$o),aTP_Lamp_kp(fun(B,A),fun(A,fun(B,$o)),F2),T2)),Xsa) = aa(list(B),list(B),aa(fun(B,$o),fun(list(B),list(B)),takeWhile(B),aa(A,fun(B,$o),aTP_Lamp_kp(fun(B,A),fun(A,fun(B,$o)),F2),T2)),Xsa) ) ) ) ).

% filter_equals_takeWhile_sorted_rev
tff(fact_4335_case__prod__Pair__iden,axiom,
    ! [B: $tType,A: $tType,P2: 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)),P2) = P2 ) ).

% case_prod_Pair_iden
tff(fact_4336_cofinal__def,axiom,
    ! [A: $tType,Aa2: set(A),R2: set(product_prod(A,A))] :
      ( bNF_Ca7293521722713021262ofinal(A,Aa2,R2)
    <=> ! [X6: A] :
          ( member(A,X6,aa(set(product_prod(A,A)),set(A),field2(A),R2))
         => ? [Xa2: A] :
              ( member(A,Xa2,Aa2)
              & ( X6 != Xa2 )
              & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X6),Xa2),R2) ) ) ) ).

% cofinal_def
tff(fact_4337_length__product__lists,axiom,
    ! [A: $tType,Xss2: list(list(A))] : ( aa(list(list(A)),nat,size_size(list(list(A))),aa(list(list(A)),list(list(A)),product_lists(A),Xss2)) = aa(nat,nat,aa(list(nat),fun(nat,nat),aa(fun(nat,fun(nat,nat)),fun(list(nat),fun(nat,nat)),foldr(nat,nat),times_times(nat)),aa(list(list(A)),list(nat),aa(fun(list(A),nat),fun(list(list(A)),list(nat)),map(list(A),nat),size_size(list(A))),Xss2)),one_one(nat)) ) ).

% length_product_lists
tff(fact_4338_sum__diff1_H__aux,axiom,
    ! [B: $tType,A: $tType] :
      ( ab_group_add(B)
     => ! [Fa: set(A),Ia2: set(A),F2: fun(A,B),Ia: A] :
          ( aa(set(A),$o,finite_finite2(A),Fa)
         => ( 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_kq(set(A),fun(fun(A,B),fun(A,$o)),Ia2),F2))),Fa)
           => ( groups1027152243600224163dd_sum(A,B,F2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Ia2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Ia),bot_bot(set(A))))) = $ite(member(A,Ia,Ia2),aa(B,B,aa(B,fun(B,B),minus_minus(B),groups1027152243600224163dd_sum(A,B,F2,Ia2)),aa(A,B,F2,Ia)),groups1027152243600224163dd_sum(A,B,F2,Ia2)) ) ) ) ) ).

% sum_diff1'_aux
tff(fact_4339_foldr__append,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,fun(A,A)),Xsa: list(B),Ys: list(B),A3: A] : ( aa(A,A,aa(list(B),fun(A,A),aa(fun(B,fun(A,A)),fun(list(B),fun(A,A)),foldr(B,A),F2),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Xsa),Ys)),A3) = aa(A,A,aa(list(B),fun(A,A),aa(fun(B,fun(A,A)),fun(list(B),fun(A,A)),foldr(B,A),F2),Xsa),aa(A,A,aa(list(B),fun(A,A),aa(fun(B,fun(A,A)),fun(list(B),fun(A,A)),foldr(B,A),F2),Ys),A3)) ) ).

% foldr_append
tff(fact_4340_sum_Oempty_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [P2: fun(B,A)] : ( groups1027152243600224163dd_sum(B,A,P2,bot_bot(set(B))) = zero_zero(A) ) ) ).

% sum.empty'
tff(fact_4341_foldr__length,axiom,
    ! [A: $tType,L: list(A)] : ( aa(nat,nat,aa(list(A),fun(nat,nat),aa(fun(A,fun(nat,nat)),fun(list(A),fun(nat,nat)),foldr(A,nat),aTP_Lamp_kr(A,fun(nat,nat))),L),zero_zero(nat)) = aa(list(A),nat,size_size(list(A)),L) ) ).

% foldr_length
tff(fact_4342_sum_Oinsert_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [Ia2: set(A),P2: fun(A,B),Ia: A] :
          ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_ks(set(A),fun(fun(A,B),fun(A,$o)),Ia2),P2)))
         => ( groups1027152243600224163dd_sum(A,B,P2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Ia),Ia2)) = $ite(member(A,Ia,Ia2),groups1027152243600224163dd_sum(A,B,P2,Ia2),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,P2,Ia)),groups1027152243600224163dd_sum(A,B,P2,Ia2))) ) ) ) ).

% sum.insert'
tff(fact_4343_foldr__cong,axiom,
    ! [B: $tType,A: $tType,A3: A,B3: A,L: list(B),K3: list(B),F2: fun(B,fun(A,A)),G: fun(B,fun(A,A))] :
      ( ( A3 = B3 )
     => ( ( L = K3 )
       => ( ! [A4: A,X: B] :
              ( member(B,X,aa(list(B),set(B),set2(B),L))
             => ( aa(A,A,aa(B,fun(A,A),F2,X),A4) = aa(A,A,aa(B,fun(A,A),G,X),A4) ) )
         => ( aa(A,A,aa(list(B),fun(A,A),aa(fun(B,fun(A,A)),fun(list(B),fun(A,A)),foldr(B,A),F2),L),A3) = aa(A,A,aa(list(B),fun(A,A),aa(fun(B,fun(A,A)),fun(list(B),fun(A,A)),foldr(B,A),G),K3),B3) ) ) ) ) ).

% foldr_cong
tff(fact_4344_foldr__snd__zip,axiom,
    ! [B: $tType,A: $tType,E: $tType,Ys: list(A),Xsa: list(B),F2: fun(A,fun(E,E)),B3: E] :
      ( aa(nat,$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)),Xsa))
     => ( aa(E,E,aa(list(product_prod(B,A)),fun(E,E),aa(fun(product_prod(B,A),fun(E,E)),fun(list(product_prod(B,A)),fun(E,E)),foldr(product_prod(B,A),E),aa(fun(B,fun(A,fun(E,E))),fun(product_prod(B,A),fun(E,E)),product_case_prod(B,A,fun(E,E)),aTP_Lamp_kt(fun(A,fun(E,E)),fun(B,fun(A,fun(E,E))),F2))),aa(list(A),list(product_prod(B,A)),aa(list(B),fun(list(A),list(product_prod(B,A))),zip(B,A),Xsa),Ys)),B3) = aa(E,E,aa(list(A),fun(E,E),aa(fun(A,fun(E,E)),fun(list(A),fun(E,E)),foldr(A,E),F2),Ys),B3) ) ) ).

% foldr_snd_zip
tff(fact_4345_foldr__conv__foldl,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,fun(A,A)),Xsa: list(B),A3: A] : ( aa(A,A,aa(list(B),fun(A,A),aa(fun(B,fun(A,A)),fun(list(B),fun(A,A)),foldr(B,A),F2),Xsa),A3) = aa(list(B),A,aa(A,fun(list(B),A),aa(fun(A,fun(B,A)),fun(A,fun(list(B),A)),foldl(A,B),aTP_Lamp_ku(fun(B,fun(A,A)),fun(A,fun(B,A)),F2)),A3),aa(list(B),list(B),rev(B),Xsa)) ) ).

% foldr_conv_foldl
tff(fact_4346_foldl__conv__foldr,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,fun(B,A)),A3: A,Xsa: list(B)] : ( aa(list(B),A,aa(A,fun(list(B),A),aa(fun(A,fun(B,A)),fun(A,fun(list(B),A)),foldl(A,B),F2),A3),Xsa) = aa(A,A,aa(list(B),fun(A,A),aa(fun(B,fun(A,A)),fun(list(B),fun(A,A)),foldr(B,A),aTP_Lamp_kv(fun(A,fun(B,A)),fun(B,fun(A,A)),F2)),aa(list(B),list(B),rev(B),Xsa)),A3) ) ).

% foldl_conv_foldr
tff(fact_4347_sum_Omono__neutral__cong__right_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [S: set(A),T4: set(A),G: fun(A,B),H2: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),T4)
         => ( ! [X: A] :
                ( member(A,X,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T4),S))
               => ( aa(A,B,G,X) = zero_zero(B) ) )
           => ( ! [X: A] :
                  ( member(A,X,S)
                 => ( aa(A,B,G,X) = aa(A,B,H2,X) ) )
             => ( groups1027152243600224163dd_sum(A,B,G,T4) = groups1027152243600224163dd_sum(A,B,H2,S) ) ) ) ) ) ).

% sum.mono_neutral_cong_right'
tff(fact_4348_sum_Omono__neutral__cong__left_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [S: set(A),T4: set(A),H2: fun(A,B),G: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),T4)
         => ( ! [I3: A] :
                ( member(A,I3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T4),S))
               => ( aa(A,B,H2,I3) = zero_zero(B) ) )
           => ( ! [X: A] :
                  ( member(A,X,S)
                 => ( aa(A,B,G,X) = aa(A,B,H2,X) ) )
             => ( groups1027152243600224163dd_sum(A,B,G,S) = groups1027152243600224163dd_sum(A,B,H2,T4) ) ) ) ) ) ).

% sum.mono_neutral_cong_left'
tff(fact_4349_sum_Omono__neutral__right_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [S: set(A),T4: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),T4)
         => ( ! [X: A] :
                ( member(A,X,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T4),S))
               => ( aa(A,B,G,X) = zero_zero(B) ) )
           => ( groups1027152243600224163dd_sum(A,B,G,T4) = groups1027152243600224163dd_sum(A,B,G,S) ) ) ) ) ).

% sum.mono_neutral_right'
tff(fact_4350_sum_Omono__neutral__left_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [S: set(A),T4: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),T4)
         => ( ! [X: A] :
                ( member(A,X,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T4),S))
               => ( aa(A,B,G,X) = zero_zero(B) ) )
           => ( groups1027152243600224163dd_sum(A,B,G,S) = groups1027152243600224163dd_sum(A,B,G,T4) ) ) ) ) ).

% sum.mono_neutral_left'
tff(fact_4351_foldr__length__aux,axiom,
    ! [A: $tType,L: list(A),A3: nat] : ( aa(nat,nat,aa(list(A),fun(nat,nat),aa(fun(A,fun(nat,nat)),fun(list(A),fun(nat,nat)),foldr(A,nat),aTP_Lamp_kr(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_4352_horner__sum__foldr,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_semiring_0(A)
     => ! [F2: fun(B,A),A3: A,Xsa: 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),F2),A3),Xsa) = aa(A,A,aa(list(B),fun(A,A),aa(fun(B,fun(A,A)),fun(list(B),fun(A,A)),foldr(B,A),aa(A,fun(B,fun(A,A)),aTP_Lamp_kw(fun(B,A),fun(A,fun(B,fun(A,A))),F2),A3)),Xsa),zero_zero(A)) ) ) ).

% horner_sum_foldr
tff(fact_4353_sum__diff1_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ab_group_add(B)
     => ! [Ia2: set(A),F2: fun(A,B),Ia: A] :
          ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_kq(set(A),fun(fun(A,B),fun(A,$o)),Ia2),F2)))
         => ( groups1027152243600224163dd_sum(A,B,F2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Ia2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Ia),bot_bot(set(A))))) = $ite(member(A,Ia,Ia2),aa(B,B,aa(B,fun(B,B),minus_minus(B),groups1027152243600224163dd_sum(A,B,F2,Ia2)),aa(A,B,F2,Ia)),groups1027152243600224163dd_sum(A,B,F2,Ia2)) ) ) ) ).

% sum_diff1'
tff(fact_4354_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_kx(num,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),L)),Qr) ) ).

% divmod_step_integer_def
tff(fact_4355_Id__on__fold,axiom,
    ! [A: $tType,Aa2: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),Aa2)
     => ( id_on(A,Aa2) = finite_fold(A,set(product_prod(A,A)),aTP_Lamp_ky(A,fun(set(product_prod(A,A)),set(product_prod(A,A)))),bot_bot(set(product_prod(A,A))),Aa2) ) ) ).

% Id_on_fold
tff(fact_4356_max__ext__def,axiom,
    ! [A: $tType,X4: set(product_prod(A,A))] : ( max_ext(A,X4) = aa(fun(product_prod(set(A),set(A)),$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),$o),max_extp(A,aTP_Lamp_cz(set(product_prod(A,A)),fun(A,fun(A,$o)),X4)))) ) ).

% max_ext_def
tff(fact_4357_Id__on__def,axiom,
    ! [A: $tType,Aa2: set(A)] : ( id_on(A,Aa2) = 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_kz(A,set(product_prod(A,A)))),Aa2)) ) ).

% Id_on_def
tff(fact_4358_Id__onI,axiom,
    ! [A: $tType,A3: A,Aa2: set(A)] :
      ( member(A,A3,Aa2)
     => 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,Aa2)) ) ).

% Id_onI
tff(fact_4359_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_4360_Id__onE,axiom,
    ! [A: $tType,C2: product_prod(A,A),Aa2: set(A)] :
      ( member(product_prod(A,A),C2,id_on(A,Aa2))
     => ~ ! [X: A] :
            ( member(A,X,Aa2)
           => ( C2 != aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),X) ) ) ) ).

% Id_onE
tff(fact_4361_Id__on__eqI,axiom,
    ! [A: $tType,A3: A,B3: A,Aa2: set(A)] :
      ( ( A3 = B3 )
     => ( member(A,A3,Aa2)
       => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B3),id_on(A,Aa2)) ) ) ).

% Id_on_eqI
tff(fact_4362_Id__on__iff,axiom,
    ! [A: $tType,Xa: A,Y: A,Aa2: set(A)] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Y),id_on(A,Aa2))
    <=> ( ( Xa = Y )
        & member(A,Xa,Aa2) ) ) ).

% Id_on_iff
tff(fact_4363_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_4364_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_4365_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_4366_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_la(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_4367_divmod__integer_H__def,axiom,
    ! [M2: num,N: num] : ( unique8689654367752047608divmod(code_integer,M2,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),divide_divide(code_integer,numeral_numeral(code_integer,M2),numeral_numeral(code_integer,N))),modulo_modulo(code_integer,numeral_numeral(code_integer,M2),numeral_numeral(code_integer,N))) ) ).

% divmod_integer'_def
tff(fact_4368_trans__Id__on,axiom,
    ! [A: $tType,Aa2: set(A)] : trans(A,id_on(A,Aa2)) ).

% trans_Id_on
tff(fact_4369_refl__on__Id__on,axiom,
    ! [A: $tType,Aa2: set(A)] : refl_on(A,Aa2,id_on(A,Aa2)) ).

% refl_on_Id_on
tff(fact_4370_Id__on__def_H,axiom,
    ! [A: $tType,Aa2: fun(A,$o)] : ( id_on(A,aa(fun(A,$o),set(A),collect(A),Aa2)) = aa(fun(product_prod(A,A),$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,$o),aTP_Lamp_lb(fun(A,$o),fun(A,fun(A,$o)),Aa2))) ) ).

% Id_on_def'
tff(fact_4371_Misc_Ofoldr__Cons,axiom,
    ! [A: $tType,Xsa: list(A)] : ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),aa(fun(A,fun(list(A),list(A))),fun(list(A),fun(list(A),list(A))),foldr(A,list(A)),cons(A)),Xsa),nil(A)) = Xsa ) ).

% Misc.foldr_Cons
tff(fact_4372_concat__conv__foldr,axiom,
    ! [A: $tType,Xss2: list(list(A))] : ( aa(list(list(A)),list(A),concat(A),Xss2) = aa(list(A),list(A),aa(list(list(A)),fun(list(A),list(A)),aa(fun(list(A),fun(list(A),list(A))),fun(list(list(A)),fun(list(A),list(A))),foldr(list(A),list(A)),append(A)),Xss2),nil(A)) ) ).

% concat_conv_foldr
tff(fact_4373_filter__conv__foldr,axiom,
    ! [A: $tType,Pa: fun(A,$o),Xsa: list(A)] : ( aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),Pa),Xsa) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),aa(fun(A,fun(list(A),list(A))),fun(list(A),fun(list(A),list(A))),foldr(A,list(A)),aTP_Lamp_lc(fun(A,$o),fun(A,fun(list(A),list(A))),Pa)),Xsa),nil(A)) ) ).

% filter_conv_foldr
tff(fact_4374_Id__on__set,axiom,
    ! [A: $tType,Xsa: list(A)] : ( id_on(A,aa(list(A),set(A),set2(A),Xsa)) = aa(list(product_prod(A,A)),set(product_prod(A,A)),set2(product_prod(A,A)),aa(list(A),list(product_prod(A,A)),aa(fun(A,product_prod(A,A)),fun(list(A),list(product_prod(A,A))),map(A,product_prod(A,A)),aTP_Lamp_ii(A,product_prod(A,A))),Xsa)) ) ).

% Id_on_set
tff(fact_4375_max__extp__max__ext__eq,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),X4: set(A),Xa4: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),max_extp(A,aTP_Lamp_cz(set(product_prod(A,A)),fun(A,fun(A,$o)),R)),X4),Xa4)
    <=> 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)),X4),Xa4),max_ext(A,R)) ) ).

% max_extp_max_ext_eq
tff(fact_4376_max__extp__eq,axiom,
    ! [A: $tType,R2: fun(A,fun(A,$o)),Xa: set(A),Y: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),max_extp(A,R2),Xa),Y)
    <=> 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)),Xa),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_4377_insort__key__remove1,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [A3: A,Xsa: list(A),F2: fun(A,B)] :
          ( member(A,A3,aa(list(A),set(A),set2(A),Xsa))
         => ( sorted_wrt(B,ord_less_eq(B),aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),Xsa))
           => ( ( aa(list(A),A,hd(A),aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_ld(A,fun(fun(A,B),fun(A,$o)),A3),F2)),Xsa)) = A3 )
             => ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,B,F2),A3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),remove1(A),A3),Xsa)) = Xsa ) ) ) ) ) ).

% insort_key_remove1
tff(fact_4378_finite__enumerate,axiom,
    ! [S: set(nat)] :
      ( aa(set(nat),$o,finite_finite2(nat),S)
     => ? [R3: fun(nat,nat)] :
          ( strict_mono_on(nat,nat,R3,aa(nat,set(nat),set_ord_lessThan(nat),finite_card(nat,S)))
          & ! [N8: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N8),finite_card(nat,S))
             => member(nat,aa(nat,nat,R3,N8),S) ) ) ) ).

% finite_enumerate
tff(fact_4379_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),$o),aTP_Lamp_le(nat,fun(list(A),fun(list(A),$o)),N)))) ) ).

% lexn.simps(2)
tff(fact_4380_set__zip,axiom,
    ! [A: $tType,B: $tType,Xsa: list(A),Ys: list(B)] : ( aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),Xsa),Ys)) = aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(list(B),fun(product_prod(A,B),$o),aTP_Lamp_lf(list(A),fun(list(B),fun(product_prod(A,B),$o)),Xsa),Ys)) ) ).

% set_zip
tff(fact_4381_finite__atLeastLessThan__integer,axiom,
    ! [L: code_integer,U: code_integer] : aa(set(code_integer),$o,finite_finite2(code_integer),set_or7035219750837199246ssThan(code_integer,L,U)) ).

% finite_atLeastLessThan_integer
tff(fact_4382_finite__atLeastAtMost__integer,axiom,
    ! [L: code_integer,U: code_integer] : aa(set(code_integer),$o,finite_finite2(code_integer),set_or1337092689740270186AtMost(code_integer,L,U)) ).

% finite_atLeastAtMost_integer
tff(fact_4383_map__prod__simp,axiom,
    ! [E: $tType,A: $tType,B: $tType,F4: $tType,F2: fun(E,A),G: fun(F4,B),A3: E,B3: F4] : ( aa(product_prod(E,F4),product_prod(A,B),product_map_prod(E,A,F4,B,F2,G),aa(F4,product_prod(E,F4),aa(E,fun(F4,product_prod(E,F4)),product_Pair(E,F4),A3),B3)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(E,A,F2,A3)),aa(F4,B,G,B3)) ) ).

% map_prod_simp
tff(fact_4384_finite__greaterThanLessThan__integer,axiom,
    ! [L: code_integer,U: code_integer] : aa(set(code_integer),$o,finite_finite2(code_integer),set_or5935395276787703475ssThan(code_integer,L,U)) ).

% finite_greaterThanLessThan_integer
tff(fact_4385_finite__greaterThanAtMost__integer,axiom,
    ! [L: code_integer,U: code_integer] : aa(set(code_integer),$o,finite_finite2(code_integer),set_or3652927894154168847AtMost(code_integer,L,U)) ).

% finite_greaterThanAtMost_integer
tff(fact_4386_remove1__insort__key,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Xa: A,F2: fun(A,B),Xsa: list(A)] : ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),remove1(A),Xa),aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,B,F2),Xa),Xsa)) = Xsa ) ) ).

% remove1_insort_key
tff(fact_4387_eq__or__mem__image__simp,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A3: B,B2: set(B)] : ( aa(fun(A,$o),set(A),collect(A),aa(set(B),fun(A,$o),aa(B,fun(set(B),fun(A,$o)),aTP_Lamp_lg(fun(B,A),fun(B,fun(set(B),fun(A,$o))),F2),A3),B2)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),aa(B,A,F2,A3)),aa(fun(A,$o),set(A),collect(A),aa(set(B),fun(A,$o),aTP_Lamp_lh(fun(B,A),fun(set(B),fun(A,$o)),F2),B2))) ) ).

% eq_or_mem_image_simp
tff(fact_4388_length__insort,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),Xa: A,Xsa: list(A)] : ( aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,B,F2),Xa),Xsa)) = aa(nat,nat,suc,aa(list(A),nat,size_size(list(A)),Xsa)) ) ) ).

% length_insort
tff(fact_4389_sort__key__simps_I2_J,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),Xa: A,Xsa: list(A)] : ( aa(list(A),list(A),aa(fun(A,B),fun(list(A),list(A)),linorder_sort_key(A,B),F2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,B,F2),Xa),aa(list(A),list(A),aa(fun(A,B),fun(list(A),list(A)),linorder_sort_key(A,B),F2),Xsa)) ) ) ).

% sort_key_simps(2)
tff(fact_4390_map__prod__imageI,axiom,
    ! [F4: $tType,E: $tType,B: $tType,A: $tType,A3: A,B3: B,R: set(product_prod(A,B)),F2: fun(A,E),G: fun(B,F4)] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B3),R)
     => member(product_prod(E,F4),aa(F4,product_prod(E,F4),aa(E,fun(F4,product_prod(E,F4)),product_Pair(E,F4),aa(A,E,F2,A3)),aa(B,F4,G,B3)),aa(set(product_prod(A,B)),set(product_prod(E,F4)),image2(product_prod(A,B),product_prod(E,F4),product_map_prod(A,E,B,F4,F2,G)),R)) ) ).

% map_prod_imageI
tff(fact_4391_sorted__list__of__set_Osorted__key__list__of__set__insert,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Aa2: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( ~ member(A,Xa,Aa2)
           => ( aa(set(A),list(A),linord4507533701916653071of_set(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),Aa2)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_ip(A,A)),Xa),aa(set(A),list(A),linord4507533701916653071of_set(A),Aa2)) ) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_insert
tff(fact_4392_sorted__list__of__set_Osorted__key__list__of__set__insert__remove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Aa2: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( aa(set(A),list(A),linord4507533701916653071of_set(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),Aa2)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_ip(A,A)),Xa),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)),Aa2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A)))))) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_insert_remove
tff(fact_4393_prod__fun__imageE,axiom,
    ! [B: $tType,A: $tType,F4: $tType,E: $tType,C2: product_prod(A,B),F2: fun(E,A),G: fun(F4,B),R: set(product_prod(E,F4))] :
      ( member(product_prod(A,B),C2,aa(set(product_prod(E,F4)),set(product_prod(A,B)),image2(product_prod(E,F4),product_prod(A,B),product_map_prod(E,A,F4,B,F2,G)),R))
     => ~ ! [X: E,Y2: F4] :
            ( ( C2 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(E,A,F2,X)),aa(F4,B,G,Y2)) )
           => ~ member(product_prod(E,F4),aa(F4,product_prod(E,F4),aa(E,fun(F4,product_prod(E,F4)),product_Pair(E,F4),X),Y2),R) ) ) ).

% prod_fun_imageE
tff(fact_4394_filter__insort__triv,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Pa: fun(A,$o),Xa: A,F2: fun(A,B),Xsa: list(A)] :
          ( ~ aa(A,$o,Pa,Xa)
         => ( aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),Pa),aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,B,F2),Xa),Xsa)) = aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),Pa),Xsa) ) ) ) ).

% filter_insort_triv
tff(fact_4395_setcompr__eq__image,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Pa: fun(B,$o)] : ( aa(fun(A,$o),set(A),collect(A),aa(fun(B,$o),fun(A,$o),aTP_Lamp_li(fun(B,A),fun(fun(B,$o),fun(A,$o)),F2),Pa)) = aa(set(B),set(A),image2(B,A,F2),aa(fun(B,$o),set(B),collect(B),Pa)) ) ).

% setcompr_eq_image
tff(fact_4396_Setcompr__eq__image,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Aa2: set(B)] : ( aa(fun(A,$o),set(A),collect(A),aa(set(B),fun(A,$o),aTP_Lamp_lh(fun(B,A),fun(set(B),fun(A,$o)),F2),Aa2)) = aa(set(B),set(A),image2(B,A,F2),Aa2) ) ).

% Setcompr_eq_image
tff(fact_4397_set__Cons__def,axiom,
    ! [A: $tType,Aa2: set(A),XS: set(list(A))] : ( aa(set(list(A)),set(list(A)),aa(set(A),fun(set(list(A)),set(list(A))),set_Cons(A),Aa2),XS) = aa(fun(list(A),$o),set(list(A)),collect(list(A)),aa(set(list(A)),fun(list(A),$o),aTP_Lamp_lj(set(A),fun(set(list(A)),fun(list(A),$o)),Aa2),XS)) ) ).

% set_Cons_def
tff(fact_4398_insort__left__comm,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Y: A,Xsa: list(A)] : ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_ip(A,A)),Xa),aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_ip(A,A)),Y),Xsa)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_ip(A,A)),Y),aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_ip(A,A)),Xa),Xsa)) ) ) ).

% insort_left_comm
tff(fact_4399_insort__key__left__comm,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F2: fun(B,A),Xa: B,Y: B,Xsa: list(B)] :
          ( ( aa(B,A,F2,Xa) != aa(B,A,F2,Y) )
         => ( aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F2),Y),aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F2),Xa),Xsa)) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F2),Xa),aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F2),Y),Xsa)) ) ) ) ).

% insort_key_left_comm
tff(fact_4400_insort__not__Nil,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),A3: A,Xsa: list(A)] : ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,B,F2),A3),Xsa) != nil(A) ) ) ).

% insort_not_Nil
tff(fact_4401_finite__atLeastZeroLessThan__integer,axiom,
    ! [U: code_integer] : aa(set(code_integer),$o,finite_finite2(code_integer),set_or7035219750837199246ssThan(code_integer,zero_zero(code_integer),U)) ).

% finite_atLeastZeroLessThan_integer
tff(fact_4402_insort__key_Osimps_I2_J,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),Xa: A,Y: A,Ys: list(A)] :
          ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,B,F2),Xa),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)) = $ite(aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,Xa)),aa(A,B,F2,Y)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,B,F2),Xa),Ys))) ) ) ).

% insort_key.simps(2)
tff(fact_4403_insort__key_Osimps_I1_J,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),Xa: A] : ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,B,F2),Xa),nil(A)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),nil(A)) ) ) ).

% insort_key.simps(1)
tff(fact_4404_set__insort__key,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),Xa: A,Xsa: list(A)] : ( aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,B,F2),Xa),Xsa)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),aa(list(A),set(A),set2(A),Xsa)) ) ) ).

% set_insort_key
tff(fact_4405_strict__mono__onD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ord(A)
        & ord(B) )
     => ! [F2: fun(A,B),Aa2: set(A),R2: A,S2: A] :
          ( strict_mono_on(A,B,F2,Aa2)
         => ( member(A,R2,Aa2)
           => ( member(A,S2,Aa2)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),R2),S2)
               => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,R2)),aa(A,B,F2,S2)) ) ) ) ) ) ).

% strict_mono_onD
tff(fact_4406_strict__mono__onI,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ord(A)
        & ord(B) )
     => ! [Aa2: set(A),F2: fun(A,B)] :
          ( ! [R3: A,S6: A] :
              ( member(A,R3,Aa2)
             => ( member(A,S6,Aa2)
               => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),R3),S6)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,R3)),aa(A,B,F2,S6)) ) ) )
         => strict_mono_on(A,B,F2,Aa2) ) ) ).

% strict_mono_onI
tff(fact_4407_strict__mono__on__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ord(A)
        & ord(B) )
     => ! [F2: fun(A,B),Aa2: set(A)] :
          ( strict_mono_on(A,B,F2,Aa2)
        <=> ! [R6: A,S7: A] :
              ( ( member(A,R6,Aa2)
                & member(A,S7,Aa2)
                & aa(A,$o,aa(A,fun(A,$o),ord_less(A),R6),S7) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,R6)),aa(A,B,F2,S7)) ) ) ) ).

% strict_mono_on_def
tff(fact_4408_distinct__insort,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),Xa: A,Xsa: list(A)] :
          ( aa(list(A),$o,distinct(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,B,F2),Xa),Xsa))
        <=> ( ~ member(A,Xa,aa(list(A),set(A),set2(A),Xsa))
            & aa(list(A),$o,distinct(A),Xsa) ) ) ) ).

% distinct_insort
tff(fact_4409_full__SetCompr__eq,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A)] : ( aa(fun(A,$o),set(A),collect(A),aTP_Lamp_lk(fun(B,A),fun(A,$o),F2)) = aa(set(B),set(A),image2(B,A,F2),top_top(set(B))) ) ).

% full_SetCompr_eq
tff(fact_4410_sort__key__def,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),Xsa: list(A)] : ( aa(list(A),list(A),aa(fun(A,B),fun(list(A),list(A)),linorder_sort_key(A,B),F2),Xsa) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),aa(fun(A,fun(list(A),list(A))),fun(list(A),fun(list(A),list(A))),foldr(A,list(A)),linorder_insort_key(A,B,F2)),Xsa),nil(A)) ) ) ).

% sort_key_def
tff(fact_4411_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_ll(product_prod(A,A),$o)) ) ).

% Id_def
tff(fact_4412_map__prod__def,axiom,
    ! [A: $tType,E: $tType,F4: $tType,B: $tType,F2: fun(A,E),G: fun(B,F4)] : ( product_map_prod(A,E,B,F4,F2,G) = aa(fun(A,fun(B,product_prod(E,F4))),fun(product_prod(A,B),product_prod(E,F4)),product_case_prod(A,B,product_prod(E,F4)),aa(fun(B,F4),fun(A,fun(B,product_prod(E,F4))),aTP_Lamp_lm(fun(A,E),fun(fun(B,F4),fun(A,fun(B,product_prod(E,F4)))),F2),G)) ) ).

% map_prod_def
tff(fact_4413_sorted__insort,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Xsa: 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_ip(A,A)),Xa),Xsa))
        <=> sorted_wrt(A,ord_less_eq(A),Xsa) ) ) ).

% sorted_insort
tff(fact_4414_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),$o),aTP_Lamp_ln(set(product_prod(A,A)),fun(list(A),fun(list(A),$o)),R2))) ) ).

% listrel1_def
tff(fact_4415_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),$o),aTP_Lamp_lo(set(product_prod(A,A)),fun(list(A),fun(list(A),$o)),R2))) ) ).

% lexord_def
tff(fact_4416_sorted__list__of__set_Ofold__insort__key_Oeq__fold,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Aa2: set(A)] : ( aa(set(A),list(A),linord4507533701916653071of_set(A),Aa2) = finite_fold(A,list(A),linorder_insort_key(A,A,aTP_Lamp_ip(A,A)),nil(A),Aa2) ) ) ).

% sorted_list_of_set.fold_insort_key.eq_fold
tff(fact_4417_insort__is__Cons,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Xsa: list(A),F2: fun(A,B),A3: A] :
          ( ! [X: A] :
              ( member(A,X,aa(list(A),set(A),set2(A),Xsa))
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,A3)),aa(A,B,F2,X)) )
         => ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,B,F2),A3),Xsa) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),Xsa) ) ) ) ).

% insort_is_Cons
tff(fact_4418_Un__interval,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(A)
     => ! [B1: A,B22: A,B32: A,F2: fun(A,B)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B1),B22)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B22),B32)
           => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),aa(fun(B,$o),set(B),collect(B),aa(fun(A,B),fun(B,$o),aa(A,fun(fun(A,B),fun(B,$o)),aTP_Lamp_lp(A,fun(A,fun(fun(A,B),fun(B,$o))),B1),B22),F2))),aa(fun(B,$o),set(B),collect(B),aa(fun(A,B),fun(B,$o),aa(A,fun(fun(A,B),fun(B,$o)),aTP_Lamp_lp(A,fun(A,fun(fun(A,B),fun(B,$o))),B22),B32),F2))) = aa(fun(B,$o),set(B),collect(B),aa(fun(A,B),fun(B,$o),aa(A,fun(fun(A,B),fun(B,$o)),aTP_Lamp_lp(A,fun(A,fun(fun(A,B),fun(B,$o))),B1),B32),F2)) ) ) ) ) ).

% Un_interval
tff(fact_4419_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),$o),aTP_Lamp_lq(set(product_prod(A,A)),fun(list(A),fun(list(A),$o)),R2))) ) ).

% lex_conv
tff(fact_4420_sorted__insort__key,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F2: fun(B,A),Xa: B,Xsa: list(B)] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F2),Xa),Xsa)))
        <=> sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Xsa)) ) ) ).

% sorted_insort_key
tff(fact_4421_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_lr(set(product_prod(A,A)),fun(nat,fun(list(A),fun(list(A),$o))),R2),N))) ) ).

% lexn_conv
tff(fact_4422_set__conv__nth,axiom,
    ! [A: $tType,Xsa: list(A)] : ( aa(list(A),set(A),set2(A),Xsa) = aa(fun(A,$o),set(A),collect(A),aTP_Lamp_ls(list(A),fun(A,$o),Xsa)) ) ).

% set_conv_nth
tff(fact_4423_distinct__insort__key,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F2: fun(B,A),Xa: B,Xsa: list(B)] :
          ( aa(list(A),$o,distinct(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F2),Xa),Xsa)))
        <=> ( ~ member(A,aa(B,A,F2,Xa),aa(set(B),set(A),image2(B,A,F2),aa(list(B),set(B),set2(B),Xsa)))
            & aa(list(A),$o,distinct(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Xsa)) ) ) ) ).

% distinct_insort_key
tff(fact_4424_filter__insort,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F2: fun(B,A),Xsa: list(B),Pa: fun(B,$o),Xa: B] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Xsa))
         => ( aa(B,$o,Pa,Xa)
           => ( aa(list(B),list(B),aa(fun(B,$o),fun(list(B),list(B)),filter2(B),Pa),aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F2),Xa),Xsa)) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F2),Xa),aa(list(B),list(B),aa(fun(B,$o),fun(list(B),list(B)),filter2(B),Pa),Xsa)) ) ) ) ) ).

% filter_insort
tff(fact_4425_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_lt(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_lu(product_prod(product_prod($o,A),product_prod($o,B)),$o))) ) ).

% brk_rel_def
tff(fact_4426_insort__remove1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,Xsa: list(A)] :
          ( member(A,A3,aa(list(A),set(A),set2(A),Xsa))
         => ( sorted_wrt(A,ord_less_eq(A),Xsa)
           => ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_ip(A,A)),A3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),remove1(A),A3),Xsa)) = Xsa ) ) ) ) ).

% insort_remove1
tff(fact_4427_sorted__list__of__set_Ofold__insort__key_Oremove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Aa2: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( member(A,Xa,Aa2)
           => ( aa(set(A),list(A),linord4507533701916653071of_set(A),Aa2) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_ip(A,A)),Xa),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)),Aa2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A)))))) ) ) ) ) ).

% sorted_list_of_set.fold_insort_key.remove
tff(fact_4428_set__drop__conv,axiom,
    ! [A: $tType,N: nat,L: list(A)] : ( aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),L)) = aa(fun(A,$o),set(A),collect(A),aa(list(A),fun(A,$o),aTP_Lamp_lv(nat,fun(list(A),fun(A,$o)),N),L)) ) ).

% set_drop_conv
tff(fact_4429_sorted__insort__is__snoc,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xsa: list(A),A3: A] :
          ( sorted_wrt(A,ord_less_eq(A),Xsa)
         => ( ! [X: A] :
                ( member(A,X,aa(list(A),set(A),set2(A),Xsa))
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),A3) )
           => ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_ip(A,A)),A3),Xsa) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),nil(A))) ) ) ) ) ).

% sorted_insort_is_snoc
tff(fact_4430_pairself__image__eq,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Pa: fun(B,fun(B,$o))] : ( aa(set(product_prod(B,B)),set(product_prod(A,A)),image2(product_prod(B,B),product_prod(A,A),pairself(B,A,F2)),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),Pa))) = 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),$o),aTP_Lamp_lw(fun(B,A),fun(fun(B,fun(B,$o)),fun(product_prod(A,A),$o)),F2),Pa)) ) ).

% pairself_image_eq
tff(fact_4431_bit__cut__integer__def,axiom,
    ! [K3: code_integer] : ( code_bit_cut_integer(K3) = aa($o,product_prod(code_integer,$o),aa(code_integer,fun($o,product_prod(code_integer,$o)),product_Pair(code_integer,$o),divide_divide(code_integer,K3,numeral_numeral(code_integer,bit0(one2)))),~ dvd_dvd(code_integer,numeral_numeral(code_integer,bit0(one2)),K3)) ) ).

% bit_cut_integer_def
tff(fact_4432_divmod__integer__def,axiom,
    ! [K3: code_integer,L: code_integer] : ( code_divmod_integer(K3,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),divide_divide(code_integer,K3,L)),modulo_modulo(code_integer,K3,L)) ) ).

% divmod_integer_def
tff(fact_4433_list__collect__set__alt,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,set(A)),L: list(B)] : ( list_collect_set(B,A,F2,L) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(fun(set(A),$o),set(set(A)),collect(set(A)),aa(list(B),fun(set(A),$o),aTP_Lamp_lx(fun(B,set(A)),fun(list(B),fun(set(A),$o)),F2),L))) ) ).

% list_collect_set_alt
tff(fact_4434_list__collect__set__simps_I2_J,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,set(A)),A3: B] : ( list_collect_set(B,A,F2,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),A3),nil(B))) = aa(B,set(A),F2,A3) ) ).

% list_collect_set_simps(2)
tff(fact_4435_list__collect__set__simps_I1_J,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,set(A))] : ( list_collect_set(B,A,F2,nil(B)) = bot_bot(set(A)) ) ).

% list_collect_set_simps(1)
tff(fact_4436_list__collect__set__simps_I3_J,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,set(A)),A3: B,L: list(B)] : ( list_collect_set(B,A,F2,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),F2,A3)),list_collect_set(B,A,F2,L)) ) ).

% list_collect_set_simps(3)
tff(fact_4437_list__collect__set__simps_I4_J,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,set(A)),L: list(B),L2: list(B)] : ( list_collect_set(B,A,F2,aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),L),L2)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),list_collect_set(B,A,F2,L)),list_collect_set(B,A,F2,L2)) ) ).

% list_collect_set_simps(4)
tff(fact_4438_list__collect__set__map__simps_I2_J,axiom,
    ! [A: $tType,B: $tType,E: $tType,F2: fun(B,set(A)),Xa: fun(E,B),A3: E] : ( list_collect_set(B,A,F2,aa(list(E),list(B),aa(fun(E,B),fun(list(E),list(B)),map(E,B),Xa),aa(list(E),list(E),aa(E,fun(list(E),list(E)),cons(E),A3),nil(E)))) = aa(B,set(A),F2,aa(E,B,Xa,A3)) ) ).

% list_collect_set_map_simps(2)
tff(fact_4439_list__collect__set__map__simps_I1_J,axiom,
    ! [E: $tType,B: $tType,A: $tType,F2: fun(B,set(A)),Xa: fun(E,B)] : ( list_collect_set(B,A,F2,aa(list(E),list(B),aa(fun(E,B),fun(list(E),list(B)),map(E,B),Xa),nil(E))) = bot_bot(set(A)) ) ).

% list_collect_set_map_simps(1)
tff(fact_4440_list__collect__set__map__simps_I3_J,axiom,
    ! [A: $tType,B: $tType,E: $tType,F2: fun(B,set(A)),Xa: fun(E,B),A3: E,L: list(E)] : ( list_collect_set(B,A,F2,aa(list(E),list(B),aa(fun(E,B),fun(list(E),list(B)),map(E,B),Xa),aa(list(E),list(E),aa(E,fun(list(E),list(E)),cons(E),A3),L))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(B,set(A),F2,aa(E,B,Xa,A3))),list_collect_set(B,A,F2,aa(list(E),list(B),aa(fun(E,B),fun(list(E),list(B)),map(E,B),Xa),L))) ) ).

% list_collect_set_map_simps(3)
tff(fact_4441_list__collect__set__map__simps_I4_J,axiom,
    ! [A: $tType,B: $tType,E: $tType,F2: fun(B,set(A)),Xa: fun(E,B),L: list(E),L2: list(E)] : ( list_collect_set(B,A,F2,aa(list(E),list(B),aa(fun(E,B),fun(list(E),list(B)),map(E,B),Xa),aa(list(E),list(E),aa(list(E),fun(list(E),list(E)),append(E),L),L2))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),list_collect_set(B,A,F2,aa(list(E),list(B),aa(fun(E,B),fun(list(E),list(B)),map(E,B),Xa),L))),list_collect_set(B,A,F2,aa(list(E),list(B),aa(fun(E,B),fun(list(E),list(B)),map(E,B),Xa),L2))) ) ).

% list_collect_set_map_simps(4)
tff(fact_4442_pairself_Oelims,axiom,
    ! [A: $tType,B: $tType,Xa: fun(B,A),Xaa: product_prod(B,B),Y: product_prod(A,A)] :
      ( ( aa(product_prod(B,B),product_prod(A,A),pairself(B,A,Xa),Xaa) = Y )
     => ~ ! [A4: B,B4: B] :
            ( ( Xaa = aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),A4),B4) )
           => ( Y != aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(B,A,Xa,A4)),aa(B,A,Xa,B4)) ) ) ) ).

% pairself.elims
tff(fact_4443_pairself_Osimps,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A3: B,B3: B] : ( aa(product_prod(B,B),product_prod(A,A),pairself(B,A,F2),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),A3),B3)) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(B,A,F2,A3)),aa(B,A,F2,B3)) ) ).

% pairself.simps
tff(fact_4444_Pow__Compl,axiom,
    ! [A: $tType,Aa2: set(A)] : ( pow(A,aa(set(A),set(A),uminus_uminus(set(A)),Aa2)) = aa(fun(set(A),$o),set(set(A)),collect(set(A)),aTP_Lamp_ly(set(A),fun(set(A),$o),Aa2)) ) ).

% Pow_Compl
tff(fact_4445_list__collect__set__def,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,set(A)),L: list(B)] : ( list_collect_set(B,A,F2,L) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(fun(set(A),$o),set(set(A)),collect(set(A)),aa(list(B),fun(set(A),$o),aTP_Lamp_lz(fun(B,set(A)),fun(list(B),fun(set(A),$o)),F2),L))) ) ).

% list_collect_set_def
tff(fact_4446_list__collect__set__as__map,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,set(A)),L: list(B)] : ( list_collect_set(B,A,F2,L) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(list(set(A)),set(set(A)),set2(set(A)),aa(list(B),list(set(A)),aa(fun(B,set(A)),fun(list(B),list(set(A))),map(B,set(A)),F2),L))) ) ).

% list_collect_set_as_map
tff(fact_4447_pairself_Opelims,axiom,
    ! [A: $tType,B: $tType,Xa: fun(B,A),Xaa: product_prod(B,B),Y: product_prod(A,A)] :
      ( ( aa(product_prod(B,B),product_prod(A,A),pairself(B,A,Xa),Xaa) = 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)),Xa),Xaa))
       => ~ ! [A4: B,B4: B] :
              ( ( Xaa = aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),A4),B4) )
             => ( ( Y = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(B,A,Xa,A4)),aa(B,A,Xa,B4)) )
               => ~ 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)),Xa),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),A4),B4))) ) ) ) ) ).

% pairself.pelims
tff(fact_4448_bit__cut__integer__code,axiom,
    ! [K3: code_integer] :
      ( code_bit_cut_integer(K3) = $ite(K3 = zero_zero(code_integer),aa($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,$o)),aTP_Lamp_ma(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,$o))),K3)),code_divmod_abs(K3,numeral_numeral(code_integer,bit0(one2))))) ) ).

% bit_cut_integer_code
tff(fact_4449_image2__def,axiom,
    ! [A: $tType,B: $tType,E: $tType,Aa2: set(E),F2: fun(E,A),G: fun(E,B)] : ( bNF_Greatest_image2(E,A,B,Aa2,F2,G) = aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(E,B),fun(product_prod(A,B),$o),aa(fun(E,A),fun(fun(E,B),fun(product_prod(A,B),$o)),aTP_Lamp_mb(set(E),fun(fun(E,A),fun(fun(E,B),fun(product_prod(A,B),$o))),Aa2),F2),G)) ) ).

% image2_def
tff(fact_4450_image__atLeastZeroLessThan__integer,axiom,
    ! [U: code_integer] :
      ( aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less_eq(code_integer),zero_zero(code_integer)),U)
     => ( set_or7035219750837199246ssThan(code_integer,zero_zero(code_integer),U) = aa(set(nat),set(code_integer),image2(nat,code_integer,semiring_1_of_nat(code_integer)),aa(nat,set(nat),set_ord_lessThan(nat),code_nat_of_integer(U))) ) ) ).

% image_atLeastZeroLessThan_integer
tff(fact_4451_image2__eqI,axiom,
    ! [A: $tType,E: $tType,B: $tType,B3: A,F2: fun(B,A),Xa: B,C2: E,G: fun(B,E),Aa2: set(B)] :
      ( ( B3 = aa(B,A,F2,Xa) )
     => ( ( C2 = aa(B,E,G,Xa) )
       => ( member(B,Xa,Aa2)
         => member(product_prod(A,E),aa(E,product_prod(A,E),aa(A,fun(E,product_prod(A,E)),product_Pair(A,E),B3),C2),bNF_Greatest_image2(B,A,E,Aa2,F2,G)) ) ) ) ).

% image2_eqI
tff(fact_4452_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_4453_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)),abs_abs(code_integer,J)) ) ).

% divmod_abs_code(5)
tff(fact_4454_nat__of__integer__less__iff,axiom,
    ! [Xa: code_integer,Y: code_integer] :
      ( aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less_eq(code_integer),zero_zero(code_integer)),Xa)
     => ( aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less_eq(code_integer),zero_zero(code_integer)),Y)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),code_nat_of_integer(Xa)),code_nat_of_integer(Y))
        <=> aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),Xa),Y) ) ) ) ).

% nat_of_integer_less_iff
tff(fact_4455_divmod__abs__def,axiom,
    ! [K3: code_integer,L: code_integer] : ( code_divmod_abs(K3,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),divide_divide(code_integer,abs_abs(code_integer,K3),abs_abs(code_integer,L))),modulo_modulo(code_integer,abs_abs(code_integer,K3),abs_abs(code_integer,L))) ) ).

% divmod_abs_def
tff(fact_4456_divmod__integer__code,axiom,
    ! [K3: code_integer,L: code_integer] :
      ( code_divmod_integer(K3,L) = $ite(
          K3 = 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)),K3),code_divmod_abs(K3,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_mc(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),L)),code_divmod_abs(K3,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)),K3),
              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),K3),zero_zero(code_integer)),code_divmod_abs(K3,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_md(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),L)),code_divmod_abs(K3,L)))) ) ) ) ) ).

% divmod_integer_code
tff(fact_4457_listrel__iff__zip,axiom,
    ! [B: $tType,A: $tType,Xsa: list(A),Ys: list(B),R2: set(product_prod(A,B))] :
      ( member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xsa),Ys),listrel(A,B,R2))
    <=> ( ( aa(list(A),nat,size_size(list(A)),Xsa) = aa(list(B),nat,size_size(list(B)),Ys) )
        & ! [X6: product_prod(A,B)] :
            ( member(product_prod(A,B),X6,aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),Xsa),Ys)))
           => aa(product_prod(A,B),$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_bi(set(product_prod(A,B)),fun(A,fun(B,$o))),R2)),X6) ) ) ) ).

% listrel_iff_zip
tff(fact_4458_comp__fun__commute__on_Ofold__set__union__disj,axiom,
    ! [B: $tType,A: $tType,S: set(A),F2: fun(A,fun(B,B)),Aa2: set(A),B2: set(A),Z2: B] :
      ( finite4664212375090638736ute_on(A,B,S,F2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),S)
       => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),S)
         => ( aa(set(A),$o,finite_finite2(A),Aa2)
           => ( aa(set(A),$o,finite_finite2(A),B2)
             => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),B2) = bot_bot(set(A)) )
               => ( finite_fold(A,B,F2,Z2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),B2)) = finite_fold(A,B,F2,finite_fold(A,B,F2,Z2,Aa2),B2) ) ) ) ) ) ) ) ).

% comp_fun_commute_on.fold_set_union_disj
tff(fact_4459_horner__sum__eq__sum__funpow,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_0(A)
     => ! [F2: fun(B,A),A3: A,Xsa: 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),F2),A3),Xsa) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(list(B),fun(nat,A),aa(A,fun(list(B),fun(nat,A)),aTP_Lamp_me(fun(B,A),fun(A,fun(list(B),fun(nat,A))),F2),A3),Xsa)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(B),nat,size_size(list(B)),Xsa))) ) ) ).

% horner_sum_eq_sum_funpow
tff(fact_4460_ball__empty,axiom,
    ! [A: $tType,Pa: fun(A,$o),X4: A] :
      ( member(A,X4,bot_bot(set(A)))
     => aa(A,$o,Pa,X4) ) ).

% ball_empty
tff(fact_4461_apsnd__conv,axiom,
    ! [A: $tType,B: $tType,E: $tType,F2: fun(E,B),Xa: A,Y: E] : ( aa(product_prod(A,E),product_prod(A,B),aa(fun(E,B),fun(product_prod(A,E),product_prod(A,B)),product_apsnd(E,B,A),F2),aa(E,product_prod(A,E),aa(A,fun(E,product_prod(A,E)),product_Pair(A,E),Xa),Y)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa),aa(E,B,F2,Y)) ) ).

% apsnd_conv
tff(fact_4462_funpow__times__power,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [F2: fun(A,nat),Xa: A] : ( compow(fun(A,A),aa(A,nat,F2,Xa),aa(A,fun(A,A),times_times(A),Xa)) = aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),aa(A,nat,F2,Xa))) ) ) ).

% funpow_times_power
tff(fact_4463_relpowp__relpow__eq,axiom,
    ! [A: $tType,N: nat,R: set(product_prod(A,A)),X4: A,Xa4: A] :
      ( aa(A,$o,aa(A,fun(A,$o),compow(fun(A,fun(A,$o)),N,aTP_Lamp_cz(set(product_prod(A,A)),fun(A,fun(A,$o)),R)),X4),Xa4)
    <=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Xa4),compow(set(product_prod(A,A)),N,R)) ) ).

% relpowp_relpow_eq
tff(fact_4464_Field__not__elem,axiom,
    ! [A: $tType,V2: A,R: set(product_prod(A,A))] :
      ( ~ member(A,V2,aa(set(product_prod(A,A)),set(A),field2(A),R))
     => ! [X4: product_prod(A,A)] :
          ( member(product_prod(A,A),X4,R)
         => aa(product_prod(A,A),$o,aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),aTP_Lamp_mf(A,fun(A,fun(A,$o)),V2)),X4) ) ) ).

% Field_not_elem
tff(fact_4465_rotate__def,axiom,
    ! [A: $tType,N: nat] : ( aa(nat,fun(list(A),list(A)),rotate(A),N) = compow(fun(list(A),list(A)),N,rotate1(A)) ) ).

% rotate_def
tff(fact_4466_irrefl__distinct,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( irrefl(A,R2)
    <=> ! [X6: product_prod(A,A)] :
          ( member(product_prod(A,A),X6,R2)
         => aa(product_prod(A,A),$o,aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),aTP_Lamp_je(A,fun(A,$o))),X6) ) ) ).

% irrefl_distinct
tff(fact_4467_relpowp__fun__conv,axiom,
    ! [A: $tType,N: nat,Pa: fun(A,fun(A,$o)),Xa: A,Y: A] :
      ( aa(A,$o,aa(A,fun(A,$o),compow(fun(A,fun(A,$o)),N,Pa),Xa),Y)
    <=> ? [F9: fun(nat,A)] :
          ( ( aa(nat,A,F9,zero_zero(nat)) = Xa )
          & ( aa(nat,A,F9,N) = Y )
          & ! [I2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),N)
             => aa(A,$o,aa(A,fun(A,$o),Pa,aa(nat,A,F9,I2)),aa(nat,A,F9,aa(nat,nat,suc,I2))) ) ) ) ).

% relpowp_fun_conv
tff(fact_4468_sorted__wrt_Oelims_I3_J,axiom,
    ! [A: $tType,Xa: fun(A,fun(A,$o)),Xaa: list(A)] :
      ( ~ sorted_wrt(A,Xa,Xaa)
     => ~ ! [X: A,Ys5: list(A)] :
            ( ( Xaa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Ys5) )
           => ( ! [Xa5: A] :
                  ( member(A,Xa5,aa(list(A),set(A),set2(A),Ys5))
                 => aa(A,$o,aa(A,fun(A,$o),Xa,X),Xa5) )
              & sorted_wrt(A,Xa,Ys5) ) ) ) ).

% sorted_wrt.elims(3)
tff(fact_4469_sorted__wrt_Osimps_I2_J,axiom,
    ! [A: $tType,Pa: fun(A,fun(A,$o)),Xa: A,Ys: list(A)] :
      ( sorted_wrt(A,Pa,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Ys))
    <=> ( ! [X6: A] :
            ( member(A,X6,aa(list(A),set(A),set2(A),Ys))
           => aa(A,$o,aa(A,fun(A,$o),Pa,Xa),X6) )
        & sorted_wrt(A,Pa,Ys) ) ) ).

% sorted_wrt.simps(2)
tff(fact_4470_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_4471_trans__join,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( trans(A,R2)
    <=> ! [X6: product_prod(A,A)] :
          ( member(product_prod(A,A),X6,R2)
         => aa(product_prod(A,A),$o,aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),aTP_Lamp_mh(set(product_prod(A,A)),fun(A,fun(A,$o)),R2)),X6) ) ) ).

% trans_join
tff(fact_4472_takeWhile__append,axiom,
    ! [A: $tType,Pa: fun(A,$o),Xsa: list(A),Ys: list(A)] :
      ( aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),takeWhile(A),Pa),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),Ys)) = $ite(
          ! [X6: A] :
            ( member(A,X6,aa(list(A),set(A),set2(A),Xsa))
           => aa(A,$o,Pa,X6) ),
          aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),takeWhile(A),Pa),Ys)),
          aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),takeWhile(A),Pa),Xsa) ) ) ).

% takeWhile_append
tff(fact_4473_refl__on__def_H,axiom,
    ! [A: $tType,Aa2: set(A),R2: set(product_prod(A,A))] :
      ( refl_on(A,Aa2,R2)
    <=> ( ! [X6: product_prod(A,A)] :
            ( member(product_prod(A,A),X6,R2)
           => aa(product_prod(A,A),$o,aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),aTP_Lamp_mi(set(A),fun(A,fun(A,$o)),Aa2)),X6) )
        & ! [X6: A] :
            ( member(A,X6,Aa2)
           => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X6),X6),R2) ) ) ) ).

% refl_on_def'
tff(fact_4474_sorted__wrt_Oelims_I1_J,axiom,
    ! [A: $tType,Xa: fun(A,fun(A,$o)),Xaa: list(A),Y: $o] :
      ( ( sorted_wrt(A,Xa,Xaa)
      <=> (Y) )
     => ( ( ( Xaa = nil(A) )
         => ~ (Y) )
       => ~ ! [X: A,Ys5: list(A)] :
              ( ( Xaa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Ys5) )
             => ( (Y)
              <=> ~ ( ! [Xa2: A] :
                        ( member(A,Xa2,aa(list(A),set(A),set2(A),Ys5))
                       => aa(A,$o,aa(A,fun(A,$o),Xa,X),Xa2) )
                    & sorted_wrt(A,Xa,Ys5) ) ) ) ) ) ).

% sorted_wrt.elims(1)
tff(fact_4475_sorted__wrt_Oelims_I2_J,axiom,
    ! [A: $tType,Xa: fun(A,fun(A,$o)),Xaa: list(A)] :
      ( sorted_wrt(A,Xa,Xaa)
     => ( ( Xaa != nil(A) )
       => ~ ! [X: A,Ys5: list(A)] :
              ( ( Xaa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Ys5) )
             => ~ ( ! [Xa4: A] :
                      ( member(A,Xa4,aa(list(A),set(A),set2(A),Ys5))
                     => aa(A,$o,aa(A,fun(A,$o),Xa,X),Xa4) )
                  & sorted_wrt(A,Xa,Ys5) ) ) ) ) ).

% sorted_wrt.elims(2)
tff(fact_4476_butlast__power,axiom,
    ! [A: $tType,N: nat,Xsa: list(A)] : ( aa(list(A),list(A),compow(fun(list(A),list(A)),N,butlast(A)),Xsa) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xsa)),N)),Xsa) ) ).

% butlast_power
tff(fact_4477_list__eq__iff__zip__eq,axiom,
    ! [A: $tType,Xsa: list(A),Ys: list(A)] :
      ( ( Xsa = Ys )
    <=> ( ( aa(list(A),nat,size_size(list(A)),Xsa) = aa(list(A),nat,size_size(list(A)),Ys) )
        & ! [X6: product_prod(A,A)] :
            ( member(product_prod(A,A),X6,aa(list(product_prod(A,A)),set(product_prod(A,A)),set2(product_prod(A,A)),aa(list(A),list(product_prod(A,A)),aa(list(A),fun(list(A),list(product_prod(A,A))),zip(A,A),Xsa),Ys)))
           => aa(product_prod(A,A),$o,aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),fequal(A)),X6) ) ) ) ).

% list_eq_iff_zip_eq
tff(fact_4478_concat__eq__concat__iff,axiom,
    ! [A: $tType,Xsa: list(list(A)),Ys: list(list(A))] :
      ( ! [X: product_prod(list(A),list(A))] :
          ( member(product_prod(list(A),list(A)),X,aa(list(product_prod(list(A),list(A))),set(product_prod(list(A),list(A))),set2(product_prod(list(A),list(A))),aa(list(list(A)),list(product_prod(list(A),list(A))),aa(list(list(A)),fun(list(list(A)),list(product_prod(list(A),list(A)))),zip(list(A),list(A)),Xsa),Ys)))
         => aa(product_prod(list(A),list(A)),$o,aa(fun(list(A),fun(list(A),$o)),fun(product_prod(list(A),list(A)),$o),product_case_prod(list(A),list(A),$o),aTP_Lamp_mj(list(A),fun(list(A),$o))),X) )
     => ( ( aa(list(list(A)),nat,size_size(list(list(A))),Xsa) = aa(list(list(A)),nat,size_size(list(list(A))),Ys) )
       => ( ( aa(list(list(A)),list(A),concat(A),Xsa) = aa(list(list(A)),list(A),concat(A),Ys) )
        <=> ( Xsa = Ys ) ) ) ) ).

% concat_eq_concat_iff
tff(fact_4479_concat__injective,axiom,
    ! [A: $tType,Xsa: list(list(A)),Ys: list(list(A))] :
      ( ( aa(list(list(A)),list(A),concat(A),Xsa) = aa(list(list(A)),list(A),concat(A),Ys) )
     => ( ( aa(list(list(A)),nat,size_size(list(list(A))),Xsa) = aa(list(list(A)),nat,size_size(list(list(A))),Ys) )
       => ( ! [X: product_prod(list(A),list(A))] :
              ( member(product_prod(list(A),list(A)),X,aa(list(product_prod(list(A),list(A))),set(product_prod(list(A),list(A))),set2(product_prod(list(A),list(A))),aa(list(list(A)),list(product_prod(list(A),list(A))),aa(list(list(A)),fun(list(list(A)),list(product_prod(list(A),list(A)))),zip(list(A),list(A)),Xsa),Ys)))
             => aa(product_prod(list(A),list(A)),$o,aa(fun(list(A),fun(list(A),$o)),fun(product_prod(list(A),list(A)),$o),product_case_prod(list(A),list(A),$o),aTP_Lamp_mj(list(A),fun(list(A),$o))),X) )
         => ( Xsa = Ys ) ) ) ) ).

% concat_injective
tff(fact_4480_comp__fun__commute__on_Ofold__fun__left__comm,axiom,
    ! [B: $tType,A: $tType,S: set(A),F2: fun(A,fun(B,B)),Xa: A,Aa2: set(A),Z2: B] :
      ( finite4664212375090638736ute_on(A,B,S,F2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),Aa2)),S)
       => ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( aa(B,B,aa(A,fun(B,B),F2,Xa),finite_fold(A,B,F2,Z2,Aa2)) = finite_fold(A,B,F2,aa(B,B,aa(A,fun(B,B),F2,Xa),Z2),Aa2) ) ) ) ) ).

% comp_fun_commute_on.fold_fun_left_comm
tff(fact_4481_comp__fun__commute__on_Ofold__insert2,axiom,
    ! [B: $tType,A: $tType,S: set(A),F2: fun(A,fun(B,B)),Xa: A,Aa2: set(A),Z2: B] :
      ( finite4664212375090638736ute_on(A,B,S,F2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),Aa2)),S)
       => ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( ~ member(A,Xa,Aa2)
           => ( finite_fold(A,B,F2,Z2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),Aa2)) = finite_fold(A,B,F2,aa(B,B,aa(A,fun(B,B),F2,Xa),Z2),Aa2) ) ) ) ) ) ).

% comp_fun_commute_on.fold_insert2
tff(fact_4482_comp__fun__commute__on_Ofold__insert,axiom,
    ! [B: $tType,A: $tType,S: set(A),F2: fun(A,fun(B,B)),Xa: A,Aa2: set(A),Z2: B] :
      ( finite4664212375090638736ute_on(A,B,S,F2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),Aa2)),S)
       => ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( ~ member(A,Xa,Aa2)
           => ( finite_fold(A,B,F2,Z2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),Aa2)) = aa(B,B,aa(A,fun(B,B),F2,Xa),finite_fold(A,B,F2,Z2,Aa2)) ) ) ) ) ) ).

% comp_fun_commute_on.fold_insert
tff(fact_4483_comp__fun__commute__on_Ofold__rec,axiom,
    ! [B: $tType,A: $tType,S: set(A),F2: fun(A,fun(B,B)),Aa2: set(A),Xa: A,Z2: B] :
      ( finite4664212375090638736ute_on(A,B,S,F2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),S)
       => ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( member(A,Xa,Aa2)
           => ( finite_fold(A,B,F2,Z2,Aa2) = aa(B,B,aa(A,fun(B,B),F2,Xa),finite_fold(A,B,F2,Z2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A)))))) ) ) ) ) ) ).

% comp_fun_commute_on.fold_rec
tff(fact_4484_comp__fun__commute__on_Ofold__insert__remove,axiom,
    ! [B: $tType,A: $tType,S: set(A),F2: fun(A,fun(B,B)),Xa: A,Aa2: set(A),Z2: B] :
      ( finite4664212375090638736ute_on(A,B,S,F2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),Aa2)),S)
       => ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( finite_fold(A,B,F2,Z2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),Aa2)) = aa(B,B,aa(A,fun(B,B),F2,Xa),finite_fold(A,B,F2,Z2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A)))))) ) ) ) ) ).

% comp_fun_commute_on.fold_insert_remove
tff(fact_4485_Under__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),Aa2: set(A)] : ( order_Under(A,R2,Aa2) = aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_mk(set(product_prod(A,A)),fun(set(A),fun(A,$o)),R2),Aa2)) ) ).

% Under_def
tff(fact_4486_UnderS__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),Aa2: set(A)] : ( order_UnderS(A,R2,Aa2) = aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_ml(set(product_prod(A,A)),fun(set(A),fun(A,$o)),R2),Aa2)) ) ).

% UnderS_def
tff(fact_4487_Above__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),Aa2: set(A)] : ( order_Above(A,R2,Aa2) = aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_mm(set(product_prod(A,A)),fun(set(A),fun(A,$o)),R2),Aa2)) ) ).

% Above_def
tff(fact_4488_Gcd__eq__Max,axiom,
    ! [M5: set(nat)] :
      ( aa(set(nat),$o,finite_finite2(nat),M5)
     => ( ( M5 != bot_bot(set(nat)) )
       => ( ~ member(nat,zero_zero(nat),M5)
         => ( gcd_Gcd(nat,M5) = 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_mn(nat,set(nat))),M5))) ) ) ) ) ).

% Gcd_eq_Max
tff(fact_4489_Inf__eq__bot__iff,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [Aa2: set(A)] :
          ( ( aa(set(A),A,complete_Inf_Inf(A),Aa2) = bot_bot(A) )
        <=> ! [X6: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),bot_bot(A)),X6)
             => ? [Xa2: A] :
                  ( member(A,Xa2,Aa2)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa2),X6) ) ) ) ) ).

% Inf_eq_bot_iff
tff(fact_4490_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_4491_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_4492_cInf__singleton,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Xa: A] : ( aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A)))) = Xa ) ) ).

% cInf_singleton
tff(fact_4493_Inf__insert,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: A,Aa2: set(A)] : ( aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),Aa2)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),aa(set(A),A,complete_Inf_Inf(A),Aa2)) ) ) ).

% Inf_insert
tff(fact_4494_Inf__atLeastLessThan,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
         => ( aa(set(A),A,complete_Inf_Inf(A),set_or7035219750837199246ssThan(A,Xa,Y)) = Xa ) ) ) ).

% Inf_atLeastLessThan
tff(fact_4495_cInf__atLeastLessThan,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Y: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xa)
         => ( aa(set(A),A,complete_Inf_Inf(A),set_or7035219750837199246ssThan(A,Y,Xa)) = Y ) ) ) ).

% cInf_atLeastLessThan
tff(fact_4496_Inf__atMost,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Xa: A] : ( aa(set(A),A,complete_Inf_Inf(A),aa(A,set(A),set_ord_atMost(A),Xa)) = bot_bot(A) ) ) ).

% Inf_atMost
tff(fact_4497_Inf__greaterThanAtMost,axiom,
    ! [A: $tType] :
      ( ( comple6319245703460814977attice(A)
        & dense_linorder(A) )
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
         => ( aa(set(A),A,complete_Inf_Inf(A),set_or3652927894154168847AtMost(A,Xa,Y)) = Xa ) ) ) ).

% Inf_greaterThanAtMost
tff(fact_4498_cInf__greaterThanAtMost,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & dense_linorder(A) )
     => ! [Y: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xa)
         => ( aa(set(A),A,complete_Inf_Inf(A),set_or3652927894154168847AtMost(A,Y,Xa)) = Y ) ) ) ).

% cInf_greaterThanAtMost
tff(fact_4499_Inf__greaterThanLessThan,axiom,
    ! [A: $tType] :
      ( ( comple6319245703460814977attice(A)
        & dense_linorder(A) )
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
         => ( aa(set(A),A,complete_Inf_Inf(A),set_or5935395276787703475ssThan(A,Xa,Y)) = Xa ) ) ) ).

% Inf_greaterThanLessThan
tff(fact_4500_cInf__greaterThanLessThan,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & dense_linorder(A) )
     => ! [Y: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xa)
         => ( aa(set(A),A,complete_Inf_Inf(A),set_or5935395276787703475ssThan(A,Y,Xa)) = Y ) ) ) ).

% cInf_greaterThanLessThan
tff(fact_4501_INF__const,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [Aa2: set(A),F2: B] :
          ( ( Aa2 != bot_bot(set(A)) )
         => ( aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,aTP_Lamp_ai(B,fun(A,B),F2)),Aa2)) = F2 ) ) ) ).

% INF_const
tff(fact_4502_cINF__const,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [Aa2: set(A),C2: B] :
          ( ( Aa2 != bot_bot(set(A)) )
         => ( aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,aTP_Lamp_ah(B,fun(A,B),C2)),Aa2)) = C2 ) ) ) ).

% cINF_const
tff(fact_4503_INF__eq__bot__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [F2: fun(B,A),Aa2: set(B)] :
          ( ( aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F2),Aa2)) = bot_bot(A) )
        <=> ! [X6: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),bot_bot(A)),X6)
             => ? [Xa2: B] :
                  ( member(B,Xa2,Aa2)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,F2,Xa2)),X6) ) ) ) ) ).

% INF_eq_bot_iff
tff(fact_4504_INT__constant,axiom,
    ! [B: $tType,A: $tType,C2: set(A),Aa2: 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_an(set(A),fun(B,set(A)),C2)),Aa2)) = $ite(Aa2 = bot_bot(set(B)),top_top(set(A)),C2) ) ).

% INT_constant
tff(fact_4505_INT__insert,axiom,
    ! [A: $tType,B: $tType,B2: fun(B,set(A)),A3: B,Aa2: set(B)] : ( aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B2),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert3(B),A3),Aa2))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(B,set(A),B2,A3)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B2),Aa2))) ) ).

% INT_insert
tff(fact_4506_Inf__atMostLessThan,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),top_top(A)),Xa)
         => ( aa(set(A),A,complete_Inf_Inf(A),aa(A,set(A),set_ord_lessThan(A),Xa)) = bot_bot(A) ) ) ) ).

% Inf_atMostLessThan
tff(fact_4507_INT__simps_I2_J,axiom,
    ! [A: $tType,B: $tType,Aa2: set(A),B2: 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_mo(set(A),fun(fun(B,set(A)),fun(B,set(A))),Aa2),B2)),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)),Aa2),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B2),C3)))) ) ).

% INT_simps(2)
tff(fact_4508_INT__simps_I1_J,axiom,
    ! [B: $tType,A: $tType,Aa2: fun(B,set(A)),B2: 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_mp(fun(B,set(A)),fun(set(A),fun(B,set(A))),Aa2),B2)),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),Aa2),C3))),B2)) ) ).

% INT_simps(1)
tff(fact_4509_INT__simps_I3_J,axiom,
    ! [B: $tType,A: $tType,Aa2: fun(B,set(A)),B2: 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_cx(fun(B,set(A)),fun(set(A),fun(B,set(A))),Aa2),B2)),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),Aa2),C3))),B2)) ) ).

% INT_simps(3)
tff(fact_4510_INT__simps_I4_J,axiom,
    ! [A: $tType,B: $tType,Aa2: set(A),B2: 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_mq(set(A),fun(fun(B,set(A)),fun(B,set(A))),Aa2),B2)),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)),Aa2),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B2),C3)))) ) ).

% INT_simps(4)
tff(fact_4511_Inf__less__iff,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [S: set(A),A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(A),A,complete_Inf_Inf(A),S)),A3)
        <=> ? [X6: A] :
              ( member(A,X6,S)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X6),A3) ) ) ) ).

% Inf_less_iff
tff(fact_4512_refl__on__INTER,axiom,
    ! [B: $tType,A: $tType,S: set(A),Aa2: fun(A,set(B)),R2: fun(A,set(product_prod(B,B)))] :
      ( ! [X: A] :
          ( member(A,X,S)
         => refl_on(B,aa(A,set(B),Aa2,X),aa(A,set(product_prod(B,B)),R2,X)) )
     => refl_on(B,aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image2(A,set(B),Aa2),S)),aa(set(set(product_prod(B,B))),set(product_prod(B,B)),complete_Inf_Inf(set(product_prod(B,B))),aa(set(A),set(set(product_prod(B,B))),image2(A,set(product_prod(B,B)),R2),S))) ) ).

% refl_on_INTER
tff(fact_4513_Inf__le__iff,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [Aa2: set(A),Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),Aa2)),Xa)
        <=> ! [Y4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y4)
             => ? [X6: A] :
                  ( member(A,X6,Aa2)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X6),Y4) ) ) ) ) ).

% Inf_le_iff
tff(fact_4514_cInf__greatest,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X3: set(A),Z2: A] :
          ( ( X3 != bot_bot(set(A)) )
         => ( ! [X: A] :
                ( member(A,X,X3)
               => aa(A,$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),aa(set(A),A,complete_Inf_Inf(A),X3)) ) ) ) ).

% cInf_greatest
tff(fact_4515_cInf__eq__non__empty,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X3: set(A),A3: A] :
          ( ( X3 != bot_bot(set(A)) )
         => ( ! [X: A] :
                ( member(A,X,X3)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),X) )
           => ( ! [Y2: A] :
                  ( ! [X4: A] :
                      ( member(A,X4,X3)
                     => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y2),X4) )
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y2),A3) )
             => ( aa(set(A),A,complete_Inf_Inf(A),X3) = A3 ) ) ) ) ) ).

% cInf_eq_non_empty
tff(fact_4516_Inf__less__eq,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Aa2: set(A),U: A] :
          ( ! [V3: A] :
              ( member(A,V3,Aa2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),V3),U) )
         => ( ( Aa2 != bot_bot(set(A)) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),Aa2)),U) ) ) ) ).

% Inf_less_eq
tff(fact_4517_cInf__lessD,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X3: set(A),Z2: A] :
          ( ( X3 != bot_bot(set(A)) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(A),A,complete_Inf_Inf(A),X3)),Z2)
           => ? [X: A] :
                ( member(A,X,X3)
                & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Z2) ) ) ) ) ).

% cInf_lessD
tff(fact_4518_finite__imp__less__Inf,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X3: set(A),Xa: A,A3: A] :
          ( aa(set(A),$o,finite_finite2(A),X3)
         => ( member(A,Xa,X3)
           => ( ! [X: A] :
                  ( member(A,X,X3)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),X) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(set(A),A,complete_Inf_Inf(A),X3)) ) ) ) ) ).

% finite_imp_less_Inf
tff(fact_4519_INF__eq__const,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [Ia2: set(A),F2: fun(A,B),Xa: B] :
          ( ( Ia2 != bot_bot(set(A)) )
         => ( ! [I3: A] :
                ( member(A,I3,Ia2)
               => ( aa(A,B,F2,I3) = Xa ) )
           => ( aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),Ia2)) = Xa ) ) ) ) ).

% INF_eq_const
tff(fact_4520_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_4521_Inf__finite__insert,axiom,
    ! [A: $tType] :
      ( finite_lattice(A)
     => ! [A3: A,Aa2: set(A)] : ( aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),Aa2)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),aa(set(A),A,complete_Inf_Inf(A),Aa2)) ) ) ).

% Inf_finite_insert
tff(fact_4522_less__INF__D,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Y: A,F2: fun(B,A),Aa2: set(B),Ia: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F2),Aa2)))
         => ( member(B,Ia,Aa2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),aa(B,A,F2,Ia)) ) ) ) ).

% less_INF_D
tff(fact_4523_INF__less__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [F2: fun(B,A),Aa2: set(B),A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F2),Aa2))),A3)
        <=> ? [X6: B] :
              ( member(B,X6,Aa2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,F2,X6)),A3) ) ) ) ).

% INF_less_iff
tff(fact_4524_INT__extend__simps_I5_J,axiom,
    ! [A: $tType,B: $tType,A3: A,B2: fun(B,set(A)),C3: set(B)] : ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B2),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_au(A,fun(fun(B,set(A)),fun(B,set(A))),A3),B2)),C3)) ) ).

% INT_extend_simps(5)
tff(fact_4525_INT__insert__distrib,axiom,
    ! [B: $tType,A: $tType,U: A,Aa2: set(A),A3: B,B2: fun(A,set(B))] :
      ( member(A,U,Aa2)
     => ( 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_cw(B,fun(fun(A,set(B)),fun(A,set(B))),A3),B2)),Aa2)) = aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert3(B),A3),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image2(A,set(B),B2),Aa2))) ) ) ).

% INT_insert_distrib
tff(fact_4526_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_4527_INF__le__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [F2: fun(B,A),Aa2: set(B),Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F2),Aa2))),Xa)
        <=> ! [Y4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y4)
             => ? [X6: B] :
                  ( member(B,X6,Aa2)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,F2,X6)),Y4) ) ) ) ) ).

% INF_le_iff
tff(fact_4528_cINF__greatest,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [Aa2: set(A),M2: B,F2: fun(A,B)] :
          ( ( Aa2 != bot_bot(set(A)) )
         => ( ! [X: A] :
                ( member(A,X,Aa2)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),M2),aa(A,B,F2,X)) )
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),M2),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),Aa2))) ) ) ) ).

% cINF_greatest
tff(fact_4529_INF__eq__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [Ia2: set(A),F2: fun(A,B),C2: B] :
          ( ( Ia2 != bot_bot(set(A)) )
         => ( ! [I3: A] :
                ( member(A,I3,Ia2)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,I3)),C2) )
           => ( ( aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),Ia2)) = C2 )
            <=> ! [X6: A] :
                  ( member(A,X6,Ia2)
                 => ( aa(A,B,F2,X6) = C2 ) ) ) ) ) ) ).

% INF_eq_iff
tff(fact_4530_finite__less__Inf__iff,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X3: set(A),A3: A] :
          ( aa(set(A),$o,finite_finite2(A),X3)
         => ( ( X3 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(set(A),A,complete_Inf_Inf(A),X3))
            <=> ! [X6: A] :
                  ( member(A,X6,X3)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),X6) ) ) ) ) ) ).

% finite_less_Inf_iff
tff(fact_4531_Inf__le__Sup,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Aa2: set(A)] :
          ( ( Aa2 != bot_bot(set(A)) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),Aa2)),aa(set(A),A,complete_Sup_Sup(A),Aa2)) ) ) ).

% Inf_le_Sup
tff(fact_4532_finite__Inf__in,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Aa2: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( ( Aa2 != bot_bot(set(A)) )
           => ( ! [X: A,Y2: A] :
                  ( member(A,X,Aa2)
                 => ( member(A,Y2,Aa2)
                   => member(A,aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y2),Aa2) ) )
             => member(A,aa(set(A),A,complete_Inf_Inf(A),Aa2),Aa2) ) ) ) ) ).

% finite_Inf_in
tff(fact_4533_cInf__abs__ge,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & linordered_idom(A) )
     => ! [S: set(A),A3: A] :
          ( ( S != bot_bot(set(A)) )
         => ( ! [X: A] :
                ( member(A,X,S)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),abs_abs(A,X)),A3) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),abs_abs(A,aa(set(A),A,complete_Inf_Inf(A),S))),A3) ) ) ) ).

% cInf_abs_ge
tff(fact_4534_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_4535_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_4536_cInf__eq__Min,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X3: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),X3)
         => ( ( X3 != bot_bot(set(A)) )
           => ( aa(set(A),A,complete_Inf_Inf(A),X3) = lattic643756798350308766er_Min(A,X3) ) ) ) ) ).

% cInf_eq_Min
tff(fact_4537_Min__Inf,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [Aa2: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( ( Aa2 != bot_bot(set(A)) )
           => ( lattic643756798350308766er_Min(A,Aa2) = aa(set(A),A,complete_Inf_Inf(A),Aa2) ) ) ) ) ).

% Min_Inf
tff(fact_4538_INF__empty,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(B,A)] : ( aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F2),bot_bot(set(B)))) = top_top(A) ) ) ).

% INF_empty
tff(fact_4539_INF__constant,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [C2: A,Aa2: set(B)] :
          ( aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,aTP_Lamp_de(A,fun(B,A),C2)),Aa2)) = $ite(Aa2 = bot_bot(set(B)),top_top(A),C2) ) ) ).

% INF_constant
tff(fact_4540_INF__inf__const2,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [Ia2: set(A),F2: fun(A,B),Xa: B] :
          ( ( Ia2 != 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_mr(fun(A,B),fun(B,fun(A,B)),F2),Xa)),Ia2)) = aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),Ia2))),Xa) ) ) ) ).

% INF_inf_const2
tff(fact_4541_INF__inf__const1,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [Ia2: set(A),Xa: B,F2: fun(A,B)] :
          ( ( Ia2 != 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_ms(B,fun(fun(A,B),fun(A,B)),Xa),F2)),Ia2)) = aa(B,B,aa(B,fun(B,B),inf_inf(B),Xa),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),Ia2))) ) ) ) ).

% INF_inf_const1
tff(fact_4542_INF__insert,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(B,A),A3: B,Aa2: set(B)] : ( aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F2),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert3(B),A3),Aa2))) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(B,A,F2,A3)),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F2),Aa2))) ) ) ).

% INF_insert
tff(fact_4543_INT__empty,axiom,
    ! [B: $tType,A: $tType,B2: 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),B2),bot_bot(set(B)))) = top_top(set(A)) ) ).

% INT_empty
tff(fact_4544_INT__extend__simps_I1_J,axiom,
    ! [A: $tType,B: $tType,Aa2: fun(B,set(A)),C3: set(B),B2: 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),Aa2),C3))),B2) = $ite(C3 = bot_bot(set(B)),B2,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_mp(fun(B,set(A)),fun(set(A),fun(B,set(A))),Aa2),B2)),C3))) ) ).

% INT_extend_simps(1)
tff(fact_4545_INT__extend__simps_I2_J,axiom,
    ! [A: $tType,B: $tType,Aa2: set(A),B2: fun(B,set(A)),C3: set(B)] :
      ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B2),C3))) = $ite(C3 = bot_bot(set(B)),Aa2,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_mo(set(A),fun(fun(B,set(A)),fun(B,set(A))),Aa2),B2)),C3))) ) ).

% INT_extend_simps(2)
tff(fact_4546_UN__extend__simps_I7_J,axiom,
    ! [A: $tType,B: $tType,Aa2: set(A),B2: fun(B,set(A)),C3: set(B)] : ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B2),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_mq(set(A),fun(fun(B,set(A)),fun(B,set(A))),Aa2),B2)),C3)) ) ).

% UN_extend_simps(7)
tff(fact_4547_Union__maximal__sets,axiom,
    ! [A: $tType,F10: set(set(A))] :
      ( aa(set(set(A)),$o,finite_finite2(set(A)),F10)
     => ( aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(fun(set(A),$o),set(set(A)),collect(set(A)),aTP_Lamp_mt(set(set(A)),fun(set(A),$o),F10))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),F10) ) ) ).

% Union_maximal_sets
tff(fact_4548_INF__le__SUP,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [Aa2: set(A),F2: fun(A,B)] :
          ( ( Aa2 != bot_bot(set(A)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),Aa2))),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),Aa2))) ) ) ).

% INF_le_SUP
tff(fact_4549_cInf__asclose,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & linordered_idom(A) )
     => ! [S: set(A),L: A,E3: A] :
          ( ( S != bot_bot(set(A)) )
         => ( ! [X: A] :
                ( member(A,X,S)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),abs_abs(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X),L))),E3) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(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))),E3) ) ) ) ).

% cInf_asclose
tff(fact_4550_Inf__insert__finite,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [S: set(A),Xa: A] :
          ( aa(set(A),$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)),insert3(A),Xa),S)) = $ite(S = bot_bot(set(A)),Xa,aa(A,A,aa(A,fun(A,A),ord_min(A),Xa),aa(set(A),A,complete_Inf_Inf(A),S))) ) ) ) ).

% Inf_insert_finite
tff(fact_4551_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_4552_INT__extend__simps_I3_J,axiom,
    ! [A: $tType,B: $tType,Aa2: fun(B,set(A)),C3: set(B),B2: 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),Aa2),C3))),B2) = $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))),B2),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_cx(fun(B,set(A)),fun(set(A),fun(B,set(A))),Aa2),B2)),C3))) ) ).

% INT_extend_simps(3)
tff(fact_4553_INT__extend__simps_I4_J,axiom,
    ! [A: $tType,B: $tType,Aa2: set(A),B2: fun(B,set(A)),C3: set(B)] :
      ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B2),C3))) = $ite(C3 = bot_bot(set(B)),Aa2,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_mq(set(A),fun(fun(B,set(A)),fun(B,set(A))),Aa2),B2)),C3))) ) ).

% INT_extend_simps(4)
tff(fact_4554_sort__mergesort,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( aa(fun(A,A),fun(list(A),list(A)),linorder_sort_key(A,A),aTP_Lamp_ip(A,A)) = mergesort(A) ) ) ).

% sort_mergesort
tff(fact_4555_times__int_Oabs__eq,axiom,
    ! [Xa: product_prod(nat,nat),Xb: product_prod(nat,nat)] : ( aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(nat,nat),int,abs_Integ,Xa)),aa(product_prod(nat,nat),int,abs_Integ,Xb)) = aa(product_prod(nat,nat),int,abs_Integ,aa(product_prod(nat,nat),product_prod(nat,nat),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat)),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_mv(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))))),Xa),Xb)) ) ).

% times_int.abs_eq
tff(fact_4556_min__list_Oelims,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Xa: list(A),Y: A] :
          ( ( min_list(A,Xa) = Y )
         => ( ! [X: A,Xs: list(A)] :
                ( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) )
               => ( Y != aa(list(A),A,aa(fun(A,fun(list(A),A)),fun(list(A),A),aa(A,fun(fun(A,fun(list(A),A)),fun(list(A),A)),case_list(A,A),X),aa(list(A),fun(A,fun(list(A),A)),aTP_Lamp_bp(A,fun(list(A),fun(A,fun(list(A),A))),X),Xs)),Xs) ) )
           => ~ ( ( Xa = nil(A) )
               => ( Y != undefined(A) ) ) ) ) ) ).

% min_list.elims
tff(fact_4557_filter__rev__alt,axiom,
    ! [A: $tType,Pa: fun(A,$o),L: list(A)] : ( aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter_rev(A),Pa),L) = aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),Pa),aa(list(A),list(A),rev(A),L)) ) ).

% filter_rev_alt
tff(fact_4558_subset__mset_OcINF__const,axiom,
    ! [A: $tType,B: $tType,Aa2: set(A),C2: multiset(B)] :
      ( ( Aa2 != 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_ap(multiset(B),fun(A,multiset(B)),C2)),Aa2)) = C2 ) ) ).

% subset_mset.cINF_const
tff(fact_4559_Inf__nat__def1,axiom,
    ! [K6: set(nat)] :
      ( ( K6 != bot_bot(set(nat)) )
     => member(nat,aa(set(nat),nat,complete_Inf_Inf(nat),K6),K6) ) ).

% Inf_nat_def1
tff(fact_4560_INF__Int__eq2,axiom,
    ! [B: $tType,A: $tType,S: set(set(product_prod(A,B))),X4: A,Xa4: B] :
      ( aa(B,$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_bi(set(product_prod(A,B)),fun(A,fun(B,$o)))),S)),X4),Xa4)
    <=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Xa4),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_4561_eq__Abs__Integ,axiom,
    ! [Z2: int] :
      ~ ! [X: 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),X),Y2)) ) ).

% eq_Abs_Integ
tff(fact_4562_INF__INT__eq2,axiom,
    ! [A: $tType,B: $tType,E: $tType,R2: fun(E,set(product_prod(A,B))),S: set(E),X4: A,Xa4: B] :
      ( aa(B,$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(E),set(fun(A,fun(B,$o))),image2(E,fun(A,fun(B,$o)),aTP_Lamp_db(fun(E,set(product_prod(A,B))),fun(E,fun(A,fun(B,$o))),R2)),S)),X4),Xa4)
    <=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Xa4),aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Inf_Inf(set(product_prod(A,B))),aa(set(E),set(set(product_prod(A,B))),image2(E,set(product_prod(A,B)),R2),S))) ) ).

% INF_INT_eq2
tff(fact_4563_trans__INTER,axiom,
    ! [B: $tType,A: $tType,S: set(A),R2: fun(A,set(product_prod(B,B)))] :
      ( ! [X: A] :
          ( member(A,X,S)
         => trans(B,aa(A,set(product_prod(B,B)),R2,X)) )
     => trans(B,aa(set(set(product_prod(B,B))),set(product_prod(B,B)),complete_Inf_Inf(set(product_prod(B,B))),aa(set(A),set(set(product_prod(B,B))),image2(A,set(product_prod(B,B)),R2),S))) ) ).

% trans_INTER
tff(fact_4564_INF__filter__not__bot,axiom,
    ! [A: $tType,B: $tType,B2: set(A),Fa: fun(A,filter(B))] :
      ( ! [X7: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X7),B2)
         => ( aa(set(A),$o,finite_finite2(A),X7)
           => ( aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image2(A,filter(B),Fa),X7)) != bot_bot(filter(B)) ) ) )
     => ( aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image2(A,filter(B),Fa),B2)) != bot_bot(filter(B)) ) ) ).

% INF_filter_not_bot
tff(fact_4565_Inf__INT__eq,axiom,
    ! [A: $tType,S: set(fun(A,$o)),X4: A] :
      ( aa(A,$o,aa(set(fun(A,$o)),fun(A,$o),complete_Inf_Inf(fun(A,$o)),S),X4)
    <=> member(A,X4,aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(fun(A,$o)),set(set(A)),image2(fun(A,$o),set(A),collect(A)),S))) ) ).

% Inf_INT_eq
tff(fact_4566_Inf__INT__eq2,axiom,
    ! [B: $tType,A: $tType,S: set(fun(A,fun(B,$o))),X4: A,Xa4: 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,$o))),S),X4),Xa4)
    <=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Xa4),aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Inf_Inf(set(product_prod(A,B))),aa(set(fun(product_prod(A,B),$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_4567_subset__mset_OcINF__greatest,axiom,
    ! [B: $tType,A: $tType,Aa2: set(A),M2: multiset(B),F2: fun(A,multiset(B))] :
      ( ( Aa2 != bot_bot(set(A)) )
     => ( ! [X: A] :
            ( member(A,X,Aa2)
           => aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),M2),aa(A,multiset(B),F2,X)) )
       => aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),M2),aa(set(multiset(B)),multiset(B),complete_Inf_Inf(multiset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),F2),Aa2))) ) ) ).

% subset_mset.cINF_greatest
tff(fact_4568_hd__def,axiom,
    ! [A: $tType,List: list(A)] : ( aa(list(A),A,hd(A),List) = aa(list(A),A,aa(fun(A,fun(list(A),A)),fun(list(A),A),aa(A,fun(fun(A,fun(list(A),A)),fun(list(A),A)),case_list(A,A),undefined(A)),aTP_Lamp_mw(A,fun(list(A),A))),List) ) ).

% hd_def
tff(fact_4569_INF__Int__eq,axiom,
    ! [A: $tType,S: set(set(A)),X4: A] :
      ( aa(A,$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,$o),aTP_Lamp_a(set(A),fun(A,$o))),S)),X4)
    <=> member(A,X4,aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),S)) ) ).

% INF_Int_eq
tff(fact_4570_INF__INT__eq,axiom,
    ! [A: $tType,B: $tType,R2: fun(B,set(A)),S: set(B),X4: A] :
      ( aa(A,$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,$o),aTP_Lamp_dd(fun(B,set(A)),fun(B,fun(A,$o)),R2)),S)),X4)
    <=> member(A,X4,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_4571_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_4572_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_4573_zipf_Oelims,axiom,
    ! [E: $tType,B: $tType,A: $tType,Xa: fun(B,fun(E,A)),Xaa: list(B),Xb: list(E),Y: list(A)] :
      ( ( zipf(B,E,A,Xa,Xaa,Xb) = Y )
     => ( ( ( Xaa = nil(B) )
         => ( ( Xb = nil(E) )
           => ( Y != nil(A) ) ) )
       => ( ! [A4: B,As: list(B)] :
              ( ( Xaa = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),A4),As) )
             => ! [B4: E,Bs2: list(E)] :
                  ( ( Xb = aa(list(E),list(E),aa(E,fun(list(E),list(E)),cons(E),B4),Bs2) )
                 => ( Y != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(E,A,aa(B,fun(E,A),Xa,A4),B4)),zipf(B,E,A,Xa,As,Bs2)) ) ) )
         => ( ( ? [V3: B,Va: list(B)] : ( Xaa = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),V3),Va) )
             => ( ( Xb = nil(E) )
               => ( Y != undefined(list(A)) ) ) )
           => ~ ( ( Xaa = nil(B) )
               => ( ? [V3: E,Va: list(E)] : ( Xb = aa(list(E),list(E),aa(E,fun(list(E),list(E)),cons(E),V3),Va) )
                 => ( Y != undefined(list(A)) ) ) ) ) ) ) ) ).

% zipf.elims
tff(fact_4574_uminus__int_Oabs__eq,axiom,
    ! [Xa: product_prod(nat,nat)] : ( aa(int,int,uminus_uminus(int),aa(product_prod(nat,nat),int,abs_Integ,Xa)) = 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_mx(nat,fun(nat,product_prod(nat,nat)))),Xa)) ) ).

% uminus_int.abs_eq
tff(fact_4575_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_4576_arg__min__list_Oelims,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Xa: fun(A,B),Xaa: list(A),Y: A] :
          ( ( arg_min_list(A,B,Xa,Xaa) = Y )
         => ( ! [X: A] :
                ( ( Xaa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)) )
               => ( Y != X ) )
           => ( ! [X: A,Y2: A,Zs2: list(A)] :
                  ( ( Xaa = 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),Y2),Zs2)) )
                 => ( Y != $let(
                        m: A,
                        m:= arg_min_list(A,B,Xa,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,Xa,X)),aa(A,B,Xa,m)),X,m) ) ) )
             => ~ ( ( Xaa = nil(A) )
                 => ( Y != undefined(A) ) ) ) ) ) ) ).

% arg_min_list.elims
tff(fact_4577_of__int_Oabs__eq,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Xa: product_prod(nat,nat)] : ( aa(int,A,ring_1_of_int(A),aa(product_prod(nat,nat),int,abs_Integ,Xa)) = 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_my(nat,fun(nat,A))),Xa) ) ) ).

% of_int.abs_eq
tff(fact_4578_less__int_Oabs__eq,axiom,
    ! [Xa: product_prod(nat,nat),Xb: product_prod(nat,nat)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(product_prod(nat,nat),int,abs_Integ,Xa)),aa(product_prod(nat,nat),int,abs_Integ,Xb))
    <=> aa(product_prod(nat,nat),$o,aa(product_prod(nat,nat),fun(product_prod(nat,nat),$o),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),$o))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)),product_case_prod(nat,nat,fun(product_prod(nat,nat),$o)),aTP_Lamp_na(nat,fun(nat,fun(product_prod(nat,nat),$o)))),Xa),Xb) ) ).

% less_int.abs_eq
tff(fact_4579_plus__int_Oabs__eq,axiom,
    ! [Xa: product_prod(nat,nat),Xb: product_prod(nat,nat)] : ( aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(product_prod(nat,nat),int,abs_Integ,Xa)),aa(product_prod(nat,nat),int,abs_Integ,Xb)) = aa(product_prod(nat,nat),int,abs_Integ,aa(product_prod(nat,nat),product_prod(nat,nat),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat)),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_nc(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))))),Xa),Xb)) ) ).

% plus_int.abs_eq
tff(fact_4580_minus__int_Oabs__eq,axiom,
    ! [Xa: product_prod(nat,nat),Xb: product_prod(nat,nat)] : ( aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(product_prod(nat,nat),int,abs_Integ,Xa)),aa(product_prod(nat,nat),int,abs_Integ,Xb)) = aa(product_prod(nat,nat),int,abs_Integ,aa(product_prod(nat,nat),product_prod(nat,nat),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat)),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_ne(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))))),Xa),Xb)) ) ).

% minus_int.abs_eq
tff(fact_4581_min__list_Opelims,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Xa: list(A),Y: A] :
          ( ( min_list(A,Xa) = Y )
         => ( accp(list(A),min_list_rel(A),Xa)
           => ( ! [X: A,Xs: list(A)] :
                  ( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) )
                 => ( ( Y = aa(list(A),A,aa(fun(A,fun(list(A),A)),fun(list(A),A),aa(A,fun(fun(A,fun(list(A),A)),fun(list(A),A)),case_list(A,A),X),aa(list(A),fun(A,fun(list(A),A)),aTP_Lamp_bp(A,fun(list(A),fun(A,fun(list(A),A))),X),Xs)),Xs) )
                   => ~ accp(list(A),min_list_rel(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) ) )
             => ~ ( ( Xa = nil(A) )
                 => ( ( Y = undefined(A) )
                   => ~ accp(list(A),min_list_rel(A),nil(A)) ) ) ) ) ) ) ).

% min_list.pelims
tff(fact_4582_mod__h__bot__normalize,axiom,
    ! [A: $tType,H2: heap_ext(product_unit),Pa: assn] :
      ( syntax7388354845996824322omatch(A,heap_ext(product_unit),undefined(A),H2)
     => ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,Pa),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),bot_bot(set(nat))))
      <=> aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,Pa),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_4583_zipf_Opelims,axiom,
    ! [A: $tType,B: $tType,E: $tType,Xa: fun(B,fun(E,A)),Xaa: list(B),Xb: list(E),Y: list(A)] :
      ( ( zipf(B,E,A,Xa,Xaa,Xb) = Y )
     => ( accp(product_prod(fun(B,fun(E,A)),product_prod(list(B),list(E))),zipf_rel(B,E,A),aa(product_prod(list(B),list(E)),product_prod(fun(B,fun(E,A)),product_prod(list(B),list(E))),aa(fun(B,fun(E,A)),fun(product_prod(list(B),list(E)),product_prod(fun(B,fun(E,A)),product_prod(list(B),list(E)))),product_Pair(fun(B,fun(E,A)),product_prod(list(B),list(E))),Xa),aa(list(E),product_prod(list(B),list(E)),aa(list(B),fun(list(E),product_prod(list(B),list(E))),product_Pair(list(B),list(E)),Xaa),Xb)))
       => ( ( ( Xaa = nil(B) )
           => ( ( Xb = nil(E) )
             => ( ( Y = nil(A) )
               => ~ accp(product_prod(fun(B,fun(E,A)),product_prod(list(B),list(E))),zipf_rel(B,E,A),aa(product_prod(list(B),list(E)),product_prod(fun(B,fun(E,A)),product_prod(list(B),list(E))),aa(fun(B,fun(E,A)),fun(product_prod(list(B),list(E)),product_prod(fun(B,fun(E,A)),product_prod(list(B),list(E)))),product_Pair(fun(B,fun(E,A)),product_prod(list(B),list(E))),Xa),aa(list(E),product_prod(list(B),list(E)),aa(list(B),fun(list(E),product_prod(list(B),list(E))),product_Pair(list(B),list(E)),nil(B)),nil(E)))) ) ) )
         => ( ! [A4: B,As: list(B)] :
                ( ( Xaa = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),A4),As) )
               => ! [B4: E,Bs2: list(E)] :
                    ( ( Xb = aa(list(E),list(E),aa(E,fun(list(E),list(E)),cons(E),B4),Bs2) )
                   => ( ( Y = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(E,A,aa(B,fun(E,A),Xa,A4),B4)),zipf(B,E,A,Xa,As,Bs2)) )
                     => ~ accp(product_prod(fun(B,fun(E,A)),product_prod(list(B),list(E))),zipf_rel(B,E,A),aa(product_prod(list(B),list(E)),product_prod(fun(B,fun(E,A)),product_prod(list(B),list(E))),aa(fun(B,fun(E,A)),fun(product_prod(list(B),list(E)),product_prod(fun(B,fun(E,A)),product_prod(list(B),list(E)))),product_Pair(fun(B,fun(E,A)),product_prod(list(B),list(E))),Xa),aa(list(E),product_prod(list(B),list(E)),aa(list(B),fun(list(E),product_prod(list(B),list(E))),product_Pair(list(B),list(E)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),A4),As)),aa(list(E),list(E),aa(E,fun(list(E),list(E)),cons(E),B4),Bs2)))) ) ) )
           => ( ! [V3: B,Va: list(B)] :
                  ( ( Xaa = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),V3),Va) )
                 => ( ( Xb = nil(E) )
                   => ( ( Y = undefined(list(A)) )
                     => ~ accp(product_prod(fun(B,fun(E,A)),product_prod(list(B),list(E))),zipf_rel(B,E,A),aa(product_prod(list(B),list(E)),product_prod(fun(B,fun(E,A)),product_prod(list(B),list(E))),aa(fun(B,fun(E,A)),fun(product_prod(list(B),list(E)),product_prod(fun(B,fun(E,A)),product_prod(list(B),list(E)))),product_Pair(fun(B,fun(E,A)),product_prod(list(B),list(E))),Xa),aa(list(E),product_prod(list(B),list(E)),aa(list(B),fun(list(E),product_prod(list(B),list(E))),product_Pair(list(B),list(E)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),V3),Va)),nil(E)))) ) ) )
             => ~ ( ( Xaa = nil(B) )
                 => ! [V3: E,Va: list(E)] :
                      ( ( Xb = aa(list(E),list(E),aa(E,fun(list(E),list(E)),cons(E),V3),Va) )
                     => ( ( Y = undefined(list(A)) )
                       => ~ accp(product_prod(fun(B,fun(E,A)),product_prod(list(B),list(E))),zipf_rel(B,E,A),aa(product_prod(list(B),list(E)),product_prod(fun(B,fun(E,A)),product_prod(list(B),list(E))),aa(fun(B,fun(E,A)),fun(product_prod(list(B),list(E)),product_prod(fun(B,fun(E,A)),product_prod(list(B),list(E)))),product_Pair(fun(B,fun(E,A)),product_prod(list(B),list(E))),Xa),aa(list(E),product_prod(list(B),list(E)),aa(list(B),fun(list(E),product_prod(list(B),list(E))),product_Pair(list(B),list(E)),nil(B)),aa(list(E),list(E),aa(E,fun(list(E),list(E)),cons(E),V3),Va)))) ) ) ) ) ) ) ) ) ).

% zipf.pelims
tff(fact_4584_divmod__integer__eq__cases,axiom,
    ! [K3: code_integer,L: code_integer] :
      ( code_divmod_integer(K3,L) = $ite(
          K3 = 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)),K3),
            aa(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer),aa(code_integer,fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)),comp(code_integer,fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)),code_integer,comp(fun(code_integer,code_integer),fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)),code_integer,product_apsnd(code_integer,code_integer,code_integer),times_times(code_integer)),sgn_sgn(code_integer)),L),
              $ite(aa(code_integer,code_integer,sgn_sgn(code_integer),K3) = aa(code_integer,code_integer,sgn_sgn(code_integer),L),code_divmod_abs(K3,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_nf(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),L)),code_divmod_abs(K3,L)))) ) ) ) ).

% divmod_integer_eq_cases
tff(fact_4585_List_Omap_Ocomp,axiom,
    ! [B: $tType,E: $tType,A: $tType,F2: fun(E,B),G: fun(A,E)] : ( comp(list(E),list(B),list(A),aa(fun(E,B),fun(list(E),list(B)),map(E,B),F2),aa(fun(A,E),fun(list(A),list(E)),map(A,E),G)) = aa(fun(A,B),fun(list(A),list(B)),map(A,B),comp(E,B,A,F2,G)) ) ).

% List.map.comp
tff(fact_4586_list_Omap__comp,axiom,
    ! [B: $tType,A: $tType,E: $tType,G: fun(B,A),F2: fun(E,B),V2: list(E)] : ( aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),G),aa(list(E),list(B),aa(fun(E,B),fun(list(E),list(B)),map(E,B),F2),V2)) = aa(list(E),list(A),aa(fun(E,A),fun(list(E),list(A)),map(E,A),comp(B,A,E,G,F2)),V2) ) ).

% list.map_comp
tff(fact_4587_List_Omap_Ocompositionality,axiom,
    ! [B: $tType,A: $tType,E: $tType,F2: fun(B,A),G: fun(E,B),List: list(E)] : ( aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),aa(list(E),list(B),aa(fun(E,B),fun(list(E),list(B)),map(E,B),G),List)) = aa(list(E),list(A),aa(fun(E,A),fun(list(E),list(A)),map(E,A),comp(B,A,E,F2,G)),List) ) ).

% List.map.compositionality
tff(fact_4588_map__map,axiom,
    ! [B: $tType,A: $tType,E: $tType,F2: fun(B,A),G: fun(E,B),Xsa: list(E)] : ( aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),aa(list(E),list(B),aa(fun(E,B),fun(list(E),list(B)),map(E,B),G),Xsa)) = aa(list(E),list(A),aa(fun(E,A),fun(list(E),list(A)),map(E,A),comp(B,A,E,F2,G)),Xsa) ) ).

% map_map
tff(fact_4589_map__comp__map,axiom,
    ! [B: $tType,E: $tType,A: $tType,F2: fun(E,B),G: fun(A,E)] : ( comp(list(E),list(B),list(A),aa(fun(E,B),fun(list(E),list(B)),map(E,B),F2),aa(fun(A,E),fun(list(A),list(E)),map(A,E),G)) = aa(fun(A,B),fun(list(A),list(B)),map(A,B),comp(E,B,A,F2,G)) ) ).

% map_comp_map
tff(fact_4590_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_ng(set(filter(A)),fun(filter(A),$o),S))) ) ).

% Inf_filter_def
tff(fact_4591_Inf__filter__not__bot,axiom,
    ! [A: $tType,B2: set(filter(A))] :
      ( ! [X7: set(filter(A))] :
          ( aa(set(filter(A)),$o,aa(set(filter(A)),fun(set(filter(A)),$o),ord_less_eq(set(filter(A))),X7),B2)
         => ( aa(set(filter(A)),$o,finite_finite2(filter(A)),X7)
           => ( aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),X7) != bot_bot(filter(A)) ) ) )
     => ( aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),B2) != bot_bot(filter(A)) ) ) ).

% Inf_filter_not_bot
tff(fact_4592_INF__filter__bot__base,axiom,
    ! [A: $tType,B: $tType,Ia2: set(A),Fa: fun(A,filter(B))] :
      ( ! [I3: A] :
          ( member(A,I3,Ia2)
         => ! [J3: A] :
              ( member(A,J3,Ia2)
             => ? [X4: A] :
                  ( member(A,X4,Ia2)
                  & aa(filter(B),$o,aa(filter(B),fun(filter(B),$o),ord_less_eq(filter(B)),aa(A,filter(B),Fa,X4)),aa(filter(B),filter(B),aa(filter(B),fun(filter(B),filter(B)),inf_inf(filter(B)),aa(A,filter(B),Fa,I3)),aa(A,filter(B),Fa,J3))) ) ) )
     => ( ( aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image2(A,filter(B),Fa),Ia2)) = bot_bot(filter(B)) )
      <=> ? [X6: A] :
            ( member(A,X6,Ia2)
            & ( aa(A,filter(B),Fa,X6) = bot_bot(filter(B)) ) ) ) ) ).

% INF_filter_bot_base
tff(fact_4593_fun__comp__eq__conv,axiom,
    ! [E: $tType,B: $tType,A: $tType,F2: fun(E,B),G: fun(A,E),Fg: fun(A,B)] :
      ( ( comp(E,B,A,F2,G) = Fg )
    <=> ! [X6: A] : ( aa(E,B,F2,aa(A,E,G,X6)) = aa(A,B,Fg,X6) ) ) ).

% fun_comp_eq_conv
tff(fact_4594_comp__cong__right,axiom,
    ! [E: $tType,B: $tType,A: $tType,Xa: fun(A,B),Y: fun(A,B),F2: fun(B,E)] :
      ( ( Xa = Y )
     => ( comp(B,E,A,F2,Xa) = comp(B,E,A,F2,Y) ) ) ).

% comp_cong_right
tff(fact_4595_comp__cong__left,axiom,
    ! [B: $tType,A: $tType,E: $tType,Xa: fun(A,B),Y: fun(A,B),F2: fun(E,A)] :
      ( ( Xa = Y )
     => ( comp(A,B,E,Xa,F2) = comp(A,B,E,Y,F2) ) ) ).

% comp_cong_left
tff(fact_4596_foldr_Ofoldr__Cons,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,fun(A,A)),Xa: B,Xsa: list(B)] : ( aa(list(B),fun(A,A),aa(fun(B,fun(A,A)),fun(list(B),fun(A,A)),foldr(B,A),F2),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Xa),Xsa)) = comp(A,A,A,aa(B,fun(A,A),F2,Xa),aa(list(B),fun(A,A),aa(fun(B,fun(A,A)),fun(list(B),fun(A,A)),foldr(B,A),F2),Xsa)) ) ).

% foldr.foldr_Cons
tff(fact_4597_foldr__map,axiom,
    ! [E: $tType,B: $tType,A: $tType,G: fun(B,fun(A,A)),F2: fun(E,B),Xsa: list(E),A3: A] : ( aa(A,A,aa(list(B),fun(A,A),aa(fun(B,fun(A,A)),fun(list(B),fun(A,A)),foldr(B,A),G),aa(list(E),list(B),aa(fun(E,B),fun(list(E),list(B)),map(E,B),F2),Xsa)),A3) = aa(A,A,aa(list(E),fun(A,A),aa(fun(E,fun(A,A)),fun(list(E),fun(A,A)),foldr(E,A),comp(B,fun(A,A),E,G,F2)),Xsa),A3) ) ).

% foldr_map
tff(fact_4598_prod_Oeq__fold,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(B,A),Aa2: set(B)] : ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),Aa2) = finite_fold(B,A,comp(A,fun(A,A),B,times_times(A),G),one_one(A),Aa2) ) ) ).

% prod.eq_fold
tff(fact_4599_SUP__fold__sup,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [Aa2: set(A),F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),Aa2)) = finite_fold(A,B,comp(B,fun(B,B),A,sup_sup(B),F2),bot_bot(B),Aa2) ) ) ) ).

% SUP_fold_sup
tff(fact_4600_mult_Osafe__commute,axiom,
    ! [A: $tType] :
      ( ab_semigroup_mult(A)
     => ! [Xa: A,Y: A,A3: A,B3: A] :
          ( syntax7388354845996824322omatch(A,A,aa(A,A,aa(A,fun(A,A),times_times(A),Xa),Y),A3)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3) = aa(A,A,aa(A,fun(A,A),times_times(A),B3),A3) ) ) ) ).

% mult.safe_commute
tff(fact_4601_less__int_Orep__eq,axiom,
    ! [Xa: int,Xaa: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Xa),Xaa)
    <=> aa(product_prod(nat,nat),$o,aa(product_prod(nat,nat),fun(product_prod(nat,nat),$o),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),$o))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)),product_case_prod(nat,nat,fun(product_prod(nat,nat),$o)),aTP_Lamp_na(nat,fun(nat,fun(product_prod(nat,nat),$o)))),aa(int,product_prod(nat,nat),rep_Integ,Xa)),aa(int,product_prod(nat,nat),rep_Integ,Xaa)) ) ).

% less_int.rep_eq
tff(fact_4602_filter__rev__aux__alt,axiom,
    ! [A: $tType,A3: list(A),Pa: fun(A,$o),L: list(A)] : ( aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter_rev_aux(A,A3),Pa),L) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),Pa),aa(list(A),list(A),rev(A),L))),A3) ) ).

% filter_rev_aux_alt
tff(fact_4603_length__filter__map,axiom,
    ! [A: $tType,B: $tType,Pa: fun(A,$o),F2: fun(B,A),Xsa: list(B)] : ( aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),Pa),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Xsa))) = aa(list(B),nat,size_size(list(B)),aa(list(B),list(B),aa(fun(B,$o),fun(list(B),list(B)),filter2(B),comp(A,$o,B,Pa,F2)),Xsa)) ) ).

% length_filter_map
tff(fact_4604_set__mset__comp__mset,axiom,
    ! [A: $tType] : ( comp(multiset(A),set(A),list(A),set_mset(A),mset(A)) = set2(A) ) ).

% set_mset_comp_mset
tff(fact_4605_sorted__list__of__set_Ofold__insort__key_Ocomp__fun__commute__on,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Y: A,Xa: A] : ( comp(list(A),list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_ip(A,A)),Y),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_ip(A,A)),Xa)) = comp(list(A),list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_ip(A,A)),Xa),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_ip(A,A)),Y)) ) ) ).

% sorted_list_of_set.fold_insort_key.comp_fun_commute_on
tff(fact_4606_empty__natural,axiom,
    ! [E: $tType,B: $tType,F4: $tType,A: $tType,F2: fun(A,E),G: fun(F4,B)] : ( comp(E,set(B),A,aTP_Lamp_nh(E,set(B)),F2) = comp(set(F4),set(B),A,image2(F4,B,G),aTP_Lamp_ni(A,set(F4))) ) ).

% empty_natural
tff(fact_4607_less__filter__def,axiom,
    ! [A: $tType,Fa: filter(A),F: filter(A)] :
      ( aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less(filter(A)),Fa),F)
    <=> ( aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),Fa),F)
        & ~ aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),F),Fa) ) ) ).

% less_filter_def
tff(fact_4608_filter__map,axiom,
    ! [A: $tType,B: $tType,Pa: fun(A,$o),F2: fun(B,A),Xsa: list(B)] : ( aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),Pa),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Xsa)) = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),aa(list(B),list(B),aa(fun(B,$o),fun(list(B),list(B)),filter2(B),comp(A,$o,B,Pa,F2)),Xsa)) ) ).

% filter_map
tff(fact_4609_takeWhile__map,axiom,
    ! [A: $tType,B: $tType,Pa: fun(A,$o),F2: fun(B,A),Xsa: list(B)] : ( aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),takeWhile(A),Pa),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Xsa)) = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),aa(list(B),list(B),aa(fun(B,$o),fun(list(B),list(B)),takeWhile(B),comp(A,$o,B,Pa,F2)),Xsa)) ) ).

% takeWhile_map
tff(fact_4610_rotate__add,axiom,
    ! [A: $tType,M2: nat,N: nat] : ( aa(nat,fun(list(A),list(A)),rotate(A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),N)) = comp(list(A),list(A),list(A),aa(nat,fun(list(A),list(A)),rotate(A),M2),aa(nat,fun(list(A),list(A)),rotate(A),N)) ) ).

% rotate_add
tff(fact_4611_filter__rev__aux_Osimps_I2_J,axiom,
    ! [A: $tType,A3: list(A),Pa: fun(A,$o),Xa: A,Xsa: list(A)] :
      ( aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter_rev_aux(A,A3),Pa),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa)) = $ite(aa(A,$o,Pa,Xa),aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter_rev_aux(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),A3)),Pa),Xsa),aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter_rev_aux(A,A3),Pa),Xsa)) ) ).

% filter_rev_aux.simps(2)
tff(fact_4612_filter__rev__aux_Osimps_I1_J,axiom,
    ! [A: $tType,A3: list(A),Pa: fun(A,$o)] : ( aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter_rev_aux(A,A3),Pa),nil(A)) = A3 ) ).

% filter_rev_aux.simps(1)
tff(fact_4613_folding__insort__key_Oinsort__key__commute,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),S: set(B),F2: fun(B,A),Xa: B,Y: B] :
      ( folding_insort_key(A,B,Less_eq,Less,S,F2)
     => ( member(B,Xa,S)
       => ( member(B,Y,S)
         => ( comp(list(B),list(B),list(B),aa(B,fun(list(B),list(B)),aa(fun(B,A),fun(B,fun(list(B),list(B))),insort_key(A,B,Less_eq),F2),Y),aa(B,fun(list(B),list(B)),aa(fun(B,A),fun(B,fun(list(B),list(B))),insort_key(A,B,Less_eq),F2),Xa)) = comp(list(B),list(B),list(B),aa(B,fun(list(B),list(B)),aa(fun(B,A),fun(B,fun(list(B),list(B))),insort_key(A,B,Less_eq),F2),Xa),aa(B,fun(list(B),list(B)),aa(fun(B,A),fun(B,fun(list(B),list(B))),insort_key(A,B,Less_eq),F2),Y)) ) ) ) ) ).

% folding_insort_key.insort_key_commute
tff(fact_4614_set__oo__map__alt,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),X4: list(A)] : ( aa(list(A),set(B),comp(list(B),set(B),list(A),set2(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2)),X4) = aa(set(A),set(B),image2(A,B,F2),aa(list(A),set(A),set2(A),X4)) ) ).

% set_oo_map_alt
tff(fact_4615_assn__aci_I11_J,axiom,
    ! [Xa: assn,Y: assn,A3: assn,B3: assn] :
      ( syntax7388354845996824322omatch(assn,assn,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Xa),Y),A3)
     => ( aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),A3),B3) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),B3),A3) ) ) ).

% assn_aci(11)
tff(fact_4616_mult_Oright__commute,axiom,
    ! [A: $tType] :
      ( ab_semigroup_mult(A)
     => ! [A3: A,B3: 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),B3)),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)),B3) ) ) ).

% mult.right_commute
tff(fact_4617_mult_Oright__assoc,axiom,
    ! [A: $tType] :
      ( ab_semigroup_mult(A)
     => ! [A3: A,B3: 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),B3)),C2) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2)) ) ) ).

% mult.right_assoc
tff(fact_4618_remove__rev__def,axiom,
    ! [A: $tType,Xa: A] : ( remove_rev(A,Xa) = aa(fun(A,$o),fun(list(A),list(A)),filter_rev(A),comp($o,$o,A,fNot,aa(A,fun(A,$o),fequal(A),Xa))) ) ).

% remove_rev_def
tff(fact_4619_remdup__sort__mergesort__remdups,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( comp(list(A),list(A),list(A),remdups(A),aa(fun(A,A),fun(list(A),list(A)),linorder_sort_key(A,A),aTP_Lamp_ip(A,A))) = mergesort_remdups(A) ) ) ).

% remdup_sort_mergesort_remdups
tff(fact_4620_sum_OUnion__disjoint,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [C3: set(set(A)),G: fun(A,B)] :
          ( ! [X: set(A)] :
              ( member(set(A),X,C3)
             => aa(set(A),$o,finite_finite2(A),X) )
         => ( ! [X: set(A)] :
                ( member(set(A),X,C3)
               => ! [Xa5: set(A)] :
                    ( member(set(A),Xa5,C3)
                   => ( ( X != Xa5 )
                     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),X),Xa5) = 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),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_4621_prod_OUnion__disjoint,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [C3: set(set(A)),G: fun(A,B)] :
          ( ! [X: set(A)] :
              ( member(set(A),X,C3)
             => aa(set(A),$o,finite_finite2(A),X) )
         => ( ! [X: set(A)] :
                ( member(set(A),X,C3)
               => ! [Xa5: set(A)] :
                    ( member(set(A),Xa5,C3)
                   => ( ( X != Xa5 )
                     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),X),Xa5) = 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),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_4622_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),comp(nat,A,nat,G,suc)),set_or7035219750837199246ssThan(nat,zero_zero(nat),N))) ) ) ).

% prod.atLeast0_lessThan_Suc_shift
tff(fact_4623_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),comp(nat,A,nat,G,suc)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N))) ) ) ).

% prod.atLeast0_atMost_Suc_shift
tff(fact_4624_of__int_Orep__eq,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Xa: int] : ( aa(int,A,ring_1_of_int(A),Xa) = 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_my(nat,fun(nat,A))),aa(int,product_prod(nat,nat),rep_Integ,Xa)) ) ) ).

% of_int.rep_eq
tff(fact_4625_filter__rev__def,axiom,
    ! [A: $tType] : ( filter_rev(A) = filter_rev_aux(A,nil(A)) ) ).

% filter_rev_def
tff(fact_4626_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_mx(nat,fun(nat,product_prod(nat,nat))))) ).

% uminus_int_def
tff(fact_4627_SUP__principal,axiom,
    ! [A: $tType,B: $tType,Aa2: fun(B,set(A)),Ia2: 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_nj(fun(B,set(A)),fun(B,filter(A)),Aa2)),Ia2)) = aa(set(A),filter(A),principal(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),Aa2),Ia2))) ) ).

% SUP_principal
tff(fact_4628_prod_Oinsert_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [Ia2: set(A),P2: fun(A,B),Ia: A] :
          ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_bb(set(A),fun(fun(A,B),fun(A,$o)),Ia2),P2)))
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1962203154675924110t_prod(A,B),P2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Ia),Ia2)) = $ite(member(A,Ia,Ia2),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1962203154675924110t_prod(A,B),P2),Ia2),aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,P2,Ia)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1962203154675924110t_prod(A,B),P2),Ia2))) ) ) ) ).

% prod.insert'
tff(fact_4629_principal__inject,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A)] :
      ( ( aa(set(A),filter(A),principal(A),Aa2) = aa(set(A),filter(A),principal(A),B2) )
    <=> ( Aa2 = B2 ) ) ).

% principal_inject
tff(fact_4630_sup__principal,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A)] : ( aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),sup_sup(filter(A)),aa(set(A),filter(A),principal(A),Aa2)),aa(set(A),filter(A),principal(A),B2)) = aa(set(A),filter(A),principal(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),B2)) ) ).

% sup_principal
tff(fact_4631_prod_Oempty_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [P2: fun(B,A)] : ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),P2),bot_bot(set(B))) = one_one(A) ) ) ).

% prod.empty'
tff(fact_4632_principal__le__iff,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A)] :
      ( aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),aa(set(A),filter(A),principal(A),Aa2)),aa(set(A),filter(A),principal(A),B2))
    <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),B2) ) ).

% principal_le_iff
tff(fact_4633_inf__principal,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A)] : ( aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),inf_inf(filter(A)),aa(set(A),filter(A),principal(A),Aa2)),aa(set(A),filter(A),principal(A),B2)) = aa(set(A),filter(A),principal(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),B2)) ) ).

% inf_principal
tff(fact_4634_top__eq__principal__UNIV,axiom,
    ! [A: $tType] : ( top_top(filter(A)) = aa(set(A),filter(A),principal(A),top_top(set(A))) ) ).

% top_eq_principal_UNIV
tff(fact_4635_bot__eq__principal__empty,axiom,
    ! [A: $tType] : ( bot_bot(filter(A)) = aa(set(A),filter(A),principal(A),bot_bot(set(A))) ) ).

% bot_eq_principal_empty
tff(fact_4636_principal__eq__bot__iff,axiom,
    ! [A: $tType,X3: set(A)] :
      ( ( aa(set(A),filter(A),principal(A),X3) = bot_bot(filter(A)) )
    <=> ( X3 = bot_bot(set(A)) ) ) ).

% principal_eq_bot_iff
tff(fact_4637_prod_Odistrib__triv_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [Ia2: set(A),G: fun(A,B),H2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),Ia2)
         => ( 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_nk(fun(A,B),fun(fun(A,B),fun(A,B)),G),H2)),Ia2) = 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),Ia2)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1962203154675924110t_prod(A,B),H2),Ia2)) ) ) ) ).

% prod.distrib_triv'
tff(fact_4638_prod_Omono__neutral__cong__right_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [S: set(A),T4: set(A),G: fun(A,B),H2: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),T4)
         => ( ! [X: A] :
                ( member(A,X,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T4),S))
               => ( aa(A,B,G,X) = one_one(B) ) )
           => ( ! [X: A] :
                  ( member(A,X,S)
                 => ( aa(A,B,G,X) = aa(A,B,H2,X) ) )
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1962203154675924110t_prod(A,B),G),T4) = 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_4639_prod_Omono__neutral__cong__left_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [S: set(A),T4: set(A),H2: fun(A,B),G: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),T4)
         => ( ! [I3: A] :
                ( member(A,I3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T4),S))
               => ( aa(A,B,H2,I3) = one_one(B) ) )
           => ( ! [X: A] :
                  ( member(A,X,S)
                 => ( aa(A,B,G,X) = aa(A,B,H2,X) ) )
             => ( 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),T4) ) ) ) ) ) ).

% prod.mono_neutral_cong_left'
tff(fact_4640_prod_Omono__neutral__right_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [S: set(A),T4: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),T4)
         => ( ! [X: A] :
                ( member(A,X,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T4),S))
               => ( aa(A,B,G,X) = one_one(B) ) )
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1962203154675924110t_prod(A,B),G),T4) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1962203154675924110t_prod(A,B),G),S) ) ) ) ) ).

% prod.mono_neutral_right'
tff(fact_4641_prod_Omono__neutral__left_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [S: set(A),T4: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),T4)
         => ( ! [X: A] :
                ( member(A,X,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T4),S))
               => ( aa(A,B,G,X) = 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),T4) ) ) ) ) ).

% prod.mono_neutral_left'
tff(fact_4642_prod_Odistrib_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [Ia2: set(A),G: fun(A,B),H2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_bb(set(A),fun(fun(A,B),fun(A,$o)),Ia2),G)))
         => ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_bb(set(A),fun(fun(A,B),fun(A,$o)),Ia2),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_nk(fun(A,B),fun(fun(A,B),fun(A,B)),G),H2)),Ia2) = 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),Ia2)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1962203154675924110t_prod(A,B),H2),Ia2)) ) ) ) ) ).

% prod.distrib'
tff(fact_4643_INF__principal__finite,axiom,
    ! [B: $tType,A: $tType,X3: set(A),F2: fun(A,set(B))] :
      ( aa(set(A),$o,finite_finite2(A),X3)
     => ( aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image2(A,filter(B),aTP_Lamp_nl(fun(A,set(B)),fun(A,filter(B)),F2)),X3)) = aa(set(B),filter(B),principal(B),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image2(A,set(B),F2),X3))) ) ) ).

% INF_principal_finite
tff(fact_4644_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_mv(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))) ).

% times_int_def
tff(fact_4645_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_ne(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))) ).

% minus_int_def
tff(fact_4646_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_nc(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))) ).

% plus_int_def
tff(fact_4647_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_nm(A,filter(A))),top_top(set(A)))) ) ) ).

% at_bot_def
tff(fact_4648_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_nn(A,filter(A))),top_top(set(A)))) ) ) ).

% at_top_def
tff(fact_4649_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_no(A,filter(A))),aa(A,set(A),set_ord_atMost(A),C2))) ) ) ).

% at_bot_sub
tff(fact_4650_trivial__limit__at__top__linorder,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( at_top(A) != bot_bot(filter(A)) ) ) ).

% trivial_limit_at_top_linorder
tff(fact_4651_trivial__limit__at__bot__linorder,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( at_bot(A) != bot_bot(filter(A)) ) ) ).

% trivial_limit_at_bot_linorder
tff(fact_4652_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_np(A,filter(A))),set_ord_atLeast(A,C2))) ) ) ).

% at_top_sub
tff(fact_4653_partition__filter__conv,axiom,
    ! [A: $tType,F2: fun(A,$o),Xsa: list(A)] : ( aa(list(A),product_prod(list(A),list(A)),aa(fun(A,$o),fun(list(A),product_prod(list(A),list(A))),partition(A),F2),Xsa) = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),F2),Xsa)),aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),comp($o,$o,A,fNot,F2)),Xsa)) ) ).

% partition_filter_conv
tff(fact_4654_partition__rev__filter__conv,axiom,
    ! [A: $tType,Pa: fun(A,$o),Yes2: list(A),No2: list(A),Xsa: list(A)] : ( partition_rev(A,Pa,aa(list(A),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),Xsa) = aa(list(A),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),aa(list(A),list(A),rev(A),aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),Pa),Xsa))),Yes2)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),rev(A),aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),comp($o,$o,A,fNot,Pa)),Xsa))),No2)) ) ).

% partition_rev_filter_conv
tff(fact_4655_transpose__max__length,axiom,
    ! [A: $tType,Xsa: list(list(A))] : ( aa(nat,nat,aa(list(list(A)),fun(nat,nat),aa(fun(list(A),fun(nat,nat)),fun(list(list(A)),fun(nat,nat)),foldr(list(A),nat),aTP_Lamp_nq(list(A),fun(nat,nat))),transpose(A,Xsa)),zero_zero(nat)) = aa(list(list(A)),nat,size_size(list(list(A))),aa(list(list(A)),list(list(A)),aa(fun(list(A),$o),fun(list(list(A)),list(list(A))),filter2(list(A)),aTP_Lamp_jg(list(A),$o)),Xsa)) ) ).

% transpose_max_length
tff(fact_4656_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_4657_max_Oleft__idem,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: 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),B3)) = aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B3) ) ) ).

% max.left_idem
tff(fact_4658_max_Oright__idem,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A] : ( aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B3)),B3) = aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B3) ) ) ).

% max.right_idem
tff(fact_4659_max_Obounded__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B3: A,C2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_max(A),B3),C2)),A3)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A3) ) ) ) ).

% max.bounded_iff
tff(fact_4660_max_Oabsorb2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
         => ( aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B3) = B3 ) ) ) ).

% max.absorb2
tff(fact_4661_max_Oabsorb1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3)
         => ( aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B3) = A3 ) ) ) ).

% max.absorb1
tff(fact_4662_max__less__iff__conj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Y: A,Z2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_max(A),Xa),Y)),Z2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Z2)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Z2) ) ) ) ).

% max_less_iff_conj
tff(fact_4663_max_Oabsorb4,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
         => ( aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B3) = B3 ) ) ) ).

% max.absorb4
tff(fact_4664_max_Oabsorb3,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A3)
         => ( aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B3) = A3 ) ) ) ).

% max.absorb3
tff(fact_4665_max__bot,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [Xa: A] : ( aa(A,A,aa(A,fun(A,A),ord_max(A),bot_bot(A)),Xa) = Xa ) ) ).

% max_bot
tff(fact_4666_max__bot2,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [Xa: A] : ( aa(A,A,aa(A,fun(A,A),ord_max(A),Xa),bot_bot(A)) = Xa ) ) ).

% max_bot2
tff(fact_4667_max__top2,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [Xa: A] : ( aa(A,A,aa(A,fun(A,A),ord_max(A),Xa),top_top(A)) = top_top(A) ) ) ).

% max_top2
tff(fact_4668_max__top,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [Xa: A] : ( aa(A,A,aa(A,fun(A,A),ord_max(A),top_top(A)),Xa) = top_top(A) ) ) ).

% max_top
tff(fact_4669_max__min__same_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Y: A] : ( aa(A,A,aa(A,fun(A,A),ord_max(A),Xa),aa(A,A,aa(A,fun(A,A),ord_min(A),Xa),Y)) = Xa ) ) ).

% max_min_same(1)
tff(fact_4670_max__min__same_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: 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),Xa),Y)),Xa) = Xa ) ) ).

% max_min_same(2)
tff(fact_4671_max__min__same_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: 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),Xa),Y)),Y) = Y ) ) ).

% max_min_same(3)
tff(fact_4672_max__min__same_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Y: A,Xa: A] : ( aa(A,A,aa(A,fun(A,A),ord_max(A),Y),aa(A,A,aa(A,fun(A,A),ord_min(A),Xa),Y)) = Y ) ) ).

% max_min_same(4)
tff(fact_4673_of__bool__or__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Pa: $o,Qa: $o] :
          ( aa($o,A,zero_neq_one_of_bool(A),
              ( (Pa)
              | (Qa) )) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa($o,A,zero_neq_one_of_bool(A),(Pa))),aa($o,A,zero_neq_one_of_bool(A),(Qa))) ) ) ).

% of_bool_or_iff
tff(fact_4674_Max__insert,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Aa2: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( ( Aa2 != bot_bot(set(A)) )
           => ( lattic643756798349783984er_Max(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),Aa2)) = aa(A,A,aa(A,fun(A,A),ord_max(A),Xa),lattic643756798349783984er_Max(A,Aa2)) ) ) ) ) ).

% Max_insert
tff(fact_4675_max_Ostrict__coboundedI2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A,B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),B3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B3)) ) ) ).

% max.strict_coboundedI2
tff(fact_4676_max_Ostrict__coboundedI1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A,A3: A,B3: A] :
          ( aa(A,$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),B3)) ) ) ).

% max.strict_coboundedI1
tff(fact_4677_max_Ostrict__order__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A3)
        <=> ( ( A3 = aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B3) )
            & ( A3 != B3 ) ) ) ) ).

% max.strict_order_iff
tff(fact_4678_max_Ostrict__boundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B3: A,C2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_max(A),B3),C2)),A3)
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A3)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),A3) ) ) ) ).

% max.strict_boundedE
tff(fact_4679_less__max__iff__disj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Z2: A,Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z2),aa(A,A,aa(A,fun(A,A),ord_max(A),Xa),Y))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z2),Xa)
            | aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z2),Y) ) ) ) ).

% less_max_iff_disj
tff(fact_4680_max_Oassoc,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: 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),B3)),C2) = aa(A,A,aa(A,fun(A,A),ord_max(A),A3),aa(A,A,aa(A,fun(A,A),ord_max(A),B3),C2)) ) ) ).

% max.assoc
tff(fact_4681_max_Ocommute,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A] : ( aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B3) = aa(A,A,aa(A,fun(A,A),ord_max(A),B3),A3) ) ) ).

% max.commute
tff(fact_4682_max_Oleft__commute,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B3: A,A3: A,C2: A] : ( aa(A,A,aa(A,fun(A,A),ord_max(A),B3),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),B3),C2)) ) ) ).

% max.left_commute
tff(fact_4683_min__max__distrib2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: 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),B3),C2)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B3)),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),C2)) ) ) ).

% min_max_distrib2
tff(fact_4684_min__max__distrib1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B3: 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),B3),C2)),A3) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),ord_min(A),B3),A3)),aa(A,A,aa(A,fun(A,A),ord_min(A),C2),A3)) ) ) ).

% min_max_distrib1
tff(fact_4685_max__min__distrib2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: 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),B3),C2)) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B3)),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),C2)) ) ) ).

% max_min_distrib2
tff(fact_4686_max__min__distrib1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B3: 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),B3),C2)),A3) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),ord_max(A),B3),A3)),aa(A,A,aa(A,fun(A,A),ord_max(A),C2),A3)) ) ) ).

% max_min_distrib1
tff(fact_4687_max__add__distrib__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [Xa: 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),Xa),Y)),Z2) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Z2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),Y),Z2)) ) ) ).

% max_add_distrib_left
tff(fact_4688_max__add__distrib__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [Xa: A,Y: A,Z2: A] : ( aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),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),Xa),Y)),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Z2)) ) ) ).

% max_add_distrib_right
tff(fact_4689_max__diff__distrib__left,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [Xa: 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),Xa),Y)),Z2) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xa),Z2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),Y),Z2)) ) ) ).

% max_diff_distrib_left
tff(fact_4690_max__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A3: A,B3: A] :
          ( aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B3) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3),B3,A3) ) ) ).

% max_def
tff(fact_4691_max__absorb1,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Y: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xa)
         => ( aa(A,A,aa(A,fun(A,A),ord_max(A),Xa),Y) = Xa ) ) ) ).

% max_absorb1
tff(fact_4692_max__absorb2,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
         => ( aa(A,A,aa(A,fun(A,A),ord_max(A),Xa),Y) = Y ) ) ) ).

% max_absorb2
tff(fact_4693_max_Omono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A,A3: A,D3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),D3),B3)
           => aa(A,$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),B3)) ) ) ) ).

% max.mono
tff(fact_4694_max_OorderE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3)
         => ( A3 = aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B3) ) ) ) ).

% max.orderE
tff(fact_4695_max_OorderI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A] :
          ( ( A3 = aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B3) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3) ) ) ).

% max.orderI
tff(fact_4696_max_OboundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B3: A,C2: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_max(A),B3),C2)),A3)
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A3) ) ) ) ).

% max.boundedE
tff(fact_4697_max_OboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B3: A,A3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_max(A),B3),C2)),A3) ) ) ) ).

% max.boundedI
tff(fact_4698_max_Oorder__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3)
        <=> ( A3 = aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B3) ) ) ) ).

% max.order_iff
tff(fact_4699_max_Ocobounded1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B3)) ) ).

% max.cobounded1
tff(fact_4700_max_Ocobounded2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B3: A,A3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B3)) ) ).

% max.cobounded2
tff(fact_4701_le__max__iff__disj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Z2: A,Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z2),aa(A,A,aa(A,fun(A,A),ord_max(A),Xa),Y))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z2),Xa)
            | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z2),Y) ) ) ) ).

% le_max_iff_disj
tff(fact_4702_max_Oabsorb__iff1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3)
        <=> ( aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B3) = A3 ) ) ) ).

% max.absorb_iff1
tff(fact_4703_max_Oabsorb__iff2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
        <=> ( aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B3) = B3 ) ) ) ).

% max.absorb_iff2
tff(fact_4704_max_OcoboundedI1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A,A3: A,B3: A] :
          ( aa(A,$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),B3)) ) ) ).

% max.coboundedI1
tff(fact_4705_max_OcoboundedI2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A,B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),B3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B3)) ) ) ).

% max.coboundedI2
tff(fact_4706_sup__max,axiom,
    ! [A: $tType] :
      ( ( semilattice_sup(A)
        & linorder(A) )
     => ( sup_sup(A) = ord_max(A) ) ) ).

% sup_max
tff(fact_4707_minus__max__eq__min,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [Xa: A,Y: A] : ( aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),ord_max(A),Xa),Y)) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,uminus_uminus(A),Xa)),aa(A,A,uminus_uminus(A),Y)) ) ) ).

% minus_max_eq_min
tff(fact_4708_minus__min__eq__max,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [Xa: A,Y: A] : ( aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),ord_min(A),Xa),Y)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,uminus_uminus(A),Xa)),aa(A,A,uminus_uminus(A),Y)) ) ) ).

% minus_min_eq_max
tff(fact_4709_partition__rev_Osimps_I2_J,axiom,
    ! [A: $tType,Pa: fun(A,$o),Yes2: list(A),No2: list(A),Xa: A,Xsa: list(A)] :
      ( partition_rev(A,Pa,aa(list(A),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),Xa),Xsa)) = partition_rev(A,Pa,
          $ite(aa(A,$o,Pa,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)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),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),Xa),No2))),
          Xsa) ) ).

% partition_rev.simps(2)
tff(fact_4710_partition__rev_Osimps_I1_J,axiom,
    ! [A: $tType,Pa: fun(A,$o),Yes2: list(A),No2: list(A)] : ( partition_rev(A,Pa,aa(list(A),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_4711_max__nat_Osemilattice__neutr__order__axioms,axiom,
    semila1105856199041335345_order(nat,ord_max(nat),zero_zero(nat),aTP_Lamp_nr(nat,fun(nat,$o)),aTP_Lamp_ac(nat,fun(nat,$o))) ).

% max_nat.semilattice_neutr_order_axioms
tff(fact_4712_partition_Osimps_I1_J,axiom,
    ! [A: $tType,Pa: fun(A,$o)] : ( aa(list(A),product_prod(list(A),list(A)),aa(fun(A,$o),fun(list(A),product_prod(list(A),list(A))),partition(A),Pa),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_4713_partition__P,axiom,
    ! [A: $tType,Pa: fun(A,$o),Xsa: list(A),Yes2: list(A),No2: list(A)] :
      ( ( aa(list(A),product_prod(list(A),list(A)),aa(fun(A,$o),fun(list(A),product_prod(list(A),list(A))),partition(A),Pa),Xsa) = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes2),No2) )
     => ( ! [X4: A] :
            ( member(A,X4,aa(list(A),set(A),set2(A),Yes2))
           => aa(A,$o,Pa,X4) )
        & ! [X4: A] :
            ( member(A,X4,aa(list(A),set(A),set2(A),No2))
           => ~ aa(A,$o,Pa,X4) ) ) ) ).

% partition_P
tff(fact_4714_min__mult__distrib__right,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A,Y: A,P2: A] :
          ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),ord_min(A),Xa),Y)),P2) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),P2),aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xa),P2)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),P2)),aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xa),P2)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),P2))) ) ) ).

% min_mult_distrib_right
tff(fact_4715_max__mult__distrib__right,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A,Y: A,P2: A] :
          ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),ord_max(A),Xa),Y)),P2) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),P2),aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xa),P2)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),P2)),aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xa),P2)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),P2))) ) ) ).

% max_mult_distrib_right
tff(fact_4716_min__mult__distrib__left,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [P2: A,Xa: A,Y: A] :
          ( aa(A,A,aa(A,fun(A,A),times_times(A),P2),aa(A,A,aa(A,fun(A,A),ord_min(A),Xa),Y)) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),P2),aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),times_times(A),P2),Xa)),aa(A,A,aa(A,fun(A,A),times_times(A),P2),Y)),aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),times_times(A),P2),Xa)),aa(A,A,aa(A,fun(A,A),times_times(A),P2),Y))) ) ) ).

% min_mult_distrib_left
tff(fact_4717_max__mult__distrib__left,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [P2: A,Xa: A,Y: A] :
          ( aa(A,A,aa(A,fun(A,A),times_times(A),P2),aa(A,A,aa(A,fun(A,A),ord_max(A),Xa),Y)) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),P2),aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),times_times(A),P2),Xa)),aa(A,A,aa(A,fun(A,A),times_times(A),P2),Y)),aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),times_times(A),P2),Xa)),aa(A,A,aa(A,fun(A,A),times_times(A),P2),Y))) ) ) ).

% max_mult_distrib_left
tff(fact_4718_Sup__insert__finite,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [S: set(A),Xa: A] :
          ( aa(set(A),$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)),insert3(A),Xa),S)) = $ite(S = bot_bot(set(A)),Xa,aa(A,A,aa(A,fun(A,A),ord_max(A),Xa),aa(set(A),A,complete_Sup_Sup(A),S))) ) ) ) ).

% Sup_insert_finite
tff(fact_4719_hom__Max__commute,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [H2: fun(A,A),N7: set(A)] :
          ( ! [X: A,Y2: A] : ( aa(A,A,H2,aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y2)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,H2,X)),aa(A,A,H2,Y2)) )
         => ( aa(set(A),$o,finite_finite2(A),N7)
           => ( ( N7 != bot_bot(set(A)) )
             => ( aa(A,A,H2,lattic643756798349783984er_Max(A,N7)) = lattic643756798349783984er_Max(A,aa(set(A),set(A),image2(A,A,H2),N7)) ) ) ) ) ) ).

% hom_Max_commute
tff(fact_4720_Max_Osubset,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Aa2: set(A),B2: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( ( B2 != bot_bot(set(A)) )
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),Aa2)
             => ( aa(A,A,aa(A,fun(A,A),ord_max(A),lattic643756798349783984er_Max(A,B2)),lattic643756798349783984er_Max(A,Aa2)) = lattic643756798349783984er_Max(A,Aa2) ) ) ) ) ) ).

% Max.subset
tff(fact_4721_Max_Oclosed,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Aa2: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( ( Aa2 != bot_bot(set(A)) )
           => ( ! [X: A,Y2: A] : member(A,aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),X),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Y2),bot_bot(set(A)))))
             => member(A,lattic643756798349783984er_Max(A,Aa2),Aa2) ) ) ) ) ).

% Max.closed
tff(fact_4722_Max_Oinsert__not__elem,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Aa2: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( ~ member(A,Xa,Aa2)
           => ( ( Aa2 != bot_bot(set(A)) )
             => ( lattic643756798349783984er_Max(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),Aa2)) = aa(A,A,aa(A,fun(A,A),ord_max(A),Xa),lattic643756798349783984er_Max(A,Aa2)) ) ) ) ) ) ).

% Max.insert_not_elem
tff(fact_4723_Max_Ounion,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Aa2: set(A),B2: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( ( Aa2 != bot_bot(set(A)) )
           => ( aa(set(A),$o,finite_finite2(A),B2)
             => ( ( B2 != bot_bot(set(A)) )
               => ( lattic643756798349783984er_Max(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),B2)) = aa(A,A,aa(A,fun(A,A),ord_max(A),lattic643756798349783984er_Max(A,Aa2)),lattic643756798349783984er_Max(A,B2)) ) ) ) ) ) ) ).

% Max.union
tff(fact_4724_Max_Oeq__fold,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Aa2: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( lattic643756798349783984er_Max(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),Aa2)) = finite_fold(A,A,ord_max(A),Xa,Aa2) ) ) ) ).

% Max.eq_fold
tff(fact_4725_partition_Osimps_I2_J,axiom,
    ! [A: $tType,Pa: fun(A,$o),Xa: A,Xsa: list(A)] : ( aa(list(A),product_prod(list(A),list(A)),aa(fun(A,$o),fun(list(A),product_prod(list(A),list(A))),partition(A),Pa),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa)) = aa(product_prod(list(A),list(A)),product_prod(list(A),list(A)),aa(fun(list(A),fun(list(A),product_prod(list(A),list(A)))),fun(product_prod(list(A),list(A)),product_prod(list(A),list(A))),product_case_prod(list(A),list(A),product_prod(list(A),list(A))),aa(A,fun(list(A),fun(list(A),product_prod(list(A),list(A)))),aTP_Lamp_ns(fun(A,$o),fun(A,fun(list(A),fun(list(A),product_prod(list(A),list(A))))),Pa),Xa)),aa(list(A),product_prod(list(A),list(A)),aa(fun(A,$o),fun(list(A),product_prod(list(A),list(A))),partition(A),Pa),Xsa)) ) ).

% partition.simps(2)
tff(fact_4726_partition__rev_Oelims,axiom,
    ! [A: $tType,Xa: fun(A,$o),Xaa: product_prod(list(A),list(A)),Xb: list(A),Y: product_prod(list(A),list(A))] :
      ( ( partition_rev(A,Xa,Xaa,Xb) = Y )
     => ( ! [Yes: list(A),No: list(A)] :
            ( ( Xaa = aa(list(A),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)] :
              ( ( Xaa = aa(list(A),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) )
             => ! [X: A,Xs: list(A)] :
                  ( ( Xb = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) )
                 => ( Y != partition_rev(A,Xa,
                        $ite(aa(A,$o,Xa,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),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),X),No))),
                        Xs) ) ) ) ) ) ).

% partition_rev.elims
tff(fact_4727_length__transpose,axiom,
    ! [A: $tType,Xsa: list(list(A))] : ( aa(list(list(A)),nat,size_size(list(list(A))),transpose(A,Xsa)) = aa(nat,nat,aa(list(list(A)),fun(nat,nat),aa(fun(list(A),fun(nat,nat)),fun(list(list(A)),fun(nat,nat)),foldr(list(A),nat),aTP_Lamp_nq(list(A),fun(nat,nat))),Xsa),zero_zero(nat)) ) ).

% length_transpose
tff(fact_4728_Max_Oinsert__remove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Aa2: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( lattic643756798349783984er_Max(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),Aa2)) = $ite(aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A)))) = bot_bot(set(A)),Xa,aa(A,A,aa(A,fun(A,A),ord_max(A),Xa),lattic643756798349783984er_Max(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A))))))) ) ) ) ).

% Max.insert_remove
tff(fact_4729_Max_Oremove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Aa2: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( member(A,Xa,Aa2)
           => ( lattic643756798349783984er_Max(A,Aa2) = $ite(aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A)))) = bot_bot(set(A)),Xa,aa(A,A,aa(A,fun(A,A),ord_max(A),Xa),lattic643756798349783984er_Max(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A))))))) ) ) ) ) ).

% Max.remove
tff(fact_4730_partition__set,axiom,
    ! [A: $tType,Pa: fun(A,$o),Xsa: list(A),Yes2: list(A),No2: list(A)] :
      ( ( aa(list(A),product_prod(list(A),list(A)),aa(fun(A,$o),fun(list(A),product_prod(list(A),list(A))),partition(A),Pa),Xsa) = aa(list(A),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),Xsa) ) ) ).

% partition_set
tff(fact_4731_transpose__aux__max,axiom,
    ! [A: $tType,B: $tType,Xsa: list(A),Xss2: list(list(B))] : ( aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,suc,aa(list(A),nat,size_size(list(A)),Xsa))),aa(nat,nat,aa(list(list(B)),fun(nat,nat),aa(fun(list(B),fun(nat,nat)),fun(list(list(B)),fun(nat,nat)),foldr(list(B),nat),aTP_Lamp_nt(list(B),fun(nat,nat))),Xss2),zero_zero(nat))) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(list(A),nat,size_size(list(A)),Xsa)),aa(nat,nat,aa(list(list(B)),fun(nat,nat),aa(fun(list(B),fun(nat,nat)),fun(list(list(B)),fun(nat,nat)),foldr(list(B),nat),aTP_Lamp_nu(list(B),fun(nat,nat))),aa(list(list(B)),list(list(B)),aa(fun(list(B),$o),fun(list(list(B)),list(list(B))),filter2(list(B)),aTP_Lamp_nv(list(B),$o)),Xss2)),zero_zero(nat)))) ) ).

% transpose_aux_max
tff(fact_4732_foldr__max__sorted,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xsa: list(A),Y: A] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),rev(A),Xsa))
         => ( aa(A,A,aa(list(A),fun(A,A),aa(fun(A,fun(A,A)),fun(list(A),fun(A,A)),foldr(A,A),ord_max(A)),Xsa),Y) = $ite(Xsa = nil(A),Y,aa(A,A,aa(A,fun(A,A),ord_max(A),aa(nat,A,nth(A,Xsa),zero_zero(nat))),Y)) ) ) ) ).

% foldr_max_sorted
tff(fact_4733_partition__rev_Opelims,axiom,
    ! [A: $tType,Xa: fun(A,$o),Xaa: product_prod(list(A),list(A)),Xb: list(A),Y: product_prod(list(A),list(A))] :
      ( ( partition_rev(A,Xa,Xaa,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,$o),product_prod(product_prod(list(A),list(A)),list(A))),Xa),aa(list(A),product_prod(product_prod(list(A),list(A)),list(A)),aa(product_prod(list(A),list(A)),fun(list(A),product_prod(product_prod(list(A),list(A)),list(A))),product_Pair(product_prod(list(A),list(A)),list(A)),Xaa),Xb)))
       => ( ! [Yes: list(A),No: list(A)] :
              ( ( Xaa = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes),No) )
             => ( ( Xb = nil(A) )
               => ( ( Y = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes),No) )
                 => ~ accp(product_prod(fun(A,$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))),Xa),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)] :
                ( ( Xaa = aa(list(A),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) )
               => ! [X: A,Xs: list(A)] :
                    ( ( Xb = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) )
                   => ( ( Y = partition_rev(A,Xa,
                            $ite(aa(A,$o,Xa,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),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),X),No))),
                            Xs) )
                     => ~ 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))),Xa),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),X),Xs)))) ) ) ) ) ) ) ).

% partition_rev.pelims
tff(fact_4734_quicksort__by__rel_Oelims,axiom,
    ! [A: $tType,Xa: fun(A,fun(A,$o)),Xaa: list(A),Xb: list(A),Y: list(A)] :
      ( ( aa(list(A),list(A),quicksort_by_rel(A,Xa,Xaa),Xb) = Y )
     => ( ( ( Xb = nil(A) )
         => ( Y != Xaa ) )
       => ~ ! [X: A,Xs: list(A)] :
              ( ( Xb = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) )
             => ( 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_nw(fun(A,fun(A,$o)),fun(list(A),fun(A,fun(list(A),fun(list(A),list(A))))),Xa),Xaa),X)),partition_rev(A,aa(A,fun(A,$o),aTP_Lamp_cn(fun(A,fun(A,$o)),fun(A,fun(A,$o)),Xa),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.elims
tff(fact_4735_quicksort__by__rel_Osimps_I2_J,axiom,
    ! [A: $tType,R: fun(A,fun(A,$o)),Sl2: list(A),Xa: A,Xsa: 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),Xa),Xsa)) = 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_nw(fun(A,fun(A,$o)),fun(list(A),fun(A,fun(list(A),fun(list(A),list(A))))),R),Sl2),Xa)),partition_rev(A,aa(A,fun(A,$o),aTP_Lamp_cn(fun(A,fun(A,$o)),fun(A,fun(A,$o)),R),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)),nil(A)),nil(A)),Xsa)) ) ).

% quicksort_by_rel.simps(2)
tff(fact_4736_set__quicksort__by__rel,axiom,
    ! [A: $tType,R: fun(A,fun(A,$o)),Sl2: list(A),Xsa: list(A)] : ( aa(list(A),set(A),set2(A),aa(list(A),list(A),quicksort_by_rel(A,R,Sl2),Xsa)) = aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),Sl2)) ) ).

% set_quicksort_by_rel
tff(fact_4737_quicksort__by__rel__permutes,axiom,
    ! [A: $tType,R: fun(A,fun(A,$o)),Sl2: list(A),Xsa: list(A)] : ( aa(list(A),multiset(A),mset(A),aa(list(A),list(A),quicksort_by_rel(A,R,Sl2),Xsa)) = aa(list(A),multiset(A),mset(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),Sl2)) ) ).

% quicksort_by_rel_permutes
tff(fact_4738_quicksort__by__rel_Osimps_I1_J,axiom,
    ! [A: $tType,R: fun(A,fun(A,$o)),Sl2: list(A)] : ( aa(list(A),list(A),quicksort_by_rel(A,R,Sl2),nil(A)) = Sl2 ) ).

% quicksort_by_rel.simps(1)
tff(fact_4739_quicksort__by__rel__remove__acc__guared,axiom,
    ! [A: $tType,Sl2: list(A),R: fun(A,fun(A,$o)),Xsa: list(A)] :
      ( ( Sl2 != nil(A) )
     => ( aa(list(A),list(A),quicksort_by_rel(A,R,Sl2),Xsa) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),quicksort_by_rel(A,R,nil(A)),Xsa)),Sl2) ) ) ).

% quicksort_by_rel_remove_acc_guared
tff(fact_4740_quicksort__by__rel__remove__acc,axiom,
    ! [A: $tType,R: fun(A,fun(A,$o)),Sl2: list(A),Xsa: list(A)] : ( aa(list(A),list(A),quicksort_by_rel(A,R,Sl2),Xsa) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),quicksort_by_rel(A,R,nil(A)),Xsa)),Sl2) ) ).

% quicksort_by_rel_remove_acc
tff(fact_4741_sorted__wrt__quicksort__by__rel,axiom,
    ! [A: $tType,R: fun(A,fun(A,$o)),Xsa: list(A)] :
      ( ! [X: A,Y2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),R,X),Y2)
          | aa(A,$o,aa(A,fun(A,$o),R,Y2),X) )
     => ( ! [X: A,Y2: A,Z3: A] :
            ( aa(A,$o,aa(A,fun(A,$o),R,X),Y2)
           => ( aa(A,$o,aa(A,fun(A,$o),R,Y2),Z3)
             => aa(A,$o,aa(A,fun(A,$o),R,X),Z3) ) )
       => sorted_wrt(A,R,aa(list(A),list(A),quicksort_by_rel(A,R,nil(A)),Xsa)) ) ) ).

% sorted_wrt_quicksort_by_rel
tff(fact_4742_sorted__quicksort__by__rel,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xsa: list(A)] : sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),quicksort_by_rel(A,ord_less_eq(A),nil(A)),Xsa)) ) ).

% sorted_quicksort_by_rel
tff(fact_4743_sort__quicksort__by__rel,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( aa(fun(A,A),fun(list(A),list(A)),linorder_sort_key(A,A),aTP_Lamp_ip(A,A)) = quicksort_by_rel(A,ord_less_eq(A),nil(A)) ) ) ).

% sort_quicksort_by_rel
tff(fact_4744_quicksort__by__rel_Opsimps_I2_J,axiom,
    ! [A: $tType,R: fun(A,fun(A,$o)),Sl2: list(A),Xa: A,Xsa: 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,$o)),product_prod(list(A),list(A))),R),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Sl2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa))))
     => ( aa(list(A),list(A),quicksort_by_rel(A,R,Sl2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa)) = aa(product_prod(list(A),list(A)),list(A),aa(fun(list(A),fun(list(A),list(A))),fun(product_prod(list(A),list(A)),list(A)),product_case_prod(list(A),list(A),list(A)),aa(A,fun(list(A),fun(list(A),list(A))),aa(list(A),fun(A,fun(list(A),fun(list(A),list(A)))),aTP_Lamp_nw(fun(A,fun(A,$o)),fun(list(A),fun(A,fun(list(A),fun(list(A),list(A))))),R),Sl2),Xa)),partition_rev(A,aa(A,fun(A,$o),aTP_Lamp_cn(fun(A,fun(A,$o)),fun(A,fun(A,$o)),R),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)),nil(A)),nil(A)),Xsa)) ) ) ).

% quicksort_by_rel.psimps(2)
tff(fact_4745_quicksort__by__rel_Opelims,axiom,
    ! [A: $tType,Xa: fun(A,fun(A,$o)),Xaa: list(A),Xb: list(A),Y: list(A)] :
      ( ( aa(list(A),list(A),quicksort_by_rel(A,Xa,Xaa),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,$o)),product_prod(list(A),list(A))),Xa),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xaa),Xb)))
       => ( ( ( Xb = nil(A) )
           => ( ( Y = Xaa )
             => ~ accp(product_prod(fun(A,fun(A,$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))),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)),Xaa),nil(A)))) ) )
         => ~ ! [X: A,Xs: list(A)] :
                ( ( Xb = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) )
               => ( ( 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_nw(fun(A,fun(A,$o)),fun(list(A),fun(A,fun(list(A),fun(list(A),list(A))))),Xa),Xaa),X)),partition_rev(A,aa(A,fun(A,$o),aTP_Lamp_cn(fun(A,fun(A,$o)),fun(A,fun(A,$o)),Xa),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)) )
                 => ~ 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))),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)),Xaa),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)))) ) ) ) ) ) ).

% quicksort_by_rel.pelims
tff(fact_4746_quicksort__by__rel_Opinduct,axiom,
    ! [A: $tType,A0: fun(A,fun(A,$o)),A1: list(A),A22: list(A),Pa: 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,$o)),product_prod(list(A),list(A))),A0),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),A1),A22)))
     => ( ! [R4: fun(A,fun(A,$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,$o)),product_prod(list(A),list(A))),R4),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Sl),nil(A))))
           => aa(list(A),$o,aa(list(A),fun(list(A),$o),aa(fun(A,fun(A,$o)),fun(list(A),fun(list(A),$o)),Pa,R4),Sl),nil(A)) )
       => ( ! [R4: fun(A,fun(A,$o)),Sl: 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,$o)),product_prod(list(A),list(A))),R4),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Sl),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs))))
             => ( ! [Xa4: product_prod(list(A),list(A)),Xb3: list(A),Y3: list(A)] :
                    ( ( Xa4 = partition_rev(A,aa(A,fun(A,$o),aTP_Lamp_cn(fun(A,fun(A,$o)),fun(A,fun(A,$o)),R4),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) )
                   => ( ( aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xb3),Y3) = Xa4 )
                     => aa(list(A),$o,aa(list(A),fun(list(A),$o),aa(fun(A,fun(A,$o)),fun(list(A),fun(list(A),$o)),Pa,R4),Sl),Y3) ) )
               => ( ! [Xa4: product_prod(list(A),list(A)),Xb3: list(A),Y3: list(A)] :
                      ( ( Xa4 = partition_rev(A,aa(A,fun(A,$o),aTP_Lamp_cn(fun(A,fun(A,$o)),fun(A,fun(A,$o)),R4),X),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),Xs) )
                     => ( ( aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xb3),Y3) = Xa4 )
                       => aa(list(A),$o,aa(list(A),fun(list(A),$o),aa(fun(A,fun(A,$o)),fun(list(A),fun(list(A),$o)),Pa,R4),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),aa(list(A),list(A),quicksort_by_rel(A,R4,Sl),Y3))),Xb3) ) )
                 => aa(list(A),$o,aa(list(A),fun(list(A),$o),aa(fun(A,fun(A,$o)),fun(list(A),fun(list(A),$o)),Pa,R4),Sl),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) ) ) )
         => aa(list(A),$o,aa(list(A),fun(list(A),$o),aa(fun(A,fun(A,$o)),fun(list(A),fun(list(A),$o)),Pa,A0),A1),A22) ) ) ) ).

% quicksort_by_rel.pinduct
tff(fact_4747_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_4748_set__nths,axiom,
    ! [A: $tType,Xsa: list(A),Ia2: set(nat)] : ( aa(list(A),set(A),set2(A),aa(set(nat),list(A),aa(list(A),fun(set(nat),list(A)),nths(A),Xsa),Ia2)) = aa(fun(A,$o),set(A),collect(A),aa(set(nat),fun(A,$o),aTP_Lamp_nx(list(A),fun(set(nat),fun(A,$o)),Xsa),Ia2)) ) ).

% set_nths
tff(fact_4749_sorted__list__of__set__def,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( linord4507533701916653071of_set(A) = linord144544945434240204of_set(A,A,aTP_Lamp_ip(A,A)) ) ) ).

% sorted_list_of_set_def
tff(fact_4750_finite__mono__remains__stable__implies__strict__prefix,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(nat,A)] :
          ( aa(set(A),$o,finite_finite2(A),aa(set(nat),set(A),image2(nat,A,F2),top_top(set(nat))))
         => ( order_mono(nat,A,F2)
           => ( ! [N5: nat] :
                  ( ( aa(nat,A,F2,N5) = aa(nat,A,F2,aa(nat,nat,suc,N5)) )
                 => ( aa(nat,A,F2,aa(nat,nat,suc,N5)) = aa(nat,A,F2,aa(nat,nat,suc,aa(nat,nat,suc,N5))) ) )
             => ? [N10: nat] :
                  ( ! [N8: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N8),N10)
                     => ! [M3: nat] :
                          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M3),N10)
                         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M3),N8)
                           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,F2,M3)),aa(nat,A,F2,N8)) ) ) )
                  & ! [N8: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N10),N8)
                     => ( aa(nat,A,F2,N10) = aa(nat,A,F2,N8) ) ) ) ) ) ) ) ).

% finite_mono_remains_stable_implies_strict_prefix
tff(fact_4751_nths__nil,axiom,
    ! [A: $tType,Aa2: set(nat)] : ( aa(set(nat),list(A),aa(list(A),fun(set(nat),list(A)),nths(A),nil(A)),Aa2) = nil(A) ) ).

% nths_nil
tff(fact_4752_nths__empty,axiom,
    ! [A: $tType,Xsa: list(A)] : ( aa(set(nat),list(A),aa(list(A),fun(set(nat),list(A)),nths(A),Xsa),bot_bot(set(nat))) = nil(A) ) ).

% nths_empty
tff(fact_4753_nths__upt__eq__take,axiom,
    ! [A: $tType,L: list(A),N: nat] : ( aa(set(nat),list(A),aa(list(A),fun(set(nat),list(A)),nths(A),L),aa(nat,set(nat),set_ord_lessThan(nat),N)) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),L) ) ).

% nths_upt_eq_take
tff(fact_4754_nths__singleton,axiom,
    ! [A: $tType,Xa: A,Aa2: set(nat)] :
      ( aa(set(nat),list(A),aa(list(A),fun(set(nat),list(A)),nths(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),nil(A))),Aa2) = $ite(member(nat,zero_zero(nat),Aa2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),nil(A)),nil(A)) ) ).

% nths_singleton
tff(fact_4755_mono__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F2: fun(A,B)] :
          ( order_mono(A,B,F2)
        <=> ! [X6: A,Y4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X6),Y4)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X6)),aa(A,B,F2,Y4)) ) ) ) ).

% mono_def
tff(fact_4756_monoI,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F2: fun(A,B)] :
          ( ! [X: A,Y2: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y2)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X)),aa(A,B,F2,Y2)) )
         => order_mono(A,B,F2) ) ) ).

% monoI
tff(fact_4757_monoE,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F2: fun(A,B),Xa: A,Y: A] :
          ( order_mono(A,B,F2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,Xa)),aa(A,B,F2,Y)) ) ) ) ).

% monoE
tff(fact_4758_monoD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F2: fun(A,B),Xa: A,Y: A] :
          ( order_mono(A,B,F2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,Xa)),aa(A,B,F2,Y)) ) ) ) ).

% monoD
tff(fact_4759_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_4760_distinct__nthsI,axiom,
    ! [A: $tType,Xsa: list(A),Ia2: set(nat)] :
      ( aa(list(A),$o,distinct(A),Xsa)
     => aa(list(A),$o,distinct(A),aa(set(nat),list(A),aa(list(A),fun(set(nat),list(A)),nths(A),Xsa),Ia2)) ) ).

% distinct_nthsI
tff(fact_4761_min__of__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & linorder(B) )
     => ! [F2: fun(A,B),M2: A,N: A] :
          ( order_mono(A,B,F2)
         => ( aa(B,B,aa(B,fun(B,B),ord_min(B),aa(A,B,F2,M2)),aa(A,B,F2,N)) = aa(A,B,F2,aa(A,A,aa(A,fun(A,A),ord_min(A),M2),N)) ) ) ) ).

% min_of_mono
tff(fact_4762_nths__map,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xsa: list(B),Ia2: set(nat)] : ( aa(set(nat),list(A),aa(list(A),fun(set(nat),list(A)),nths(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Xsa)),Ia2) = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),aa(set(nat),list(B),aa(list(B),fun(set(nat),list(B)),nths(B),Xsa),Ia2)) ) ).

% nths_map
tff(fact_4763_in__set__nthsD,axiom,
    ! [A: $tType,Xa: A,Xsa: list(A),Ia2: set(nat)] :
      ( member(A,Xa,aa(list(A),set(A),set2(A),aa(set(nat),list(A),aa(list(A),fun(set(nat),list(A)),nths(A),Xsa),Ia2)))
     => member(A,Xa,aa(list(A),set(A),set2(A),Xsa)) ) ).

% in_set_nthsD
tff(fact_4764_notin__set__nthsI,axiom,
    ! [A: $tType,Xa: A,Xsa: list(A),Ia2: set(nat)] :
      ( ~ member(A,Xa,aa(list(A),set(A),set2(A),Xsa))
     => ~ member(A,Xa,aa(list(A),set(A),set2(A),aa(set(nat),list(A),aa(list(A),fun(set(nat),list(A)),nths(A),Xsa),Ia2))) ) ).

% notin_set_nthsI
tff(fact_4765_mono__funpow,axiom,
    ! [A: $tType] :
      ( ( lattice(A)
        & order_bot(A) )
     => ! [Qa: fun(A,A)] :
          ( order_mono(A,A,Qa)
         => order_mono(nat,A,aTP_Lamp_ny(fun(A,A),fun(nat,A),Qa)) ) ) ).

% mono_funpow
tff(fact_4766_mono__strict__invE,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & order(B) )
     => ! [F2: fun(A,B),Xa: A,Y: A] :
          ( order_mono(A,B,F2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,Xa)),aa(A,B,F2,Y))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y) ) ) ) ).

% mono_strict_invE
tff(fact_4767_max__of__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & linorder(B) )
     => ! [F2: fun(A,B),M2: A,N: A] :
          ( order_mono(A,B,F2)
         => ( aa(B,B,aa(B,fun(B,B),ord_max(B),aa(A,B,F2,M2)),aa(A,B,F2,N)) = aa(A,B,F2,aa(A,A,aa(A,fun(A,A),ord_max(A),M2),N)) ) ) ) ).

% max_of_mono
tff(fact_4768_mono__invE,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & order(B) )
     => ! [F2: fun(A,B),Xa: A,Y: A] :
          ( order_mono(A,B,F2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,Xa)),aa(A,B,F2,Y))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y) ) ) ) ).

% mono_invE
tff(fact_4769_mono__inf,axiom,
    ! [B: $tType,A: $tType] :
      ( ( semilattice_inf(A)
        & semilattice_inf(B) )
     => ! [F2: fun(A,B),Aa2: A,B2: A] :
          ( order_mono(A,B,F2)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,aa(A,A,aa(A,fun(A,A),inf_inf(A),Aa2),B2))),aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(A,B,F2,Aa2)),aa(A,B,F2,B2))) ) ) ).

% mono_inf
tff(fact_4770_mono__sup,axiom,
    ! [B: $tType,A: $tType] :
      ( ( semilattice_sup(A)
        & semilattice_sup(B) )
     => ! [F2: fun(A,B),Aa2: A,B2: A] :
          ( order_mono(A,B,F2)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),sup_sup(B),aa(A,B,F2,Aa2)),aa(A,B,F2,B2))),aa(A,B,F2,aa(A,A,aa(A,fun(A,A),sup_sup(A),Aa2),B2))) ) ) ).

% mono_sup
tff(fact_4771_set__nths__subset,axiom,
    ! [A: $tType,Xsa: list(A),Ia2: set(nat)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),aa(set(nat),list(A),aa(list(A),fun(set(nat),list(A)),nths(A),Xsa),Ia2))),aa(list(A),set(A),set2(A),Xsa)) ).

% set_nths_subset
tff(fact_4772_nths__all,axiom,
    ! [A: $tType,Xsa: list(A),Ia2: set(nat)] :
      ( ! [I3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),Xsa))
         => member(nat,I3,Ia2) )
     => ( aa(set(nat),list(A),aa(list(A),fun(set(nat),list(A)),nths(A),Xsa),Ia2) = Xsa ) ) ).

% nths_all
tff(fact_4773_sorted__nths,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xsa: list(A),Ia2: set(nat)] :
          ( sorted_wrt(A,ord_less_eq(A),Xsa)
         => sorted_wrt(A,ord_less_eq(A),aa(set(nat),list(A),aa(list(A),fun(set(nat),list(A)),nths(A),Xsa),Ia2)) ) ) ).

% sorted_nths
tff(fact_4774_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_4775_mono__times__nat,axiom,
    ! [N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
     => order_mono(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N)) ) ).

% mono_times_nat
tff(fact_4776_mono__image__least,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [F2: fun(A,B),M2: A,N: A,M8: B,N6: B] :
          ( order_mono(A,B,F2)
         => ( ( aa(set(A),set(B),image2(A,B,F2),set_or7035219750837199246ssThan(A,M2,N)) = set_or7035219750837199246ssThan(B,M8,N6) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),M2),N)
             => ( aa(A,B,F2,M2) = M8 ) ) ) ) ) ).

% mono_image_least
tff(fact_4777_Kleene__iter__lpfp,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [F2: fun(A,A),P2: A,K3: nat] :
          ( order_mono(A,A,F2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,F2,P2)),P2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,compow(fun(A,A),K3,F2),bot_bot(A))),P2) ) ) ) ).

% Kleene_iter_lpfp
tff(fact_4778_drop__eq__nths,axiom,
    ! [A: $tType,N: nat,Xsa: list(A)] : ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),Xsa) = aa(set(nat),list(A),aa(list(A),fun(set(nat),list(A)),nths(A),Xsa),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),ord_less_eq(nat),N))) ) ).

% drop_eq_nths
tff(fact_4779_funpow__decreasing,axiom,
    ! [A: $tType] :
      ( ( lattice(A)
        & order_bot(A) )
     => ! [M2: nat,N: nat,F2: fun(A,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),N)
         => ( order_mono(A,A,F2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,compow(fun(A,A),M2,F2),bot_bot(A))),aa(A,A,compow(fun(A,A),N,F2),bot_bot(A))) ) ) ) ).

% funpow_decreasing
tff(fact_4780_mono__Max__commute,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & linorder(B) )
     => ! [F2: fun(A,B),Aa2: set(A)] :
          ( order_mono(A,B,F2)
         => ( aa(set(A),$o,finite_finite2(A),Aa2)
           => ( ( Aa2 != bot_bot(set(A)) )
             => ( aa(A,B,F2,lattic643756798349783984er_Max(A,Aa2)) = lattic643756798349783984er_Max(B,aa(set(A),set(B),image2(A,B,F2),Aa2)) ) ) ) ) ) ).

% mono_Max_commute
tff(fact_4781_mono__Min__commute,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & linorder(B) )
     => ! [F2: fun(A,B),Aa2: set(A)] :
          ( order_mono(A,B,F2)
         => ( aa(set(A),$o,finite_finite2(A),Aa2)
           => ( ( Aa2 != bot_bot(set(A)) )
             => ( aa(A,B,F2,lattic643756798350308766er_Min(A,Aa2)) = lattic643756798350308766er_Min(B,aa(set(A),set(B),image2(A,B,F2),Aa2)) ) ) ) ) ) ).

% mono_Min_commute
tff(fact_4782_nths__append,axiom,
    ! [A: $tType,L: list(A),L2: list(A),Aa2: set(nat)] : ( aa(set(nat),list(A),aa(list(A),fun(set(nat),list(A)),nths(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L),L2)),Aa2) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(set(nat),list(A),aa(list(A),fun(set(nat),list(A)),nths(A),L),Aa2)),aa(set(nat),list(A),aa(list(A),fun(set(nat),list(A)),nths(A),L2),aa(fun(nat,$o),set(nat),collect(nat),aa(set(nat),fun(nat,$o),aTP_Lamp_nz(list(A),fun(set(nat),fun(nat,$o)),L),Aa2)))) ) ).

% nths_append
tff(fact_4783_nths__drop,axiom,
    ! [A: $tType,N: nat,Xsa: list(A),Ia2: set(nat)] : ( aa(set(nat),list(A),aa(list(A),fun(set(nat),list(A)),nths(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),Xsa)),Ia2) = aa(set(nat),list(A),aa(list(A),fun(set(nat),list(A)),nths(A),Xsa),aa(set(nat),set(nat),image2(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N)),Ia2)) ) ).

% nths_drop
tff(fact_4784_filter__in__nths,axiom,
    ! [A: $tType,Xsa: list(A),S2: set(nat)] :
      ( aa(list(A),$o,distinct(A),Xsa)
     => ( aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),aa(set(nat),fun(A,$o),aTP_Lamp_oa(list(A),fun(set(nat),fun(A,$o)),Xsa),S2)),Xsa) = aa(set(nat),list(A),aa(list(A),fun(set(nat),list(A)),nths(A),Xsa),S2) ) ) ).

% filter_in_nths
tff(fact_4785_length__nths,axiom,
    ! [A: $tType,Xsa: list(A),Ia2: set(nat)] : ( aa(list(A),nat,size_size(list(A)),aa(set(nat),list(A),aa(list(A),fun(set(nat),list(A)),nths(A),Xsa),Ia2)) = finite_card(nat,aa(fun(nat,$o),set(nat),collect(nat),aa(set(nat),fun(nat,$o),aTP_Lamp_ob(list(A),fun(set(nat),fun(nat,$o)),Xsa),Ia2))) ) ).

% length_nths
tff(fact_4786_filter__eq__nths,axiom,
    ! [A: $tType,Pa: fun(A,$o),Xsa: list(A)] : ( aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),Pa),Xsa) = aa(set(nat),list(A),aa(list(A),fun(set(nat),list(A)),nths(A),Xsa),aa(fun(nat,$o),set(nat),collect(nat),aa(list(A),fun(nat,$o),aTP_Lamp_jk(fun(A,$o),fun(list(A),fun(nat,$o)),Pa),Xsa))) ) ).

% filter_eq_nths
tff(fact_4787_nths__Cons,axiom,
    ! [A: $tType,Xa: A,L: list(A),Aa2: set(nat)] :
      ( aa(set(nat),list(A),aa(list(A),fun(set(nat),list(A)),nths(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),L)),Aa2) = aa(list(A),list(A),
          aa(list(A),fun(list(A),list(A)),append(A),
            $ite(member(nat,zero_zero(nat),Aa2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),nil(A)),nil(A))),
          aa(set(nat),list(A),aa(list(A),fun(set(nat),list(A)),nths(A),L),aa(fun(nat,$o),set(nat),collect(nat),aTP_Lamp_oc(set(nat),fun(nat,$o),Aa2)))) ) ).

% nths_Cons
tff(fact_4788_remdups__adj__altdef,axiom,
    ! [A: $tType,Xsa: list(A),Ys: list(A)] :
      ( ( aa(list(A),list(A),remdups_adj(A),Xsa) = Ys )
    <=> ? [F9: fun(nat,nat)] :
          ( order_mono(nat,nat,F9)
          & ( aa(set(nat),set(nat),image2(nat,nat,F9),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xsa))) = set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(A),nat,size_size(list(A)),Ys)) )
          & ! [I2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xsa))
             => ( aa(nat,A,nth(A,Xsa),I2) = aa(nat,A,nth(A,Ys),aa(nat,nat,F9,I2)) ) )
          & ! [I2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),one_one(nat))),aa(list(A),nat,size_size(list(A)),Xsa))
             => ( ( aa(nat,A,nth(A,Xsa),I2) = aa(nat,A,nth(A,Xsa),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),one_one(nat))) )
              <=> ( aa(nat,nat,F9,I2) = aa(nat,nat,F9,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),one_one(nat))) ) ) ) ) ) ).

% remdups_adj_altdef
tff(fact_4789_enumerate__replicate__eq,axiom,
    ! [A: $tType,N: nat,M2: nat,A3: A] : ( enumerate(A,N,aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),M2),A3)) = aa(list(nat),list(product_prod(nat,A)),aa(fun(nat,product_prod(nat,A)),fun(list(nat),list(product_prod(nat,A))),map(nat,product_prod(nat,A)),aTP_Lamp_od(A,fun(nat,product_prod(nat,A)),A3)),upt(N,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M2))) ) ).

% enumerate_replicate_eq
tff(fact_4790_finite__subsets__at__top__finite,axiom,
    ! [A: $tType,Aa2: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),Aa2)
     => ( finite5375528669736107172at_top(A,Aa2) = aa(set(set(A)),filter(set(A)),principal(set(A)),aa(set(set(A)),set(set(A)),aa(set(A),fun(set(set(A)),set(set(A))),insert3(set(A)),Aa2),bot_bot(set(set(A))))) ) ) ).

% finite_subsets_at_top_finite
tff(fact_4791_replicate__eq__replicate,axiom,
    ! [A: $tType,M2: nat,Xa: A,N: nat,Y: A] :
      ( ( aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),M2),Xa) = aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),Y) )
    <=> ( ( M2 = N )
        & ( ( M2 != zero_zero(nat) )
         => ( Xa = Y ) ) ) ) ).

% replicate_eq_replicate
tff(fact_4792_length__replicate,axiom,
    ! [A: $tType,N: nat,Xa: A] : ( aa(list(A),nat,size_size(list(A)),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),Xa)) = N ) ).

% length_replicate
tff(fact_4793_concat__replicate__trivial,axiom,
    ! [A: $tType,Ia: nat] : ( aa(list(list(A)),list(A),concat(A),aa(list(A),list(list(A)),aa(nat,fun(list(A),list(list(A))),replicate(list(A)),Ia),nil(A))) = nil(A) ) ).

% concat_replicate_trivial
tff(fact_4794_map__replicate,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),N: nat,Xa: B] : ( aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),aa(B,list(B),aa(nat,fun(B,list(B)),replicate(B),N),Xa)) = aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),aa(B,A,F2,Xa)) ) ).

% map_replicate
tff(fact_4795_remdups__adj__Nil__iff,axiom,
    ! [A: $tType,Xsa: list(A)] :
      ( ( aa(list(A),list(A),remdups_adj(A),Xsa) = nil(A) )
    <=> ( Xsa = nil(A) ) ) ).

% remdups_adj_Nil_iff
tff(fact_4796_remdups__adj__set,axiom,
    ! [A: $tType,Xsa: list(A)] : ( aa(list(A),set(A),set2(A),aa(list(A),list(A),remdups_adj(A),Xsa)) = aa(list(A),set(A),set2(A),Xsa) ) ).

% remdups_adj_set
tff(fact_4797_rev__replicate,axiom,
    ! [A: $tType,N: nat,Xa: A] : ( aa(list(A),list(A),rev(A),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),Xa)) = aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),Xa) ) ).

% rev_replicate
tff(fact_4798_remdups__adj__rev,axiom,
    ! [A: $tType,Xsa: list(A)] : ( aa(list(A),list(A),remdups_adj(A),aa(list(A),list(A),rev(A),Xsa)) = aa(list(A),list(A),rev(A),aa(list(A),list(A),remdups_adj(A),Xsa)) ) ).

% remdups_adj_rev
tff(fact_4799_hd__remdups__adj,axiom,
    ! [A: $tType,Xsa: list(A)] : ( aa(list(A),A,hd(A),aa(list(A),list(A),remdups_adj(A),Xsa)) = aa(list(A),A,hd(A),Xsa) ) ).

% hd_remdups_adj
tff(fact_4800_last__remdups__adj,axiom,
    ! [A: $tType,Xsa: list(A)] : ( last(A,aa(list(A),list(A),remdups_adj(A),Xsa)) = last(A,Xsa) ) ).

% last_remdups_adj
tff(fact_4801_replicate__empty,axiom,
    ! [A: $tType,N: nat,Xa: A] :
      ( ( aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),Xa) = nil(A) )
    <=> ( N = zero_zero(nat) ) ) ).

% replicate_empty
tff(fact_4802_empty__replicate,axiom,
    ! [A: $tType,N: nat,Xa: A] :
      ( ( nil(A) = aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),Xa) )
    <=> ( N = zero_zero(nat) ) ) ).

% empty_replicate
tff(fact_4803_in__set__replicate,axiom,
    ! [A: $tType,Xa: A,N: nat,Y: A] :
      ( member(A,Xa,aa(list(A),set(A),set2(A),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),Y)))
    <=> ( ( Xa = Y )
        & ( N != zero_zero(nat) ) ) ) ).

% in_set_replicate
tff(fact_4804_Bex__set__replicate,axiom,
    ! [A: $tType,N: nat,A3: A,Pa: fun(A,$o)] :
      ( ? [X6: A] :
          ( member(A,X6,aa(list(A),set(A),set2(A),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),A3)))
          & aa(A,$o,Pa,X6) )
    <=> ( aa(A,$o,Pa,A3)
        & ( N != zero_zero(nat) ) ) ) ).

% Bex_set_replicate
tff(fact_4805_Ball__set__replicate,axiom,
    ! [A: $tType,N: nat,A3: A,Pa: fun(A,$o)] :
      ( ! [X6: A] :
          ( member(A,X6,aa(list(A),set(A),set2(A),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),A3)))
         => aa(A,$o,Pa,X6) )
    <=> ( aa(A,$o,Pa,A3)
        | ( N = zero_zero(nat) ) ) ) ).

% Ball_set_replicate
tff(fact_4806_nth__replicate,axiom,
    ! [A: $tType,Ia: nat,N: nat,Xa: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),N)
     => ( aa(nat,A,nth(A,aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),Xa)),Ia) = Xa ) ) ).

% nth_replicate
tff(fact_4807_hd__replicate,axiom,
    ! [A: $tType,N: nat,Xa: A] :
      ( ( N != zero_zero(nat) )
     => ( aa(list(A),A,hd(A),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),Xa)) = Xa ) ) ).

% hd_replicate
tff(fact_4808_drop__replicate,axiom,
    ! [A: $tType,Ia: nat,K3: nat,Xa: A] : ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),Ia),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),K3),Xa)) = aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K3),Ia)),Xa) ) ).

% drop_replicate
tff(fact_4809_takeWhile__replicate,axiom,
    ! [A: $tType,Pa: fun(A,$o),N: nat,Xa: A] :
      ( aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),takeWhile(A),Pa),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),Xa)) = $ite(aa(A,$o,Pa,Xa),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),Xa),nil(A)) ) ).

% takeWhile_replicate
tff(fact_4810_remdups__adj__Cons__alt,axiom,
    ! [A: $tType,Xa: A,Xsa: list(A)] : ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),aa(list(A),list(A),tl(A),aa(list(A),list(A),remdups_adj(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa)))) = aa(list(A),list(A),remdups_adj(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa)) ) ).

% remdups_adj_Cons_alt
tff(fact_4811_last__replicate,axiom,
    ! [A: $tType,N: nat,Xa: A] :
      ( ( N != zero_zero(nat) )
     => ( last(A,aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),Xa)) = Xa ) ) ).

% last_replicate
tff(fact_4812_foldr__replicate,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,fun(A,A)),N: nat,Xa: B] : ( aa(list(B),fun(A,A),aa(fun(B,fun(A,A)),fun(list(B),fun(A,A)),foldr(B,A),F2),aa(B,list(B),aa(nat,fun(B,list(B)),replicate(B),N),Xa)) = compow(fun(A,A),N,aa(B,fun(A,A),F2,Xa)) ) ).

% foldr_replicate
tff(fact_4813_take__replicate,axiom,
    ! [A: $tType,Ia: nat,K3: nat,Xa: A] : ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),Ia),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),K3),Xa)) = aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),Ia),K3)),Xa) ) ).

% take_replicate
tff(fact_4814_tl__replicate,axiom,
    ! [A: $tType,N: nat,Xa: A] : ( aa(list(A),list(A),tl(A),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),Xa)) = aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))),Xa) ) ).

% tl_replicate
tff(fact_4815_zip__replicate,axiom,
    ! [A: $tType,B: $tType,Ia: nat,Xa: A,J: nat,Y: B] : ( aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),Ia),Xa)),aa(B,list(B),aa(nat,fun(B,list(B)),replicate(B),J),Y)) = aa(product_prod(A,B),list(product_prod(A,B)),aa(nat,fun(product_prod(A,B),list(product_prod(A,B))),replicate(product_prod(A,B)),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),Ia),J)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa),Y)) ) ).

% zip_replicate
tff(fact_4816_set__replicate,axiom,
    ! [A: $tType,N: nat,Xa: A] :
      ( ( N != zero_zero(nat) )
     => ( aa(list(A),set(A),set2(A),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),Xa)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A))) ) ) ).

% set_replicate
tff(fact_4817_finite__subsets__at__top__neq__bot,axiom,
    ! [A: $tType,Aa2: set(A)] : ( finite5375528669736107172at_top(A,Aa2) != bot_bot(filter(set(A))) ) ).

% finite_subsets_at_top_neq_bot
tff(fact_4818_remdups__adj__replicate,axiom,
    ! [A: $tType,N: nat,Xa: A] :
      ( aa(list(A),list(A),remdups_adj(A),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),Xa)) = $ite(N = zero_zero(nat),nil(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),nil(A))) ) ).

% remdups_adj_replicate
tff(fact_4819_append__replicate__commute,axiom,
    ! [A: $tType,N: nat,Xa: A,K3: nat] : ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),Xa)),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),K3),Xa)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),K3),Xa)),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),Xa)) ) ).

% append_replicate_commute
tff(fact_4820_remdups__adj_Osimps_I1_J,axiom,
    ! [A: $tType] : ( aa(list(A),list(A),remdups_adj(A),nil(A)) = nil(A) ) ).

% remdups_adj.simps(1)
tff(fact_4821_remdups__adj__singleton,axiom,
    ! [A: $tType,Xsa: list(A),Xa: A] :
      ( ( aa(list(A),list(A),remdups_adj(A),Xsa) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),nil(A)) )
     => ( Xsa = aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),aa(list(A),nat,size_size(list(A)),Xsa)),Xa) ) ) ).

% remdups_adj_singleton
tff(fact_4822_remdups__adj_Osimps_I3_J,axiom,
    ! [A: $tType,Xa: A,Y: A,Xsa: list(A)] :
      ( aa(list(A),list(A),remdups_adj(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Xsa))) = $ite(Xa = Y,aa(list(A),list(A),remdups_adj(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),aa(list(A),list(A),remdups_adj(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Xsa)))) ) ).

% remdups_adj.simps(3)
tff(fact_4823_remdups__adj__distinct,axiom,
    ! [A: $tType,Xsa: list(A)] :
      ( aa(list(A),$o,distinct(A),Xsa)
     => ( aa(list(A),list(A),remdups_adj(A),Xsa) = Xsa ) ) ).

% remdups_adj_distinct
tff(fact_4824_mono__Int,axiom,
    ! [B: $tType,A: $tType,F2: fun(set(A),set(B)),Aa2: set(A),B2: set(A)] :
      ( order_mono(set(A),set(B),F2)
     => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),F2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),B2))),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(set(A),set(B),F2,Aa2)),aa(set(A),set(B),F2,B2))) ) ).

% mono_Int
tff(fact_4825_mono__Un,axiom,
    ! [B: $tType,A: $tType,F2: fun(set(A),set(B)),Aa2: set(A),B2: set(A)] :
      ( order_mono(set(A),set(B),F2)
     => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),aa(set(A),set(B),F2,Aa2)),aa(set(A),set(B),F2,B2))),aa(set(A),set(B),F2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),B2))) ) ).

% mono_Un
tff(fact_4826_replicate__Suc,axiom,
    ! [A: $tType,N: nat,Xa: A] : ( aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),aa(nat,nat,suc,N)),Xa) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),Xa)) ) ).

% replicate_Suc
tff(fact_4827_replicate__0,axiom,
    ! [A: $tType,Xa: A] : ( aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),zero_zero(nat)),Xa) = nil(A) ) ).

% replicate_0
tff(fact_4828_replicate__app__Cons__same,axiom,
    ! [A: $tType,N: nat,Xa: A,Xsa: list(A)] : ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),Xa)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),Xa)),Xsa)) ) ).

% replicate_app_Cons_same
tff(fact_4829_replicate__length__same,axiom,
    ! [A: $tType,Xsa: list(A),Xa: A] :
      ( ! [X: A] :
          ( member(A,X,aa(list(A),set(A),set2(A),Xsa))
         => ( X = Xa ) )
     => ( aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),aa(list(A),nat,size_size(list(A)),Xsa)),Xa) = Xsa ) ) ).

% replicate_length_same
tff(fact_4830_replicate__eqI,axiom,
    ! [A: $tType,Xsa: list(A),N: nat,Xa: A] :
      ( ( aa(list(A),nat,size_size(list(A)),Xsa) = N )
     => ( ! [Y2: A] :
            ( member(A,Y2,aa(list(A),set(A),set2(A),Xsa))
           => ( Y2 = Xa ) )
       => ( Xsa = aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),Xa) ) ) ) ).

% replicate_eqI
tff(fact_4831_sorted__replicate,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [N: nat,Xa: A] : sorted_wrt(A,ord_less_eq(A),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),Xa)) ) ).

% sorted_replicate
tff(fact_4832_replicate__add,axiom,
    ! [A: $tType,N: nat,M2: nat,Xa: A] : ( aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M2)),Xa) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),Xa)),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),M2),Xa)) ) ).

% replicate_add
tff(fact_4833_filter__replicate,axiom,
    ! [A: $tType,Pa: fun(A,$o),N: nat,Xa: A] :
      ( aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),Pa),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),Xa)) = $ite(aa(A,$o,Pa,Xa),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),Xa),nil(A)) ) ).

% filter_replicate
tff(fact_4834_comm__append__are__replicate,axiom,
    ! [A: $tType,Xsa: list(A),Ys: list(A)] :
      ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),Ys) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Xsa) )
     => ? [M6: nat,N5: nat,Zs2: list(A)] :
          ( ( aa(list(list(A)),list(A),concat(A),aa(list(A),list(list(A)),aa(nat,fun(list(A),list(list(A))),replicate(list(A)),M6),Zs2)) = Xsa )
          & ( aa(list(list(A)),list(A),concat(A),aa(list(A),list(list(A)),aa(nat,fun(list(A),list(list(A))),replicate(list(A)),N5),Zs2)) = Ys ) ) ) ).

% comm_append_are_replicate
tff(fact_4835_remdups__adj_Oelims,axiom,
    ! [A: $tType,Xa: list(A),Y: list(A)] :
      ( ( aa(list(A),list(A),remdups_adj(A),Xa) = Y )
     => ( ( ( Xa = nil(A) )
         => ( Y != nil(A) ) )
       => ( ! [X: A] :
              ( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)) )
             => ( Y != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)) ) )
         => ~ ! [X: A,Y2: A,Xs: list(A)] :
                ( ( Xa = 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),Y2),Xs)) )
               => ( Y != $ite(X = Y2,aa(list(A),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),aa(list(A),list(A),remdups_adj(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Xs)))) ) ) ) ) ) ).

% remdups_adj.elims
tff(fact_4836_remdups__adj_Osimps_I2_J,axiom,
    ! [A: $tType,Xa: A] : ( aa(list(A),list(A),remdups_adj(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),nil(A))) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),nil(A)) ) ).

% remdups_adj.simps(2)
tff(fact_4837_remdups__adj__length,axiom,
    ! [A: $tType,Xsa: list(A)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),remdups_adj(A),Xsa))),aa(list(A),nat,size_size(list(A)),Xsa)) ).

% remdups_adj_length
tff(fact_4838_sorted__remdups__adj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xsa: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),Xsa)
         => sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),remdups_adj(A),Xsa)) ) ) ).

% sorted_remdups_adj
tff(fact_4839_map__replicate__const,axiom,
    ! [B: $tType,A: $tType,K3: A,Lst: list(B)] : ( aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),aTP_Lamp_as(A,fun(B,A),K3)),Lst) = aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),aa(list(B),nat,size_size(list(B)),Lst)),K3) ) ).

% map_replicate_const
tff(fact_4840_replicate__length__filter,axiom,
    ! [A: $tType,Xa: A,Xsa: list(A)] : ( aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),aa(A,fun(A,$o),fequal(A),Xa)),Xsa))),Xa) = aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),aa(A,fun(A,$o),fequal(A),Xa)),Xsa) ) ).

% replicate_length_filter
tff(fact_4841_remdups__adj__singleton__iff,axiom,
    ! [A: $tType,Xsa: list(A)] :
      ( ( aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),remdups_adj(A),Xsa)) = aa(nat,nat,suc,zero_zero(nat)) )
    <=> ( ( Xsa != nil(A) )
        & ( Xsa = aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),aa(list(A),nat,size_size(list(A)),Xsa)),aa(list(A),A,hd(A),Xsa)) ) ) ) ).

% remdups_adj_singleton_iff
tff(fact_4842_replicate__append__same,axiom,
    ! [A: $tType,Ia: nat,Xa: A] : ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),Ia),Xa)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),nil(A))) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),Ia),Xa)) ) ).

% replicate_append_same
tff(fact_4843_sum__list__replicate,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [N: nat,C2: A] : ( groups8242544230860333062m_list(A,aa(A,list(A),aa(nat,fun(A,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_4844_remdups__adj__append__two,axiom,
    ! [A: $tType,Xsa: list(A),Xa: A,Y: A] :
      ( aa(list(A),list(A),remdups_adj(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),nil(A))))) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),remdups_adj(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),nil(A))))),
          $ite(Xa = Y,nil(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),nil(A)))) ) ).

% remdups_adj_append_two
tff(fact_4845_map__replicate__trivial,axiom,
    ! [A: $tType,Xa: A,Ia: nat] : ( aa(list(nat),list(A),aa(fun(nat,A),fun(list(nat),list(A)),map(nat,A),aTP_Lamp_oe(A,fun(nat,A),Xa)),upt(zero_zero(nat),Ia)) = aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),Ia),Xa) ) ).

% map_replicate_trivial
tff(fact_4846_remdups__adj__Cons,axiom,
    ! [A: $tType,Xa: A,Xsa: list(A)] : ( aa(list(A),list(A),remdups_adj(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa)) = aa(list(A),list(A),aa(fun(A,fun(list(A),list(A))),fun(list(A),list(A)),aa(list(A),fun(fun(A,fun(list(A),list(A))),fun(list(A),list(A))),case_list(list(A),A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),nil(A))),aTP_Lamp_of(A,fun(A,fun(list(A),list(A))),Xa)),aa(list(A),list(A),remdups_adj(A),Xsa)) ) ).

% remdups_adj_Cons
tff(fact_4847_set__replicate__conv__if,axiom,
    ! [A: $tType,N: nat,Xa: A] :
      ( aa(list(A),set(A),set2(A),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),Xa)) = $ite(N = zero_zero(nat),bot_bot(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A)))) ) ).

% set_replicate_conv_if
tff(fact_4848_set__replicate__Suc,axiom,
    ! [A: $tType,N: nat,Xa: A] : ( aa(list(A),set(A),set2(A),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),aa(nat,nat,suc,N)),Xa)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A))) ) ).

% set_replicate_Suc
tff(fact_4849_replicate__Suc__conv__snoc,axiom,
    ! [A: $tType,N: nat,Xa: A] : ( aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),aa(nat,nat,suc,N)),Xa) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),Xa)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),nil(A))) ) ).

% replicate_Suc_conv_snoc
tff(fact_4850_zip__replicate1,axiom,
    ! [A: $tType,B: $tType,N: nat,Xa: A,Ys: list(B)] : ( aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),Xa)),Ys) = aa(list(B),list(product_prod(A,B)),aa(fun(B,product_prod(A,B)),fun(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),Xa)),aa(list(B),list(B),aa(nat,fun(list(B),list(B)),take(B),N),Ys)) ) ).

% zip_replicate1
tff(fact_4851_remdups__adj__adjacent,axiom,
    ! [A: $tType,Ia: nat,Xsa: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,Ia)),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),remdups_adj(A),Xsa)))
     => ( aa(nat,A,nth(A,aa(list(A),list(A),remdups_adj(A),Xsa)),Ia) != aa(nat,A,nth(A,aa(list(A),list(A),remdups_adj(A),Xsa)),aa(nat,nat,suc,Ia)) ) ) ).

% remdups_adj_adjacent
tff(fact_4852_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_og(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_4853_remdups__adj__append,axiom,
    ! [A: $tType,Xs_1: list(A),Xa: A,Xs_2: list(A)] : ( aa(list(A),list(A),remdups_adj(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs_1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xs_2))) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),remdups_adj(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs_1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),nil(A))))),aa(list(A),list(A),tl(A),aa(list(A),list(A),remdups_adj(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xs_2)))) ) ).

% remdups_adj_append
tff(fact_4854_finite_Omono,axiom,
    ! [A: $tType] : order_mono(fun(set(A),$o),fun(set(A),$o),aTP_Lamp_oh(fun(set(A),$o),fun(set(A),$o))) ).

% finite.mono
tff(fact_4855_map__zip2,axiom,
    ! [A: $tType,B: $tType,K3: A,L: list(B)] : ( aa(list(B),list(product_prod(A,B)),aa(fun(B,product_prod(A,B)),fun(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),K3)),L) = aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),aa(list(B),nat,size_size(list(B)),L)),K3)),L) ) ).

% map_zip2
tff(fact_4856_map__zip1,axiom,
    ! [A: $tType,B: $tType,K3: B,L: list(A)] : ( aa(list(A),list(product_prod(A,B)),aa(fun(A,product_prod(A,B)),fun(list(A),list(product_prod(A,B))),map(A,product_prod(A,B)),aa(B,fun(A,product_prod(A,B)),aTP_Lamp_io(B,fun(A,product_prod(A,B))),K3)),L) = aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),L),aa(B,list(B),aa(nat,fun(B,list(B)),replicate(B),aa(list(A),nat,size_size(list(A)),L)),K3)) ) ).

% map_zip1
tff(fact_4857_remdups__adj__append_H,axiom,
    ! [A: $tType,Xsa: list(A),Ys: list(A)] :
      ( ( ( Xsa = nil(A) )
        | ( Ys = nil(A) )
        | ( last(A,Xsa) != aa(list(A),A,hd(A),Ys) ) )
     => ( aa(list(A),list(A),remdups_adj(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),Ys)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),remdups_adj(A),Xsa)),aa(list(A),list(A),remdups_adj(A),Ys)) ) ) ).

% remdups_adj_append'
tff(fact_4858_zip__replicate2,axiom,
    ! [B: $tType,A: $tType,Xsa: list(A),N: nat,Y: B] : ( aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),Xsa),aa(B,list(B),aa(nat,fun(B,list(B)),replicate(B),N),Y)) = aa(list(A),list(product_prod(A,B)),aa(fun(A,product_prod(A,B)),fun(list(A),list(product_prod(A,B))),map(A,product_prod(A,B)),aa(B,fun(A,product_prod(A,B)),aTP_Lamp_io(B,fun(A,product_prod(A,B))),Y)),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),Xsa)) ) ).

% zip_replicate2
tff(fact_4859_finite__subsets__at__top__def,axiom,
    ! [A: $tType,Aa2: set(A)] : ( finite5375528669736107172at_top(A,Aa2) = aa(set(filter(set(A))),filter(set(A)),complete_Inf_Inf(filter(set(A))),aa(set(set(A)),set(filter(set(A))),image2(set(A),filter(set(A)),aTP_Lamp_oj(set(A),fun(set(A),filter(set(A))),Aa2)),aa(fun(set(A),$o),set(set(A)),collect(set(A)),aTP_Lamp_ok(set(A),fun(set(A),$o),Aa2)))) ) ).

% finite_subsets_at_top_def
tff(fact_4860_Cons__replicate__eq,axiom,
    ! [A: $tType,Xa: A,Xsa: list(A),N: nat,Y: A] :
      ( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa) = aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),Y) )
    <=> ( ( Xa = Y )
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
        & ( Xsa = aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))),Xa) ) ) ) ).

% Cons_replicate_eq
tff(fact_4861_comm__append__is__replicate,axiom,
    ! [A: $tType,Xsa: list(A),Ys: list(A)] :
      ( ( Xsa != nil(A) )
     => ( ( Ys != nil(A) )
       => ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),Ys) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Xsa) )
         => ? [N5: nat,Zs2: list(A)] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),one_one(nat)),N5)
              & ( aa(list(list(A)),list(A),concat(A),aa(list(A),list(list(A)),aa(nat,fun(list(A),list(list(A))),replicate(list(A)),N5),Zs2)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),Ys) ) ) ) ) ) ).

% comm_append_is_replicate
tff(fact_4862_lexordp_Omono,axiom,
    ! [A: $tType] :
      ( ord(A)
     => order_mono(fun(list(A),fun(list(A),$o)),fun(list(A),fun(list(A),$o)),aTP_Lamp_ol(fun(list(A),fun(list(A),$o)),fun(list(A),fun(list(A),$o)))) ) ).

% lexordp.mono
tff(fact_4863_remdups__adj__length__ge1,axiom,
    ! [A: $tType,Xsa: list(A)] :
      ( ( Xsa != nil(A) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),remdups_adj(A),Xsa))) ) ).

% remdups_adj_length_ge1
tff(fact_4864_remdups__adj_Opelims,axiom,
    ! [A: $tType,Xa: list(A),Y: list(A)] :
      ( ( aa(list(A),list(A),remdups_adj(A),Xa) = Y )
     => ( accp(list(A),remdups_adj_rel(A),Xa)
       => ( ( ( Xa = nil(A) )
           => ( ( Y = nil(A) )
             => ~ accp(list(A),remdups_adj_rel(A),nil(A)) ) )
         => ( ! [X: A] :
                ( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)) )
               => ( ( Y = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)) )
                 => ~ accp(list(A),remdups_adj_rel(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A))) ) )
           => ~ ! [X: A,Y2: A,Xs: list(A)] :
                  ( ( Xa = 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),Y2),Xs)) )
                 => ( ( Y = $ite(X = Y2,aa(list(A),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),aa(list(A),list(A),remdups_adj(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Xs)))) )
                   => ~ accp(list(A),remdups_adj_rel(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Xs))) ) ) ) ) ) ) ).

% remdups_adj.pelims
tff(fact_4865_listrel__def,axiom,
    ! [B: $tType,A: $tType,X4: set(product_prod(A,B))] : ( listrel(A,B,X4) = aa(fun(product_prod(list(A),list(B)),$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_bi(set(product_prod(A,B)),fun(A,fun(B,$o))),X4)))) ) ).

% listrel_def
tff(fact_4866_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_my(nat,fun(nat,A)))) ) ) ).

% ring_1_class.of_int_def
tff(fact_4867_list_Omap__id0,axiom,
    ! [A: $tType] : ( aa(fun(A,A),fun(list(A),list(A)),map(A,A),id(A)) = id(list(A)) ) ).

% list.map_id0
tff(fact_4868_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_4869_rotate0,axiom,
    ! [A: $tType] : ( aa(nat,fun(list(A),list(A)),rotate(A),zero_zero(nat)) = id(list(A)) ) ).

% rotate0
tff(fact_4870_mono__compose,axiom,
    ! [A: $tType,E: $tType,B: $tType,F4: $tType] :
      ( ( order(E)
        & order(A) )
     => ! [Qa: fun(A,fun(B,E)),F2: fun(F4,B)] :
          ( order_mono(A,fun(B,E),Qa)
         => order_mono(A,fun(F4,E),aa(fun(F4,B),fun(A,fun(F4,E)),aTP_Lamp_om(fun(A,fun(B,E)),fun(fun(F4,B),fun(A,fun(F4,E))),Qa),F2)) ) ) ).

% mono_compose
tff(fact_4871_list_Omap__id,axiom,
    ! [A: $tType,T2: list(A)] : ( aa(list(A),list(A),aa(fun(A,A),fun(list(A),list(A)),map(A,A),id(A)),T2) = T2 ) ).

% list.map_id
tff(fact_4872_List_Omap_Oidentity,axiom,
    ! [A: $tType] : ( aa(fun(A,A),fun(list(A),list(A)),map(A,A),aTP_Lamp_ab(A,A)) = id(list(A)) ) ).

% List.map.identity
tff(fact_4873_listrelp_OCons,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(B,$o)),Xa: A,Y: B,Xsa: list(A),Ys: list(B)] :
      ( aa(B,$o,aa(A,fun(B,$o),R2,Xa),Y)
     => ( aa(list(B),$o,aa(list(A),fun(list(B),$o),listrelp(A,B,R2),Xsa),Ys)
       => aa(list(B),$o,aa(list(A),fun(list(B),$o),listrelp(A,B,R2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),Ys)) ) ) ).

% listrelp.Cons
tff(fact_4874_listrelp_ONil,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(B,$o))] : aa(list(B),$o,aa(list(A),fun(list(B),$o),listrelp(A,B,R2),nil(A)),nil(B)) ).

% listrelp.Nil
tff(fact_4875_foldr__Nil,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,fun(A,A))] : ( aa(list(B),fun(A,A),aa(fun(B,fun(A,A)),fun(list(B),fun(A,A)),foldr(B,A),F2),nil(B)) = id(A) ) ).

% foldr_Nil
tff(fact_4876_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_na(nat,fun(nat,fun(product_prod(nat,nat),$o))))) ).

% less_int_def
tff(fact_4877_foldr__filter,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,fun(A,A)),Pa: fun(B,$o),Xsa: list(B)] : ( aa(list(B),fun(A,A),aa(fun(B,fun(A,A)),fun(list(B),fun(A,A)),foldr(B,A),F2),aa(list(B),list(B),aa(fun(B,$o),fun(list(B),list(B)),filter2(B),Pa),Xsa)) = aa(list(B),fun(A,A),aa(fun(B,fun(A,A)),fun(list(B),fun(A,A)),foldr(B,A),aa(fun(B,$o),fun(B,fun(A,A)),aTP_Lamp_on(fun(B,fun(A,A)),fun(fun(B,$o),fun(B,fun(A,A))),F2),Pa)),Xsa) ) ).

% foldr_filter
tff(fact_4878_listrelp_Osimps,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(B,$o)),A1: list(A),A22: list(B)] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),listrelp(A,B,R2),A1),A22)
    <=> ( ( ( A1 = nil(A) )
          & ( A22 = nil(B) ) )
        | ? [X6: A,Y4: B,Xs2: list(A),Ys4: list(B)] :
            ( ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X6),Xs2) )
            & ( A22 = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y4),Ys4) )
            & aa(B,$o,aa(A,fun(B,$o),R2,X6),Y4)
            & aa(list(B),$o,aa(list(A),fun(list(B),$o),listrelp(A,B,R2),Xs2),Ys4) ) ) ) ).

% listrelp.simps
tff(fact_4879_listrelp_Ocases,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(B,$o)),A1: list(A),A22: list(B)] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),listrelp(A,B,R2),A1),A22)
     => ( ( ( A1 = nil(A) )
         => ( A22 != nil(B) ) )
       => ~ ! [X: A,Y2: B,Xs: list(A)] :
              ( ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) )
             => ! [Ys5: list(B)] :
                  ( ( A22 = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y2),Ys5) )
                 => ( aa(B,$o,aa(A,fun(B,$o),R2,X),Y2)
                   => ~ aa(list(B),$o,aa(list(A),fun(list(B),$o),listrelp(A,B,R2),Xs),Ys5) ) ) ) ) ) ).

% listrelp.cases
tff(fact_4880_listrelp__listrel__eq,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,B)),X4: list(A),Xa4: list(B)] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),listrelp(A,B,aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_bi(set(product_prod(A,B)),fun(A,fun(B,$o))),R2)),X4),Xa4)
    <=> 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)),X4),Xa4),listrel(A,B,R2)) ) ).

% listrelp_listrel_eq
tff(fact_4881_map__fst__mk__fst,axiom,
    ! [B: $tType,A: $tType,K3: A,L: list(B)] : ( aa(list(product_prod(A,B)),list(A),aa(fun(product_prod(A,B),A),fun(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)),aa(fun(B,product_prod(A,B)),fun(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),K3)),L)) = aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),aa(list(B),nat,size_size(list(B)),L)),K3) ) ).

% map_fst_mk_fst
tff(fact_4882_finite__def,axiom,
    ! [A: $tType] : ( finite_finite2(A) = complete_lattice_lfp(fun(set(A),$o),aTP_Lamp_oh(fun(set(A),$o),fun(set(A),$o))) ) ).

% finite_def
tff(fact_4883_comp__fun__idem__on_Ofold__insert__idem2,axiom,
    ! [B: $tType,A: $tType,S: set(A),F2: fun(A,fun(B,B)),Xa: A,Aa2: set(A),Z2: B] :
      ( finite673082921795544331dem_on(A,B,S,F2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),Aa2)),S)
       => ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( finite_fold(A,B,F2,Z2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),Aa2)) = finite_fold(A,B,F2,aa(B,B,aa(A,fun(B,B),F2,Xa),Z2),Aa2) ) ) ) ) ).

% comp_fun_idem_on.fold_insert_idem2
tff(fact_4884_img__fst,axiom,
    ! [B: $tType,A: $tType,A3: A,B3: B,S: set(product_prod(A,B))] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B3),S)
     => 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_4885_map__fst__zip,axiom,
    ! [B: $tType,A: $tType,Xsa: list(A),Ys: list(B)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xsa) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( aa(list(product_prod(A,B)),list(A),aa(fun(product_prod(A,B),A),fun(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)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),Xsa),Ys)) = Xsa ) ) ).

% map_fst_zip
tff(fact_4886_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_4887_map__fst__enumerate,axiom,
    ! [A: $tType,N: nat,Xsa: list(A)] : ( aa(list(product_prod(nat,A)),list(nat),aa(fun(product_prod(nat,A),nat),fun(list(product_prod(nat,A)),list(nat)),map(product_prod(nat,A),nat),product_fst(nat,A)),enumerate(A,N,Xsa)) = upt(N,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(list(A),nat,size_size(list(A)),Xsa))) ) ).

% map_fst_enumerate
tff(fact_4888_sorted__wrt__map__linord,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(A)
     => ! [L: list(product_prod(A,B))] :
          ( sorted_wrt(product_prod(A,B),aTP_Lamp_oo(product_prod(A,B),fun(product_prod(A,B),$o)),L)
        <=> sorted_wrt(A,ord_less_eq(A),aa(list(product_prod(A,B)),list(A),aa(fun(product_prod(A,B),A),fun(list(product_prod(A,B)),list(A)),map(product_prod(A,B),A),product_fst(A,B)),L)) ) ) ).

% sorted_wrt_map_linord
tff(fact_4889_map__fst__mk__snd,axiom,
    ! [B: $tType,A: $tType,K3: B,L: list(A)] : ( aa(list(product_prod(A,B)),list(A),aa(fun(product_prod(A,B),A),fun(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)),aa(fun(A,product_prod(A,B)),fun(list(A),list(product_prod(A,B))),map(A,product_prod(A,B)),aa(B,fun(A,product_prod(A,B)),aTP_Lamp_io(B,fun(A,product_prod(A,B))),K3)),L)) = L ) ).

% map_fst_mk_snd
tff(fact_4890_sorted__wrt__map__rev__linord,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(A)
     => ! [L: list(product_prod(A,B))] :
          ( sorted_wrt(product_prod(A,B),aTP_Lamp_op(product_prod(A,B),fun(product_prod(A,B),$o)),L)
        <=> sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),rev(A),aa(list(product_prod(A,B)),list(A),aa(fun(product_prod(A,B),A),fun(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_4891_fst__diag__fst,axiom,
    ! [B: $tType,A: $tType] : ( comp(product_prod(A,A),A,product_prod(A,B),product_fst(A,A),comp(A,product_prod(A,A),product_prod(A,B),aTP_Lamp_ii(A,product_prod(A,A)),product_fst(A,B))) = product_fst(A,B) ) ).

% fst_diag_fst
tff(fact_4892_fn__fst__conv,axiom,
    ! [B: $tType,E: $tType,A: $tType,F2: fun(A,E)] : ( aTP_Lamp_oq(fun(A,E),fun(product_prod(A,B),E),F2) = aa(fun(A,fun(B,E)),fun(product_prod(A,B),E),product_case_prod(A,B,E),aTP_Lamp_or(fun(A,E),fun(A,fun(B,E)),F2)) ) ).

% fn_fst_conv
tff(fact_4893_fstE,axiom,
    ! [B: $tType,A: $tType,Xa: product_prod(A,B),A3: A,B3: B,Pa: fun(A,$o)] :
      ( ( Xa = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B3) )
     => ( aa(A,$o,Pa,aa(product_prod(A,B),A,product_fst(A,B),Xa))
       => aa(A,$o,Pa,A3) ) ) ).

% fstE
tff(fact_4894_fst__eqD,axiom,
    ! [B: $tType,A: $tType,Xa: 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),Xa),Y)) = A3 )
     => ( Xa = A3 ) ) ).

% fst_eqD
tff(fact_4895_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_4896_in__fst__imageE,axiom,
    ! [B: $tType,A: $tType,Xa: A,S: set(product_prod(A,B))] :
      ( member(A,Xa,aa(set(product_prod(A,B)),set(A),image2(product_prod(A,B),A,product_fst(A,B)),S))
     => ~ ! [Y2: B] : ~ member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa),Y2),S) ) ).

% in_fst_imageE
tff(fact_4897_distinct__map__fst__filterI,axiom,
    ! [B: $tType,A: $tType,Xsa: list(product_prod(A,B)),Pa: fun(product_prod(A,B),$o)] :
      ( aa(list(A),$o,distinct(A),aa(list(product_prod(A,B)),list(A),aa(fun(product_prod(A,B),A),fun(list(product_prod(A,B)),list(A)),map(product_prod(A,B),A),product_fst(A,B)),Xsa))
     => aa(list(A),$o,distinct(A),aa(list(product_prod(A,B)),list(A),aa(fun(product_prod(A,B),A),fun(list(product_prod(A,B)),list(A)),map(product_prod(A,B),A),product_fst(A,B)),aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(fun(product_prod(A,B),$o),fun(list(product_prod(A,B)),list(product_prod(A,B))),filter2(product_prod(A,B)),Pa),Xsa))) ) ).

% distinct_map_fst_filterI
tff(fact_4898_zip__takeWhile__fst,axiom,
    ! [A: $tType,B: $tType,Pa: fun(A,$o),Xsa: list(A),Ys: list(B)] : ( aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),takeWhile(A),Pa),Xsa)),Ys) = aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(fun(product_prod(A,B),$o),fun(list(product_prod(A,B)),list(product_prod(A,B))),takeWhile(product_prod(A,B)),comp(A,$o,product_prod(A,B),Pa,product_fst(A,B))),aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),Xsa),Ys)) ) ).

% zip_takeWhile_fst
tff(fact_4899_fst__diag__id,axiom,
    ! [A: $tType,Z2: A] : ( aa(A,A,comp(product_prod(A,A),A,A,product_fst(A,A),aTP_Lamp_ii(A,product_prod(A,A))),Z2) = aa(A,A,id(A),Z2) ) ).

% fst_diag_id
tff(fact_4900_partition__filter1,axiom,
    ! [A: $tType,Pa: fun(A,$o),Xsa: list(A)] : ( aa(product_prod(list(A),list(A)),list(A),product_fst(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(fun(A,$o),fun(list(A),product_prod(list(A),list(A))),partition(A),Pa),Xsa)) = aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),Pa),Xsa) ) ).

% partition_filter1
tff(fact_4901_fst__image__mp,axiom,
    ! [B: $tType,A: $tType,Aa2: set(product_prod(A,B)),B2: set(A),Xa: A,Y: B] :
      ( aa(set(A),$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)),Aa2)),B2)
     => ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa),Y),Aa2)
       => member(A,Xa,B2) ) ) ).

% fst_image_mp
tff(fact_4902_fst__in__Field,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(product_prod(A,A)),set(A),image2(product_prod(A,A),A,product_fst(A,A)),R)),aa(set(product_prod(A,A)),set(A),field2(A),R)) ).

% fst_in_Field
tff(fact_4903_sorted__enumerate,axiom,
    ! [A: $tType,N: nat,Xsa: list(A)] : sorted_wrt(nat,ord_less_eq(nat),aa(list(product_prod(nat,A)),list(nat),aa(fun(product_prod(nat,A),nat),fun(list(product_prod(nat,A)),list(nat)),map(product_prod(nat,A),nat),product_fst(nat,A)),enumerate(A,N,Xsa))) ).

% sorted_enumerate
tff(fact_4904_fst__foldl,axiom,
    ! [B: $tType,A: $tType,E: $tType,F2: fun(A,fun(E,A)),G: fun(A,fun(B,fun(E,B))),A3: A,B3: B,Xsa: list(E)] : ( aa(product_prod(A,B),A,product_fst(A,B),aa(list(E),product_prod(A,B),aa(product_prod(A,B),fun(list(E),product_prod(A,B)),aa(fun(product_prod(A,B),fun(E,product_prod(A,B))),fun(product_prod(A,B),fun(list(E),product_prod(A,B))),foldl(product_prod(A,B),E),aa(fun(A,fun(B,fun(E,product_prod(A,B)))),fun(product_prod(A,B),fun(E,product_prod(A,B))),product_case_prod(A,B,fun(E,product_prod(A,B))),aa(fun(A,fun(B,fun(E,B))),fun(A,fun(B,fun(E,product_prod(A,B)))),aTP_Lamp_os(fun(A,fun(E,A)),fun(fun(A,fun(B,fun(E,B))),fun(A,fun(B,fun(E,product_prod(A,B))))),F2),G))),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B3)),Xsa)) = aa(list(E),A,aa(A,fun(list(E),A),aa(fun(A,fun(E,A)),fun(A,fun(list(E),A)),foldl(A,E),F2),A3),Xsa) ) ).

% fst_foldl
tff(fact_4905_distinct__map__fstD,axiom,
    ! [A: $tType,B: $tType,Xsa: list(product_prod(A,B)),Xa: A,Y: B,Z2: B] :
      ( aa(list(A),$o,distinct(A),aa(list(product_prod(A,B)),list(A),aa(fun(product_prod(A,B),A),fun(list(product_prod(A,B)),list(A)),map(product_prod(A,B),A),product_fst(A,B)),Xsa))
     => ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa),Y),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Xsa))
       => ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa),Z2),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Xsa))
         => ( Y = Z2 ) ) ) ) ).

% distinct_map_fstD
tff(fact_4906_lfp__Kleene__iter,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(A,A),K3: nat] :
          ( order_mono(A,A,F2)
         => ( ( aa(A,A,compow(fun(A,A),aa(nat,nat,suc,K3),F2),bot_bot(A)) = aa(A,A,compow(fun(A,A),K3,F2),bot_bot(A)) )
           => ( complete_lattice_lfp(A,F2) = aa(A,A,compow(fun(A,A),K3,F2),bot_bot(A)) ) ) ) ) ).

% lfp_Kleene_iter
tff(fact_4907_map__fst__zip__take,axiom,
    ! [B: $tType,A: $tType,Xsa: list(A),Ys: list(B)] : ( aa(list(product_prod(A,B)),list(A),aa(fun(product_prod(A,B),A),fun(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)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),Xsa),Ys)) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(list(A),nat,size_size(list(A)),Xsa)),aa(list(B),nat,size_size(list(B)),Ys))),Xsa) ) ).

% map_fst_zip_take
tff(fact_4908_comp__fun__idem__on_Ofold__insert__idem,axiom,
    ! [B: $tType,A: $tType,S: set(A),F2: fun(A,fun(B,B)),Xa: A,Aa2: set(A),Z2: B] :
      ( finite673082921795544331dem_on(A,B,S,F2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),Aa2)),S)
       => ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( finite_fold(A,B,F2,Z2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),Aa2)) = aa(B,B,aa(A,fun(B,B),F2,Xa),finite_fold(A,B,F2,Z2,Aa2)) ) ) ) ) ).

% comp_fun_idem_on.fold_insert_idem
tff(fact_4909_eq__key__imp__eq__value,axiom,
    ! [A: $tType,B: $tType,Xsa: list(product_prod(A,B)),K3: A,V1: B,V22: B] :
      ( aa(list(A),$o,distinct(A),aa(list(product_prod(A,B)),list(A),aa(fun(product_prod(A,B),A),fun(list(product_prod(A,B)),list(A)),map(product_prod(A,B),A),product_fst(A,B)),Xsa))
     => ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K3),V1),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Xsa))
       => ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K3),V22),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Xsa))
         => ( V1 = V22 ) ) ) ) ).

% eq_key_imp_eq_value
tff(fact_4910_bezw_Oelims,axiom,
    ! [Xa: nat,Xaa: nat,Y: product_prod(int,int)] :
      ( ( bezw(Xa,Xaa) = Y )
     => ( Y = $ite(Xaa = zero_zero(nat),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),zero_zero(int)),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Xaa,modulo_modulo(nat,Xa,Xaa)))),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(product_prod(int,int),int,product_fst(int,int),bezw(Xaa,modulo_modulo(nat,Xa,Xaa)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Xaa,modulo_modulo(nat,Xa,Xaa)))),aa(nat,int,semiring_1_of_nat(int),divide_divide(nat,Xa,Xaa)))))) ) ) ).

% bezw.elims
tff(fact_4911_bezw_Osimps,axiom,
    ! [Xa: nat,Y: nat] :
      ( bezw(Xa,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,Xa,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,Xa,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,Xa,Y)))),aa(nat,int,semiring_1_of_nat(int),divide_divide(nat,Xa,Y)))))) ) ).

% bezw.simps
tff(fact_4912_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_4913_img__snd,axiom,
    ! [B: $tType,A: $tType,A3: A,B3: B,S: set(product_prod(A,B))] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B3),S)
     => member(B,B3,aa(set(product_prod(A,B)),set(B),image2(product_prod(A,B),B,product_snd(A,B)),S)) ) ).

% img_snd
tff(fact_4914_map__snd__zip,axiom,
    ! [A: $tType,B: $tType,Xsa: list(A),Ys: list(B)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xsa) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( aa(list(product_prod(A,B)),list(B),aa(fun(product_prod(A,B),B),fun(list(product_prod(A,B)),list(B)),map(product_prod(A,B),B),product_snd(A,B)),aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),Xsa),Ys)) = Ys ) ) ).

% map_snd_zip
tff(fact_4915_zip__map__fst__snd,axiom,
    ! [B: $tType,A: $tType,Zs: list(product_prod(A,B))] : ( aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),aa(list(product_prod(A,B)),list(A),aa(fun(product_prod(A,B),A),fun(list(product_prod(A,B)),list(A)),map(product_prod(A,B),A),product_fst(A,B)),Zs)),aa(list(product_prod(A,B)),list(B),aa(fun(product_prod(A,B),B),fun(list(product_prod(A,B)),list(B)),map(product_prod(A,B),B),product_snd(A,B)),Zs)) = Zs ) ).

% zip_map_fst_snd
tff(fact_4916_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_4917_map__snd__mk__fst,axiom,
    ! [B: $tType,A: $tType,K3: B,L: list(A)] : ( aa(list(product_prod(B,A)),list(A),aa(fun(product_prod(B,A),A),fun(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)),aa(fun(A,product_prod(B,A)),fun(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),K3)),L)) = L ) ).

% map_snd_mk_fst
tff(fact_4918_map__snd__mk__snd,axiom,
    ! [B: $tType,A: $tType,K3: A,L: list(B)] : ( aa(list(product_prod(B,A)),list(A),aa(fun(product_prod(B,A),A),fun(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)),aa(fun(B,product_prod(B,A)),fun(list(B),list(product_prod(B,A))),map(B,product_prod(B,A)),aa(A,fun(B,product_prod(B,A)),aTP_Lamp_ot(A,fun(B,product_prod(B,A))),K3)),L)) = aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),aa(list(B),nat,size_size(list(B)),L)),K3) ) ).

% map_snd_mk_snd
tff(fact_4919_Ex__prod__contract,axiom,
    ! [B: $tType,A: $tType,Pa: fun(A,fun(B,$o))] :
      ( ? [A7: A,X_12: B] : aa(B,$o,aa(A,fun(B,$o),Pa,A7),X_12)
    <=> ? [Z5: product_prod(A,B)] : aa(B,$o,aa(A,fun(B,$o),Pa,aa(product_prod(A,B),A,product_fst(A,B),Z5)),aa(product_prod(A,B),B,product_snd(A,B),Z5)) ) ).

% Ex_prod_contract
tff(fact_4920_All__prod__contract,axiom,
    ! [B: $tType,A: $tType,Pa: fun(A,fun(B,$o))] :
      ( ! [A7: A,X_12: B] : aa(B,$o,aa(A,fun(B,$o),Pa,A7),X_12)
    <=> ! [Z5: product_prod(A,B)] : aa(B,$o,aa(A,fun(B,$o),Pa,aa(product_prod(A,B),A,product_fst(A,B),Z5)),aa(product_prod(A,B),B,product_snd(A,B),Z5)) ) ).

% All_prod_contract
tff(fact_4921_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_4922_snd__eqD,axiom,
    ! [B: $tType,A: $tType,Xa: 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),Xa),Y)) = A3 )
     => ( Y = A3 ) ) ).

% snd_eqD
tff(fact_4923_sndE,axiom,
    ! [A: $tType,B: $tType,Xa: product_prod(A,B),A3: A,B3: B,Pa: fun(B,$o)] :
      ( ( Xa = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B3) )
     => ( aa(B,$o,Pa,aa(product_prod(A,B),B,product_snd(A,B),Xa))
       => aa(B,$o,Pa,B3) ) ) ).

% sndE
tff(fact_4924_fn__snd__conv,axiom,
    ! [A: $tType,E: $tType,B: $tType,F2: fun(B,E)] : ( aTP_Lamp_ou(fun(B,E),fun(product_prod(A,B),E),F2) = aa(fun(A,fun(B,E)),fun(product_prod(A,B),E),product_case_prod(A,B,E),aTP_Lamp_ov(fun(B,E),fun(A,fun(B,E)),F2)) ) ).

% fn_snd_conv
tff(fact_4925_snd__diag__snd,axiom,
    ! [B: $tType,A: $tType] : ( comp(product_prod(B,B),B,product_prod(A,B),product_snd(B,B),comp(B,product_prod(B,B),product_prod(A,B),aTP_Lamp_ow(B,product_prod(B,B)),product_snd(A,B))) = product_snd(A,B) ) ).

% snd_diag_snd
tff(fact_4926_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [B: $tType,A: $tType,Pa: fun(A,fun(B,$o)),Xa: A,Y: B,A3: product_prod(A,B)] :
      ( aa(B,$o,aa(A,fun(B,$o),Pa,Xa),Y)
     => ( ( A3 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa),Y) )
       => aa(B,$o,aa(A,fun(B,$o),Pa,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_4927_surjective__pairing,axiom,
    ! [B: $tType,A: $tType,T2: product_prod(A,B)] : ( T2 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(product_prod(A,B),A,product_fst(A,B),T2)),aa(product_prod(A,B),B,product_snd(A,B),T2)) ) ).

% surjective_pairing
tff(fact_4928_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_4929_in__snd__imageE,axiom,
    ! [A: $tType,B: $tType,Y: A,S: set(product_prod(B,A))] :
      ( member(A,Y,aa(set(product_prod(B,A)),set(A),image2(product_prod(B,A),A,product_snd(B,A)),S))
     => ~ ! [X: B] : ~ 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),S) ) ).

% in_snd_imageE
tff(fact_4930_pair__list__eqI,axiom,
    ! [B: $tType,A: $tType,Xsa: list(product_prod(A,B)),Ys: list(product_prod(A,B))] :
      ( ( aa(list(product_prod(A,B)),list(A),aa(fun(product_prod(A,B),A),fun(list(product_prod(A,B)),list(A)),map(product_prod(A,B),A),product_fst(A,B)),Xsa) = aa(list(product_prod(A,B)),list(A),aa(fun(product_prod(A,B),A),fun(list(product_prod(A,B)),list(A)),map(product_prod(A,B),A),product_fst(A,B)),Ys) )
     => ( ( aa(list(product_prod(A,B)),list(B),aa(fun(product_prod(A,B),B),fun(list(product_prod(A,B)),list(B)),map(product_prod(A,B),B),product_snd(A,B)),Xsa) = aa(list(product_prod(A,B)),list(B),aa(fun(product_prod(A,B),B),fun(list(product_prod(A,B)),list(B)),map(product_prod(A,B),B),product_snd(A,B)),Ys) )
       => ( Xsa = Ys ) ) ) ).

% pair_list_eqI
tff(fact_4931_zip__takeWhile__snd,axiom,
    ! [A: $tType,B: $tType,Xsa: list(A),Pa: fun(B,$o),Ys: list(B)] : ( aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),Xsa),aa(list(B),list(B),aa(fun(B,$o),fun(list(B),list(B)),takeWhile(B),Pa),Ys)) = aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(fun(product_prod(A,B),$o),fun(list(product_prod(A,B)),list(product_prod(A,B))),takeWhile(product_prod(A,B)),comp(B,$o,product_prod(A,B),Pa,product_snd(A,B))),aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),Xsa),Ys)) ) ).

% zip_takeWhile_snd
tff(fact_4932_snd__diag__fst,axiom,
    ! [B: $tType,A: $tType] : ( comp(product_prod(A,A),A,product_prod(A,B),product_snd(A,A),comp(A,product_prod(A,A),product_prod(A,B),aTP_Lamp_ii(A,product_prod(A,A)),product_fst(A,B))) = product_fst(A,B) ) ).

% snd_diag_fst
tff(fact_4933_fst__diag__snd,axiom,
    ! [B: $tType,A: $tType] : ( comp(product_prod(B,B),B,product_prod(A,B),product_fst(B,B),comp(B,product_prod(B,B),product_prod(A,B),aTP_Lamp_ow(B,product_prod(B,B)),product_snd(A,B))) = product_snd(A,B) ) ).

% fst_diag_snd
tff(fact_4934_snd__diag__id,axiom,
    ! [A: $tType,Z2: A] : ( aa(A,A,comp(product_prod(A,A),A,A,product_snd(A,A),aTP_Lamp_ii(A,product_prod(A,A))),Z2) = aa(A,A,id(A),Z2) ) ).

% snd_diag_id
tff(fact_4935_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_ox(product_prod(A,A),$o)) ) ).

% Id_fstsnd_eq
tff(fact_4936_prod_Osplit__sel,axiom,
    ! [A: $tType,E: $tType,B: $tType,Pa: fun(A,$o),F2: fun(B,fun(E,A)),Prod: product_prod(B,E)] :
      ( aa(A,$o,Pa,aa(product_prod(B,E),A,aa(fun(B,fun(E,A)),fun(product_prod(B,E),A),product_case_prod(B,E,A),F2),Prod))
    <=> ( ( Prod = aa(E,product_prod(B,E),aa(B,fun(E,product_prod(B,E)),product_Pair(B,E),aa(product_prod(B,E),B,product_fst(B,E),Prod)),aa(product_prod(B,E),E,product_snd(B,E),Prod)) )
       => aa(A,$o,Pa,aa(E,A,aa(B,fun(E,A),F2,aa(product_prod(B,E),B,product_fst(B,E),Prod)),aa(product_prod(B,E),E,product_snd(B,E),Prod))) ) ) ).

% prod.split_sel
tff(fact_4937_prod_Osplit__sel__asm,axiom,
    ! [A: $tType,E: $tType,B: $tType,Pa: fun(A,$o),F2: fun(B,fun(E,A)),Prod: product_prod(B,E)] :
      ( aa(A,$o,Pa,aa(product_prod(B,E),A,aa(fun(B,fun(E,A)),fun(product_prod(B,E),A),product_case_prod(B,E,A),F2),Prod))
    <=> ~ ( ( Prod = aa(E,product_prod(B,E),aa(B,fun(E,product_prod(B,E)),product_Pair(B,E),aa(product_prod(B,E),B,product_fst(B,E),Prod)),aa(product_prod(B,E),E,product_snd(B,E),Prod)) )
          & ~ aa(A,$o,Pa,aa(E,A,aa(B,fun(E,A),F2,aa(product_prod(B,E),B,product_fst(B,E),Prod)),aa(product_prod(B,E),E,product_snd(B,E),Prod))) ) ) ).

% prod.split_sel_asm
tff(fact_4938_snd__image__mp,axiom,
    ! [B: $tType,A: $tType,Aa2: set(product_prod(B,A)),B2: set(A),Xa: B,Y: A] :
      ( aa(set(A),$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)),Aa2)),B2)
     => ( member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Xa),Y),Aa2)
       => member(A,Y,B2) ) ) ).

% snd_image_mp
tff(fact_4939_snd__in__Field,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(product_prod(A,A)),set(A),image2(product_prod(A,A),A,product_snd(A,A)),R)),aa(set(product_prod(A,A)),set(A),field2(A),R)) ).

% snd_in_Field
tff(fact_4940_fs__contract,axiom,
    ! [A: $tType,B: $tType,E: $tType,F2: fun(A,fun(B,E)),S: set(E)] : ( aa(set(product_prod(A,B)),set(A),image2(product_prod(A,B),A,product_fst(A,B)),aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(set(E),fun(product_prod(A,B),$o),aTP_Lamp_oy(fun(A,fun(B,E)),fun(set(E),fun(product_prod(A,B),$o)),F2),S))) = aa(fun(A,$o),set(A),collect(A),aa(set(E),fun(A,$o),aTP_Lamp_oz(fun(A,fun(B,E)),fun(set(E),fun(A,$o)),F2),S)) ) ).

% fs_contract
tff(fact_4941_lfp__induct2,axiom,
    ! [A: $tType,B: $tType,A3: A,B3: B,F2: fun(set(product_prod(A,B)),set(product_prod(A,B))),Pa: fun(A,fun(B,$o))] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B3),complete_lattice_lfp(set(product_prod(A,B)),F2))
     => ( order_mono(set(product_prod(A,B)),set(product_prod(A,B)),F2)
       => ( ! [A4: A,B4: B] :
              ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B4),aa(set(product_prod(A,B)),set(product_prod(A,B)),F2,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)),F2)),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),Pa)))))
             => aa(B,$o,aa(A,fun(B,$o),Pa,A4),B4) )
         => aa(B,$o,aa(A,fun(B,$o),Pa,A3),B3) ) ) ) ).

% lfp_induct2
tff(fact_4942_zip__eq__conv,axiom,
    ! [A: $tType,B: $tType,Xsa: list(A),Ys: list(B),Zs: list(product_prod(A,B))] :
      ( ( aa(list(A),nat,size_size(list(A)),Xsa) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( ( aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),Xsa),Ys) = Zs )
      <=> ( ( aa(list(product_prod(A,B)),list(A),aa(fun(product_prod(A,B),A),fun(list(product_prod(A,B)),list(A)),map(product_prod(A,B),A),product_fst(A,B)),Zs) = Xsa )
          & ( aa(list(product_prod(A,B)),list(B),aa(fun(product_prod(A,B),B),fun(list(product_prod(A,B)),list(B)),map(product_prod(A,B),B),product_snd(A,B)),Zs) = Ys ) ) ) ) ).

% zip_eq_conv
tff(fact_4943_update__zip,axiom,
    ! [B: $tType,A: $tType,Xsa: list(A),Ys: list(B),Ia: nat,Xy: product_prod(A,B)] : ( aa(product_prod(A,B),list(product_prod(A,B)),aa(nat,fun(product_prod(A,B),list(product_prod(A,B))),aa(list(product_prod(A,B)),fun(nat,fun(product_prod(A,B),list(product_prod(A,B)))),list_update(product_prod(A,B)),aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),Xsa),Ys)),Ia),Xy) = aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xsa),Ia),aa(product_prod(A,B),A,product_fst(A,B),Xy))),aa(B,list(B),aa(nat,fun(B,list(B)),aa(list(B),fun(nat,fun(B,list(B))),list_update(B),Ys),Ia),aa(product_prod(A,B),B,product_snd(A,B),Xy))) ) ).

% update_zip
tff(fact_4944_map__snd__zip__take,axiom,
    ! [B: $tType,A: $tType,Xsa: list(B),Ys: list(A)] : ( aa(list(product_prod(B,A)),list(A),aa(fun(product_prod(B,A),A),fun(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)),aa(list(B),fun(list(A),list(product_prod(B,A))),zip(B,A),Xsa),Ys)) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(list(B),nat,size_size(list(B)),Xsa)),aa(list(A),nat,size_size(list(A)),Ys))),Ys) ) ).

% map_snd_zip_take
tff(fact_4945_in__set__zip,axiom,
    ! [A: $tType,B: $tType,P2: product_prod(A,B),Xsa: list(A),Ys: list(B)] :
      ( member(product_prod(A,B),P2,aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),Xsa),Ys)))
    <=> ? [N2: nat] :
          ( ( aa(nat,A,nth(A,Xsa),N2) = aa(product_prod(A,B),A,product_fst(A,B),P2) )
          & ( aa(nat,B,nth(B,Ys),N2) = aa(product_prod(A,B),B,product_snd(A,B),P2) )
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N2),aa(list(A),nat,size_size(list(A)),Xsa))
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N2),aa(list(B),nat,size_size(list(B)),Ys)) ) ) ).

% in_set_zip
tff(fact_4946_finite__range__prod,axiom,
    ! [A: $tType,E: $tType,B: $tType,F2: fun(B,product_prod(A,E))] :
      ( aa(set(A),$o,finite_finite2(A),aa(set(B),set(A),image2(B,A,comp(product_prod(A,E),A,B,product_fst(A,E),F2)),top_top(set(B))))
     => ( aa(set(E),$o,finite_finite2(E),aa(set(B),set(E),image2(B,E,comp(product_prod(A,E),E,B,product_snd(A,E),F2)),top_top(set(B))))
       => aa(set(product_prod(A,E)),$o,finite_finite2(product_prod(A,E)),aa(set(B),set(product_prod(A,E)),image2(B,product_prod(A,E),F2),top_top(set(B)))) ) ) ).

% finite_range_prod
tff(fact_4947_bezw__non__0,axiom,
    ! [Y: nat,Xa: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Y)
     => ( bezw(Xa,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,Xa,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,Xa,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,Xa,Y)))),aa(nat,int,semiring_1_of_nat(int),divide_divide(nat,Xa,Y))))) ) ) ).

% bezw_non_0
tff(fact_4948_fst__snd__flip,axiom,
    ! [B: $tType,A: $tType,Xy: product_prod(A,B)] : ( aa(product_prod(A,B),A,product_fst(A,B),Xy) = aa(product_prod(A,B),A,comp(product_prod(B,A),A,product_prod(A,B),product_snd(B,A),aa(fun(A,fun(B,product_prod(B,A))),fun(product_prod(A,B),product_prod(B,A)),product_case_prod(A,B,product_prod(B,A)),aTP_Lamp_ot(A,fun(B,product_prod(B,A))))),Xy) ) ).

% fst_snd_flip
tff(fact_4949_snd__fst__flip,axiom,
    ! [A: $tType,B: $tType,Xy: product_prod(B,A)] : ( aa(product_prod(B,A),A,product_snd(B,A),Xy) = aa(product_prod(B,A),A,comp(product_prod(A,B),A,product_prod(B,A),product_fst(A,B),aa(fun(B,fun(A,product_prod(A,B))),fun(product_prod(B,A),product_prod(A,B)),product_case_prod(B,A,product_prod(A,B)),aTP_Lamp_io(B,fun(A,product_prod(A,B))))),Xy) ) ).

% snd_fst_flip
tff(fact_4950_bezw_Opelims,axiom,
    ! [Xa: nat,Xaa: nat,Y: product_prod(int,int)] :
      ( ( bezw(Xa,Xaa) = 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),Xa),Xaa))
       => ~ ( ( Y = $ite(Xaa = zero_zero(nat),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),zero_zero(int)),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Xaa,modulo_modulo(nat,Xa,Xaa)))),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(product_prod(int,int),int,product_fst(int,int),bezw(Xaa,modulo_modulo(nat,Xa,Xaa)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Xaa,modulo_modulo(nat,Xa,Xaa)))),aa(nat,int,semiring_1_of_nat(int),divide_divide(nat,Xa,Xaa)))))) )
           => ~ accp(product_prod(nat,nat),bezw_rel,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xa),Xaa)) ) ) ) ).

% bezw.pelims
tff(fact_4951_map__snd__enumerate,axiom,
    ! [A: $tType,N: nat,Xsa: list(A)] : ( aa(list(product_prod(nat,A)),list(A),aa(fun(product_prod(nat,A),A),fun(list(product_prod(nat,A)),list(A)),map(product_prod(nat,A),A),product_snd(nat,A)),enumerate(A,N,Xsa)) = Xsa ) ).

% map_snd_enumerate
tff(fact_4952_mod__pure,axiom,
    ! [B3: $o,H2: product_prod(heap_ext(product_unit),set(nat))] :
      ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,pure_assn((B3))),H2)
    <=> ( ( aa(product_prod(heap_ext(product_unit),set(nat)),set(nat),product_snd(heap_ext(product_unit),set(nat)),H2) = bot_bot(set(nat)) )
        & (B3) ) ) ).

% mod_pure
tff(fact_4953_nths__shift__lemma__Suc,axiom,
    ! [A: $tType,Pa: fun(nat,$o),Xsa: list(A),Is: list(nat)] : ( aa(list(product_prod(A,nat)),list(A),aa(fun(product_prod(A,nat),A),fun(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)),aa(fun(product_prod(A,nat),$o),fun(list(product_prod(A,nat)),list(product_prod(A,nat))),filter2(product_prod(A,nat)),aTP_Lamp_pa(fun(nat,$o),fun(product_prod(A,nat),$o),Pa)),aa(list(nat),list(product_prod(A,nat)),aa(list(A),fun(list(nat),list(product_prod(A,nat))),zip(A,nat),Xsa),Is))) = aa(list(product_prod(A,nat)),list(A),aa(fun(product_prod(A,nat),A),fun(list(product_prod(A,nat)),list(A)),map(product_prod(A,nat),A),product_fst(A,nat)),aa(list(product_prod(A,nat)),list(product_prod(A,nat)),aa(fun(product_prod(A,nat),$o),fun(list(product_prod(A,nat)),list(product_prod(A,nat))),filter2(product_prod(A,nat)),aTP_Lamp_pb(fun(nat,$o),fun(product_prod(A,nat),$o),Pa)),aa(list(nat),list(product_prod(A,nat)),aa(list(A),fun(list(nat),list(product_prod(A,nat))),zip(A,nat),Xsa),aa(list(nat),list(nat),aa(fun(nat,nat),fun(list(nat),list(nat)),map(nat,nat),suc),Is)))) ) ).

% nths_shift_lemma_Suc
tff(fact_4954_mod__emp,axiom,
    ! [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,one_one(assn)),H2)
    <=> ( aa(product_prod(heap_ext(product_unit),set(nat)),set(nat),product_snd(heap_ext(product_unit),set(nat)),H2) = bot_bot(set(nat)) ) ) ).

% mod_emp
tff(fact_4955_fi__match__entails,axiom,
    ! [M2: list(product_prod(assn,assn))] :
      ( ! [X: product_prod(assn,assn)] :
          ( member(product_prod(assn,assn),X,aa(list(product_prod(assn,assn)),set(product_prod(assn,assn)),set2(product_prod(assn,assn)),M2))
         => aa(product_prod(assn,assn),$o,aa(fun(assn,fun(assn,$o)),fun(product_prod(assn,assn),$o),product_case_prod(assn,assn,$o),entails),X) )
     => aa(assn,$o,aa(assn,fun(assn,$o),entails,aa(assn,assn,aa(list(assn),fun(assn,assn),aa(fun(assn,fun(assn,assn)),fun(list(assn),fun(assn,assn)),foldr(assn,assn),times_times(assn)),aa(list(product_prod(assn,assn)),list(assn),aa(fun(product_prod(assn,assn),assn),fun(list(product_prod(assn,assn)),list(assn)),map(product_prod(assn,assn),assn),product_fst(assn,assn)),M2)),one_one(assn))),aa(assn,assn,aa(list(assn),fun(assn,assn),aa(fun(assn,fun(assn,assn)),fun(list(assn),fun(assn,assn)),foldr(assn,assn),times_times(assn)),aa(list(product_prod(assn,assn)),list(assn),aa(fun(product_prod(assn,assn),assn),fun(list(product_prod(assn,assn)),list(assn)),map(product_prod(assn,assn),assn),product_snd(assn,assn)),M2)),one_one(assn))) ) ).

% fi_match_entails
tff(fact_4956_partition__filter2,axiom,
    ! [A: $tType,Pa: fun(A,$o),Xsa: list(A)] : ( aa(product_prod(list(A),list(A)),list(A),product_snd(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(fun(A,$o),fun(list(A),product_prod(list(A),list(A))),partition(A),Pa),Xsa)) = aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),comp($o,$o,A,fNot,Pa)),Xsa) ) ).

% partition_filter2
tff(fact_4957_nths__shift__lemma,axiom,
    ! [A: $tType,Aa2: set(nat),Xsa: list(A),Ia: nat] : ( aa(list(product_prod(A,nat)),list(A),aa(fun(product_prod(A,nat),A),fun(list(product_prod(A,nat)),list(A)),map(product_prod(A,nat),A),product_fst(A,nat)),aa(list(product_prod(A,nat)),list(product_prod(A,nat)),aa(fun(product_prod(A,nat),$o),fun(list(product_prod(A,nat)),list(product_prod(A,nat))),filter2(product_prod(A,nat)),aTP_Lamp_pc(set(nat),fun(product_prod(A,nat),$o),Aa2)),aa(list(nat),list(product_prod(A,nat)),aa(list(A),fun(list(nat),list(product_prod(A,nat))),zip(A,nat),Xsa),upt(Ia,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ia),aa(list(A),nat,size_size(list(A)),Xsa)))))) = aa(list(product_prod(A,nat)),list(A),aa(fun(product_prod(A,nat),A),fun(list(product_prod(A,nat)),list(A)),map(product_prod(A,nat),A),product_fst(A,nat)),aa(list(product_prod(A,nat)),list(product_prod(A,nat)),aa(fun(product_prod(A,nat),$o),fun(list(product_prod(A,nat)),list(product_prod(A,nat))),filter2(product_prod(A,nat)),aa(nat,fun(product_prod(A,nat),$o),aTP_Lamp_pd(set(nat),fun(nat,fun(product_prod(A,nat),$o)),Aa2),Ia)),aa(list(nat),list(product_prod(A,nat)),aa(list(A),fun(list(nat),list(product_prod(A,nat))),zip(A,nat),Xsa),upt(zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xsa))))) ) ).

% nths_shift_lemma
tff(fact_4958_mod__star__trueE_H,axiom,
    ! [Pa: 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),Pa),top_top(assn))),H2)
     => ~ ! [H6: product_prod(heap_ext(product_unit),set(nat))] :
            ( ( aa(product_prod(heap_ext(product_unit),set(nat)),heap_ext(product_unit),product_fst(heap_ext(product_unit),set(nat)),H6) = aa(product_prod(heap_ext(product_unit),set(nat)),heap_ext(product_unit),product_fst(heap_ext(product_unit),set(nat)),H2) )
           => ( aa(set(nat),$o,aa(set(nat),fun(set(nat),$o),ord_less_eq(set(nat)),aa(product_prod(heap_ext(product_unit),set(nat)),set(nat),product_snd(heap_ext(product_unit),set(nat)),H6)),aa(product_prod(heap_ext(product_unit),set(nat)),set(nat),product_snd(heap_ext(product_unit),set(nat)),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,Pa),H6) ) ) ) ).

% mod_star_trueE'
tff(fact_4959_nths__def,axiom,
    ! [A: $tType,Xsa: list(A),Aa2: set(nat)] : ( aa(set(nat),list(A),aa(list(A),fun(set(nat),list(A)),nths(A),Xsa),Aa2) = aa(list(product_prod(A,nat)),list(A),aa(fun(product_prod(A,nat),A),fun(list(product_prod(A,nat)),list(A)),map(product_prod(A,nat),A),product_fst(A,nat)),aa(list(product_prod(A,nat)),list(product_prod(A,nat)),aa(fun(product_prod(A,nat),$o),fun(list(product_prod(A,nat)),list(product_prod(A,nat))),filter2(product_prod(A,nat)),aTP_Lamp_pc(set(nat),fun(product_prod(A,nat),$o),Aa2)),aa(list(nat),list(product_prod(A,nat)),aa(list(A),fun(list(nat),list(product_prod(A,nat))),zip(A,nat),Xsa),upt(zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xsa))))) ) ).

% nths_def
tff(fact_4960_FI__RESULT__def,axiom,
    ! [M5: list(product_prod(assn,assn)),UP: assn,UQ: assn,Fa: assn] :
      ( fI_RESULT(M5,UP,UQ,Fa)
    <=> ( ! [X6: product_prod(assn,assn)] :
            ( member(product_prod(assn,assn),X6,aa(list(product_prod(assn,assn)),set(product_prod(assn,assn)),set2(product_prod(assn,assn)),M5))
           => aa(product_prod(assn,assn),$o,aa(fun(assn,fun(assn,$o)),fun(product_prod(assn,assn),$o),product_case_prod(assn,assn,$o),entails),X6) )
       => aa(assn,$o,aa(assn,fun(assn,$o),entails,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(list(assn),fun(assn,assn),aa(fun(assn,fun(assn,assn)),fun(list(assn),fun(assn,assn)),foldr(assn,assn),times_times(assn)),aa(list(product_prod(assn,assn)),list(assn),aa(fun(product_prod(assn,assn),assn),fun(list(product_prod(assn,assn)),list(assn)),map(product_prod(assn,assn),assn),product_fst(assn,assn)),M5)),one_one(assn))),UP)),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(list(assn),fun(assn,assn),aa(fun(assn,fun(assn,assn)),fun(list(assn),fun(assn,assn)),foldr(assn,assn),times_times(assn)),aa(list(product_prod(assn,assn)),list(assn),aa(fun(product_prod(assn,assn),assn),fun(list(product_prod(assn,assn)),list(assn)),map(product_prod(assn,assn),assn),product_snd(assn,assn)),M5)),one_one(assn))),UQ)),Fa)) ) ) ).

% FI_RESULT_def
tff(fact_4961_fstI,axiom,
    ! [B: $tType,A: $tType,Xa: product_prod(A,B),Y: A,Z2: B] :
      ( ( Xa = 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),Xa) = Y ) ) ).

% fstI
tff(fact_4962_sndI,axiom,
    ! [A: $tType,B: $tType,Xa: product_prod(A,B),Y: A,Z2: B] :
      ( ( Xa = 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),Xa) = Z2 ) ) ).

% sndI
tff(fact_4963_in__set__enumerate__eq,axiom,
    ! [A: $tType,P2: product_prod(nat,A),N: nat,Xsa: list(A)] :
      ( member(product_prod(nat,A),P2,aa(list(product_prod(nat,A)),set(product_prod(nat,A)),set2(product_prod(nat,A)),enumerate(A,N,Xsa)))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),aa(product_prod(nat,A),nat,product_fst(nat,A),P2))
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(product_prod(nat,A),nat,product_fst(nat,A),P2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),Xsa)),N))
        & ( aa(nat,A,nth(A,Xsa),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(product_prod(nat,A),nat,product_fst(nat,A),P2)),N)) = aa(product_prod(nat,A),A,product_snd(nat,A),P2) ) ) ) ).

% in_set_enumerate_eq
tff(fact_4964_prod__decode__aux_Opelims,axiom,
    ! [Xa: nat,Xaa: nat,Y: product_prod(nat,nat)] :
      ( ( nat_prod_decode_aux(Xa,Xaa) = 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),Xa),Xaa))
       => ~ ( ( Y = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xaa),Xa),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xaa),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Xa),Xaa)),nat_prod_decode_aux(aa(nat,nat,suc,Xa),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Xaa),aa(nat,nat,suc,Xa)))) )
           => ~ 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),Xa),Xaa)) ) ) ) ).

% prod_decode_aux.pelims
tff(fact_4965_mergesort__by__rel__split__length,axiom,
    ! [A: $tType,Xs12: list(A),Xs23: list(A),Xsa: 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)),Xs12),Xs23),Xsa))) = 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)),Xs12)),divide_divide(nat,aa(list(A),nat,size_size(list(A)),Xsa),numeral_numeral(nat,bit0(one2))))),modulo_modulo(nat,aa(list(A),nat,size_size(list(A)),Xsa),numeral_numeral(nat,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)),Xs12),Xs23),Xsa))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),Xs23)),divide_divide(nat,aa(list(A),nat,size_size(list(A)),Xsa),numeral_numeral(nat,bit0(one2)))) ) ) ).

% mergesort_by_rel_split_length
tff(fact_4966_mset__mergesort__by__rel__split,axiom,
    ! [A: $tType,Xs12: list(A),Xs23: list(A),Xsa: 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)),Xs12),Xs23),Xsa)))),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)),Xs12),Xs23),Xsa)))) = 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),Xsa)),aa(list(A),multiset(A),mset(A),Xs12))),aa(list(A),multiset(A),mset(A),Xs23)) ) ).

% mset_mergesort_by_rel_split
tff(fact_4967_mergesort__by__rel__split_Osimps_I3_J,axiom,
    ! [A: $tType,Xs12: list(A),Xs23: list(A),X1: A,X22: A,Xsa: 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)),Xs12),Xs23),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),Xsa))) = 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),Xs12)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X22),Xs23)),Xsa) ) ).

% mergesort_by_rel_split.simps(3)
tff(fact_4968_mergesort__by__rel__split_Osimps_I1_J,axiom,
    ! [A: $tType,Xs12: list(A),Xs23: 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)),Xs12),Xs23),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)),Xs12),Xs23) ) ).

% mergesort_by_rel_split.simps(1)
tff(fact_4969_mergesort__by__rel__split_Osimps_I2_J,axiom,
    ! [A: $tType,Xs12: list(A),Xs23: list(A),Xa: 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)),Xs12),Xs23),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),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),Xa),Xs12)),Xs23) ) ).

% mergesort_by_rel_split.simps(2)
tff(fact_4970_mergesort__by__rel__split_Oelims,axiom,
    ! [A: $tType,Xa: product_prod(list(A),list(A)),Xaa: list(A),Y: product_prod(list(A),list(A))] :
      ( ( merges295452479951948502_split(A,Xa,Xaa) = Y )
     => ( ! [Xs1: list(A),Xs22: 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)),Xs1),Xs22) )
           => ( ( Xaa = 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)),Xs1),Xs22) ) ) )
       => ( ! [Xs1: list(A),Xs22: 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)),Xs1),Xs22) )
             => ! [X: A] :
                  ( ( Xaa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),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),X),Xs1)),Xs22) ) ) )
         => ~ ! [Xs1: list(A),Xs22: 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)),Xs1),Xs22) )
               => ! [X12: A,X23: A,Xs: list(A)] :
                    ( ( Xaa = 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),Xs)) )
                   => ( 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),Xs1)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X23),Xs22)),Xs) ) ) ) ) ) ) ).

% mergesort_by_rel_split.elims
tff(fact_4971_FI__def,axiom,
    ! [M2: list(product_prod(assn,assn)),P2: assn,Q2: assn,Up: assn,Uq: assn,F2: assn] :
      ( fi(M2,P2,Q2,Up,Uq,F2)
    <=> ( ! [X6: product_prod(assn,assn)] :
            ( member(product_prod(assn,assn),X6,aa(list(product_prod(assn,assn)),set(product_prod(assn,assn)),set2(product_prod(assn,assn)),M2))
           => aa(product_prod(assn,assn),$o,aa(fun(assn,fun(assn,$o)),fun(product_prod(assn,assn),$o),product_case_prod(assn,assn,$o),entails),X6) )
       => aa(assn,$o,aa(assn,fun(assn,$o),entails,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(list(assn),fun(assn,assn),aa(fun(assn,fun(assn,assn)),fun(list(assn),fun(assn,assn)),foldr(assn,assn),times_times(assn)),aa(list(product_prod(assn,assn)),list(assn),aa(fun(product_prod(assn,assn),assn),fun(list(product_prod(assn,assn)),list(assn)),map(product_prod(assn,assn),assn),product_fst(assn,assn)),M2)),one_one(assn))),P2)),Up)),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(list(assn),fun(assn,assn),aa(fun(assn,fun(assn,assn)),fun(list(assn),fun(assn,assn)),foldr(assn,assn),times_times(assn)),aa(list(product_prod(assn,assn)),list(assn),aa(fun(product_prod(assn,assn),assn),fun(list(product_prod(assn,assn)),list(assn)),map(product_prod(assn,assn),assn),product_snd(assn,assn)),M2)),one_one(assn))),Q2)),Uq)),F2)) ) ) ).

% FI_def
tff(fact_4972_conjI__realizer,axiom,
    ! [A: $tType,B: $tType,Pa: fun(A,$o),P2: A,Qa: fun(B,$o),Q2: B] :
      ( aa(A,$o,Pa,P2)
     => ( aa(B,$o,Qa,Q2)
       => ( aa(A,$o,Pa,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),P2),Q2)))
          & aa(B,$o,Qa,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),P2),Q2))) ) ) ) ).

% conjI_realizer
tff(fact_4973_exI__realizer,axiom,
    ! [B: $tType,A: $tType,Pa: fun(A,fun(B,$o)),Y: A,Xa: B] :
      ( aa(B,$o,aa(A,fun(B,$o),Pa,Y),Xa)
     => aa(B,$o,aa(A,fun(B,$o),Pa,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),Xa),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),Xa),Y))) ) ).

% exI_realizer
tff(fact_4974_FI__p__nomatch,axiom,
    ! [M2: list(product_prod(assn,assn)),Ps3: assn,Qs: assn,Q2: assn,P2: assn,Up: assn,Uq: assn,F2: assn] :
      ( fi(M2,Ps3,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Qs),Q2),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P2),Up),Uq,F2)
     => fi(M2,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Ps3),P2),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Qs),Q2),Up,Uq,F2) ) ).

% FI_p_nomatch
tff(fact_4975_FI__finalize,axiom,
    ! [M2: list(product_prod(assn,assn)),P2: assn,Up: assn,Q2: assn,Uq: assn,F2: assn] :
      ( fI_RESULT(M2,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P2),Up),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Q2),Uq),F2)
     => fi(M2,P2,Q2,Up,Uq,F2) ) ).

% FI_finalize
tff(fact_4976_set__relcomp,axiom,
    ! [B: $tType,E: $tType,A: $tType,Xys: list(product_prod(A,E)),Yzs: list(product_prod(E,B))] : ( relcomp(A,E,B,aa(list(product_prod(A,E)),set(product_prod(A,E)),set2(product_prod(A,E)),Xys),aa(list(product_prod(E,B)),set(product_prod(E,B)),set2(product_prod(E,B)),Yzs)) = aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),aa(list(list(product_prod(A,B))),list(product_prod(A,B)),concat(product_prod(A,B)),aa(list(product_prod(A,E)),list(list(product_prod(A,B))),aa(fun(product_prod(A,E),list(product_prod(A,B))),fun(list(product_prod(A,E)),list(list(product_prod(A,B)))),map(product_prod(A,E),list(product_prod(A,B))),aTP_Lamp_pf(list(product_prod(E,B)),fun(product_prod(A,E),list(product_prod(A,B))),Yzs)),Xys))) ) ).

% set_relcomp
tff(fact_4977_mergesort__by__rel_Opinduct,axiom,
    ! [A: $tType,A0: fun(A,fun(A,$o)),A1: list(A),Pa: 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,$o)),list(A)),A0),A1))
     => ( ! [R4: fun(A,fun(A,$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,$o)),list(A)),R4),Xs))
           => ( ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),Xs)),numeral_numeral(nat,bit0(one2)))
               => aa(list(A),$o,aa(fun(A,fun(A,$o)),fun(list(A),$o),Pa,R4),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(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),Xs)),numeral_numeral(nat,bit0(one2)))
                 => aa(list(A),$o,aa(fun(A,fun(A,$o)),fun(list(A),$o),Pa,R4),aa(product_prod(list(A),list(A)),list(A),product_snd(list(A),list(A)),merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),Xs))) )
               => aa(list(A),$o,aa(fun(A,fun(A,$o)),fun(list(A),$o),Pa,R4),Xs) ) ) )
       => aa(list(A),$o,aa(fun(A,fun(A,$o)),fun(list(A),$o),Pa,A0),A1) ) ) ).

% mergesort_by_rel.pinduct
tff(fact_4978_insert__relcomp__union__fold,axiom,
    ! [E: $tType,B: $tType,A: $tType,S: set(product_prod(A,B)),Xa: product_prod(E,A),X3: set(product_prod(E,B))] :
      ( aa(set(product_prod(A,B)),$o,finite_finite2(product_prod(A,B)),S)
     => ( aa(set(product_prod(E,B)),set(product_prod(E,B)),aa(set(product_prod(E,B)),fun(set(product_prod(E,B)),set(product_prod(E,B))),sup_sup(set(product_prod(E,B))),relcomp(E,A,B,aa(set(product_prod(E,A)),set(product_prod(E,A)),aa(product_prod(E,A),fun(set(product_prod(E,A)),set(product_prod(E,A))),insert3(product_prod(E,A)),Xa),bot_bot(set(product_prod(E,A)))),S)),X3) = finite_fold(product_prod(A,B),set(product_prod(E,B)),aa(fun(A,fun(B,fun(set(product_prod(E,B)),set(product_prod(E,B))))),fun(product_prod(A,B),fun(set(product_prod(E,B)),set(product_prod(E,B)))),product_case_prod(A,B,fun(set(product_prod(E,B)),set(product_prod(E,B)))),aTP_Lamp_pg(product_prod(E,A),fun(A,fun(B,fun(set(product_prod(E,B)),set(product_prod(E,B))))),Xa)),X3,S) ) ) ).

% insert_relcomp_union_fold
tff(fact_4979_relcomp__empty1,axiom,
    ! [E: $tType,B: $tType,A: $tType,R: set(product_prod(E,B))] : ( relcomp(A,E,B,bot_bot(set(product_prod(A,E))),R) = bot_bot(set(product_prod(A,B))) ) ).

% relcomp_empty1
tff(fact_4980_relcomp__empty2,axiom,
    ! [E: $tType,B: $tType,A: $tType,R: set(product_prod(A,E))] : ( relcomp(A,E,B,R,bot_bot(set(product_prod(E,B)))) = bot_bot(set(product_prod(A,B))) ) ).

% relcomp_empty2
tff(fact_4981_relcomp__distrib2,axiom,
    ! [A: $tType,B: $tType,E: $tType,S: set(product_prod(A,E)),T4: set(product_prod(A,E)),R: set(product_prod(E,B))] : ( relcomp(A,E,B,aa(set(product_prod(A,E)),set(product_prod(A,E)),aa(set(product_prod(A,E)),fun(set(product_prod(A,E)),set(product_prod(A,E))),sup_sup(set(product_prod(A,E))),S),T4),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,E,B,S,R)),relcomp(A,E,B,T4,R)) ) ).

% relcomp_distrib2
tff(fact_4982_relcomp__distrib,axiom,
    ! [A: $tType,B: $tType,E: $tType,R: set(product_prod(A,E)),S: set(product_prod(E,B)),T4: set(product_prod(E,B))] : ( relcomp(A,E,B,R,aa(set(product_prod(E,B)),set(product_prod(E,B)),aa(set(product_prod(E,B)),fun(set(product_prod(E,B)),set(product_prod(E,B))),sup_sup(set(product_prod(E,B))),S),T4)) = 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,E,B,R,S)),relcomp(A,E,B,R,T4)) ) ).

% relcomp_distrib
tff(fact_4983_Id__O__R,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,B))] : ( relcomp(A,A,B,id2(A),R) = R ) ).

% Id_O_R
tff(fact_4984_R__O__Id,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,B))] : ( relcomp(A,B,B,R,id2(B)) = R ) ).

% R_O_Id
tff(fact_4985_relcomp__mono,axiom,
    ! [A: $tType,E: $tType,B: $tType,R5: set(product_prod(A,B)),R2: set(product_prod(A,B)),S4: set(product_prod(B,E)),S2: set(product_prod(B,E))] :
      ( aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),R5),R2)
     => ( aa(set(product_prod(B,E)),$o,aa(set(product_prod(B,E)),fun(set(product_prod(B,E)),$o),ord_less_eq(set(product_prod(B,E))),S4),S2)
       => aa(set(product_prod(A,E)),$o,aa(set(product_prod(A,E)),fun(set(product_prod(A,E)),$o),ord_less_eq(set(product_prod(A,E))),relcomp(A,B,E,R5,S4)),relcomp(A,B,E,R2,S2)) ) ) ).

% relcomp_mono
tff(fact_4986_relcompEpair,axiom,
    ! [A: $tType,B: $tType,E: $tType,A3: A,C2: B,R2: set(product_prod(A,E)),S2: set(product_prod(E,B))] :
      ( 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,E,B,R2,S2))
     => ~ ! [B4: E] :
            ( member(product_prod(A,E),aa(E,product_prod(A,E),aa(A,fun(E,product_prod(A,E)),product_Pair(A,E),A3),B4),R2)
           => ~ member(product_prod(E,B),aa(B,product_prod(E,B),aa(E,fun(B,product_prod(E,B)),product_Pair(E,B),B4),C2),S2) ) ) ).

% relcompEpair
tff(fact_4987_relcompE,axiom,
    ! [A: $tType,B: $tType,E: $tType,Xz: product_prod(A,B),R2: set(product_prod(A,E)),S2: set(product_prod(E,B))] :
      ( member(product_prod(A,B),Xz,relcomp(A,E,B,R2,S2))
     => ~ ! [X: A,Y2: E,Z3: B] :
            ( ( Xz = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Z3) )
           => ( member(product_prod(A,E),aa(E,product_prod(A,E),aa(A,fun(E,product_prod(A,E)),product_Pair(A,E),X),Y2),R2)
             => ~ member(product_prod(E,B),aa(B,product_prod(E,B),aa(E,fun(B,product_prod(E,B)),product_Pair(E,B),Y2),Z3),S2) ) ) ) ).

% relcompE
tff(fact_4988_relcomp_OrelcompI,axiom,
    ! [A: $tType,E: $tType,B: $tType,A3: A,B3: B,R2: set(product_prod(A,B)),C2: E,S2: set(product_prod(B,E))] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B3),R2)
     => ( member(product_prod(B,E),aa(E,product_prod(B,E),aa(B,fun(E,product_prod(B,E)),product_Pair(B,E),B3),C2),S2)
       => member(product_prod(A,E),aa(E,product_prod(A,E),aa(A,fun(E,product_prod(A,E)),product_Pair(A,E),A3),C2),relcomp(A,B,E,R2,S2)) ) ) ).

% relcomp.relcompI
tff(fact_4989_relcomp_Osimps,axiom,
    ! [A: $tType,B: $tType,E: $tType,A1: A,A22: B,R2: set(product_prod(A,E)),S2: set(product_prod(E,B))] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A1),A22),relcomp(A,E,B,R2,S2))
    <=> ? [A7: A,B8: E,C6: B] :
          ( ( A1 = A7 )
          & ( A22 = C6 )
          & member(product_prod(A,E),aa(E,product_prod(A,E),aa(A,fun(E,product_prod(A,E)),product_Pair(A,E),A7),B8),R2)
          & member(product_prod(E,B),aa(B,product_prod(E,B),aa(E,fun(B,product_prod(E,B)),product_Pair(E,B),B8),C6),S2) ) ) ).

% relcomp.simps
tff(fact_4990_relcomp_Ocases,axiom,
    ! [A: $tType,B: $tType,E: $tType,A1: A,A22: B,R2: set(product_prod(A,E)),S2: set(product_prod(E,B))] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A1),A22),relcomp(A,E,B,R2,S2))
     => ~ ! [B4: E] :
            ( member(product_prod(A,E),aa(E,product_prod(A,E),aa(A,fun(E,product_prod(A,E)),product_Pair(A,E),A1),B4),R2)
           => ~ member(product_prod(E,B),aa(B,product_prod(E,B),aa(E,fun(B,product_prod(E,B)),product_Pair(E,B),B4),A22),S2) ) ) ).

% relcomp.cases
tff(fact_4991_O__assoc,axiom,
    ! [A: $tType,F4: $tType,B: $tType,E: $tType,R: set(product_prod(A,F4)),S: set(product_prod(F4,E)),T4: set(product_prod(E,B))] : ( relcomp(A,E,B,relcomp(A,F4,E,R,S),T4) = relcomp(A,F4,B,R,relcomp(F4,E,B,S,T4)) ) ).

% O_assoc
tff(fact_4992_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)))),insert3(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_4993_trans__O__subset,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( trans(A,R2)
     => aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),relcomp(A,A,A,R2,R2)),R2) ) ).

% trans_O_subset
tff(fact_4994_relcomp__UNION__distrib2,axiom,
    ! [E: $tType,B: $tType,A: $tType,F4: $tType,R2: fun(F4,set(product_prod(A,E))),Ia2: set(F4),S2: set(product_prod(E,B))] : ( relcomp(A,E,B,aa(set(set(product_prod(A,E))),set(product_prod(A,E)),complete_Sup_Sup(set(product_prod(A,E))),aa(set(F4),set(set(product_prod(A,E))),image2(F4,set(product_prod(A,E)),R2),Ia2)),S2) = aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),aa(set(F4),set(set(product_prod(A,B))),image2(F4,set(product_prod(A,B)),aa(set(product_prod(E,B)),fun(F4,set(product_prod(A,B))),aTP_Lamp_ph(fun(F4,set(product_prod(A,E))),fun(set(product_prod(E,B)),fun(F4,set(product_prod(A,B)))),R2),S2)),Ia2)) ) ).

% relcomp_UNION_distrib2
tff(fact_4995_relcomp__UNION__distrib,axiom,
    ! [E: $tType,B: $tType,A: $tType,F4: $tType,S2: set(product_prod(A,E)),R2: fun(F4,set(product_prod(E,B))),Ia2: set(F4)] : ( relcomp(A,E,B,S2,aa(set(set(product_prod(E,B))),set(product_prod(E,B)),complete_Sup_Sup(set(product_prod(E,B))),aa(set(F4),set(set(product_prod(E,B))),image2(F4,set(product_prod(E,B)),R2),Ia2))) = aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),aa(set(F4),set(set(product_prod(A,B))),image2(F4,set(product_prod(A,B)),aa(fun(F4,set(product_prod(E,B))),fun(F4,set(product_prod(A,B))),aTP_Lamp_pi(set(product_prod(A,E)),fun(fun(F4,set(product_prod(E,B))),fun(F4,set(product_prod(A,B)))),S2),R2)),Ia2)) ) ).

% relcomp_UNION_distrib
tff(fact_4996_relcomp__unfold,axiom,
    ! [A: $tType,B: $tType,E: $tType,R2: set(product_prod(A,E)),S2: set(product_prod(E,B))] : ( relcomp(A,E,B,R2,S2) = aa(fun(product_prod(A,B),$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,$o),aa(set(product_prod(E,B)),fun(A,fun(B,$o)),aTP_Lamp_pj(set(product_prod(A,E)),fun(set(product_prod(E,B)),fun(A,fun(B,$o))),R2),S2))) ) ).

% relcomp_unfold
tff(fact_4997_relcomp__fold,axiom,
    ! [E: $tType,B: $tType,A: $tType,R: set(product_prod(A,B)),S: set(product_prod(B,E))] :
      ( aa(set(product_prod(A,B)),$o,finite_finite2(product_prod(A,B)),R)
     => ( aa(set(product_prod(B,E)),$o,finite_finite2(product_prod(B,E)),S)
       => ( relcomp(A,B,E,R,S) = finite_fold(product_prod(A,B),set(product_prod(A,E)),aa(fun(A,fun(B,fun(set(product_prod(A,E)),set(product_prod(A,E))))),fun(product_prod(A,B),fun(set(product_prod(A,E)),set(product_prod(A,E)))),product_case_prod(A,B,fun(set(product_prod(A,E)),set(product_prod(A,E)))),aTP_Lamp_pl(set(product_prod(B,E)),fun(A,fun(B,fun(set(product_prod(A,E)),set(product_prod(A,E))))),S)),bot_bot(set(product_prod(A,E))),R) ) ) ) ).

% relcomp_fold
tff(fact_4998_insert__relcomp__fold,axiom,
    ! [E: $tType,B: $tType,A: $tType,S: set(product_prod(A,B)),Xa: product_prod(E,A),R: set(product_prod(E,A))] :
      ( aa(set(product_prod(A,B)),$o,finite_finite2(product_prod(A,B)),S)
     => ( relcomp(E,A,B,aa(set(product_prod(E,A)),set(product_prod(E,A)),aa(product_prod(E,A),fun(set(product_prod(E,A)),set(product_prod(E,A))),insert3(product_prod(E,A)),Xa),R),S) = finite_fold(product_prod(A,B),set(product_prod(E,B)),aa(fun(A,fun(B,fun(set(product_prod(E,B)),set(product_prod(E,B))))),fun(product_prod(A,B),fun(set(product_prod(E,B)),set(product_prod(E,B)))),product_case_prod(A,B,fun(set(product_prod(E,B)),set(product_prod(E,B)))),aTP_Lamp_pg(product_prod(E,A),fun(A,fun(B,fun(set(product_prod(E,B)),set(product_prod(E,B))))),Xa)),relcomp(E,A,B,R,S),S) ) ) ).

% insert_relcomp_fold
tff(fact_4999_arg__min__list_Opelims,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Xa: fun(A,B),Xaa: list(A),Y: A] :
          ( ( arg_min_list(A,B,Xa,Xaa) = 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)),Xa),Xaa))
           => ( ! [X: A] :
                  ( ( Xaa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)) )
                 => ( ( Y = X )
                   => ~ 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)),Xa),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)))) ) )
             => ( ! [X: A,Y2: A,Zs2: list(A)] :
                    ( ( Xaa = 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),Y2),Zs2)) )
                   => ( ( Y = $let(
                            m: A,
                            m:= arg_min_list(A,B,Xa,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,Xa,X)),aa(A,B,Xa,m)),X,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)),Xa),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),Y2),Zs2)))) ) )
               => ~ ( ( Xaa = 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)),Xa),nil(A))) ) ) ) ) ) ) ) ).

% arg_min_list.pelims
tff(fact_5000_sorted__wrt_Opelims_I1_J,axiom,
    ! [A: $tType,Xa: fun(A,fun(A,$o)),Xaa: list(A),Y: $o] :
      ( ( sorted_wrt(A,Xa,Xaa)
      <=> (Y) )
     => ( accp(product_prod(fun(A,fun(A,$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)),Xa),Xaa))
       => ( ( ( Xaa = 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,$o)),list(A)),Xa),nil(A))) ) )
         => ~ ! [X: A,Ys5: list(A)] :
                ( ( Xaa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Ys5) )
               => ( ( (Y)
                  <=> ( ! [Xa2: A] :
                          ( member(A,Xa2,aa(list(A),set(A),set2(A),Ys5))
                         => aa(A,$o,aa(A,fun(A,$o),Xa,X),Xa2) )
                      & sorted_wrt(A,Xa,Ys5) ) )
                 => ~ accp(product_prod(fun(A,fun(A,$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)),Xa),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Ys5))) ) ) ) ) ) ).

% sorted_wrt.pelims(1)
tff(fact_5001_sorted__wrt_Opelims_I2_J,axiom,
    ! [A: $tType,Xa: fun(A,fun(A,$o)),Xaa: list(A)] :
      ( sorted_wrt(A,Xa,Xaa)
     => ( 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,$o)),list(A)),Xa),Xaa))
       => ( ( ( Xaa = nil(A) )
           => ~ accp(product_prod(fun(A,fun(A,$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)),Xa),nil(A))) )
         => ~ ! [X: A,Ys5: list(A)] :
                ( ( Xaa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Ys5) )
               => ( 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,$o)),list(A)),Xa),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Ys5)))
                 => ~ ( ! [Xa4: A] :
                          ( member(A,Xa4,aa(list(A),set(A),set2(A),Ys5))
                         => aa(A,$o,aa(A,fun(A,$o),Xa,X),Xa4) )
                      & sorted_wrt(A,Xa,Ys5) ) ) ) ) ) ) ).

% sorted_wrt.pelims(2)
tff(fact_5002_sorted__wrt_Opelims_I3_J,axiom,
    ! [A: $tType,Xa: fun(A,fun(A,$o)),Xaa: list(A)] :
      ( ~ sorted_wrt(A,Xa,Xaa)
     => ( 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,$o)),list(A)),Xa),Xaa))
       => ~ ! [X: A,Ys5: list(A)] :
              ( ( Xaa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Ys5) )
             => ( accp(product_prod(fun(A,fun(A,$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)),Xa),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Ys5)))
               => ( ! [Xa5: A] :
                      ( member(A,Xa5,aa(list(A),set(A),set2(A),Ys5))
                     => aa(A,$o,aa(A,fun(A,$o),Xa,X),Xa5) )
                  & sorted_wrt(A,Xa,Ys5) ) ) ) ) ) ).

% sorted_wrt.pelims(3)
tff(fact_5003_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)))),insert3(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_5004_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_5005_mergesort__by__rel_Opsimps,axiom,
    ! [A: $tType,R: fun(A,fun(A,$o)),Xsa: 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),Xsa))
     => ( aa(list(A),list(A),mergesort_by_rel(A,R),Xsa) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),Xsa)),numeral_numeral(nat,bit0(one2))),Xsa,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)),Xsa))),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)),Xsa))))) ) ) ).

% mergesort_by_rel.psimps
tff(fact_5006_mergesort__by__rel__simps_I1_J,axiom,
    ! [A: $tType,R: fun(A,fun(A,$o))] : ( aa(list(A),list(A),mergesort_by_rel(A,R),nil(A)) = nil(A) ) ).

% mergesort_by_rel_simps(1)
tff(fact_5007_set__mergesort__by__rel,axiom,
    ! [A: $tType,R: fun(A,fun(A,$o)),Xsa: list(A)] : ( aa(list(A),set(A),set2(A),aa(list(A),list(A),mergesort_by_rel(A,R),Xsa)) = aa(list(A),set(A),set2(A),Xsa) ) ).

% set_mergesort_by_rel
tff(fact_5008_mergesort__by__rel__merge__simps_I3_J,axiom,
    ! [A: $tType,R: fun(A,fun(A,$o)),Ys: list(A)] : ( merges9089515139780605204_merge(A,R,nil(A),Ys) = Ys ) ).

% mergesort_by_rel_merge_simps(3)
tff(fact_5009_sorted__wrt__mergesort__by__rel__merge,axiom,
    ! [A: $tType,R: fun(A,fun(A,$o)),Xsa: list(A),Ys: list(A)] :
      ( ! [X: A,Y2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),R,X),Y2)
          | aa(A,$o,aa(A,fun(A,$o),R,Y2),X) )
     => ( ! [X: A,Y2: A,Z3: A] :
            ( aa(A,$o,aa(A,fun(A,$o),R,X),Y2)
           => ( aa(A,$o,aa(A,fun(A,$o),R,Y2),Z3)
             => aa(A,$o,aa(A,fun(A,$o),R,X),Z3) ) )
       => ( sorted_wrt(A,R,merges9089515139780605204_merge(A,R,Xsa,Ys))
        <=> ( sorted_wrt(A,R,Xsa)
            & sorted_wrt(A,R,Ys) ) ) ) ) ).

% sorted_wrt_mergesort_by_rel_merge
tff(fact_5010_mergesort__by__rel__permutes,axiom,
    ! [A: $tType,R: fun(A,fun(A,$o)),Xsa: list(A)] : ( aa(list(A),multiset(A),mset(A),aa(list(A),list(A),mergesort_by_rel(A,R),Xsa)) = aa(list(A),multiset(A),mset(A),Xsa) ) ).

% mergesort_by_rel_permutes
tff(fact_5011_mergesort__by__rel__simps_I2_J,axiom,
    ! [A: $tType,R: fun(A,fun(A,$o)),Xa: 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),Xa),nil(A))) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),nil(A)) ) ).

% mergesort_by_rel_simps(2)
tff(fact_5012_set__mergesort__by__rel__merge,axiom,
    ! [A: $tType,R: fun(A,fun(A,$o)),Xsa: list(A),Ys: list(A)] : ( aa(list(A),set(A),set2(A),merges9089515139780605204_merge(A,R,Xsa,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),Xsa)),aa(list(A),set(A),set2(A),Ys)) ) ).

% set_mergesort_by_rel_merge
tff(fact_5013_mset__mergesort__by__rel__merge,axiom,
    ! [A: $tType,R: fun(A,fun(A,$o)),Xsa: list(A),Ys: list(A)] : ( aa(list(A),multiset(A),mset(A),merges9089515139780605204_merge(A,R,Xsa,Ys)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(list(A),multiset(A),mset(A),Xsa)),aa(list(A),multiset(A),mset(A),Ys)) ) ).

% mset_mergesort_by_rel_merge
tff(fact_5014_mergesort__by__rel__simps_I3_J,axiom,
    ! [A: $tType,R: fun(A,fun(A,$o)),X1: A,X22: A,Xsa: 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),Xsa))) = 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_pm(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))),Xsa)) ) ).

% mergesort_by_rel_simps(3)
tff(fact_5015_mergesort__by__rel__merge__simps_I1_J,axiom,
    ! [A: $tType,R: fun(A,fun(A,$o)),Xa: A,Xsa: list(A),Y: A,Ys: list(A)] :
      ( merges9089515139780605204_merge(A,R,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)) = $ite(aa(A,$o,aa(A,fun(A,$o),R,Xa),Y),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),merges9089515139780605204_merge(A,R,Xsa,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys))),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),merges9089515139780605204_merge(A,R,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa),Ys))) ) ).

% mergesort_by_rel_merge_simps(1)
tff(fact_5016_mergesort__by__rel__merge__simps_I2_J,axiom,
    ! [A: $tType,R: fun(A,fun(A,$o)),Xsa: list(A)] : ( merges9089515139780605204_merge(A,R,Xsa,nil(A)) = Xsa ) ).

% mergesort_by_rel_merge_simps(2)
tff(fact_5017_sorted__wrt__mergesort__by__rel,axiom,
    ! [A: $tType,R: fun(A,fun(A,$o)),Xsa: list(A)] :
      ( ! [X: A,Y2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),R,X),Y2)
          | aa(A,$o,aa(A,fun(A,$o),R,Y2),X) )
     => ( ! [X: A,Y2: A,Z3: A] :
            ( aa(A,$o,aa(A,fun(A,$o),R,X),Y2)
           => ( aa(A,$o,aa(A,fun(A,$o),R,Y2),Z3)
             => aa(A,$o,aa(A,fun(A,$o),R,X),Z3) ) )
       => sorted_wrt(A,R,aa(list(A),list(A),mergesort_by_rel(A,R),Xsa)) ) ) ).

% sorted_wrt_mergesort_by_rel
tff(fact_5018_sorted__mergesort__by__rel,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xsa: list(A)] : sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),mergesort_by_rel(A,ord_less_eq(A)),Xsa)) ) ).

% sorted_mergesort_by_rel
tff(fact_5019_mergesort__by__rel__merge_Osimps_I3_J,axiom,
    ! [A: $tType,R: fun(A,fun(A,$o)),V2: A,Va2: list(A)] : ( merges9089515139780605204_merge(A,R,nil(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V2),Va2)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V2),Va2) ) ).

% mergesort_by_rel_merge.simps(3)
tff(fact_5020_mergesort__by__rel__merge_Oelims,axiom,
    ! [A: $tType,Xa: fun(A,fun(A,$o)),Xaa: list(A),Xb: list(A),Y: list(A)] :
      ( ( merges9089515139780605204_merge(A,Xa,Xaa,Xb) = Y )
     => ( ! [X: A,Xs: list(A)] :
            ( ( Xaa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) )
           => ! [Y2: A,Ys5: list(A)] :
                ( ( Xb = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys5) )
               => ( Y != $ite(aa(A,$o,aa(A,fun(A,$o),Xa,X),Y2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),merges9089515139780605204_merge(A,Xa,Xs,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys5))),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),merges9089515139780605204_merge(A,Xa,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs),Ys5))) ) ) )
       => ( ( ( Xb = nil(A) )
           => ( Y != Xaa ) )
         => ~ ( ( Xaa = nil(A) )
             => ! [V3: A,Va: list(A)] :
                  ( ( Xb = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V3),Va) )
                 => ( Y != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V3),Va) ) ) ) ) ) ) ).

% mergesort_by_rel_merge.elims
tff(fact_5021_sort__mergesort__by__rel,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( aa(fun(A,A),fun(list(A),list(A)),linorder_sort_key(A,A),aTP_Lamp_ip(A,A)) = mergesort_by_rel(A,ord_less_eq(A)) ) ) ).

% sort_mergesort_by_rel
tff(fact_5022_mergesort__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ( mergesort(A) = mergesort_by_rel(A,ord_less_eq(A)) ) ) ).

% mergesort_def
tff(fact_5023_mergesort__by__rel_Oelims,axiom,
    ! [A: $tType,Xa: fun(A,fun(A,$o)),Xaa: list(A),Y: list(A)] :
      ( ( aa(list(A),list(A),mergesort_by_rel(A,Xa),Xaa) = Y )
     => ( Y = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),Xaa)),numeral_numeral(nat,bit0(one2))),Xaa,merges9089515139780605204_merge(A,Xa,aa(list(A),list(A),mergesort_by_rel(A,Xa),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)),Xaa))),aa(list(A),list(A),mergesort_by_rel(A,Xa),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)),Xaa))))) ) ) ).

% mergesort_by_rel.elims
tff(fact_5024_mergesort__by__rel_Osimps,axiom,
    ! [A: $tType,R: fun(A,fun(A,$o)),Xsa: list(A)] :
      ( aa(list(A),list(A),mergesort_by_rel(A,R),Xsa) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),Xsa)),numeral_numeral(nat,bit0(one2))),Xsa,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)),Xsa))),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)),Xsa))))) ) ).

% mergesort_by_rel.simps
tff(fact_5025_mergesort__by__rel_Opelims,axiom,
    ! [A: $tType,Xa: fun(A,fun(A,$o)),Xaa: list(A),Y: list(A)] :
      ( ( aa(list(A),list(A),mergesort_by_rel(A,Xa),Xaa) = 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)),Xa),Xaa))
       => ~ ( ( Y = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),Xaa)),numeral_numeral(nat,bit0(one2))),Xaa,merges9089515139780605204_merge(A,Xa,aa(list(A),list(A),mergesort_by_rel(A,Xa),aa(product_prod(list(A),list(A)),list(A),product_fst(list(A),list(A)),merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),Xaa))),aa(list(A),list(A),mergesort_by_rel(A,Xa),aa(product_prod(list(A),list(A)),list(A),product_snd(list(A),list(A)),merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),Xaa))))) )
           => ~ accp(product_prod(fun(A,fun(A,$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)),Xa),Xaa)) ) ) ) ).

% mergesort_by_rel.pelims
tff(fact_5026_mergesort__by__rel__merge_Opelims,axiom,
    ! [A: $tType,Xa: fun(A,fun(A,$o)),Xaa: list(A),Xb: list(A),Y: list(A)] :
      ( ( merges9089515139780605204_merge(A,Xa,Xaa,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))),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)),Xaa),Xb)))
       => ( ! [X: A,Xs: list(A)] :
              ( ( Xaa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) )
             => ! [Y2: A,Ys5: list(A)] :
                  ( ( Xb = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys5) )
                 => ( ( Y = $ite(aa(A,$o,aa(A,fun(A,$o),Xa,X),Y2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),merges9089515139780605204_merge(A,Xa,Xs,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys5))),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),merges9089515139780605204_merge(A,Xa,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs),Ys5))) )
                   => ~ accp(product_prod(fun(A,fun(A,$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))),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)),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),Y2),Ys5)))) ) ) )
         => ( ( ( Xb = nil(A) )
             => ( ( Y = Xaa )
               => ~ 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,$o)),product_prod(list(A),list(A))),Xa),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xaa),nil(A)))) ) )
           => ~ ( ( Xaa = nil(A) )
               => ! [V3: A,Va: list(A)] :
                    ( ( Xb = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V3),Va) )
                   => ( ( Y = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V3),Va) )
                     => ~ accp(product_prod(fun(A,fun(A,$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))),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)),nil(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V3),Va)))) ) ) ) ) ) ) ) ).

% mergesort_by_rel_merge.pelims
tff(fact_5027_comp__fun__commute__relcomp__fold,axiom,
    ! [E: $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(E,A),set(product_prod(E,B)),aa(fun(E,fun(A,fun(set(product_prod(E,B)),set(product_prod(E,B))))),fun(product_prod(E,A),fun(set(product_prod(E,B)),set(product_prod(E,B)))),product_case_prod(E,A,fun(set(product_prod(E,B)),set(product_prod(E,B)))),aTP_Lamp_po(set(product_prod(A,B)),fun(E,fun(A,fun(set(product_prod(E,B)),set(product_prod(E,B))))),S))) ) ).

% comp_fun_commute_relcomp_fold
tff(fact_5028_FI__init,axiom,
    ! [Pa: assn,Qa: assn,Fa: assn] :
      ( fi(nil(product_prod(assn,assn)),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),sln),Pa),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),sln),Qa),sln,sln,Fa)
     => fI_QUERY(Pa,Qa,Fa) ) ).

% FI_init
tff(fact_5029_comp__fun__commute__filter__fold,axiom,
    ! [A: $tType,Pa: fun(A,$o)] : finite6289374366891150609ommute(A,set(A),aTP_Lamp_km(fun(A,$o),fun(A,fun(set(A),set(A))),Pa)) ).

% comp_fun_commute_filter_fold
tff(fact_5030_SLN__right,axiom,
    ! [Pa: assn] : ( aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Pa),sln) = Pa ) ).

% SLN_right
tff(fact_5031_SLN__left,axiom,
    ! [Pa: assn] : ( aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),sln),Pa) = Pa ) ).

% SLN_left
tff(fact_5032_comp__fun__commute_Ofoldl__f__commute,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,fun(B,B)),A3: A,B3: B,Xsa: list(A)] :
      ( finite6289374366891150609ommute(A,B,F2)
     => ( aa(B,B,aa(A,fun(B,B),F2,A3),aa(list(A),B,aa(B,fun(list(A),B),aa(fun(B,fun(A,B)),fun(B,fun(list(A),B)),foldl(B,A),aTP_Lamp_pp(fun(A,fun(B,B)),fun(B,fun(A,B)),F2)),B3),Xsa)) = aa(list(A),B,aa(B,fun(list(A),B),aa(fun(B,fun(A,B)),fun(B,fun(list(A),B)),foldl(B,A),aTP_Lamp_pp(fun(A,fun(B,B)),fun(B,fun(A,B)),F2)),aa(B,B,aa(A,fun(B,B),F2,A3),B3)),Xsa) ) ) ).

% comp_fun_commute.foldl_f_commute
tff(fact_5033_SLN__def,axiom,
    sln = one_one(assn) ).

% SLN_def
tff(fact_5034_comp__fun__commute_Ofoldr__conv__foldl,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,fun(B,B)),Xsa: list(A),A3: B] :
      ( finite6289374366891150609ommute(A,B,F2)
     => ( aa(B,B,aa(list(A),fun(B,B),aa(fun(A,fun(B,B)),fun(list(A),fun(B,B)),foldr(A,B),F2),Xsa),A3) = aa(list(A),B,aa(B,fun(list(A),B),aa(fun(B,fun(A,B)),fun(B,fun(list(A),B)),foldl(B,A),aTP_Lamp_pp(fun(A,fun(B,B)),fun(B,fun(A,B)),F2)),A3),Xsa) ) ) ).

% comp_fun_commute.foldr_conv_foldl
tff(fact_5035_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_pq(set(A),fun(A,fun(B,fun(set(B),set(B)))),S))) ).

% comp_fun_commute_Image_fold
tff(fact_5036_comp__fun__commute__fold__insert,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,fun(B,B)),Aa2: set(A),Xa: A,Z2: B] :
      ( finite6289374366891150609ommute(A,B,F2)
     => ( aa(set(A),$o,finite_finite2(A),Aa2)
       => ( ~ member(A,Xa,Aa2)
         => ( finite_fold(A,B,F2,Z2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),Aa2)) = aa(B,B,aa(A,fun(B,B),F2,Xa),finite_fold(A,B,F2,Z2,Aa2)) ) ) ) ) ).

% comp_fun_commute_fold_insert
tff(fact_5037_FI__q__nomatch,axiom,
    ! [M2: list(product_prod(assn,assn)),Up: assn,Qs: assn,Q2: assn,Uq: assn,F2: assn] :
      ( fi(M2,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),sln),Up),Qs,sln,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Q2),Uq),F2)
     => fi(M2,sln,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Qs),Q2),Up,Uq,F2) ) ).

% FI_q_nomatch
tff(fact_5038_comp__fun__commute__product__fold,axiom,
    ! [B: $tType,A: $tType,B2: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),B2)
     => finite6289374366891150609ommute(B,set(product_prod(B,A)),aTP_Lamp_pr(set(A),fun(B,fun(set(product_prod(B,A)),set(product_prod(B,A)))),B2)) ) ).

% comp_fun_commute_product_fold
tff(fact_5039_FI__match,axiom,
    ! [P2: assn,Q2: assn,M2: list(product_prod(assn,assn)),Ps3: assn,Up: assn,Qs: assn,Uq: assn,F2: assn] :
      ( aa(assn,$o,aa(assn,fun(assn,$o),entails,P2),Q2)
     => ( fi(aa(list(product_prod(assn,assn)),list(product_prod(assn,assn)),aa(product_prod(assn,assn),fun(list(product_prod(assn,assn)),list(product_prod(assn,assn))),cons(product_prod(assn,assn)),aa(assn,product_prod(assn,assn),aa(assn,fun(assn,product_prod(assn,assn)),product_Pair(assn,assn),P2),Q2)),M2),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Ps3),Up),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Qs),Uq),sln,sln,F2)
       => fi(M2,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Ps3),P2),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Qs),Q2),Up,Uq,F2) ) ) ).

% FI_match
tff(fact_5040_range__prod,axiom,
    ! [B: $tType,A: $tType,E: $tType,F2: fun(E,product_prod(A,B))] : aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),aa(set(E),set(product_prod(A,B)),image2(E,product_prod(A,B),F2),top_top(set(E)))),product_Sigma(A,B,aa(set(E),set(A),image2(E,A,comp(product_prod(A,B),A,E,product_fst(A,B),F2)),top_top(set(E))),aTP_Lamp_ps(fun(E,product_prod(A,B)),fun(A,set(B)),F2))) ).

% range_prod
tff(fact_5041_apsnd__apfst,axiom,
    ! [A: $tType,B: $tType,E: $tType,F4: $tType,F2: fun(E,B),G: fun(F4,A),Xa: product_prod(F4,E)] : ( aa(product_prod(A,E),product_prod(A,B),aa(fun(E,B),fun(product_prod(A,E),product_prod(A,B)),product_apsnd(E,B,A),F2),aa(product_prod(F4,E),product_prod(A,E),product_apfst(F4,A,E,G),Xa)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(F4,A,G,aa(product_prod(F4,E),F4,product_fst(F4,E),Xa))),aa(E,B,F2,aa(product_prod(F4,E),E,product_snd(F4,E),Xa))) ) ).

% apsnd_apfst
tff(fact_5042_apfst__apsnd,axiom,
    ! [A: $tType,B: $tType,F4: $tType,E: $tType,F2: fun(E,A),G: fun(F4,B),Xa: product_prod(E,F4)] : ( aa(product_prod(E,B),product_prod(A,B),product_apfst(E,A,B,F2),aa(product_prod(E,F4),product_prod(E,B),aa(fun(F4,B),fun(product_prod(E,F4),product_prod(E,B)),product_apsnd(F4,B,E),G),Xa)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(E,A,F2,aa(product_prod(E,F4),E,product_fst(E,F4),Xa))),aa(F4,B,G,aa(product_prod(E,F4),F4,product_snd(E,F4),Xa))) ) ).

% apfst_apsnd
tff(fact_5043_SigmaI,axiom,
    ! [B: $tType,A: $tType,A3: A,Aa2: set(A),B3: B,B2: fun(A,set(B))] :
      ( member(A,A3,Aa2)
     => ( member(B,B3,aa(A,set(B),B2,A3))
       => member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B3),product_Sigma(A,B,Aa2,B2)) ) ) ).

% SigmaI
tff(fact_5044_mem__Sigma__iff,axiom,
    ! [B: $tType,A: $tType,A3: A,B3: B,Aa2: set(A),B2: fun(A,set(B))] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B3),product_Sigma(A,B,Aa2,B2))
    <=> ( member(A,A3,Aa2)
        & member(B,B3,aa(A,set(B),B2,A3)) ) ) ).

% mem_Sigma_iff
tff(fact_5045_apfst__conv,axiom,
    ! [E: $tType,A: $tType,B: $tType,F2: fun(E,A),Xa: E,Y: B] : ( aa(product_prod(E,B),product_prod(A,B),product_apfst(E,A,B,F2),aa(B,product_prod(E,B),aa(E,fun(B,product_prod(E,B)),product_Pair(E,B),Xa),Y)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(E,A,F2,Xa)),Y) ) ).

% apfst_conv
tff(fact_5046_Field__square,axiom,
    ! [A: $tType,Xa: set(A)] : ( aa(set(product_prod(A,A)),set(A),field2(A),product_Sigma(A,A,Xa,aTP_Lamp_pt(set(A),fun(A,set(A)),Xa))) = Xa ) ).

% Field_square
tff(fact_5047_Sigma__empty1,axiom,
    ! [B: $tType,A: $tType,B2: fun(A,set(B))] : ( product_Sigma(A,B,bot_bot(set(A)),B2) = bot_bot(set(product_prod(A,B))) ) ).

% Sigma_empty1
tff(fact_5048_Sigma__empty2,axiom,
    ! [B: $tType,A: $tType,Aa2: set(A)] : ( product_Sigma(A,B,Aa2,aTP_Lamp_pu(A,set(B))) = bot_bot(set(product_prod(A,B))) ) ).

% Sigma_empty2
tff(fact_5049_Times__empty,axiom,
    ! [A: $tType,B: $tType,Aa2: set(A),B2: set(B)] :
      ( ( product_Sigma(A,B,Aa2,aTP_Lamp_pv(set(B),fun(A,set(B)),B2)) = bot_bot(set(product_prod(A,B))) )
    <=> ( ( Aa2 = bot_bot(set(A)) )
        | ( B2 = bot_bot(set(B)) ) ) ) ).

% Times_empty
tff(fact_5050_fst__image__times,axiom,
    ! [B: $tType,A: $tType,Aa2: set(A),B2: set(B)] :
      ( aa(set(product_prod(A,B)),set(A),image2(product_prod(A,B),A,product_fst(A,B)),product_Sigma(A,B,Aa2,aTP_Lamp_pv(set(B),fun(A,set(B)),B2))) = $ite(B2 = bot_bot(set(B)),bot_bot(set(A)),Aa2) ) ).

% fst_image_times
tff(fact_5051_snd__image__times,axiom,
    ! [B: $tType,A: $tType,Aa2: set(B),B2: set(A)] :
      ( aa(set(product_prod(B,A)),set(A),image2(product_prod(B,A),A,product_snd(B,A)),product_Sigma(B,A,Aa2,aTP_Lamp_an(set(A),fun(B,set(A)),B2))) = $ite(Aa2 = bot_bot(set(B)),bot_bot(set(A)),B2) ) ).

% snd_image_times
tff(fact_5052_Sigma__UNIV__cancel,axiom,
    ! [B: $tType,A: $tType,Aa2: set(A),X3: 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,Aa2,aTP_Lamp_pv(set(B),fun(A,set(B)),X3))),product_Sigma(A,B,Aa2,aTP_Lamp_pw(A,set(B)))) = bot_bot(set(product_prod(A,B))) ) ).

% Sigma_UNIV_cancel
tff(fact_5053_set__product,axiom,
    ! [A: $tType,B: $tType,Xsa: list(A),Ys: list(B)] : ( aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),product(A,B),Xsa),Ys)) = product_Sigma(A,B,aa(list(A),set(A),set2(A),Xsa),aTP_Lamp_px(list(B),fun(A,set(B)),Ys)) ) ).

% set_product
tff(fact_5054_pairself__image__cart,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Aa2: set(B),B2: 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,F2)),product_Sigma(B,B,Aa2,aTP_Lamp_py(set(B),fun(B,set(B)),B2))) = product_Sigma(A,A,aa(set(B),set(A),image2(B,A,F2),Aa2),aa(set(B),fun(A,set(A)),aTP_Lamp_pz(fun(B,A),fun(set(B),fun(A,set(A))),F2),B2)) ) ).

% pairself_image_cart
tff(fact_5055_insert__Times__insert,axiom,
    ! [A: $tType,B: $tType,A3: A,Aa2: set(A),B3: B,B2: set(B)] : ( product_Sigma(A,B,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),Aa2),aa(set(B),fun(A,set(B)),aTP_Lamp_qa(B,fun(set(B),fun(A,set(B))),B3),B2)) = 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))),insert3(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),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,Aa2,aa(set(B),fun(A,set(B)),aTP_Lamp_qa(B,fun(set(B),fun(A,set(B))),B3),B2))),product_Sigma(A,B,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),Aa2),aTP_Lamp_pv(set(B),fun(A,set(B)),B2)))) ) ).

% insert_Times_insert
tff(fact_5056_times__eq__iff,axiom,
    ! [A: $tType,B: $tType,Aa2: set(A),B2: set(B),C3: set(A),D2: set(B)] :
      ( ( product_Sigma(A,B,Aa2,aTP_Lamp_pv(set(B),fun(A,set(B)),B2)) = product_Sigma(A,B,C3,aTP_Lamp_pv(set(B),fun(A,set(B)),D2)) )
    <=> ( ( ( Aa2 = C3 )
          & ( B2 = D2 ) )
        | ( ( ( Aa2 = bot_bot(set(A)) )
            | ( B2 = bot_bot(set(B)) ) )
          & ( ( C3 = bot_bot(set(A)) )
            | ( D2 = bot_bot(set(B)) ) ) ) ) ) ).

% times_eq_iff
tff(fact_5057_SigmaE,axiom,
    ! [A: $tType,B: $tType,C2: product_prod(A,B),Aa2: set(A),B2: fun(A,set(B))] :
      ( member(product_prod(A,B),C2,product_Sigma(A,B,Aa2,B2))
     => ~ ! [X: A] :
            ( member(A,X,Aa2)
           => ! [Y2: B] :
                ( member(B,Y2,aa(A,set(B),B2,X))
               => ( C2 != aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y2) ) ) ) ) ).

% SigmaE
tff(fact_5058_SigmaD1,axiom,
    ! [B: $tType,A: $tType,A3: A,B3: B,Aa2: set(A),B2: fun(A,set(B))] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B3),product_Sigma(A,B,Aa2,B2))
     => member(A,A3,Aa2) ) ).

% SigmaD1
tff(fact_5059_SigmaD2,axiom,
    ! [B: $tType,A: $tType,A3: A,B3: B,Aa2: set(A),B2: fun(A,set(B))] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B3),product_Sigma(A,B,Aa2,B2))
     => member(B,B3,aa(A,set(B),B2,A3)) ) ).

% SigmaD2
tff(fact_5060_SigmaE2,axiom,
    ! [B: $tType,A: $tType,A3: A,B3: B,Aa2: set(A),B2: fun(A,set(B))] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B3),product_Sigma(A,B,Aa2,B2))
     => ~ ( member(A,A3,Aa2)
         => ~ member(B,B3,aa(A,set(B),B2,A3)) ) ) ).

% SigmaE2
tff(fact_5061_mult__fun__commute,axiom,
    ! [A: $tType,F2: fun(A,assn)] : finite6289374366891150609ommute(A,assn,aTP_Lamp_qb(fun(A,assn),fun(A,fun(assn,assn)),F2)) ).

% mult_fun_commute
tff(fact_5062_Sigma__Diff__distrib1,axiom,
    ! [B: $tType,A: $tType,Ia2: set(A),J4: set(A),C3: fun(A,set(B))] : ( product_Sigma(A,B,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Ia2),J4),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,Ia2,C3)),product_Sigma(A,B,J4,C3)) ) ).

% Sigma_Diff_distrib1
tff(fact_5063_comp__fun__commute__insort,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => finite6289374366891150609ommute(A,list(A),linorder_insort_key(A,A,aTP_Lamp_ip(A,A))) ) ).

% comp_fun_commute_insort
tff(fact_5064_in__prod__fst__sndI,axiom,
    ! [A: $tType,B: $tType,Xa: product_prod(A,B),Aa2: set(A),B2: set(B)] :
      ( member(A,aa(product_prod(A,B),A,product_fst(A,B),Xa),Aa2)
     => ( member(B,aa(product_prod(A,B),B,product_snd(A,B),Xa),B2)
       => member(product_prod(A,B),Xa,product_Sigma(A,B,Aa2,aTP_Lamp_pv(set(B),fun(A,set(B)),B2))) ) ) ).

% in_prod_fst_sndI
tff(fact_5065_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,aa(set(product_prod(A,A)),set(A),field2(A),R),aTP_Lamp_qc(set(product_prod(A,A)),fun(A,set(A)),R))) ).

% R_subset_Field
tff(fact_5066_Sigma__empty__iff,axiom,
    ! [A: $tType,B: $tType,Ia2: set(A),X3: fun(A,set(B))] :
      ( ( product_Sigma(A,B,Ia2,X3) = bot_bot(set(product_prod(A,B))) )
    <=> ! [X6: A] :
          ( member(A,X6,Ia2)
         => ( aa(A,set(B),X3,X6) = bot_bot(set(B)) ) ) ) ).

% Sigma_empty_iff
tff(fact_5067_listrel__subset,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),Aa2: set(A)] :
      ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R2),product_Sigma(A,A,Aa2,aTP_Lamp_pt(set(A),fun(A,set(A)),Aa2)))
     => aa(set(product_prod(list(A),list(A))),$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),aa(set(A),set(list(A)),lists(A),Aa2),aTP_Lamp_qd(set(A),fun(list(A),set(list(A))),Aa2))) ) ).

% listrel_subset
tff(fact_5068_relcomp__subset__Sigma,axiom,
    ! [B: $tType,A: $tType,E: $tType,R2: set(product_prod(A,B)),Aa2: set(A),B2: set(B),S2: set(product_prod(B,E)),C3: set(E)] :
      ( aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),R2),product_Sigma(A,B,Aa2,aTP_Lamp_pv(set(B),fun(A,set(B)),B2)))
     => ( aa(set(product_prod(B,E)),$o,aa(set(product_prod(B,E)),fun(set(product_prod(B,E)),$o),ord_less_eq(set(product_prod(B,E))),S2),product_Sigma(B,E,B2,aTP_Lamp_qe(set(E),fun(B,set(E)),C3)))
       => aa(set(product_prod(A,E)),$o,aa(set(product_prod(A,E)),fun(set(product_prod(A,E)),$o),ord_less_eq(set(product_prod(A,E))),relcomp(A,B,E,R2,S2)),product_Sigma(A,E,Aa2,aTP_Lamp_qf(set(E),fun(A,set(E)),C3))) ) ) ).

% relcomp_subset_Sigma
tff(fact_5069_Times__Diff__distrib1,axiom,
    ! [A: $tType,B: $tType,Aa2: set(A),B2: 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)),Aa2),B2),aTP_Lamp_pv(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,Aa2,aTP_Lamp_pv(set(B),fun(A,set(B)),C3))),product_Sigma(A,B,B2,aTP_Lamp_pv(set(B),fun(A,set(B)),C3))) ) ).

% Times_Diff_distrib1
tff(fact_5070_Sigma__Diff__distrib2,axiom,
    ! [B: $tType,A: $tType,Ia2: set(A),Aa2: fun(A,set(B)),B2: fun(A,set(B))] : ( product_Sigma(A,B,Ia2,aa(fun(A,set(B)),fun(A,set(B)),aTP_Lamp_qg(fun(A,set(B)),fun(fun(A,set(B)),fun(A,set(B))),Aa2),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))),minus_minus(set(product_prod(A,B))),product_Sigma(A,B,Ia2,Aa2)),product_Sigma(A,B,Ia2,B2)) ) ).

% Sigma_Diff_distrib2
tff(fact_5071_Id__on__subset__Times,axiom,
    ! [A: $tType,Aa2: set(A)] : aa(set(product_prod(A,A)),$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,Aa2)),product_Sigma(A,A,Aa2,aTP_Lamp_pt(set(A),fun(A,set(A)),Aa2))) ).

% Id_on_subset_Times
tff(fact_5072_enumerate__Suc__eq,axiom,
    ! [A: $tType,N: nat,Xsa: list(A)] : ( enumerate(A,aa(nat,nat,suc,N),Xsa) = aa(list(product_prod(nat,A)),list(product_prod(nat,A)),aa(fun(product_prod(nat,A),product_prod(nat,A)),fun(list(product_prod(nat,A)),list(product_prod(nat,A))),map(product_prod(nat,A),product_prod(nat,A)),product_apfst(nat,nat,A,suc)),enumerate(A,N,Xsa)) ) ).

% enumerate_Suc_eq
tff(fact_5073_times__subset__iff,axiom,
    ! [A: $tType,B: $tType,Aa2: set(A),C3: set(B),B2: set(A),D2: set(B)] :
      ( aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),product_Sigma(A,B,Aa2,aTP_Lamp_pv(set(B),fun(A,set(B)),C3))),product_Sigma(A,B,B2,aTP_Lamp_pv(set(B),fun(A,set(B)),D2)))
    <=> ( ( Aa2 = bot_bot(set(A)) )
        | ( C3 = bot_bot(set(B)) )
        | ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),B2)
          & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),C3),D2) ) ) ) ).

% times_subset_iff
tff(fact_5074_wfI,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),Aa2: set(A),B2: set(A)] :
      ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R2),product_Sigma(A,A,Aa2,aTP_Lamp_pt(set(A),fun(A,set(A)),B2)))
     => ( ! [X: A,P3: fun(A,$o)] :
            ( ! [Xa4: A] :
                ( ! [Y2: A] :
                    ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),Xa4),R2)
                   => aa(A,$o,P3,Y2) )
               => aa(A,$o,P3,Xa4) )
           => ( member(A,X,Aa2)
             => ( member(A,X,B2)
               => aa(A,$o,P3,X) ) ) )
       => wf(A,R2) ) ) ).

% wfI
tff(fact_5075_finite__SigmaI2,axiom,
    ! [B: $tType,A: $tType,Aa2: set(A),B2: fun(A,set(B))] :
      ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,set(B)),fun(A,$o),aTP_Lamp_qh(set(A),fun(fun(A,set(B)),fun(A,$o)),Aa2),B2)))
     => ( ! [A4: A] :
            ( member(A,A4,Aa2)
           => aa(set(B),$o,finite_finite2(B),aa(A,set(B),B2,A4)) )
       => aa(set(product_prod(A,B)),$o,finite_finite2(product_prod(A,B)),product_Sigma(A,B,Aa2,B2)) ) ) ).

% finite_SigmaI2
tff(fact_5076_finite__cartesian__productD1,axiom,
    ! [B: $tType,A: $tType,Aa2: set(A),B2: set(B)] :
      ( aa(set(product_prod(A,B)),$o,finite_finite2(product_prod(A,B)),product_Sigma(A,B,Aa2,aTP_Lamp_pv(set(B),fun(A,set(B)),B2)))
     => ( ( B2 != bot_bot(set(B)) )
       => aa(set(A),$o,finite_finite2(A),Aa2) ) ) ).

% finite_cartesian_productD1
tff(fact_5077_finite__cartesian__productD2,axiom,
    ! [A: $tType,B: $tType,Aa2: set(A),B2: set(B)] :
      ( aa(set(product_prod(A,B)),$o,finite_finite2(product_prod(A,B)),product_Sigma(A,B,Aa2,aTP_Lamp_pv(set(B),fun(A,set(B)),B2)))
     => ( ( Aa2 != bot_bot(set(A)) )
       => aa(set(B),$o,finite_finite2(B),B2) ) ) ).

% finite_cartesian_productD2
tff(fact_5078_finite__cartesian__product__iff,axiom,
    ! [A: $tType,B: $tType,Aa2: set(A),B2: set(B)] :
      ( aa(set(product_prod(A,B)),$o,finite_finite2(product_prod(A,B)),product_Sigma(A,B,Aa2,aTP_Lamp_pv(set(B),fun(A,set(B)),B2)))
    <=> ( ( Aa2 = bot_bot(set(A)) )
        | ( B2 = bot_bot(set(B)) )
        | ( aa(set(A),$o,finite_finite2(A),Aa2)
          & aa(set(B),$o,finite_finite2(B),B2) ) ) ) ).

% finite_cartesian_product_iff
tff(fact_5079_fst__image__Sigma,axiom,
    ! [B: $tType,A: $tType,Aa2: set(A),B2: fun(A,set(B))] : ( aa(set(product_prod(A,B)),set(A),image2(product_prod(A,B),A,product_fst(A,B)),product_Sigma(A,B,Aa2,B2)) = aa(fun(A,$o),set(A),collect(A),aa(fun(A,set(B)),fun(A,$o),aTP_Lamp_qh(set(A),fun(fun(A,set(B)),fun(A,$o)),Aa2),B2)) ) ).

% fst_image_Sigma
tff(fact_5080_Restr__trancl__mono,axiom,
    ! [A: $tType,V2: A,W2: A,E5: set(product_prod(A,A)),U2: set(A)] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),V2),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))),E5),product_Sigma(A,A,U2,aTP_Lamp_pt(set(A),fun(A,set(A)),U2)))))
     => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),V2),W2),transitive_trancl(A,E5)) ) ).

% Restr_trancl_mono
tff(fact_5081_refl__onI,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),Aa2: set(A)] :
      ( aa(set(product_prod(A,A)),$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,Aa2,aTP_Lamp_pt(set(A),fun(A,set(A)),Aa2)))
     => ( ! [X: A] :
            ( member(A,X,Aa2)
           => 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),R2) )
       => refl_on(A,Aa2,R2) ) ) ).

% refl_onI
tff(fact_5082_refl__on__def,axiom,
    ! [A: $tType,Aa2: set(A),R2: set(product_prod(A,A))] :
      ( refl_on(A,Aa2,R2)
    <=> ( aa(set(product_prod(A,A)),$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,Aa2,aTP_Lamp_pt(set(A),fun(A,set(A)),Aa2)))
        & ! [X6: A] :
            ( member(A,X6,Aa2)
           => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X6),X6),R2) ) ) ) ).

% refl_on_def
tff(fact_5083_set__zip__cart,axiom,
    ! [A: $tType,B: $tType,Xa: product_prod(A,B),L: list(A),L2: list(B)] :
      ( member(product_prod(A,B),Xa,aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),L),L2)))
     => member(product_prod(A,B),Xa,product_Sigma(A,B,aa(list(A),set(A),set2(A),L),aTP_Lamp_px(list(B),fun(A,set(B)),L2))) ) ).

% set_zip_cart
tff(fact_5084_swap__product,axiom,
    ! [A: $tType,B: $tType,Aa2: set(B),B2: 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_io(B,fun(A,product_prod(A,B))))),product_Sigma(B,A,Aa2,aTP_Lamp_an(set(A),fun(B,set(A)),B2))) = product_Sigma(A,B,B2,aTP_Lamp_pv(set(B),fun(A,set(B)),Aa2)) ) ).

% swap_product
tff(fact_5085_image__paired__Times,axiom,
    ! [E: $tType,B: $tType,A: $tType,F4: $tType,F2: fun(E,A),G: fun(F4,B),Aa2: set(E),B2: set(F4)] : ( aa(set(product_prod(E,F4)),set(product_prod(A,B)),image2(product_prod(E,F4),product_prod(A,B),aa(fun(E,fun(F4,product_prod(A,B))),fun(product_prod(E,F4),product_prod(A,B)),product_case_prod(E,F4,product_prod(A,B)),aa(fun(F4,B),fun(E,fun(F4,product_prod(A,B))),aTP_Lamp_ho(fun(E,A),fun(fun(F4,B),fun(E,fun(F4,product_prod(A,B)))),F2),G))),product_Sigma(E,F4,Aa2,aTP_Lamp_qi(set(F4),fun(E,set(F4)),B2))) = product_Sigma(A,B,aa(set(E),set(A),image2(E,A,F2),Aa2),aa(set(F4),fun(A,set(B)),aTP_Lamp_qj(fun(F4,B),fun(set(F4),fun(A,set(B))),G),B2)) ) ).

% image_paired_Times
tff(fact_5086_card__cartesian__product__singleton,axiom,
    ! [A: $tType,B: $tType,Xa: A,Aa2: set(B)] : ( finite_card(product_prod(A,B),product_Sigma(A,B,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A))),aTP_Lamp_pv(set(B),fun(A,set(B)),Aa2))) = finite_card(B,Aa2) ) ).

% card_cartesian_product_singleton
tff(fact_5087_comp__fun__commute__Pow__fold,axiom,
    ! [A: $tType] : finite6289374366891150609ommute(A,set(set(A)),aTP_Lamp_ha(A,fun(set(set(A)),set(set(A))))) ).

% comp_fun_commute_Pow_fold
tff(fact_5088_Sigma__def,axiom,
    ! [B: $tType,A: $tType,Aa2: set(A),B2: fun(A,set(B))] : ( product_Sigma(A,B,Aa2,B2) = 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_ql(fun(A,set(B)),fun(A,set(product_prod(A,B))),B2)),Aa2)) ) ).

% Sigma_def
tff(fact_5089_product__fold,axiom,
    ! [B: $tType,A: $tType,Aa2: set(A),B2: set(B)] :
      ( aa(set(A),$o,finite_finite2(A),Aa2)
     => ( aa(set(B),$o,finite_finite2(B),B2)
       => ( product_Sigma(A,B,Aa2,aTP_Lamp_pv(set(B),fun(A,set(B)),B2)) = finite_fold(A,set(product_prod(A,B)),aTP_Lamp_qn(set(B),fun(A,fun(set(product_prod(A,B)),set(product_prod(A,B)))),B2),bot_bot(set(product_prod(A,B))),Aa2) ) ) ) ).

% product_fold
tff(fact_5090_lists__length__Suc__eq,axiom,
    ! [A: $tType,Aa2: set(A),N: nat] : ( aa(fun(list(A),$o),set(list(A)),collect(list(A)),aa(nat,fun(list(A),$o),aTP_Lamp_qo(set(A),fun(nat,fun(list(A),$o)),Aa2),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_if(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_dp(set(A),fun(nat,fun(list(A),$o)),Aa2),N)),aTP_Lamp_qp(set(A),fun(list(A),set(A)),Aa2))) ) ).

% lists_length_Suc_eq
tff(fact_5091_apfst__convE,axiom,
    ! [E: $tType,A: $tType,B: $tType,Q2: product_prod(A,B),F2: fun(E,A),P2: product_prod(E,B)] :
      ( ( Q2 = aa(product_prod(E,B),product_prod(A,B),product_apfst(E,A,B,F2),P2) )
     => ~ ! [X: E,Y2: B] :
            ( ( P2 = aa(B,product_prod(E,B),aa(E,fun(B,product_prod(E,B)),product_Pair(E,B),X),Y2) )
           => ( Q2 != aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(E,A,F2,X)),Y2) ) ) ) ).

% apfst_convE
tff(fact_5092_rtrancl__last__visit__node,axiom,
    ! [A: $tType,S2: A,S4: A,R: set(product_prod(A,A)),Sh: A] :
      ( 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),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R))
     => ( ( ( S2 != Sh )
          & 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),aa(set(product_prod(A,A)),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))),inf_inf(set(product_prod(A,A))),R),product_Sigma(A,A,top_top(set(A)),aTP_Lamp_qq(A,fun(A,set(A)),Sh))))) )
        | ( 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),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R))
          & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Sh),S4),aa(set(product_prod(A,A)),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))),inf_inf(set(product_prod(A,A))),R),product_Sigma(A,A,top_top(set(A)),aTP_Lamp_qq(A,fun(A,set(A)),Sh))))) ) ) ) ).

% rtrancl_last_visit_node
tff(fact_5093_eq__snd__iff,axiom,
    ! [B: $tType,A: $tType,B3: A,P2: product_prod(B,A)] :
      ( ( B3 = aa(product_prod(B,A),A,product_snd(B,A),P2) )
    <=> ? [A7: B] : ( P2 = aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),A7),B3) ) ) ).

% eq_snd_iff
tff(fact_5094_listrel__rtrancl__refl,axiom,
    ! [A: $tType,Xsa: list(A),R2: set(product_prod(A,A))] : member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xsa),Xsa),listrel(A,A,aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R2))) ).

% listrel_rtrancl_refl
tff(fact_5095_wf__insert,axiom,
    ! [A: $tType,Y: A,Xa: 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))),insert3(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Xa)),R2))
    <=> ( wf(A,R2)
        & ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Y),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R2)) ) ) ).

% wf_insert
tff(fact_5096_converse__rtranclE_H,axiom,
    ! [A: $tType,U: A,V2: A,R: set(product_prod(A,A))] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),U),V2),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R))
     => ( ( U != V2 )
       => ~ ! [Vh: A] :
              ( ( U != Vh )
             => ( 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)
               => ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Vh),V2),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R)) ) ) ) ) ).

% converse_rtranclE'
tff(fact_5097_converse__rtrancl__into__rtrancl,axiom,
    ! [A: $tType,A3: A,B3: A,R2: set(product_prod(A,A)),C2: A] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B3),R2)
     => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B3),C2),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R2))
       => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),C2),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R2)) ) ) ).

% converse_rtrancl_into_rtrancl
tff(fact_5098_converse__rtrancl__induct,axiom,
    ! [A: $tType,A3: A,B3: A,R2: set(product_prod(A,A)),Pa: fun(A,$o)] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B3),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R2))
     => ( aa(A,$o,Pa,B3)
       => ( ! [Y2: A,Z3: A] :
              ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),Z3),R2)
             => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z3),B3),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R2))
               => ( aa(A,$o,Pa,Z3)
                 => aa(A,$o,Pa,Y2) ) ) )
         => aa(A,$o,Pa,A3) ) ) ) ).

% converse_rtrancl_induct
tff(fact_5099_converse__rtranclE,axiom,
    ! [A: $tType,Xa: A,Z2: A,R2: set(product_prod(A,A))] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Z2),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R2))
     => ( ( Xa != Z2 )
       => ~ ! [Y2: A] :
              ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Y2),R2)
             => ~ 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),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R2)) ) ) ) ).

% converse_rtranclE
tff(fact_5100_rtrancl__induct,axiom,
    ! [A: $tType,A3: A,B3: A,R2: set(product_prod(A,A)),Pa: fun(A,$o)] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B3),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R2))
     => ( aa(A,$o,Pa,A3)
       => ( ! [Y2: A,Z3: A] :
              ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),Y2),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R2))
             => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),Z3),R2)
               => ( aa(A,$o,Pa,Y2)
                 => aa(A,$o,Pa,Z3) ) ) )
         => aa(A,$o,Pa,B3) ) ) ) ).

% rtrancl_induct
tff(fact_5101_rtrancl__trans,axiom,
    ! [A: $tType,Xa: A,Y: A,R2: set(product_prod(A,A)),Z2: A] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Y),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R2))
     => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z2),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R2))
       => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Z2),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R2)) ) ) ).

% rtrancl_trans
tff(fact_5102_rtranclE,axiom,
    ! [A: $tType,A3: A,B3: A,R2: set(product_prod(A,A))] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B3),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R2))
     => ( ( A3 != B3 )
       => ~ ! [Y2: A] :
              ( 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),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R2))
             => ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),B3),R2) ) ) ) ).

% rtranclE
tff(fact_5103_rtrancl_Ortrancl__into__rtrancl,axiom,
    ! [A: $tType,A3: A,B3: A,R2: set(product_prod(A,A)),C2: A] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B3),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R2))
     => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B3),C2),R2)
       => 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),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R2)) ) ) ).

% rtrancl.rtrancl_into_rtrancl
tff(fact_5104_rtrancl_Ortrancl__refl,axiom,
    ! [A: $tType,A3: A,R2: set(product_prod(A,A))] : member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),A3),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R2)) ).

% rtrancl.rtrancl_refl
tff(fact_5105_rtrancl_Osimps,axiom,
    ! [A: $tType,A1: A,A22: A,R2: set(product_prod(A,A))] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A1),A22),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R2))
    <=> ( ? [A7: A] :
            ( ( A1 = A7 )
            & ( A22 = A7 ) )
        | ? [A7: A,B8: A,C6: A] :
            ( ( A1 = A7 )
            & ( A22 = C6 )
            & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A7),B8),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R2))
            & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B8),C6),R2) ) ) ) ).

% rtrancl.simps
tff(fact_5106_rtrancl_Ocases,axiom,
    ! [A: $tType,A1: A,A22: A,R2: set(product_prod(A,A))] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A1),A22),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R2))
     => ( ( A22 != A1 )
       => ~ ! [B4: A] :
              ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A1),B4),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R2))
             => ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B4),A22),R2) ) ) ) ).

% rtrancl.cases
tff(fact_5107_listrel__rtrancl__eq__rtrancl__listrel1,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : ( listrel(A,A,aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R2)) = aa(set(product_prod(list(A),list(A))),set(product_prod(list(A),list(A))),transitive_rtrancl(list(A)),listrel1(A,R2)) ) ).

% listrel_rtrancl_eq_rtrancl_listrel1
tff(fact_5108_listrel1__rtrancl__subset__rtrancl__listrel1,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : aa(set(product_prod(list(A),list(A))),$o,aa(set(product_prod(list(A),list(A))),fun(set(product_prod(list(A),list(A))),$o),ord_less_eq(set(product_prod(list(A),list(A)))),listrel1(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R2))),aa(set(product_prod(list(A),list(A))),set(product_prod(list(A),list(A))),transitive_rtrancl(list(A)),listrel1(A,R2))) ).

% listrel1_rtrancl_subset_rtrancl_listrel1
tff(fact_5109_rtrancl__mono__rightI,axiom,
    ! [A: $tType,S: set(product_prod(A,A)),S3: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),S),S3)
     => aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),S),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),S3)) ) ).

% rtrancl_mono_rightI
tff(fact_5110_rtrancl__mono__mp,axiom,
    ! [A: $tType,U2: set(product_prod(A,A)),V: set(product_prod(A,A)),Xa: product_prod(A,A)] :
      ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),U2),V)
     => ( member(product_prod(A,A),Xa,aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),U2))
       => member(product_prod(A,A),Xa,aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),V)) ) ) ).

% rtrancl_mono_mp
tff(fact_5111_r__le__rtrancl,axiom,
    ! [A: $tType,S: set(product_prod(A,A))] : aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),S),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),S)) ).

% r_le_rtrancl
tff(fact_5112_in__rtrancl__insert,axiom,
    ! [A: $tType,Xa: product_prod(A,A),R: set(product_prod(A,A)),R2: product_prod(A,A)] :
      ( member(product_prod(A,A),Xa,aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R))
     => member(product_prod(A,A),Xa,aa(set(product_prod(A,A)),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))),insert3(product_prod(A,A)),R2),R))) ) ).

% in_rtrancl_insert
tff(fact_5113_rtrancl__listrel1__ConsI2,axiom,
    ! [A: $tType,Xa: A,Y: A,R2: set(product_prod(A,A)),Xsa: list(A),Ys: list(A)] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Y),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R2))
     => ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xsa),Ys),aa(set(product_prod(list(A),list(A))),set(product_prod(list(A),list(A))),transitive_rtrancl(list(A)),listrel1(A,R2)))
       => member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)),aa(set(product_prod(list(A),list(A))),set(product_prod(list(A),list(A))),transitive_rtrancl(list(A)),listrel1(A,R2))) ) ) ).

% rtrancl_listrel1_ConsI2
tff(fact_5114_tranclD,axiom,
    ! [A: $tType,Xa: A,Y: A,R: set(product_prod(A,A))] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Y),transitive_trancl(A,R))
     => ? [Z3: A] :
          ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Z3),R)
          & 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),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R)) ) ) ).

% tranclD
tff(fact_5115_rtranclD,axiom,
    ! [A: $tType,A3: A,B3: A,R: set(product_prod(A,A))] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B3),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R))
     => ( ( A3 = B3 )
        | ( ( A3 != B3 )
          & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B3),transitive_trancl(A,R)) ) ) ) ).

% rtranclD
tff(fact_5116_tranclD2,axiom,
    ! [A: $tType,Xa: A,Y: A,R: set(product_prod(A,A))] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Y),transitive_trancl(A,R))
     => ? [Z3: A] :
          ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Z3),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R))
          & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z3),Y),R) ) ) ).

% tranclD2
tff(fact_5117_trancl__into__rtrancl,axiom,
    ! [A: $tType,A3: A,B3: A,R2: set(product_prod(A,A))] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B3),transitive_trancl(A,R2))
     => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B3),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R2)) ) ).

% trancl_into_rtrancl
tff(fact_5118_rtrancl__eq__or__trancl,axiom,
    ! [A: $tType,Xa: A,Y: A,R: set(product_prod(A,A))] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Y),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R))
    <=> ( ( Xa = Y )
        | ( ( Xa != Y )
          & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Y),transitive_trancl(A,R)) ) ) ) ).

% rtrancl_eq_or_trancl
tff(fact_5119_rtrancl__into__trancl1,axiom,
    ! [A: $tType,A3: A,B3: A,R2: set(product_prod(A,A)),C2: A] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B3),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R2))
     => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B3),C2),R2)
       => 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_5120_rtrancl__into__trancl2,axiom,
    ! [A: $tType,A3: A,B3: A,R2: set(product_prod(A,A)),C2: A] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B3),R2)
     => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B3),C2),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R2))
       => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),C2),transitive_trancl(A,R2)) ) ) ).

% rtrancl_into_trancl2
tff(fact_5121_rtrancl__trancl__trancl,axiom,
    ! [A: $tType,Xa: A,Y: A,R2: set(product_prod(A,A)),Z2: A] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Y),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R2))
     => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z2),transitive_trancl(A,R2))
       => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Z2),transitive_trancl(A,R2)) ) ) ).

% rtrancl_trancl_trancl
tff(fact_5122_trancl__rtrancl__trancl,axiom,
    ! [A: $tType,A3: A,B3: A,R2: set(product_prod(A,A)),C2: A] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B3),transitive_trancl(A,R2))
     => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B3),C2),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R2))
       => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),C2),transitive_trancl(A,R2)) ) ) ).

% trancl_rtrancl_trancl
tff(fact_5123_rtrancl__Un__separator__converseE,axiom,
    ! [A: $tType,A3: A,B3: A,Pa: set(product_prod(A,A)),Qa: set(product_prod(A,A))] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B3),aa(set(product_prod(A,A)),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))),Pa),Qa)))
     => ( ! [X: A,Y2: A] :
            ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),B3),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),Pa))
           => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),X),Qa)
             => ( Y2 = X ) ) )
       => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B3),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),Pa)) ) ) ).

% rtrancl_Un_separator_converseE
tff(fact_5124_rtrancl__Un__separatorE,axiom,
    ! [A: $tType,A3: A,B3: A,Pa: set(product_prod(A,A)),Qa: set(product_prod(A,A))] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B3),aa(set(product_prod(A,A)),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))),Pa),Qa)))
     => ( ! [X: A,Y2: A] :
            ( 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),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),Pa))
           => ( 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),Qa)
             => ( X = Y2 ) ) )
       => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B3),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),Pa)) ) ) ).

% rtrancl_Un_separatorE
tff(fact_5125_rtrancl__induct2,axiom,
    ! [A: $tType,B: $tType,Ax: A,Ay: B,Bx: A,By: B,R2: set(product_prod(product_prod(A,B),product_prod(A,B))),Pa: fun(A,fun(B,$o))] :
      ( member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Bx),By)),aa(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(product_prod(A,B),product_prod(A,B))),transitive_rtrancl(product_prod(A,B)),R2))
     => ( aa(B,$o,aa(A,fun(B,$o),Pa,Ax),Ay)
       => ( ! [A4: A,B4: B,Aa3: A,Ba: B] :
              ( member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B4)),aa(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(product_prod(A,B),product_prod(A,B))),transitive_rtrancl(product_prod(A,B)),R2))
             => ( member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B4)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa3),Ba)),R2)
               => ( aa(B,$o,aa(A,fun(B,$o),Pa,A4),B4)
                 => aa(B,$o,aa(A,fun(B,$o),Pa,Aa3),Ba) ) ) )
         => aa(B,$o,aa(A,fun(B,$o),Pa,Bx),By) ) ) ) ).

% rtrancl_induct2
tff(fact_5126_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)))] :
      ( 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)),aa(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(product_prod(A,B),product_prod(A,B))),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) )
       => ~ ! [A4: A,B4: B] :
              ( member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa),Xb)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B4)),R2)
             => ~ member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B4)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Za2),Zb)),aa(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(product_prod(A,B),product_prod(A,B))),transitive_rtrancl(product_prod(A,B)),R2)) ) ) ) ).

% converse_rtranclE2
tff(fact_5127_converse__rtrancl__induct2,axiom,
    ! [A: $tType,B: $tType,Ax: A,Ay: B,Bx: A,By: B,R2: set(product_prod(product_prod(A,B),product_prod(A,B))),Pa: fun(A,fun(B,$o))] :
      ( member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Bx),By)),aa(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(product_prod(A,B),product_prod(A,B))),transitive_rtrancl(product_prod(A,B)),R2))
     => ( aa(B,$o,aa(A,fun(B,$o),Pa,Bx),By)
       => ( ! [A4: A,B4: B,Aa3: A,Ba: B] :
              ( member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B4)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa3),Ba)),R2)
             => ( member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa3),Ba)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Bx),By)),aa(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(product_prod(A,B),product_prod(A,B))),transitive_rtrancl(product_prod(A,B)),R2))
               => ( aa(B,$o,aa(A,fun(B,$o),Pa,Aa3),Ba)
                 => aa(B,$o,aa(A,fun(B,$o),Pa,A4),B4) ) ) )
         => aa(B,$o,aa(A,fun(B,$o),Pa,Ax),Ay) ) ) ) ).

% converse_rtrancl_induct2
tff(fact_5128_rtrancl__sub__insert__rtrancl,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),Xa: product_prod(A,A)] : aa(set(product_prod(A,A)),$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)),transitive_rtrancl(A),R)),aa(set(product_prod(A,A)),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))),insert3(product_prod(A,A)),Xa),R))) ).

% rtrancl_sub_insert_rtrancl
tff(fact_5129_listrel__rtrancl__trans,axiom,
    ! [A: $tType,Xsa: list(A),Ys: list(A),R2: set(product_prod(A,A)),Zs: list(A)] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xsa),Ys),listrel(A,A,aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R2)))
     => ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Ys),Zs),listrel(A,A,aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R2)))
       => member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xsa),Zs),listrel(A,A,aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R2))) ) ) ).

% listrel_rtrancl_trans
tff(fact_5130_wf__linord__ex__has__least,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,A)),Pa: fun(B,$o),K3: B,M2: fun(B,A)] :
      ( wf(A,R2)
     => ( ! [X: A,Y2: A] :
            ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y2),transitive_trancl(A,R2))
          <=> ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),X),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R2)) )
       => ( aa(B,$o,Pa,K3)
         => ? [X: B] :
              ( aa(B,$o,Pa,X)
              & ! [Y3: B] :
                  ( aa(B,$o,Pa,Y3)
                 => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(B,A,M2,X)),aa(B,A,M2,Y3)),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R2)) ) ) ) ) ) ).

% wf_linord_ex_has_least
tff(fact_5131_trancl__union__outside,axiom,
    ! [A: $tType,V2: A,W2: A,E5: set(product_prod(A,A)),U2: set(product_prod(A,A))] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),V2),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))),E5),U2)))
     => ( ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),V2),W2),transitive_trancl(A,E5))
       => ? [X: A,Y2: A] :
            ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),V2),X),aa(set(product_prod(A,A)),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))),E5),U2)))
            & 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),U2)
            & 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),aa(set(product_prod(A,A)),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))),E5),U2))) ) ) ) ).

% trancl_union_outside
tff(fact_5132_rtrancl__listrel1__ConsI1,axiom,
    ! [A: $tType,Xsa: list(A),Ys: list(A),R2: set(product_prod(A,A)),Xa: A] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xsa),Ys),aa(set(product_prod(list(A),list(A))),set(product_prod(list(A),list(A))),transitive_rtrancl(list(A)),listrel1(A,R2)))
     => member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Ys)),aa(set(product_prod(list(A),list(A))),set(product_prod(list(A),list(A))),transitive_rtrancl(list(A)),listrel1(A,R2))) ) ).

% rtrancl_listrel1_ConsI1
tff(fact_5133_trancl__over__edgeE,axiom,
    ! [A: $tType,U: A,W2: A,V1: A,V22: A,E5: set(product_prod(A,A))] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),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))),insert3(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),V1),V22)),E5)))
     => ( ~ 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,E5))
       => ~ ( 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),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),E5))
           => ~ 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),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),E5)) ) ) ) ).

% trancl_over_edgeE
tff(fact_5134_rtrancl__listrel1__eq__len,axiom,
    ! [A: $tType,Xa: list(A),Y: list(A),R2: set(product_prod(A,A))] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xa),Y),aa(set(product_prod(list(A),list(A))),set(product_prod(list(A),list(A))),transitive_rtrancl(list(A)),listrel1(A,R2)))
     => ( aa(list(A),nat,size_size(list(A)),Xa) = aa(list(A),nat,size_size(list(A)),Y) ) ) ).

% rtrancl_listrel1_eq_len
tff(fact_5135_trans__rtrancl__eq__reflcl,axiom,
    ! [A: $tType,Aa2: set(product_prod(A,A))] :
      ( trans(A,Aa2)
     => ( aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),Aa2) = 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))),Aa2),id2(A)) ) ) ).

% trans_rtrancl_eq_reflcl
tff(fact_5136_listrel__reflcl__if__listrel1,axiom,
    ! [A: $tType,Xsa: list(A),Ys: list(A),R2: set(product_prod(A,A))] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xsa),Ys),listrel1(A,R2))
     => member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xsa),Ys),listrel(A,A,aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R2))) ) ).

% listrel_reflcl_if_listrel1
tff(fact_5137_rtrancl__listrel1__if__listrel,axiom,
    ! [A: $tType,Xsa: list(A),Ys: list(A),R2: set(product_prod(A,A))] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xsa),Ys),listrel(A,A,R2))
     => member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xsa),Ys),aa(set(product_prod(list(A),list(A))),set(product_prod(list(A),list(A))),transitive_rtrancl(list(A)),listrel1(A,R2))) ) ).

% rtrancl_listrel1_if_listrel
tff(fact_5138_trancl__subset__Sigma__aux,axiom,
    ! [A: $tType,A3: A,B3: A,R2: set(product_prod(A,A)),Aa2: set(A)] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B3),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R2))
     => ( aa(set(product_prod(A,A)),$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,Aa2,aTP_Lamp_pt(set(A),fun(A,set(A)),Aa2)))
       => ( ( A3 = B3 )
          | member(A,A3,Aa2) ) ) ) ).

% trancl_subset_Sigma_aux
tff(fact_5139_Restr__rtrancl__mono,axiom,
    ! [A: $tType,V2: A,W2: A,E5: set(product_prod(A,A)),U2: set(A)] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),V2),W2),aa(set(product_prod(A,A)),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))),inf_inf(set(product_prod(A,A))),E5),product_Sigma(A,A,U2,aTP_Lamp_pt(set(A),fun(A,set(A)),U2)))))
     => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),V2),W2),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),E5)) ) ).

% Restr_rtrancl_mono
tff(fact_5140_pred__nat__trancl__eq__le,axiom,
    ! [M2: nat,N: nat] :
      ( member(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),M2),N),aa(set(product_prod(nat,nat)),set(product_prod(nat,nat)),transitive_rtrancl(nat),pred_nat))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),N) ) ).

% pred_nat_trancl_eq_le
tff(fact_5141_rtrancl__mapI,axiom,
    ! [B: $tType,A: $tType,A3: A,B3: A,E5: set(product_prod(A,A)),F2: fun(A,B)] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B3),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),E5))
     => 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,F2,A3)),aa(A,B,F2,B3)),aa(set(product_prod(B,B)),set(product_prod(B,B)),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,F2)),E5))) ) ).

% rtrancl_mapI
tff(fact_5142_listrel__subset__rtrancl__listrel1,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : aa(set(product_prod(list(A),list(A))),$o,aa(set(product_prod(list(A),list(A))),fun(set(product_prod(list(A),list(A))),$o),ord_less_eq(set(product_prod(list(A),list(A)))),listrel(A,A,R2)),aa(set(product_prod(list(A),list(A))),set(product_prod(list(A),list(A))),transitive_rtrancl(list(A)),listrel1(A,R2))) ).

% listrel_subset_rtrancl_listrel1
tff(fact_5143_rtrancl__last__touch,axiom,
    ! [A: $tType,Q2: A,Q5: A,R: set(product_prod(A,A)),S: set(A)] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q2),Q5),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R))
     => ( member(A,Q2,S)
       => ~ ! [Qt: A] :
              ( member(A,Qt,S)
             => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q2),Qt),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R))
               => ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Qt),Q5),aa(set(product_prod(A,A)),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))),minus_minus(set(product_prod(A,A))),R),product_Sigma(A,A,top_top(set(A)),aTP_Lamp_pt(set(A),fun(A,set(A)),S))))) ) ) ) ) ).

% rtrancl_last_touch
tff(fact_5144_rtrancl__last__visit_H,axiom,
    ! [A: $tType,Q2: A,Q5: A,R: set(product_prod(A,A)),S: set(A)] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q2),Q5),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R))
     => ( ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q2),Q5),aa(set(product_prod(A,A)),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))),minus_minus(set(product_prod(A,A))),R),product_Sigma(A,A,top_top(set(A)),aTP_Lamp_pt(set(A),fun(A,set(A)),S)))))
       => ~ ! [Qt: A] :
              ( member(A,Qt,S)
             => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q2),Qt),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R))
               => ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Qt),Q5),aa(set(product_prod(A,A)),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))),minus_minus(set(product_prod(A,A))),R),product_Sigma(A,A,top_top(set(A)),aTP_Lamp_pt(set(A),fun(A,set(A)),S))))) ) ) ) ) ).

% rtrancl_last_visit'
tff(fact_5145_rtrancl__insert,axiom,
    ! [A: $tType,A3: A,B3: A,R2: set(product_prod(A,A))] : ( aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(product_prod(A,A),fun(set(product_prod(A,A)),set(product_prod(A,A))),insert3(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B3)),R2)) = aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R2)),aa(fun(product_prod(A,A),$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,$o))),aTP_Lamp_qr(A,fun(A,fun(set(product_prod(A,A)),fun(A,fun(A,$o)))),A3),B3),R2)))) ) ).

% rtrancl_insert
tff(fact_5146_rtrancl__last__visit,axiom,
    ! [A: $tType,Q2: A,Q5: A,R: set(product_prod(A,A)),S: set(A)] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q2),Q5),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R))
     => ( ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q2),Q5),aa(set(product_prod(A,A)),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))),minus_minus(set(product_prod(A,A))),R),product_Sigma(A,A,top_top(set(A)),aTP_Lamp_pt(set(A),fun(A,set(A)),S)))))
       => ~ ! [Qt: A] :
              ( member(A,Qt,S)
             => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q2),Qt),transitive_trancl(A,R))
               => ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Qt),Q5),aa(set(product_prod(A,A)),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))),minus_minus(set(product_prod(A,A))),R),product_Sigma(A,A,top_top(set(A)),aTP_Lamp_pt(set(A),fun(A,set(A)),S))))) ) ) ) ) ).

% rtrancl_last_visit
tff(fact_5147_trancl__insert,axiom,
    ! [A: $tType,Y: A,Xa: A,R2: set(product_prod(A,A))] : ( transitive_trancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(product_prod(A,A),fun(set(product_prod(A,A)),set(product_prod(A,A))),insert3(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Xa)),R2)) = aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),transitive_trancl(A,R2)),aa(fun(product_prod(A,A),$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,$o))),aTP_Lamp_qr(A,fun(A,fun(set(product_prod(A,A)),fun(A,fun(A,$o)))),Y),Xa),R2)))) ) ).

% trancl_insert
tff(fact_5148_trancl__multi__insert2,axiom,
    ! [A: $tType,A3: A,B3: A,R2: set(product_prod(A,A)),M2: A,X3: set(A)] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B3),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)),insert3(A),M2),bot_bot(set(A))),aTP_Lamp_pt(set(A),fun(A,set(A)),X3)))))
     => ( ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B3),transitive_trancl(A,R2))
       => ~ ! [X: A] :
              ( member(A,X,X3)
             => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),M2),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R2))
               => ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),B3),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R2)) ) ) ) ) ).

% trancl_multi_insert2
tff(fact_5149_trancl__multi__insert,axiom,
    ! [A: $tType,A3: A,B3: A,R2: set(product_prod(A,A)),X3: set(A),M2: A] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B3),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,X3,aTP_Lamp_qs(A,fun(A,set(A)),M2)))))
     => ( ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B3),transitive_trancl(A,R2))
       => ~ ! [X: A] :
              ( member(A,X,X3)
             => ( 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),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R2))
               => ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),M2),B3),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R2)) ) ) ) ) ).

% trancl_multi_insert
tff(fact_5150_eq__fst__iff,axiom,
    ! [A: $tType,B: $tType,A3: A,P2: product_prod(A,B)] :
      ( ( A3 = aa(product_prod(A,B),A,product_fst(A,B),P2) )
    <=> ? [B8: B] : ( P2 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B8) ) ) ).

% eq_fst_iff
tff(fact_5151_rtrancl__restrictI,axiom,
    ! [A: $tType,U: A,V2: A,E5: set(product_prod(A,A)),R: set(A)] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),U),V2),aa(set(product_prod(A,A)),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))),minus_minus(set(product_prod(A,A))),E5),product_Sigma(A,A,top_top(set(A)),aTP_Lamp_pt(set(A),fun(A,set(A)),R)))))
     => ( ~ member(A,U,R)
       => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),U),V2),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),rel_restrict(A,E5,R))) ) ) ).

% rtrancl_restrictI
tff(fact_5152_image__split__eq__Sigma,axiom,
    ! [B: $tType,A: $tType,E: $tType,F2: fun(E,A),G: fun(E,B),Aa2: set(E)] : ( aa(set(E),set(product_prod(A,B)),image2(E,product_prod(A,B),aa(fun(E,B),fun(E,product_prod(A,B)),aTP_Lamp_qt(fun(E,A),fun(fun(E,B),fun(E,product_prod(A,B))),F2),G)),Aa2) = product_Sigma(A,B,aa(set(E),set(A),image2(E,A,F2),Aa2),aa(set(E),fun(A,set(B)),aa(fun(E,B),fun(set(E),fun(A,set(B))),aTP_Lamp_qu(fun(E,A),fun(fun(E,B),fun(set(E),fun(A,set(B)))),F2),G),Aa2)) ) ).

% image_split_eq_Sigma
tff(fact_5153_ord__class_Olexordp__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ( ord_lexordp(A) = complete_lattice_lfp(fun(list(A),fun(list(A),$o)),aTP_Lamp_ol(fun(list(A),fun(list(A),$o)),fun(list(A),fun(list(A),$o)))) ) ) ).

% ord_class.lexordp_def
tff(fact_5154_vimage__eq,axiom,
    ! [A: $tType,B: $tType,A3: A,F2: fun(A,B),B2: set(B)] :
      ( member(A,A3,vimage(A,B,F2,B2))
    <=> member(B,aa(A,B,F2,A3),B2) ) ).

% vimage_eq
tff(fact_5155_vimageI,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),A3: B,B3: A,B2: set(A)] :
      ( ( aa(B,A,F2,A3) = B3 )
     => ( member(A,B3,B2)
       => member(B,A3,vimage(B,A,F2,B2)) ) ) ).

% vimageI
tff(fact_5156_vimage__Collect__eq,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),Pa: fun(B,$o)] : ( vimage(A,B,F2,aa(fun(B,$o),set(B),collect(B),Pa)) = aa(fun(A,$o),set(A),collect(A),aa(fun(B,$o),fun(A,$o),aTP_Lamp_qv(fun(A,B),fun(fun(B,$o),fun(A,$o)),F2),Pa)) ) ).

% vimage_Collect_eq
tff(fact_5157_vimage__ident,axiom,
    ! [A: $tType,Y5: set(A)] : ( vimage(A,A,aTP_Lamp_ab(A,A),Y5) = Y5 ) ).

% vimage_ident
tff(fact_5158_vimage__empty,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B)] : ( vimage(A,B,F2,bot_bot(set(B))) = bot_bot(set(A)) ) ).

% vimage_empty
tff(fact_5159_vimage__UNIV,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B)] : ( vimage(A,B,F2,top_top(set(B))) = top_top(set(A)) ) ).

% vimage_UNIV
tff(fact_5160_vimage__Int,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),Aa2: set(B),B2: set(B)] : ( vimage(A,B,F2,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),Aa2),B2)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),vimage(A,B,F2,Aa2)),vimage(A,B,F2,B2)) ) ).

% vimage_Int
tff(fact_5161_vimage__Un,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),Aa2: set(B),B2: set(B)] : ( vimage(A,B,F2,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),Aa2),B2)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),vimage(A,B,F2,Aa2)),vimage(A,B,F2,B2)) ) ).

% vimage_Un
tff(fact_5162_lexordp__simps_I1_J,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Ys: list(A)] :
          ( aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),nil(A)),Ys)
        <=> ( Ys != nil(A) ) ) ) ).

% lexordp_simps(1)
tff(fact_5163_lexordp__simps_I2_J,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Xsa: list(A)] : ~ aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),Xsa),nil(A)) ) ).

% lexordp_simps(2)
tff(fact_5164_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_5165_lexordp__simps_I3_J,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Xa: A,Xsa: list(A),Y: A,Ys: list(A)] :
          ( aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
            | ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xa)
              & aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),Xsa),Ys) ) ) ) ) ).

% lexordp_simps(3)
tff(fact_5166_vimage__const,axiom,
    ! [B: $tType,A: $tType,C2: B,Aa2: set(B)] :
      ( vimage(A,B,aTP_Lamp_bd(B,fun(A,B),C2),Aa2) = $ite(member(B,C2,Aa2),top_top(set(A)),bot_bot(set(A))) ) ).

% vimage_const
tff(fact_5167_image__vimage__eq,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Aa2: set(A)] : ( aa(set(B),set(A),image2(B,A,F2),vimage(B,A,F2,Aa2)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),aa(set(B),set(A),image2(B,A,F2),top_top(set(B)))) ) ).

% image_vimage_eq
tff(fact_5168_vimage__if,axiom,
    ! [B: $tType,A: $tType,B2: set(A),C2: B,D3: B,Aa2: set(B)] :
      ( vimage(A,B,aa(B,fun(A,B),aa(B,fun(B,fun(A,B)),aTP_Lamp_qw(set(A),fun(B,fun(B,fun(A,B))),B2),C2),D3),Aa2) = $ite(
          member(B,C2,Aa2),
          $ite(member(B,D3,Aa2),top_top(set(A)),B2),
          $ite(member(B,D3,Aa2),aa(set(A),set(A),uminus_uminus(set(A)),B2),bot_bot(set(A))) ) ) ).

% vimage_if
tff(fact_5169_image__subset__iff__subset__vimage,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),Aa2: set(B),B2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(B),set(A),image2(B,A,F2),Aa2)),B2)
    <=> aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),Aa2),vimage(B,A,F2,B2)) ) ).

% image_subset_iff_subset_vimage
tff(fact_5170_image__vimage__subset,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),Aa2: set(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(B),set(A),image2(B,A,F2),vimage(B,A,F2,Aa2))),Aa2) ).

% image_vimage_subset
tff(fact_5171_vimage__mono,axiom,
    ! [B: $tType,A: $tType,Aa2: set(A),B2: set(A),F2: fun(B,A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),B2)
     => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),vimage(B,A,F2,Aa2)),vimage(B,A,F2,B2)) ) ).

% vimage_mono
tff(fact_5172_subset__vimage__iff,axiom,
    ! [A: $tType,B: $tType,Aa2: set(A),F2: fun(A,B),B2: set(B)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),vimage(A,B,F2,B2))
    <=> ! [X6: A] :
          ( member(A,X6,Aa2)
         => member(B,aa(A,B,F2,X6),B2) ) ) ).

% subset_vimage_iff
tff(fact_5173_rel__restrict__sub,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),Aa2: set(A)] : aa(set(product_prod(A,A)),$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,R,Aa2)),R) ).

% rel_restrict_sub
tff(fact_5174_rel__restrict__mono,axiom,
    ! [A: $tType,Aa2: set(product_prod(A,A)),B2: set(product_prod(A,A)),R: set(A)] :
      ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),Aa2),B2)
     => aa(set(product_prod(A,A)),$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,Aa2,R)),rel_restrict(A,B2,R)) ) ).

% rel_restrict_mono
tff(fact_5175_rel__restrict__union,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),Aa2: set(A),B2: set(A)] : ( rel_restrict(A,R,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),B2)) = rel_restrict(A,rel_restrict(A,R,Aa2),B2) ) ).

% rel_restrict_union
tff(fact_5176_vimage__inter__cong,axiom,
    ! [B: $tType,A: $tType,S: set(A),F2: fun(A,B),G: fun(A,B),Y: set(B)] :
      ( ! [W: A] :
          ( member(A,W,S)
         => ( aa(A,B,F2,W) = aa(A,B,G,W) ) )
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),vimage(A,B,F2,Y)),S) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),vimage(A,B,G,Y)),S) ) ) ).

% vimage_inter_cong
tff(fact_5177_finite__rel__restrict,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),Aa2: set(A)] :
      ( aa(set(product_prod(A,A)),$o,finite_finite2(product_prod(A,A)),R)
     => aa(set(product_prod(A,A)),$o,finite_finite2(product_prod(A,A)),rel_restrict(A,R,Aa2)) ) ).

% finite_rel_restrict
tff(fact_5178_vimage__Diff,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),Aa2: set(B),B2: set(B)] : ( vimage(A,B,F2,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),Aa2),B2)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),vimage(A,B,F2,Aa2)),vimage(A,B,F2,B2)) ) ).

% vimage_Diff
tff(fact_5179_lexordp__irreflexive_H,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xsa: list(A)] : ~ aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),Xsa),Xsa) ) ).

% lexordp_irreflexive'
tff(fact_5180_lexordp__linear,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xsa: list(A),Ys: list(A)] :
          ( aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),Xsa),Ys)
          | ( Xsa = Ys )
          | aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),Ys),Xsa) ) ) ).

% lexordp_linear
tff(fact_5181_lexordp__trans,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xsa: list(A),Ys: list(A),Zs: list(A)] :
          ( aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),Xsa),Ys)
         => ( aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),Ys),Zs)
           => aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),Xsa),Zs) ) ) ) ).

% lexordp_trans
tff(fact_5182_lexordp__antisym,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xsa: list(A),Ys: list(A)] :
          ( aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),Xsa),Ys)
         => ~ aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),Ys),Xsa) ) ) ).

% lexordp_antisym
tff(fact_5183_vimage__Collect,axiom,
    ! [B: $tType,A: $tType,Pa: fun(B,$o),F2: fun(A,B),Qa: fun(A,$o)] :
      ( ! [X: A] :
          ( aa(B,$o,Pa,aa(A,B,F2,X))
        <=> aa(A,$o,Qa,X) )
     => ( vimage(A,B,F2,aa(fun(B,$o),set(B),collect(B),Pa)) = aa(fun(A,$o),set(A),collect(A),Qa) ) ) ).

% vimage_Collect
tff(fact_5184_vimageI2,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),A3: B,Aa2: set(A)] :
      ( member(A,aa(B,A,F2,A3),Aa2)
     => member(B,A3,vimage(B,A,F2,Aa2)) ) ).

% vimageI2
tff(fact_5185_vimageE,axiom,
    ! [A: $tType,B: $tType,A3: A,F2: fun(A,B),B2: set(B)] :
      ( member(A,A3,vimage(A,B,F2,B2))
     => member(B,aa(A,B,F2,A3),B2) ) ).

% vimageE
tff(fact_5186_vimageD,axiom,
    ! [A: $tType,B: $tType,A3: A,F2: fun(A,B),Aa2: set(B)] :
      ( member(A,A3,vimage(A,B,F2,Aa2))
     => member(B,aa(A,B,F2,A3),Aa2) ) ).

% vimageD
tff(fact_5187_lexordp__append__leftI,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Us: list(A),Vs: list(A),Xsa: list(A)] :
          ( aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),Us),Vs)
         => aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),Us)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),Vs)) ) ) ).

% lexordp_append_leftI
tff(fact_5188_vimage__Compl,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),Aa2: set(B)] : ( vimage(A,B,F2,aa(set(B),set(B),uminus_uminus(set(B)),Aa2)) = aa(set(A),set(A),uminus_uminus(set(A)),vimage(A,B,F2,Aa2)) ) ).

% vimage_Compl
tff(fact_5189_lexordp__irreflexive,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Xsa: list(A)] :
          ( ! [X: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),X)
         => ~ aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),Xsa),Xsa) ) ) ).

% lexordp_irreflexive
tff(fact_5190_vimage__def,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),B2: set(B)] : ( vimage(A,B,F2,B2) = aa(fun(A,$o),set(A),collect(A),aa(set(B),fun(A,$o),aTP_Lamp_qx(fun(A,B),fun(set(B),fun(A,$o)),F2),B2)) ) ).

% vimage_def
tff(fact_5191_vimage__singleton__eq,axiom,
    ! [A: $tType,B: $tType,A3: A,F2: fun(A,B),B3: B] :
      ( member(A,A3,vimage(A,B,F2,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert3(B),B3),bot_bot(set(B)))))
    <=> ( aa(A,B,F2,A3) = B3 ) ) ).

% vimage_singleton_eq
tff(fact_5192_rel__restrict__lift,axiom,
    ! [A: $tType,Xa: A,Y: A,E5: set(product_prod(A,A)),R: set(A)] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Y),rel_restrict(A,E5,R))
     => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Y),E5) ) ).

% rel_restrict_lift
tff(fact_5193_rel__restrictI,axiom,
    ! [A: $tType,Xa: A,R: set(A),Y: A,E5: set(product_prod(A,A))] :
      ( ~ member(A,Xa,R)
     => ( ~ member(A,Y,R)
       => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Y),E5)
         => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Y),rel_restrict(A,E5,R)) ) ) ) ).

% rel_restrictI
tff(fact_5194_rel__restrict__notR_I1_J,axiom,
    ! [A: $tType,Xa: A,Y: A,Aa2: set(product_prod(A,A)),R: set(A)] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Y),rel_restrict(A,Aa2,R))
     => ~ member(A,Xa,R) ) ).

% rel_restrict_notR(1)
tff(fact_5195_rel__restrict__notR_I2_J,axiom,
    ! [A: $tType,Xa: A,Y: A,Aa2: set(product_prod(A,A)),R: set(A)] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Y),rel_restrict(A,Aa2,R))
     => ~ member(A,Y,R) ) ).

% rel_restrict_notR(2)
tff(fact_5196_vimage__Suc__insert__Suc,axiom,
    ! [N: nat,Aa2: set(nat)] : ( vimage(nat,nat,suc,aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert3(nat),aa(nat,nat,suc,N)),Aa2)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert3(nat),N),vimage(nat,nat,suc,Aa2)) ) ).

% vimage_Suc_insert_Suc
tff(fact_5197_lexordp_OCons__eq,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Xa: A,Y: A,Xsa: list(A),Ys: list(A)] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
         => ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xa)
           => ( aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),Xsa),Ys)
             => aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)) ) ) ) ) ).

% lexordp.Cons_eq
tff(fact_5198_lexordp_OCons,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Xa: A,Y: A,Xsa: list(A),Ys: list(A)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
         => aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)) ) ) ).

% lexordp.Cons
tff(fact_5199_lexordp_ONil,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Y: A,Ys: list(A)] : aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),nil(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)) ) ).

% lexordp.Nil
tff(fact_5200_lexordp__append__leftD,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Xsa: list(A),Us: list(A),Vs: list(A)] :
          ( aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),Us)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),Vs))
         => ( ! [A4: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),A4)
           => aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),Us),Vs) ) ) ) ).

% lexordp_append_leftD
tff(fact_5201_lexordp__append__rightI,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Ys: list(A),Xsa: list(A)] :
          ( ( Ys != nil(A) )
         => aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),Xsa),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),Ys)) ) ) ).

% lexordp_append_rightI
tff(fact_5202_rel__restrict__trancl__notR_I2_J,axiom,
    ! [A: $tType,V2: A,W2: A,E5: set(product_prod(A,A)),R: set(A)] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),V2),W2),transitive_trancl(A,rel_restrict(A,E5,R)))
     => ~ member(A,W2,R) ) ).

% rel_restrict_trancl_notR(2)
tff(fact_5203_rel__restrict__trancl__notR_I1_J,axiom,
    ! [A: $tType,V2: A,W2: A,E5: set(product_prod(A,A)),R: set(A)] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),V2),W2),transitive_trancl(A,rel_restrict(A,E5,R)))
     => ~ member(A,V2,R) ) ).

% rel_restrict_trancl_notR(1)
tff(fact_5204_rel__restrict__trancl__mem,axiom,
    ! [A: $tType,A3: A,B3: A,Aa2: set(product_prod(A,A)),R: set(A)] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B3),transitive_trancl(A,rel_restrict(A,Aa2,R)))
     => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B3),rel_restrict(A,transitive_trancl(A,Aa2),R)) ) ).

% rel_restrict_trancl_mem
tff(fact_5205_rel__restrict__mono2,axiom,
    ! [A: $tType,R: set(A),S: set(A),Aa2: set(product_prod(A,A))] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),R),S)
     => aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),rel_restrict(A,Aa2,S)),rel_restrict(A,Aa2,R)) ) ).

% rel_restrict_mono2
tff(fact_5206_Pair__vimage__Sigma,axiom,
    ! [B: $tType,A: $tType,Xa: B,Aa2: set(B),F2: fun(B,set(A))] :
      ( vimage(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Xa),product_Sigma(B,A,Aa2,F2)) = $ite(member(B,Xa,Aa2),aa(B,set(A),F2,Xa),bot_bot(set(A))) ) ).

% Pair_vimage_Sigma
tff(fact_5207_surj__vimage__empty,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),Aa2: set(A)] :
      ( ( aa(set(B),set(A),image2(B,A,F2),top_top(set(B))) = top_top(set(A)) )
     => ( ( vimage(B,A,F2,Aa2) = bot_bot(set(B)) )
      <=> ( Aa2 = bot_bot(set(A)) ) ) ) ).

% surj_vimage_empty
tff(fact_5208_rel__restrict__trancl__sub,axiom,
    ! [A: $tType,Aa2: set(product_prod(A,A)),R: set(A)] : aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),transitive_trancl(A,rel_restrict(A,Aa2,R))),rel_restrict(A,transitive_trancl(A,Aa2),R)) ).

% rel_restrict_trancl_sub
tff(fact_5209_vimage__insert,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),A3: B,B2: set(B)] : ( vimage(A,B,F2,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert3(B),A3),B2)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),vimage(A,B,F2,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert3(B),A3),bot_bot(set(B))))),vimage(A,B,F2,B2)) ) ).

% vimage_insert
tff(fact_5210_vimage__Suc__insert__0,axiom,
    ! [Aa2: set(nat)] : ( vimage(nat,nat,suc,aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert3(nat),zero_zero(nat)),Aa2)) = vimage(nat,nat,suc,Aa2) ) ).

% vimage_Suc_insert_0
tff(fact_5211_rel__restrict__def,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),Aa2: set(A)] : ( rel_restrict(A,R,Aa2) = aa(fun(product_prod(A,A),$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_qy(set(product_prod(A,A)),fun(set(A),fun(A,fun(A,$o))),R),Aa2))) ) ).

% rel_restrict_def
tff(fact_5212_lexordp__induct,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xsa: list(A),Ys: list(A),Pa: fun(list(A),fun(list(A),$o))] :
          ( aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),Xsa),Ys)
         => ( ! [Y2: A,Ys5: list(A)] : aa(list(A),$o,aa(list(A),fun(list(A),$o),Pa,nil(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys5))
           => ( ! [X: A,Xs: list(A),Y2: A,Ys5: list(A)] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y2)
                 => aa(list(A),$o,aa(list(A),fun(list(A),$o),Pa,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),Y2),Ys5)) )
             => ( ! [X: A,Xs: list(A),Ys5: list(A)] :
                    ( aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),Xs),Ys5)
                   => ( aa(list(A),$o,aa(list(A),fun(list(A),$o),Pa,Xs),Ys5)
                     => aa(list(A),$o,aa(list(A),fun(list(A),$o),Pa,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),Ys5)) ) )
               => aa(list(A),$o,aa(list(A),fun(list(A),$o),Pa,Xsa),Ys) ) ) ) ) ) ).

% lexordp_induct
tff(fact_5213_lexordp__cases,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xsa: list(A),Ys: list(A)] :
          ( aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),Xsa),Ys)
         => ( ( ( Xsa = nil(A) )
             => ! [Y2: A,Ys3: list(A)] : ( Ys != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys3) ) )
           => ( ! [X: A] :
                  ( ? [Xs4: list(A)] : ( Xsa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs4) )
                 => ! [Y2: A] :
                      ( ? [Ys3: list(A)] : ( Ys = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys3) )
                     => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y2) ) )
             => ~ ! [X: A,Xs4: list(A)] :
                    ( ( Xsa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs4) )
                   => ! [Ys3: list(A)] :
                        ( ( Ys = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Ys3) )
                       => ~ aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),Xs4),Ys3) ) ) ) ) ) ) ).

% lexordp_cases
tff(fact_5214_lexordp_Osimps,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A1: list(A),A22: list(A)] :
          ( aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),A1),A22)
        <=> ( ? [Y4: A,Ys4: list(A)] :
                ( ( A1 = nil(A) )
                & ( A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys4) ) )
            | ? [X6: A,Y4: A,Xs2: list(A),Ys4: list(A)] :
                ( ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X6),Xs2) )
                & ( A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys4) )
                & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X6),Y4) )
            | ? [X6: A,Y4: A,Xs2: list(A),Ys4: list(A)] :
                ( ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X6),Xs2) )
                & ( A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys4) )
                & ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X6),Y4)
                & ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y4),X6)
                & aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),Xs2),Ys4) ) ) ) ) ).

% lexordp.simps
tff(fact_5215_lexordp_Ocases,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A1: list(A),A22: list(A)] :
          ( aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),A1),A22)
         => ( ( ( A1 = nil(A) )
             => ! [Y2: A,Ys5: list(A)] : ( A22 != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys5) ) )
           => ( ! [X: A] :
                  ( ? [Xs: list(A)] : ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) )
                 => ! [Y2: A] :
                      ( ? [Ys5: list(A)] : ( A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys5) )
                     => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y2) ) )
             => ~ ! [X: A,Y2: A,Xs: list(A)] :
                    ( ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) )
                   => ! [Ys5: list(A)] :
                        ( ( A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys5) )
                       => ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y2)
                         => ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y2),X)
                           => ~ aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),Xs),Ys5) ) ) ) ) ) ) ) ) ).

% lexordp.cases
tff(fact_5216_lexordp__append__left__rightI,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Xa: A,Y: A,Us: list(A),Xsa: list(A),Ys: list(A)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
         => aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa))),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys))) ) ) ).

% lexordp_append_left_rightI
tff(fact_5217_lexordp__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xsa: list(A),Ys: list(A)] :
          ( aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),Xsa),Ys)
        <=> ( ? [X6: A,Vs3: list(A)] : ( Ys = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X6),Vs3)) )
            | ? [Us2: list(A),A7: A,B8: A,Vs3: list(A),Ws3: list(A)] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A7),B8)
                & ( Xsa = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A7),Vs3)) )
                & ( Ys = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),B8),Ws3)) ) ) ) ) ) ).

% lexordp_iff
tff(fact_5218_inf__img__fin__dom,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),Aa2: set(B)] :
      ( aa(set(A),$o,finite_finite2(A),aa(set(B),set(A),image2(B,A,F2),Aa2))
     => ( ~ aa(set(B),$o,finite_finite2(B),Aa2)
       => ? [X: A] :
            ( member(A,X,aa(set(B),set(A),image2(B,A,F2),Aa2))
            & ~ aa(set(B),$o,finite_finite2(B),vimage(B,A,F2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),X),bot_bot(set(A))))) ) ) ) ).

% inf_img_fin_dom
tff(fact_5219_inf__img__fin__domE,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),Aa2: set(B)] :
      ( aa(set(A),$o,finite_finite2(A),aa(set(B),set(A),image2(B,A,F2),Aa2))
     => ( ~ aa(set(B),$o,finite_finite2(B),Aa2)
       => ~ ! [Y2: A] :
              ( member(A,Y2,aa(set(B),set(A),image2(B,A,F2),Aa2))
             => aa(set(B),$o,finite_finite2(B),vimage(B,A,F2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Y2),bot_bot(set(A))))) ) ) ) ).

% inf_img_fin_domE
tff(fact_5220_finite__finite__vimage__IntI,axiom,
    ! [A: $tType,B: $tType,Fa: set(A),H2: fun(B,A),Aa2: set(B)] :
      ( aa(set(A),$o,finite_finite2(A),Fa)
     => ( ! [Y2: A] :
            ( member(A,Y2,Fa)
           => aa(set(B),$o,finite_finite2(B),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),vimage(B,A,H2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Y2),bot_bot(set(A))))),Aa2)) )
       => aa(set(B),$o,finite_finite2(B),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),vimage(B,A,H2,Fa)),Aa2)) ) ) ).

% finite_finite_vimage_IntI
tff(fact_5221_rel__restrict__Int__empty,axiom,
    ! [A: $tType,Aa2: 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)),Aa2),aa(set(product_prod(A,A)),set(A),field2(A),R)) = bot_bot(set(A)) )
     => ( rel_restrict(A,R,Aa2) = R ) ) ).

% rel_restrict_Int_empty
tff(fact_5222_Field__rel__restrict,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),Aa2: set(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(product_prod(A,A)),set(A),field2(A),rel_restrict(A,R,Aa2))),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),field2(A),R)),Aa2)) ).

% Field_rel_restrict
tff(fact_5223_vimage__eq__UN,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),B2: set(B)] : ( vimage(A,B,F2,B2) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aTP_Lamp_qz(fun(A,B),fun(B,set(A)),F2)),B2)) ) ).

% vimage_eq_UN
tff(fact_5224_rel__restrict__compl,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),Aa2: 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,Aa2)),rel_restrict(A,R,aa(set(A),set(A),uminus_uminus(set(A)),Aa2))) = bot_bot(set(product_prod(A,A))) ) ).

% rel_restrict_compl
tff(fact_5225_homo__rel__restrict__mono,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),B2: set(A),Aa2: set(A)] :
      ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R),product_Sigma(A,A,B2,aTP_Lamp_pt(set(A),fun(A,set(A)),B2)))
     => aa(set(product_prod(A,A)),$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,Aa2)),product_Sigma(A,A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B2),Aa2),aa(set(A),fun(A,set(A)),aTP_Lamp_ra(set(A),fun(set(A),fun(A,set(A))),B2),Aa2))) ) ).

% homo_rel_restrict_mono
tff(fact_5226_inf__img__fin__dom_H,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Aa2: set(B)] :
      ( aa(set(A),$o,finite_finite2(A),aa(set(B),set(A),image2(B,A,F2),Aa2))
     => ( ~ aa(set(B),$o,finite_finite2(B),Aa2)
       => ? [X: A] :
            ( member(A,X,aa(set(B),set(A),image2(B,A,F2),Aa2))
            & ~ aa(set(B),$o,finite_finite2(B),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),vimage(B,A,F2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),X),bot_bot(set(A))))),Aa2)) ) ) ) ).

% inf_img_fin_dom'
tff(fact_5227_inf__img__fin__domE_H,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Aa2: set(B)] :
      ( aa(set(A),$o,finite_finite2(A),aa(set(B),set(A),image2(B,A,F2),Aa2))
     => ( ~ aa(set(B),$o,finite_finite2(B),Aa2)
       => ~ ! [Y2: A] :
              ( member(A,Y2,aa(set(B),set(A),image2(B,A,F2),Aa2))
             => aa(set(B),$o,finite_finite2(B),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),vimage(B,A,F2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Y2),bot_bot(set(A))))),Aa2)) ) ) ) ).

% inf_img_fin_domE'
tff(fact_5228_rel__restrict__alt__def,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),Aa2: set(A)] : ( rel_restrict(A,R,Aa2) = 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)),Aa2),aTP_Lamp_rb(set(A),fun(A,set(A)),Aa2))) ) ).

% rel_restrict_alt_def
tff(fact_5229_lexordp__conv__lexord,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xsa: list(A),Ys: list(A)] :
          ( aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),Xsa),Ys)
        <=> member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xsa),Ys),lexord(A,aa(fun(product_prod(A,A),$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_5230_rel__restrict__Sigma__sub,axiom,
    ! [A: $tType,Aa2: set(A),R: set(A)] : aa(set(product_prod(A,A)),$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,Aa2,aTP_Lamp_pt(set(A),fun(A,set(A)),Aa2))),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)),Aa2),R),aa(set(A),fun(A,set(A)),aTP_Lamp_ra(set(A),fun(set(A),fun(A,set(A))),Aa2),R)))) ).

% rel_restrict_Sigma_sub
tff(fact_5231_inv__image__partition,axiom,
    ! [A: $tType,Xsa: list(A),Pa: fun(A,$o),Ys: list(A)] :
      ( ! [X: A] :
          ( member(A,X,aa(list(A),set(A),set2(A),Xsa))
         => aa(A,$o,Pa,X) )
     => ( ! [Y2: A] :
            ( member(A,Y2,aa(list(A),set(A),set2(A),Ys))
           => ~ aa(A,$o,Pa,Y2) )
       => ( vimage(list(A),product_prod(list(A),list(A)),aa(fun(A,$o),fun(list(A),product_prod(list(A),list(A))),partition(A),Pa),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)))),insert3(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)),Xsa),Ys)),bot_bot(set(product_prod(list(A),list(A)))))) = aa(list(A),set(list(A)),aa(list(A),fun(list(A),set(list(A))),shuffles(A),Xsa),Ys) ) ) ) ).

% inv_image_partition
tff(fact_5232_ord_Olexordp__def,axiom,
    ! [A: $tType,Less: fun(A,fun(A,$o))] : ( lexordp2(A,Less) = complete_lattice_lfp(fun(list(A),fun(list(A),$o)),aTP_Lamp_og(fun(A,fun(A,$o)),fun(fun(list(A),fun(list(A),$o)),fun(list(A),fun(list(A),$o))),Less)) ) ).

% ord.lexordp_def
tff(fact_5233_sorted__key__list__of__set__def,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B)] : ( linord144544945434240204of_set(A,B,F2) = finite_folding_F(A,list(A),linorder_insort_key(A,B,F2),nil(A)) ) ) ).

% sorted_key_list_of_set_def
tff(fact_5234_size__list__conv__sum__list,axiom,
    ! [A: $tType,F2: fun(A,nat),Xsa: list(A)] : ( aa(list(A),nat,size_list(A,F2),Xsa) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),groups8242544230860333062m_list(nat,aa(list(A),list(nat),aa(fun(A,nat),fun(list(A),list(nat)),map(A,nat),F2),Xsa))),aa(list(A),nat,size_size(list(A)),Xsa)) ) ).

% size_list_conv_sum_list
tff(fact_5235_ord_Olexordp__simps_I3_J,axiom,
    ! [A: $tType,Less: fun(A,fun(A,$o)),Xa: A,Xsa: list(A),Y: A,Ys: list(A)] :
      ( aa(list(A),$o,aa(list(A),fun(list(A),$o),lexordp2(A,Less),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys))
    <=> ( aa(A,$o,aa(A,fun(A,$o),Less,Xa),Y)
        | ( ~ aa(A,$o,aa(A,fun(A,$o),Less,Y),Xa)
          & aa(list(A),$o,aa(list(A),fun(list(A),$o),lexordp2(A,Less),Xsa),Ys) ) ) ) ).

% ord.lexordp_simps(3)
tff(fact_5236_ord_Olexordp__simps_I1_J,axiom,
    ! [A: $tType,Less: fun(A,fun(A,$o)),Ys: list(A)] :
      ( aa(list(A),$o,aa(list(A),fun(list(A),$o),lexordp2(A,Less),nil(A)),Ys)
    <=> ( Ys != nil(A) ) ) ).

% ord.lexordp_simps(1)
tff(fact_5237_ord_Olexordp__simps_I2_J,axiom,
    ! [A: $tType,Less: fun(A,fun(A,$o)),Xsa: list(A)] : ~ aa(list(A),$o,aa(list(A),fun(list(A),$o),lexordp2(A,Less),Xsa),nil(A)) ).

% ord.lexordp_simps(2)
tff(fact_5238_size__list__map,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,nat),G: fun(B,A),Xsa: list(B)] : ( aa(list(A),nat,size_list(A,F2),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),G),Xsa)) = aa(list(B),nat,size_list(B,comp(A,nat,B,F2,G)),Xsa) ) ).

% size_list_map
tff(fact_5239_size__list__append,axiom,
    ! [A: $tType,F2: fun(A,nat),Xsa: list(A),Ys: list(A)] : ( aa(list(A),nat,size_list(A,F2),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),Ys)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_list(A,F2),Xsa)),aa(list(A),nat,size_list(A,F2),Ys)) ) ).

% size_list_append
tff(fact_5240_ord_Olexordp_OCons,axiom,
    ! [A: $tType,Less: fun(A,fun(A,$o)),Xa: A,Y: A,Xsa: list(A),Ys: list(A)] :
      ( aa(A,$o,aa(A,fun(A,$o),Less,Xa),Y)
     => aa(list(A),$o,aa(list(A),fun(list(A),$o),lexordp2(A,Less),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)) ) ).

% ord.lexordp.Cons
tff(fact_5241_ord_Olexordp_OCons__eq,axiom,
    ! [A: $tType,Less: fun(A,fun(A,$o)),Xa: A,Y: A,Xsa: list(A),Ys: list(A)] :
      ( ~ aa(A,$o,aa(A,fun(A,$o),Less,Xa),Y)
     => ( ~ aa(A,$o,aa(A,fun(A,$o),Less,Y),Xa)
       => ( aa(list(A),$o,aa(list(A),fun(list(A),$o),lexordp2(A,Less),Xsa),Ys)
         => aa(list(A),$o,aa(list(A),fun(list(A),$o),lexordp2(A,Less),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)) ) ) ) ).

% ord.lexordp.Cons_eq
tff(fact_5242_ord_Olexordp_Ocong,axiom,
    ! [A: $tType,Less: fun(A,fun(A,$o))] : ( lexordp2(A,Less) = lexordp2(A,Less) ) ).

% ord.lexordp.cong
tff(fact_5243_ord_Olexordp__irreflexive,axiom,
    ! [A: $tType,Less: fun(A,fun(A,$o)),Xsa: list(A)] :
      ( ! [X: A] : ~ aa(A,$o,aa(A,fun(A,$o),Less,X),X)
     => ~ aa(list(A),$o,aa(list(A),fun(list(A),$o),lexordp2(A,Less),Xsa),Xsa) ) ).

% ord.lexordp_irreflexive
tff(fact_5244_ord_Olexordp__append__leftI,axiom,
    ! [A: $tType,Less: fun(A,fun(A,$o)),Us: list(A),Vs: list(A),Xsa: list(A)] :
      ( aa(list(A),$o,aa(list(A),fun(list(A),$o),lexordp2(A,Less),Us),Vs)
     => aa(list(A),$o,aa(list(A),fun(list(A),$o),lexordp2(A,Less),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),Us)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),Vs)) ) ).

% ord.lexordp_append_leftI
tff(fact_5245_ord_Olexordp__append__leftD,axiom,
    ! [A: $tType,Less: fun(A,fun(A,$o)),Xsa: list(A),Us: list(A),Vs: list(A)] :
      ( aa(list(A),$o,aa(list(A),fun(list(A),$o),lexordp2(A,Less),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),Us)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),Vs))
     => ( ! [A4: A] : ~ aa(A,$o,aa(A,fun(A,$o),Less,A4),A4)
       => aa(list(A),$o,aa(list(A),fun(list(A),$o),lexordp2(A,Less),Us),Vs) ) ) ).

% ord.lexordp_append_leftD
tff(fact_5246_list_Osize__gen__o__map,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,nat),G: fun(A,B)] : ( comp(list(B),nat,list(A),size_list(B,F2),aa(fun(A,B),fun(list(A),list(B)),map(A,B),G)) = size_list(A,comp(B,nat,A,F2,G)) ) ).

% list.size_gen_o_map
tff(fact_5247_ord_Olexordp_Osimps,axiom,
    ! [A: $tType,Less: fun(A,fun(A,$o)),A1: list(A),A22: list(A)] :
      ( aa(list(A),$o,aa(list(A),fun(list(A),$o),lexordp2(A,Less),A1),A22)
    <=> ( ? [Y4: A,Ys4: list(A)] :
            ( ( A1 = nil(A) )
            & ( A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys4) ) )
        | ? [X6: A,Y4: A,Xs2: list(A),Ys4: list(A)] :
            ( ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X6),Xs2) )
            & ( A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys4) )
            & aa(A,$o,aa(A,fun(A,$o),Less,X6),Y4) )
        | ? [X6: A,Y4: A,Xs2: list(A),Ys4: list(A)] :
            ( ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X6),Xs2) )
            & ( A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys4) )
            & ~ aa(A,$o,aa(A,fun(A,$o),Less,X6),Y4)
            & ~ aa(A,$o,aa(A,fun(A,$o),Less,Y4),X6)
            & aa(list(A),$o,aa(list(A),fun(list(A),$o),lexordp2(A,Less),Xs2),Ys4) ) ) ) ).

% ord.lexordp.simps
tff(fact_5248_ord_Olexordp_Ocases,axiom,
    ! [A: $tType,Less: fun(A,fun(A,$o)),A1: list(A),A22: list(A)] :
      ( aa(list(A),$o,aa(list(A),fun(list(A),$o),lexordp2(A,Less),A1),A22)
     => ( ( ( A1 = nil(A) )
         => ! [Y2: A,Ys5: list(A)] : ( A22 != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys5) ) )
       => ( ! [X: A] :
              ( ? [Xs: list(A)] : ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) )
             => ! [Y2: A] :
                  ( ? [Ys5: list(A)] : ( A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys5) )
                 => ~ aa(A,$o,aa(A,fun(A,$o),Less,X),Y2) ) )
         => ~ ! [X: A,Y2: A,Xs: list(A)] :
                ( ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) )
               => ! [Ys5: list(A)] :
                    ( ( A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys5) )
                   => ( ~ aa(A,$o,aa(A,fun(A,$o),Less,X),Y2)
                     => ( ~ aa(A,$o,aa(A,fun(A,$o),Less,Y2),X)
                       => ~ aa(list(A),$o,aa(list(A),fun(list(A),$o),lexordp2(A,Less),Xs),Ys5) ) ) ) ) ) ) ) ).

% ord.lexordp.cases
tff(fact_5249_ord_Olexordp_ONil,axiom,
    ! [A: $tType,Less: fun(A,fun(A,$o)),Y: A,Ys: list(A)] : aa(list(A),$o,aa(list(A),fun(list(A),$o),lexordp2(A,Less),nil(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)) ).

% ord.lexordp.Nil
tff(fact_5250_ord_Olexordp__append__left__rightI,axiom,
    ! [A: $tType,Less: fun(A,fun(A,$o)),Xa: A,Y: A,Us: list(A),Xsa: list(A),Ys: list(A)] :
      ( aa(A,$o,aa(A,fun(A,$o),Less,Xa),Y)
     => aa(list(A),$o,aa(list(A),fun(list(A),$o),lexordp2(A,Less),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa))),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys))) ) ).

% ord.lexordp_append_left_rightI
tff(fact_5251_ord_Olexordp__append__rightI,axiom,
    ! [A: $tType,Ys: list(A),Less: fun(A,fun(A,$o)),Xsa: list(A)] :
      ( ( Ys != nil(A) )
     => aa(list(A),$o,aa(list(A),fun(list(A),$o),lexordp2(A,Less),Xsa),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),Ys)) ) ).

% ord.lexordp_append_rightI
tff(fact_5252_list_Osize__gen_I1_J,axiom,
    ! [A: $tType,Xa: fun(A,nat)] : ( aa(list(A),nat,size_list(A,Xa),nil(A)) = zero_zero(nat) ) ).

% list.size_gen(1)
tff(fact_5253_size__list__estimation,axiom,
    ! [A: $tType,Xa: A,Xsa: list(A),Y: nat,F2: fun(A,nat)] :
      ( member(A,Xa,aa(list(A),set(A),set2(A),Xsa))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y),aa(A,nat,F2,Xa))
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y),aa(list(A),nat,size_list(A,F2),Xsa)) ) ) ).

% size_list_estimation
tff(fact_5254_size__list__pointwise,axiom,
    ! [A: $tType,Xsa: list(A),F2: fun(A,nat),G: fun(A,nat)] :
      ( ! [X: A] :
          ( member(A,X,aa(list(A),set(A),set2(A),Xsa))
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,F2,X)),aa(A,nat,G,X)) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_list(A,F2),Xsa)),aa(list(A),nat,size_list(A,G),Xsa)) ) ).

% size_list_pointwise
tff(fact_5255_size__list__estimation_H,axiom,
    ! [A: $tType,Xa: A,Xsa: list(A),Y: nat,F2: fun(A,nat)] :
      ( member(A,Xa,aa(list(A),set(A),set2(A),Xsa))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y),aa(A,nat,F2,Xa))
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y),aa(list(A),nat,size_list(A,F2),Xsa)) ) ) ).

% size_list_estimation'
tff(fact_5256_list_Osize__gen_I2_J,axiom,
    ! [A: $tType,Xa: fun(A,nat),X21: A,X222: list(A)] : ( aa(list(A),nat,size_list(A,Xa),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,Xa,X21)),aa(list(A),nat,size_list(A,Xa),X222))),aa(nat,nat,suc,zero_zero(nat))) ) ).

% list.size_gen(2)
tff(fact_5257_folding__on_Oremove,axiom,
    ! [B: $tType,A: $tType,S: set(A),F2: fun(A,fun(B,B)),Aa2: set(A),Xa: A,Z2: B] :
      ( finite_folding_on(A,B,S,F2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),S)
       => ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( member(A,Xa,Aa2)
           => ( aa(set(A),B,finite_folding_F(A,B,F2,Z2),Aa2) = aa(B,B,aa(A,fun(B,B),F2,Xa),aa(set(A),B,finite_folding_F(A,B,F2,Z2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A)))))) ) ) ) ) ) ).

% folding_on.remove
tff(fact_5258_folding__on_Oinsert__remove,axiom,
    ! [B: $tType,A: $tType,S: set(A),F2: fun(A,fun(B,B)),Xa: A,Aa2: set(A),Z2: B] :
      ( finite_folding_on(A,B,S,F2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),Aa2)),S)
       => ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( aa(set(A),B,finite_folding_F(A,B,F2,Z2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),Aa2)) = aa(B,B,aa(A,fun(B,B),F2,Xa),aa(set(A),B,finite_folding_F(A,B,F2,Z2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A)))))) ) ) ) ) ).

% folding_on.insert_remove
tff(fact_5259_folding__idem__on_Oinsert__idem,axiom,
    ! [B: $tType,A: $tType,S: set(A),F2: fun(A,fun(B,B)),Xa: A,Aa2: set(A),Z2: B] :
      ( finite1890593828518410140dem_on(A,B,S,F2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),Aa2)),S)
       => ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( aa(set(A),B,finite_folding_F(A,B,F2,Z2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),Aa2)) = aa(B,B,aa(A,fun(B,B),F2,Xa),aa(set(A),B,finite_folding_F(A,B,F2,Z2),Aa2)) ) ) ) ) ).

% folding_idem_on.insert_idem
tff(fact_5260_folding__on_Oempty,axiom,
    ! [A: $tType,B: $tType,S: set(A),F2: fun(A,fun(B,B)),Z2: B] :
      ( finite_folding_on(A,B,S,F2)
     => ( aa(set(A),B,finite_folding_F(A,B,F2,Z2),bot_bot(set(A))) = Z2 ) ) ).

% folding_on.empty
tff(fact_5261_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_ip(A,A))) ) ).

% sorted_list_of_set.fold_insort_key.folding_on_axioms
tff(fact_5262_folding__on_Oinsert,axiom,
    ! [B: $tType,A: $tType,S: set(A),F2: fun(A,fun(B,B)),Xa: A,Aa2: set(A),Z2: B] :
      ( finite_folding_on(A,B,S,F2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),Aa2)),S)
       => ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( ~ member(A,Xa,Aa2)
           => ( aa(set(A),B,finite_folding_F(A,B,F2,Z2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),Aa2)) = aa(B,B,aa(A,fun(B,B),F2,Xa),aa(set(A),B,finite_folding_F(A,B,F2,Z2),Aa2)) ) ) ) ) ) ).

% folding_on.insert
tff(fact_5263_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),$o),aTP_Lamp_rc(set(product_prod(A,A)),fun(multiset(A),fun(multiset(A),$o)),R2))) ) ).

% mult1_def
tff(fact_5264_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_rd(set(product_prod(A,A)),fun(product_prod(set(A),set(A)),$o),R2)) ) ).

% min_ext_def
tff(fact_5265_comp__fun__commute__on_Ofold__graph__insertE__aux,axiom,
    ! [A: $tType,B: $tType,S: set(A),F2: fun(A,fun(B,B)),Aa2: set(A),Z2: B,Y: B,A3: A] :
      ( finite4664212375090638736ute_on(A,B,S,F2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),S)
       => ( finite_fold_graph(A,B,F2,Z2,Aa2,Y)
         => ( member(A,A3,Aa2)
           => ? [Y7: B] :
                ( ( Y = aa(B,B,aa(A,fun(B,B),F2,A3),Y7) )
                & finite_fold_graph(A,B,F2,Z2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),bot_bot(set(A)))),Y7) ) ) ) ) ) ).

% comp_fun_commute_on.fold_graph_insertE_aux
tff(fact_5266_bex__empty,axiom,
    ! [A: $tType,Pa: fun(A,$o)] :
      ~ ? [X4: A] :
          ( member(A,X4,bot_bot(set(A)))
          & aa(A,$o,Pa,X4) ) ).

% bex_empty
tff(fact_5267_ball__UNIV,axiom,
    ! [A: $tType,Pa: fun(A,$o)] :
      ( ! [X6: A] :
          ( member(A,X6,top_top(set(A)))
         => aa(A,$o,Pa,X6) )
    <=> ! [X_12: A] : aa(A,$o,Pa,X_12) ) ).

% ball_UNIV
tff(fact_5268_bex__UNIV,axiom,
    ! [A: $tType,Pa: fun(A,$o)] :
      ( ? [X6: A] :
          ( member(A,X6,top_top(set(A)))
          & aa(A,$o,Pa,X6) )
    <=> ? [X_12: A] : aa(A,$o,Pa,X_12) ) ).

% bex_UNIV
tff(fact_5269_empty__fold__graphE,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,fun(B,B)),Z2: B,Xa: B] :
      ( finite_fold_graph(A,B,F2,Z2,bot_bot(set(A)),Xa)
     => ( Xa = Z2 ) ) ).

% empty_fold_graphE
tff(fact_5270_fold__graph_OemptyI,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,fun(B,B)),Z2: B] : finite_fold_graph(A,B,F2,Z2,bot_bot(set(A)),Z2) ).

% fold_graph.emptyI
tff(fact_5271_image__def,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Aa2: set(B)] : ( aa(set(B),set(A),image2(B,A,F2),Aa2) = aa(fun(A,$o),set(A),collect(A),aa(set(B),fun(A,$o),aTP_Lamp_re(fun(B,A),fun(set(B),fun(A,$o)),F2),Aa2)) ) ).

% image_def
tff(fact_5272_fold__graph_OinsertI,axiom,
    ! [A: $tType,B: $tType,Xa: A,Aa2: set(A),F2: fun(A,fun(B,B)),Z2: B,Y: B] :
      ( ~ member(A,Xa,Aa2)
     => ( finite_fold_graph(A,B,F2,Z2,Aa2,Y)
       => finite_fold_graph(A,B,F2,Z2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),Aa2),aa(B,B,aa(A,fun(B,B),F2,Xa),Y)) ) ) ).

% fold_graph.insertI
tff(fact_5273_Bex__set__list__ex,axiom,
    ! [A: $tType,Xsa: list(A),Pa: fun(A,$o)] :
      ( ? [X6: A] :
          ( member(A,X6,aa(list(A),set(A),set2(A),Xsa))
          & aa(A,$o,Pa,X6) )
    <=> aa(list(A),$o,aa(fun(A,$o),fun(list(A),$o),list_ex(A),Pa),Xsa) ) ).

% Bex_set_list_ex
tff(fact_5274_vimage__image__eq,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),Aa2: set(A)] : ( vimage(A,B,F2,aa(set(A),set(B),image2(A,B,F2),Aa2)) = aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_rf(fun(A,B),fun(set(A),fun(A,$o)),F2),Aa2)) ) ).

% vimage_image_eq
tff(fact_5275_fold__graph_Ocases,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,fun(B,B)),Z2: B,A1: set(A),A22: B] :
      ( finite_fold_graph(A,B,F2,Z2,A1,A22)
     => ( ( ( A1 = bot_bot(set(A)) )
         => ( A22 != Z2 ) )
       => ~ ! [X: A,A12: set(A)] :
              ( ( A1 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),X),A12) )
             => ! [Y2: B] :
                  ( ( A22 = aa(B,B,aa(A,fun(B,B),F2,X),Y2) )
                 => ( ~ member(A,X,A12)
                   => ~ finite_fold_graph(A,B,F2,Z2,A12,Y2) ) ) ) ) ) ).

% fold_graph.cases
tff(fact_5276_fold__graph_Osimps,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,fun(B,B)),Z2: B,A1: set(A),A22: B] :
      ( finite_fold_graph(A,B,F2,Z2,A1,A22)
    <=> ( ( ( A1 = bot_bot(set(A)) )
          & ( A22 = Z2 ) )
        | ? [X6: A,A11: set(A),Y4: B] :
            ( ( A1 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),X6),A11) )
            & ( A22 = aa(B,B,aa(A,fun(B,B),F2,X6),Y4) )
            & ~ member(A,X6,A11)
            & finite_fold_graph(A,B,F2,Z2,A11,Y4) ) ) ) ).

% fold_graph.simps
tff(fact_5277_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),$o),aTP_Lamp_rg(set(product_prod(A,A)),fun(set(A),fun(set(A),$o)),R))) ) ).

% max_ext_eq
tff(fact_5278_nths__nths,axiom,
    ! [A: $tType,Xsa: list(A),Aa2: set(nat),B2: set(nat)] : ( aa(set(nat),list(A),aa(list(A),fun(set(nat),list(A)),nths(A),aa(set(nat),list(A),aa(list(A),fun(set(nat),list(A)),nths(A),Xsa),Aa2)),B2) = aa(set(nat),list(A),aa(list(A),fun(set(nat),list(A)),nths(A),Xsa),aa(fun(nat,$o),set(nat),collect(nat),aa(set(nat),fun(nat,$o),aTP_Lamp_ri(set(nat),fun(set(nat),fun(nat,$o)),Aa2),B2))) ) ).

% nths_nths
tff(fact_5279_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))) )
             => ~ ! [X4: A] :
                    ( member(A,X4,A1)
                   => ? [Xa5: A] :
                        ( member(A,Xa5,A22)
                        & aa(A,$o,aa(A,fun(A,$o),R,X4),Xa5) ) ) ) ) ) ) ).

% max_extp.cases
tff(fact_5280_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))) )
        & ! [X6: A] :
            ( member(A,X6,A1)
           => ? [Xa2: A] :
                ( member(A,Xa2,A22)
                & aa(A,$o,aa(A,fun(A,$o),R,X6),Xa2) ) ) ) ) ).

% max_extp.simps
tff(fact_5281_max__extp_Omax__extI,axiom,
    ! [A: $tType,X3: set(A),Y5: set(A),R: fun(A,fun(A,$o))] :
      ( aa(set(A),$o,finite_finite2(A),X3)
     => ( aa(set(A),$o,finite_finite2(A),Y5)
       => ( ( Y5 != aa(fun(A,$o),set(A),collect(A),bot_bot(fun(A,$o))) )
         => ( ! [X: A] :
                ( member(A,X,X3)
               => ? [Xa4: A] :
                    ( member(A,Xa4,Y5)
                    & aa(A,$o,aa(A,fun(A,$o),R,X),Xa4) ) )
           => aa(set(A),$o,aa(set(A),fun(set(A),$o),max_extp(A,R),X3),Y5) ) ) ) ) ).

% max_extp.max_extI
tff(fact_5282_comp__fun__commute__on_Ofold__graph__insertE,axiom,
    ! [A: $tType,B: $tType,S: set(A),F2: fun(A,fun(B,B)),Xa: A,Aa2: set(A),Z2: B,V2: B] :
      ( finite4664212375090638736ute_on(A,B,S,F2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),Aa2)),S)
       => ( finite_fold_graph(A,B,F2,Z2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),Aa2),V2)
         => ( ~ member(A,Xa,Aa2)
           => ~ ! [Y2: B] :
                  ( ( V2 = aa(B,B,aa(A,fun(B,B),F2,Xa),Y2) )
                 => ~ finite_fold_graph(A,B,F2,Z2,Aa2,Y2) ) ) ) ) ) ).

% comp_fun_commute_on.fold_graph_insertE
tff(fact_5283_prod_Oset__conv__list,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(B,A),Xsa: list(B)] : ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),aa(list(B),set(B),set2(B),Xsa)) = aa(list(A),A,groups5270119922927024881d_list(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),G),aa(list(B),list(B),remdups(B),Xsa))) ) ) ).

% prod.set_conv_list
tff(fact_5284_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),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))))
           => ( 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) = infini527867602293511546merate(A,S,N) ) ) ) ) ).

% finite_enumerate_initial_segment
tff(fact_5285_INF__set__fold,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(B,A),Xsa: list(B)] : ( aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F2),aa(list(B),set(B),set2(B),Xsa))) = aa(A,A,aa(list(B),fun(A,A),aa(fun(B,fun(A,A)),fun(list(B),fun(A,A)),fold(B,A),comp(A,fun(A,A),B,inf_inf(A),F2)),Xsa),top_top(A)) ) ) ).

% INF_set_fold
tff(fact_5286_fold__append,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,fun(A,A)),Xsa: list(B),Ys: list(B)] : ( aa(list(B),fun(A,A),aa(fun(B,fun(A,A)),fun(list(B),fun(A,A)),fold(B,A),F2),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Xsa),Ys)) = comp(A,A,A,aa(list(B),fun(A,A),aa(fun(B,fun(A,A)),fun(list(B),fun(A,A)),fold(B,A),F2),Ys),aa(list(B),fun(A,A),aa(fun(B,fun(A,A)),fun(list(B),fun(A,A)),fold(B,A),F2),Xsa)) ) ).

% fold_append
tff(fact_5287_prod__list_OCons,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Xa: A,Xsa: list(A)] : ( aa(list(A),A,groups5270119922927024881d_list(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa)) = aa(A,A,aa(A,fun(A,A),times_times(A),Xa),aa(list(A),A,groups5270119922927024881d_list(A),Xsa)) ) ) ).

% prod_list.Cons
tff(fact_5288_prod__list_Oappend,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Xsa: 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),Xsa),Ys)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(list(A),A,groups5270119922927024881d_list(A),Xsa)),aa(list(A),A,groups5270119922927024881d_list(A),Ys)) ) ) ).

% prod_list.append
tff(fact_5289_fold__replicate,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,fun(A,A)),N: nat,Xa: B] : ( aa(list(B),fun(A,A),aa(fun(B,fun(A,A)),fun(list(B),fun(A,A)),fold(B,A),F2),aa(B,list(B),aa(nat,fun(B,list(B)),replicate(B),N),Xa)) = compow(fun(A,A),N,aa(B,fun(A,A),F2,Xa)) ) ).

% fold_replicate
tff(fact_5290_enumerate__mono__iff,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S: set(A),M2: nat,N: nat] :
          ( ~ aa(set(A),$o,finite_finite2(A),S)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),infini527867602293511546merate(A,S,M2)),infini527867602293511546merate(A,S,N))
          <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),N) ) ) ) ).

% enumerate_mono_iff
tff(fact_5291_finite__enumerate__mono__iff,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S: set(A),M2: nat,N: nat] :
          ( aa(set(A),$o,finite_finite2(A),S)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),finite_card(A,S))
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),finite_card(A,S))
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),infini527867602293511546merate(A,S,M2)),infini527867602293511546merate(A,S,N))
              <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),N) ) ) ) ) ) ).

% finite_enumerate_mono_iff
tff(fact_5292_fold__simps_I1_J,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,fun(A,A)),S2: A] : ( aa(A,A,aa(list(B),fun(A,A),aa(fun(B,fun(A,A)),fun(list(B),fun(A,A)),fold(B,A),F2),nil(B)),S2) = S2 ) ).

% fold_simps(1)
tff(fact_5293_foldl__conv__fold,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,fun(B,A)),S2: A,Xsa: list(B)] : ( aa(list(B),A,aa(A,fun(list(B),A),aa(fun(A,fun(B,A)),fun(A,fun(list(B),A)),foldl(A,B),F2),S2),Xsa) = aa(A,A,aa(list(B),fun(A,A),aa(fun(B,fun(A,A)),fun(list(B),fun(A,A)),fold(B,A),aTP_Lamp_kv(fun(A,fun(B,A)),fun(B,fun(A,A)),F2)),Xsa),S2) ) ).

% foldl_conv_fold
tff(fact_5294_fold__invariant,axiom,
    ! [A: $tType,B: $tType,Xsa: list(A),Qa: fun(A,$o),Pa: fun(B,$o),S2: B,F2: fun(A,fun(B,B))] :
      ( ! [X: A] :
          ( member(A,X,aa(list(A),set(A),set2(A),Xsa))
         => aa(A,$o,Qa,X) )
     => ( aa(B,$o,Pa,S2)
       => ( ! [X: A,S6: B] :
              ( aa(A,$o,Qa,X)
             => ( aa(B,$o,Pa,S6)
               => aa(B,$o,Pa,aa(B,B,aa(A,fun(B,B),F2,X),S6)) ) )
         => aa(B,$o,Pa,aa(B,B,aa(list(A),fun(B,B),aa(fun(A,fun(B,B)),fun(list(A),fun(B,B)),fold(A,B),F2),Xsa),S2)) ) ) ) ).

% fold_invariant
tff(fact_5295_List_Ofold__cong,axiom,
    ! [B: $tType,A: $tType,A3: A,B3: A,Xsa: list(B),Ys: list(B),F2: fun(B,fun(A,A)),G: fun(B,fun(A,A))] :
      ( ( A3 = B3 )
     => ( ( Xsa = Ys )
       => ( ! [X: B] :
              ( member(B,X,aa(list(B),set(B),set2(B),Xsa))
             => ( aa(B,fun(A,A),F2,X) = aa(B,fun(A,A),G,X) ) )
         => ( aa(A,A,aa(list(B),fun(A,A),aa(fun(B,fun(A,A)),fun(list(B),fun(A,A)),fold(B,A),F2),Xsa),A3) = aa(A,A,aa(list(B),fun(A,A),aa(fun(B,fun(A,A)),fun(list(B),fun(A,A)),fold(B,A),G),Ys),B3) ) ) ) ) ).

% List.fold_cong
tff(fact_5296_fold__simps_I2_J,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,fun(A,A)),Xa: B,Xsa: list(B),S2: A] : ( aa(A,A,aa(list(B),fun(A,A),aa(fun(B,fun(A,A)),fun(list(B),fun(A,A)),fold(B,A),F2),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Xa),Xsa)),S2) = aa(A,A,aa(list(B),fun(A,A),aa(fun(B,fun(A,A)),fun(list(B),fun(A,A)),fold(B,A),F2),Xsa),aa(A,A,aa(B,fun(A,A),F2,Xa),S2)) ) ).

% fold_simps(2)
tff(fact_5297_fold__Cons,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,fun(A,A)),Xa: B,Xsa: list(B)] : ( aa(list(B),fun(A,A),aa(fun(B,fun(A,A)),fun(list(B),fun(A,A)),fold(B,A),F2),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Xa),Xsa)) = comp(A,A,A,aa(list(B),fun(A,A),aa(fun(B,fun(A,A)),fun(list(B),fun(A,A)),fold(B,A),F2),Xsa),aa(B,fun(A,A),F2,Xa)) ) ).

% fold_Cons
tff(fact_5298_fold__commute__apply,axiom,
    ! [A: $tType,E: $tType,B: $tType,Xsa: list(A),H2: fun(B,E),G: fun(A,fun(B,B)),F2: fun(A,fun(E,E)),S2: B] :
      ( ! [X: A] :
          ( member(A,X,aa(list(A),set(A),set2(A),Xsa))
         => ( comp(B,E,B,H2,aa(A,fun(B,B),G,X)) = comp(E,E,B,aa(A,fun(E,E),F2,X),H2) ) )
     => ( aa(B,E,H2,aa(B,B,aa(list(A),fun(B,B),aa(fun(A,fun(B,B)),fun(list(A),fun(B,B)),fold(A,B),G),Xsa),S2)) = aa(E,E,aa(list(A),fun(E,E),aa(fun(A,fun(E,E)),fun(list(A),fun(E,E)),fold(A,E),F2),Xsa),aa(B,E,H2,S2)) ) ) ).

% fold_commute_apply
tff(fact_5299_fold__commute,axiom,
    ! [A: $tType,E: $tType,B: $tType,Xsa: list(A),H2: fun(B,E),G: fun(A,fun(B,B)),F2: fun(A,fun(E,E))] :
      ( ! [X: A] :
          ( member(A,X,aa(list(A),set(A),set2(A),Xsa))
         => ( comp(B,E,B,H2,aa(A,fun(B,B),G,X)) = comp(E,E,B,aa(A,fun(E,E),F2,X),H2) ) )
     => ( comp(B,E,B,H2,aa(list(A),fun(B,B),aa(fun(A,fun(B,B)),fun(list(A),fun(B,B)),fold(A,B),G),Xsa)) = comp(E,E,B,aa(list(A),fun(E,E),aa(fun(A,fun(E,E)),fun(list(A),fun(E,E)),fold(A,E),F2),Xsa),H2) ) ) ).

% fold_commute
tff(fact_5300_fold__Nil,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,fun(A,A))] : ( aa(list(B),fun(A,A),aa(fun(B,fun(A,A)),fun(list(B),fun(A,A)),fold(B,A),F2),nil(B)) = id(A) ) ).

% fold_Nil
tff(fact_5301_fold__id,axiom,
    ! [A: $tType,B: $tType,Xsa: list(A),F2: fun(A,fun(B,B))] :
      ( ! [X: A] :
          ( member(A,X,aa(list(A),set(A),set2(A),Xsa))
         => ( aa(A,fun(B,B),F2,X) = id(B) ) )
     => ( aa(list(A),fun(B,B),aa(fun(A,fun(B,B)),fun(list(A),fun(B,B)),fold(A,B),F2),Xsa) = id(B) ) ) ).

% fold_id
tff(fact_5302_fold__permuted__eq,axiom,
    ! [A: $tType,B: $tType,Xsa: list(A),Ys: list(A),Pa: fun(B,$o),Z2: B,F2: fun(A,fun(B,B))] :
      ( ( aa(list(A),multiset(A),mset(A),Xsa) = aa(list(A),multiset(A),mset(A),Ys) )
     => ( aa(B,$o,Pa,Z2)
       => ( ! [X: A,Z3: B] :
              ( member(A,X,aa(list(A),set(A),set2(A),Xsa))
             => ( aa(B,$o,Pa,Z3)
               => aa(B,$o,Pa,aa(B,B,aa(A,fun(B,B),F2,X),Z3)) ) )
         => ( ! [X: A,Y2: A,Z3: B] :
                ( member(A,X,aa(list(A),set(A),set2(A),Xsa))
               => ( member(A,Y2,aa(list(A),set(A),set2(A),Xsa))
                 => ( aa(B,$o,Pa,Z3)
                   => ( aa(B,B,aa(A,fun(B,B),F2,X),aa(B,B,aa(A,fun(B,B),F2,Y2),Z3)) = aa(B,B,aa(A,fun(B,B),F2,Y2),aa(B,B,aa(A,fun(B,B),F2,X),Z3)) ) ) ) )
           => ( aa(B,B,aa(list(A),fun(B,B),aa(fun(A,fun(B,B)),fun(list(A),fun(B,B)),fold(A,B),F2),Xsa),Z2) = aa(B,B,aa(list(A),fun(B,B),aa(fun(A,fun(B,B)),fun(list(A),fun(B,B)),fold(A,B),F2),Ys),Z2) ) ) ) ) ) ).

% fold_permuted_eq
tff(fact_5303_fold__map,axiom,
    ! [B: $tType,A: $tType,E: $tType,G: fun(B,fun(A,A)),F2: fun(E,B),Xsa: list(E)] : ( aa(list(B),fun(A,A),aa(fun(B,fun(A,A)),fun(list(B),fun(A,A)),fold(B,A),G),aa(list(E),list(B),aa(fun(E,B),fun(list(E),list(B)),map(E,B),F2),Xsa)) = aa(list(E),fun(A,A),aa(fun(E,fun(A,A)),fun(list(E),fun(A,A)),fold(E,A),comp(B,fun(A,A),E,G,F2)),Xsa) ) ).

% fold_map
tff(fact_5304_foldr__conv__fold,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,fun(A,A)),Xsa: list(B)] : ( aa(list(B),fun(A,A),aa(fun(B,fun(A,A)),fun(list(B),fun(A,A)),foldr(B,A),F2),Xsa) = aa(list(B),fun(A,A),aa(fun(B,fun(A,A)),fun(list(B),fun(A,A)),fold(B,A),F2),aa(list(B),list(B),rev(B),Xsa)) ) ).

% foldr_conv_fold
tff(fact_5305_fold__filter,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,fun(A,A)),Pa: fun(B,$o),Xsa: list(B)] : ( aa(list(B),fun(A,A),aa(fun(B,fun(A,A)),fun(list(B),fun(A,A)),fold(B,A),F2),aa(list(B),list(B),aa(fun(B,$o),fun(list(B),list(B)),filter2(B),Pa),Xsa)) = aa(list(B),fun(A,A),aa(fun(B,fun(A,A)),fun(list(B),fun(A,A)),fold(B,A),aa(fun(B,$o),fun(B,fun(A,A)),aTP_Lamp_on(fun(B,fun(A,A)),fun(fun(B,$o),fun(B,fun(A,A))),F2),Pa)),Xsa) ) ).

% fold_filter
tff(fact_5306_prod__list__zero__iff,axiom,
    ! [A: $tType] :
      ( ( semiring_1(A)
        & semiri3467727345109120633visors(A) )
     => ! [Xsa: list(A)] :
          ( ( aa(list(A),A,groups5270119922927024881d_list(A),Xsa) = zero_zero(A) )
        <=> member(A,zero_zero(A),aa(list(A),set(A),set2(A),Xsa)) ) ) ).

% prod_list_zero_iff
tff(fact_5307_union__set__fold,axiom,
    ! [A: $tType,Xsa: list(A),Aa2: 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),Xsa)),Aa2) = aa(set(A),set(A),aa(list(A),fun(set(A),set(A)),aa(fun(A,fun(set(A),set(A))),fun(list(A),fun(set(A),set(A))),fold(A,set(A)),insert3(A)),Xsa),Aa2) ) ).

% union_set_fold
tff(fact_5308_rev__conv__fold,axiom,
    ! [A: $tType,Xsa: list(A)] : ( aa(list(A),list(A),rev(A),Xsa) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),aa(fun(A,fun(list(A),list(A))),fun(list(A),fun(list(A),list(A))),fold(A,list(A)),cons(A)),Xsa),nil(A)) ) ).

% rev_conv_fold
tff(fact_5309_fold__Cons__rev,axiom,
    ! [A: $tType,Xsa: list(A)] : ( aa(list(A),fun(list(A),list(A)),aa(fun(A,fun(list(A),list(A))),fun(list(A),fun(list(A),list(A))),fold(A,list(A)),cons(A)),Xsa) = aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),rev(A),Xsa)) ) ).

% fold_Cons_rev
tff(fact_5310_fold__multiset__equiv,axiom,
    ! [B: $tType,A: $tType,Xsa: list(A),F2: fun(A,fun(B,B)),Ys: list(A)] :
      ( ! [X: A,Y2: A] :
          ( member(A,X,aa(list(A),set(A),set2(A),Xsa))
         => ( member(A,Y2,aa(list(A),set(A),set2(A),Xsa))
           => ( comp(B,B,B,aa(A,fun(B,B),F2,X),aa(A,fun(B,B),F2,Y2)) = comp(B,B,B,aa(A,fun(B,B),F2,Y2),aa(A,fun(B,B),F2,X)) ) ) )
     => ( ( aa(list(A),multiset(A),mset(A),Xsa) = aa(list(A),multiset(A),mset(A),Ys) )
       => ( aa(list(A),fun(B,B),aa(fun(A,fun(B,B)),fun(list(A),fun(B,B)),fold(A,B),F2),Xsa) = aa(list(A),fun(B,B),aa(fun(A,fun(B,B)),fun(list(A),fun(B,B)),fold(A,B),F2),Ys) ) ) ) ).

% fold_multiset_equiv
tff(fact_5311_fold__rev,axiom,
    ! [B: $tType,A: $tType,Xsa: list(A),F2: fun(A,fun(B,B))] :
      ( ! [X: A,Y2: A] :
          ( member(A,X,aa(list(A),set(A),set2(A),Xsa))
         => ( member(A,Y2,aa(list(A),set(A),set2(A),Xsa))
           => ( comp(B,B,B,aa(A,fun(B,B),F2,Y2),aa(A,fun(B,B),F2,X)) = comp(B,B,B,aa(A,fun(B,B),F2,X),aa(A,fun(B,B),F2,Y2)) ) ) )
     => ( aa(list(A),fun(B,B),aa(fun(A,fun(B,B)),fun(list(A),fun(B,B)),fold(A,B),F2),aa(list(A),list(A),rev(A),Xsa)) = aa(list(A),fun(B,B),aa(fun(A,fun(B,B)),fun(list(A),fun(B,B)),fold(A,B),F2),Xsa) ) ) ).

% fold_rev
tff(fact_5312_List_Ounion__def,axiom,
    ! [A: $tType] : ( union(A) = aa(fun(A,fun(list(A),list(A))),fun(list(A),fun(list(A),list(A))),fold(A,list(A)),insert(A)) ) ).

% List.union_def
tff(fact_5313_foldr__fold,axiom,
    ! [B: $tType,A: $tType,Xsa: list(A),F2: fun(A,fun(B,B))] :
      ( ! [X: A,Y2: A] :
          ( member(A,X,aa(list(A),set(A),set2(A),Xsa))
         => ( member(A,Y2,aa(list(A),set(A),set2(A),Xsa))
           => ( comp(B,B,B,aa(A,fun(B,B),F2,Y2),aa(A,fun(B,B),F2,X)) = comp(B,B,B,aa(A,fun(B,B),F2,X),aa(A,fun(B,B),F2,Y2)) ) ) )
     => ( aa(list(A),fun(B,B),aa(fun(A,fun(B,B)),fun(list(A),fun(B,B)),foldr(A,B),F2),Xsa) = aa(list(A),fun(B,B),aa(fun(A,fun(B,B)),fun(list(A),fun(B,B)),fold(A,B),F2),Xsa) ) ) ).

% foldr_fold
tff(fact_5314_fold__remove1__split,axiom,
    ! [B: $tType,A: $tType,Xsa: list(A),F2: fun(A,fun(B,B)),Xa: A] :
      ( ! [X: A,Y2: A] :
          ( member(A,X,aa(list(A),set(A),set2(A),Xsa))
         => ( member(A,Y2,aa(list(A),set(A),set2(A),Xsa))
           => ( comp(B,B,B,aa(A,fun(B,B),F2,X),aa(A,fun(B,B),F2,Y2)) = comp(B,B,B,aa(A,fun(B,B),F2,Y2),aa(A,fun(B,B),F2,X)) ) ) )
     => ( member(A,Xa,aa(list(A),set(A),set2(A),Xsa))
       => ( aa(list(A),fun(B,B),aa(fun(A,fun(B,B)),fun(list(A),fun(B,B)),fold(A,B),F2),Xsa) = comp(B,B,B,aa(list(A),fun(B,B),aa(fun(A,fun(B,B)),fun(list(A),fun(B,B)),fold(A,B),F2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),remove1(A),Xa),Xsa)),aa(A,fun(B,B),F2,Xa)) ) ) ) ).

% fold_remove1_split
tff(fact_5315_minus__set__fold,axiom,
    ! [A: $tType,Aa2: set(A),Xsa: list(A)] : ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(list(A),set(A),set2(A),Xsa)) = aa(set(A),set(A),aa(list(A),fun(set(A),set(A)),aa(fun(A,fun(set(A),set(A))),fun(list(A),fun(set(A),set(A))),fold(A,set(A)),remove(A)),Xsa),Aa2) ) ).

% minus_set_fold
tff(fact_5316_enumerate__step,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S: set(A),N: nat] :
          ( ~ aa(set(A),$o,finite_finite2(A),S)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),infini527867602293511546merate(A,S,N)),infini527867602293511546merate(A,S,aa(nat,nat,suc,N))) ) ) ).

% enumerate_step
tff(fact_5317_enumerate__mono,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [M2: nat,N: nat,S: set(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),N)
         => ( ~ aa(set(A),$o,finite_finite2(A),S)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),infini527867602293511546merate(A,S,M2)),infini527867602293511546merate(A,S,N)) ) ) ) ).

% enumerate_mono
tff(fact_5318_sort__conv__fold,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xsa: list(A)] : ( aa(list(A),list(A),aa(fun(A,A),fun(list(A),list(A)),linorder_sort_key(A,A),aTP_Lamp_ip(A,A)),Xsa) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),aa(fun(A,fun(list(A),list(A))),fun(list(A),fun(list(A),list(A))),fold(A,list(A)),linorder_insort_key(A,A,aTP_Lamp_ip(A,A))),Xsa),nil(A)) ) ) ).

% sort_conv_fold
tff(fact_5319_finite__enumerate__in__set,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),N),finite_card(A,S))
           => member(A,infini527867602293511546merate(A,S,N),S) ) ) ) ).

% finite_enumerate_in_set
tff(fact_5320_finite__enumerate__Ex,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S: set(A),S2: A] :
          ( aa(set(A),$o,finite_finite2(A),S)
         => ( member(A,S2,S)
           => ? [N5: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N5),finite_card(A,S))
                & ( infini527867602293511546merate(A,S,N5) = S2 ) ) ) ) ) ).

% finite_enumerate_Ex
tff(fact_5321_finite__enum__ext,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [X3: set(A),Y5: set(A)] :
          ( ! [I3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),finite_card(A,X3))
             => ( infini527867602293511546merate(A,X3,I3) = infini527867602293511546merate(A,Y5,I3) ) )
         => ( aa(set(A),$o,finite_finite2(A),X3)
           => ( aa(set(A),$o,finite_finite2(A),Y5)
             => ( ( finite_card(A,X3) = finite_card(A,Y5) )
               => ( X3 = Y5 ) ) ) ) ) ) ).

% finite_enum_ext
tff(fact_5322_inter__coset__fold,axiom,
    ! [A: $tType,Aa2: set(A),Xsa: list(A)] : ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),coset(A,Xsa)) = aa(set(A),set(A),aa(list(A),fun(set(A),set(A)),aa(fun(A,fun(set(A),set(A))),fun(list(A),fun(set(A),set(A))),fold(A,set(A)),remove(A)),Xsa),Aa2) ) ).

% inter_coset_fold
tff(fact_5323_fold__append__concat__rev,axiom,
    ! [A: $tType,Xss2: list(list(A))] : ( aa(list(list(A)),fun(list(A),list(A)),aa(fun(list(A),fun(list(A),list(A))),fun(list(list(A)),fun(list(A),list(A))),fold(list(A),list(A)),append(A)),Xss2) = aa(list(A),fun(list(A),list(A)),append(A),aa(list(list(A)),list(A),concat(A),aa(list(list(A)),list(list(A)),rev(list(A)),Xss2))) ) ).

% fold_append_concat_rev
tff(fact_5324_Sup__set__fold,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Xsa: list(A)] : ( aa(set(A),A,complete_Sup_Sup(A),aa(list(A),set(A),set2(A),Xsa)) = aa(A,A,aa(list(A),fun(A,A),aa(fun(A,fun(A,A)),fun(list(A),fun(A,A)),fold(A,A),sup_sup(A)),Xsa),bot_bot(A)) ) ) ).

% Sup_set_fold
tff(fact_5325_Inf__set__fold,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Xsa: list(A)] : ( aa(set(A),A,complete_Inf_Inf(A),aa(list(A),set(A),set2(A),Xsa)) = aa(A,A,aa(list(A),fun(A,A),aa(fun(A,fun(A,A)),fun(list(A),fun(A,A)),fold(A,A),inf_inf(A)),Xsa),top_top(A)) ) ) ).

% Inf_set_fold
tff(fact_5326_Max_Oset__eq__fold,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Xsa: list(A)] : ( lattic643756798349783984er_Max(A,aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa))) = aa(A,A,aa(list(A),fun(A,A),aa(fun(A,fun(A,A)),fun(list(A),fun(A,A)),fold(A,A),ord_max(A)),Xsa),Xa) ) ) ).

% Max.set_eq_fold
tff(fact_5327_Min_Oset__eq__fold,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Xsa: list(A)] : ( lattic643756798350308766er_Min(A,aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa))) = aa(A,A,aa(list(A),fun(A,A),aa(fun(A,fun(A,A)),fun(list(A),fun(A,A)),fold(A,A),ord_min(A)),Xsa),Xa) ) ) ).

% Min.set_eq_fold
tff(fact_5328_comp__fun__idem__on_Ofold__set__fold,axiom,
    ! [A: $tType,B: $tType,S: set(A),F2: fun(A,fun(B,B)),Xsa: list(A),Y: B] :
      ( finite673082921795544331dem_on(A,B,S,F2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xsa)),S)
       => ( finite_fold(A,B,F2,Y,aa(list(A),set(A),set2(A),Xsa)) = aa(B,B,aa(list(A),fun(B,B),aa(fun(A,fun(B,B)),fun(list(A),fun(B,B)),fold(A,B),F2),Xsa),Y) ) ) ) ).

% comp_fun_idem_on.fold_set_fold
tff(fact_5329_prod__list_Oeq__foldr,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Xsa: list(A)] : ( aa(list(A),A,groups5270119922927024881d_list(A),Xsa) = aa(A,A,aa(list(A),fun(A,A),aa(fun(A,fun(A,A)),fun(list(A),fun(A,A)),foldr(A,A),times_times(A)),Xsa),one_one(A)) ) ) ).

% prod_list.eq_foldr
tff(fact_5330_finite__enumerate__mono,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [M2: nat,N: nat,S: set(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),N)
         => ( aa(set(A),$o,finite_finite2(A),S)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),finite_card(A,S))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),infini527867602293511546merate(A,S,M2)),infini527867602293511546merate(A,S,N)) ) ) ) ) ).

% finite_enumerate_mono
tff(fact_5331_finite__le__enumerate,axiom,
    ! [S: set(nat),N: nat] :
      ( aa(set(nat),$o,finite_finite2(nat),S)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),finite_card(nat,S))
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),infini527867602293511546merate(nat,S,N)) ) ) ).

% finite_le_enumerate
tff(fact_5332_comp__fun__commute__on_Ofold__set__fold__remdups,axiom,
    ! [A: $tType,B: $tType,S: set(A),F2: fun(A,fun(B,B)),Xsa: list(A),Y: B] :
      ( finite4664212375090638736ute_on(A,B,S,F2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xsa)),S)
       => ( finite_fold(A,B,F2,Y,aa(list(A),set(A),set2(A),Xsa)) = aa(B,B,aa(list(A),fun(B,B),aa(fun(A,fun(B,B)),fun(list(A),fun(B,B)),fold(A,B),F2),aa(list(A),list(A),remdups(A),Xsa)),Y) ) ) ) ).

% comp_fun_commute_on.fold_set_fold_remdups
tff(fact_5333_finite__enumerate__step,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)),finite_card(A,S))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),infini527867602293511546merate(A,S,N)),infini527867602293511546merate(A,S,aa(nat,nat,suc,N))) ) ) ) ).

% finite_enumerate_step
tff(fact_5334_enumerate__Suc_H,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S: set(A),N: nat] : ( infini527867602293511546merate(A,S,aa(nat,nat,suc,N)) = 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)),insert3(A),infini527867602293511546merate(A,S,zero_zero(nat))),bot_bot(set(A)))),N) ) ) ).

% enumerate_Suc'
tff(fact_5335_prod_Odistinct__set__conv__list,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [Xsa: list(A),G: fun(A,B)] :
          ( aa(list(A),$o,distinct(A),Xsa)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(list(A),set(A),set2(A),Xsa)) = aa(list(B),B,groups5270119922927024881d_list(B),aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),G),Xsa)) ) ) ) ).

% prod.distinct_set_conv_list
tff(fact_5336_finite__enum__subset,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [X3: set(A),Y5: set(A)] :
          ( ! [I3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),finite_card(A,X3))
             => ( infini527867602293511546merate(A,X3,I3) = infini527867602293511546merate(A,Y5,I3) ) )
         => ( aa(set(A),$o,finite_finite2(A),X3)
           => ( aa(set(A),$o,finite_finite2(A),Y5)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),finite_card(A,X3)),finite_card(A,Y5))
               => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X3),Y5) ) ) ) ) ) ).

% finite_enum_subset
tff(fact_5337_SUP__set__fold,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(B,A),Xsa: list(B)] : ( aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F2),aa(list(B),set(B),set2(B),Xsa))) = aa(A,A,aa(list(B),fun(A,A),aa(fun(B,fun(A,A)),fun(list(B),fun(A,A)),fold(B,A),comp(A,fun(A,A),B,sup_sup(A),F2)),Xsa),bot_bot(A)) ) ) ).

% SUP_set_fold
tff(fact_5338_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)),finite_card(A,S))
           => ( infini527867602293511546merate(A,S,aa(nat,nat,suc,N)) = ord_Least(A,aa(nat,fun(A,$o),aTP_Lamp_rj(set(A),fun(nat,fun(A,$o)),S),N)) ) ) ) ) ).

% finite_enumerate_Suc''
tff(fact_5339_enumerate__Suc,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S: set(A),N: nat] : ( infini527867602293511546merate(A,S,aa(nat,nat,suc,N)) = infini527867602293511546merate(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),ord_Least(A,aTP_Lamp_rk(set(A),fun(A,$o),S))),bot_bot(set(A)))),N) ) ) ).

% enumerate_Suc
tff(fact_5340_subset__mset_OcINF__union,axiom,
    ! [B: $tType,A: $tType,Aa2: set(A),F2: fun(A,multiset(B)),B2: set(A)] :
      ( ( Aa2 != bot_bot(set(A)) )
     => ( condit8119078960628432327_below(multiset(B),subseteq_mset(B),aa(set(A),set(multiset(B)),image2(A,multiset(B),F2),Aa2))
       => ( ( B2 != bot_bot(set(A)) )
         => ( condit8119078960628432327_below(multiset(B),subseteq_mset(B),aa(set(A),set(multiset(B)),image2(A,multiset(B),F2),B2))
           => ( aa(set(multiset(B)),multiset(B),complete_Inf_Inf(multiset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),B2))) = inter_mset(B,aa(set(multiset(B)),multiset(B),complete_Inf_Inf(multiset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),F2),Aa2)),aa(set(multiset(B)),multiset(B),complete_Inf_Inf(multiset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),F2),B2))) ) ) ) ) ) ).

% subset_mset.cINF_union
tff(fact_5341_Least__le,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [Pa: fun(A,$o),K3: A] :
          ( aa(A,$o,Pa,K3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),ord_Least(A,Pa)),K3) ) ) ).

% Least_le
tff(fact_5342_not__less__Least,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [K3: A,Pa: fun(A,$o)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),K3),ord_Least(A,Pa))
         => ~ aa(A,$o,Pa,K3) ) ) ).

% not_less_Least
tff(fact_5343_LeastI,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [Pa: fun(A,$o),K3: A] :
          ( aa(A,$o,Pa,K3)
         => aa(A,$o,Pa,ord_Least(A,Pa)) ) ) ).

% LeastI
tff(fact_5344_LeastI2,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [Pa: fun(A,$o),A3: A,Qa: fun(A,$o)] :
          ( aa(A,$o,Pa,A3)
         => ( ! [X: A] :
                ( aa(A,$o,Pa,X)
               => aa(A,$o,Qa,X) )
           => aa(A,$o,Qa,ord_Least(A,Pa)) ) ) ) ).

% LeastI2
tff(fact_5345_LeastI__ex,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [Pa: fun(A,$o)] :
          ( ? [X_13: A] : aa(A,$o,Pa,X_13)
         => aa(A,$o,Pa,ord_Least(A,Pa)) ) ) ).

% LeastI_ex
tff(fact_5346_LeastI2__ex,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [Pa: fun(A,$o),Qa: fun(A,$o)] :
          ( ? [X_13: A] : aa(A,$o,Pa,X_13)
         => ( ! [X: A] :
                ( aa(A,$o,Pa,X)
               => aa(A,$o,Qa,X) )
           => aa(A,$o,Qa,ord_Least(A,Pa)) ) ) ) ).

% LeastI2_ex
tff(fact_5347_Least1I,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Pa: fun(A,$o)] :
          ( ? [X4: A] :
              ( aa(A,$o,Pa,X4)
              & ! [Y2: A] :
                  ( aa(A,$o,Pa,Y2)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),Y2) )
              & ! [Y2: A] :
                  ( ( aa(A,$o,Pa,Y2)
                    & ! [Ya2: A] :
                        ( aa(A,$o,Pa,Ya2)
                       => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y2),Ya2) ) )
                 => ( Y2 = X4 ) ) )
         => aa(A,$o,Pa,ord_Least(A,Pa)) ) ) ).

% Least1I
tff(fact_5348_Least1__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Pa: fun(A,$o),Z2: A] :
          ( ? [X4: A] :
              ( aa(A,$o,Pa,X4)
              & ! [Y2: A] :
                  ( aa(A,$o,Pa,Y2)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),Y2) )
              & ! [Y2: A] :
                  ( ( aa(A,$o,Pa,Y2)
                    & ! [Ya2: A] :
                        ( aa(A,$o,Pa,Ya2)
                       => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y2),Ya2) ) )
                 => ( Y2 = X4 ) ) )
         => ( aa(A,$o,Pa,Z2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),ord_Least(A,Pa)),Z2) ) ) ) ).

% Least1_le
tff(fact_5349_LeastI2__order,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Pa: fun(A,$o),Xa: A,Qa: fun(A,$o)] :
          ( aa(A,$o,Pa,Xa)
         => ( ! [Y2: A] :
                ( aa(A,$o,Pa,Y2)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y2) )
           => ( ! [X: A] :
                  ( aa(A,$o,Pa,X)
                 => ( ! [Y3: A] :
                        ( aa(A,$o,Pa,Y3)
                       => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y3) )
                   => aa(A,$o,Qa,X) ) )
             => aa(A,$o,Qa,ord_Least(A,Pa)) ) ) ) ) ).

% LeastI2_order
tff(fact_5350_Least__equality,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Pa: fun(A,$o),Xa: A] :
          ( aa(A,$o,Pa,Xa)
         => ( ! [Y2: A] :
                ( aa(A,$o,Pa,Y2)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y2) )
           => ( ord_Least(A,Pa) = Xa ) ) ) ) ).

% Least_equality
tff(fact_5351_LeastI2__wellorder,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [Pa: fun(A,$o),A3: A,Qa: fun(A,$o)] :
          ( aa(A,$o,Pa,A3)
         => ( ! [A4: A] :
                ( aa(A,$o,Pa,A4)
               => ( ! [B9: A] :
                      ( aa(A,$o,Pa,B9)
                     => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),B9) )
                 => aa(A,$o,Qa,A4) ) )
           => aa(A,$o,Qa,ord_Least(A,Pa)) ) ) ) ).

% LeastI2_wellorder
tff(fact_5352_LeastI2__wellorder__ex,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [Pa: fun(A,$o),Qa: fun(A,$o)] :
          ( ? [X_13: A] : aa(A,$o,Pa,X_13)
         => ( ! [A4: A] :
                ( aa(A,$o,Pa,A4)
               => ( ! [B9: A] :
                      ( aa(A,$o,Pa,B9)
                     => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),B9) )
                 => aa(A,$o,Qa,A4) ) )
           => aa(A,$o,Qa,ord_Least(A,Pa)) ) ) ) ).

% LeastI2_wellorder_ex
tff(fact_5353_abort__Bleast__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [S: set(A),Pa: fun(A,$o)] : ( abort_Bleast(A,S,Pa) = ord_Least(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_rl(set(A),fun(fun(A,$o),fun(A,$o)),S),Pa)) ) ) ).

% abort_Bleast_def
tff(fact_5354_Bleast__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [S: set(A),Pa: fun(A,$o)] : ( bleast(A,S,Pa) = ord_Least(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_rl(set(A),fun(fun(A,$o),fun(A,$o)),S),Pa)) ) ) ).

% Bleast_def
tff(fact_5355_subset__mset_Ole__cINF__iff,axiom,
    ! [B: $tType,A: $tType,Aa2: set(A),F2: fun(A,multiset(B)),U: multiset(B)] :
      ( ( Aa2 != bot_bot(set(A)) )
     => ( condit8119078960628432327_below(multiset(B),subseteq_mset(B),aa(set(A),set(multiset(B)),image2(A,multiset(B),F2),Aa2))
       => ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),U),aa(set(multiset(B)),multiset(B),complete_Inf_Inf(multiset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),F2),Aa2)))
        <=> ! [X6: A] :
              ( member(A,X6,Aa2)
             => aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),U),aa(A,multiset(B),F2,X6)) ) ) ) ) ).

% subset_mset.le_cINF_iff
tff(fact_5356_subset__mset_OcINF__mono,axiom,
    ! [E: $tType,B: $tType,A: $tType,B2: set(A),F2: fun(E,multiset(B)),Aa2: set(E),G: fun(A,multiset(B))] :
      ( ( B2 != bot_bot(set(A)) )
     => ( condit8119078960628432327_below(multiset(B),subseteq_mset(B),aa(set(E),set(multiset(B)),image2(E,multiset(B),F2),Aa2))
       => ( ! [M6: A] :
              ( member(A,M6,B2)
             => ? [X4: E] :
                  ( member(E,X4,Aa2)
                  & aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),aa(E,multiset(B),F2,X4)),aa(A,multiset(B),G,M6)) ) )
         => aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),aa(set(multiset(B)),multiset(B),complete_Inf_Inf(multiset(B)),aa(set(E),set(multiset(B)),image2(E,multiset(B),F2),Aa2))),aa(set(multiset(B)),multiset(B),complete_Inf_Inf(multiset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),G),B2))) ) ) ) ).

% subset_mset.cINF_mono
tff(fact_5357_Least__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F2: fun(A,B),S: set(A)] :
          ( order_mono(A,B,F2)
         => ( ? [X4: A] :
                ( member(A,X4,S)
                & ! [Xa5: A] :
                    ( member(A,Xa5,S)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),Xa5) ) )
           => ( ord_Least(B,aa(set(A),fun(B,$o),aTP_Lamp_rm(fun(A,B),fun(set(A),fun(B,$o)),F2),S)) = aa(A,B,F2,ord_Least(A,aTP_Lamp_rn(set(A),fun(A,$o),S))) ) ) ) ) ).

% Least_mono
tff(fact_5358_subset__mset_OcINF__superset__mono,axiom,
    ! [B: $tType,A: $tType,Aa2: set(A),G: fun(A,multiset(B)),B2: set(A),F2: fun(A,multiset(B))] :
      ( ( Aa2 != bot_bot(set(A)) )
     => ( condit8119078960628432327_below(multiset(B),subseteq_mset(B),aa(set(A),set(multiset(B)),image2(A,multiset(B),G),B2))
       => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),B2)
         => ( ! [X: A] :
                ( member(A,X,B2)
               => aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),aa(A,multiset(B),G,X)),aa(A,multiset(B),F2,X)) )
           => aa(multiset(B),$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),B2))),aa(set(multiset(B)),multiset(B),complete_Inf_Inf(multiset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),F2),Aa2))) ) ) ) ) ).

% subset_mset.cINF_superset_mono
tff(fact_5359_subset__mset_OcINF__inf__distrib,axiom,
    ! [B: $tType,A: $tType,Aa2: set(A),F2: fun(A,multiset(B)),G: fun(A,multiset(B))] :
      ( ( Aa2 != bot_bot(set(A)) )
     => ( condit8119078960628432327_below(multiset(B),subseteq_mset(B),aa(set(A),set(multiset(B)),image2(A,multiset(B),F2),Aa2))
       => ( condit8119078960628432327_below(multiset(B),subseteq_mset(B),aa(set(A),set(multiset(B)),image2(A,multiset(B),G),Aa2))
         => ( inter_mset(B,aa(set(multiset(B)),multiset(B),complete_Inf_Inf(multiset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),F2),Aa2)),aa(set(multiset(B)),multiset(B),complete_Inf_Inf(multiset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),G),Aa2))) = 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_ro(fun(A,multiset(B)),fun(fun(A,multiset(B)),fun(A,multiset(B))),F2),G)),Aa2)) ) ) ) ) ).

% subset_mset.cINF_inf_distrib
tff(fact_5360_enumerate__Suc_H_H,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S: set(A),N: nat] :
          ( ~ aa(set(A),$o,finite_finite2(A),S)
         => ( infini527867602293511546merate(A,S,aa(nat,nat,suc,N)) = ord_Least(A,aa(nat,fun(A,$o),aTP_Lamp_rj(set(A),fun(nat,fun(A,$o)),S),N)) ) ) ) ).

% enumerate_Suc''
tff(fact_5361_subset__mset_OcINF__insert,axiom,
    ! [B: $tType,A: $tType,Aa2: set(A),F2: fun(A,multiset(B)),A3: A] :
      ( ( Aa2 != bot_bot(set(A)) )
     => ( condit8119078960628432327_below(multiset(B),subseteq_mset(B),aa(set(A),set(multiset(B)),image2(A,multiset(B),F2),Aa2))
       => ( aa(set(multiset(B)),multiset(B),complete_Inf_Inf(multiset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),F2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),Aa2))) = inter_mset(B,aa(A,multiset(B),F2,A3),aa(set(multiset(B)),multiset(B),complete_Inf_Inf(multiset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),F2),Aa2))) ) ) ) ).

% subset_mset.cINF_insert
tff(fact_5362_subset__mset_Omono__cINF,axiom,
    ! [A: $tType,B: $tType,E: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [F2: fun(multiset(A),B),Aa2: fun(E,multiset(A)),Ia2: set(E)] :
          ( aa(fun(multiset(A),B),$o,mono(multiset(A),B,subseteq_mset(A)),F2)
         => ( condit8119078960628432327_below(multiset(A),subseteq_mset(A),aa(set(E),set(multiset(A)),image2(E,multiset(A),Aa2),Ia2))
           => ( ( Ia2 != bot_bot(set(E)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(multiset(A),B,F2,aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),aa(set(E),set(multiset(A)),image2(E,multiset(A),Aa2),Ia2)))),aa(set(B),B,complete_Inf_Inf(B),aa(set(E),set(B),image2(E,B,aa(fun(E,multiset(A)),fun(E,B),aTP_Lamp_rp(fun(multiset(A),B),fun(fun(E,multiset(A)),fun(E,B)),F2),Aa2)),Ia2))) ) ) ) ) ).

% subset_mset.mono_cINF
tff(fact_5363_list_Oin__rel,axiom,
    ! [A: $tType,B: $tType,R: fun(A,fun(B,$o)),A3: list(A),B3: list(B)] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),R),A3),B3)
    <=> ? [Z5: list(product_prod(A,B))] :
          ( member(list(product_prod(A,B)),Z5,aa(fun(list(product_prod(A,B)),$o),set(list(product_prod(A,B))),collect(list(product_prod(A,B))),aTP_Lamp_rq(fun(A,fun(B,$o)),fun(list(product_prod(A,B)),$o),R)))
          & ( aa(list(product_prod(A,B)),list(A),aa(fun(product_prod(A,B),A),fun(list(product_prod(A,B)),list(A)),map(product_prod(A,B),A),product_fst(A,B)),Z5) = A3 )
          & ( aa(list(product_prod(A,B)),list(B),aa(fun(product_prod(A,B),B),fun(list(product_prod(A,B)),list(B)),map(product_prod(A,B),B),product_snd(A,B)),Z5) = B3 ) ) ) ).

% list.in_rel
tff(fact_5364_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)
     => ( 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_pq(set(A),fun(A,fun(B,fun(set(B),set(B)))),S)),bot_bot(set(B)),R) ) ) ).

% Image_fold
tff(fact_5365_ImageI,axiom,
    ! [B: $tType,A: $tType,A3: A,B3: B,R2: set(product_prod(A,B)),Aa2: set(A)] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B3),R2)
     => ( member(A,A3,Aa2)
       => member(B,B3,image(A,B,R2,Aa2)) ) ) ).

% ImageI
tff(fact_5366_list__all2__Nil2,axiom,
    ! [B: $tType,A: $tType,Pa: fun(A,fun(B,$o)),Xsa: list(A)] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Pa),Xsa),nil(B))
    <=> ( Xsa = nil(A) ) ) ).

% list_all2_Nil2
tff(fact_5367_list__all2__Nil,axiom,
    ! [A: $tType,B: $tType,Pa: fun(A,fun(B,$o)),Ys: list(B)] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Pa),nil(A)),Ys)
    <=> ( Ys = nil(B) ) ) ).

% list_all2_Nil
tff(fact_5368_Image__empty2,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(B,A))] : ( image(B,A,R,bot_bot(set(B))) = bot_bot(set(A)) ) ).

% Image_empty2
tff(fact_5369_list__all2__rev,axiom,
    ! [A: $tType,B: $tType,Pa: fun(A,fun(B,$o)),Xsa: list(A),Ys: list(B)] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Pa),aa(list(A),list(A),rev(A),Xsa)),aa(list(B),list(B),rev(B),Ys))
    <=> aa(list(B),$o,aa(list(A),fun(list(B),$o),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Pa),Xsa),Ys) ) ).

% list_all2_rev
tff(fact_5370_Image__Id,axiom,
    ! [A: $tType,Aa2: set(A)] : ( image(A,A,id2(A),Aa2) = Aa2 ) ).

% Image_Id
tff(fact_5371_Image__empty1,axiom,
    ! [B: $tType,A: $tType,X3: set(B)] : ( image(B,A,bot_bot(set(product_prod(B,A))),X3) = bot_bot(set(A)) ) ).

% Image_empty1
tff(fact_5372_Image__Id__on,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A)] : ( image(A,A,id_on(A,Aa2),B2) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),B2) ) ).

% Image_Id_on
tff(fact_5373_Image__Collect__case__prod,axiom,
    ! [A: $tType,B: $tType,Pa: fun(B,fun(A,$o)),Aa2: set(B)] : ( 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),Pa)),Aa2) = aa(fun(A,$o),set(A),collect(A),aa(set(B),fun(A,$o),aTP_Lamp_rr(fun(B,fun(A,$o)),fun(set(B),fun(A,$o)),Pa),Aa2)) ) ).

% Image_Collect_case_prod
tff(fact_5374_Image__singleton__iff,axiom,
    ! [A: $tType,B: $tType,B3: A,R2: set(product_prod(B,A)),A3: B] :
      ( member(A,B3,image(B,A,R2,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert3(B),A3),bot_bot(set(B)))))
    <=> member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),A3),B3),R2) ) ).

% Image_singleton_iff
tff(fact_5375_listrel__Nil,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(B,A))] : ( image(list(B),list(A),listrel(B,A,R2),aa(set(list(B)),set(list(B)),aa(list(B),fun(set(list(B)),set(list(B))),insert3(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))),insert3(list(A)),nil(A)),bot_bot(set(list(A)))) ) ).

% listrel_Nil
tff(fact_5376_pair__vimage__is__Image,axiom,
    ! [A: $tType,B: $tType,U: B,E5: set(product_prod(B,A))] : ( vimage(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),U),E5) = image(B,A,E5,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert3(B),U),bot_bot(set(B)))) ) ).

% pair_vimage_is_Image
tff(fact_5377_Image__Int__subset,axiom,
    ! [A: $tType,B: $tType,R: set(product_prod(B,A)),Aa2: set(B),B2: set(B)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),image(B,A,R,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),Aa2),B2))),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),image(B,A,R,Aa2)),image(B,A,R,B2))) ).

% Image_Int_subset
tff(fact_5378_rtrancl__image__advance,axiom,
    ! [A: $tType,Q2: A,R: set(product_prod(A,A)),Q0: set(A),Xa: A] :
      ( member(A,Q2,image(A,A,aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R),Q0))
     => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q2),Xa),R)
       => member(A,Xa,image(A,A,aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R),Q0)) ) ) ).

% rtrancl_image_advance
tff(fact_5379_rtrancl__image__advance__rtrancl,axiom,
    ! [A: $tType,Q2: A,R: set(product_prod(A,A)),Q0: set(A),Xa: A] :
      ( member(A,Q2,image(A,A,aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R),Q0))
     => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q2),Xa),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R))
       => member(A,Xa,image(A,A,aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R),Q0)) ) ) ).

% rtrancl_image_advance_rtrancl
tff(fact_5380_list__all2__append2,axiom,
    ! [A: $tType,B: $tType,Pa: fun(A,fun(B,$o)),Xsa: list(A),Ys: list(B),Zs: list(B)] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Pa),Xsa),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Ys),Zs))
    <=> ? [Us2: list(A),Vs3: list(A)] :
          ( ( Xsa = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us2),Vs3) )
          & ( aa(list(A),nat,size_size(list(A)),Us2) = aa(list(B),nat,size_size(list(B)),Ys) )
          & ( aa(list(A),nat,size_size(list(A)),Vs3) = aa(list(B),nat,size_size(list(B)),Zs) )
          & aa(list(B),$o,aa(list(A),fun(list(B),$o),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Pa),Us2),Ys)
          & aa(list(B),$o,aa(list(A),fun(list(B),$o),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Pa),Vs3),Zs) ) ) ).

% list_all2_append2
tff(fact_5381_list__all2__append1,axiom,
    ! [A: $tType,B: $tType,Pa: fun(A,fun(B,$o)),Xsa: list(A),Ys: list(A),Zs: list(B)] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Pa),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),Ys)),Zs)
    <=> ? [Us2: list(B),Vs3: list(B)] :
          ( ( Zs = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Us2),Vs3) )
          & ( aa(list(B),nat,size_size(list(B)),Us2) = aa(list(A),nat,size_size(list(A)),Xsa) )
          & ( aa(list(B),nat,size_size(list(B)),Vs3) = aa(list(A),nat,size_size(list(A)),Ys) )
          & aa(list(B),$o,aa(list(A),fun(list(B),$o),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Pa),Xsa),Us2)
          & aa(list(B),$o,aa(list(A),fun(list(B),$o),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Pa),Ys),Vs3) ) ) ).

% list_all2_append1
tff(fact_5382_list__all2__append,axiom,
    ! [A: $tType,B: $tType,Xsa: list(A),Ys: list(B),Pa: fun(A,fun(B,$o)),Us: list(A),Vs: list(B)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xsa) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( aa(list(B),$o,aa(list(A),fun(list(B),$o),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Pa),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),Us)),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Ys),Vs))
      <=> ( aa(list(B),$o,aa(list(A),fun(list(B),$o),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Pa),Xsa),Ys)
          & aa(list(B),$o,aa(list(A),fun(list(B),$o),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Pa),Us),Vs) ) ) ) ).

% list_all2_append
tff(fact_5383_list_Orel__map_I2_J,axiom,
    ! [A: $tType,B: $tType,E: $tType,Sa: fun(A,fun(B,$o)),Xa: list(A),G: fun(E,B),Y: list(E)] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Sa),Xa),aa(list(E),list(B),aa(fun(E,B),fun(list(E),list(B)),map(E,B),G),Y))
    <=> aa(list(E),$o,aa(list(A),fun(list(E),$o),aa(fun(A,fun(E,$o)),fun(list(A),fun(list(E),$o)),list_all2(A,E),aa(fun(E,B),fun(A,fun(E,$o)),aTP_Lamp_rs(fun(A,fun(B,$o)),fun(fun(E,B),fun(A,fun(E,$o))),Sa),G)),Xa),Y) ) ).

% list.rel_map(2)
tff(fact_5384_list_Orel__map_I1_J,axiom,
    ! [E: $tType,A: $tType,B: $tType,Sb: fun(A,fun(B,$o)),Ia: fun(E,A),Xa: list(E),Y: list(B)] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Sb),aa(list(E),list(A),aa(fun(E,A),fun(list(E),list(A)),map(E,A),Ia),Xa)),Y)
    <=> aa(list(B),$o,aa(list(E),fun(list(B),$o),aa(fun(E,fun(B,$o)),fun(list(E),fun(list(B),$o)),list_all2(E,B),aa(fun(E,A),fun(E,fun(B,$o)),aTP_Lamp_rt(fun(A,fun(B,$o)),fun(fun(E,A),fun(E,fun(B,$o))),Sb),Ia)),Xa),Y) ) ).

% list.rel_map(1)
tff(fact_5385_list__all2__map1,axiom,
    ! [E: $tType,A: $tType,B: $tType,Pa: fun(A,fun(B,$o)),F2: fun(E,A),As2: list(E),Bs: list(B)] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Pa),aa(list(E),list(A),aa(fun(E,A),fun(list(E),list(A)),map(E,A),F2),As2)),Bs)
    <=> aa(list(B),$o,aa(list(E),fun(list(B),$o),aa(fun(E,fun(B,$o)),fun(list(E),fun(list(B),$o)),list_all2(E,B),aa(fun(E,A),fun(E,fun(B,$o)),aTP_Lamp_rt(fun(A,fun(B,$o)),fun(fun(E,A),fun(E,fun(B,$o))),Pa),F2)),As2),Bs) ) ).

% list_all2_map1
tff(fact_5386_list__all2__map2,axiom,
    ! [A: $tType,B: $tType,E: $tType,Pa: fun(A,fun(B,$o)),As2: list(A),F2: fun(E,B),Bs: list(E)] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Pa),As2),aa(list(E),list(B),aa(fun(E,B),fun(list(E),list(B)),map(E,B),F2),Bs))
    <=> aa(list(E),$o,aa(list(A),fun(list(E),$o),aa(fun(A,fun(E,$o)),fun(list(A),fun(list(E),$o)),list_all2(A,E),aa(fun(E,B),fun(A,fun(E,$o)),aTP_Lamp_rs(fun(A,fun(B,$o)),fun(fun(E,B),fun(A,fun(E,$o))),Pa),F2)),As2),Bs) ) ).

% list_all2_map2
tff(fact_5387_List_Olist__all2__induct,axiom,
    ! [A: $tType,B: $tType,Pa: fun(A,fun(B,$o)),Xsa: list(A),Ys: list(B),R: fun(list(A),fun(list(B),$o))] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Pa),Xsa),Ys)
     => ( aa(list(B),$o,aa(list(A),fun(list(B),$o),R,nil(A)),nil(B))
       => ( ! [X: A,Xs: list(A),Y2: B,Ys5: list(B)] :
              ( aa(B,$o,aa(A,fun(B,$o),Pa,X),Y2)
             => ( aa(list(B),$o,aa(list(A),fun(list(B),$o),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Pa),Xs),Ys5)
               => ( aa(list(B),$o,aa(list(A),fun(list(B),$o),R,Xs),Ys5)
                 => aa(list(B),$o,aa(list(A),fun(list(B),$o),R,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y2),Ys5)) ) ) )
         => aa(list(B),$o,aa(list(A),fun(list(B),$o),R,Xsa),Ys) ) ) ) ).

% List.list_all2_induct
tff(fact_5388_list_Orel__induct,axiom,
    ! [A: $tType,B: $tType,R: fun(A,fun(B,$o)),Xa: list(A),Y: list(B),Qa: fun(list(A),fun(list(B),$o))] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),R),Xa),Y)
     => ( aa(list(B),$o,aa(list(A),fun(list(B),$o),Qa,nil(A)),nil(B))
       => ( ! [A21: A,A222: list(A),B21: B,B222: list(B)] :
              ( aa(B,$o,aa(A,fun(B,$o),R,A21),B21)
             => ( aa(list(B),$o,aa(list(A),fun(list(B),$o),Qa,A222),B222)
               => aa(list(B),$o,aa(list(A),fun(list(B),$o),Qa,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A21),A222)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B21),B222)) ) )
         => aa(list(B),$o,aa(list(A),fun(list(B),$o),Qa,Xa),Y) ) ) ) ).

% list.rel_induct
tff(fact_5389_list_Orel__cases,axiom,
    ! [A: $tType,B: $tType,R: fun(A,fun(B,$o)),A3: list(A),B3: list(B)] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),R),A3),B3)
     => ( ( ( A3 = nil(A) )
         => ( B3 != nil(B) ) )
       => ~ ! [X12: A,X23: list(A)] :
              ( ( A3 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X12),X23) )
             => ! [Y12: B,Y23: list(B)] :
                  ( ( B3 = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y12),Y23) )
                 => ( aa(B,$o,aa(A,fun(B,$o),R,X12),Y12)
                   => ~ aa(list(B),$o,aa(list(A),fun(list(B),$o),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),R),X23),Y23) ) ) ) ) ) ).

% list.rel_cases
tff(fact_5390_list_Orel__distinct_I1_J,axiom,
    ! [A: $tType,B: $tType,R: fun(A,fun(B,$o)),Y21: B,Y222: list(B)] : ~ aa(list(B),$o,aa(list(A),fun(list(B),$o),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),R),nil(A)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y21),Y222)) ).

% list.rel_distinct(1)
tff(fact_5391_list_Orel__distinct_I2_J,axiom,
    ! [A: $tType,B: $tType,R: fun(A,fun(B,$o)),Y21: A,Y222: list(A)] : ~ aa(list(B),$o,aa(list(A),fun(list(B),$o),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),R),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y21),Y222)),nil(B)) ).

% list.rel_distinct(2)
tff(fact_5392_Misc_Olist__all2__induct,axiom,
    ! [A: $tType,B: $tType,Pa: fun(A,fun(B,$o)),L: list(A),L2: list(B),Qa: fun(list(A),fun(list(B),$o))] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Pa),L),L2)
     => ( aa(list(B),$o,aa(list(A),fun(list(B),$o),Qa,nil(A)),nil(B))
       => ( ! [X: A,X2: B,Ls: list(A),Ls4: list(B)] :
              ( aa(B,$o,aa(A,fun(B,$o),Pa,X),X2)
             => ( aa(list(B),$o,aa(list(A),fun(list(B),$o),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Pa),Ls),Ls4)
               => ( aa(list(B),$o,aa(list(A),fun(list(B),$o),Qa,Ls),Ls4)
                 => aa(list(B),$o,aa(list(A),fun(list(B),$o),Qa,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Ls)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X2),Ls4)) ) ) )
         => aa(list(B),$o,aa(list(A),fun(list(B),$o),Qa,L),L2) ) ) ) ).

% Misc.list_all2_induct
tff(fact_5393_list_Octr__transfer_I1_J,axiom,
    ! [A: $tType,B: $tType,R: fun(A,fun(B,$o))] : aa(list(B),$o,aa(list(A),fun(list(B),$o),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),R),nil(A)),nil(B)) ).

% list.ctr_transfer(1)
tff(fact_5394_relcomp__Image,axiom,
    ! [A: $tType,E: $tType,B: $tType,X3: set(product_prod(B,E)),Y5: set(product_prod(E,A)),Z6: set(B)] : ( image(B,A,relcomp(B,E,A,X3,Y5),Z6) = image(E,A,Y5,image(B,E,X3,Z6)) ) ).

% relcomp_Image
tff(fact_5395_list_Orel__mono,axiom,
    ! [B: $tType,A: $tType,R: fun(A,fun(B,$o)),Ra2: fun(A,fun(B,$o))] :
      ( aa(fun(A,fun(B,$o)),$o,aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),$o),ord_less_eq(fun(A,fun(B,$o))),R),Ra2)
     => aa(fun(list(A),fun(list(B),$o)),$o,aa(fun(list(A),fun(list(B),$o)),fun(fun(list(A),fun(list(B),$o)),$o),ord_less_eq(fun(list(A),fun(list(B),$o))),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),R)),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Ra2)) ) ).

% list.rel_mono
tff(fact_5396_Image__Un,axiom,
    ! [A: $tType,B: $tType,R: set(product_prod(B,A)),Aa2: set(B),B2: set(B)] : ( image(B,A,R,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),Aa2),B2)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),image(B,A,R,Aa2)),image(B,A,R,B2)) ) ).

% Image_Un
tff(fact_5397_list__all2__dropI,axiom,
    ! [A: $tType,B: $tType,Pa: fun(A,fun(B,$o)),Xsa: list(A),Ys: list(B),N: nat] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Pa),Xsa),Ys)
     => aa(list(B),$o,aa(list(A),fun(list(B),$o),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Pa),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),Xsa)),aa(list(B),list(B),aa(nat,fun(list(B),list(B)),drop(B),N),Ys)) ) ).

% list_all2_dropI
tff(fact_5398_list__all2__takeI,axiom,
    ! [A: $tType,B: $tType,Pa: fun(A,fun(B,$o)),Xsa: list(A),Ys: list(B),N: nat] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Pa),Xsa),Ys)
     => aa(list(B),$o,aa(list(A),fun(list(B),$o),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Pa),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),Xsa)),aa(list(B),list(B),aa(nat,fun(list(B),list(B)),take(B),N),Ys)) ) ).

% list_all2_takeI
tff(fact_5399_list__all2__lengthD,axiom,
    ! [A: $tType,B: $tType,Pa: fun(A,fun(B,$o)),Xsa: list(A),Ys: list(B)] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Pa),Xsa),Ys)
     => ( aa(list(A),nat,size_size(list(A)),Xsa) = aa(list(B),nat,size_size(list(B)),Ys) ) ) ).

% list_all2_lengthD
tff(fact_5400_list__all2__antisym,axiom,
    ! [A: $tType,Pa: fun(A,fun(A,$o)),Qa: fun(A,fun(A,$o)),Xsa: list(A),Ys: list(A)] :
      ( ! [X: A,Y2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),Pa,X),Y2)
         => ( aa(A,$o,aa(A,fun(A,$o),Qa,Y2),X)
           => ( X = Y2 ) ) )
     => ( aa(list(A),$o,aa(list(A),fun(list(A),$o),aa(fun(A,fun(A,$o)),fun(list(A),fun(list(A),$o)),list_all2(A,A),Pa),Xsa),Ys)
       => ( aa(list(A),$o,aa(list(A),fun(list(A),$o),aa(fun(A,fun(A,$o)),fun(list(A),fun(list(A),$o)),list_all2(A,A),Qa),Ys),Xsa)
         => ( Xsa = Ys ) ) ) ) ).

% list_all2_antisym
tff(fact_5401_list__all2__trans,axiom,
    ! [B: $tType,A: $tType,E: $tType,P1: fun(A,fun(B,$o)),P22: fun(B,fun(E,$o)),P32: fun(A,fun(E,$o)),As2: list(A),Bs: list(B),Cs: list(E)] :
      ( ! [A4: A,B4: B,C5: E] :
          ( aa(B,$o,aa(A,fun(B,$o),P1,A4),B4)
         => ( aa(E,$o,aa(B,fun(E,$o),P22,B4),C5)
           => aa(E,$o,aa(A,fun(E,$o),P32,A4),C5) ) )
     => ( aa(list(B),$o,aa(list(A),fun(list(B),$o),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),P1),As2),Bs)
       => ( aa(list(E),$o,aa(list(B),fun(list(E),$o),aa(fun(B,fun(E,$o)),fun(list(B),fun(list(E),$o)),list_all2(B,E),P22),Bs),Cs)
         => aa(list(E),$o,aa(list(A),fun(list(E),$o),aa(fun(A,fun(E,$o)),fun(list(A),fun(list(E),$o)),list_all2(A,E),P32),As2),Cs) ) ) ) ).

% list_all2_trans
tff(fact_5402_list__all2__refl,axiom,
    ! [A: $tType,Pa: fun(A,fun(A,$o)),Xsa: list(A)] :
      ( ! [X: A] : aa(A,$o,aa(A,fun(A,$o),Pa,X),X)
     => aa(list(A),$o,aa(list(A),fun(list(A),$o),aa(fun(A,fun(A,$o)),fun(list(A),fun(list(A),$o)),list_all2(A,A),Pa),Xsa),Xsa) ) ).

% list_all2_refl
tff(fact_5403_list__all2__mono,axiom,
    ! [A: $tType,B: $tType,Pa: fun(A,fun(B,$o)),Xsa: list(A),Ys: list(B),Qa: fun(A,fun(B,$o))] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Pa),Xsa),Ys)
     => ( ! [Xs: A,Ys5: B] :
            ( aa(B,$o,aa(A,fun(B,$o),Pa,Xs),Ys5)
           => aa(B,$o,aa(A,fun(B,$o),Qa,Xs),Ys5) )
       => aa(list(B),$o,aa(list(A),fun(list(B),$o),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Qa),Xsa),Ys) ) ) ).

% list_all2_mono
tff(fact_5404_list__all2__eq,axiom,
    ! [A: $tType,Xsa: list(A),Ys: list(A)] :
      ( ( Xsa = Ys )
    <=> aa(list(A),$o,aa(list(A),fun(list(A),$o),aa(fun(A,fun(A,$o)),fun(list(A),fun(list(A),$o)),list_all2(A,A),fequal(A)),Xsa),Ys) ) ).

% list_all2_eq
tff(fact_5405_list_Orel__refl,axiom,
    ! [A: $tType,Ra2: fun(A,fun(A,$o)),Xa: list(A)] :
      ( ! [X: A] : aa(A,$o,aa(A,fun(A,$o),Ra2,X),X)
     => aa(list(A),$o,aa(list(A),fun(list(A),$o),aa(fun(A,fun(A,$o)),fun(list(A),fun(list(A),$o)),list_all2(A,A),Ra2),Xa),Xa) ) ).

% list.rel_refl
tff(fact_5406_list_Orel__eq,axiom,
    ! [A: $tType] : ( aa(fun(A,fun(A,$o)),fun(list(A),fun(list(A),$o)),list_all2(A,A),fequal(A)) = fequal(list(A)) ) ).

% list.rel_eq
tff(fact_5407_list__all2__update__cong,axiom,
    ! [A: $tType,B: $tType,Pa: fun(A,fun(B,$o)),Xsa: list(A),Ys: list(B),Xa: A,Y: B,Ia: nat] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Pa),Xsa),Ys)
     => ( aa(B,$o,aa(A,fun(B,$o),Pa,Xa),Y)
       => aa(list(B),$o,aa(list(A),fun(list(B),$o),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Pa),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xsa),Ia),Xa)),aa(B,list(B),aa(nat,fun(B,list(B)),aa(list(B),fun(nat,fun(B,list(B))),list_update(B),Ys),Ia),Y)) ) ) ).

% list_all2_update_cong
tff(fact_5408_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_5409_list__all2__appendI,axiom,
    ! [A: $tType,B: $tType,Pa: fun(A,fun(B,$o)),A3: list(A),B3: list(B),C2: list(A),D3: list(B)] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Pa),A3),B3)
     => ( aa(list(B),$o,aa(list(A),fun(list(B),$o),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Pa),C2),D3)
       => aa(list(B),$o,aa(list(A),fun(list(B),$o),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Pa),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),A3),C2)),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),B3),D3)) ) ) ).

% list_all2_appendI
tff(fact_5410_list__all2__rev1,axiom,
    ! [A: $tType,B: $tType,Pa: fun(A,fun(B,$o)),Xsa: list(A),Ys: list(B)] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Pa),aa(list(A),list(A),rev(A),Xsa)),Ys)
    <=> aa(list(B),$o,aa(list(A),fun(list(B),$o),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Pa),Xsa),aa(list(B),list(B),rev(B),Ys)) ) ).

% list_all2_rev1
tff(fact_5411_list_Orel__cong,axiom,
    ! [A: $tType,B: $tType,Xa: list(A),Ya: list(A),Y: list(B),Xaa: list(B),R: fun(A,fun(B,$o)),Ra2: fun(A,fun(B,$o))] :
      ( ( Xa = Ya )
     => ( ( Y = Xaa )
       => ( ! [Z3: A,Yb: B] :
              ( member(A,Z3,aa(list(A),set(A),set2(A),Ya))
             => ( member(B,Yb,aa(list(B),set(B),set2(B),Xaa))
               => ( aa(B,$o,aa(A,fun(B,$o),R,Z3),Yb)
                <=> aa(B,$o,aa(A,fun(B,$o),Ra2,Z3),Yb) ) ) )
         => ( aa(list(B),$o,aa(list(A),fun(list(B),$o),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),R),Xa),Y)
          <=> aa(list(B),$o,aa(list(A),fun(list(B),$o),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Ra2),Ya),Xaa) ) ) ) ) ).

% list.rel_cong
tff(fact_5412_list_Orel__mono__strong,axiom,
    ! [A: $tType,B: $tType,R: fun(A,fun(B,$o)),Xa: list(A),Y: list(B),Ra2: fun(A,fun(B,$o))] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),R),Xa),Y)
     => ( ! [Z3: A,Yb: B] :
            ( member(A,Z3,aa(list(A),set(A),set2(A),Xa))
           => ( member(B,Yb,aa(list(B),set(B),set2(B),Y))
             => ( aa(B,$o,aa(A,fun(B,$o),R,Z3),Yb)
               => aa(B,$o,aa(A,fun(B,$o),Ra2,Z3),Yb) ) ) )
       => aa(list(B),$o,aa(list(A),fun(list(B),$o),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Ra2),Xa),Y) ) ) ).

% list.rel_mono_strong
tff(fact_5413_list_Orel__refl__strong,axiom,
    ! [A: $tType,Xa: list(A),Ra2: fun(A,fun(A,$o))] :
      ( ! [Z3: A] :
          ( member(A,Z3,aa(list(A),set(A),set2(A),Xa))
         => aa(A,$o,aa(A,fun(A,$o),Ra2,Z3),Z3) )
     => aa(list(A),$o,aa(list(A),fun(list(A),$o),aa(fun(A,fun(A,$o)),fun(list(A),fun(list(A),$o)),list_all2(A,A),Ra2),Xa),Xa) ) ).

% list.rel_refl_strong
tff(fact_5414_list__all2__same,axiom,
    ! [A: $tType,Pa: fun(A,fun(A,$o)),Xsa: list(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)),list_all2(A,A),Pa),Xsa),Xsa)
    <=> ! [X6: A] :
          ( member(A,X6,aa(list(A),set(A),set2(A),Xsa))
         => aa(A,$o,aa(A,fun(A,$o),Pa,X6),X6) ) ) ).

% list_all2_same
tff(fact_5415_rev__ImageI,axiom,
    ! [B: $tType,A: $tType,A3: A,Aa2: set(A),B3: B,R2: set(product_prod(A,B))] :
      ( member(A,A3,Aa2)
     => ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B3),R2)
       => member(B,B3,image(A,B,R2,Aa2)) ) ) ).

% rev_ImageI
tff(fact_5416_Image__iff,axiom,
    ! [A: $tType,B: $tType,B3: A,R2: set(product_prod(B,A)),Aa2: set(B)] :
      ( member(A,B3,image(B,A,R2,Aa2))
    <=> ? [X6: B] :
          ( member(B,X6,Aa2)
          & member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X6),B3),R2) ) ) ).

% Image_iff
tff(fact_5417_ImageE,axiom,
    ! [A: $tType,B: $tType,B3: A,R2: set(product_prod(B,A)),Aa2: set(B)] :
      ( member(A,B3,image(B,A,R2,Aa2))
     => ~ ! [X: B] :
            ( member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X),B3),R2)
           => ~ member(B,X,Aa2) ) ) ).

% ImageE
tff(fact_5418_list__all2__Cons2,axiom,
    ! [A: $tType,B: $tType,Pa: fun(A,fun(B,$o)),Xsa: list(A),Y: B,Ys: list(B)] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Pa),Xsa),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),Ys))
    <=> ? [Z5: A,Zs3: list(A)] :
          ( ( Xsa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Z5),Zs3) )
          & aa(B,$o,aa(A,fun(B,$o),Pa,Z5),Y)
          & aa(list(B),$o,aa(list(A),fun(list(B),$o),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Pa),Zs3),Ys) ) ) ).

% list_all2_Cons2
tff(fact_5419_list__all2__Cons1,axiom,
    ! [A: $tType,B: $tType,Pa: fun(A,fun(B,$o)),Xa: A,Xsa: list(A),Ys: list(B)] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Pa),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa)),Ys)
    <=> ? [Z5: B,Zs3: list(B)] :
          ( ( Ys = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Z5),Zs3) )
          & aa(B,$o,aa(A,fun(B,$o),Pa,Xa),Z5)
          & aa(list(B),$o,aa(list(A),fun(list(B),$o),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Pa),Xsa),Zs3) ) ) ).

% list_all2_Cons1
tff(fact_5420_list__all2__Cons,axiom,
    ! [A: $tType,B: $tType,Pa: fun(A,fun(B,$o)),Xa: A,Xsa: list(A),Y: B,Ys: list(B)] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Pa),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),Ys))
    <=> ( aa(B,$o,aa(A,fun(B,$o),Pa,Xa),Y)
        & aa(list(B),$o,aa(list(A),fun(list(B),$o),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Pa),Xsa),Ys) ) ) ).

% list_all2_Cons
tff(fact_5421_list_Orel__intros_I2_J,axiom,
    ! [A: $tType,B: $tType,R: fun(A,fun(B,$o)),X21: A,Y21: B,X222: list(A),Y222: list(B)] :
      ( aa(B,$o,aa(A,fun(B,$o),R,X21),Y21)
     => ( aa(list(B),$o,aa(list(A),fun(list(B),$o),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),R),X222),Y222)
       => aa(list(B),$o,aa(list(A),fun(list(B),$o),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),R),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X21),X222)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y21),Y222)) ) ) ).

% list.rel_intros(2)
tff(fact_5422_list_Orel__inject_I2_J,axiom,
    ! [A: $tType,B: $tType,R: fun(A,fun(B,$o)),X21: A,X222: list(A),Y21: B,Y222: list(B)] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),R),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X21),X222)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y21),Y222))
    <=> ( aa(B,$o,aa(A,fun(B,$o),R,X21),Y21)
        & aa(list(B),$o,aa(list(A),fun(list(B),$o),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),R),X222),Y222) ) ) ).

% list.rel_inject(2)
tff(fact_5423_finite__Image__subset,axiom,
    ! [A: $tType,B: $tType,Aa2: set(product_prod(B,A)),B2: set(B),C3: set(product_prod(B,A))] :
      ( aa(set(A),$o,finite_finite2(A),image(B,A,Aa2,B2))
     => ( 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))),C3),Aa2)
       => aa(set(A),$o,finite_finite2(A),image(B,A,C3,B2)) ) ) ).

% finite_Image_subset
tff(fact_5424_rtrancl__reachable__induct,axiom,
    ! [A: $tType,Ia2: set(A),INV: set(A),E5: set(product_prod(A,A))] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Ia2),INV)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),image(A,A,E5,INV)),INV)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),image(A,A,aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),E5),Ia2)),INV) ) ) ).

% rtrancl_reachable_induct
tff(fact_5425_rtrancl__image__unfold__right,axiom,
    ! [A: $tType,E5: set(product_prod(A,A)),V: set(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),image(A,A,E5,image(A,A,aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),E5),V))),image(A,A,aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),E5),V)) ).

% rtrancl_image_unfold_right
tff(fact_5426_Image__mono,axiom,
    ! [B: $tType,A: $tType,R5: set(product_prod(A,B)),R2: set(product_prod(A,B)),A5: set(A),Aa2: set(A)] :
      ( aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),R5),R2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),Aa2)
       => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),image(A,B,R5,A5)),image(A,B,R2,Aa2)) ) ) ).

% Image_mono
tff(fact_5427_trancl__Image__unfold__left,axiom,
    ! [A: $tType,E5: set(product_prod(A,A)),S: set(A)] : ( image(A,A,transitive_trancl(A,E5),S) = image(A,A,aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),E5),image(A,A,E5,S)) ) ).

% trancl_Image_unfold_left
tff(fact_5428_trancl__Image__unfold__right,axiom,
    ! [A: $tType,E5: set(product_prod(A,A)),S: set(A)] : ( image(A,A,transitive_trancl(A,E5),S) = image(A,A,E5,image(A,A,aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),E5),S)) ) ).

% trancl_Image_unfold_right
tff(fact_5429_finite__Image,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,B)),Aa2: set(A)] :
      ( aa(set(product_prod(A,B)),$o,finite_finite2(product_prod(A,B)),R)
     => aa(set(B),$o,finite_finite2(B),image(A,B,R,Aa2)) ) ).

% finite_Image
tff(fact_5430_Un__Image,axiom,
    ! [A: $tType,B: $tType,R: set(product_prod(B,A)),S: set(product_prod(B,A)),Aa2: set(B)] : ( 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),Aa2) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),image(B,A,R,Aa2)),image(B,A,S,Aa2)) ) ).

% Un_Image
tff(fact_5431_Image__def,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(B,A)),S2: set(B)] : ( image(B,A,R2,S2) = aa(fun(A,$o),set(A),collect(A),aa(set(B),fun(A,$o),aTP_Lamp_ru(set(product_prod(B,A)),fun(set(B),fun(A,$o)),R2),S2)) ) ).

% Image_def
tff(fact_5432_Image__UN,axiom,
    ! [B: $tType,A: $tType,E: $tType,R2: set(product_prod(B,A)),B2: fun(E,set(B)),Aa2: set(E)] : ( image(B,A,R2,aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(E),set(set(B)),image2(E,set(B),B2),Aa2))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(E),set(set(A)),image2(E,set(A),aa(fun(E,set(B)),fun(E,set(A)),aTP_Lamp_rv(set(product_prod(B,A)),fun(fun(E,set(B)),fun(E,set(A))),R2),B2)),Aa2)) ) ).

% Image_UN
tff(fact_5433_list__all2__nthD,axiom,
    ! [A: $tType,B: $tType,Pa: fun(A,fun(B,$o)),Xsa: list(A),Ys: list(B),P2: nat] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Pa),Xsa),Ys)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),P2),aa(list(A),nat,size_size(list(A)),Xsa))
       => aa(B,$o,aa(A,fun(B,$o),Pa,aa(nat,A,nth(A,Xsa),P2)),aa(nat,B,nth(B,Ys),P2)) ) ) ).

% list_all2_nthD
tff(fact_5434_list__all2__nthD2,axiom,
    ! [A: $tType,B: $tType,Pa: fun(A,fun(B,$o)),Xsa: list(A),Ys: list(B),P2: nat] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Pa),Xsa),Ys)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),P2),aa(list(B),nat,size_size(list(B)),Ys))
       => aa(B,$o,aa(A,fun(B,$o),Pa,aa(nat,A,nth(A,Xsa),P2)),aa(nat,B,nth(B,Ys),P2)) ) ) ).

% list_all2_nthD2
tff(fact_5435_list__all2__all__nthI,axiom,
    ! [A: $tType,B: $tType,A3: list(A),B3: list(B),Pa: fun(A,fun(B,$o))] :
      ( ( aa(list(A),nat,size_size(list(A)),A3) = aa(list(B),nat,size_size(list(B)),B3) )
     => ( ! [N5: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N5),aa(list(A),nat,size_size(list(A)),A3))
           => aa(B,$o,aa(A,fun(B,$o),Pa,aa(nat,A,nth(A,A3),N5)),aa(nat,B,nth(B,B3),N5)) )
       => aa(list(B),$o,aa(list(A),fun(list(B),$o),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Pa),A3),B3) ) ) ).

% list_all2_all_nthI
tff(fact_5436_list__all2__conv__all__nth,axiom,
    ! [A: $tType,B: $tType,Pa: fun(A,fun(B,$o)),Xsa: list(A),Ys: list(B)] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Pa),Xsa),Ys)
    <=> ( ( aa(list(A),nat,size_size(list(A)),Xsa) = aa(list(B),nat,size_size(list(B)),Ys) )
        & ! [I2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xsa))
           => aa(B,$o,aa(A,fun(B,$o),Pa,aa(nat,A,nth(A,Xsa),I2)),aa(nat,B,nth(B,Ys),I2)) ) ) ) ).

% list_all2_conv_all_nth
tff(fact_5437_Image__empty__rtrancl__Image__id,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),V2: A] :
      ( ( image(A,A,R,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),V2),bot_bot(set(A)))) = bot_bot(set(A)) )
     => ( image(A,A,aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),V2),bot_bot(set(A)))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),V2),bot_bot(set(A))) ) ) ).

% Image_empty_rtrancl_Image_id
tff(fact_5438_wfE__pf,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),Aa2: set(A)] :
      ( wf(A,R)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),image(A,A,R,Aa2))
       => ( Aa2 = bot_bot(set(A)) ) ) ) ).

% wfE_pf
tff(fact_5439_wfI__pf,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( ! [A12: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A12),image(A,A,R,A12))
         => ( A12 = bot_bot(set(A)) ) )
     => wf(A,R) ) ).

% wfI_pf
tff(fact_5440_Image__empty__trancl__Image__empty,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),V2: A] :
      ( ( image(A,A,R,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),V2),bot_bot(set(A)))) = bot_bot(set(A)) )
     => ( image(A,A,transitive_trancl(A,R),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),V2),bot_bot(set(A)))) = bot_bot(set(A)) ) ) ).

% Image_empty_trancl_Image_empty
tff(fact_5441_reachable__mono,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),R8: set(product_prod(A,A)),X3: set(A),X8: set(A)] :
      ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R),R8)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X3),X8)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),image(A,A,aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R),X3)),image(A,A,aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R8),X8)) ) ) ).

% reachable_mono
tff(fact_5442_list_Orel__sel,axiom,
    ! [A: $tType,B: $tType,R: fun(A,fun(B,$o)),A3: list(A),B3: list(B)] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),R),A3),B3)
    <=> ( ( ( A3 = nil(A) )
        <=> ( B3 = nil(B) ) )
        & ( ( A3 != nil(A) )
         => ( ( B3 != nil(B) )
           => ( aa(B,$o,aa(A,fun(B,$o),R,aa(list(A),A,hd(A),A3)),aa(list(B),B,hd(B),B3))
              & aa(list(B),$o,aa(list(A),fun(list(B),$o),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),R),aa(list(A),list(A),tl(A),A3)),aa(list(B),list(B),tl(B),B3)) ) ) ) ) ) ).

% list.rel_sel
tff(fact_5443_Image__subset__snd__image,axiom,
    ! [A: $tType,B: $tType,Aa2: set(product_prod(B,A)),B2: set(B)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),image(B,A,Aa2,B2)),aa(set(product_prod(B,A)),set(A),image2(product_prod(B,A),A,product_snd(B,A)),Aa2)) ).

% Image_subset_snd_image
tff(fact_5444_trancl__image__by__rtrancl,axiom,
    ! [A: $tType,E5: 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)),image(A,A,transitive_trancl(A,E5),Vi)),Vi) = image(A,A,aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),E5),Vi) ) ).

% trancl_image_by_rtrancl
tff(fact_5445_Image__singleton,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(B,A)),A3: B] : ( image(B,A,R2,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert3(B),A3),bot_bot(set(B)))) = aa(fun(A,$o),set(A),collect(A),aa(B,fun(A,$o),aTP_Lamp_rw(set(product_prod(B,A)),fun(B,fun(A,$o)),R2),A3)) ) ).

% Image_singleton
tff(fact_5446_Image__subset,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(A,B)),Aa2: set(A),B2: set(B),C3: set(A)] :
      ( aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),R2),product_Sigma(A,B,Aa2,aTP_Lamp_pv(set(B),fun(A,set(B)),B2)))
     => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),image(A,B,R2,C3)),B2) ) ).

% Image_subset
tff(fact_5447_Image__INT__subset,axiom,
    ! [B: $tType,A: $tType,E: $tType,R2: set(product_prod(B,A)),B2: fun(E,set(B)),Aa2: set(E)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),image(B,A,R2,aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(E),set(set(B)),image2(E,set(B),B2),Aa2)))),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(E),set(set(A)),image2(E,set(A),aa(fun(E,set(B)),fun(E,set(A)),aTP_Lamp_rv(set(product_prod(B,A)),fun(fun(E,set(B)),fun(E,set(A))),R2),B2)),Aa2))) ).

% Image_INT_subset
tff(fact_5448_product__lists__set,axiom,
    ! [A: $tType,Xss2: list(list(A))] : ( aa(list(list(A)),set(list(A)),set2(list(A)),aa(list(list(A)),list(list(A)),product_lists(A),Xss2)) = aa(fun(list(A),$o),set(list(A)),collect(list(A)),aTP_Lamp_ry(list(list(A)),fun(list(A),$o),Xss2)) ) ).

% product_lists_set
tff(fact_5449_listrel__Cons,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(B,A)),Xa: B,Xsa: list(B)] : ( image(list(B),list(A),listrel(B,A,R2),aa(set(list(B)),set(list(B)),aa(list(B),fun(set(list(B)),set(list(B))),insert3(list(B)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Xa),Xsa)),bot_bot(set(list(B))))) = aa(set(list(A)),set(list(A)),aa(set(A),fun(set(list(A)),set(list(A))),set_Cons(A),image(B,A,R2,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert3(B),Xa),bot_bot(set(B))))),image(list(B),list(A),listrel(B,A,R2),aa(set(list(B)),set(list(B)),aa(list(B),fun(set(list(B)),set(list(B))),insert3(list(B)),Xsa),bot_bot(set(list(B)))))) ) ).

% listrel_Cons
tff(fact_5450_rtrancl__apply__insert,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),Xa: A,S: set(A)] : ( image(A,A,aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),S)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),image(A,A,aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),S),image(A,A,R,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A))))))) ) ).

% rtrancl_apply_insert
tff(fact_5451_rtrancl__Image__in__Field,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),V: set(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),image(A,A,aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R),V)),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),field2(A),R)),V)) ).

% rtrancl_Image_in_Field
tff(fact_5452_list__all2I,axiom,
    ! [A: $tType,B: $tType,A3: list(A),B3: list(B),Pa: fun(A,fun(B,$o))] :
      ( ! [X: product_prod(A,B)] :
          ( member(product_prod(A,B),X,aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),A3),B3)))
         => aa(product_prod(A,B),$o,aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),Pa),X) )
     => ( ( aa(list(A),nat,size_size(list(A)),A3) = aa(list(B),nat,size_size(list(B)),B3) )
       => aa(list(B),$o,aa(list(A),fun(list(B),$o),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Pa),A3),B3) ) ) ).

% list_all2I
tff(fact_5453_E__closed__restr__reach__cases,axiom,
    ! [A: $tType,U: A,V2: A,E5: set(product_prod(A,A)),R: set(A)] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),U),V2),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),E5))
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),image(A,A,E5,R)),R)
       => ( ~ member(A,V2,R)
         => ~ ( ~ member(A,U,R)
             => ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),U),V2),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),rel_restrict(A,E5,R))) ) ) ) ) ).

% E_closed_restr_reach_cases
tff(fact_5454_rel__restrict__tranclI,axiom,
    ! [A: $tType,Xa: A,Y: A,E5: set(product_prod(A,A)),R: set(A)] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Y),transitive_trancl(A,E5))
     => ( ~ member(A,Xa,R)
       => ( ~ member(A,Y,R)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),image(A,A,E5,R)),R)
           => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Y),transitive_trancl(A,rel_restrict(A,E5,R))) ) ) ) ) ).

% rel_restrict_tranclI
tff(fact_5455_Image__absorb__rtrancl,axiom,
    ! [A: $tType,Aa2: set(product_prod(A,A)),B2: set(A),C3: set(A)] :
      ( trans(A,Aa2)
     => ( refl_on(A,B2,Aa2)
       => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C3),B2)
         => ( image(A,A,aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),Aa2),C3) = image(A,A,Aa2,C3) ) ) ) ) ).

% Image_absorb_rtrancl
tff(fact_5456_refl__on__reflcl__Image,axiom,
    ! [A: $tType,B2: set(A),Aa2: set(product_prod(A,A)),C3: set(A)] :
      ( refl_on(A,B2,Aa2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C3),B2)
       => ( 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))),Aa2),id2(A)),C3) = image(A,A,Aa2,C3) ) ) ) ).

% refl_on_reflcl_Image
tff(fact_5457_Image__eq__UN,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(B,A)),B2: set(B)] : ( image(B,A,R2,B2) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aTP_Lamp_rz(set(product_prod(B,A)),fun(B,set(A)),R2)),B2)) ) ).

% Image_eq_UN
tff(fact_5458_Sigma__Image,axiom,
    ! [A: $tType,B: $tType,Aa2: set(B),B2: fun(B,set(A)),X3: set(B)] : ( image(B,A,product_Sigma(B,A,Aa2,B2),X3) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),X3),Aa2))) ) ).

% Sigma_Image
tff(fact_5459_UN__Image,axiom,
    ! [B: $tType,A: $tType,E: $tType,X3: fun(E,set(product_prod(B,A))),Ia2: set(E),S: set(B)] : ( 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(E),set(set(product_prod(B,A))),image2(E,set(product_prod(B,A)),X3),Ia2)),S) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(E),set(set(A)),image2(E,set(A),aa(set(B),fun(E,set(A)),aTP_Lamp_sa(fun(E,set(product_prod(B,A))),fun(set(B),fun(E,set(A))),X3),S)),Ia2)) ) ).

% UN_Image
tff(fact_5460_finite__reachable__advance,axiom,
    ! [A: $tType,E5: set(product_prod(A,A)),V0: A,V2: A] :
      ( aa(set(A),$o,finite_finite2(A),image(A,A,aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),E5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),V0),bot_bot(set(A)))))
     => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),V0),V2),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),E5))
       => aa(set(A),$o,finite_finite2(A),image(A,A,aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),E5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),V2),bot_bot(set(A))))) ) ) ).

% finite_reachable_advance
tff(fact_5461_rtrancl__Image__advance__ss,axiom,
    ! [A: $tType,U: A,V2: A,E5: set(product_prod(A,A))] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),U),V2),E5)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),image(A,A,aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),E5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),V2),bot_bot(set(A))))),image(A,A,aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),E5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),U),bot_bot(set(A))))) ) ).

% rtrancl_Image_advance_ss
tff(fact_5462_trancl__Image__advance__ss,axiom,
    ! [A: $tType,U: A,V2: A,E5: set(product_prod(A,A))] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),U),V2),E5)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),image(A,A,transitive_trancl(A,E5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),V2),bot_bot(set(A))))),image(A,A,transitive_trancl(A,E5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),U),bot_bot(set(A))))) ) ).

% trancl_Image_advance_ss
tff(fact_5463_trancl__restrict__reachable,axiom,
    ! [A: $tType,U: A,V2: A,E5: set(product_prod(A,A)),S: set(A)] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),U),V2),transitive_trancl(A,E5))
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),image(A,A,E5,S)),S)
       => ( member(A,U,S)
         => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),U),V2),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))),E5),product_Sigma(A,A,S,aTP_Lamp_pt(set(A),fun(A,set(A)),S))))) ) ) ) ).

% trancl_restrict_reachable
tff(fact_5464_list__all2__iff,axiom,
    ! [B: $tType,A: $tType,Pa: fun(A,fun(B,$o)),Xsa: list(A),Ys: list(B)] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Pa),Xsa),Ys)
    <=> ( ( aa(list(A),nat,size_size(list(A)),Xsa) = aa(list(B),nat,size_size(list(B)),Ys) )
        & ! [X6: product_prod(A,B)] :
            ( member(product_prod(A,B),X6,aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),Xsa),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),Pa),X6) ) ) ) ).

% list_all2_iff
tff(fact_5465_subset__Image1__Image1__iff,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A3: A,B3: A] :
      ( order_preorder_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => ( member(A,A3,aa(set(product_prod(A,A)),set(A),field2(A),R2))
       => ( member(A,B3,aa(set(product_prod(A,A)),set(A),field2(A),R2))
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),bot_bot(set(A))))),image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),B3),bot_bot(set(A)))))
          <=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B3),A3),R2) ) ) ) ) ).

% subset_Image1_Image1_iff
tff(fact_5466_subset__mset_Omono__cSUP,axiom,
    ! [B: $tType,A: $tType,E: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [F2: fun(multiset(A),B),Aa2: fun(E,multiset(A)),Ia2: set(E)] :
          ( aa(fun(multiset(A),B),$o,mono(multiset(A),B,subseteq_mset(A)),F2)
         => ( condit8047198070973881523_above(multiset(A),subseteq_mset(A),aa(set(E),set(multiset(A)),image2(E,multiset(A),Aa2),Ia2))
           => ( ( Ia2 != bot_bot(set(E)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(E),set(B),image2(E,B,aa(fun(E,multiset(A)),fun(E,B),aTP_Lamp_rp(fun(multiset(A),B),fun(fun(E,multiset(A)),fun(E,B)),F2),Aa2)),Ia2))),aa(multiset(A),B,F2,aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),aa(set(E),set(multiset(A)),image2(E,multiset(A),Aa2),Ia2)))) ) ) ) ) ).

% subset_mset.mono_cSUP
tff(fact_5467_trans__wf__iff,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( trans(A,R2)
     => ( wf(A,R2)
      <=> ! [A7: 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,image(A,A,converse(A,A,R2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A7),bot_bot(set(A)))),aa(A,fun(A,set(A)),aTP_Lamp_sb(set(product_prod(A,A)),fun(A,fun(A,set(A))),R2),A7)))) ) ) ).

% trans_wf_iff
tff(fact_5468_converse__inject,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(B,A)),S2: set(product_prod(B,A))] :
      ( ( converse(B,A,R2) = converse(B,A,S2) )
    <=> ( R2 = S2 ) ) ).

% converse_inject
tff(fact_5469_converse__converse,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,B))] : ( converse(B,A,converse(A,B,R2)) = R2 ) ).

% converse_converse
tff(fact_5470_converse__iff,axiom,
    ! [A: $tType,B: $tType,A3: A,B3: B,R2: set(product_prod(B,A))] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B3),converse(B,A,R2))
    <=> member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),B3),A3),R2) ) ).

% converse_iff
tff(fact_5471_Field__converse,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : ( aa(set(product_prod(A,A)),set(A),field2(A),converse(A,A,R2)) = aa(set(product_prod(A,A)),set(A),field2(A),R2) ) ).

% Field_converse
tff(fact_5472_converse__mono,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(B,A)),S2: set(product_prod(B,A))] :
      ( aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),converse(B,A,R2)),converse(B,A,S2))
    <=> aa(set(product_prod(B,A)),$o,aa(set(product_prod(B,A)),fun(set(product_prod(B,A)),$o),ord_less_eq(set(product_prod(B,A))),R2),S2) ) ).

% converse_mono
tff(fact_5473_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_5474_trans__converse,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( trans(A,converse(A,A,R2))
    <=> trans(A,R2) ) ).

% trans_converse
tff(fact_5475_converse__Id,axiom,
    ! [A: $tType] : ( converse(A,A,id2(A)) = id2(A) ) ).

% converse_Id
tff(fact_5476_refl__on__converse,axiom,
    ! [A: $tType,Aa2: set(A),R2: set(product_prod(A,A))] :
      ( refl_on(A,Aa2,converse(A,A,R2))
    <=> refl_on(A,Aa2,R2) ) ).

% refl_on_converse
tff(fact_5477_finite__converse,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(B,A))] :
      ( aa(set(product_prod(A,B)),$o,finite_finite2(product_prod(A,B)),converse(B,A,R2))
    <=> aa(set(product_prod(B,A)),$o,finite_finite2(product_prod(B,A)),R2) ) ).

% finite_converse
tff(fact_5478_total__on__converse,axiom,
    ! [A: $tType,Aa2: set(A),R2: set(product_prod(A,A))] :
      ( total_on(A,Aa2,converse(A,A,R2))
    <=> total_on(A,Aa2,R2) ) ).

% total_on_converse
tff(fact_5479_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_5480_converse__Id__on,axiom,
    ! [A: $tType,Aa2: set(A)] : ( converse(A,A,id_on(A,Aa2)) = id_on(A,Aa2) ) ).

% converse_Id_on
tff(fact_5481_card__inverse,axiom,
    ! [A: $tType,B: $tType,R: set(product_prod(B,A))] : ( finite_card(product_prod(A,B),converse(B,A,R)) = finite_card(product_prod(B,A),R) ) ).

% card_inverse
tff(fact_5482_pair__set__inverse,axiom,
    ! [A: $tType,B: $tType,Pa: fun(B,fun(A,$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),Pa))) = 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_sc(fun(B,fun(A,$o)),fun(A,fun(B,$o)),Pa))) ) ).

% pair_set_inverse
tff(fact_5483_below__Id__inv,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)),$o),ord_less_eq(set(product_prod(A,A))),converse(A,A,R)),id2(A))
    <=> aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R),id2(A)) ) ).

% below_Id_inv
tff(fact_5484_in__listrel1__converse,axiom,
    ! [A: $tType,Xa: list(A),Y: list(A),R2: set(product_prod(A,A))] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xa),Y),listrel1(A,converse(A,A,R2)))
    <=> member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xa),Y),converse(list(A),list(A),listrel1(A,R2))) ) ).

% in_listrel1_converse
tff(fact_5485_converse_Ocases,axiom,
    ! [A: $tType,B: $tType,A1: A,A22: B,R2: set(product_prod(B,A))] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A1),A22),converse(B,A,R2))
     => 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_5486_converse_Osimps,axiom,
    ! [A: $tType,B: $tType,A1: A,A22: B,R2: set(product_prod(B,A))] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A1),A22),converse(B,A,R2))
    <=> ? [A7: B,B8: A] :
          ( ( A1 = B8 )
          & ( A22 = A7 )
          & member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),A7),B8),R2) ) ) ).

% converse.simps
tff(fact_5487_converseD,axiom,
    ! [A: $tType,B: $tType,A3: A,B3: B,R2: set(product_prod(B,A))] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B3),converse(B,A,R2))
     => member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),B3),A3),R2) ) ).

% converseD
tff(fact_5488_converseE,axiom,
    ! [A: $tType,B: $tType,Yx: product_prod(A,B),R2: set(product_prod(B,A))] :
      ( member(product_prod(A,B),Yx,converse(B,A,R2))
     => ~ ! [X: B,Y2: A] :
            ( ( Yx = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Y2),X) )
           => ~ member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X),Y2),R2) ) ) ).

% converseE
tff(fact_5489_converseI,axiom,
    ! [B: $tType,A: $tType,A3: A,B3: B,R2: set(product_prod(A,B))] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B3),R2)
     => member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),B3),A3),converse(A,B,R2)) ) ).

% converseI
tff(fact_5490_listrel1__converse,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : ( listrel1(A,converse(A,A,R2)) = converse(list(A),list(A),listrel1(A,R2)) ) ).

% listrel1_converse
tff(fact_5491_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_5492_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_5493_converse__subset__swap,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(A,B)),S2: set(product_prod(B,A))] :
      ( aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),R2),converse(B,A,S2))
    <=> aa(set(product_prod(B,A)),$o,aa(set(product_prod(B,A)),fun(set(product_prod(B,A)),$o),ord_less_eq(set(product_prod(B,A))),converse(A,B,R2)),S2) ) ).

% converse_subset_swap
tff(fact_5494_converse__relcomp,axiom,
    ! [A: $tType,E: $tType,B: $tType,R2: set(product_prod(B,E)),S2: set(product_prod(E,A))] : ( converse(B,A,relcomp(B,E,A,R2,S2)) = relcomp(A,E,B,converse(E,A,S2),converse(B,E,R2)) ) ).

% converse_relcomp
tff(fact_5495_rtrancl__converseI,axiom,
    ! [A: $tType,Y: A,Xa: A,R2: set(product_prod(A,A))] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Xa),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R2))
     => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Y),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),converse(A,A,R2))) ) ).

% rtrancl_converseI
tff(fact_5496_rtrancl__converseD,axiom,
    ! [A: $tType,Xa: A,Y: A,R2: set(product_prod(A,A))] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Y),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),converse(A,A,R2)))
     => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Xa),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R2)) ) ).

% rtrancl_converseD
tff(fact_5497_trancl__converseD,axiom,
    ! [A: $tType,Xa: A,Y: A,R2: set(product_prod(A,A))] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Y),transitive_trancl(A,converse(A,A,R2)))
     => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Y),converse(A,A,transitive_trancl(A,R2))) ) ).

% trancl_converseD
tff(fact_5498_trancl__converseI,axiom,
    ! [A: $tType,Xa: A,Y: A,R2: set(product_prod(A,A))] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Y),converse(A,A,transitive_trancl(A,R2)))
     => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Y),transitive_trancl(A,converse(A,A,R2))) ) ).

% trancl_converseI
tff(fact_5499_finite__wf__eq__wf__converse,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,finite_finite2(product_prod(A,A)),R)
     => ( wf(A,converse(A,A,R))
      <=> wf(A,R) ) ) ).

% finite_wf_eq_wf_converse
tff(fact_5500_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,$o),aTP_Lamp_sd(set(product_prod(B,A)),fun(A,fun(B,$o)),R2))) ) ).

% converse_unfold
tff(fact_5501_converse__UNION,axiom,
    ! [B: $tType,A: $tType,E: $tType,R2: fun(E,set(product_prod(B,A))),S: set(E)] : ( 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(E),set(set(product_prod(B,A))),image2(E,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(E),set(set(product_prod(A,B))),image2(E,set(product_prod(A,B)),aTP_Lamp_se(fun(E,set(product_prod(B,A))),fun(E,set(product_prod(A,B))),R2)),S)) ) ).

% converse_UNION
tff(fact_5502_converse__INTER,axiom,
    ! [B: $tType,A: $tType,E: $tType,R2: fun(E,set(product_prod(B,A))),S: set(E)] : ( 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(E),set(set(product_prod(B,A))),image2(E,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(E),set(set(product_prod(A,B))),image2(E,set(product_prod(A,B)),aTP_Lamp_se(fun(E,set(product_prod(B,A))),fun(E,set(product_prod(A,B))),R2)),S)) ) ).

% converse_INTER
tff(fact_5503_Image__subset__eq,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(B,A)),Aa2: set(B),B2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),image(B,A,R2,Aa2)),B2)
    <=> aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),Aa2),aa(set(B),set(B),uminus_uminus(set(B)),image(A,B,converse(B,A,R2),aa(set(A),set(A),uminus_uminus(set(A)),B2)))) ) ).

% Image_subset_eq
tff(fact_5504_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_5505_subset__mset_OcSUP__le__iff,axiom,
    ! [A: $tType,B: $tType,Aa2: set(A),F2: fun(A,multiset(B)),U: multiset(B)] :
      ( ( Aa2 != bot_bot(set(A)) )
     => ( condit8047198070973881523_above(multiset(B),subseteq_mset(B),aa(set(A),set(multiset(B)),image2(A,multiset(B),F2),Aa2))
       => ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),aa(set(multiset(B)),multiset(B),complete_Sup_Sup(multiset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),F2),Aa2))),U)
        <=> ! [X6: A] :
              ( member(A,X6,Aa2)
             => aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),aa(A,multiset(B),F2,X6)),U) ) ) ) ) ).

% subset_mset.cSUP_le_iff
tff(fact_5506_subset__mset_OcSUP__mono,axiom,
    ! [A: $tType,B: $tType,E: $tType,Aa2: set(A),G: fun(E,multiset(B)),B2: set(E),F2: fun(A,multiset(B))] :
      ( ( Aa2 != bot_bot(set(A)) )
     => ( condit8047198070973881523_above(multiset(B),subseteq_mset(B),aa(set(E),set(multiset(B)),image2(E,multiset(B),G),B2))
       => ( ! [N5: A] :
              ( member(A,N5,Aa2)
             => ? [X4: E] :
                  ( member(E,X4,B2)
                  & aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),aa(A,multiset(B),F2,N5)),aa(E,multiset(B),G,X4)) ) )
         => aa(multiset(B),$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),F2),Aa2))),aa(set(multiset(B)),multiset(B),complete_Sup_Sup(multiset(B)),aa(set(E),set(multiset(B)),image2(E,multiset(B),G),B2))) ) ) ) ).

% subset_mset.cSUP_mono
tff(fact_5507_irrefl__tranclI,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),Xa: 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)),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R2)) = bot_bot(set(product_prod(A,A))) )
     => ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Xa),transitive_trancl(A,R2)) ) ).

% irrefl_tranclI
tff(fact_5508_subset__mset_OcSUP__subset__mono,axiom,
    ! [B: $tType,A: $tType,Aa2: set(A),G: fun(A,multiset(B)),B2: set(A),F2: fun(A,multiset(B))] :
      ( ( Aa2 != bot_bot(set(A)) )
     => ( condit8047198070973881523_above(multiset(B),subseteq_mset(B),aa(set(A),set(multiset(B)),image2(A,multiset(B),G),B2))
       => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),B2)
         => ( ! [X: A] :
                ( member(A,X,Aa2)
               => aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),aa(A,multiset(B),F2,X)),aa(A,multiset(B),G,X)) )
           => aa(multiset(B),$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),F2),Aa2))),aa(set(multiset(B)),multiset(B),complete_Sup_Sup(multiset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),G),B2))) ) ) ) ) ).

% subset_mset.cSUP_subset_mono
tff(fact_5509_subset__Image__Image__iff,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),Aa2: set(A),B2: set(A)] :
      ( order_preorder_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),aa(set(product_prod(A,A)),set(A),field2(A),R2))
       => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),aa(set(product_prod(A,A)),set(A),field2(A),R2))
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),image(A,A,R2,Aa2)),image(A,A,R2,B2))
          <=> ! [X6: A] :
                ( member(A,X6,Aa2)
               => ? [Xa2: A] :
                    ( member(A,Xa2,B2)
                    & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa2),X6),R2) ) ) ) ) ) ) ).

% subset_Image_Image_iff
tff(fact_5510_Image__INT__eq,axiom,
    ! [B: $tType,A: $tType,E: $tType,R2: set(product_prod(B,A)),Aa2: set(E),B2: fun(E,set(B))] :
      ( single_valued(A,B,converse(B,A,R2))
     => ( ( Aa2 != bot_bot(set(E)) )
       => ( image(B,A,R2,aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(E),set(set(B)),image2(E,set(B),B2),Aa2))) = aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(E),set(set(A)),image2(E,set(A),aa(fun(E,set(B)),fun(E,set(A)),aTP_Lamp_rv(set(product_prod(B,A)),fun(fun(E,set(B)),fun(E,set(A))),R2),B2)),Aa2)) ) ) ) ).

% Image_INT_eq
tff(fact_5511_Refl__antisym__eq__Image1__Image1__iff,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A3: A,B3: A] :
      ( refl_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => ( antisym(A,R2)
       => ( member(A,A3,aa(set(product_prod(A,A)),set(A),field2(A),R2))
         => ( member(A,B3,aa(set(product_prod(A,A)),set(A),field2(A),R2))
           => ( ( image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),bot_bot(set(A)))) = image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),B3),bot_bot(set(A)))) )
            <=> ( A3 = B3 ) ) ) ) ) ) ).

% Refl_antisym_eq_Image1_Image1_iff
tff(fact_5512_Sup__multiset__in__multiset,axiom,
    ! [A: $tType,Aa2: set(multiset(A))] :
      ( ( Aa2 != bot_bot(set(multiset(A))) )
     => ( condit8047198070973881523_above(multiset(A),subseteq_mset(A),Aa2)
       => aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_sg(set(multiset(A)),fun(A,$o),Aa2))) ) ) ).

% Sup_multiset_in_multiset
tff(fact_5513_antisym__converse,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( antisym(A,converse(A,A,R2))
    <=> antisym(A,R2) ) ).

% antisym_converse
tff(fact_5514_mset__empty__count,axiom,
    ! [A: $tType,M5: multiset(A)] :
      ( ! [P4: A] : ( aa(A,nat,aa(multiset(A),fun(A,nat),count(A),M5),P4) = zero_zero(nat) )
    <=> ( M5 = zero_zero(multiset(A)) ) ) ).

% mset_empty_count
tff(fact_5515_count__greater__zero__iff,axiom,
    ! [A: $tType,M5: multiset(A),Xa: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(A,nat,aa(multiset(A),fun(A,nat),count(A),M5),Xa))
    <=> member(A,Xa,aa(multiset(A),set(A),set_mset(A),M5)) ) ).

% count_greater_zero_iff
tff(fact_5516_count__mset__0__iff,axiom,
    ! [A: $tType,Xsa: list(A),Xa: A] :
      ( ( aa(A,nat,aa(multiset(A),fun(A,nat),count(A),aa(list(A),multiset(A),mset(A),Xsa)),Xa) = zero_zero(nat) )
    <=> ~ member(A,Xa,aa(list(A),set(A),set2(A),Xsa)) ) ).

% count_mset_0_iff
tff(fact_5517_antisym__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)
     => ( antisym(A,S2)
       => antisym(A,R2) ) ) ).

% antisym_subset
tff(fact_5518_single__valued__subset,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,B)),S2: set(product_prod(A,B))] :
      ( aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),R2),S2)
     => ( single_valued(A,B,S2)
       => single_valued(A,B,R2) ) ) ).

% single_valued_subset
tff(fact_5519_single__valued__relcomp,axiom,
    ! [A: $tType,E: $tType,B: $tType,R2: set(product_prod(A,B)),S2: set(product_prod(B,E))] :
      ( single_valued(A,B,R2)
     => ( single_valued(B,E,S2)
       => single_valued(A,E,relcomp(A,B,E,R2,S2)) ) ) ).

% single_valued_relcomp
tff(fact_5520_antisym__empty,axiom,
    ! [A: $tType] : antisym(A,bot_bot(set(product_prod(A,A)))) ).

% antisym_empty
tff(fact_5521_single__valued__empty,axiom,
    ! [B: $tType,A: $tType] : single_valued(A,B,bot_bot(set(product_prod(A,B)))) ).

% single_valued_empty
tff(fact_5522_antisym__Id__on,axiom,
    ! [A: $tType,Aa2: set(A)] : antisym(A,id_on(A,Aa2)) ).

% antisym_Id_on
tff(fact_5523_antisym__Id,axiom,
    ! [A: $tType] : antisym(A,id2(A)) ).

% antisym_Id
tff(fact_5524_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_5525_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_5526_single__valued__Id,axiom,
    ! [A: $tType] : single_valued(A,A,id2(A)) ).

% single_valued_Id
tff(fact_5527_antisymD,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A3: A,B3: A] :
      ( antisym(A,R2)
     => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B3),R2)
       => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B3),A3),R2)
         => ( A3 = B3 ) ) ) ) ).

% antisymD
tff(fact_5528_antisymI,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( ! [X: A,Y2: A] :
          ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y2),R2)
         => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),X),R2)
           => ( X = Y2 ) ) )
     => antisym(A,R2) ) ).

% antisymI
tff(fact_5529_antisym__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( antisym(A,R2)
    <=> ! [X6: A,Y4: A] :
          ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X6),Y4),R2)
         => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),X6),R2)
           => ( X6 = Y4 ) ) ) ) ).

% antisym_def
tff(fact_5530_single__valuedD,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(A,B)),Xa: A,Y: B,Z2: B] :
      ( single_valued(A,B,R2)
     => ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa),Y),R2)
       => ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa),Z2),R2)
         => ( Y = Z2 ) ) ) ) ).

% single_valuedD
tff(fact_5531_single__valuedI,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,B))] :
      ( ! [X: A,Y2: B,Z3: B] :
          ( 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),R2)
         => ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Z3),R2)
           => ( Y2 = Z3 ) ) )
     => single_valued(A,B,R2) ) ).

% single_valuedI
tff(fact_5532_single__valued__def,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(A,B))] :
      ( single_valued(A,B,R2)
    <=> ! [X6: A,Y4: B] :
          ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X6),Y4),R2)
         => ! [Z5: B] :
              ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X6),Z5),R2)
             => ( Y4 = Z5 ) ) ) ) ).

% single_valued_def
tff(fact_5533_single__valued__Id__on,axiom,
    ! [A: $tType,Aa2: set(A)] : single_valued(A,A,id_on(A,Aa2)) ).

% single_valued_Id_on
tff(fact_5534_single__valued__confluent,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),Xa: A,Y: A,Z2: A] :
      ( single_valued(A,A,R2)
     => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Y),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R2))
       => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Z2),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R2))
         => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z2),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R2))
            | member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z2),Y),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R2)) ) ) ) ) ).

% single_valued_confluent
tff(fact_5535_in__diff__count,axiom,
    ! [A: $tType,A3: A,M5: multiset(A),N7: multiset(A)] :
      ( member(A,A3,aa(multiset(A),set(A),set_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),M5),N7)))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,aa(multiset(A),fun(A,nat),count(A),N7),A3)),aa(A,nat,aa(multiset(A),fun(A,nat),count(A),M5),A3)) ) ).

% in_diff_count
tff(fact_5536_count__ne__remove,axiom,
    ! [A: $tType,Xa: A,T2: A,S: multiset(A)] :
      ( ( Xa != T2 )
     => ( aa(A,nat,aa(multiset(A),fun(A,nat),count(A),S),Xa) = aa(A,nat,aa(multiset(A),fun(A,nat),count(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),S),add_mset(A,T2,zero_zero(multiset(A))))),Xa) ) ) ).

% count_ne_remove
tff(fact_5537_antisym__singleton,axiom,
    ! [A: $tType,Xa: 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))),insert3(product_prod(A,A)),Xa),bot_bot(set(product_prod(A,A))))) ).

% antisym_singleton
tff(fact_5538_single__valued__below__Id,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),id2(A))
     => single_valued(A,A,R) ) ).

% single_valued_below_Id
tff(fact_5539_set__mset__def,axiom,
    ! [A: $tType,M5: multiset(A)] : ( aa(multiset(A),set(A),set_mset(A),M5) = aa(fun(A,$o),set(A),collect(A),aTP_Lamp_sh(multiset(A),fun(A,$o),M5)) ) ).

% set_mset_def
tff(fact_5540_set__mset__diff,axiom,
    ! [A: $tType,M5: multiset(A),N7: multiset(A)] : ( aa(multiset(A),set(A),set_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),M5),N7)) = aa(fun(A,$o),set(A),collect(A),aa(multiset(A),fun(A,$o),aTP_Lamp_si(multiset(A),fun(multiset(A),fun(A,$o)),M5),N7)) ) ).

% set_mset_diff
tff(fact_5541_count__mset,axiom,
    ! [A: $tType,Xsa: list(A),Xa: A] : ( aa(A,nat,aa(multiset(A),fun(A,nat),count(A),aa(list(A),multiset(A),mset(A),Xsa)),Xa) = aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),aa(A,fun(A,$o),fequal(A),Xa)),Xsa)) ) ).

% count_mset
tff(fact_5542_count,axiom,
    ! [A: $tType,Xa: multiset(A)] : member(fun(A,nat),aa(multiset(A),fun(A,nat),count(A),Xa),aa(fun(fun(A,nat),$o),set(fun(A,nat)),collect(fun(A,nat)),aTP_Lamp_sj(fun(A,nat),$o))) ).

% count
tff(fact_5543_count__cases,axiom,
    ! [A: $tType,Y: fun(A,nat)] :
      ( member(fun(A,nat),Y,aa(fun(fun(A,nat),$o),set(fun(A,nat)),collect(fun(A,nat)),aTP_Lamp_sj(fun(A,nat),$o)))
     => ~ ! [X: multiset(A)] : ( Y != aa(multiset(A),fun(A,nat),count(A),X) ) ) ).

% count_cases
tff(fact_5544_count__induct,axiom,
    ! [A: $tType,Y: fun(A,nat),Pa: fun(fun(A,nat),$o)] :
      ( member(fun(A,nat),Y,aa(fun(fun(A,nat),$o),set(fun(A,nat)),collect(fun(A,nat)),aTP_Lamp_sj(fun(A,nat),$o)))
     => ( ! [X: multiset(A)] : aa(fun(A,nat),$o,Pa,aa(multiset(A),fun(A,nat),count(A),X))
       => aa(fun(A,nat),$o,Pa,Y) ) ) ).

% count_induct
tff(fact_5545_count__mset__gt__0,axiom,
    ! [A: $tType,Xa: A,Xsa: list(A)] :
      ( member(A,Xa,aa(list(A),set(A),set2(A),Xsa))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(A,nat,aa(multiset(A),fun(A,nat),count(A),aa(list(A),multiset(A),mset(A),Xsa)),Xa)) ) ).

% count_mset_gt_0
tff(fact_5546_bijective__alt,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,B))] :
      ( bijective(A,B,R)
    <=> ( single_valued(A,B,R)
        & single_valued(B,A,converse(A,B,R)) ) ) ).

% bijective_alt
tff(fact_5547_Image__Int__eq,axiom,
    ! [A: $tType,B: $tType,R: set(product_prod(B,A)),Aa2: set(B),B2: set(B)] :
      ( single_valued(A,B,converse(B,A,R))
     => ( image(B,A,R,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),Aa2),B2)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),image(B,A,R,Aa2)),image(B,A,R,B2)) ) ) ).

% Image_Int_eq
tff(fact_5548_trans__diff__Id,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( trans(A,R2)
     => ( antisym(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))),minus_minus(set(product_prod(A,A))),R2),id2(A))) ) ) ).

% trans_diff_Id
tff(fact_5549_distinct__count__atmost__1,axiom,
    ! [A: $tType,Xa: list(A)] :
      ( aa(list(A),$o,distinct(A),Xa)
    <=> ! [A7: A] :
          ( aa(A,nat,aa(multiset(A),fun(A,nat),count(A),aa(list(A),multiset(A),mset(A),Xa)),A7) = $ite(member(A,A7,aa(list(A),set(A),set2(A),Xa)),one_one(nat),zero_zero(nat)) ) ) ).

% distinct_count_atmost_1
tff(fact_5550_bdd__above__multiset__imp__finite__support,axiom,
    ! [A: $tType,Aa2: set(multiset(A))] :
      ( ( Aa2 != bot_bot(set(multiset(A))) )
     => ( condit8047198070973881523_above(multiset(A),subseteq_mset(A),Aa2)
       => aa(set(A),$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_sk(multiset(A),set(A))),Aa2))) ) ) ).

% bdd_above_multiset_imp_finite_support
tff(fact_5551_count__image__mset,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Aa2: multiset(B),Xa: A] : ( aa(A,nat,aa(multiset(A),fun(A,nat),count(A),image_mset(B,A,F2,Aa2)),Xa) = aa(set(B),nat,aa(fun(B,nat),fun(set(B),nat),groups7311177749621191930dd_sum(B,nat),aa(multiset(B),fun(B,nat),count(B),Aa2)),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),vimage(B,A,F2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A))))),aa(multiset(B),set(B),set_mset(B),Aa2))) ) ).

% count_image_mset
tff(fact_5552_count__Abs__multiset,axiom,
    ! [A: $tType,F2: fun(A,nat)] :
      ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_be(fun(A,nat),fun(A,$o),F2)))
     => ( aa(multiset(A),fun(A,nat),count(A),aa(fun(A,nat),multiset(A),abs_multiset(A),F2)) = F2 ) ) ).

% count_Abs_multiset
tff(fact_5553_subset__mset_OcSUP__union,axiom,
    ! [B: $tType,A: $tType,Aa2: set(A),F2: fun(A,multiset(B)),B2: set(A)] :
      ( ( Aa2 != bot_bot(set(A)) )
     => ( condit8047198070973881523_above(multiset(B),subseteq_mset(B),aa(set(A),set(multiset(B)),image2(A,multiset(B),F2),Aa2))
       => ( ( B2 != bot_bot(set(A)) )
         => ( condit8047198070973881523_above(multiset(B),subseteq_mset(B),aa(set(A),set(multiset(B)),image2(A,multiset(B),F2),B2))
           => ( aa(set(multiset(B)),multiset(B),complete_Sup_Sup(multiset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),B2))) = union_mset(B,aa(set(multiset(B)),multiset(B),complete_Sup_Sup(multiset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),F2),Aa2)),aa(set(multiset(B)),multiset(B),complete_Sup_Sup(multiset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),F2),B2))) ) ) ) ) ) ).

% subset_mset.cSUP_union
tff(fact_5554_mset__map__split__orig__le,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),Pa: multiset(B),M1: multiset(A),M22: multiset(A)] :
      ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),image_mset(B,A,F2,Pa)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M1),M22))
     => ~ ! [P12: multiset(B),P23: multiset(B)] :
            ( ( Pa = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),P12),P23) )
           => ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),image_mset(B,A,F2,P12)),M1)
             => ~ aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),image_mset(B,A,F2,P23)),M22) ) ) ) ).

% mset_map_split_orig_le
tff(fact_5555_mset__map__id,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),G: fun(A,B),X3: multiset(A)] :
      ( ! [X: A] : ( aa(B,A,F2,aa(A,B,G,X)) = X )
     => ( image_mset(B,A,F2,image_mset(A,B,G,X3)) = X3 ) ) ).

% mset_map_id
tff(fact_5556_mset__map__split__orig,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),Pa: multiset(B),M1: multiset(A),M22: multiset(A)] :
      ( ( image_mset(B,A,F2,Pa) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M1),M22) )
     => ~ ! [P12: multiset(B),P23: multiset(B)] :
            ( ( Pa = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),P12),P23) )
           => ( ( image_mset(B,A,F2,P12) = M1 )
             => ( image_mset(B,A,F2,P23) != M22 ) ) ) ) ).

% mset_map_split_orig
tff(fact_5557_image__mset__cong__pair,axiom,
    ! [E: $tType,B: $tType,A: $tType,M5: multiset(product_prod(A,B)),F2: fun(A,fun(B,E)),G: fun(A,fun(B,E))] :
      ( ! [X: A,Y2: B] :
          ( 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(multiset(product_prod(A,B)),set(product_prod(A,B)),set_mset(product_prod(A,B)),M5))
         => ( aa(B,E,aa(A,fun(B,E),F2,X),Y2) = aa(B,E,aa(A,fun(B,E),G,X),Y2) ) )
     => ( image_mset(product_prod(A,B),E,aa(fun(A,fun(B,E)),fun(product_prod(A,B),E),product_case_prod(A,B,E),F2),M5) = image_mset(product_prod(A,B),E,aa(fun(A,fun(B,E)),fun(product_prod(A,B),E),product_case_prod(A,B,E),G),M5) ) ) ).

% image_mset_cong_pair
tff(fact_5558_Abs__multiset__inject,axiom,
    ! [A: $tType,Xa: fun(A,nat),Y: fun(A,nat)] :
      ( member(fun(A,nat),Xa,aa(fun(fun(A,nat),$o),set(fun(A,nat)),collect(fun(A,nat)),aTP_Lamp_sj(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_sj(fun(A,nat),$o)))
       => ( ( aa(fun(A,nat),multiset(A),abs_multiset(A),Xa) = aa(fun(A,nat),multiset(A),abs_multiset(A),Y) )
        <=> ( Xa = Y ) ) ) ) ).

% Abs_multiset_inject
tff(fact_5559_Abs__multiset__induct,axiom,
    ! [A: $tType,Pa: fun(multiset(A),$o),Xa: multiset(A)] :
      ( ! [Y2: fun(A,nat)] :
          ( member(fun(A,nat),Y2,aa(fun(fun(A,nat),$o),set(fun(A,nat)),collect(fun(A,nat)),aTP_Lamp_sj(fun(A,nat),$o)))
         => aa(multiset(A),$o,Pa,aa(fun(A,nat),multiset(A),abs_multiset(A),Y2)) )
     => aa(multiset(A),$o,Pa,Xa) ) ).

% Abs_multiset_induct
tff(fact_5560_Abs__multiset__cases,axiom,
    ! [A: $tType,Xa: multiset(A)] :
      ~ ! [Y2: fun(A,nat)] :
          ( ( Xa = aa(fun(A,nat),multiset(A),abs_multiset(A),Y2) )
         => ~ member(fun(A,nat),Y2,aa(fun(fun(A,nat),$o),set(fun(A,nat)),collect(fun(A,nat)),aTP_Lamp_sj(fun(A,nat),$o))) ) ).

% Abs_multiset_cases
tff(fact_5561_subset__mset_OSUP__sup__distrib,axiom,
    ! [B: $tType,A: $tType,Aa2: set(A),F2: fun(A,multiset(B)),G: fun(A,multiset(B))] :
      ( ( Aa2 != bot_bot(set(A)) )
     => ( condit8047198070973881523_above(multiset(B),subseteq_mset(B),aa(set(A),set(multiset(B)),image2(A,multiset(B),F2),Aa2))
       => ( condit8047198070973881523_above(multiset(B),subseteq_mset(B),aa(set(A),set(multiset(B)),image2(A,multiset(B),G),Aa2))
         => ( union_mset(B,aa(set(multiset(B)),multiset(B),complete_Sup_Sup(multiset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),F2),Aa2)),aa(set(multiset(B)),multiset(B),complete_Sup_Sup(multiset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),G),Aa2))) = 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_sl(fun(A,multiset(B)),fun(fun(A,multiset(B)),fun(A,multiset(B))),F2),G)),Aa2)) ) ) ) ) ).

% subset_mset.SUP_sup_distrib
tff(fact_5562_Abs__multiset__inverse,axiom,
    ! [A: $tType,Y: fun(A,nat)] :
      ( member(fun(A,nat),Y,aa(fun(fun(A,nat),$o),set(fun(A,nat)),collect(fun(A,nat)),aTP_Lamp_sj(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_5563_subset__mset_OcSUP__insert,axiom,
    ! [B: $tType,A: $tType,Aa2: set(A),F2: fun(A,multiset(B)),A3: A] :
      ( ( Aa2 != bot_bot(set(A)) )
     => ( condit8047198070973881523_above(multiset(B),subseteq_mset(B),aa(set(A),set(multiset(B)),image2(A,multiset(B),F2),Aa2))
       => ( aa(set(multiset(B)),multiset(B),complete_Sup_Sup(multiset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),F2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),Aa2))) = union_mset(B,aa(A,multiset(B),F2,A3),aa(set(multiset(B)),multiset(B),complete_Sup_Sup(multiset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),F2),Aa2))) ) ) ) ).

% subset_mset.cSUP_insert
tff(fact_5564_sum__mset__constant,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(A)
     => ! [Y: A,Aa2: multiset(B)] : ( comm_m7189776963980413722m_mset(A,image_mset(B,A,aTP_Lamp_ey(A,fun(B,A),Y),Aa2)) = 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)),Aa2))),Y) ) ) ).

% sum_mset_constant
tff(fact_5565_takeWhile__neq__rev,axiom,
    ! [A: $tType,Xsa: list(A),Xa: A] :
      ( aa(list(A),$o,distinct(A),Xsa)
     => ( member(A,Xa,aa(list(A),set(A),set2(A),Xsa))
       => ( aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),takeWhile(A),aa(A,fun(A,$o),aTP_Lamp_cg(A,fun(A,$o)),Xa)),aa(list(A),list(A),rev(A),Xsa)) = aa(list(A),list(A),rev(A),aa(list(A),list(A),tl(A),aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),dropWhile(A),aa(A,fun(A,$o),aTP_Lamp_cg(A,fun(A,$o)),Xa)),Xsa))) ) ) ) ).

% takeWhile_neq_rev
tff(fact_5566_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_sj(fun(A,nat),$o))) ).

% type_definition_multiset
tff(fact_5567_dropWhile__idem,axiom,
    ! [A: $tType,Pa: fun(A,$o),Xsa: list(A)] : ( aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),dropWhile(A),Pa),aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),dropWhile(A),Pa),Xsa)) = aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),dropWhile(A),Pa),Xsa) ) ).

% dropWhile_idem
tff(fact_5568_dropWhile__eq__Nil__conv,axiom,
    ! [A: $tType,Pa: fun(A,$o),Xsa: list(A)] :
      ( ( aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),dropWhile(A),Pa),Xsa) = nil(A) )
    <=> ! [X6: A] :
          ( member(A,X6,aa(list(A),set(A),set2(A),Xsa))
         => aa(A,$o,Pa,X6) ) ) ).

% dropWhile_eq_Nil_conv
tff(fact_5569_dropWhile__append1,axiom,
    ! [A: $tType,Xa: A,Xsa: list(A),Pa: fun(A,$o),Ys: list(A)] :
      ( member(A,Xa,aa(list(A),set(A),set2(A),Xsa))
     => ( ~ aa(A,$o,Pa,Xa)
       => ( aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),dropWhile(A),Pa),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),Ys)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),dropWhile(A),Pa),Xsa)),Ys) ) ) ) ).

% dropWhile_append1
tff(fact_5570_dropWhile__append2,axiom,
    ! [A: $tType,Xsa: list(A),Pa: fun(A,$o),Ys: list(A)] :
      ( ! [X: A] :
          ( member(A,X,aa(list(A),set(A),set2(A),Xsa))
         => aa(A,$o,Pa,X) )
     => ( aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),dropWhile(A),Pa),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),Ys)) = aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),dropWhile(A),Pa),Ys) ) ) ).

% dropWhile_append2
tff(fact_5571_dropWhile__replicate,axiom,
    ! [A: $tType,Pa: fun(A,$o),N: nat,Xa: A] :
      ( aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),dropWhile(A),Pa),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),Xa)) = $ite(aa(A,$o,Pa,Xa),nil(A),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),Xa)) ) ).

% dropWhile_replicate
tff(fact_5572_takeWhile__dropWhile__id,axiom,
    ! [A: $tType,Pa: fun(A,$o),Xsa: list(A)] : ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),takeWhile(A),Pa),Xsa)),aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),dropWhile(A),Pa),Xsa)) = Xsa ) ).

% takeWhile_dropWhile_id
tff(fact_5573_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_5574_dropWhile_Osimps_I1_J,axiom,
    ! [A: $tType,Pa: fun(A,$o)] : ( aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),dropWhile(A),Pa),nil(A)) = nil(A) ) ).

% dropWhile.simps(1)
tff(fact_5575_distinct__dropWhile,axiom,
    ! [A: $tType,Xsa: list(A),Pa: fun(A,$o)] :
      ( aa(list(A),$o,distinct(A),Xsa)
     => aa(list(A),$o,distinct(A),aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),dropWhile(A),Pa),Xsa)) ) ).

% distinct_dropWhile
tff(fact_5576_set__dropWhileD,axiom,
    ! [A: $tType,Xa: A,Pa: fun(A,$o),Xsa: list(A)] :
      ( member(A,Xa,aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),dropWhile(A),Pa),Xsa)))
     => member(A,Xa,aa(list(A),set(A),set2(A),Xsa)) ) ).

% set_dropWhileD
tff(fact_5577_dropWhile__cong,axiom,
    ! [A: $tType,L: list(A),K3: list(A),Pa: fun(A,$o),Qa: fun(A,$o)] :
      ( ( L = K3 )
     => ( ! [X: A] :
            ( member(A,X,aa(list(A),set(A),set2(A),L))
           => ( aa(A,$o,Pa,X)
            <=> aa(A,$o,Qa,X) ) )
       => ( aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),dropWhile(A),Pa),L) = aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),dropWhile(A),Qa),K3) ) ) ) ).

% dropWhile_cong
tff(fact_5578_dropWhile_Osimps_I2_J,axiom,
    ! [A: $tType,Pa: fun(A,$o),Xa: A,Xsa: list(A)] :
      ( aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),dropWhile(A),Pa),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa)) = $ite(aa(A,$o,Pa,Xa),aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),dropWhile(A),Pa),Xsa),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa)) ) ).

% dropWhile.simps(2)
tff(fact_5579_dropWhile__append3,axiom,
    ! [A: $tType,Pa: fun(A,$o),Y: A,Xsa: list(A),Ys: list(A)] :
      ( ~ aa(A,$o,Pa,Y)
     => ( aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),dropWhile(A),Pa),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),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),aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),dropWhile(A),Pa),Xsa)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)) ) ) ).

% dropWhile_append3
tff(fact_5580_length__dropWhile__le,axiom,
    ! [A: $tType,Pa: fun(A,$o),Xsa: list(A)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),dropWhile(A),Pa),Xsa))),aa(list(A),nat,size_size(list(A)),Xsa)) ).

% length_dropWhile_le
tff(fact_5581_sorted__dropWhile,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xsa: list(A),Pa: fun(A,$o)] :
          ( sorted_wrt(A,ord_less_eq(A),Xsa)
         => sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),dropWhile(A),Pa),Xsa)) ) ) ).

% sorted_dropWhile
tff(fact_5582_dropWhile__eq__self__iff,axiom,
    ! [A: $tType,Pa: fun(A,$o),Xsa: list(A)] :
      ( ( aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),dropWhile(A),Pa),Xsa) = Xsa )
    <=> ( ( Xsa = nil(A) )
        | ~ aa(A,$o,Pa,aa(list(A),A,hd(A),Xsa)) ) ) ).

% dropWhile_eq_self_iff
tff(fact_5583_hd__dropWhile,axiom,
    ! [A: $tType,Pa: fun(A,$o),Xsa: list(A)] :
      ( ( aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),dropWhile(A),Pa),Xsa) != nil(A) )
     => ~ aa(A,$o,Pa,aa(list(A),A,hd(A),aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),dropWhile(A),Pa),Xsa))) ) ).

% hd_dropWhile
tff(fact_5584_dropWhile__last,axiom,
    ! [A: $tType,Xa: A,Xsa: list(A),Pa: fun(A,$o)] :
      ( member(A,Xa,aa(list(A),set(A),set2(A),Xsa))
     => ( ~ aa(A,$o,Pa,Xa)
       => ( last(A,aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),dropWhile(A),Pa),Xsa)) = last(A,Xsa) ) ) ) ).

% dropWhile_last
tff(fact_5585_dropWhile__map,axiom,
    ! [A: $tType,B: $tType,Pa: fun(A,$o),F2: fun(B,A),Xsa: list(B)] : ( aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),dropWhile(A),Pa),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Xsa)) = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),aa(list(B),list(B),aa(fun(B,$o),fun(list(B),list(B)),dropWhile(B),comp(A,$o,B,Pa,F2)),Xsa)) ) ).

% dropWhile_map
tff(fact_5586_remdups__adj__Cons_H,axiom,
    ! [A: $tType,Xa: A,Xsa: list(A)] : ( aa(list(A),list(A),remdups_adj(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),aa(list(A),list(A),remdups_adj(A),aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),dropWhile(A),aTP_Lamp_ad(A,fun(A,$o),Xa)),Xsa))) ) ).

% remdups_adj_Cons'
tff(fact_5587_sum__mset__product,axiom,
    ! [E: $tType,A: $tType,B: $tType] :
      ( ( comm_monoid_add(B)
        & times(B)
        & semiring_0(A) )
     => ! [F2: fun(B,A),Aa2: multiset(B),G: fun(E,A),B2: multiset(E)] : ( aa(A,A,aa(A,fun(A,A),times_times(A),comm_m7189776963980413722m_mset(A,image_mset(B,A,F2,Aa2))),comm_m7189776963980413722m_mset(A,image_mset(E,A,G,B2))) = comm_m7189776963980413722m_mset(A,image_mset(B,A,aa(multiset(E),fun(B,A),aa(fun(E,A),fun(multiset(E),fun(B,A)),aTP_Lamp_sn(fun(B,A),fun(fun(E,A),fun(multiset(E),fun(B,A))),F2),G),B2),Aa2)) ) ) ).

% sum_mset_product
tff(fact_5588_sum__mset__distrib__right,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_0(A)
     => ! [F2: fun(B,A),M5: multiset(B),C2: A] : ( aa(A,A,aa(A,fun(A,A),times_times(A),comm_m7189776963980413722m_mset(A,image_mset(B,A,F2,M5))),C2) = comm_m7189776963980413722m_mset(A,image_mset(B,A,aa(A,fun(B,A),aTP_Lamp_fe(fun(B,A),fun(A,fun(B,A)),F2),C2),M5)) ) ) ).

% sum_mset_distrib_right
tff(fact_5589_sum__mset__distrib__left,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_0(A)
     => ! [C2: A,F2: fun(B,A),M5: multiset(B)] : ( aa(A,A,aa(A,fun(A,A),times_times(A),C2),comm_m7189776963980413722m_mset(A,image_mset(B,A,F2,M5))) = comm_m7189776963980413722m_mset(A,image_mset(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_fd(A,fun(fun(B,A),fun(B,A)),C2),F2),M5)) ) ) ).

% sum_mset_distrib_left
tff(fact_5590_dropWhile__eq__Cons__conv,axiom,
    ! [A: $tType,Pa: fun(A,$o),Xsa: list(A),Y: A,Ys: list(A)] :
      ( ( aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),dropWhile(A),Pa),Xsa) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys) )
    <=> ( ( Xsa = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),takeWhile(A),Pa),Xsa)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)) )
        & ~ aa(A,$o,Pa,Y) ) ) ).

% dropWhile_eq_Cons_conv
tff(fact_5591_sum__mset__diff,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [N7: multiset(A),M5: multiset(A)] :
          ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),N7),M5)
         => ( comm_m7189776963980413722m_mset(A,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),M5),N7)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),comm_m7189776963980413722m_mset(A,M5)),comm_m7189776963980413722m_mset(A,N7)) ) ) ) ).

% sum_mset_diff
tff(fact_5592_takeWhile__eq__filter,axiom,
    ! [A: $tType,Pa: fun(A,$o),Xsa: list(A)] :
      ( ! [X: A] :
          ( member(A,X,aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),dropWhile(A),Pa),Xsa)))
         => ~ aa(A,$o,Pa,X) )
     => ( aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),takeWhile(A),Pa),Xsa) = aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),Pa),Xsa) ) ) ).

% takeWhile_eq_filter
tff(fact_5593_dropWhile__eq__drop,axiom,
    ! [A: $tType,Pa: fun(A,$o),Xsa: list(A)] : ( aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),dropWhile(A),Pa),Xsa) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),takeWhile(A),Pa),Xsa))),Xsa) ) ).

% dropWhile_eq_drop
tff(fact_5594_dropWhile__append,axiom,
    ! [A: $tType,Pa: fun(A,$o),Xsa: list(A),Ys: list(A)] :
      ( aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),dropWhile(A),Pa),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),Ys)) = $ite(
          ! [X6: A] :
            ( member(A,X6,aa(list(A),set(A),set2(A),Xsa))
           => aa(A,$o,Pa,X6) ),
          aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),dropWhile(A),Pa),Ys),
          aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),dropWhile(A),Pa),Xsa)),Ys) ) ) ).

% dropWhile_append
tff(fact_5595_length__dropWhile__takeWhile,axiom,
    ! [A: $tType,Xa: nat,Pa: fun(A,$o),Xsa: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),dropWhile(A),Pa),Xsa)))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Xa),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),takeWhile(A),Pa),Xsa)))),aa(list(A),nat,size_size(list(A)),Xsa)) ) ).

% length_dropWhile_takeWhile
tff(fact_5596_remdups__adj__append__dropWhile,axiom,
    ! [A: $tType,Xsa: list(A),Y: A,Ys: list(A)] : ( aa(list(A),list(A),remdups_adj(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys))) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),remdups_adj(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),nil(A))))),aa(list(A),list(A),remdups_adj(A),aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),dropWhile(A),aTP_Lamp_ad(A,fun(A,$o),Y)),Ys))) ) ).

% remdups_adj_append_dropWhile
tff(fact_5597_remdups__adj__append_H_H,axiom,
    ! [A: $tType,Xsa: list(A),Ys: list(A)] :
      ( ( Xsa != nil(A) )
     => ( aa(list(A),list(A),remdups_adj(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),Ys)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),remdups_adj(A),Xsa)),aa(list(A),list(A),remdups_adj(A),aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),dropWhile(A),aTP_Lamp_so(list(A),fun(A,$o),Xsa)),Ys))) ) ) ).

% remdups_adj_append''
tff(fact_5598_tl__remdups__adj,axiom,
    ! [A: $tType,Ys: list(A)] :
      ( ( Ys != nil(A) )
     => ( aa(list(A),list(A),tl(A),aa(list(A),list(A),remdups_adj(A),Ys)) = aa(list(A),list(A),remdups_adj(A),aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),dropWhile(A),aTP_Lamp_sp(list(A),fun(A,$o),Ys)),aa(list(A),list(A),tl(A),Ys))) ) ) ).

% tl_remdups_adj
tff(fact_5599_dropWhile__nth,axiom,
    ! [A: $tType,J: nat,Pa: fun(A,$o),Xsa: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),dropWhile(A),Pa),Xsa)))
     => ( aa(nat,A,nth(A,aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),dropWhile(A),Pa),Xsa)),J) = aa(nat,A,nth(A,Xsa),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),takeWhile(A),Pa),Xsa)))) ) ) ).

% dropWhile_nth
tff(fact_5600_dropWhile__neq__rev,axiom,
    ! [A: $tType,Xsa: list(A),Xa: A] :
      ( aa(list(A),$o,distinct(A),Xsa)
     => ( member(A,Xa,aa(list(A),set(A),set2(A),Xsa))
       => ( aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),dropWhile(A),aa(A,fun(A,$o),aTP_Lamp_cg(A,fun(A,$o)),Xa)),aa(list(A),list(A),rev(A),Xsa)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),aa(list(A),list(A),rev(A),aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),takeWhile(A),aa(A,fun(A,$o),aTP_Lamp_cg(A,fun(A,$o)),Xa)),Xsa))) ) ) ) ).

% dropWhile_neq_rev
tff(fact_5601_sum__mset__delta_H,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_1(A)
     => ! [Y: B,C2: A,Aa2: multiset(B)] : ( comm_m7189776963980413722m_mset(A,image_mset(B,A,aa(A,fun(B,A),aTP_Lamp_sq(B,fun(A,fun(B,A)),Y),C2),Aa2)) = 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),Aa2),Y))) ) ) ).

% sum_mset_delta'
tff(fact_5602_sum__mset__delta,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_1(A)
     => ! [Y: B,C2: A,Aa2: multiset(B)] : ( comm_m7189776963980413722m_mset(A,image_mset(B,A,aa(A,fun(B,A),aTP_Lamp_sr(B,fun(A,fun(B,A)),Y),C2),Aa2)) = 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),Aa2),Y))) ) ) ).

% sum_mset_delta
tff(fact_5603_minus__multiset_Oabs__eq,axiom,
    ! [A: $tType,Xa: fun(A,nat),Xb: fun(A,nat)] :
      ( aa(fun(A,nat),$o,aa(fun(A,nat),fun(fun(A,nat),$o),bNF_eq_onp(fun(A,nat),aTP_Lamp_sj(fun(A,nat),$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_sj(fun(A,nat),$o)),Xb),Xb)
       => ( 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),Xb)) = 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_ss(fun(A,nat),fun(fun(A,nat),fun(A,nat))),Xa),Xb)) ) ) ) ).

% minus_multiset.abs_eq
tff(fact_5604_relation__of__def,axiom,
    ! [A: $tType,Pa: fun(A,fun(A,$o)),Aa2: set(A)] : ( order_relation_of(A,Pa,Aa2) = 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,$o)),aTP_Lamp_st(fun(A,fun(A,$o)),fun(set(A),fun(A,fun(A,$o))),Pa),Aa2))) ) ).

% relation_of_def
tff(fact_5605_stable__sort__key__def,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Sk: fun(fun(A,B),fun(list(A),list(A)))] :
          ( linord3483353639454293061rt_key(A,B,Sk)
        <=> ! [F9: fun(A,B),Xs2: list(A),K4: B] : ( aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),aa(B,fun(A,$o),aTP_Lamp_jo(fun(A,B),fun(B,fun(A,$o)),F9),K4)),aa(list(A),list(A),aa(fun(A,B),fun(list(A),list(A)),Sk,F9),Xs2)) = aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),aa(B,fun(A,$o),aTP_Lamp_jo(fun(A,B),fun(B,fun(A,$o)),F9),K4)),Xs2) ) ) ) ).

% stable_sort_key_def
tff(fact_5606_zero__multiset_Orsp,axiom,
    ! [A: $tType] : aa(fun(A,nat),$o,aa(fun(A,nat),fun(fun(A,nat),$o),bNF_eq_onp(fun(A,nat),aTP_Lamp_sj(fun(A,nat),$o)),aTP_Lamp_su(A,nat)),aTP_Lamp_su(A,nat)) ).

% zero_multiset.rsp
tff(fact_5607_stable__sort__key__sort__key,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => linord3483353639454293061rt_key(A,B,linorder_sort_key(A,B)) ) ).

% stable_sort_key_sort_key
tff(fact_5608_add__mset_Oabs__eq,axiom,
    ! [A: $tType,Xa: fun(A,nat),Xaa: A] :
      ( aa(fun(A,nat),$o,aa(fun(A,nat),fun(fun(A,nat),$o),bNF_eq_onp(fun(A,nat),aTP_Lamp_sj(fun(A,nat),$o)),Xa),Xa)
     => ( add_mset(A,Xaa,aa(fun(A,nat),multiset(A),abs_multiset(A),Xa)) = aa(fun(A,nat),multiset(A),abs_multiset(A),aa(A,fun(A,nat),aTP_Lamp_sv(fun(A,nat),fun(A,fun(A,nat)),Xa),Xaa)) ) ) ).

% add_mset.abs_eq
tff(fact_5609_plus__multiset_Oabs__eq,axiom,
    ! [A: $tType,Xa: fun(A,nat),Xb: fun(A,nat)] :
      ( aa(fun(A,nat),$o,aa(fun(A,nat),fun(fun(A,nat),$o),bNF_eq_onp(fun(A,nat),aTP_Lamp_sj(fun(A,nat),$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_sj(fun(A,nat),$o)),Xb),Xb)
       => ( 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),Xb)) = 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_sw(fun(A,nat),fun(fun(A,nat),fun(A,nat))),Xa),Xb)) ) ) ) ).

% plus_multiset.abs_eq
tff(fact_5610_repeat__mset_Oabs__eq,axiom,
    ! [A: $tType,Xa: fun(A,nat),Xaa: nat] :
      ( aa(fun(A,nat),$o,aa(fun(A,nat),fun(fun(A,nat),$o),bNF_eq_onp(fun(A,nat),aTP_Lamp_sj(fun(A,nat),$o)),Xa),Xa)
     => ( repeat_mset(A,Xaa,aa(fun(A,nat),multiset(A),abs_multiset(A),Xa)) = aa(fun(A,nat),multiset(A),abs_multiset(A),aa(nat,fun(A,nat),aTP_Lamp_sx(fun(A,nat),fun(nat,fun(A,nat)),Xa),Xaa)) ) ) ).

% repeat_mset.abs_eq
tff(fact_5611_Inf__multiset_Oabs__eq,axiom,
    ! [A: $tType,Xa: set(fun(A,nat))] :
      ( aa(set(fun(A,nat)),$o,aa(set(fun(A,nat)),fun(set(fun(A,nat)),$o),bNF_rel_set(fun(A,nat),fun(A,nat),bNF_eq_onp(fun(A,nat),aTP_Lamp_sj(fun(A,nat),$o))),Xa),Xa)
     => ( 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)),Xa)) = aa(fun(A,nat),multiset(A),abs_multiset(A),aa(set(fun(A,nat)),fun(A,nat),aTP_Lamp_sz(set(fun(A,nat)),fun(A,nat)),Xa)) ) ) ).

% Inf_multiset.abs_eq
tff(fact_5612_filter__mset_Oabs__eq,axiom,
    ! [A: $tType,Xa: fun(A,nat),Xaa: fun(A,$o)] :
      ( aa(fun(A,nat),$o,aa(fun(A,nat),fun(fun(A,nat),$o),bNF_eq_onp(fun(A,nat),aTP_Lamp_sj(fun(A,nat),$o)),Xa),Xa)
     => ( filter_mset(A,Xaa,aa(fun(A,nat),multiset(A),abs_multiset(A),Xa)) = aa(fun(A,nat),multiset(A),abs_multiset(A),aa(fun(A,$o),fun(A,nat),aTP_Lamp_ta(fun(A,nat),fun(fun(A,$o),fun(A,nat)),Xa),Xaa)) ) ) ).

% filter_mset.abs_eq
tff(fact_5613_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)),$o),bNF_rel_fun(set(fun(A,nat)),set(fun(A,nat)),fun(A,nat),fun(A,nat),bNF_rel_set(fun(A,nat),fun(A,nat),bNF_eq_onp(fun(A,nat),aTP_Lamp_sj(fun(A,nat),$o))),bNF_eq_onp(fun(A,nat),aTP_Lamp_sj(fun(A,nat),$o))),aTP_Lamp_sz(set(fun(A,nat)),fun(A,nat))),aTP_Lamp_sz(set(fun(A,nat)),fun(A,nat))) ).

% Inf_multiset.rsp
tff(fact_5614_empty__transfer,axiom,
    ! [A: $tType,B: $tType,Aa2: fun(A,fun(B,$o))] : aa(set(B),$o,aa(set(A),fun(set(B),$o),bNF_rel_set(A,B,Aa2),bot_bot(set(A))),bot_bot(set(B))) ).

% empty_transfer
tff(fact_5615_shuffles_Opelims,axiom,
    ! [A: $tType,Xa: list(A),Xaa: list(A),Y: set(list(A))] :
      ( ( aa(list(A),set(list(A)),aa(list(A),fun(list(A),set(list(A))),shuffles(A),Xa),Xaa) = 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)),Xa),Xaa))
       => ( ( ( Xa = nil(A) )
           => ( ( Y = aa(set(list(A)),set(list(A)),aa(list(A),fun(set(list(A)),set(list(A))),insert3(list(A)),Xaa),bot_bot(set(list(A)))) )
             => ~ accp(product_prod(list(A),list(A)),shuffles_rel(A),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),Xaa)) ) )
         => ( ( ( Xaa = nil(A) )
             => ( ( Y = aa(set(list(A)),set(list(A)),aa(list(A),fun(set(list(A)),set(list(A))),insert3(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)),Xa),nil(A))) ) )
           => ~ ! [X: A,Xs: list(A)] :
                  ( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) )
                 => ! [Y2: A,Ys5: list(A)] :
                      ( ( Xaa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys5) )
                     => ( ( 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),X)),aa(list(A),set(list(A)),aa(list(A),fun(list(A),set(list(A))),shuffles(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys5)))),aa(set(list(A)),set(list(A)),image2(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2)),aa(list(A),set(list(A)),aa(list(A),fun(list(A),set(list(A))),shuffles(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),Ys5))) )
                       => ~ 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),Y2),Ys5))) ) ) ) ) ) ) ) ).

% shuffles.pelims
tff(fact_5616_shuffles_Opinduct,axiom,
    ! [A: $tType,A0: list(A),A1: list(A),Pa: fun(list(A),fun(list(A),$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))
     => ( ! [Ys5: 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)),Ys5))
           => aa(list(A),$o,aa(list(A),fun(list(A),$o),Pa,nil(A)),Ys5) )
       => ( ! [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)))
             => aa(list(A),$o,aa(list(A),fun(list(A),$o),Pa,Xs),nil(A)) )
         => ( ! [X: A,Xs: list(A),Y2: A,Ys5: 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),Y2),Ys5)))
               => ( aa(list(A),$o,aa(list(A),fun(list(A),$o),Pa,Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys5))
                 => ( aa(list(A),$o,aa(list(A),fun(list(A),$o),Pa,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),Ys5)
                   => aa(list(A),$o,aa(list(A),fun(list(A),$o),Pa,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),Y2),Ys5)) ) ) )
           => aa(list(A),$o,aa(list(A),fun(list(A),$o),Pa,A0),A1) ) ) ) ) ).

% shuffles.pinduct
tff(fact_5617_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_5618_prod__list__transfer,axiom,
    ! [A: $tType,B: $tType] :
      ( ( monoid_mult(B)
        & monoid_mult(A) )
     => ! [Aa2: fun(A,fun(B,$o))] :
          ( aa(B,$o,aa(A,fun(B,$o),Aa2,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),Aa2,bNF_rel_fun(A,B,A,B,Aa2,Aa2)),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,aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Aa2),Aa2),groups5270119922927024881d_list(A)),groups5270119922927024881d_list(B)) ) ) ) ).

% prod_list_transfer
tff(fact_5619_foldr__transfer,axiom,
    ! [A: $tType,B: $tType,F4: $tType,E: $tType,Aa2: fun(A,fun(E,$o)),B2: fun(B,fun(F4,$o))] : aa(fun(fun(E,fun(F4,F4)),fun(list(E),fun(F4,F4))),$o,aa(fun(fun(A,fun(B,B)),fun(list(A),fun(B,B))),fun(fun(fun(E,fun(F4,F4)),fun(list(E),fun(F4,F4))),$o),bNF_rel_fun(fun(A,fun(B,B)),fun(E,fun(F4,F4)),fun(list(A),fun(B,B)),fun(list(E),fun(F4,F4)),bNF_rel_fun(A,E,fun(B,B),fun(F4,F4),Aa2,bNF_rel_fun(B,F4,B,F4,B2,B2)),bNF_rel_fun(list(A),list(E),fun(B,B),fun(F4,F4),aa(fun(A,fun(E,$o)),fun(list(A),fun(list(E),$o)),list_all2(A,E),Aa2),bNF_rel_fun(B,F4,B,F4,B2,B2))),foldr(A,B)),foldr(E,F4)) ).

% foldr_transfer
tff(fact_5620_foldl__transfer,axiom,
    ! [A: $tType,B: $tType,F4: $tType,E: $tType,B2: fun(A,fun(E,$o)),Aa2: fun(B,fun(F4,$o))] : aa(fun(fun(E,fun(F4,E)),fun(E,fun(list(F4),E))),$o,aa(fun(fun(A,fun(B,A)),fun(A,fun(list(B),A))),fun(fun(fun(E,fun(F4,E)),fun(E,fun(list(F4),E))),$o),bNF_rel_fun(fun(A,fun(B,A)),fun(E,fun(F4,E)),fun(A,fun(list(B),A)),fun(E,fun(list(F4),E)),bNF_rel_fun(A,E,fun(B,A),fun(F4,E),B2,bNF_rel_fun(B,F4,A,E,Aa2,B2)),bNF_rel_fun(A,E,fun(list(B),A),fun(list(F4),E),B2,bNF_rel_fun(list(B),list(F4),A,E,aa(fun(B,fun(F4,$o)),fun(list(B),fun(list(F4),$o)),list_all2(B,F4),Aa2),B2))),foldl(A,B)),foldl(E,F4)) ).

% foldl_transfer
tff(fact_5621_horner__sum__transfer,axiom,
    ! [E: $tType,A: $tType,B: $tType,F4: $tType] :
      ( ( comm_semiring_0(B)
        & comm_semiring_0(A) )
     => ! [Aa2: fun(A,fun(B,$o)),B2: fun(E,fun(F4,$o))] :
          ( aa(B,$o,aa(A,fun(B,$o),Aa2,zero_zero(A)),zero_zero(B))
         => ( aa(fun(B,fun(B,B)),$o,aa(fun(A,fun(A,A)),fun(fun(B,fun(B,B)),$o),bNF_rel_fun(A,B,fun(A,A),fun(B,B),Aa2,bNF_rel_fun(A,B,A,B,Aa2,Aa2)),plus_plus(A)),plus_plus(B))
           => ( aa(fun(B,fun(B,B)),$o,aa(fun(A,fun(A,A)),fun(fun(B,fun(B,B)),$o),bNF_rel_fun(A,B,fun(A,A),fun(B,B),Aa2,bNF_rel_fun(A,B,A,B,Aa2,Aa2)),times_times(A)),times_times(B))
             => aa(fun(fun(F4,B),fun(B,fun(list(F4),B))),$o,aa(fun(fun(E,A),fun(A,fun(list(E),A))),fun(fun(fun(F4,B),fun(B,fun(list(F4),B))),$o),bNF_rel_fun(fun(E,A),fun(F4,B),fun(A,fun(list(E),A)),fun(B,fun(list(F4),B)),bNF_rel_fun(E,F4,A,B,B2,Aa2),bNF_rel_fun(A,B,fun(list(E),A),fun(list(F4),B),Aa2,bNF_rel_fun(list(E),list(F4),A,B,aa(fun(E,fun(F4,$o)),fun(list(E),fun(list(F4),$o)),list_all2(E,F4),B2),Aa2))),groups4207007520872428315er_sum(E,A)),groups4207007520872428315er_sum(F4,B)) ) ) ) ) ).

% horner_sum_transfer
tff(fact_5622_list_Omap__transfer,axiom,
    ! [A: $tType,B: $tType,F4: $tType,E: $tType,Rb2: fun(A,fun(E,$o)),Sd: fun(B,fun(F4,$o))] : aa(fun(fun(E,F4),fun(list(E),list(F4))),$o,aa(fun(fun(A,B),fun(list(A),list(B))),fun(fun(fun(E,F4),fun(list(E),list(F4))),$o),bNF_rel_fun(fun(A,B),fun(E,F4),fun(list(A),list(B)),fun(list(E),list(F4)),bNF_rel_fun(A,E,B,F4,Rb2,Sd),bNF_rel_fun(list(A),list(E),list(B),list(F4),aa(fun(A,fun(E,$o)),fun(list(A),fun(list(E),$o)),list_all2(A,E),Rb2),aa(fun(B,fun(F4,$o)),fun(list(B),fun(list(F4),$o)),list_all2(B,F4),Sd))),map(A,B)),map(E,F4)) ).

% list.map_transfer
tff(fact_5623_List_Ofold__transfer,axiom,
    ! [A: $tType,B: $tType,F4: $tType,E: $tType,Aa2: fun(A,fun(E,$o)),B2: fun(B,fun(F4,$o))] : aa(fun(fun(E,fun(F4,F4)),fun(list(E),fun(F4,F4))),$o,aa(fun(fun(A,fun(B,B)),fun(list(A),fun(B,B))),fun(fun(fun(E,fun(F4,F4)),fun(list(E),fun(F4,F4))),$o),bNF_rel_fun(fun(A,fun(B,B)),fun(E,fun(F4,F4)),fun(list(A),fun(B,B)),fun(list(E),fun(F4,F4)),bNF_rel_fun(A,E,fun(B,B),fun(F4,F4),Aa2,bNF_rel_fun(B,F4,B,F4,B2,B2)),bNF_rel_fun(list(A),list(E),fun(B,B),fun(F4,F4),aa(fun(A,fun(E,$o)),fun(list(A),fun(list(E),$o)),list_all2(A,E),Aa2),bNF_rel_fun(B,F4,B,F4,B2,B2))),fold(A,B)),fold(E,F4)) ).

% List.fold_transfer
tff(fact_5624_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))
     => ( aa(list(A),set(list(A)),aa(list(A),fun(list(A),set(list(A))),shuffles(A),nil(A)),Ys) = aa(set(list(A)),set(list(A)),aa(list(A),fun(set(list(A)),set(list(A))),insert3(list(A)),Ys),bot_bot(set(list(A)))) ) ) ).

% shuffles.psimps(1)
tff(fact_5625_shuffles_Opsimps_I2_J,axiom,
    ! [A: $tType,Xsa: 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)),Xsa),nil(A)))
     => ( aa(list(A),set(list(A)),aa(list(A),fun(list(A),set(list(A))),shuffles(A),Xsa),nil(A)) = aa(set(list(A)),set(list(A)),aa(list(A),fun(set(list(A)),set(list(A))),insert3(list(A)),Xsa),bot_bot(set(list(A)))) ) ) ).

% shuffles.psimps(2)
tff(fact_5626_shuffles_Opsimps_I3_J,axiom,
    ! [A: $tType,Xa: A,Xsa: 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),Xa),Xsa)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)))
     => ( aa(list(A),set(list(A)),aa(list(A),fun(list(A),set(list(A))),shuffles(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa)),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),Xa)),aa(list(A),set(list(A)),aa(list(A),fun(list(A),set(list(A))),shuffles(A),Xsa),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)),aa(list(A),set(list(A)),aa(list(A),fun(list(A),set(list(A))),shuffles(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa)),Ys))) ) ) ).

% shuffles.psimps(3)
tff(fact_5627_revg_Opelims,axiom,
    ! [A: $tType,Xa: list(A),Xaa: list(A),Y: list(A)] :
      ( ( revg(A,Xa,Xaa) = 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)),Xa),Xaa))
       => ( ( ( Xa = nil(A) )
           => ( ( Y = Xaa )
             => ~ 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)),Xaa)) ) )
         => ~ ! [A4: A,As: list(A)] :
                ( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A4),As) )
               => ( ( Y = revg(A,As,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A4),Xaa)) )
                 => ~ 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),A4),As)),Xaa)) ) ) ) ) ) ).

% revg.pelims
tff(fact_5628_merge_Opelims,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: list(A),Xaa: list(A),Y: list(A)] :
          ( ( merge(A,Xa,Xaa) = 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)),Xa),Xaa))
           => ( ( ( Xa = nil(A) )
               => ( ( Y = Xaa )
                 => ~ 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)),Xaa)) ) )
             => ( ! [V3: A,Va: list(A)] :
                    ( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V3),Va) )
                   => ( ( Xaa = nil(A) )
                     => ( ( Y = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V3),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),V3),Va)),nil(A))) ) ) )
               => ~ ! [X12: A,L1: list(A)] :
                      ( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X12),L1) )
                     => ! [X23: A,L22: list(A)] :
                          ( ( Xaa = 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_5629_merge__list_Opelims,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: list(list(A)),Xaa: list(list(A)),Y: list(A)] :
          ( ( merge_list(A,Xa,Xaa) = 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))),Xa),Xaa))
           => ( ( ( Xa = nil(list(A)) )
               => ( ( Xaa = 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)))) ) ) )
             => ( ( ( Xa = nil(list(A)) )
                 => ! [L3: list(A)] :
                      ( ( Xaa = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L3),nil(list(A))) )
                     => ( ( Y = L3 )
                       => ~ 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)),L3),nil(list(A))))) ) ) )
               => ( ! [La: list(A),Acc2: list(list(A))] :
                      ( ( Xa = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc2) )
                     => ( ( Xaa = 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),Acc2)) )
                         => ~ 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),Acc2)),nil(list(A)))) ) ) )
                 => ( ! [La: list(A),Acc2: list(list(A))] :
                        ( ( Xa = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc2) )
                       => ! [L3: list(A)] :
                            ( ( Xaa = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L3),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)),L3),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc2))) )
                             => ~ 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),Acc2)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L3),nil(list(A))))) ) ) )
                   => ~ ! [L1: list(A),L22: list(A),Ls: list(list(A))] :
                          ( ( Xaa = 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)),Xa),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))),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)))) ) ) ) ) ) ) ) ) ) ).

% merge_list.pelims
tff(fact_5630_shuffles__transfer,axiom,
    ! [A: $tType,B: $tType,Aa2: fun(A,fun(B,$o))] : aa(fun(list(B),fun(list(B),set(list(B)))),$o,aa(fun(list(A),fun(list(A),set(list(A)))),fun(fun(list(B),fun(list(B),set(list(B)))),$o),bNF_rel_fun(list(A),list(B),fun(list(A),set(list(A))),fun(list(B),set(list(B))),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Aa2),bNF_rel_fun(list(A),list(B),set(list(A)),set(list(B)),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Aa2),bNF_rel_set(list(A),list(B),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Aa2)))),shuffles(A)),shuffles(B)) ).

% shuffles_transfer
tff(fact_5631_set__Cons__transfer,axiom,
    ! [A: $tType,B: $tType,Aa2: fun(A,fun(B,$o))] : aa(fun(set(B),fun(set(list(B)),set(list(B)))),$o,aa(fun(set(A),fun(set(list(A)),set(list(A)))),fun(fun(set(B),fun(set(list(B)),set(list(B)))),$o),bNF_rel_fun(set(A),set(B),fun(set(list(A)),set(list(A))),fun(set(list(B)),set(list(B))),bNF_rel_set(A,B,Aa2),bNF_rel_fun(set(list(A)),set(list(B)),set(list(A)),set(list(B)),bNF_rel_set(list(A),list(B),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Aa2)),bNF_rel_set(list(A),list(B),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Aa2)))),set_Cons(A)),set_Cons(B)) ).

% set_Cons_transfer
tff(fact_5632_nths__transfer,axiom,
    ! [A: $tType,B: $tType,Aa2: fun(A,fun(B,$o))] : aa(fun(list(B),fun(set(nat),list(B))),$o,aa(fun(list(A),fun(set(nat),list(A))),fun(fun(list(B),fun(set(nat),list(B))),$o),bNF_rel_fun(list(A),list(B),fun(set(nat),list(A)),fun(set(nat),list(B)),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Aa2),bNF_rel_fun(set(nat),set(nat),list(A),list(B),bNF_rel_set(nat,nat,fequal(nat)),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Aa2))),nths(A)),nths(B)) ).

% nths_transfer
tff(fact_5633_Lifting__Set_Oinsert__transfer,axiom,
    ! [A: $tType,B: $tType,Aa2: fun(A,fun(B,$o))] : aa(fun(B,fun(set(B),set(B))),$o,aa(fun(A,fun(set(A),set(A))),fun(fun(B,fun(set(B),set(B))),$o),bNF_rel_fun(A,B,fun(set(A),set(A)),fun(set(B),set(B)),Aa2,bNF_rel_fun(set(A),set(B),set(A),set(B),bNF_rel_set(A,B,Aa2),bNF_rel_set(A,B,Aa2))),insert3(A)),insert3(B)) ).

% Lifting_Set.insert_transfer
tff(fact_5634_dropWhile__transfer,axiom,
    ! [A: $tType,B: $tType,Aa2: fun(A,fun(B,$o))] : aa(fun(fun(B,$o),fun(list(B),list(B))),$o,aa(fun(fun(A,$o),fun(list(A),list(A))),fun(fun(fun(B,$o),fun(list(B),list(B))),$o),bNF_rel_fun(fun(A,$o),fun(B,$o),fun(list(A),list(A)),fun(list(B),list(B)),bNF_rel_fun(A,B,$o,$o,Aa2,fequal($o)),bNF_rel_fun(list(A),list(B),list(A),list(B),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Aa2),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Aa2))),dropWhile(A)),dropWhile(B)) ).

% dropWhile_transfer
tff(fact_5635_tl__transfer,axiom,
    ! [A: $tType,B: $tType,Aa2: fun(A,fun(B,$o))] : aa(fun(list(B),list(B)),$o,aa(fun(list(A),list(A)),fun(fun(list(B),list(B)),$o),bNF_rel_fun(list(A),list(B),list(A),list(B),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Aa2),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Aa2)),tl(A)),tl(B)) ).

% tl_transfer
tff(fact_5636_less__natural_Orsp,axiom,
    aa(fun(nat,fun(nat,$o)),$o,aa(fun(nat,fun(nat,$o)),fun(fun(nat,fun(nat,$o)),$o),bNF_rel_fun(nat,nat,fun(nat,$o),fun(nat,$o),fequal(nat),bNF_rel_fun(nat,nat,$o,$o,fequal(nat),fequal($o))),ord_less(nat)),ord_less(nat)) ).

% less_natural.rsp
tff(fact_5637_rev__transfer,axiom,
    ! [A: $tType,B: $tType,Aa2: fun(A,fun(B,$o))] : aa(fun(list(B),list(B)),$o,aa(fun(list(A),list(A)),fun(fun(list(B),list(B)),$o),bNF_rel_fun(list(A),list(B),list(A),list(B),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Aa2),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Aa2)),rev(A)),rev(B)) ).

% rev_transfer
tff(fact_5638_list_Octr__transfer_I2_J,axiom,
    ! [A: $tType,B: $tType,R: fun(A,fun(B,$o))] : aa(fun(B,fun(list(B),list(B))),$o,aa(fun(A,fun(list(A),list(A))),fun(fun(B,fun(list(B),list(B))),$o),bNF_rel_fun(A,B,fun(list(A),list(A)),fun(list(B),list(B)),R,bNF_rel_fun(list(A),list(B),list(A),list(B),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),R),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),R))),cons(A)),cons(B)) ).

% list.ctr_transfer(2)
tff(fact_5639_List_Ofilter__transfer,axiom,
    ! [A: $tType,B: $tType,Aa2: fun(A,fun(B,$o))] : aa(fun(fun(B,$o),fun(list(B),list(B))),$o,aa(fun(fun(A,$o),fun(list(A),list(A))),fun(fun(fun(B,$o),fun(list(B),list(B))),$o),bNF_rel_fun(fun(A,$o),fun(B,$o),fun(list(A),list(A)),fun(list(B),list(B)),bNF_rel_fun(A,B,$o,$o,Aa2,fequal($o)),bNF_rel_fun(list(A),list(B),list(A),list(B),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Aa2),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Aa2))),filter2(A)),filter2(B)) ).

% List.filter_transfer
tff(fact_5640_takeWhile__transfer,axiom,
    ! [A: $tType,B: $tType,Aa2: fun(A,fun(B,$o))] : aa(fun(fun(B,$o),fun(list(B),list(B))),$o,aa(fun(fun(A,$o),fun(list(A),list(A))),fun(fun(fun(B,$o),fun(list(B),list(B))),$o),bNF_rel_fun(fun(A,$o),fun(B,$o),fun(list(A),list(A)),fun(list(B),list(B)),bNF_rel_fun(A,B,$o,$o,Aa2,fequal($o)),bNF_rel_fun(list(A),list(B),list(A),list(B),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Aa2),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Aa2))),takeWhile(A)),takeWhile(B)) ).

% takeWhile_transfer
tff(fact_5641_replicate__transfer,axiom,
    ! [A: $tType,B: $tType,Aa2: fun(A,fun(B,$o))] : aa(fun(nat,fun(B,list(B))),$o,aa(fun(nat,fun(A,list(A))),fun(fun(nat,fun(B,list(B))),$o),bNF_rel_fun(nat,nat,fun(A,list(A)),fun(B,list(B)),fequal(nat),bNF_rel_fun(A,B,list(A),list(B),Aa2,aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Aa2))),replicate(A)),replicate(B)) ).

% replicate_transfer
tff(fact_5642_list_Ocase__transfer,axiom,
    ! [A: $tType,E: $tType,F4: $tType,B: $tType,S: fun(A,fun(B,$o)),R: fun(E,fun(F4,$o))] : aa(fun(B,fun(fun(F4,fun(list(F4),B)),fun(list(F4),B))),$o,aa(fun(A,fun(fun(E,fun(list(E),A)),fun(list(E),A))),fun(fun(B,fun(fun(F4,fun(list(F4),B)),fun(list(F4),B))),$o),bNF_rel_fun(A,B,fun(fun(E,fun(list(E),A)),fun(list(E),A)),fun(fun(F4,fun(list(F4),B)),fun(list(F4),B)),S,bNF_rel_fun(fun(E,fun(list(E),A)),fun(F4,fun(list(F4),B)),fun(list(E),A),fun(list(F4),B),bNF_rel_fun(E,F4,fun(list(E),A),fun(list(F4),B),R,bNF_rel_fun(list(E),list(F4),A,B,aa(fun(E,fun(F4,$o)),fun(list(E),fun(list(F4),$o)),list_all2(E,F4),R),S)),bNF_rel_fun(list(E),list(F4),A,B,aa(fun(E,fun(F4,$o)),fun(list(E),fun(list(F4),$o)),list_all2(E,F4),R),S))),case_list(A,E)),case_list(B,F4)) ).

% list.case_transfer
tff(fact_5643_append__transfer,axiom,
    ! [A: $tType,B: $tType,Aa2: fun(A,fun(B,$o))] : aa(fun(list(B),fun(list(B),list(B))),$o,aa(fun(list(A),fun(list(A),list(A))),fun(fun(list(B),fun(list(B),list(B))),$o),bNF_rel_fun(list(A),list(B),fun(list(A),list(A)),fun(list(B),list(B)),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Aa2),bNF_rel_fun(list(A),list(B),list(A),list(B),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Aa2),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Aa2))),append(A)),append(B)) ).

% append_transfer
tff(fact_5644_rotate__transfer,axiom,
    ! [A: $tType,B: $tType,Aa2: fun(A,fun(B,$o))] : aa(fun(nat,fun(list(B),list(B))),$o,aa(fun(nat,fun(list(A),list(A))),fun(fun(nat,fun(list(B),list(B))),$o),bNF_rel_fun(nat,nat,fun(list(A),list(A)),fun(list(B),list(B)),fequal(nat),bNF_rel_fun(list(A),list(B),list(A),list(B),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Aa2),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Aa2))),rotate(A)),rotate(B)) ).

% rotate_transfer
tff(fact_5645_list__update__transfer,axiom,
    ! [A: $tType,B: $tType,Aa2: fun(A,fun(B,$o))] : aa(fun(list(B),fun(nat,fun(B,list(B)))),$o,aa(fun(list(A),fun(nat,fun(A,list(A)))),fun(fun(list(B),fun(nat,fun(B,list(B)))),$o),bNF_rel_fun(list(A),list(B),fun(nat,fun(A,list(A))),fun(nat,fun(B,list(B))),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Aa2),bNF_rel_fun(nat,nat,fun(A,list(A)),fun(B,list(B)),fequal(nat),bNF_rel_fun(A,B,list(A),list(B),Aa2,aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Aa2)))),list_update(A)),list_update(B)) ).

% list_update_transfer
tff(fact_5646_list__ex__transfer,axiom,
    ! [A: $tType,B: $tType,Aa2: fun(A,fun(B,$o))] : aa(fun(fun(B,$o),fun(list(B),$o)),$o,aa(fun(fun(A,$o),fun(list(A),$o)),fun(fun(fun(B,$o),fun(list(B),$o)),$o),bNF_rel_fun(fun(A,$o),fun(B,$o),fun(list(A),$o),fun(list(B),$o),bNF_rel_fun(A,B,$o,$o,Aa2,fequal($o)),bNF_rel_fun(list(A),list(B),$o,$o,aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Aa2),fequal($o))),list_ex(A)),list_ex(B)) ).

% list_ex_transfer
tff(fact_5647_list_Orel__transfer,axiom,
    ! [A: $tType,B: $tType,F4: $tType,E: $tType,Sa: fun(A,fun(E,$o)),Sc: fun(B,fun(F4,$o))] : aa(fun(fun(E,fun(F4,$o)),fun(list(E),fun(list(F4),$o))),$o,aa(fun(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o))),fun(fun(fun(E,fun(F4,$o)),fun(list(E),fun(list(F4),$o))),$o),bNF_rel_fun(fun(A,fun(B,$o)),fun(E,fun(F4,$o)),fun(list(A),fun(list(B),$o)),fun(list(E),fun(list(F4),$o)),bNF_rel_fun(A,E,fun(B,$o),fun(F4,$o),Sa,bNF_rel_fun(B,F4,$o,$o,Sc,fequal($o))),bNF_rel_fun(list(A),list(E),fun(list(B),$o),fun(list(F4),$o),aa(fun(A,fun(E,$o)),fun(list(A),fun(list(E),$o)),list_all2(A,E),Sa),bNF_rel_fun(list(B),list(F4),$o,$o,aa(fun(B,fun(F4,$o)),fun(list(B),fun(list(F4),$o)),list_all2(B,F4),Sc),fequal($o)))),list_all2(A,B)),list_all2(E,F4)) ).

% list.rel_transfer
tff(fact_5648_length__transfer,axiom,
    ! [A: $tType,B: $tType,Aa2: fun(A,fun(B,$o))] : aa(fun(list(B),nat),$o,aa(fun(list(A),nat),fun(fun(list(B),nat),$o),bNF_rel_fun(list(A),list(B),nat,nat,aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Aa2),fequal(nat)),size_size(list(A))),size_size(list(B))) ).

% length_transfer
tff(fact_5649_take__transfer,axiom,
    ! [A: $tType,B: $tType,Aa2: fun(A,fun(B,$o))] : aa(fun(nat,fun(list(B),list(B))),$o,aa(fun(nat,fun(list(A),list(A))),fun(fun(nat,fun(list(B),list(B))),$o),bNF_rel_fun(nat,nat,fun(list(A),list(A)),fun(list(B),list(B)),fequal(nat),bNF_rel_fun(list(A),list(B),list(A),list(B),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Aa2),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Aa2))),take(A)),take(B)) ).

% take_transfer
tff(fact_5650_drop__transfer,axiom,
    ! [A: $tType,B: $tType,Aa2: fun(A,fun(B,$o))] : aa(fun(nat,fun(list(B),list(B))),$o,aa(fun(nat,fun(list(A),list(A))),fun(fun(nat,fun(list(B),list(B))),$o),bNF_rel_fun(nat,nat,fun(list(A),list(A)),fun(list(B),list(B)),fequal(nat),bNF_rel_fun(list(A),list(B),list(A),list(B),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Aa2),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Aa2))),drop(A)),drop(B)) ).

% drop_transfer
tff(fact_5651_concat__transfer,axiom,
    ! [A: $tType,B: $tType,Aa2: fun(A,fun(B,$o))] : aa(fun(list(list(B)),list(B)),$o,aa(fun(list(list(A)),list(A)),fun(fun(list(list(B)),list(B)),$o),bNF_rel_fun(list(list(A)),list(list(B)),list(A),list(B),aa(fun(list(A),fun(list(B),$o)),fun(list(list(A)),fun(list(list(B)),$o)),list_all2(list(A),list(B)),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Aa2)),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Aa2)),concat(A)),concat(B)) ).

% concat_transfer
tff(fact_5652_butlast__transfer,axiom,
    ! [A: $tType,B: $tType,Aa2: fun(A,fun(B,$o))] : aa(fun(list(B),list(B)),$o,aa(fun(list(A),list(A)),fun(fun(list(B),list(B)),$o),bNF_rel_fun(list(A),list(B),list(A),list(B),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Aa2),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Aa2)),butlast(A)),butlast(B)) ).

% butlast_transfer
tff(fact_5653_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,$o)),bNF_rel_fun(fun(A,nat),fun(A,nat),fun(A,nat),fun(A,nat),bNF_eq_onp(fun(A,nat),aTP_Lamp_sj(fun(A,nat),$o)),bNF_eq_onp(fun(A,nat),aTP_Lamp_sj(fun(A,nat),$o)))),aTP_Lamp_tb(fun(A,$o),fun(fun(A,nat),fun(A,nat)))),aTP_Lamp_tb(fun(A,$o),fun(fun(A,nat),fun(A,nat)))) ).

% filter_mset.rsp
tff(fact_5654_rotate1__transfer,axiom,
    ! [A: $tType,B: $tType,Aa2: fun(A,fun(B,$o))] : aa(fun(list(B),list(B)),$o,aa(fun(list(A),list(A)),fun(fun(list(B),list(B)),$o),bNF_rel_fun(list(A),list(B),list(A),list(B),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Aa2),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Aa2)),rotate1(A)),rotate1(B)) ).

% rotate1_transfer
tff(fact_5655_subseqs__transfer,axiom,
    ! [A: $tType,B: $tType,Aa2: fun(A,fun(B,$o))] : aa(fun(list(B),list(list(B))),$o,aa(fun(list(A),list(list(A))),fun(fun(list(B),list(list(B))),$o),bNF_rel_fun(list(A),list(B),list(list(A)),list(list(B)),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Aa2),aa(fun(list(A),fun(list(B),$o)),fun(list(list(A)),fun(list(list(B)),$o)),list_all2(list(A),list(B)),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Aa2))),subseqs(A)),subseqs(B)) ).

% subseqs_transfer
tff(fact_5656_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,$o,aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),R),fequal($o)),aTP_Lamp_tc(list(A),$o)),aTP_Lamp_td(list(B),$o)) ).

% list.disc_transfer(1)
tff(fact_5657_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,$o,aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),R),fequal($o)),aTP_Lamp_jg(list(A),$o)),aTP_Lamp_nv(list(B),$o)) ).

% list.disc_transfer(2)
tff(fact_5658_product__lists__transfer,axiom,
    ! [A: $tType,B: $tType,Aa2: fun(A,fun(B,$o))] : aa(fun(list(list(B)),list(list(B))),$o,aa(fun(list(list(A)),list(list(A))),fun(fun(list(list(B)),list(list(B))),$o),bNF_rel_fun(list(list(A)),list(list(B)),list(list(A)),list(list(B)),aa(fun(list(A),fun(list(B),$o)),fun(list(list(A)),fun(list(list(B)),$o)),list_all2(list(A),list(B)),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Aa2)),aa(fun(list(A),fun(list(B),$o)),fun(list(list(A)),fun(list(list(B)),$o)),list_all2(list(A),list(B)),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Aa2))),product_lists(A)),product_lists(B)) ).

% product_lists_transfer
tff(fact_5659_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))),$o),bNF_rel_fun(A,A,fun(fun(A,nat),fun(A,nat)),fun(fun(A,nat),fun(A,nat)),fequal(A),bNF_rel_fun(fun(A,nat),fun(A,nat),fun(A,nat),fun(A,nat),bNF_eq_onp(fun(A,nat),aTP_Lamp_sj(fun(A,nat),$o)),bNF_eq_onp(fun(A,nat),aTP_Lamp_sj(fun(A,nat),$o)))),aTP_Lamp_te(A,fun(fun(A,nat),fun(A,nat)))),aTP_Lamp_te(A,fun(fun(A,nat),fun(A,nat)))) ).

% add_mset.rsp
tff(fact_5660_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))),$o),bNF_rel_fun(fun(A,nat),fun(A,nat),fun(fun(A,nat),fun(A,nat)),fun(fun(A,nat),fun(A,nat)),bNF_eq_onp(fun(A,nat),aTP_Lamp_sj(fun(A,nat),$o)),bNF_rel_fun(fun(A,nat),fun(A,nat),fun(A,nat),fun(A,nat),bNF_eq_onp(fun(A,nat),aTP_Lamp_sj(fun(A,nat),$o)),bNF_eq_onp(fun(A,nat),aTP_Lamp_sj(fun(A,nat),$o)))),aTP_Lamp_sw(fun(A,nat),fun(fun(A,nat),fun(A,nat)))),aTP_Lamp_sw(fun(A,nat),fun(fun(A,nat),fun(A,nat)))) ).

% plus_multiset.rsp
tff(fact_5661_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))),$o),bNF_rel_fun(fun(A,nat),fun(A,nat),fun(fun(A,nat),fun(A,nat)),fun(fun(A,nat),fun(A,nat)),bNF_eq_onp(fun(A,nat),aTP_Lamp_sj(fun(A,nat),$o)),bNF_rel_fun(fun(A,nat),fun(A,nat),fun(A,nat),fun(A,nat),bNF_eq_onp(fun(A,nat),aTP_Lamp_sj(fun(A,nat),$o)),bNF_eq_onp(fun(A,nat),aTP_Lamp_sj(fun(A,nat),$o)))),aTP_Lamp_ss(fun(A,nat),fun(fun(A,nat),fun(A,nat)))),aTP_Lamp_ss(fun(A,nat),fun(fun(A,nat),fun(A,nat)))) ).

% minus_multiset.rsp
tff(fact_5662_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))),$o),bNF_rel_fun(nat,nat,fun(fun(A,nat),fun(A,nat)),fun(fun(A,nat),fun(A,nat)),fequal(nat),bNF_rel_fun(fun(A,nat),fun(A,nat),fun(A,nat),fun(A,nat),bNF_eq_onp(fun(A,nat),aTP_Lamp_sj(fun(A,nat),$o)),bNF_eq_onp(fun(A,nat),aTP_Lamp_sj(fun(A,nat),$o)))),aTP_Lamp_tf(nat,fun(fun(A,nat),fun(A,nat)))),aTP_Lamp_tf(nat,fun(fun(A,nat),fun(A,nat)))) ).

% repeat_mset.rsp
tff(fact_5663_null__transfer,axiom,
    ! [A: $tType,B: $tType,Aa2: fun(A,fun(B,$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,$o,aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Aa2),fequal($o)),null(A)),null(B)) ).

% null_transfer
tff(fact_5664_list_Oset__transfer,axiom,
    ! [A: $tType,B: $tType,R: fun(A,fun(B,$o))] : aa(fun(list(B),set(B)),$o,aa(fun(list(A),set(A)),fun(fun(list(B),set(B)),$o),bNF_rel_fun(list(A),list(B),set(A),set(B),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),R),bNF_rel_set(A,B,R)),set2(A)),set2(B)) ).

% list.set_transfer
tff(fact_5665_lists__transfer,axiom,
    ! [A: $tType,B: $tType,Aa2: fun(A,fun(B,$o))] : aa(fun(set(B),set(list(B))),$o,aa(fun(set(A),set(list(A))),fun(fun(set(B),set(list(B))),$o),bNF_rel_fun(set(A),set(B),set(list(A)),set(list(B)),bNF_rel_set(A,B,Aa2),bNF_rel_set(list(A),list(B),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Aa2))),lists(A)),lists(B)) ).

% lists_transfer
tff(fact_5666_listset__transfer,axiom,
    ! [A: $tType,B: $tType,Aa2: fun(A,fun(B,$o))] : aa(fun(list(set(B)),set(list(B))),$o,aa(fun(list(set(A)),set(list(A))),fun(fun(list(set(B)),set(list(B))),$o),bNF_rel_fun(list(set(A)),list(set(B)),set(list(A)),set(list(B)),aa(fun(set(A),fun(set(B),$o)),fun(list(set(A)),fun(list(set(B)),$o)),list_all2(set(A),set(B)),bNF_rel_set(A,B,Aa2)),bNF_rel_set(list(A),list(B),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Aa2))),listset(A)),listset(B)) ).

% listset_transfer
tff(fact_5667_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_5668_merge__list_Opsimps_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [La2: list(A),Acc22: 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),Acc22)),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),Acc22),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),Acc22))) ) ) ) ).

% merge_list.psimps(4)
tff(fact_5669_merge__list_Opsimps_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [La2: 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)),La2),Acc22)),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),Acc22),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),Acc22)) ) ) ) ).

% merge_list.psimps(3)
tff(fact_5670_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_5671_merge__list_Opinduct,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A0: list(list(A)),A1: list(list(A)),Pa: fun(list(list(A)),fun(list(list(A)),$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),Pa,nil(list(A))),nil(list(A))) )
           => ( ! [L3: 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)),L3),nil(list(A)))))
                 => aa(list(list(A)),$o,aa(list(list(A)),fun(list(list(A)),$o),Pa,nil(list(A))),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L3),nil(list(A)))) )
             => ( ! [La: list(A),Acc2: 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),Acc2)),nil(list(A))))
                   => ( aa(list(list(A)),$o,aa(list(list(A)),fun(list(list(A)),$o),Pa,nil(list(A))),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc2))
                     => aa(list(list(A)),$o,aa(list(list(A)),fun(list(list(A)),$o),Pa,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc2)),nil(list(A))) ) )
               => ( ! [La: list(A),Acc2: list(list(A)),L3: 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),Acc2)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L3),nil(list(A)))))
                     => ( aa(list(list(A)),$o,aa(list(list(A)),fun(list(list(A)),$o),Pa,nil(list(A))),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L3),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc2)))
                       => aa(list(list(A)),$o,aa(list(list(A)),fun(list(list(A)),$o),Pa,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc2)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L3),nil(list(A)))) ) )
                 => ( ! [Acc2: 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))),Acc2),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),Pa,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),merge(A,L1,L22)),Acc2)),Ls)
                         => aa(list(list(A)),$o,aa(list(list(A)),fun(list(list(A)),$o),Pa,Acc2),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),Pa,A0),A1) ) ) ) ) ) ) ) ).

% merge_list.pinduct
tff(fact_5672_merge__list_Opsimps_I5_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Acc22: list(list(A)),L12: list(A),L23: list(A),Ls3: 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)),L12),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L23),Ls3))))
         => ( merge_list(A,Acc22,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),Ls3))) = 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)),Acc22),Ls3) ) ) ) ).

% merge_list.psimps(5)
tff(fact_5673_list_Orec__transfer,axiom,
    ! [A: $tType,E: $tType,F4: $tType,B: $tType,S: fun(A,fun(B,$o)),R: fun(E,fun(F4,$o))] : aa(fun(B,fun(fun(F4,fun(list(F4),fun(B,B))),fun(list(F4),B))),$o,aa(fun(A,fun(fun(E,fun(list(E),fun(A,A))),fun(list(E),A))),fun(fun(B,fun(fun(F4,fun(list(F4),fun(B,B))),fun(list(F4),B))),$o),bNF_rel_fun(A,B,fun(fun(E,fun(list(E),fun(A,A))),fun(list(E),A)),fun(fun(F4,fun(list(F4),fun(B,B))),fun(list(F4),B)),S,bNF_rel_fun(fun(E,fun(list(E),fun(A,A))),fun(F4,fun(list(F4),fun(B,B))),fun(list(E),A),fun(list(F4),B),bNF_rel_fun(E,F4,fun(list(E),fun(A,A)),fun(list(F4),fun(B,B)),R,bNF_rel_fun(list(E),list(F4),fun(A,A),fun(B,B),aa(fun(E,fun(F4,$o)),fun(list(E),fun(list(F4),$o)),list_all2(E,F4),R),bNF_rel_fun(A,B,A,B,S,S))),bNF_rel_fun(list(E),list(F4),A,B,aa(fun(E,fun(F4,$o)),fun(list(E),fun(list(F4),$o)),list_all2(E,F4),R),S))),rec_list(A,E)),rec_list(B,F4)) ).

% list.rec_transfer
tff(fact_5674_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_mv(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))),times_times(int)) ).

% times_int.transfer
tff(fact_5675_splice_Opinduct,axiom,
    ! [A: $tType,A0: list(A),A1: list(A),Pa: fun(list(A),fun(list(A),$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))
     => ( ! [Ys5: 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)),Ys5))
           => aa(list(A),$o,aa(list(A),fun(list(A),$o),Pa,nil(A)),Ys5) )
       => ( ! [X: A,Xs: list(A),Ys5: 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)),Ys5))
             => ( aa(list(A),$o,aa(list(A),fun(list(A),$o),Pa,Ys5),Xs)
               => aa(list(A),$o,aa(list(A),fun(list(A),$o),Pa,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),Ys5) ) )
         => aa(list(A),$o,aa(list(A),fun(list(A),$o),Pa,A0),A1) ) ) ) ).

% splice.pinduct
tff(fact_5676_list_Osimps_I6_J,axiom,
    ! [B: $tType,A: $tType,F1: A,F23: fun(B,fun(list(B),fun(A,A)))] : ( aa(list(B),A,aa(fun(B,fun(list(B),fun(A,A))),fun(list(B),A),aa(A,fun(fun(B,fun(list(B),fun(A,A))),fun(list(B),A)),rec_list(A,B),F1),F23),nil(B)) = F1 ) ).

% list.simps(6)
tff(fact_5677_list_Osimps_I7_J,axiom,
    ! [A: $tType,B: $tType,F1: A,F23: fun(B,fun(list(B),fun(A,A))),X21: B,X222: list(B)] : ( aa(list(B),A,aa(fun(B,fun(list(B),fun(A,A))),fun(list(B),A),aa(A,fun(fun(B,fun(list(B),fun(A,A))),fun(list(B),A)),rec_list(A,B),F1),F23),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X21),X222)) = aa(A,A,aa(list(B),fun(A,A),aa(B,fun(list(B),fun(A,A)),F23,X21),X222),aa(list(B),A,aa(fun(B,fun(list(B),fun(A,A))),fun(list(B),A),aa(A,fun(fun(B,fun(list(B),fun(A,A))),fun(list(B),A)),rec_list(A,B),F1),F23),X222)) ) ).

% list.simps(7)
tff(fact_5678_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_5679_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_tg(nat,product_prod(nat,nat))),semiring_1_of_nat(int)) ).

% int_transfer
tff(fact_5680_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_mx(nat,fun(nat,product_prod(nat,nat))))),uminus_uminus(int)) ).

% uminus_int.transfer
tff(fact_5681_list_Orec__o__map,axiom,
    ! [B: $tType,E: $tType,A: $tType,G: B,Ga: fun(E,fun(list(E),fun(B,B))),F2: fun(A,E)] : ( comp(list(E),B,list(A),aa(fun(E,fun(list(E),fun(B,B))),fun(list(E),B),aa(B,fun(fun(E,fun(list(E),fun(B,B))),fun(list(E),B)),rec_list(B,E),G),Ga),aa(fun(A,E),fun(list(A),list(E)),map(A,E),F2)) = aa(fun(A,fun(list(A),fun(B,B))),fun(list(A),B),aa(B,fun(fun(A,fun(list(A),fun(B,B))),fun(list(A),B)),rec_list(B,A),G),aa(fun(A,E),fun(A,fun(list(A),fun(B,B))),aTP_Lamp_th(fun(E,fun(list(E),fun(B,B))),fun(fun(A,E),fun(A,fun(list(A),fun(B,B)))),Ga),F2)) ) ).

% list.rec_o_map
tff(fact_5682_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_5683_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_my(nat,fun(nat,A)))),ring_1_of_int(A)) ) ).

% of_int.transfer
tff(fact_5684_set__rec,axiom,
    ! [A: $tType,Xsa: list(A)] : ( aa(list(A),set(A),set2(A),Xsa) = aa(list(A),set(A),aa(fun(A,fun(list(A),fun(set(A),set(A)))),fun(list(A),set(A)),aa(set(A),fun(fun(A,fun(list(A),fun(set(A),set(A)))),fun(list(A),set(A))),rec_list(set(A),A),bot_bot(set(A))),aTP_Lamp_ti(A,fun(list(A),fun(set(A),set(A))))),Xsa) ) ).

% set_rec
tff(fact_5685_map__rec,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xsa: list(B)] : ( aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Xsa) = aa(list(B),list(A),aa(fun(B,fun(list(B),fun(list(A),list(A)))),fun(list(B),list(A)),aa(list(A),fun(fun(B,fun(list(B),fun(list(A),list(A)))),fun(list(B),list(A))),rec_list(list(A),B),nil(A)),aTP_Lamp_tj(fun(B,A),fun(B,fun(list(B),fun(list(A),list(A)))),F2)),Xsa) ) ).

% map_rec
tff(fact_5686_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),$o)),aTP_Lamp_na(nat,fun(nat,fun(product_prod(nat,nat),$o))))),ord_less(int)) ).

% less_int.transfer
tff(fact_5687_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_nc(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))),plus_plus(int)) ).

% plus_int.transfer
tff(fact_5688_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_ne(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))),minus_minus(int)) ).

% minus_int.transfer
tff(fact_5689_splice_Opelims,axiom,
    ! [A: $tType,Xa: list(A),Xaa: list(A),Y: list(A)] :
      ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),splice(A),Xa),Xaa) = 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)),Xa),Xaa))
       => ( ( ( Xa = nil(A) )
           => ( ( Y = Xaa )
             => ~ accp(product_prod(list(A),list(A)),splice_rel(A),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),Xaa)) ) )
         => ~ ! [X: A,Xs: list(A)] :
                ( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) )
               => ( ( Y = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),splice(A),Xaa),Xs)) )
                 => ~ 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)),Xaa)) ) ) ) ) ) ).

% splice.pelims
tff(fact_5690_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))
     => ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),splice(A),nil(A)),Ys) = Ys ) ) ).

% splice.psimps(1)
tff(fact_5691_splice_Opsimps_I2_J,axiom,
    ! [A: $tType,Xa: A,Xsa: 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),Xa),Xsa)),Ys))
     => ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),splice(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa)),Ys) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),splice(A),Ys),Xsa)) ) ) ).

% splice.psimps(2)
tff(fact_5692_split__Nil__iff,axiom,
    ! [A: $tType,Xsa: list(A),Ys: list(A)] :
      ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),splice(A),Xsa),Ys) = nil(A) )
    <=> ( ( Xsa = nil(A) )
        & ( Ys = nil(A) ) ) ) ).

% split_Nil_iff
tff(fact_5693_splice__Nil2,axiom,
    ! [A: $tType,Xsa: list(A)] : ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),splice(A),Xsa),nil(A)) = Xsa ) ).

% splice_Nil2
tff(fact_5694_splice__in__shuffles,axiom,
    ! [A: $tType,Xsa: list(A),Ys: list(A)] : member(list(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),splice(A),Xsa),Ys),aa(list(A),set(list(A)),aa(list(A),fun(list(A),set(list(A))),shuffles(A),Xsa),Ys)) ).

% splice_in_shuffles
tff(fact_5695_length__splice,axiom,
    ! [A: $tType,Xsa: list(A),Ys: list(A)] : ( aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),splice(A),Xsa),Ys)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),Xsa)),aa(list(A),nat,size_size(list(A)),Ys)) ) ).

% length_splice
tff(fact_5696_splice__replicate,axiom,
    ! [A: $tType,M2: nat,Xa: A,N: nat] : ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),splice(A),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),M2),Xa)),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),Xa)) = aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),N)),Xa) ) ).

% splice_replicate
tff(fact_5697_splice_Osimps_I1_J,axiom,
    ! [A: $tType,Ys: list(A)] : ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),splice(A),nil(A)),Ys) = Ys ) ).

% splice.simps(1)
tff(fact_5698_splice_Osimps_I2_J,axiom,
    ! [A: $tType,Xa: A,Xsa: list(A),Ys: list(A)] : ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),splice(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa)),Ys) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),splice(A),Ys),Xsa)) ) ).

% splice.simps(2)
tff(fact_5699_splice_Oelims,axiom,
    ! [A: $tType,Xa: list(A),Xaa: list(A),Y: list(A)] :
      ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),splice(A),Xa),Xaa) = Y )
     => ( ( ( Xa = nil(A) )
         => ( Y != Xaa ) )
       => ~ ! [X: A,Xs: list(A)] :
              ( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) )
             => ( Y != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),splice(A),Xaa),Xs)) ) ) ) ) ).

% splice.elims
tff(fact_5700_splice__transfer,axiom,
    ! [A: $tType,B: $tType,Aa2: fun(A,fun(B,$o))] : aa(fun(list(B),fun(list(B),list(B))),$o,aa(fun(list(A),fun(list(A),list(A))),fun(fun(list(B),fun(list(B),list(B))),$o),bNF_rel_fun(list(A),list(B),fun(list(A),list(A)),fun(list(B),list(B)),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Aa2),bNF_rel_fun(list(A),list(B),list(A),list(B),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Aa2),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Aa2))),splice(A)),splice(B)) ).

% splice_transfer
tff(fact_5701_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_mv(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_mv(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))) ).

% times_int.rsp
tff(fact_5702_funpow__inj__finite,axiom,
    ! [A: $tType,P2: fun(A,A),Xa: A] :
      ( inj_on(A,A,P2,top_top(set(A)))
     => ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(A,fun(A,$o),aTP_Lamp_tk(fun(A,A),fun(A,fun(A,$o)),P2),Xa)))
       => ~ ! [N5: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N5)
             => ( aa(A,A,compow(fun(A,A),N5,P2),Xa) != Xa ) ) ) ) ).

% funpow_inj_finite
tff(fact_5703_filterlim__base__iff,axiom,
    ! [A: $tType,B: $tType,E: $tType,F4: $tType,Ia2: set(A),Fa: fun(A,set(B)),F2: fun(B,E),G4: fun(F4,set(E)),J4: set(F4)] :
      ( ( Ia2 != bot_bot(set(A)) )
     => ( ! [I3: A] :
            ( member(A,I3,Ia2)
           => ! [J3: A] :
                ( member(A,J3,Ia2)
               => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(A,set(B),Fa,I3)),aa(A,set(B),Fa,J3))
                  | aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(A,set(B),Fa,J3)),aa(A,set(B),Fa,I3)) ) ) )
       => ( filterlim(B,E,F2,aa(set(filter(E)),filter(E),complete_Inf_Inf(filter(E)),aa(set(F4),set(filter(E)),image2(F4,filter(E),aTP_Lamp_tl(fun(F4,set(E)),fun(F4,filter(E)),G4)),J4)),aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image2(A,filter(B),aTP_Lamp_nl(fun(A,set(B)),fun(A,filter(B)),Fa)),Ia2)))
        <=> ! [X6: F4] :
              ( member(F4,X6,J4)
             => ? [Xa2: A] :
                  ( member(A,Xa2,Ia2)
                  & ! [Xb5: B] :
                      ( member(B,Xb5,aa(A,set(B),Fa,Xa2))
                     => member(E,aa(B,E,F2,Xb5),aa(F4,set(E),G4,X6)) ) ) ) ) ) ) ).

% filterlim_base_iff
tff(fact_5704_inj__on__rev,axiom,
    ! [A: $tType,Aa2: set(list(A))] : inj_on(list(A),list(A),rev(A),Aa2) ).

% inj_on_rev
tff(fact_5705_inj__on__empty,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B)] : inj_on(A,B,F2,bot_bot(set(A))) ).

% inj_on_empty
tff(fact_5706_inj__map__eq__map,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),Xsa: list(A),Ys: list(A)] :
      ( inj_on(A,B,F2,top_top(set(A)))
     => ( ( aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),Xsa) = aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),Ys) )
      <=> ( Xsa = Ys ) ) ) ).

% inj_map_eq_map
tff(fact_5707_intrel__iff,axiom,
    ! [Xa: nat,Y: nat,U: nat,V2: 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),Xa),Y)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),U),V2))
    <=> ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Xa),V2) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),U),Y) ) ) ).

% intrel_iff
tff(fact_5708_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_5709_inj__mapI,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B)] :
      ( inj_on(A,B,F2,top_top(set(A)))
     => inj_on(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),top_top(set(list(A)))) ) ).

% inj_mapI
tff(fact_5710_inj__map,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B)] :
      ( inj_on(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),top_top(set(list(A))))
    <=> inj_on(A,B,F2,top_top(set(A))) ) ).

% inj_map
tff(fact_5711_inj__on__insert,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A3: A,Aa2: set(A)] :
      ( inj_on(A,B,F2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),Aa2))
    <=> ( inj_on(A,B,F2,Aa2)
        & ~ member(B,aa(A,B,F2,A3),aa(set(A),set(B),image2(A,B,F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),bot_bot(set(A)))))) ) ) ).

% inj_on_insert
tff(fact_5712_inj__on__mapI,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),Aa2: set(list(A))] :
      ( inj_on(A,B,F2,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(list(A)),set(set(A)),image2(list(A),set(A),set2(A)),Aa2)))
     => inj_on(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),Aa2) ) ).

% inj_on_mapI
tff(fact_5713_inj__img__insertE,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),Aa2: set(A),Xa: B,B2: set(B)] :
      ( inj_on(A,B,F2,Aa2)
     => ( ~ member(B,Xa,B2)
       => ( ( aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert3(B),Xa),B2) = aa(set(A),set(B),image2(A,B,F2),Aa2) )
         => ~ ! [X2: A,A10: set(A)] :
                ( ~ member(A,X2,A10)
               => ( ( Aa2 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),X2),A10) )
                 => ( ( Xa = aa(A,B,F2,X2) )
                   => ( B2 != aa(set(A),set(B),image2(A,B,F2),A10) ) ) ) ) ) ) ) ).

% inj_img_insertE
tff(fact_5714_inj__on__strict__subset,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),B2: set(A),Aa2: set(A)] :
      ( inj_on(A,B,F2,B2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),Aa2),B2)
       => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less(set(B)),aa(set(A),set(B),image2(A,B,F2),Aa2)),aa(set(A),set(B),image2(A,B,F2),B2)) ) ) ).

% inj_on_strict_subset
tff(fact_5715_map__injective,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xsa: list(B),Ys: list(B)] :
      ( ( aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Xsa) = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Ys) )
     => ( inj_on(B,A,F2,top_top(set(B)))
       => ( Xsa = Ys ) ) ) ).

% map_injective
tff(fact_5716_linorder__injI,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(A)
     => ! [F2: fun(A,B)] :
          ( ! [X: A,Y2: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y2)
             => ( aa(A,B,F2,X) != aa(A,B,F2,Y2) ) )
         => inj_on(A,B,F2,top_top(set(A))) ) ) ).

% linorder_injI
tff(fact_5717_inj__on__Cons1,axiom,
    ! [A: $tType,Xa: A,Aa2: set(list(A))] : inj_on(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Aa2) ).

% inj_on_Cons1
tff(fact_5718_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_tm(fun(A,B),fun(A,product_prod(B,A)),C2),S) ).

% inj_Pair(2)
tff(fact_5719_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_tn(fun(A,B),fun(A,product_prod(A,B)),C2),S) ).

% inj_Pair(1)
tff(fact_5720_inj__on__convol__ident,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),X3: set(A)] : inj_on(A,product_prod(A,B),aTP_Lamp_tn(fun(A,B),fun(A,product_prod(A,B)),F2),X3) ).

% inj_on_convol_ident
tff(fact_5721_sorted__list__of__set_Oinj__on,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => inj_on(A,A,aTP_Lamp_ip(A,A),top_top(set(A))) ) ).

% sorted_list_of_set.inj_on
tff(fact_5722_filterlim__ident,axiom,
    ! [A: $tType,Fa: filter(A)] : filterlim(A,A,aTP_Lamp_ab(A,A),Fa,Fa) ).

% filterlim_ident
tff(fact_5723_filterlim__compose,axiom,
    ! [B: $tType,A: $tType,E: $tType,G: fun(A,B),F32: filter(B),F22: filter(A),F2: fun(E,A),F12: filter(E)] :
      ( filterlim(A,B,G,F32,F22)
     => ( filterlim(E,A,F2,F22,F12)
       => filterlim(E,B,aa(fun(E,A),fun(E,B),aTP_Lamp_to(fun(A,B),fun(fun(E,A),fun(E,B)),G),F2),F32,F12) ) ) ).

% filterlim_compose
tff(fact_5724_filterlim__sequentially__Suc,axiom,
    ! [A: $tType,F2: fun(nat,A),Fa: filter(A)] :
      ( filterlim(nat,A,aTP_Lamp_kh(fun(nat,A),fun(nat,A),F2),Fa,at_top(nat))
    <=> filterlim(nat,A,F2,Fa,at_top(nat)) ) ).

% filterlim_sequentially_Suc
tff(fact_5725_filterlim__Suc,axiom,
    filterlim(nat,nat,suc,at_top(nat),at_top(nat)) ).

% filterlim_Suc
tff(fact_5726_folding__insort__key_Oinj__on,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),S: set(B),F2: fun(B,A)] :
      ( folding_insort_key(A,B,Less_eq,Less,S,F2)
     => inj_on(B,A,F2,S) ) ).

% folding_insort_key.inj_on
tff(fact_5727_inj__on__diff,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),Aa2: set(A),B2: set(A)] :
      ( inj_on(A,B,F2,Aa2)
     => inj_on(A,B,F2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),B2)) ) ).

% inj_on_diff
tff(fact_5728_filterlim__top,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),Fa: filter(A)] : filterlim(A,B,F2,top_top(filter(B)),Fa) ).

% filterlim_top
tff(fact_5729_filterlim__sup,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),Fa: filter(B),F12: filter(A),F22: filter(A)] :
      ( filterlim(A,B,F2,Fa,F12)
     => ( filterlim(A,B,F2,Fa,F22)
       => filterlim(A,B,F2,Fa,aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),sup_sup(filter(A)),F12),F22)) ) ) ).

% filterlim_sup
tff(fact_5730_filterlim__inf,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),F22: filter(B),F32: filter(B),F12: filter(A)] :
      ( filterlim(A,B,F2,aa(filter(B),filter(B),aa(filter(B),fun(filter(B),filter(B)),inf_inf(filter(B)),F22),F32),F12)
    <=> ( filterlim(A,B,F2,F22,F12)
        & filterlim(A,B,F2,F32,F12) ) ) ).

% filterlim_inf
tff(fact_5731_inj__on__mult,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A3: A,Aa2: set(A)] :
          ( ( A3 != zero_zero(A) )
         => inj_on(A,A,aa(A,fun(A,A),times_times(A),A3),Aa2) ) ) ).

% inj_on_mult
tff(fact_5732_inj__mapD,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B)] :
      ( inj_on(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),top_top(set(list(A))))
     => inj_on(A,B,F2,top_top(set(A))) ) ).

% inj_mapD
tff(fact_5733_linorder__inj__onI,axiom,
    ! [B: $tType,A: $tType] :
      ( order(A)
     => ! [Aa2: set(A),F2: fun(A,B)] :
          ( ! [X: A,Y2: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y2)
             => ( member(A,X,Aa2)
               => ( member(A,Y2,Aa2)
                 => ( aa(A,B,F2,X) != aa(A,B,F2,Y2) ) ) ) )
         => ( ! [X: A,Y2: A] :
                ( member(A,X,Aa2)
               => ( member(A,Y2,Aa2)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y2)
                    | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y2),X) ) ) )
           => inj_on(A,B,F2,Aa2) ) ) ) ).

% linorder_inj_onI
tff(fact_5734_filterlim__mono,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),F22: filter(B),F12: filter(A),F24: filter(B),F13: filter(A)] :
      ( filterlim(A,B,F2,F22,F12)
     => ( aa(filter(B),$o,aa(filter(B),fun(filter(B),$o),ord_less_eq(filter(B)),F22),F24)
       => ( aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),F13),F12)
         => filterlim(A,B,F2,F24,F13) ) ) ) ).

% filterlim_mono
tff(fact_5735_inj__diff__right,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A] : inj_on(A,A,aTP_Lamp_af(A,fun(A,A),A3),top_top(set(A))) ) ).

% inj_diff_right
tff(fact_5736_inj__singleton,axiom,
    ! [A: $tType,Aa2: set(A)] : inj_on(A,set(A),aTP_Lamp_at(A,set(A)),Aa2) ).

% inj_singleton
tff(fact_5737_finite__Collect,axiom,
    ! [A: $tType,B: $tType,S: set(A),F2: fun(B,A)] :
      ( aa(set(A),$o,finite_finite2(A),S)
     => ( inj_on(B,A,F2,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,$o),aTP_Lamp_tp(set(A),fun(fun(B,A),fun(B,$o)),S),F2))) ) ) ).

% finite_Collect
tff(fact_5738_swap__inj__on,axiom,
    ! [B: $tType,A: $tType,Aa2: 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_ot(A,fun(B,product_prod(B,A)))),Aa2) ).

% swap_inj_on
tff(fact_5739_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_5740_inj__split__Cons,axiom,
    ! [A: $tType,X3: 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_if(list(A),fun(A,list(A)))),X3) ).

% inj_split_Cons
tff(fact_5741_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_5742_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_5743_inj__on__iff__surj,axiom,
    ! [B: $tType,A: $tType,Aa2: set(A),A5: set(B)] :
      ( ( Aa2 != bot_bot(set(A)) )
     => ( ? [F9: fun(A,B)] :
            ( inj_on(A,B,F9,Aa2)
            & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F9),Aa2)),A5) )
      <=> ? [G5: fun(B,A)] : ( aa(set(B),set(A),image2(B,A,G5),A5) = Aa2 ) ) ) ).

% inj_on_iff_surj
tff(fact_5744_inj__on__image__set__diff,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),C3: set(A),Aa2: set(A),B2: set(A)] :
      ( inj_on(A,B,F2,C3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),B2)),C3)
       => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),C3)
         => ( aa(set(A),set(B),image2(A,B,F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),B2)) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),aa(set(A),set(B),image2(A,B,F2),Aa2)),aa(set(A),set(B),image2(A,B,F2),B2)) ) ) ) ) ).

% inj_on_image_set_diff
tff(fact_5745_image__set__diff,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),Aa2: set(A),B2: set(A)] :
      ( inj_on(A,B,F2,top_top(set(A)))
     => ( aa(set(A),set(B),image2(A,B,F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),B2)) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),aa(set(A),set(B),image2(A,B,F2),Aa2)),aa(set(A),set(B),image2(A,B,F2),B2)) ) ) ).

% image_set_diff
tff(fact_5746_pigeonhole,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Aa2: set(B)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),finite_card(A,aa(set(B),set(A),image2(B,A,F2),Aa2))),finite_card(B,Aa2))
     => ~ inj_on(B,A,F2,Aa2) ) ).

% pigeonhole
tff(fact_5747_map__inj__on,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xsa: list(B),Ys: list(B)] :
      ( ( aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Xsa) = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Ys) )
     => ( inj_on(B,A,F2,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),aa(list(B),set(B),set2(B),Xsa)),aa(list(B),set(B),set2(B),Ys)))
       => ( Xsa = Ys ) ) ) ).

% map_inj_on
tff(fact_5748_inj__on__map__eq__map,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),Xsa: list(A),Ys: list(A)] :
      ( inj_on(A,B,F2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(list(A),set(A),set2(A),Xsa)),aa(list(A),set(A),set2(A),Ys)))
     => ( ( aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),Xsa) = aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),Ys) )
      <=> ( Xsa = Ys ) ) ) ).

% inj_on_map_eq_map
tff(fact_5749_filterlim__INF_H,axiom,
    ! [E: $tType,B: $tType,A: $tType,Xa: A,Aa2: set(A),F2: fun(B,E),Fa: filter(E),G4: fun(A,filter(B))] :
      ( member(A,Xa,Aa2)
     => ( filterlim(B,E,F2,Fa,aa(A,filter(B),G4,Xa))
       => filterlim(B,E,F2,Fa,aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image2(A,filter(B),G4),Aa2))) ) ) ).

% filterlim_INF'
tff(fact_5750_filterlim__INF,axiom,
    ! [E: $tType,B: $tType,A: $tType,F2: fun(A,B),G4: fun(E,filter(B)),B2: set(E),Fa: filter(A)] :
      ( filterlim(A,B,F2,aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(E),set(filter(B)),image2(E,filter(B),G4),B2)),Fa)
    <=> ! [X6: E] :
          ( member(E,X6,B2)
         => filterlim(A,B,F2,aa(E,filter(B),G4,X6),Fa) ) ) ).

% filterlim_INF
tff(fact_5751_distinct__map,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xsa: list(B)] :
      ( aa(list(A),$o,distinct(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Xsa))
    <=> ( aa(list(B),$o,distinct(B),Xsa)
        & inj_on(B,A,F2,aa(list(B),set(B),set2(B),Xsa)) ) ) ).

% distinct_map
tff(fact_5752_remdups__adj__map__injective,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),Xsa: list(A)] :
      ( inj_on(A,B,F2,top_top(set(A)))
     => ( aa(list(B),list(B),remdups_adj(B),aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),Xsa)) = aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),aa(list(A),list(A),remdups_adj(A),Xsa)) ) ) ).

% remdups_adj_map_injective
tff(fact_5753_filterlim__If,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),G4: filter(B),Fa: filter(A),Pa: fun(A,$o),G: fun(A,B)] :
      ( filterlim(A,B,F2,G4,aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),inf_inf(filter(A)),Fa),aa(set(A),filter(A),principal(A),aa(fun(A,$o),set(A),collect(A),Pa))))
     => ( filterlim(A,B,G,G4,aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),inf_inf(filter(A)),Fa),aa(set(A),filter(A),principal(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_ce(fun(A,$o),fun(A,$o),Pa)))))
       => filterlim(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,$o),fun(fun(A,B),fun(A,B)),aTP_Lamp_tq(fun(A,B),fun(fun(A,$o),fun(fun(A,B),fun(A,B))),F2),Pa),G),G4,Fa) ) ) ).

% filterlim_If
tff(fact_5754_map__removeAll__inj,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),Xa: A,Xsa: list(A)] :
      ( inj_on(A,B,F2,top_top(set(A)))
     => ( aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),removeAll(A),Xa),Xsa)) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),removeAll(B),aa(A,B,F2,Xa)),aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),Xsa)) ) ) ).

% map_removeAll_inj
tff(fact_5755_sort__key__eq__sort__key,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Xsa: list(A),Ys: list(A),F2: fun(A,B)] :
          ( ( aa(list(A),multiset(A),mset(A),Xsa) = aa(list(A),multiset(A),mset(A),Ys) )
         => ( inj_on(A,B,F2,aa(list(A),set(A),set2(A),Xsa))
           => ( aa(list(A),list(A),aa(fun(A,B),fun(list(A),list(A)),linorder_sort_key(A,B),F2),Xsa) = aa(list(A),list(A),aa(fun(A,B),fun(list(A),list(A)),linorder_sort_key(A,B),F2),Ys) ) ) ) ) ).

% sort_key_eq_sort_key
tff(fact_5756_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_mx(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_mx(nat,fun(nat,product_prod(nat,nat))))) ).

% uminus_int.rsp
tff(fact_5757_inj__on__filter__key__eq,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),Y: A,Xsa: list(A)] :
      ( inj_on(A,B,F2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Y),aa(list(A),set(A),set2(A),Xsa)))
     => ( aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),aa(A,fun(A,$o),aTP_Lamp_tr(fun(A,B),fun(A,fun(A,$o)),F2),Y)),Xsa) = aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter2(A),aa(A,fun(A,$o),fequal(A),Y)),Xsa) ) ) ).

% inj_on_filter_key_eq
tff(fact_5758_inj__on__INTER,axiom,
    ! [E: $tType,B: $tType,A: $tType,Ia2: set(A),F2: fun(B,E),Aa2: fun(A,set(B))] :
      ( ( Ia2 != bot_bot(set(A)) )
     => ( ! [I3: A] :
            ( member(A,I3,Ia2)
           => inj_on(B,E,F2,aa(A,set(B),Aa2,I3)) )
       => inj_on(B,E,F2,aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image2(A,set(B),Aa2),Ia2))) ) ) ).

% inj_on_INTER
tff(fact_5759_finite__imp__nat__seg__image__inj__on,axiom,
    ! [A: $tType,Aa2: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),Aa2)
     => ? [N5: nat,F3: fun(nat,A)] :
          ( ( Aa2 = aa(set(nat),set(A),image2(nat,A,F3),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_ac(nat,fun(nat,$o)),N5))) )
          & inj_on(nat,A,F3,aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_ac(nat,fun(nat,$o)),N5))) ) ) ).

% finite_imp_nat_seg_image_inj_on
tff(fact_5760_finite__imp__inj__to__nat__seg,axiom,
    ! [A: $tType,Aa2: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),Aa2)
     => ? [F3: fun(A,nat),N5: nat] :
          ( ( aa(set(A),set(nat),image2(A,nat,F3),Aa2) = aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_ac(nat,fun(nat,$o)),N5)) )
          & inj_on(A,nat,F3,Aa2) ) ) ).

% finite_imp_inj_to_nat_seg
tff(fact_5761_finite__imp__inj__to__nat__seg_H,axiom,
    ! [A: $tType,Aa2: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),Aa2)
     => ~ ! [F3: fun(A,nat)] :
            ( ? [N5: nat] : ( aa(set(A),set(nat),image2(A,nat,F3),Aa2) = aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_ac(nat,fun(nat,$o)),N5)) )
           => ~ inj_on(A,nat,F3,Aa2) ) ) ).

% finite_imp_inj_to_nat_seg'
tff(fact_5762_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_5763_inj__on__nth,axiom,
    ! [A: $tType,Xsa: list(A),Ia2: set(nat)] :
      ( aa(list(A),$o,distinct(A),Xsa)
     => ( ! [X: nat] :
            ( member(nat,X,Ia2)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),aa(list(A),nat,size_size(list(A)),Xsa)) )
       => inj_on(nat,A,nth(A,Xsa),Ia2) ) ) ).

% inj_on_nth
tff(fact_5764_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_my(nat,fun(nat,A)))),aa(fun(nat,fun(nat,A)),fun(product_prod(nat,nat),A),product_case_prod(nat,nat,A),aTP_Lamp_my(nat,fun(nat,A)))) ) ).

% of_int.rsp
tff(fact_5765_map__removeAll__inj__on,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),Xa: A,Xsa: list(A)] :
      ( inj_on(A,B,F2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),aa(list(A),set(A),set2(A),Xsa)))
     => ( aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),removeAll(A),Xa),Xsa)) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),removeAll(B),aa(A,B,F2,Xa)),aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),Xsa)) ) ) ).

% map_removeAll_inj_on
tff(fact_5766_filterlim__base,axiom,
    ! [B: $tType,A: $tType,G2: $tType,F4: $tType,E: $tType,J4: set(A),Ia: fun(A,E),Ia2: set(E),Fa: fun(E,set(F4)),F2: fun(F4,G2),G4: fun(A,set(G2))] :
      ( ! [M6: A,X: B] :
          ( member(A,M6,J4)
         => member(E,aa(A,E,Ia,M6),Ia2) )
     => ( ! [M6: A,X: F4] :
            ( member(A,M6,J4)
           => ( member(F4,X,aa(E,set(F4),Fa,aa(A,E,Ia,M6)))
             => member(G2,aa(F4,G2,F2,X),aa(A,set(G2),G4,M6)) ) )
       => filterlim(F4,G2,F2,aa(set(filter(G2)),filter(G2),complete_Inf_Inf(filter(G2)),aa(set(A),set(filter(G2)),image2(A,filter(G2),aTP_Lamp_ts(fun(A,set(G2)),fun(A,filter(G2)),G4)),J4)),aa(set(filter(F4)),filter(F4),complete_Inf_Inf(filter(F4)),aa(set(E),set(filter(F4)),image2(E,filter(F4),aTP_Lamp_tt(fun(E,set(F4)),fun(E,filter(F4)),Fa)),Ia2))) ) ) ).

% filterlim_base
tff(fact_5767_inj__on__disjoint__Un,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),Aa2: set(A),G: fun(A,B),B2: set(A)] :
      ( inj_on(A,B,F2,Aa2)
     => ( inj_on(A,B,G,B2)
       => ( ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(set(A),set(B),image2(A,B,F2),Aa2)),aa(set(A),set(B),image2(A,B,G),B2)) = 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_tu(fun(A,B),fun(set(A),fun(fun(A,B),fun(A,B))),F2),Aa2),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),B2)) ) ) ) ).

% inj_on_disjoint_Un
tff(fact_5768_inj__on__funpow__least,axiom,
    ! [A: $tType,N: nat,F2: fun(A,A),S2: A] :
      ( ( aa(A,A,compow(fun(A,A),N,F2),S2) = S2 )
     => ( ! [M6: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M6)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M6),N)
             => ( aa(A,A,compow(fun(A,A),M6,F2),S2) != S2 ) ) )
       => inj_on(nat,A,aa(A,fun(nat,A),aTP_Lamp_tv(fun(A,A),fun(A,fun(nat,A)),F2),S2),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)) ) ) ).

% inj_on_funpow_least
tff(fact_5769_inj__on__Un,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),Aa2: set(A),B2: set(A)] :
      ( inj_on(A,B,F2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),B2))
    <=> ( inj_on(A,B,F2,Aa2)
        & inj_on(A,B,F2,B2)
        & ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(set(A),set(B),image2(A,B,F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),B2))),aa(set(A),set(B),image2(A,B,F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B2),Aa2))) = bot_bot(set(B)) ) ) ) ).

% inj_on_Un
tff(fact_5770_sort__key__conv__fold,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),Xsa: list(A)] :
          ( inj_on(A,B,F2,aa(list(A),set(A),set2(A),Xsa))
         => ( aa(list(A),list(A),aa(fun(A,B),fun(list(A),list(A)),linorder_sort_key(A,B),F2),Xsa) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),aa(fun(A,fun(list(A),list(A))),fun(list(A),fun(list(A),list(A))),fold(A,list(A)),linorder_insort_key(A,B,F2)),Xsa),nil(A)) ) ) ) ).

% sort_key_conv_fold
tff(fact_5771_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),$o)),aTP_Lamp_na(nat,fun(nat,fun(product_prod(nat,nat),$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),$o)),aTP_Lamp_na(nat,fun(nat,fun(product_prod(nat,nat),$o))))) ).

% less_int.rsp
tff(fact_5772_map__sorted__distinct__set__unique,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),Xsa: list(A),Ys: list(A)] :
          ( inj_on(A,B,F2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(list(A),set(A),set2(A),Xsa)),aa(list(A),set(A),set2(A),Ys)))
         => ( sorted_wrt(B,ord_less_eq(B),aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),Xsa))
           => ( aa(list(B),$o,distinct(B),aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),Xsa))
             => ( sorted_wrt(B,ord_less_eq(B),aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),Ys))
               => ( aa(list(B),$o,distinct(B),aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),Ys))
                 => ( ( aa(list(A),set(A),set2(A),Xsa) = aa(list(A),set(A),set2(A),Ys) )
                   => ( Xsa = Ys ) ) ) ) ) ) ) ) ).

% map_sorted_distinct_set_unique
tff(fact_5773_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_ne(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_ne(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))) ).

% minus_int.rsp
tff(fact_5774_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_nc(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_nc(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))) ).

% plus_int.rsp
tff(fact_5775_sort__key__inj__key__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Xsa: list(A),Ys: list(A),F2: fun(A,B)] :
          ( ( aa(list(A),multiset(A),mset(A),Xsa) = aa(list(A),multiset(A),mset(A),Ys) )
         => ( inj_on(A,B,F2,aa(list(A),set(A),set2(A),Xsa))
           => ( sorted_wrt(B,ord_less_eq(B),aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),Ys))
             => ( aa(list(A),list(A),aa(fun(A,B),fun(list(A),list(A)),linorder_sort_key(A,B),F2),Xsa) = Ys ) ) ) ) ) ).

% sort_key_inj_key_eq
tff(fact_5776_sum__mult__sum__if__inj,axiom,
    ! [A: $tType,E: $tType,B: $tType] :
      ( semiring_0(E)
     => ! [F2: fun(A,E),G: fun(B,E),Aa2: set(A),B2: set(B)] :
          ( inj_on(product_prod(A,B),E,aa(fun(A,fun(B,E)),fun(product_prod(A,B),E),product_case_prod(A,B,E),aa(fun(B,E),fun(A,fun(B,E)),aTP_Lamp_tw(fun(A,E),fun(fun(B,E),fun(A,fun(B,E))),F2),G)),product_Sigma(A,B,Aa2,aTP_Lamp_pv(set(B),fun(A,set(B)),B2)))
         => ( aa(E,E,aa(E,fun(E,E),times_times(E),aa(set(A),E,aa(fun(A,E),fun(set(A),E),groups7311177749621191930dd_sum(A,E),F2),Aa2)),aa(set(B),E,aa(fun(B,E),fun(set(B),E),groups7311177749621191930dd_sum(B,E),G),B2)) = aa(set(E),E,aa(fun(E,E),fun(set(E),E),groups7311177749621191930dd_sum(E,E),id(E)),aa(fun(E,$o),set(E),collect(E),aa(set(B),fun(E,$o),aa(set(A),fun(set(B),fun(E,$o)),aa(fun(B,E),fun(set(A),fun(set(B),fun(E,$o))),aTP_Lamp_tx(fun(A,E),fun(fun(B,E),fun(set(A),fun(set(B),fun(E,$o)))),F2),G),Aa2),B2))) ) ) ) ).

% sum_mult_sum_if_inj
tff(fact_5777_inj__on__map__inv__f,axiom,
    ! [B: $tType,A: $tType,L: list(A),Aa2: set(A),F2: fun(A,B)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),L)),Aa2)
     => ( inj_on(A,B,F2,Aa2)
       => ( aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),inv_on(A,B,F2,Aa2)),aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),L)) = L ) ) ) ).

% inj_on_map_inv_f
tff(fact_5778_inj__on__vimage__singleton,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),Aa2: set(A),A3: B] :
      ( inj_on(A,B,F2,Aa2)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),vimage(A,B,F2,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert3(B),A3),bot_bot(set(B))))),Aa2)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),the(A,aa(B,fun(A,$o),aa(set(A),fun(B,fun(A,$o)),aTP_Lamp_ty(fun(A,B),fun(set(A),fun(B,fun(A,$o))),F2),Aa2),A3))),bot_bot(set(A)))) ) ).

% inj_on_vimage_singleton
tff(fact_5779_inj__vimage__singleton,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A3: B] :
      ( inj_on(A,B,F2,top_top(set(A)))
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),vimage(A,B,F2,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert3(B),A3),bot_bot(set(B))))),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),the(A,aa(B,fun(A,$o),aTP_Lamp_tz(fun(A,B),fun(B,fun(A,$o)),F2),A3))),bot_bot(set(A)))) ) ).

% inj_vimage_singleton
tff(fact_5780_trivial__limit__sequentially,axiom,
    at_top(nat) != bot_bot(filter(nat)) ).

% trivial_limit_sequentially
tff(fact_5781_The__split__eq,axiom,
    ! [A: $tType,B: $tType,Xa: A,Y: B] : ( the(product_prod(A,B),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aa(B,fun(A,fun(B,$o)),aTP_Lamp_ua(A,fun(B,fun(A,fun(B,$o))),Xa),Y))) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa),Y) ) ).

% The_split_eq
tff(fact_5782_inv__on__f__f,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),Aa2: set(A),Xa: A] :
      ( inj_on(A,B,F2,Aa2)
     => ( member(A,Xa,Aa2)
       => ( aa(B,A,inv_on(A,B,F2,Aa2),aa(A,B,F2,Xa)) = Xa ) ) ) ).

% inv_on_f_f
tff(fact_5783_inv__on__f__range,axiom,
    ! [A: $tType,B: $tType,Y: A,F2: fun(B,A),Aa2: set(B)] :
      ( member(A,Y,aa(set(B),set(A),image2(B,A,F2),Aa2))
     => member(B,aa(A,B,inv_on(B,A,F2,Aa2),Y),Aa2) ) ).

% inv_on_f_range
tff(fact_5784_f__inv__on__f,axiom,
    ! [B: $tType,A: $tType,Y: A,F2: fun(B,A),Aa2: set(B)] :
      ( member(A,Y,aa(set(B),set(A),image2(B,A,F2),Aa2))
     => ( aa(B,A,F2,aa(A,B,inv_on(B,A,F2,Aa2),Y)) = Y ) ) ).

% f_inv_on_f
tff(fact_5785_Least__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Pa: fun(A,$o)] : ( ord_Least(A,Pa) = the(A,aTP_Lamp_ub(fun(A,$o),fun(A,$o),Pa)) ) ) ).

% Least_def
tff(fact_5786_the__elem__def,axiom,
    ! [A: $tType,X3: set(A)] : ( the_elem(A,X3) = the(A,aTP_Lamp_uc(set(A),fun(A,$o),X3)) ) ).

% the_elem_def
tff(fact_5787_Greatest__def,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Pa: fun(A,$o)] : ( order_Greatest(A,Pa) = the(A,aTP_Lamp_ud(fun(A,$o),fun(A,$o),Pa)) ) ) ).

% Greatest_def
tff(fact_5788_ord_OLeast__def,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,$o)),Pa: fun(A,$o)] : ( aa(fun(A,$o),A,least(A,Less_eq),Pa) = the(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ue(fun(A,fun(A,$o)),fun(fun(A,$o),fun(A,$o)),Less_eq),Pa)) ) ).

% ord.Least_def
tff(fact_5789_card__quotient__disjoint,axiom,
    ! [A: $tType,Aa2: set(A),R2: set(product_prod(A,A))] :
      ( aa(set(A),$o,finite_finite2(A),Aa2)
     => ( inj_on(A,set(set(A)),aTP_Lamp_uf(set(product_prod(A,A)),fun(A,set(set(A))),R2),Aa2)
       => ( finite_card(set(A),equiv_quotient(A,Aa2,R2)) = finite_card(A,Aa2) ) ) ) ).

% card_quotient_disjoint
tff(fact_5790_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_5791_GreatestI2__order,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Pa: fun(A,$o),Xa: A,Qa: fun(A,$o)] :
          ( aa(A,$o,Pa,Xa)
         => ( ! [Y2: A] :
                ( aa(A,$o,Pa,Y2)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y2),Xa) )
           => ( ! [X: A] :
                  ( aa(A,$o,Pa,X)
                 => ( ! [Y3: A] :
                        ( aa(A,$o,Pa,Y3)
                       => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y3),X) )
                   => aa(A,$o,Qa,X) ) )
             => aa(A,$o,Qa,order_Greatest(A,Pa)) ) ) ) ) ).

% GreatestI2_order
tff(fact_5792_Greatest__equality,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Pa: fun(A,$o),Xa: A] :
          ( aa(A,$o,Pa,Xa)
         => ( ! [Y2: A] :
                ( aa(A,$o,Pa,Y2)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y2),Xa) )
           => ( order_Greatest(A,Pa) = Xa ) ) ) ) ).

% Greatest_equality
tff(fact_5793_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_5794_quotient__is__empty,axiom,
    ! [A: $tType,Aa2: set(A),R2: set(product_prod(A,A))] :
      ( ( equiv_quotient(A,Aa2,R2) = bot_bot(set(set(A))) )
    <=> ( Aa2 = bot_bot(set(A)) ) ) ).

% quotient_is_empty
tff(fact_5795_quotient__is__empty2,axiom,
    ! [A: $tType,Aa2: set(A),R2: set(product_prod(A,A))] :
      ( ( bot_bot(set(set(A))) = equiv_quotient(A,Aa2,R2) )
    <=> ( Aa2 = bot_bot(set(A)) ) ) ).

% quotient_is_empty2
tff(fact_5796_quotientE,axiom,
    ! [A: $tType,X3: set(A),Aa2: set(A),R2: set(product_prod(A,A))] :
      ( member(set(A),X3,equiv_quotient(A,Aa2,R2))
     => ~ ! [X: A] :
            ( ( X3 = image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),X),bot_bot(set(A)))) )
           => ~ member(A,X,Aa2) ) ) ).

% quotientE
tff(fact_5797_quotientI,axiom,
    ! [A: $tType,Xa: A,Aa2: set(A),R2: set(product_prod(A,A))] :
      ( member(A,Xa,Aa2)
     => member(set(A),image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A)))),equiv_quotient(A,Aa2,R2)) ) ).

% quotientI
tff(fact_5798_quotient__diff1,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),Aa2: set(A),A3: A] :
      ( inj_on(A,set(set(A)),aTP_Lamp_uf(set(product_prod(A,A)),fun(A,set(set(A))),R2),Aa2)
     => ( member(A,A3,Aa2)
       => ( equiv_quotient(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(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,Aa2,R2)),equiv_quotient(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),bot_bot(set(A))),R2)) ) ) ) ).

% quotient_diff1
tff(fact_5799_singleton__quotient,axiom,
    ! [A: $tType,Xa: A,R2: set(product_prod(A,A))] : ( equiv_quotient(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A))),R2) = aa(set(set(A)),set(set(A)),aa(set(A),fun(set(set(A)),set(set(A))),insert3(set(A)),image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A))))),bot_bot(set(set(A)))) ) ).

% singleton_quotient
tff(fact_5800_quotient__def,axiom,
    ! [A: $tType,Aa2: set(A),R2: set(product_prod(A,A))] : ( equiv_quotient(A,Aa2,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_ug(set(product_prod(A,A)),fun(A,set(set(A))),R2)),Aa2)) ) ).

% quotient_def
tff(fact_5801_list__all__transfer,axiom,
    ! [A: $tType,B: $tType,Aa2: fun(A,fun(B,$o))] : aa(fun(fun(B,$o),fun(list(B),$o)),$o,aa(fun(fun(A,$o),fun(list(A),$o)),fun(fun(fun(B,$o),fun(list(B),$o)),$o),bNF_rel_fun(fun(A,$o),fun(B,$o),fun(list(A),$o),fun(list(B),$o),bNF_rel_fun(A,B,$o,$o,Aa2,fequal($o)),bNF_rel_fun(list(A),list(B),$o,$o,aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Aa2),fequal($o))),list_all(A)),list_all(B)) ).

% list_all_transfer
tff(fact_5802_list_Opred__transfer,axiom,
    ! [A: $tType,B: $tType,R: fun(A,fun(B,$o))] : aa(fun(fun(B,$o),fun(list(B),$o)),$o,aa(fun(fun(A,$o),fun(list(A),$o)),fun(fun(fun(B,$o),fun(list(B),$o)),$o),bNF_rel_fun(fun(A,$o),fun(B,$o),fun(list(A),$o),fun(list(B),$o),bNF_rel_fun(A,B,$o,$o,R,fequal($o)),bNF_rel_fun(list(A),list(B),$o,$o,aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),R),fequal($o))),list_all(A)),list_all(B)) ).

% list.pred_transfer
tff(fact_5803_Gcd__fin__0__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [Aa2: set(A)] :
          ( ( semiring_gcd_Gcd_fin(A,Aa2) = zero_zero(A) )
        <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),zero_zero(A)),bot_bot(set(A))))
            & aa(set(A),$o,finite_finite2(A),Aa2) ) ) ) ).

% Gcd_fin_0_iff
tff(fact_5804_list__all__simps_I1_J,axiom,
    ! [A: $tType,Pa: fun(A,$o),Xa: A,Xsa: list(A)] :
      ( aa(list(A),$o,aa(fun(A,$o),fun(list(A),$o),list_all(A),Pa),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa))
    <=> ( aa(A,$o,Pa,Xa)
        & aa(list(A),$o,aa(fun(A,$o),fun(list(A),$o),list_all(A),Pa),Xsa) ) ) ).

% list_all_simps(1)
tff(fact_5805_list_Opred__inject_I2_J,axiom,
    ! [A: $tType,Pa: fun(A,$o),A3: A,Aa4: list(A)] :
      ( aa(list(A),$o,aa(fun(A,$o),fun(list(A),$o),list_all(A),Pa),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),Aa4))
    <=> ( aa(A,$o,Pa,A3)
        & aa(list(A),$o,aa(fun(A,$o),fun(list(A),$o),list_all(A),Pa),Aa4) ) ) ).

% list.pred_inject(2)
tff(fact_5806_list__all__simps_I2_J,axiom,
    ! [A: $tType,Pa: fun(A,$o)] : aa(list(A),$o,aa(fun(A,$o),fun(list(A),$o),list_all(A),Pa),nil(A)) ).

% list_all_simps(2)
tff(fact_5807_list__all__append,axiom,
    ! [A: $tType,Pa: fun(A,$o),Xsa: list(A),Ys: list(A)] :
      ( aa(list(A),$o,aa(fun(A,$o),fun(list(A),$o),list_all(A),Pa),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),Ys))
    <=> ( aa(list(A),$o,aa(fun(A,$o),fun(list(A),$o),list_all(A),Pa),Xsa)
        & aa(list(A),$o,aa(fun(A,$o),fun(list(A),$o),list_all(A),Pa),Ys) ) ) ).

% list_all_append
tff(fact_5808_list__all__rev,axiom,
    ! [A: $tType,Pa: fun(A,$o),Xsa: list(A)] :
      ( aa(list(A),$o,aa(fun(A,$o),fun(list(A),$o),list_all(A),Pa),aa(list(A),list(A),rev(A),Xsa))
    <=> aa(list(A),$o,aa(fun(A,$o),fun(list(A),$o),list_all(A),Pa),Xsa) ) ).

% list_all_rev
tff(fact_5809_Gcd__fin_Oempty,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ( semiring_gcd_Gcd_fin(A,bot_bot(set(A))) = zero_zero(A) ) ) ).

% Gcd_fin.empty
tff(fact_5810_list_Opred__mono__strong,axiom,
    ! [A: $tType,Pa: fun(A,$o),Xa: list(A),Paa: fun(A,$o)] :
      ( aa(list(A),$o,aa(fun(A,$o),fun(list(A),$o),list_all(A),Pa),Xa)
     => ( ! [Z3: A] :
            ( member(A,Z3,aa(list(A),set(A),set2(A),Xa))
           => ( aa(A,$o,Pa,Z3)
             => aa(A,$o,Paa,Z3) ) )
       => aa(list(A),$o,aa(fun(A,$o),fun(list(A),$o),list_all(A),Paa),Xa) ) ) ).

% list.pred_mono_strong
tff(fact_5811_list__all__cong,axiom,
    ! [A: $tType,Xa: list(A),Ya: list(A),Pa: fun(A,$o),Paa: fun(A,$o)] :
      ( ( Xa = Ya )
     => ( ! [Z3: A] :
            ( member(A,Z3,aa(list(A),set(A),set2(A),Ya))
           => ( aa(A,$o,Pa,Z3)
            <=> aa(A,$o,Paa,Z3) ) )
       => ( aa(list(A),$o,aa(fun(A,$o),fun(list(A),$o),list_all(A),Pa),Xa)
        <=> aa(list(A),$o,aa(fun(A,$o),fun(list(A),$o),list_all(A),Paa),Ya) ) ) ) ).

% list_all_cong
tff(fact_5812_list_Opred__True,axiom,
    ! [A: $tType,X4: list(A)] : aa(list(A),$o,aa(fun(A,$o),fun(list(A),$o),list_all(A),aTP_Lamp_cc(A,$o)),X4) ).

% list.pred_True
tff(fact_5813_list_Opred__inject_I1_J,axiom,
    ! [A: $tType,Pa: fun(A,$o)] : aa(list(A),$o,aa(fun(A,$o),fun(list(A),$o),list_all(A),Pa),nil(A)) ).

% list.pred_inject(1)
tff(fact_5814_list_Omap__cong__pred,axiom,
    ! [B: $tType,A: $tType,Xa: list(A),Ya: list(A),F2: fun(A,B),G: fun(A,B)] :
      ( ( Xa = Ya )
     => ( aa(list(A),$o,aa(fun(A,$o),fun(list(A),$o),list_all(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_uh(fun(A,B),fun(fun(A,B),fun(A,$o)),F2),G)),Ya)
       => ( aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),Xa) = aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),G),Ya) ) ) ) ).

% list.map_cong_pred
tff(fact_5815_list_Opred__mono,axiom,
    ! [A: $tType,Pa: fun(A,$o),Paa: fun(A,$o)] :
      ( aa(fun(A,$o),$o,aa(fun(A,$o),fun(fun(A,$o),$o),ord_less_eq(fun(A,$o)),Pa),Paa)
     => aa(fun(list(A),$o),$o,aa(fun(list(A),$o),fun(fun(list(A),$o),$o),ord_less_eq(fun(list(A),$o)),aa(fun(A,$o),fun(list(A),$o),list_all(A),Pa)),aa(fun(A,$o),fun(list(A),$o),list_all(A),Paa)) ) ).

% list.pred_mono
tff(fact_5816_prop__matchD,axiom,
    ! [A: $tType,Al: list(A),E3: A,Bl: list(A),Al2: list(A),E4: A,Bl2: list(A),Pa: fun(A,$o)] :
      ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Al),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E3),Bl)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Al2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E4),Bl2)) )
     => ( aa(list(A),$o,aa(fun(A,$o),fun(list(A),$o),list_all(A),Pa),Al)
       => ( ~ aa(A,$o,Pa,E3)
         => ( ~ aa(A,$o,Pa,E4)
           => ( aa(list(A),$o,aa(fun(A,$o),fun(list(A),$o),list_all(A),Pa),Bl)
             => ( ( Al = Al2 )
                & ( E3 = E4 )
                & ( Bl = Bl2 ) ) ) ) ) ) ) ).

% prop_matchD
tff(fact_5817_prop__match,axiom,
    ! [A: $tType,Pa: fun(A,$o),Al: list(A),E3: A,E4: A,Bl: list(A),Al2: list(A),Bl2: list(A)] :
      ( aa(list(A),$o,aa(fun(A,$o),fun(list(A),$o),list_all(A),Pa),Al)
     => ( ~ aa(A,$o,Pa,E3)
       => ( ~ aa(A,$o,Pa,E4)
         => ( aa(list(A),$o,aa(fun(A,$o),fun(list(A),$o),list_all(A),Pa),Bl)
           => ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Al),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E3),Bl)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Al2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E4),Bl2)) )
            <=> ( ( Al = Al2 )
                & ( E3 = E4 )
                & ( Bl = Bl2 ) ) ) ) ) ) ) ).

% prop_match
tff(fact_5818_Ball__set__list__all,axiom,
    ! [A: $tType,Xsa: list(A),Pa: fun(A,$o)] :
      ( ! [X6: A] :
          ( member(A,X6,aa(list(A),set(A),set2(A),Xsa))
         => aa(A,$o,Pa,X6) )
    <=> aa(list(A),$o,aa(fun(A,$o),fun(list(A),$o),list_all(A),Pa),Xsa) ) ).

% Ball_set_list_all
tff(fact_5819_list__all__iff,axiom,
    ! [A: $tType,Pa: fun(A,$o),Xa: list(A)] :
      ( aa(list(A),$o,aa(fun(A,$o),fun(list(A),$o),list_all(A),Pa),Xa)
    <=> ! [X6: A] :
          ( member(A,X6,aa(list(A),set(A),set2(A),Xa))
         => aa(A,$o,Pa,X6) ) ) ).

% list_all_iff
tff(fact_5820_list_Opred__set,axiom,
    ! [A: $tType,Pa: fun(A,$o),X4: list(A)] :
      ( aa(list(A),$o,aa(fun(A,$o),fun(list(A),$o),list_all(A),Pa),X4)
    <=> ! [Xa2: A] :
          ( member(A,Xa2,aa(list(A),set(A),set2(A),X4))
         => aa(A,$o,Pa,Xa2) ) ) ).

% list.pred_set
tff(fact_5821_list_Opred__rel,axiom,
    ! [A: $tType,Pa: fun(A,$o),Xa: list(A)] :
      ( aa(list(A),$o,aa(fun(A,$o),fun(list(A),$o),list_all(A),Pa),Xa)
    <=> aa(list(A),$o,aa(list(A),fun(list(A),$o),aa(fun(A,fun(A,$o)),fun(list(A),fun(list(A),$o)),list_all2(A,A),bNF_eq_onp(A,Pa)),Xa),Xa) ) ).

% list.pred_rel
tff(fact_5822_list_Orel__eq__onp,axiom,
    ! [A: $tType,Pa: fun(A,$o)] : ( aa(fun(A,fun(A,$o)),fun(list(A),fun(list(A),$o)),list_all2(A,A),bNF_eq_onp(A,Pa)) = bNF_eq_onp(list(A),aa(fun(A,$o),fun(list(A),$o),list_all(A),Pa)) ) ).

% list.rel_eq_onp
tff(fact_5823_dvd__gcd__list__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [B3: A,Xsa: list(A)] :
          ( dvd_dvd(A,B3,semiring_gcd_Gcd_fin(A,aa(list(A),set(A),set2(A),Xsa)))
        <=> ! [X6: A] :
              ( member(A,X6,aa(list(A),set(A),set2(A),Xsa))
             => dvd_dvd(A,B3,X6) ) ) ) ).

% dvd_gcd_list_iff
tff(fact_5824_gcd__list__greatest,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [Bs: list(A),A3: A] :
          ( ! [B4: A] :
              ( member(A,B4,aa(list(A),set(A),set2(A),Bs))
             => dvd_dvd(A,A3,B4) )
         => dvd_dvd(A,A3,semiring_gcd_Gcd_fin(A,aa(list(A),set(A),set2(A),Bs))) ) ) ).

% gcd_list_greatest
tff(fact_5825_list_Opred__map,axiom,
    ! [A: $tType,B: $tType,Qa: fun(A,$o),F2: fun(B,A),Xa: list(B)] :
      ( aa(list(A),$o,aa(fun(A,$o),fun(list(A),$o),list_all(A),Qa),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Xa))
    <=> aa(list(B),$o,aa(fun(B,$o),fun(list(B),$o),list_all(B),comp(A,$o,B,Qa,F2)),Xa) ) ).

% list.pred_map
tff(fact_5826_list__all__length,axiom,
    ! [A: $tType,Pa: fun(A,$o),Xsa: list(A)] :
      ( aa(list(A),$o,aa(fun(A,$o),fun(list(A),$o),list_all(A),Pa),Xsa)
    <=> ! [N2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N2),aa(list(A),nat,size_size(list(A)),Xsa))
         => aa(A,$o,Pa,aa(nat,A,nth(A,Xsa),N2)) ) ) ).

% list_all_length
tff(fact_5827_list__ex1__simps_I2_J,axiom,
    ! [A: $tType,Pa: fun(A,$o),Xa: A,Xsa: list(A)] :
      ( list_ex1(A,Pa,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa))
    <=> $ite(aa(A,$o,Pa,Xa),aa(list(A),$o,aa(fun(A,$o),fun(list(A),$o),list_all(A),aa(A,fun(A,$o),aTP_Lamp_ui(fun(A,$o),fun(A,fun(A,$o)),Pa),Xa)),Xsa),list_ex1(A,Pa,Xsa)) ) ).

% list_ex1_simps(2)
tff(fact_5828_map__of__rev__distinct,axiom,
    ! [B: $tType,A: $tType,M2: list(product_prod(A,B))] :
      ( aa(list(A),$o,distinct(A),aa(list(product_prod(A,B)),list(A),aa(fun(product_prod(A,B),A),fun(list(product_prod(A,B)),list(A)),map(product_prod(A,B),A),product_fst(A,B)),M2))
     => ( map_of(A,B,aa(list(product_prod(A,B)),list(product_prod(A,B)),rev(product_prod(A,B)),M2)) = map_of(A,B,M2) ) ) ).

% map_of_rev_distinct
tff(fact_5829_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)),$o),aTP_Lamp_uj(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o)))) ) ).

% init_seg_of_def
tff(fact_5830_list__ex1__simps_I1_J,axiom,
    ! [A: $tType,Pa: fun(A,$o)] : ~ list_ex1(A,Pa,nil(A)) ).

% list_ex1_simps(1)
tff(fact_5831_trans__init__seg__of,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),S2: set(product_prod(A,A)),T2: set(product_prod(A,A))] :
      ( 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))
     => ( 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),T2),init_seg_of(A))
       => 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),T2),init_seg_of(A)) ) ) ).

% trans_init_seg_of
tff(fact_5832_antisym__init__seg__of,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),S2: set(product_prod(A,A))] :
      ( 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))
     => ( 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_5833_refl__on__init__seg__of,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : 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_5834_list__ex1__iff,axiom,
    ! [A: $tType,Pa: fun(A,$o),Xsa: list(A)] :
      ( list_ex1(A,Pa,Xsa)
    <=> ? [X6: A] :
          ( member(A,X6,aa(list(A),set(A),set2(A),Xsa))
          & aa(A,$o,Pa,X6)
          & ! [Y4: A] :
              ( ( member(A,Y4,aa(list(A),set(A),set2(A),Xsa))
                & aa(A,$o,Pa,Y4) )
             => ( Y4 = X6 ) ) ) ) ).

% list_ex1_iff
tff(fact_5835_map__of__eqI,axiom,
    ! [B: $tType,A: $tType,Xsa: list(product_prod(A,B)),Ys: list(product_prod(A,B))] :
      ( ( aa(list(A),set(A),set2(A),aa(list(product_prod(A,B)),list(A),aa(fun(product_prod(A,B),A),fun(list(product_prod(A,B)),list(A)),map(product_prod(A,B),A),product_fst(A,B)),Xsa)) = aa(list(A),set(A),set2(A),aa(list(product_prod(A,B)),list(A),aa(fun(product_prod(A,B),A),fun(list(product_prod(A,B)),list(A)),map(product_prod(A,B),A),product_fst(A,B)),Ys)) )
     => ( ! [X: A] :
            ( member(A,X,aa(list(A),set(A),set2(A),aa(list(product_prod(A,B)),list(A),aa(fun(product_prod(A,B),A),fun(list(product_prod(A,B)),list(A)),map(product_prod(A,B),A),product_fst(A,B)),Xsa)))
           => ( aa(A,option(B),map_of(A,B,Xsa),X) = aa(A,option(B),map_of(A,B,Ys),X) ) )
       => ( map_of(A,B,Xsa) = map_of(A,B,Ys) ) ) ) ).

% map_of_eqI
tff(fact_5836_map__of__zip__inject,axiom,
    ! [B: $tType,A: $tType,Ys: list(A),Xsa: list(B),Zs: list(A)] :
      ( ( aa(list(A),nat,size_size(list(A)),Ys) = aa(list(B),nat,size_size(list(B)),Xsa) )
     => ( ( aa(list(A),nat,size_size(list(A)),Zs) = aa(list(B),nat,size_size(list(B)),Xsa) )
       => ( aa(list(B),$o,distinct(B),Xsa)
         => ( ( map_of(B,A,aa(list(A),list(product_prod(B,A)),aa(list(B),fun(list(A),list(product_prod(B,A))),zip(B,A),Xsa),Ys)) = map_of(B,A,aa(list(A),list(product_prod(B,A)),aa(list(B),fun(list(A),list(product_prod(B,A))),zip(B,A),Xsa),Zs)) )
           => ( Ys = Zs ) ) ) ) ) ).

% map_of_zip_inject
tff(fact_5837_map__of__eq__dom,axiom,
    ! [B: $tType,A: $tType,Xsa: list(product_prod(A,B)),Ys: list(product_prod(A,B))] :
      ( ( map_of(A,B,Xsa) = map_of(A,B,Ys) )
     => ( aa(set(product_prod(A,B)),set(A),image2(product_prod(A,B),A,product_fst(A,B)),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Xsa)) = aa(set(product_prod(A,B)),set(A),image2(product_prod(A,B),A,product_fst(A,B)),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Ys)) ) ) ).

% map_of_eq_dom
tff(fact_5838_initial__segment__of__Diff,axiom,
    ! [A: $tType,P2: set(product_prod(A,A)),Q2: set(product_prod(A,A)),S2: set(product_prod(A,A))] :
      ( 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))),P2),Q2),init_seg_of(A))
     => 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))),P2),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))),Q2),S2)),init_seg_of(A)) ) ).

% initial_segment_of_Diff
tff(fact_5839_map__of__inject__set,axiom,
    ! [B: $tType,A: $tType,Xsa: list(product_prod(A,B)),Ys: list(product_prod(A,B))] :
      ( aa(list(A),$o,distinct(A),aa(list(product_prod(A,B)),list(A),aa(fun(product_prod(A,B),A),fun(list(product_prod(A,B)),list(A)),map(product_prod(A,B),A),product_fst(A,B)),Xsa))
     => ( aa(list(A),$o,distinct(A),aa(list(product_prod(A,B)),list(A),aa(fun(product_prod(A,B),A),fun(list(product_prod(A,B)),list(A)),map(product_prod(A,B),A),product_fst(A,B)),Ys))
       => ( ( map_of(A,B,Xsa) = map_of(A,B,Ys) )
        <=> ( aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Xsa) = aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Ys) ) ) ) ) ).

% map_of_inject_set
tff(fact_5840_can__select__set__list__ex1,axiom,
    ! [A: $tType,Pa: fun(A,$o),Aa2: list(A)] :
      ( can_select(A,Pa,aa(list(A),set(A),set2(A),Aa2))
    <=> list_ex1(A,Pa,Aa2) ) ).

% can_select_set_list_ex1
tff(fact_5841_image__mset__map__of,axiom,
    ! [B: $tType,A: $tType,Xsa: list(product_prod(A,B))] :
      ( aa(list(A),$o,distinct(A),aa(list(product_prod(A,B)),list(A),aa(fun(product_prod(A,B),A),fun(list(product_prod(A,B)),list(A)),map(product_prod(A,B),A),product_fst(A,B)),Xsa))
     => ( image_mset(A,B,aTP_Lamp_uk(list(product_prod(A,B)),fun(A,B),Xsa),aa(list(A),multiset(A),mset(A),aa(list(product_prod(A,B)),list(A),aa(fun(product_prod(A,B),A),fun(list(product_prod(A,B)),list(A)),map(product_prod(A,B),A),product_fst(A,B)),Xsa))) = aa(list(B),multiset(B),mset(B),aa(list(product_prod(A,B)),list(B),aa(fun(product_prod(A,B),B),fun(list(product_prod(A,B)),list(B)),map(product_prod(A,B),B),product_snd(A,B)),Xsa)) ) ) ).

% image_mset_map_of
tff(fact_5842_Misc_Oran__distinct,axiom,
    ! [B: $tType,A: $tType,Al: list(product_prod(A,B))] :
      ( aa(list(A),$o,distinct(A),aa(list(product_prod(A,B)),list(A),aa(fun(product_prod(A,B),A),fun(list(product_prod(A,B)),list(A)),map(product_prod(A,B),A),product_fst(A,B)),Al))
     => ( ran(A,B,map_of(A,B,Al)) = aa(set(product_prod(A,B)),set(B),image2(product_prod(A,B),B,product_snd(A,B)),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Al)) ) ) ).

% Misc.ran_distinct
tff(fact_5843_can__select__def,axiom,
    ! [A: $tType,Pa: fun(A,$o),Aa2: set(A)] :
      ( can_select(A,Pa,Aa2)
    <=> ? [X6: A] :
          ( member(A,X6,Aa2)
          & aa(A,$o,Pa,X6)
          & ! [Y4: A] :
              ( ( member(A,Y4,Aa2)
                & aa(A,$o,Pa,Y4) )
             => ( Y4 = X6 ) ) ) ) ).

% can_select_def
tff(fact_5844_ran__map__of__zip,axiom,
    ! [A: $tType,B: $tType,Xsa: list(A),Ys: list(B)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xsa) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( aa(list(A),$o,distinct(A),Xsa)
       => ( ran(A,B,map_of(A,B,aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),Xsa),Ys))) = aa(list(B),set(B),set2(B),Ys) ) ) ) ).

% ran_map_of_zip
tff(fact_5845_ran__map__of,axiom,
    ! [A: $tType,B: $tType,Xsa: list(product_prod(B,A))] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),ran(B,A,map_of(B,A,Xsa))),aa(set(product_prod(B,A)),set(A),image2(product_prod(B,A),A,product_snd(B,A)),aa(list(product_prod(B,A)),set(product_prod(B,A)),set2(product_prod(B,A)),Xsa))) ).

% ran_map_of
tff(fact_5846_map__to__set__map__of,axiom,
    ! [B: $tType,A: $tType,L: list(product_prod(A,B))] :
      ( aa(list(A),$o,distinct(A),aa(list(product_prod(A,B)),list(A),aa(fun(product_prod(A,B),A),fun(list(product_prod(A,B)),list(A)),map(product_prod(A,B),A),product_fst(A,B)),L))
     => ( map_to_set(A,B,map_of(A,B,L)) = aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),L) ) ) ).

% map_to_set_map_of
tff(fact_5847_map__of__map__to__set,axiom,
    ! [B: $tType,A: $tType,L: list(product_prod(A,B)),M2: fun(A,option(B))] :
      ( aa(list(A),$o,distinct(A),aa(list(product_prod(A,B)),list(A),aa(fun(product_prod(A,B),A),fun(list(product_prod(A,B)),list(A)),map(product_prod(A,B),A),product_fst(A,B)),L))
     => ( ( map_of(A,B,L) = M2 )
      <=> ( aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),L) = map_to_set(A,B,M2) ) ) ) ).

% map_of_map_to_set
tff(fact_5848_map__of__distinct__upd4,axiom,
    ! [A: $tType,B: $tType,Xa: A,Xsa: list(product_prod(A,B)),Ys: list(product_prod(A,B)),Y: B] :
      ( ~ member(A,Xa,aa(list(A),set(A),set2(A),aa(list(product_prod(A,B)),list(A),aa(fun(product_prod(A,B),A),fun(list(product_prod(A,B)),list(A)),map(product_prod(A,B),A),product_fst(A,B)),Xsa)))
     => ( ~ member(A,Xa,aa(list(A),set(A),set2(A),aa(list(product_prod(A,B)),list(A),aa(fun(product_prod(A,B),A),fun(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)),Xsa),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)),Xsa),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),Xa),Y)),Ys))),Xa,none(B)) ) ) ) ).

% map_of_distinct_upd4
tff(fact_5849_ran__empty,axiom,
    ! [B: $tType,A: $tType] : ( ran(B,A,aTP_Lamp_ul(B,option(A))) = bot_bot(set(A)) ) ).

% ran_empty
tff(fact_5850_map__to__set__empty,axiom,
    ! [B: $tType,A: $tType] : ( map_to_set(A,B,aTP_Lamp_um(A,option(B))) = bot_bot(set(product_prod(A,B))) ) ).

% map_to_set_empty
tff(fact_5851_map__of__zip__is__None,axiom,
    ! [B: $tType,A: $tType,Xsa: list(A),Ys: list(B),Xa: A] :
      ( ( aa(list(A),nat,size_size(list(A)),Xsa) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( ( aa(A,option(B),map_of(A,B,aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),Xsa),Ys)),Xa) = none(B) )
      <=> ~ member(A,Xa,aa(list(A),set(A),set2(A),Xsa)) ) ) ).

% map_of_zip_is_None
tff(fact_5852_map__to__set__empty__iff_I1_J,axiom,
    ! [A: $tType,B: $tType,M2: fun(A,option(B))] :
      ( ( map_to_set(A,B,M2) = bot_bot(set(product_prod(A,B))) )
    <=> ! [X6: A] : ( aa(A,option(B),M2,X6) = none(B) ) ) ).

% map_to_set_empty_iff(1)
tff(fact_5853_map__to__set__empty__iff_I2_J,axiom,
    ! [A: $tType,B: $tType,M2: fun(A,option(B))] :
      ( ( bot_bot(set(product_prod(A,B))) = map_to_set(A,B,M2) )
    <=> ! [X6: A] : ( aa(A,option(B),M2,X6) = none(B) ) ) ).

% map_to_set_empty_iff(2)
tff(fact_5854_map__of__None__filterD,axiom,
    ! [B: $tType,A: $tType,Xsa: list(product_prod(B,A)),Xa: B,Pa: fun(product_prod(B,A),$o)] :
      ( ( aa(B,option(A),map_of(B,A,Xsa),Xa) = none(A) )
     => ( aa(B,option(A),map_of(B,A,aa(list(product_prod(B,A)),list(product_prod(B,A)),aa(fun(product_prod(B,A),$o),fun(list(product_prod(B,A)),list(product_prod(B,A))),filter2(product_prod(B,A)),Pa),Xsa)),Xa) = none(A) ) ) ).

% map_of_None_filterD
tff(fact_5855_inj__on__fst__map__to__set,axiom,
    ! [B: $tType,A: $tType,M2: fun(A,option(B))] : inj_on(product_prod(A,B),A,product_fst(A,B),map_to_set(A,B,M2)) ).

% inj_on_fst_map_to_set
tff(fact_5856_map__of__eq__None__iff,axiom,
    ! [A: $tType,B: $tType,Xys: list(product_prod(B,A)),Xa: B] :
      ( ( aa(B,option(A),map_of(B,A,Xys),Xa) = none(A) )
    <=> ~ member(B,Xa,aa(set(product_prod(B,A)),set(B),image2(product_prod(B,A),B,product_fst(B,A)),aa(list(product_prod(B,A)),set(product_prod(B,A)),set2(product_prod(B,A)),Xys))) ) ).

% map_of_eq_None_iff
tff(fact_5857_map__to__set__ran,axiom,
    ! [A: $tType,B: $tType,M2: fun(B,option(A))] : ( ran(B,A,M2) = aa(set(product_prod(B,A)),set(A),image2(product_prod(B,A),A,product_snd(B,A)),map_to_set(B,A,M2)) ) ).

% map_to_set_ran
tff(fact_5858_rel__of__empty,axiom,
    ! [B: $tType,A: $tType,Pa: fun(product_prod(A,B),$o)] : ( rel_of(A,B,aTP_Lamp_um(A,option(B)),Pa) = bot_bot(set(product_prod(A,B))) ) ).

% rel_of_empty
tff(fact_5859_map__filter__def,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,option(A)),Xsa: list(B)] : ( map_filter(B,A,F2,Xsa) = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),comp(option(A),A,B,the2(A),F2)),aa(list(B),list(B),aa(fun(B,$o),fun(list(B),list(B)),filter2(B),aTP_Lamp_un(fun(B,option(A)),fun(B,$o),F2)),Xsa)) ) ).

% map_filter_def
tff(fact_5860_map__of__distinct__upd3,axiom,
    ! [A: $tType,B: $tType,Xa: A,Xsa: list(product_prod(A,B)),Ys: list(product_prod(A,B)),Y: B,Y6: B] :
      ( ~ member(A,Xa,aa(list(A),set(A),set2(A),aa(list(product_prod(A,B)),list(A),aa(fun(product_prod(A,B),A),fun(list(product_prod(A,B)),list(A)),map(product_prod(A,B),A),product_fst(A,B)),Xsa)))
     => ( ~ member(A,Xa,aa(list(A),set(A),set2(A),aa(list(product_prod(A,B)),list(A),aa(fun(product_prod(A,B),A),fun(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)),Xsa),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),Xa),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)),Xsa),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),Xa),Y6)),Ys))),Xa,aa(B,option(B),some(B),Y)) ) ) ) ).

% map_of_distinct_upd3
tff(fact_5861_not__Some__eq2,axiom,
    ! [B: $tType,A: $tType,V2: option(product_prod(A,B))] :
      ( ! [X6: A,Y4: B] : ( V2 != 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),X6),Y4)) )
    <=> ( V2 = none(product_prod(A,B)) ) ) ).

% not_Some_eq2
tff(fact_5862_map__update__eta__repair_I2_J,axiom,
    ! [A: $tType,B: $tType,M2: fun(B,option(A)),K3: B,V2: A] :
      ( ( aa(B,option(A),M2,K3) = none(A) )
     => ( ran(B,A,aa(A,fun(B,option(A)),aa(B,fun(A,fun(B,option(A))),aTP_Lamp_uo(fun(B,option(A)),fun(B,fun(A,fun(B,option(A)))),M2),K3),V2)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),V2),ran(B,A,M2)) ) ) ).

% map_update_eta_repair(2)
tff(fact_5863_ran__map__upd,axiom,
    ! [A: $tType,B: $tType,M2: fun(B,option(A)),A3: B,B3: A] :
      ( ( aa(B,option(A),M2,A3) = none(A) )
     => ( ran(B,A,fun_upd(B,option(A),M2,A3,aa(A,option(A),some(A),B3))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),B3),ran(B,A,M2)) ) ) ).

% ran_map_upd
tff(fact_5864_ran__nth__set__encoding__conv,axiom,
    ! [A: $tType,L: list(A)] : ( ran(nat,A,aTP_Lamp_up(list(A),fun(nat,option(A)),L)) = aa(list(A),set(A),set2(A),L) ) ).

% ran_nth_set_encoding_conv
tff(fact_5865_map__of__is__SomeI,axiom,
    ! [A: $tType,B: $tType,Xys: list(product_prod(A,B)),Xa: A,Y: B] :
      ( aa(list(A),$o,distinct(A),aa(list(product_prod(A,B)),list(A),aa(fun(product_prod(A,B),A),fun(list(product_prod(A,B)),list(A)),map(product_prod(A,B),A),product_fst(A,B)),Xys))
     => ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa),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),Xa) = aa(B,option(B),some(B),Y) ) ) ) ).

% map_of_is_SomeI
tff(fact_5866_Some__eq__map__of__iff,axiom,
    ! [B: $tType,A: $tType,Xys: list(product_prod(A,B)),Y: B,Xa: A] :
      ( aa(list(A),$o,distinct(A),aa(list(product_prod(A,B)),list(A),aa(fun(product_prod(A,B),A),fun(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),Xa) )
      <=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa),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_5867_map__of__eq__Some__iff,axiom,
    ! [B: $tType,A: $tType,Xys: list(product_prod(A,B)),Xa: A,Y: B] :
      ( aa(list(A),$o,distinct(A),aa(list(product_prod(A,B)),list(A),aa(fun(product_prod(A,B),A),fun(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),Xa) = aa(B,option(B),some(B),Y) )
      <=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa),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_5868_map__to__set__upd,axiom,
    ! [B: $tType,A: $tType,M2: fun(A,option(B)),K3: A,V2: B] : ( map_to_set(A,B,fun_upd(A,option(B),M2,K3,aa(B,option(B),some(B),V2))) = aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(product_prod(A,B),fun(set(product_prod(A,B)),set(product_prod(A,B))),insert3(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K3),V2)),aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),minus_minus(set(product_prod(A,B))),map_to_set(A,B,M2)),aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aTP_Lamp_uq(A,fun(product_prod(A,B),$o),K3)))) ) ).

% map_to_set_upd
tff(fact_5869_map__filter__map__filter,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Pa: fun(B,$o),Xsa: list(B)] : ( aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),aa(list(B),list(B),aa(fun(B,$o),fun(list(B),list(B)),filter2(B),Pa),Xsa)) = map_filter(B,A,aa(fun(B,$o),fun(B,option(A)),aTP_Lamp_ur(fun(B,A),fun(fun(B,$o),fun(B,option(A))),F2),Pa),Xsa) ) ).

% map_filter_map_filter
tff(fact_5870_le__some__optE,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [M2: A,Xa: option(A)] :
          ( aa(option(A),$o,aa(option(A),fun(option(A),$o),ord_less_eq(option(A)),aa(A,option(A),some(A),M2)),Xa)
         => ~ ! [M9: A] :
                ( ( Xa = aa(A,option(A),some(A),M9) )
               => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),M2),M9) ) ) ) ).

% le_some_optE
tff(fact_5871_rel__of__def,axiom,
    ! [B: $tType,A: $tType,M2: fun(A,option(B)),Pa: fun(product_prod(A,B),$o)] : ( rel_of(A,B,M2,Pa) = 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_us(fun(A,option(B)),fun(fun(product_prod(A,B),$o),fun(A,fun(B,$o))),M2),Pa))) ) ).

% rel_of_def
tff(fact_5872_set__map__filter,axiom,
    ! [A: $tType,B: $tType,G: fun(B,option(A)),Xsa: list(B)] : ( aa(list(A),set(A),set2(A),map_filter(B,A,G,Xsa)) = aa(fun(A,$o),set(A),collect(A),aa(list(B),fun(A,$o),aTP_Lamp_ut(fun(B,option(A)),fun(list(B),fun(A,$o)),G),Xsa)) ) ).

% set_map_filter
tff(fact_5873_map__filter__simps_I2_J,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,option(A))] : ( map_filter(B,A,F2,nil(B)) = nil(A) ) ).

% map_filter_simps(2)
tff(fact_5874_map__to__set__def,axiom,
    ! [B: $tType,A: $tType,M2: fun(A,option(B))] : ( map_to_set(A,B,M2) = aa(fun(product_prod(A,B),$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,$o),aTP_Lamp_uu(fun(A,option(B)),fun(A,fun(B,$o)),M2))) ) ).

% map_to_set_def
tff(fact_5875_map__of__SomeD,axiom,
    ! [A: $tType,B: $tType,Xsa: list(product_prod(B,A)),K3: B,Y: A] :
      ( ( aa(B,option(A),map_of(B,A,Xsa),K3) = aa(A,option(A),some(A),Y) )
     => member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),K3),Y),aa(list(product_prod(B,A)),set(product_prod(B,A)),set2(product_prod(B,A)),Xsa)) ) ).

% map_of_SomeD
tff(fact_5876_weak__map__of__SomeI,axiom,
    ! [A: $tType,B: $tType,K3: A,Xa: B,L: list(product_prod(A,B))] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K3),Xa),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),L))
     => ? [X: B] : ( aa(A,option(B),map_of(A,B,L),K3) = aa(B,option(B),some(B),X) ) ) ).

% weak_map_of_SomeI
tff(fact_5877_map__of__Cons__code_I2_J,axiom,
    ! [A: $tType,B: $tType,L: B,V2: A,Ps3: list(product_prod(B,A)),K3: 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),V2)),Ps3)),K3) = $ite(L = K3,aa(A,option(A),some(A),V2),aa(B,option(A),map_of(B,A,Ps3),K3)) ) ).

% map_of_Cons_code(2)
tff(fact_5878_map__of__zip__is__Some,axiom,
    ! [A: $tType,B: $tType,Xsa: list(A),Ys: list(B),Xa: A] :
      ( ( aa(list(A),nat,size_size(list(A)),Xsa) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( member(A,Xa,aa(list(A),set(A),set2(A),Xsa))
      <=> ? [Y4: B] : ( aa(A,option(B),map_of(A,B,aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),Xsa),Ys)),Xa) = aa(B,option(B),some(B),Y4) ) ) ) ).

% map_of_zip_is_Some
tff(fact_5879_map__of__zip__map,axiom,
    ! [A: $tType,B: $tType,Xsa: list(A),F2: fun(A,B),X4: A] :
      ( aa(A,option(B),map_of(A,B,aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),Xsa),aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),Xsa))),X4) = $ite(member(A,X4,aa(list(A),set(A),set2(A),Xsa)),aa(B,option(B),some(B),aa(A,B,F2,X4)),none(B)) ) ).

% map_of_zip_map
tff(fact_5880_map__of__zip__upd,axiom,
    ! [B: $tType,A: $tType,Ys: list(A),Xsa: list(B),Zs: list(A),Xa: B,Y: A,Z2: A] :
      ( ( aa(list(A),nat,size_size(list(A)),Ys) = aa(list(B),nat,size_size(list(B)),Xsa) )
     => ( ( aa(list(A),nat,size_size(list(A)),Zs) = aa(list(B),nat,size_size(list(B)),Xsa) )
       => ( ~ member(B,Xa,aa(list(B),set(B),set2(B),Xsa))
         => ( ( fun_upd(B,option(A),map_of(B,A,aa(list(A),list(product_prod(B,A)),aa(list(B),fun(list(A),list(product_prod(B,A))),zip(B,A),Xsa),Ys)),Xa,aa(A,option(A),some(A),Y)) = fun_upd(B,option(A),map_of(B,A,aa(list(A),list(product_prod(B,A)),aa(list(B),fun(list(A),list(product_prod(B,A))),zip(B,A),Xsa),Zs)),Xa,aa(A,option(A),some(A),Z2)) )
           => ( map_of(B,A,aa(list(A),list(product_prod(B,A)),aa(list(B),fun(list(A),list(product_prod(B,A))),zip(B,A),Xsa),Ys)) = map_of(B,A,aa(list(A),list(product_prod(B,A)),aa(list(B),fun(list(A),list(product_prod(B,A))),zip(B,A),Xsa),Zs)) ) ) ) ) ) ).

% map_of_zip_upd
tff(fact_5881_map__of__Some__split,axiom,
    ! [B: $tType,A: $tType,Xsa: list(product_prod(B,A)),K3: B,V2: A] :
      ( ( aa(B,option(A),map_of(B,A,Xsa),K3) = aa(A,option(A),some(A),V2) )
     => ? [Ys5: list(product_prod(B,A)),Zs2: list(product_prod(B,A))] :
          ( ( Xsa = 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)),Ys5),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),K3),V2)),Zs2)) )
          & ( aa(B,option(A),map_of(B,A,Ys5),K3) = none(A) ) ) ) ).

% map_of_Some_split
tff(fact_5882_map__of__zip__nth,axiom,
    ! [A: $tType,B: $tType,Xsa: list(A),Ys: list(B),Ia: nat] :
      ( ( aa(list(A),nat,size_size(list(A)),Xsa) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( aa(list(A),$o,distinct(A),Xsa)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),aa(list(B),nat,size_size(list(B)),Ys))
         => ( aa(A,option(B),map_of(A,B,aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),Xsa),Ys)),aa(nat,A,nth(A,Xsa),Ia)) = aa(B,option(B),some(B),aa(nat,B,nth(B,Ys),Ia)) ) ) ) ) ).

% map_of_zip_nth
tff(fact_5883_set__map__of__compr,axiom,
    ! [B: $tType,A: $tType,Xsa: list(product_prod(A,B))] :
      ( aa(list(A),$o,distinct(A),aa(list(product_prod(A,B)),list(A),aa(fun(product_prod(A,B),A),fun(list(product_prod(A,B)),list(A)),map(product_prod(A,B),A),product_fst(A,B)),Xsa))
     => ( aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Xsa) = 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_uv(list(product_prod(A,B)),fun(A,fun(B,$o)),Xsa))) ) ) ).

% set_map_of_compr
tff(fact_5884_map__of__mapk__SomeI,axiom,
    ! [A: $tType,B: $tType,E: $tType,F2: fun(A,B),T2: list(product_prod(A,E)),K3: A,Xa: E] :
      ( inj_on(A,B,F2,top_top(set(A)))
     => ( ( aa(A,option(E),map_of(A,E,T2),K3) = aa(E,option(E),some(E),Xa) )
       => ( aa(B,option(E),map_of(B,E,aa(list(product_prod(A,E)),list(product_prod(B,E)),aa(fun(product_prod(A,E),product_prod(B,E)),fun(list(product_prod(A,E)),list(product_prod(B,E))),map(product_prod(A,E),product_prod(B,E)),aa(fun(A,fun(E,product_prod(B,E))),fun(product_prod(A,E),product_prod(B,E)),product_case_prod(A,E,product_prod(B,E)),aTP_Lamp_uw(fun(A,B),fun(A,fun(E,product_prod(B,E))),F2))),T2)),aa(A,B,F2,K3)) = aa(E,option(E),some(E),Xa) ) ) ) ).

% map_of_mapk_SomeI
tff(fact_5885_map__of__Some__filter__not__in,axiom,
    ! [B: $tType,A: $tType,Xsa: list(product_prod(B,A)),K3: B,V2: A,Pa: fun(product_prod(B,A),$o)] :
      ( ( aa(B,option(A),map_of(B,A,Xsa),K3) = aa(A,option(A),some(A),V2) )
     => ( ~ aa(product_prod(B,A),$o,Pa,aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),K3),V2))
       => ( aa(list(B),$o,distinct(B),aa(list(product_prod(B,A)),list(B),aa(fun(product_prod(B,A),B),fun(list(product_prod(B,A)),list(B)),map(product_prod(B,A),B),product_fst(B,A)),Xsa))
         => ( aa(B,option(A),map_of(B,A,aa(list(product_prod(B,A)),list(product_prod(B,A)),aa(fun(product_prod(B,A),$o),fun(list(product_prod(B,A)),list(product_prod(B,A))),filter2(product_prod(B,A)),Pa),Xsa)),K3) = none(A) ) ) ) ) ).

% map_of_Some_filter_not_in
tff(fact_5886_map__of__distinct__lookup,axiom,
    ! [A: $tType,B: $tType,Xa: A,Xsa: list(product_prod(A,B)),Ys: list(product_prod(A,B)),Y: B] :
      ( ~ member(A,Xa,aa(list(A),set(A),set2(A),aa(list(product_prod(A,B)),list(A),aa(fun(product_prod(A,B),A),fun(list(product_prod(A,B)),list(A)),map(product_prod(A,B),A),product_fst(A,B)),Xsa)))
     => ( ~ member(A,Xa,aa(list(A),set(A),set2(A),aa(list(product_prod(A,B)),list(A),aa(fun(product_prod(A,B),A),fun(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)),Xsa),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),Xa),Y)),Ys))),Xa) = aa(B,option(B),some(B),Y) ) ) ) ).

% map_of_distinct_lookup
tff(fact_5887_map__of__distinct__upd2,axiom,
    ! [A: $tType,B: $tType,Xa: A,Xsa: list(product_prod(A,B)),Ys: list(product_prod(A,B)),Y: B] :
      ( ~ member(A,Xa,aa(list(A),set(A),set2(A),aa(list(product_prod(A,B)),list(A),aa(fun(product_prod(A,B),A),fun(list(product_prod(A,B)),list(A)),map(product_prod(A,B),A),product_fst(A,B)),Xsa)))
     => ( ~ member(A,Xa,aa(list(A),set(A),set2(A),aa(list(product_prod(A,B)),list(A),aa(fun(product_prod(A,B),A),fun(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)),Xsa),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),Xa),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)),Xsa),Ys)),Xa,aa(B,option(B),some(B),Y)) ) ) ) ).

% map_of_distinct_upd2
tff(fact_5888_less__option__Some,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xa: A,Y: A] :
          ( aa(option(A),$o,aa(option(A),fun(option(A),$o),ord_less(option(A)),aa(A,option(A),some(A),Xa)),aa(A,option(A),some(A),Y))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y) ) ) ).

% less_option_Some
tff(fact_5889_map__project__def,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,option(A)),Aa2: set(B)] : ( map_project(B,A,F2,Aa2) = aa(fun(A,$o),set(A),collect(A),aa(set(B),fun(A,$o),aTP_Lamp_ux(fun(B,option(A)),fun(set(B),fun(A,$o)),F2),Aa2)) ) ).

% map_project_def
tff(fact_5890_Some__SUP,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [Aa2: set(A),F2: fun(A,B)] :
          ( ( Aa2 != 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,F2),Aa2))) = aa(set(option(B)),option(B),complete_Sup_Sup(option(B)),aa(set(A),set(option(B)),image2(A,option(B),aTP_Lamp_uy(fun(A,B),fun(A,option(B)),F2)),Aa2)) ) ) ) ).

% Some_SUP
tff(fact_5891_Some__Sup,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Aa2: set(A)] :
          ( ( Aa2 != bot_bot(set(A)) )
         => ( aa(A,option(A),some(A),aa(set(A),A,complete_Sup_Sup(A),Aa2)) = aa(set(option(A)),option(A),complete_Sup_Sup(option(A)),aa(set(A),set(option(A)),image2(A,option(A),some(A)),Aa2)) ) ) ) ).

% Some_Sup
tff(fact_5892_map__upds__append1,axiom,
    ! [B: $tType,A: $tType,Xsa: list(A),Ys: list(B),M2: fun(A,option(B)),Xa: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),Xsa)),aa(list(B),nat,size_size(list(B)),Ys))
     => ( map_upds(A,B,M2,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),nil(A))),Ys) = fun_upd(A,option(B),map_upds(A,B,M2,Xsa,Ys),Xa,aa(B,option(B),some(B),aa(nat,B,nth(B,Ys),aa(list(A),nat,size_size(list(A)),Xsa)))) ) ) ).

% map_upds_append1
tff(fact_5893_set__to__map__insert,axiom,
    ! [B: $tType,A: $tType,Kv: product_prod(A,B),S: set(product_prod(A,B))] :
      ( ~ member(A,aa(product_prod(A,B),A,product_fst(A,B),Kv),aa(set(product_prod(A,B)),set(A),image2(product_prod(A,B),A,product_fst(A,B)),S))
     => ( set_to_map(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))),insert3(product_prod(A,B)),Kv),S)) = fun_upd(A,option(B),set_to_map(A,B,S),aa(product_prod(A,B),A,product_fst(A,B),Kv),aa(B,option(B),some(B),aa(product_prod(A,B),B,product_snd(A,B),Kv))) ) ) ).

% set_to_map_insert
tff(fact_5894_extract__def,axiom,
    ! [A: $tType,Pa: fun(A,$o),Xsa: list(A)] : ( extract(A,Pa,Xsa) = aa(list(A),option(product_prod(list(A),product_prod(A,list(A)))),aa(fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A)))))),fun(list(A),option(product_prod(list(A),product_prod(A,list(A))))),aa(option(product_prod(list(A),product_prod(A,list(A)))),fun(fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A)))))),fun(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_uz(fun(A,$o),fun(list(A),fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A))))))),Pa),Xsa)),aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),dropWhile(A),comp($o,$o,A,fNot,Pa)),Xsa)) ) ).

% extract_def
tff(fact_5895_map__upds__apply__nontin,axiom,
    ! [B: $tType,A: $tType,Xa: A,Xsa: list(A),F2: fun(A,option(B)),Ys: list(B)] :
      ( ~ member(A,Xa,aa(list(A),set(A),set2(A),Xsa))
     => ( aa(A,option(B),map_upds(A,B,F2,Xsa,Ys),Xa) = aa(A,option(B),F2,Xa) ) ) ).

% map_upds_apply_nontin
tff(fact_5896_fun__upds__append2__drop,axiom,
    ! [A: $tType,B: $tType,Xsa: list(A),Ys: list(B),M2: fun(A,option(B)),Zs: list(B)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xsa) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( map_upds(A,B,M2,Xsa,aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Ys),Zs)) = map_upds(A,B,M2,Xsa,Ys) ) ) ).

% fun_upds_append2_drop
tff(fact_5897_fun__upds__append__drop,axiom,
    ! [A: $tType,B: $tType,Xsa: list(A),Ys: list(B),M2: fun(A,option(B)),Zs: list(A)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xsa) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( map_upds(A,B,M2,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),Zs),Ys) = map_upds(A,B,M2,Xsa,Ys) ) ) ).

% fun_upds_append_drop
tff(fact_5898_set__to__map__empty,axiom,
    ! [A: $tType,B: $tType,X4: A] : ( aa(A,option(B),set_to_map(A,B,bot_bot(set(product_prod(A,B)))),X4) = none(B) ) ).

% set_to_map_empty
tff(fact_5899_map__upds__list__update2__drop,axiom,
    ! [A: $tType,B: $tType,Xsa: list(A),Ia: nat,M2: fun(A,option(B)),Ys: list(B),Y: B] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xsa)),Ia)
     => ( map_upds(A,B,M2,Xsa,aa(B,list(B),aa(nat,fun(B,list(B)),aa(list(B),fun(nat,fun(B,list(B))),list_update(B),Ys),Ia),Y)) = map_upds(A,B,M2,Xsa,Ys) ) ) ).

% map_upds_list_update2_drop
tff(fact_5900_map__upds__twist,axiom,
    ! [A: $tType,B: $tType,A3: A,As2: list(A),M2: fun(A,option(B)),B3: B,Bs: list(B)] :
      ( ~ member(A,A3,aa(list(A),set(A),set2(A),As2))
     => ( map_upds(A,B,fun_upd(A,option(B),M2,A3,aa(B,option(B),some(B),B3)),As2,Bs) = fun_upd(A,option(B),map_upds(A,B,M2,As2,Bs),A3,aa(B,option(B),some(B),B3)) ) ) ).

% map_upds_twist
tff(fact_5901_extract__None__iff,axiom,
    ! [A: $tType,Pa: fun(A,$o),Xsa: list(A)] :
      ( ( extract(A,Pa,Xsa) = none(product_prod(list(A),product_prod(A,list(A)))) )
    <=> ~ ? [X6: A] :
            ( member(A,X6,aa(list(A),set(A),set2(A),Xsa))
            & aa(A,$o,Pa,X6) ) ) ).

% extract_None_iff
tff(fact_5902_extract__Nil__code,axiom,
    ! [A: $tType,Pa: fun(A,$o)] : ( extract(A,Pa,nil(A)) = none(product_prod(list(A),product_prod(A,list(A)))) ) ).

% extract_Nil_code
tff(fact_5903_map__to__set__inverse,axiom,
    ! [B: $tType,A: $tType,M2: fun(A,option(B))] : ( set_to_map(A,B,map_to_set(A,B,M2)) = M2 ) ).

% map_to_set_inverse
tff(fact_5904_set__to__map__empty__iff_I1_J,axiom,
    ! [B: $tType,A: $tType,S: set(product_prod(A,B))] :
      ( ! [X6: A] : ( aa(A,option(B),set_to_map(A,B,S),X6) = none(B) )
    <=> ( S = bot_bot(set(product_prod(A,B))) ) ) ).

% set_to_map_empty_iff(1)
tff(fact_5905_set__to__map__empty__iff_I2_J,axiom,
    ! [B: $tType,A: $tType,S: set(product_prod(A,B))] :
      ( ( aTP_Lamp_um(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_5906_map__upd__upds__conv__if,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,option(B)),Xa: A,Y: B,Xsa: list(A),Ys: list(B)] :
      ( map_upds(A,B,fun_upd(A,option(B),F2,Xa,aa(B,option(B),some(B),Y)),Xsa,Ys) = $ite(member(A,Xa,aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),aa(list(B),nat,size_size(list(B)),Ys)),Xsa))),map_upds(A,B,F2,Xsa,Ys),fun_upd(A,option(B),map_upds(A,B,F2,Xsa,Ys),Xa,aa(B,option(B),some(B),Y))) ) ).

% map_upd_upds_conv_if
tff(fact_5907_set__to__map__inverse,axiom,
    ! [B: $tType,A: $tType,S: set(product_prod(A,B))] :
      ( inj_on(product_prod(A,B),A,product_fst(A,B),S)
     => ( map_to_set(A,B,set_to_map(A,B,S)) = S ) ) ).

% set_to_map_inverse
tff(fact_5908_set__to__map__simp,axiom,
    ! [B: $tType,A: $tType,S: set(product_prod(A,B)),K3: A,V2: B] :
      ( inj_on(product_prod(A,B),A,product_fst(A,B),S)
     => ( ( aa(A,option(B),set_to_map(A,B,S),K3) = aa(B,option(B),some(B),V2) )
      <=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K3),V2),S) ) ) ).

% set_to_map_simp
tff(fact_5909_set__to__map__ran,axiom,
    ! [A: $tType,B: $tType,S: set(product_prod(B,A))] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),ran(B,A,set_to_map(B,A,S))),aa(set(product_prod(B,A)),set(A),image2(product_prod(B,A),A,product_snd(B,A)),S)) ).

% set_to_map_ran
tff(fact_5910_extract__SomeE,axiom,
    ! [A: $tType,Pa: fun(A,$o),Xsa: list(A),Ys: list(A),Y: A,Zs: list(A)] :
      ( ( extract(A,Pa,Xsa) = 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))) )
     => ( ( Xsa = 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,$o,Pa,Y)
        & ~ ? [X4: A] :
              ( member(A,X4,aa(list(A),set(A),set2(A),Ys))
              & aa(A,$o,Pa,X4) ) ) ) ).

% extract_SomeE
tff(fact_5911_extract__Some__iff,axiom,
    ! [A: $tType,Pa: fun(A,$o),Xsa: list(A),Ys: list(A),Y: A,Zs: list(A)] :
      ( ( extract(A,Pa,Xsa) = 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))) )
    <=> ( ( Xsa = 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,$o,Pa,Y)
        & ~ ? [X6: A] :
              ( member(A,X6,aa(list(A),set(A),set2(A),Ys))
              & aa(A,$o,Pa,X6) ) ) ) ).

% extract_Some_iff
tff(fact_5912_extract__Cons__code,axiom,
    ! [A: $tType,Pa: fun(A,$o),Xa: A,Xsa: list(A)] :
      ( extract(A,Pa,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa)) = $ite(aa(A,$o,Pa,Xa),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)),Xa),Xsa))),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_vb(A,fun(list(A),fun(product_prod(A,list(A)),option(product_prod(list(A),product_prod(A,list(A)))))),Xa)),extract(A,Pa,Xsa))) ) ).

% extract_Cons_code
tff(fact_5913_find__dropWhile,axiom,
    ! [A: $tType,Pa: fun(A,$o),Xsa: list(A)] : ( find(A,Pa,Xsa) = aa(list(A),option(A),aa(fun(A,fun(list(A),option(A))),fun(list(A),option(A)),aa(option(A),fun(fun(A,fun(list(A),option(A))),fun(list(A),option(A))),case_list(option(A),A),none(A)),aTP_Lamp_vc(A,fun(list(A),option(A)))),aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),dropWhile(A),comp($o,$o,A,fNot,Pa)),Xsa)) ) ).

% find_dropWhile
tff(fact_5914_graph__map__upd,axiom,
    ! [A: $tType,B: $tType,M2: fun(A,option(B)),K3: A,V2: B] : ( graph(A,B,fun_upd(A,option(B),M2,K3,aa(B,option(B),some(B),V2))) = 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))),insert3(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K3),V2)),graph(A,B,fun_upd(A,option(B),M2,K3,none(B)))) ) ).

% graph_map_upd
tff(fact_5915_find__SomeD_I1_J,axiom,
    ! [A: $tType,Pa: fun(A,$o),Xsa: list(A),Xa: A] :
      ( ( find(A,Pa,Xsa) = aa(A,option(A),some(A),Xa) )
     => aa(A,$o,Pa,Xa) ) ).

% find_SomeD(1)
tff(fact_5916_find__cong,axiom,
    ! [A: $tType,Xsa: list(A),Ys: list(A),Pa: fun(A,$o),Qa: fun(A,$o)] :
      ( ( Xsa = Ys )
     => ( ! [X: A] :
            ( member(A,X,aa(list(A),set(A),set2(A),Ys))
           => ( aa(A,$o,Pa,X)
            <=> aa(A,$o,Qa,X) ) )
       => ( find(A,Pa,Xsa) = find(A,Qa,Ys) ) ) ) ).

% find_cong
tff(fact_5917_in__graphI,axiom,
    ! [A: $tType,B: $tType,M2: fun(B,option(A)),K3: B,V2: A] :
      ( ( aa(B,option(A),M2,K3) = aa(A,option(A),some(A),V2) )
     => member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),K3),V2),graph(B,A,M2)) ) ).

% in_graphI
tff(fact_5918_in__graphD,axiom,
    ! [A: $tType,B: $tType,K3: A,V2: B,M2: fun(A,option(B))] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K3),V2),graph(A,B,M2))
     => ( aa(A,option(B),M2,K3) = aa(B,option(B),some(B),V2) ) ) ).

% in_graphD
tff(fact_5919_find_Osimps_I2_J,axiom,
    ! [A: $tType,Pa: fun(A,$o),Xa: A,Xsa: list(A)] :
      ( find(A,Pa,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa)) = $ite(aa(A,$o,Pa,Xa),aa(A,option(A),some(A),Xa),find(A,Pa,Xsa)) ) ).

% find.simps(2)
tff(fact_5920_find_Osimps_I1_J,axiom,
    ! [A: $tType,Uu: fun(A,$o)] : ( find(A,Uu,nil(A)) = none(A) ) ).

% find.simps(1)
tff(fact_5921_find__SomeD_I2_J,axiom,
    ! [A: $tType,Pa: fun(A,$o),Xsa: list(A),Xa: A] :
      ( ( find(A,Pa,Xsa) = aa(A,option(A),some(A),Xa) )
     => member(A,Xa,aa(list(A),set(A),set2(A),Xsa)) ) ).

% find_SomeD(2)
tff(fact_5922_find__None__iff,axiom,
    ! [A: $tType,Pa: fun(A,$o),Xsa: list(A)] :
      ( ( find(A,Pa,Xsa) = none(A) )
    <=> ~ ? [X6: A] :
            ( member(A,X6,aa(list(A),set(A),set2(A),Xsa))
            & aa(A,$o,Pa,X6) ) ) ).

% find_None_iff
tff(fact_5923_find__None__iff2,axiom,
    ! [A: $tType,Pa: fun(A,$o),Xsa: list(A)] :
      ( ( none(A) = find(A,Pa,Xsa) )
    <=> ~ ? [X6: A] :
            ( member(A,X6,aa(list(A),set(A),set2(A),Xsa))
            & aa(A,$o,Pa,X6) ) ) ).

% find_None_iff2
tff(fact_5924_graph__def,axiom,
    ! [B: $tType,A: $tType,M2: fun(A,option(B))] : ( graph(A,B,M2) = aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aTP_Lamp_vd(fun(A,option(B)),fun(product_prod(A,B),$o),M2)) ) ).

% graph_def
tff(fact_5925_find__Some__iff2,axiom,
    ! [A: $tType,Xa: A,Pa: fun(A,$o),Xsa: list(A)] :
      ( ( aa(A,option(A),some(A),Xa) = find(A,Pa,Xsa) )
    <=> ? [I2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xsa))
          & aa(A,$o,Pa,aa(nat,A,nth(A,Xsa),I2))
          & ( Xa = aa(nat,A,nth(A,Xsa),I2) )
          & ! [J2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J2),I2)
             => ~ aa(A,$o,Pa,aa(nat,A,nth(A,Xsa),J2)) ) ) ) ).

% find_Some_iff2
tff(fact_5926_find__Some__iff,axiom,
    ! [A: $tType,Pa: fun(A,$o),Xsa: list(A),Xa: A] :
      ( ( find(A,Pa,Xsa) = aa(A,option(A),some(A),Xa) )
    <=> ? [I2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xsa))
          & aa(A,$o,Pa,aa(nat,A,nth(A,Xsa),I2))
          & ( Xa = aa(nat,A,nth(A,Xsa),I2) )
          & ! [J2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J2),I2)
             => ~ aa(A,$o,Pa,aa(nat,A,nth(A,Xsa),J2)) ) ) ) ).

% find_Some_iff
tff(fact_5927_graph__map__of__if__distinct__dom,axiom,
    ! [B: $tType,A: $tType,Al: list(product_prod(A,B))] :
      ( aa(list(A),$o,distinct(A),aa(list(product_prod(A,B)),list(A),aa(fun(product_prod(A,B),A),fun(list(product_prod(A,B)),list(A)),map(product_prod(A,B),A),product_fst(A,B)),Al))
     => ( graph(A,B,map_of(A,B,Al)) = aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Al) ) ) ).

% graph_map_of_if_distinct_dom
tff(fact_5928_sorted__find__Min,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xsa: list(A),Pa: fun(A,$o)] :
          ( sorted_wrt(A,ord_less_eq(A),Xsa)
         => ( ? [X4: A] :
                ( member(A,X4,aa(list(A),set(A),set2(A),Xsa))
                & aa(A,$o,Pa,X4) )
           => ( find(A,Pa,Xsa) = aa(A,option(A),some(A),lattic643756798350308766er_Min(A,aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_ve(list(A),fun(fun(A,$o),fun(A,$o)),Xsa),Pa)))) ) ) ) ) ).

% sorted_find_Min
tff(fact_5929_map__of__distinct__upd,axiom,
    ! [A: $tType,B: $tType,Xa: A,Xsa: list(product_prod(A,B)),Y: B] :
      ( ~ member(A,Xa,aa(list(A),set(A),set2(A),aa(list(product_prod(A,B)),list(A),aa(fun(product_prod(A,B),A),fun(list(product_prod(A,B)),list(A)),map(product_prod(A,B),A),product_fst(A,B)),Xsa)))
     => ( map_add(A,B,fun_upd(A,option(B),aTP_Lamp_um(A,option(B)),Xa,aa(B,option(B),some(B),Y)),map_of(A,B,Xsa)) = fun_upd(A,option(B),map_of(A,B,Xsa),Xa,aa(B,option(B),some(B),Y)) ) ) ).

% map_of_distinct_upd
tff(fact_5930_restrict__upd__same,axiom,
    ! [B: $tType,A: $tType,M2: fun(A,option(B)),Xa: A,Y: B] : ( restrict_map(A,B,fun_upd(A,option(B),M2,Xa,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)),insert3(A),Xa),bot_bot(set(A))))) = restrict_map(A,B,M2,aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A))))) ) ).

% restrict_upd_same
tff(fact_5931_restrict__map__upds,axiom,
    ! [A: $tType,B: $tType,Xsa: list(A),Ys: list(B),D2: set(A),M2: fun(A,option(B))] :
      ( ( aa(list(A),nat,size_size(list(A)),Xsa) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xsa)),D2)
       => ( restrict_map(A,B,map_upds(A,B,M2,Xsa,Ys),D2) = map_upds(A,B,restrict_map(A,B,M2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),D2),aa(list(A),set(A),set2(A),Xsa))),Xsa,Ys) ) ) ) ).

% restrict_map_upds
tff(fact_5932_restrict__map__UNIV,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,option(B))] : ( restrict_map(A,B,F2,top_top(set(A))) = F2 ) ).

% restrict_map_UNIV
tff(fact_5933_restrict__map__to__empty,axiom,
    ! [A: $tType,B: $tType,M2: fun(A,option(B)),X4: A] : ( aa(A,option(B),restrict_map(A,B,M2,bot_bot(set(A))),X4) = none(B) ) ).

% restrict_map_to_empty
tff(fact_5934_fun__upd__restrict__conv,axiom,
    ! [A: $tType,B: $tType,Xa: A,D2: set(A),M2: fun(A,option(B)),Y: option(B)] :
      ( member(A,Xa,D2)
     => ( fun_upd(A,option(B),restrict_map(A,B,M2,D2),Xa,Y) = fun_upd(A,option(B),restrict_map(A,B,M2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),D2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A))))),Xa,Y) ) ) ).

% fun_upd_restrict_conv
tff(fact_5935_restrict__fun__upd,axiom,
    ! [B: $tType,A: $tType,M2: fun(A,option(B)),Xa: A,Y: option(B),D2: set(A)] :
      ( restrict_map(A,B,fun_upd(A,option(B),M2,Xa,Y),D2) = $ite(member(A,Xa,D2),fun_upd(A,option(B),restrict_map(A,B,M2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),D2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A))))),Xa,Y),restrict_map(A,B,M2,D2)) ) ).

% restrict_fun_upd
tff(fact_5936_fun__upd__None__restrict,axiom,
    ! [B: $tType,A: $tType,M2: fun(A,option(B)),D2: set(A),Xa: A] :
      ( fun_upd(A,option(B),restrict_map(A,B,M2,D2),Xa,none(B)) = $ite(member(A,Xa,D2),restrict_map(A,B,M2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),D2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A))))),restrict_map(A,B,M2,D2)) ) ).

% fun_upd_None_restrict
tff(fact_5937_map__add__first__le,axiom,
    ! [B: $tType,A: $tType] :
      ( order(B)
     => ! [M2: fun(A,option(B)),M8: fun(A,option(B)),N: fun(A,option(B))] :
          ( aa(fun(A,option(B)),$o,aa(fun(A,option(B)),fun(fun(A,option(B)),$o),ord_less_eq(fun(A,option(B))),M2),M8)
         => aa(fun(A,option(B)),$o,aa(fun(A,option(B)),fun(fun(A,option(B)),$o),ord_less_eq(fun(A,option(B))),map_add(A,B,M2,N)),map_add(A,B,M8,N)) ) ) ).

% map_add_first_le
tff(fact_5938_le__map__restrict,axiom,
    ! [B: $tType,A: $tType] :
      ( order(B)
     => ! [M2: fun(A,option(B)),X3: set(A)] : aa(fun(A,option(B)),$o,aa(fun(A,option(B)),fun(fun(A,option(B)),$o),ord_less_eq(fun(A,option(B))),restrict_map(A,B,M2,X3)),M2) ) ).

% le_map_restrict
tff(fact_5939_restrict__map__subset__eq,axiom,
    ! [B: $tType,A: $tType,M2: fun(A,option(B)),R: set(A),M8: fun(A,option(B)),R8: set(A)] :
      ( ( restrict_map(A,B,M2,R) = M8 )
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),R8),R)
       => ( restrict_map(A,B,M2,R8) = restrict_map(A,B,M8,R8) ) ) ) ).

% restrict_map_subset_eq
tff(fact_5940_map__add__left__None,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,option(A)),K3: B,G: fun(B,option(A))] :
      ( ( aa(B,option(A),F2,K3) = none(A) )
     => ( aa(B,option(A),map_add(B,A,F2,G),K3) = aa(B,option(A),G,K3) ) ) ).

% map_add_left_None
tff(fact_5941_map__add__find__left,axiom,
    ! [A: $tType,B: $tType,G: fun(B,option(A)),K3: B,F2: fun(B,option(A))] :
      ( ( aa(B,option(A),G,K3) = none(A) )
     => ( aa(B,option(A),map_add(B,A,F2,G),K3) = aa(B,option(A),F2,K3) ) ) ).

% map_add_find_left
tff(fact_5942_restrict__map__eq_I2_J,axiom,
    ! [A: $tType,B: $tType,M2: fun(B,option(A)),Aa2: set(B),K3: B,V2: A] :
      ( ( aa(B,option(A),restrict_map(B,A,M2,Aa2),K3) = aa(A,option(A),some(A),V2) )
    <=> ( ( aa(B,option(A),M2,K3) = aa(A,option(A),some(A),V2) )
        & member(B,K3,Aa2) ) ) ).

% restrict_map_eq(2)
tff(fact_5943_restrict__map__insert,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,option(B)),A3: A,Aa2: set(A)] : ( restrict_map(A,B,F2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),Aa2)) = fun_upd(A,option(B),restrict_map(A,B,F2,Aa2),A3,aa(A,option(B),F2,A3)) ) ).

% restrict_map_insert
tff(fact_5944_graph__restrictD_I1_J,axiom,
    ! [B: $tType,A: $tType,K3: A,V2: B,M2: fun(A,option(B)),Aa2: set(A)] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K3),V2),graph(A,B,restrict_map(A,B,M2,Aa2)))
     => member(A,K3,Aa2) ) ).

% graph_restrictD(1)
tff(fact_5945_map__restrict__insert__none__simp,axiom,
    ! [A: $tType,B: $tType,M2: fun(B,option(A)),Xa: B,S2: set(B)] :
      ( ( aa(B,option(A),M2,Xa) = none(A) )
     => ( restrict_map(B,A,M2,aa(set(B),set(B),uminus_uminus(set(B)),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert3(B),Xa),S2))) = restrict_map(B,A,M2,aa(set(B),set(B),uminus_uminus(set(B)),S2)) ) ) ).

% map_restrict_insert_none_simp
tff(fact_5946_restrict__map__upd,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,option(B)),S: set(A),K3: A,V2: B] : ( fun_upd(A,option(B),restrict_map(A,B,F2,S),K3,aa(B,option(B),some(B),V2)) = restrict_map(A,B,fun_upd(A,option(B),F2,K3,aa(B,option(B),some(B),V2)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),K3),S)) ) ).

% restrict_map_upd
tff(fact_5947_graph__restrictD_I2_J,axiom,
    ! [A: $tType,B: $tType,K3: A,V2: B,M2: fun(A,option(B)),Aa2: set(A)] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K3),V2),graph(A,B,restrict_map(A,B,M2,Aa2)))
     => ( aa(A,option(B),M2,K3) = aa(B,option(B),some(B),V2) ) ) ).

% graph_restrictD(2)
tff(fact_5948_fun__upd__restrict,axiom,
    ! [A: $tType,B: $tType,M2: fun(A,option(B)),D2: set(A),Xa: A,Y: option(B)] : ( fun_upd(A,option(B),restrict_map(A,B,M2,D2),Xa,Y) = fun_upd(A,option(B),restrict_map(A,B,M2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),D2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A))))),Xa,Y) ) ).

% fun_upd_restrict
tff(fact_5949_map__of__concat,axiom,
    ! [B: $tType,A: $tType,Xss2: list(list(product_prod(A,B)))] : ( map_of(A,B,aa(list(list(product_prod(A,B))),list(product_prod(A,B)),concat(product_prod(A,B)),Xss2)) = aa(fun(A,option(B)),fun(A,option(B)),aa(list(list(product_prod(A,B))),fun(fun(A,option(B)),fun(A,option(B))),aa(fun(list(product_prod(A,B)),fun(fun(A,option(B)),fun(A,option(B)))),fun(list(list(product_prod(A,B))),fun(fun(A,option(B)),fun(A,option(B)))),foldr(list(product_prod(A,B)),fun(A,option(B))),aTP_Lamp_vf(list(product_prod(A,B)),fun(fun(A,option(B)),fun(A,option(B))))),Xss2),aTP_Lamp_um(A,option(B))) ) ).

% map_of_concat
tff(fact_5950_map__upd__eq__restrict,axiom,
    ! [B: $tType,A: $tType,M2: fun(A,option(B)),Xa: A] : ( fun_upd(A,option(B),M2,Xa,none(B)) = restrict_map(A,B,M2,aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A))))) ) ).

% map_upd_eq_restrict
tff(fact_5951_restrict__complement__singleton__eq,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,option(B)),Xa: A] : ( restrict_map(A,B,F2,aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A))))) = fun_upd(A,option(B),F2,Xa,none(B)) ) ).

% restrict_complement_singleton_eq
tff(fact_5952_map__of__map__restrict,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),Ks: list(A)] : ( map_of(A,B,aa(list(A),list(product_prod(A,B)),aa(fun(A,product_prod(A,B)),fun(list(A),list(product_prod(A,B))),map(A,product_prod(A,B)),aTP_Lamp_tn(fun(A,B),fun(A,product_prod(A,B)),F2)),Ks)) = restrict_map(A,B,comp(B,option(B),A,some(B),F2),aa(list(A),set(A),set2(A),Ks)) ) ).

% map_of_map_restrict
tff(fact_5953_ran__map__upd__Some,axiom,
    ! [B: $tType,A: $tType,M2: fun(B,option(A)),Xa: B,Y: A,Z2: A] :
      ( ( aa(B,option(A),M2,Xa) = aa(A,option(A),some(A),Y) )
     => ( inj_on(B,option(A),M2,dom(B,A,M2))
       => ( ~ member(A,Z2,ran(B,A,M2))
         => ( ran(B,A,fun_upd(B,option(A),M2,Xa,aa(A,option(A),some(A),Z2))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),ran(B,A,M2)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Y),bot_bot(set(A))))),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Z2),bot_bot(set(A)))) ) ) ) ) ).

% ran_map_upd_Some
tff(fact_5954_Eps__Opt__eq__None,axiom,
    ! [A: $tType,Pa: fun(A,$o)] :
      ( ( eps_Opt(A,Pa) = none(A) )
    <=> ~ ? [X_12: A] : aa(A,$o,Pa,X_12) ) ).

% Eps_Opt_eq_None
tff(fact_5955_Chains__init__seg__of__Union,axiom,
    ! [A: $tType,R: set(set(product_prod(A,A))),R2: set(product_prod(A,A))] :
      ( member(set(set(product_prod(A,A))),R,chains(set(product_prod(A,A)),init_seg_of(A)))
     => ( member(set(product_prod(A,A)),R2,R)
       => 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_5956_restrict__map__self,axiom,
    ! [B: $tType,A: $tType,M2: fun(A,option(B))] : ( restrict_map(A,B,M2,dom(A,B,M2)) = M2 ) ).

% restrict_map_self
tff(fact_5957_some__opt__sym__eq__trivial,axiom,
    ! [A: $tType,Xa: A] : ( eps_Opt(A,aa(A,fun(A,$o),fequal(A),Xa)) = aa(A,option(A),some(A),Xa) ) ).

% some_opt_sym_eq_trivial
tff(fact_5958_some__opt__eq__trivial,axiom,
    ! [A: $tType,Xa: A] : ( eps_Opt(A,aTP_Lamp_ad(A,fun(A,$o),Xa)) = aa(A,option(A),some(A),Xa) ) ).

% some_opt_eq_trivial
tff(fact_5959_some__opt__false__trivial,axiom,
    ! [A: $tType] : ( eps_Opt(A,aTP_Lamp_cq(A,$o)) = none(A) ) ).

% some_opt_false_trivial
tff(fact_5960_dom__eq__empty__conv,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,option(B))] :
      ( ( dom(A,B,F2) = bot_bot(set(A)) )
    <=> ! [X6: A] : ( aa(A,option(B),F2,X6) = none(B) ) ) ).

% dom_eq_empty_conv
tff(fact_5961_dom__empty,axiom,
    ! [B: $tType,A: $tType] : ( dom(A,B,aTP_Lamp_um(A,option(B))) = bot_bot(set(A)) ) ).

% dom_empty
tff(fact_5962_map__update__eta__repair_I1_J,axiom,
    ! [B: $tType,A: $tType,K3: A,V2: B,M2: 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_vg(A,fun(B,fun(fun(A,option(B)),fun(A,option(B)))),K3),V2),M2)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),K3),dom(A,B,M2)) ) ).

% map_update_eta_repair(1)
tff(fact_5963_dom__const_H,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B)] : ( dom(A,B,aTP_Lamp_vh(fun(A,B),fun(A,option(B)),F2)) = top_top(set(A)) ) ).

% dom_const'
tff(fact_5964_restrict__map__inv,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,option(B)),X4: A] : ( aa(A,option(B),restrict_map(A,B,F2,aa(set(A),set(A),uminus_uminus(set(A)),dom(A,B,F2))),X4) = none(B) ) ).

% restrict_map_inv
tff(fact_5965_dom__map__of__zip,axiom,
    ! [B: $tType,A: $tType,Xsa: list(A),Ys: list(B)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xsa) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( dom(A,B,map_of(A,B,aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),Xsa),Ys))) = aa(list(A),set(A),set2(A),Xsa) ) ) ).

% dom_map_of_zip
tff(fact_5966_dom__fun__upd,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,option(B)),Xa: A,Y: option(B)] :
      ( dom(A,B,fun_upd(A,option(B),F2,Xa,Y)) = $ite(Y = none(B),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),dom(A,B,F2)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A)))),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),dom(A,B,F2))) ) ).

% dom_fun_upd
tff(fact_5967_dom__map__upds,axiom,
    ! [B: $tType,A: $tType,M2: fun(A,option(B)),Xsa: list(A),Ys: list(B)] : ( dom(A,B,map_upds(A,B,M2,Xsa,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),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),aa(list(B),nat,size_size(list(B)),Ys)),Xsa))),dom(A,B,M2)) ) ).

% dom_map_upds
tff(fact_5968_ran__add,axiom,
    ! [B: $tType,A: $tType,F2: 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,F2)),dom(A,B,G)) = bot_bot(set(A)) )
     => ( ran(A,B,map_add(A,B,F2,G)) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),ran(A,B,F2)),ran(A,B,G)) ) ) ).

% ran_add
tff(fact_5969_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_5970_insert__dom,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,option(A)),Xa: B,Y: A] :
      ( ( aa(B,option(A),F2,Xa) = aa(A,option(A),some(A),Y) )
     => ( aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert3(B),Xa),dom(B,A,F2)) = dom(B,A,F2) ) ) ).

% insert_dom
tff(fact_5971_nempty__dom,axiom,
    ! [B: $tType,A: $tType,E3: fun(A,option(B))] :
      ( ~ ! [X4: A] : ( aa(A,option(B),E3,X4) = none(B) )
     => ~ ! [M6: A] : ~ member(A,M6,dom(A,B,E3)) ) ).

% nempty_dom
tff(fact_5972_le__map__dom__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( preorder(B)
     => ! [M2: fun(A,option(B)),M8: fun(A,option(B))] :
          ( aa(fun(A,option(B)),$o,aa(fun(A,option(B)),fun(fun(A,option(B)),$o),ord_less_eq(fun(A,option(B))),M2),M8)
         => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),dom(A,B,M2)),dom(A,B,M8)) ) ) ).

% le_map_dom_mono
tff(fact_5973_map__dom__ran__finite,axiom,
    ! [B: $tType,A: $tType,M5: fun(A,option(B))] :
      ( aa(set(A),$o,finite_finite2(A),dom(A,B,M5))
     => aa(set(B),$o,finite_finite2(B),ran(A,B,M5)) ) ).

% map_dom_ran_finite
tff(fact_5974_Eps__Opt__eq__Some__implies,axiom,
    ! [A: $tType,Pa: fun(A,$o),Xa: A] :
      ( ( eps_Opt(A,Pa) = aa(A,option(A),some(A),Xa) )
     => aa(A,$o,Pa,Xa) ) ).

% Eps_Opt_eq_Some_implies
tff(fact_5975_Eps__Opt__eq__Some,axiom,
    ! [A: $tType,Pa: fun(A,$o),Xa: A] :
      ( ! [X2: A] :
          ( aa(A,$o,Pa,Xa)
         => ( aa(A,$o,Pa,X2)
           => ( X2 = Xa ) ) )
     => ( ( eps_Opt(A,Pa) = aa(A,option(A),some(A),Xa) )
      <=> aa(A,$o,Pa,Xa) ) ) ).

% Eps_Opt_eq_Some
tff(fact_5976_dom__minus,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,option(A)),Xa: B,Aa2: set(B)] :
      ( ( aa(B,option(A),F2,Xa) = none(A) )
     => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),dom(B,A,F2)),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert3(B),Xa),Aa2)) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),dom(B,A,F2)),Aa2) ) ) ).

% dom_minus
tff(fact_5977_map__add__left__comm,axiom,
    ! [B: $tType,A: $tType,Aa2: fun(A,option(B)),B2: 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,Aa2)),dom(A,B,B2)) = bot_bot(set(A)) )
     => ( map_add(A,B,Aa2,map_add(A,B,B2,C3)) = map_add(A,B,B2,map_add(A,B,Aa2,C3)) ) ) ).

% map_add_left_comm
tff(fact_5978_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_5979_map__add__distinct__le,axiom,
    ! [B: $tType,A: $tType] :
      ( preorder(B)
     => ! [M2: fun(A,option(B)),M8: fun(A,option(B)),N: fun(A,option(B)),N6: fun(A,option(B))] :
          ( aa(fun(A,option(B)),$o,aa(fun(A,option(B)),fun(fun(A,option(B)),$o),ord_less_eq(fun(A,option(B))),M2),M8)
         => ( aa(fun(A,option(B)),$o,aa(fun(A,option(B)),fun(fun(A,option(B)),$o),ord_less_eq(fun(A,option(B))),N),N6)
           => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),dom(A,B,M8)),dom(A,B,N6)) = bot_bot(set(A)) )
             => aa(fun(A,option(B)),$o,aa(fun(A,option(B)),fun(fun(A,option(B)),$o),ord_less_eq(fun(A,option(B))),map_add(A,B,M2,N)),map_add(A,B,M8,N6)) ) ) ) ) ).

% map_add_distinct_le
tff(fact_5980_restrict__map__eq_I1_J,axiom,
    ! [A: $tType,B: $tType,M2: fun(B,option(A)),Aa2: set(B),K3: B] :
      ( ( aa(B,option(A),restrict_map(B,A,M2,Aa2),K3) = none(A) )
    <=> ~ member(B,K3,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),dom(B,A,M2)),Aa2)) ) ).

% restrict_map_eq(1)
tff(fact_5981_map__card__eq__iff,axiom,
    ! [B: $tType,A: $tType,M5: fun(A,option(B)),Xa: A,Y: A] :
      ( aa(set(A),$o,finite_finite2(A),dom(A,B,M5))
     => ( ( finite_card(A,dom(A,B,M5)) = finite_card(B,ran(A,B,M5)) )
       => ( member(A,Xa,dom(A,B,M5))
         => ( ( aa(A,option(B),M5,Xa) = aa(A,option(B),M5,Y) )
          <=> ( Xa = Y ) ) ) ) ) ).

% map_card_eq_iff
tff(fact_5982_finite__map__to__set,axiom,
    ! [B: $tType,A: $tType,M2: fun(A,option(B))] :
      ( aa(set(product_prod(A,B)),$o,finite_finite2(product_prod(A,B)),map_to_set(A,B,M2))
    <=> aa(set(A),$o,finite_finite2(A),dom(A,B,M2)) ) ).

% finite_map_to_set
tff(fact_5983_map__to__set__dom,axiom,
    ! [B: $tType,A: $tType,M2: fun(A,option(B))] : ( dom(A,B,M2) = aa(set(product_prod(A,B)),set(A),image2(product_prod(A,B),A,product_fst(A,B)),map_to_set(A,B,M2)) ) ).

% map_to_set_dom
tff(fact_5984_set__to__map__dom,axiom,
    ! [B: $tType,A: $tType,S: set(product_prod(A,B))] : ( dom(A,B,set_to_map(A,B,S)) = aa(set(product_prod(A,B)),set(A),image2(product_prod(A,B),A,product_fst(A,B)),S) ) ).

% set_to_map_dom
tff(fact_5985_card__map__to__set,axiom,
    ! [B: $tType,A: $tType,M2: fun(A,option(B))] : ( finite_card(product_prod(A,B),map_to_set(A,B,M2)) = finite_card(A,dom(A,B,M2)) ) ).

% card_map_to_set
tff(fact_5986_dom__map__of__conv__image__fst,axiom,
    ! [B: $tType,A: $tType,Xys: list(product_prod(A,B))] : ( dom(A,B,map_of(A,B,Xys)) = aa(set(product_prod(A,B)),set(A),image2(product_prod(A,B),A,product_fst(A,B)),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Xys)) ) ).

% dom_map_of_conv_image_fst
tff(fact_5987_inj__on__map__the,axiom,
    ! [B: $tType,A: $tType,D2: set(A),M2: fun(A,option(B))] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),D2),dom(A,B,M2))
     => ( inj_on(A,option(B),M2,D2)
       => inj_on(A,B,comp(option(B),B,A,the2(B),M2),D2) ) ) ).

% inj_on_map_the
tff(fact_5988_ran__is__image,axiom,
    ! [A: $tType,B: $tType,M5: fun(B,option(A))] : ( ran(B,A,M5) = aa(set(B),set(A),image2(B,A,comp(option(A),A,B,the2(A),M5)),dom(B,A,M5)) ) ).

% ran_is_image
tff(fact_5989_graph__eq__to__snd__dom,axiom,
    ! [B: $tType,A: $tType,M2: fun(A,option(B))] : ( graph(A,B,M2) = aa(set(A),set(product_prod(A,B)),image2(A,product_prod(A,B),aTP_Lamp_vi(fun(A,option(B)),fun(A,product_prod(A,B)),M2)),dom(A,B,M2)) ) ).

% graph_eq_to_snd_dom
tff(fact_5990_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_vj(set(product_prod(A,A)),fun(set(A),$o),R2)) ) ).

% Chains_def
tff(fact_5991_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_5992_dom__eq__singleton__conv,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,option(B)),Xa: A] :
      ( ( dom(A,B,F2) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A))) )
    <=> ? [V4: B] : ( F2 = fun_upd(A,option(B),aTP_Lamp_um(A,option(B)),Xa,aa(B,option(B),some(B),V4)) ) ) ).

% dom_eq_singleton_conv
tff(fact_5993_map__of__map__keys,axiom,
    ! [B: $tType,A: $tType,Xsa: list(A),M2: fun(A,option(B))] :
      ( ( aa(list(A),set(A),set2(A),Xsa) = dom(A,B,M2) )
     => ( map_of(A,B,aa(list(A),list(product_prod(A,B)),aa(fun(A,product_prod(A,B)),fun(list(A),list(product_prod(A,B))),map(A,product_prod(A,B)),aTP_Lamp_vi(fun(A,option(B)),fun(A,product_prod(A,B)),M2)),Xsa)) = M2 ) ) ).

% map_of_map_keys
tff(fact_5994_set__to__map__def,axiom,
    ! [A: $tType,B: $tType,S: set(product_prod(B,A)),K3: B] : ( aa(B,option(A),set_to_map(B,A,S),K3) = eps_Opt(A,aa(B,fun(A,$o),aTP_Lamp_rw(set(product_prod(B,A)),fun(B,fun(A,$o)),S),K3)) ) ).

% set_to_map_def
tff(fact_5995_dom__override__on,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,option(B)),G: fun(A,option(B)),Aa2: set(A)] : ( dom(A,B,override_on(A,option(B),F2,G,Aa2)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),dom(A,B,F2)),aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_vk(fun(A,option(B)),fun(set(A),fun(A,$o)),G),Aa2)))),aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_vl(fun(A,option(B)),fun(set(A),fun(A,$o)),G),Aa2))) ) ).

% dom_override_on
tff(fact_5996_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_cz(set(product_prod(A,A)),fun(A,fun(A,$o)),R2)))),chains(A,R2)) ) ).

% Chains_subset'
tff(fact_5997_override__on__emptyset,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),G: fun(A,B)] : ( override_on(A,B,F2,G,bot_bot(set(A))) = F2 ) ).

% override_on_emptyset
tff(fact_5998_subset__Zorn,axiom,
    ! [A: $tType,Aa2: set(set(A))] :
      ( ! [C9: set(set(A))] :
          ( aa(set(set(A)),$o,pred_chain(set(A),Aa2,ord_less(set(A))),C9)
         => ? [X4: set(A)] :
              ( member(set(A),X4,Aa2)
              & ! [Xa5: set(A)] :
                  ( member(set(A),Xa5,C9)
                 => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Xa5),X4) ) ) )
     => ? [X: set(A)] :
          ( member(set(A),X,Aa2)
          & ! [Xa4: set(A)] :
              ( member(set(A),Xa4,Aa2)
             => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X),Xa4)
               => ( Xa4 = X ) ) ) ) ) ).

% subset_Zorn
tff(fact_5999_subset_Ochain__def,axiom,
    ! [A: $tType,Aa2: set(set(A)),C3: set(set(A))] :
      ( aa(set(set(A)),$o,pred_chain(set(A),Aa2,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),Aa2)
        & ! [X6: set(A)] :
            ( member(set(A),X6,C3)
           => ! [Xa2: set(A)] :
                ( 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))),X6),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),X6) ) ) ) ) ) ).

% subset.chain_def
tff(fact_6000_subset_OchainI,axiom,
    ! [A: $tType,C3: set(set(A)),Aa2: set(set(A))] :
      ( aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),C3),Aa2)
     => ( ! [X: set(A),Y2: set(A)] :
            ( member(set(A),X,C3)
           => ( member(set(A),Y2,C3)
             => ( aa(set(A),$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),$o))),ord_less(set(A))),fequal(set(A))),X),Y2)
                | aa(set(A),$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),$o))),ord_less(set(A))),fequal(set(A))),Y2),X) ) ) )
       => aa(set(set(A)),$o,pred_chain(set(A),Aa2,ord_less(set(A))),C3) ) ) ).

% subset.chainI
tff(fact_6001_subset_Ochain__total,axiom,
    ! [A: $tType,Aa2: set(set(A)),C3: set(set(A)),Xa: set(A),Y: set(A)] :
      ( aa(set(set(A)),$o,pred_chain(set(A),Aa2,ord_less(set(A))),C3)
     => ( member(set(A),Xa,C3)
       => ( 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))),Xa),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),Xa) ) ) ) ) ).

% subset.chain_total
tff(fact_6002_pred__on_Ochain__empty,axiom,
    ! [A: $tType,Aa2: set(A),Pa: fun(A,fun(A,$o))] : aa(set(A),$o,pred_chain(A,Aa2,Pa),bot_bot(set(A))) ).

% pred_on.chain_empty
tff(fact_6003_subset_Ochain__empty,axiom,
    ! [A: $tType,Aa2: set(set(A))] : aa(set(set(A)),$o,pred_chain(set(A),Aa2,ord_less(set(A))),bot_bot(set(set(A)))) ).

% subset.chain_empty
tff(fact_6004_subset__Zorn_H,axiom,
    ! [A: $tType,Aa2: set(set(A))] :
      ( ! [C9: set(set(A))] :
          ( aa(set(set(A)),$o,pred_chain(set(A),Aa2,ord_less(set(A))),C9)
         => member(set(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C9),Aa2) )
     => ? [X: set(A)] :
          ( member(set(A),X,Aa2)
          & ! [Xa4: set(A)] :
              ( member(set(A),Xa4,Aa2)
             => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X),Xa4)
               => ( Xa4 = X ) ) ) ) ) ).

% subset_Zorn'
tff(fact_6005_subset__chain__def,axiom,
    ! [A: $tType,A14: set(set(A)),C10: set(set(A))] :
      ( aa(set(set(A)),$o,pred_chain(set(A),A14,ord_less(set(A))),C10)
    <=> ( aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),C10),A14)
        & ! [X6: set(A)] :
            ( member(set(A),X6,C10)
           => ! [Xa2: set(A)] :
                ( member(set(A),Xa2,C10)
               => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X6),Xa2)
                  | aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Xa2),X6) ) ) ) ) ) ).

% subset_chain_def
tff(fact_6006_subset__chain__insert,axiom,
    ! [A: $tType,A14: set(set(A)),B2: set(A),B14: set(set(A))] :
      ( aa(set(set(A)),$o,pred_chain(set(A),A14,ord_less(set(A))),aa(set(set(A)),set(set(A)),aa(set(A),fun(set(set(A)),set(set(A))),insert3(set(A)),B2),B14))
    <=> ( member(set(A),B2,A14)
        & ! [X6: set(A)] :
            ( member(set(A),X6,B14)
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X6),B2)
              | aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),X6) ) )
        & aa(set(set(A)),$o,pred_chain(set(A),A14,ord_less(set(A))),B14) ) ) ).

% subset_chain_insert
tff(fact_6007_subset__Zorn__nonempty,axiom,
    ! [A: $tType,A14: set(set(A))] :
      ( ( A14 != bot_bot(set(set(A))) )
     => ( ! [C11: set(set(A))] :
            ( ( C11 != bot_bot(set(set(A))) )
           => ( aa(set(set(A)),$o,pred_chain(set(A),A14,ord_less(set(A))),C11)
             => member(set(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C11),A14) ) )
       => ? [X: set(A)] :
            ( member(set(A),X,A14)
            & ! [Xa4: set(A)] :
                ( member(set(A),Xa4,A14)
               => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X),Xa4)
                 => ( Xa4 = X ) ) ) ) ) ) ).

% subset_Zorn_nonempty
tff(fact_6008_override__on__insert_H,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),G: fun(A,B),Xa: A,X3: set(A)] : ( override_on(A,B,F2,G,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),X3)) = override_on(A,B,fun_upd(A,B,F2,Xa,aa(A,B,G,Xa)),G,X3) ) ).

% override_on_insert'
tff(fact_6009_override__on__insert,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),G: fun(A,B),Xa: A,X3: set(A)] : ( override_on(A,B,F2,G,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),X3)) = fun_upd(A,B,override_on(A,B,F2,G,X3),Xa,aa(A,B,G,Xa)) ) ).

% override_on_insert
tff(fact_6010_Union__in__chain,axiom,
    ! [A: $tType,B14: set(set(A)),A14: set(set(A))] :
      ( aa(set(set(A)),$o,finite_finite2(set(A)),B14)
     => ( ( B14 != bot_bot(set(set(A))) )
       => ( aa(set(set(A)),$o,pred_chain(set(A),A14,ord_less(set(A))),B14)
         => member(set(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),B14),B14) ) ) ) ).

% Union_in_chain
tff(fact_6011_Inter__in__chain,axiom,
    ! [A: $tType,B14: set(set(A)),A14: set(set(A))] :
      ( aa(set(set(A)),$o,finite_finite2(set(A)),B14)
     => ( ( B14 != bot_bot(set(set(A))) )
       => ( aa(set(set(A)),$o,pred_chain(set(A),A14,ord_less(set(A))),B14)
         => member(set(A),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),B14),B14) ) ) ) ).

% Inter_in_chain
tff(fact_6012_subset_Ochain__extend,axiom,
    ! [A: $tType,Aa2: set(set(A)),C3: set(set(A)),Z2: set(A)] :
      ( aa(set(set(A)),$o,pred_chain(set(A),Aa2,ord_less(set(A))),C3)
     => ( member(set(A),Z2,Aa2)
       => ( ! [X: set(A)] :
              ( member(set(A),X,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),Z2) )
         => aa(set(set(A)),$o,pred_chain(set(A),Aa2,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))),insert3(set(A)),Z2),bot_bot(set(set(A))))),C3)) ) ) ) ).

% subset.chain_extend
tff(fact_6013_pred__on_Ochain__extend,axiom,
    ! [A: $tType,Aa2: set(A),Pa: fun(A,fun(A,$o)),C3: set(A),Z2: A] :
      ( aa(set(A),$o,pred_chain(A,Aa2,Pa),C3)
     => ( member(A,Z2,Aa2)
       => ( ! [X: A] :
              ( member(A,X,C3)
             => aa(A,$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,$o))),Pa),fequal(A)),X),Z2) )
         => aa(set(A),$o,pred_chain(A,Aa2,Pa),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)),insert3(A),Z2),bot_bot(set(A)))),C3)) ) ) ) ).

% pred_on.chain_extend
tff(fact_6014_finite__subset__Union__chain,axiom,
    ! [A: $tType,Aa2: set(A),B14: set(set(A)),A14: set(set(A))] :
      ( aa(set(A),$o,finite_finite2(A),Aa2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),B14))
       => ( ( B14 != bot_bot(set(set(A))) )
         => ( aa(set(set(A)),$o,pred_chain(set(A),A14,ord_less(set(A))),B14)
           => ~ ! [B5: set(A)] :
                  ( member(set(A),B5,B14)
                 => ~ aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),B5) ) ) ) ) ) ).

% finite_subset_Union_chain
tff(fact_6015_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_cz(set(product_prod(A,A)),fun(A,fun(A,$o)),R2))) ) ) ).

% Chains_alt_def
tff(fact_6016_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_cz(set(product_prod(A,A)),fun(A,fun(A,$o)),R2)))) ).

% Chains_subset
tff(fact_6017_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_6018_eq__f__restr__ss__eq,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F2: fun(fun(A,option(B)),fun(A,option(B))),Aa2: fun(A,option(B))] :
      ( aa(set(A),$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)),F2,Aa2)))
     => ( ( Aa2 = restrict_map(A,B,aa(fun(A,option(B)),fun(A,option(B)),F2,Aa2),aa(set(A),set(A),uminus_uminus(set(A)),S2)) )
      <=> ( map_le(A,B,Aa2,aa(fun(A,option(B)),fun(A,option(B)),F2,Aa2))
          & ( 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)),F2,Aa2))),dom(A,B,Aa2)) ) ) ) ) ).

% eq_f_restr_ss_eq
tff(fact_6019_map__leD,axiom,
    ! [A: $tType,B: $tType,M12: fun(A,option(B)),M23: fun(A,option(B)),K3: A,V2: B] :
      ( map_le(A,B,M12,M23)
     => ( ( aa(A,option(B),M12,K3) = aa(B,option(B),some(B),V2) )
       => ( aa(A,option(B),M23,K3) = aa(B,option(B),some(B),V2) ) ) ) ).

% map_leD
tff(fact_6020_map__leI,axiom,
    ! [B: $tType,A: $tType,M12: fun(A,option(B)),M23: fun(A,option(B))] :
      ( ! [X: A,V3: B] :
          ( ( aa(A,option(B),M12,X) = aa(B,option(B),some(B),V3) )
         => ( aa(A,option(B),M23,X) = aa(B,option(B),some(B),V3) ) )
     => map_le(A,B,M12,M23) ) ).

% map_leI
tff(fact_6021_eq__f__restr__conv,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F2: fun(fun(A,option(B)),fun(A,option(B))),Aa2: fun(A,option(B))] :
      ( ( aa(set(A),$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)),F2,Aa2)))
        & ( Aa2 = restrict_map(A,B,aa(fun(A,option(B)),fun(A,option(B)),F2,Aa2),aa(set(A),set(A),uminus_uminus(set(A)),S2)) ) )
    <=> ( map_le(A,B,Aa2,aa(fun(A,option(B)),fun(A,option(B)),F2,Aa2))
        & ( 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)),F2,Aa2))),dom(A,B,Aa2)) ) ) ) ).

% eq_f_restr_conv
tff(fact_6022_le__map__mmupd__not__dom,axiom,
    ! [A: $tType,B: $tType,M2: fun(A,option(B)),K6: set(A),V2: B] : map_le(A,B,M2,map_mmupd(A,B,M2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),K6),dom(A,B,M2)),V2)) ).

% le_map_mmupd_not_dom
tff(fact_6023_map__mmupd__update__less,axiom,
    ! [A: $tType,B: $tType,K6: set(A),K7: set(A),M2: fun(A,option(B)),V2: B] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),K6),K7)
     => map_le(A,B,map_mmupd(A,B,M2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),K6),dom(A,B,M2)),V2),map_mmupd(A,B,M2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),K7),dom(A,B,M2)),V2)) ) ).

% map_mmupd_update_less
tff(fact_6024_mmupd__notin__upd,axiom,
    ! [B: $tType,A: $tType,K3: A,K6: set(A),M2: fun(A,option(B)),V2: B] :
      ( ~ member(A,K3,K6)
     => ( aa(A,option(B),map_mmupd(A,B,M2,K6,V2),K3) = aa(A,option(B),M2,K3) ) ) ).

% mmupd_notin_upd
tff(fact_6025_map__mmupd__empty,axiom,
    ! [B: $tType,A: $tType,M2: fun(A,option(B)),V2: B] : ( map_mmupd(A,B,M2,bot_bot(set(A)),V2) = M2 ) ).

% map_mmupd_empty
tff(fact_6026_mmupd__in__upd,axiom,
    ! [A: $tType,B: $tType,K3: A,K6: set(A),M2: fun(A,option(B)),V2: B] :
      ( member(A,K3,K6)
     => ( aa(A,option(B),map_mmupd(A,B,M2,K6,V2),K3) = aa(B,option(B),some(B),V2) ) ) ).

% mmupd_in_upd
tff(fact_6027_dom__mmupd,axiom,
    ! [B: $tType,A: $tType,M2: fun(A,option(B)),K6: set(A),V2: B] : ( dom(A,B,map_mmupd(A,B,M2,K6,V2)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),dom(A,B,M2)),K6) ) ).

% dom_mmupd
tff(fact_6028_map__mmupd__def,axiom,
    ! [A: $tType,B: $tType,M2: fun(B,option(A)),K6: set(B),V2: A,K3: B] :
      ( aa(B,option(A),map_mmupd(B,A,M2,K6,V2),K3) = $ite(member(B,K3,K6),aa(A,option(A),some(A),V2),aa(B,option(A),M2,K3)) ) ).

% map_mmupd_def
tff(fact_6029_map__mmupdE,axiom,
    ! [B: $tType,A: $tType,M2: fun(B,option(A)),K6: set(B),V2: A,K3: B,Xa: A] :
      ( ( aa(B,option(A),map_mmupd(B,A,M2,K6,V2),K3) = aa(A,option(A),some(A),Xa) )
     => ( ( ~ member(B,K3,K6)
         => ( aa(B,option(A),M2,K3) != aa(A,option(A),some(A),Xa) ) )
       => ~ ( member(B,K3,K6)
           => ( Xa != V2 ) ) ) ) ).

% map_mmupdE
tff(fact_6030_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_6031_the__dflt__None__empty,axiom,
    ! [A: $tType] : ( dflt_None_set(A,bot_bot(set(A))) = none(set(A)) ) ).

% the_dflt_None_empty
tff(fact_6032_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_6033_the__dflt__None__set,axiom,
    ! [A: $tType,Xa: set(A)] : ( the_default(set(A),bot_bot(set(A)),dflt_None_set(A,Xa)) = Xa ) ).

% the_dflt_None_set
tff(fact_6034_Func__empty,axiom,
    ! [B: $tType,A: $tType,B2: set(B)] : ( bNF_Wellorder_Func(A,B,bot_bot(set(A)),B2) = aa(set(fun(A,B)),set(fun(A,B)),aa(fun(A,B),fun(set(fun(A,B)),set(fun(A,B))),insert3(fun(A,B)),aTP_Lamp_vm(A,B)),bot_bot(set(fun(A,B)))) ) ).

% Func_empty
tff(fact_6035_Func__non__emp,axiom,
    ! [A: $tType,B: $tType,B2: set(A),Aa2: set(B)] :
      ( ( B2 != bot_bot(set(A)) )
     => ( bNF_Wellorder_Func(B,A,Aa2,B2) != bot_bot(set(fun(B,A))) ) ) ).

% Func_non_emp
tff(fact_6036_Func__is__emp,axiom,
    ! [A: $tType,B: $tType,Aa2: set(A),B2: set(B)] :
      ( ( bNF_Wellorder_Func(A,B,Aa2,B2) = bot_bot(set(fun(A,B))) )
    <=> ( ( Aa2 != bot_bot(set(A)) )
        & ( B2 = bot_bot(set(B)) ) ) ) ).

% Func_is_emp
tff(fact_6037_the__default_Osimps_I1_J,axiom,
    ! [A: $tType,Uu: A,Xa: A] : ( the_default(A,Uu,aa(A,option(A),some(A),Xa)) = Xa ) ).

% the_default.simps(1)
tff(fact_6038_the__default_Osimps_I2_J,axiom,
    ! [A: $tType,Xa: A] : ( the_default(A,Xa,none(A)) = Xa ) ).

% the_default.simps(2)
tff(fact_6039_Func__map__surj,axiom,
    ! [E: $tType,A: $tType,F4: $tType,B: $tType,F1: fun(B,A),A15: set(B),B15: set(A),F23: fun(E,F4),B23: set(E),A23: set(F4)] :
      ( ( aa(set(B),set(A),image2(B,A,F1),A15) = B15 )
     => ( inj_on(E,F4,F23,B23)
       => ( aa(set(F4),$o,aa(set(F4),fun(set(F4),$o),ord_less_eq(set(F4)),aa(set(E),set(F4),image2(E,F4,F23),B23)),A23)
         => ( ( ( B23 = bot_bot(set(E)) )
             => ( A23 = bot_bot(set(F4)) ) )
           => ( bNF_Wellorder_Func(E,A,B23,B15) = aa(set(fun(F4,B)),set(fun(E,A)),image2(fun(F4,B),fun(E,A),bNF_We4925052301507509544nc_map(E,B,A,F4,B23,F1,F23)),bNF_Wellorder_Func(F4,B,A23,A15)) ) ) ) ) ) ).

% Func_map_surj
tff(fact_6040_If__the__inv__into__in__Func,axiom,
    ! [B: $tType,A: $tType,G: fun(A,B),C3: set(A),B2: set(A),Xa: A] :
      ( inj_on(A,B,G,C3)
     => ( aa(set(A),$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)),B2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A)))))
       => member(fun(B,A),aa(A,fun(B,A),aa(set(A),fun(A,fun(B,A)),aTP_Lamp_vn(fun(A,B),fun(set(A),fun(A,fun(B,A))),G),C3),Xa),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)),B2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A)))))) ) ) ).

% If_the_inv_into_in_Func
tff(fact_6041_flat__lub__def,axiom,
    ! [A: $tType,B3: A,Aa2: set(A)] :
      ( aa(set(A),A,partial_flat_lub(A,B3),Aa2) = $ite(aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),B3),bot_bot(set(A)))),B3,the(A,aa(set(A),fun(A,$o),aTP_Lamp_vo(A,fun(set(A),fun(A,$o)),B3),Aa2))) ) ).

% flat_lub_def
tff(fact_6042_insort__insert__key__def,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),Xa: A,Xsa: list(A)] :
          ( linord329482645794927042rt_key(A,B,F2,Xa,Xsa) = $ite(member(B,aa(A,B,F2,Xa),aa(set(A),set(B),image2(A,B,F2),aa(list(A),set(A),set2(A),Xsa))),Xsa,aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,B,F2),Xa),Xsa)) ) ) ).

% insort_insert_key_def
tff(fact_6043_distinct__insort__insert,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Xsa: list(A),F2: fun(A,B),Xa: A] :
          ( aa(list(A),$o,distinct(A),Xsa)
         => aa(list(A),$o,distinct(A),linord329482645794927042rt_key(A,B,F2,Xa,Xsa)) ) ) ).

% distinct_insort_insert
tff(fact_6044_insort__insert__triv,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Xsa: list(A)] :
          ( member(A,Xa,aa(list(A),set(A),set2(A),Xsa))
         => ( linord329482645794927042rt_key(A,A,aTP_Lamp_ip(A,A),Xa,Xsa) = Xsa ) ) ) ).

% insort_insert_triv
tff(fact_6045_insort__insert__key__triv,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F2: fun(B,A),Xa: B,Xsa: list(B)] :
          ( member(A,aa(B,A,F2,Xa),aa(set(B),set(A),image2(B,A,F2),aa(list(B),set(B),set2(B),Xsa)))
         => ( linord329482645794927042rt_key(B,A,F2,Xa,Xsa) = Xsa ) ) ) ).

% insort_insert_key_triv
tff(fact_6046_set__insort__insert,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Xsa: list(A)] : ( aa(list(A),set(A),set2(A),linord329482645794927042rt_key(A,A,aTP_Lamp_ip(A,A),Xa,Xsa)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),aa(list(A),set(A),set2(A),Xsa)) ) ) ).

% set_insort_insert
tff(fact_6047_sorted__insort__insert,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xsa: list(A),Xa: A] :
          ( sorted_wrt(A,ord_less_eq(A),Xsa)
         => sorted_wrt(A,ord_less_eq(A),linord329482645794927042rt_key(A,A,aTP_Lamp_ip(A,A),Xa,Xsa)) ) ) ).

% sorted_insort_insert
tff(fact_6048_insort__insert__insort,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Xsa: list(A)] :
          ( ~ member(A,Xa,aa(list(A),set(A),set2(A),Xsa))
         => ( linord329482645794927042rt_key(A,A,aTP_Lamp_ip(A,A),Xa,Xsa) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_ip(A,A)),Xa),Xsa) ) ) ) ).

% insort_insert_insort
tff(fact_6049_sorted__insort__insert__key,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F2: fun(B,A),Xsa: list(B),Xa: B] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Xsa))
         => sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),linord329482645794927042rt_key(B,A,F2,Xa,Xsa))) ) ) ).

% sorted_insort_insert_key
tff(fact_6050_insort__insert__insort__key,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F2: fun(B,A),Xa: B,Xsa: list(B)] :
          ( ~ member(A,aa(B,A,F2,Xa),aa(set(B),set(A),image2(B,A,F2),aa(list(B),set(B),set2(B),Xsa)))
         => ( linord329482645794927042rt_key(B,A,F2,Xa,Xsa) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F2),Xa),Xsa) ) ) ) ).

% insort_insert_insort_key
tff(fact_6051_maps__def,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,list(A)),Xsa: list(B)] : ( maps(B,A,F2,Xsa) = aa(list(list(A)),list(A),concat(A),aa(list(B),list(list(A)),aa(fun(B,list(A)),fun(list(B),list(list(A))),map(B,list(A)),F2),Xsa)) ) ).

% maps_def
tff(fact_6052_concat__map__maps,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,list(A)),Xsa: list(B)] : ( aa(list(list(A)),list(A),concat(A),aa(list(B),list(list(A)),aa(fun(B,list(A)),fun(list(B),list(list(A))),map(B,list(A)),F2),Xsa)) = maps(B,A,F2,Xsa) ) ).

% concat_map_maps
tff(fact_6053_maps__simps_I2_J,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,list(A))] : ( maps(B,A,F2,nil(B)) = nil(A) ) ).

% maps_simps(2)
tff(fact_6054_maps__simps_I1_J,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,list(A)),Xa: B,Xsa: list(B)] : ( maps(B,A,F2,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Xa),Xsa)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(B,list(A),F2,Xa)),maps(B,A,F2,Xsa)) ) ).

% maps_simps(1)
tff(fact_6055_mset__set__Union,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),Aa2)
     => ( aa(set(A),$o,finite_finite2(A),B2)
       => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),B2) = bot_bot(set(A)) )
         => ( mset_set(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),B2)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),mset_set(A,Aa2)),mset_set(A,B2)) ) ) ) ) ).

% mset_set_Union
tff(fact_6056_Partial__order__eq__Image1__Image1__iff,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A3: A,B3: A] :
      ( order_7125193373082350890der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => ( member(A,A3,aa(set(product_prod(A,A)),set(A),field2(A),R2))
       => ( member(A,B3,aa(set(product_prod(A,A)),set(A),field2(A),R2))
         => ( ( image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),bot_bot(set(A)))) = image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),B3),bot_bot(set(A)))) )
          <=> ( A3 = B3 ) ) ) ) ) ).

% Partial_order_eq_Image1_Image1_iff
tff(fact_6057_mset__set_Oempty,axiom,
    ! [A: $tType] : ( mset_set(A,bot_bot(set(A))) = zero_zero(multiset(A)) ) ).

% mset_set.empty
tff(fact_6058_mset__set_Oinsert,axiom,
    ! [A: $tType,Aa2: set(A),Xa: A] :
      ( aa(set(A),$o,finite_finite2(A),Aa2)
     => ( ~ member(A,Xa,Aa2)
       => ( mset_set(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),Aa2)) = add_mset(A,Xa,mset_set(A,Aa2)) ) ) ) ).

% mset_set.insert
tff(fact_6059_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_6060_mset__set__empty__iff,axiom,
    ! [A: $tType,Aa2: set(A)] :
      ( ( mset_set(A,Aa2) = zero_zero(multiset(A)) )
    <=> ( ( Aa2 = bot_bot(set(A)) )
        | ~ aa(set(A),$o,finite_finite2(A),Aa2) ) ) ).

% mset_set_empty_iff
tff(fact_6061_infinite__set__mset__mset__set,axiom,
    ! [A: $tType,Aa2: set(A)] :
      ( ~ aa(set(A),$o,finite_finite2(A),Aa2)
     => ( aa(multiset(A),set(A),set_mset(A),mset_set(A,Aa2)) = bot_bot(set(A)) ) ) ).

% infinite_set_mset_mset_set
tff(fact_6062_mset__set__set,axiom,
    ! [A: $tType,Xsa: list(A)] :
      ( aa(list(A),$o,distinct(A),Xsa)
     => ( mset_set(A,aa(list(A),set(A),set2(A),Xsa)) = aa(list(A),multiset(A),mset(A),Xsa) ) ) ).

% mset_set_set
tff(fact_6063_mset__set__Diff,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),Aa2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),Aa2)
       => ( mset_set(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),B2)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),mset_set(A,Aa2)),mset_set(A,B2)) ) ) ) ).

% mset_set_Diff
tff(fact_6064_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(member(A,A3,S),one_one(nat),zero_zero(nat)) ) ) ).

% count_mset_set_finite_iff
tff(fact_6065_Zorns__po__lemma,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( order_7125193373082350890der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => ( ! [C9: set(A)] :
            ( member(set(A),C9,chains(A,R2))
           => ? [X4: A] :
                ( member(A,X4,aa(set(product_prod(A,A)),set(A),field2(A),R2))
                & ! [Xa5: A] :
                    ( member(A,Xa5,C9)
                   => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa5),X4),R2) ) ) )
       => ? [X: A] :
            ( member(A,X,aa(set(product_prod(A,A)),set(A),field2(A),R2))
            & ! [Xa4: A] :
                ( member(A,Xa4,aa(set(product_prod(A,A)),set(A),field2(A),R2))
               => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Xa4),R2)
                 => ( Xa4 = X ) ) ) ) ) ) ).

% Zorns_po_lemma
tff(fact_6066_mset__set_Oremove,axiom,
    ! [A: $tType,Aa2: set(A),Xa: A] :
      ( aa(set(A),$o,finite_finite2(A),Aa2)
     => ( member(A,Xa,Aa2)
       => ( mset_set(A,Aa2) = add_mset(A,Xa,mset_set(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A)))))) ) ) ) ).

% mset_set.remove
tff(fact_6067_mset__set_Oinsert__remove,axiom,
    ! [A: $tType,Aa2: set(A),Xa: A] :
      ( aa(set(A),$o,finite_finite2(A),Aa2)
     => ( mset_set(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),Aa2)) = add_mset(A,Xa,mset_set(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A)))))) ) ) ).

% mset_set.insert_remove
tff(fact_6068_Sup__fin_Oremove,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Aa2: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( member(A,Xa,Aa2)
           => ( lattic5882676163264333800up_fin(A,Aa2) = $ite(aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A)))) = bot_bot(set(A)),Xa,aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),lattic5882676163264333800up_fin(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A))))))) ) ) ) ) ).

% Sup_fin.remove
tff(fact_6069_Sup__fin_Oinsert__remove,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Aa2: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( lattic5882676163264333800up_fin(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),Aa2)) = $ite(aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A)))) = bot_bot(set(A)),Xa,aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),lattic5882676163264333800up_fin(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A))))))) ) ) ) ).

% Sup_fin.insert_remove
tff(fact_6070_Sup__fin_Osingleton,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xa: A] : ( lattic5882676163264333800up_fin(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A)))) = Xa ) ) ).

% Sup_fin.singleton
tff(fact_6071_Sup__fin_Oinsert,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Aa2: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( ( Aa2 != bot_bot(set(A)) )
           => ( lattic5882676163264333800up_fin(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),Aa2)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),lattic5882676163264333800up_fin(A,Aa2)) ) ) ) ) ).

% Sup_fin.insert
tff(fact_6072_Sup__fin_Obounded__iff,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Aa2: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( ( Aa2 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),lattic5882676163264333800up_fin(A,Aa2)),Xa)
            <=> ! [X6: A] :
                  ( member(A,X6,Aa2)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X6),Xa) ) ) ) ) ) ).

% Sup_fin.bounded_iff
tff(fact_6073_Sup__fin_OboundedI,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Aa2: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( ( Aa2 != bot_bot(set(A)) )
           => ( ! [A4: A] :
                  ( member(A,A4,Aa2)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),Xa) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),lattic5882676163264333800up_fin(A,Aa2)),Xa) ) ) ) ) ).

% Sup_fin.boundedI
tff(fact_6074_Sup__fin_OboundedE,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Aa2: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( ( Aa2 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),lattic5882676163264333800up_fin(A,Aa2)),Xa)
             => ! [A13: A] :
                  ( member(A,A13,Aa2)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A13),Xa) ) ) ) ) ) ).

% Sup_fin.boundedE
tff(fact_6075_cSup__eq__Sup__fin,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X3: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),X3)
         => ( ( X3 != bot_bot(set(A)) )
           => ( aa(set(A),A,complete_Sup_Sup(A),X3) = lattic5882676163264333800up_fin(A,X3) ) ) ) ) ).

% cSup_eq_Sup_fin
tff(fact_6076_Sup__fin__Sup,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Aa2: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( ( Aa2 != bot_bot(set(A)) )
           => ( lattic5882676163264333800up_fin(A,Aa2) = aa(set(A),A,complete_Sup_Sup(A),Aa2) ) ) ) ) ).

% Sup_fin_Sup
tff(fact_6077_Sup__fin_Osubset__imp,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Aa2: set(A),B2: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),B2)
         => ( ( Aa2 != bot_bot(set(A)) )
           => ( aa(set(A),$o,finite_finite2(A),B2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),lattic5882676163264333800up_fin(A,Aa2)),lattic5882676163264333800up_fin(A,B2)) ) ) ) ) ).

% Sup_fin.subset_imp
tff(fact_6078_Sup__fin_Ohom__commute,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [H2: fun(A,A),N7: set(A)] :
          ( ! [X: A,Y2: A] : ( aa(A,A,H2,aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y2)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,H2,X)),aa(A,A,H2,Y2)) )
         => ( aa(set(A),$o,finite_finite2(A),N7)
           => ( ( N7 != bot_bot(set(A)) )
             => ( aa(A,A,H2,lattic5882676163264333800up_fin(A,N7)) = lattic5882676163264333800up_fin(A,aa(set(A),set(A),image2(A,A,H2),N7)) ) ) ) ) ) ).

% Sup_fin.hom_commute
tff(fact_6079_Sup__fin_Osubset,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Aa2: set(A),B2: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( ( B2 != bot_bot(set(A)) )
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),Aa2)
             => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),lattic5882676163264333800up_fin(A,B2)),lattic5882676163264333800up_fin(A,Aa2)) = lattic5882676163264333800up_fin(A,Aa2) ) ) ) ) ) ).

% Sup_fin.subset
tff(fact_6080_Sup__fin_Oclosed,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Aa2: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( ( Aa2 != bot_bot(set(A)) )
           => ( ! [X: A,Y2: A] : member(A,aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),X),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Y2),bot_bot(set(A)))))
             => member(A,lattic5882676163264333800up_fin(A,Aa2),Aa2) ) ) ) ) ).

% Sup_fin.closed
tff(fact_6081_Sup__fin_Oinsert__not__elem,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Aa2: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( ~ member(A,Xa,Aa2)
           => ( ( Aa2 != bot_bot(set(A)) )
             => ( lattic5882676163264333800up_fin(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),Aa2)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),lattic5882676163264333800up_fin(A,Aa2)) ) ) ) ) ) ).

% Sup_fin.insert_not_elem
tff(fact_6082_Sup__fin_Ounion,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Aa2: set(A),B2: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( ( Aa2 != bot_bot(set(A)) )
           => ( aa(set(A),$o,finite_finite2(A),B2)
             => ( ( B2 != bot_bot(set(A)) )
               => ( lattic5882676163264333800up_fin(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),B2)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),lattic5882676163264333800up_fin(A,Aa2)),lattic5882676163264333800up_fin(A,B2)) ) ) ) ) ) ) ).

% Sup_fin.union
tff(fact_6083_Sup__fin_Oeq__fold,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Aa2: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( lattic5882676163264333800up_fin(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),Aa2)) = finite_fold(A,A,sup_sup(A),Xa,Aa2) ) ) ) ).

% Sup_fin.eq_fold
tff(fact_6084_Sup__fin_Oset__eq__fold,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xa: A,Xsa: list(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),Xa),Xsa))) = aa(A,A,aa(list(A),fun(A,A),aa(fun(A,fun(A,A)),fun(list(A),fun(A,A)),fold(A,A),sup_sup(A)),Xsa),Xa) ) ) ).

% Sup_fin.set_eq_fold
tff(fact_6085_inf__Sup1__distrib,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [Aa2: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( ( Aa2 != bot_bot(set(A)) )
           => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),lattic5882676163264333800up_fin(A,Aa2)) = lattic5882676163264333800up_fin(A,aa(fun(A,$o),set(A),collect(A),aa(A,fun(A,$o),aTP_Lamp_vp(set(A),fun(A,fun(A,$o)),Aa2),Xa))) ) ) ) ) ).

% inf_Sup1_distrib
tff(fact_6086_inf__Sup2__distrib,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [Aa2: set(A),B2: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( ( Aa2 != bot_bot(set(A)) )
           => ( aa(set(A),$o,finite_finite2(A),B2)
             => ( ( B2 != bot_bot(set(A)) )
               => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),lattic5882676163264333800up_fin(A,Aa2)),lattic5882676163264333800up_fin(A,B2)) = lattic5882676163264333800up_fin(A,aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_vq(set(A),fun(set(A),fun(A,$o)),Aa2),B2))) ) ) ) ) ) ) ).

% inf_Sup2_distrib
tff(fact_6087_Inf__fin_Oinsert__remove,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Aa2: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( lattic7752659483105999362nf_fin(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),Aa2)) = $ite(aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A)))) = bot_bot(set(A)),Xa,aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),lattic7752659483105999362nf_fin(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A))))))) ) ) ) ).

% Inf_fin.insert_remove
tff(fact_6088_Inf__fin_Oremove,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Aa2: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( member(A,Xa,Aa2)
           => ( lattic7752659483105999362nf_fin(A,Aa2) = $ite(aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A)))) = bot_bot(set(A)),Xa,aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),lattic7752659483105999362nf_fin(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A))))))) ) ) ) ) ).

% Inf_fin.remove
tff(fact_6089_Inf__fin_Osingleton,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xa: A] : ( lattic7752659483105999362nf_fin(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A)))) = Xa ) ) ).

% Inf_fin.singleton
tff(fact_6090_Inf__fin_Oinsert,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Aa2: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( ( Aa2 != bot_bot(set(A)) )
           => ( lattic7752659483105999362nf_fin(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),Aa2)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),lattic7752659483105999362nf_fin(A,Aa2)) ) ) ) ) ).

% Inf_fin.insert
tff(fact_6091_Inf__fin_OboundedE,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Aa2: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( ( Aa2 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),lattic7752659483105999362nf_fin(A,Aa2))
             => ! [A13: A] :
                  ( member(A,A13,Aa2)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),A13) ) ) ) ) ) ).

% Inf_fin.boundedE
tff(fact_6092_Inf__fin_OboundedI,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Aa2: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( ( Aa2 != bot_bot(set(A)) )
           => ( ! [A4: A] :
                  ( member(A,A4,Aa2)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),A4) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),lattic7752659483105999362nf_fin(A,Aa2)) ) ) ) ) ).

% Inf_fin.boundedI
tff(fact_6093_Inf__fin_Obounded__iff,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Aa2: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( ( Aa2 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),lattic7752659483105999362nf_fin(A,Aa2))
            <=> ! [X6: A] :
                  ( member(A,X6,Aa2)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),X6) ) ) ) ) ) ).

% Inf_fin.bounded_iff
tff(fact_6094_Inf__fin__Inf,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Aa2: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( ( Aa2 != bot_bot(set(A)) )
           => ( lattic7752659483105999362nf_fin(A,Aa2) = aa(set(A),A,complete_Inf_Inf(A),Aa2) ) ) ) ) ).

% Inf_fin_Inf
tff(fact_6095_cInf__eq__Inf__fin,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X3: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),X3)
         => ( ( X3 != bot_bot(set(A)) )
           => ( aa(set(A),A,complete_Inf_Inf(A),X3) = lattic7752659483105999362nf_fin(A,X3) ) ) ) ) ).

% cInf_eq_Inf_fin
tff(fact_6096_Inf__fin_Osubset__imp,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Aa2: set(A),B2: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),B2)
         => ( ( Aa2 != bot_bot(set(A)) )
           => ( aa(set(A),$o,finite_finite2(A),B2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),lattic7752659483105999362nf_fin(A,B2)),lattic7752659483105999362nf_fin(A,Aa2)) ) ) ) ) ).

% Inf_fin.subset_imp
tff(fact_6097_Inf__fin_Ohom__commute,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [H2: fun(A,A),N7: set(A)] :
          ( ! [X: A,Y2: A] : ( aa(A,A,H2,aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y2)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,H2,X)),aa(A,A,H2,Y2)) )
         => ( aa(set(A),$o,finite_finite2(A),N7)
           => ( ( N7 != bot_bot(set(A)) )
             => ( aa(A,A,H2,lattic7752659483105999362nf_fin(A,N7)) = lattic7752659483105999362nf_fin(A,aa(set(A),set(A),image2(A,A,H2),N7)) ) ) ) ) ) ).

% Inf_fin.hom_commute
tff(fact_6098_Inf__fin_Osubset,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Aa2: set(A),B2: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( ( B2 != bot_bot(set(A)) )
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),Aa2)
             => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),lattic7752659483105999362nf_fin(A,B2)),lattic7752659483105999362nf_fin(A,Aa2)) = lattic7752659483105999362nf_fin(A,Aa2) ) ) ) ) ) ).

% Inf_fin.subset
tff(fact_6099_Inf__fin_Oinsert__not__elem,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Aa2: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( ~ member(A,Xa,Aa2)
           => ( ( Aa2 != bot_bot(set(A)) )
             => ( lattic7752659483105999362nf_fin(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),Aa2)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),lattic7752659483105999362nf_fin(A,Aa2)) ) ) ) ) ) ).

% Inf_fin.insert_not_elem
tff(fact_6100_Inf__fin_Oclosed,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Aa2: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( ( Aa2 != bot_bot(set(A)) )
           => ( ! [X: A,Y2: A] : member(A,aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),X),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Y2),bot_bot(set(A)))))
             => member(A,lattic7752659483105999362nf_fin(A,Aa2),Aa2) ) ) ) ) ).

% Inf_fin.closed
tff(fact_6101_Inf__fin_Ounion,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Aa2: set(A),B2: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( ( Aa2 != bot_bot(set(A)) )
           => ( aa(set(A),$o,finite_finite2(A),B2)
             => ( ( B2 != bot_bot(set(A)) )
               => ( lattic7752659483105999362nf_fin(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),B2)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),lattic7752659483105999362nf_fin(A,Aa2)),lattic7752659483105999362nf_fin(A,B2)) ) ) ) ) ) ) ).

% Inf_fin.union
tff(fact_6102_Inf__fin_Oeq__fold,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Aa2: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( lattic7752659483105999362nf_fin(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),Aa2)) = finite_fold(A,A,inf_inf(A),Xa,Aa2) ) ) ) ).

% Inf_fin.eq_fold
tff(fact_6103_Inf__fin__le__Sup__fin,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Aa2: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( ( Aa2 != bot_bot(set(A)) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),lattic7752659483105999362nf_fin(A,Aa2)),lattic5882676163264333800up_fin(A,Aa2)) ) ) ) ).

% Inf_fin_le_Sup_fin
tff(fact_6104_Inf__fin_Oset__eq__fold,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xa: A,Xsa: list(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),Xa),Xsa))) = aa(A,A,aa(list(A),fun(A,A),aa(fun(A,fun(A,A)),fun(list(A),fun(A,A)),fold(A,A),inf_inf(A)),Xsa),Xa) ) ) ).

% Inf_fin.set_eq_fold
tff(fact_6105_sup__Inf1__distrib,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [Aa2: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( ( Aa2 != bot_bot(set(A)) )
           => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),lattic7752659483105999362nf_fin(A,Aa2)) = lattic7752659483105999362nf_fin(A,aa(fun(A,$o),set(A),collect(A),aa(A,fun(A,$o),aTP_Lamp_vr(set(A),fun(A,fun(A,$o)),Aa2),Xa))) ) ) ) ) ).

% sup_Inf1_distrib
tff(fact_6106_sup__Inf2__distrib,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [Aa2: set(A),B2: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( ( Aa2 != bot_bot(set(A)) )
           => ( aa(set(A),$o,finite_finite2(A),B2)
             => ( ( B2 != bot_bot(set(A)) )
               => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),lattic7752659483105999362nf_fin(A,Aa2)),lattic7752659483105999362nf_fin(A,B2)) = lattic7752659483105999362nf_fin(A,aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_vs(set(A),fun(set(A),fun(A,$o)),Aa2),B2))) ) ) ) ) ) ) ).

% sup_Inf2_distrib
tff(fact_6107_eventually__INF__base,axiom,
    ! [B: $tType,A: $tType,B2: set(A),Fa: fun(A,filter(B)),Pa: fun(B,$o)] :
      ( ( B2 != bot_bot(set(A)) )
     => ( ! [A4: A] :
            ( member(A,A4,B2)
           => ! [B4: A] :
                ( member(A,B4,B2)
               => ? [X4: A] :
                    ( member(A,X4,B2)
                    & aa(filter(B),$o,aa(filter(B),fun(filter(B),$o),ord_less_eq(filter(B)),aa(A,filter(B),Fa,X4)),aa(filter(B),filter(B),aa(filter(B),fun(filter(B),filter(B)),inf_inf(filter(B)),aa(A,filter(B),Fa,A4)),aa(A,filter(B),Fa,B4))) ) ) )
       => ( aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),eventually(B),Pa),aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image2(A,filter(B),Fa),B2)))
        <=> ? [X6: A] :
              ( member(A,X6,B2)
              & aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),eventually(B),Pa),aa(A,filter(B),Fa,X6)) ) ) ) ) ).

% eventually_INF_base
tff(fact_6108_prod__mset_Ounion__disjoint,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [Aa2: multiset(A),B2: multiset(A),G: fun(A,B)] :
          ( ( inter_mset(A,Aa2,B2) = zero_zero(multiset(A)) )
         => ( aa(multiset(B),B,comm_m9189036328036947845d_mset(B),image_mset(A,B,G,union_mset(A,Aa2,B2))) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(multiset(B),B,comm_m9189036328036947845d_mset(B),image_mset(A,B,G,Aa2))),aa(multiset(B),B,comm_m9189036328036947845d_mset(B),image_mset(A,B,G,B2))) ) ) ) ).

% prod_mset.union_disjoint
tff(fact_6109_eventually__top,axiom,
    ! [A: $tType,Pa: fun(A,$o)] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Pa),top_top(filter(A)))
    <=> ! [X_12: A] : aa(A,$o,Pa,X_12) ) ).

% eventually_top
tff(fact_6110_eventually__const,axiom,
    ! [A: $tType,Fa: filter(A),Pa: $o] :
      ( ( Fa != bot_bot(filter(A)) )
     => ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aTP_Lamp_ae($o,fun(A,$o),(Pa))),Fa)
      <=> (Pa) ) ) ).

% eventually_const
tff(fact_6111_eventually__sequentially__Suc,axiom,
    ! [Pa: fun(nat,$o)] :
      ( aa(filter(nat),$o,aa(fun(nat,$o),fun(filter(nat),$o),eventually(nat),aTP_Lamp_vt(fun(nat,$o),fun(nat,$o),Pa)),at_top(nat))
    <=> aa(filter(nat),$o,aa(fun(nat,$o),fun(filter(nat),$o),eventually(nat),Pa),at_top(nat)) ) ).

% eventually_sequentially_Suc
tff(fact_6112_eventually__sequentially__seg,axiom,
    ! [Pa: fun(nat,$o),K3: nat] :
      ( aa(filter(nat),$o,aa(fun(nat,$o),fun(filter(nat),$o),eventually(nat),aa(nat,fun(nat,$o),aTP_Lamp_vu(fun(nat,$o),fun(nat,fun(nat,$o)),Pa),K3)),at_top(nat))
    <=> aa(filter(nat),$o,aa(fun(nat,$o),fun(filter(nat),$o),eventually(nat),Pa),at_top(nat)) ) ).

% eventually_sequentially_seg
tff(fact_6113_prod__mset_Oadd__mset,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Xa: A,N7: multiset(A)] : ( aa(multiset(A),A,comm_m9189036328036947845d_mset(A),add_mset(A,Xa,N7)) = aa(A,A,aa(A,fun(A,A),times_times(A),Xa),aa(multiset(A),A,comm_m9189036328036947845d_mset(A),N7)) ) ) ).

% prod_mset.add_mset
tff(fact_6114_prod__mset__Un,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Aa2: multiset(A),B2: 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)),Aa2),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(multiset(A),A,comm_m9189036328036947845d_mset(A),Aa2)),aa(multiset(A),A,comm_m9189036328036947845d_mset(A),B2)) ) ) ).

% prod_mset_Un
tff(fact_6115_prod__mset_Ounion,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M5: multiset(A),N7: 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)),M5),N7)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(multiset(A),A,comm_m9189036328036947845d_mset(A),M5)),aa(multiset(A),A,comm_m9189036328036947845d_mset(A),N7)) ) ) ).

% prod_mset.union
tff(fact_6116_eventually__finite__subsets__at__top__weakI,axiom,
    ! [A: $tType,Aa2: set(A),Pa: fun(set(A),$o)] :
      ( ! [X7: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),X7)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X7),Aa2)
           => aa(set(A),$o,Pa,X7) ) )
     => aa(filter(set(A)),$o,aa(fun(set(A),$o),fun(filter(set(A)),$o),eventually(set(A)),Pa),finite5375528669736107172at_top(A,Aa2)) ) ).

% eventually_finite_subsets_at_top_weakI
tff(fact_6117_prod__mset_Oinsert,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(B,A),Xa: B,Aa2: multiset(B)] : ( aa(multiset(A),A,comm_m9189036328036947845d_mset(A),image_mset(B,A,G,add_mset(B,Xa,Aa2))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,G,Xa)),aa(multiset(A),A,comm_m9189036328036947845d_mset(A),image_mset(B,A,G,Aa2))) ) ) ).

% prod_mset.insert
tff(fact_6118_trivial__limit__def,axiom,
    ! [A: $tType,Fa: filter(A)] :
      ( ( Fa = bot_bot(filter(A)) )
    <=> aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aTP_Lamp_cq(A,$o)),Fa) ) ).

% trivial_limit_def
tff(fact_6119_eventually__const__iff,axiom,
    ! [A: $tType,Pa: $o,Fa: filter(A)] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aTP_Lamp_ae($o,fun(A,$o),(Pa))),Fa)
    <=> ( (Pa)
        | ( Fa = bot_bot(filter(A)) ) ) ) ).

% eventually_const_iff
tff(fact_6120_False__imp__not__eventually,axiom,
    ! [A: $tType,Pa: fun(A,$o),Net: filter(A)] :
      ( ! [X: A] : ~ aa(A,$o,Pa,X)
     => ( ( Net != bot_bot(filter(A)) )
       => ~ aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Pa),Net) ) ) ).

% False_imp_not_eventually
tff(fact_6121_eventually__happens_H,axiom,
    ! [A: $tType,Fa: filter(A),Pa: fun(A,$o)] :
      ( ( Fa != bot_bot(filter(A)) )
     => ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Pa),Fa)
       => ? [X_1: A] : aa(A,$o,Pa,X_1) ) ) ).

% eventually_happens'
tff(fact_6122_eventually__happens,axiom,
    ! [A: $tType,Pa: fun(A,$o),Net: filter(A)] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Pa),Net)
     => ( ( Net = bot_bot(filter(A)) )
        | ? [X_1: A] : aa(A,$o,Pa,X_1) ) ) ).

% eventually_happens
tff(fact_6123_eventually__bot,axiom,
    ! [A: $tType,Pa: fun(A,$o)] : aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Pa),bot_bot(filter(A))) ).

% eventually_bot
tff(fact_6124_eventually__finite__subsets__at__top__finite,axiom,
    ! [A: $tType,Aa2: set(A),Pa: fun(set(A),$o)] :
      ( aa(set(A),$o,finite_finite2(A),Aa2)
     => ( aa(filter(set(A)),$o,aa(fun(set(A),$o),fun(filter(set(A)),$o),eventually(set(A)),Pa),finite5375528669736107172at_top(A,Aa2))
      <=> aa(set(A),$o,Pa,Aa2) ) ) ).

% eventually_finite_subsets_at_top_finite
tff(fact_6125_eventually__ball__finite,axiom,
    ! [A: $tType,B: $tType,Aa2: set(A),Pa: fun(B,fun(A,$o)),Net: filter(B)] :
      ( aa(set(A),$o,finite_finite2(A),Aa2)
     => ( ! [X: A] :
            ( member(A,X,Aa2)
           => aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),eventually(B),aa(A,fun(B,$o),aTP_Lamp_sc(fun(B,fun(A,$o)),fun(A,fun(B,$o)),Pa),X)),Net) )
       => aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),eventually(B),aa(fun(B,fun(A,$o)),fun(B,$o),aTP_Lamp_vv(set(A),fun(fun(B,fun(A,$o)),fun(B,$o)),Aa2),Pa)),Net) ) ) ).

% eventually_ball_finite
tff(fact_6126_eventually__ball__finite__distrib,axiom,
    ! [A: $tType,B: $tType,Aa2: set(A),Pa: fun(B,fun(A,$o)),Net: filter(B)] :
      ( aa(set(A),$o,finite_finite2(A),Aa2)
     => ( aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),eventually(B),aa(fun(B,fun(A,$o)),fun(B,$o),aTP_Lamp_vv(set(A),fun(fun(B,fun(A,$o)),fun(B,$o)),Aa2),Pa)),Net)
      <=> ! [X6: A] :
            ( member(A,X6,Aa2)
           => aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),eventually(B),aa(A,fun(B,$o),aTP_Lamp_sc(fun(B,fun(A,$o)),fun(A,fun(B,$o)),Pa),X6)),Net) ) ) ) ).

% eventually_ball_finite_distrib
tff(fact_6127_eventually__all__finite,axiom,
    ! [A: $tType,B: $tType] :
      ( finite_finite(A)
     => ! [Pa: fun(B,fun(A,$o)),Net: filter(B)] :
          ( ! [Y2: A] : aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),eventually(B),aa(A,fun(B,$o),aTP_Lamp_vw(fun(B,fun(A,$o)),fun(A,fun(B,$o)),Pa),Y2)),Net)
         => aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),eventually(B),aTP_Lamp_vx(fun(B,fun(A,$o)),fun(B,$o),Pa)),Net) ) ) ).

% eventually_all_finite
tff(fact_6128_eventually__ex,axiom,
    ! [B: $tType,A: $tType,Pa: fun(A,fun(B,$o)),Fa: filter(A)] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aTP_Lamp_vy(fun(A,fun(B,$o)),fun(A,$o),Pa)),Fa)
    <=> ? [Y8: fun(A,B)] : aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_vz(fun(A,fun(B,$o)),fun(fun(A,B),fun(A,$o)),Pa),Y8)),Fa) ) ).

% eventually_ex
tff(fact_6129_eventually__principal,axiom,
    ! [A: $tType,Pa: fun(A,$o),S: set(A)] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Pa),aa(set(A),filter(A),principal(A),S))
    <=> ! [X6: A] :
          ( member(A,X6,S)
         => aa(A,$o,Pa,X6) ) ) ).

% eventually_principal
tff(fact_6130_eventually__inf,axiom,
    ! [A: $tType,Pa: fun(A,$o),Fa: filter(A),F: filter(A)] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Pa),aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),inf_inf(filter(A)),Fa),F))
    <=> ? [Q7: fun(A,$o),R7: fun(A,$o)] :
          ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Q7),Fa)
          & aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),R7),F)
          & ! [X6: A] :
              ( ( aa(A,$o,Q7,X6)
                & aa(A,$o,R7,X6) )
             => aa(A,$o,Pa,X6) ) ) ) ).

% eventually_inf
tff(fact_6131_always__eventually,axiom,
    ! [A: $tType,Pa: fun(A,$o),Fa: filter(A)] :
      ( ! [X_1: A] : aa(A,$o,Pa,X_1)
     => aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Pa),Fa) ) ).

% always_eventually
tff(fact_6132_not__eventuallyD,axiom,
    ! [A: $tType,Pa: fun(A,$o),Fa: filter(A)] :
      ( ~ aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Pa),Fa)
     => ? [X: A] : ~ aa(A,$o,Pa,X) ) ).

% not_eventuallyD
tff(fact_6133_eventually__mono,axiom,
    ! [A: $tType,Pa: fun(A,$o),Fa: filter(A),Qa: fun(A,$o)] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Pa),Fa)
     => ( ! [X: A] :
            ( aa(A,$o,Pa,X)
           => aa(A,$o,Qa,X) )
       => aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Qa),Fa) ) ) ).

% eventually_mono
tff(fact_6134_filter__eq__iff,axiom,
    ! [A: $tType,Fa: filter(A),F: filter(A)] :
      ( ( Fa = F )
    <=> ! [P6: fun(A,$o)] :
          ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P6),Fa)
        <=> aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P6),F) ) ) ).

% filter_eq_iff
tff(fact_6135_eventuallyI,axiom,
    ! [A: $tType,Pa: fun(A,$o),Fa: filter(A)] :
      ( ! [X: A] : aa(A,$o,Pa,X)
     => aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Pa),Fa) ) ).

% eventuallyI
tff(fact_6136_eventually__at__bot__not__equal,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_bot(A) )
     => ! [C2: A] : aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aTP_Lamp_wa(A,fun(A,$o),C2)),at_bot(A)) ) ).

% eventually_at_bot_not_equal
tff(fact_6137_not__eventually__impI,axiom,
    ! [A: $tType,Pa: fun(A,$o),Fa: filter(A),Qa: fun(A,$o)] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Pa),Fa)
     => ( ~ aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Qa),Fa)
       => ~ aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_bt(fun(A,$o),fun(fun(A,$o),fun(A,$o)),Pa),Qa)),Fa) ) ) ).

% not_eventually_impI
tff(fact_6138_eventually__conj__iff,axiom,
    ! [A: $tType,Pa: fun(A,$o),Qa: fun(A,$o),Fa: filter(A)] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_bx(fun(A,$o),fun(fun(A,$o),fun(A,$o)),Pa),Qa)),Fa)
    <=> ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Pa),Fa)
        & aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Qa),Fa) ) ) ).

% eventually_conj_iff
tff(fact_6139_eventually__rev__mp,axiom,
    ! [A: $tType,Pa: fun(A,$o),Fa: filter(A),Qa: fun(A,$o)] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Pa),Fa)
     => ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_bt(fun(A,$o),fun(fun(A,$o),fun(A,$o)),Pa),Qa)),Fa)
       => aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Qa),Fa) ) ) ).

% eventually_rev_mp
tff(fact_6140_eventually__subst,axiom,
    ! [A: $tType,Pa: fun(A,$o),Qa: fun(A,$o),Fa: filter(A)] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_wb(fun(A,$o),fun(fun(A,$o),fun(A,$o)),Pa),Qa)),Fa)
     => ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Pa),Fa)
      <=> aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Qa),Fa) ) ) ).

% eventually_subst
tff(fact_6141_eventually__elim2,axiom,
    ! [A: $tType,Pa: fun(A,$o),Fa: filter(A),Qa: fun(A,$o),R: fun(A,$o)] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Pa),Fa)
     => ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Qa),Fa)
       => ( ! [I3: A] :
              ( aa(A,$o,Pa,I3)
             => ( aa(A,$o,Qa,I3)
               => aa(A,$o,R,I3) ) )
         => aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),R),Fa) ) ) ) ).

% eventually_elim2
tff(fact_6142_eventually__conj,axiom,
    ! [A: $tType,Pa: fun(A,$o),Fa: filter(A),Qa: fun(A,$o)] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Pa),Fa)
     => ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Qa),Fa)
       => aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_bx(fun(A,$o),fun(fun(A,$o),fun(A,$o)),Pa),Qa)),Fa) ) ) ).

% eventually_conj
tff(fact_6143_eventually__True,axiom,
    ! [A: $tType,Fa: filter(A)] : aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aTP_Lamp_cc(A,$o)),Fa) ).

% eventually_True
tff(fact_6144_eventually__mp,axiom,
    ! [A: $tType,Pa: fun(A,$o),Qa: fun(A,$o),Fa: filter(A)] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_bt(fun(A,$o),fun(fun(A,$o),fun(A,$o)),Pa),Qa)),Fa)
     => ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Pa),Fa)
       => aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Qa),Fa) ) ) ).

% eventually_mp
tff(fact_6145_eventually__frequently__const__simps_I3_J,axiom,
    ! [A: $tType,Pa: fun(A,$o),C3: $o,Fa: filter(A)] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa($o,fun(A,$o),aTP_Lamp_wc(fun(A,$o),fun($o,fun(A,$o)),Pa),(C3))),Fa)
    <=> ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Pa),Fa)
        | (C3) ) ) ).

% eventually_frequently_const_simps(3)
tff(fact_6146_eventually__frequently__const__simps_I4_J,axiom,
    ! [A: $tType,C3: $o,Pa: fun(A,$o),Fa: filter(A)] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_wd($o,fun(fun(A,$o),fun(A,$o)),(C3)),Pa)),Fa)
    <=> ( (C3)
        | aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Pa),Fa) ) ) ).

% eventually_frequently_const_simps(4)
tff(fact_6147_eventually__frequently__const__simps_I6_J,axiom,
    ! [A: $tType,C3: $o,Pa: fun(A,$o),Fa: filter(A)] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_we($o,fun(fun(A,$o),fun(A,$o)),(C3)),Pa)),Fa)
    <=> ( (C3)
       => aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Pa),Fa) ) ) ).

% eventually_frequently_const_simps(6)
tff(fact_6148_eventually__False__sequentially,axiom,
    ~ aa(filter(nat),$o,aa(fun(nat,$o),fun(filter(nat),$o),eventually(nat),aTP_Lamp_wf(nat,$o)),at_top(nat)) ).

% eventually_False_sequentially
tff(fact_6149_eventually__at__top__not__equal,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_top(A) )
     => ! [C2: A] : aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aTP_Lamp_wg(A,fun(A,$o),C2)),at_top(A)) ) ).

% eventually_at_top_not_equal
tff(fact_6150_eventually__compose__filterlim,axiom,
    ! [A: $tType,B: $tType,Pa: fun(A,$o),Fa: filter(A),F2: fun(B,A),G4: filter(B)] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Pa),Fa)
     => ( filterlim(B,A,F2,Fa,G4)
       => aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),eventually(B),aa(fun(B,A),fun(B,$o),aTP_Lamp_wh(fun(A,$o),fun(fun(B,A),fun(B,$o)),Pa),F2)),G4) ) ) ).

% eventually_compose_filterlim
tff(fact_6151_filterlim__cong,axiom,
    ! [A: $tType,B: $tType,F12: filter(A),F13: filter(A),F22: filter(B),F24: filter(B),F2: fun(B,A),G: fun(B,A)] :
      ( ( F12 = F13 )
     => ( ( F22 = F24 )
       => ( aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),eventually(B),aa(fun(B,A),fun(B,$o),aTP_Lamp_wi(fun(B,A),fun(fun(B,A),fun(B,$o)),F2),G)),F22)
         => ( filterlim(B,A,F2,F12,F22)
          <=> filterlim(B,A,G,F13,F24) ) ) ) ) ).

% filterlim_cong
tff(fact_6152_filterlim__iff,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),F22: filter(B),F12: filter(A)] :
      ( filterlim(A,B,F2,F22,F12)
    <=> ! [P6: fun(B,$o)] :
          ( aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),eventually(B),P6),F22)
         => aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(fun(B,$o),fun(A,$o),aTP_Lamp_qv(fun(A,B),fun(fun(B,$o),fun(A,$o)),F2),P6)),F12) ) ) ).

% filterlim_iff
tff(fact_6153_eventually__at__top__dense,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_top(A) )
     => ! [Pa: fun(A,$o)] :
          ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Pa),at_top(A))
        <=> ? [N11: A] :
            ! [N2: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),N11),N2)
             => aa(A,$o,Pa,N2) ) ) ) ).

% eventually_at_top_dense
tff(fact_6154_eventually__at__bot__dense,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_bot(A) )
     => ! [Pa: fun(A,$o)] :
          ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Pa),at_bot(A))
        <=> ? [N11: A] :
            ! [N2: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),N2),N11)
             => aa(A,$o,Pa,N2) ) ) ) ).

% eventually_at_bot_dense
tff(fact_6155_eventually__Sup,axiom,
    ! [A: $tType,Pa: fun(A,$o),S: set(filter(A))] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Pa),aa(set(filter(A)),filter(A),complete_Sup_Sup(filter(A)),S))
    <=> ! [X6: filter(A)] :
          ( member(filter(A),X6,S)
         => aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Pa),X6) ) ) ).

% eventually_Sup
tff(fact_6156_eventually__sequentiallyI,axiom,
    ! [C2: nat,Pa: fun(nat,$o)] :
      ( ! [X: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),C2),X)
         => aa(nat,$o,Pa,X) )
     => aa(filter(nat),$o,aa(fun(nat,$o),fun(filter(nat),$o),eventually(nat),Pa),at_top(nat)) ) ).

% eventually_sequentiallyI
tff(fact_6157_eventually__sequentially,axiom,
    ! [Pa: fun(nat,$o)] :
      ( aa(filter(nat),$o,aa(fun(nat,$o),fun(filter(nat),$o),eventually(nat),Pa),at_top(nat))
    <=> ? [N11: nat] :
        ! [N2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N11),N2)
         => aa(nat,$o,Pa,N2) ) ) ).

% eventually_sequentially
tff(fact_6158_eventually__at__top__linorderI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A,Pa: fun(A,$o)] :
          ( ! [X: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),X)
             => aa(A,$o,Pa,X) )
         => aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Pa),at_top(A)) ) ) ).

% eventually_at_top_linorderI
tff(fact_6159_eventually__at__top__linorder,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Pa: fun(A,$o)] :
          ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Pa),at_top(A))
        <=> ? [N11: A] :
            ! [N2: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),N11),N2)
             => aa(A,$o,Pa,N2) ) ) ) ).

% eventually_at_top_linorder
tff(fact_6160_le__filter__def,axiom,
    ! [A: $tType,Fa: filter(A),F: filter(A)] :
      ( aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),Fa),F)
    <=> ! [P6: fun(A,$o)] :
          ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P6),F)
         => aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P6),Fa) ) ) ).

% le_filter_def
tff(fact_6161_filter__leI,axiom,
    ! [A: $tType,F: filter(A),Fa: filter(A)] :
      ( ! [P3: fun(A,$o)] :
          ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P3),F)
         => aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P3),Fa) )
     => aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),Fa),F) ) ).

% filter_leI
tff(fact_6162_filter__leD,axiom,
    ! [A: $tType,Fa: filter(A),F: filter(A),Pa: fun(A,$o)] :
      ( aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),Fa),F)
     => ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Pa),F)
       => aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Pa),Fa) ) ) ).

% filter_leD
tff(fact_6163_eventually__at__bot__linorder,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Pa: fun(A,$o)] :
          ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Pa),at_bot(A))
        <=> ? [N11: A] :
            ! [N2: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),N2),N11)
             => aa(A,$o,Pa,N2) ) ) ) ).

% eventually_at_bot_linorder
tff(fact_6164_eventually__sup,axiom,
    ! [A: $tType,Pa: fun(A,$o),Fa: filter(A),F: filter(A)] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Pa),aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),sup_sup(filter(A)),Fa),F))
    <=> ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Pa),Fa)
        & aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Pa),F) ) ) ).

% eventually_sup
tff(fact_6165_eventually__ge__at__top,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A] : aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(A,fun(A,$o),ord_less_eq(A),C2)),at_top(A)) ) ).

% eventually_ge_at_top
tff(fact_6166_eventually__gt__at__top,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_top(A) )
     => ! [C2: A] : aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(A,fun(A,$o),ord_less(A),C2)),at_top(A)) ) ).

% eventually_gt_at_top
tff(fact_6167_eventually__le__at__bot,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A] : aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(A,fun(A,$o),aTP_Lamp_ao(A,fun(A,$o)),C2)),at_bot(A)) ) ).

% eventually_le_at_bot
tff(fact_6168_eventually__gt__at__bot,axiom,
    ! [A: $tType] :
      ( unboun7993243217541854897norder(A)
     => ! [C2: A] : aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aTP_Lamp_wj(A,fun(A,$o),C2)),at_bot(A)) ) ).

% eventually_gt_at_bot
tff(fact_6169_le__sequentially,axiom,
    ! [Fa: filter(nat)] :
      ( aa(filter(nat),$o,aa(filter(nat),fun(filter(nat),$o),ord_less_eq(filter(nat)),Fa),at_top(nat))
    <=> ! [N11: nat] : aa(filter(nat),$o,aa(fun(nat,$o),fun(filter(nat),$o),eventually(nat),aa(nat,fun(nat,$o),ord_less_eq(nat),N11)),Fa) ) ).

% le_sequentially
tff(fact_6170_filterlim__mono__eventually,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),Fa: filter(B),G4: filter(A),F: filter(B),G6: filter(A),F11: fun(A,B)] :
      ( filterlim(A,B,F2,Fa,G4)
     => ( aa(filter(B),$o,aa(filter(B),fun(filter(B),$o),ord_less_eq(filter(B)),Fa),F)
       => ( aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),G6),G4)
         => ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_uh(fun(A,B),fun(fun(A,B),fun(A,$o)),F2),F11)),G6)
           => filterlim(A,B,F11,F,G6) ) ) ) ) ).

% filterlim_mono_eventually
tff(fact_6171_filterlim__principal,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),S: set(B),Fa: filter(A)] :
      ( filterlim(A,B,F2,aa(set(B),filter(B),principal(B),S),Fa)
    <=> aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(set(B),fun(A,$o),aTP_Lamp_qx(fun(A,B),fun(set(B),fun(A,$o)),F2),S)),Fa) ) ).

% filterlim_principal
tff(fact_6172_le__principal,axiom,
    ! [A: $tType,Fa: filter(A),Aa2: set(A)] :
      ( aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),Fa),aa(set(A),filter(A),principal(A),Aa2))
    <=> aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(set(A),fun(A,$o),aTP_Lamp_a(set(A),fun(A,$o)),Aa2)),Fa) ) ).

% le_principal
tff(fact_6173_eventually__INF1,axiom,
    ! [B: $tType,A: $tType,Ia: A,Ia2: set(A),Pa: fun(B,$o),Fa: fun(A,filter(B))] :
      ( member(A,Ia,Ia2)
     => ( aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),eventually(B),Pa),aa(A,filter(B),Fa,Ia))
       => aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),eventually(B),Pa),aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image2(A,filter(B),Fa),Ia2))) ) ) ).

% eventually_INF1
tff(fact_6174_eventually__inf__principal,axiom,
    ! [A: $tType,Pa: fun(A,$o),Fa: filter(A),S2: set(A)] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Pa),aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),inf_inf(filter(A)),Fa),aa(set(A),filter(A),principal(A),S2)))
    <=> aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(set(A),fun(A,$o),aTP_Lamp_wk(fun(A,$o),fun(set(A),fun(A,$o)),Pa),S2)),Fa) ) ).

% eventually_inf_principal
tff(fact_6175_prod__mset_Odistrib,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(B,A),H2: fun(B,A),Aa2: multiset(B)] : ( aa(multiset(A),A,comm_m9189036328036947845d_mset(A),image_mset(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_ba(fun(B,A),fun(fun(B,A),fun(B,A)),G),H2),Aa2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(multiset(A),A,comm_m9189036328036947845d_mset(A),image_mset(B,A,G,Aa2))),aa(multiset(A),A,comm_m9189036328036947845d_mset(A),image_mset(B,A,H2,Aa2))) ) ) ).

% prod_mset.distrib
tff(fact_6176_filterlim__at__top__at__top,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & linorder(B) )
     => ! [Qa: fun(A,$o),F2: fun(A,B),Pa: fun(B,$o),G: fun(B,A)] :
          ( ! [X: A,Y2: A] :
              ( aa(A,$o,Qa,X)
             => ( aa(A,$o,Qa,Y2)
               => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y2)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X)),aa(A,B,F2,Y2)) ) ) )
         => ( ! [X: B] :
                ( aa(B,$o,Pa,X)
               => ( aa(A,B,F2,aa(B,A,G,X)) = X ) )
           => ( ! [X: B] :
                  ( aa(B,$o,Pa,X)
                 => aa(A,$o,Qa,aa(B,A,G,X)) )
             => ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Qa),at_top(A))
               => ( aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),eventually(B),Pa),at_top(B))
                 => filterlim(A,B,F2,at_top(B),at_top(A)) ) ) ) ) ) ) ).

% filterlim_at_top_at_top
tff(fact_6177_eventually__finite__subsets__at__top,axiom,
    ! [A: $tType,Pa: fun(set(A),$o),Aa2: set(A)] :
      ( aa(filter(set(A)),$o,aa(fun(set(A),$o),fun(filter(set(A)),$o),eventually(set(A)),Pa),finite5375528669736107172at_top(A,Aa2))
    <=> ? [X9: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),X9)
          & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X9),Aa2)
          & ! [Y8: set(A)] :
              ( ( aa(set(A),$o,finite_finite2(A),Y8)
                & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X9),Y8)
                & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Y8),Aa2) )
             => aa(set(A),$o,Pa,Y8) ) ) ) ).

% eventually_finite_subsets_at_top
tff(fact_6178_eventually__all__ge__at__top,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Pa: fun(A,$o)] :
          ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Pa),at_top(A))
         => aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aTP_Lamp_wl(fun(A,$o),fun(A,$o),Pa)),at_top(A)) ) ) ).

% eventually_all_ge_at_top
tff(fact_6179_eventually__Inf__base,axiom,
    ! [A: $tType,B2: set(filter(A)),Pa: fun(A,$o)] :
      ( ( B2 != bot_bot(set(filter(A))) )
     => ( ! [F7: filter(A)] :
            ( member(filter(A),F7,B2)
           => ! [G7: filter(A)] :
                ( member(filter(A),G7,B2)
               => ? [X4: filter(A)] :
                    ( member(filter(A),X4,B2)
                    & aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),X4),aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),inf_inf(filter(A)),F7),G7)) ) ) )
       => ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Pa),aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),B2))
        <=> ? [X6: filter(A)] :
              ( member(filter(A),X6,B2)
              & aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Pa),X6) ) ) ) ) ).

% eventually_Inf_base
tff(fact_6180_filterlim__at__top__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),Fa: filter(A),G: fun(A,B)] :
          ( filterlim(A,B,F2,at_top(B),Fa)
         => ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_wm(fun(A,B),fun(fun(A,B),fun(A,$o)),F2),G)),Fa)
           => filterlim(A,B,G,at_top(B),Fa) ) ) ) ).

% filterlim_at_top_mono
tff(fact_6181_filterlim__at__top__ge,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),Fa: filter(A),C2: B] :
          ( filterlim(A,B,F2,at_top(B),Fa)
        <=> ! [Z7: B] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),C2),Z7)
             => aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(B,fun(A,$o),aTP_Lamp_wn(fun(A,B),fun(B,fun(A,$o)),F2),Z7)),Fa) ) ) ) ).

% filterlim_at_top_ge
tff(fact_6182_filterlim__at__top,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),Fa: filter(A)] :
          ( filterlim(A,B,F2,at_top(B),Fa)
        <=> ! [Z7: B] : aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(B,fun(A,$o),aTP_Lamp_wn(fun(A,B),fun(B,fun(A,$o)),F2),Z7)),Fa) ) ) ).

% filterlim_at_top
tff(fact_6183_filterlim__at__top__dense,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [F2: fun(A,B),Fa: filter(A)] :
          ( filterlim(A,B,F2,at_top(B),Fa)
        <=> ! [Z7: B] : aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(B,fun(A,$o),aTP_Lamp_wo(fun(A,B),fun(B,fun(A,$o)),F2),Z7)),Fa) ) ) ).

% filterlim_at_top_dense
tff(fact_6184_eventually__INF__finite,axiom,
    ! [A: $tType,B: $tType,Aa2: set(A),Pa: fun(B,$o),Fa: fun(A,filter(B))] :
      ( aa(set(A),$o,finite_finite2(A),Aa2)
     => ( aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),eventually(B),Pa),aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image2(A,filter(B),Fa),Aa2)))
      <=> ? [Q7: fun(A,fun(B,$o))] :
            ( ! [X6: A] :
                ( member(A,X6,Aa2)
               => aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),eventually(B),aa(A,fun(B,$o),Q7,X6)),aa(A,filter(B),Fa,X6)) )
            & ! [Y4: B] :
                ( ! [X6: A] :
                    ( member(A,X6,Aa2)
                   => aa(B,$o,aa(A,fun(B,$o),Q7,X6),Y4) )
               => aa(B,$o,Pa,Y4) ) ) ) ) ).

% eventually_INF_finite
tff(fact_6185_filterlim__at__bot__le,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),Fa: filter(A),C2: B] :
          ( filterlim(A,B,F2,at_bot(B),Fa)
        <=> ! [Z7: B] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Z7),C2)
             => aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(B,fun(A,$o),aTP_Lamp_wp(fun(A,B),fun(B,fun(A,$o)),F2),Z7)),Fa) ) ) ) ).

% filterlim_at_bot_le
tff(fact_6186_filterlim__at__bot,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),Fa: filter(A)] :
          ( filterlim(A,B,F2,at_bot(B),Fa)
        <=> ! [Z7: B] : aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(B,fun(A,$o),aTP_Lamp_wp(fun(A,B),fun(B,fun(A,$o)),F2),Z7)),Fa) ) ) ).

% filterlim_at_bot
tff(fact_6187_filterlim__at__bot__dense,axiom,
    ! [B: $tType,A: $tType] :
      ( ( dense_linorder(B)
        & no_bot(B) )
     => ! [F2: fun(A,B),Fa: filter(A)] :
          ( filterlim(A,B,F2,at_bot(B),Fa)
        <=> ! [Z7: B] : aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(B,fun(A,$o),aTP_Lamp_wq(fun(A,B),fun(B,fun(A,$o)),F2),Z7)),Fa) ) ) ).

% filterlim_at_bot_dense
tff(fact_6188_filterlim__at__top__gt,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [F2: fun(A,B),Fa: filter(A),C2: B] :
          ( filterlim(A,B,F2,at_top(B),Fa)
        <=> ! [Z7: B] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),C2),Z7)
             => aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(B,fun(A,$o),aTP_Lamp_wr(fun(A,B),fun(B,fun(A,$o)),F2),Z7)),Fa) ) ) ) ).

% filterlim_at_top_gt
tff(fact_6189_eventually__INF,axiom,
    ! [A: $tType,B: $tType,Pa: fun(A,$o),Fa: fun(B,filter(A)),B2: set(B)] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Pa),aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(B),set(filter(A)),image2(B,filter(A),Fa),B2)))
    <=> ? [X9: set(B)] :
          ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),X9),B2)
          & aa(set(B),$o,finite_finite2(B),X9)
          & aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Pa),aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(B),set(filter(A)),image2(B,filter(A),Fa),X9))) ) ) ).

% eventually_INF
tff(fact_6190_filterlim__at__bot__lt,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [F2: fun(A,B),Fa: filter(A),C2: B] :
          ( filterlim(A,B,F2,at_bot(B),Fa)
        <=> ! [Z7: B] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),Z7),C2)
             => aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(B,fun(A,$o),aTP_Lamp_ws(fun(A,B),fun(B,fun(A,$o)),F2),Z7)),Fa) ) ) ) ).

% filterlim_at_bot_lt
tff(fact_6191_eventually__Inf,axiom,
    ! [A: $tType,Pa: fun(A,$o),B2: set(filter(A))] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Pa),aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),B2))
    <=> ? [X9: set(filter(A))] :
          ( aa(set(filter(A)),$o,aa(set(filter(A)),fun(set(filter(A)),$o),ord_less_eq(set(filter(A))),X9),B2)
          & aa(set(filter(A)),$o,finite_finite2(filter(A)),X9)
          & aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Pa),aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),X9)) ) ) ).

% eventually_Inf
tff(fact_6192_filterlim__finite__subsets__at__top,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,set(B)),Aa2: set(B),Fa: filter(A)] :
      ( filterlim(A,set(B),F2,finite5375528669736107172at_top(B,Aa2),Fa)
    <=> ! [X9: set(B)] :
          ( ( aa(set(B),$o,finite_finite2(B),X9)
            & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),X9),Aa2) )
         => aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(set(B),fun(A,$o),aa(set(B),fun(set(B),fun(A,$o)),aTP_Lamp_wt(fun(A,set(B)),fun(set(B),fun(set(B),fun(A,$o))),F2),Aa2),X9)),Fa) ) ) ).

% filterlim_finite_subsets_at_top
tff(fact_6193_prod__mset_Oremove,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Xa: A,Aa2: multiset(A)] :
          ( member(A,Xa,aa(multiset(A),set(A),set_mset(A),Aa2))
         => ( aa(multiset(A),A,comm_m9189036328036947845d_mset(A),Aa2) = aa(A,A,aa(A,fun(A,A),times_times(A),Xa),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)),Aa2),add_mset(A,Xa,zero_zero(multiset(A)))))) ) ) ) ).

% prod_mset.remove
tff(fact_6194_map__filter__on__comp,axiom,
    ! [A: $tType,E: $tType,B: $tType,G: fun(B,A),Y5: set(B),X3: set(A),Fa: filter(B),F2: fun(A,E)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(B),set(A),image2(B,A,G),Y5)),X3)
     => ( aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),eventually(B),aTP_Lamp_wu(set(B),fun(B,$o),Y5)),Fa)
       => ( aa(filter(A),filter(E),aa(fun(A,E),fun(filter(A),filter(E)),map_filter_on(A,E,X3),F2),aa(filter(B),filter(A),aa(fun(B,A),fun(filter(B),filter(A)),map_filter_on(B,A,Y5),G),Fa)) = aa(filter(B),filter(E),aa(fun(B,E),fun(filter(B),filter(E)),map_filter_on(B,E,Y5),comp(A,E,B,F2,G)),Fa) ) ) ) ).

% map_filter_on_comp
tff(fact_6195_Sup__filter__def,axiom,
    ! [A: $tType,S: set(filter(A))] : ( aa(set(filter(A)),filter(A),complete_Sup_Sup(filter(A)),S) = aa(fun(fun(A,$o),$o),filter(A),abs_filter(A),aTP_Lamp_wv(set(filter(A)),fun(fun(A,$o),$o),S)) ) ).

% Sup_filter_def
tff(fact_6196_map__filter__on__def,axiom,
    ! [A: $tType,B: $tType,X3: set(B),F2: fun(B,A),Fa: filter(B)] : ( aa(filter(B),filter(A),aa(fun(B,A),fun(filter(B),filter(A)),map_filter_on(B,A,X3),F2),Fa) = aa(fun(fun(A,$o),$o),filter(A),abs_filter(A),aa(filter(B),fun(fun(A,$o),$o),aa(fun(B,A),fun(filter(B),fun(fun(A,$o),$o)),aTP_Lamp_wx(set(B),fun(fun(B,A),fun(filter(B),fun(fun(A,$o),$o))),X3),F2),Fa)) ) ).

% map_filter_on_def
tff(fact_6197_bot__filter__def,axiom,
    ! [A: $tType] : ( bot_bot(filter(A)) = aa(fun(fun(A,$o),$o),filter(A),abs_filter(A),aTP_Lamp_wy(fun(A,$o),$o)) ) ).

% bot_filter_def
tff(fact_6198_eventually__map__filter__on,axiom,
    ! [B: $tType,A: $tType,X3: set(A),Fa: filter(A),Pa: fun(B,$o),F2: fun(A,B)] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(set(A),fun(A,$o),aTP_Lamp_a(set(A),fun(A,$o)),X3)),Fa)
     => ( aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),eventually(B),Pa),aa(filter(A),filter(B),aa(fun(A,B),fun(filter(A),filter(B)),map_filter_on(A,B,X3),F2),Fa))
      <=> aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(fun(A,B),fun(A,$o),aa(fun(B,$o),fun(fun(A,B),fun(A,$o)),aTP_Lamp_wz(set(A),fun(fun(B,$o),fun(fun(A,B),fun(A,$o))),X3),Pa),F2)),Fa) ) ) ).

% eventually_map_filter_on
tff(fact_6199_sup__filter__def,axiom,
    ! [A: $tType,Fa: filter(A),F: filter(A)] : ( aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),sup_sup(filter(A)),Fa),F) = aa(fun(fun(A,$o),$o),filter(A),abs_filter(A),aa(filter(A),fun(fun(A,$o),$o),aTP_Lamp_xa(filter(A),fun(filter(A),fun(fun(A,$o),$o)),Fa),F)) ) ).

% sup_filter_def
tff(fact_6200_principal__def,axiom,
    ! [A: $tType,S: set(A)] : ( aa(set(A),filter(A),principal(A),S) = aa(fun(fun(A,$o),$o),filter(A),abs_filter(A),ball(A,S)) ) ).

% principal_def
tff(fact_6201_top__filter__def,axiom,
    ! [A: $tType] : ( top_top(filter(A)) = aa(fun(fun(A,$o),$o),filter(A),abs_filter(A),fAll(A)) ) ).

% top_filter_def
tff(fact_6202_inf__filter__def,axiom,
    ! [A: $tType,Fa: filter(A),F: filter(A)] : ( aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),inf_inf(filter(A)),Fa),F) = aa(fun(fun(A,$o),$o),filter(A),abs_filter(A),aa(filter(A),fun(fun(A,$o),$o),aTP_Lamp_xb(filter(A),fun(filter(A),fun(fun(A,$o),$o)),Fa),F)) ) ).

% inf_filter_def
tff(fact_6203_bot_Oordering__top__axioms,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ordering_top(A,aTP_Lamp_xc(A,fun(A,$o)),aTP_Lamp_xd(A,fun(A,$o)),bot_bot(A)) ) ).

% bot.ordering_top_axioms
tff(fact_6204_bit__concat__bit__iff,axiom,
    ! [M2: nat,K3: int,L: int,N: nat] :
      ( bit_se5641148757651400278ts_bit(int,bit_concat_bit(M2,K3,L),N)
    <=> ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),M2)
          & bit_se5641148757651400278ts_bit(int,K3,N) )
        | ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),N)
          & bit_se5641148757651400278ts_bit(int,L,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M2)) ) ) ) ).

% bit_concat_bit_iff
tff(fact_6205_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_6206_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_6207_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_6208_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_6209_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_6210_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_6211_bot__nat__0_Oordering__top__axioms,axiom,
    ordering_top(nat,aTP_Lamp_nr(nat,fun(nat,$o)),aTP_Lamp_ac(nat,fun(nat,$o)),zero_zero(nat)) ).

% bot_nat_0.ordering_top_axioms
tff(fact_6212_filtercomap__def,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),Fa: filter(B)] : ( filtercomap(A,B,F2,Fa) = aa(fun(fun(A,$o),$o),filter(A),abs_filter(A),aa(filter(B),fun(fun(A,$o),$o),aTP_Lamp_xe(fun(A,B),fun(filter(B),fun(fun(A,$o),$o)),F2),Fa)) ) ).

% filtercomap_def
tff(fact_6213_multp__def,axiom,
    ! [A: $tType,R2: fun(A,fun(A,$o)),M5: multiset(A),N7: multiset(A)] :
      ( multp(A,R2,M5,N7)
    <=> 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)),M5),N7),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_6214_filterlim__filtercomap,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),Fa: filter(B)] : filterlim(A,B,F2,Fa,filtercomap(A,B,F2,Fa)) ).

% filterlim_filtercomap
tff(fact_6215_filtercomap__bot,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B)] : ( filtercomap(A,B,F2,bot_bot(filter(B))) = bot_bot(filter(A)) ) ).

% filtercomap_bot
tff(fact_6216_eventually__filtercomapI,axiom,
    ! [B: $tType,A: $tType,Pa: fun(A,$o),Fa: filter(A),F2: fun(B,A)] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Pa),Fa)
     => aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),eventually(B),aa(fun(B,A),fun(B,$o),aTP_Lamp_wh(fun(A,$o),fun(fun(B,A),fun(B,$o)),Pa),F2)),filtercomap(B,A,F2,Fa)) ) ).

% eventually_filtercomapI
tff(fact_6217_filtercomap__top,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B)] : ( filtercomap(A,B,F2,top_top(filter(B))) = top_top(filter(A)) ) ).

% filtercomap_top
tff(fact_6218_filtercomap__principal,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),Aa2: set(B)] : ( filtercomap(A,B,F2,aa(set(B),filter(B),principal(B),Aa2)) = aa(set(A),filter(A),principal(A),vimage(A,B,F2,Aa2)) ) ).

% filtercomap_principal
tff(fact_6219_less__multiset__def,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [M5: multiset(A),N7: multiset(A)] :
          ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),ord_less(multiset(A)),M5),N7)
        <=> multp(A,ord_less(A),M5,N7) ) ) ).

% less_multiset_def
tff(fact_6220_filtercomap__ident,axiom,
    ! [A: $tType,Fa: filter(A)] : ( filtercomap(A,A,aTP_Lamp_ab(A,A),Fa) = Fa ) ).

% filtercomap_ident
tff(fact_6221_filtercomap__filtercomap,axiom,
    ! [B: $tType,A: $tType,E: $tType,F2: fun(A,B),G: fun(B,E),Fa: filter(E)] : ( filtercomap(A,B,F2,filtercomap(B,E,G,Fa)) = filtercomap(A,E,aa(fun(B,E),fun(A,E),aTP_Lamp_xf(fun(A,B),fun(fun(B,E),fun(A,E)),F2),G),Fa) ) ).

% filtercomap_filtercomap
tff(fact_6222_filtercomap__inf,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),F12: filter(B),F22: filter(B)] : ( filtercomap(A,B,F2,aa(filter(B),filter(B),aa(filter(B),fun(filter(B),filter(B)),inf_inf(filter(B)),F12),F22)) = aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),inf_inf(filter(A)),filtercomap(A,B,F2,F12)),filtercomap(A,B,F2,F22)) ) ).

% filtercomap_inf
tff(fact_6223_filtercomap__mono,axiom,
    ! [B: $tType,A: $tType,Fa: filter(A),F: filter(A),F2: fun(B,A)] :
      ( aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),Fa),F)
     => aa(filter(B),$o,aa(filter(B),fun(filter(B),$o),ord_less_eq(filter(B)),filtercomap(B,A,F2,Fa)),filtercomap(B,A,F2,F)) ) ).

% filtercomap_mono
tff(fact_6224_eventually__filtercomap,axiom,
    ! [B: $tType,A: $tType,Pa: fun(A,$o),F2: fun(A,B),Fa: filter(B)] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Pa),filtercomap(A,B,F2,Fa))
    <=> ? [Q7: fun(B,$o)] :
          ( aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),eventually(B),Q7),Fa)
          & ! [X6: A] :
              ( aa(B,$o,Q7,aa(A,B,F2,X6))
             => aa(A,$o,Pa,X6) ) ) ) ).

% eventually_filtercomap
tff(fact_6225_filterlim__filtercomap__iff,axiom,
    ! [E: $tType,B: $tType,A: $tType,F2: fun(A,B),G: fun(B,E),G4: filter(E),Fa: filter(A)] :
      ( filterlim(A,B,F2,filtercomap(B,E,G,G4),Fa)
    <=> filterlim(A,E,comp(B,E,A,G,F2),G4,Fa) ) ).

% filterlim_filtercomap_iff
tff(fact_6226_filtercomap__neq__bot,axiom,
    ! [A: $tType,B: $tType,Fa: filter(A),F2: fun(B,A)] :
      ( ! [P3: fun(A,$o)] :
          ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P3),Fa)
         => ? [X4: B] : aa(A,$o,P3,aa(B,A,F2,X4)) )
     => ( filtercomap(B,A,F2,Fa) != bot_bot(filter(B)) ) ) ).

% filtercomap_neq_bot
tff(fact_6227_filterlim__iff__le__filtercomap,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),Fa: filter(B),G4: filter(A)] :
      ( filterlim(A,B,F2,Fa,G4)
    <=> aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),G4),filtercomap(A,B,F2,Fa)) ) ).

% filterlim_iff_le_filtercomap
tff(fact_6228_filtercomap__sup,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),F12: filter(B),F22: filter(B)] : aa(filter(A),$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,F2,F12)),filtercomap(A,B,F2,F22))),filtercomap(A,B,F2,aa(filter(B),filter(B),aa(filter(B),fun(filter(B),filter(B)),sup_sup(filter(B)),F12),F22))) ).

% filtercomap_sup
tff(fact_6229_filtercomap__INF,axiom,
    ! [B: $tType,A: $tType,E: $tType,F2: fun(A,B),Fa: fun(E,filter(B)),B2: set(E)] : ( filtercomap(A,B,F2,aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(E),set(filter(B)),image2(E,filter(B),Fa),B2))) = aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(E),set(filter(A)),image2(E,filter(A),aa(fun(E,filter(B)),fun(E,filter(A)),aTP_Lamp_xg(fun(A,B),fun(fun(E,filter(B)),fun(E,filter(A))),F2),Fa)),B2)) ) ).

% filtercomap_INF
tff(fact_6230_filtercomap__neq__bot__surj,axiom,
    ! [A: $tType,B: $tType,Fa: filter(A),F2: fun(B,A)] :
      ( ( Fa != bot_bot(filter(A)) )
     => ( ( aa(set(B),set(A),image2(B,A,F2),top_top(set(B))) = top_top(set(A)) )
       => ( filtercomap(B,A,F2,Fa) != bot_bot(filter(B)) ) ) ) ).

% filtercomap_neq_bot_surj
tff(fact_6231_eventually__filtercomap__at__top__linorder,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Pa: fun(A,$o),F2: fun(A,B)] :
          ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Pa),filtercomap(A,B,F2,at_top(B)))
        <=> ? [N11: B] :
            ! [X6: A] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),N11),aa(A,B,F2,X6))
             => aa(A,$o,Pa,X6) ) ) ) ).

% eventually_filtercomap_at_top_linorder
tff(fact_6232_eventually__filtercomap__at__top__dense,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(B)
        & no_top(B) )
     => ! [Pa: fun(A,$o),F2: fun(A,B)] :
          ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Pa),filtercomap(A,B,F2,at_top(B)))
        <=> ? [N11: B] :
            ! [X6: A] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),N11),aa(A,B,F2,X6))
             => aa(A,$o,Pa,X6) ) ) ) ).

% eventually_filtercomap_at_top_dense
tff(fact_6233_eventually__filtercomap__at__bot__linorder,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Pa: fun(A,$o),F2: fun(A,B)] :
          ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Pa),filtercomap(A,B,F2,at_bot(B)))
        <=> ? [N11: B] :
            ! [X6: A] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X6)),N11)
             => aa(A,$o,Pa,X6) ) ) ) ).

% eventually_filtercomap_at_bot_linorder
tff(fact_6234_eventually__filtercomap__at__bot__dense,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(B)
        & no_bot(B) )
     => ! [Pa: fun(A,$o),F2: fun(A,B)] :
          ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Pa),filtercomap(A,B,F2,at_bot(B)))
        <=> ? [N11: B] :
            ! [X6: A] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,X6)),N11)
             => aa(A,$o,Pa,X6) ) ) ) ).

% eventually_filtercomap_at_bot_dense
tff(fact_6235_filtercomap__SUP,axiom,
    ! [A: $tType,E: $tType,B: $tType,F2: fun(A,E),Fa: fun(B,filter(E)),B2: set(B)] : aa(filter(A),$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(E)),fun(B,filter(A)),aTP_Lamp_xh(fun(A,E),fun(fun(B,filter(E)),fun(B,filter(A))),F2),Fa)),B2))),filtercomap(A,E,F2,aa(set(filter(E)),filter(E),complete_Sup_Sup(filter(E)),aa(set(B),set(filter(E)),image2(B,filter(E),Fa),B2)))) ).

% filtercomap_SUP
tff(fact_6236_pair__lessI2,axiom,
    ! [A3: nat,B3: nat,S2: nat,T2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A3),B3)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),S2),T2)
       => 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),B3),T2)),fun_pair_less) ) ) ).

% pair_lessI2
tff(fact_6237_euclidean__size__times__nonunit,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [A3: A,B3: A] :
          ( ( A3 != zero_zero(A) )
         => ( ( B3 != zero_zero(A) )
           => ( ~ dvd_dvd(A,A3,one_one(A))
             => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),euclid6346220572633701492n_size(A,B3)),euclid6346220572633701492n_size(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3))) ) ) ) ) ).

% euclidean_size_times_nonunit
tff(fact_6238_euclidean__size__greater__0__iff,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [B3: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),euclid6346220572633701492n_size(A,B3))
        <=> ( B3 != zero_zero(A) ) ) ) ).

% euclidean_size_greater_0_iff
tff(fact_6239_pair__less__iff1,axiom,
    ! [Xa: nat,Y: nat,Z2: nat] :
      ( member(product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xa),Y)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xa),Z2)),fun_pair_less)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y),Z2) ) ).

% pair_less_iff1
tff(fact_6240_euclidean__size__mult,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [A3: A,B3: A] : ( euclid6346220572633701492n_size(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),euclid6346220572633701492n_size(A,A3)),euclid6346220572633701492n_size(A,B3)) ) ) ).

% euclidean_size_mult
tff(fact_6241_size__mult__mono,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [B3: A,A3: A] :
          ( ( B3 != zero_zero(A) )
         => aa(nat,$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),B3))) ) ) ).

% size_mult_mono
tff(fact_6242_size__mult__mono_H,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [B3: A,A3: A] :
          ( ( B3 != zero_zero(A) )
         => aa(nat,$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),B3),A3))) ) ) ).

% size_mult_mono'
tff(fact_6243_euclidean__size__times__unit,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [A3: A,B3: A] :
          ( dvd_dvd(A,A3,one_one(A))
         => ( euclid6346220572633701492n_size(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)) = euclid6346220572633701492n_size(A,B3) ) ) ) ).

% euclidean_size_times_unit
tff(fact_6244_dvd__proper__imp__size__less,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [A3: A,B3: A] :
          ( dvd_dvd(A,A3,B3)
         => ( ~ dvd_dvd(A,B3,A3)
           => ( ( B3 != zero_zero(A) )
             => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),euclid6346220572633701492n_size(A,A3)),euclid6346220572633701492n_size(A,B3)) ) ) ) ) ).

% dvd_proper_imp_size_less
tff(fact_6245_mod__size__less,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [B3: A,A3: A] :
          ( ( B3 != zero_zero(A) )
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),euclid6346220572633701492n_size(A,modulo_modulo(A,A3,B3))),euclid6346220572633701492n_size(A,B3)) ) ) ).

% mod_size_less
tff(fact_6246_pair__lessI1,axiom,
    ! [A3: nat,B3: nat,S2: nat,T2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A3),B3)
     => 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),B3),T2)),fun_pair_less) ) ).

% pair_lessI1
tff(fact_6247_divmod__cases,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [B3: A,A3: A] :
          ( ( ( B3 != zero_zero(A) )
           => ( ( modulo_modulo(A,A3,B3) = zero_zero(A) )
             => ( A3 != aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,A3,B3)),B3) ) ) )
         => ( ( ( B3 != zero_zero(A) )
             => ! [Q4: A,R3: A] :
                  ( ( euclid7384307370059645450egment(A,R3) = euclid7384307370059645450egment(A,B3) )
                 => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),euclid6346220572633701492n_size(A,R3)),euclid6346220572633701492n_size(A,B3))
                   => ( ( R3 != zero_zero(A) )
                     => ( ( divide_divide(A,A3,B3) = Q4 )
                       => ( ( modulo_modulo(A,A3,B3) = R3 )
                         => ( A3 != aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Q4),B3)),R3) ) ) ) ) ) ) )
           => ( B3 = zero_zero(A) ) ) ) ) ).

% divmod_cases
tff(fact_6248_mod__eqI,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [B3: A,R2: A,Q2: A,A3: A] :
          ( ( B3 != zero_zero(A) )
         => ( ( euclid7384307370059645450egment(A,R2) = euclid7384307370059645450egment(A,B3) )
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),euclid6346220572633701492n_size(A,R2)),euclid6346220572633701492n_size(A,B3))
             => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Q2),B3)),R2) = A3 )
               => ( modulo_modulo(A,A3,B3) = R2 ) ) ) ) ) ) ).

% mod_eqI
tff(fact_6249_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_6250_division__segment__mult,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [A3: A,B3: A] :
          ( ( A3 != zero_zero(A) )
         => ( ( B3 != zero_zero(A) )
           => ( euclid7384307370059645450egment(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)) = aa(A,A,aa(A,fun(A,A),times_times(A),euclid7384307370059645450egment(A,A3)),euclid7384307370059645450egment(A,B3)) ) ) ) ) ).

% division_segment_mult
tff(fact_6251_unique__euclidean__semiring__class_Odiv__eq__0__iff,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [A3: A,B3: A] :
          ( ( euclid7384307370059645450egment(A,A3) = euclid7384307370059645450egment(A,B3) )
         => ( ( divide_divide(A,A3,B3) = zero_zero(A) )
          <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),euclid6346220572633701492n_size(A,A3)),euclid6346220572633701492n_size(A,B3))
              | ( B3 = zero_zero(A) ) ) ) ) ) ).

% unique_euclidean_semiring_class.div_eq_0_iff
tff(fact_6252_div__eqI,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [B3: A,R2: A,Q2: A,A3: A] :
          ( ( B3 != zero_zero(A) )
         => ( ( euclid7384307370059645450egment(A,R2) = euclid7384307370059645450egment(A,B3) )
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),euclid6346220572633701492n_size(A,R2)),euclid6346220572633701492n_size(A,B3))
             => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Q2),B3)),R2) = A3 )
               => ( divide_divide(A,A3,B3) = Q2 ) ) ) ) ) ) ).

% div_eqI
tff(fact_6253_div__bounded,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [B3: A,R2: A,Q2: A] :
          ( ( B3 != zero_zero(A) )
         => ( ( euclid7384307370059645450egment(A,R2) = euclid7384307370059645450egment(A,B3) )
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),euclid6346220572633701492n_size(A,R2)),euclid6346220572633701492n_size(A,B3))
             => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Q2),B3)),R2),B3) = Q2 ) ) ) ) ) ).

% div_bounded
tff(fact_6254_smin__insertI,axiom,
    ! [Xa: product_prod(nat,nat),XS: set(product_prod(nat,nat)),Y: product_prod(nat,nat),YS: set(product_prod(nat,nat))] :
      ( member(product_prod(nat,nat),Xa,XS)
     => ( member(product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),Xa),Y),fun_pair_less)
       => ( member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),XS),YS),fun_min_strict)
         => member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),XS),aa(set(product_prod(nat,nat)),set(product_prod(nat,nat)),aa(product_prod(nat,nat),fun(set(product_prod(nat,nat)),set(product_prod(nat,nat))),insert3(product_prod(nat,nat)),Y),YS)),fun_min_strict) ) ) ) ).

% smin_insertI
tff(fact_6255_pair__leqI2,axiom,
    ! [A3: nat,B3: nat,S2: nat,T2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A3),B3)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),S2),T2)
       => 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),B3),T2)),fun_pair_leq) ) ) ).

% pair_leqI2
tff(fact_6256_smin__emptyI,axiom,
    ! [X3: set(product_prod(nat,nat))] :
      ( ( X3 != bot_bot(set(product_prod(nat,nat))) )
     => member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),X3),bot_bot(set(product_prod(nat,nat)))),fun_min_strict) ) ).

% smin_emptyI
tff(fact_6257_pair__leqI1,axiom,
    ! [A3: nat,B3: nat,S2: nat,T2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A3),B3)
     => 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),B3),T2)),fun_pair_leq) ) ).

% pair_leqI1
tff(fact_6258_wmax__insertI,axiom,
    ! [Y: product_prod(nat,nat),YS: set(product_prod(nat,nat)),Xa: product_prod(nat,nat),XS: set(product_prod(nat,nat))] :
      ( member(product_prod(nat,nat),Y,YS)
     => ( member(product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),Xa),Y),fun_pair_leq)
       => ( member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),XS),YS),fun_max_weak)
         => member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),set(product_prod(nat,nat)),aa(product_prod(nat,nat),fun(set(product_prod(nat,nat)),set(product_prod(nat,nat))),insert3(product_prod(nat,nat)),Xa),XS)),YS),fun_max_weak) ) ) ) ).

% wmax_insertI
tff(fact_6259_wmin__insertI,axiom,
    ! [Xa: product_prod(nat,nat),XS: set(product_prod(nat,nat)),Y: product_prod(nat,nat),YS: set(product_prod(nat,nat))] :
      ( member(product_prod(nat,nat),Xa,XS)
     => ( member(product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),Xa),Y),fun_pair_leq)
       => ( member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),XS),YS),fun_min_weak)
         => member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),XS),aa(set(product_prod(nat,nat)),set(product_prod(nat,nat)),aa(product_prod(nat,nat),fun(set(product_prod(nat,nat)),set(product_prod(nat,nat))),insert3(product_prod(nat,nat)),Y),YS)),fun_min_weak) ) ) ) ).

% wmin_insertI
tff(fact_6260_wmax__emptyI,axiom,
    ! [X3: set(product_prod(nat,nat))] :
      ( aa(set(product_prod(nat,nat)),$o,finite_finite2(product_prod(nat,nat)),X3)
     => 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)))),X3),fun_max_weak) ) ).

% wmax_emptyI
tff(fact_6261_wmin__emptyI,axiom,
    ! [X3: set(product_prod(nat,nat))] : member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),X3),bot_bot(set(product_prod(nat,nat)))),fun_min_weak) ).

% wmin_emptyI
tff(fact_6262_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_6263_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))))),insert3(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_6264_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))))),insert3(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_6265_smax__insertI,axiom,
    ! [Y: product_prod(nat,nat),Y5: set(product_prod(nat,nat)),Xa: product_prod(nat,nat),X3: set(product_prod(nat,nat))] :
      ( member(product_prod(nat,nat),Y,Y5)
     => ( member(product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),Xa),Y),fun_pair_less)
       => ( member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),X3),Y5),fun_max_strict)
         => member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),set(product_prod(nat,nat)),aa(product_prod(nat,nat),fun(set(product_prod(nat,nat)),set(product_prod(nat,nat))),insert3(product_prod(nat,nat)),Xa),X3)),Y5),fun_max_strict) ) ) ) ).

% smax_insertI
tff(fact_6266_distinct__adj__append__iff,axiom,
    ! [A: $tType,Xsa: list(A),Ys: list(A)] :
      ( aa(list(A),$o,distinct_adj(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),Ys))
    <=> ( aa(list(A),$o,distinct_adj(A),Xsa)
        & aa(list(A),$o,distinct_adj(A),Ys)
        & ( ( Xsa = nil(A) )
          | ( Ys = nil(A) )
          | ( last(A,Xsa) != aa(list(A),A,hd(A),Ys) ) ) ) ) ).

% distinct_adj_append_iff
tff(fact_6267_distinct__adj__Cons__Cons,axiom,
    ! [A: $tType,Xa: A,Y: A,Xsa: list(A)] :
      ( aa(list(A),$o,distinct_adj(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Xsa)))
    <=> ( ( Xa != Y )
        & aa(list(A),$o,distinct_adj(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Xsa)) ) ) ).

% distinct_adj_Cons_Cons
tff(fact_6268_distinct__adj__rev,axiom,
    ! [A: $tType,Xsa: list(A)] :
      ( aa(list(A),$o,distinct_adj(A),aa(list(A),list(A),rev(A),Xsa))
    <=> aa(list(A),$o,distinct_adj(A),Xsa) ) ).

% distinct_adj_rev
tff(fact_6269_distinct__adj__mapD,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xsa: list(B)] :
      ( aa(list(A),$o,distinct_adj(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Xsa))
     => aa(list(B),$o,distinct_adj(B),Xsa) ) ).

% distinct_adj_mapD
tff(fact_6270_distinct__adj__Nil,axiom,
    ! [A: $tType] : aa(list(A),$o,distinct_adj(A),nil(A)) ).

% distinct_adj_Nil
tff(fact_6271_distinct__adj__altdef,axiom,
    ! [A: $tType,Xsa: list(A)] :
      ( aa(list(A),$o,distinct_adj(A),Xsa)
    <=> ( aa(list(A),list(A),remdups_adj(A),Xsa) = Xsa ) ) ).

% distinct_adj_altdef
tff(fact_6272_distinct__adj__remdups__adj,axiom,
    ! [A: $tType,Xsa: list(A)] : aa(list(A),$o,distinct_adj(A),aa(list(A),list(A),remdups_adj(A),Xsa)) ).

% distinct_adj_remdups_adj
tff(fact_6273_distinct__adj__appendD1,axiom,
    ! [A: $tType,Xsa: list(A),Ys: list(A)] :
      ( aa(list(A),$o,distinct_adj(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),Ys))
     => aa(list(A),$o,distinct_adj(A),Xsa) ) ).

% distinct_adj_appendD1
tff(fact_6274_distinct__adj__appendD2,axiom,
    ! [A: $tType,Xsa: list(A),Ys: list(A)] :
      ( aa(list(A),$o,distinct_adj(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),Ys))
     => aa(list(A),$o,distinct_adj(A),Ys) ) ).

% distinct_adj_appendD2
tff(fact_6275_distinct__adj__ConsD,axiom,
    ! [A: $tType,Xa: A,Xsa: list(A)] :
      ( aa(list(A),$o,distinct_adj(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa))
     => aa(list(A),$o,distinct_adj(A),Xsa) ) ).

% distinct_adj_ConsD
tff(fact_6276_distinct__adj__singleton,axiom,
    ! [A: $tType,Xa: A] : aa(list(A),$o,distinct_adj(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),nil(A))) ).

% distinct_adj_singleton
tff(fact_6277_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_6278_distinct__adj__mapI,axiom,
    ! [B: $tType,A: $tType,Xsa: list(A),F2: fun(A,B)] :
      ( aa(list(A),$o,distinct_adj(A),Xsa)
     => ( inj_on(A,B,F2,aa(list(A),set(A),set2(A),Xsa))
       => aa(list(B),$o,distinct_adj(B),aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),Xsa)) ) ) ).

% distinct_adj_mapI
tff(fact_6279_distinct__adj__map__iff,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),Xsa: list(A)] :
      ( inj_on(A,B,F2,aa(list(A),set(A),set2(A),Xsa))
     => ( aa(list(B),$o,distinct_adj(B),aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),Xsa))
      <=> aa(list(A),$o,distinct_adj(A),Xsa) ) ) ).

% distinct_adj_map_iff
tff(fact_6280_distinct__adj__Cons,axiom,
    ! [A: $tType,Xa: A,Xsa: list(A)] :
      ( aa(list(A),$o,distinct_adj(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa))
    <=> ( ( Xsa = nil(A) )
        | ( ( Xa != aa(list(A),A,hd(A),Xsa) )
          & aa(list(A),$o,distinct_adj(A),Xsa) ) ) ) ).

% distinct_adj_Cons
tff(fact_6281_smax__emptyI,axiom,
    ! [Y5: set(product_prod(nat,nat))] :
      ( aa(set(product_prod(nat,nat)),$o,finite_finite2(product_prod(nat,nat)),Y5)
     => ( ( Y5 != bot_bot(set(product_prod(nat,nat))) )
       => member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),bot_bot(set(product_prod(nat,nat)))),Y5),fun_max_strict) ) ) ).

% smax_emptyI
tff(fact_6282_wmsI,axiom,
    ! [Aa2: multiset(product_prod(nat,nat)),B2: multiset(product_prod(nat,nat)),Z6: multiset(product_prod(nat,nat))] :
      ( ( member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),set(product_prod(nat,nat)),set_mset(product_prod(nat,nat)),Aa2)),aa(multiset(product_prod(nat,nat)),set(product_prod(nat,nat)),set_mset(product_prod(nat,nat)),B2)),fun_max_strict)
        | ( ( Aa2 = zero_zero(multiset(product_prod(nat,nat))) )
          & ( B2 = zero_zero(multiset(product_prod(nat,nat))) ) ) )
     => 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))),Z6),Aa2)),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))),Z6),B2)),ms_weak) ) ).

% wmsI
tff(fact_6283_smsI,axiom,
    ! [Aa2: multiset(product_prod(nat,nat)),B2: multiset(product_prod(nat,nat)),Z6: multiset(product_prod(nat,nat))] :
      ( member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),set(product_prod(nat,nat)),set_mset(product_prod(nat,nat)),Aa2)),aa(multiset(product_prod(nat,nat)),set(product_prod(nat,nat)),set_mset(product_prod(nat,nat)),B2)),fun_max_strict)
     => 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))),Z6),Aa2)),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))),Z6),B2)),ms_strict) ) ).

% smsI
tff(fact_6284_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_6285_ms__weakI1,axiom,
    ! [Z6: multiset(product_prod(nat,nat)),Z8: multiset(product_prod(nat,nat)),Aa2: multiset(product_prod(nat,nat)),B2: multiset(product_prod(nat,nat))] :
      ( pw_leq(Z6,Z8)
     => ( 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)),Aa2)),aa(multiset(product_prod(nat,nat)),set(product_prod(nat,nat)),set_mset(product_prod(nat,nat)),B2)),fun_max_strict)
       => 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))),Z6),Aa2)),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))),Z8),B2)),ms_weak) ) ) ).

% ms_weakI1
tff(fact_6286_ms__strictI,axiom,
    ! [Z6: multiset(product_prod(nat,nat)),Z8: multiset(product_prod(nat,nat)),Aa2: multiset(product_prod(nat,nat)),B2: multiset(product_prod(nat,nat))] :
      ( pw_leq(Z6,Z8)
     => ( 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)),Aa2)),aa(multiset(product_prod(nat,nat)),set(product_prod(nat,nat)),set_mset(product_prod(nat,nat)),B2)),fun_max_strict)
       => 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))),Z6),Aa2)),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))),Z8),B2)),ms_strict) ) ) ).

% ms_strictI
tff(fact_6287_pw__leq__lstep,axiom,
    ! [Xa: product_prod(nat,nat),Y: product_prod(nat,nat)] :
      ( member(product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),Xa),Y),fun_pair_leq)
     => pw_leq(add_mset(product_prod(nat,nat),Xa,zero_zero(multiset(product_prod(nat,nat)))),add_mset(product_prod(nat,nat),Y,zero_zero(multiset(product_prod(nat,nat))))) ) ).

% pw_leq_lstep
tff(fact_6288_pw__leq__step,axiom,
    ! [Xa: product_prod(nat,nat),Y: product_prod(nat,nat),X3: multiset(product_prod(nat,nat)),Y5: multiset(product_prod(nat,nat))] :
      ( member(product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),Xa),Y),fun_pair_leq)
     => ( pw_leq(X3,Y5)
       => 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))),add_mset(product_prod(nat,nat),Xa,zero_zero(multiset(product_prod(nat,nat))))),X3),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))),add_mset(product_prod(nat,nat),Y,zero_zero(multiset(product_prod(nat,nat))))),Y5)) ) ) ).

% pw_leq_step
tff(fact_6289_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))) ) )
        | ? [X6: product_prod(nat,nat),Y4: product_prod(nat,nat),X9: 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))),add_mset(product_prod(nat,nat),X6,zero_zero(multiset(product_prod(nat,nat))))),X9) )
            & ( 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))),add_mset(product_prod(nat,nat),Y4,zero_zero(multiset(product_prod(nat,nat))))),Y8) )
            & 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)),X6),Y4),fun_pair_leq)
            & pw_leq(X9,Y8) ) ) ) ).

% pw_leq.simps
tff(fact_6290_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))) ) )
       => ~ ! [X: product_prod(nat,nat),Y2: product_prod(nat,nat),X7: 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))),add_mset(product_prod(nat,nat),X,zero_zero(multiset(product_prod(nat,nat))))),X7) )
             => ! [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))),add_mset(product_prod(nat,nat),Y2,zero_zero(multiset(product_prod(nat,nat))))),Y9) )
                 => ( 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),Y2),fun_pair_leq)
                   => ~ pw_leq(X7,Y9) ) ) ) ) ) ).

% pw_leq.cases
tff(fact_6291_ms__weakI2,axiom,
    ! [Z6: multiset(product_prod(nat,nat)),Z8: multiset(product_prod(nat,nat))] :
      ( pw_leq(Z6,Z8)
     => 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))),Z6),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))),Z8),zero_zero(multiset(product_prod(nat,nat))))),ms_weak) ) ).

% ms_weakI2
tff(fact_6292_pw__leq__split,axiom,
    ! [X3: multiset(product_prod(nat,nat)),Y5: multiset(product_prod(nat,nat))] :
      ( pw_leq(X3,Y5)
     => ? [A12: multiset(product_prod(nat,nat)),B5: multiset(product_prod(nat,nat)),Z9: multiset(product_prod(nat,nat))] :
          ( ( X3 = 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))),A12),Z9) )
          & ( Y5 = 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))),B5),Z9) )
          & ( 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)),A12)),aa(multiset(product_prod(nat,nat)),set(product_prod(nat,nat)),set_mset(product_prod(nat,nat)),B5)),fun_max_strict)
            | ( ( B5 = zero_zero(multiset(product_prod(nat,nat))) )
              & ( A12 = zero_zero(multiset(product_prod(nat,nat))) ) ) ) ) ) ).

% pw_leq_split
tff(fact_6293_proj__def,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(B,A)),Xa: B] : ( equiv_proj(B,A,R2,Xa) = image(B,A,R2,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert3(B),Xa),bot_bot(set(B)))) ) ).

% proj_def
tff(fact_6294_AboveS__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),Aa2: set(A)] : ( order_AboveS(A,R2,Aa2) = aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_xi(set(product_prod(A,A)),fun(set(A),fun(A,$o)),R2),Aa2)) ) ).

% AboveS_def
tff(fact_6295_AboveS__disjoint,axiom,
    ! [A: $tType,Aa2: 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)),Aa2),order_AboveS(A,R2,Aa2)) = bot_bot(set(A)) ) ).

% AboveS_disjoint
tff(fact_6296_wo__rel_Osuc__greater,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),B2: set(A),B3: A] :
      ( bNF_Wellorder_wo_rel(A,R2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),aa(set(product_prod(A,A)),set(A),field2(A),R2))
       => ( ( order_AboveS(A,R2,B2) != bot_bot(set(A)) )
         => ( member(A,B3,B2)
           => ( ( bNF_Wellorder_wo_suc(A,R2,B2) != B3 )
              & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B3),bNF_Wellorder_wo_suc(A,R2,B2)),R2) ) ) ) ) ) ).

% wo_rel.suc_greater
tff(fact_6297_wo__rel_Osuc__AboveS,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),B2: set(A)] :
      ( bNF_Wellorder_wo_rel(A,R2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),aa(set(product_prod(A,A)),set(A),field2(A),R2))
       => ( ( order_AboveS(A,R2,B2) != bot_bot(set(A)) )
         => member(A,bNF_Wellorder_wo_suc(A,R2,B2),order_AboveS(A,R2,B2)) ) ) ) ).

% wo_rel.suc_AboveS
tff(fact_6298_wo__rel_Osuc__least__AboveS,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A3: A,B2: set(A)] :
      ( bNF_Wellorder_wo_rel(A,R2)
     => ( member(A,A3,order_AboveS(A,R2,B2))
       => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),bNF_Wellorder_wo_suc(A,R2,B2)),A3),R2) ) ) ).

% wo_rel.suc_least_AboveS
tff(fact_6299_wo__rel_Oequals__suc__AboveS,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),B2: set(A),A3: A] :
      ( bNF_Wellorder_wo_rel(A,R2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),aa(set(product_prod(A,A)),set(A),field2(A),R2))
       => ( member(A,A3,order_AboveS(A,R2,B2))
         => ( ! [A9: A] :
                ( member(A,A9,order_AboveS(A,R2,B2))
               => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),A9),R2) )
           => ( A3 = bNF_Wellorder_wo_suc(A,R2,B2) ) ) ) ) ) ).

% wo_rel.equals_suc_AboveS
tff(fact_6300_wo__rel_Osuc__inField,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),B2: set(A)] :
      ( bNF_Wellorder_wo_rel(A,R2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),aa(set(product_prod(A,A)),set(A),field2(A),R2))
       => ( ( order_AboveS(A,R2,B2) != bot_bot(set(A)) )
         => member(A,bNF_Wellorder_wo_suc(A,R2,B2),aa(set(product_prod(A,A)),set(A),field2(A),R2)) ) ) ) ).

% wo_rel.suc_inField
tff(fact_6301_wo__rel_Osuc__ofilter__in,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),Aa2: set(A),B3: A] :
      ( bNF_Wellorder_wo_rel(A,R2)
     => ( order_ofilter(A,R2,Aa2)
       => ( ( order_AboveS(A,R2,Aa2) != bot_bot(set(A)) )
         => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B3),bNF_Wellorder_wo_suc(A,R2,Aa2)),R2)
           => ( ( B3 != bNF_Wellorder_wo_suc(A,R2,Aa2) )
             => member(A,B3,Aa2) ) ) ) ) ) ).

% wo_rel.suc_ofilter_in
tff(fact_6302_proj__iff,axiom,
    ! [A: $tType,Aa2: set(A),R2: set(product_prod(A,A)),Xa: A,Y: A] :
      ( equiv_equiv(A,Aa2,R2)
     => ( aa(set(A),$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)),insert3(A),Xa),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Y),bot_bot(set(A))))),Aa2)
       => ( ( equiv_proj(A,A,R2,Xa) = equiv_proj(A,A,R2,Y) )
        <=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Y),R2) ) ) ) ).

% proj_iff
tff(fact_6303_quotient__eqI,axiom,
    ! [A: $tType,Aa2: set(A),R2: set(product_prod(A,A)),X3: set(A),Y5: set(A),Xa: A,Y: A] :
      ( equiv_equiv(A,Aa2,R2)
     => ( member(set(A),X3,equiv_quotient(A,Aa2,R2))
       => ( member(set(A),Y5,equiv_quotient(A,Aa2,R2))
         => ( member(A,Xa,X3)
           => ( member(A,Y,Y5)
             => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Y),R2)
               => ( X3 = Y5 ) ) ) ) ) ) ) ).

% quotient_eqI
tff(fact_6304_quotient__eq__iff,axiom,
    ! [A: $tType,Aa2: set(A),R2: set(product_prod(A,A)),X3: set(A),Y5: set(A),Xa: A,Y: A] :
      ( equiv_equiv(A,Aa2,R2)
     => ( member(set(A),X3,equiv_quotient(A,Aa2,R2))
       => ( member(set(A),Y5,equiv_quotient(A,Aa2,R2))
         => ( member(A,Xa,X3)
           => ( member(A,Y,Y5)
             => ( ( X3 = Y5 )
              <=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Y),R2) ) ) ) ) ) ) ).

% quotient_eq_iff
tff(fact_6305_in__quotient__imp__closed,axiom,
    ! [A: $tType,Aa2: set(A),R2: set(product_prod(A,A)),X3: set(A),Xa: A,Y: A] :
      ( equiv_equiv(A,Aa2,R2)
     => ( member(set(A),X3,equiv_quotient(A,Aa2,R2))
       => ( member(A,Xa,X3)
         => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Y),R2)
           => member(A,Y,X3) ) ) ) ) ).

% in_quotient_imp_closed
tff(fact_6306_in__quotient__imp__non__empty,axiom,
    ! [A: $tType,Aa2: set(A),R2: set(product_prod(A,A)),X3: set(A)] :
      ( equiv_equiv(A,Aa2,R2)
     => ( member(set(A),X3,equiv_quotient(A,Aa2,R2))
       => ( X3 != bot_bot(set(A)) ) ) ) ).

% in_quotient_imp_non_empty
tff(fact_6307_equiv__listrel,axiom,
    ! [A: $tType,Aa2: set(A),R2: set(product_prod(A,A))] :
      ( equiv_equiv(A,Aa2,R2)
     => equiv_equiv(list(A),aa(set(A),set(list(A)),lists(A),Aa2),listrel(A,A,R2)) ) ).

% equiv_listrel
tff(fact_6308_equiv__class__self,axiom,
    ! [A: $tType,Aa2: set(A),R2: set(product_prod(A,A)),A3: A] :
      ( equiv_equiv(A,Aa2,R2)
     => ( member(A,A3,Aa2)
       => member(A,A3,image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),bot_bot(set(A))))) ) ) ).

% equiv_class_self
tff(fact_6309_quotient__disj,axiom,
    ! [A: $tType,Aa2: set(A),R2: set(product_prod(A,A)),X3: set(A),Y5: set(A)] :
      ( equiv_equiv(A,Aa2,R2)
     => ( member(set(A),X3,equiv_quotient(A,Aa2,R2))
       => ( member(set(A),Y5,equiv_quotient(A,Aa2,R2))
         => ( ( X3 = Y5 )
            | ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),X3),Y5) = bot_bot(set(A)) ) ) ) ) ) ).

% quotient_disj
tff(fact_6310_equiv__class__eq__iff,axiom,
    ! [A: $tType,Aa2: set(A),R2: set(product_prod(A,A)),Xa: A,Y: A] :
      ( equiv_equiv(A,Aa2,R2)
     => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Y),R2)
      <=> ( ( image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A)))) = image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Y),bot_bot(set(A)))) )
          & member(A,Xa,Aa2)
          & member(A,Y,Aa2) ) ) ) ).

% equiv_class_eq_iff
tff(fact_6311_eq__equiv__class__iff,axiom,
    ! [A: $tType,Aa2: set(A),R2: set(product_prod(A,A)),Xa: A,Y: A] :
      ( equiv_equiv(A,Aa2,R2)
     => ( member(A,Xa,Aa2)
       => ( member(A,Y,Aa2)
         => ( ( image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A)))) = image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Y),bot_bot(set(A)))) )
          <=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Y),R2) ) ) ) ) ).

% eq_equiv_class_iff
tff(fact_6312_equiv__class__eq,axiom,
    ! [A: $tType,Aa2: set(A),R2: set(product_prod(A,A)),A3: A,B3: A] :
      ( equiv_equiv(A,Aa2,R2)
     => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B3),R2)
       => ( image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),bot_bot(set(A)))) = image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),B3),bot_bot(set(A)))) ) ) ) ).

% equiv_class_eq
tff(fact_6313_eq__equiv__class,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A3: A,B3: A,Aa2: set(A)] :
      ( ( image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),bot_bot(set(A)))) = image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),B3),bot_bot(set(A)))) )
     => ( equiv_equiv(A,Aa2,R2)
       => ( member(A,B3,Aa2)
         => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B3),R2) ) ) ) ).

% eq_equiv_class
tff(fact_6314_eq__equiv__class__iff2,axiom,
    ! [A: $tType,Aa2: set(A),R2: set(product_prod(A,A)),Xa: A,Y: A] :
      ( equiv_equiv(A,Aa2,R2)
     => ( member(A,Xa,Aa2)
       => ( member(A,Y,Aa2)
         => ( ( equiv_quotient(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A))),R2) = equiv_quotient(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Y),bot_bot(set(A))),R2) )
          <=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Y),R2) ) ) ) ) ).

% eq_equiv_class_iff2
tff(fact_6315_refines__equiv__class__eq2,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),S: set(product_prod(A,A)),Aa2: set(A),A3: A] :
      ( aa(set(product_prod(A,A)),$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,Aa2,R)
       => ( equiv_equiv(A,Aa2,S)
         => ( image(A,A,S,image(A,A,R,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),bot_bot(set(A))))) = image(A,A,S,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),bot_bot(set(A)))) ) ) ) ) ).

% refines_equiv_class_eq2
tff(fact_6316_refines__equiv__class__eq,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),S: set(product_prod(A,A)),Aa2: set(A),A3: A] :
      ( aa(set(product_prod(A,A)),$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,Aa2,R)
       => ( equiv_equiv(A,Aa2,S)
         => ( image(A,A,R,image(A,A,S,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),bot_bot(set(A))))) = image(A,A,S,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),bot_bot(set(A)))) ) ) ) ) ).

% refines_equiv_class_eq
tff(fact_6317_equiv__class__subset,axiom,
    ! [A: $tType,Aa2: set(A),R2: set(product_prod(A,A)),A3: A,B3: A] :
      ( equiv_equiv(A,Aa2,R2)
     => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B3),R2)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),bot_bot(set(A))))),image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),B3),bot_bot(set(A))))) ) ) ).

% equiv_class_subset
tff(fact_6318_subset__equiv__class,axiom,
    ! [A: $tType,Aa2: set(A),R2: set(product_prod(A,A)),B3: A,A3: A] :
      ( equiv_equiv(A,Aa2,R2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),B3),bot_bot(set(A))))),image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),bot_bot(set(A)))))
       => ( member(A,B3,Aa2)
         => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B3),R2) ) ) ) ).

% subset_equiv_class
tff(fact_6319_equiv__class__nondisjoint,axiom,
    ! [A: $tType,Aa2: set(A),R2: set(product_prod(A,A)),Xa: A,A3: A,B3: A] :
      ( equiv_equiv(A,Aa2,R2)
     => ( member(A,Xa,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),bot_bot(set(A))))),image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),B3),bot_bot(set(A))))))
       => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B3),R2) ) ) ).

% equiv_class_nondisjoint
tff(fact_6320_in__quotient__imp__in__rel,axiom,
    ! [A: $tType,Aa2: set(A),R2: set(product_prod(A,A)),X3: set(A),Xa: A,Y: A] :
      ( equiv_equiv(A,Aa2,R2)
     => ( member(set(A),X3,equiv_quotient(A,Aa2,R2))
       => ( aa(set(A),$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)),insert3(A),Xa),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Y),bot_bot(set(A))))),X3)
         => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Y),R2) ) ) ) ).

% in_quotient_imp_in_rel
tff(fact_6321_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),$o),aTP_Lamp_xj(set(product_prod(A,A)),fun(set(A),fun(set(A),$o)),R2))) ) ).

% ofilterIncl_def
tff(fact_6322_UN__equiv__class2,axiom,
    ! [A: $tType,E: $tType,B: $tType,A15: set(A),R12: set(product_prod(A,A)),A23: set(B),R23: set(product_prod(B,B)),F2: fun(A,fun(B,set(E))),A1: A,A22: B] :
      ( equiv_equiv(A,A15,R12)
     => ( equiv_equiv(B,A23,R23)
       => ( equiv_congruent2(A,B,set(E),R12,R23,F2)
         => ( member(A,A1,A15)
           => ( member(B,A22,A23)
             => ( aa(set(set(E)),set(E),complete_Sup_Sup(set(E)),aa(set(A),set(set(E)),image2(A,set(E),aa(B,fun(A,set(E)),aa(fun(A,fun(B,set(E))),fun(B,fun(A,set(E))),aTP_Lamp_xk(set(product_prod(B,B)),fun(fun(A,fun(B,set(E))),fun(B,fun(A,set(E)))),R23),F2),A22)),image(A,A,R12,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A1),bot_bot(set(A)))))) = aa(B,set(E),aa(A,fun(B,set(E)),F2,A1),A22) ) ) ) ) ) ) ).

% UN_equiv_class2
tff(fact_6323_congruent2I_H,axiom,
    ! [E: $tType,B: $tType,A: $tType,R12: set(product_prod(A,A)),R23: set(product_prod(B,B)),F2: fun(A,fun(B,E))] :
      ( ! [Y12: A,Z1: A,Y23: B,Z22: B] :
          ( 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)
         => ( 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,E,aa(A,fun(B,E),F2,Y12),Y23) = aa(B,E,aa(A,fun(B,E),F2,Z1),Z22) ) ) )
     => equiv_congruent2(A,B,E,R12,R23,F2) ) ).

% congruent2I'
tff(fact_6324_congruent2D,axiom,
    ! [A: $tType,E: $tType,B: $tType,R12: set(product_prod(A,A)),R23: set(product_prod(B,B)),F2: fun(A,fun(B,E)),Y1: A,Z12: A,Y22: B,Z23: B] :
      ( equiv_congruent2(A,B,E,R12,R23,F2)
     => ( 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)
       => ( 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,E,aa(A,fun(B,E),F2,Y1),Y22) = aa(B,E,aa(A,fun(B,E),F2,Z12),Z23) ) ) ) ) ).

% congruent2D
tff(fact_6325_congruent2__commuteI,axiom,
    ! [B: $tType,A: $tType,Aa2: set(A),R2: set(product_prod(A,A)),F2: fun(A,fun(A,B))] :
      ( equiv_equiv(A,Aa2,R2)
     => ( ! [Y2: A,Z3: A] :
            ( member(A,Y2,Aa2)
           => ( member(A,Z3,Aa2)
             => ( aa(A,B,aa(A,fun(A,B),F2,Y2),Z3) = aa(A,B,aa(A,fun(A,B),F2,Z3),Y2) ) ) )
       => ( ! [Y2: A,Z3: A,W: A] :
              ( member(A,W,Aa2)
             => ( 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),F2,W),Y2) = aa(A,B,aa(A,fun(A,B),F2,W),Z3) ) ) )
         => equiv_congruent2(A,A,B,R2,R2,F2) ) ) ) ).

% congruent2_commuteI
tff(fact_6326_congruent2I,axiom,
    ! [E: $tType,B: $tType,A: $tType,A15: set(A),R12: set(product_prod(A,A)),A23: set(B),R23: set(product_prod(B,B)),F2: fun(A,fun(B,E))] :
      ( equiv_equiv(A,A15,R12)
     => ( equiv_equiv(B,A23,R23)
       => ( ! [Y2: A,Z3: A,W: B] :
              ( member(B,W,A23)
             => ( 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,E,aa(A,fun(B,E),F2,Y2),W) = aa(B,E,aa(A,fun(B,E),F2,Z3),W) ) ) )
         => ( ! [Y2: B,Z3: B,W: A] :
                ( member(A,W,A15)
               => ( 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,E,aa(A,fun(B,E),F2,W),Y2) = aa(B,E,aa(A,fun(B,E),F2,W),Z3) ) ) )
           => equiv_congruent2(A,B,E,R12,R23,F2) ) ) ) ) ).

% congruent2I
tff(fact_6327_congruent2__implies__congruent__UN,axiom,
    ! [A: $tType,E: $tType,B: $tType,A15: set(A),R12: set(product_prod(A,A)),A23: set(B),R23: set(product_prod(B,B)),F2: fun(A,fun(B,set(E))),A3: B] :
      ( equiv_equiv(A,A15,R12)
     => ( equiv_equiv(B,A23,R23)
       => ( equiv_congruent2(A,B,set(E),R12,R23,F2)
         => ( member(B,A3,A23)
           => equiv_congruent(A,set(E),R12,aa(B,fun(A,set(E)),aa(fun(A,fun(B,set(E))),fun(B,fun(A,set(E))),aTP_Lamp_xk(set(product_prod(B,B)),fun(fun(A,fun(B,set(E))),fun(B,fun(A,set(E)))),R23),F2),A3)) ) ) ) ) ).

% congruent2_implies_congruent_UN
tff(fact_6328_UN__equiv__class,axiom,
    ! [B: $tType,A: $tType,Aa2: set(A),R2: set(product_prod(A,A)),F2: fun(A,set(B)),A3: A] :
      ( equiv_equiv(A,Aa2,R2)
     => ( equiv_congruent(A,set(B),R2,F2)
       => ( member(A,A3,Aa2)
         => ( aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),F2),image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),bot_bot(set(A)))))) = aa(A,set(B),F2,A3) ) ) ) ) ).

% UN_equiv_class
tff(fact_6329_congruentD,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,A)),F2: fun(A,B),Y: A,Z2: A] :
      ( equiv_congruent(A,B,R2,F2)
     => ( 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,F2,Y) = aa(A,B,F2,Z2) ) ) ) ).

% congruentD
tff(fact_6330_congruentI,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,A)),F2: fun(A,B)] :
      ( ! [Y2: A,Z3: A] :
          ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),Z3),R2)
         => ( aa(A,B,F2,Y2) = aa(A,B,F2,Z3) ) )
     => equiv_congruent(A,B,R2,F2) ) ).

% congruentI
tff(fact_6331_UN__equiv__class__inject,axiom,
    ! [B: $tType,A: $tType,Aa2: set(A),R2: set(product_prod(A,A)),F2: fun(A,set(B)),X3: set(A),Y5: set(A)] :
      ( equiv_equiv(A,Aa2,R2)
     => ( equiv_congruent(A,set(B),R2,F2)
       => ( ( aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),F2),X3)) = aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),F2),Y5)) )
         => ( member(set(A),X3,equiv_quotient(A,Aa2,R2))
           => ( member(set(A),Y5,equiv_quotient(A,Aa2,R2))
             => ( ! [X: A,Y2: A] :
                    ( member(A,X,Aa2)
                   => ( member(A,Y2,Aa2)
                     => ( ( aa(A,set(B),F2,X) = aa(A,set(B),F2,Y2) )
                       => 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) ) ) )
               => ( X3 = Y5 ) ) ) ) ) ) ) ).

% UN_equiv_class_inject
tff(fact_6332_disjnt__equiv__class,axiom,
    ! [A: $tType,Aa2: set(A),R2: set(product_prod(A,A)),A3: A,B3: A] :
      ( equiv_equiv(A,Aa2,R2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),disjnt(A),image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),bot_bot(set(A))))),image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),B3),bot_bot(set(A)))))
      <=> ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B3),R2) ) ) ).

% disjnt_equiv_class
tff(fact_6333_chains__alt__def,axiom,
    ! [A: $tType,Aa2: set(set(A))] : ( chains2(A,Aa2) = aa(fun(set(set(A)),$o),set(set(set(A))),collect(set(set(A))),pred_chain(set(A),Aa2,ord_less(set(A)))) ) ).

% chains_alt_def
tff(fact_6334_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_6335_disjnt__insert1,axiom,
    ! [A: $tType,A3: A,X3: set(A),Y5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),disjnt(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),X3)),Y5)
    <=> ( ~ member(A,A3,Y5)
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),disjnt(A),X3),Y5) ) ) ).

% disjnt_insert1
tff(fact_6336_disjnt__insert2,axiom,
    ! [A: $tType,Y5: set(A),A3: A,X3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),disjnt(A),Y5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),X3))
    <=> ( ~ member(A,A3,Y5)
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),disjnt(A),Y5),X3) ) ) ).

% disjnt_insert2
tff(fact_6337_disjnt__Un1,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A),C3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),disjnt(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),B2)),C3)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),disjnt(A),Aa2),C3)
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),disjnt(A),B2),C3) ) ) ).

% disjnt_Un1
tff(fact_6338_disjnt__Un2,axiom,
    ! [A: $tType,C3: set(A),Aa2: set(A),B2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),disjnt(A),C3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),B2))
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),disjnt(A),C3),Aa2)
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),disjnt(A),C3),B2) ) ) ).

% disjnt_Un2
tff(fact_6339_disjnt__Times2__iff,axiom,
    ! [B: $tType,A: $tType,Aa2: set(A),C3: set(B),B2: set(A)] :
      ( aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),disjnt(product_prod(A,B)),product_Sigma(A,B,Aa2,aTP_Lamp_pv(set(B),fun(A,set(B)),C3))),product_Sigma(A,B,B2,aTP_Lamp_pv(set(B),fun(A,set(B)),C3)))
    <=> ( ( C3 = bot_bot(set(B)) )
        | aa(set(A),$o,aa(set(A),fun(set(A),$o),disjnt(A),Aa2),B2) ) ) ).

% disjnt_Times2_iff
tff(fact_6340_disjnt__Times1__iff,axiom,
    ! [A: $tType,B: $tType,C3: set(A),Aa2: set(B),B2: set(B)] :
      ( aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),disjnt(product_prod(A,B)),product_Sigma(A,B,C3,aTP_Lamp_pv(set(B),fun(A,set(B)),Aa2))),product_Sigma(A,B,C3,aTP_Lamp_pv(set(B),fun(A,set(B)),B2)))
    <=> ( ( C3 = bot_bot(set(A)) )
        | aa(set(B),$o,aa(set(B),fun(set(B),$o),disjnt(B),Aa2),B2) ) ) ).

% disjnt_Times1_iff
tff(fact_6341_disjnt__subset1,axiom,
    ! [A: $tType,X3: set(A),Y5: set(A),Z6: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),disjnt(A),X3),Y5)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Z6),X3)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),disjnt(A),Z6),Y5) ) ) ).

% disjnt_subset1
tff(fact_6342_disjnt__subset2,axiom,
    ! [A: $tType,X3: set(A),Y5: set(A),Z6: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),disjnt(A),X3),Y5)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Z6),Y5)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),disjnt(A),X3),Z6) ) ) ).

% disjnt_subset2
tff(fact_6343_disjnt__sym,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),disjnt(A),Aa2),B2)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),disjnt(A),B2),Aa2) ) ).

% disjnt_sym
tff(fact_6344_disjnt__iff,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),disjnt(A),Aa2),B2)
    <=> ! [X6: A] :
          ~ ( member(A,X6,Aa2)
            & member(A,X6,B2) ) ) ).

% disjnt_iff
tff(fact_6345_disjnt__insert,axiom,
    ! [A: $tType,Xa: A,N7: set(A),M5: set(A)] :
      ( ~ member(A,Xa,N7)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),disjnt(A),M5),N7)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),disjnt(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),M5)),N7) ) ) ).

% disjnt_insert
tff(fact_6346_disjnt__empty2,axiom,
    ! [A: $tType,Aa2: set(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),disjnt(A),Aa2),bot_bot(set(A))) ).

% disjnt_empty2
tff(fact_6347_disjnt__empty1,axiom,
    ! [A: $tType,Aa2: set(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),disjnt(A),bot_bot(set(A))),Aa2) ).

% disjnt_empty1
tff(fact_6348_disjnt__Sigma__iff,axiom,
    ! [B: $tType,A: $tType,Aa2: set(A),C3: fun(A,set(B)),B2: set(A)] :
      ( aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),disjnt(product_prod(A,B)),product_Sigma(A,B,Aa2,C3)),product_Sigma(A,B,B2,C3))
    <=> ( ! [X6: A] :
            ( member(A,X6,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),B2))
           => ( aa(A,set(B),C3,X6) = bot_bot(set(B)) ) )
        | aa(set(A),$o,aa(set(A),fun(set(A),$o),disjnt(A),Aa2),B2) ) ) ).

% disjnt_Sigma_iff
tff(fact_6349_disjnt__def,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),disjnt(A),Aa2),B2)
    <=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),B2) = bot_bot(set(A)) ) ) ).

% disjnt_def
tff(fact_6350_disjoint__image__subset,axiom,
    ! [A: $tType,A14: set(set(A)),F2: fun(set(A),set(A))] :
      ( pairwise(set(A),disjnt(A),A14)
     => ( ! [X7: set(A)] :
            ( member(set(A),X7,A14)
           => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),F2,X7)),X7) )
       => pairwise(set(A),disjnt(A),aa(set(set(A)),set(set(A)),image2(set(A),set(A),F2),A14)) ) ) ).

% disjoint_image_subset
tff(fact_6351_disjnt__ge__max,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Y5: set(A),X3: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),Y5)
         => ( ! [X: A] :
                ( member(A,X,X3)
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),lattic643756798349783984er_Max(A,Y5)),X) )
           => aa(set(A),$o,aa(set(A),fun(set(A),$o),disjnt(A),X3),Y5) ) ) ) ).

% disjnt_ge_max
tff(fact_6352_total__inv__image,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),R2: set(product_prod(B,B))] :
      ( inj_on(A,B,F2,top_top(set(A)))
     => ( total_on(B,top_top(set(B)),R2)
       => total_on(A,top_top(set(A)),inv_image(B,A,R2,F2)) ) ) ).

% total_inv_image
tff(fact_6353_cSUP__UNION,axiom,
    ! [B: $tType,E: $tType,A: $tType] :
      ( condit1219197933456340205attice(E)
     => ! [Aa2: set(A),B2: fun(A,set(B)),F2: fun(B,E)] :
          ( ( Aa2 != bot_bot(set(A)) )
         => ( ! [X: A] :
                ( member(A,X,Aa2)
               => ( aa(A,set(B),B2,X) != bot_bot(set(B)) ) )
           => ( condit941137186595557371_above(E,aa(set(set(E)),set(E),complete_Sup_Sup(set(E)),aa(set(A),set(set(E)),image2(A,set(E),aa(fun(B,E),fun(A,set(E)),aTP_Lamp_xl(fun(A,set(B)),fun(fun(B,E),fun(A,set(E))),B2),F2)),Aa2)))
             => ( aa(set(E),E,complete_Sup_Sup(E),aa(set(B),set(E),image2(B,E,F2),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),B2),Aa2)))) = aa(set(E),E,complete_Sup_Sup(E),aa(set(A),set(E),image2(A,E,aa(fun(B,E),fun(A,E),aTP_Lamp_xm(fun(A,set(B)),fun(fun(B,E),fun(A,E)),B2),F2)),Aa2)) ) ) ) ) ) ).

% cSUP_UNION
tff(fact_6354_bdd__above__empty,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => condit941137186595557371_above(A,bot_bot(set(A))) ) ).

% bdd_above_empty
tff(fact_6355_bdd__above__insert,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [A3: A,Aa2: set(A)] :
          ( condit941137186595557371_above(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),Aa2))
        <=> condit941137186595557371_above(A,Aa2) ) ) ).

% bdd_above_insert
tff(fact_6356_in__inv__image,axiom,
    ! [A: $tType,B: $tType,Xa: A,Y: A,R2: set(product_prod(B,B)),F2: fun(A,B)] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Y),inv_image(B,A,R2,F2))
    <=> 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,F2,Xa)),aa(A,B,F2,Y)),R2) ) ).

% in_inv_image
tff(fact_6357_converse__inv__image,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(B,B)),F2: fun(A,B)] : ( converse(A,A,inv_image(B,A,R,F2)) = inv_image(B,A,converse(B,B,R),F2) ) ).

% converse_inv_image
tff(fact_6358_trans__inv__image,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(A,A)),F2: fun(B,A)] :
      ( trans(A,R2)
     => trans(B,inv_image(A,B,R2,F2)) ) ).

% trans_inv_image
tff(fact_6359_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,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),S)),A3)
            <=> ! [X6: A] :
                  ( member(A,X6,S)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X6),A3) ) ) ) ) ) ).

% cSup_le_iff
tff(fact_6360_cSup__mono,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [B2: set(A),Aa2: set(A)] :
          ( ( B2 != bot_bot(set(A)) )
         => ( condit941137186595557371_above(A,Aa2)
           => ( ! [B4: A] :
                  ( member(A,B4,B2)
                 => ? [X4: A] :
                      ( member(A,X4,Aa2)
                      & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B4),X4) ) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),B2)),aa(set(A),A,complete_Sup_Sup(A),Aa2)) ) ) ) ) ).

% cSup_mono
tff(fact_6361_less__cSup__iff,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X3: set(A),Y: A] :
          ( ( X3 != bot_bot(set(A)) )
         => ( condit941137186595557371_above(A,X3)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),aa(set(A),A,complete_Sup_Sup(A),X3))
            <=> ? [X6: A] :
                  ( member(A,X6,X3)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),X6) ) ) ) ) ) ).

% less_cSup_iff
tff(fact_6362_cSUP__lessD,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [F2: fun(B,A),Aa2: set(B),Y: A,Ia: B] :
          ( condit941137186595557371_above(A,aa(set(B),set(A),image2(B,A,F2),Aa2))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F2),Aa2))),Y)
           => ( member(B,Ia,Aa2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,F2,Ia)),Y) ) ) ) ) ).

% cSUP_lessD
tff(fact_6363_cSUP__le__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [Aa2: set(A),F2: fun(A,B),U: B] :
          ( ( Aa2 != bot_bot(set(A)) )
         => ( condit941137186595557371_above(B,aa(set(A),set(B),image2(A,B,F2),Aa2))
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),Aa2))),U)
            <=> ! [X6: A] :
                  ( member(A,X6,Aa2)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X6)),U) ) ) ) ) ) ).

% cSUP_le_iff
tff(fact_6364_cSUP__mono,axiom,
    ! [A: $tType,B: $tType,E: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [Aa2: set(A),G: fun(E,B),B2: set(E),F2: fun(A,B)] :
          ( ( Aa2 != bot_bot(set(A)) )
         => ( condit941137186595557371_above(B,aa(set(E),set(B),image2(E,B,G),B2))
           => ( ! [N5: A] :
                  ( member(A,N5,Aa2)
                 => ? [X4: E] :
                      ( member(E,X4,B2)
                      & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,N5)),aa(E,B,G,X4)) ) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),Aa2))),aa(set(B),B,complete_Sup_Sup(B),aa(set(E),set(B),image2(E,B,G),B2))) ) ) ) ) ).

% cSUP_mono
tff(fact_6365_cSup__subset__mono,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Aa2: set(A),B2: set(A)] :
          ( ( Aa2 != bot_bot(set(A)) )
         => ( condit941137186595557371_above(A,B2)
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),B2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),Aa2)),aa(set(A),A,complete_Sup_Sup(A),B2)) ) ) ) ) ).

% cSup_subset_mono
tff(fact_6366_cSup__insert,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X3: set(A),A3: A] :
          ( ( X3 != bot_bot(set(A)) )
         => ( condit941137186595557371_above(A,X3)
           => ( aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),X3)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),aa(set(A),A,complete_Sup_Sup(A),X3)) ) ) ) ) ).

% cSup_insert
tff(fact_6367_cSup__insert__If,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X3: set(A),A3: A] :
          ( condit941137186595557371_above(A,X3)
         => ( aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),X3)) = $ite(X3 = 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),X3))) ) ) ) ).

% cSup_insert_If
tff(fact_6368_cSup__union__distrib,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Aa2: set(A),B2: set(A)] :
          ( ( Aa2 != bot_bot(set(A)) )
         => ( condit941137186595557371_above(A,Aa2)
           => ( ( B2 != bot_bot(set(A)) )
             => ( condit941137186595557371_above(A,B2)
               => ( 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)),Aa2),B2)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Sup_Sup(A),Aa2)),aa(set(A),A,complete_Sup_Sup(A),B2)) ) ) ) ) ) ) ).

% cSup_union_distrib
tff(fact_6369_inv__image__def,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(B,B)),F2: fun(A,B)] : ( inv_image(B,A,R2,F2) = aa(fun(product_prod(A,A),$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_xn(set(product_prod(B,B)),fun(fun(A,B),fun(A,fun(A,$o))),R2),F2))) ) ).

% inv_image_def
tff(fact_6370_less__cSUP__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( condit6923001295902523014norder(B)
     => ! [Aa2: set(A),F2: fun(A,B),A3: B] :
          ( ( Aa2 != bot_bot(set(A)) )
         => ( condit941137186595557371_above(B,aa(set(A),set(B),image2(A,B,F2),Aa2))
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A3),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),Aa2)))
            <=> ? [X6: A] :
                  ( member(A,X6,Aa2)
                  & aa(B,$o,aa(B,fun(B,$o),ord_less(B),A3),aa(A,B,F2,X6)) ) ) ) ) ) ).

% less_cSUP_iff
tff(fact_6371_conditionally__complete__lattice__class_OSUP__sup__distrib,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [Aa2: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( ( Aa2 != bot_bot(set(A)) )
         => ( condit941137186595557371_above(B,aa(set(A),set(B),image2(A,B,F2),Aa2))
           => ( condit941137186595557371_above(B,aa(set(A),set(B),image2(A,B,G),Aa2))
             => ( aa(B,B,aa(B,fun(B,B),sup_sup(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),Aa2))),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,G),Aa2))) = 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_xo(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)),Aa2)) ) ) ) ) ) ).

% conditionally_complete_lattice_class.SUP_sup_distrib
tff(fact_6372_cSUP__subset__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [Aa2: set(A),G: fun(A,B),B2: set(A),F2: fun(A,B)] :
          ( ( Aa2 != bot_bot(set(A)) )
         => ( condit941137186595557371_above(B,aa(set(A),set(B),image2(A,B,G),B2))
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),B2)
             => ( ! [X: A] :
                    ( member(A,X,Aa2)
                   => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X)),aa(A,B,G,X)) )
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),Aa2))),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,G),B2))) ) ) ) ) ) ).

% cSUP_subset_mono
tff(fact_6373_cSup__inter__less__eq,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Aa2: set(A),B2: set(A)] :
          ( condit941137186595557371_above(A,Aa2)
         => ( condit941137186595557371_above(A,B2)
           => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),B2) != bot_bot(set(A)) )
             => aa(A,$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)),Aa2),B2))),aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Sup_Sup(A),Aa2)),aa(set(A),A,complete_Sup_Sup(A),B2))) ) ) ) ) ).

% cSup_inter_less_eq
tff(fact_6374_cSUP__insert,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [Aa2: set(A),F2: fun(A,B),A3: A] :
          ( ( Aa2 != bot_bot(set(A)) )
         => ( condit941137186595557371_above(B,aa(set(A),set(B),image2(A,B,F2),Aa2))
           => ( aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),Aa2))) = aa(B,B,aa(B,fun(B,B),sup_sup(B),aa(A,B,F2,A3)),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),Aa2))) ) ) ) ) ).

% cSUP_insert
tff(fact_6375_cSUP__union,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [Aa2: set(A),F2: fun(A,B),B2: set(A)] :
          ( ( Aa2 != bot_bot(set(A)) )
         => ( condit941137186595557371_above(B,aa(set(A),set(B),image2(A,B,F2),Aa2))
           => ( ( B2 != bot_bot(set(A)) )
             => ( condit941137186595557371_above(B,aa(set(A),set(B),image2(A,B,F2),B2))
               => ( aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),B2))) = aa(B,B,aa(B,fun(B,B),sup_sup(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),Aa2))),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),B2))) ) ) ) ) ) ) ).

% cSUP_union
tff(fact_6376_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_xp(set(A),fun(A,$o),S))) ) ) ) ) ).

% cSup_cInf
tff(fact_6377_mono__cSUP,axiom,
    ! [B: $tType,A: $tType,E: $tType] :
      ( ( condit1219197933456340205attice(A)
        & condit1219197933456340205attice(B) )
     => ! [F2: fun(A,B),Aa2: fun(E,A),Ia2: set(E)] :
          ( order_mono(A,B,F2)
         => ( condit941137186595557371_above(A,aa(set(E),set(A),image2(E,A,Aa2),Ia2))
           => ( ( Ia2 != bot_bot(set(E)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(E),set(B),image2(E,B,aa(fun(E,A),fun(E,B),aTP_Lamp_xq(fun(A,B),fun(fun(E,A),fun(E,B)),F2),Aa2)),Ia2))),aa(A,B,F2,aa(set(A),A,complete_Sup_Sup(A),aa(set(E),set(A),image2(E,A,Aa2),Ia2)))) ) ) ) ) ).

% mono_cSUP
tff(fact_6378_mono__cSup,axiom,
    ! [B: $tType,A: $tType] :
      ( ( condit1219197933456340205attice(A)
        & condit1219197933456340205attice(B) )
     => ! [F2: fun(A,B),Aa2: set(A)] :
          ( order_mono(A,B,F2)
         => ( condit941137186595557371_above(A,Aa2)
           => ( ( Aa2 != bot_bot(set(A)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),Aa2))),aa(A,B,F2,aa(set(A),A,complete_Sup_Sup(A),Aa2))) ) ) ) ) ).

% mono_cSup
tff(fact_6379_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_xr(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_6380_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_xs(list(A),product_prod(nat,list(A)))) ) ).

% lenlex_def
tff(fact_6381_less__than__iff,axiom,
    ! [Xa: nat,Y: nat] :
      ( member(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xa),Y),less_than)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Y) ) ).

% less_than_iff
tff(fact_6382_measures__def,axiom,
    ! [A: $tType,Fs: list(fun(A,nat))] : ( measures(A,Fs) = inv_image(list(nat),A,lex(nat,less_than),aTP_Lamp_xt(list(fun(A,nat)),fun(A,list(nat)),Fs)) ) ).

% measures_def
tff(fact_6383_mlex__prod__def,axiom,
    ! [A: $tType,F2: fun(A,nat),R: set(product_prod(A,A))] : ( mlex_prod(A,F2,R) = inv_image(product_prod(nat,A),A,lex_prod(nat,A,less_than,R),aTP_Lamp_xu(fun(A,nat),fun(A,product_prod(nat,A)),F2)) ) ).

% mlex_prod_def
tff(fact_6384_cINF__UNION,axiom,
    ! [B: $tType,E: $tType,A: $tType] :
      ( condit1219197933456340205attice(E)
     => ! [Aa2: set(A),B2: fun(A,set(B)),F2: fun(B,E)] :
          ( ( Aa2 != bot_bot(set(A)) )
         => ( ! [X: A] :
                ( member(A,X,Aa2)
               => ( aa(A,set(B),B2,X) != bot_bot(set(B)) ) )
           => ( condit1013018076250108175_below(E,aa(set(set(E)),set(E),complete_Sup_Sup(set(E)),aa(set(A),set(set(E)),image2(A,set(E),aa(fun(B,E),fun(A,set(E)),aTP_Lamp_xl(fun(A,set(B)),fun(fun(B,E),fun(A,set(E))),B2),F2)),Aa2)))
             => ( aa(set(E),E,complete_Inf_Inf(E),aa(set(B),set(E),image2(B,E,F2),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),B2),Aa2)))) = aa(set(E),E,complete_Inf_Inf(E),aa(set(A),set(E),image2(A,E,aa(fun(B,E),fun(A,E),aTP_Lamp_xv(fun(A,set(B)),fun(fun(B,E),fun(A,E)),B2),F2)),Aa2)) ) ) ) ) ) ).

% cINF_UNION
tff(fact_6385_mono__cINF,axiom,
    ! [A: $tType,B: $tType,E: $tType] :
      ( ( condit1219197933456340205attice(B)
        & condit1219197933456340205attice(A) )
     => ! [F2: fun(A,B),Aa2: fun(E,A),Ia2: set(E)] :
          ( order_mono(A,B,F2)
         => ( condit1013018076250108175_below(A,aa(set(E),set(A),image2(E,A,Aa2),Ia2))
           => ( ( Ia2 != bot_bot(set(E)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,aa(set(A),A,complete_Inf_Inf(A),aa(set(E),set(A),image2(E,A,Aa2),Ia2)))),aa(set(B),B,complete_Inf_Inf(B),aa(set(E),set(B),image2(E,B,aa(fun(E,A),fun(E,B),aTP_Lamp_xq(fun(A,B),fun(fun(E,A),fun(E,B)),F2),Aa2)),Ia2))) ) ) ) ) ).

% mono_cINF
tff(fact_6386_bdd__below__empty,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => condit1013018076250108175_below(A,bot_bot(set(A))) ) ).

% bdd_below_empty
tff(fact_6387_bdd__below__insert,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [A3: A,Aa2: set(A)] :
          ( condit1013018076250108175_below(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),Aa2))
        <=> condit1013018076250108175_below(A,Aa2) ) ) ).

% bdd_below_insert
tff(fact_6388_le__cInf__iff,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [S: set(A),A3: A] :
          ( ( S != bot_bot(set(A)) )
         => ( condit1013018076250108175_below(A,S)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(set(A),A,complete_Inf_Inf(A),S))
            <=> ! [X6: A] :
                  ( member(A,X6,S)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),X6) ) ) ) ) ) ).

% le_cInf_iff
tff(fact_6389_cInf__mono,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [B2: set(A),Aa2: set(A)] :
          ( ( B2 != bot_bot(set(A)) )
         => ( condit1013018076250108175_below(A,Aa2)
           => ( ! [B4: A] :
                  ( member(A,B4,B2)
                 => ? [X4: A] :
                      ( member(A,X4,Aa2)
                      & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),B4) ) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),Aa2)),aa(set(A),A,complete_Inf_Inf(A),B2)) ) ) ) ) ).

% cInf_mono
tff(fact_6390_cInf__less__iff,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X3: set(A),Y: A] :
          ( ( X3 != bot_bot(set(A)) )
         => ( condit1013018076250108175_below(A,X3)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(A),A,complete_Inf_Inf(A),X3)),Y)
            <=> ? [X6: A] :
                  ( member(A,X6,X3)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X6),Y) ) ) ) ) ) ).

% cInf_less_iff
tff(fact_6391_less__cINF__D,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [F2: fun(B,A),Aa2: set(B),Y: A,Ia: B] :
          ( condit1013018076250108175_below(A,aa(set(B),set(A),image2(B,A,F2),Aa2))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F2),Aa2)))
           => ( member(B,Ia,Aa2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),aa(B,A,F2,Ia)) ) ) ) ) ).

% less_cINF_D
tff(fact_6392_cINF__mono,axiom,
    ! [E: $tType,B: $tType,A: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [B2: set(A),F2: fun(E,B),Aa2: set(E),G: fun(A,B)] :
          ( ( B2 != bot_bot(set(A)) )
         => ( condit1013018076250108175_below(B,aa(set(E),set(B),image2(E,B,F2),Aa2))
           => ( ! [M6: A] :
                  ( member(A,M6,B2)
                 => ? [X4: E] :
                      ( member(E,X4,Aa2)
                      & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(E,B,F2,X4)),aa(A,B,G,M6)) ) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(E),set(B),image2(E,B,F2),Aa2))),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,G),B2))) ) ) ) ) ).

% cINF_mono
tff(fact_6393_le__cINF__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [Aa2: set(A),F2: fun(A,B),U: B] :
          ( ( Aa2 != bot_bot(set(A)) )
         => ( condit1013018076250108175_below(B,aa(set(A),set(B),image2(A,B,F2),Aa2))
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),U),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),Aa2)))
            <=> ! [X6: A] :
                  ( member(A,X6,Aa2)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),U),aa(A,B,F2,X6)) ) ) ) ) ) ).

% le_cINF_iff
tff(fact_6394_cInf__superset__mono,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Aa2: set(A),B2: set(A)] :
          ( ( Aa2 != bot_bot(set(A)) )
         => ( condit1013018076250108175_below(A,B2)
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),B2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),B2)),aa(set(A),A,complete_Inf_Inf(A),Aa2)) ) ) ) ) ).

% cInf_superset_mono
tff(fact_6395_cInf__insert,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X3: set(A),A3: A] :
          ( ( X3 != bot_bot(set(A)) )
         => ( condit1013018076250108175_below(A,X3)
           => ( aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),X3)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),aa(set(A),A,complete_Inf_Inf(A),X3)) ) ) ) ) ).

% cInf_insert
tff(fact_6396_cInf__insert__If,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X3: set(A),A3: A] :
          ( condit1013018076250108175_below(A,X3)
         => ( aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),X3)) = $ite(X3 = 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),X3))) ) ) ) ).

% cInf_insert_If
tff(fact_6397_cInf__union__distrib,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Aa2: set(A),B2: set(A)] :
          ( ( Aa2 != bot_bot(set(A)) )
         => ( condit1013018076250108175_below(A,Aa2)
           => ( ( B2 != bot_bot(set(A)) )
             => ( condit1013018076250108175_below(A,B2)
               => ( 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)),Aa2),B2)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Inf_Inf(A),Aa2)),aa(set(A),A,complete_Inf_Inf(A),B2)) ) ) ) ) ) ) ).

% cInf_union_distrib
tff(fact_6398_cINF__less__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( condit6923001295902523014norder(B)
     => ! [Aa2: set(A),F2: fun(A,B),A3: B] :
          ( ( Aa2 != bot_bot(set(A)) )
         => ( condit1013018076250108175_below(B,aa(set(A),set(B),image2(A,B,F2),Aa2))
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),Aa2))),A3)
            <=> ? [X6: A] :
                  ( member(A,X6,Aa2)
                  & aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,X6)),A3) ) ) ) ) ) ).

% cINF_less_iff
tff(fact_6399_cINF__inf__distrib,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [Aa2: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( ( Aa2 != bot_bot(set(A)) )
         => ( condit1013018076250108175_below(B,aa(set(A),set(B),image2(A,B,F2),Aa2))
           => ( condit1013018076250108175_below(B,aa(set(A),set(B),image2(A,B,G),Aa2))
             => ( aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),Aa2))),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,G),Aa2))) = 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_xw(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)),Aa2)) ) ) ) ) ) ).

% cINF_inf_distrib
tff(fact_6400_cINF__superset__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [Aa2: set(A),G: fun(A,B),B2: set(A),F2: fun(A,B)] :
          ( ( Aa2 != bot_bot(set(A)) )
         => ( condit1013018076250108175_below(B,aa(set(A),set(B),image2(A,B,G),B2))
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),B2)
             => ( ! [X: A] :
                    ( member(A,X,B2)
                   => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,G,X)),aa(A,B,F2,X)) )
               => aa(B,$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),B2))),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),Aa2))) ) ) ) ) ) ).

% cINF_superset_mono
tff(fact_6401_less__eq__cInf__inter,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Aa2: set(A),B2: set(A)] :
          ( condit1013018076250108175_below(A,Aa2)
         => ( condit1013018076250108175_below(A,B2)
           => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),B2) != bot_bot(set(A)) )
             => aa(A,$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),Aa2)),aa(set(A),A,complete_Inf_Inf(A),B2))),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)),Aa2),B2))) ) ) ) ) ).

% less_eq_cInf_inter
tff(fact_6402_cINF__insert,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [Aa2: set(A),F2: fun(A,B),A3: A] :
          ( ( Aa2 != bot_bot(set(A)) )
         => ( condit1013018076250108175_below(B,aa(set(A),set(B),image2(A,B,F2),Aa2))
           => ( aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),Aa2))) = aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(A,B,F2,A3)),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),Aa2))) ) ) ) ) ).

% cINF_insert
tff(fact_6403_cINF__union,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [Aa2: set(A),F2: fun(A,B),B2: set(A)] :
          ( ( Aa2 != bot_bot(set(A)) )
         => ( condit1013018076250108175_below(B,aa(set(A),set(B),image2(A,B,F2),Aa2))
           => ( ( B2 != bot_bot(set(A)) )
             => ( condit1013018076250108175_below(B,aa(set(A),set(B),image2(A,B,F2),B2))
               => ( aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),B2))) = aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),Aa2))),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),B2))) ) ) ) ) ) ) ).

% cINF_union
tff(fact_6404_cInf__le__cSup,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Aa2: set(A)] :
          ( ( Aa2 != bot_bot(set(A)) )
         => ( condit941137186595557371_above(A,Aa2)
           => ( condit1013018076250108175_below(A,Aa2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),Aa2)),aa(set(A),A,complete_Sup_Sup(A),Aa2)) ) ) ) ) ).

% cInf_le_cSup
tff(fact_6405_cInf__cSup,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [S: set(A)] :
          ( ( S != bot_bot(set(A)) )
         => ( condit1013018076250108175_below(A,S)
           => ( aa(set(A),A,complete_Inf_Inf(A),S) = aa(set(A),A,complete_Sup_Sup(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_xx(set(A),fun(A,$o),S))) ) ) ) ) ).

% cInf_cSup
tff(fact_6406_mono__cInf,axiom,
    ! [B: $tType,A: $tType] :
      ( ( condit1219197933456340205attice(A)
        & condit1219197933456340205attice(B) )
     => ! [F2: fun(A,B),Aa2: set(A)] :
          ( order_mono(A,B,F2)
         => ( condit1013018076250108175_below(A,Aa2)
           => ( ( Aa2 != bot_bot(set(A)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,aa(set(A),A,complete_Inf_Inf(A),Aa2))),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),Aa2))) ) ) ) ) ).

% mono_cInf
tff(fact_6407_filterlim__INF__INF,axiom,
    ! [A: $tType,E: $tType,F4: $tType,B: $tType,J4: set(A),Ia2: set(B),F2: fun(F4,E),Fa: fun(B,filter(F4)),G4: fun(A,filter(E))] :
      ( ! [M6: A] :
          ( member(A,M6,J4)
         => ? [X4: B] :
              ( member(B,X4,Ia2)
              & aa(filter(E),$o,aa(filter(E),fun(filter(E),$o),ord_less_eq(filter(E)),aa(filter(F4),filter(E),aa(fun(F4,E),fun(filter(F4),filter(E)),filtermap(F4,E),F2),aa(B,filter(F4),Fa,X4))),aa(A,filter(E),G4,M6)) ) )
     => filterlim(F4,E,F2,aa(set(filter(E)),filter(E),complete_Inf_Inf(filter(E)),aa(set(A),set(filter(E)),image2(A,filter(E),G4),J4)),aa(set(filter(F4)),filter(F4),complete_Inf_Inf(filter(F4)),aa(set(B),set(filter(F4)),image2(B,filter(F4),Fa),Ia2))) ) ).

% filterlim_INF_INF
tff(fact_6408_bsqr__max2,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A1: A,A22: A,B1: A,B22: A] :
      ( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => ( 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),B1),B22)),bNF_Wellorder_bsqr(A,R2))
       => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),bNF_We1388413361240627857o_max2(A,R2,A1,A22)),bNF_We1388413361240627857o_max2(A,R2,B1,B22)),R2) ) ) ).

% bsqr_max2
tff(fact_6409_filtermap__id_H,axiom,
    ! [A: $tType,X4: filter(A)] : ( aa(filter(A),filter(A),aa(fun(A,A),fun(filter(A),filter(A)),filtermap(A,A),aTP_Lamp_ab(A,A)),X4) = X4 ) ).

% filtermap_id'
tff(fact_6410_filtermap__id,axiom,
    ! [A: $tType] : ( aa(fun(A,A),fun(filter(A),filter(A)),filtermap(A,A),id(A)) = id(filter(A)) ) ).

% filtermap_id
tff(fact_6411_filtermap__bot,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A)] : ( aa(filter(B),filter(A),aa(fun(B,A),fun(filter(B),filter(A)),filtermap(B,A),F2),bot_bot(filter(B))) = bot_bot(filter(A)) ) ).

% filtermap_bot
tff(fact_6412_filtermap__principal,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Aa2: set(B)] : ( aa(filter(B),filter(A),aa(fun(B,A),fun(filter(B),filter(A)),filtermap(B,A),F2),aa(set(B),filter(B),principal(B),Aa2)) = aa(set(A),filter(A),principal(A),aa(set(B),set(A),image2(B,A,F2),Aa2)) ) ).

% filtermap_principal
tff(fact_6413_eventually__filtermap,axiom,
    ! [A: $tType,B: $tType,Pa: fun(A,$o),F2: fun(B,A),Fa: filter(B)] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Pa),aa(filter(B),filter(A),aa(fun(B,A),fun(filter(B),filter(A)),filtermap(B,A),F2),Fa))
    <=> aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),eventually(B),aa(fun(B,A),fun(B,$o),aTP_Lamp_wh(fun(A,$o),fun(fun(B,A),fun(B,$o)),Pa),F2)),Fa) ) ).

% eventually_filtermap
tff(fact_6414_filtermap__mono,axiom,
    ! [B: $tType,A: $tType,Fa: filter(A),F: filter(A),F2: fun(A,B)] :
      ( aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),Fa),F)
     => aa(filter(B),$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),F2),Fa)),aa(filter(A),filter(B),aa(fun(A,B),fun(filter(A),filter(B)),filtermap(A,B),F2),F)) ) ).

% filtermap_mono
tff(fact_6415_filterlim__filtermap,axiom,
    ! [B: $tType,A: $tType,E: $tType,F2: fun(A,B),F12: filter(B),G: fun(E,A),F22: filter(E)] :
      ( filterlim(A,B,F2,F12,aa(filter(E),filter(A),aa(fun(E,A),fun(filter(E),filter(A)),filtermap(E,A),G),F22))
    <=> filterlim(E,B,aa(fun(E,A),fun(E,B),aTP_Lamp_to(fun(A,B),fun(fun(E,A),fun(E,B)),F2),G),F12,F22) ) ).

% filterlim_filtermap
tff(fact_6416_filtermap__filtermap,axiom,
    ! [B: $tType,A: $tType,E: $tType,F2: fun(B,A),G: fun(E,B),Fa: filter(E)] : ( aa(filter(B),filter(A),aa(fun(B,A),fun(filter(B),filter(A)),filtermap(B,A),F2),aa(filter(E),filter(B),aa(fun(E,B),fun(filter(E),filter(B)),filtermap(E,B),G),Fa)) = aa(filter(E),filter(A),aa(fun(E,A),fun(filter(E),filter(A)),filtermap(E,A),aa(fun(E,B),fun(E,A),aTP_Lamp_bn(fun(B,A),fun(fun(E,B),fun(E,A)),F2),G)),Fa) ) ).

% filtermap_filtermap
tff(fact_6417_filtermap__ident,axiom,
    ! [A: $tType,Fa: filter(A)] : ( aa(filter(A),filter(A),aa(fun(A,A),fun(filter(A),filter(A)),filtermap(A,A),aTP_Lamp_ab(A,A)),Fa) = Fa ) ).

% filtermap_ident
tff(fact_6418_filtermap__eq__strong,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),Fa: filter(A),G4: filter(A)] :
      ( inj_on(A,B,F2,top_top(set(A)))
     => ( ( aa(filter(A),filter(B),aa(fun(A,B),fun(filter(A),filter(B)),filtermap(A,B),F2),Fa) = aa(filter(A),filter(B),aa(fun(A,B),fun(filter(A),filter(B)),filtermap(A,B),F2),G4) )
      <=> ( Fa = G4 ) ) ) ).

% filtermap_eq_strong
tff(fact_6419_filtermap__sup,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),F12: filter(B),F22: filter(B)] : ( aa(filter(B),filter(A),aa(fun(B,A),fun(filter(B),filter(A)),filtermap(B,A),F2),aa(filter(B),filter(B),aa(filter(B),fun(filter(B),filter(B)),sup_sup(filter(B)),F12),F22)) = 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),F2),F12)),aa(filter(B),filter(A),aa(fun(B,A),fun(filter(B),filter(A)),filtermap(B,A),F2),F22)) ) ).

% filtermap_sup
tff(fact_6420_filtermap__bot__iff,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Fa: filter(B)] :
      ( ( aa(filter(B),filter(A),aa(fun(B,A),fun(filter(B),filter(A)),filtermap(B,A),F2),Fa) = bot_bot(filter(A)) )
    <=> ( Fa = bot_bot(filter(B)) ) ) ).

% filtermap_bot_iff
tff(fact_6421_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_6422_well__order__on__domain,axiom,
    ! [A: $tType,Aa2: set(A),R2: set(product_prod(A,A)),A3: A,B3: A] :
      ( order_well_order_on(A,Aa2,R2)
     => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B3),R2)
       => ( member(A,A3,Aa2)
          & member(A,B3,Aa2) ) ) ) ).

% well_order_on_domain
tff(fact_6423_filterlim__def,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),F22: filter(B),F12: filter(A)] :
      ( filterlim(A,B,F2,F22,F12)
    <=> aa(filter(B),$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),F2),F12)),F22) ) ).

% filterlim_def
tff(fact_6424_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_6425_filtercomap__filtermap,axiom,
    ! [B: $tType,A: $tType,Fa: filter(A),F2: fun(A,B)] : aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),Fa),filtercomap(A,B,F2,aa(filter(A),filter(B),aa(fun(A,B),fun(filter(A),filter(B)),filtermap(A,B),F2),Fa))) ).

% filtercomap_filtermap
tff(fact_6426_filtermap__filtercomap,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),Fa: filter(A)] : aa(filter(A),$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),F2),filtercomap(B,A,F2,Fa))),Fa) ).

% filtermap_filtercomap
tff(fact_6427_filtermap__le__iff__le__filtercomap,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),Fa: filter(B),G4: filter(A)] :
      ( aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),aa(filter(B),filter(A),aa(fun(B,A),fun(filter(B),filter(A)),filtermap(B,A),F2),Fa)),G4)
    <=> aa(filter(B),$o,aa(filter(B),fun(filter(B),$o),ord_less_eq(filter(B)),Fa),filtercomap(B,A,F2,G4)) ) ).

% filtermap_le_iff_le_filtercomap
tff(fact_6428_filtermap__sequentually__ne__bot,axiom,
    ! [A: $tType,F2: fun(nat,A)] : ( aa(filter(nat),filter(A),aa(fun(nat,A),fun(filter(nat),filter(A)),filtermap(nat,A),F2),at_top(nat)) != bot_bot(filter(A)) ) ).

% filtermap_sequentually_ne_bot
tff(fact_6429_filtermap__inf,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),F12: filter(B),F22: filter(B)] : aa(filter(A),$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),F2),aa(filter(B),filter(B),aa(filter(B),fun(filter(B),filter(B)),inf_inf(filter(B)),F12),F22))),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),F2),F12)),aa(filter(B),filter(A),aa(fun(B,A),fun(filter(B),filter(A)),filtermap(B,A),F2),F22))) ).

% filtermap_inf
tff(fact_6430_filtermap__fun__inverse,axiom,
    ! [B: $tType,A: $tType,G: fun(A,B),Fa: filter(B),G4: filter(A),F2: fun(B,A)] :
      ( filterlim(A,B,G,Fa,G4)
     => ( filterlim(B,A,F2,G4,Fa)
       => ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(fun(B,A),fun(A,$o),aTP_Lamp_xy(fun(A,B),fun(fun(B,A),fun(A,$o)),G),F2)),G4)
         => ( aa(filter(B),filter(A),aa(fun(B,A),fun(filter(B),filter(A)),filtermap(B,A),F2),Fa) = G4 ) ) ) ) ).

% filtermap_fun_inverse
tff(fact_6431_filtermap__SUP,axiom,
    ! [B: $tType,A: $tType,E: $tType,F2: fun(B,A),Fa: fun(E,filter(B)),B2: set(E)] : ( aa(filter(B),filter(A),aa(fun(B,A),fun(filter(B),filter(A)),filtermap(B,A),F2),aa(set(filter(B)),filter(B),complete_Sup_Sup(filter(B)),aa(set(E),set(filter(B)),image2(E,filter(B),Fa),B2))) = aa(set(filter(A)),filter(A),complete_Sup_Sup(filter(A)),aa(set(E),set(filter(A)),image2(E,filter(A),aa(fun(E,filter(B)),fun(E,filter(A)),aTP_Lamp_xz(fun(B,A),fun(fun(E,filter(B)),fun(E,filter(A))),F2),Fa)),B2)) ) ).

% filtermap_SUP
tff(fact_6432_filtermap__def,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Fa: filter(B)] : ( aa(filter(B),filter(A),aa(fun(B,A),fun(filter(B),filter(A)),filtermap(B,A),F2),Fa) = aa(fun(fun(A,$o),$o),filter(A),abs_filter(A),aa(filter(B),fun(fun(A,$o),$o),aTP_Lamp_yb(fun(B,A),fun(filter(B),fun(fun(A,$o),$o)),F2),Fa)) ) ).

% filtermap_def
tff(fact_6433_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_ac(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,$o),aTP_Lamp_ix(nat,fun(nat,fun(nat,$o)),N)))) ).

% natLeq_on_well_order_on
tff(fact_6434_filtermap__mono__strong,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),Fa: filter(A),G4: filter(A)] :
      ( inj_on(A,B,F2,top_top(set(A)))
     => ( aa(filter(B),$o,aa(filter(B),fun(filter(B),$o),ord_less_eq(filter(B)),aa(filter(A),filter(B),aa(fun(A,B),fun(filter(A),filter(B)),filtermap(A,B),F2),Fa)),aa(filter(A),filter(B),aa(fun(A,B),fun(filter(A),filter(B)),filtermap(A,B),F2),G4))
      <=> aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),Fa),G4) ) ) ).

% filtermap_mono_strong
tff(fact_6435_filtermap__image__finite__subsets__at__top,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),Aa2: set(A)] :
      ( inj_on(A,B,F2,Aa2)
     => ( aa(filter(set(A)),filter(set(B)),aa(fun(set(A),set(B)),fun(filter(set(A)),filter(set(B))),filtermap(set(A),set(B)),image2(A,B,F2)),finite5375528669736107172at_top(A,Aa2)) = finite5375528669736107172at_top(B,aa(set(A),set(B),image2(A,B,F2),Aa2)) ) ) ).

% filtermap_image_finite_subsets_at_top
tff(fact_6436_filtermap__INF,axiom,
    ! [B: $tType,A: $tType,E: $tType,F2: fun(B,A),Fa: fun(E,filter(B)),B2: set(E)] : aa(filter(A),$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),F2),aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(E),set(filter(B)),image2(E,filter(B),Fa),B2)))),aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(E),set(filter(A)),image2(E,filter(A),aa(fun(E,filter(B)),fun(E,filter(A)),aTP_Lamp_xz(fun(B,A),fun(fun(E,filter(B)),fun(E,filter(A))),F2),Fa)),B2))) ).

% filtermap_INF
tff(fact_6437_natLeq__on__Well__order,axiom,
    ! [N: nat] : order_well_order_on(nat,aa(set(product_prod(nat,nat)),set(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_ix(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,$o),aTP_Lamp_ix(nat,fun(nat,fun(nat,$o)),N)))) ).

% natLeq_on_Well_order
tff(fact_6438_Linear__order__Well__order__iff,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( order_679001287576687338der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => ( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
      <=> ! [A11: set(A)] :
            ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A11),aa(set(product_prod(A,A)),set(A),field2(A),R2))
           => ( ( A11 != bot_bot(set(A)) )
             => ? [X6: A] :
                  ( member(A,X6,A11)
                  & ! [Xa2: A] :
                      ( member(A,Xa2,A11)
                     => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X6),Xa2),R2) ) ) ) ) ) ) ).

% Linear_order_Well_order_iff
tff(fact_6439_ofilter__subset__embedS,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),Aa2: set(A),B2: set(A)] :
      ( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => ( order_ofilter(A,R2,Aa2)
       => ( order_ofilter(A,R2,B2)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),Aa2),B2)
          <=> 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,Aa2,aTP_Lamp_pt(set(A),fun(A,set(A)),Aa2))),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,B2,aTP_Lamp_pt(set(A),fun(A,set(A)),B2))),id(A)) ) ) ) ) ).

% ofilter_subset_embedS
tff(fact_6440_bsqr__ofilter,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),D2: set(product_prod(A,A))] :
      ( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => ( order_ofilter(product_prod(A,A),bNF_Wellorder_bsqr(A,R2),D2)
       => ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less(set(product_prod(A,A))),D2),product_Sigma(A,A,aa(set(product_prod(A,A)),set(A),field2(A),R2),aTP_Lamp_qc(set(product_prod(A,A)),fun(A,set(A)),R2)))
         => ( ~ ? [A4: A] : ( aa(set(product_prod(A,A)),set(A),field2(A),R2) = order_under(A,R2,A4) )
           => ? [A12: set(A)] :
                ( order_ofilter(A,R2,A12)
                & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A12),aa(set(product_prod(A,A)),set(A),field2(A),R2))
                & aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),D2),product_Sigma(A,A,A12,aTP_Lamp_pt(set(A),fun(A,set(A)),A12))) ) ) ) ) ) ).

% bsqr_ofilter
tff(fact_6441_under__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A3: A] : ( order_under(A,R2,A3) = aa(fun(A,$o),set(A),collect(A),aa(A,fun(A,$o),aTP_Lamp_dq(set(product_prod(A,A)),fun(A,fun(A,$o)),R2),A3)) ) ).

% under_def
tff(fact_6442_under__incr,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A3: A,B3: A] :
      ( trans(A,R2)
     => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B3),R2)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),order_under(A,R2,A3)),order_under(A,R2,B3)) ) ) ).

% under_incr
tff(fact_6443_embedS__Field,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(A,A)),R5: set(product_prod(B,B)),F2: fun(A,B)] :
      ( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => ( bNF_Wellorder_embedS(A,B,R2,R5,F2)
       => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less(set(B)),aa(set(A),set(B),image2(A,B,F2),aa(set(product_prod(A,A)),set(A),field2(A),R2))),aa(set(product_prod(B,B)),set(B),field2(B),R5)) ) ) ).

% embedS_Field
tff(fact_6444_ofilter__subset__embedS__iso,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),Aa2: set(A),B2: set(A)] :
      ( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => ( order_ofilter(A,R2,Aa2)
       => ( order_ofilter(A,R2,B2)
         => ( ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),Aa2),B2)
            <=> 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,Aa2,aTP_Lamp_pt(set(A),fun(A,set(A)),Aa2))),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,B2,aTP_Lamp_pt(set(A),fun(A,set(A)),B2))),id(A)) )
            & ( ( Aa2 = B2 )
            <=> 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,Aa2,aTP_Lamp_pt(set(A),fun(A,set(A)),Aa2))),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,B2,aTP_Lamp_pt(set(A),fun(A,set(A)),B2))),id(A)) ) ) ) ) ) ).

% ofilter_subset_embedS_iso
tff(fact_6445_iso__forward,axiom,
    ! [A: $tType,B: $tType,Xa: A,Y: A,R2: set(product_prod(A,A)),R5: set(product_prod(B,B)),F2: fun(A,B)] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Y),R2)
     => ( bNF_Wellorder_iso(A,B,R2,R5,F2)
       => 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,F2,Xa)),aa(A,B,F2,Y)),R5) ) ) ).

% iso_forward
tff(fact_6446_ofilter__ordLess,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),Aa2: set(A)] :
      ( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => ( order_ofilter(A,R2,Aa2)
       => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),Aa2),aa(set(product_prod(A,A)),set(A),field2(A),R2))
        <=> 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,Aa2,aTP_Lamp_pt(set(A),fun(A,set(A)),Aa2)))),R2),bNF_We4044943003108391690rdLess(A,A)) ) ) ) ).

% ofilter_ordLess
tff(fact_6447_ofilter__subset__ordLess,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),Aa2: set(A),B2: set(A)] :
      ( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => ( order_ofilter(A,R2,Aa2)
       => ( order_ofilter(A,R2,B2)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),Aa2),B2)
          <=> 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,Aa2,aTP_Lamp_pt(set(A),fun(A,set(A)),Aa2)))),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,B2,aTP_Lamp_pt(set(A),fun(A,set(A)),B2)))),bNF_We4044943003108391690rdLess(A,A)) ) ) ) ) ).

% ofilter_subset_ordLess
tff(fact_6448_ordLess__irreflexive,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : ~ 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_6449_ordLess__transitive,axiom,
    ! [B: $tType,E: $tType,A: $tType,R2: set(product_prod(A,A)),R5: set(product_prod(B,B)),R9: set(product_prod(E,E))] :
      ( 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),R5),bNF_We4044943003108391690rdLess(A,B))
     => ( 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))),R5),R9),bNF_We4044943003108391690rdLess(B,E))
       => member(product_prod(set(product_prod(A,A)),set(product_prod(E,E))),aa(set(product_prod(E,E)),product_prod(set(product_prod(A,A)),set(product_prod(E,E))),aa(set(product_prod(A,A)),fun(set(product_prod(E,E)),product_prod(set(product_prod(A,A)),set(product_prod(E,E)))),product_Pair(set(product_prod(A,A)),set(product_prod(E,E))),R2),R9),bNF_We4044943003108391690rdLess(A,E)) ) ) ).

% ordLess_transitive
tff(fact_6450_finite__ordLess__infinite,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,A)),R5: set(product_prod(B,B))] :
      ( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => ( order_well_order_on(B,aa(set(product_prod(B,B)),set(B),field2(B),R5),R5)
       => ( aa(set(A),$o,finite_finite2(A),aa(set(product_prod(A,A)),set(A),field2(A),R2))
         => ( ~ aa(set(B),$o,finite_finite2(B),aa(set(product_prod(B,B)),set(B),field2(B),R5))
           => 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),R5),bNF_We4044943003108391690rdLess(A,B)) ) ) ) ) ).

% finite_ordLess_infinite
tff(fact_6451_underS__Restr__ordLess,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A3: A] :
      ( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => ( ( aa(set(product_prod(A,A)),set(A),field2(A),R2) != bot_bot(set(A)) )
       => 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_yc(set(product_prod(A,A)),fun(A,fun(A,set(A))),R2),A3)))),R2),bNF_We4044943003108391690rdLess(A,A)) ) ) ).

% underS_Restr_ordLess
tff(fact_6452_ofilter__subset__ordLeq,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),Aa2: set(A),B2: set(A)] :
      ( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => ( order_ofilter(A,R2,Aa2)
       => ( order_ofilter(A,R2,B2)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),B2)
          <=> 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,Aa2,aTP_Lamp_pt(set(A),fun(A,set(A)),Aa2)))),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,B2,aTP_Lamp_pt(set(A),fun(A,set(A)),B2)))),bNF_Wellorder_ordLeq(A,A)) ) ) ) ) ).

% ofilter_subset_ordLeq
tff(fact_6453_ordLess__ordLeq__trans,axiom,
    ! [B: $tType,E: $tType,A: $tType,R2: set(product_prod(A,A)),R5: set(product_prod(B,B)),R9: set(product_prod(E,E))] :
      ( 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),R5),bNF_We4044943003108391690rdLess(A,B))
     => ( 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))),R5),R9),bNF_Wellorder_ordLeq(B,E))
       => member(product_prod(set(product_prod(A,A)),set(product_prod(E,E))),aa(set(product_prod(E,E)),product_prod(set(product_prod(A,A)),set(product_prod(E,E))),aa(set(product_prod(A,A)),fun(set(product_prod(E,E)),product_prod(set(product_prod(A,A)),set(product_prod(E,E)))),product_Pair(set(product_prod(A,A)),set(product_prod(E,E))),R2),R9),bNF_We4044943003108391690rdLess(A,E)) ) ) ).

% ordLess_ordLeq_trans
tff(fact_6454_ordLeq__ordLess__trans,axiom,
    ! [B: $tType,E: $tType,A: $tType,R2: set(product_prod(A,A)),R5: set(product_prod(B,B)),R9: set(product_prod(E,E))] :
      ( 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),R5),bNF_Wellorder_ordLeq(A,B))
     => ( 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))),R5),R9),bNF_We4044943003108391690rdLess(B,E))
       => member(product_prod(set(product_prod(A,A)),set(product_prod(E,E))),aa(set(product_prod(E,E)),product_prod(set(product_prod(A,A)),set(product_prod(E,E))),aa(set(product_prod(A,A)),fun(set(product_prod(E,E)),product_prod(set(product_prod(A,A)),set(product_prod(E,E)))),product_Pair(set(product_prod(A,A)),set(product_prod(E,E))),R2),R9),bNF_We4044943003108391690rdLess(A,E)) ) ) ).

% ordLeq_ordLess_trans
tff(fact_6455_ordLess__imp__ordLeq,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,A)),R5: set(product_prod(B,B))] :
      ( 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),R5),bNF_We4044943003108391690rdLess(A,B))
     => 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),R5),bNF_Wellorder_ordLeq(A,B)) ) ).

% ordLess_imp_ordLeq
tff(fact_6456_not__ordLess__ordLeq,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(A,A)),R5: set(product_prod(B,B))] :
      ( 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),R5),bNF_We4044943003108391690rdLess(A,B))
     => ~ 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))),R5),R2),bNF_Wellorder_ordLeq(B,A)) ) ).

% not_ordLess_ordLeq
tff(fact_6457_underS__Field2,axiom,
    ! [A: $tType,A3: A,R2: set(product_prod(A,A))] :
      ( member(A,A3,aa(set(product_prod(A,A)),set(A),field2(A),R2))
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),order_underS(A,R2,A3)),aa(set(product_prod(A,A)),set(A),field2(A),R2)) ) ).

% underS_Field2
tff(fact_6458_underS__empty,axiom,
    ! [A: $tType,A3: A,R2: set(product_prod(A,A))] :
      ( ~ member(A,A3,aa(set(product_prod(A,A)),set(A),field2(A),R2))
     => ( order_underS(A,R2,A3) = bot_bot(set(A)) ) ) ).

% underS_empty
tff(fact_6459_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,$o),aTP_Lamp_yd(set(product_prod(A,A)),fun(A,fun(A,$o)),R2),A3)) ) ).

% underS_def
tff(fact_6460_underS__I,axiom,
    ! [A: $tType,Ia: A,J: A,R: set(product_prod(A,A))] :
      ( ( Ia != J )
     => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Ia),J),R)
       => member(A,Ia,order_underS(A,R,J)) ) ) ).

% underS_I
tff(fact_6461_underS__E,axiom,
    ! [A: $tType,Ia: A,R: set(product_prod(A,A)),J: A] :
      ( member(A,Ia,order_underS(A,R,J))
     => ( ( Ia != J )
        & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Ia),J),R) ) ) ).

% underS_E
tff(fact_6462_ordLeq__transitive,axiom,
    ! [B: $tType,E: $tType,A: $tType,R2: set(product_prod(A,A)),R5: set(product_prod(B,B)),R9: set(product_prod(E,E))] :
      ( 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),R5),bNF_Wellorder_ordLeq(A,B))
     => ( 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))),R5),R9),bNF_Wellorder_ordLeq(B,E))
       => member(product_prod(set(product_prod(A,A)),set(product_prod(E,E))),aa(set(product_prod(E,E)),product_prod(set(product_prod(A,A)),set(product_prod(E,E))),aa(set(product_prod(A,A)),fun(set(product_prod(E,E)),product_prod(set(product_prod(A,A)),set(product_prod(E,E)))),product_Pair(set(product_prod(A,A)),set(product_prod(E,E))),R2),R9),bNF_Wellorder_ordLeq(A,E)) ) ) ).

% ordLeq_transitive
tff(fact_6463_exists__minim__Well__order,axiom,
    ! [A: $tType,R: set(set(product_prod(A,A)))] :
      ( ( R != bot_bot(set(set(product_prod(A,A)))) )
     => ( ! [X: set(product_prod(A,A))] :
            ( member(set(product_prod(A,A)),X,R)
           => order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),X),X) )
       => ? [X: set(product_prod(A,A))] :
            ( member(set(product_prod(A,A)),X,R)
            & ! [Xa4: set(product_prod(A,A))] :
                ( member(set(product_prod(A,A)),Xa4,R)
               => 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))),X),Xa4),bNF_Wellorder_ordLeq(A,A)) ) ) ) ) ).

% exists_minim_Well_order
tff(fact_6464_ordLeq__total,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(A,A)),R5: set(product_prod(B,B))] :
      ( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => ( order_well_order_on(B,aa(set(product_prod(B,B)),set(B),field2(B),R5),R5)
       => ( 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),R5),bNF_Wellorder_ordLeq(A,B))
          | 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))),R5),R2),bNF_Wellorder_ordLeq(B,A)) ) ) ) ).

% ordLeq_total
tff(fact_6465_ordLeq__reflexive,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => 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_6466_ordLeq__Well__order__simp,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(A,A)),R5: set(product_prod(B,B))] :
      ( 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),R5),bNF_Wellorder_ordLeq(A,B))
     => ( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
        & order_well_order_on(B,aa(set(product_prod(B,B)),set(B),field2(B),R5),R5) ) ) ).

% ordLeq_Well_order_simp
tff(fact_6467_ordLeq__iff__ordLess__Restr,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,A)),R5: set(product_prod(B,B))] :
      ( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => ( order_well_order_on(B,aa(set(product_prod(B,B)),set(B),field2(B),R5),R5)
       => ( 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),R5),bNF_Wellorder_ordLeq(A,B))
        <=> ! [X6: A] :
              ( member(A,X6,aa(set(product_prod(A,A)),set(A),field2(A),R2))
             => 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,X6),aa(A,fun(A,set(A)),aTP_Lamp_yc(set(product_prod(A,A)),fun(A,fun(A,set(A))),R2),X6)))),R5),bNF_We4044943003108391690rdLess(A,B)) ) ) ) ) ).

% ordLeq_iff_ordLess_Restr
tff(fact_6468_underS__Field3,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A3: A] :
      ( ( aa(set(product_prod(A,A)),set(A),field2(A),R2) != bot_bot(set(A)) )
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),order_underS(A,R2,A3)),aa(set(product_prod(A,A)),set(A),field2(A),R2)) ) ).

% underS_Field3
tff(fact_6469_ordLess__or__ordLeq,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(A,A)),R5: set(product_prod(B,B))] :
      ( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => ( order_well_order_on(B,aa(set(product_prod(B,B)),set(B),field2(B),R5),R5)
       => ( 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),R5),bNF_We4044943003108391690rdLess(A,B))
          | 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))),R5),R2),bNF_Wellorder_ordLeq(B,A)) ) ) ) ).

% ordLess_or_ordLeq
tff(fact_6470_not__ordLeq__iff__ordLess,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,A)),R5: set(product_prod(B,B))] :
      ( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => ( order_well_order_on(B,aa(set(product_prod(B,B)),set(B),field2(B),R5),R5)
       => ( ~ 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))),R5),R2),bNF_Wellorder_ordLeq(B,A))
        <=> 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),R5),bNF_We4044943003108391690rdLess(A,B)) ) ) ) ).

% not_ordLeq_iff_ordLess
tff(fact_6471_not__ordLess__iff__ordLeq,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,A)),R5: set(product_prod(B,B))] :
      ( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => ( order_well_order_on(B,aa(set(product_prod(B,B)),set(B),field2(B),R5),R5)
       => ( ~ 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))),R5),R2),bNF_We4044943003108391690rdLess(B,A))
        <=> 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),R5),bNF_Wellorder_ordLeq(A,B)) ) ) ) ).

% not_ordLess_iff_ordLeq
tff(fact_6472_underS__incl__iff,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A3: A,B3: A] :
      ( order_679001287576687338der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => ( member(A,A3,aa(set(product_prod(A,A)),set(A),field2(A),R2))
       => ( member(A,B3,aa(set(product_prod(A,A)),set(A),field2(A),R2))
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),order_underS(A,R2,A3)),order_underS(A,R2,B3))
          <=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B3),R2) ) ) ) ) ).

% underS_incl_iff
tff(fact_6473_underS__incr,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A3: A,B3: A] :
      ( trans(A,R2)
     => ( antisym(A,R2)
       => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B3),R2)
         => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),order_underS(A,R2,A3)),order_underS(A,R2,B3)) ) ) ) ).

% underS_incr
tff(fact_6474_Refl__under__underS,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A3: A] :
      ( refl_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => ( member(A,A3,aa(set(product_prod(A,A)),set(A),field2(A),R2))
       => ( 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)),insert3(A),A3),bot_bot(set(A)))) ) ) ) ).

% Refl_under_underS
tff(fact_6475_ordLess__iff__ordIso__Restr,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(A,A)),R5: set(product_prod(B,B))] :
      ( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => ( order_well_order_on(B,aa(set(product_prod(B,B)),set(B),field2(B),R5),R5)
       => ( 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))),R5),R2),bNF_We4044943003108391690rdLess(B,A))
        <=> ? [X6: A] :
              ( member(A,X6,aa(set(product_prod(A,A)),set(A),field2(A),R2))
              & 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))),R5),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,X6),aa(A,fun(A,set(A)),aTP_Lamp_yc(set(product_prod(A,A)),fun(A,fun(A,set(A))),R2),X6)))),bNF_Wellorder_ordIso(B,A)) ) ) ) ) ).

% ordLess_iff_ordIso_Restr
tff(fact_6476_embed__ordLess__ofilterIncl,axiom,
    ! [B: $tType,A: $tType,E: $tType,R12: set(product_prod(A,A)),R23: set(product_prod(B,B)),R32: set(product_prod(E,E)),F132: fun(A,E),F232: fun(B,E)] :
      ( 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))
     => ( 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),R32),bNF_We4044943003108391690rdLess(B,E))
       => ( bNF_Wellorder_embed(A,E,R12,R32,F132)
         => ( bNF_Wellorder_embed(B,E,R23,R32,F232)
           => member(product_prod(set(E),set(E)),aa(set(E),product_prod(set(E),set(E)),aa(set(E),fun(set(E),product_prod(set(E),set(E))),product_Pair(set(E),set(E)),aa(set(A),set(E),image2(A,E,F132),aa(set(product_prod(A,A)),set(A),field2(A),R12))),aa(set(B),set(E),image2(B,E,F232),aa(set(product_prod(B,B)),set(B),field2(B),R23))),bNF_We413866401316099525erIncl(E,R32)) ) ) ) ) ).

% embed_ordLess_ofilterIncl
tff(fact_6477_ordIso__iff__ordLeq,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(A,A)),R5: set(product_prod(B,B))] :
      ( 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),R5),bNF_Wellorder_ordIso(A,B))
    <=> ( 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),R5),bNF_Wellorder_ordLeq(A,B))
        & 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))),R5),R2),bNF_Wellorder_ordLeq(B,A)) ) ) ).

% ordIso_iff_ordLeq
tff(fact_6478_ordIso__imp__ordLeq,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,A)),R5: set(product_prod(B,B))] :
      ( 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),R5),bNF_Wellorder_ordIso(A,B))
     => 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),R5),bNF_Wellorder_ordLeq(A,B)) ) ).

% ordIso_imp_ordLeq
tff(fact_6479_ordIso__ordLeq__trans,axiom,
    ! [B: $tType,E: $tType,A: $tType,R2: set(product_prod(A,A)),R5: set(product_prod(B,B)),R9: set(product_prod(E,E))] :
      ( 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),R5),bNF_Wellorder_ordIso(A,B))
     => ( 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))),R5),R9),bNF_Wellorder_ordLeq(B,E))
       => member(product_prod(set(product_prod(A,A)),set(product_prod(E,E))),aa(set(product_prod(E,E)),product_prod(set(product_prod(A,A)),set(product_prod(E,E))),aa(set(product_prod(A,A)),fun(set(product_prod(E,E)),product_prod(set(product_prod(A,A)),set(product_prod(E,E)))),product_Pair(set(product_prod(A,A)),set(product_prod(E,E))),R2),R9),bNF_Wellorder_ordLeq(A,E)) ) ) ).

% ordIso_ordLeq_trans
tff(fact_6480_ordLeq__ordIso__trans,axiom,
    ! [B: $tType,E: $tType,A: $tType,R2: set(product_prod(A,A)),R5: set(product_prod(B,B)),R9: set(product_prod(E,E))] :
      ( 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),R5),bNF_Wellorder_ordLeq(A,B))
     => ( 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))),R5),R9),bNF_Wellorder_ordIso(B,E))
       => member(product_prod(set(product_prod(A,A)),set(product_prod(E,E))),aa(set(product_prod(E,E)),product_prod(set(product_prod(A,A)),set(product_prod(E,E))),aa(set(product_prod(A,A)),fun(set(product_prod(E,E)),product_prod(set(product_prod(A,A)),set(product_prod(E,E)))),product_Pair(set(product_prod(A,A)),set(product_prod(E,E))),R2),R9),bNF_Wellorder_ordLeq(A,E)) ) ) ).

% ordLeq_ordIso_trans
tff(fact_6481_ordIso__transitive,axiom,
    ! [B: $tType,E: $tType,A: $tType,R2: set(product_prod(A,A)),R5: set(product_prod(B,B)),R9: set(product_prod(E,E))] :
      ( 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),R5),bNF_Wellorder_ordIso(A,B))
     => ( 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))),R5),R9),bNF_Wellorder_ordIso(B,E))
       => member(product_prod(set(product_prod(A,A)),set(product_prod(E,E))),aa(set(product_prod(E,E)),product_prod(set(product_prod(A,A)),set(product_prod(E,E))),aa(set(product_prod(A,A)),fun(set(product_prod(E,E)),product_prod(set(product_prod(A,A)),set(product_prod(E,E)))),product_Pair(set(product_prod(A,A)),set(product_prod(E,E))),R2),R9),bNF_Wellorder_ordIso(A,E)) ) ) ).

% ordIso_transitive
tff(fact_6482_ordIso__symmetric,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(A,A)),R5: set(product_prod(B,B))] :
      ( 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),R5),bNF_Wellorder_ordIso(A,B))
     => 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))),R5),R2),bNF_Wellorder_ordIso(B,A)) ) ).

% ordIso_symmetric
tff(fact_6483_ordLess__not__embed,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(A,A)),R5: set(product_prod(B,B))] :
      ( 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),R5),bNF_We4044943003108391690rdLess(A,B))
     => ~ ? [X_13: fun(B,A)] : bNF_Wellorder_embed(B,A,R5,R2,X_13) ) ).

% ordLess_not_embed
tff(fact_6484_not__ordLess__ordIso,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,A)),R5: set(product_prod(B,B))] :
      ( 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),R5),bNF_We4044943003108391690rdLess(A,B))
     => ~ 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),R5),bNF_Wellorder_ordIso(A,B)) ) ).

% not_ordLess_ordIso
tff(fact_6485_ordIso__ordLess__trans,axiom,
    ! [B: $tType,E: $tType,A: $tType,R2: set(product_prod(A,A)),R5: set(product_prod(B,B)),R9: set(product_prod(E,E))] :
      ( 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),R5),bNF_Wellorder_ordIso(A,B))
     => ( 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))),R5),R9),bNF_We4044943003108391690rdLess(B,E))
       => member(product_prod(set(product_prod(A,A)),set(product_prod(E,E))),aa(set(product_prod(E,E)),product_prod(set(product_prod(A,A)),set(product_prod(E,E))),aa(set(product_prod(A,A)),fun(set(product_prod(E,E)),product_prod(set(product_prod(A,A)),set(product_prod(E,E)))),product_Pair(set(product_prod(A,A)),set(product_prod(E,E))),R2),R9),bNF_We4044943003108391690rdLess(A,E)) ) ) ).

% ordIso_ordLess_trans
tff(fact_6486_ordLess__ordIso__trans,axiom,
    ! [B: $tType,E: $tType,A: $tType,R2: set(product_prod(A,A)),R5: set(product_prod(B,B)),R9: set(product_prod(E,E))] :
      ( 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),R5),bNF_We4044943003108391690rdLess(A,B))
     => ( 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))),R5),R9),bNF_Wellorder_ordIso(B,E))
       => member(product_prod(set(product_prod(A,A)),set(product_prod(E,E))),aa(set(product_prod(E,E)),product_prod(set(product_prod(A,A)),set(product_prod(E,E))),aa(set(product_prod(A,A)),fun(set(product_prod(E,E)),product_prod(set(product_prod(A,A)),set(product_prod(E,E)))),product_Pair(set(product_prod(A,A)),set(product_prod(E,E))),R2),R9),bNF_We4044943003108391690rdLess(A,E)) ) ) ).

% ordLess_ordIso_trans
tff(fact_6487_finite__well__order__on__ordIso,axiom,
    ! [A: $tType,Aa2: set(A),R2: set(product_prod(A,A)),R5: set(product_prod(A,A))] :
      ( aa(set(A),$o,finite_finite2(A),Aa2)
     => ( order_well_order_on(A,Aa2,R2)
       => ( order_well_order_on(A,Aa2,R5)
         => 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),R5),bNF_Wellorder_ordIso(A,A)) ) ) ) ).

% finite_well_order_on_ordIso
tff(fact_6488_ordIso__reflexive,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => 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_6489_ordLeq__iff__ordLess__or__ordIso,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,A)),R5: set(product_prod(B,B))] :
      ( 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),R5),bNF_Wellorder_ordLeq(A,B))
    <=> ( 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),R5),bNF_We4044943003108391690rdLess(A,B))
        | 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),R5),bNF_Wellorder_ordIso(A,B)) ) ) ).

% ordLeq_iff_ordLess_or_ordIso
tff(fact_6490_internalize__ordLeq,axiom,
    ! [A: $tType,B: $tType,R5: set(product_prod(A,A)),R2: set(product_prod(B,B))] :
      ( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R5),R2),bNF_Wellorder_ordLeq(A,B))
    <=> ? [P4: set(product_prod(B,B))] :
          ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(product_prod(B,B)),set(B),field2(B),P4)),aa(set(product_prod(B,B)),set(B),field2(B),R2))
          & 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))),R5),P4),bNF_Wellorder_ordIso(A,B))
          & 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))),P4),R2),bNF_Wellorder_ordLeq(B,B)) ) ) ).

% internalize_ordLeq
tff(fact_6491_BNF__Wellorder__Constructions_OordLess__Field,axiom,
    ! [A: $tType,B: $tType,R12: set(product_prod(A,A)),R23: set(product_prod(B,B)),F2: fun(A,B)] :
      ( 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))
     => ( bNF_Wellorder_embed(A,B,R12,R23,F2)
       => ( aa(set(A),set(B),image2(A,B,F2),aa(set(product_prod(A,A)),set(A),field2(A),R12)) != aa(set(product_prod(B,B)),set(B),field2(B),R23) ) ) ) ).

% BNF_Wellorder_Constructions.ordLess_Field
tff(fact_6492_internalize__ordLess,axiom,
    ! [A: $tType,B: $tType,R5: set(product_prod(A,A)),R2: set(product_prod(B,B))] :
      ( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R5),R2),bNF_We4044943003108391690rdLess(A,B))
    <=> ? [P4: set(product_prod(B,B))] :
          ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less(set(B)),aa(set(product_prod(B,B)),set(B),field2(B),P4)),aa(set(product_prod(B,B)),set(B),field2(B),R2))
          & 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))),R5),P4),bNF_Wellorder_ordIso(A,B))
          & 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))),P4),R2),bNF_We4044943003108391690rdLess(B,B)) ) ) ).

% internalize_ordLess
tff(fact_6493_ordLess__iff,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(A,A)),R5: set(product_prod(B,B))] :
      ( 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),R5),bNF_We4044943003108391690rdLess(A,B))
    <=> ( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
        & order_well_order_on(B,aa(set(product_prod(B,B)),set(B),field2(B),R5),R5)
        & ~ ? [X_12: fun(B,A)] : bNF_Wellorder_embed(B,A,R5,R2,X_12) ) ) ).

% ordLess_iff
tff(fact_6494_embedS__iff,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(A,A)),R5: set(product_prod(B,B)),F2: fun(A,B)] :
      ( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => ( bNF_Wellorder_embed(A,B,R2,R5,F2)
       => ( bNF_Wellorder_embedS(A,B,R2,R5,F2)
        <=> aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less(set(B)),aa(set(A),set(B),image2(A,B,F2),aa(set(product_prod(A,A)),set(A),field2(A),R2))),aa(set(product_prod(B,B)),set(B),field2(B),R5)) ) ) ) ).

% embedS_iff
tff(fact_6495_dir__image__ordIso,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,A)),F2: fun(A,B)] :
      ( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => ( inj_on(A,B,F2,aa(set(product_prod(A,A)),set(A),field2(A),R2))
       => 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,F2)),bNF_Wellorder_ordIso(A,B)) ) ) ).

% dir_image_ordIso
tff(fact_6496_prod__filter__principal__singleton2,axiom,
    ! [B: $tType,A: $tType,Fa: filter(A),Xa: B] : ( aa(filter(B),filter(product_prod(A,B)),aa(filter(A),fun(filter(B),filter(product_prod(A,B))),prod_filter(A,B),Fa),aa(set(B),filter(B),principal(B),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert3(B),Xa),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_io(B,fun(A,product_prod(A,B))),Xa)),Fa) ) ).

% prod_filter_principal_singleton2
tff(fact_6497_prod__filter__eq__bot,axiom,
    ! [A: $tType,B: $tType,Aa2: filter(A),B2: filter(B)] :
      ( ( aa(filter(B),filter(product_prod(A,B)),aa(filter(A),fun(filter(B),filter(product_prod(A,B))),prod_filter(A,B),Aa2),B2) = bot_bot(filter(product_prod(A,B))) )
    <=> ( ( Aa2 = bot_bot(filter(A)) )
        | ( B2 = bot_bot(filter(B)) ) ) ) ).

% prod_filter_eq_bot
tff(fact_6498_eventually__prod__same,axiom,
    ! [A: $tType,Pa: fun(product_prod(A,A),$o),Fa: filter(A)] :
      ( aa(filter(product_prod(A,A)),$o,aa(fun(product_prod(A,A),$o),fun(filter(product_prod(A,A)),$o),eventually(product_prod(A,A)),Pa),aa(filter(A),filter(product_prod(A,A)),aa(filter(A),fun(filter(A),filter(product_prod(A,A))),prod_filter(A,A),Fa),Fa))
    <=> ? [Q7: fun(A,$o)] :
          ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Q7),Fa)
          & ! [X6: A,Y4: A] :
              ( aa(A,$o,Q7,X6)
             => ( aa(A,$o,Q7,Y4)
               => aa(product_prod(A,A),$o,Pa,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X6),Y4)) ) ) ) ) ).

% eventually_prod_same
tff(fact_6499_eventually__prod__filter,axiom,
    ! [A: $tType,B: $tType,Pa: fun(product_prod(A,B),$o),Fa: filter(A),G4: filter(B)] :
      ( aa(filter(product_prod(A,B)),$o,aa(fun(product_prod(A,B),$o),fun(filter(product_prod(A,B)),$o),eventually(product_prod(A,B)),Pa),aa(filter(B),filter(product_prod(A,B)),aa(filter(A),fun(filter(B),filter(product_prod(A,B))),prod_filter(A,B),Fa),G4))
    <=> ? [Pf: fun(A,$o),Pg: fun(B,$o)] :
          ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Pf),Fa)
          & aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),eventually(B),Pg),G4)
          & ! [X6: A,Y4: B] :
              ( aa(A,$o,Pf,X6)
             => ( aa(B,$o,Pg,Y4)
               => aa(product_prod(A,B),$o,Pa,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X6),Y4)) ) ) ) ) ).

% eventually_prod_filter
tff(fact_6500_prod__filter__assoc,axiom,
    ! [A: $tType,B: $tType,E: $tType,Fa: filter(A),G4: filter(B),H7: filter(E)] : ( aa(filter(E),filter(product_prod(product_prod(A,B),E)),aa(filter(product_prod(A,B)),fun(filter(E),filter(product_prod(product_prod(A,B),E))),prod_filter(product_prod(A,B),E),aa(filter(B),filter(product_prod(A,B)),aa(filter(A),fun(filter(B),filter(product_prod(A,B))),prod_filter(A,B),Fa),G4)),H7) = aa(filter(product_prod(A,product_prod(B,E))),filter(product_prod(product_prod(A,B),E)),aa(fun(product_prod(A,product_prod(B,E)),product_prod(product_prod(A,B),E)),fun(filter(product_prod(A,product_prod(B,E))),filter(product_prod(product_prod(A,B),E))),filtermap(product_prod(A,product_prod(B,E)),product_prod(product_prod(A,B),E)),aa(fun(A,fun(product_prod(B,E),product_prod(product_prod(A,B),E))),fun(product_prod(A,product_prod(B,E)),product_prod(product_prod(A,B),E)),product_case_prod(A,product_prod(B,E),product_prod(product_prod(A,B),E)),aTP_Lamp_yf(A,fun(product_prod(B,E),product_prod(product_prod(A,B),E))))),aa(filter(product_prod(B,E)),filter(product_prod(A,product_prod(B,E))),aa(filter(A),fun(filter(product_prod(B,E)),filter(product_prod(A,product_prod(B,E)))),prod_filter(A,product_prod(B,E)),Fa),aa(filter(E),filter(product_prod(B,E)),aa(filter(B),fun(filter(E),filter(product_prod(B,E))),prod_filter(B,E),G4),H7))) ) ).

% prod_filter_assoc
tff(fact_6501_prod__filter__mono,axiom,
    ! [A: $tType,B: $tType,Fa: filter(A),F: filter(A),G4: filter(B),G6: filter(B)] :
      ( aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),Fa),F)
     => ( aa(filter(B),$o,aa(filter(B),fun(filter(B),$o),ord_less_eq(filter(B)),G4),G6)
       => aa(filter(product_prod(A,B)),$o,aa(filter(product_prod(A,B)),fun(filter(product_prod(A,B)),$o),ord_less_eq(filter(product_prod(A,B))),aa(filter(B),filter(product_prod(A,B)),aa(filter(A),fun(filter(B),filter(product_prod(A,B))),prod_filter(A,B),Fa),G4)),aa(filter(B),filter(product_prod(A,B)),aa(filter(A),fun(filter(B),filter(product_prod(A,B))),prod_filter(A,B),F),G6)) ) ) ).

% prod_filter_mono
tff(fact_6502_filtermap__fst__prod__filter,axiom,
    ! [B: $tType,A: $tType,Aa2: filter(A),B2: filter(B)] : aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),aa(filter(product_prod(A,B)),filter(A),aa(fun(product_prod(A,B),A),fun(filter(product_prod(A,B)),filter(A)),filtermap(product_prod(A,B),A),product_fst(A,B)),aa(filter(B),filter(product_prod(A,B)),aa(filter(A),fun(filter(B),filter(product_prod(A,B))),prod_filter(A,B),Aa2),B2))),Aa2) ).

% filtermap_fst_prod_filter
tff(fact_6503_filtermap__snd__prod__filter,axiom,
    ! [B: $tType,A: $tType,Aa2: filter(B),B2: filter(A)] : aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),aa(filter(product_prod(B,A)),filter(A),aa(fun(product_prod(B,A),A),fun(filter(product_prod(B,A)),filter(A)),filtermap(product_prod(B,A),A),product_snd(B,A)),aa(filter(A),filter(product_prod(B,A)),aa(filter(B),fun(filter(A),filter(product_prod(B,A))),prod_filter(B,A),Aa2),B2))),B2) ).

% filtermap_snd_prod_filter
tff(fact_6504_prod__filter__mono__iff,axiom,
    ! [A: $tType,B: $tType,Aa2: filter(A),B2: filter(B),C3: filter(A),D2: filter(B)] :
      ( ( Aa2 != bot_bot(filter(A)) )
     => ( ( B2 != bot_bot(filter(B)) )
       => ( aa(filter(product_prod(A,B)),$o,aa(filter(product_prod(A,B)),fun(filter(product_prod(A,B)),$o),ord_less_eq(filter(product_prod(A,B))),aa(filter(B),filter(product_prod(A,B)),aa(filter(A),fun(filter(B),filter(product_prod(A,B))),prod_filter(A,B),Aa2),B2)),aa(filter(B),filter(product_prod(A,B)),aa(filter(A),fun(filter(B),filter(product_prod(A,B))),prod_filter(A,B),C3),D2))
        <=> ( aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),Aa2),C3)
            & aa(filter(B),$o,aa(filter(B),fun(filter(B),$o),ord_less_eq(filter(B)),B2),D2) ) ) ) ) ).

% prod_filter_mono_iff
tff(fact_6505_prod__filtermap1,axiom,
    ! [A: $tType,E: $tType,B: $tType,F2: fun(E,A),Fa: filter(E),G4: filter(B)] : ( aa(filter(B),filter(product_prod(A,B)),aa(filter(A),fun(filter(B),filter(product_prod(A,B))),prod_filter(A,B),aa(filter(E),filter(A),aa(fun(E,A),fun(filter(E),filter(A)),filtermap(E,A),F2),Fa)),G4) = aa(filter(product_prod(E,B)),filter(product_prod(A,B)),aa(fun(product_prod(E,B),product_prod(A,B)),fun(filter(product_prod(E,B)),filter(product_prod(A,B))),filtermap(product_prod(E,B),product_prod(A,B)),product_apfst(E,A,B,F2)),aa(filter(B),filter(product_prod(E,B)),aa(filter(E),fun(filter(B),filter(product_prod(E,B))),prod_filter(E,B),Fa),G4)) ) ).

% prod_filtermap1
tff(fact_6506_prod__filtermap2,axiom,
    ! [B: $tType,A: $tType,E: $tType,Fa: filter(A),G: fun(E,B),G4: filter(E)] : ( aa(filter(B),filter(product_prod(A,B)),aa(filter(A),fun(filter(B),filter(product_prod(A,B))),prod_filter(A,B),Fa),aa(filter(E),filter(B),aa(fun(E,B),fun(filter(E),filter(B)),filtermap(E,B),G),G4)) = aa(filter(product_prod(A,E)),filter(product_prod(A,B)),aa(fun(product_prod(A,E),product_prod(A,B)),fun(filter(product_prod(A,E)),filter(product_prod(A,B))),filtermap(product_prod(A,E),product_prod(A,B)),aa(fun(E,B),fun(product_prod(A,E),product_prod(A,B)),product_apsnd(E,B,A),G)),aa(filter(E),filter(product_prod(A,E)),aa(filter(A),fun(filter(E),filter(product_prod(A,E))),prod_filter(A,E),Fa),G4)) ) ).

% prod_filtermap2
tff(fact_6507_dir__image__def,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(B,B)),F2: fun(B,A)] : ( bNF_We2720479622203943262_image(B,A,R2,F2) = aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(B,A),fun(product_prod(A,A),$o),aTP_Lamp_yg(set(product_prod(B,B)),fun(fun(B,A),fun(product_prod(A,A),$o)),R2),F2)) ) ).

% dir_image_def
tff(fact_6508_filterlim__Pair,axiom,
    ! [E: $tType,B: $tType,A: $tType,F2: fun(A,B),G4: filter(B),Fa: filter(A),G: fun(A,E),H7: filter(E)] :
      ( filterlim(A,B,F2,G4,Fa)
     => ( filterlim(A,E,G,H7,Fa)
       => filterlim(A,product_prod(B,E),aa(fun(A,E),fun(A,product_prod(B,E)),aTP_Lamp_yh(fun(A,B),fun(fun(A,E),fun(A,product_prod(B,E))),F2),G),aa(filter(E),filter(product_prod(B,E)),aa(filter(B),fun(filter(E),filter(product_prod(B,E))),prod_filter(B,E),G4),H7),Fa) ) ) ).

% filterlim_Pair
tff(fact_6509_filtermap__Pair,axiom,
    ! [A: $tType,B: $tType,E: $tType,F2: fun(E,A),G: fun(E,B),Fa: filter(E)] : aa(filter(product_prod(A,B)),$o,aa(filter(product_prod(A,B)),fun(filter(product_prod(A,B)),$o),ord_less_eq(filter(product_prod(A,B))),aa(filter(E),filter(product_prod(A,B)),aa(fun(E,product_prod(A,B)),fun(filter(E),filter(product_prod(A,B))),filtermap(E,product_prod(A,B)),aa(fun(E,B),fun(E,product_prod(A,B)),aTP_Lamp_qt(fun(E,A),fun(fun(E,B),fun(E,product_prod(A,B))),F2),G)),Fa)),aa(filter(B),filter(product_prod(A,B)),aa(filter(A),fun(filter(B),filter(product_prod(A,B))),prod_filter(A,B),aa(filter(E),filter(A),aa(fun(E,A),fun(filter(E),filter(A)),filtermap(E,A),F2),Fa)),aa(filter(E),filter(B),aa(fun(E,B),fun(filter(E),filter(B)),filtermap(E,B),G),Fa))) ).

% filtermap_Pair
tff(fact_6510_principal__prod__principal,axiom,
    ! [A: $tType,B: $tType,Aa2: set(A),B2: set(B)] : ( aa(filter(B),filter(product_prod(A,B)),aa(filter(A),fun(filter(B),filter(product_prod(A,B))),prod_filter(A,B),aa(set(A),filter(A),principal(A),Aa2)),aa(set(B),filter(B),principal(B),B2)) = aa(set(product_prod(A,B)),filter(product_prod(A,B)),principal(product_prod(A,B)),product_Sigma(A,B,Aa2,aTP_Lamp_pv(set(B),fun(A,set(B)),B2))) ) ).

% principal_prod_principal
tff(fact_6511_prod__filter__def,axiom,
    ! [A: $tType,B: $tType,Fa: filter(A),G4: filter(B)] : ( aa(filter(B),filter(product_prod(A,B)),aa(filter(A),fun(filter(B),filter(product_prod(A,B))),prod_filter(A,B),Fa),G4) = aa(set(filter(product_prod(A,B))),filter(product_prod(A,B)),complete_Inf_Inf(filter(product_prod(A,B))),aa(set(product_prod(fun(A,$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,$o),filter(product_prod(A,B))),aTP_Lamp_yj(fun(A,$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_yk(filter(A),fun(filter(B),fun(fun(A,$o),fun(fun(B,$o),$o))),Fa),G4))))) ) ).

% prod_filter_def
tff(fact_6512_le__prod__filterI,axiom,
    ! [A: $tType,B: $tType,Fa: filter(product_prod(A,B)),Aa2: filter(A),B2: filter(B)] :
      ( aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),aa(filter(product_prod(A,B)),filter(A),aa(fun(product_prod(A,B),A),fun(filter(product_prod(A,B)),filter(A)),filtermap(product_prod(A,B),A),product_fst(A,B)),Fa)),Aa2)
     => ( aa(filter(B),$o,aa(filter(B),fun(filter(B),$o),ord_less_eq(filter(B)),aa(filter(product_prod(A,B)),filter(B),aa(fun(product_prod(A,B),B),fun(filter(product_prod(A,B)),filter(B)),filtermap(product_prod(A,B),B),product_snd(A,B)),Fa)),B2)
       => aa(filter(product_prod(A,B)),$o,aa(filter(product_prod(A,B)),fun(filter(product_prod(A,B)),$o),ord_less_eq(filter(product_prod(A,B))),Fa),aa(filter(B),filter(product_prod(A,B)),aa(filter(A),fun(filter(B),filter(product_prod(A,B))),prod_filter(A,B),Aa2),B2)) ) ) ).

% le_prod_filterI
tff(fact_6513_eventually__prod__sequentially,axiom,
    ! [Pa: fun(product_prod(nat,nat),$o)] :
      ( aa(filter(product_prod(nat,nat)),$o,aa(fun(product_prod(nat,nat),$o),fun(filter(product_prod(nat,nat)),$o),eventually(product_prod(nat,nat)),Pa),aa(filter(nat),filter(product_prod(nat,nat)),aa(filter(nat),fun(filter(nat),filter(product_prod(nat,nat))),prod_filter(nat,nat),at_top(nat)),at_top(nat)))
    <=> ? [N11: nat] :
        ! [M: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N11),M)
         => ! [N2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N11),N2)
             => aa(product_prod(nat,nat),$o,Pa,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),N2),M)) ) ) ) ).

% eventually_prod_sequentially
tff(fact_6514_eventually__prodI,axiom,
    ! [A: $tType,B: $tType,Pa: fun(A,$o),Fa: filter(A),Qa: fun(B,$o),G4: filter(B)] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Pa),Fa)
     => ( aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),eventually(B),Qa),G4)
       => aa(filter(product_prod(A,B)),$o,aa(fun(product_prod(A,B),$o),fun(filter(product_prod(A,B)),$o),eventually(product_prod(A,B)),aa(fun(B,$o),fun(product_prod(A,B),$o),aTP_Lamp_yl(fun(A,$o),fun(fun(B,$o),fun(product_prod(A,B),$o)),Pa),Qa)),aa(filter(B),filter(product_prod(A,B)),aa(filter(A),fun(filter(B),filter(product_prod(A,B))),prod_filter(A,B),Fa),G4)) ) ) ).

% eventually_prodI
tff(fact_6515_eventually__prod2,axiom,
    ! [A: $tType,B: $tType,Aa2: filter(A),Pa: fun(B,$o),B2: filter(B)] :
      ( ( Aa2 != bot_bot(filter(A)) )
     => ( aa(filter(product_prod(A,B)),$o,aa(fun(product_prod(A,B),$o),fun(filter(product_prod(A,B)),$o),eventually(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aTP_Lamp_ym(fun(B,$o),fun(A,fun(B,$o)),Pa))),aa(filter(B),filter(product_prod(A,B)),aa(filter(A),fun(filter(B),filter(product_prod(A,B))),prod_filter(A,B),Aa2),B2))
      <=> aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),eventually(B),Pa),B2) ) ) ).

% eventually_prod2
tff(fact_6516_eventually__prod1,axiom,
    ! [A: $tType,B: $tType,B2: filter(A),Pa: fun(B,$o),Aa2: filter(B)] :
      ( ( B2 != bot_bot(filter(A)) )
     => ( aa(filter(product_prod(B,A)),$o,aa(fun(product_prod(B,A),$o),fun(filter(product_prod(B,A)),$o),eventually(product_prod(B,A)),aa(fun(B,fun(A,$o)),fun(product_prod(B,A),$o),product_case_prod(B,A,$o),aTP_Lamp_yn(fun(B,$o),fun(B,fun(A,$o)),Pa))),aa(filter(A),filter(product_prod(B,A)),aa(filter(B),fun(filter(A),filter(product_prod(B,A))),prod_filter(B,A),Aa2),B2))
      <=> aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),eventually(B),Pa),Aa2) ) ) ).

% eventually_prod1
tff(fact_6517_prod__filter__INF2,axiom,
    ! [E: $tType,B: $tType,A: $tType,J4: set(A),Aa2: filter(B),B2: fun(A,filter(E))] :
      ( ( J4 != bot_bot(set(A)) )
     => ( aa(filter(E),filter(product_prod(B,E)),aa(filter(B),fun(filter(E),filter(product_prod(B,E))),prod_filter(B,E),Aa2),aa(set(filter(E)),filter(E),complete_Inf_Inf(filter(E)),aa(set(A),set(filter(E)),image2(A,filter(E),B2),J4))) = aa(set(filter(product_prod(B,E))),filter(product_prod(B,E)),complete_Inf_Inf(filter(product_prod(B,E))),aa(set(A),set(filter(product_prod(B,E))),image2(A,filter(product_prod(B,E)),aa(fun(A,filter(E)),fun(A,filter(product_prod(B,E))),aTP_Lamp_yo(filter(B),fun(fun(A,filter(E)),fun(A,filter(product_prod(B,E)))),Aa2),B2)),J4)) ) ) ).

% prod_filter_INF2
tff(fact_6518_prod__filter__INF1,axiom,
    ! [E: $tType,B: $tType,A: $tType,Ia2: set(A),Aa2: fun(A,filter(B)),B2: filter(E)] :
      ( ( Ia2 != bot_bot(set(A)) )
     => ( aa(filter(E),filter(product_prod(B,E)),aa(filter(B),fun(filter(E),filter(product_prod(B,E))),prod_filter(B,E),aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image2(A,filter(B),Aa2),Ia2))),B2) = aa(set(filter(product_prod(B,E))),filter(product_prod(B,E)),complete_Inf_Inf(filter(product_prod(B,E))),aa(set(A),set(filter(product_prod(B,E))),image2(A,filter(product_prod(B,E)),aa(filter(E),fun(A,filter(product_prod(B,E))),aTP_Lamp_yp(fun(A,filter(B)),fun(filter(E),fun(A,filter(product_prod(B,E)))),Aa2),B2)),Ia2)) ) ) ).

% prod_filter_INF1
tff(fact_6519_prod__filter__INF,axiom,
    ! [B: $tType,F4: $tType,E: $tType,A: $tType,Ia2: set(A),J4: set(B),Aa2: fun(A,filter(E)),B2: fun(B,filter(F4))] :
      ( ( Ia2 != bot_bot(set(A)) )
     => ( ( J4 != bot_bot(set(B)) )
       => ( aa(filter(F4),filter(product_prod(E,F4)),aa(filter(E),fun(filter(F4),filter(product_prod(E,F4))),prod_filter(E,F4),aa(set(filter(E)),filter(E),complete_Inf_Inf(filter(E)),aa(set(A),set(filter(E)),image2(A,filter(E),Aa2),Ia2))),aa(set(filter(F4)),filter(F4),complete_Inf_Inf(filter(F4)),aa(set(B),set(filter(F4)),image2(B,filter(F4),B2),J4))) = aa(set(filter(product_prod(E,F4))),filter(product_prod(E,F4)),complete_Inf_Inf(filter(product_prod(E,F4))),aa(set(A),set(filter(product_prod(E,F4))),image2(A,filter(product_prod(E,F4)),aa(fun(B,filter(F4)),fun(A,filter(product_prod(E,F4))),aa(fun(A,filter(E)),fun(fun(B,filter(F4)),fun(A,filter(product_prod(E,F4)))),aTP_Lamp_yr(set(B),fun(fun(A,filter(E)),fun(fun(B,filter(F4)),fun(A,filter(product_prod(E,F4))))),J4),Aa2),B2)),Ia2)) ) ) ) ).

% prod_filter_INF
tff(fact_6520_prod__filter__principal__singleton,axiom,
    ! [A: $tType,B: $tType,Xa: A,Fa: filter(B)] : ( aa(filter(B),filter(product_prod(A,B)),aa(filter(A),fun(filter(B),filter(product_prod(A,B))),prod_filter(A,B),aa(set(A),filter(A),principal(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A))))),Fa) = 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),Xa)),Fa) ) ).

% prod_filter_principal_singleton
tff(fact_6521_Max_Osemilattice__order__set__axioms,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => lattic4895041142388067077er_set(A,ord_max(A),aTP_Lamp_ao(A,fun(A,$o)),aTP_Lamp_ys(A,fun(A,$o))) ) ).

% Max.semilattice_order_set_axioms
tff(fact_6522_subset__singleton__iff__Uniq,axiom,
    ! [A: $tType,Aa2: set(A)] :
      ( ? [A7: A] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A7),bot_bot(set(A))))
    <=> uniq(A,aa(set(A),fun(A,$o),aTP_Lamp_a(set(A),fun(A,$o)),Aa2)) ) ).

% subset_singleton_iff_Uniq
tff(fact_6523_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_6524_Min_Osemilattice__order__set__axioms,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => lattic4895041142388067077er_set(A,ord_min(A),ord_less_eq(A),ord_less(A)) ) ).

% Min.semilattice_order_set_axioms
tff(fact_6525_pairwise__disjnt__iff,axiom,
    ! [A: $tType,A14: set(set(A))] :
      ( pairwise(set(A),disjnt(A),A14)
    <=> ! [X6: A] : uniq(set(A),aa(A,fun(set(A),$o),aTP_Lamp_yt(set(set(A)),fun(A,fun(set(A),$o)),A14),X6)) ) ).

% pairwise_disjnt_iff
tff(fact_6526_Sup__fin_Osemilattice__order__set__axioms,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => lattic4895041142388067077er_set(A,sup_sup(A),aTP_Lamp_yu(A,fun(A,$o)),aTP_Lamp_yv(A,fun(A,$o))) ) ).

% Sup_fin.semilattice_order_set_axioms
tff(fact_6527_strict__sorted__equal__Uniq,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Aa2: set(A)] : uniq(list(A),aTP_Lamp_yw(set(A),fun(list(A),$o),Aa2)) ) ).

% strict_sorted_equal_Uniq
tff(fact_6528_gen__length__def,axiom,
    ! [A: $tType,N: nat,Xsa: list(A)] : ( aa(list(A),nat,gen_length(A,N),Xsa) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(list(A),nat,size_size(list(A)),Xsa)) ) ).

% gen_length_def
tff(fact_6529_ex__is__arg__min__if__finite,axiom,
    ! [B: $tType,A: $tType] :
      ( order(B)
     => ! [S: set(A),F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),S)
         => ( ( S != bot_bot(set(A)) )
           => ? [X_1: A] : lattic501386751177426532rg_min(A,B,F2,aa(set(A),fun(A,$o),aTP_Lamp_a(set(A),fun(A,$o)),S),X_1) ) ) ) ).

% ex_is_arg_min_if_finite
tff(fact_6530_is__arg__min__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F2: fun(A,B),Pa: fun(A,$o),Xa: A] :
          ( lattic501386751177426532rg_min(A,B,F2,Pa,Xa)
        <=> ( aa(A,$o,Pa,Xa)
            & ~ ? [Y4: A] :
                  ( aa(A,$o,Pa,Y4)
                  & aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,Y4)),aa(A,B,F2,Xa)) ) ) ) ) ).

% is_arg_min_def
tff(fact_6531_gen__length__code_I1_J,axiom,
    ! [A: $tType,N: nat] : ( aa(list(A),nat,gen_length(A,N),nil(A)) = N ) ).

% gen_length_code(1)
tff(fact_6532_gen__length__code_I2_J,axiom,
    ! [A: $tType,N: nat,Xa: A,Xsa: list(A)] : ( aa(list(A),nat,gen_length(A,N),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa)) = aa(list(A),nat,gen_length(A,aa(nat,nat,suc,N)),Xsa) ) ).

% gen_length_code(2)
tff(fact_6533_length__code,axiom,
    ! [A: $tType] : ( size_size(list(A)) = gen_length(A,zero_zero(nat)) ) ).

% length_code
tff(fact_6534_card__of__UNION__ordLeq__infinite,axiom,
    ! [B: $tType,A: $tType,E: $tType,B2: set(A),Ia2: set(B),Aa2: fun(B,set(E))] :
      ( ~ aa(set(A),$o,finite_finite2(A),B2)
     => ( 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,Ia2)),bNF_Ca6860139660246222851ard_of(A,B2)),bNF_Wellorder_ordLeq(B,A))
       => ( ! [X: B] :
              ( member(B,X,Ia2)
             => member(product_prod(set(product_prod(E,E)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(E,E)),set(product_prod(A,A))),aa(set(product_prod(E,E)),fun(set(product_prod(A,A)),product_prod(set(product_prod(E,E)),set(product_prod(A,A)))),product_Pair(set(product_prod(E,E)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(E,aa(B,set(E),Aa2,X))),bNF_Ca6860139660246222851ard_of(A,B2)),bNF_Wellorder_ordLeq(E,A)) )
         => member(product_prod(set(product_prod(E,E)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(E,E)),set(product_prod(A,A))),aa(set(product_prod(E,E)),fun(set(product_prod(A,A)),product_prod(set(product_prod(E,E)),set(product_prod(A,A)))),product_Pair(set(product_prod(E,E)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(E,aa(set(set(E)),set(E),complete_Sup_Sup(set(E)),aa(set(B),set(set(E)),image2(B,set(E),Aa2),Ia2)))),bNF_Ca6860139660246222851ard_of(A,B2)),bNF_Wellorder_ordLeq(E,A)) ) ) ) ).

% card_of_UNION_ordLeq_infinite
tff(fact_6535_relImage__def,axiom,
    ! [A: $tType,B: $tType,R: set(product_prod(B,B)),F2: fun(B,A)] : ( bNF_Gr4221423524335903396lImage(B,A,R,F2) = aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(B,A),fun(product_prod(A,A),$o),aTP_Lamp_yg(set(product_prod(B,B)),fun(fun(B,A),fun(product_prod(A,A),$o)),R),F2)) ) ).

% relImage_def
tff(fact_6536_card__of__Times__infinite__simps_I4_J,axiom,
    ! [B: $tType,A: $tType,Aa2: set(A),B2: set(B)] :
      ( ~ aa(set(A),$o,finite_finite2(A),Aa2)
     => ( ( B2 != bot_bot(set(B)) )
       => ( 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,B2)),bNF_Ca6860139660246222851ard_of(A,Aa2)),bNF_Wellorder_ordLeq(B,A))
         => 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,Aa2)),bNF_Ca6860139660246222851ard_of(product_prod(B,A),product_Sigma(B,A,B2,aTP_Lamp_an(set(A),fun(B,set(A)),Aa2)))),bNF_Wellorder_ordIso(A,product_prod(B,A))) ) ) ) ).

% card_of_Times_infinite_simps(4)
tff(fact_6537_card__of__Times__infinite__simps_I3_J,axiom,
    ! [A: $tType,B: $tType,Aa2: set(A),B2: set(B)] :
      ( ~ aa(set(A),$o,finite_finite2(A),Aa2)
     => ( ( B2 != bot_bot(set(B)) )
       => ( 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,B2)),bNF_Ca6860139660246222851ard_of(A,Aa2)),bNF_Wellorder_ordLeq(B,A))
         => 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,B2,aTP_Lamp_an(set(A),fun(B,set(A)),Aa2)))),bNF_Ca6860139660246222851ard_of(A,Aa2)),bNF_Wellorder_ordIso(product_prod(B,A),A)) ) ) ) ).

% card_of_Times_infinite_simps(3)
tff(fact_6538_card__of__Times__infinite__simps_I2_J,axiom,
    ! [B: $tType,A: $tType,Aa2: set(A),B2: set(B)] :
      ( ~ aa(set(A),$o,finite_finite2(A),Aa2)
     => ( ( B2 != bot_bot(set(B)) )
       => ( 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,B2)),bNF_Ca6860139660246222851ard_of(A,Aa2)),bNF_Wellorder_ordLeq(B,A))
         => 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,Aa2)),bNF_Ca6860139660246222851ard_of(product_prod(A,B),product_Sigma(A,B,Aa2,aTP_Lamp_pv(set(B),fun(A,set(B)),B2)))),bNF_Wellorder_ordIso(A,product_prod(A,B))) ) ) ) ).

% card_of_Times_infinite_simps(2)
tff(fact_6539_card__of__Times__infinite__simps_I1_J,axiom,
    ! [B: $tType,A: $tType,Aa2: set(A),B2: set(B)] :
      ( ~ aa(set(A),$o,finite_finite2(A),Aa2)
     => ( ( B2 != bot_bot(set(B)) )
       => ( 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,B2)),bNF_Ca6860139660246222851ard_of(A,Aa2)),bNF_Wellorder_ordLeq(B,A))
         => 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,Aa2,aTP_Lamp_pv(set(B),fun(A,set(B)),B2)))),bNF_Ca6860139660246222851ard_of(A,Aa2)),bNF_Wellorder_ordIso(product_prod(A,B),A)) ) ) ) ).

% card_of_Times_infinite_simps(1)
tff(fact_6540_card__of__Times__infinite,axiom,
    ! [A: $tType,B: $tType,Aa2: set(A),B2: set(B)] :
      ( ~ aa(set(A),$o,finite_finite2(A),Aa2)
     => ( ( B2 != bot_bot(set(B)) )
       => ( 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,B2)),bNF_Ca6860139660246222851ard_of(A,Aa2)),bNF_Wellorder_ordLeq(B,A))
         => ( 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,Aa2,aTP_Lamp_pv(set(B),fun(A,set(B)),B2)))),bNF_Ca6860139660246222851ard_of(A,Aa2)),bNF_Wellorder_ordIso(product_prod(A,B),A))
            & 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,B2,aTP_Lamp_an(set(A),fun(B,set(A)),Aa2)))),bNF_Ca6860139660246222851ard_of(A,Aa2)),bNF_Wellorder_ordIso(product_prod(B,A),A)) ) ) ) ) ).

% card_of_Times_infinite
tff(fact_6541_card__of__Times1,axiom,
    ! [A: $tType,B: $tType,Aa2: set(A),B2: set(B)] :
      ( ( Aa2 != bot_bot(set(A)) )
     => 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,B2)),bNF_Ca6860139660246222851ard_of(product_prod(B,A),product_Sigma(B,A,B2,aTP_Lamp_an(set(A),fun(B,set(A)),Aa2)))),bNF_Wellorder_ordLeq(B,product_prod(B,A))) ) ).

% card_of_Times1
tff(fact_6542_card__of__Times2,axiom,
    ! [A: $tType,B: $tType,Aa2: set(A),B2: set(B)] :
      ( ( Aa2 != bot_bot(set(A)) )
     => 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,B2)),bNF_Ca6860139660246222851ard_of(product_prod(A,B),product_Sigma(A,B,Aa2,aTP_Lamp_pv(set(B),fun(A,set(B)),B2)))),bNF_Wellorder_ordLeq(B,product_prod(A,B))) ) ).

% card_of_Times2
tff(fact_6543_card__of__Times__same__infinite,axiom,
    ! [A: $tType,Aa2: set(A)] :
      ( ~ aa(set(A),$o,finite_finite2(A),Aa2)
     => 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,Aa2,aTP_Lamp_pt(set(A),fun(A,set(A)),Aa2)))),bNF_Ca6860139660246222851ard_of(A,Aa2)),bNF_Wellorder_ordIso(product_prod(A,A),A)) ) ).

% card_of_Times_same_infinite
tff(fact_6544_card__of__Times__commute,axiom,
    ! [B: $tType,A: $tType,Aa2: set(A),B2: set(B)] : 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,Aa2,aTP_Lamp_pv(set(B),fun(A,set(B)),B2)))),bNF_Ca6860139660246222851ard_of(product_prod(B,A),product_Sigma(B,A,B2,aTP_Lamp_an(set(A),fun(B,set(A)),Aa2)))),bNF_Wellorder_ordIso(product_prod(A,B),product_prod(B,A))) ).

% card_of_Times_commute
tff(fact_6545_card__of__refl,axiom,
    ! [A: $tType,Aa2: set(A)] : 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,Aa2)),bNF_Ca6860139660246222851ard_of(A,Aa2)),bNF_Wellorder_ordIso(A,A)) ).

% card_of_refl
tff(fact_6546_card__of__Pow__Func,axiom,
    ! [A: $tType,Aa2: set(A)] : member(product_prod(set(product_prod(set(A),set(A))),set(product_prod(fun(A,$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,$o)))),bNF_Ca6860139660246222851ard_of(set(A),pow(A,Aa2))),bNF_Ca6860139660246222851ard_of(fun(A,$o),bNF_Wellorder_Func(A,$o,Aa2,top_top(set($o))))),bNF_Wellorder_ordIso(set(A),fun(A,$o))) ).

% card_of_Pow_Func
tff(fact_6547_card__of__Func__Times,axiom,
    ! [E: $tType,B: $tType,A: $tType,Aa2: set(A),B2: set(B),C3: set(E)] : member(product_prod(set(product_prod(fun(product_prod(A,B),E),fun(product_prod(A,B),E))),set(product_prod(fun(A,fun(B,E)),fun(A,fun(B,E))))),aa(set(product_prod(fun(A,fun(B,E)),fun(A,fun(B,E)))),product_prod(set(product_prod(fun(product_prod(A,B),E),fun(product_prod(A,B),E))),set(product_prod(fun(A,fun(B,E)),fun(A,fun(B,E))))),aa(set(product_prod(fun(product_prod(A,B),E),fun(product_prod(A,B),E))),fun(set(product_prod(fun(A,fun(B,E)),fun(A,fun(B,E)))),product_prod(set(product_prod(fun(product_prod(A,B),E),fun(product_prod(A,B),E))),set(product_prod(fun(A,fun(B,E)),fun(A,fun(B,E)))))),product_Pair(set(product_prod(fun(product_prod(A,B),E),fun(product_prod(A,B),E))),set(product_prod(fun(A,fun(B,E)),fun(A,fun(B,E))))),bNF_Ca6860139660246222851ard_of(fun(product_prod(A,B),E),bNF_Wellorder_Func(product_prod(A,B),E,product_Sigma(A,B,Aa2,aTP_Lamp_pv(set(B),fun(A,set(B)),B2)),C3))),bNF_Ca6860139660246222851ard_of(fun(A,fun(B,E)),bNF_Wellorder_Func(A,fun(B,E),Aa2,bNF_Wellorder_Func(B,E,B2,C3)))),bNF_Wellorder_ordIso(fun(product_prod(A,B),E),fun(A,fun(B,E)))) ).

% card_of_Func_Times
tff(fact_6548_Func__Times__Range,axiom,
    ! [E: $tType,B: $tType,A: $tType,Aa2: set(A),B2: set(B),C3: set(E)] : member(product_prod(set(product_prod(fun(A,product_prod(B,E)),fun(A,product_prod(B,E)))),set(product_prod(product_prod(fun(A,B),fun(A,E)),product_prod(fun(A,B),fun(A,E))))),aa(set(product_prod(product_prod(fun(A,B),fun(A,E)),product_prod(fun(A,B),fun(A,E)))),product_prod(set(product_prod(fun(A,product_prod(B,E)),fun(A,product_prod(B,E)))),set(product_prod(product_prod(fun(A,B),fun(A,E)),product_prod(fun(A,B),fun(A,E))))),aa(set(product_prod(fun(A,product_prod(B,E)),fun(A,product_prod(B,E)))),fun(set(product_prod(product_prod(fun(A,B),fun(A,E)),product_prod(fun(A,B),fun(A,E)))),product_prod(set(product_prod(fun(A,product_prod(B,E)),fun(A,product_prod(B,E)))),set(product_prod(product_prod(fun(A,B),fun(A,E)),product_prod(fun(A,B),fun(A,E)))))),product_Pair(set(product_prod(fun(A,product_prod(B,E)),fun(A,product_prod(B,E)))),set(product_prod(product_prod(fun(A,B),fun(A,E)),product_prod(fun(A,B),fun(A,E))))),bNF_Ca6860139660246222851ard_of(fun(A,product_prod(B,E)),bNF_Wellorder_Func(A,product_prod(B,E),Aa2,product_Sigma(B,E,B2,aTP_Lamp_qe(set(E),fun(B,set(E)),C3))))),bNF_Ca6860139660246222851ard_of(product_prod(fun(A,B),fun(A,E)),product_Sigma(fun(A,B),fun(A,E),bNF_Wellorder_Func(A,B,Aa2,B2),aa(set(E),fun(fun(A,B),set(fun(A,E))),aTP_Lamp_yx(set(A),fun(set(E),fun(fun(A,B),set(fun(A,E)))),Aa2),C3)))),bNF_Wellorder_ordIso(fun(A,product_prod(B,E)),product_prod(fun(A,B),fun(A,E)))) ).

% Func_Times_Range
tff(fact_6549_card__of__bool,axiom,
    ! [A: $tType,A1: A,A22: A] :
      ( ( A1 != A22 )
     => member(product_prod(set(product_prod($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($o)))),bNF_Ca6860139660246222851ard_of(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A1),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A22),bot_bot(set(A)))))),bNF_Wellorder_ordIso($o,A)) ) ).

% card_of_bool
tff(fact_6550_card__of__Pow,axiom,
    ! [A: $tType,Aa2: set(A)] : 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,Aa2)),bNF_Ca6860139660246222851ard_of(set(A),pow(A,Aa2))),bNF_We4044943003108391690rdLess(A,set(A))) ).

% card_of_Pow
tff(fact_6551_card__of__UNION__Sigma,axiom,
    ! [B: $tType,A: $tType,Aa2: fun(B,set(A)),Ia2: set(B)] : 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),Aa2),Ia2)))),bNF_Ca6860139660246222851ard_of(product_prod(B,A),product_Sigma(B,A,Ia2,Aa2))),bNF_Wellorder_ordLeq(A,product_prod(B,A))) ).

% card_of_UNION_Sigma
tff(fact_6552_card__of__Sigma__ordLeq__infinite,axiom,
    ! [A: $tType,E: $tType,B: $tType,B2: set(A),Ia2: set(B),Aa2: fun(B,set(E))] :
      ( ~ aa(set(A),$o,finite_finite2(A),B2)
     => ( 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,Ia2)),bNF_Ca6860139660246222851ard_of(A,B2)),bNF_Wellorder_ordLeq(B,A))
       => ( ! [X: B] :
              ( member(B,X,Ia2)
             => member(product_prod(set(product_prod(E,E)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(E,E)),set(product_prod(A,A))),aa(set(product_prod(E,E)),fun(set(product_prod(A,A)),product_prod(set(product_prod(E,E)),set(product_prod(A,A)))),product_Pair(set(product_prod(E,E)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(E,aa(B,set(E),Aa2,X))),bNF_Ca6860139660246222851ard_of(A,B2)),bNF_Wellorder_ordLeq(E,A)) )
         => member(product_prod(set(product_prod(product_prod(B,E),product_prod(B,E))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(B,E),product_prod(B,E))),set(product_prod(A,A))),aa(set(product_prod(product_prod(B,E),product_prod(B,E))),fun(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(B,E),product_prod(B,E))),set(product_prod(A,A)))),product_Pair(set(product_prod(product_prod(B,E),product_prod(B,E))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(product_prod(B,E),product_Sigma(B,E,Ia2,Aa2))),bNF_Ca6860139660246222851ard_of(A,B2)),bNF_Wellorder_ordLeq(product_prod(B,E),A)) ) ) ) ).

% card_of_Sigma_ordLeq_infinite
tff(fact_6553_card__of__Times3,axiom,
    ! [A: $tType,Aa2: set(A)] : 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,Aa2)),bNF_Ca6860139660246222851ard_of(product_prod(A,A),product_Sigma(A,A,Aa2,aTP_Lamp_pt(set(A),fun(A,set(A)),Aa2)))),bNF_Wellorder_ordLeq(A,product_prod(A,A))) ).

% card_of_Times3
tff(fact_6554_card__of__Sigma__mono1,axiom,
    ! [E: $tType,B: $tType,A: $tType,Ia2: set(A),Aa2: fun(A,set(B)),B2: fun(A,set(E))] :
      ( ! [X: A] :
          ( member(A,X,Ia2)
         => 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(A,set(B),Aa2,X))),bNF_Ca6860139660246222851ard_of(E,aa(A,set(E),B2,X))),bNF_Wellorder_ordLeq(B,E)) )
     => member(product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(product_prod(A,E),product_prod(A,E)))),aa(set(product_prod(product_prod(A,E),product_prod(A,E))),product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(product_prod(A,E),product_prod(A,E)))),aa(set(product_prod(product_prod(A,B),product_prod(A,B))),fun(set(product_prod(product_prod(A,E),product_prod(A,E))),product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(product_prod(A,E),product_prod(A,E))))),product_Pair(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(product_prod(A,E),product_prod(A,E)))),bNF_Ca6860139660246222851ard_of(product_prod(A,B),product_Sigma(A,B,Ia2,Aa2))),bNF_Ca6860139660246222851ard_of(product_prod(A,E),product_Sigma(A,E,Ia2,B2))),bNF_Wellorder_ordLeq(product_prod(A,B),product_prod(A,E))) ) ).

% card_of_Sigma_mono1
tff(fact_6555_card__of__Times__mono1,axiom,
    ! [B: $tType,E: $tType,A: $tType,Aa2: set(A),B2: set(B),C3: set(E)] :
      ( 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,Aa2)),bNF_Ca6860139660246222851ard_of(B,B2)),bNF_Wellorder_ordLeq(A,B))
     => member(product_prod(set(product_prod(product_prod(A,E),product_prod(A,E))),set(product_prod(product_prod(B,E),product_prod(B,E)))),aa(set(product_prod(product_prod(B,E),product_prod(B,E))),product_prod(set(product_prod(product_prod(A,E),product_prod(A,E))),set(product_prod(product_prod(B,E),product_prod(B,E)))),aa(set(product_prod(product_prod(A,E),product_prod(A,E))),fun(set(product_prod(product_prod(B,E),product_prod(B,E))),product_prod(set(product_prod(product_prod(A,E),product_prod(A,E))),set(product_prod(product_prod(B,E),product_prod(B,E))))),product_Pair(set(product_prod(product_prod(A,E),product_prod(A,E))),set(product_prod(product_prod(B,E),product_prod(B,E)))),bNF_Ca6860139660246222851ard_of(product_prod(A,E),product_Sigma(A,E,Aa2,aTP_Lamp_qf(set(E),fun(A,set(E)),C3)))),bNF_Ca6860139660246222851ard_of(product_prod(B,E),product_Sigma(B,E,B2,aTP_Lamp_qe(set(E),fun(B,set(E)),C3)))),bNF_Wellorder_ordLeq(product_prod(A,E),product_prod(B,E))) ) ).

% card_of_Times_mono1
tff(fact_6556_card__of__Times__mono2,axiom,
    ! [B: $tType,A: $tType,E: $tType,Aa2: set(A),B2: set(B),C3: set(E)] :
      ( 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,Aa2)),bNF_Ca6860139660246222851ard_of(B,B2)),bNF_Wellorder_ordLeq(A,B))
     => member(product_prod(set(product_prod(product_prod(E,A),product_prod(E,A))),set(product_prod(product_prod(E,B),product_prod(E,B)))),aa(set(product_prod(product_prod(E,B),product_prod(E,B))),product_prod(set(product_prod(product_prod(E,A),product_prod(E,A))),set(product_prod(product_prod(E,B),product_prod(E,B)))),aa(set(product_prod(product_prod(E,A),product_prod(E,A))),fun(set(product_prod(product_prod(E,B),product_prod(E,B))),product_prod(set(product_prod(product_prod(E,A),product_prod(E,A))),set(product_prod(product_prod(E,B),product_prod(E,B))))),product_Pair(set(product_prod(product_prod(E,A),product_prod(E,A))),set(product_prod(product_prod(E,B),product_prod(E,B)))),bNF_Ca6860139660246222851ard_of(product_prod(E,A),product_Sigma(E,A,C3,aTP_Lamp_yy(set(A),fun(E,set(A)),Aa2)))),bNF_Ca6860139660246222851ard_of(product_prod(E,B),product_Sigma(E,B,C3,aTP_Lamp_yz(set(B),fun(E,set(B)),B2)))),bNF_Wellorder_ordLeq(product_prod(E,A),product_prod(E,B))) ) ).

% card_of_Times_mono2
tff(fact_6557_infinite__iff__card__of__nat,axiom,
    ! [A: $tType,Aa2: set(A)] :
      ( ~ aa(set(A),$o,finite_finite2(A),Aa2)
    <=> 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,Aa2)),bNF_Wellorder_ordLeq(nat,A)) ) ).

% infinite_iff_card_of_nat
tff(fact_6558_card__of__image,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),Aa2: set(B)] : 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,F2),Aa2))),bNF_Ca6860139660246222851ard_of(B,Aa2)),bNF_Wellorder_ordLeq(A,B)) ).

% card_of_image
tff(fact_6559_card__of__empty3,axiom,
    ! [B: $tType,A: $tType,Aa2: set(A)] :
      ( 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,Aa2)),bNF_Ca6860139660246222851ard_of(B,bot_bot(set(B)))),bNF_Wellorder_ordLeq(A,B))
     => ( Aa2 = bot_bot(set(A)) ) ) ).

% card_of_empty3
tff(fact_6560_card__of__empty,axiom,
    ! [B: $tType,A: $tType,Aa2: set(B)] : 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,Aa2)),bNF_Wellorder_ordLeq(A,B)) ).

% card_of_empty
tff(fact_6561_card__of__ordLeq__infinite,axiom,
    ! [A: $tType,B: $tType,Aa2: set(A),B2: set(B)] :
      ( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,Aa2)),bNF_Ca6860139660246222851ard_of(B,B2)),bNF_Wellorder_ordLeq(A,B))
     => ( ~ aa(set(A),$o,finite_finite2(A),Aa2)
       => ~ aa(set(B),$o,finite_finite2(B),B2) ) ) ).

% card_of_ordLeq_infinite
tff(fact_6562_card__of__ordLeq__finite,axiom,
    ! [B: $tType,A: $tType,Aa2: set(A),B2: set(B)] :
      ( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,Aa2)),bNF_Ca6860139660246222851ard_of(B,B2)),bNF_Wellorder_ordLeq(A,B))
     => ( aa(set(B),$o,finite_finite2(B),B2)
       => aa(set(A),$o,finite_finite2(A),Aa2) ) ) ).

% card_of_ordLeq_finite
tff(fact_6563_card__of__mono1,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),B2)
     => 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,Aa2)),bNF_Ca6860139660246222851ard_of(A,B2)),bNF_Wellorder_ordLeq(A,A)) ) ).

% card_of_mono1
tff(fact_6564_card__of__empty__ordIso,axiom,
    ! [B: $tType,A: $tType] : 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_6565_card__of__empty2,axiom,
    ! [B: $tType,A: $tType,Aa2: set(A)] :
      ( 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,Aa2)),bNF_Ca6860139660246222851ard_of(B,bot_bot(set(B)))),bNF_Wellorder_ordIso(A,B))
     => ( Aa2 = bot_bot(set(A)) ) ) ).

% card_of_empty2
tff(fact_6566_card__of__ordIso__finite,axiom,
    ! [A: $tType,B: $tType,Aa2: set(A),B2: set(B)] :
      ( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,Aa2)),bNF_Ca6860139660246222851ard_of(B,B2)),bNF_Wellorder_ordIso(A,B))
     => ( aa(set(A),$o,finite_finite2(A),Aa2)
      <=> aa(set(B),$o,finite_finite2(B),B2) ) ) ).

% card_of_ordIso_finite
tff(fact_6567_card__of__mono2,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,A)),R5: set(product_prod(B,B))] :
      ( 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),R5),bNF_Wellorder_ordLeq(A,B))
     => 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(product_prod(A,A)),set(A),field2(A),R2))),bNF_Ca6860139660246222851ard_of(B,aa(set(product_prod(B,B)),set(B),field2(B),R5))),bNF_Wellorder_ordLeq(A,B)) ) ).

% card_of_mono2
tff(fact_6568_card__of__cong,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,A)),R5: set(product_prod(B,B))] :
      ( 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),R5),bNF_Wellorder_ordIso(A,B))
     => 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(product_prod(A,A)),set(A),field2(A),R2))),bNF_Ca6860139660246222851ard_of(B,aa(set(product_prod(B,B)),set(B),field2(B),R5))),bNF_Wellorder_ordIso(A,B)) ) ).

% card_of_cong
tff(fact_6569_card__of__ordLeqI,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),Aa2: set(A),B2: set(B)] :
      ( inj_on(A,B,F2,Aa2)
     => ( ! [A4: A] :
            ( member(A,A4,Aa2)
           => member(B,aa(A,B,F2,A4),B2) )
       => 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,Aa2)),bNF_Ca6860139660246222851ard_of(B,B2)),bNF_Wellorder_ordLeq(A,B)) ) ) ).

% card_of_ordLeqI
tff(fact_6570_card__of__least,axiom,
    ! [A: $tType,Aa2: set(A),R2: set(product_prod(A,A))] :
      ( order_well_order_on(A,Aa2,R2)
     => 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,Aa2)),R2),bNF_Wellorder_ordLeq(A,A)) ) ).

% card_of_least
tff(fact_6571_BNF__Cardinal__Order__Relation_OordLess__Field,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,A)),R5: set(product_prod(B,B))] :
      ( 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),R5),bNF_We4044943003108391690rdLess(A,B))
     => 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(product_prod(A,A)),set(A),field2(A),R2))),R5),bNF_We4044943003108391690rdLess(A,B)) ) ).

% BNF_Cardinal_Order_Relation.ordLess_Field
tff(fact_6572_type__copy__set__bd,axiom,
    ! [A: $tType,F4: $tType,E: $tType,B: $tType,S: fun(A,set(B)),Bd: set(product_prod(E,E)),Rep: fun(F4,A),Xa: F4] :
      ( ! [Y2: A] : 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(A,set(B),S,Y2))),Bd),bNF_Wellorder_ordLeq(B,E))
     => 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(F4,set(B),comp(A,set(B),F4,S,Rep),Xa))),Bd),bNF_Wellorder_ordLeq(B,E)) ) ).

% type_copy_set_bd
tff(fact_6573_card__of__ordLeq2,axiom,
    ! [B: $tType,A: $tType,Aa2: set(A),B2: set(B)] :
      ( ( Aa2 != bot_bot(set(A)) )
     => ( ? [G5: fun(B,A)] : ( aa(set(B),set(A),image2(B,A,G5),B2) = Aa2 )
      <=> 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,Aa2)),bNF_Ca6860139660246222851ard_of(B,B2)),bNF_Wellorder_ordLeq(A,B)) ) ) ).

% card_of_ordLeq2
tff(fact_6574_surj__imp__ordLeq,axiom,
    ! [B: $tType,A: $tType,B2: set(A),F2: fun(B,A),Aa2: set(B)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),aa(set(B),set(A),image2(B,A,F2),Aa2))
     => 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,B2)),bNF_Ca6860139660246222851ard_of(B,Aa2)),bNF_Wellorder_ordLeq(A,B)) ) ).

% surj_imp_ordLeq
tff(fact_6575_card__of__singl__ordLeq,axiom,
    ! [A: $tType,B: $tType,Aa2: set(A),B3: B] :
      ( ( Aa2 != bot_bot(set(A)) )
     => 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)),insert3(B),B3),bot_bot(set(B))))),bNF_Ca6860139660246222851ard_of(A,Aa2)),bNF_Wellorder_ordLeq(B,A)) ) ).

% card_of_singl_ordLeq
tff(fact_6576_card__of__ordLess2,axiom,
    ! [A: $tType,B: $tType,B2: set(A),Aa2: set(B)] :
      ( ( B2 != bot_bot(set(A)) )
     => ( ~ ? [F9: fun(B,A)] : ( aa(set(B),set(A),image2(B,A,F9),Aa2) = B2 )
      <=> 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,Aa2)),bNF_Ca6860139660246222851ard_of(A,B2)),bNF_We4044943003108391690rdLess(B,A)) ) ) ).

% card_of_ordLess2
tff(fact_6577_internalize__card__of__ordLeq2,axiom,
    ! [A: $tType,B: $tType,Aa2: set(A),C3: set(B)] :
      ( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,Aa2)),bNF_Ca6860139660246222851ard_of(B,C3)),bNF_Wellorder_ordLeq(A,B))
    <=> ? [B12: set(B)] :
          ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),B12),C3)
          & 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,Aa2)),bNF_Ca6860139660246222851ard_of(B,B12)),bNF_Wellorder_ordIso(A,B))
          & 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,B12)),bNF_Ca6860139660246222851ard_of(B,C3)),bNF_Wellorder_ordLeq(B,B)) ) ) ).

% internalize_card_of_ordLeq2
tff(fact_6578_card__of__Field__ordLess,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => 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,aa(set(product_prod(A,A)),set(A),field2(A),R2))),R2),bNF_Wellorder_ordLeq(A,A)) ) ).

% card_of_Field_ordLess
tff(fact_6579_card__of__Func__UNIV,axiom,
    ! [B: $tType,A: $tType,B2: set(B)] : member(product_prod(set(product_prod(fun(A,B),fun(A,B))),set(product_prod(fun(A,B),fun(A,B)))),aa(set(product_prod(fun(A,B),fun(A,B))),product_prod(set(product_prod(fun(A,B),fun(A,B))),set(product_prod(fun(A,B),fun(A,B)))),aa(set(product_prod(fun(A,B),fun(A,B))),fun(set(product_prod(fun(A,B),fun(A,B))),product_prod(set(product_prod(fun(A,B),fun(A,B))),set(product_prod(fun(A,B),fun(A,B))))),product_Pair(set(product_prod(fun(A,B),fun(A,B))),set(product_prod(fun(A,B),fun(A,B)))),bNF_Ca6860139660246222851ard_of(fun(A,B),bNF_Wellorder_Func(A,B,top_top(set(A)),B2))),bNF_Ca6860139660246222851ard_of(fun(A,B),aa(fun(fun(A,B),$o),set(fun(A,B)),collect(fun(A,B)),aTP_Lamp_za(set(B),fun(fun(A,B),$o),B2)))),bNF_Wellorder_ordIso(fun(A,B),fun(A,B))) ).

% card_of_Func_UNIV
tff(fact_6580_ordLeq__Times__mono1,axiom,
    ! [B: $tType,E: $tType,A: $tType,R2: set(product_prod(A,A)),R5: set(product_prod(B,B)),C3: set(E)] :
      ( 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),R5),bNF_Wellorder_ordLeq(A,B))
     => member(product_prod(set(product_prod(product_prod(A,E),product_prod(A,E))),set(product_prod(product_prod(B,E),product_prod(B,E)))),aa(set(product_prod(product_prod(B,E),product_prod(B,E))),product_prod(set(product_prod(product_prod(A,E),product_prod(A,E))),set(product_prod(product_prod(B,E),product_prod(B,E)))),aa(set(product_prod(product_prod(A,E),product_prod(A,E))),fun(set(product_prod(product_prod(B,E),product_prod(B,E))),product_prod(set(product_prod(product_prod(A,E),product_prod(A,E))),set(product_prod(product_prod(B,E),product_prod(B,E))))),product_Pair(set(product_prod(product_prod(A,E),product_prod(A,E))),set(product_prod(product_prod(B,E),product_prod(B,E)))),bNF_Ca6860139660246222851ard_of(product_prod(A,E),product_Sigma(A,E,aa(set(product_prod(A,A)),set(A),field2(A),R2),aTP_Lamp_qf(set(E),fun(A,set(E)),C3)))),bNF_Ca6860139660246222851ard_of(product_prod(B,E),product_Sigma(B,E,aa(set(product_prod(B,B)),set(B),field2(B),R5),aTP_Lamp_qe(set(E),fun(B,set(E)),C3)))),bNF_Wellorder_ordLeq(product_prod(A,E),product_prod(B,E))) ) ).

% ordLeq_Times_mono1
tff(fact_6581_ordLeq__Times__mono2,axiom,
    ! [B: $tType,A: $tType,E: $tType,R2: set(product_prod(A,A)),R5: set(product_prod(B,B)),Aa2: set(E)] :
      ( 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),R5),bNF_Wellorder_ordLeq(A,B))
     => member(product_prod(set(product_prod(product_prod(E,A),product_prod(E,A))),set(product_prod(product_prod(E,B),product_prod(E,B)))),aa(set(product_prod(product_prod(E,B),product_prod(E,B))),product_prod(set(product_prod(product_prod(E,A),product_prod(E,A))),set(product_prod(product_prod(E,B),product_prod(E,B)))),aa(set(product_prod(product_prod(E,A),product_prod(E,A))),fun(set(product_prod(product_prod(E,B),product_prod(E,B))),product_prod(set(product_prod(product_prod(E,A),product_prod(E,A))),set(product_prod(product_prod(E,B),product_prod(E,B))))),product_Pair(set(product_prod(product_prod(E,A),product_prod(E,A))),set(product_prod(product_prod(E,B),product_prod(E,B)))),bNF_Ca6860139660246222851ard_of(product_prod(E,A),product_Sigma(E,A,Aa2,aTP_Lamp_zb(set(product_prod(A,A)),fun(E,set(A)),R2)))),bNF_Ca6860139660246222851ard_of(product_prod(E,B),product_Sigma(E,B,Aa2,aTP_Lamp_zc(set(product_prod(B,B)),fun(E,set(B)),R5)))),bNF_Wellorder_ordLeq(product_prod(E,A),product_prod(E,B))) ) ).

% ordLeq_Times_mono2
tff(fact_6582_card__of__ordLeq,axiom,
    ! [B: $tType,A: $tType,Aa2: set(A),B2: set(B)] :
      ( ? [F9: fun(A,B)] :
          ( inj_on(A,B,F9,Aa2)
          & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F9),Aa2)),B2) )
    <=> 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,Aa2)),bNF_Ca6860139660246222851ard_of(B,B2)),bNF_Wellorder_ordLeq(A,B)) ) ).

% card_of_ordLeq
tff(fact_6583_internalize__card__of__ordLeq,axiom,
    ! [A: $tType,B: $tType,Aa2: set(A),R2: set(product_prod(B,B))] :
      ( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,Aa2)),R2),bNF_Wellorder_ordLeq(A,B))
    <=> ? [B12: set(B)] :
          ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),B12),aa(set(product_prod(B,B)),set(B),field2(B),R2))
          & 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,Aa2)),bNF_Ca6860139660246222851ard_of(B,B12)),bNF_Wellorder_ordIso(A,B))
          & 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,B12)),R2),bNF_Wellorder_ordLeq(B,B)) ) ) ).

% internalize_card_of_ordLeq
tff(fact_6584_card__of__ordLess,axiom,
    ! [A: $tType,B: $tType,Aa2: set(A),B2: set(B)] :
      ( ~ ? [F9: fun(A,B)] :
            ( inj_on(A,B,F9,Aa2)
            & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F9),Aa2)),B2) )
    <=> 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,B2)),bNF_Ca6860139660246222851ard_of(A,Aa2)),bNF_We4044943003108391690rdLess(B,A)) ) ).

% card_of_ordLess
tff(fact_6585_regularCard__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( bNF_Ca7133664381575040944arCard(A,R2)
    <=> ! [K8: set(A)] :
          ( ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),K8),aa(set(product_prod(A,A)),set(A),field2(A),R2))
            & bNF_Ca7293521722713021262ofinal(A,K8,R2) )
         => 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,K8)),R2),bNF_Wellorder_ordIso(A,A)) ) ) ).

% regularCard_def
tff(fact_6586_comp__set__bd__Union__o__collect,axiom,
    ! [E: $tType,B: $tType,A: $tType,Xa: E,X3: set(fun(E,set(set(A)))),Hbd: set(product_prod(B,B))] :
      ( 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(E,set(set(A)))),set(set(set(A))),image2(fun(E,set(set(A))),set(set(A)),aTP_Lamp_zd(E,fun(fun(E,set(set(A))),set(set(A))),Xa)),X3))))),Hbd),bNF_Wellorder_ordLeq(A,B))
     => 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(E,set(A),comp(set(set(A)),set(A),E,complete_Sup_Sup(set(A)),bNF_collect(E,set(A),X3)),Xa))),Hbd),bNF_Wellorder_ordLeq(A,B)) ) ).

% comp_set_bd_Union_o_collect
tff(fact_6587_UNIV__bool,axiom,
    top_top(set($o)) = aa(set($o),set($o),aa($o,fun(set($o),set($o)),insert3($o),$false),aa(set($o),set($o),aa($o,fun(set($o),set($o)),insert3($o),$true),bot_bot(set($o)))) ).

% UNIV_bool
tff(fact_6588_card__of__ordIso__subst,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A)] :
      ( ( Aa2 = B2 )
     => 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,Aa2)),bNF_Ca6860139660246222851ard_of(A,B2)),bNF_Wellorder_ordIso(A,A)) ) ).

% card_of_ordIso_subst
tff(fact_6589_card__of__Times__ordLeq__infinite__Field,axiom,
    ! [A: $tType,E: $tType,B: $tType,R2: set(product_prod(A,A)),Aa2: set(B),B2: set(E)] :
      ( ~ aa(set(A),$o,finite_finite2(A),aa(set(product_prod(A,A)),set(A),field2(A),R2))
     => ( 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,Aa2)),R2),bNF_Wellorder_ordLeq(B,A))
       => ( member(product_prod(set(product_prod(E,E)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(E,E)),set(product_prod(A,A))),aa(set(product_prod(E,E)),fun(set(product_prod(A,A)),product_prod(set(product_prod(E,E)),set(product_prod(A,A)))),product_Pair(set(product_prod(E,E)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(E,B2)),R2),bNF_Wellorder_ordLeq(E,A))
         => ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
           => member(product_prod(set(product_prod(product_prod(B,E),product_prod(B,E))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(B,E),product_prod(B,E))),set(product_prod(A,A))),aa(set(product_prod(product_prod(B,E),product_prod(B,E))),fun(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(B,E),product_prod(B,E))),set(product_prod(A,A)))),product_Pair(set(product_prod(product_prod(B,E),product_prod(B,E))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(product_prod(B,E),product_Sigma(B,E,Aa2,aTP_Lamp_qe(set(E),fun(B,set(E)),B2)))),R2),bNF_Wellorder_ordLeq(product_prod(B,E),A)) ) ) ) ) ).

% card_of_Times_ordLeq_infinite_Field
tff(fact_6590_ordLeq__refl,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => 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_6591_Card__order__trans,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),Xa: A,Y: A,Z2: A] :
      ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => ( ( Xa != Y )
       => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Y),R2)
         => ( ( Y != Z2 )
           => ( 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)
             => ( ( Xa != Z2 )
                & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Z2),R2) ) ) ) ) ) ) ).

% Card_order_trans
tff(fact_6592_ordIso__refl,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => 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_6593_card__order__on__ordIso,axiom,
    ! [A: $tType,Aa2: set(A),R2: set(product_prod(A,A)),R5: set(product_prod(A,A))] :
      ( bNF_Ca8970107618336181345der_on(A,Aa2,R2)
     => ( bNF_Ca8970107618336181345der_on(A,Aa2,R5)
       => 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),R5),bNF_Wellorder_ordIso(A,A)) ) ) ).

% card_order_on_ordIso
tff(fact_6594_infinite__Card__order__limit,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A3: A] :
      ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => ( ~ aa(set(A),$o,finite_finite2(A),aa(set(product_prod(A,A)),set(A),field2(A),R2))
       => ( member(A,A3,aa(set(product_prod(A,A)),set(A),field2(A),R2))
         => ? [X: A] :
              ( member(A,X,aa(set(product_prod(A,A)),set(A),field2(A),R2))
              & ( A3 != X )
              & 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) ) ) ) ) ).

% infinite_Card_order_limit
tff(fact_6595_dir__image,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),R2: set(product_prod(A,A))] :
      ( ! [X: A,Y2: A] :
          ( ( aa(A,B,F2,X) = aa(A,B,F2,Y2) )
        <=> ( X = Y2 ) )
     => ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
       => 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,F2)),bNF_Wellorder_ordIso(A,B)) ) ) ).

% dir_image
tff(fact_6596_Card__order__ordIso2,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(A,A)),R5: set(product_prod(B,B))] :
      ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => ( 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),R5),bNF_Wellorder_ordIso(A,B))
       => bNF_Ca8970107618336181345der_on(B,aa(set(product_prod(B,B)),set(B),field2(B),R5),R5) ) ) ).

% Card_order_ordIso2
tff(fact_6597_Card__order__ordIso,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(A,A)),R5: set(product_prod(B,B))] :
      ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => ( 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))),R5),R2),bNF_Wellorder_ordIso(B,A))
       => bNF_Ca8970107618336181345der_on(B,aa(set(product_prod(B,B)),set(B),field2(B),R5),R5) ) ) ).

% Card_order_ordIso
tff(fact_6598_card__of__unique,axiom,
    ! [A: $tType,Aa2: set(A),R2: set(product_prod(A,A))] :
      ( bNF_Ca8970107618336181345der_on(A,Aa2,R2)
     => 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,Aa2)),bNF_Wellorder_ordIso(A,A)) ) ).

% card_of_unique
tff(fact_6599_card__order__on__def,axiom,
    ! [A: $tType,Aa2: set(A),R2: set(product_prod(A,A))] :
      ( bNF_Ca8970107618336181345der_on(A,Aa2,R2)
    <=> ( order_well_order_on(A,Aa2,R2)
        & ! [R10: set(product_prod(A,A))] :
            ( order_well_order_on(A,Aa2,R10)
           => 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_6600_Card__order__iff__ordLeq__card__of,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
    <=> 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,aa(set(product_prod(A,A)),set(A),field2(A),R2))),bNF_Wellorder_ordLeq(A,A)) ) ).

% Card_order_iff_ordLeq_card_of
tff(fact_6601_ordIso__card__of__imp__Card__order,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,A)),Aa2: set(B)] :
      ( 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,Aa2)),bNF_Wellorder_ordIso(A,B))
     => bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2) ) ).

% ordIso_card_of_imp_Card_order
tff(fact_6602_Card__order__iff__ordIso__card__of,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
    <=> 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,aa(set(product_prod(A,A)),set(A),field2(A),R2))),bNF_Wellorder_ordIso(A,A)) ) ).

% Card_order_iff_ordIso_card_of
tff(fact_6603_card__of__Field__ordIso,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => 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,aa(set(product_prod(A,A)),set(A),field2(A),R2))),R2),bNF_Wellorder_ordIso(A,A)) ) ).

% card_of_Field_ordIso
tff(fact_6604_exists__minim__Card__order,axiom,
    ! [A: $tType,R: set(set(product_prod(A,A)))] :
      ( ( R != bot_bot(set(set(product_prod(A,A)))) )
     => ( ! [X: set(product_prod(A,A))] :
            ( member(set(product_prod(A,A)),X,R)
           => bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),X),X) )
       => ? [X: set(product_prod(A,A))] :
            ( member(set(product_prod(A,A)),X,R)
            & ! [Xa4: set(product_prod(A,A))] :
                ( member(set(product_prod(A,A)),Xa4,R)
               => 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))),X),Xa4),bNF_Wellorder_ordLeq(A,A)) ) ) ) ) ).

% exists_minim_Card_order
tff(fact_6605_Card__order__empty,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(A,A))] :
      ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => 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_6606_card__of__ordIso__finite__Field,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(A,A)),Aa2: set(B)] :
      ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => ( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R2),bNF_Ca6860139660246222851ard_of(B,Aa2)),bNF_Wellorder_ordIso(A,B))
       => ( aa(set(A),$o,finite_finite2(A),aa(set(product_prod(A,A)),set(A),field2(A),R2))
        <=> aa(set(B),$o,finite_finite2(B),Aa2) ) ) ) ).

% card_of_ordIso_finite_Field
tff(fact_6607_card__of__underS,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A3: A] :
      ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => ( member(A,A3,aa(set(product_prod(A,A)),set(A),field2(A),R2))
       => 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_6608_Card__order__singl__ordLeq,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(A,A)),B3: B] :
      ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => ( ( aa(set(product_prod(A,A)),set(A),field2(A),R2) != bot_bot(set(A)) )
       => 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)),insert3(B),B3),bot_bot(set(B))))),R2),bNF_Wellorder_ordLeq(B,A)) ) ) ).

% Card_order_singl_ordLeq
tff(fact_6609_card__of__Un__ordLeq__infinite__Field,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(A,A)),Aa2: set(B),B2: set(B)] :
      ( ~ aa(set(A),$o,finite_finite2(A),aa(set(product_prod(A,A)),set(A),field2(A),R2))
     => ( 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,Aa2)),R2),bNF_Wellorder_ordLeq(B,A))
       => ( 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,B2)),R2),bNF_Wellorder_ordLeq(B,A))
         => ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
           => 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)),Aa2),B2))),R2),bNF_Wellorder_ordLeq(B,A)) ) ) ) ) ).

% card_of_Un_ordLeq_infinite_Field
tff(fact_6610_card__of__empty1,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(A,A))] :
      ( ( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
        | bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2) )
     => 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_6611_Card__order__Pow,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => 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),pow(A,aa(set(product_prod(A,A)),set(A),field2(A),R2)))),bNF_We4044943003108391690rdLess(A,set(A))) ) ).

% Card_order_Pow
tff(fact_6612_Card__order__Times1,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,A)),B2: set(B)] :
      ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => ( ( B2 != bot_bot(set(B)) )
       => 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,aa(set(product_prod(A,A)),set(A),field2(A),R2),aTP_Lamp_pv(set(B),fun(A,set(B)),B2)))),bNF_Wellorder_ordLeq(A,product_prod(A,B))) ) ) ).

% Card_order_Times1
tff(fact_6613_Card__order__Times2,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,A)),Aa2: set(B)] :
      ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => ( ( Aa2 != bot_bot(set(B)) )
       => 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,Aa2,aTP_Lamp_ze(set(product_prod(A,A)),fun(B,set(A)),R2)))),bNF_Wellorder_ordLeq(A,product_prod(B,A))) ) ) ).

% Card_order_Times2
tff(fact_6614_Card__order__Times__same__infinite,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => ( ~ aa(set(A),$o,finite_finite2(A),aa(set(product_prod(A,A)),set(A),field2(A),R2))
       => 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,aa(set(product_prod(A,A)),set(A),field2(A),R2),aTP_Lamp_qc(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_6615_card__of__UNION__ordLeq__infinite__Field,axiom,
    ! [B: $tType,A: $tType,E: $tType,R2: set(product_prod(A,A)),Ia2: set(B),Aa2: fun(B,set(E))] :
      ( ~ aa(set(A),$o,finite_finite2(A),aa(set(product_prod(A,A)),set(A),field2(A),R2))
     => ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
       => ( 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,Ia2)),R2),bNF_Wellorder_ordLeq(B,A))
         => ( ! [X: B] :
                ( member(B,X,Ia2)
               => member(product_prod(set(product_prod(E,E)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(E,E)),set(product_prod(A,A))),aa(set(product_prod(E,E)),fun(set(product_prod(A,A)),product_prod(set(product_prod(E,E)),set(product_prod(A,A)))),product_Pair(set(product_prod(E,E)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(E,aa(B,set(E),Aa2,X))),R2),bNF_Wellorder_ordLeq(E,A)) )
           => member(product_prod(set(product_prod(E,E)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(E,E)),set(product_prod(A,A))),aa(set(product_prod(E,E)),fun(set(product_prod(A,A)),product_prod(set(product_prod(E,E)),set(product_prod(A,A)))),product_Pair(set(product_prod(E,E)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(E,aa(set(set(E)),set(E),complete_Sup_Sup(set(E)),aa(set(B),set(set(E)),image2(B,set(E),Aa2),Ia2)))),R2),bNF_Wellorder_ordLeq(E,A)) ) ) ) ) ).

% card_of_UNION_ordLeq_infinite_Field
tff(fact_6616_Card__order__iff__Restr__underS,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
      <=> ! [X6: A] :
            ( member(A,X6,aa(set(product_prod(A,A)),set(A),field2(A),R2))
           => 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,X6),aa(A,fun(A,set(A)),aTP_Lamp_yc(set(product_prod(A,A)),fun(A,fun(A,set(A))),R2),X6)))),bNF_Ca6860139660246222851ard_of(A,aa(set(product_prod(A,A)),set(A),field2(A),R2))),bNF_We4044943003108391690rdLess(A,A)) ) ) ) ).

% Card_order_iff_Restr_underS
tff(fact_6617_regularCard__UNION,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,A)),As4: fun(A,set(B)),B2: set(B)] :
      ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => ( bNF_Ca7133664381575040944arCard(A,R2)
       => ( bNF_Ca3754400796208372196lChain(A,set(B),R2,As4)
         => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),B2),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),As4),aa(set(product_prod(A,A)),set(A),field2(A),R2))))
           => ( member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,B2)),R2),bNF_We4044943003108391690rdLess(B,A))
             => ? [X: A] :
                  ( member(A,X,aa(set(product_prod(A,A)),set(A),field2(A),R2))
                  & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),B2),aa(A,set(B),As4,X)) ) ) ) ) ) ) ).

% regularCard_UNION
tff(fact_6618_Card__order__Times__infinite,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(A,A)),P2: set(product_prod(B,B))] :
      ( ~ aa(set(A),$o,finite_finite2(A),aa(set(product_prod(A,A)),set(A),field2(A),R2))
     => ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
       => ( ( aa(set(product_prod(B,B)),set(B),field2(B),P2) != bot_bot(set(B)) )
         => ( 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))),P2),R2),bNF_Wellorder_ordLeq(B,A))
           => ( 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,aa(set(product_prod(A,A)),set(A),field2(A),R2),aTP_Lamp_zf(set(product_prod(B,B)),fun(A,set(B)),P2)))),R2),bNF_Wellorder_ordIso(product_prod(A,B),A))
              & 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,aa(set(product_prod(B,B)),set(B),field2(B),P2),aTP_Lamp_ze(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_6619_card__of__Sigma__ordLeq__infinite__Field,axiom,
    ! [A: $tType,E: $tType,B: $tType,R2: set(product_prod(A,A)),Ia2: set(B),Aa2: fun(B,set(E))] :
      ( ~ aa(set(A),$o,finite_finite2(A),aa(set(product_prod(A,A)),set(A),field2(A),R2))
     => ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
       => ( 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,Ia2)),R2),bNF_Wellorder_ordLeq(B,A))
         => ( ! [X: B] :
                ( member(B,X,Ia2)
               => member(product_prod(set(product_prod(E,E)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(E,E)),set(product_prod(A,A))),aa(set(product_prod(E,E)),fun(set(product_prod(A,A)),product_prod(set(product_prod(E,E)),set(product_prod(A,A)))),product_Pair(set(product_prod(E,E)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(E,aa(B,set(E),Aa2,X))),R2),bNF_Wellorder_ordLeq(E,A)) )
           => member(product_prod(set(product_prod(product_prod(B,E),product_prod(B,E))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(B,E),product_prod(B,E))),set(product_prod(A,A))),aa(set(product_prod(product_prod(B,E),product_prod(B,E))),fun(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(B,E),product_prod(B,E))),set(product_prod(A,A)))),product_Pair(set(product_prod(product_prod(B,E),product_prod(B,E))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(product_prod(B,E),product_Sigma(B,E,Ia2,Aa2))),R2),bNF_Wellorder_ordLeq(product_prod(B,E),A)) ) ) ) ) ).

% card_of_Sigma_ordLeq_infinite_Field
tff(fact_6620_ex__toCard__pred,axiom,
    ! [B: $tType,A: $tType,Aa2: set(A),R2: set(product_prod(B,B))] :
      ( 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,Aa2)),R2),bNF_Wellorder_ordLeq(A,B))
     => ( bNF_Ca8970107618336181345der_on(B,aa(set(product_prod(B,B)),set(B),field2(B),R2),R2)
       => ? [X_1: fun(A,B)] : bNF_Gr1419584066657907630d_pred(A,B,Aa2,R2,X_1) ) ) ).

% ex_toCard_pred
tff(fact_6621_cardSuc__UNION,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,A)),As4: fun(set(A),set(B)),B2: set(B)] :
      ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => ( ~ aa(set(A),$o,finite_finite2(A),aa(set(product_prod(A,A)),set(A),field2(A),R2))
       => ( bNF_Ca3754400796208372196lChain(set(A),set(B),bNF_Ca8387033319878233205ardSuc(A,R2),As4)
         => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),B2),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(set(A)),set(set(B)),image2(set(A),set(B),As4),aa(set(product_prod(set(A),set(A))),set(set(A)),field2(set(A)),bNF_Ca8387033319878233205ardSuc(A,R2)))))
           => ( 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,B2)),R2),bNF_Wellorder_ordLeq(B,A))
             => ? [X: set(A)] :
                  ( member(set(A),X,aa(set(product_prod(set(A),set(A))),set(set(A)),field2(set(A)),bNF_Ca8387033319878233205ardSuc(A,R2)))
                  & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),B2),aa(set(A),set(B),As4,X)) ) ) ) ) ) ) ).

% cardSuc_UNION
tff(fact_6622_cardSuc__ordLeq,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => 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_6623_cardSuc__greater,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => 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_6624_cardSuc__least,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,A)),R5: set(product_prod(B,B))] :
      ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => ( bNF_Ca8970107618336181345der_on(B,aa(set(product_prod(B,B)),set(B),field2(B),R5),R5)
       => ( 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),R5),bNF_We4044943003108391690rdLess(A,B))
         => 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)),R5),bNF_Wellorder_ordLeq(set(A),B)) ) ) ) ).

% cardSuc_least
tff(fact_6625_cardSuc__ordLess__ordLeq,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,A)),R5: set(product_prod(B,B))] :
      ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => ( bNF_Ca8970107618336181345der_on(B,aa(set(product_prod(B,B)),set(B),field2(B),R5),R5)
       => ( 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),R5),bNF_We4044943003108391690rdLess(A,B))
        <=> 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)),R5),bNF_Wellorder_ordLeq(set(A),B)) ) ) ) ).

% cardSuc_ordLess_ordLeq
tff(fact_6626_cardSuc__mono__ordLeq,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,A)),R5: set(product_prod(B,B))] :
      ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => ( bNF_Ca8970107618336181345der_on(B,aa(set(product_prod(B,B)),set(B),field2(B),R5),R5)
       => ( 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,R5)),bNF_Wellorder_ordLeq(set(A),set(B)))
        <=> 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),R5),bNF_Wellorder_ordLeq(A,B)) ) ) ) ).

% cardSuc_mono_ordLeq
tff(fact_6627_cardSuc__invar__ordIso,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,A)),R5: set(product_prod(B,B))] :
      ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => ( bNF_Ca8970107618336181345der_on(B,aa(set(product_prod(B,B)),set(B),field2(B),R5),R5)
       => ( 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,R5)),bNF_Wellorder_ordIso(set(A),set(B)))
        <=> 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),R5),bNF_Wellorder_ordIso(A,B)) ) ) ) ).

% cardSuc_invar_ordIso
tff(fact_6628_cardSuc__least__aux,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),R5: set(product_prod(set(A),set(A)))] :
      ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => ( bNF_Ca8970107618336181345der_on(set(A),aa(set(product_prod(set(A),set(A))),set(set(A)),field2(set(A)),R5),R5)
       => ( 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),R5),bNF_We4044943003108391690rdLess(A,set(A)))
         => 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)),R5),bNF_Wellorder_ordLeq(set(A),set(A))) ) ) ) ).

% cardSuc_least_aux
tff(fact_6629_cardSuc__ordLeq__ordLess,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(A,A)),R5: set(product_prod(B,B))] :
      ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => ( bNF_Ca8970107618336181345der_on(B,aa(set(product_prod(B,B)),set(B),field2(B),R5),R5)
       => ( 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)))),R5),bNF_Ca8387033319878233205ardSuc(A,R2)),bNF_We4044943003108391690rdLess(B,set(A)))
        <=> 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))),R5),R2),bNF_Wellorder_ordLeq(B,A)) ) ) ) ).

% cardSuc_ordLeq_ordLess
tff(fact_6630_isCardSuc__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),R5: set(product_prod(set(A),set(A)))] :
      ( bNF_Ca6246979054910435723ardSuc(A,R2,R5)
    <=> ( bNF_Ca8970107618336181345der_on(set(A),aa(set(product_prod(set(A),set(A))),set(set(A)),field2(set(A)),R5),R5)
        & 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),R5),bNF_We4044943003108391690rdLess(A,set(A)))
        & ! [R11: set(product_prod(set(A),set(A)))] :
            ( ( bNF_Ca8970107618336181345der_on(set(A),aa(set(product_prod(set(A),set(A))),set(set(A)),field2(set(A)),R11),R11)
              & 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))) )
           => 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)))),R5),R11),bNF_Wellorder_ordLeq(set(A),set(A))) ) ) ) ).

% isCardSuc_def
tff(fact_6631_toCard__pred__toCard,axiom,
    ! [A: $tType,B: $tType,Aa2: set(A),R2: set(product_prod(B,B))] :
      ( 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,Aa2)),R2),bNF_Wellorder_ordLeq(A,B))
     => ( bNF_Ca8970107618336181345der_on(B,aa(set(product_prod(B,B)),set(B),field2(B),R2),R2)
       => bNF_Gr1419584066657907630d_pred(A,B,Aa2,R2,bNF_Greatest_toCard(A,B,Aa2,R2)) ) ) ).

% toCard_pred_toCard
tff(fact_6632_toCard__inj,axiom,
    ! [B: $tType,A: $tType,Aa2: set(A),R2: set(product_prod(B,B)),Xa: A,Y: A] :
      ( 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,Aa2)),R2),bNF_Wellorder_ordLeq(A,B))
     => ( bNF_Ca8970107618336181345der_on(B,aa(set(product_prod(B,B)),set(B),field2(B),R2),R2)
       => ( member(A,Xa,Aa2)
         => ( member(A,Y,Aa2)
           => ( ( aa(A,B,bNF_Greatest_toCard(A,B,Aa2,R2),Xa) = aa(A,B,bNF_Greatest_toCard(A,B,Aa2,R2),Y) )
            <=> ( Xa = Y ) ) ) ) ) ) ).

% toCard_inj
tff(fact_6633_fromCard__toCard,axiom,
    ! [B: $tType,A: $tType,Aa2: set(A),R2: set(product_prod(B,B)),B3: A] :
      ( 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,Aa2)),R2),bNF_Wellorder_ordLeq(A,B))
     => ( bNF_Ca8970107618336181345der_on(B,aa(set(product_prod(B,B)),set(B),field2(B),R2),R2)
       => ( member(A,B3,Aa2)
         => ( bNF_Gr5436034075474128252omCard(A,B,Aa2,R2,aa(A,B,bNF_Greatest_toCard(A,B,Aa2,R2),B3)) = B3 ) ) ) ) ).

% fromCard_toCard
tff(fact_6634_cardSuc__UNION__Cinfinite,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,A)),As4: fun(set(A),set(B)),B2: set(B)] :
      ( ( bNF_Ca4139267488887388095finite(A,R2)
        & bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2) )
     => ( bNF_Ca3754400796208372196lChain(set(A),set(B),bNF_Ca8387033319878233205ardSuc(A,R2),As4)
       => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),B2),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(set(A)),set(set(B)),image2(set(A),set(B),As4),aa(set(product_prod(set(A),set(A))),set(set(A)),field2(set(A)),bNF_Ca8387033319878233205ardSuc(A,R2)))))
         => ( member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,B2)),R2),bNF_Wellorder_ordLeq(B,A))
           => ? [X: set(A)] :
                ( member(set(A),X,aa(set(product_prod(set(A),set(A))),set(set(A)),field2(set(A)),bNF_Ca8387033319878233205ardSuc(A,R2)))
                & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),B2),aa(set(A),set(B),As4,X)) ) ) ) ) ) ).

% cardSuc_UNION_Cinfinite
tff(fact_6635_cinfinite__mono,axiom,
    ! [A: $tType,B: $tType,R12: set(product_prod(A,A)),R23: set(product_prod(B,B))] :
      ( 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_6636_Cinfinite__limit2,axiom,
    ! [A: $tType,X1: A,R2: set(product_prod(A,A)),X22: A] :
      ( member(A,X1,aa(set(product_prod(A,A)),set(A),field2(A),R2))
     => ( member(A,X22,aa(set(product_prod(A,A)),set(A),field2(A),R2))
       => ( ( bNF_Ca4139267488887388095finite(A,R2)
            & bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2) )
         => ? [X: A] :
              ( member(A,X,aa(set(product_prod(A,A)),set(A),field2(A),R2))
              & ( X1 != X )
              & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X1),X),R2)
              & ( X22 != X )
              & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X22),X),R2) ) ) ) ) ).

% Cinfinite_limit2
tff(fact_6637_Cinfinite__limit,axiom,
    ! [A: $tType,Xa: A,R2: set(product_prod(A,A))] :
      ( member(A,Xa,aa(set(product_prod(A,A)),set(A),field2(A),R2))
     => ( ( bNF_Ca4139267488887388095finite(A,R2)
          & bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2) )
       => ? [X: A] :
            ( member(A,X,aa(set(product_prod(A,A)),set(A),field2(A),R2))
            & ( Xa != X )
            & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),X),R2) ) ) ) ).

% Cinfinite_limit
tff(fact_6638_Cinfinite__cong,axiom,
    ! [A: $tType,B: $tType,R12: set(product_prod(A,A)),R23: set(product_prod(B,B))] :
      ( 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)
          & bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R12),R12) )
       => ( bNF_Ca4139267488887388095finite(B,R23)
          & bNF_Ca8970107618336181345der_on(B,aa(set(product_prod(B,B)),set(B),field2(B),R23),R23) ) ) ) ).

% Cinfinite_cong
tff(fact_6639_Cinfinite__limit__finite,axiom,
    ! [A: $tType,X3: set(A),R2: set(product_prod(A,A))] :
      ( aa(set(A),$o,finite_finite2(A),X3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X3),aa(set(product_prod(A,A)),set(A),field2(A),R2))
       => ( ( bNF_Ca4139267488887388095finite(A,R2)
            & bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2) )
         => ? [X: A] :
              ( member(A,X,aa(set(product_prod(A,A)),set(A),field2(A),R2))
              & ! [Xa4: A] :
                  ( member(A,Xa4,X3)
                 => ( ( Xa4 != X )
                    & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa4),X),R2) ) ) ) ) ) ) ).

% Cinfinite_limit_finite
tff(fact_6640_Un__Cinfinite__bound,axiom,
    ! [B: $tType,A: $tType,Aa2: set(A),R2: set(product_prod(B,B)),B2: set(A)] :
      ( 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,Aa2)),R2),bNF_Wellorder_ordLeq(A,B))
     => ( 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,B2)),R2),bNF_Wellorder_ordLeq(A,B))
       => ( ( bNF_Ca4139267488887388095finite(B,R2)
            & bNF_Ca8970107618336181345der_on(B,aa(set(product_prod(B,B)),set(B),field2(B),R2),R2) )
         => 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)),Aa2),B2))),R2),bNF_Wellorder_ordLeq(A,B)) ) ) ) ).

% Un_Cinfinite_bound
tff(fact_6641_UNION__Cinfinite__bound,axiom,
    ! [A: $tType,B: $tType,E: $tType,Ia2: set(A),R2: set(product_prod(B,B)),Aa2: fun(A,set(E))] :
      ( 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,Ia2)),R2),bNF_Wellorder_ordLeq(A,B))
     => ( ! [X: A] :
            ( member(A,X,Ia2)
           => member(product_prod(set(product_prod(E,E)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(E,E)),set(product_prod(B,B))),aa(set(product_prod(E,E)),fun(set(product_prod(B,B)),product_prod(set(product_prod(E,E)),set(product_prod(B,B)))),product_Pair(set(product_prod(E,E)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(E,aa(A,set(E),Aa2,X))),R2),bNF_Wellorder_ordLeq(E,B)) )
       => ( ( bNF_Ca4139267488887388095finite(B,R2)
            & bNF_Ca8970107618336181345der_on(B,aa(set(product_prod(B,B)),set(B),field2(B),R2),R2) )
         => member(product_prod(set(product_prod(E,E)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(E,E)),set(product_prod(B,B))),aa(set(product_prod(E,E)),fun(set(product_prod(B,B)),product_prod(set(product_prod(E,E)),set(product_prod(B,B)))),product_Pair(set(product_prod(E,E)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(E,aa(set(set(E)),set(E),complete_Sup_Sup(set(E)),aa(set(A),set(set(E)),image2(A,set(E),Aa2),Ia2)))),R2),bNF_Wellorder_ordLeq(E,B)) ) ) ) ).

% UNION_Cinfinite_bound
tff(fact_6642_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)
        & bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2) )
     => ( ( bNF_Ca4139267488887388095finite(B,S2)
          & bNF_Ca8970107618336181345der_on(B,aa(set(product_prod(B,B)),set(B),field2(B),S2),S2) )
       => 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_6643_comp__single__set__bd,axiom,
    ! [B: $tType,F4: $tType,A: $tType,G2: $tType,E: $tType,Fbd: set(product_prod(A,A)),Fset: fun(B,set(E)),Gset: fun(F4,set(B)),Gbd: set(product_prod(G2,G2)),Xa: F4] :
      ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),Fbd),Fbd)
     => ( ! [X: B] : member(product_prod(set(product_prod(E,E)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(E,E)),set(product_prod(A,A))),aa(set(product_prod(E,E)),fun(set(product_prod(A,A)),product_prod(set(product_prod(E,E)),set(product_prod(A,A)))),product_Pair(set(product_prod(E,E)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(E,aa(B,set(E),Fset,X))),Fbd),bNF_Wellorder_ordLeq(E,A))
       => ( ! [X: F4] : member(product_prod(set(product_prod(B,B)),set(product_prod(G2,G2))),aa(set(product_prod(G2,G2)),product_prod(set(product_prod(B,B)),set(product_prod(G2,G2))),aa(set(product_prod(B,B)),fun(set(product_prod(G2,G2)),product_prod(set(product_prod(B,B)),set(product_prod(G2,G2)))),product_Pair(set(product_prod(B,B)),set(product_prod(G2,G2))),bNF_Ca6860139660246222851ard_of(B,aa(F4,set(B),Gset,X))),Gbd),bNF_Wellorder_ordLeq(B,G2))
         => member(product_prod(set(product_prod(E,E)),set(product_prod(product_prod(G2,A),product_prod(G2,A)))),aa(set(product_prod(product_prod(G2,A),product_prod(G2,A))),product_prod(set(product_prod(E,E)),set(product_prod(product_prod(G2,A),product_prod(G2,A)))),aa(set(product_prod(E,E)),fun(set(product_prod(product_prod(G2,A),product_prod(G2,A))),product_prod(set(product_prod(E,E)),set(product_prod(product_prod(G2,A),product_prod(G2,A))))),product_Pair(set(product_prod(E,E)),set(product_prod(product_prod(G2,A),product_prod(G2,A)))),bNF_Ca6860139660246222851ard_of(E,aa(set(set(E)),set(E),complete_Sup_Sup(set(E)),aa(set(B),set(set(E)),image2(B,set(E),Fset),aa(F4,set(B),Gset,Xa))))),bNF_Cardinal_cprod(G2,A,Gbd,Fbd)),bNF_Wellorder_ordLeq(E,product_prod(G2,A))) ) ) ) ).

% comp_single_set_bd
tff(fact_6644_cprod__com,axiom,
    ! [B: $tType,A: $tType,P13: set(product_prod(A,A)),P24: set(product_prod(B,B))] : 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,P13,P24)),bNF_Cardinal_cprod(B,A,P24,P13)),bNF_Wellorder_ordIso(product_prod(A,B),product_prod(B,A))) ).

% cprod_com
tff(fact_6645_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)
        & bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2) )
     => ( ( bNF_Ca4139267488887388095finite(B,S2)
          & bNF_Ca8970107618336181345der_on(B,aa(set(product_prod(B,B)),set(B),field2(B),S2),S2) )
       => 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_6646_cprod__mono,axiom,
    ! [F4: $tType,B: $tType,E: $tType,A: $tType,P13: set(product_prod(A,A)),R12: set(product_prod(B,B)),P24: set(product_prod(E,E)),R23: set(product_prod(F4,F4))] :
      ( 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))),P13),R12),bNF_Wellorder_ordLeq(A,B))
     => ( member(product_prod(set(product_prod(E,E)),set(product_prod(F4,F4))),aa(set(product_prod(F4,F4)),product_prod(set(product_prod(E,E)),set(product_prod(F4,F4))),aa(set(product_prod(E,E)),fun(set(product_prod(F4,F4)),product_prod(set(product_prod(E,E)),set(product_prod(F4,F4)))),product_Pair(set(product_prod(E,E)),set(product_prod(F4,F4))),P24),R23),bNF_Wellorder_ordLeq(E,F4))
       => member(product_prod(set(product_prod(product_prod(A,E),product_prod(A,E))),set(product_prod(product_prod(B,F4),product_prod(B,F4)))),aa(set(product_prod(product_prod(B,F4),product_prod(B,F4))),product_prod(set(product_prod(product_prod(A,E),product_prod(A,E))),set(product_prod(product_prod(B,F4),product_prod(B,F4)))),aa(set(product_prod(product_prod(A,E),product_prod(A,E))),fun(set(product_prod(product_prod(B,F4),product_prod(B,F4))),product_prod(set(product_prod(product_prod(A,E),product_prod(A,E))),set(product_prod(product_prod(B,F4),product_prod(B,F4))))),product_Pair(set(product_prod(product_prod(A,E),product_prod(A,E))),set(product_prod(product_prod(B,F4),product_prod(B,F4)))),bNF_Cardinal_cprod(A,E,P13,P24)),bNF_Cardinal_cprod(B,F4,R12,R23)),bNF_Wellorder_ordLeq(product_prod(A,E),product_prod(B,F4))) ) ) ).

% cprod_mono
tff(fact_6647_cprod__mono1,axiom,
    ! [B: $tType,E: $tType,A: $tType,P13: set(product_prod(A,A)),R12: set(product_prod(B,B)),Q2: set(product_prod(E,E))] :
      ( 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))),P13),R12),bNF_Wellorder_ordLeq(A,B))
     => member(product_prod(set(product_prod(product_prod(A,E),product_prod(A,E))),set(product_prod(product_prod(B,E),product_prod(B,E)))),aa(set(product_prod(product_prod(B,E),product_prod(B,E))),product_prod(set(product_prod(product_prod(A,E),product_prod(A,E))),set(product_prod(product_prod(B,E),product_prod(B,E)))),aa(set(product_prod(product_prod(A,E),product_prod(A,E))),fun(set(product_prod(product_prod(B,E),product_prod(B,E))),product_prod(set(product_prod(product_prod(A,E),product_prod(A,E))),set(product_prod(product_prod(B,E),product_prod(B,E))))),product_Pair(set(product_prod(product_prod(A,E),product_prod(A,E))),set(product_prod(product_prod(B,E),product_prod(B,E)))),bNF_Cardinal_cprod(A,E,P13,Q2)),bNF_Cardinal_cprod(B,E,R12,Q2)),bNF_Wellorder_ordLeq(product_prod(A,E),product_prod(B,E))) ) ).

% cprod_mono1
tff(fact_6648_cprod__mono2,axiom,
    ! [B: $tType,A: $tType,E: $tType,P24: set(product_prod(A,A)),R23: set(product_prod(B,B)),Q2: set(product_prod(E,E))] :
      ( 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))),P24),R23),bNF_Wellorder_ordLeq(A,B))
     => member(product_prod(set(product_prod(product_prod(E,A),product_prod(E,A))),set(product_prod(product_prod(E,B),product_prod(E,B)))),aa(set(product_prod(product_prod(E,B),product_prod(E,B))),product_prod(set(product_prod(product_prod(E,A),product_prod(E,A))),set(product_prod(product_prod(E,B),product_prod(E,B)))),aa(set(product_prod(product_prod(E,A),product_prod(E,A))),fun(set(product_prod(product_prod(E,B),product_prod(E,B))),product_prod(set(product_prod(product_prod(E,A),product_prod(E,A))),set(product_prod(product_prod(E,B),product_prod(E,B))))),product_Pair(set(product_prod(product_prod(E,A),product_prod(E,A))),set(product_prod(product_prod(E,B),product_prod(E,B)))),bNF_Cardinal_cprod(E,A,Q2,P24)),bNF_Cardinal_cprod(E,B,Q2,R23)),bNF_Wellorder_ordLeq(product_prod(E,A),product_prod(E,B))) ) ).

% cprod_mono2
tff(fact_6649_cprod__cong,axiom,
    ! [F4: $tType,B: $tType,E: $tType,A: $tType,P13: set(product_prod(A,A)),R12: set(product_prod(B,B)),P24: set(product_prod(E,E)),R23: set(product_prod(F4,F4))] :
      ( 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))),P13),R12),bNF_Wellorder_ordIso(A,B))
     => ( member(product_prod(set(product_prod(E,E)),set(product_prod(F4,F4))),aa(set(product_prod(F4,F4)),product_prod(set(product_prod(E,E)),set(product_prod(F4,F4))),aa(set(product_prod(E,E)),fun(set(product_prod(F4,F4)),product_prod(set(product_prod(E,E)),set(product_prod(F4,F4)))),product_Pair(set(product_prod(E,E)),set(product_prod(F4,F4))),P24),R23),bNF_Wellorder_ordIso(E,F4))
       => member(product_prod(set(product_prod(product_prod(A,E),product_prod(A,E))),set(product_prod(product_prod(B,F4),product_prod(B,F4)))),aa(set(product_prod(product_prod(B,F4),product_prod(B,F4))),product_prod(set(product_prod(product_prod(A,E),product_prod(A,E))),set(product_prod(product_prod(B,F4),product_prod(B,F4)))),aa(set(product_prod(product_prod(A,E),product_prod(A,E))),fun(set(product_prod(product_prod(B,F4),product_prod(B,F4))),product_prod(set(product_prod(product_prod(A,E),product_prod(A,E))),set(product_prod(product_prod(B,F4),product_prod(B,F4))))),product_Pair(set(product_prod(product_prod(A,E),product_prod(A,E))),set(product_prod(product_prod(B,F4),product_prod(B,F4)))),bNF_Cardinal_cprod(A,E,P13,P24)),bNF_Cardinal_cprod(B,F4,R12,R23)),bNF_Wellorder_ordIso(product_prod(A,E),product_prod(B,F4))) ) ) ).

% cprod_cong
tff(fact_6650_cprod__cong1,axiom,
    ! [B: $tType,E: $tType,A: $tType,P13: set(product_prod(A,A)),R12: set(product_prod(B,B)),P24: set(product_prod(E,E))] :
      ( 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))),P13),R12),bNF_Wellorder_ordIso(A,B))
     => member(product_prod(set(product_prod(product_prod(A,E),product_prod(A,E))),set(product_prod(product_prod(B,E),product_prod(B,E)))),aa(set(product_prod(product_prod(B,E),product_prod(B,E))),product_prod(set(product_prod(product_prod(A,E),product_prod(A,E))),set(product_prod(product_prod(B,E),product_prod(B,E)))),aa(set(product_prod(product_prod(A,E),product_prod(A,E))),fun(set(product_prod(product_prod(B,E),product_prod(B,E))),product_prod(set(product_prod(product_prod(A,E),product_prod(A,E))),set(product_prod(product_prod(B,E),product_prod(B,E))))),product_Pair(set(product_prod(product_prod(A,E),product_prod(A,E))),set(product_prod(product_prod(B,E),product_prod(B,E)))),bNF_Cardinal_cprod(A,E,P13,P24)),bNF_Cardinal_cprod(B,E,R12,P24)),bNF_Wellorder_ordIso(product_prod(A,E),product_prod(B,E))) ) ).

% cprod_cong1
tff(fact_6651_cprod__cong2,axiom,
    ! [B: $tType,A: $tType,E: $tType,P24: set(product_prod(A,A)),R23: set(product_prod(B,B)),Q2: set(product_prod(E,E))] :
      ( 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))),P24),R23),bNF_Wellorder_ordIso(A,B))
     => member(product_prod(set(product_prod(product_prod(E,A),product_prod(E,A))),set(product_prod(product_prod(E,B),product_prod(E,B)))),aa(set(product_prod(product_prod(E,B),product_prod(E,B))),product_prod(set(product_prod(product_prod(E,A),product_prod(E,A))),set(product_prod(product_prod(E,B),product_prod(E,B)))),aa(set(product_prod(product_prod(E,A),product_prod(E,A))),fun(set(product_prod(product_prod(E,B),product_prod(E,B))),product_prod(set(product_prod(product_prod(E,A),product_prod(E,A))),set(product_prod(product_prod(E,B),product_prod(E,B))))),product_Pair(set(product_prod(product_prod(E,A),product_prod(E,A))),set(product_prod(product_prod(E,B),product_prod(E,B)))),bNF_Cardinal_cprod(E,A,Q2,P24)),bNF_Cardinal_cprod(E,B,Q2,R23)),bNF_Wellorder_ordIso(product_prod(E,A),product_prod(E,B))) ) ).

% cprod_cong2
tff(fact_6652_cprod__infinite,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( ( bNF_Ca4139267488887388095finite(A,R2)
        & bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2) )
     => 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_6653_cprod__cinfinite__bound,axiom,
    ! [B: $tType,E: $tType,A: $tType,P2: set(product_prod(A,A)),R2: set(product_prod(B,B)),Q2: set(product_prod(E,E))] :
      ( 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))),P2),R2),bNF_Wellorder_ordLeq(A,B))
     => ( member(product_prod(set(product_prod(E,E)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(E,E)),set(product_prod(B,B))),aa(set(product_prod(E,E)),fun(set(product_prod(B,B)),product_prod(set(product_prod(E,E)),set(product_prod(B,B)))),product_Pair(set(product_prod(E,E)),set(product_prod(B,B))),Q2),R2),bNF_Wellorder_ordLeq(E,B))
       => ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),P2),P2)
         => ( bNF_Ca8970107618336181345der_on(E,aa(set(product_prod(E,E)),set(E),field2(E),Q2),Q2)
           => ( ( bNF_Ca4139267488887388095finite(B,R2)
                & bNF_Ca8970107618336181345der_on(B,aa(set(product_prod(B,B)),set(B),field2(B),R2),R2) )
             => member(product_prod(set(product_prod(product_prod(A,E),product_prod(A,E))),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(product_prod(A,E),product_prod(A,E))),set(product_prod(B,B))),aa(set(product_prod(product_prod(A,E),product_prod(A,E))),fun(set(product_prod(B,B)),product_prod(set(product_prod(product_prod(A,E),product_prod(A,E))),set(product_prod(B,B)))),product_Pair(set(product_prod(product_prod(A,E),product_prod(A,E))),set(product_prod(B,B))),bNF_Cardinal_cprod(A,E,P2,Q2)),R2),bNF_Wellorder_ordLeq(product_prod(A,E),B)) ) ) ) ) ) ).

% cprod_cinfinite_bound
tff(fact_6654_cprod__dup,axiom,
    ! [A: $tType,E: $tType,B: $tType,R2: set(product_prod(A,A)),P2: set(product_prod(B,B)),P7: set(product_prod(E,E))] :
      ( bNF_Ca4139267488887388095finite(A,R2)
     => ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
       => ( member(product_prod(set(product_prod(product_prod(B,E),product_prod(B,E))),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,E),product_prod(B,E))),set(product_prod(product_prod(A,A),product_prod(A,A)))),aa(set(product_prod(product_prod(B,E),product_prod(B,E))),fun(set(product_prod(product_prod(A,A),product_prod(A,A))),product_prod(set(product_prod(product_prod(B,E),product_prod(B,E))),set(product_prod(product_prod(A,A),product_prod(A,A))))),product_Pair(set(product_prod(product_prod(B,E),product_prod(B,E))),set(product_prod(product_prod(A,A),product_prod(A,A)))),bNF_Cardinal_cprod(B,E,P2,P7)),bNF_Cardinal_cprod(A,A,R2,R2)),bNF_Wellorder_ordIso(product_prod(B,E),product_prod(A,A)))
         => member(product_prod(set(product_prod(product_prod(B,E),product_prod(B,E))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(B,E),product_prod(B,E))),set(product_prod(A,A))),aa(set(product_prod(product_prod(B,E),product_prod(B,E))),fun(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(B,E),product_prod(B,E))),set(product_prod(A,A)))),product_Pair(set(product_prod(product_prod(B,E),product_prod(B,E))),set(product_prod(A,A))),bNF_Cardinal_cprod(B,E,P2,P7)),R2),bNF_Wellorder_ordIso(product_prod(B,E),A)) ) ) ) ).

% cprod_dup
tff(fact_6655_card__of__Csum__Times_H,axiom,
    ! [A: $tType,E: $tType,B: $tType,R2: set(product_prod(A,A)),Ia2: set(B),Aa2: fun(B,set(E))] :
      ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => ( ! [X: B] :
            ( member(B,X,Ia2)
           => member(product_prod(set(product_prod(E,E)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(E,E)),set(product_prod(A,A))),aa(set(product_prod(E,E)),fun(set(product_prod(A,A)),product_prod(set(product_prod(E,E)),set(product_prod(A,A)))),product_Pair(set(product_prod(E,E)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(E,aa(B,set(E),Aa2,X))),R2),bNF_Wellorder_ordLeq(E,A)) )
       => member(product_prod(set(product_prod(product_prod(B,E),product_prod(B,E))),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,E),product_prod(B,E))),set(product_prod(product_prod(B,A),product_prod(B,A)))),aa(set(product_prod(product_prod(B,E),product_prod(B,E))),fun(set(product_prod(product_prod(B,A),product_prod(B,A))),product_prod(set(product_prod(product_prod(B,E),product_prod(B,E))),set(product_prod(product_prod(B,A),product_prod(B,A))))),product_Pair(set(product_prod(product_prod(B,E),product_prod(B,E))),set(product_prod(product_prod(B,A),product_prod(B,A)))),bNF_Cardinal_Csum(B,E,bNF_Ca6860139660246222851ard_of(B,Ia2),aTP_Lamp_zg(fun(B,set(E)),fun(B,set(product_prod(E,E))),Aa2))),bNF_Cardinal_cprod(B,A,bNF_Ca6860139660246222851ard_of(B,Ia2),R2)),bNF_Wellorder_ordLeq(product_prod(B,E),product_prod(B,A))) ) ) ).

% card_of_Csum_Times'
tff(fact_6656_card__of__Csum__Times,axiom,
    ! [E: $tType,B: $tType,A: $tType,Ia2: set(A),Aa2: fun(A,set(B)),B2: set(E)] :
      ( ! [X: A] :
          ( member(A,X,Ia2)
         => 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(A,set(B),Aa2,X))),bNF_Ca6860139660246222851ard_of(E,B2)),bNF_Wellorder_ordLeq(B,E)) )
     => member(product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(product_prod(A,E),product_prod(A,E)))),aa(set(product_prod(product_prod(A,E),product_prod(A,E))),product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(product_prod(A,E),product_prod(A,E)))),aa(set(product_prod(product_prod(A,B),product_prod(A,B))),fun(set(product_prod(product_prod(A,E),product_prod(A,E))),product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(product_prod(A,E),product_prod(A,E))))),product_Pair(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(product_prod(A,E),product_prod(A,E)))),bNF_Cardinal_Csum(A,B,bNF_Ca6860139660246222851ard_of(A,Ia2),aTP_Lamp_zh(fun(A,set(B)),fun(A,set(product_prod(B,B))),Aa2))),bNF_Cardinal_cprod(A,E,bNF_Ca6860139660246222851ard_of(A,Ia2),bNF_Ca6860139660246222851ard_of(E,B2))),bNF_Wellorder_ordLeq(product_prod(A,B),product_prod(A,E))) ) ).

% card_of_Csum_Times
tff(fact_6657_cprod__infinite1_H,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,A)),P2: set(product_prod(B,B))] :
      ( ( bNF_Ca4139267488887388095finite(A,R2)
        & bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2) )
     => ( ( ~ 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))),P2),bNF_Cardinal_czero(B)),bNF_Wellorder_ordIso(B,B))
          & bNF_Ca8970107618336181345der_on(B,aa(set(product_prod(B,B)),set(B),field2(B),P2),P2) )
       => ( 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))),P2),R2),bNF_Wellorder_ordLeq(B,A))
         => 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,P2)),R2),bNF_Wellorder_ordIso(product_prod(A,B),A)) ) ) ) ).

% cprod_infinite1'
tff(fact_6658_ordLeq__cprod2,axiom,
    ! [A: $tType,B: $tType,P13: set(product_prod(A,A)),P24: set(product_prod(B,B))] :
      ( ( ~ 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))),P13),bNF_Cardinal_czero(A)),bNF_Wellorder_ordIso(A,A))
        & bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),P13),P13) )
     => ( bNF_Ca8970107618336181345der_on(B,aa(set(product_prod(B,B)),set(B),field2(B),P24),P24)
       => 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)))),P24),bNF_Cardinal_cprod(A,B,P13,P24)),bNF_Wellorder_ordLeq(B,product_prod(A,B))) ) ) ).

% ordLeq_cprod2
tff(fact_6659_czero__def,axiom,
    ! [A: $tType] : ( bNF_Cardinal_czero(A) = bNF_Ca6860139660246222851ard_of(A,bot_bot(set(A))) ) ).

% czero_def
tff(fact_6660_czero__ordIso,axiom,
    ! [B: $tType,A: $tType] : 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_6661_cinfinite__not__czero,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,A))] :
      ( bNF_Ca4139267488887388095finite(A,R2)
     => ~ 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_6662_czeroE,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,A))] :
      ( 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))
     => ( aa(set(product_prod(A,A)),set(A),field2(A),R2) = bot_bot(set(A)) ) ) ).

% czeroE
tff(fact_6663_card__of__ordIso__czero__iff__empty,axiom,
    ! [B: $tType,A: $tType,Aa2: set(A)] :
      ( 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,Aa2)),bNF_Cardinal_czero(B)),bNF_Wellorder_ordIso(A,B))
    <=> ( Aa2 = bot_bot(set(A)) ) ) ).

% card_of_ordIso_czero_iff_empty
tff(fact_6664_czeroI,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,A))] :
      ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => ( ( aa(set(product_prod(A,A)),set(A),field2(A),R2) = bot_bot(set(A)) )
       => 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_6665_Cnotzero__imp__not__empty,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( ( ~ 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))
        & bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2) )
     => ( aa(set(product_prod(A,A)),set(A),field2(A),R2) != bot_bot(set(A)) ) ) ).

% Cnotzero_imp_not_empty
tff(fact_6666_Cnotzero__mono,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(A,A)),Q2: set(product_prod(B,B))] :
      ( ( ~ 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))
        & bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2) )
     => ( bNF_Ca8970107618336181345der_on(B,aa(set(product_prod(B,B)),set(B),field2(B),Q2),Q2)
       => ( 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),Q2),bNF_Wellorder_ordLeq(A,B))
         => ( ~ 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))),Q2),bNF_Cardinal_czero(B)),bNF_Wellorder_ordIso(B,B))
            & bNF_Ca8970107618336181345der_on(B,aa(set(product_prod(B,B)),set(B),field2(B),Q2),Q2) ) ) ) ) ).

% Cnotzero_mono
tff(fact_6667_Cinfinite__Cnotzero,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( ( bNF_Ca4139267488887388095finite(A,R2)
        & bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2) )
     => ( ~ 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))
        & bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2) ) ) ).

% Cinfinite_Cnotzero
tff(fact_6668_Cnotzero__UNIV,axiom,
    ! [A: $tType] :
      ( ~ 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))
      & bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),bNF_Ca6860139660246222851ard_of(A,top_top(set(A)))),bNF_Ca6860139660246222851ard_of(A,top_top(set(A)))) ) ).

% Cnotzero_UNIV
tff(fact_6669_cinfinite__cprod2,axiom,
    ! [A: $tType,B: $tType,R12: set(product_prod(A,A)),R23: set(product_prod(B,B))] :
      ( ( ~ 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))
        & bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R12),R12) )
     => ( ( bNF_Ca4139267488887388095finite(B,R23)
          & bNF_Ca8970107618336181345der_on(B,aa(set(product_prod(B,B)),set(B),field2(B),R23),R23) )
       => bNF_Ca4139267488887388095finite(product_prod(A,B),bNF_Cardinal_cprod(A,B,R12,R23)) ) ) ).

% cinfinite_cprod2
tff(fact_6670_Cinfinite__cprod2,axiom,
    ! [A: $tType,B: $tType,R12: set(product_prod(A,A)),R23: set(product_prod(B,B))] :
      ( ( ~ 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))
        & bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R12),R12) )
     => ( ( bNF_Ca4139267488887388095finite(B,R23)
          & bNF_Ca8970107618336181345der_on(B,aa(set(product_prod(B,B)),set(B),field2(B),R23),R23) )
       => ( bNF_Ca4139267488887388095finite(product_prod(A,B),bNF_Cardinal_cprod(A,B,R12,R23))
          & bNF_Ca8970107618336181345der_on(product_prod(A,B),aa(set(product_prod(product_prod(A,B),product_prod(A,B))),set(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_6671_cone__ordLeq__Cnotzero,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( ( ~ 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))
        & bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2) )
     => 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_6672_cexp__mono2__Cnotzero,axiom,
    ! [B: $tType,E: $tType,A: $tType,P24: set(product_prod(A,A)),R23: set(product_prod(B,B)),Q2: set(product_prod(E,E))] :
      ( 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))),P24),R23),bNF_Wellorder_ordLeq(A,B))
     => ( bNF_Ca8970107618336181345der_on(E,aa(set(product_prod(E,E)),set(E),field2(E),Q2),Q2)
       => ( ( ~ 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))),P24),bNF_Cardinal_czero(A)),bNF_Wellorder_ordIso(A,A))
            & bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),P24),P24) )
         => member(product_prod(set(product_prod(fun(A,E),fun(A,E))),set(product_prod(fun(B,E),fun(B,E)))),aa(set(product_prod(fun(B,E),fun(B,E))),product_prod(set(product_prod(fun(A,E),fun(A,E))),set(product_prod(fun(B,E),fun(B,E)))),aa(set(product_prod(fun(A,E),fun(A,E))),fun(set(product_prod(fun(B,E),fun(B,E))),product_prod(set(product_prod(fun(A,E),fun(A,E))),set(product_prod(fun(B,E),fun(B,E))))),product_Pair(set(product_prod(fun(A,E),fun(A,E))),set(product_prod(fun(B,E),fun(B,E)))),bNF_Cardinal_cexp(E,A,Q2,P24)),bNF_Cardinal_cexp(E,B,Q2,R23)),bNF_Wellorder_ordLeq(fun(A,E),fun(B,E))) ) ) ) ).

% cexp_mono2_Cnotzero
tff(fact_6673_cexp__cprod,axiom,
    ! [A: $tType,E: $tType,B: $tType,R12: set(product_prod(A,A)),R23: set(product_prod(E,E)),R32: set(product_prod(B,B))] :
      ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R12),R12)
     => member(product_prod(set(product_prod(fun(B,fun(E,A)),fun(B,fun(E,A)))),set(product_prod(fun(product_prod(E,B),A),fun(product_prod(E,B),A)))),aa(set(product_prod(fun(product_prod(E,B),A),fun(product_prod(E,B),A))),product_prod(set(product_prod(fun(B,fun(E,A)),fun(B,fun(E,A)))),set(product_prod(fun(product_prod(E,B),A),fun(product_prod(E,B),A)))),aa(set(product_prod(fun(B,fun(E,A)),fun(B,fun(E,A)))),fun(set(product_prod(fun(product_prod(E,B),A),fun(product_prod(E,B),A))),product_prod(set(product_prod(fun(B,fun(E,A)),fun(B,fun(E,A)))),set(product_prod(fun(product_prod(E,B),A),fun(product_prod(E,B),A))))),product_Pair(set(product_prod(fun(B,fun(E,A)),fun(B,fun(E,A)))),set(product_prod(fun(product_prod(E,B),A),fun(product_prod(E,B),A)))),bNF_Cardinal_cexp(fun(E,A),B,bNF_Cardinal_cexp(A,E,R12,R23),R32)),bNF_Cardinal_cexp(A,product_prod(E,B),R12,bNF_Cardinal_cprod(E,B,R23,R32))),bNF_Wellorder_ordIso(fun(B,fun(E,A)),fun(product_prod(E,B),A))) ) ).

% cexp_cprod
tff(fact_6674_cexp__cone,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => 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_6675_cone__not__czero,axiom,
    ! [A: $tType] : ~ 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_6676_cprod__cexp,axiom,
    ! [E: $tType,B: $tType,A: $tType,R2: set(product_prod(B,B)),S2: set(product_prod(E,E)),T2: set(product_prod(A,A))] : member(product_prod(set(product_prod(fun(A,product_prod(B,E)),fun(A,product_prod(B,E)))),set(product_prod(product_prod(fun(A,B),fun(A,E)),product_prod(fun(A,B),fun(A,E))))),aa(set(product_prod(product_prod(fun(A,B),fun(A,E)),product_prod(fun(A,B),fun(A,E)))),product_prod(set(product_prod(fun(A,product_prod(B,E)),fun(A,product_prod(B,E)))),set(product_prod(product_prod(fun(A,B),fun(A,E)),product_prod(fun(A,B),fun(A,E))))),aa(set(product_prod(fun(A,product_prod(B,E)),fun(A,product_prod(B,E)))),fun(set(product_prod(product_prod(fun(A,B),fun(A,E)),product_prod(fun(A,B),fun(A,E)))),product_prod(set(product_prod(fun(A,product_prod(B,E)),fun(A,product_prod(B,E)))),set(product_prod(product_prod(fun(A,B),fun(A,E)),product_prod(fun(A,B),fun(A,E)))))),product_Pair(set(product_prod(fun(A,product_prod(B,E)),fun(A,product_prod(B,E)))),set(product_prod(product_prod(fun(A,B),fun(A,E)),product_prod(fun(A,B),fun(A,E))))),bNF_Cardinal_cexp(product_prod(B,E),A,bNF_Cardinal_cprod(B,E,R2,S2),T2)),bNF_Cardinal_cprod(fun(A,B),fun(A,E),bNF_Cardinal_cexp(B,A,R2,T2),bNF_Cardinal_cexp(E,A,S2,T2))),bNF_Wellorder_ordIso(fun(A,product_prod(B,E)),product_prod(fun(A,B),fun(A,E)))) ).

% cprod_cexp
tff(fact_6677_cexp__cprod__ordLeq,axiom,
    ! [A: $tType,B: $tType,E: $tType,R12: set(product_prod(A,A)),R23: set(product_prod(B,B)),R32: set(product_prod(E,E))] :
      ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R12),R12)
     => ( ( bNF_Ca4139267488887388095finite(B,R23)
          & bNF_Ca8970107618336181345der_on(B,aa(set(product_prod(B,B)),set(B),field2(B),R23),R23) )
       => ( ( ~ member(product_prod(set(product_prod(E,E)),set(product_prod(E,E))),aa(set(product_prod(E,E)),product_prod(set(product_prod(E,E)),set(product_prod(E,E))),aa(set(product_prod(E,E)),fun(set(product_prod(E,E)),product_prod(set(product_prod(E,E)),set(product_prod(E,E)))),product_Pair(set(product_prod(E,E)),set(product_prod(E,E))),R32),bNF_Cardinal_czero(E)),bNF_Wellorder_ordIso(E,E))
            & bNF_Ca8970107618336181345der_on(E,aa(set(product_prod(E,E)),set(E),field2(E),R32),R32) )
         => ( member(product_prod(set(product_prod(E,E)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(E,E)),set(product_prod(B,B))),aa(set(product_prod(E,E)),fun(set(product_prod(B,B)),product_prod(set(product_prod(E,E)),set(product_prod(B,B)))),product_Pair(set(product_prod(E,E)),set(product_prod(B,B))),R32),R23),bNF_Wellorder_ordLeq(E,B))
           => member(product_prod(set(product_prod(fun(E,fun(B,A)),fun(E,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(E,fun(B,A)),fun(E,fun(B,A)))),set(product_prod(fun(B,A),fun(B,A)))),aa(set(product_prod(fun(E,fun(B,A)),fun(E,fun(B,A)))),fun(set(product_prod(fun(B,A),fun(B,A))),product_prod(set(product_prod(fun(E,fun(B,A)),fun(E,fun(B,A)))),set(product_prod(fun(B,A),fun(B,A))))),product_Pair(set(product_prod(fun(E,fun(B,A)),fun(E,fun(B,A)))),set(product_prod(fun(B,A),fun(B,A)))),bNF_Cardinal_cexp(fun(B,A),E,bNF_Cardinal_cexp(A,B,R12,R23),R32)),bNF_Cardinal_cexp(A,B,R12,R23)),bNF_Wellorder_ordIso(fun(E,fun(B,A)),fun(B,A))) ) ) ) ) ).

% cexp_cprod_ordLeq
tff(fact_6678_cexp__mono_H,axiom,
    ! [B: $tType,F4: $tType,A: $tType,E: $tType,P13: set(product_prod(A,A)),R12: set(product_prod(B,B)),P24: set(product_prod(E,E)),R23: set(product_prod(F4,F4))] :
      ( 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))),P13),R12),bNF_Wellorder_ordLeq(A,B))
     => ( member(product_prod(set(product_prod(E,E)),set(product_prod(F4,F4))),aa(set(product_prod(F4,F4)),product_prod(set(product_prod(E,E)),set(product_prod(F4,F4))),aa(set(product_prod(E,E)),fun(set(product_prod(F4,F4)),product_prod(set(product_prod(E,E)),set(product_prod(F4,F4)))),product_Pair(set(product_prod(E,E)),set(product_prod(F4,F4))),P24),R23),bNF_Wellorder_ordLeq(E,F4))
       => ( ( ( aa(set(product_prod(E,E)),set(E),field2(E),P24) = bot_bot(set(E)) )
           => ( aa(set(product_prod(F4,F4)),set(F4),field2(F4),R23) = bot_bot(set(F4)) ) )
         => member(product_prod(set(product_prod(fun(E,A),fun(E,A))),set(product_prod(fun(F4,B),fun(F4,B)))),aa(set(product_prod(fun(F4,B),fun(F4,B))),product_prod(set(product_prod(fun(E,A),fun(E,A))),set(product_prod(fun(F4,B),fun(F4,B)))),aa(set(product_prod(fun(E,A),fun(E,A))),fun(set(product_prod(fun(F4,B),fun(F4,B))),product_prod(set(product_prod(fun(E,A),fun(E,A))),set(product_prod(fun(F4,B),fun(F4,B))))),product_Pair(set(product_prod(fun(E,A),fun(E,A))),set(product_prod(fun(F4,B),fun(F4,B)))),bNF_Cardinal_cexp(A,E,P13,P24)),bNF_Cardinal_cexp(B,F4,R12,R23)),bNF_Wellorder_ordLeq(fun(E,A),fun(F4,B))) ) ) ) ).

% cexp_mono'
tff(fact_6679_cexp__mono1,axiom,
    ! [B: $tType,A: $tType,E: $tType,P13: set(product_prod(A,A)),R12: set(product_prod(B,B)),Q2: set(product_prod(E,E))] :
      ( 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))),P13),R12),bNF_Wellorder_ordLeq(A,B))
     => ( bNF_Ca8970107618336181345der_on(E,aa(set(product_prod(E,E)),set(E),field2(E),Q2),Q2)
       => member(product_prod(set(product_prod(fun(E,A),fun(E,A))),set(product_prod(fun(E,B),fun(E,B)))),aa(set(product_prod(fun(E,B),fun(E,B))),product_prod(set(product_prod(fun(E,A),fun(E,A))),set(product_prod(fun(E,B),fun(E,B)))),aa(set(product_prod(fun(E,A),fun(E,A))),fun(set(product_prod(fun(E,B),fun(E,B))),product_prod(set(product_prod(fun(E,A),fun(E,A))),set(product_prod(fun(E,B),fun(E,B))))),product_Pair(set(product_prod(fun(E,A),fun(E,A))),set(product_prod(fun(E,B),fun(E,B)))),bNF_Cardinal_cexp(A,E,P13,Q2)),bNF_Cardinal_cexp(B,E,R12,Q2)),bNF_Wellorder_ordLeq(fun(E,A),fun(E,B))) ) ) ).

% cexp_mono1
tff(fact_6680_cexp__mono2_H,axiom,
    ! [B: $tType,E: $tType,A: $tType,P24: set(product_prod(A,A)),R23: set(product_prod(B,B)),Q2: set(product_prod(E,E))] :
      ( 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))),P24),R23),bNF_Wellorder_ordLeq(A,B))
     => ( bNF_Ca8970107618336181345der_on(E,aa(set(product_prod(E,E)),set(E),field2(E),Q2),Q2)
       => ( ( ( aa(set(product_prod(A,A)),set(A),field2(A),P24) = bot_bot(set(A)) )
           => ( aa(set(product_prod(B,B)),set(B),field2(B),R23) = bot_bot(set(B)) ) )
         => member(product_prod(set(product_prod(fun(A,E),fun(A,E))),set(product_prod(fun(B,E),fun(B,E)))),aa(set(product_prod(fun(B,E),fun(B,E))),product_prod(set(product_prod(fun(A,E),fun(A,E))),set(product_prod(fun(B,E),fun(B,E)))),aa(set(product_prod(fun(A,E),fun(A,E))),fun(set(product_prod(fun(B,E),fun(B,E))),product_prod(set(product_prod(fun(A,E),fun(A,E))),set(product_prod(fun(B,E),fun(B,E))))),product_Pair(set(product_prod(fun(A,E),fun(A,E))),set(product_prod(fun(B,E),fun(B,E)))),bNF_Cardinal_cexp(E,A,Q2,P24)),bNF_Cardinal_cexp(E,B,Q2,R23)),bNF_Wellorder_ordLeq(fun(A,E),fun(B,E))) ) ) ) ).

% cexp_mono2'
tff(fact_6681_ordLeq__cexp1,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(A,A)),Q2: set(product_prod(B,B))] :
      ( ( ~ 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))
        & bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2) )
     => ( bNF_Ca8970107618336181345der_on(B,aa(set(product_prod(B,B)),set(B),field2(B),Q2),Q2)
       => 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)))),Q2),bNF_Cardinal_cexp(B,A,Q2,R2)),bNF_Wellorder_ordLeq(B,fun(A,B))) ) ) ).

% ordLeq_cexp1
tff(fact_6682_cexp__cong2,axiom,
    ! [B: $tType,E: $tType,A: $tType,P24: set(product_prod(A,A)),R23: set(product_prod(B,B)),Q2: set(product_prod(E,E))] :
      ( 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))),P24),R23),bNF_Wellorder_ordIso(A,B))
     => ( bNF_Ca8970107618336181345der_on(E,aa(set(product_prod(E,E)),set(E),field2(E),Q2),Q2)
       => ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),P24),P24)
         => member(product_prod(set(product_prod(fun(A,E),fun(A,E))),set(product_prod(fun(B,E),fun(B,E)))),aa(set(product_prod(fun(B,E),fun(B,E))),product_prod(set(product_prod(fun(A,E),fun(A,E))),set(product_prod(fun(B,E),fun(B,E)))),aa(set(product_prod(fun(A,E),fun(A,E))),fun(set(product_prod(fun(B,E),fun(B,E))),product_prod(set(product_prod(fun(A,E),fun(A,E))),set(product_prod(fun(B,E),fun(B,E))))),product_Pair(set(product_prod(fun(A,E),fun(A,E))),set(product_prod(fun(B,E),fun(B,E)))),bNF_Cardinal_cexp(E,A,Q2,P24)),bNF_Cardinal_cexp(E,B,Q2,R23)),bNF_Wellorder_ordIso(fun(A,E),fun(B,E))) ) ) ) ).

% cexp_cong2
tff(fact_6683_cexp__cong1,axiom,
    ! [B: $tType,A: $tType,E: $tType,P13: set(product_prod(A,A)),R12: set(product_prod(B,B)),Q2: set(product_prod(E,E))] :
      ( 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))),P13),R12),bNF_Wellorder_ordIso(A,B))
     => ( bNF_Ca8970107618336181345der_on(E,aa(set(product_prod(E,E)),set(E),field2(E),Q2),Q2)
       => member(product_prod(set(product_prod(fun(E,A),fun(E,A))),set(product_prod(fun(E,B),fun(E,B)))),aa(set(product_prod(fun(E,B),fun(E,B))),product_prod(set(product_prod(fun(E,A),fun(E,A))),set(product_prod(fun(E,B),fun(E,B)))),aa(set(product_prod(fun(E,A),fun(E,A))),fun(set(product_prod(fun(E,B),fun(E,B))),product_prod(set(product_prod(fun(E,A),fun(E,A))),set(product_prod(fun(E,B),fun(E,B))))),product_Pair(set(product_prod(fun(E,A),fun(E,A))),set(product_prod(fun(E,B),fun(E,B)))),bNF_Cardinal_cexp(A,E,P13,Q2)),bNF_Cardinal_cexp(B,E,R12,Q2)),bNF_Wellorder_ordIso(fun(E,A),fun(E,B))) ) ) ).

% cexp_cong1
tff(fact_6684_cexp__cong,axiom,
    ! [B: $tType,F4: $tType,A: $tType,E: $tType,P13: set(product_prod(A,A)),R12: set(product_prod(B,B)),P24: set(product_prod(E,E)),R23: set(product_prod(F4,F4))] :
      ( 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))),P13),R12),bNF_Wellorder_ordIso(A,B))
     => ( member(product_prod(set(product_prod(E,E)),set(product_prod(F4,F4))),aa(set(product_prod(F4,F4)),product_prod(set(product_prod(E,E)),set(product_prod(F4,F4))),aa(set(product_prod(E,E)),fun(set(product_prod(F4,F4)),product_prod(set(product_prod(E,E)),set(product_prod(F4,F4)))),product_Pair(set(product_prod(E,E)),set(product_prod(F4,F4))),P24),R23),bNF_Wellorder_ordIso(E,F4))
       => ( bNF_Ca8970107618336181345der_on(F4,aa(set(product_prod(F4,F4)),set(F4),field2(F4),R23),R23)
         => ( bNF_Ca8970107618336181345der_on(E,aa(set(product_prod(E,E)),set(E),field2(E),P24),P24)
           => member(product_prod(set(product_prod(fun(E,A),fun(E,A))),set(product_prod(fun(F4,B),fun(F4,B)))),aa(set(product_prod(fun(F4,B),fun(F4,B))),product_prod(set(product_prod(fun(E,A),fun(E,A))),set(product_prod(fun(F4,B),fun(F4,B)))),aa(set(product_prod(fun(E,A),fun(E,A))),fun(set(product_prod(fun(F4,B),fun(F4,B))),product_prod(set(product_prod(fun(E,A),fun(E,A))),set(product_prod(fun(F4,B),fun(F4,B))))),product_Pair(set(product_prod(fun(E,A),fun(E,A))),set(product_prod(fun(F4,B),fun(F4,B)))),bNF_Cardinal_cexp(A,E,P13,P24)),bNF_Cardinal_cexp(B,F4,R12,R23)),bNF_Wellorder_ordIso(fun(E,A),fun(F4,B))) ) ) ) ) ).

% cexp_cong
tff(fact_6685_cexp__mono,axiom,
    ! [G2: $tType,H5: $tType,B: $tType,F4: $tType,A: $tType,E: $tType,P13: set(product_prod(A,A)),R12: set(product_prod(B,B)),P24: set(product_prod(E,E)),R23: set(product_prod(F4,F4))] :
      ( 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))),P13),R12),bNF_Wellorder_ordLeq(A,B))
     => ( member(product_prod(set(product_prod(E,E)),set(product_prod(F4,F4))),aa(set(product_prod(F4,F4)),product_prod(set(product_prod(E,E)),set(product_prod(F4,F4))),aa(set(product_prod(E,E)),fun(set(product_prod(F4,F4)),product_prod(set(product_prod(E,E)),set(product_prod(F4,F4)))),product_Pair(set(product_prod(E,E)),set(product_prod(F4,F4))),P24),R23),bNF_Wellorder_ordLeq(E,F4))
       => ( ( member(product_prod(set(product_prod(E,E)),set(product_prod(G2,G2))),aa(set(product_prod(G2,G2)),product_prod(set(product_prod(E,E)),set(product_prod(G2,G2))),aa(set(product_prod(E,E)),fun(set(product_prod(G2,G2)),product_prod(set(product_prod(E,E)),set(product_prod(G2,G2)))),product_Pair(set(product_prod(E,E)),set(product_prod(G2,G2))),P24),bNF_Cardinal_czero(G2)),bNF_Wellorder_ordIso(E,G2))
           => member(product_prod(set(product_prod(F4,F4)),set(product_prod(H5,H5))),aa(set(product_prod(H5,H5)),product_prod(set(product_prod(F4,F4)),set(product_prod(H5,H5))),aa(set(product_prod(F4,F4)),fun(set(product_prod(H5,H5)),product_prod(set(product_prod(F4,F4)),set(product_prod(H5,H5)))),product_Pair(set(product_prod(F4,F4)),set(product_prod(H5,H5))),R23),bNF_Cardinal_czero(H5)),bNF_Wellorder_ordIso(F4,H5)) )
         => ( bNF_Ca8970107618336181345der_on(E,aa(set(product_prod(E,E)),set(E),field2(E),P24),P24)
           => member(product_prod(set(product_prod(fun(E,A),fun(E,A))),set(product_prod(fun(F4,B),fun(F4,B)))),aa(set(product_prod(fun(F4,B),fun(F4,B))),product_prod(set(product_prod(fun(E,A),fun(E,A))),set(product_prod(fun(F4,B),fun(F4,B)))),aa(set(product_prod(fun(E,A),fun(E,A))),fun(set(product_prod(fun(F4,B),fun(F4,B))),product_prod(set(product_prod(fun(E,A),fun(E,A))),set(product_prod(fun(F4,B),fun(F4,B))))),product_Pair(set(product_prod(fun(E,A),fun(E,A))),set(product_prod(fun(F4,B),fun(F4,B)))),bNF_Cardinal_cexp(A,E,P13,P24)),bNF_Cardinal_cexp(B,F4,R12,R23)),bNF_Wellorder_ordLeq(fun(E,A),fun(F4,B))) ) ) ) ) ).

% cexp_mono
tff(fact_6686_cexp__mono2,axiom,
    ! [F4: $tType,G2: $tType,B: $tType,E: $tType,A: $tType,P24: set(product_prod(A,A)),R23: set(product_prod(B,B)),Q2: set(product_prod(E,E))] :
      ( 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))),P24),R23),bNF_Wellorder_ordLeq(A,B))
     => ( bNF_Ca8970107618336181345der_on(E,aa(set(product_prod(E,E)),set(E),field2(E),Q2),Q2)
       => ( ( member(product_prod(set(product_prod(A,A)),set(product_prod(F4,F4))),aa(set(product_prod(F4,F4)),product_prod(set(product_prod(A,A)),set(product_prod(F4,F4))),aa(set(product_prod(A,A)),fun(set(product_prod(F4,F4)),product_prod(set(product_prod(A,A)),set(product_prod(F4,F4)))),product_Pair(set(product_prod(A,A)),set(product_prod(F4,F4))),P24),bNF_Cardinal_czero(F4)),bNF_Wellorder_ordIso(A,F4))
           => member(product_prod(set(product_prod(B,B)),set(product_prod(G2,G2))),aa(set(product_prod(G2,G2)),product_prod(set(product_prod(B,B)),set(product_prod(G2,G2))),aa(set(product_prod(B,B)),fun(set(product_prod(G2,G2)),product_prod(set(product_prod(B,B)),set(product_prod(G2,G2)))),product_Pair(set(product_prod(B,B)),set(product_prod(G2,G2))),R23),bNF_Cardinal_czero(G2)),bNF_Wellorder_ordIso(B,G2)) )
         => ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),P24),P24)
           => member(product_prod(set(product_prod(fun(A,E),fun(A,E))),set(product_prod(fun(B,E),fun(B,E)))),aa(set(product_prod(fun(B,E),fun(B,E))),product_prod(set(product_prod(fun(A,E),fun(A,E))),set(product_prod(fun(B,E),fun(B,E)))),aa(set(product_prod(fun(A,E),fun(A,E))),fun(set(product_prod(fun(B,E),fun(B,E))),product_prod(set(product_prod(fun(A,E),fun(A,E))),set(product_prod(fun(B,E),fun(B,E))))),product_Pair(set(product_prod(fun(A,E),fun(A,E))),set(product_prod(fun(B,E),fun(B,E)))),bNF_Cardinal_cexp(E,A,Q2,P24)),bNF_Cardinal_cexp(E,B,Q2,R23)),bNF_Wellorder_ordLeq(fun(A,E),fun(B,E))) ) ) ) ) ).

% cexp_mono2
tff(fact_6687_ordLeq__cexp2,axiom,
    ! [A: $tType,B: $tType,Q2: set(product_prod(A,A)),R2: set(product_prod(B,B))] :
      ( member(product_prod(set(product_prod($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,$o)),set(product_prod(A,A))),bNF_Cardinal_ctwo),Q2),bNF_Wellorder_ordLeq($o,A))
     => ( bNF_Ca8970107618336181345der_on(B,aa(set(product_prod(B,B)),set(B),field2(B),R2),R2)
       => 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,Q2,R2)),bNF_Wellorder_ordLeq(B,fun(B,A))) ) ) ).

% ordLeq_cexp2
tff(fact_6688_Cfinite__cexp__Cinfinite,axiom,
    ! [A: $tType,B: $tType,S2: set(product_prod(A,A)),T2: set(product_prod(B,B))] :
      ( ( bNF_Cardinal_cfinite(A,S2)
        & bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),S2),S2) )
     => ( ( bNF_Ca4139267488887388095finite(B,T2)
          & bNF_Ca8970107618336181345der_on(B,aa(set(product_prod(B,B)),set(B),field2(B),T2),T2) )
       => member(product_prod(set(product_prod(fun(B,A),fun(B,A))),set(product_prod(fun(B,$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,$o)))),bNF_Cardinal_cexp(A,B,S2,T2)),bNF_Cardinal_cexp($o,B,bNF_Cardinal_ctwo,T2)),bNF_Wellorder_ordLeq(fun(B,A),fun(B,$o))) ) ) ).

% Cfinite_cexp_Cinfinite
tff(fact_6689_ctwo__Cnotzero,axiom,
    ( ~ 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,$o))
    & bNF_Ca8970107618336181345der_on($o,aa(set(product_prod($o,$o)),set($o),field2($o),bNF_Cardinal_ctwo),bNF_Cardinal_ctwo) ) ).

% ctwo_Cnotzero
tff(fact_6690_ordLess__ctwo__cexp,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => member(product_prod(set(product_prod(A,A)),set(product_prod(fun(A,$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_6691_ctwo__not__czero,axiom,
    ! [A: $tType] : ~ member(product_prod(set(product_prod($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_6692_ctwo__ordLess__Cinfinite,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( ( bNF_Ca4139267488887388095finite(A,R2)
        & bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2) )
     => member(product_prod(set(product_prod($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_6693_ctwo__ordLeq__Cinfinite,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( ( bNF_Ca4139267488887388095finite(A,R2)
        & bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2) )
     => member(product_prod(set(product_prod($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_6694_cinfinite__cexp,axiom,
    ! [A: $tType,B: $tType,Q2: set(product_prod(A,A)),R2: set(product_prod(B,B))] :
      ( member(product_prod(set(product_prod($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,$o)),set(product_prod(A,A))),bNF_Cardinal_ctwo),Q2),bNF_Wellorder_ordLeq($o,A))
     => ( ( bNF_Ca4139267488887388095finite(B,R2)
          & bNF_Ca8970107618336181345der_on(B,aa(set(product_prod(B,B)),set(B),field2(B),R2),R2) )
       => bNF_Ca4139267488887388095finite(fun(B,A),bNF_Cardinal_cexp(A,B,Q2,R2)) ) ) ).

% cinfinite_cexp
tff(fact_6695_Cinfinite__cexp,axiom,
    ! [A: $tType,B: $tType,Q2: set(product_prod(A,A)),R2: set(product_prod(B,B))] :
      ( member(product_prod(set(product_prod($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,$o)),set(product_prod(A,A))),bNF_Cardinal_ctwo),Q2),bNF_Wellorder_ordLeq($o,A))
     => ( ( bNF_Ca4139267488887388095finite(B,R2)
          & bNF_Ca8970107618336181345der_on(B,aa(set(product_prod(B,B)),set(B),field2(B),R2),R2) )
       => ( bNF_Ca4139267488887388095finite(fun(B,A),bNF_Cardinal_cexp(A,B,Q2,R2))
          & bNF_Ca8970107618336181345der_on(fun(B,A),aa(set(product_prod(fun(B,A),fun(B,A))),set(fun(B,A)),field2(fun(B,A)),bNF_Cardinal_cexp(A,B,Q2,R2)),bNF_Cardinal_cexp(A,B,Q2,R2)) ) ) ) ).

% Cinfinite_cexp
tff(fact_6696_card__of__Plus__Times__aux,axiom,
    ! [B: $tType,A: $tType,A1: A,A22: A,Aa2: set(A),B2: set(B)] :
      ( ( ( A1 != A22 )
        & aa(set(A),$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)),insert3(A),A1),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A22),bot_bot(set(A))))),Aa2) )
     => ( 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,Aa2)),bNF_Ca6860139660246222851ard_of(B,B2)),bNF_Wellorder_ordLeq(A,B))
       => 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,Aa2,B2))),bNF_Ca6860139660246222851ard_of(product_prod(A,B),product_Sigma(A,B,Aa2,aTP_Lamp_pv(set(B),fun(A,set(B)),B2)))),bNF_Wellorder_ordLeq(sum_sum(A,B),product_prod(A,B))) ) ) ).

% card_of_Plus_Times_aux
tff(fact_6697_natLeq__ordLeq__cinfinite,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( ( bNF_Ca4139267488887388095finite(A,R2)
        & bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2) )
     => 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_6698_Card__order__Plus2,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,A)),Aa2: set(B)] :
      ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => 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,Aa2,aa(set(product_prod(A,A)),set(A),field2(A),R2)))),bNF_Wellorder_ordLeq(A,sum_sum(B,A))) ) ).

% Card_order_Plus2
tff(fact_6699_Card__order__Plus1,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,A)),B2: set(B)] :
      ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => 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,aa(set(product_prod(A,A)),set(A),field2(A),R2),B2))),bNF_Wellorder_ordLeq(A,sum_sum(A,B))) ) ).

% Card_order_Plus1
tff(fact_6700_card__of__Plus__Times__bool,axiom,
    ! [A: $tType,Aa2: set(A)] : member(product_prod(set(product_prod(sum_sum(A,A),sum_sum(A,A))),set(product_prod(product_prod(A,$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,$o)))),bNF_Ca6860139660246222851ard_of(sum_sum(A,A),sum_Plus(A,A,Aa2,Aa2))),bNF_Ca6860139660246222851ard_of(product_prod(A,$o),product_Sigma(A,$o,Aa2,aTP_Lamp_zi(A,set($o))))),bNF_Wellorder_ordIso(sum_sum(A,A),product_prod(A,$o))) ).

% card_of_Plus_Times_bool
tff(fact_6701_card__of__Times__Plus__distrib,axiom,
    ! [E: $tType,B: $tType,A: $tType,Aa2: set(A),B2: set(B),C3: set(E)] : member(product_prod(set(product_prod(product_prod(A,sum_sum(B,E)),product_prod(A,sum_sum(B,E)))),set(product_prod(sum_sum(product_prod(A,B),product_prod(A,E)),sum_sum(product_prod(A,B),product_prod(A,E))))),aa(set(product_prod(sum_sum(product_prod(A,B),product_prod(A,E)),sum_sum(product_prod(A,B),product_prod(A,E)))),product_prod(set(product_prod(product_prod(A,sum_sum(B,E)),product_prod(A,sum_sum(B,E)))),set(product_prod(sum_sum(product_prod(A,B),product_prod(A,E)),sum_sum(product_prod(A,B),product_prod(A,E))))),aa(set(product_prod(product_prod(A,sum_sum(B,E)),product_prod(A,sum_sum(B,E)))),fun(set(product_prod(sum_sum(product_prod(A,B),product_prod(A,E)),sum_sum(product_prod(A,B),product_prod(A,E)))),product_prod(set(product_prod(product_prod(A,sum_sum(B,E)),product_prod(A,sum_sum(B,E)))),set(product_prod(sum_sum(product_prod(A,B),product_prod(A,E)),sum_sum(product_prod(A,B),product_prod(A,E)))))),product_Pair(set(product_prod(product_prod(A,sum_sum(B,E)),product_prod(A,sum_sum(B,E)))),set(product_prod(sum_sum(product_prod(A,B),product_prod(A,E)),sum_sum(product_prod(A,B),product_prod(A,E))))),bNF_Ca6860139660246222851ard_of(product_prod(A,sum_sum(B,E)),product_Sigma(A,sum_sum(B,E),Aa2,aa(set(E),fun(A,set(sum_sum(B,E))),aTP_Lamp_zj(set(B),fun(set(E),fun(A,set(sum_sum(B,E)))),B2),C3)))),bNF_Ca6860139660246222851ard_of(sum_sum(product_prod(A,B),product_prod(A,E)),sum_Plus(product_prod(A,B),product_prod(A,E),product_Sigma(A,B,Aa2,aTP_Lamp_pv(set(B),fun(A,set(B)),B2)),product_Sigma(A,E,Aa2,aTP_Lamp_qf(set(E),fun(A,set(E)),C3))))),bNF_Wellorder_ordIso(product_prod(A,sum_sum(B,E)),sum_sum(product_prod(A,B),product_prod(A,E)))) ).

% card_of_Times_Plus_distrib
tff(fact_6702_card__of__Plus__commute,axiom,
    ! [B: $tType,A: $tType,Aa2: set(A),B2: set(B)] : 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,Aa2,B2))),bNF_Ca6860139660246222851ard_of(sum_sum(B,A),sum_Plus(B,A,B2,Aa2))),bNF_Wellorder_ordIso(sum_sum(A,B),sum_sum(B,A))) ).

% card_of_Plus_commute
tff(fact_6703_card__of__Plus__assoc,axiom,
    ! [E: $tType,B: $tType,A: $tType,Aa2: set(A),B2: set(B),C3: set(E)] : member(product_prod(set(product_prod(sum_sum(sum_sum(A,B),E),sum_sum(sum_sum(A,B),E))),set(product_prod(sum_sum(A,sum_sum(B,E)),sum_sum(A,sum_sum(B,E))))),aa(set(product_prod(sum_sum(A,sum_sum(B,E)),sum_sum(A,sum_sum(B,E)))),product_prod(set(product_prod(sum_sum(sum_sum(A,B),E),sum_sum(sum_sum(A,B),E))),set(product_prod(sum_sum(A,sum_sum(B,E)),sum_sum(A,sum_sum(B,E))))),aa(set(product_prod(sum_sum(sum_sum(A,B),E),sum_sum(sum_sum(A,B),E))),fun(set(product_prod(sum_sum(A,sum_sum(B,E)),sum_sum(A,sum_sum(B,E)))),product_prod(set(product_prod(sum_sum(sum_sum(A,B),E),sum_sum(sum_sum(A,B),E))),set(product_prod(sum_sum(A,sum_sum(B,E)),sum_sum(A,sum_sum(B,E)))))),product_Pair(set(product_prod(sum_sum(sum_sum(A,B),E),sum_sum(sum_sum(A,B),E))),set(product_prod(sum_sum(A,sum_sum(B,E)),sum_sum(A,sum_sum(B,E))))),bNF_Ca6860139660246222851ard_of(sum_sum(sum_sum(A,B),E),sum_Plus(sum_sum(A,B),E,sum_Plus(A,B,Aa2,B2),C3))),bNF_Ca6860139660246222851ard_of(sum_sum(A,sum_sum(B,E)),sum_Plus(A,sum_sum(B,E),Aa2,sum_Plus(B,E,B2,C3)))),bNF_Wellorder_ordIso(sum_sum(sum_sum(A,B),E),sum_sum(A,sum_sum(B,E)))) ).

% card_of_Plus_assoc
tff(fact_6704_card__of__Un__Plus__ordLeq,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A)] : 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)),Aa2),B2))),bNF_Ca6860139660246222851ard_of(sum_sum(A,A),sum_Plus(A,A,Aa2,B2))),bNF_Wellorder_ordLeq(A,sum_sum(A,A))) ).

% card_of_Un_Plus_ordLeq
tff(fact_6705_card__of__Plus2,axiom,
    ! [B: $tType,A: $tType,B2: set(A),Aa2: set(B)] : 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,B2)),bNF_Ca6860139660246222851ard_of(sum_sum(B,A),sum_Plus(B,A,Aa2,B2))),bNF_Wellorder_ordLeq(A,sum_sum(B,A))) ).

% card_of_Plus2
tff(fact_6706_card__of__Plus1,axiom,
    ! [B: $tType,A: $tType,Aa2: set(A),B2: set(B)] : 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,Aa2)),bNF_Ca6860139660246222851ard_of(sum_sum(A,B),sum_Plus(A,B,Aa2,B2))),bNF_Wellorder_ordLeq(A,sum_sum(A,B))) ).

% card_of_Plus1
tff(fact_6707_ctwo__ordLess__natLeq,axiom,
    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_6708_card__of__Plus__empty1,axiom,
    ! [B: $tType,A: $tType,Aa2: set(A)] : 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,Aa2)),bNF_Ca6860139660246222851ard_of(sum_sum(A,B),sum_Plus(A,B,Aa2,bot_bot(set(B))))),bNF_Wellorder_ordIso(A,sum_sum(A,B))) ).

% card_of_Plus_empty1
tff(fact_6709_card__of__Plus__empty2,axiom,
    ! [B: $tType,A: $tType,Aa2: set(A)] : 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,Aa2)),bNF_Ca6860139660246222851ard_of(sum_sum(B,A),sum_Plus(B,A,bot_bot(set(B)),Aa2))),bNF_Wellorder_ordIso(A,sum_sum(B,A))) ).

% card_of_Plus_empty2
tff(fact_6710_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_ac(nat,fun(nat,$o)),N)) ) ).

% natLeq_underS_less
tff(fact_6711_ordLeq__Plus__mono,axiom,
    ! [F4: $tType,B: $tType,E: $tType,A: $tType,R2: set(product_prod(A,A)),R5: set(product_prod(B,B)),P2: set(product_prod(E,E)),P7: set(product_prod(F4,F4))] :
      ( 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),R5),bNF_Wellorder_ordLeq(A,B))
     => ( member(product_prod(set(product_prod(E,E)),set(product_prod(F4,F4))),aa(set(product_prod(F4,F4)),product_prod(set(product_prod(E,E)),set(product_prod(F4,F4))),aa(set(product_prod(E,E)),fun(set(product_prod(F4,F4)),product_prod(set(product_prod(E,E)),set(product_prod(F4,F4)))),product_Pair(set(product_prod(E,E)),set(product_prod(F4,F4))),P2),P7),bNF_Wellorder_ordLeq(E,F4))
       => member(product_prod(set(product_prod(sum_sum(A,E),sum_sum(A,E))),set(product_prod(sum_sum(B,F4),sum_sum(B,F4)))),aa(set(product_prod(sum_sum(B,F4),sum_sum(B,F4))),product_prod(set(product_prod(sum_sum(A,E),sum_sum(A,E))),set(product_prod(sum_sum(B,F4),sum_sum(B,F4)))),aa(set(product_prod(sum_sum(A,E),sum_sum(A,E))),fun(set(product_prod(sum_sum(B,F4),sum_sum(B,F4))),product_prod(set(product_prod(sum_sum(A,E),sum_sum(A,E))),set(product_prod(sum_sum(B,F4),sum_sum(B,F4))))),product_Pair(set(product_prod(sum_sum(A,E),sum_sum(A,E))),set(product_prod(sum_sum(B,F4),sum_sum(B,F4)))),bNF_Ca6860139660246222851ard_of(sum_sum(A,E),sum_Plus(A,E,aa(set(product_prod(A,A)),set(A),field2(A),R2),aa(set(product_prod(E,E)),set(E),field2(E),P2)))),bNF_Ca6860139660246222851ard_of(sum_sum(B,F4),sum_Plus(B,F4,aa(set(product_prod(B,B)),set(B),field2(B),R5),aa(set(product_prod(F4,F4)),set(F4),field2(F4),P7)))),bNF_Wellorder_ordLeq(sum_sum(A,E),sum_sum(B,F4))) ) ) ).

% ordLeq_Plus_mono
tff(fact_6712_ordLeq__Plus__mono1,axiom,
    ! [B: $tType,E: $tType,A: $tType,R2: set(product_prod(A,A)),R5: set(product_prod(B,B)),C3: set(E)] :
      ( 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),R5),bNF_Wellorder_ordLeq(A,B))
     => member(product_prod(set(product_prod(sum_sum(A,E),sum_sum(A,E))),set(product_prod(sum_sum(B,E),sum_sum(B,E)))),aa(set(product_prod(sum_sum(B,E),sum_sum(B,E))),product_prod(set(product_prod(sum_sum(A,E),sum_sum(A,E))),set(product_prod(sum_sum(B,E),sum_sum(B,E)))),aa(set(product_prod(sum_sum(A,E),sum_sum(A,E))),fun(set(product_prod(sum_sum(B,E),sum_sum(B,E))),product_prod(set(product_prod(sum_sum(A,E),sum_sum(A,E))),set(product_prod(sum_sum(B,E),sum_sum(B,E))))),product_Pair(set(product_prod(sum_sum(A,E),sum_sum(A,E))),set(product_prod(sum_sum(B,E),sum_sum(B,E)))),bNF_Ca6860139660246222851ard_of(sum_sum(A,E),sum_Plus(A,E,aa(set(product_prod(A,A)),set(A),field2(A),R2),C3))),bNF_Ca6860139660246222851ard_of(sum_sum(B,E),sum_Plus(B,E,aa(set(product_prod(B,B)),set(B),field2(B),R5),C3))),bNF_Wellorder_ordLeq(sum_sum(A,E),sum_sum(B,E))) ) ).

% ordLeq_Plus_mono1
tff(fact_6713_ordLeq__Plus__mono2,axiom,
    ! [B: $tType,A: $tType,E: $tType,R2: set(product_prod(A,A)),R5: set(product_prod(B,B)),Aa2: set(E)] :
      ( 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),R5),bNF_Wellorder_ordLeq(A,B))
     => member(product_prod(set(product_prod(sum_sum(E,A),sum_sum(E,A))),set(product_prod(sum_sum(E,B),sum_sum(E,B)))),aa(set(product_prod(sum_sum(E,B),sum_sum(E,B))),product_prod(set(product_prod(sum_sum(E,A),sum_sum(E,A))),set(product_prod(sum_sum(E,B),sum_sum(E,B)))),aa(set(product_prod(sum_sum(E,A),sum_sum(E,A))),fun(set(product_prod(sum_sum(E,B),sum_sum(E,B))),product_prod(set(product_prod(sum_sum(E,A),sum_sum(E,A))),set(product_prod(sum_sum(E,B),sum_sum(E,B))))),product_Pair(set(product_prod(sum_sum(E,A),sum_sum(E,A))),set(product_prod(sum_sum(E,B),sum_sum(E,B)))),bNF_Ca6860139660246222851ard_of(sum_sum(E,A),sum_Plus(E,A,Aa2,aa(set(product_prod(A,A)),set(A),field2(A),R2)))),bNF_Ca6860139660246222851ard_of(sum_sum(E,B),sum_Plus(E,B,Aa2,aa(set(product_prod(B,B)),set(B),field2(B),R5)))),bNF_Wellorder_ordLeq(sum_sum(E,A),sum_sum(E,B))) ) ).

% ordLeq_Plus_mono2
tff(fact_6714_ordIso__Plus__cong,axiom,
    ! [F4: $tType,B: $tType,E: $tType,A: $tType,R2: set(product_prod(A,A)),R5: set(product_prod(B,B)),P2: set(product_prod(E,E)),P7: set(product_prod(F4,F4))] :
      ( 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),R5),bNF_Wellorder_ordIso(A,B))
     => ( member(product_prod(set(product_prod(E,E)),set(product_prod(F4,F4))),aa(set(product_prod(F4,F4)),product_prod(set(product_prod(E,E)),set(product_prod(F4,F4))),aa(set(product_prod(E,E)),fun(set(product_prod(F4,F4)),product_prod(set(product_prod(E,E)),set(product_prod(F4,F4)))),product_Pair(set(product_prod(E,E)),set(product_prod(F4,F4))),P2),P7),bNF_Wellorder_ordIso(E,F4))
       => member(product_prod(set(product_prod(sum_sum(A,E),sum_sum(A,E))),set(product_prod(sum_sum(B,F4),sum_sum(B,F4)))),aa(set(product_prod(sum_sum(B,F4),sum_sum(B,F4))),product_prod(set(product_prod(sum_sum(A,E),sum_sum(A,E))),set(product_prod(sum_sum(B,F4),sum_sum(B,F4)))),aa(set(product_prod(sum_sum(A,E),sum_sum(A,E))),fun(set(product_prod(sum_sum(B,F4),sum_sum(B,F4))),product_prod(set(product_prod(sum_sum(A,E),sum_sum(A,E))),set(product_prod(sum_sum(B,F4),sum_sum(B,F4))))),product_Pair(set(product_prod(sum_sum(A,E),sum_sum(A,E))),set(product_prod(sum_sum(B,F4),sum_sum(B,F4)))),bNF_Ca6860139660246222851ard_of(sum_sum(A,E),sum_Plus(A,E,aa(set(product_prod(A,A)),set(A),field2(A),R2),aa(set(product_prod(E,E)),set(E),field2(E),P2)))),bNF_Ca6860139660246222851ard_of(sum_sum(B,F4),sum_Plus(B,F4,aa(set(product_prod(B,B)),set(B),field2(B),R5),aa(set(product_prod(F4,F4)),set(F4),field2(F4),P7)))),bNF_Wellorder_ordIso(sum_sum(A,E),sum_sum(B,F4))) ) ) ).

% ordIso_Plus_cong
tff(fact_6715_ordIso__Plus__cong1,axiom,
    ! [B: $tType,E: $tType,A: $tType,R2: set(product_prod(A,A)),R5: set(product_prod(B,B)),C3: set(E)] :
      ( 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),R5),bNF_Wellorder_ordIso(A,B))
     => member(product_prod(set(product_prod(sum_sum(A,E),sum_sum(A,E))),set(product_prod(sum_sum(B,E),sum_sum(B,E)))),aa(set(product_prod(sum_sum(B,E),sum_sum(B,E))),product_prod(set(product_prod(sum_sum(A,E),sum_sum(A,E))),set(product_prod(sum_sum(B,E),sum_sum(B,E)))),aa(set(product_prod(sum_sum(A,E),sum_sum(A,E))),fun(set(product_prod(sum_sum(B,E),sum_sum(B,E))),product_prod(set(product_prod(sum_sum(A,E),sum_sum(A,E))),set(product_prod(sum_sum(B,E),sum_sum(B,E))))),product_Pair(set(product_prod(sum_sum(A,E),sum_sum(A,E))),set(product_prod(sum_sum(B,E),sum_sum(B,E)))),bNF_Ca6860139660246222851ard_of(sum_sum(A,E),sum_Plus(A,E,aa(set(product_prod(A,A)),set(A),field2(A),R2),C3))),bNF_Ca6860139660246222851ard_of(sum_sum(B,E),sum_Plus(B,E,aa(set(product_prod(B,B)),set(B),field2(B),R5),C3))),bNF_Wellorder_ordIso(sum_sum(A,E),sum_sum(B,E))) ) ).

% ordIso_Plus_cong1
tff(fact_6716_ordIso__Plus__cong2,axiom,
    ! [B: $tType,A: $tType,E: $tType,R2: set(product_prod(A,A)),R5: set(product_prod(B,B)),Aa2: set(E)] :
      ( 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),R5),bNF_Wellorder_ordIso(A,B))
     => member(product_prod(set(product_prod(sum_sum(E,A),sum_sum(E,A))),set(product_prod(sum_sum(E,B),sum_sum(E,B)))),aa(set(product_prod(sum_sum(E,B),sum_sum(E,B))),product_prod(set(product_prod(sum_sum(E,A),sum_sum(E,A))),set(product_prod(sum_sum(E,B),sum_sum(E,B)))),aa(set(product_prod(sum_sum(E,A),sum_sum(E,A))),fun(set(product_prod(sum_sum(E,B),sum_sum(E,B))),product_prod(set(product_prod(sum_sum(E,A),sum_sum(E,A))),set(product_prod(sum_sum(E,B),sum_sum(E,B))))),product_Pair(set(product_prod(sum_sum(E,A),sum_sum(E,A))),set(product_prod(sum_sum(E,B),sum_sum(E,B)))),bNF_Ca6860139660246222851ard_of(sum_sum(E,A),sum_Plus(E,A,Aa2,aa(set(product_prod(A,A)),set(A),field2(A),R2)))),bNF_Ca6860139660246222851ard_of(sum_sum(E,B),sum_Plus(E,B,Aa2,aa(set(product_prod(B,B)),set(B),field2(B),R5)))),bNF_Wellorder_ordIso(sum_sum(E,A),sum_sum(E,B))) ) ).

% ordIso_Plus_cong2
tff(fact_6717_card__of__Plus__mono,axiom,
    ! [F4: $tType,B: $tType,E: $tType,A: $tType,Aa2: set(A),B2: set(B),C3: set(E),D2: set(F4)] :
      ( 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,Aa2)),bNF_Ca6860139660246222851ard_of(B,B2)),bNF_Wellorder_ordLeq(A,B))
     => ( member(product_prod(set(product_prod(E,E)),set(product_prod(F4,F4))),aa(set(product_prod(F4,F4)),product_prod(set(product_prod(E,E)),set(product_prod(F4,F4))),aa(set(product_prod(E,E)),fun(set(product_prod(F4,F4)),product_prod(set(product_prod(E,E)),set(product_prod(F4,F4)))),product_Pair(set(product_prod(E,E)),set(product_prod(F4,F4))),bNF_Ca6860139660246222851ard_of(E,C3)),bNF_Ca6860139660246222851ard_of(F4,D2)),bNF_Wellorder_ordLeq(E,F4))
       => member(product_prod(set(product_prod(sum_sum(A,E),sum_sum(A,E))),set(product_prod(sum_sum(B,F4),sum_sum(B,F4)))),aa(set(product_prod(sum_sum(B,F4),sum_sum(B,F4))),product_prod(set(product_prod(sum_sum(A,E),sum_sum(A,E))),set(product_prod(sum_sum(B,F4),sum_sum(B,F4)))),aa(set(product_prod(sum_sum(A,E),sum_sum(A,E))),fun(set(product_prod(sum_sum(B,F4),sum_sum(B,F4))),product_prod(set(product_prod(sum_sum(A,E),sum_sum(A,E))),set(product_prod(sum_sum(B,F4),sum_sum(B,F4))))),product_Pair(set(product_prod(sum_sum(A,E),sum_sum(A,E))),set(product_prod(sum_sum(B,F4),sum_sum(B,F4)))),bNF_Ca6860139660246222851ard_of(sum_sum(A,E),sum_Plus(A,E,Aa2,C3))),bNF_Ca6860139660246222851ard_of(sum_sum(B,F4),sum_Plus(B,F4,B2,D2))),bNF_Wellorder_ordLeq(sum_sum(A,E),sum_sum(B,F4))) ) ) ).

% card_of_Plus_mono
tff(fact_6718_card__of__Plus__mono1,axiom,
    ! [B: $tType,E: $tType,A: $tType,Aa2: set(A),B2: set(B),C3: set(E)] :
      ( 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,Aa2)),bNF_Ca6860139660246222851ard_of(B,B2)),bNF_Wellorder_ordLeq(A,B))
     => member(product_prod(set(product_prod(sum_sum(A,E),sum_sum(A,E))),set(product_prod(sum_sum(B,E),sum_sum(B,E)))),aa(set(product_prod(sum_sum(B,E),sum_sum(B,E))),product_prod(set(product_prod(sum_sum(A,E),sum_sum(A,E))),set(product_prod(sum_sum(B,E),sum_sum(B,E)))),aa(set(product_prod(sum_sum(A,E),sum_sum(A,E))),fun(set(product_prod(sum_sum(B,E),sum_sum(B,E))),product_prod(set(product_prod(sum_sum(A,E),sum_sum(A,E))),set(product_prod(sum_sum(B,E),sum_sum(B,E))))),product_Pair(set(product_prod(sum_sum(A,E),sum_sum(A,E))),set(product_prod(sum_sum(B,E),sum_sum(B,E)))),bNF_Ca6860139660246222851ard_of(sum_sum(A,E),sum_Plus(A,E,Aa2,C3))),bNF_Ca6860139660246222851ard_of(sum_sum(B,E),sum_Plus(B,E,B2,C3))),bNF_Wellorder_ordLeq(sum_sum(A,E),sum_sum(B,E))) ) ).

% card_of_Plus_mono1
tff(fact_6719_card__of__Plus__mono2,axiom,
    ! [B: $tType,A: $tType,E: $tType,Aa2: set(A),B2: set(B),C3: set(E)] :
      ( 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,Aa2)),bNF_Ca6860139660246222851ard_of(B,B2)),bNF_Wellorder_ordLeq(A,B))
     => member(product_prod(set(product_prod(sum_sum(E,A),sum_sum(E,A))),set(product_prod(sum_sum(E,B),sum_sum(E,B)))),aa(set(product_prod(sum_sum(E,B),sum_sum(E,B))),product_prod(set(product_prod(sum_sum(E,A),sum_sum(E,A))),set(product_prod(sum_sum(E,B),sum_sum(E,B)))),aa(set(product_prod(sum_sum(E,A),sum_sum(E,A))),fun(set(product_prod(sum_sum(E,B),sum_sum(E,B))),product_prod(set(product_prod(sum_sum(E,A),sum_sum(E,A))),set(product_prod(sum_sum(E,B),sum_sum(E,B))))),product_Pair(set(product_prod(sum_sum(E,A),sum_sum(E,A))),set(product_prod(sum_sum(E,B),sum_sum(E,B)))),bNF_Ca6860139660246222851ard_of(sum_sum(E,A),sum_Plus(E,A,C3,Aa2))),bNF_Ca6860139660246222851ard_of(sum_sum(E,B),sum_Plus(E,B,C3,B2))),bNF_Wellorder_ordLeq(sum_sum(E,A),sum_sum(E,B))) ) ).

% card_of_Plus_mono2
tff(fact_6720_card__of__Plus__cong,axiom,
    ! [F4: $tType,B: $tType,E: $tType,A: $tType,Aa2: set(A),B2: set(B),C3: set(E),D2: set(F4)] :
      ( 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,Aa2)),bNF_Ca6860139660246222851ard_of(B,B2)),bNF_Wellorder_ordIso(A,B))
     => ( member(product_prod(set(product_prod(E,E)),set(product_prod(F4,F4))),aa(set(product_prod(F4,F4)),product_prod(set(product_prod(E,E)),set(product_prod(F4,F4))),aa(set(product_prod(E,E)),fun(set(product_prod(F4,F4)),product_prod(set(product_prod(E,E)),set(product_prod(F4,F4)))),product_Pair(set(product_prod(E,E)),set(product_prod(F4,F4))),bNF_Ca6860139660246222851ard_of(E,C3)),bNF_Ca6860139660246222851ard_of(F4,D2)),bNF_Wellorder_ordIso(E,F4))
       => member(product_prod(set(product_prod(sum_sum(A,E),sum_sum(A,E))),set(product_prod(sum_sum(B,F4),sum_sum(B,F4)))),aa(set(product_prod(sum_sum(B,F4),sum_sum(B,F4))),product_prod(set(product_prod(sum_sum(A,E),sum_sum(A,E))),set(product_prod(sum_sum(B,F4),sum_sum(B,F4)))),aa(set(product_prod(sum_sum(A,E),sum_sum(A,E))),fun(set(product_prod(sum_sum(B,F4),sum_sum(B,F4))),product_prod(set(product_prod(sum_sum(A,E),sum_sum(A,E))),set(product_prod(sum_sum(B,F4),sum_sum(B,F4))))),product_Pair(set(product_prod(sum_sum(A,E),sum_sum(A,E))),set(product_prod(sum_sum(B,F4),sum_sum(B,F4)))),bNF_Ca6860139660246222851ard_of(sum_sum(A,E),sum_Plus(A,E,Aa2,C3))),bNF_Ca6860139660246222851ard_of(sum_sum(B,F4),sum_Plus(B,F4,B2,D2))),bNF_Wellorder_ordIso(sum_sum(A,E),sum_sum(B,F4))) ) ) ).

% card_of_Plus_cong
tff(fact_6721_card__of__Plus__cong1,axiom,
    ! [B: $tType,E: $tType,A: $tType,Aa2: set(A),B2: set(B),C3: set(E)] :
      ( 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,Aa2)),bNF_Ca6860139660246222851ard_of(B,B2)),bNF_Wellorder_ordIso(A,B))
     => member(product_prod(set(product_prod(sum_sum(A,E),sum_sum(A,E))),set(product_prod(sum_sum(B,E),sum_sum(B,E)))),aa(set(product_prod(sum_sum(B,E),sum_sum(B,E))),product_prod(set(product_prod(sum_sum(A,E),sum_sum(A,E))),set(product_prod(sum_sum(B,E),sum_sum(B,E)))),aa(set(product_prod(sum_sum(A,E),sum_sum(A,E))),fun(set(product_prod(sum_sum(B,E),sum_sum(B,E))),product_prod(set(product_prod(sum_sum(A,E),sum_sum(A,E))),set(product_prod(sum_sum(B,E),sum_sum(B,E))))),product_Pair(set(product_prod(sum_sum(A,E),sum_sum(A,E))),set(product_prod(sum_sum(B,E),sum_sum(B,E)))),bNF_Ca6860139660246222851ard_of(sum_sum(A,E),sum_Plus(A,E,Aa2,C3))),bNF_Ca6860139660246222851ard_of(sum_sum(B,E),sum_Plus(B,E,B2,C3))),bNF_Wellorder_ordIso(sum_sum(A,E),sum_sum(B,E))) ) ).

% card_of_Plus_cong1
tff(fact_6722_card__of__Plus__cong2,axiom,
    ! [B: $tType,A: $tType,E: $tType,Aa2: set(A),B2: set(B),C3: set(E)] :
      ( 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,Aa2)),bNF_Ca6860139660246222851ard_of(B,B2)),bNF_Wellorder_ordIso(A,B))
     => member(product_prod(set(product_prod(sum_sum(E,A),sum_sum(E,A))),set(product_prod(sum_sum(E,B),sum_sum(E,B)))),aa(set(product_prod(sum_sum(E,B),sum_sum(E,B))),product_prod(set(product_prod(sum_sum(E,A),sum_sum(E,A))),set(product_prod(sum_sum(E,B),sum_sum(E,B)))),aa(set(product_prod(sum_sum(E,A),sum_sum(E,A))),fun(set(product_prod(sum_sum(E,B),sum_sum(E,B))),product_prod(set(product_prod(sum_sum(E,A),sum_sum(E,A))),set(product_prod(sum_sum(E,B),sum_sum(E,B))))),product_Pair(set(product_prod(sum_sum(E,A),sum_sum(E,A))),set(product_prod(sum_sum(E,B),sum_sum(E,B)))),bNF_Ca6860139660246222851ard_of(sum_sum(E,A),sum_Plus(E,A,C3,Aa2))),bNF_Ca6860139660246222851ard_of(sum_sum(E,B),sum_Plus(E,B,C3,B2))),bNF_Wellorder_ordIso(sum_sum(E,A),sum_sum(E,B))) ) ).

% card_of_Plus_cong2
tff(fact_6723_card__of__Plus__infinite2,axiom,
    ! [A: $tType,B: $tType,Aa2: set(A),B2: set(B)] :
      ( ~ aa(set(A),$o,finite_finite2(A),Aa2)
     => ( 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,B2)),bNF_Ca6860139660246222851ard_of(A,Aa2)),bNF_Wellorder_ordLeq(B,A))
       => 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,B2,Aa2))),bNF_Ca6860139660246222851ard_of(A,Aa2)),bNF_Wellorder_ordIso(sum_sum(B,A),A)) ) ) ).

% card_of_Plus_infinite2
tff(fact_6724_card__of__Plus__infinite1,axiom,
    ! [B: $tType,A: $tType,Aa2: set(A),B2: set(B)] :
      ( ~ aa(set(A),$o,finite_finite2(A),Aa2)
     => ( 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,B2)),bNF_Ca6860139660246222851ard_of(A,Aa2)),bNF_Wellorder_ordLeq(B,A))
       => 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,Aa2,B2))),bNF_Ca6860139660246222851ard_of(A,Aa2)),bNF_Wellorder_ordIso(sum_sum(A,B),A)) ) ) ).

% card_of_Plus_infinite1
tff(fact_6725_card__of__Plus__infinite,axiom,
    ! [A: $tType,B: $tType,Aa2: set(A),B2: set(B)] :
      ( ~ aa(set(A),$o,finite_finite2(A),Aa2)
     => ( 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,B2)),bNF_Ca6860139660246222851ard_of(A,Aa2)),bNF_Wellorder_ordLeq(B,A))
       => ( 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,Aa2,B2))),bNF_Ca6860139660246222851ard_of(A,Aa2)),bNF_Wellorder_ordIso(sum_sum(A,B),A))
          & 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,B2,Aa2))),bNF_Ca6860139660246222851ard_of(A,Aa2)),bNF_Wellorder_ordIso(sum_sum(B,A),A)) ) ) ) ).

% card_of_Plus_infinite
tff(fact_6726_card__of__Plus__ordLess__infinite,axiom,
    ! [A: $tType,E: $tType,B: $tType,C3: set(A),Aa2: set(B),B2: set(E)] :
      ( ~ aa(set(A),$o,finite_finite2(A),C3)
     => ( 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,Aa2)),bNF_Ca6860139660246222851ard_of(A,C3)),bNF_We4044943003108391690rdLess(B,A))
       => ( member(product_prod(set(product_prod(E,E)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(E,E)),set(product_prod(A,A))),aa(set(product_prod(E,E)),fun(set(product_prod(A,A)),product_prod(set(product_prod(E,E)),set(product_prod(A,A)))),product_Pair(set(product_prod(E,E)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(E,B2)),bNF_Ca6860139660246222851ard_of(A,C3)),bNF_We4044943003108391690rdLess(E,A))
         => member(product_prod(set(product_prod(sum_sum(B,E),sum_sum(B,E))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(B,E),sum_sum(B,E))),set(product_prod(A,A))),aa(set(product_prod(sum_sum(B,E),sum_sum(B,E))),fun(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(B,E),sum_sum(B,E))),set(product_prod(A,A)))),product_Pair(set(product_prod(sum_sum(B,E),sum_sum(B,E))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(sum_sum(B,E),sum_Plus(B,E,Aa2,B2))),bNF_Ca6860139660246222851ard_of(A,C3)),bNF_We4044943003108391690rdLess(sum_sum(B,E),A)) ) ) ) ).

% card_of_Plus_ordLess_infinite
tff(fact_6727_finite__iff__ordLess__natLeq,axiom,
    ! [A: $tType,Aa2: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),Aa2)
    <=> 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,Aa2)),bNF_Ca8665028551170535155natLeq),bNF_We4044943003108391690rdLess(A,nat)) ) ).

% finite_iff_ordLess_natLeq
tff(fact_6728_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_zk(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_ix(nat,fun(nat,fun(nat,$o)),N))) ) ).

% Restr_natLeq2
tff(fact_6729_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_ac(nat,fun(nat,$o)),N)),aTP_Lamp_zl(nat,fun(nat,set(nat)),N))) = aa(fun(product_prod(nat,nat),$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,$o),aTP_Lamp_ix(nat,fun(nat,fun(nat,$o)),N))) ) ).

% Restr_natLeq
tff(fact_6730_Card__order__Plus__infinite,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(A,A)),P2: set(product_prod(B,B))] :
      ( ~ aa(set(A),$o,finite_finite2(A),aa(set(product_prod(A,A)),set(A),field2(A),R2))
     => ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
       => ( 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))),P2),R2),bNF_Wellorder_ordLeq(B,A))
         => ( 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,aa(set(product_prod(A,A)),set(A),field2(A),R2),aa(set(product_prod(B,B)),set(B),field2(B),P2)))),R2),bNF_Wellorder_ordIso(sum_sum(A,B),A))
            & 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,aa(set(product_prod(B,B)),set(B),field2(B),P2),aa(set(product_prod(A,A)),set(A),field2(A),R2)))),R2),bNF_Wellorder_ordIso(sum_sum(B,A),A)) ) ) ) ) ).

% Card_order_Plus_infinite
tff(fact_6731_card__of__nat,axiom,
    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_6732_card__of__Plus__Times,axiom,
    ! [B: $tType,A: $tType,A1: A,A22: A,Aa2: set(A),B1: B,B22: B,B2: set(B)] :
      ( ( ( A1 != A22 )
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A1),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A22),bot_bot(set(A))))),Aa2) )
     => ( ( ( B1 != B22 )
          & aa(set(B),$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)),insert3(B),B1),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert3(B),B22),bot_bot(set(B))))),B2) )
       => 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,Aa2,B2))),bNF_Ca6860139660246222851ard_of(product_prod(A,B),product_Sigma(A,B,Aa2,aTP_Lamp_pv(set(B),fun(A,set(B)),B2)))),bNF_Wellorder_ordLeq(sum_sum(A,B),product_prod(A,B))) ) ) ).

% card_of_Plus_Times
tff(fact_6733_card__of__Field__natLeq,axiom,
    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,aa(set(product_prod(nat,nat)),set(nat),field2(nat),bNF_Ca8665028551170535155natLeq))),bNF_Ca8665028551170535155natLeq),bNF_Wellorder_ordIso(nat,nat)) ).

% card_of_Field_natLeq
tff(fact_6734_card__of__Plus__ordLeq__infinite__Field,axiom,
    ! [A: $tType,E: $tType,B: $tType,R2: set(product_prod(A,A)),Aa2: set(B),B2: set(E)] :
      ( ~ aa(set(A),$o,finite_finite2(A),aa(set(product_prod(A,A)),set(A),field2(A),R2))
     => ( 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,Aa2)),R2),bNF_Wellorder_ordLeq(B,A))
       => ( member(product_prod(set(product_prod(E,E)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(E,E)),set(product_prod(A,A))),aa(set(product_prod(E,E)),fun(set(product_prod(A,A)),product_prod(set(product_prod(E,E)),set(product_prod(A,A)))),product_Pair(set(product_prod(E,E)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(E,B2)),R2),bNF_Wellorder_ordLeq(E,A))
         => ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
           => member(product_prod(set(product_prod(sum_sum(B,E),sum_sum(B,E))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(B,E),sum_sum(B,E))),set(product_prod(A,A))),aa(set(product_prod(sum_sum(B,E),sum_sum(B,E))),fun(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(B,E),sum_sum(B,E))),set(product_prod(A,A)))),product_Pair(set(product_prod(sum_sum(B,E),sum_sum(B,E))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(sum_sum(B,E),sum_Plus(B,E,Aa2,B2))),R2),bNF_Wellorder_ordLeq(sum_sum(B,E),A)) ) ) ) ) ).

% card_of_Plus_ordLeq_infinite_Field
tff(fact_6735_card__of__Plus__ordLess__infinite__Field,axiom,
    ! [A: $tType,E: $tType,B: $tType,R2: set(product_prod(A,A)),Aa2: set(B),B2: set(E)] :
      ( ~ aa(set(A),$o,finite_finite2(A),aa(set(product_prod(A,A)),set(A),field2(A),R2))
     => ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
       => ( 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,Aa2)),R2),bNF_We4044943003108391690rdLess(B,A))
         => ( member(product_prod(set(product_prod(E,E)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(E,E)),set(product_prod(A,A))),aa(set(product_prod(E,E)),fun(set(product_prod(A,A)),product_prod(set(product_prod(E,E)),set(product_prod(A,A)))),product_Pair(set(product_prod(E,E)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(E,B2)),R2),bNF_We4044943003108391690rdLess(E,A))
           => member(product_prod(set(product_prod(sum_sum(B,E),sum_sum(B,E))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(B,E),sum_sum(B,E))),set(product_prod(A,A))),aa(set(product_prod(sum_sum(B,E),sum_sum(B,E))),fun(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(B,E),sum_sum(B,E))),set(product_prod(A,A)))),product_Pair(set(product_prod(sum_sum(B,E),sum_sum(B,E))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(sum_sum(B,E),sum_Plus(B,E,Aa2,B2))),R2),bNF_We4044943003108391690rdLess(sum_sum(B,E),A)) ) ) ) ) ).

% card_of_Plus_ordLess_infinite_Field
tff(fact_6736_infinite__iff__natLeq__ordLeq,axiom,
    ! [A: $tType,Aa2: set(A)] :
      ~ ( aa(set(A),$o,finite_finite2(A),Aa2)
      <=> 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,Aa2)),bNF_Wellorder_ordLeq(nat,A)) ) ).

% infinite_iff_natLeq_ordLeq
tff(fact_6737_Plus__eq__empty__conv,axiom,
    ! [A: $tType,B: $tType,Aa2: set(A),B2: set(B)] :
      ( ( sum_Plus(A,B,Aa2,B2) = bot_bot(set(sum_sum(A,B))) )
    <=> ( ( Aa2 = bot_bot(set(A)) )
        & ( B2 = bot_bot(set(B)) ) ) ) ).

% Plus_eq_empty_conv
tff(fact_6738_csum__cinfinite__bound,axiom,
    ! [B: $tType,E: $tType,A: $tType,P2: set(product_prod(A,A)),R2: set(product_prod(B,B)),Q2: set(product_prod(E,E))] :
      ( 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))),P2),R2),bNF_Wellorder_ordLeq(A,B))
     => ( member(product_prod(set(product_prod(E,E)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(E,E)),set(product_prod(B,B))),aa(set(product_prod(E,E)),fun(set(product_prod(B,B)),product_prod(set(product_prod(E,E)),set(product_prod(B,B)))),product_Pair(set(product_prod(E,E)),set(product_prod(B,B))),Q2),R2),bNF_Wellorder_ordLeq(E,B))
       => ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),P2),P2)
         => ( bNF_Ca8970107618336181345der_on(E,aa(set(product_prod(E,E)),set(E),field2(E),Q2),Q2)
           => ( ( bNF_Ca4139267488887388095finite(B,R2)
                & bNF_Ca8970107618336181345der_on(B,aa(set(product_prod(B,B)),set(B),field2(B),R2),R2) )
             => member(product_prod(set(product_prod(sum_sum(A,E),sum_sum(A,E))),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(sum_sum(A,E),sum_sum(A,E))),set(product_prod(B,B))),aa(set(product_prod(sum_sum(A,E),sum_sum(A,E))),fun(set(product_prod(B,B)),product_prod(set(product_prod(sum_sum(A,E),sum_sum(A,E))),set(product_prod(B,B)))),product_Pair(set(product_prod(sum_sum(A,E),sum_sum(A,E))),set(product_prod(B,B))),bNF_Cardinal_csum(A,E,P2,Q2)),R2),bNF_Wellorder_ordLeq(sum_sum(A,E),B)) ) ) ) ) ) ).

% csum_cinfinite_bound
tff(fact_6739_cprod__csum__distrib1,axiom,
    ! [E: $tType,B: $tType,A: $tType,R12: set(product_prod(A,A)),R23: set(product_prod(B,B)),R32: set(product_prod(E,E))] : member(product_prod(set(product_prod(sum_sum(product_prod(A,B),product_prod(A,E)),sum_sum(product_prod(A,B),product_prod(A,E)))),set(product_prod(product_prod(A,sum_sum(B,E)),product_prod(A,sum_sum(B,E))))),aa(set(product_prod(product_prod(A,sum_sum(B,E)),product_prod(A,sum_sum(B,E)))),product_prod(set(product_prod(sum_sum(product_prod(A,B),product_prod(A,E)),sum_sum(product_prod(A,B),product_prod(A,E)))),set(product_prod(product_prod(A,sum_sum(B,E)),product_prod(A,sum_sum(B,E))))),aa(set(product_prod(sum_sum(product_prod(A,B),product_prod(A,E)),sum_sum(product_prod(A,B),product_prod(A,E)))),fun(set(product_prod(product_prod(A,sum_sum(B,E)),product_prod(A,sum_sum(B,E)))),product_prod(set(product_prod(sum_sum(product_prod(A,B),product_prod(A,E)),sum_sum(product_prod(A,B),product_prod(A,E)))),set(product_prod(product_prod(A,sum_sum(B,E)),product_prod(A,sum_sum(B,E)))))),product_Pair(set(product_prod(sum_sum(product_prod(A,B),product_prod(A,E)),sum_sum(product_prod(A,B),product_prod(A,E)))),set(product_prod(product_prod(A,sum_sum(B,E)),product_prod(A,sum_sum(B,E))))),bNF_Cardinal_csum(product_prod(A,B),product_prod(A,E),bNF_Cardinal_cprod(A,B,R12,R23),bNF_Cardinal_cprod(A,E,R12,R32))),bNF_Cardinal_cprod(A,sum_sum(B,E),R12,bNF_Cardinal_csum(B,E,R23,R32))),bNF_Wellorder_ordIso(sum_sum(product_prod(A,B),product_prod(A,E)),product_prod(A,sum_sum(B,E)))) ).

% cprod_csum_distrib1
tff(fact_6740_ordLeq__csum1,axiom,
    ! [B: $tType,A: $tType,P13: set(product_prod(A,A)),P24: set(product_prod(B,B))] :
      ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),P13),P13)
     => 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)))),P13),bNF_Cardinal_csum(A,B,P13,P24)),bNF_Wellorder_ordLeq(A,sum_sum(A,B))) ) ).

% ordLeq_csum1
tff(fact_6741_ordLeq__csum2,axiom,
    ! [B: $tType,A: $tType,P24: set(product_prod(A,A)),P13: set(product_prod(B,B))] :
      ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),P24),P24)
     => 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)))),P24),bNF_Cardinal_csum(B,A,P13,P24)),bNF_Wellorder_ordLeq(A,sum_sum(B,A))) ) ).

% ordLeq_csum2
tff(fact_6742_csum__com,axiom,
    ! [B: $tType,A: $tType,P13: set(product_prod(A,A)),P24: set(product_prod(B,B))] : 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,P13,P24)),bNF_Cardinal_csum(B,A,P24,P13)),bNF_Wellorder_ordIso(sum_sum(A,B),sum_sum(B,A))) ).

% csum_com
tff(fact_6743_csum__cong,axiom,
    ! [F4: $tType,B: $tType,E: $tType,A: $tType,P13: set(product_prod(A,A)),R12: set(product_prod(B,B)),P24: set(product_prod(E,E)),R23: set(product_prod(F4,F4))] :
      ( 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))),P13),R12),bNF_Wellorder_ordIso(A,B))
     => ( member(product_prod(set(product_prod(E,E)),set(product_prod(F4,F4))),aa(set(product_prod(F4,F4)),product_prod(set(product_prod(E,E)),set(product_prod(F4,F4))),aa(set(product_prod(E,E)),fun(set(product_prod(F4,F4)),product_prod(set(product_prod(E,E)),set(product_prod(F4,F4)))),product_Pair(set(product_prod(E,E)),set(product_prod(F4,F4))),P24),R23),bNF_Wellorder_ordIso(E,F4))
       => member(product_prod(set(product_prod(sum_sum(A,E),sum_sum(A,E))),set(product_prod(sum_sum(B,F4),sum_sum(B,F4)))),aa(set(product_prod(sum_sum(B,F4),sum_sum(B,F4))),product_prod(set(product_prod(sum_sum(A,E),sum_sum(A,E))),set(product_prod(sum_sum(B,F4),sum_sum(B,F4)))),aa(set(product_prod(sum_sum(A,E),sum_sum(A,E))),fun(set(product_prod(sum_sum(B,F4),sum_sum(B,F4))),product_prod(set(product_prod(sum_sum(A,E),sum_sum(A,E))),set(product_prod(sum_sum(B,F4),sum_sum(B,F4))))),product_Pair(set(product_prod(sum_sum(A,E),sum_sum(A,E))),set(product_prod(sum_sum(B,F4),sum_sum(B,F4)))),bNF_Cardinal_csum(A,E,P13,P24)),bNF_Cardinal_csum(B,F4,R12,R23)),bNF_Wellorder_ordIso(sum_sum(A,E),sum_sum(B,F4))) ) ) ).

% csum_cong
tff(fact_6744_csum__csum,axiom,
    ! [F4: $tType,E: $tType,B: $tType,A: $tType,R12: set(product_prod(A,A)),R23: set(product_prod(B,B)),R32: set(product_prod(E,E)),R42: set(product_prod(F4,F4))] : member(product_prod(set(product_prod(sum_sum(sum_sum(A,B),sum_sum(E,F4)),sum_sum(sum_sum(A,B),sum_sum(E,F4)))),set(product_prod(sum_sum(sum_sum(A,E),sum_sum(B,F4)),sum_sum(sum_sum(A,E),sum_sum(B,F4))))),aa(set(product_prod(sum_sum(sum_sum(A,E),sum_sum(B,F4)),sum_sum(sum_sum(A,E),sum_sum(B,F4)))),product_prod(set(product_prod(sum_sum(sum_sum(A,B),sum_sum(E,F4)),sum_sum(sum_sum(A,B),sum_sum(E,F4)))),set(product_prod(sum_sum(sum_sum(A,E),sum_sum(B,F4)),sum_sum(sum_sum(A,E),sum_sum(B,F4))))),aa(set(product_prod(sum_sum(sum_sum(A,B),sum_sum(E,F4)),sum_sum(sum_sum(A,B),sum_sum(E,F4)))),fun(set(product_prod(sum_sum(sum_sum(A,E),sum_sum(B,F4)),sum_sum(sum_sum(A,E),sum_sum(B,F4)))),product_prod(set(product_prod(sum_sum(sum_sum(A,B),sum_sum(E,F4)),sum_sum(sum_sum(A,B),sum_sum(E,F4)))),set(product_prod(sum_sum(sum_sum(A,E),sum_sum(B,F4)),sum_sum(sum_sum(A,E),sum_sum(B,F4)))))),product_Pair(set(product_prod(sum_sum(sum_sum(A,B),sum_sum(E,F4)),sum_sum(sum_sum(A,B),sum_sum(E,F4)))),set(product_prod(sum_sum(sum_sum(A,E),sum_sum(B,F4)),sum_sum(sum_sum(A,E),sum_sum(B,F4))))),bNF_Cardinal_csum(sum_sum(A,B),sum_sum(E,F4),bNF_Cardinal_csum(A,B,R12,R23),bNF_Cardinal_csum(E,F4,R32,R42))),bNF_Cardinal_csum(sum_sum(A,E),sum_sum(B,F4),bNF_Cardinal_csum(A,E,R12,R32),bNF_Cardinal_csum(B,F4,R23,R42))),bNF_Wellorder_ordIso(sum_sum(sum_sum(A,B),sum_sum(E,F4)),sum_sum(sum_sum(A,E),sum_sum(B,F4)))) ).

% csum_csum
tff(fact_6745_csum__assoc,axiom,
    ! [E: $tType,B: $tType,A: $tType,P13: set(product_prod(A,A)),P24: set(product_prod(B,B)),P33: set(product_prod(E,E))] : member(product_prod(set(product_prod(sum_sum(sum_sum(A,B),E),sum_sum(sum_sum(A,B),E))),set(product_prod(sum_sum(A,sum_sum(B,E)),sum_sum(A,sum_sum(B,E))))),aa(set(product_prod(sum_sum(A,sum_sum(B,E)),sum_sum(A,sum_sum(B,E)))),product_prod(set(product_prod(sum_sum(sum_sum(A,B),E),sum_sum(sum_sum(A,B),E))),set(product_prod(sum_sum(A,sum_sum(B,E)),sum_sum(A,sum_sum(B,E))))),aa(set(product_prod(sum_sum(sum_sum(A,B),E),sum_sum(sum_sum(A,B),E))),fun(set(product_prod(sum_sum(A,sum_sum(B,E)),sum_sum(A,sum_sum(B,E)))),product_prod(set(product_prod(sum_sum(sum_sum(A,B),E),sum_sum(sum_sum(A,B),E))),set(product_prod(sum_sum(A,sum_sum(B,E)),sum_sum(A,sum_sum(B,E)))))),product_Pair(set(product_prod(sum_sum(sum_sum(A,B),E),sum_sum(sum_sum(A,B),E))),set(product_prod(sum_sum(A,sum_sum(B,E)),sum_sum(A,sum_sum(B,E))))),bNF_Cardinal_csum(sum_sum(A,B),E,bNF_Cardinal_csum(A,B,P13,P24),P33)),bNF_Cardinal_csum(A,sum_sum(B,E),P13,bNF_Cardinal_csum(B,E,P24,P33))),bNF_Wellorder_ordIso(sum_sum(sum_sum(A,B),E),sum_sum(A,sum_sum(B,E)))) ).

% csum_assoc
tff(fact_6746_csum__cong1,axiom,
    ! [B: $tType,E: $tType,A: $tType,P13: set(product_prod(A,A)),R12: set(product_prod(B,B)),Q2: set(product_prod(E,E))] :
      ( 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))),P13),R12),bNF_Wellorder_ordIso(A,B))
     => member(product_prod(set(product_prod(sum_sum(A,E),sum_sum(A,E))),set(product_prod(sum_sum(B,E),sum_sum(B,E)))),aa(set(product_prod(sum_sum(B,E),sum_sum(B,E))),product_prod(set(product_prod(sum_sum(A,E),sum_sum(A,E))),set(product_prod(sum_sum(B,E),sum_sum(B,E)))),aa(set(product_prod(sum_sum(A,E),sum_sum(A,E))),fun(set(product_prod(sum_sum(B,E),sum_sum(B,E))),product_prod(set(product_prod(sum_sum(A,E),sum_sum(A,E))),set(product_prod(sum_sum(B,E),sum_sum(B,E))))),product_Pair(set(product_prod(sum_sum(A,E),sum_sum(A,E))),set(product_prod(sum_sum(B,E),sum_sum(B,E)))),bNF_Cardinal_csum(A,E,P13,Q2)),bNF_Cardinal_csum(B,E,R12,Q2)),bNF_Wellorder_ordIso(sum_sum(A,E),sum_sum(B,E))) ) ).

% csum_cong1
tff(fact_6747_csum__cong2,axiom,
    ! [B: $tType,A: $tType,E: $tType,P24: set(product_prod(A,A)),R23: set(product_prod(B,B)),Q2: set(product_prod(E,E))] :
      ( 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))),P24),R23),bNF_Wellorder_ordIso(A,B))
     => member(product_prod(set(product_prod(sum_sum(E,A),sum_sum(E,A))),set(product_prod(sum_sum(E,B),sum_sum(E,B)))),aa(set(product_prod(sum_sum(E,B),sum_sum(E,B))),product_prod(set(product_prod(sum_sum(E,A),sum_sum(E,A))),set(product_prod(sum_sum(E,B),sum_sum(E,B)))),aa(set(product_prod(sum_sum(E,A),sum_sum(E,A))),fun(set(product_prod(sum_sum(E,B),sum_sum(E,B))),product_prod(set(product_prod(sum_sum(E,A),sum_sum(E,A))),set(product_prod(sum_sum(E,B),sum_sum(E,B))))),product_Pair(set(product_prod(sum_sum(E,A),sum_sum(E,A))),set(product_prod(sum_sum(E,B),sum_sum(E,B)))),bNF_Cardinal_csum(E,A,Q2,P24)),bNF_Cardinal_csum(E,B,Q2,R23)),bNF_Wellorder_ordIso(sum_sum(E,A),sum_sum(E,B))) ) ).

% csum_cong2
tff(fact_6748_csum__mono,axiom,
    ! [F4: $tType,B: $tType,E: $tType,A: $tType,P13: set(product_prod(A,A)),R12: set(product_prod(B,B)),P24: set(product_prod(E,E)),R23: set(product_prod(F4,F4))] :
      ( 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))),P13),R12),bNF_Wellorder_ordLeq(A,B))
     => ( member(product_prod(set(product_prod(E,E)),set(product_prod(F4,F4))),aa(set(product_prod(F4,F4)),product_prod(set(product_prod(E,E)),set(product_prod(F4,F4))),aa(set(product_prod(E,E)),fun(set(product_prod(F4,F4)),product_prod(set(product_prod(E,E)),set(product_prod(F4,F4)))),product_Pair(set(product_prod(E,E)),set(product_prod(F4,F4))),P24),R23),bNF_Wellorder_ordLeq(E,F4))
       => member(product_prod(set(product_prod(sum_sum(A,E),sum_sum(A,E))),set(product_prod(sum_sum(B,F4),sum_sum(B,F4)))),aa(set(product_prod(sum_sum(B,F4),sum_sum(B,F4))),product_prod(set(product_prod(sum_sum(A,E),sum_sum(A,E))),set(product_prod(sum_sum(B,F4),sum_sum(B,F4)))),aa(set(product_prod(sum_sum(A,E),sum_sum(A,E))),fun(set(product_prod(sum_sum(B,F4),sum_sum(B,F4))),product_prod(set(product_prod(sum_sum(A,E),sum_sum(A,E))),set(product_prod(sum_sum(B,F4),sum_sum(B,F4))))),product_Pair(set(product_prod(sum_sum(A,E),sum_sum(A,E))),set(product_prod(sum_sum(B,F4),sum_sum(B,F4)))),bNF_Cardinal_csum(A,E,P13,P24)),bNF_Cardinal_csum(B,F4,R12,R23)),bNF_Wellorder_ordLeq(sum_sum(A,E),sum_sum(B,F4))) ) ) ).

% csum_mono
tff(fact_6749_csum__mono1,axiom,
    ! [B: $tType,E: $tType,A: $tType,P13: set(product_prod(A,A)),R12: set(product_prod(B,B)),Q2: set(product_prod(E,E))] :
      ( 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))),P13),R12),bNF_Wellorder_ordLeq(A,B))
     => member(product_prod(set(product_prod(sum_sum(A,E),sum_sum(A,E))),set(product_prod(sum_sum(B,E),sum_sum(B,E)))),aa(set(product_prod(sum_sum(B,E),sum_sum(B,E))),product_prod(set(product_prod(sum_sum(A,E),sum_sum(A,E))),set(product_prod(sum_sum(B,E),sum_sum(B,E)))),aa(set(product_prod(sum_sum(A,E),sum_sum(A,E))),fun(set(product_prod(sum_sum(B,E),sum_sum(B,E))),product_prod(set(product_prod(sum_sum(A,E),sum_sum(A,E))),set(product_prod(sum_sum(B,E),sum_sum(B,E))))),product_Pair(set(product_prod(sum_sum(A,E),sum_sum(A,E))),set(product_prod(sum_sum(B,E),sum_sum(B,E)))),bNF_Cardinal_csum(A,E,P13,Q2)),bNF_Cardinal_csum(B,E,R12,Q2)),bNF_Wellorder_ordLeq(sum_sum(A,E),sum_sum(B,E))) ) ).

% csum_mono1
tff(fact_6750_csum__mono2,axiom,
    ! [B: $tType,A: $tType,E: $tType,P24: set(product_prod(A,A)),R23: set(product_prod(B,B)),Q2: set(product_prod(E,E))] :
      ( 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))),P24),R23),bNF_Wellorder_ordLeq(A,B))
     => member(product_prod(set(product_prod(sum_sum(E,A),sum_sum(E,A))),set(product_prod(sum_sum(E,B),sum_sum(E,B)))),aa(set(product_prod(sum_sum(E,B),sum_sum(E,B))),product_prod(set(product_prod(sum_sum(E,A),sum_sum(E,A))),set(product_prod(sum_sum(E,B),sum_sum(E,B)))),aa(set(product_prod(sum_sum(E,A),sum_sum(E,A))),fun(set(product_prod(sum_sum(E,B),sum_sum(E,B))),product_prod(set(product_prod(sum_sum(E,A),sum_sum(E,A))),set(product_prod(sum_sum(E,B),sum_sum(E,B))))),product_Pair(set(product_prod(sum_sum(E,A),sum_sum(E,A))),set(product_prod(sum_sum(E,B),sum_sum(E,B)))),bNF_Cardinal_csum(E,A,Q2,P24)),bNF_Cardinal_csum(E,B,Q2,R23)),bNF_Wellorder_ordLeq(sum_sum(E,A),sum_sum(E,B))) ) ).

% csum_mono2
tff(fact_6751_Un__csum,axiom,
    ! [A: $tType,Aa2: set(A),B2: set(A)] : 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)),Aa2),B2))),bNF_Cardinal_csum(A,A,bNF_Ca6860139660246222851ard_of(A,Aa2),bNF_Ca6860139660246222851ard_of(A,B2))),bNF_Wellorder_ordLeq(A,sum_sum(A,A))) ).

% Un_csum
tff(fact_6752_cprod__csum__cexp,axiom,
    ! [B: $tType,A: $tType,R12: set(product_prod(A,A)),R23: set(product_prod(B,B))] : member(product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(fun($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_6753_csum__Cfinite__cexp__Cinfinite,axiom,
    ! [B: $tType,A: $tType,E: $tType,R2: set(product_prod(A,A)),S2: set(product_prod(B,B)),T2: set(product_prod(E,E))] :
      ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => ( ( bNF_Cardinal_cfinite(B,S2)
          & bNF_Ca8970107618336181345der_on(B,aa(set(product_prod(B,B)),set(B),field2(B),S2),S2) )
       => ( ( bNF_Ca4139267488887388095finite(E,T2)
            & bNF_Ca8970107618336181345der_on(E,aa(set(product_prod(E,E)),set(E),field2(E),T2),T2) )
         => member(product_prod(set(product_prod(fun(E,sum_sum(A,B)),fun(E,sum_sum(A,B)))),set(product_prod(fun(E,sum_sum(A,$o)),fun(E,sum_sum(A,$o))))),aa(set(product_prod(fun(E,sum_sum(A,$o)),fun(E,sum_sum(A,$o)))),product_prod(set(product_prod(fun(E,sum_sum(A,B)),fun(E,sum_sum(A,B)))),set(product_prod(fun(E,sum_sum(A,$o)),fun(E,sum_sum(A,$o))))),aa(set(product_prod(fun(E,sum_sum(A,B)),fun(E,sum_sum(A,B)))),fun(set(product_prod(fun(E,sum_sum(A,$o)),fun(E,sum_sum(A,$o)))),product_prod(set(product_prod(fun(E,sum_sum(A,B)),fun(E,sum_sum(A,B)))),set(product_prod(fun(E,sum_sum(A,$o)),fun(E,sum_sum(A,$o)))))),product_Pair(set(product_prod(fun(E,sum_sum(A,B)),fun(E,sum_sum(A,B)))),set(product_prod(fun(E,sum_sum(A,$o)),fun(E,sum_sum(A,$o))))),bNF_Cardinal_cexp(sum_sum(A,B),E,bNF_Cardinal_csum(A,B,R2,S2),T2)),bNF_Cardinal_cexp(sum_sum(A,$o),E,bNF_Cardinal_csum(A,$o,R2,bNF_Cardinal_ctwo),T2)),bNF_Wellorder_ordLeq(fun(E,sum_sum(A,B)),fun(E,sum_sum(A,$o)))) ) ) ) ).

% csum_Cfinite_cexp_Cinfinite
tff(fact_6754_Plus__csum,axiom,
    ! [B: $tType,A: $tType,Aa2: set(A),B2: set(B)] : 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,Aa2,B2))),bNF_Cardinal_csum(A,B,bNF_Ca6860139660246222851ard_of(A,Aa2),bNF_Ca6860139660246222851ard_of(B,B2))),bNF_Wellorder_ordIso(sum_sum(A,B),sum_sum(A,B))) ).

% Plus_csum
tff(fact_6755_csum__dup,axiom,
    ! [A: $tType,E: $tType,B: $tType,R2: set(product_prod(A,A)),P2: set(product_prod(B,B)),P7: set(product_prod(E,E))] :
      ( bNF_Ca4139267488887388095finite(A,R2)
     => ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
       => ( member(product_prod(set(product_prod(sum_sum(B,E),sum_sum(B,E))),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,E),sum_sum(B,E))),set(product_prod(sum_sum(A,A),sum_sum(A,A)))),aa(set(product_prod(sum_sum(B,E),sum_sum(B,E))),fun(set(product_prod(sum_sum(A,A),sum_sum(A,A))),product_prod(set(product_prod(sum_sum(B,E),sum_sum(B,E))),set(product_prod(sum_sum(A,A),sum_sum(A,A))))),product_Pair(set(product_prod(sum_sum(B,E),sum_sum(B,E))),set(product_prod(sum_sum(A,A),sum_sum(A,A)))),bNF_Cardinal_csum(B,E,P2,P7)),bNF_Cardinal_csum(A,A,R2,R2)),bNF_Wellorder_ordIso(sum_sum(B,E),sum_sum(A,A)))
         => member(product_prod(set(product_prod(sum_sum(B,E),sum_sum(B,E))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(B,E),sum_sum(B,E))),set(product_prod(A,A))),aa(set(product_prod(sum_sum(B,E),sum_sum(B,E))),fun(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(B,E),sum_sum(B,E))),set(product_prod(A,A)))),product_Pair(set(product_prod(sum_sum(B,E),sum_sum(B,E))),set(product_prod(A,A))),bNF_Cardinal_csum(B,E,P2,P7)),R2),bNF_Wellorder_ordIso(sum_sum(B,E),A)) ) ) ) ).

% csum_dup
tff(fact_6756_csum__Cnotzero1,axiom,
    ! [A: $tType,B: $tType,R12: set(product_prod(A,A)),R23: set(product_prod(B,B))] :
      ( ( ~ 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))
        & bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R12),R12) )
     => ( ~ 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)))
        & bNF_Ca8970107618336181345der_on(sum_sum(A,B),aa(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(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_6757_csum__absorb1,axiom,
    ! [B: $tType,A: $tType,R23: set(product_prod(A,A)),R12: set(product_prod(B,B))] :
      ( ( bNF_Ca4139267488887388095finite(A,R23)
        & bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R23),R23) )
     => ( 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))
       => 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_6758_csum__absorb1_H,axiom,
    ! [B: $tType,A: $tType,R23: set(product_prod(A,A)),R12: set(product_prod(B,B))] :
      ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R23),R23)
     => ( 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) )
         => 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_6759_csum__absorb2_H,axiom,
    ! [A: $tType,B: $tType,R23: set(product_prod(A,A)),R12: set(product_prod(B,B))] :
      ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R23),R23)
     => ( 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) )
         => 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_6760_cprod__cexp__csum__cexp__Cinfinite,axiom,
    ! [E: $tType,B: $tType,A: $tType,T2: set(product_prod(A,A)),R2: set(product_prod(B,B)),S2: set(product_prod(E,E))] :
      ( ( bNF_Ca4139267488887388095finite(A,T2)
        & bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),T2),T2) )
     => member(product_prod(set(product_prod(fun(A,product_prod(B,E)),fun(A,product_prod(B,E)))),set(product_prod(fun(A,sum_sum(B,E)),fun(A,sum_sum(B,E))))),aa(set(product_prod(fun(A,sum_sum(B,E)),fun(A,sum_sum(B,E)))),product_prod(set(product_prod(fun(A,product_prod(B,E)),fun(A,product_prod(B,E)))),set(product_prod(fun(A,sum_sum(B,E)),fun(A,sum_sum(B,E))))),aa(set(product_prod(fun(A,product_prod(B,E)),fun(A,product_prod(B,E)))),fun(set(product_prod(fun(A,sum_sum(B,E)),fun(A,sum_sum(B,E)))),product_prod(set(product_prod(fun(A,product_prod(B,E)),fun(A,product_prod(B,E)))),set(product_prod(fun(A,sum_sum(B,E)),fun(A,sum_sum(B,E)))))),product_Pair(set(product_prod(fun(A,product_prod(B,E)),fun(A,product_prod(B,E)))),set(product_prod(fun(A,sum_sum(B,E)),fun(A,sum_sum(B,E))))),bNF_Cardinal_cexp(product_prod(B,E),A,bNF_Cardinal_cprod(B,E,R2,S2),T2)),bNF_Cardinal_cexp(sum_sum(B,E),A,bNF_Cardinal_csum(B,E,R2,S2),T2)),bNF_Wellorder_ordLeq(fun(A,product_prod(B,E)),fun(A,sum_sum(B,E)))) ) ).

% cprod_cexp_csum_cexp_Cinfinite
tff(fact_6761_subset__mset_Osum__strict__mono,axiom,
    ! [B: $tType,A: $tType,Aa2: set(A),F2: fun(A,multiset(B)),G: fun(A,multiset(B))] :
      ( aa(set(A),$o,finite_finite2(A),Aa2)
     => ( ( Aa2 != bot_bot(set(A)) )
       => ( ! [X: A] :
              ( member(A,X,Aa2)
             => aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subset_mset(B),aa(A,multiset(B),F2,X)),aa(A,multiset(B),G,X)) )
         => 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)),F2,Aa2)),groups3894954378712506084id_sum(multiset(B),A,plus_plus(multiset(B)),zero_zero(multiset(B)),G,Aa2)) ) ) ) ).

% subset_mset.sum_strict_mono
tff(fact_6762_subset__mset_Osum__pos,axiom,
    ! [B: $tType,A: $tType,Ia2: set(A),F2: fun(A,multiset(B))] :
      ( aa(set(A),$o,finite_finite2(A),Ia2)
     => ( ( Ia2 != bot_bot(set(A)) )
       => ( ! [I3: A] :
              ( member(A,I3,Ia2)
             => aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subset_mset(B),zero_zero(multiset(B))),aa(A,multiset(B),F2,I3)) )
         => 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)),F2,Ia2)) ) ) ) ).

% subset_mset.sum_pos
tff(fact_6763_subset__implies__mult,axiom,
    ! [A: $tType,Aa2: multiset(A),B2: multiset(A),R2: set(product_prod(A,A))] :
      ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subset_mset(A),Aa2),B2)
     => 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)),Aa2),B2),mult(A,R2)) ) ).

% subset_implies_mult
tff(fact_6764_mset__subset__size,axiom,
    ! [A: $tType,Aa2: multiset(A),B2: multiset(A)] :
      ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subset_mset(A),Aa2),B2)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(multiset(A),nat,size_size(multiset(A)),Aa2)),aa(multiset(A),nat,size_size(multiset(A)),B2)) ) ).

% mset_subset_size
tff(fact_6765_subset__mset_Olift__Suc__mono__less,axiom,
    ! [A: $tType,F2: fun(nat,multiset(A)),N: nat,N6: nat] :
      ( ! [N5: nat] : aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subset_mset(A),aa(nat,multiset(A),F2,N5)),aa(nat,multiset(A),F2,aa(nat,nat,suc,N5)))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),N6)
       => aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subset_mset(A),aa(nat,multiset(A),F2,N)),aa(nat,multiset(A),F2,N6)) ) ) ).

% subset_mset.lift_Suc_mono_less
tff(fact_6766_subset__mset_Olift__Suc__mono__less__iff,axiom,
    ! [A: $tType,F2: fun(nat,multiset(A)),N: nat,M2: nat] :
      ( ! [N5: nat] : aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subset_mset(A),aa(nat,multiset(A),F2,N5)),aa(nat,multiset(A),F2,aa(nat,nat,suc,N5)))
     => ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subset_mset(A),aa(nat,multiset(A),F2,N)),aa(nat,multiset(A),F2,M2))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),M2) ) ) ).

% subset_mset.lift_Suc_mono_less_iff
tff(fact_6767_size__psubset,axiom,
    ! [A: $tType,M5: multiset(A),M10: multiset(A)] :
      ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),M5),M10)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(multiset(A),nat,size_size(multiset(A)),M5)),aa(multiset(A),nat,size_size(multiset(A)),M10))
       => aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subset_mset(A),M5),M10) ) ) ).

% size_psubset
tff(fact_6768_rel__filter_Ocases,axiom,
    ! [A: $tType,B: $tType,R: fun(A,fun(B,$o)),Fa: filter(A),G4: filter(B)] :
      ( aa(filter(B),$o,aa(filter(A),fun(filter(B),$o),rel_filter(A,B,R),Fa),G4)
     => ~ ! [Z9: filter(product_prod(A,B))] :
            ( aa(filter(product_prod(A,B)),$o,aa(fun(product_prod(A,B),$o),fun(filter(product_prod(A,B)),$o),eventually(product_prod(A,B)),aa(fun(A,fun(B,$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) = Fa )
             => ( 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_6769_rel__filter_Osimps,axiom,
    ! [A: $tType,B: $tType,R: fun(A,fun(B,$o)),Fa: filter(A),G4: filter(B)] :
      ( aa(filter(B),$o,aa(filter(A),fun(filter(B),$o),rel_filter(A,B,R),Fa),G4)
    <=> ? [Z7: filter(product_prod(A,B))] :
          ( aa(filter(product_prod(A,B)),$o,aa(fun(product_prod(A,B),$o),fun(filter(product_prod(A,B)),$o),eventually(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),R)),Z7)
          & ( 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)),Z7) = Fa )
          & ( 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)),Z7) = G4 ) ) ) ).

% rel_filter.simps
tff(fact_6770_sup__filter__parametric,axiom,
    ! [A: $tType,B: $tType,Aa2: fun(A,fun(B,$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,Aa2),bNF_rel_fun(filter(A),filter(B),filter(A),filter(B),rel_filter(A,B,Aa2),rel_filter(A,B,Aa2))),sup_sup(filter(A))),sup_sup(filter(B))) ).

% sup_filter_parametric
tff(fact_6771_Sup__filter__parametric,axiom,
    ! [A: $tType,B: $tType,Aa2: fun(A,fun(B,$o))] : aa(fun(set(filter(B)),filter(B)),$o,aa(fun(set(filter(A)),filter(A)),fun(fun(set(filter(B)),filter(B)),$o),bNF_rel_fun(set(filter(A)),set(filter(B)),filter(A),filter(B),bNF_rel_set(filter(A),filter(B),rel_filter(A,B,Aa2)),rel_filter(A,B,Aa2)),complete_Sup_Sup(filter(A))),complete_Sup_Sup(filter(B))) ).

% Sup_filter_parametric
tff(fact_6772_rel__filter__eq,axiom,
    ! [A: $tType] : ( rel_filter(A,A,fequal(A)) = fequal(filter(A)) ) ).

% rel_filter_eq
tff(fact_6773_eventually__parametric,axiom,
    ! [A: $tType,B: $tType,Aa2: fun(A,fun(B,$o))] : aa(fun(fun(B,$o),fun(filter(B),$o)),$o,aa(fun(fun(A,$o),fun(filter(A),$o)),fun(fun(fun(B,$o),fun(filter(B),$o)),$o),bNF_rel_fun(fun(A,$o),fun(B,$o),fun(filter(A),$o),fun(filter(B),$o),bNF_rel_fun(A,B,$o,$o,Aa2,fequal($o)),bNF_rel_fun(filter(A),filter(B),$o,$o,rel_filter(A,B,Aa2),fequal($o))),eventually(A)),eventually(B)) ).

% eventually_parametric
tff(fact_6774_rel__filter__mono,axiom,
    ! [B: $tType,A: $tType,Aa2: fun(A,fun(B,$o)),B2: fun(A,fun(B,$o))] :
      ( aa(fun(A,fun(B,$o)),$o,aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),$o),ord_less_eq(fun(A,fun(B,$o))),Aa2),B2)
     => aa(fun(filter(A),fun(filter(B),$o)),$o,aa(fun(filter(A),fun(filter(B),$o)),fun(fun(filter(A),fun(filter(B),$o)),$o),ord_less_eq(fun(filter(A),fun(filter(B),$o))),rel_filter(A,B,Aa2)),rel_filter(A,B,B2)) ) ).

% rel_filter_mono
tff(fact_6775_principal__parametric,axiom,
    ! [A: $tType,B: $tType,Aa2: fun(A,fun(B,$o))] : aa(fun(set(B),filter(B)),$o,aa(fun(set(A),filter(A)),fun(fun(set(B),filter(B)),$o),bNF_rel_fun(set(A),set(B),filter(A),filter(B),bNF_rel_set(A,B,Aa2),rel_filter(A,B,Aa2)),principal(A)),principal(B)) ).

% principal_parametric
tff(fact_6776_filtermap__parametric,axiom,
    ! [A: $tType,B: $tType,F4: $tType,E: $tType,Aa2: fun(A,fun(E,$o)),B2: fun(B,fun(F4,$o))] : aa(fun(fun(E,F4),fun(filter(E),filter(F4))),$o,aa(fun(fun(A,B),fun(filter(A),filter(B))),fun(fun(fun(E,F4),fun(filter(E),filter(F4))),$o),bNF_rel_fun(fun(A,B),fun(E,F4),fun(filter(A),filter(B)),fun(filter(E),filter(F4)),bNF_rel_fun(A,E,B,F4,Aa2,B2),bNF_rel_fun(filter(A),filter(E),filter(B),filter(F4),rel_filter(A,E,Aa2),rel_filter(B,F4,B2))),filtermap(A,B)),filtermap(E,F4)) ).

% filtermap_parametric
tff(fact_6777_bot__filter__parametric,axiom,
    ! [A: $tType,B: $tType,Aa2: fun(A,fun(B,$o))] : aa(filter(B),$o,aa(filter(A),fun(filter(B),$o),rel_filter(A,B,Aa2),bot_bot(filter(A))),bot_bot(filter(B))) ).

% bot_filter_parametric
tff(fact_6778_rel__filter_Ointros,axiom,
    ! [A: $tType,B: $tType,R: fun(A,fun(B,$o)),Z6: filter(product_prod(A,B)),Fa: filter(A),G4: filter(B)] :
      ( aa(filter(product_prod(A,B)),$o,aa(fun(product_prod(A,B),$o),fun(filter(product_prod(A,B)),$o),eventually(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),R)),Z6)
     => ( ( 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)),Z6) = Fa )
       => ( ( 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)),Z6) = G4 )
         => aa(filter(B),$o,aa(filter(A),fun(filter(B),$o),rel_filter(A,B,R),Fa),G4) ) ) ) ).

% rel_filter.intros
tff(fact_6779_mod__h__bot__iff_I8_J,axiom,
    ! [A: $tType,R: fun(A,assn),H2: heap_ext(product_unit)] :
      ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,ex_assn(A,R)),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),bot_bot(set(nat))))
    <=> ? [X6: A] : aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,aa(A,assn,R,X6)),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)))) ) ).

% mod_h_bot_iff(8)
tff(fact_6780_take__Cons,axiom,
    ! [A: $tType,N: nat,Xa: A,Xsa: list(A)] : ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa)) = case_nat(list(A),nil(A),aa(list(A),fun(nat,list(A)),aTP_Lamp_zm(A,fun(list(A),fun(nat,list(A))),Xa),Xsa),N) ) ).

% take_Cons
tff(fact_6781_ex__assn__const,axiom,
    ! [A: $tType,C2: assn] : ( ex_assn(A,aTP_Lamp_zn(assn,fun(A,assn),C2)) = C2 ) ).

% ex_assn_const
tff(fact_6782_norm__assertion__simps_I17_J,axiom,
    ! [A: $tType,R: assn,Qa: fun(A,assn)] : ( aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),R),ex_assn(A,Qa)) = ex_assn(A,aa(fun(A,assn),fun(A,assn),aTP_Lamp_zo(assn,fun(fun(A,assn),fun(A,assn)),R),Qa)) ) ).

% norm_assertion_simps(17)
tff(fact_6783_norm__assertion__simps_I16_J,axiom,
    ! [A: $tType,Qa: fun(A,assn),R: assn] : ( aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),ex_assn(A,Qa)),R) = ex_assn(A,aa(assn,fun(A,assn),aTP_Lamp_zp(fun(A,assn),fun(assn,fun(A,assn)),Qa),R)) ) ).

% norm_assertion_simps(16)
tff(fact_6784_mod__ex__dist,axiom,
    ! [A: $tType,Pa: fun(A,assn),H2: product_prod(heap_ext(product_unit),set(nat))] :
      ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,ex_assn(A,Pa)),H2)
    <=> ? [X6: A] : aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,aa(A,assn,Pa,X6)),H2) ) ).

% mod_ex_dist
tff(fact_6785_triv__exI,axiom,
    ! [A: $tType,Qa: fun(A,assn),Xa: A] : aa(assn,$o,aa(assn,fun(assn,$o),entails,aa(A,assn,Qa,Xa)),ex_assn(A,Qa)) ).

% triv_exI
tff(fact_6786_norm__assertion__simps_I20_J,axiom,
    ! [A: $tType,Qa: fun(A,assn),Pa: assn] : ( aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),ex_assn(A,Qa)),Pa) = ex_assn(A,aa(assn,fun(A,assn),aTP_Lamp_zq(fun(A,assn),fun(assn,fun(A,assn)),Qa),Pa)) ) ).

% norm_assertion_simps(20)
tff(fact_6787_norm__assertion__simps_I21_J,axiom,
    ! [A: $tType,Qa: assn,Pa: fun(A,assn)] : ( aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),Qa),ex_assn(A,Pa)) = ex_assn(A,aa(fun(A,assn),fun(A,assn),aTP_Lamp_zr(assn,fun(fun(A,assn),fun(A,assn)),Qa),Pa)) ) ).

% norm_assertion_simps(21)
tff(fact_6788_norm__assertion__simps_I18_J,axiom,
    ! [A: $tType,Qa: fun(A,assn),Pa: assn] : ( aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),ex_assn(A,Qa)),Pa) = ex_assn(A,aa(assn,fun(A,assn),aTP_Lamp_zs(fun(A,assn),fun(assn,fun(A,assn)),Qa),Pa)) ) ).

% norm_assertion_simps(18)
tff(fact_6789_norm__assertion__simps_I19_J,axiom,
    ! [A: $tType,Qa: assn,Pa: fun(A,assn)] : ( aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),Qa),ex_assn(A,Pa)) = ex_assn(A,aa(fun(A,assn),fun(A,assn),aTP_Lamp_zt(assn,fun(fun(A,assn),fun(A,assn)),Qa),Pa)) ) ).

% norm_assertion_simps(19)
tff(fact_6790_ex__distrib__and,axiom,
    ! [A: $tType,Pa: fun(A,assn),Qa: assn] : ( ex_assn(A,aa(assn,fun(A,assn),aTP_Lamp_zs(fun(A,assn),fun(assn,fun(A,assn)),Pa),Qa)) = aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),ex_assn(A,Pa)),Qa) ) ).

% ex_distrib_and
tff(fact_6791_mod__exE,axiom,
    ! [A: $tType,Pa: fun(A,assn),H2: product_prod(heap_ext(product_unit),set(nat))] :
      ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,ex_assn(A,Pa)),H2)
     => ~ ! [X: A] : ~ aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,aa(A,assn,Pa,X)),H2) ) ).

% mod_exE
tff(fact_6792_mod__exI,axiom,
    ! [A: $tType,Pa: fun(A,assn),H2: product_prod(heap_ext(product_unit),set(nat))] :
      ( ? [X4: A] : aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,aa(A,assn,Pa,X4)),H2)
     => aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,ex_assn(A,Pa)),H2) ) ).

% mod_exI
tff(fact_6793_ex__one__point__gen,axiom,
    ! [A: $tType,Pa: fun(A,assn),V2: A] :
      ( ! [H: product_prod(heap_ext(product_unit),set(nat)),X: A] :
          ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,aa(A,assn,Pa,X)),H)
         => ( X = V2 ) )
     => ( ex_assn(A,Pa) = aa(A,assn,Pa,V2) ) ) ).

% ex_one_point_gen
tff(fact_6794_ex__distrib__star,axiom,
    ! [A: $tType,Pa: fun(A,assn),Qa: assn] : ( ex_assn(A,aa(assn,fun(A,assn),aTP_Lamp_zp(fun(A,assn),fun(assn,fun(A,assn)),Pa),Qa)) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),ex_assn(A,Pa)),Qa) ) ).

% ex_distrib_star
tff(fact_6795_enorm__exI_H,axiom,
    ! [A: $tType,Z6: fun(A,$o),Pa: assn,Qa: fun(A,assn)] :
      ( ! [X: A] :
          ( aa(A,$o,Z6,X)
         => aa(assn,$o,aa(assn,fun(assn,$o),entails,Pa),aa(A,assn,Qa,X)) )
     => ( ? [X_13: A] : aa(A,$o,Z6,X_13)
       => aa(assn,$o,aa(assn,fun(assn,$o),entails,Pa),ex_assn(A,Qa)) ) ) ).

% enorm_exI'
tff(fact_6796_ent__ex__preI,axiom,
    ! [A: $tType,Pa: fun(A,assn),Qa: assn] :
      ( ! [X: A] : aa(assn,$o,aa(assn,fun(assn,$o),entails,aa(A,assn,Pa,X)),Qa)
     => aa(assn,$o,aa(assn,fun(assn,$o),entails,ex_assn(A,Pa)),Qa) ) ).

% ent_ex_preI
tff(fact_6797_ent__ex__postI,axiom,
    ! [A: $tType,Pa: assn,Qa: fun(A,assn),Xa: A] :
      ( aa(assn,$o,aa(assn,fun(assn,$o),entails,Pa),aa(A,assn,Qa,Xa))
     => aa(assn,$o,aa(assn,fun(assn,$o),entails,Pa),ex_assn(A,Qa)) ) ).

% ent_ex_postI
tff(fact_6798_ex__join__or,axiom,
    ! [A: $tType,Pa: fun(A,assn),Qa: fun(A,assn)] : ( ex_assn(A,aa(fun(A,assn),fun(A,assn),aTP_Lamp_zu(fun(A,assn),fun(fun(A,assn),fun(A,assn)),Pa),Qa)) = ex_assn(A,aa(fun(A,assn),fun(A,assn),aTP_Lamp_zv(fun(A,assn),fun(fun(A,assn),fun(A,assn)),Pa),Qa)) ) ).

% ex_join_or
tff(fact_6799_ex__distrib__or,axiom,
    ! [A: $tType,Pa: fun(A,assn),Qa: assn] : ( ex_assn(A,aa(assn,fun(A,assn),aTP_Lamp_zq(fun(A,assn),fun(assn,fun(A,assn)),Pa),Qa)) = aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),ex_assn(A,Pa)),Qa) ) ).

% ex_distrib_or
tff(fact_6800_drop__Cons,axiom,
    ! [A: $tType,N: nat,Xa: A,Xsa: list(A)] : ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa)) = case_nat(list(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa),aTP_Lamp_zw(list(A),fun(nat,list(A)),Xsa),N) ) ).

% drop_Cons
tff(fact_6801_list__update_Osimps_I2_J,axiom,
    ! [A: $tType,Xa: A,Xsa: list(A),Ia: nat,V2: A] : ( aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa)),Ia),V2) = case_nat(list(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V2),Xsa),aa(A,fun(nat,list(A)),aa(list(A),fun(A,fun(nat,list(A))),aTP_Lamp_zx(A,fun(list(A),fun(A,fun(nat,list(A)))),Xa),Xsa),V2),Ia) ) ).

% list_update.simps(2)
tff(fact_6802_nth__Cons,axiom,
    ! [A: $tType,Xa: A,Xsa: list(A),N: nat] : ( aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa)),N) = case_nat(A,Xa,nth(A,Xsa),N) ) ).

% nth_Cons
tff(fact_6803_frequently__parametric,axiom,
    ! [A: $tType,B: $tType,Aa2: fun(A,fun(B,$o))] : aa(fun(fun(B,$o),fun(filter(B),$o)),$o,aa(fun(fun(A,$o),fun(filter(A),$o)),fun(fun(fun(B,$o),fun(filter(B),$o)),$o),bNF_rel_fun(fun(A,$o),fun(B,$o),fun(filter(A),$o),fun(filter(B),$o),bNF_rel_fun(A,B,$o,$o,Aa2,fequal($o)),bNF_rel_fun(filter(A),filter(B),$o,$o,rel_filter(A,B,Aa2),fequal($o))),frequently(A)),frequently(B)) ).

% frequently_parametric
tff(fact_6804_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_zy(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))),plus_plus(rat)) ).

% plus_rat.transfer
tff(fact_6805_frequently__const,axiom,
    ! [A: $tType,Fa: filter(A),Pa: $o] :
      ( ( Fa != bot_bot(filter(A)) )
     => ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),aTP_Lamp_ae($o,fun(A,$o),(Pa))),Fa)
      <=> (Pa) ) ) ).

% frequently_const
tff(fact_6806_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_6807_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_6808_frequently__bex__finite,axiom,
    ! [A: $tType,B: $tType,Aa2: set(A),Pa: fun(B,fun(A,$o)),Fa: filter(B)] :
      ( aa(set(A),$o,finite_finite2(A),Aa2)
     => ( aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),frequently(B),aa(fun(B,fun(A,$o)),fun(B,$o),aTP_Lamp_zz(set(A),fun(fun(B,fun(A,$o)),fun(B,$o)),Aa2),Pa)),Fa)
       => ? [X: A] :
            ( member(A,X,Aa2)
            & aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),frequently(B),aa(A,fun(B,$o),aTP_Lamp_sc(fun(B,fun(A,$o)),fun(A,fun(B,$o)),Pa),X)),Fa) ) ) ) ).

% frequently_bex_finite
tff(fact_6809_frequently__bex__finite__distrib,axiom,
    ! [A: $tType,B: $tType,Aa2: set(A),Pa: fun(B,fun(A,$o)),Fa: filter(B)] :
      ( aa(set(A),$o,finite_finite2(A),Aa2)
     => ( aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),frequently(B),aa(fun(B,fun(A,$o)),fun(B,$o),aTP_Lamp_zz(set(A),fun(fun(B,fun(A,$o)),fun(B,$o)),Aa2),Pa)),Fa)
      <=> ? [X6: A] :
            ( member(A,X6,Aa2)
            & aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),frequently(B),aa(A,fun(B,$o),aTP_Lamp_sc(fun(B,fun(A,$o)),fun(A,fun(B,$o)),Pa),X6)),Fa) ) ) ) ).

% frequently_bex_finite_distrib
tff(fact_6810_eventually__frequently__const__simps_I2_J,axiom,
    ! [A: $tType,C3: $o,Pa: fun(A,$o),Fa: filter(A)] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_aaa($o,fun(fun(A,$o),fun(A,$o)),(C3)),Pa)),Fa)
    <=> ( (C3)
        & aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),Pa),Fa) ) ) ).

% eventually_frequently_const_simps(2)
tff(fact_6811_eventually__frequently__const__simps_I1_J,axiom,
    ! [A: $tType,Pa: fun(A,$o),C3: $o,Fa: filter(A)] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),aa($o,fun(A,$o),aTP_Lamp_aab(fun(A,$o),fun($o,fun(A,$o)),Pa),(C3))),Fa)
    <=> ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),Pa),Fa)
        & (C3) ) ) ).

% eventually_frequently_const_simps(1)
tff(fact_6812_frequently__ex,axiom,
    ! [A: $tType,Pa: fun(A,$o),Fa: filter(A)] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),Pa),Fa)
     => ? [X_1: A] : aa(A,$o,Pa,X_1) ) ).

% frequently_ex
tff(fact_6813_frequently__disj,axiom,
    ! [A: $tType,Pa: fun(A,$o),Fa: filter(A),Qa: fun(A,$o)] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),Pa),Fa)
     => ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),Qa),Fa)
       => aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_bv(fun(A,$o),fun(fun(A,$o),fun(A,$o)),Pa),Qa)),Fa) ) ) ).

% frequently_disj
tff(fact_6814_frequently__elim1,axiom,
    ! [A: $tType,Pa: fun(A,$o),Fa: filter(A),Qa: fun(A,$o)] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),Pa),Fa)
     => ( ! [I3: A] :
            ( aa(A,$o,Pa,I3)
           => aa(A,$o,Qa,I3) )
       => aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),Qa),Fa) ) ) ).

% frequently_elim1
tff(fact_6815_frequently__disj__iff,axiom,
    ! [A: $tType,Pa: fun(A,$o),Qa: fun(A,$o),Fa: filter(A)] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_bv(fun(A,$o),fun(fun(A,$o),fun(A,$o)),Pa),Qa)),Fa)
    <=> ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),Pa),Fa)
        | aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),Qa),Fa) ) ) ).

% frequently_disj_iff
tff(fact_6816_not__frequently__False,axiom,
    ! [A: $tType,Fa: filter(A)] : ~ aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),aTP_Lamp_cq(A,$o)),Fa) ).

% not_frequently_False
tff(fact_6817_frequentlyE,axiom,
    ! [A: $tType,Pa: fun(A,$o),Fa: filter(A)] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),Pa),Fa)
     => ~ ! [X: A] : ~ aa(A,$o,Pa,X) ) ).

% frequentlyE
tff(fact_6818_frequently__mono,axiom,
    ! [A: $tType,Pa: fun(A,$o),Qa: fun(A,$o),Fa: filter(A)] :
      ( ! [X: A] :
          ( aa(A,$o,Pa,X)
         => aa(A,$o,Qa,X) )
     => ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),Pa),Fa)
       => aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),Qa),Fa) ) ) ).

% frequently_mono
tff(fact_6819_frequently__all,axiom,
    ! [B: $tType,A: $tType,Pa: fun(A,fun(B,$o)),Fa: filter(A)] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),aTP_Lamp_aac(fun(A,fun(B,$o)),fun(A,$o),Pa)),Fa)
    <=> ! [Y8: fun(A,B)] : aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_vz(fun(A,fun(B,$o)),fun(fun(A,B),fun(A,$o)),Pa),Y8)),Fa) ) ).

% frequently_all
tff(fact_6820_frequently__imp__iff,axiom,
    ! [A: $tType,Pa: fun(A,$o),Qa: fun(A,$o),Fa: filter(A)] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_bt(fun(A,$o),fun(fun(A,$o),fun(A,$o)),Pa),Qa)),Fa)
    <=> ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Pa),Fa)
       => aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),Qa),Fa) ) ) ).

% frequently_imp_iff
tff(fact_6821_frequently__rev__mp,axiom,
    ! [A: $tType,Pa: fun(A,$o),Fa: filter(A),Qa: fun(A,$o)] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),Pa),Fa)
     => ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_bt(fun(A,$o),fun(fun(A,$o),fun(A,$o)),Pa),Qa)),Fa)
       => aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),Qa),Fa) ) ) ).

% frequently_rev_mp
tff(fact_6822_not__frequently,axiom,
    ! [A: $tType,Pa: fun(A,$o),Fa: filter(A)] :
      ( ~ aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),Pa),Fa)
    <=> aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aTP_Lamp_ce(fun(A,$o),fun(A,$o),Pa)),Fa) ) ).

% not_frequently
tff(fact_6823_not__eventually,axiom,
    ! [A: $tType,Pa: fun(A,$o),Fa: filter(A)] :
      ( ~ aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Pa),Fa)
    <=> aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),aTP_Lamp_ce(fun(A,$o),fun(A,$o),Pa)),Fa) ) ).

% not_eventually
tff(fact_6824_frequently__def,axiom,
    ! [A: $tType,Pa: fun(A,$o),Fa: filter(A)] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),Pa),Fa)
    <=> ~ aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aTP_Lamp_ce(fun(A,$o),fun(A,$o),Pa)),Fa) ) ).

% frequently_def
tff(fact_6825_frequently__mp,axiom,
    ! [A: $tType,Pa: fun(A,$o),Qa: fun(A,$o),Fa: filter(A)] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_bt(fun(A,$o),fun(fun(A,$o),fun(A,$o)),Pa),Qa)),Fa)
     => ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),Pa),Fa)
       => aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),Qa),Fa) ) ) ).

% frequently_mp
tff(fact_6826_eventually__frequently__const__simps_I5_J,axiom,
    ! [A: $tType,Pa: fun(A,$o),C3: $o,Fa: filter(A)] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa($o,fun(A,$o),aTP_Lamp_aad(fun(A,$o),fun($o,fun(A,$o)),Pa),(C3))),Fa)
    <=> ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),Pa),Fa)
       => (C3) ) ) ).

% eventually_frequently_const_simps(5)
tff(fact_6827_eventually__frequentlyE,axiom,
    ! [A: $tType,Pa: fun(A,$o),Fa: filter(A),Qa: fun(A,$o)] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Pa),Fa)
     => ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_aae(fun(A,$o),fun(fun(A,$o),fun(A,$o)),Pa),Qa)),Fa)
       => ( ( Fa != bot_bot(filter(A)) )
         => aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),Qa),Fa) ) ) ) ).

% eventually_frequentlyE
tff(fact_6828_frequently__const__iff,axiom,
    ! [A: $tType,Pa: $o,Fa: filter(A)] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),aTP_Lamp_ae($o,fun(A,$o),(Pa))),Fa)
    <=> ( (Pa)
        & ( Fa != bot_bot(filter(A)) ) ) ) ).

% frequently_const_iff
tff(fact_6829_eventually__frequently,axiom,
    ! [A: $tType,Fa: filter(A),Pa: fun(A,$o)] :
      ( ( Fa != bot_bot(filter(A)) )
     => ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Pa),Fa)
       => aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),Pa),Fa) ) ) ).

% eventually_frequently
tff(fact_6830_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_aaf(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))),times_times(rat)) ).

% times_rat.transfer
tff(fact_6831_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_aag(product_prod(int,int),product_prod(int,int))),uminus_uminus(rat)) ).

% uminus_rat.transfer
tff(fact_6832_normalize__negative,axiom,
    ! [Q2: int,P2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Q2),zero_zero(int))
     => ( normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P2),Q2)) = normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,uminus_uminus(int),P2)),aa(int,int,uminus_uminus(int),Q2))) ) ) ).

% normalize_negative
tff(fact_6833_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_aah(product_prod(int,int),product_prod(int,int))),inverse_inverse(rat)) ).

% inverse_rat.transfer
tff(fact_6834_inverse__mult__distrib,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A,B3: A] : ( aa(A,A,inverse_inverse(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B3)) ) ) ).

% inverse_mult_distrib
tff(fact_6835_inverse__positive__iff__positive,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,inverse_inverse(A),A3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3) ) ) ).

% inverse_positive_iff_positive
tff(fact_6836_inverse__negative__iff__negative,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),A3)),zero_zero(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A)) ) ) ).

% inverse_negative_iff_negative
tff(fact_6837_inverse__less__iff__less__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),zero_zero(A))
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B3))
            <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A3) ) ) ) ) ).

% inverse_less_iff_less_neg
tff(fact_6838_inverse__less__iff__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B3)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B3))
            <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A3) ) ) ) ) ).

% inverse_less_iff_less
tff(fact_6839_inverse__le__iff__le__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),zero_zero(A))
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B3))
            <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3) ) ) ) ) ).

% inverse_le_iff_le_neg
tff(fact_6840_inverse__le__iff__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B3)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B3))
            <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3) ) ) ) ) ).

% inverse_le_iff_le
tff(fact_6841_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_6842_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_6843_normalize__denom__zero,axiom,
    ! [P2: int] : ( normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P2),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_6844_inverse__less__imp__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B3))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A3) ) ) ) ).

% inverse_less_imp_less
tff(fact_6845_less__imp__inverse__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
         => ( aa(A,$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,inverse_inverse(A),B3)),aa(A,A,inverse_inverse(A),A3)) ) ) ) ).

% less_imp_inverse_less
tff(fact_6846_inverse__less__imp__less__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B3))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A3) ) ) ) ).

% inverse_less_imp_less_neg
tff(fact_6847_less__imp__inverse__less__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),B3)),aa(A,A,inverse_inverse(A),A3)) ) ) ) ).

% less_imp_inverse_less_neg
tff(fact_6848_inverse__negative__imp__negative,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),A3)),zero_zero(A))
         => ( ( A3 != zero_zero(A) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A)) ) ) ) ).

% inverse_negative_imp_negative
tff(fact_6849_inverse__positive__imp__positive,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,inverse_inverse(A),A3))
         => ( ( A3 != zero_zero(A) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3) ) ) ) ).

% inverse_positive_imp_positive
tff(fact_6850_negative__imp__inverse__negative,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( aa(A,$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,inverse_inverse(A),A3)),zero_zero(A)) ) ) ).

% negative_imp_inverse_negative
tff(fact_6851_positive__imp__inverse__positive,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),zero_zero(A)),aa(A,A,inverse_inverse(A),A3)) ) ) ).

% positive_imp_inverse_positive
tff(fact_6852_nonzero__inverse__mult__distrib,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B3: A] :
          ( ( A3 != zero_zero(A) )
         => ( ( B3 != zero_zero(A) )
           => ( aa(A,A,inverse_inverse(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),B3)),aa(A,A,inverse_inverse(A),A3)) ) ) ) ) ).

% nonzero_inverse_mult_distrib
tff(fact_6853_mult__inverse__of__int__commute,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xa: int,Xb: A] : ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),aa(int,A,ring_1_of_int(A),Xa))),Xb) = aa(A,A,aa(A,fun(A,A),times_times(A),Xb),aa(A,A,inverse_inverse(A),aa(int,A,ring_1_of_int(A),Xa))) ) ) ).

% mult_inverse_of_int_commute
tff(fact_6854_field__class_Ofield__divide__inverse,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A,B3: A] : ( divide_divide(A,A3,B3) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,inverse_inverse(A),B3)) ) ) ).

% field_class.field_divide_inverse
tff(fact_6855_divide__inverse,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B3: A] : ( divide_divide(A,A3,B3) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,inverse_inverse(A),B3)) ) ) ).

% divide_inverse
tff(fact_6856_divide__inverse__commute,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A,B3: A] : ( divide_divide(A,A3,B3) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),B3)),A3) ) ) ).

% divide_inverse_commute
tff(fact_6857_power__mult__inverse__distrib,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xa: A,M2: nat] : ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),M2)),aa(A,A,inverse_inverse(A),Xa)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),Xa)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),M2)) ) ) ).

% power_mult_inverse_distrib
tff(fact_6858_power__mult__power__inverse__commute,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xa: A,M2: nat,N: nat] : ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),M2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,inverse_inverse(A),Xa)),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),Xa)),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),M2)) ) ) ).

% power_mult_power_inverse_commute
tff(fact_6859_mult__inverse__of__nat__commute,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xa: nat,Xb: A] : ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),aa(nat,A,semiring_1_of_nat(A),Xa))),Xb) = aa(A,A,aa(A,fun(A,A),times_times(A),Xb),aa(A,A,inverse_inverse(A),aa(nat,A,semiring_1_of_nat(A),Xa))) ) ) ).

% mult_inverse_of_nat_commute
tff(fact_6860_mult__commute__imp__mult__inverse__commute,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Y: A,Xa: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),Y),Xa) = aa(A,A,aa(A,fun(A,A),times_times(A),Xa),Y) )
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),Y)),Xa) = aa(A,A,aa(A,fun(A,A),times_times(A),Xa),aa(A,A,inverse_inverse(A),Y)) ) ) ) ).

% mult_commute_imp_mult_inverse_commute
tff(fact_6861_inverse__unique,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3) = one_one(A) )
         => ( aa(A,A,inverse_inverse(A),A3) = B3 ) ) ) ).

% inverse_unique
tff(fact_6862_le__imp__inverse__le__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,inverse_inverse(A),B3)),aa(A,A,inverse_inverse(A),A3)) ) ) ) ).

% le_imp_inverse_le_neg
tff(fact_6863_inverse__le__imp__le__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B3))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3) ) ) ) ).

% inverse_le_imp_le_neg
tff(fact_6864_le__imp__inverse__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
         => ( aa(A,$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,inverse_inverse(A),B3)),aa(A,A,inverse_inverse(A),A3)) ) ) ) ).

% le_imp_inverse_le
tff(fact_6865_inverse__le__imp__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B3))
         => ( aa(A,$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),B3),A3) ) ) ) ).

% inverse_le_imp_le
tff(fact_6866_one__less__inverse__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(A,A,inverse_inverse(A),Xa))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xa)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),one_one(A)) ) ) ) ).

% one_less_inverse_iff
tff(fact_6867_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_6868_division__ring__inverse__add,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B3: A] :
          ( ( A3 != zero_zero(A) )
         => ( ( B3 != 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),B3)) = 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),B3))),aa(A,A,inverse_inverse(A),B3)) ) ) ) ) ).

% division_ring_inverse_add
tff(fact_6869_inverse__add,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A,B3: A] :
          ( ( A3 != zero_zero(A) )
         => ( ( B3 != 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),B3)) = 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),B3)),aa(A,A,inverse_inverse(A),A3))),aa(A,A,inverse_inverse(A),B3)) ) ) ) ) ).

% inverse_add
tff(fact_6870_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_6871_division__ring__inverse__diff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B3: A] :
          ( ( A3 != zero_zero(A) )
         => ( ( B3 != 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),B3)) = 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),B3),A3))),aa(A,A,inverse_inverse(A),B3)) ) ) ) ) ).

% division_ring_inverse_diff
tff(fact_6872_inverse__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B3))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A3) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3) ) ) ) ) ).

% inverse_less_iff
tff(fact_6873_inverse__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B3))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3) ) ) ) ) ).

% inverse_le_iff
tff(fact_6874_one__le__inverse__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(A,A,inverse_inverse(A),Xa))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xa)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),one_one(A)) ) ) ) ).

% one_le_inverse_iff
tff(fact_6875_inverse__less__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),Xa)),one_one(A))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),zero_zero(A))
            | aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),Xa) ) ) ) ).

% inverse_less_1_iff
tff(fact_6876_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_6877_reals__Archimedean,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xa)
         => ? [N5: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,N5)))),Xa) ) ) ).

% reals_Archimedean
tff(fact_6878_ex__inverse__of__nat__less,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xa)
         => ? [N5: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N5)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),aa(nat,A,semiring_1_of_nat(A),N5))),Xa) ) ) ) ).

% ex_inverse_of_nat_less
tff(fact_6879_power__diff__conv__inverse,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xa: A,M2: nat,N: nat] :
          ( ( Xa != zero_zero(A) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),N)
           => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,inverse_inverse(A),Xa)),M2)) ) ) ) ) ).

% power_diff_conv_inverse
tff(fact_6880_normalize__denom__pos,axiom,
    ! [R2: product_prod(int,int),P2: int,Q2: int] :
      ( ( normalize(R2) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P2),Q2) )
     => aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Q2) ) ).

% normalize_denom_pos
tff(fact_6881_normalize__crossproduct,axiom,
    ! [Q2: int,S2: int,P2: int,R2: int] :
      ( ( Q2 != zero_zero(int) )
     => ( ( S2 != zero_zero(int) )
       => ( ( normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P2),Q2)) = 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),P2),S2) = aa(int,int,aa(int,fun(int,int),times_times(int),R2),Q2) ) ) ) ) ).

% normalize_crossproduct
tff(fact_6882_rat__inverse__code,axiom,
    ! [P2: rat] : ( quotient_of(aa(rat,rat,inverse_inverse(rat),P2)) = 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_aai(int,fun(int,product_prod(int,int)))),quotient_of(P2)) ) ).

% rat_inverse_code
tff(fact_6883_normalize__def,axiom,
    ! [P2: product_prod(int,int)] :
      ( normalize(P2) = $ite(
          aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),aa(product_prod(int,int),int,product_snd(int,int),P2)),
          $let(
            a4: int,
            a4:= aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(product_prod(int,int),int,product_fst(int,int),P2)),aa(product_prod(int,int),int,product_snd(int,int),P2)),
            aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),divide_divide(int,aa(product_prod(int,int),int,product_fst(int,int),P2),a4)),divide_divide(int,aa(product_prod(int,int),int,product_snd(int,int),P2),a4)) ),
          $ite(
            aa(product_prod(int,int),int,product_snd(int,int),P2) = 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(
              a4: int,
              a4:= 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),P2)),aa(product_prod(int,int),int,product_snd(int,int),P2))),
              aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),divide_divide(int,aa(product_prod(int,int),int,product_fst(int,int),P2),a4)),divide_divide(int,aa(product_prod(int,int),int,product_snd(int,int),P2),a4)) ) ) ) ) ).

% normalize_def
tff(fact_6884_Gcd__insert,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A3: A,Aa2: set(A)] : ( gcd_Gcd(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),Aa2)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),gcd_Gcd(A,Aa2)) ) ) ).

% Gcd_insert
tff(fact_6885_Gcd__fin_Oinsert,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,Aa2: set(A)] : ( semiring_gcd_Gcd_fin(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),Aa2)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),semiring_gcd_Gcd_fin(A,Aa2)) ) ) ).

% Gcd_fin.insert
tff(fact_6886_Gcd__2,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A3: A,B3: A] : ( gcd_Gcd(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),B3),bot_bot(set(A))))) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B3) ) ) ).

% Gcd_2
tff(fact_6887_quotient__of__number_I3_J,axiom,
    ! [K3: num] : ( quotient_of(numeral_numeral(rat,K3)) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),numeral_numeral(int,K3)),one_one(int)) ) ).

% quotient_of_number(3)
tff(fact_6888_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_6889_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_6890_quotient__of__number_I5_J,axiom,
    ! [K3: num] : ( quotient_of(aa(rat,rat,uminus_uminus(rat),numeral_numeral(rat,K3))) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,uminus_uminus(int),numeral_numeral(int,K3))),one_one(int)) ) ).

% quotient_of_number(5)
tff(fact_6891_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_6892_gcd__add__mult,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [M2: A,K3: A,N: A] : ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),M2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),K3),M2)),N)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),M2),N) ) ) ).

% gcd_add_mult
tff(fact_6893_Gcd__int__set__eq__fold,axiom,
    ! [Xsa: list(int)] : ( gcd_Gcd(int,aa(list(int),set(int),set2(int),Xsa)) = aa(int,int,aa(list(int),fun(int,int),aa(fun(int,fun(int,int)),fun(list(int),fun(int,int)),fold(int,int),gcd_gcd(int)),Xsa),zero_zero(int)) ) ).

% Gcd_int_set_eq_fold
tff(fact_6894_gcd__dvd__prod,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B3: A,K3: A] : dvd_dvd(A,aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B3),aa(A,A,aa(A,fun(A,A),times_times(A),K3),B3)) ) ).

% gcd_dvd_prod
tff(fact_6895_gcd__diff2,axiom,
    ! [A: $tType] :
      ( ring_gcd(A)
     => ! [N: A,M2: A] : ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),N),M2)),N) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),M2),N) ) ) ).

% gcd_diff2
tff(fact_6896_gcd__diff1,axiom,
    ! [A: $tType] :
      ( ring_gcd(A)
     => ! [M2: A,N: A] : ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),M2),N)),N) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),M2),N) ) ) ).

% gcd_diff1
tff(fact_6897_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 = 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_6898_gcd__mult__unit1,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B3: A,C2: A] :
          ( 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),B3),A3)),C2) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),B3),C2) ) ) ) ).

% gcd_mult_unit1
tff(fact_6899_gcd__mult__unit2,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B3: A,C2: A] :
          ( dvd_dvd(A,A3,one_one(A))
         => ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),B3),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),B3),C2) ) ) ) ).

% gcd_mult_unit2
tff(fact_6900_quotient__of__denom__pos,axiom,
    ! [R2: rat,P2: int,Q2: int] :
      ( ( quotient_of(R2) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P2),Q2) )
     => aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Q2) ) ).

% quotient_of_denom_pos
tff(fact_6901_Gcd__fin_Oinsert__remove,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,Aa2: set(A)] : ( semiring_gcd_Gcd_fin(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),Aa2)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),semiring_gcd_Gcd_fin(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),bot_bot(set(A)))))) ) ) ).

% Gcd_fin.insert_remove
tff(fact_6902_Gcd__fin_Oremove,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,Aa2: set(A)] :
          ( member(A,A3,Aa2)
         => ( semiring_gcd_Gcd_fin(A,Aa2) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),semiring_gcd_Gcd_fin(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),bot_bot(set(A)))))) ) ) ) ).

% Gcd_fin.remove
tff(fact_6903_Gcd__set__eq__fold,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [Xsa: list(A)] : ( gcd_Gcd(A,aa(list(A),set(A),set2(A),Xsa)) = aa(A,A,aa(list(A),fun(A,A),aa(fun(A,fun(A,A)),fun(list(A),fun(A,A)),fold(A,A),gcd_gcd(A)),Xsa),zero_zero(A)) ) ) ).

% Gcd_set_eq_fold
tff(fact_6904_Gcd__fin_Oset__eq__fold,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [Xsa: list(A)] : ( semiring_gcd_Gcd_fin(A,aa(list(A),set(A),set2(A),Xsa)) = aa(A,A,aa(list(A),fun(A,A),aa(fun(A,fun(A,A)),fun(list(A),fun(A,A)),fold(A,A),gcd_gcd(A)),Xsa),zero_zero(A)) ) ) ).

% Gcd_fin.set_eq_fold
tff(fact_6905_rat__abs__code,axiom,
    ! [P2: rat] : ( quotient_of(abs_abs(rat,P2)) = 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_aaj(int,fun(int,product_prod(int,int)))),quotient_of(P2)) ) ).

% rat_abs_code
tff(fact_6906_rat__sgn__code,axiom,
    ! [P2: rat] : ( quotient_of(aa(rat,rat,sgn_sgn(rat),P2)) = 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(P2)))),one_one(int)) ) ).

% rat_sgn_code
tff(fact_6907_rat__uminus__code,axiom,
    ! [P2: rat] : ( quotient_of(aa(rat,rat,uminus_uminus(rat),P2)) = 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_aak(int,fun(int,product_prod(int,int)))),quotient_of(P2)) ) ).

% rat_uminus_code
tff(fact_6908_rat__divide__code,axiom,
    ! [P2: rat,Q2: rat] : ( quotient_of(divide_divide(rat,P2,Q2)) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aTP_Lamp_aam(rat,fun(int,fun(int,product_prod(int,int))),Q2)),quotient_of(P2)) ) ).

% rat_divide_code
tff(fact_6909_rat__minus__code,axiom,
    ! [P2: rat,Q2: rat] : ( quotient_of(aa(rat,rat,aa(rat,fun(rat,rat),minus_minus(rat),P2),Q2)) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aTP_Lamp_aao(rat,fun(int,fun(int,product_prod(int,int))),Q2)),quotient_of(P2)) ) ).

% rat_minus_code
tff(fact_6910_rat__times__code,axiom,
    ! [P2: rat,Q2: rat] : ( quotient_of(aa(rat,rat,aa(rat,fun(rat,rat),times_times(rat),P2),Q2)) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aTP_Lamp_aaq(rat,fun(int,fun(int,product_prod(int,int))),Q2)),quotient_of(P2)) ) ).

% rat_times_code
tff(fact_6911_rat__plus__code,axiom,
    ! [P2: rat,Q2: rat] : ( quotient_of(aa(rat,rat,aa(rat,fun(rat,rat),plus_plus(rat),P2),Q2)) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aTP_Lamp_aas(rat,fun(int,fun(int,product_prod(int,int))),Q2)),quotient_of(P2)) ) ).

% rat_plus_code
tff(fact_6912_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_6913_Frct__code__post_I5_J,axiom,
    ! [K3: num] : ( frct(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),numeral_numeral(int,K3))) = divide_divide(rat,one_one(rat),numeral_numeral(rat,K3)) ) ).

% Frct_code_post(5)
tff(fact_6914_gcd__pos__nat,axiom,
    ! [M2: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),M2),N))
    <=> ( ( M2 != zero_zero(nat) )
        | ( N != zero_zero(nat) ) ) ) ).

% gcd_pos_nat
tff(fact_6915_Gcd__in,axiom,
    ! [Aa2: set(nat)] :
      ( ! [A4: nat,B4: nat] :
          ( member(nat,A4,Aa2)
         => ( member(nat,B4,Aa2)
           => member(nat,aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A4),B4),Aa2) ) )
     => ( ( Aa2 != bot_bot(set(nat)) )
       => member(nat,gcd_Gcd(nat,Aa2),Aa2) ) ) ).

% Gcd_in
tff(fact_6916_Gcd__nat__set__eq__fold,axiom,
    ! [Xsa: list(nat)] : ( gcd_Gcd(nat,aa(list(nat),set(nat),set2(nat),Xsa)) = aa(nat,nat,aa(list(nat),fun(nat,nat),aa(fun(nat,fun(nat,nat)),fun(list(nat),fun(nat,nat)),fold(nat,nat),gcd_gcd(nat)),Xsa),zero_zero(nat)) ) ).

% Gcd_nat_set_eq_fold
tff(fact_6917_Frct__code__post_I6_J,axiom,
    ! [K3: num,L: num] : ( frct(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),numeral_numeral(int,K3)),numeral_numeral(int,L))) = divide_divide(rat,numeral_numeral(rat,K3),numeral_numeral(rat,L)) ) ).

% Frct_code_post(6)
tff(fact_6918_gcd__is__Max__divisors__nat,axiom,
    ! [N: nat,M2: 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),M2),N) = lattic643756798349783984er_Max(nat,aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_aat(nat,fun(nat,fun(nat,$o)),N),M2))) ) ) ).

% gcd_is_Max_divisors_nat
tff(fact_6919_Frct__code__post_I4_J,axiom,
    ! [K3: num] : ( frct(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),numeral_numeral(int,K3)),one_one(int))) = numeral_numeral(rat,K3) ) ).

% Frct_code_post(4)
tff(fact_6920_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_6921_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_6922_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_6923_Frct__code__post_I8_J,axiom,
    ! [A3: int,B3: 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),B3))) = 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),B3))) ) ).

% Frct_code_post(8)
tff(fact_6924_Frct__code__post_I7_J,axiom,
    ! [A3: int,B3: 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)),B3)) = 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),B3))) ) ).

% Frct_code_post(7)
tff(fact_6925_gcd__nat_Opelims,axiom,
    ! [Xa: nat,Xaa: nat,Y: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Xa),Xaa) = 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),Xa),Xaa))
       => ~ ( ( Y = $ite(Xaa = zero_zero(nat),Xa,aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Xaa),modulo_modulo(nat,Xa,Xaa))) )
           => ~ accp(product_prod(nat,nat),gcd_nat_rel,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xa),Xaa)) ) ) ) ).

% gcd_nat.pelims
tff(fact_6926_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_zy(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))),aTP_Lamp_zy(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))) ).

% plus_rat.rsp
tff(fact_6927_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_6928_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_6929_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_aau(int,fun(int,product_prod(int,int)))),aTP_Lamp_aau(int,fun(int,product_prod(int,int)))) ).

% Fract.rsp
tff(fact_6930_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_aaf(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))),aTP_Lamp_aaf(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))) ).

% times_rat.rsp
tff(fact_6931_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_aag(product_prod(int,int),product_prod(int,int))),aTP_Lamp_aag(product_prod(int,int),product_prod(int,int))) ).

% uminus_rat.rsp
tff(fact_6932_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_aah(product_prod(int,int),product_prod(int,int))),aTP_Lamp_aah(product_prod(int,int),product_prod(int,int))) ).

% inverse_rat.rsp
tff(fact_6933_plus__rat_Oabs__eq,axiom,
    ! [Xa: product_prod(int,int),Xb: 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,Xb),Xb)
       => ( 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,Xb)) = 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),Xb))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),Xb)),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),Xb)))) ) ) ) ).

% plus_rat.abs_eq
tff(fact_6934_inverse__rat_Oabs__eq,axiom,
    ! [Xa: 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(rat,rat,inverse_inverse(rat),aa(product_prod(int,int),rat,abs_Rat,Xa)) = aa(product_prod(int,int),rat,abs_Rat,
            $ite(aa(product_prod(int,int),int,product_fst(int,int),Xa) = 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),Xa)),aa(product_prod(int,int),int,product_fst(int,int),Xa)))) ) ) ).

% inverse_rat.abs_eq
tff(fact_6935_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_6936_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_6937_uminus__rat_Oabs__eq,axiom,
    ! [Xa: 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(rat,rat,uminus_uminus(rat),aa(product_prod(int,int),rat,abs_Rat,Xa)) = 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),Xa))),aa(product_prod(int,int),int,product_snd(int,int),Xa))) ) ) ).

% uminus_rat.abs_eq
tff(fact_6938_times__rat_Oabs__eq,axiom,
    ! [Xa: product_prod(int,int),Xb: 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,Xb),Xb)
       => ( 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,Xb)) = 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),Xb))),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),Xb)))) ) ) ) ).

% times_rat.abs_eq
tff(fact_6939_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_aah(product_prod(int,int),product_prod(int,int))) ).

% inverse_rat_def
tff(fact_6940_flip__pred,axiom,
    ! [A: $tType,B: $tType,Aa2: set(product_prod(A,B)),R: fun(B,fun(A,$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))),Aa2),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_ot(A,fun(B,product_prod(B,A))))),Aa2)),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_6941_conversep__eq,axiom,
    ! [A: $tType] : ( conversep(A,A,fequal(A)) = fequal(A) ) ).

% conversep_eq
tff(fact_6942_conversep__iff,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(B,$o)),A3: B,B3: A] :
      ( aa(A,$o,aa(B,fun(A,$o),conversep(A,B,R2),A3),B3)
    <=> aa(B,$o,aa(A,fun(B,$o),R2,B3),A3) ) ).

% conversep_iff
tff(fact_6943_conversep__inject,axiom,
    ! [A: $tType,B: $tType,R2: fun(B,fun(A,$o)),S2: fun(B,fun(A,$o))] :
      ( ( conversep(B,A,R2) = conversep(B,A,S2) )
    <=> ( R2 = S2 ) ) ).

% conversep_inject
tff(fact_6944_conversep__conversep,axiom,
    ! [B: $tType,A: $tType,R2: fun(A,fun(B,$o))] : ( conversep(B,A,conversep(A,B,R2)) = R2 ) ).

% conversep_conversep
tff(fact_6945_conversep__noteq,axiom,
    ! [A: $tType,X4: A,Xa4: A] :
      ( aa(A,$o,aa(A,fun(A,$o),conversep(A,A,aTP_Lamp_je(A,fun(A,$o))),X4),Xa4)
    <=> ( X4 != Xa4 ) ) ).

% conversep_noteq
tff(fact_6946_conversep__mono,axiom,
    ! [A: $tType,B: $tType,R2: fun(B,fun(A,$o)),S2: fun(B,fun(A,$o))] :
      ( aa(fun(A,fun(B,$o)),$o,aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),$o),ord_less_eq(fun(A,fun(B,$o))),conversep(B,A,R2)),conversep(B,A,S2))
    <=> aa(fun(B,fun(A,$o)),$o,aa(fun(B,fun(A,$o)),fun(fun(B,fun(A,$o)),$o),ord_less_eq(fun(B,fun(A,$o))),R2),S2) ) ).

% conversep_mono
tff(fact_6947_conversep__le__swap,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(B,$o)),S2: fun(B,fun(A,$o))] :
      ( aa(fun(A,fun(B,$o)),$o,aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),$o),ord_less_eq(fun(A,fun(B,$o))),R2),conversep(B,A,S2))
    <=> aa(fun(B,fun(A,$o)),$o,aa(fun(B,fun(A,$o)),fun(fun(B,fun(A,$o)),$o),ord_less_eq(fun(B,fun(A,$o))),conversep(A,B,R2)),S2) ) ).

% conversep_le_swap
tff(fact_6948_conversep__converse__eq,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(B,A)),X4: A,Xa4: B] :
      ( aa(B,$o,aa(A,fun(B,$o),conversep(B,A,aTP_Lamp_rw(set(product_prod(B,A)),fun(B,fun(A,$o)),R2)),X4),Xa4)
    <=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Xa4),converse(B,A,R2)) ) ).

% conversep_converse_eq
tff(fact_6949_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_6950_conversep_Osimps,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(B,$o)),A1: B,A22: A] :
      ( aa(A,$o,aa(B,fun(A,$o),conversep(A,B,R2),A1),A22)
    <=> ? [A7: A,B8: B] :
          ( ( A1 = B8 )
          & ( A22 = A7 )
          & aa(B,$o,aa(A,fun(B,$o),R2,A7),B8) ) ) ).

% conversep.simps
tff(fact_6951_conversepD,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(B,$o)),B3: B,A3: A] :
      ( aa(A,$o,aa(B,fun(A,$o),conversep(A,B,R2),B3),A3)
     => aa(B,$o,aa(A,fun(B,$o),R2,A3),B3) ) ).

% conversepD
tff(fact_6952_conversepE,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(B,$o)),A1: B,A22: A] :
      ( aa(A,$o,aa(B,fun(A,$o),conversep(A,B,R2),A1),A22)
     => aa(B,$o,aa(A,fun(B,$o),R2,A22),A1) ) ).

% conversepE
tff(fact_6953_conversepI,axiom,
    ! [B: $tType,A: $tType,R2: fun(A,fun(B,$o)),A3: A,B3: B] :
      ( aa(B,$o,aa(A,fun(B,$o),R2,A3),B3)
     => aa(A,$o,aa(B,fun(A,$o),conversep(A,B,R2),B3),A3) ) ).

% conversepI
tff(fact_6954_list_Orel__conversep,axiom,
    ! [A: $tType,B: $tType,R: fun(B,fun(A,$o))] : ( aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),conversep(B,A,R)) = conversep(list(B),list(A),aa(fun(B,fun(A,$o)),fun(list(B),fun(list(A),$o)),list_all2(B,A),R)) ) ).

% list.rel_conversep
tff(fact_6955_list_Orel__flip,axiom,
    ! [B: $tType,A: $tType,R: fun(B,fun(A,$o)),A3: list(A),B3: list(B)] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),conversep(B,A,R)),A3),B3)
    <=> aa(list(A),$o,aa(list(B),fun(list(A),$o),aa(fun(B,fun(A,$o)),fun(list(B),fun(list(A),$o)),list_all2(B,A),R),B3),A3) ) ).

% list.rel_flip
tff(fact_6956_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_6957_rel__filter__conversep,axiom,
    ! [A: $tType,B: $tType,Aa2: fun(B,fun(A,$o))] : ( rel_filter(A,B,conversep(B,A,Aa2)) = conversep(filter(B),filter(A),rel_filter(B,A,Aa2)) ) ).

% rel_filter_conversep
tff(fact_6958_converse__def,axiom,
    ! [B: $tType,A: $tType,X4: set(product_prod(A,B))] : ( converse(A,B,X4) = aa(fun(product_prod(B,A),$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_bi(set(product_prod(A,B)),fun(A,fun(B,$o))),X4)))) ) ).

% converse_def
tff(fact_6959_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_aag(product_prod(int,int),product_prod(int,int))) ).

% uminus_rat_def
tff(fact_6960_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_zy(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))) ).

% plus_rat_def
tff(fact_6961_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_aaf(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))) ).

% times_rat_def
tff(fact_6962_sorted__wrt__iff__nth__Suc__transp,axiom,
    ! [A: $tType,Pa: fun(A,fun(A,$o)),Xsa: list(A)] :
      ( transp(A,Pa)
     => ( sorted_wrt(A,Pa,Xsa)
      <=> ! [I2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,I2)),aa(list(A),nat,size_size(list(A)),Xsa))
           => aa(A,$o,aa(A,fun(A,$o),Pa,aa(nat,A,nth(A,Xsa),I2)),aa(nat,A,nth(A,Xsa),aa(nat,nat,suc,I2))) ) ) ) ).

% sorted_wrt_iff_nth_Suc_transp
tff(fact_6963_prod__set__simps_I1_J,axiom,
    ! [B: $tType,A: $tType,Xa: 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),Xa),Y)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A))) ) ).

% prod_set_simps(1)
tff(fact_6964_transp__ge,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => transp(A,aTP_Lamp_aav(A,fun(A,$o))) ) ).

% transp_ge
tff(fact_6965_transp__le,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => transp(A,ord_less_eq(A)) ) ).

% transp_le
tff(fact_6966_transp__trans__eq,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( transp(A,aTP_Lamp_cz(set(product_prod(A,A)),fun(A,fun(A,$o)),R2))
    <=> trans(A,R2) ) ).

% transp_trans_eq
tff(fact_6967_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_6968_transp__empty,axiom,
    ! [A: $tType] : transp(A,aTP_Lamp_aaw(A,fun(A,$o))) ).

% transp_empty
tff(fact_6969_transp__singleton,axiom,
    ! [A: $tType,A3: A] : transp(A,aTP_Lamp_aax(A,fun(A,fun(A,$o)),A3)) ).

% transp_singleton
tff(fact_6970_transpD,axiom,
    ! [A: $tType,R2: fun(A,fun(A,$o)),Xa: A,Y: A,Z2: A] :
      ( transp(A,R2)
     => ( aa(A,$o,aa(A,fun(A,$o),R2,Xa),Y)
       => ( aa(A,$o,aa(A,fun(A,$o),R2,Y),Z2)
         => aa(A,$o,aa(A,fun(A,$o),R2,Xa),Z2) ) ) ) ).

% transpD
tff(fact_6971_transpE,axiom,
    ! [A: $tType,R2: fun(A,fun(A,$o)),Xa: A,Y: A,Z2: A] :
      ( transp(A,R2)
     => ( aa(A,$o,aa(A,fun(A,$o),R2,Xa),Y)
       => ( aa(A,$o,aa(A,fun(A,$o),R2,Y),Z2)
         => aa(A,$o,aa(A,fun(A,$o),R2,Xa),Z2) ) ) ) ).

% transpE
tff(fact_6972_transpI,axiom,
    ! [A: $tType,R2: fun(A,fun(A,$o))] :
      ( ! [X: A,Y2: A,Z3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),R2,X),Y2)
         => ( aa(A,$o,aa(A,fun(A,$o),R2,Y2),Z3)
           => aa(A,$o,aa(A,fun(A,$o),R2,X),Z3) ) )
     => transp(A,R2) ) ).

% transpI
tff(fact_6973_transp__def,axiom,
    ! [A: $tType,R2: fun(A,fun(A,$o))] :
      ( transp(A,R2)
    <=> ! [X6: A,Y4: A,Z5: A] :
          ( aa(A,$o,aa(A,fun(A,$o),R2,X6),Y4)
         => ( aa(A,$o,aa(A,fun(A,$o),R2,Y4),Z5)
           => aa(A,$o,aa(A,fun(A,$o),R2,X6),Z5) ) ) ) ).

% transp_def
tff(fact_6974_transp__equality,axiom,
    ! [A: $tType] : transp(A,fequal(A)) ).

% transp_equality
tff(fact_6975_list_Orel__transp,axiom,
    ! [A: $tType,R: fun(A,fun(A,$o))] :
      ( transp(A,R)
     => transp(list(A),aa(fun(A,fun(A,$o)),fun(list(A),fun(list(A),$o)),list_all2(A,A),R)) ) ).

% list.rel_transp
tff(fact_6976_sorted__wrt2,axiom,
    ! [A: $tType,Pa: fun(A,fun(A,$o)),Xa: A,Y: A,Zs: list(A)] :
      ( transp(A,Pa)
     => ( sorted_wrt(A,Pa,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Zs)))
      <=> ( aa(A,$o,aa(A,fun(A,$o),Pa,Xa),Y)
          & sorted_wrt(A,Pa,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Zs)) ) ) ) ).

% sorted_wrt2
tff(fact_6977_transp__less,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => transp(A,ord_less(A)) ) ).

% transp_less
tff(fact_6978_transp__gr,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => transp(A,aTP_Lamp_ij(A,fun(A,$o))) ) ).

% transp_gr
tff(fact_6979_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_6980_transp__INF,axiom,
    ! [B: $tType,A: $tType,S: set(A),R2: fun(A,fun(B,fun(B,$o)))] :
      ( ! [X: A] :
          ( member(A,X,S)
         => transp(B,aa(A,fun(B,fun(B,$o)),R2,X)) )
     => transp(B,aa(set(fun(B,fun(B,$o))),fun(B,fun(B,$o)),complete_Inf_Inf(fun(B,fun(B,$o))),aa(set(A),set(fun(B,fun(B,$o))),image2(A,fun(B,fun(B,$o)),R2),S))) ) ).

% transp_INF
tff(fact_6981_prod__set__defs_I1_J,axiom,
    ! [B: $tType,A: $tType,X4: product_prod(A,B)] : ( basic_fsts(A,B,X4) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),aa(product_prod(A,B),A,product_fst(A,B),X4)),bot_bot(set(A))) ) ).

% prod_set_defs(1)
tff(fact_6982_prod__set__simps_I2_J,axiom,
    ! [B: $tType,A: $tType,Xa: 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),Xa),Y)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Y),bot_bot(set(A))) ) ).

% prod_set_simps(2)
tff(fact_6983_prod__set__defs_I2_J,axiom,
    ! [A: $tType,B: $tType,X4: product_prod(A,B)] : ( basic_snds(A,B,X4) = aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert3(B),aa(product_prod(A,B),B,product_snd(A,B),X4)),bot_bot(set(B))) ) ).

% prod_set_defs(2)
tff(fact_6984_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_bi(set(product_prod(A,B)),fun(A,fun(B,$o))),R2))
    <=> single_valued(A,B,R2) ) ).

% single_valuedp_single_valued_eq
tff(fact_6985_Eps__Opt__def,axiom,
    ! [A: $tType,Pa: fun(A,$o)] :
      ( eps_Opt(A,Pa) = $ite(
          ? [X9: A] : aa(A,$o,Pa,X9),
          aa(A,option(A),some(A),fChoice(A,Pa)),
          none(A) ) ) ).

% Eps_Opt_def
tff(fact_6986_Eps__case__prod__eq,axiom,
    ! [A: $tType,B: $tType,Xa: A,Y: B] : ( fChoice(product_prod(A,B),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aa(B,fun(A,fun(B,$o)),aTP_Lamp_ua(A,fun(B,fun(A,fun(B,$o))),Xa),Y))) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa),Y) ) ).

% Eps_case_prod_eq
tff(fact_6987_some__insert__self,axiom,
    ! [A: $tType,S: set(A)] :
      ( ( S != bot_bot(set(A)) )
     => ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),fChoice(A,aa(set(A),fun(A,$o),aTP_Lamp_a(set(A),fun(A,$o)),S))),S) = S ) ) ).

% some_insert_self
tff(fact_6988_single__valuedp__less__eq,axiom,
    ! [B: $tType,A: $tType,R2: fun(A,fun(B,$o)),S2: fun(A,fun(B,$o))] :
      ( aa(fun(A,fun(B,$o)),$o,aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),$o),ord_less_eq(fun(A,fun(B,$o))),R2),S2)
     => ( single_valuedp(A,B,S2)
       => single_valuedp(A,B,R2) ) ) ).

% single_valuedp_less_eq
tff(fact_6989_split__paired__Eps,axiom,
    ! [B: $tType,A: $tType,Pa: fun(product_prod(A,B),$o)] : ( fChoice(product_prod(A,B),Pa) = fChoice(product_prod(A,B),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aTP_Lamp_aay(fun(product_prod(A,B),$o),fun(A,fun(B,$o)),Pa))) ) ).

% split_paired_Eps
tff(fact_6990_inv__on__def,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),Aa2: set(A),Xa: B] : ( aa(B,A,inv_on(A,B,F2,Aa2),Xa) = fChoice(A,aa(B,fun(A,$o),aa(set(A),fun(B,fun(A,$o)),aTP_Lamp_ty(fun(A,B),fun(set(A),fun(B,fun(A,$o))),F2),Aa2),Xa)) ) ).

% inv_on_def
tff(fact_6991_single__valuedpD,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(B,$o)),Xa: A,Y: B,Z2: B] :
      ( single_valuedp(A,B,R2)
     => ( aa(B,$o,aa(A,fun(B,$o),R2,Xa),Y)
       => ( aa(B,$o,aa(A,fun(B,$o),R2,Xa),Z2)
         => ( Y = Z2 ) ) ) ) ).

% single_valuedpD
tff(fact_6992_single__valuedpI,axiom,
    ! [B: $tType,A: $tType,R2: fun(A,fun(B,$o))] :
      ( ! [X: A,Y2: B,Z3: B] :
          ( aa(B,$o,aa(A,fun(B,$o),R2,X),Y2)
         => ( aa(B,$o,aa(A,fun(B,$o),R2,X),Z3)
           => ( Y2 = Z3 ) ) )
     => single_valuedp(A,B,R2) ) ).

% single_valuedpI
tff(fact_6993_single__valuedp__def,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(B,$o))] :
      ( single_valuedp(A,B,R2)
    <=> ! [X6: A,Y4: B] :
          ( aa(B,$o,aa(A,fun(B,$o),R2,X6),Y4)
         => ! [Z5: B] :
              ( aa(B,$o,aa(A,fun(B,$o),R2,X6),Z5)
             => ( Y4 = Z5 ) ) ) ) ).

% single_valuedp_def
tff(fact_6994_some__theI,axiom,
    ! [B: $tType,A: $tType,Pa: fun(A,fun(B,$o))] :
      ( ? [A13: A,X_13: B] : aa(B,$o,aa(A,fun(B,$o),Pa,A13),X_13)
     => ( ! [B16: B,B24: B] :
            ( ? [A4: A] : aa(B,$o,aa(A,fun(B,$o),Pa,A4),B16)
           => ( ? [A4: A] : aa(B,$o,aa(A,fun(B,$o),Pa,A4),B24)
             => ( B16 = B24 ) ) )
       => aa(B,$o,aa(A,fun(B,$o),Pa,fChoice(A,aTP_Lamp_vy(fun(A,fun(B,$o)),fun(A,$o),Pa))),the(B,aTP_Lamp_aaz(fun(A,fun(B,$o)),fun(B,$o),Pa))) ) ) ).

% some_theI
tff(fact_6995_some__elem,axiom,
    ! [A: $tType,S: set(A)] :
      ( ( S != bot_bot(set(A)) )
     => member(A,fChoice(A,aa(set(A),fun(A,$o),aTP_Lamp_a(set(A),fun(A,$o)),S)),S) ) ).

% some_elem
tff(fact_6996_some__in__eq,axiom,
    ! [A: $tType,Aa2: set(A)] :
      ( member(A,fChoice(A,aa(set(A),fun(A,$o),aTP_Lamp_a(set(A),fun(A,$o)),Aa2)),Aa2)
    <=> ( Aa2 != bot_bot(set(A)) ) ) ).

% some_in_eq
tff(fact_6997_single__valuedp__bot,axiom,
    ! [B: $tType,A: $tType] : single_valuedp(A,B,bot_bot(fun(A,fun(B,$o)))) ).

% single_valuedp_bot
tff(fact_6998_single__valuedp__iff__Uniq,axiom,
    ! [B: $tType,A: $tType,R2: fun(A,fun(B,$o))] :
      ( single_valuedp(A,B,R2)
    <=> ! [X6: A] : uniq(B,aa(A,fun(B,$o),R2,X6)) ) ).

% single_valuedp_iff_Uniq
tff(fact_6999_fun__of__rel__def,axiom,
    ! [A: $tType,B: $tType,R: set(product_prod(B,A)),Xa: B] : ( fun_of_rel(B,A,R,Xa) = fChoice(A,aa(B,fun(A,$o),aTP_Lamp_rw(set(product_prod(B,A)),fun(B,fun(A,$o)),R),Xa)) ) ).

% fun_of_rel_def
tff(fact_7000_pair__in__swap__image,axiom,
    ! [A: $tType,B: $tType,Y: A,Xa: B,Aa2: set(product_prod(B,A))] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Y),Xa),aa(set(product_prod(B,A)),set(product_prod(A,B)),image2(product_prod(B,A),product_prod(A,B),product_swap(B,A)),Aa2))
    <=> member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Xa),Y),Aa2) ) ).

% pair_in_swap_image
tff(fact_7001_swap__simp,axiom,
    ! [A: $tType,B: $tType,Xa: 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),Xa),Y)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Y),Xa) ) ).

% swap_simp
tff(fact_7002_prod__filter__commute,axiom,
    ! [B: $tType,A: $tType,Fa: filter(A),G4: filter(B)] : ( aa(filter(B),filter(product_prod(A,B)),aa(filter(A),fun(filter(B),filter(product_prod(A,B))),prod_filter(A,B),Fa),G4) = aa(filter(product_prod(B,A)),filter(product_prod(A,B)),aa(fun(product_prod(B,A),product_prod(A,B)),fun(filter(product_prod(B,A)),filter(product_prod(A,B))),filtermap(product_prod(B,A),product_prod(A,B)),product_swap(B,A)),aa(filter(A),filter(product_prod(B,A)),aa(filter(B),fun(filter(A),filter(product_prod(B,A))),prod_filter(B,A),G4),Fa)) ) ).

% prod_filter_commute
tff(fact_7003_prod_Oswap__def,axiom,
    ! [A: $tType,B: $tType,P2: product_prod(B,A)] : ( aa(product_prod(B,A),product_prod(A,B),product_swap(B,A),P2) = 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),P2)),aa(product_prod(B,A),B,product_fst(B,A),P2)) ) ).

% prod.swap_def
tff(fact_7004_acyclic__insert,axiom,
    ! [A: $tType,Y: A,Xa: 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))),insert3(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Xa)),R2))
    <=> ( transitive_acyclic(A,R2)
        & ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Y),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R2)) ) ) ).

% acyclic_insert
tff(fact_7005_successively__transfer,axiom,
    ! [A: $tType,B: $tType,Aa2: fun(A,fun(B,$o))] : aa(fun(fun(B,fun(B,$o)),fun(list(B),$o)),$o,aa(fun(fun(A,fun(A,$o)),fun(list(A),$o)),fun(fun(fun(B,fun(B,$o)),fun(list(B),$o)),$o),bNF_rel_fun(fun(A,fun(A,$o)),fun(B,fun(B,$o)),fun(list(A),$o),fun(list(B),$o),bNF_rel_fun(A,B,fun(A,$o),fun(B,$o),Aa2,bNF_rel_fun(A,B,$o,$o,Aa2,fequal($o))),bNF_rel_fun(list(A),list(B),$o,$o,aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Aa2),fequal($o))),successively(A)),successively(B)) ).

% successively_transfer
tff(fact_7006_acyclic__empty,axiom,
    ! [A: $tType] : transitive_acyclic(A,bot_bot(set(product_prod(A,A)))) ).

% acyclic_empty
tff(fact_7007_successively__rev,axiom,
    ! [A: $tType,Pa: fun(A,fun(A,$o)),Xsa: list(A)] :
      ( aa(list(A),$o,aa(fun(A,fun(A,$o)),fun(list(A),$o),successively(A),Pa),aa(list(A),list(A),rev(A),Xsa))
    <=> aa(list(A),$o,aa(fun(A,fun(A,$o)),fun(list(A),$o),successively(A),aTP_Lamp_cn(fun(A,fun(A,$o)),fun(A,fun(A,$o)),Pa)),Xsa) ) ).

% successively_rev
tff(fact_7008_successively__conv__sorted__wrt,axiom,
    ! [A: $tType,Pa: fun(A,fun(A,$o)),Xsa: list(A)] :
      ( transp(A,Pa)
     => ( aa(list(A),$o,aa(fun(A,fun(A,$o)),fun(list(A),$o),successively(A),Pa),Xsa)
      <=> sorted_wrt(A,Pa,Xsa) ) ) ).

% successively_conv_sorted_wrt
tff(fact_7009_cyclic__subset,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),S: set(product_prod(A,A))] :
      ( ~ transitive_acyclic(A,R)
     => ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R),S)
       => ~ transitive_acyclic(A,S) ) ) ).

% cyclic_subset
tff(fact_7010_subset__mset_OacyclicI__order,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,A)),F2: fun(A,multiset(B))] :
      ( ! [A4: A,B4: A] :
          ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),B4),R2)
         => aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subset_mset(B),aa(A,multiset(B),F2,B4)),aa(A,multiset(B),F2,A4)) )
     => transitive_acyclic(A,R2) ) ).

% subset_mset.acyclicI_order
tff(fact_7011_acyclicI__order,axiom,
    ! [B: $tType,A: $tType] :
      ( preorder(B)
     => ! [R2: set(product_prod(A,A)),F2: fun(A,B)] :
          ( ! [A4: A,B4: A] :
              ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),B4),R2)
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,B4)),aa(A,B,F2,A4)) )
         => transitive_acyclic(A,R2) ) ) ).

% acyclicI_order
tff(fact_7012_cyclicE,axiom,
    ! [A: $tType,G: set(product_prod(A,A))] :
      ( ~ transitive_acyclic(A,G)
     => ~ ! [X: A] : ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),X),transitive_trancl(A,G)) ) ).

% cyclicE
tff(fact_7013_acyclic__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( transitive_acyclic(A,R2)
    <=> ! [X6: A] : ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X6),X6),transitive_trancl(A,R2)) ) ).

% acyclic_def
tff(fact_7014_acyclicI,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( ! [X: A] : ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),X),transitive_trancl(A,R2))
     => transitive_acyclic(A,R2) ) ).

% acyclicI
tff(fact_7015_successively__altdef,axiom,
    ! [A: $tType,X4: fun(A,fun(A,$o))] : ( aa(fun(A,fun(A,$o)),fun(list(A),$o),successively(A),X4) = aa(fun(A,fun(list(A),fun($o,$o))),fun(list(A),$o),aa($o,fun(fun(A,fun(list(A),fun($o,$o))),fun(list(A),$o)),rec_list($o,A),$true),aTP_Lamp_abb(fun(A,fun(A,$o)),fun(A,fun(list(A),fun($o,$o))),X4)) ) ).

% successively_altdef
tff(fact_7016_successively__remdups__adj__iff,axiom,
    ! [A: $tType,Xsa: list(A),Pa: fun(A,fun(A,$o))] :
      ( ! [X: A] :
          ( member(A,X,aa(list(A),set(A),set2(A),Xsa))
         => aa(A,$o,aa(A,fun(A,$o),Pa,X),X) )
     => ( aa(list(A),$o,aa(fun(A,fun(A,$o)),fun(list(A),$o),successively(A),Pa),aa(list(A),list(A),remdups_adj(A),Xsa))
      <=> aa(list(A),$o,aa(fun(A,fun(A,$o)),fun(list(A),$o),successively(A),Pa),Xsa) ) ) ).

% successively_remdups_adj_iff
tff(fact_7017_distinct__adj__def,axiom,
    ! [A: $tType] : ( distinct_adj(A) = aa(fun(A,fun(A,$o)),fun(list(A),$o),successively(A),aTP_Lamp_je(A,fun(A,$o))) ) ).

% distinct_adj_def
tff(fact_7018_acyclic__union_I1_J,axiom,
    ! [A: $tType,Aa2: set(product_prod(A,A)),B2: 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))),Aa2),B2))
     => transitive_acyclic(A,Aa2) ) ).

% acyclic_union(1)
tff(fact_7019_acyclic__union_I2_J,axiom,
    ! [A: $tType,Aa2: set(product_prod(A,A)),B2: 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))),Aa2),B2))
     => transitive_acyclic(A,B2) ) ).

% acyclic_union(2)
tff(fact_7020_successively_Osimps_I3_J,axiom,
    ! [A: $tType,Pa: fun(A,fun(A,$o)),Xa: A,Y: A,Xsa: list(A)] :
      ( aa(list(A),$o,aa(fun(A,fun(A,$o)),fun(list(A),$o),successively(A),Pa),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Xsa)))
    <=> ( aa(A,$o,aa(A,fun(A,$o),Pa,Xa),Y)
        & aa(list(A),$o,aa(fun(A,fun(A,$o)),fun(list(A),$o),successively(A),Pa),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Xsa)) ) ) ).

% successively.simps(3)
tff(fact_7021_successively_Oelims_I3_J,axiom,
    ! [A: $tType,Xa: fun(A,fun(A,$o)),Xaa: list(A)] :
      ( ~ aa(list(A),$o,aa(fun(A,fun(A,$o)),fun(list(A),$o),successively(A),Xa),Xaa)
     => ~ ! [X: A,Y2: A,Xs: list(A)] :
            ( ( Xaa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Xs)) )
           => ( aa(A,$o,aa(A,fun(A,$o),Xa,X),Y2)
              & aa(list(A),$o,aa(fun(A,fun(A,$o)),fun(list(A),$o),successively(A),Xa),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Xs)) ) ) ) ).

% successively.elims(3)
tff(fact_7022_successively__cong,axiom,
    ! [A: $tType,Xsa: list(A),Pa: fun(A,fun(A,$o)),Qa: fun(A,fun(A,$o)),Ys: list(A)] :
      ( ! [X: A,Y2: A] :
          ( member(A,X,aa(list(A),set(A),set2(A),Xsa))
         => ( member(A,Y2,aa(list(A),set(A),set2(A),Xsa))
           => ( aa(A,$o,aa(A,fun(A,$o),Pa,X),Y2)
            <=> aa(A,$o,aa(A,fun(A,$o),Qa,X),Y2) ) ) )
     => ( ( Xsa = Ys )
       => ( aa(list(A),$o,aa(fun(A,fun(A,$o)),fun(list(A),$o),successively(A),Pa),Xsa)
        <=> aa(list(A),$o,aa(fun(A,fun(A,$o)),fun(list(A),$o),successively(A),Qa),Ys) ) ) ) ).

% successively_cong
tff(fact_7023_successively__mono,axiom,
    ! [A: $tType,Pa: fun(A,fun(A,$o)),Xsa: list(A),Qa: fun(A,fun(A,$o))] :
      ( aa(list(A),$o,aa(fun(A,fun(A,$o)),fun(list(A),$o),successively(A),Pa),Xsa)
     => ( ! [X: A,Y2: A] :
            ( member(A,X,aa(list(A),set(A),set2(A),Xsa))
           => ( member(A,Y2,aa(list(A),set(A),set2(A),Xsa))
             => ( aa(A,$o,aa(A,fun(A,$o),Pa,X),Y2)
               => aa(A,$o,aa(A,fun(A,$o),Qa,X),Y2) ) ) )
       => aa(list(A),$o,aa(fun(A,fun(A,$o)),fun(list(A),$o),successively(A),Qa),Xsa) ) ) ).

% successively_mono
tff(fact_7024_successively__if__sorted__wrt,axiom,
    ! [A: $tType,Pa: fun(A,fun(A,$o)),Xsa: list(A)] :
      ( sorted_wrt(A,Pa,Xsa)
     => aa(list(A),$o,aa(fun(A,fun(A,$o)),fun(list(A),$o),successively(A),Pa),Xsa) ) ).

% successively_if_sorted_wrt
tff(fact_7025_successively__remdups__adjI,axiom,
    ! [A: $tType,Pa: fun(A,fun(A,$o)),Xsa: list(A)] :
      ( aa(list(A),$o,aa(fun(A,fun(A,$o)),fun(list(A),$o),successively(A),Pa),Xsa)
     => aa(list(A),$o,aa(fun(A,fun(A,$o)),fun(list(A),$o),successively(A),Pa),aa(list(A),list(A),remdups_adj(A),Xsa)) ) ).

% successively_remdups_adjI
tff(fact_7026_successively_Osimps_I1_J,axiom,
    ! [A: $tType,Pa: fun(A,fun(A,$o))] : aa(list(A),$o,aa(fun(A,fun(A,$o)),fun(list(A),$o),successively(A),Pa),nil(A)) ).

% successively.simps(1)
tff(fact_7027_successively_Oelims_I2_J,axiom,
    ! [A: $tType,Xa: fun(A,fun(A,$o)),Xaa: list(A)] :
      ( aa(list(A),$o,aa(fun(A,fun(A,$o)),fun(list(A),$o),successively(A),Xa),Xaa)
     => ( ( Xaa != nil(A) )
       => ( ! [X: A] : ( Xaa != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)) )
         => ~ ! [X: A,Y2: A,Xs: list(A)] :
                ( ( Xaa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Xs)) )
               => ~ ( aa(A,$o,aa(A,fun(A,$o),Xa,X),Y2)
                    & aa(list(A),$o,aa(fun(A,fun(A,$o)),fun(list(A),$o),successively(A),Xa),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Xs)) ) ) ) ) ) ).

% successively.elims(2)
tff(fact_7028_successively_Oelims_I1_J,axiom,
    ! [A: $tType,Xa: fun(A,fun(A,$o)),Xaa: list(A),Y: $o] :
      ( ( aa(list(A),$o,aa(fun(A,fun(A,$o)),fun(list(A),$o),successively(A),Xa),Xaa)
      <=> (Y) )
     => ( ( ( Xaa = nil(A) )
         => ~ (Y) )
       => ( ( ? [X: A] : ( Xaa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)) )
           => ~ (Y) )
         => ~ ! [X: A,Y2: A,Xs: list(A)] :
                ( ( Xaa = 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),Y2),Xs)) )
               => ( (Y)
                <=> ~ ( aa(A,$o,aa(A,fun(A,$o),Xa,X),Y2)
                      & aa(list(A),$o,aa(fun(A,fun(A,$o)),fun(list(A),$o),successively(A),Xa),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Xs)) ) ) ) ) ) ) ).

% successively.elims(1)
tff(fact_7029_successively_Osimps_I2_J,axiom,
    ! [A: $tType,Pa: fun(A,fun(A,$o)),Xa: A] : aa(list(A),$o,aa(fun(A,fun(A,$o)),fun(list(A),$o),successively(A),Pa),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),nil(A))) ).

% successively.simps(2)
tff(fact_7030_successively__iff__sorted__wrt__strong,axiom,
    ! [A: $tType,Xsa: list(A),Pa: fun(A,fun(A,$o))] :
      ( ! [X: A,Y2: A,Z3: A] :
          ( member(A,X,aa(list(A),set(A),set2(A),Xsa))
         => ( member(A,Y2,aa(list(A),set(A),set2(A),Xsa))
           => ( member(A,Z3,aa(list(A),set(A),set2(A),Xsa))
             => ( aa(A,$o,aa(A,fun(A,$o),Pa,X),Y2)
               => ( aa(A,$o,aa(A,fun(A,$o),Pa,Y2),Z3)
                 => aa(A,$o,aa(A,fun(A,$o),Pa,X),Z3) ) ) ) ) )
     => ( aa(list(A),$o,aa(fun(A,fun(A,$o)),fun(list(A),$o),successively(A),Pa),Xsa)
      <=> sorted_wrt(A,Pa,Xsa) ) ) ).

% successively_iff_sorted_wrt_strong
tff(fact_7031_successively__map,axiom,
    ! [A: $tType,B: $tType,Pa: fun(A,fun(A,$o)),F2: fun(B,A),Xsa: list(B)] :
      ( aa(list(A),$o,aa(fun(A,fun(A,$o)),fun(list(A),$o),successively(A),Pa),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Xsa))
    <=> aa(list(B),$o,aa(fun(B,fun(B,$o)),fun(list(B),$o),successively(B),aa(fun(B,A),fun(B,fun(B,$o)),aTP_Lamp_hi(fun(A,fun(A,$o)),fun(fun(B,A),fun(B,fun(B,$o))),Pa),F2)),Xsa) ) ).

% successively_map
tff(fact_7032_wf__set,axiom,
    ! [A: $tType,Xsa: list(product_prod(A,A))] :
      ( wf(A,aa(list(product_prod(A,A)),set(product_prod(A,A)),set2(product_prod(A,A)),Xsa))
    <=> transitive_acyclic(A,aa(list(product_prod(A,A)),set(product_prod(A,A)),set2(product_prod(A,A)),Xsa)) ) ).

% wf_set
tff(fact_7033_successively__Cons,axiom,
    ! [A: $tType,Pa: fun(A,fun(A,$o)),Xa: A,Xsa: list(A)] :
      ( aa(list(A),$o,aa(fun(A,fun(A,$o)),fun(list(A),$o),successively(A),Pa),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa))
    <=> ( ( Xsa = nil(A) )
        | ( aa(A,$o,aa(A,fun(A,$o),Pa,Xa),aa(list(A),A,hd(A),Xsa))
          & aa(list(A),$o,aa(fun(A,fun(A,$o)),fun(list(A),$o),successively(A),Pa),Xsa) ) ) ) ).

% successively_Cons
tff(fact_7034_acyclic__insert__cyclic,axiom,
    ! [A: $tType,G: set(product_prod(A,A)),Xa: 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))),insert3(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Y)),G))
       => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Xa),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),G)) ) ) ).

% acyclic_insert_cyclic
tff(fact_7035_successively__append__iff,axiom,
    ! [A: $tType,Pa: fun(A,fun(A,$o)),Xsa: list(A),Ys: list(A)] :
      ( aa(list(A),$o,aa(fun(A,fun(A,$o)),fun(list(A),$o),successively(A),Pa),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),Ys))
    <=> ( aa(list(A),$o,aa(fun(A,fun(A,$o)),fun(list(A),$o),successively(A),Pa),Xsa)
        & aa(list(A),$o,aa(fun(A,fun(A,$o)),fun(list(A),$o),successively(A),Pa),Ys)
        & ( ( Xsa = nil(A) )
          | ( Ys = nil(A) )
          | aa(A,$o,aa(A,fun(A,$o),Pa,last(A,Xsa)),aa(list(A),A,hd(A),Ys)) ) ) ) ).

% successively_append_iff
tff(fact_7036_successively_Opelims_I1_J,axiom,
    ! [A: $tType,Xa: fun(A,fun(A,$o)),Xaa: list(A),Y: $o] :
      ( ( aa(list(A),$o,aa(fun(A,fun(A,$o)),fun(list(A),$o),successively(A),Xa),Xaa)
      <=> (Y) )
     => ( accp(product_prod(fun(A,fun(A,$o)),list(A)),successively_rel(A),aa(list(A),product_prod(fun(A,fun(A,$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,$o)),list(A)),Xa),Xaa))
       => ( ( ( Xaa = nil(A) )
           => ( (Y)
             => ~ accp(product_prod(fun(A,fun(A,$o)),list(A)),successively_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)),Xa),nil(A))) ) )
         => ( ! [X: A] :
                ( ( Xaa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)) )
               => ( (Y)
                 => ~ accp(product_prod(fun(A,fun(A,$o)),list(A)),successively_rel(A),aa(list(A),product_prod(fun(A,fun(A,$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,$o)),list(A)),Xa),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)))) ) )
           => ~ ! [X: A,Y2: A,Xs: list(A)] :
                  ( ( Xaa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Xs)) )
                 => ( ( (Y)
                    <=> ( aa(A,$o,aa(A,fun(A,$o),Xa,X),Y2)
                        & aa(list(A),$o,aa(fun(A,fun(A,$o)),fun(list(A),$o),successively(A),Xa),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Xs)) ) )
                   => ~ accp(product_prod(fun(A,fun(A,$o)),list(A)),successively_rel(A),aa(list(A),product_prod(fun(A,fun(A,$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)),Xa),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),Y2),Xs)))) ) ) ) ) ) ) ).

% successively.pelims(1)
tff(fact_7037_successively_Opelims_I2_J,axiom,
    ! [A: $tType,Xa: fun(A,fun(A,$o)),Xaa: list(A)] :
      ( aa(list(A),$o,aa(fun(A,fun(A,$o)),fun(list(A),$o),successively(A),Xa),Xaa)
     => ( accp(product_prod(fun(A,fun(A,$o)),list(A)),successively_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)),Xa),Xaa))
       => ( ( ( Xaa = nil(A) )
           => ~ accp(product_prod(fun(A,fun(A,$o)),list(A)),successively_rel(A),aa(list(A),product_prod(fun(A,fun(A,$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,$o)),list(A)),Xa),nil(A))) )
         => ( ! [X: A] :
                ( ( Xaa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)) )
               => ~ accp(product_prod(fun(A,fun(A,$o)),list(A)),successively_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)),Xa),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)))) )
           => ~ ! [X: A,Y2: A,Xs: list(A)] :
                  ( ( Xaa = 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),Y2),Xs)) )
                 => ( accp(product_prod(fun(A,fun(A,$o)),list(A)),successively_rel(A),aa(list(A),product_prod(fun(A,fun(A,$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,$o)),list(A)),Xa),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Xs))))
                   => ~ ( aa(A,$o,aa(A,fun(A,$o),Xa,X),Y2)
                        & aa(list(A),$o,aa(fun(A,fun(A,$o)),fun(list(A),$o),successively(A),Xa),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Xs)) ) ) ) ) ) ) ) ).

% successively.pelims(2)
tff(fact_7038_successively_Opelims_I3_J,axiom,
    ! [A: $tType,Xa: fun(A,fun(A,$o)),Xaa: list(A)] :
      ( ~ aa(list(A),$o,aa(fun(A,fun(A,$o)),fun(list(A),$o),successively(A),Xa),Xaa)
     => ( accp(product_prod(fun(A,fun(A,$o)),list(A)),successively_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)),Xa),Xaa))
       => ~ ! [X: A,Y2: A,Xs: list(A)] :
              ( ( Xaa = 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),Y2),Xs)) )
             => ( accp(product_prod(fun(A,fun(A,$o)),list(A)),successively_rel(A),aa(list(A),product_prod(fun(A,fun(A,$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,$o)),list(A)),Xa),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Xs))))
               => ( aa(A,$o,aa(A,fun(A,$o),Xa,X),Y2)
                  & aa(list(A),$o,aa(fun(A,fun(A,$o)),fun(list(A),$o),successively(A),Xa),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Xs)) ) ) ) ) ) ).

% successively.pelims(3)
tff(fact_7039_su__rel__fun__def,axiom,
    ! [B: $tType,A: $tType,Fa: set(product_prod(A,B)),F2: fun(A,B)] :
      ( su_rel_fun(A,B,Fa,F2)
    <=> ( ! [A11: A,B12: B,B17: B] :
            ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A11),B12),Fa)
           => ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A11),B17),Fa)
             => ( B12 = B17 ) ) )
        & ! [A11: A,P6: $o] :
            ( ! [B12: B] :
                ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A11),B12),Fa)
               => (P6) )
           => (P6) )
        & ! [A11: A] : ( aa(A,B,F2,A11) = the(B,aa(A,fun(B,$o),aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_bi(set(product_prod(A,B)),fun(A,fun(B,$o))),Fa),A11)) ) ) ) ).

% su_rel_fun_def
tff(fact_7040_su__rel__fun_Ointro,axiom,
    ! [B: $tType,A: $tType,Fa: set(product_prod(A,B)),F2: fun(A,B)] :
      ( ! [A12: A,B5: B,B7: B] :
          ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A12),B5),Fa)
         => ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A12),B7),Fa)
           => ( B5 = B7 ) ) )
     => ( ! [A12: A,P3: $o] :
            ( ! [B13: B] :
                ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A12),B13),Fa)
               => (P3) )
           => (P3) )
       => ( ! [A12: A] : ( aa(A,B,F2,A12) = the(B,aa(A,fun(B,$o),aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_bi(set(product_prod(A,B)),fun(A,fun(B,$o))),Fa),A12)) )
         => su_rel_fun(A,B,Fa,F2) ) ) ) ).

% su_rel_fun.intro
tff(fact_7041_su__rel__fun_Orepr,axiom,
    ! [B: $tType,A: $tType,Fa: set(product_prod(A,B)),F2: fun(A,B),Aa2: A,B2: B] :
      ( su_rel_fun(A,B,Fa,F2)
     => ( ( aa(A,B,F2,Aa2) = B2 )
      <=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa2),B2),Fa) ) ) ).

% su_rel_fun.repr
tff(fact_7042_su__rel__fun_Orepr1,axiom,
    ! [B: $tType,A: $tType,Fa: set(product_prod(A,B)),F2: fun(A,B),Aa2: A] :
      ( su_rel_fun(A,B,Fa,F2)
     => member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa2),aa(A,B,F2,Aa2)),Fa) ) ).

% su_rel_fun.repr1
tff(fact_7043_su__rel__fun_Orepr2,axiom,
    ! [B: $tType,A: $tType,Fa: set(product_prod(A,B)),F2: fun(A,B),Aa2: A,B2: B] :
      ( su_rel_fun(A,B,Fa,F2)
     => ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa2),B2),Fa)
       => ( B2 = aa(A,B,F2,Aa2) ) ) ) ).

% su_rel_fun.repr2
tff(fact_7044_su__rel__fun_Ounique,axiom,
    ! [A: $tType,B: $tType,Fa: set(product_prod(A,B)),F2: fun(A,B),Aa2: A,B2: B,B10: B] :
      ( su_rel_fun(A,B,Fa,F2)
     => ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa2),B2),Fa)
       => ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa2),B10),Fa)
         => ( B2 = B10 ) ) ) ) ).

% su_rel_fun.unique
tff(fact_7045_su__rel__fun_Osurjective,axiom,
    ! [B: $tType,A: $tType,Fa: set(product_prod(A,B)),F2: fun(A,B),Aa2: A] :
      ( su_rel_fun(A,B,Fa,F2)
     => ~ ! [B5: B] : ~ member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa2),B5),Fa) ) ).

% su_rel_fun.surjective
tff(fact_7046_su__rel__fun_Of__def,axiom,
    ! [B: $tType,A: $tType,Fa: set(product_prod(A,B)),F2: fun(A,B),Aa2: A] :
      ( su_rel_fun(A,B,Fa,F2)
     => ( aa(A,B,F2,Aa2) = the(B,aa(A,fun(B,$o),aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_bi(set(product_prod(A,B)),fun(A,fun(B,$o))),Fa),Aa2)) ) ) ).

% su_rel_fun.f_def
tff(fact_7047_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,$o),aTP_Lamp_abc(set(product_prod(A,A)),fun(A,fun(A,$o)),R2),A3)) ) ).

% aboveS_def
tff(fact_7048_map__of__map,axiom,
    ! [B: $tType,E: $tType,A: $tType,F2: fun(E,B),Xsa: list(product_prod(A,E))] : ( map_of(A,B,aa(list(product_prod(A,E)),list(product_prod(A,B)),aa(fun(product_prod(A,E),product_prod(A,B)),fun(list(product_prod(A,E)),list(product_prod(A,B))),map(product_prod(A,E),product_prod(A,B)),aa(fun(A,fun(E,product_prod(A,B))),fun(product_prod(A,E),product_prod(A,B)),product_case_prod(A,E,product_prod(A,B)),aTP_Lamp_hm(fun(E,B),fun(A,fun(E,product_prod(A,B))),F2))),Xsa)) = comp(option(E),option(B),A,map_option(E,B,F2),map_of(A,E,Xsa)) ) ).

% map_of_map
tff(fact_7049_ex__assn__def,axiom,
    ! [A: $tType,Pa: fun(A,assn)] : ( ex_assn(A,Pa) = aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),assn,abs_assn,aTP_Lamp_abd(fun(A,assn),fun(product_prod(heap_ext(product_unit),set(nat)),$o),Pa)) ) ).

% ex_assn_def
tff(fact_7050_relInvImage__def,axiom,
    ! [B: $tType,A: $tType,Aa2: set(A),R: set(product_prod(B,B)),F2: fun(A,B)] : ( bNF_Gr7122648621184425601vImage(A,B,Aa2,R,F2) = aa(fun(product_prod(A,A),$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_abe(set(A),fun(set(product_prod(B,B)),fun(fun(A,B),fun(product_prod(A,A),$o))),Aa2),R),F2)) ) ).

% relInvImage_def
tff(fact_7051_Rep__assn__inverse,axiom,
    ! [Xa: 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,Xa)) = Xa ) ).

% Rep_assn_inverse
tff(fact_7052_Abs__assn__eqI_I2_J,axiom,
    ! [Pa: 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,Pa,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,Pa) ) ) ).

% Abs_assn_eqI(2)
tff(fact_7053_Abs__assn__eqI_I1_J,axiom,
    ! [Pa: 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,Pa,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,Pa) = Pr ) ) ).

% Abs_assn_eqI(1)
tff(fact_7054_pure__assn__def,axiom,
    ! [B3: $o] : ( pure_assn((B3)) = aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),assn,abs_assn,pure_assn_raw(heap_ext(product_unit),nat,(B3))) ) ).

% pure_assn_def
tff(fact_7055_bot__assn__def,axiom,
    bot_bot(assn) = aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),assn,abs_assn,aTP_Lamp_abf(product_prod(heap_ext(product_unit),set(nat)),$o)) ).

% bot_assn_def
tff(fact_7056_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_7057_sup__assn__def,axiom,
    ! [Pa: assn,Qa: assn] : ( aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),Pa),Qa) = 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_abg(assn,fun(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o)),Pa),Qa)) ) ).

% sup_assn_def
tff(fact_7058_inf__assn__def,axiom,
    ! [Pa: assn,Qa: assn] : ( aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),Pa),Qa) = 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_abh(assn,fun(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o)),Pa),Qa)) ) ).

% inf_assn_def
tff(fact_7059_times__assn__def,axiom,
    ! [Pa: assn,Qa: assn] : ( aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Pa),Qa) = 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,Pa),aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,Qa))) ) ).

% times_assn_def
tff(fact_7060_wand__assn__def,axiom,
    ! [Pa: assn,Qa: assn] : ( wand_assn(Pa,Qa) = 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,Pa),aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,Qa))) ) ).

% wand_assn_def
tff(fact_7061_sngr__assn__def,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [R2: ref(A),Xa: A] : ( sngr_assn(A,R2,Xa) = aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),assn,abs_assn,sngr_assn_raw(A,R2,Xa)) ) ) ).

% sngr_assn_def
tff(fact_7062_snga__assn__def,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [R2: array(A),A3: list(A)] : ( snga_assn(A,R2,A3) = aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),assn,abs_assn,snga_assn_raw(A,R2,A3)) ) ) ).

% snga_assn_def
tff(fact_7063_uminus__assn__def,axiom,
    ! [Pa: assn] : ( aa(assn,assn,uminus_uminus(assn),Pa) = aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),assn,abs_assn,aTP_Lamp_abi(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),Pa)) ) ).

% uminus_assn_def
tff(fact_7064_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_7065_mod__true,axiom,
    ! [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,top_top(assn)),H2)
    <=> aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,H2) ) ).

% mod_true
tff(fact_7066_in__range__dist__union,axiom,
    ! [H2: heap_ext(product_unit),As2: set(nat),As3: set(nat)] :
      ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As2),As3)))
    <=> ( aa(product_prod(heap_ext(product_unit),set(nat)),$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),As2))
        & 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_7067_mod__not__dist,axiom,
    ! [Pa: 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,uminus_uminus(assn),Pa)),H2)
    <=> ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,H2)
        & ~ aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,Pa),H2) ) ) ).

% mod_not_dist
tff(fact_7068_in__range__subset,axiom,
    ! [As2: 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)),As2),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),As2)) ) ) ).

% in_range_subset
tff(fact_7069_models__in__range,axiom,
    ! [Pa: 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,Pa),H2)
     => aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,H2) ) ).

% models_in_range
tff(fact_7070_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_7071_wand__assnI,axiom,
    ! [H2: heap_ext(product_unit),As2: set(nat),Qa: assn,R: assn] :
      ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As2))
     => ( ! [H6: heap_ext(product_unit),As5: set(nat)] :
            ( ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As2),As5) = bot_bot(set(nat)) )
           => ( relH(As2,H2,H6)
             => ( aa(product_prod(heap_ext(product_unit),set(nat)),$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),As2))
               => ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,Qa),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),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)),H6),aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As2),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(Qa,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),As2)) ) ) ).

% wand_assnI
tff(fact_7072_wand__raw_Oelims_I3_J,axiom,
    ! [Xa: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xaa: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xb: product_prod(heap_ext(product_unit),set(nat))] :
      ( ~ aa(product_prod(heap_ext(product_unit),set(nat)),$o,wand_raw(Xa,Xaa),Xb)
     => ~ ! [H: heap_ext(product_unit),As: set(nat)] :
            ( ( Xb = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As) )
           => ( aa(product_prod(heap_ext(product_unit),set(nat)),$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)),H),As))
              & ! [H6: 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,H,H6)
                    & aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H6),As))
                    & aa(product_prod(heap_ext(product_unit),set(nat)),$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)),H6),As5)) )
                 => aa(product_prod(heap_ext(product_unit),set(nat)),$o,Xaa,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),aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As),As5))) ) ) ) ) ).

% wand_raw.elims(3)
tff(fact_7073_relH__dist__union,axiom,
    ! [As2: set(nat),As3: set(nat),H2: heap_ext(product_unit),H4: heap_ext(product_unit)] :
      ( relH(aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As2),As3),H2,H4)
    <=> ( relH(As2,H2,H4)
        & relH(As3,H2,H4) ) ) ).

% relH_dist_union
tff(fact_7074_relH__refl,axiom,
    ! [H2: heap_ext(product_unit),As2: 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),As2))
     => relH(As2,H2,H2) ) ).

% relH_refl
tff(fact_7075_relH__in__rangeI_I1_J,axiom,
    ! [As2: set(nat),H2: heap_ext(product_unit),H4: heap_ext(product_unit)] :
      ( relH(As2,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)),H2),As2)) ) ).

% relH_in_rangeI(1)
tff(fact_7076_relH__in__rangeI_I2_J,axiom,
    ! [As2: set(nat),H2: heap_ext(product_unit),H4: heap_ext(product_unit)] :
      ( relH(As2,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),As2)) ) ).

% relH_in_rangeI(2)
tff(fact_7077_mod__relH,axiom,
    ! [As2: set(nat),H2: heap_ext(product_unit),H4: heap_ext(product_unit),Pa: assn] :
      ( relH(As2,H2,H4)
     => ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,Pa),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))
      <=> aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,Pa),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),As2)) ) ) ).

% mod_relH
tff(fact_7078_relH__trans,axiom,
    ! [As2: set(nat),H12: heap_ext(product_unit),H23: heap_ext(product_unit),H32: heap_ext(product_unit)] :
      ( relH(As2,H12,H23)
     => ( relH(As2,H23,H32)
       => relH(As2,H12,H32) ) ) ).

% relH_trans
tff(fact_7079_relH__sym,axiom,
    ! [As2: set(nat),H2: heap_ext(product_unit),H4: heap_ext(product_unit)] :
      ( relH(As2,H2,H4)
     => relH(As2,H4,H2) ) ).

% relH_sym
tff(fact_7080_relH__subset,axiom,
    ! [Bs: set(nat),H2: heap_ext(product_unit),H4: heap_ext(product_unit),As2: set(nat)] :
      ( relH(Bs,H2,H4)
     => ( aa(set(nat),$o,aa(set(nat),fun(set(nat),$o),ord_less_eq(set(nat)),As2),Bs)
       => relH(As2,H2,H4) ) ) ).

% relH_subset
tff(fact_7081_wand__raw_Osimps,axiom,
    ! [Pa: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Qa: fun(product_prod(heap_ext(product_unit),set(nat)),$o),H2: heap_ext(product_unit),As2: set(nat)] :
      ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,wand_raw(Pa,Qa),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As2))
    <=> ( aa(product_prod(heap_ext(product_unit),set(nat)),$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),As2))
        & ! [H8: heap_ext(product_unit),As6: set(nat)] :
            ( ( ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As2),As6) = bot_bot(set(nat)) )
              & relH(As2,H2,H8)
              & aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H8),As2))
              & aa(product_prod(heap_ext(product_unit),set(nat)),$o,Pa,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)),H8),As6)) )
           => aa(product_prod(heap_ext(product_unit),set(nat)),$o,Qa,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)),H8),aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As2),As6))) ) ) ) ).

% wand_raw.simps
tff(fact_7082_wand__raw_Oelims_I1_J,axiom,
    ! [Xa: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xaa: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xb: product_prod(heap_ext(product_unit),set(nat)),Y: $o] :
      ( ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,wand_raw(Xa,Xaa),Xb)
      <=> (Y) )
     => ~ ! [H: heap_ext(product_unit),As: 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),As) )
           => ( (Y)
            <=> ~ ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As))
                  & ! [H8: heap_ext(product_unit),As6: set(nat)] :
                      ( ( ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As),As6) = bot_bot(set(nat)) )
                        & relH(As,H,H8)
                        & aa(product_prod(heap_ext(product_unit),set(nat)),$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)),H8),As))
                        & aa(product_prod(heap_ext(product_unit),set(nat)),$o,Xa,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H8),As6)) )
                     => aa(product_prod(heap_ext(product_unit),set(nat)),$o,Xaa,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)),H8),aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As),As6))) ) ) ) ) ) ).

% wand_raw.elims(1)
tff(fact_7083_wand__raw_Oelims_I2_J,axiom,
    ! [Xa: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xaa: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xb: product_prod(heap_ext(product_unit),set(nat))] :
      ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,wand_raw(Xa,Xaa),Xb)
     => ~ ! [H: heap_ext(product_unit),As: set(nat)] :
            ( ( Xb = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As) )
           => ~ ( aa(product_prod(heap_ext(product_unit),set(nat)),$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)),H),As))
                & ! [H9: heap_ext(product_unit),As7: set(nat)] :
                    ( ( ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As),As7) = bot_bot(set(nat)) )
                      & relH(As,H,H9)
                      & aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H9),As))
                      & aa(product_prod(heap_ext(product_unit),set(nat)),$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)),H9),As7)) )
                   => aa(product_prod(heap_ext(product_unit),set(nat)),$o,Xaa,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)),H9),aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As),As7))) ) ) ) ) ).

% wand_raw.elims(2)
tff(fact_7084_wand__raw_Opelims_I3_J,axiom,
    ! [Xa: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xaa: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xb: product_prod(heap_ext(product_unit),set(nat))] :
      ( ~ aa(product_prod(heap_ext(product_unit),set(nat)),$o,wand_raw(Xa,Xaa),Xb)
     => ( accp(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$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)))),Xa),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))),Xaa),Xb)))
       => ~ ! [H: heap_ext(product_unit),As: 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),As) )
             => ( 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)),$o),product_prod(heap_ext(product_unit),set(nat)))),Xa),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$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)),$o),product_prod(heap_ext(product_unit),set(nat))),Xaa),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As))))
               => ( aa(product_prod(heap_ext(product_unit),set(nat)),$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)),H),As))
                  & ! [H6: 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,H,H6)
                        & aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H6),As))
                        & aa(product_prod(heap_ext(product_unit),set(nat)),$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)),H6),As5)) )
                     => aa(product_prod(heap_ext(product_unit),set(nat)),$o,Xaa,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),aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As),As5))) ) ) ) ) ) ) ).

% wand_raw.pelims(3)
tff(fact_7085_wand__raw_Opelims_I2_J,axiom,
    ! [Xa: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xaa: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xb: product_prod(heap_ext(product_unit),set(nat))] :
      ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,wand_raw(Xa,Xaa),Xb)
     => ( accp(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$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)))),Xa),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))),Xaa),Xb)))
       => ~ ! [H: heap_ext(product_unit),As: 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),As) )
             => ( 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)),$o),product_prod(heap_ext(product_unit),set(nat)))),Xa),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$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)),$o),product_prod(heap_ext(product_unit),set(nat))),Xaa),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As))))
               => ~ ( aa(product_prod(heap_ext(product_unit),set(nat)),$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)),H),As))
                    & ! [H9: heap_ext(product_unit),As7: set(nat)] :
                        ( ( ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As),As7) = bot_bot(set(nat)) )
                          & relH(As,H,H9)
                          & aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H9),As))
                          & aa(product_prod(heap_ext(product_unit),set(nat)),$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)),H9),As7)) )
                       => aa(product_prod(heap_ext(product_unit),set(nat)),$o,Xaa,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)),H9),aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As),As7))) ) ) ) ) ) ) ).

% wand_raw.pelims(2)
tff(fact_7086_wand__raw_Opelims_I1_J,axiom,
    ! [Xa: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xaa: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xb: product_prod(heap_ext(product_unit),set(nat)),Y: $o] :
      ( ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,wand_raw(Xa,Xaa),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)),$o),product_prod(heap_ext(product_unit),set(nat)))),Xa),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$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)),$o),product_prod(heap_ext(product_unit),set(nat))),Xaa),Xb)))
       => ~ ! [H: heap_ext(product_unit),As: set(nat)] :
              ( ( Xb = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As) )
             => ( ( (Y)
                <=> ( aa(product_prod(heap_ext(product_unit),set(nat)),$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),As))
                    & ! [H8: heap_ext(product_unit),As6: set(nat)] :
                        ( ( ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As),As6) = bot_bot(set(nat)) )
                          & relH(As,H,H8)
                          & aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H8),As))
                          & aa(product_prod(heap_ext(product_unit),set(nat)),$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)),H8),As6)) )
                       => aa(product_prod(heap_ext(product_unit),set(nat)),$o,Xaa,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H8),aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As),As6))) ) ) )
               => ~ accp(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$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)))),Xa),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))),Xaa),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),As)))) ) ) ) ) ).

% wand_raw.pelims(1)
tff(fact_7087_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_7088_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_7089_Range__Id__on,axiom,
    ! [A: $tType,Aa2: set(A)] : ( aa(set(product_prod(A,A)),set(A),range2(A,A),id_on(A,Aa2)) = Aa2 ) ).

% Range_Id_on
tff(fact_7090_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_7091_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_7092_Range__Collect__case__prod,axiom,
    ! [A: $tType,B: $tType,Pa: fun(B,fun(A,$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),Pa))) = aa(fun(A,$o),set(A),collect(A),aTP_Lamp_abj(fun(B,fun(A,$o)),fun(A,$o),Pa)) ) ).

% Range_Collect_case_prod
tff(fact_7093_Range__insert,axiom,
    ! [A: $tType,B: $tType,A3: B,B3: 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))),insert3(product_prod(B,A)),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),A3),B3)),R2)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),B3),aa(set(product_prod(B,A)),set(A),range2(B,A),R2)) ) ).

% Range_insert
tff(fact_7094_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_7095_finite__Range,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,B))] :
      ( aa(set(product_prod(A,B)),$o,finite_finite2(product_prod(A,B)),R2)
     => aa(set(B),$o,finite_finite2(B),aa(set(product_prod(A,B)),set(B),range2(A,B),R2)) ) ).

% finite_Range
tff(fact_7096_Range__Un__eq,axiom,
    ! [A: $tType,B: $tType,Aa2: set(product_prod(B,A)),B2: 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))),Aa2),B2)) = 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),Aa2)),aa(set(product_prod(B,A)),set(A),range2(B,A),B2)) ) ).

% Range_Un_eq
tff(fact_7097_snd__eq__Range,axiom,
    ! [A: $tType,B: $tType,R: set(product_prod(B,A))] : ( aa(set(product_prod(B,A)),set(A),image2(product_prod(B,A),A,product_snd(B,A)),R) = aa(set(product_prod(B,A)),set(A),range2(B,A),R) ) ).

% snd_eq_Range
tff(fact_7098_Range__snd,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(B,A))] : ( aa(set(product_prod(B,A)),set(A),range2(B,A),R2) = aa(set(product_prod(B,A)),set(A),image2(product_prod(B,A),A,product_snd(B,A)),R2) ) ).

% Range_snd
tff(fact_7099_Range_Ocases,axiom,
    ! [A: $tType,B: $tType,A3: A,R2: set(product_prod(B,A))] :
      ( member(A,A3,aa(set(product_prod(B,A)),set(A),range2(B,A),R2))
     => ~ ! [A4: B] : ~ member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),A4),A3),R2) ) ).

% Range.cases
tff(fact_7100_Range_Osimps,axiom,
    ! [A: $tType,B: $tType,A3: A,R2: set(product_prod(B,A))] :
      ( member(A,A3,aa(set(product_prod(B,A)),set(A),range2(B,A),R2))
    <=> ? [A7: B,B8: A] :
          ( ( A3 = B8 )
          & member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),A7),B8),R2) ) ) ).

% Range.simps
tff(fact_7101_Range_Ointros,axiom,
    ! [B: $tType,A: $tType,A3: A,B3: B,R2: set(product_prod(A,B))] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B3),R2)
     => member(B,B3,aa(set(product_prod(A,B)),set(B),range2(A,B),R2)) ) ).

% Range.intros
tff(fact_7102_RangeE,axiom,
    ! [A: $tType,B: $tType,B3: A,R2: set(product_prod(B,A))] :
      ( member(A,B3,aa(set(product_prod(B,A)),set(A),range2(B,A),R2))
     => ~ ! [A4: B] : ~ member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),A4),B3),R2) ) ).

% RangeE
tff(fact_7103_Range__iff,axiom,
    ! [A: $tType,B: $tType,A3: A,R2: set(product_prod(B,A))] :
      ( member(A,A3,aa(set(product_prod(B,A)),set(A),range2(B,A),R2))
    <=> ? [Y4: B] : member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Y4),A3),R2) ) ).

% Range_iff
tff(fact_7104_Range__mono,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,B)),S2: set(product_prod(A,B))] :
      ( aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),R2),S2)
     => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(product_prod(A,B)),set(B),range2(A,B),R2)),aa(set(product_prod(A,B)),set(B),range2(A,B),S2)) ) ).

% Range_mono
tff(fact_7105_Range__Int__subset,axiom,
    ! [A: $tType,B: $tType,Aa2: set(product_prod(B,A)),B2: set(product_prod(B,A))] : aa(set(A),$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))),Aa2),B2))),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),Aa2)),aa(set(product_prod(B,A)),set(A),range2(B,A),B2))) ).

% Range_Int_subset
tff(fact_7106_trancl__Image__in__Range,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),V: set(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),image(A,A,transitive_trancl(A,R),V)),aa(set(product_prod(A,A)),set(A),range2(A,A),R)) ).

% trancl_Image_in_Range
tff(fact_7107_Range__Diff__subset,axiom,
    ! [A: $tType,B: $tType,Aa2: set(product_prod(B,A)),B2: set(product_prod(B,A))] : aa(set(A),$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)),aa(set(product_prod(B,A)),set(A),range2(B,A),Aa2)),aa(set(product_prod(B,A)),set(A),range2(B,A),B2))),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))),minus_minus(set(product_prod(B,A))),Aa2),B2))) ).

% Range_Diff_subset
tff(fact_7108_Range__rel__restrict,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),Aa2: set(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(product_prod(A,A)),set(A),range2(A,A),rel_restrict(A,R,Aa2))),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)),Aa2)) ).

% Range_rel_restrict
tff(fact_7109_finite__reachable__restrictedI,axiom,
    ! [A: $tType,Qa: set(A),Ia2: set(A),E5: set(product_prod(A,A))] :
      ( aa(set(A),$o,finite_finite2(A),Qa)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Ia2),Qa)
       => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(product_prod(A,A)),set(A),range2(A,A),E5)),Qa)
         => aa(set(A),$o,finite_finite2(A),image(A,A,aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),E5),Ia2)) ) ) ) ).

% finite_reachable_restrictedI
tff(fact_7110_wf__UN,axiom,
    ! [B: $tType,A: $tType,Ia2: set(A),R2: fun(A,set(product_prod(B,B)))] :
      ( ! [I3: A] :
          ( member(A,I3,Ia2)
         => wf(B,aa(A,set(product_prod(B,B)),R2,I3)) )
     => ( ! [I3: A,J3: A] :
            ( member(A,I3,Ia2)
           => ( member(A,J3,Ia2)
             => ( ( aa(A,set(product_prod(B,B)),R2,I3) != aa(A,set(product_prod(B,B)),R2,J3) )
               => ( 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,I3))),aa(set(product_prod(B,B)),set(B),range2(B,B),aa(A,set(product_prod(B,B)),R2,J3))) = 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),Ia2))) ) ) ).

% wf_UN
tff(fact_7111_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_7112_Domain__Id__on,axiom,
    ! [A: $tType,Aa2: set(A)] : ( aa(set(product_prod(A,A)),set(A),domain(A,A),id_on(A,Aa2)) = Aa2 ) ).

% Domain_Id_on
tff(fact_7113_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_7114_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_7115_Range__converse,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,B))] : ( aa(set(product_prod(B,A)),set(A),range2(B,A),converse(A,B,R2)) = aa(set(product_prod(A,B)),set(A),domain(A,B),R2) ) ).

% Range_converse
tff(fact_7116_Domain__converse,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(B,A))] : ( aa(set(product_prod(A,B)),set(A),domain(A,B),converse(B,A,R2)) = aa(set(product_prod(B,A)),set(A),range2(B,A),R2) ) ).

% Domain_converse
tff(fact_7117_Domain__Collect__case__prod,axiom,
    ! [B: $tType,A: $tType,Pa: fun(A,fun(B,$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),Pa))) = aa(fun(A,$o),set(A),collect(A),aTP_Lamp_vy(fun(A,fun(B,$o)),fun(A,$o),Pa)) ) ).

% Domain_Collect_case_prod
tff(fact_7118_Domain__insert,axiom,
    ! [B: $tType,A: $tType,A3: A,B3: 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))),insert3(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B3)),R2)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),aa(set(product_prod(A,B)),set(A),domain(A,B),R2)) ) ).

% Domain_insert
tff(fact_7119_Domain__mono,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,B)),S2: set(product_prod(A,B))] :
      ( aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),R2),S2)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(product_prod(A,B)),set(A),domain(A,B),R2)),aa(set(product_prod(A,B)),set(A),domain(A,B),S2)) ) ).

% Domain_mono
tff(fact_7120_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_abk(set(product_prod(A,B)),fun(A,$o),R2)) ) ).

% Domain_unfold
tff(fact_7121_Domain__iff,axiom,
    ! [B: $tType,A: $tType,A3: A,R2: set(product_prod(A,B))] :
      ( member(A,A3,aa(set(product_prod(A,B)),set(A),domain(A,B),R2))
    <=> ? [Y4: B] : member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),Y4),R2) ) ).

% Domain_iff
tff(fact_7122_DomainE,axiom,
    ! [B: $tType,A: $tType,A3: A,R2: set(product_prod(A,B))] :
      ( member(A,A3,aa(set(product_prod(A,B)),set(A),domain(A,B),R2))
     => ~ ! [B4: B] : ~ member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B4),R2) ) ).

% DomainE
tff(fact_7123_Domain_ODomainI,axiom,
    ! [B: $tType,A: $tType,A3: A,B3: B,R2: set(product_prod(A,B))] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B3),R2)
     => member(A,A3,aa(set(product_prod(A,B)),set(A),domain(A,B),R2)) ) ).

% Domain.DomainI
tff(fact_7124_Domain_Osimps,axiom,
    ! [B: $tType,A: $tType,A3: A,R2: set(product_prod(A,B))] :
      ( member(A,A3,aa(set(product_prod(A,B)),set(A),domain(A,B),R2))
    <=> ? [A7: A,B8: B] :
          ( ( A3 = A7 )
          & member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A7),B8),R2) ) ) ).

% Domain.simps
tff(fact_7125_Domain_Ocases,axiom,
    ! [B: $tType,A: $tType,A3: A,R2: set(product_prod(A,B))] :
      ( member(A,A3,aa(set(product_prod(A,B)),set(A),domain(A,B),R2))
     => ~ ! [B4: B] : ~ member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B4),R2) ) ).

% Domain.cases
tff(fact_7126_Not__Domain__rtrancl,axiom,
    ! [A: $tType,Xa: A,R: set(product_prod(A,A)),Y: A] :
      ( ~ member(A,Xa,aa(set(product_prod(A,A)),set(A),domain(A,A),R))
     => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Y),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R))
      <=> ( Xa = Y ) ) ) ).

% Not_Domain_rtrancl
tff(fact_7127_Domain__fst,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,B))] : ( aa(set(product_prod(A,B)),set(A),domain(A,B),R2) = aa(set(product_prod(A,B)),set(A),image2(product_prod(A,B),A,product_fst(A,B)),R2) ) ).

% Domain_fst
tff(fact_7128_fst__eq__Domain,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,B))] : ( aa(set(product_prod(A,B)),set(A),image2(product_prod(A,B),A,product_fst(A,B)),R) = aa(set(product_prod(A,B)),set(A),domain(A,B),R) ) ).

% fst_eq_Domain
tff(fact_7129_Domain__Un__eq,axiom,
    ! [B: $tType,A: $tType,Aa2: set(product_prod(A,B)),B2: 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))),Aa2),B2)) = 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),Aa2)),aa(set(product_prod(A,B)),set(A),domain(A,B),B2)) ) ).

% Domain_Un_eq
tff(fact_7130_finite__Domain,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,B))] :
      ( aa(set(product_prod(A,B)),$o,finite_finite2(product_prod(A,B)),R2)
     => aa(set(A),$o,finite_finite2(A),aa(set(product_prod(A,B)),set(A),domain(A,B),R2)) ) ).

% finite_Domain
tff(fact_7131_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_7132_Domain__Int__subset,axiom,
    ! [B: $tType,A: $tType,Aa2: set(product_prod(A,B)),B2: set(product_prod(A,B))] : aa(set(A),$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))),Aa2),B2))),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),Aa2)),aa(set(product_prod(A,B)),set(A),domain(A,B),B2))) ).

% Domain_Int_subset
tff(fact_7133_for__in__RI,axiom,
    ! [B: $tType,A: $tType,Xa: A,R: set(product_prod(A,B))] :
      ( member(A,Xa,aa(set(product_prod(A,B)),set(A),domain(A,B),R))
     => member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa),fun_of_rel(A,B,R,Xa)),R) ) ).

% for_in_RI
tff(fact_7134_Field__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : ( aa(set(product_prod(A,A)),set(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_7135_Domain__Diff__subset,axiom,
    ! [B: $tType,A: $tType,Aa2: set(product_prod(A,B)),B2: set(product_prod(A,B))] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(product_prod(A,B)),set(A),domain(A,B),Aa2)),aa(set(product_prod(A,B)),set(A),domain(A,B),B2))),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))),minus_minus(set(product_prod(A,B))),Aa2),B2))) ).

% Domain_Diff_subset
tff(fact_7136_Domain__rel__restrict,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),Aa2: set(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(product_prod(A,A)),set(A),domain(A,A),rel_restrict(A,R,Aa2))),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)),Aa2)) ).

% Domain_rel_restrict
tff(fact_7137_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_7138_wf__min,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( wf(A,R)
     => ( ( R != bot_bot(set(product_prod(A,A))) )
       => ~ ! [M6: A] : ~ member(A,M6,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_7139_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_7140_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_7141_wf__Union,axiom,
    ! [A: $tType,R: set(set(product_prod(A,A)))] :
      ( ! [X: set(product_prod(A,A))] :
          ( member(set(product_prod(A,A)),X,R)
         => wf(A,X) )
     => ( ! [X: set(product_prod(A,A))] :
            ( member(set(product_prod(A,A)),X,R)
           => ! [Xa5: set(product_prod(A,A))] :
                ( member(set(product_prod(A,A)),Xa5,R)
               => ( ( X != Xa5 )
                 => ( 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),X)),aa(set(product_prod(A,A)),set(A),range2(A,A),Xa5)) = 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_7142_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))) )
       => ~ ! [M6: A] : ~ member(A,M6,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_7143_Range__def,axiom,
    ! [B: $tType,A: $tType,X4: set(product_prod(A,B))] : ( aa(set(product_prod(A,B)),set(B),range2(A,B),X4) = aa(fun(B,$o),set(B),collect(B),rangep(A,B,aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_bi(set(product_prod(A,B)),fun(A,fun(B,$o))),X4))) ) ).

% Range_def
tff(fact_7144_Rangep__Range__eq,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(B,A)),X4: A] :
      ( aa(A,$o,rangep(B,A,aTP_Lamp_rw(set(product_prod(B,A)),fun(B,fun(A,$o)),R2)),X4)
    <=> member(A,X4,aa(set(product_prod(B,A)),set(A),range2(B,A),R2)) ) ).

% Rangep_Range_eq
tff(fact_7145_RangepE,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(B,$o)),B3: B] :
      ( aa(B,$o,rangep(A,B,R2),B3)
     => ~ ! [A4: A] : ~ aa(B,$o,aa(A,fun(B,$o),R2,A4),B3) ) ).

% RangepE
tff(fact_7146_RangePI,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(B,$o)),A3: A,B3: B] :
      ( aa(B,$o,aa(A,fun(B,$o),R2,A3),B3)
     => aa(B,$o,rangep(A,B,R2),B3) ) ).

% RangePI
tff(fact_7147_Rangep_Osimps,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(B,$o)),A3: B] :
      ( aa(B,$o,rangep(A,B,R2),A3)
    <=> ? [A7: A,B8: B] :
          ( ( A3 = B8 )
          & aa(B,$o,aa(A,fun(B,$o),R2,A7),B8) ) ) ).

% Rangep.simps
tff(fact_7148_Rangep_Ocases,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(B,$o)),A3: B] :
      ( aa(B,$o,rangep(A,B,R2),A3)
     => ~ ! [A4: A] : ~ aa(B,$o,aa(A,fun(B,$o),R2,A4),A3) ) ).

% Rangep.cases
tff(fact_7149_normalize__stable,axiom,
    ! [Q2: int,P2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Q2)
     => ( algebr8660921524188924756oprime(int,P2,Q2)
       => ( normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P2),Q2)) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P2),Q2) ) ) ) ).

% normalize_stable
tff(fact_7150_semilattice__order__set_Osubset__imp,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),Aa2: set(A),B2: set(A)] :
      ( lattic4895041142388067077er_set(A,F2,Less_eq,Less)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),B2)
       => ( ( Aa2 != bot_bot(set(A)) )
         => ( aa(set(A),$o,finite_finite2(A),B2)
           => aa(A,$o,aa(A,fun(A,$o),Less_eq,lattic1715443433743089157tice_F(A,F2,B2)),lattic1715443433743089157tice_F(A,F2,Aa2)) ) ) ) ) ).

% semilattice_order_set.subset_imp
tff(fact_7151_coprime__mult__left__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B3: A,C2: A] :
          ( algebr8660921524188924756oprime(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3),C2)
        <=> ( algebr8660921524188924756oprime(A,A3,C2)
            & algebr8660921524188924756oprime(A,B3,C2) ) ) ) ).

% coprime_mult_left_iff
tff(fact_7152_coprime__mult__right__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [C2: A,A3: A,B3: A] :
          ( algebr8660921524188924756oprime(A,C2,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3))
        <=> ( algebr8660921524188924756oprime(A,C2,A3)
            & algebr8660921524188924756oprime(A,C2,B3) ) ) ) ).

% coprime_mult_right_iff
tff(fact_7153_coprime__self,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A] :
          ( algebr8660921524188924756oprime(A,A3,A3)
        <=> dvd_dvd(A,A3,one_one(A)) ) ) ).

% coprime_self
tff(fact_7154_coprime__0__left__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A] :
          ( algebr8660921524188924756oprime(A,zero_zero(A),A3)
        <=> dvd_dvd(A,A3,one_one(A)) ) ) ).

% coprime_0_left_iff
tff(fact_7155_coprime__0__right__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A] :
          ( algebr8660921524188924756oprime(A,A3,zero_zero(A))
        <=> dvd_dvd(A,A3,one_one(A)) ) ) ).

% coprime_0_right_iff
tff(fact_7156_coprime__mult__self__left__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,A3: A,B3: 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),B3))
        <=> ( dvd_dvd(A,C2,one_one(A))
            & algebr8660921524188924756oprime(A,A3,B3) ) ) ) ).

% coprime_mult_self_left_iff
tff(fact_7157_coprime__mult__self__right__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,C2: A,B3: 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),B3),C2))
        <=> ( dvd_dvd(A,C2,one_one(A))
            & algebr8660921524188924756oprime(A,A3,B3) ) ) ) ).

% coprime_mult_self_right_iff
tff(fact_7158_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_7159_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_7160_normalize__coprime,axiom,
    ! [R2: product_prod(int,int),P2: int,Q2: int] :
      ( ( normalize(R2) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P2),Q2) )
     => algebr8660921524188924756oprime(int,P2,Q2) ) ).

% normalize_coprime
tff(fact_7161_coprimeI,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B3: A] :
          ( ! [C5: A] :
              ( dvd_dvd(A,C5,A3)
             => ( dvd_dvd(A,C5,B3)
               => dvd_dvd(A,C5,one_one(A)) ) )
         => algebr8660921524188924756oprime(A,A3,B3) ) ) ).

% coprimeI
tff(fact_7162_coprime__def,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B3: A] :
          ( algebr8660921524188924756oprime(A,A3,B3)
        <=> ! [C6: A] :
              ( dvd_dvd(A,C6,A3)
             => ( dvd_dvd(A,C6,B3)
               => dvd_dvd(A,C6,one_one(A)) ) ) ) ) ).

% coprime_def
tff(fact_7163_not__coprimeE,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B3: A] :
          ( ~ algebr8660921524188924756oprime(A,A3,B3)
         => ~ ! [C5: A] :
                ( dvd_dvd(A,C5,A3)
               => ( dvd_dvd(A,C5,B3)
                 => dvd_dvd(A,C5,one_one(A)) ) ) ) ) ).

% not_coprimeE
tff(fact_7164_not__coprimeI,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,A3: A,B3: A] :
          ( dvd_dvd(A,C2,A3)
         => ( dvd_dvd(A,C2,B3)
           => ( ~ dvd_dvd(A,C2,one_one(A))
             => ~ algebr8660921524188924756oprime(A,A3,B3) ) ) ) ) ).

% not_coprimeI
tff(fact_7165_coprime__absorb__left,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [Xa: A,Y: A] :
          ( dvd_dvd(A,Xa,Y)
         => ( algebr8660921524188924756oprime(A,Xa,Y)
          <=> dvd_dvd(A,Xa,one_one(A)) ) ) ) ).

% coprime_absorb_left
tff(fact_7166_coprime__imp__coprime,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,D3: A,A3: A,B3: A] :
          ( algebr8660921524188924756oprime(A,C2,D3)
         => ( ! [E2: A] :
                ( ~ dvd_dvd(A,E2,one_one(A))
               => ( dvd_dvd(A,E2,A3)
                 => ( dvd_dvd(A,E2,B3)
                   => dvd_dvd(A,E2,C2) ) ) )
           => ( ! [E2: A] :
                  ( ~ dvd_dvd(A,E2,one_one(A))
                 => ( dvd_dvd(A,E2,A3)
                   => ( dvd_dvd(A,E2,B3)
                     => dvd_dvd(A,E2,D3) ) ) )
             => algebr8660921524188924756oprime(A,A3,B3) ) ) ) ) ).

% coprime_imp_coprime
tff(fact_7167_coprime__absorb__right,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [Y: A,Xa: A] :
          ( dvd_dvd(A,Y,Xa)
         => ( algebr8660921524188924756oprime(A,Xa,Y)
          <=> dvd_dvd(A,Y,one_one(A)) ) ) ) ).

% coprime_absorb_right
tff(fact_7168_coprime__common__divisor,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B3: A,C2: A] :
          ( algebr8660921524188924756oprime(A,A3,B3)
         => ( dvd_dvd(A,C2,A3)
           => ( dvd_dvd(A,C2,B3)
             => dvd_dvd(A,C2,one_one(A)) ) ) ) ) ).

% coprime_common_divisor
tff(fact_7169_is__unit__left__imp__coprime,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B3: A] :
          ( dvd_dvd(A,A3,one_one(A))
         => algebr8660921524188924756oprime(A,A3,B3) ) ) ).

% is_unit_left_imp_coprime
tff(fact_7170_is__unit__right__imp__coprime,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B3: A,A3: A] :
          ( dvd_dvd(A,B3,one_one(A))
         => algebr8660921524188924756oprime(A,A3,B3) ) ) ).

% is_unit_right_imp_coprime
tff(fact_7171_mult__mod__cancel__right,axiom,
    ! [A: $tType] :
      ( ( euclid8851590272496341667cancel(A)
        & semiring_gcd(A) )
     => ! [A3: A,N: A,M2: A,B3: A] :
          ( ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),N),M2) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),B3),N),M2) )
         => ( algebr8660921524188924756oprime(A,M2,N)
           => ( modulo_modulo(A,A3,M2) = modulo_modulo(A,B3,M2) ) ) ) ) ).

% mult_mod_cancel_right
tff(fact_7172_mult__mod__cancel__left,axiom,
    ! [A: $tType] :
      ( ( euclid8851590272496341667cancel(A)
        & semiring_gcd(A) )
     => ! [N: A,A3: A,M2: A,B3: A] :
          ( ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),N),A3),M2) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),N),B3),M2) )
         => ( algebr8660921524188924756oprime(A,M2,N)
           => ( modulo_modulo(A,A3,M2) = modulo_modulo(A,B3,M2) ) ) ) ) ).

% mult_mod_cancel_left
tff(fact_7173_coprime__1__right,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A] : algebr8660921524188924756oprime(A,A3,one_one(A)) ) ).

% coprime_1_right
tff(fact_7174_coprime__1__left,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A] : algebr8660921524188924756oprime(A,one_one(A),A3) ) ).

% coprime_1_left
tff(fact_7175_coprime__dvd__mult__right__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,C2: A,B3: A] :
          ( algebr8660921524188924756oprime(A,A3,C2)
         => ( dvd_dvd(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3))
          <=> dvd_dvd(A,A3,B3) ) ) ) ).

% coprime_dvd_mult_right_iff
tff(fact_7176_coprime__dvd__mult__left__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,C2: A,B3: A] :
          ( algebr8660921524188924756oprime(A,A3,C2)
         => ( dvd_dvd(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2))
          <=> dvd_dvd(A,A3,B3) ) ) ) ).

% coprime_dvd_mult_left_iff
tff(fact_7177_divides__mult,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,C2: A,B3: A] :
          ( dvd_dvd(A,A3,C2)
         => ( dvd_dvd(A,B3,C2)
           => ( algebr8660921524188924756oprime(A,A3,B3)
             => dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3),C2) ) ) ) ) ).

% divides_mult
tff(fact_7178_coprime__commute,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B3: A,A3: A] :
          ( algebr8660921524188924756oprime(A,B3,A3)
        <=> algebr8660921524188924756oprime(A,A3,B3) ) ) ).

% coprime_commute
tff(fact_7179_coprime__divisors,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,C2: A,B3: A,D3: A] :
          ( dvd_dvd(A,A3,C2)
         => ( dvd_dvd(A,B3,D3)
           => ( algebr8660921524188924756oprime(A,C2,D3)
             => algebr8660921524188924756oprime(A,A3,B3) ) ) ) ) ).

% coprime_divisors
tff(fact_7180_prod__list__coprime__right,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [Xsa: list(A),A3: A] :
          ( ! [X: A] :
              ( member(A,X,aa(list(A),set(A),set2(A),Xsa))
             => algebr8660921524188924756oprime(A,A3,X) )
         => algebr8660921524188924756oprime(A,A3,aa(list(A),A,groups5270119922927024881d_list(A),Xsa)) ) ) ).

% prod_list_coprime_right
tff(fact_7181_prod__list__coprime__left,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [Xsa: list(A),A3: A] :
          ( ! [X: A] :
              ( member(A,X,aa(list(A),set(A),set2(A),Xsa))
             => algebr8660921524188924756oprime(A,X,A3) )
         => algebr8660921524188924756oprime(A,aa(list(A),A,groups5270119922927024881d_list(A),Xsa),A3) ) ) ).

% prod_list_coprime_left
tff(fact_7182_quotient__of__coprime,axiom,
    ! [R2: rat,P2: int,Q2: int] :
      ( ( quotient_of(R2) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P2),Q2) )
     => algebr8660921524188924756oprime(int,P2,Q2) ) ).

% quotient_of_coprime
tff(fact_7183_gcd__mult__left__left__cancel,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [B3: A,C2: A,A3: A] :
          ( algebr8660921524188924756oprime(A,B3,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)),B3) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B3) ) ) ) ).

% gcd_mult_left_left_cancel
tff(fact_7184_gcd__mult__left__right__cancel,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [B3: A,C2: A,A3: A] :
          ( algebr8660921524188924756oprime(A,B3,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)),B3) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B3) ) ) ) ).

% gcd_mult_left_right_cancel
tff(fact_7185_gcd__mult__right__left__cancel,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,C2: A,B3: 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),B3)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B3) ) ) ) ).

% gcd_mult_right_left_cancel
tff(fact_7186_gcd__mult__right__right__cancel,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,C2: A,B3: 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),B3),C2)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B3) ) ) ) ).

% gcd_mult_right_right_cancel
tff(fact_7187_gcd__coprime,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B3: A,A6: A,B6: A] :
          ( ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B3) != zero_zero(A) )
         => ( ( A3 = aa(A,A,aa(A,fun(A,A),times_times(A),A6),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B3)) )
           => ( ( B3 = aa(A,A,aa(A,fun(A,A),times_times(A),B6),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B3)) )
             => algebr8660921524188924756oprime(A,A6,B6) ) ) ) ) ).

% gcd_coprime
tff(fact_7188_gcd__coprime__exists,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B3) != zero_zero(A) )
         => ? [A9: A,B11: A] :
              ( ( A3 = aa(A,A,aa(A,fun(A,A),times_times(A),A9),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B3)) )
              & ( B3 = aa(A,A,aa(A,fun(A,A),times_times(A),B11),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B3)) )
              & algebr8660921524188924756oprime(A,A9,B11) ) ) ) ).

% gcd_coprime_exists
tff(fact_7189_invertible__coprime,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A3: A,B3: A,C2: A] :
          ( ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3),C2) = one_one(A) )
         => algebr8660921524188924756oprime(A,A3,C2) ) ) ).

% invertible_coprime
tff(fact_7190_semilattice__order__set_Obounded__iff,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),Aa2: set(A),Xa: A] :
      ( lattic4895041142388067077er_set(A,F2,Less_eq,Less)
     => ( aa(set(A),$o,finite_finite2(A),Aa2)
       => ( ( Aa2 != bot_bot(set(A)) )
         => ( aa(A,$o,aa(A,fun(A,$o),Less_eq,Xa),lattic1715443433743089157tice_F(A,F2,Aa2))
          <=> ! [X6: A] :
                ( member(A,X6,Aa2)
               => aa(A,$o,aa(A,fun(A,$o),Less_eq,Xa),X6) ) ) ) ) ) ).

% semilattice_order_set.bounded_iff
tff(fact_7191_semilattice__order__set_OboundedI,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),Aa2: set(A),Xa: A] :
      ( lattic4895041142388067077er_set(A,F2,Less_eq,Less)
     => ( aa(set(A),$o,finite_finite2(A),Aa2)
       => ( ( Aa2 != bot_bot(set(A)) )
         => ( ! [A4: A] :
                ( member(A,A4,Aa2)
               => aa(A,$o,aa(A,fun(A,$o),Less_eq,Xa),A4) )
           => aa(A,$o,aa(A,fun(A,$o),Less_eq,Xa),lattic1715443433743089157tice_F(A,F2,Aa2)) ) ) ) ) ).

% semilattice_order_set.boundedI
tff(fact_7192_semilattice__order__set_OboundedE,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),Aa2: set(A),Xa: A] :
      ( lattic4895041142388067077er_set(A,F2,Less_eq,Less)
     => ( aa(set(A),$o,finite_finite2(A),Aa2)
       => ( ( Aa2 != bot_bot(set(A)) )
         => ( aa(A,$o,aa(A,fun(A,$o),Less_eq,Xa),lattic1715443433743089157tice_F(A,F2,Aa2))
           => ! [A13: A] :
                ( member(A,A13,Aa2)
               => aa(A,$o,aa(A,fun(A,$o),Less_eq,Xa),A13) ) ) ) ) ) ).

% semilattice_order_set.boundedE
tff(fact_7193_semilattice__set_Oremove,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Aa2: set(A),Xa: A] :
      ( lattic149705377957585745ce_set(A,F2)
     => ( aa(set(A),$o,finite_finite2(A),Aa2)
       => ( member(A,Xa,Aa2)
         => ( lattic1715443433743089157tice_F(A,F2,Aa2) = $ite(aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A)))) = bot_bot(set(A)),Xa,aa(A,A,aa(A,fun(A,A),F2,Xa),lattic1715443433743089157tice_F(A,F2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A))))))) ) ) ) ) ).

% semilattice_set.remove
tff(fact_7194_semilattice__set_Oinsert__remove,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Aa2: set(A),Xa: A] :
      ( lattic149705377957585745ce_set(A,F2)
     => ( aa(set(A),$o,finite_finite2(A),Aa2)
       => ( lattic1715443433743089157tice_F(A,F2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),Aa2)) = $ite(aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A)))) = bot_bot(set(A)),Xa,aa(A,A,aa(A,fun(A,A),F2,Xa),lattic1715443433743089157tice_F(A,F2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A))))))) ) ) ) ).

% semilattice_set.insert_remove
tff(fact_7195_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_7196_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_7197_semilattice__set_Osingleton,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Xa: A] :
      ( lattic149705377957585745ce_set(A,F2)
     => ( lattic1715443433743089157tice_F(A,F2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A)))) = Xa ) ) ).

% semilattice_set.singleton
tff(fact_7198_semilattice__set_Ohom__commute,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),H2: fun(A,A),N7: set(A)] :
      ( lattic149705377957585745ce_set(A,F2)
     => ( ! [X: A,Y2: A] : ( aa(A,A,H2,aa(A,A,aa(A,fun(A,A),F2,X),Y2)) = aa(A,A,aa(A,fun(A,A),F2,aa(A,A,H2,X)),aa(A,A,H2,Y2)) )
       => ( aa(set(A),$o,finite_finite2(A),N7)
         => ( ( N7 != bot_bot(set(A)) )
           => ( aa(A,A,H2,lattic1715443433743089157tice_F(A,F2,N7)) = lattic1715443433743089157tice_F(A,F2,aa(set(A),set(A),image2(A,A,H2),N7)) ) ) ) ) ) ).

% semilattice_set.hom_commute
tff(fact_7199_semilattice__set_Osubset,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Aa2: set(A),B2: set(A)] :
      ( lattic149705377957585745ce_set(A,F2)
     => ( aa(set(A),$o,finite_finite2(A),Aa2)
       => ( ( B2 != bot_bot(set(A)) )
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),Aa2)
           => ( aa(A,A,aa(A,fun(A,A),F2,lattic1715443433743089157tice_F(A,F2,B2)),lattic1715443433743089157tice_F(A,F2,Aa2)) = lattic1715443433743089157tice_F(A,F2,Aa2) ) ) ) ) ) ).

% semilattice_set.subset
tff(fact_7200_semilattice__set_Oclosed,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Aa2: set(A)] :
      ( lattic149705377957585745ce_set(A,F2)
     => ( aa(set(A),$o,finite_finite2(A),Aa2)
       => ( ( Aa2 != bot_bot(set(A)) )
         => ( ! [X: A,Y2: A] : member(A,aa(A,A,aa(A,fun(A,A),F2,X),Y2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),X),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Y2),bot_bot(set(A)))))
           => member(A,lattic1715443433743089157tice_F(A,F2,Aa2),Aa2) ) ) ) ) ).

% semilattice_set.closed
tff(fact_7201_semilattice__set_Oinsert,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Aa2: set(A),Xa: A] :
      ( lattic149705377957585745ce_set(A,F2)
     => ( aa(set(A),$o,finite_finite2(A),Aa2)
       => ( ( Aa2 != bot_bot(set(A)) )
         => ( lattic1715443433743089157tice_F(A,F2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),Aa2)) = aa(A,A,aa(A,fun(A,A),F2,Xa),lattic1715443433743089157tice_F(A,F2,Aa2)) ) ) ) ) ).

% semilattice_set.insert
tff(fact_7202_semilattice__set_Oinsert__not__elem,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Aa2: set(A),Xa: A] :
      ( lattic149705377957585745ce_set(A,F2)
     => ( aa(set(A),$o,finite_finite2(A),Aa2)
       => ( ~ member(A,Xa,Aa2)
         => ( ( Aa2 != bot_bot(set(A)) )
           => ( lattic1715443433743089157tice_F(A,F2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),Aa2)) = aa(A,A,aa(A,fun(A,A),F2,Xa),lattic1715443433743089157tice_F(A,F2,Aa2)) ) ) ) ) ) ).

% semilattice_set.insert_not_elem
tff(fact_7203_semilattice__set_Ounion,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Aa2: set(A),B2: set(A)] :
      ( lattic149705377957585745ce_set(A,F2)
     => ( aa(set(A),$o,finite_finite2(A),Aa2)
       => ( ( Aa2 != bot_bot(set(A)) )
         => ( aa(set(A),$o,finite_finite2(A),B2)
           => ( ( B2 != bot_bot(set(A)) )
             => ( lattic1715443433743089157tice_F(A,F2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),B2)) = aa(A,A,aa(A,fun(A,A),F2,lattic1715443433743089157tice_F(A,F2,Aa2)),lattic1715443433743089157tice_F(A,F2,B2)) ) ) ) ) ) ) ).

% semilattice_set.union
tff(fact_7204_semilattice__set_Oeq__fold,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Aa2: set(A),Xa: A] :
      ( lattic149705377957585745ce_set(A,F2)
     => ( aa(set(A),$o,finite_finite2(A),Aa2)
       => ( lattic1715443433743089157tice_F(A,F2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),Aa2)) = finite_fold(A,A,F2,Xa,Aa2) ) ) ) ).

% semilattice_set.eq_fold
tff(fact_7205_semilattice__set_Oset__eq__fold,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Xa: A,Xsa: list(A)] :
      ( lattic149705377957585745ce_set(A,F2)
     => ( lattic1715443433743089157tice_F(A,F2,aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa))) = aa(A,A,aa(list(A),fun(A,A),aa(fun(A,fun(A,A)),fun(list(A),fun(A,A)),fold(A,A),F2),Xsa),Xa) ) ) ).

% semilattice_set.set_eq_fold
tff(fact_7206_pred__on_Onot__maxchain__Some,axiom,
    ! [A: $tType,Aa2: set(A),Pa: fun(A,fun(A,$o)),C3: set(A)] :
      ( aa(set(A),$o,pred_chain(A,Aa2,Pa),C3)
     => ( ~ pred_maxchain(A,Aa2,Pa,C3)
       => ( aa(set(A),$o,pred_chain(A,Aa2,Pa),fChoice(set(A),aa(set(A),fun(set(A),$o),aa(fun(A,fun(A,$o)),fun(set(A),fun(set(A),$o)),aTP_Lamp_abl(set(A),fun(fun(A,fun(A,$o)),fun(set(A),fun(set(A),$o))),Aa2),Pa),C3)))
          & aa(set(A),$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_abl(set(A),fun(fun(A,fun(A,$o)),fun(set(A),fun(set(A),$o))),Aa2),Pa),C3))) ) ) ) ).

% pred_on.not_maxchain_Some
tff(fact_7207_antisymp__antisym__eq,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( antisymp(A,aTP_Lamp_cz(set(product_prod(A,A)),fun(A,fun(A,$o)),R2))
    <=> antisym(A,R2) ) ).

% antisymp_antisym_eq
tff(fact_7208_antisymp__less__eq,axiom,
    ! [A: $tType,R2: fun(A,fun(A,$o)),S2: 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))),R2),S2)
     => ( antisymp(A,S2)
       => antisymp(A,R2) ) ) ).

% antisymp_less_eq
tff(fact_7209_antisympD,axiom,
    ! [A: $tType,R2: fun(A,fun(A,$o)),A3: A,B3: A] :
      ( antisymp(A,R2)
     => ( aa(A,$o,aa(A,fun(A,$o),R2,A3),B3)
       => ( aa(A,$o,aa(A,fun(A,$o),R2,B3),A3)
         => ( A3 = B3 ) ) ) ) ).

% antisympD
tff(fact_7210_antisympI,axiom,
    ! [A: $tType,R2: fun(A,fun(A,$o))] :
      ( ! [X: A,Y2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),R2,X),Y2)
         => ( aa(A,$o,aa(A,fun(A,$o),R2,Y2),X)
           => ( X = Y2 ) ) )
     => antisymp(A,R2) ) ).

% antisympI
tff(fact_7211_antisymp__def,axiom,
    ! [A: $tType,R2: fun(A,fun(A,$o))] :
      ( antisymp(A,R2)
    <=> ! [X6: A,Y4: A] :
          ( aa(A,$o,aa(A,fun(A,$o),R2,X6),Y4)
         => ( aa(A,$o,aa(A,fun(A,$o),R2,Y4),X6)
           => ( X6 = Y4 ) ) ) ) ).

% antisymp_def
tff(fact_7212_antisymp__equality,axiom,
    ! [A: $tType] : antisymp(A,fequal(A)) ).

% antisymp_equality
tff(fact_7213_subset_OHausdorff,axiom,
    ! [A: $tType,Aa2: set(set(A))] :
    ? [X_1: set(set(A))] : pred_maxchain(set(A),Aa2,ord_less(set(A)),X_1) ).

% subset.Hausdorff
tff(fact_7214_antisym__bot,axiom,
    ! [A: $tType] : antisymp(A,bot_bot(fun(A,fun(A,$o)))) ).

% antisym_bot
tff(fact_7215_subset_Omaxchain__def,axiom,
    ! [A: $tType,Aa2: set(set(A)),C3: set(set(A))] :
      ( pred_maxchain(set(A),Aa2,ord_less(set(A)),C3)
    <=> ( aa(set(set(A)),$o,pred_chain(set(A),Aa2,ord_less(set(A))),C3)
        & ~ ? [S8: set(set(A))] :
              ( aa(set(set(A)),$o,pred_chain(set(A),Aa2,ord_less(set(A))),S8)
              & aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less(set(set(A))),C3),S8) ) ) ) ).

% subset.maxchain_def
tff(fact_7216_subset_Omaxchain__imp__chain,axiom,
    ! [A: $tType,Aa2: set(set(A)),C3: set(set(A))] :
      ( pred_maxchain(set(A),Aa2,ord_less(set(A)),C3)
     => aa(set(set(A)),$o,pred_chain(set(A),Aa2,ord_less(set(A))),C3) ) ).

% subset.maxchain_imp_chain
tff(fact_7217_subset_Onot__maxchain__Some,axiom,
    ! [A: $tType,Aa2: set(set(A)),C3: set(set(A))] :
      ( aa(set(set(A)),$o,pred_chain(set(A),Aa2,ord_less(set(A))),C3)
     => ( ~ pred_maxchain(set(A),Aa2,ord_less(set(A)),C3)
       => ( aa(set(set(A)),$o,pred_chain(set(A),Aa2,ord_less(set(A))),fChoice(set(set(A)),aa(set(set(A)),fun(set(set(A)),$o),aTP_Lamp_abm(set(set(A)),fun(set(set(A)),fun(set(set(A)),$o)),Aa2),C3)))
          & aa(set(set(A)),$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_abm(set(set(A)),fun(set(set(A)),fun(set(set(A)),$o)),Aa2),C3))) ) ) ) ).

% subset.not_maxchain_Some
tff(fact_7218_pred__on_Omaxchain__def,axiom,
    ! [A: $tType,Aa2: set(A),Pa: fun(A,fun(A,$o)),C3: set(A)] :
      ( pred_maxchain(A,Aa2,Pa,C3)
    <=> ( aa(set(A),$o,pred_chain(A,Aa2,Pa),C3)
        & ~ ? [S8: set(A)] :
              ( aa(set(A),$o,pred_chain(A,Aa2,Pa),S8)
              & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),C3),S8) ) ) ) ).

% pred_on.maxchain_def
tff(fact_7219_subset__maxchain__max,axiom,
    ! [A: $tType,Aa2: set(set(A)),C3: set(set(A)),X3: set(A)] :
      ( pred_maxchain(set(A),Aa2,ord_less(set(A)),C3)
     => ( member(set(A),X3,Aa2)
       => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C3)),X3)
         => ( aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C3) = X3 ) ) ) ) ).

% subset_maxchain_max
tff(fact_7220_pred__on_Osuc__def,axiom,
    ! [A: $tType,Aa2: set(A),Pa: fun(A,fun(A,$o)),C3: set(A)] :
      ( pred_suc(A,Aa2,Pa,C3) = $ite(
          ( ~ aa(set(A),$o,pred_chain(A,Aa2,Pa),C3)
          | pred_maxchain(A,Aa2,Pa,C3) ),
          C3,
          fChoice(set(A),aa(set(A),fun(set(A),$o),aa(fun(A,fun(A,$o)),fun(set(A),fun(set(A),$o)),aTP_Lamp_abl(set(A),fun(fun(A,fun(A,$o)),fun(set(A),fun(set(A),$o))),Aa2),Pa),C3)) ) ) ).

% pred_on.suc_def
tff(fact_7221_Quotient__multiset,axiom,
    ! [A: $tType] : quotient(fun(A,nat),multiset(A),bNF_eq_onp(fun(A,nat),aTP_Lamp_sj(fun(A,nat),$o)),abs_multiset(A),count(A),cr_multiset(A)) ).

% Quotient_multiset
tff(fact_7222_subset_Onot__chain__suc,axiom,
    ! [A: $tType,Aa2: set(set(A)),X3: set(set(A))] :
      ( ~ aa(set(set(A)),$o,pred_chain(set(A),Aa2,ord_less(set(A))),X3)
     => ( pred_suc(set(A),Aa2,ord_less(set(A)),X3) = X3 ) ) ).

% subset.not_chain_suc
tff(fact_7223_subset_Omaxchain__suc,axiom,
    ! [A: $tType,Aa2: set(set(A)),X3: set(set(A))] :
      ( pred_maxchain(set(A),Aa2,ord_less(set(A)),X3)
     => ( pred_suc(set(A),Aa2,ord_less(set(A)),X3) = X3 ) ) ).

% subset.maxchain_suc
tff(fact_7224_subset_Osuc__def,axiom,
    ! [A: $tType,Aa2: set(set(A)),C3: set(set(A))] :
      ( pred_suc(set(A),Aa2,ord_less(set(A)),C3) = $ite(
          ( ~ aa(set(set(A)),$o,pred_chain(set(A),Aa2,ord_less(set(A))),C3)
          | pred_maxchain(set(A),Aa2,ord_less(set(A)),C3) ),
          C3,
          fChoice(set(set(A)),aa(set(set(A)),fun(set(set(A)),$o),aTP_Lamp_abm(set(set(A)),fun(set(set(A)),fun(set(set(A)),$o)),Aa2),C3)) ) ) ).

% subset.suc_def
tff(fact_7225_subset_Osuc__not__equals,axiom,
    ! [A: $tType,Aa2: set(set(A)),C3: set(set(A))] :
      ( aa(set(set(A)),$o,pred_chain(set(A),Aa2,ord_less(set(A))),C3)
     => ( ~ pred_maxchain(set(A),Aa2,ord_less(set(A)),C3)
       => ( pred_suc(set(A),Aa2,ord_less(set(A)),C3) != C3 ) ) ) ).

% subset.suc_not_equals
tff(fact_7226_Quotient__filter,axiom,
    ! [B: $tType,A: $tType,R: fun(A,fun(A,$o)),Abs: fun(A,B),Rep: fun(B,A),T4: fun(A,fun(B,$o))] :
      ( quotient(A,B,R,Abs,Rep,T4)
     => quotient(filter(A),filter(B),rel_filter(A,A,R),aa(fun(A,B),fun(filter(A),filter(B)),filtermap(A,B),Abs),aa(fun(B,A),fun(filter(B),filter(A)),filtermap(B,A),Rep),rel_filter(A,B,T4)) ) ).

% Quotient_filter
tff(fact_7227_subset_Ochain__suc,axiom,
    ! [A: $tType,Aa2: set(set(A)),X3: set(set(A))] :
      ( aa(set(set(A)),$o,pred_chain(set(A),Aa2,ord_less(set(A))),X3)
     => aa(set(set(A)),$o,pred_chain(set(A),Aa2,ord_less(set(A))),pred_suc(set(A),Aa2,ord_less(set(A)),X3)) ) ).

% subset.chain_suc
tff(fact_7228_list_OQuotient,axiom,
    ! [B: $tType,A: $tType,R: fun(A,fun(A,$o)),Abs: fun(A,B),Rep: fun(B,A),T4: fun(A,fun(B,$o))] :
      ( quotient(A,B,R,Abs,Rep,T4)
     => quotient(list(A),list(B),aa(fun(A,fun(A,$o)),fun(list(A),fun(list(A),$o)),list_all2(A,A),R),aa(fun(A,B),fun(list(A),list(B)),map(A,B),Abs),aa(fun(B,A),fun(list(B),list(A)),map(B,A),Rep),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),T4)) ) ).

% list.Quotient
tff(fact_7229_subset_Osuc__in__carrier,axiom,
    ! [A: $tType,X3: set(set(A)),Aa2: set(set(A))] :
      ( aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),X3),Aa2)
     => aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),pred_suc(set(A),Aa2,ord_less(set(A)),X3)),Aa2) ) ).

% subset.suc_in_carrier
tff(fact_7230_subset_Osuc__subset,axiom,
    ! [A: $tType,X3: set(set(A)),Aa2: set(set(A))] : aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),X3),pred_suc(set(A),Aa2,ord_less(set(A)),X3)) ).

% subset.suc_subset
tff(fact_7231_subset_Osubset__suc,axiom,
    ! [A: $tType,X3: set(set(A)),Y5: set(set(A)),Aa2: set(set(A))] :
      ( aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),X3),Y5)
     => aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),X3),pred_suc(set(A),Aa2,ord_less(set(A)),Y5)) ) ).

% subset.subset_suc
tff(fact_7232_subset_Ochain__sucD,axiom,
    ! [A: $tType,Aa2: set(set(A)),X3: set(set(A))] :
      ( aa(set(set(A)),$o,pred_chain(set(A),Aa2,ord_less(set(A))),X3)
     => ( aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),pred_suc(set(A),Aa2,ord_less(set(A)),X3)),Aa2)
        & aa(set(set(A)),$o,pred_chain(set(A),Aa2,ord_less(set(A))),pred_suc(set(A),Aa2,ord_less(set(A)),X3)) ) ) ).

% subset.chain_sucD
tff(fact_7233_bind__singleton__conv__image,axiom,
    ! [A: $tType,B: $tType,Aa2: set(B),F2: fun(B,A)] : ( bind3(B,A,Aa2,aTP_Lamp_dk(fun(B,A),fun(B,set(A)),F2)) = aa(set(B),set(A),image2(B,A,F2),Aa2) ) ).

% bind_singleton_conv_image
tff(fact_7234_prod__mset_Oeq__fold,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M5: multiset(A)] : ( aa(multiset(A),A,comm_m9189036328036947845d_mset(A),M5) = fold_mset(A,A,times_times(A),one_one(A),M5) ) ) ).

% prod_mset.eq_fold
tff(fact_7235_empty__bind,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,set(A))] : ( bind3(B,A,bot_bot(set(B)),F2) = bot_bot(set(A)) ) ).

% empty_bind
tff(fact_7236_Set_Obind__def,axiom,
    ! [A: $tType,B: $tType,Aa2: set(B),F2: fun(B,set(A))] : ( bind3(B,A,Aa2,F2) = aa(fun(A,$o),set(A),collect(A),aa(fun(B,set(A)),fun(A,$o),aTP_Lamp_abn(set(B),fun(fun(B,set(A)),fun(A,$o)),Aa2),F2)) ) ).

% Set.bind_def
tff(fact_7237_Set_Obind__bind,axiom,
    ! [E: $tType,A: $tType,B: $tType,Aa2: set(E),B2: fun(E,set(B)),C3: fun(B,set(A))] : ( bind3(B,A,bind3(E,B,Aa2,B2),C3) = bind3(E,A,Aa2,aa(fun(B,set(A)),fun(E,set(A)),aTP_Lamp_abo(fun(E,set(B)),fun(fun(B,set(A)),fun(E,set(A))),B2),C3)) ) ).

% Set.bind_bind
tff(fact_7238_nonempty__bind__const,axiom,
    ! [A: $tType,B: $tType,Aa2: set(A),B2: set(B)] :
      ( ( Aa2 != bot_bot(set(A)) )
     => ( bind3(A,B,Aa2,aTP_Lamp_pv(set(B),fun(A,set(B)),B2)) = B2 ) ) ).

% nonempty_bind_const
tff(fact_7239_bind__const,axiom,
    ! [B: $tType,A: $tType,Aa2: set(B),B2: set(A)] :
      ( bind3(B,A,Aa2,aTP_Lamp_an(set(A),fun(B,set(A)),B2)) = $ite(Aa2 = bot_bot(set(B)),bot_bot(set(A)),B2) ) ).

% bind_const
tff(fact_7240_Diff__transfer,axiom,
    ! [A: $tType,B: $tType,Aa2: fun(A,fun(B,$o))] :
      ( bi_unique(A,B,Aa2)
     => 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,Aa2),bNF_rel_fun(set(A),set(B),set(A),set(B),bNF_rel_set(A,B,Aa2),bNF_rel_set(A,B,Aa2))),minus_minus(set(A))),minus_minus(set(B))) ) ).

% Diff_transfer
tff(fact_7241_bij__betw__nth,axiom,
    ! [A: $tType,Xsa: list(A),Aa2: set(nat),B2: set(A)] :
      ( aa(list(A),$o,distinct(A),Xsa)
     => ( ( Aa2 = aa(nat,set(nat),set_ord_lessThan(nat),aa(list(A),nat,size_size(list(A)),Xsa)) )
       => ( ( B2 = aa(list(A),set(A),set2(A),Xsa) )
         => bij_betw(nat,A,nth(A,Xsa),Aa2,B2) ) ) ) ).

% bij_betw_nth
tff(fact_7242_iso__iff2,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(A,A)),R5: set(product_prod(B,B)),F2: fun(A,B)] :
      ( bNF_Wellorder_iso(A,B,R2,R5,F2)
    <=> ( bij_betw(A,B,F2,aa(set(product_prod(A,A)),set(A),field2(A),R2),aa(set(product_prod(B,B)),set(B),field2(B),R5))
        & ! [X6: A] :
            ( member(A,X6,aa(set(product_prod(A,A)),set(A),field2(A),R2))
           => ! [Xa2: A] :
                ( member(A,Xa2,aa(set(product_prod(A,A)),set(A),field2(A),R2))
               => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X6),Xa2),R2)
                <=> 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,F2,X6)),aa(A,B,F2,Xa2)),R5) ) ) ) ) ) ).

% iso_iff2
tff(fact_7243_notIn__Un__bij__betw,axiom,
    ! [A: $tType,B: $tType,B3: A,Aa2: set(A),F2: fun(A,B),A5: set(B)] :
      ( ~ member(A,B3,Aa2)
     => ( ~ member(B,aa(A,B,F2,B3),A5)
       => ( bij_betw(A,B,F2,Aa2,A5)
         => bij_betw(A,B,F2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),B3),bot_bot(set(A)))),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A5),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert3(B),aa(A,B,F2,B3)),bot_bot(set(B))))) ) ) ) ).

% notIn_Un_bij_betw
tff(fact_7244_notIn__Un__bij__betw3,axiom,
    ! [A: $tType,B: $tType,B3: A,Aa2: set(A),F2: fun(A,B),A5: set(B)] :
      ( ~ member(A,B3,Aa2)
     => ( ~ member(B,aa(A,B,F2,B3),A5)
       => ( bij_betw(A,B,F2,Aa2,A5)
        <=> bij_betw(A,B,F2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),B3),bot_bot(set(A)))),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A5),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert3(B),aa(A,B,F2,B3)),bot_bot(set(B))))) ) ) ) ).

% notIn_Un_bij_betw3
tff(fact_7245_bi__unique__rel__filter,axiom,
    ! [B: $tType,A: $tType,Aa2: fun(A,fun(B,$o))] :
      ( bi_unique(A,B,Aa2)
     => bi_unique(filter(A),filter(B),rel_filter(A,B,Aa2)) ) ).

% bi_unique_rel_filter
tff(fact_7246_bij__betw__empty1,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),Aa2: set(B)] :
      ( bij_betw(A,B,F2,bot_bot(set(A)),Aa2)
     => ( Aa2 = bot_bot(set(B)) ) ) ).

% bij_betw_empty1
tff(fact_7247_bij__betw__empty2,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),Aa2: set(A)] :
      ( bij_betw(A,B,F2,Aa2,bot_bot(set(B)))
     => ( Aa2 = bot_bot(set(A)) ) ) ).

% bij_betw_empty2
tff(fact_7248_list_Obi__unique__rel,axiom,
    ! [B: $tType,A: $tType,R: fun(A,fun(B,$o))] :
      ( bi_unique(A,B,R)
     => bi_unique(list(A),list(B),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),R)) ) ).

% list.bi_unique_rel
tff(fact_7249_bij__betw__partition,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),Aa2: set(A),C3: set(A),B2: set(B),D2: set(B)] :
      ( bij_betw(A,B,F2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),C3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),B2),D2))
     => ( bij_betw(A,B,F2,C3,D2)
       => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),C3) = bot_bot(set(A)) )
         => ( ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),B2),D2) = bot_bot(set(B)) )
           => bij_betw(A,B,F2,Aa2,B2) ) ) ) ) ).

% bij_betw_partition
tff(fact_7250_bij__betw__combine,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),Aa2: set(A),B2: set(B),C3: set(A),D2: set(B)] :
      ( bij_betw(A,B,F2,Aa2,B2)
     => ( bij_betw(A,B,F2,C3,D2)
       => ( ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),B2),D2) = bot_bot(set(B)) )
         => bij_betw(A,B,F2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),C3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),B2),D2)) ) ) ) ).

% bij_betw_combine
tff(fact_7251_ex__bij__betw,axiom,
    ! [B: $tType,A: $tType,Aa2: set(A),R2: set(product_prod(B,B))] :
      ( 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,Aa2)),R2),bNF_Wellorder_ordLeq(A,B))
     => ? [F3: fun(B,A),B5: set(B)] : bij_betw(B,A,F3,B5,Aa2) ) ).

% ex_bij_betw
tff(fact_7252_card__of__ordIsoI,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),Aa2: set(A),B2: set(B)] :
      ( bij_betw(A,B,F2,Aa2,B2)
     => 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,Aa2)),bNF_Ca6860139660246222851ard_of(B,B2)),bNF_Wellorder_ordIso(A,B)) ) ).

% card_of_ordIsoI
tff(fact_7253_card__of__ordIso,axiom,
    ! [B: $tType,A: $tType,Aa2: set(A),B2: set(B)] :
      ( ? [F9: fun(A,B)] : bij_betw(A,B,F9,Aa2,B2)
    <=> 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,Aa2)),bNF_Ca6860139660246222851ard_of(B,B2)),bNF_Wellorder_ordIso(A,B)) ) ).

% card_of_ordIso
tff(fact_7254_Well__order__iso__copy,axiom,
    ! [B: $tType,A: $tType,Aa2: set(A),R2: set(product_prod(A,A)),F2: fun(A,B),A5: set(B)] :
      ( order_well_order_on(A,Aa2,R2)
     => ( bij_betw(A,B,F2,Aa2,A5)
       => ? [R13: set(product_prod(B,B))] :
            ( order_well_order_on(B,A5,R13)
            & 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),R13),bNF_Wellorder_ordIso(A,B)) ) ) ) ).

% Well_order_iso_copy
tff(fact_7255_strict__subset__transfer,axiom,
    ! [A: $tType,B: $tType,Aa2: fun(A,fun(B,$o))] :
      ( bi_unique(A,B,Aa2)
     => aa(fun(set(B),fun(set(B),$o)),$o,aa(fun(set(A),fun(set(A),$o)),fun(fun(set(B),fun(set(B),$o)),$o),bNF_rel_fun(set(A),set(B),fun(set(A),$o),fun(set(B),$o),bNF_rel_set(A,B,Aa2),bNF_rel_fun(set(A),set(B),$o,$o,bNF_rel_set(A,B,Aa2),fequal($o))),ord_less(set(A))),ord_less(set(B))) ) ).

% strict_subset_transfer
tff(fact_7256_bij__betw__disjoint__Un,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),Aa2: set(A),C3: set(B),G: fun(A,B),B2: set(A),D2: set(B)] :
      ( bij_betw(A,B,F2,Aa2,C3)
     => ( bij_betw(A,B,G,B2,D2)
       => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),B2) = bot_bot(set(A)) )
         => ( ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),C3),D2) = 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_tu(fun(A,B),fun(set(A),fun(fun(A,B),fun(A,B))),F2),Aa2),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),B2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),C3),D2)) ) ) ) ) ).

% bij_betw_disjoint_Un
tff(fact_7257_remdups__adj__transfer,axiom,
    ! [A: $tType,B: $tType,Aa2: fun(A,fun(B,$o))] :
      ( bi_unique(A,B,Aa2)
     => aa(fun(list(B),list(B)),$o,aa(fun(list(A),list(A)),fun(fun(list(B),list(B)),$o),bNF_rel_fun(list(A),list(B),list(A),list(B),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Aa2),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Aa2)),remdups_adj(A)),remdups_adj(B)) ) ).

% remdups_adj_transfer
tff(fact_7258_remdups__transfer,axiom,
    ! [A: $tType,B: $tType,Aa2: fun(A,fun(B,$o))] :
      ( bi_unique(A,B,Aa2)
     => aa(fun(list(B),list(B)),$o,aa(fun(list(A),list(A)),fun(fun(list(B),list(B)),$o),bNF_rel_fun(list(A),list(B),list(A),list(B),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Aa2),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Aa2)),remdups(A)),remdups(B)) ) ).

% remdups_transfer
tff(fact_7259_distinct__transfer,axiom,
    ! [A: $tType,B: $tType,Aa2: fun(A,fun(B,$o))] :
      ( bi_unique(A,B,Aa2)
     => 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,$o,aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Aa2),fequal($o)),distinct(A)),distinct(B)) ) ).

% distinct_transfer
tff(fact_7260_distinct__adj__transfer,axiom,
    ! [A: $tType,B: $tType,Aa2: fun(A,fun(B,$o))] :
      ( bi_unique(A,B,Aa2)
     => 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,$o,aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Aa2),fequal($o)),distinct_adj(A)),distinct_adj(B)) ) ).

% distinct_adj_transfer
tff(fact_7261_infinite__imp__bij__betw2,axiom,
    ! [A: $tType,Aa2: set(A),A3: A] :
      ( ~ aa(set(A),$o,finite_finite2(A),Aa2)
     => ? [H: fun(A,A)] : bij_betw(A,A,H,Aa2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),bot_bot(set(A))))) ) ).

% infinite_imp_bij_betw2
tff(fact_7262_infinite__imp__bij__betw,axiom,
    ! [A: $tType,Aa2: set(A),A3: A] :
      ( ~ aa(set(A),$o,finite_finite2(A),Aa2)
     => ? [H: fun(A,A)] : bij_betw(A,A,H,Aa2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),bot_bot(set(A))))) ) ).

% infinite_imp_bij_betw
tff(fact_7263_le__filter__parametric,axiom,
    ! [A: $tType,B: $tType,Aa2: fun(A,fun(B,$o))] :
      ( bi_unique(A,B,Aa2)
     => aa(fun(filter(B),fun(filter(B),$o)),$o,aa(fun(filter(A),fun(filter(A),$o)),fun(fun(filter(B),fun(filter(B),$o)),$o),bNF_rel_fun(filter(A),filter(B),fun(filter(A),$o),fun(filter(B),$o),rel_filter(A,B,Aa2),bNF_rel_fun(filter(A),filter(B),$o,$o,rel_filter(A,B,Aa2),fequal($o))),ord_less_eq(filter(A))),ord_less_eq(filter(B))) ) ).

% le_filter_parametric
tff(fact_7264_sum_Oreindex__bij__betw__not__neutral,axiom,
    ! [A: $tType,E: $tType,B: $tType] :
      ( comm_monoid_add(E)
     => ! [S3: set(A),T6: set(B),H2: fun(A,B),S: set(A),T4: set(B),G: fun(B,E)] :
          ( aa(set(A),$o,finite_finite2(A),S3)
         => ( aa(set(B),$o,finite_finite2(B),T6)
           => ( 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)),T4),T6))
             => ( ! [A4: A] :
                    ( member(A,A4,S3)
                   => ( aa(B,E,G,aa(A,B,H2,A4)) = zero_zero(E) ) )
               => ( ! [B4: B] :
                      ( member(B,B4,T6)
                     => ( aa(B,E,G,B4) = zero_zero(E) ) )
                 => ( aa(set(A),E,aa(fun(A,E),fun(set(A),E),groups7311177749621191930dd_sum(A,E),aa(fun(B,E),fun(A,E),aTP_Lamp_abp(fun(A,B),fun(fun(B,E),fun(A,E)),H2),G)),S) = aa(set(B),E,aa(fun(B,E),fun(set(B),E),groups7311177749621191930dd_sum(B,E),G),T4) ) ) ) ) ) ) ) ).

% sum.reindex_bij_betw_not_neutral
tff(fact_7265_prod_Oreindex__bij__betw__not__neutral,axiom,
    ! [A: $tType,E: $tType,B: $tType] :
      ( comm_monoid_mult(E)
     => ! [S3: set(A),T6: set(B),H2: fun(A,B),S: set(A),T4: set(B),G: fun(B,E)] :
          ( aa(set(A),$o,finite_finite2(A),S3)
         => ( aa(set(B),$o,finite_finite2(B),T6)
           => ( 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)),T4),T6))
             => ( ! [A4: A] :
                    ( member(A,A4,S3)
                   => ( aa(B,E,G,aa(A,B,H2,A4)) = one_one(E) ) )
               => ( ! [B4: B] :
                      ( member(B,B4,T6)
                     => ( aa(B,E,G,B4) = one_one(E) ) )
                 => ( aa(set(A),E,aa(fun(A,E),fun(set(A),E),groups7121269368397514597t_prod(A,E),aa(fun(B,E),fun(A,E),aTP_Lamp_abq(fun(A,B),fun(fun(B,E),fun(A,E)),H2),G)),S) = aa(set(B),E,aa(fun(B,E),fun(set(B),E),groups7121269368397514597t_prod(B,E),G),T4) ) ) ) ) ) ) ) ).

% prod.reindex_bij_betw_not_neutral
tff(fact_7266_remove1__transfer,axiom,
    ! [A: $tType,B: $tType,Aa2: fun(A,fun(B,$o))] :
      ( bi_unique(A,B,Aa2)
     => aa(fun(B,fun(list(B),list(B))),$o,aa(fun(A,fun(list(A),list(A))),fun(fun(B,fun(list(B),list(B))),$o),bNF_rel_fun(A,B,fun(list(A),list(A)),fun(list(B),list(B)),Aa2,bNF_rel_fun(list(A),list(B),list(A),list(B),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Aa2),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Aa2))),remove1(A)),remove1(B)) ) ).

% remove1_transfer
tff(fact_7267_less__filter__parametric,axiom,
    ! [A: $tType,B: $tType,Aa2: fun(A,fun(B,$o))] :
      ( bi_unique(A,B,Aa2)
     => aa(fun(filter(B),fun(filter(B),$o)),$o,aa(fun(filter(A),fun(filter(A),$o)),fun(fun(filter(B),fun(filter(B),$o)),$o),bNF_rel_fun(filter(A),filter(B),fun(filter(A),$o),fun(filter(B),$o),rel_filter(A,B,Aa2),bNF_rel_fun(filter(A),filter(B),$o,$o,rel_filter(A,B,Aa2),fequal($o))),ord_less(filter(A))),ord_less(filter(B))) ) ).

% less_filter_parametric
tff(fact_7268_removeAll__transfer,axiom,
    ! [A: $tType,B: $tType,Aa2: fun(A,fun(B,$o))] :
      ( bi_unique(A,B,Aa2)
     => aa(fun(B,fun(list(B),list(B))),$o,aa(fun(A,fun(list(A),list(A))),fun(fun(B,fun(list(B),list(B))),$o),bNF_rel_fun(A,B,fun(list(A),list(A)),fun(list(B),list(B)),Aa2,bNF_rel_fun(list(A),list(B),list(A),list(B),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Aa2),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Aa2))),removeAll(A)),removeAll(B)) ) ).

% removeAll_transfer
tff(fact_7269_List_Oinsert__transfer,axiom,
    ! [A: $tType,B: $tType,Aa2: fun(A,fun(B,$o))] :
      ( bi_unique(A,B,Aa2)
     => aa(fun(B,fun(list(B),list(B))),$o,aa(fun(A,fun(list(A),list(A))),fun(fun(B,fun(list(B),list(B))),$o),bNF_rel_fun(A,B,fun(list(A),list(A)),fun(list(B),list(B)),Aa2,bNF_rel_fun(list(A),list(B),list(A),list(B),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Aa2),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Aa2))),insert(A)),insert(B)) ) ).

% List.insert_transfer
tff(fact_7270_Inf__filter__parametric,axiom,
    ! [A: $tType,B: $tType,Aa2: fun(A,fun(B,$o))] :
      ( bi_unique(A,B,Aa2)
     => ( bi_total(A,B,Aa2)
       => aa(fun(set(filter(B)),filter(B)),$o,aa(fun(set(filter(A)),filter(A)),fun(fun(set(filter(B)),filter(B)),$o),bNF_rel_fun(set(filter(A)),set(filter(B)),filter(A),filter(B),bNF_rel_set(filter(A),filter(B),rel_filter(A,B,Aa2)),rel_filter(A,B,Aa2)),complete_Inf_Inf(filter(A))),complete_Inf_Inf(filter(B))) ) ) ).

% Inf_filter_parametric
tff(fact_7271_inf__filter__parametric,axiom,
    ! [A: $tType,B: $tType,Aa2: fun(A,fun(B,$o))] :
      ( bi_unique(A,B,Aa2)
     => ( bi_total(A,B,Aa2)
       => 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,Aa2),bNF_rel_fun(filter(A),filter(B),filter(A),filter(B),rel_filter(A,B,Aa2),rel_filter(A,B,Aa2))),inf_inf(filter(A))),inf_inf(filter(B))) ) ) ).

% inf_filter_parametric
tff(fact_7272_list_Obi__total__rel,axiom,
    ! [B: $tType,A: $tType,R: fun(A,fun(B,$o))] :
      ( bi_total(A,B,R)
     => bi_total(list(A),list(B),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),R)) ) ).

% list.bi_total_rel
tff(fact_7273_bi__total__rel__filter,axiom,
    ! [B: $tType,A: $tType,Aa2: fun(A,fun(B,$o))] :
      ( bi_total(A,B,Aa2)
     => bi_total(filter(A),filter(B),rel_filter(A,B,Aa2)) ) ).

% bi_total_rel_filter
tff(fact_7274_top__filter__parametric,axiom,
    ! [A: $tType,B: $tType,Aa2: fun(A,fun(B,$o))] :
      ( bi_total(A,B,Aa2)
     => aa(filter(B),$o,aa(filter(A),fun(filter(B),$o),rel_filter(A,B,Aa2),top_top(filter(A))),top_top(filter(B))) ) ).

% top_filter_parametric
tff(fact_7275_fun__lub__parametric,axiom,
    ! [A: $tType,E: $tType,G2: $tType,H5: $tType,F4: $tType,B: $tType,Aa2: fun(A,fun(B,$o)),B2: fun(E,fun(F4,$o)),C3: fun(G2,fun(H5,$o))] :
      ( bi_total(A,B,Aa2)
     => ( bi_unique(A,B,Aa2)
       => aa(fun(fun(set(B),F4),fun(set(fun(H5,B)),fun(H5,F4))),$o,aa(fun(fun(set(A),E),fun(set(fun(G2,A)),fun(G2,E))),fun(fun(fun(set(B),F4),fun(set(fun(H5,B)),fun(H5,F4))),$o),bNF_rel_fun(fun(set(A),E),fun(set(B),F4),fun(set(fun(G2,A)),fun(G2,E)),fun(set(fun(H5,B)),fun(H5,F4)),bNF_rel_fun(set(A),set(B),E,F4,bNF_rel_set(A,B,Aa2),B2),bNF_rel_fun(set(fun(G2,A)),set(fun(H5,B)),fun(G2,E),fun(H5,F4),bNF_rel_set(fun(G2,A),fun(H5,B),bNF_rel_fun(G2,H5,A,B,C3,Aa2)),bNF_rel_fun(G2,H5,E,F4,C3,B2))),partial_fun_lub(A,E,G2)),partial_fun_lub(B,F4,H5)) ) ) ).

% fun_lub_parametric
tff(fact_7276_monotone__parametric,axiom,
    ! [A: $tType,E: $tType,F4: $tType,B: $tType,Aa2: fun(A,fun(B,$o)),B2: fun(E,fun(F4,$o))] :
      ( bi_total(A,B,Aa2)
     => aa(fun(fun(B,fun(B,$o)),fun(fun(F4,fun(F4,$o)),fun(fun(B,F4),$o))),$o,aa(fun(fun(A,fun(A,$o)),fun(fun(E,fun(E,$o)),fun(fun(A,E),$o))),fun(fun(fun(B,fun(B,$o)),fun(fun(F4,fun(F4,$o)),fun(fun(B,F4),$o))),$o),bNF_rel_fun(fun(A,fun(A,$o)),fun(B,fun(B,$o)),fun(fun(E,fun(E,$o)),fun(fun(A,E),$o)),fun(fun(F4,fun(F4,$o)),fun(fun(B,F4),$o)),bNF_rel_fun(A,B,fun(A,$o),fun(B,$o),Aa2,bNF_rel_fun(A,B,$o,$o,Aa2,fequal($o))),bNF_rel_fun(fun(E,fun(E,$o)),fun(F4,fun(F4,$o)),fun(fun(A,E),$o),fun(fun(B,F4),$o),bNF_rel_fun(E,F4,fun(E,$o),fun(F4,$o),B2,bNF_rel_fun(E,F4,$o,$o,B2,fequal($o))),bNF_rel_fun(fun(A,E),fun(B,F4),$o,$o,bNF_rel_fun(A,B,E,F4,Aa2,B2),fequal($o)))),comple7038119648293358887notone(A,E)),comple7038119648293358887notone(B,F4)) ) ).

% monotone_parametric
tff(fact_7277_fun__ord__parametric,axiom,
    ! [E: $tType,F4: $tType,A: $tType,B: $tType,H5: $tType,G2: $tType,C3: fun(A,fun(B,$o)),Aa2: fun(E,fun(G2,$o)),B2: fun(F4,fun(H5,$o))] :
      ( bi_total(A,B,C3)
     => aa(fun(fun(G2,fun(H5,$o)),fun(fun(B,G2),fun(fun(B,H5),$o))),$o,aa(fun(fun(E,fun(F4,$o)),fun(fun(A,E),fun(fun(A,F4),$o))),fun(fun(fun(G2,fun(H5,$o)),fun(fun(B,G2),fun(fun(B,H5),$o))),$o),bNF_rel_fun(fun(E,fun(F4,$o)),fun(G2,fun(H5,$o)),fun(fun(A,E),fun(fun(A,F4),$o)),fun(fun(B,G2),fun(fun(B,H5),$o)),bNF_rel_fun(E,G2,fun(F4,$o),fun(H5,$o),Aa2,bNF_rel_fun(F4,H5,$o,$o,B2,fequal($o))),bNF_rel_fun(fun(A,E),fun(B,G2),fun(fun(A,F4),$o),fun(fun(B,H5),$o),bNF_rel_fun(A,B,E,G2,C3,Aa2),bNF_rel_fun(fun(A,F4),fun(B,H5),$o,$o,bNF_rel_fun(A,B,F4,H5,C3,B2),fequal($o)))),partial_fun_ord(E,F4,A)),partial_fun_ord(G2,H5,B)) ) ).

% fun_ord_parametric
tff(fact_7278_power__int__add__1_H,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xa: A,M2: int] :
          ( ( ( Xa != zero_zero(A) )
            | ( M2 != aa(int,int,uminus_uminus(int),one_one(int)) ) )
         => ( power_int(A,Xa,aa(int,int,aa(int,fun(int,int),plus_plus(int),M2),one_one(int))) = aa(A,A,aa(A,fun(A,A),times_times(A),Xa),power_int(A,Xa,M2)) ) ) ) ).

% power_int_add_1'
tff(fact_7279_power__int__mult__distrib__numeral1,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [W2: num,Y: A,M2: int] : ( power_int(A,aa(A,A,aa(A,fun(A,A),times_times(A),numeral_numeral(A,W2)),Y),M2) = aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,numeral_numeral(A,W2),M2)),power_int(A,Y,M2)) ) ) ).

% power_int_mult_distrib_numeral1
tff(fact_7280_power__int__mult__distrib__numeral2,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Xa: A,W2: num,M2: int] : ( power_int(A,aa(A,A,aa(A,fun(A,A),times_times(A),Xa),numeral_numeral(A,W2)),M2) = aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,Xa,M2)),power_int(A,numeral_numeral(A,W2),M2)) ) ) ).

% power_int_mult_distrib_numeral2
tff(fact_7281_power__int__minus__one__mult__self,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [M2: int] : ( aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,aa(A,A,uminus_uminus(A),one_one(A)),M2)),power_int(A,aa(A,A,uminus_uminus(A),one_one(A)),M2)) = one_one(A) ) ) ).

% power_int_minus_one_mult_self
tff(fact_7282_power__int__minus__one__mult__self_H,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [M2: int,B3: A] : ( aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,aa(A,A,uminus_uminus(A),one_one(A)),M2)),aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,aa(A,A,uminus_uminus(A),one_one(A)),M2)),B3)) = B3 ) ) ).

% power_int_minus_one_mult_self'
tff(fact_7283_power__int__add__numeral,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xa: A,M2: num,N: num] : ( aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,Xa,numeral_numeral(int,M2))),power_int(A,Xa,numeral_numeral(int,N))) = power_int(A,Xa,numeral_numeral(int,aa(num,num,aa(num,fun(num,num),plus_plus(num),M2),N))) ) ) ).

% power_int_add_numeral
tff(fact_7284_power__int__add__numeral2,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xa: A,M2: num,N: num,B3: A] : ( aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,Xa,numeral_numeral(int,M2))),aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,Xa,numeral_numeral(int,N))),B3)) = aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,Xa,numeral_numeral(int,aa(num,num,aa(num,fun(num,num),plus_plus(num),M2),N)))),B3) ) ) ).

% power_int_add_numeral2
tff(fact_7285_power__int__strict__increasing,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [N: int,N7: int,A3: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),N),N7)
         => ( aa(A,$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,N7)) ) ) ) ).

% power_int_strict_increasing
tff(fact_7286_zero__less__power__int,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A,N: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xa)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),power_int(A,Xa,N)) ) ) ).

% zero_less_power_int
tff(fact_7287_power__int__mult__distrib,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Xa: A,Y: A,M2: int] : ( power_int(A,aa(A,A,aa(A,fun(A,A),times_times(A),Xa),Y),M2) = aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,Xa,M2)),power_int(A,Y,M2)) ) ) ).

% power_int_mult_distrib
tff(fact_7288_power__int__commutes,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xa: A,N: int] : ( aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,Xa,N)),Xa) = aa(A,A,aa(A,fun(A,A),times_times(A),Xa),power_int(A,Xa,N)) ) ) ).

% power_int_commutes
tff(fact_7289_power__int__strict__decreasing,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [N: int,N7: int,A3: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),N),N7)
         => ( aa(A,$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,N7)),power_int(A,A3,N)) ) ) ) ) ).

% power_int_strict_decreasing
tff(fact_7290_power__int__strict__antimono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A,N: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),N),zero_zero(int))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),power_int(A,B3,N)),power_int(A,A3,N)) ) ) ) ) ).

% power_int_strict_antimono
tff(fact_7291_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_7292_power__int__add,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xa: A,M2: int,N: int] :
          ( ( ( Xa != zero_zero(A) )
            | ( aa(int,int,aa(int,fun(int,int),plus_plus(int),M2),N) != zero_zero(int) ) )
         => ( power_int(A,Xa,aa(int,int,aa(int,fun(int,int),plus_plus(int),M2),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,Xa,M2)),power_int(A,Xa,N)) ) ) ) ).

% power_int_add
tff(fact_7293_power__int__minus__left__distrib,axiom,
    ! [B: $tType,A: $tType,E: $tType] :
      ( ( division_ring(E)
        & one(A)
        & uminus(A) )
     => ! [Xa: B,A3: E,N: int] :
          ( nO_MATCH(A,B,aa(A,A,uminus_uminus(A),one_one(A)),Xa)
         => ( power_int(E,aa(E,E,uminus_uminus(E),A3),N) = aa(E,E,aa(E,fun(E,E),times_times(E),power_int(E,aa(E,E,uminus_uminus(E),one_one(E)),N)),power_int(E,A3,N)) ) ) ) ).

% power_int_minus_left_distrib
tff(fact_7294_power__int__antimono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A,N: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),N),zero_zero(int))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),power_int(A,B3,N)),power_int(A,A3,N)) ) ) ) ) ).

% power_int_antimono
tff(fact_7295_power__int__strict__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A,N: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),N)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),power_int(A,A3,N)),power_int(A,B3,N)) ) ) ) ) ).

% power_int_strict_mono
tff(fact_7296_power__int__le__imp__le__exp,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A,M2: int,N: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),Xa)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),power_int(A,Xa,M2)),power_int(A,Xa,N))
           => ( aa(int,$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),M2),N) ) ) ) ) ).

% power_int_le_imp_le_exp
tff(fact_7297_power__int__le__imp__less__exp,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A,M2: int,N: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),Xa)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),power_int(A,Xa,M2)),power_int(A,Xa,N))
           => ( aa(int,$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),M2),N) ) ) ) ) ).

% power_int_le_imp_less_exp
tff(fact_7298_power__int__minus__mult,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Xa: A,N: int] :
          ( ( ( Xa != zero_zero(A) )
            | ( N != zero_zero(int) ) )
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,Xa,aa(int,int,aa(int,fun(int,int),minus_minus(int),N),one_one(int)))),Xa) = power_int(A,Xa,N) ) ) ) ).

% power_int_minus_mult
tff(fact_7299_power__int__add__1,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xa: A,M2: int] :
          ( ( ( Xa != zero_zero(A) )
            | ( M2 != aa(int,int,uminus_uminus(int),one_one(int)) ) )
         => ( power_int(A,Xa,aa(int,int,aa(int,fun(int,int),plus_plus(int),M2),one_one(int))) = aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,Xa,M2)),Xa) ) ) ) ).

% power_int_add_1
tff(fact_7300_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_aau(int,fun(int,product_prod(int,int)))),fract) ).

% Fract.transfer
tff(fact_7301_subset__mset_Ostrict__mono__def,axiom,
    ! [B: $tType,A: $tType] :
      ( order(B)
     => ! [F2: fun(multiset(A),B)] :
          ( aa(fun(multiset(A),B),$o,strict_mono(multiset(A),B,subset_mset(A)),F2)
        <=> ! [X6: multiset(A),Y4: multiset(A)] :
              ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subset_mset(A),X6),Y4)
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(multiset(A),B,F2,X6)),aa(multiset(A),B,F2,Y4)) ) ) ) ).

% subset_mset.strict_mono_def
tff(fact_7302_quotient__of__eq,axiom,
    ! [A3: int,B3: int,P2: int,Q2: int] :
      ( ( quotient_of(aa(int,rat,aa(int,fun(int,rat),fract,A3),B3)) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P2),Q2) )
     => ( aa(int,rat,aa(int,fun(int,rat),fract,P2),Q2) = aa(int,rat,aa(int,fun(int,rat),fract,A3),B3) ) ) ).

% quotient_of_eq
tff(fact_7303_order_Ostrict__mono_Ocong,axiom,
    ! [B: $tType,A: $tType] :
      ( order(B)
     => ! [Less: fun(A,fun(A,$o))] : ( strict_mono(A,B,Less) = strict_mono(A,B,Less) ) ) ).

% order.strict_mono.cong
tff(fact_7304_normalize__eq,axiom,
    ! [A3: int,B3: int,P2: int,Q2: int] :
      ( ( normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),A3),B3)) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P2),Q2) )
     => ( aa(int,rat,aa(int,fun(int,rat),fract,P2),Q2) = aa(int,rat,aa(int,fun(int,rat),fract,A3),B3) ) ) ).

% normalize_eq
tff(fact_7305_quotient__of__Fract,axiom,
    ! [A3: int,B3: int] : ( quotient_of(aa(int,rat,aa(int,fun(int,rat),fract,A3),B3)) = normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),A3),B3)) ) ).

% quotient_of_Fract
tff(fact_7306_Fract_Oabs__eq,axiom,
    ! [Xa: int,Xb: int] :
      ( aa(int,rat,aa(int,fun(int,rat),fract,Xa),Xb) = aa(product_prod(int,int),rat,abs_Rat,
          $ite(Xb = 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),Xb))) ) ).

% Fract.abs_eq
tff(fact_7307_subset__mset_Ostrict__monoD,axiom,
    ! [B: $tType,A: $tType] :
      ( order(B)
     => ! [F2: fun(multiset(A),B),Xa: multiset(A),Y: multiset(A)] :
          ( aa(fun(multiset(A),B),$o,strict_mono(multiset(A),B,subset_mset(A)),F2)
         => ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subset_mset(A),Xa),Y)
           => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(multiset(A),B,F2,Xa)),aa(multiset(A),B,F2,Y)) ) ) ) ).

% subset_mset.strict_monoD
tff(fact_7308_subset__mset_Ostrict__monoI,axiom,
    ! [B: $tType,A: $tType] :
      ( order(B)
     => ! [F2: fun(multiset(A),B)] :
          ( ! [X: multiset(A),Y2: multiset(A)] :
              ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subset_mset(A),X),Y2)
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(multiset(A),B,F2,X)),aa(multiset(A),B,F2,Y2)) )
         => aa(fun(multiset(A),B),$o,strict_mono(multiset(A),B,subset_mset(A)),F2) ) ) ).

% subset_mset.strict_monoI
tff(fact_7309_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_7310_folding__insort__key_Oaxioms_I2_J,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),S: set(B),F2: fun(B,A)] :
      ( folding_insort_key(A,B,Less_eq,Less,S,F2)
     => foldin3648464208017769352axioms(B,A,S,F2) ) ).

% folding_insort_key.axioms(2)
tff(fact_7311_group_Oleft__cancel,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A),A3: A,B3: A,C2: A] :
      ( group(A,F2,Z2,Inverse)
     => ( ( aa(A,A,aa(A,fun(A,A),F2,A3),B3) = aa(A,A,aa(A,fun(A,A),F2,A3),C2) )
      <=> ( B3 = C2 ) ) ) ).

% group.left_cancel
tff(fact_7312_group_Oleft__inverse,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A),A3: A] :
      ( group(A,F2,Z2,Inverse)
     => ( aa(A,A,aa(A,fun(A,A),F2,aa(A,A,Inverse,A3)),A3) = Z2 ) ) ).

% group.left_inverse
tff(fact_7313_group_Oright__cancel,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A),B3: A,A3: A,C2: A] :
      ( group(A,F2,Z2,Inverse)
     => ( ( aa(A,A,aa(A,fun(A,A),F2,B3),A3) = aa(A,A,aa(A,fun(A,A),F2,C2),A3) )
      <=> ( B3 = C2 ) ) ) ).

% group.right_cancel
tff(fact_7314_group_Oright__inverse,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A),A3: A] :
      ( group(A,F2,Z2,Inverse)
     => ( aa(A,A,aa(A,fun(A,A),F2,A3),aa(A,A,Inverse,A3)) = Z2 ) ) ).

% group.right_inverse
tff(fact_7315_group_Oinverse__unique,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A),A3: A,B3: A] :
      ( group(A,F2,Z2,Inverse)
     => ( ( aa(A,A,aa(A,fun(A,A),F2,A3),B3) = Z2 )
       => ( aa(A,A,Inverse,A3) = B3 ) ) ) ).

% group.inverse_unique
tff(fact_7316_group_Oinverse__inverse,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A),A3: A] :
      ( group(A,F2,Z2,Inverse)
     => ( aa(A,A,Inverse,aa(A,A,Inverse,A3)) = A3 ) ) ).

% group.inverse_inverse
tff(fact_7317_group_Oinverse__neutral,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A)] :
      ( group(A,F2,Z2,Inverse)
     => ( aa(A,A,Inverse,Z2) = Z2 ) ) ).

% group.inverse_neutral
tff(fact_7318_group_Ogroup__left__neutral,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A),A3: A] :
      ( group(A,F2,Z2,Inverse)
     => ( aa(A,A,aa(A,fun(A,A),F2,Z2),A3) = A3 ) ) ).

% group.group_left_neutral
tff(fact_7319_group_Oinverse__distrib__swap,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A),A3: A,B3: A] :
      ( group(A,F2,Z2,Inverse)
     => ( aa(A,A,Inverse,aa(A,A,aa(A,fun(A,A),F2,A3),B3)) = aa(A,A,aa(A,fun(A,A),F2,aa(A,A,Inverse,B3)),aa(A,A,Inverse,A3)) ) ) ).

% group.inverse_distrib_swap
tff(fact_7320_folding__insort__key__axioms_Ointro,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),S: set(A)] :
      ( inj_on(A,B,F2,S)
     => foldin3648464208017769352axioms(A,B,S,F2) ) ).

% folding_insort_key_axioms.intro
tff(fact_7321_folding__insort__key__axioms__def,axiom,
    ! [B: $tType,A: $tType,S: set(A),F2: fun(A,B)] :
      ( foldin3648464208017769352axioms(A,B,S,F2)
    <=> inj_on(A,B,F2,S) ) ).

% folding_insort_key_axioms_def
tff(fact_7322_Powp__Pow__eq,axiom,
    ! [A: $tType,Aa2: set(A),X4: set(A)] :
      ( aa(set(A),$o,powp(A,aa(set(A),fun(A,$o),aTP_Lamp_a(set(A),fun(A,$o)),Aa2)),X4)
    <=> member(set(A),X4,pow(A,Aa2)) ) ).

% Powp_Pow_eq
tff(fact_7323_mergesort__by__rel__split_Opelims,axiom,
    ! [A: $tType,Xa: product_prod(list(A),list(A)),Xaa: list(A),Y: product_prod(list(A),list(A))] :
      ( ( merges295452479951948502_split(A,Xa,Xaa) = 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)),Xa),Xaa))
       => ( ! [Xs1: list(A),Xs22: 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)),Xs1),Xs22) )
             => ( ( Xaa = 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)),Xs1),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)),Xs1),Xs22)),nil(A))) ) ) )
         => ( ! [Xs1: list(A),Xs22: 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)),Xs1),Xs22) )
               => ! [X: A] :
                    ( ( Xaa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),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),X),Xs1)),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)),Xs1),Xs22)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)))) ) ) )
           => ~ ! [Xs1: list(A),Xs22: 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)),Xs1),Xs22) )
                 => ! [X12: A,X23: A,Xs: list(A)] :
                      ( ( Xaa = 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),Xs)) )
                     => ( ( 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),Xs1)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X23),Xs22)),Xs) )
                       => ~ 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)),Xs1),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),Xs)))) ) ) ) ) ) ) ) ).

% mergesort_by_rel_split.pelims
tff(fact_7324_Powp__mono,axiom,
    ! [A: $tType,Aa2: fun(A,$o),B2: fun(A,$o)] :
      ( aa(fun(A,$o),$o,aa(fun(A,$o),fun(fun(A,$o),$o),ord_less_eq(fun(A,$o)),Aa2),B2)
     => aa(fun(set(A),$o),$o,aa(fun(set(A),$o),fun(fun(set(A),$o),$o),ord_less_eq(fun(set(A),$o)),powp(A,Aa2)),powp(A,B2)) ) ).

% Powp_mono
tff(fact_7325_Powp__def,axiom,
    ! [A: $tType,Aa2: fun(A,$o),X4: set(A)] :
      ( aa(set(A),$o,powp(A,Aa2),X4)
    <=> ! [Xa2: A] :
          ( member(A,Xa2,X4)
         => aa(A,$o,Aa2,Xa2) ) ) ).

% Powp_def
tff(fact_7326_exhaustive__integer_H_Ocases,axiom,
    ! [Xa: product_prod(fun(code_integer,option(product_prod($o,list(code_term)))),product_prod(code_integer,code_integer))] :
      ~ ! [F3: fun(code_integer,option(product_prod($o,list(code_term)))),D4: code_integer,I3: code_integer] : ( Xa != 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)),F3),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),D4),I3)) ) ).

% exhaustive_integer'.cases
tff(fact_7327_subset_Osuc__Union__closedp_Ointros_I2_J,axiom,
    ! [A: $tType,Aa2: set(set(A)),Xa6: set(set(set(A)))] :
      ( aa(set(set(set(A))),$o,powp(set(set(A)),pred_s7749564232668923593losedp(set(A),Aa2,ord_less(set(A)))),Xa6)
     => aa(set(set(A)),$o,pred_s7749564232668923593losedp(set(A),Aa2,ord_less(set(A))),aa(set(set(set(A))),set(set(A)),complete_Sup_Sup(set(set(A))),Xa6)) ) ).

% subset.suc_Union_closedp.intros(2)
tff(fact_7328_subset_Osuc__Union__closedp_OUnion,axiom,
    ! [A: $tType,Aa2: set(set(A)),X3: set(set(set(A)))] :
      ( aa(set(set(set(A))),$o,powp(set(set(A)),pred_s7749564232668923593losedp(set(A),Aa2,ord_less(set(A)))),X3)
     => aa(set(set(A)),$o,pred_s7749564232668923593losedp(set(A),Aa2,ord_less(set(A))),aa(set(set(set(A))),set(set(A)),complete_Sup_Sup(set(set(A))),X3)) ) ).

% subset.suc_Union_closedp.Union
tff(fact_7329_subset_Osuc__Union__closedp_Osuc,axiom,
    ! [A: $tType,Aa2: set(set(A)),X3: set(set(A))] :
      ( aa(set(set(A)),$o,pred_s7749564232668923593losedp(set(A),Aa2,ord_less(set(A))),X3)
     => aa(set(set(A)),$o,pred_s7749564232668923593losedp(set(A),Aa2,ord_less(set(A))),pred_suc(set(A),Aa2,ord_less(set(A)),X3)) ) ).

% subset.suc_Union_closedp.suc
tff(fact_7330_exhaustive__int_H_Ocases,axiom,
    ! [Xa: product_prod(fun(int,option(product_prod($o,list(code_term)))),product_prod(int,int))] :
      ~ ! [F3: fun(int,option(product_prod($o,list(code_term)))),D4: int,I3: int] : ( Xa != 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)),F3),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),D4),I3)) ) ).

% exhaustive_int'.cases
tff(fact_7331_full__exhaustive__int_H_Ocases,axiom,
    ! [Xa: product_prod(fun(product_prod(int,fun(product_unit,code_term)),option(product_prod($o,list(code_term)))),product_prod(int,int))] :
      ~ ! [F3: fun(product_prod(int,fun(product_unit,code_term)),option(product_prod($o,list(code_term)))),D4: int,I3: int] : ( Xa != 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)),F3),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),D4),I3)) ) ).

% full_exhaustive_int'.cases
tff(fact_7332_subset_Osuc__Union__closedp_Ocases,axiom,
    ! [A: $tType,Aa2: set(set(A)),A3: set(set(A))] :
      ( aa(set(set(A)),$o,pred_s7749564232668923593losedp(set(A),Aa2,ord_less(set(A))),A3)
     => ( ! [X7: set(set(A))] :
            ( ( A3 = pred_suc(set(A),Aa2,ord_less(set(A)),X7) )
           => ~ aa(set(set(A)),$o,pred_s7749564232668923593losedp(set(A),Aa2,ord_less(set(A))),X7) )
       => ~ ! [X7: set(set(set(A)))] :
              ( ( A3 = aa(set(set(set(A))),set(set(A)),complete_Sup_Sup(set(set(A))),X7) )
             => ~ aa(set(set(set(A))),$o,powp(set(set(A)),pred_s7749564232668923593losedp(set(A),Aa2,ord_less(set(A)))),X7) ) ) ) ).

% subset.suc_Union_closedp.cases
tff(fact_7333_subset_Osuc__Union__closedp_Osimps,axiom,
    ! [A: $tType,Aa2: set(set(A)),A3: set(set(A))] :
      ( aa(set(set(A)),$o,pred_s7749564232668923593losedp(set(A),Aa2,ord_less(set(A))),A3)
    <=> ( ? [X9: set(set(A))] :
            ( ( A3 = pred_suc(set(A),Aa2,ord_less(set(A)),X9) )
            & aa(set(set(A)),$o,pred_s7749564232668923593losedp(set(A),Aa2,ord_less(set(A))),X9) )
        | ? [X9: set(set(set(A)))] :
            ( ( A3 = aa(set(set(set(A))),set(set(A)),complete_Sup_Sup(set(set(A))),X9) )
            & aa(set(set(set(A))),$o,powp(set(set(A)),pred_s7749564232668923593losedp(set(A),Aa2,ord_less(set(A)))),X9) ) ) ) ).

% subset.suc_Union_closedp.simps
tff(fact_7334_full__exhaustive__integer_H_Ocases,axiom,
    ! [Xa: product_prod(fun(product_prod(code_integer,fun(product_unit,code_term)),option(product_prod($o,list(code_term)))),product_prod(code_integer,code_integer))] :
      ~ ! [F3: fun(product_prod(code_integer,fun(product_unit,code_term)),option(product_prod($o,list(code_term)))),D4: code_integer,I3: code_integer] : ( Xa != 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)),F3),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),D4),I3)) ) ).

% full_exhaustive_integer'.cases
tff(fact_7335_list__encode_Oelims,axiom,
    ! [Xa: list(nat),Y: nat] :
      ( ( nat_list_encode(Xa) = Y )
     => ( ( ( Xa = nil(nat) )
         => ( Y != zero_zero(nat) ) )
       => ~ ! [X: nat,Xs: list(nat)] :
              ( ( Xa = aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),X),Xs) )
             => ( Y != aa(nat,nat,suc,nat_prod_encode(aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X),nat_list_encode(Xs)))) ) ) ) ) ).

% list_encode.elims
tff(fact_7336_irreflp__irrefl__eq,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( irreflp(A,aTP_Lamp_cz(set(product_prod(A,A)),fun(A,fun(A,$o)),R))
    <=> irrefl(A,R) ) ).

% irreflp_irrefl_eq
tff(fact_7337_list__encode_Osimps_I2_J,axiom,
    ! [Xa: nat,Xsa: list(nat)] : ( nat_list_encode(aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),Xa),Xsa)) = aa(nat,nat,suc,nat_prod_encode(aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xa),nat_list_encode(Xsa)))) ) ).

% list_encode.simps(2)
tff(fact_7338_irreflp__def,axiom,
    ! [A: $tType,R: fun(A,fun(A,$o))] :
      ( irreflp(A,R)
    <=> ! [A7: A] : ~ aa(A,$o,aa(A,fun(A,$o),R,A7),A7) ) ).

% irreflp_def
tff(fact_7339_irreflpI,axiom,
    ! [A: $tType,R: fun(A,fun(A,$o))] :
      ( ! [A4: A] : ~ aa(A,$o,aa(A,fun(A,$o),R,A4),A4)
     => irreflp(A,R) ) ).

% irreflpI
tff(fact_7340_irreflp__greater,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => irreflp(A,aTP_Lamp_ij(A,fun(A,$o))) ) ).

% irreflp_greater
tff(fact_7341_irreflp__less,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => irreflp(A,ord_less(A)) ) ).

% irreflp_less
tff(fact_7342_le__prod__encode__1,axiom,
    ! [A3: nat,B3: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A3),nat_prod_encode(aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),A3),B3))) ).

% le_prod_encode_1
tff(fact_7343_le__prod__encode__2,axiom,
    ! [B3: nat,A3: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),B3),nat_prod_encode(aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),A3),B3))) ).

% le_prod_encode_2
tff(fact_7344_list__encode_Opelims,axiom,
    ! [Xa: list(nat),Y: nat] :
      ( ( nat_list_encode(Xa) = Y )
     => ( accp(list(nat),nat_list_encode_rel,Xa)
       => ( ( ( Xa = nil(nat) )
           => ( ( Y = zero_zero(nat) )
             => ~ accp(list(nat),nat_list_encode_rel,nil(nat)) ) )
         => ~ ! [X: nat,Xs: list(nat)] :
                ( ( Xa = aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),X),Xs) )
               => ( ( Y = aa(nat,nat,suc,nat_prod_encode(aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X),nat_list_encode(Xs)))) )
                 => ~ accp(list(nat),nat_list_encode_rel,aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),X),Xs)) ) ) ) ) ) ).

% list_encode.pelims
tff(fact_7345_equivp__equiv,axiom,
    ! [A: $tType,Aa2: set(product_prod(A,A))] :
      ( equiv_equiv(A,top_top(set(A)),Aa2)
    <=> equiv_equivp(A,aTP_Lamp_cz(set(product_prod(A,A)),fun(A,fun(A,$o)),Aa2)) ) ).

% equivp_equiv
tff(fact_7346_is__filter__parametric,axiom,
    ! [A: $tType,B: $tType,Aa2: fun(A,fun(B,$o))] :
      ( bi_total(A,B,Aa2)
     => ( bi_unique(A,B,Aa2)
       => aa(fun(fun(fun(B,$o),$o),$o),$o,aa(fun(fun(fun(A,$o),$o),$o),fun(fun(fun(fun(B,$o),$o),$o),$o),bNF_rel_fun(fun(fun(A,$o),$o),fun(fun(B,$o),$o),$o,$o,bNF_rel_fun(fun(A,$o),fun(B,$o),$o,$o,bNF_rel_fun(A,B,$o,$o,Aa2,fequal($o)),fequal($o)),fequal($o)),is_filter(A)),is_filter(B)) ) ) ).

% is_filter_parametric
tff(fact_7347_asymp__asym__eq,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( asymp(A,aTP_Lamp_cz(set(product_prod(A,A)),fun(A,fun(A,$o)),R))
    <=> asym(A,R) ) ).

% asymp_asym_eq
tff(fact_7348_is__filter__map__filter__on,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),X3: set(B),Fa: filter(B)] :
      ( aa(fun(fun(A,$o),$o),$o,is_filter(A),aa(filter(B),fun(fun(A,$o),$o),aa(set(B),fun(filter(B),fun(fun(A,$o),$o)),aTP_Lamp_abs(fun(B,A),fun(set(B),fun(filter(B),fun(fun(A,$o),$o))),F2),X3),Fa))
    <=> aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),eventually(B),aTP_Lamp_wu(set(B),fun(B,$o),X3)),Fa) ) ).

% is_filter_map_filter_on
tff(fact_7349_asym__inv__image,axiom,
    ! [A: $tType,B: $tType,R: set(product_prod(A,A)),F2: fun(B,A)] :
      ( asym(A,R)
     => asym(B,inv_image(A,B,R,F2)) ) ).

% asym_inv_image
tff(fact_7350_asym__lex,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( asym(A,R)
     => asym(list(A),lex(A,R)) ) ).

% asym_lex
tff(fact_7351_Abs__filter__cases,axiom,
    ! [A: $tType,Xa: filter(A)] :
      ~ ! [Y2: fun(fun(A,$o),$o)] :
          ( ( Xa = aa(fun(fun(A,$o),$o),filter(A),abs_filter(A),Y2) )
         => ~ member(fun(fun(A,$o),$o),Y2,aa(fun(fun(fun(A,$o),$o),$o),set(fun(fun(A,$o),$o)),collect(fun(fun(A,$o),$o)),is_filter(A))) ) ).

% Abs_filter_cases
tff(fact_7352_Abs__filter__induct,axiom,
    ! [A: $tType,Pa: fun(filter(A),$o),Xa: filter(A)] :
      ( ! [Y2: fun(fun(A,$o),$o)] :
          ( member(fun(fun(A,$o),$o),Y2,aa(fun(fun(fun(A,$o),$o),$o),set(fun(fun(A,$o),$o)),collect(fun(fun(A,$o),$o)),is_filter(A)))
         => aa(filter(A),$o,Pa,aa(fun(fun(A,$o),$o),filter(A),abs_filter(A),Y2)) )
     => aa(filter(A),$o,Pa,Xa) ) ).

% Abs_filter_induct
tff(fact_7353_Abs__filter__inject,axiom,
    ! [A: $tType,Xa: fun(fun(A,$o),$o),Y: fun(fun(A,$o),$o)] :
      ( member(fun(fun(A,$o),$o),Xa,aa(fun(fun(fun(A,$o),$o),$o),set(fun(fun(A,$o),$o)),collect(fun(fun(A,$o),$o)),is_filter(A)))
     => ( member(fun(fun(A,$o),$o),Y,aa(fun(fun(fun(A,$o),$o),$o),set(fun(fun(A,$o),$o)),collect(fun(fun(A,$o),$o)),is_filter(A)))
       => ( ( aa(fun(fun(A,$o),$o),filter(A),abs_filter(A),Xa) = aa(fun(fun(A,$o),$o),filter(A),abs_filter(A),Y) )
        <=> ( Xa = Y ) ) ) ) ).

% Abs_filter_inject
tff(fact_7354_asym__lenlex,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( asym(A,R)
     => asym(list(A),lenlex(A,R)) ) ).

% asym_lenlex
tff(fact_7355_lexord__asym,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( asym(A,R)
     => asym(list(A),lexord(A,R)) ) ).

% lexord_asym
tff(fact_7356_is__filter__def,axiom,
    ! [A: $tType,Fa: fun(fun(A,$o),$o)] :
      ( aa(fun(fun(A,$o),$o),$o,is_filter(A),Fa)
    <=> ( aa(fun(A,$o),$o,Fa,aTP_Lamp_cc(A,$o))
        & ! [P6: fun(A,$o)] :
            ( aa(fun(A,$o),$o,Fa,P6)
           => ! [Q7: fun(A,$o)] :
                ( aa(fun(A,$o),$o,Fa,Q7)
               => aa(fun(A,$o),$o,Fa,aa(fun(A,$o),fun(A,$o),aTP_Lamp_bx(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P6),Q7)) ) )
        & ! [P6: fun(A,$o),Q7: fun(A,$o)] :
            ( ! [X6: A] :
                ( aa(A,$o,P6,X6)
               => aa(A,$o,Q7,X6) )
           => ( aa(fun(A,$o),$o,Fa,P6)
             => aa(fun(A,$o),$o,Fa,Q7) ) ) ) ) ).

% is_filter_def
tff(fact_7357_is__filter_Ointro,axiom,
    ! [A: $tType,Fa: fun(fun(A,$o),$o)] :
      ( aa(fun(A,$o),$o,Fa,aTP_Lamp_cc(A,$o))
     => ( ! [P3: fun(A,$o)] :
            ( aa(fun(A,$o),$o,Fa,P3)
           => ! [Q3: fun(A,$o)] :
                ( aa(fun(A,$o),$o,Fa,Q3)
               => aa(fun(A,$o),$o,Fa,aa(fun(A,$o),fun(A,$o),aTP_Lamp_bx(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P3),Q3)) ) )
       => ( ! [P3: fun(A,$o),Q3: fun(A,$o)] :
              ( ! [X4: A] :
                  ( aa(A,$o,P3,X4)
                 => aa(A,$o,Q3,X4) )
             => ( aa(fun(A,$o),$o,Fa,P3)
               => aa(fun(A,$o),$o,Fa,Q3) ) )
         => aa(fun(fun(A,$o),$o),$o,is_filter(A),Fa) ) ) ) ).

% is_filter.intro
tff(fact_7358_is__filter_Omono,axiom,
    ! [A: $tType,Fa: fun(fun(A,$o),$o),Pa: fun(A,$o),Qa: fun(A,$o)] :
      ( aa(fun(fun(A,$o),$o),$o,is_filter(A),Fa)
     => ( ! [X: A] :
            ( aa(A,$o,Pa,X)
           => aa(A,$o,Qa,X) )
       => ( aa(fun(A,$o),$o,Fa,Pa)
         => aa(fun(A,$o),$o,Fa,Qa) ) ) ) ).

% is_filter.mono
tff(fact_7359_is__filter_Oconj,axiom,
    ! [A: $tType,Fa: fun(fun(A,$o),$o),Pa: fun(A,$o),Qa: fun(A,$o)] :
      ( aa(fun(fun(A,$o),$o),$o,is_filter(A),Fa)
     => ( aa(fun(A,$o),$o,Fa,Pa)
       => ( aa(fun(A,$o),$o,Fa,Qa)
         => aa(fun(A,$o),$o,Fa,aa(fun(A,$o),fun(A,$o),aTP_Lamp_bx(fun(A,$o),fun(fun(A,$o),fun(A,$o)),Pa),Qa)) ) ) ) ).

% is_filter.conj
tff(fact_7360_is__filter_OTrue,axiom,
    ! [A: $tType,Fa: fun(fun(A,$o),$o)] :
      ( aa(fun(fun(A,$o),$o),$o,is_filter(A),Fa)
     => aa(fun(A,$o),$o,Fa,aTP_Lamp_cc(A,$o)) ) ).

% is_filter.True
tff(fact_7361_asym_Ocases,axiom,
    ! [A: $tType,A3: set(product_prod(A,A))] :
      ( asym(A,A3)
     => ! [A13: A,B9: A] :
          ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A13),B9),A3)
         => ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B9),A13),A3) ) ) ).

% asym.cases
tff(fact_7362_asym_Osimps,axiom,
    ! [A: $tType,A3: set(product_prod(A,A))] :
      ( asym(A,A3)
    <=> ? [R7: set(product_prod(A,A))] :
          ( ( A3 = R7 )
          & ! [X6: A,Xa2: A] :
              ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X6),Xa2),R7)
             => ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa2),X6),R7) ) ) ) ).

% asym.simps
tff(fact_7363_asym_Ointros,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( ! [A4: A,B4: A] :
          ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),B4),R)
         => ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B4),A4),R) )
     => asym(A,R) ) ).

% asym.intros
tff(fact_7364_asymD,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),Xa: A,Y: A] :
      ( asym(A,R)
     => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Y),R)
       => ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Xa),R) ) ) ).

% asymD
tff(fact_7365_asym__iff,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( asym(A,R)
    <=> ! [X6: A,Y4: A] :
          ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X6),Y4),R)
         => ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),X6),R) ) ) ).

% asym_iff
tff(fact_7366_eventually__Abs__filter,axiom,
    ! [A: $tType,Fa: fun(fun(A,$o),$o),Pa: fun(A,$o)] :
      ( aa(fun(fun(A,$o),$o),$o,is_filter(A),Fa)
     => ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Pa),aa(fun(fun(A,$o),$o),filter(A),abs_filter(A),Fa))
      <=> aa(fun(A,$o),$o,Fa,Pa) ) ) ).

% eventually_Abs_filter
tff(fact_7367_lexord__asymmetric,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A3: list(A),B3: list(A)] :
      ( asym(A,R)
     => ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),A3),B3),lexord(A,R))
       => ~ member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),B3),A3),lexord(A,R)) ) ) ).

% lexord_asymmetric
tff(fact_7368_Abs__filter__inverse,axiom,
    ! [A: $tType,Y: fun(fun(A,$o),$o)] :
      ( member(fun(fun(A,$o),$o),Y,aa(fun(fun(fun(A,$o),$o),$o),set(fun(fun(A,$o),$o)),collect(fun(fun(A,$o),$o)),is_filter(A)))
     => ( aa(filter(A),fun(fun(A,$o),$o),rep_filter(A),aa(fun(fun(A,$o),$o),filter(A),abs_filter(A),Y)) = Y ) ) ).

% Abs_filter_inverse
tff(fact_7369_Abs__filter__inverse_H,axiom,
    ! [A: $tType,Fa: fun(fun(A,$o),$o)] :
      ( aa(fun(fun(A,$o),$o),$o,is_filter(A),Fa)
     => ( aa(filter(A),fun(fun(A,$o),$o),rep_filter(A),aa(fun(fun(A,$o),$o),filter(A),abs_filter(A),Fa)) = Fa ) ) ).

% Abs_filter_inverse'
tff(fact_7370_Rep__filter__iff__eventually,axiom,
    ! [A: $tType,Fa: filter(A),Pa: fun(A,$o)] :
      ( aa(fun(A,$o),$o,aa(filter(A),fun(fun(A,$o),$o),rep_filter(A),Fa),Pa)
    <=> aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Pa),Fa) ) ).

% Rep_filter_iff_eventually
tff(fact_7371_is__filter__Rep__filter,axiom,
    ! [A: $tType,Fa: filter(A)] : aa(fun(fun(A,$o),$o),$o,is_filter(A),aa(filter(A),fun(fun(A,$o),$o),rep_filter(A),Fa)) ).

% is_filter_Rep_filter
tff(fact_7372_Rep__filter__induct,axiom,
    ! [A: $tType,Y: fun(fun(A,$o),$o),Pa: fun(fun(fun(A,$o),$o),$o)] :
      ( member(fun(fun(A,$o),$o),Y,aa(fun(fun(fun(A,$o),$o),$o),set(fun(fun(A,$o),$o)),collect(fun(fun(A,$o),$o)),is_filter(A)))
     => ( ! [X: filter(A)] : aa(fun(fun(A,$o),$o),$o,Pa,aa(filter(A),fun(fun(A,$o),$o),rep_filter(A),X))
       => aa(fun(fun(A,$o),$o),$o,Pa,Y) ) ) ).

% Rep_filter_induct
tff(fact_7373_Rep__filter__cases,axiom,
    ! [A: $tType,Y: fun(fun(A,$o),$o)] :
      ( member(fun(fun(A,$o),$o),Y,aa(fun(fun(fun(A,$o),$o),$o),set(fun(fun(A,$o),$o)),collect(fun(fun(A,$o),$o)),is_filter(A)))
     => ~ ! [X: filter(A)] : ( Y != aa(filter(A),fun(fun(A,$o),$o),rep_filter(A),X) ) ) ).

% Rep_filter_cases
tff(fact_7374_Rep__filter,axiom,
    ! [A: $tType,Xa: filter(A)] : member(fun(fun(A,$o),$o),aa(filter(A),fun(fun(A,$o),$o),rep_filter(A),Xa),aa(fun(fun(fun(A,$o),$o),$o),set(fun(fun(A,$o),$o)),collect(fun(fun(A,$o),$o)),is_filter(A))) ).

% Rep_filter
tff(fact_7375_Rep__filter__inverse,axiom,
    ! [A: $tType,Xa: filter(A)] : ( aa(fun(fun(A,$o),$o),filter(A),abs_filter(A),aa(filter(A),fun(fun(A,$o),$o),rep_filter(A),Xa)) = Xa ) ).

% Rep_filter_inverse
tff(fact_7376_Rep__filter__inject,axiom,
    ! [A: $tType,Xa: filter(A),Y: filter(A)] :
      ( ( aa(filter(A),fun(fun(A,$o),$o),rep_filter(A),Xa) = aa(filter(A),fun(fun(A,$o),$o),rep_filter(A),Y) )
    <=> ( Xa = Y ) ) ).

% Rep_filter_inject
tff(fact_7377_min__of__antimono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & linorder(B) )
     => ! [F2: fun(A,B),Xa: A,Y: A] :
          ( order_antimono(A,B,F2)
         => ( aa(B,B,aa(B,fun(B,B),ord_min(B),aa(A,B,F2,Xa)),aa(A,B,F2,Y)) = aa(A,B,F2,aa(A,A,aa(A,fun(A,A),ord_max(A),Xa),Y)) ) ) ) ).

% min_of_antimono
tff(fact_7378_max__of__antimono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & linorder(B) )
     => ! [F2: fun(A,B),Xa: A,Y: A] :
          ( order_antimono(A,B,F2)
         => ( aa(B,B,aa(B,fun(B,B),ord_max(B),aa(A,B,F2,Xa)),aa(A,B,F2,Y)) = aa(A,B,F2,aa(A,A,aa(A,fun(A,A),ord_min(A),Xa),Y)) ) ) ) ).

% max_of_antimono
tff(fact_7379_antimono__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F2: fun(A,B)] :
          ( order_antimono(A,B,F2)
        <=> ! [X6: A,Y4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X6),Y4)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,Y4)),aa(A,B,F2,X6)) ) ) ) ).

% antimono_def
tff(fact_7380_antimonoI,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F2: fun(A,B)] :
          ( ! [X: A,Y2: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y2)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,Y2)),aa(A,B,F2,X)) )
         => order_antimono(A,B,F2) ) ) ).

% antimonoI
tff(fact_7381_antimonoE,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F2: fun(A,B),Xa: A,Y: A] :
          ( order_antimono(A,B,F2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,Y)),aa(A,B,F2,Xa)) ) ) ) ).

% antimonoE
tff(fact_7382_antimonoD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F2: fun(A,B),Xa: A,Y: A] :
          ( order_antimono(A,B,F2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,Y)),aa(A,B,F2,Xa)) ) ) ) ).

% antimonoD
tff(fact_7383_in__chain__finite,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [Aa2: set(A)] :
          ( comple1602240252501008431_chain(A,ord_less_eq(A),Aa2)
         => ( aa(set(A),$o,finite_finite2(A),Aa2)
           => ( ( Aa2 != bot_bot(set(A)) )
             => member(A,aa(set(A),A,complete_Sup_Sup(A),Aa2),Aa2) ) ) ) ) ).

% in_chain_finite
tff(fact_7384_chain__empty,axiom,
    ! [A: $tType,Ord: fun(A,fun(A,$o))] : comple1602240252501008431_chain(A,Ord,bot_bot(set(A))) ).

% chain_empty
tff(fact_7385_ccpo__Sup__mono,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [Aa2: set(A),B2: set(A)] :
          ( comple1602240252501008431_chain(A,ord_less_eq(A),Aa2)
         => ( comple1602240252501008431_chain(A,ord_less_eq(A),B2)
           => ( ! [X: A] :
                  ( member(A,X,Aa2)
                 => ? [Xa4: A] :
                      ( member(A,Xa4,B2)
                      & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Xa4) ) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),Aa2)),aa(set(A),A,complete_Sup_Sup(A),B2)) ) ) ) ) ).

% ccpo_Sup_mono
tff(fact_7386_chain__singleton,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [Xa: A] : comple1602240252501008431_chain(A,ord_less_eq(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A)))) ) ).

% chain_singleton
tff(fact_7387_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_7388_wfP__SUP,axiom,
    ! [B: $tType,A: $tType,R2: fun(A,fun(B,fun(B,$o)))] :
      ( ! [I3: A] : wfP(B,aa(A,fun(B,fun(B,$o)),R2,I3))
     => ( ! [I3: A,J3: A] :
            ( ( aa(A,fun(B,fun(B,$o)),R2,I3) != aa(A,fun(B,fun(B,$o)),R2,J3) )
           => ( aa(fun(B,$o),fun(B,$o),aa(fun(B,$o),fun(fun(B,$o),fun(B,$o)),inf_inf(fun(B,$o)),domainp(B,B,aa(A,fun(B,fun(B,$o)),R2,I3))),rangep(B,B,aa(A,fun(B,fun(B,$o)),R2,J3))) = bot_bot(fun(B,$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_7389_Domainp__Domain__eq,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,B)),X4: A] :
      ( aa(A,$o,domainp(A,B,aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_bi(set(product_prod(A,B)),fun(A,fun(B,$o))),R2)),X4)
    <=> member(A,X4,aa(set(product_prod(A,B)),set(A),domain(A,B),R2)) ) ).

% Domainp_Domain_eq
tff(fact_7390_Domain__def,axiom,
    ! [B: $tType,A: $tType,X4: set(product_prod(A,B))] : ( aa(set(product_prod(A,B)),set(A),domain(A,B),X4) = aa(fun(A,$o),set(A),collect(A),domainp(A,B,aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_bi(set(product_prod(A,B)),fun(A,fun(B,$o))),X4))) ) ).

% Domain_def
tff(fact_7391_INFM__nat,axiom,
    ! [Pa: fun(nat,$o)] :
      ( aa(filter(nat),$o,aa(fun(nat,$o),fun(filter(nat),$o),frequently(nat),Pa),cofinite(nat))
    <=> ! [M: nat] :
        ? [N2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),N2)
          & aa(nat,$o,Pa,N2) ) ) ).

% INFM_nat
tff(fact_7392_INFM__nat__inductI,axiom,
    ! [Pa: fun(nat,$o),Qa: fun(nat,$o)] :
      ( aa(nat,$o,Pa,zero_zero(nat))
     => ( ! [I3: nat] :
            ( aa(nat,$o,Pa,I3)
           => ? [J6: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),J6)
                & aa(nat,$o,Pa,J6)
                & aa(nat,$o,Qa,J6) ) )
       => aa(filter(nat),$o,aa(fun(nat,$o),fun(filter(nat),$o),frequently(nat),Qa),cofinite(nat)) ) ) ).

% INFM_nat_inductI
tff(fact_7393_MOST__nat,axiom,
    ! [Pa: fun(nat,$o)] :
      ( aa(filter(nat),$o,aa(fun(nat,$o),fun(filter(nat),$o),eventually(nat),Pa),cofinite(nat))
    <=> ? [M: nat] :
        ! [N2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),N2)
         => aa(nat,$o,Pa,N2) ) ) ).

% MOST_nat
tff(fact_7394_frequently__cofinite,axiom,
    ! [A: $tType,Pa: fun(A,$o)] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),Pa),cofinite(A))
    <=> ~ aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),Pa)) ) ).

% frequently_cofinite
tff(fact_7395_eventually__cofinite,axiom,
    ! [A: $tType,Pa: fun(A,$o)] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Pa),cofinite(A))
    <=> aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_ce(fun(A,$o),fun(A,$o),Pa))) ) ).

% eventually_cofinite
tff(fact_7396_list_ODomainp__rel,axiom,
    ! [B: $tType,A: $tType,R: fun(A,fun(B,$o))] : ( domainp(list(A),list(B),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),R)) = aa(fun(A,$o),fun(list(A),$o),list_all(A),domainp(A,B,R)) ) ).

% list.Domainp_rel
tff(fact_7397_DomainpE,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(B,$o)),A3: A] :
      ( aa(A,$o,domainp(A,B,R2),A3)
     => ~ ! [B4: B] : ~ aa(B,$o,aa(A,fun(B,$o),R2,A3),B4) ) ).

% DomainpE
tff(fact_7398_DomainPI,axiom,
    ! [B: $tType,A: $tType,R2: fun(A,fun(B,$o)),A3: A,B3: B] :
      ( aa(B,$o,aa(A,fun(B,$o),R2,A3),B3)
     => aa(A,$o,domainp(A,B,R2),A3) ) ).

% DomainPI
tff(fact_7399_Domainp_Osimps,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(B,$o)),A3: A] :
      ( aa(A,$o,domainp(A,B,R2),A3)
    <=> ? [A7: A,B8: B] :
          ( ( A3 = A7 )
          & aa(B,$o,aa(A,fun(B,$o),R2,A7),B8) ) ) ).

% Domainp.simps
tff(fact_7400_Domainp_Ocases,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(B,$o)),A3: A] :
      ( aa(A,$o,domainp(A,B,R2),A3)
     => ~ ! [B4: B] : ~ aa(B,$o,aa(A,fun(B,$o),R2,A3),B4) ) ).

% Domainp.cases
tff(fact_7401_cofinite__eq__sequentially,axiom,
    cofinite(nat) = at_top(nat) ).

% cofinite_eq_sequentially
tff(fact_7402_wfP__less,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => wfP(A,ord_less(A)) ) ).

% wfP_less
tff(fact_7403_wfP__less__multiset,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ( wfP(A,ord_less(A))
       => wfP(multiset(A),ord_less(multiset(A))) ) ) ).

% wfP_less_multiset
tff(fact_7404_wfP__wf__eq,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( wfP(A,aTP_Lamp_cz(set(product_prod(A,A)),fun(A,fun(A,$o)),R2))
    <=> wf(A,R2) ) ).

% wfP_wf_eq
tff(fact_7405_cofinite__def,axiom,
    ! [A: $tType] : ( cofinite(A) = aa(fun(fun(A,$o),$o),filter(A),abs_filter(A),aTP_Lamp_abt(fun(A,$o),$o)) ) ).

% cofinite_def
tff(fact_7406_multiset_Odomain,axiom,
    ! [A: $tType,B: $tType,T4: fun(A,fun(B,$o)),X4: fun(A,nat)] :
      ( aa(fun(A,nat),$o,domainp(fun(A,nat),multiset(B),pcr_multiset(A,B,T4)),X4)
    <=> ? [Y4: fun(B,nat)] :
          ( aa(fun(B,nat),$o,aa(fun(A,nat),fun(fun(B,nat),$o),bNF_rel_fun(A,B,nat,nat,T4,fequal(nat)),X4),Y4)
          & aa(set(B),$o,finite_finite2(B),aa(fun(B,$o),set(B),collect(B),aTP_Lamp_abu(fun(B,nat),fun(B,$o),Y4))) ) ) ).

% multiset.domain
tff(fact_7407_bit__double__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,N: nat] :
          ( bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),times_times(A),numeral_numeral(A,bit0(one2))),A3),N)
        <=> ( 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_7408_bit__minus__2__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat] :
          ( bit_se5641148757651400278ts_bit(A,aa(A,A,uminus_uminus(A),numeral_numeral(A,bit0(one2))),N)
        <=> ( bit_se6407376104438227557le_bit(A,type2(A),N)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N) ) ) ) ).

% bit_minus_2_iff
tff(fact_7409_bit__mask__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M2: nat,N: nat] :
          ( bit_se5641148757651400278ts_bit(A,bit_se2239418461657761734s_mask(A,M2),N)
        <=> ( bit_se6407376104438227557le_bit(A,type2(A),N)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),M2) ) ) ) ).

% bit_mask_iff
tff(fact_7410_bit__minus__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A3: A,N: nat] :
          ( bit_se5641148757651400278ts_bit(A,aa(A,A,uminus_uminus(A),A3),N)
        <=> ( bit_se6407376104438227557le_bit(A,type2(A),N)
            & ~ 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_7411_bit__mask__sub__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M2: nat,N: nat] :
          ( bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),numeral_numeral(A,bit0(one2))),M2)),one_one(A)),N)
        <=> ( bit_se6407376104438227557le_bit(A,type2(A),N)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),M2) ) ) ) ).

% bit_mask_sub_iff
tff(fact_7412_multiset_Odomain__eq,axiom,
    ! [A: $tType,X4: fun(A,nat)] :
      ( aa(fun(A,nat),$o,domainp(fun(A,nat),multiset(A),pcr_multiset(A,A,fequal(A))),X4)
    <=> aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_be(fun(A,nat),fun(A,$o),X4))) ) ).

% multiset.domain_eq
tff(fact_7413_drop__bit__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M2: nat,N: nat] :
          ( bit_se4197421643247451524op_bit(A,M2,aa(nat,A,aa(A,fun(nat,A),power_power(A),numeral_numeral(A,bit0(one2))),N)) = aa(A,A,
              aa(A,fun(A,A),times_times(A),
                aa($o,A,zero_neq_one_of_bool(A),
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),N)
                  & bit_se6407376104438227557le_bit(A,type2(A),N) ))),
              aa(nat,A,aa(A,fun(nat,A),power_power(A),numeral_numeral(A,bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M2))) ) ) ).

% drop_bit_exp_eq
tff(fact_7414_CHAR__pos__iff,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),semiri4206861660011772517g_char(A,type2(A)))
      <=> ? [N2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N2)
            & ( aa(nat,A,semiring_1_of_nat(A),N2) = zero_zero(A) ) ) ) ) ).

% CHAR_pos_iff
tff(fact_7415_CHAR__eq0__iff,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ( ( semiri4206861660011772517g_char(A,type2(A)) = zero_zero(nat) )
      <=> ! [N2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N2)
           => ( aa(nat,A,semiring_1_of_nat(A),N2) != zero_zero(A) ) ) ) ) ).

% CHAR_eq0_iff
tff(fact_7416_CHAR__eq__posI,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [C2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),C2)
         => ( ( aa(nat,A,semiring_1_of_nat(A),C2) = zero_zero(A) )
           => ( ! [X: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),X)
                 => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),C2)
                   => ( aa(nat,A,semiring_1_of_nat(A),X) != zero_zero(A) ) ) )
             => ( semiri4206861660011772517g_char(A,type2(A)) = C2 ) ) ) ) ) ).

% CHAR_eq_posI
tff(fact_7417_multiset_Odomain__par__left__total,axiom,
    ! [B: $tType,A: $tType,T4: fun(A,fun(B,$o)),P: fun(fun(A,nat),$o)] :
      ( left_total(fun(A,nat),fun(B,nat),bNF_rel_fun(A,B,nat,nat,T4,fequal(nat)))
     => ( aa(fun(fun(B,nat),$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,T4,fequal(nat)),fequal($o)),P),aTP_Lamp_abv(fun(B,nat),$o))
       => ( domainp(fun(A,nat),multiset(B),pcr_multiset(A,B,T4)) = P ) ) ) ).

% multiset.domain_par_left_total
tff(fact_7418_multiset_Odomain__par,axiom,
    ! [B: $tType,A: $tType,T4: fun(A,fun(B,$o)),DT: fun(A,$o),DS: fun(nat,$o),P25: fun(fun(A,nat),$o)] :
      ( ( domainp(A,B,T4) = DT )
     => ( ( domainp(nat,nat,fequal(nat)) = DS )
       => ( left_unique(A,B,T4)
         => ( aa(fun(fun(B,nat),$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,T4,fequal(nat)),fequal($o)),P25),aTP_Lamp_abv(fun(B,nat),$o))
           => ( domainp(fun(A,nat),multiset(B),pcr_multiset(A,B,T4)) = 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)),basic_pred_fun(A,nat,DT,DS)),P25) ) ) ) ) ) ).

% multiset.domain_par
tff(fact_7419_list_Oleft__unique__rel,axiom,
    ! [B: $tType,A: $tType,R: fun(A,fun(B,$o))] :
      ( left_unique(A,B,R)
     => left_unique(list(A),list(B),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),R)) ) ).

% list.left_unique_rel
tff(fact_7420_list_Oleft__total__rel,axiom,
    ! [B: $tType,A: $tType,R: fun(A,fun(B,$o))] :
      ( left_total(A,B,R)
     => left_total(list(A),list(B),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),R)) ) ).

% list.left_total_rel
tff(fact_7421_left__unique__rel__filter,axiom,
    ! [B: $tType,A: $tType,Aa2: fun(A,fun(B,$o))] :
      ( left_unique(A,B,Aa2)
     => left_unique(filter(A),filter(B),rel_filter(A,B,Aa2)) ) ).

% left_unique_rel_filter
tff(fact_7422_left__total__rel__filter,axiom,
    ! [B: $tType,A: $tType,Aa2: fun(A,fun(B,$o))] :
      ( left_total(A,B,Aa2)
     => left_total(filter(A),filter(B),rel_filter(A,B,Aa2)) ) ).

% left_total_rel_filter
tff(fact_7423_cut__def,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),R: set(product_prod(A,A)),Xa: A,X4: A] :
      ( aa(A,B,cut(A,B,F2,R,Xa),X4) = $ite(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Xa),R),aa(A,B,F2,X4),undefined(B)) ) ).

% cut_def
tff(fact_7424_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,$o),aTP_Lamp_cz(set(product_prod(A,A)),fun(A,fun(A,$o)),R2),A3)) ) ).

% above_def
tff(fact_7425_cuts__eq,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),R: set(product_prod(A,A)),Xa: A,G: fun(A,B)] :
      ( ( cut(A,B,F2,R,Xa) = cut(A,B,G,R,Xa) )
    <=> ! [Y4: A] :
          ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),Xa),R)
         => ( aa(A,B,F2,Y4) = aa(A,B,G,Y4) ) ) ) ).

% cuts_eq
tff(fact_7426_cut__apply,axiom,
    ! [B: $tType,A: $tType,Xa: A,A3: A,R: set(product_prod(A,A)),F2: fun(A,B)] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),A3),R)
     => ( aa(A,B,cut(A,B,F2,R,A3),Xa) = aa(A,B,F2,Xa) ) ) ).

% cut_apply
tff(fact_7427_type__definition__filter,axiom,
    ! [A: $tType] : type_definition(filter(A),fun(fun(A,$o),$o),rep_filter(A),abs_filter(A),aa(fun(fun(fun(A,$o),$o),$o),set(fun(fun(A,$o),$o)),collect(fun(fun(A,$o),$o)),is_filter(A))) ).

% type_definition_filter
tff(fact_7428_list__ex__iff__not__all__inverval__int,axiom,
    ! [Pa: fun(int,$o),Ia: int,J: int] :
      ( aa(list(int),$o,aa(fun(int,$o),fun(list(int),$o),list_ex(int),Pa),upto(Ia,J))
    <=> ~ all_interval_int(comp($o,$o,int,fNot,Pa),Ia,J) ) ).

% list_ex_iff_not_all_inverval_int
tff(fact_7429_all__interval__int__def,axiom,
    ! [Pa: fun(int,$o),Ia: int,J: int] :
      ( all_interval_int(Pa,Ia,J)
    <=> ! [X6: int] :
          ( member(int,X6,set_or1337092689740270186AtMost(int,Ia,J))
         => aa(int,$o,Pa,X6) ) ) ).

% all_interval_int_def
tff(fact_7430_list__all__iff__all__interval__int,axiom,
    ! [Pa: fun(int,$o),Ia: int,J: int] :
      ( aa(list(int),$o,aa(fun(int,$o),fun(list(int),$o),list_all(int),Pa),upto(Ia,J))
    <=> all_interval_int(Pa,Ia,J) ) ).

% list_all_iff_all_interval_int
tff(fact_7431_fixp__mono,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [F2: fun(A,A),G: fun(A,A)] :
          ( aa(fun(A,A),$o,aa(fun(A,fun(A,$o)),fun(fun(A,A),$o),aa(fun(A,fun(A,$o)),fun(fun(A,fun(A,$o)),fun(fun(A,A),$o)),comple7038119648293358887notone(A,A),ord_less_eq(A)),ord_less_eq(A)),F2)
         => ( aa(fun(A,A),$o,aa(fun(A,fun(A,$o)),fun(fun(A,A),$o),aa(fun(A,fun(A,$o)),fun(fun(A,fun(A,$o)),fun(fun(A,A),$o)),comple7038119648293358887notone(A,A),ord_less_eq(A)),ord_less_eq(A)),G)
           => ( ! [Z9: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,F2,Z9)),aa(A,A,G,Z9))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),comple115746919287870866o_fixp(A,F2)),comple115746919287870866o_fixp(A,G)) ) ) ) ) ).

% fixp_mono
tff(fact_7432_map__filter__simps_I1_J,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,option(A)),Xa: B,Xsa: list(B)] : ( map_filter(B,A,F2,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Xa),Xsa)) = case_option(list(A),A,map_filter(B,A,F2,Xsa),aa(list(B),fun(A,list(A)),aTP_Lamp_abw(fun(B,option(A)),fun(list(B),fun(A,list(A))),F2),Xsa),aa(B,option(A),F2,Xa)) ) ).

% map_filter_simps(1)
tff(fact_7433_fixp__induct,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [Pa: fun(A,$o),F2: fun(A,A)] :
          ( comple1908693960933563346ssible(A,complete_Sup_Sup(A),ord_less_eq(A),Pa)
         => ( aa(fun(A,A),$o,aa(fun(A,fun(A,$o)),fun(fun(A,A),$o),aa(fun(A,fun(A,$o)),fun(fun(A,fun(A,$o)),fun(fun(A,A),$o)),comple7038119648293358887notone(A,A),ord_less_eq(A)),ord_less_eq(A)),F2)
           => ( aa(A,$o,Pa,aa(set(A),A,complete_Sup_Sup(A),bot_bot(set(A))))
             => ( ! [X: A] :
                    ( aa(A,$o,Pa,X)
                   => aa(A,$o,Pa,aa(A,A,F2,X)) )
               => aa(A,$o,Pa,comple115746919287870866o_fixp(A,F2)) ) ) ) ) ) ).

% fixp_induct
tff(fact_7434_arg__max__nat__le,axiom,
    ! [A: $tType,Pa: fun(A,$o),Xa: A,F2: fun(A,nat),B3: nat] :
      ( aa(A,$o,Pa,Xa)
     => ( ! [Y2: A] :
            ( aa(A,$o,Pa,Y2)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,F2,Y2)),B3) )
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,F2,Xa)),aa(A,nat,F2,lattices_ord_arg_max(A,nat,F2,Pa))) ) ) ).

% arg_max_nat_le
tff(fact_7435_arg__max__natI,axiom,
    ! [A: $tType,Pa: fun(A,$o),K3: A,F2: fun(A,nat),B3: nat] :
      ( aa(A,$o,Pa,K3)
     => ( ! [Y2: A] :
            ( aa(A,$o,Pa,Y2)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,F2,Y2)),B3) )
       => aa(A,$o,Pa,lattices_ord_arg_max(A,nat,F2,Pa)) ) ) ).

% arg_max_natI
tff(fact_7436_arg__maxI,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [Pa: fun(A,$o),Xa: A,F2: fun(A,B),Qa: fun(A,$o)] :
          ( aa(A,$o,Pa,Xa)
         => ( ! [Y2: A] :
                ( aa(A,$o,Pa,Y2)
               => ~ aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,Xa)),aa(A,B,F2,Y2)) )
           => ( ! [X: A] :
                  ( aa(A,$o,Pa,X)
                 => ( ! [Y3: A] :
                        ( aa(A,$o,Pa,Y3)
                       => ~ aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,X)),aa(A,B,F2,Y3)) )
                   => aa(A,$o,Qa,X) ) )
             => aa(A,$o,Qa,lattices_ord_arg_max(A,B,F2,Pa)) ) ) ) ) ).

% arg_maxI
tff(fact_7437_ccpo_OadmissibleD,axiom,
    ! [A: $tType,Lub: fun(set(A),A),Ord: fun(A,fun(A,$o)),Pa: fun(A,$o),Aa2: set(A)] :
      ( comple1908693960933563346ssible(A,Lub,Ord,Pa)
     => ( comple1602240252501008431_chain(A,Ord,Aa2)
       => ( ( Aa2 != bot_bot(set(A)) )
         => ( ! [X: A] :
                ( member(A,X,Aa2)
               => aa(A,$o,Pa,X) )
           => aa(A,$o,Pa,aa(set(A),A,Lub,Aa2)) ) ) ) ) ).

% ccpo.admissibleD
tff(fact_7438_ccpo_OadmissibleI,axiom,
    ! [A: $tType,Ord: fun(A,fun(A,$o)),Pa: fun(A,$o),Lub: fun(set(A),A)] :
      ( ! [A12: set(A)] :
          ( comple1602240252501008431_chain(A,Ord,A12)
         => ( ( A12 != bot_bot(set(A)) )
           => ( ! [X4: A] :
                  ( member(A,X4,A12)
                 => aa(A,$o,Pa,X4) )
             => aa(A,$o,Pa,aa(set(A),A,Lub,A12)) ) ) )
     => comple1908693960933563346ssible(A,Lub,Ord,Pa) ) ).

% ccpo.admissibleI
tff(fact_7439_ccpo_Oadmissible__def,axiom,
    ! [A: $tType,Lub: fun(set(A),A),Ord: fun(A,fun(A,$o)),Pa: fun(A,$o)] :
      ( comple1908693960933563346ssible(A,Lub,Ord,Pa)
    <=> ! [A11: set(A)] :
          ( comple1602240252501008431_chain(A,Ord,A11)
         => ( ( A11 != bot_bot(set(A)) )
           => ( ! [X6: A] :
                  ( member(A,X6,A11)
                 => aa(A,$o,Pa,X6) )
             => aa(A,$o,Pa,aa(set(A),A,Lub,A11)) ) ) ) ) ).

% ccpo.admissible_def
tff(fact_7440_arg__max__nat__lemma,axiom,
    ! [A: $tType,Pa: fun(A,$o),K3: A,F2: fun(A,nat),B3: nat] :
      ( aa(A,$o,Pa,K3)
     => ( ! [Y2: A] :
            ( aa(A,$o,Pa,Y2)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,F2,Y2)),B3) )
       => ( aa(A,$o,Pa,lattices_ord_arg_max(A,nat,F2,Pa))
          & ! [Y3: A] :
              ( aa(A,$o,Pa,Y3)
             => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,F2,Y3)),aa(A,nat,F2,lattices_ord_arg_max(A,nat,F2,Pa))) ) ) ) ) ).

% arg_max_nat_lemma
tff(fact_7441_heap__step__admissible,axiom,
    ! [A: $tType,E: $tType,B: $tType,F4: $tType,G2: $tType,Pa: fun(G2,fun(A,fun(E,fun(B,fun(F4,$o))))),Xa: G2] : comple1908693960933563346ssible(fun(A,option(product_prod(B,product_prod(E,F4)))),aa(fun(set(option(product_prod(B,product_prod(E,F4)))),option(product_prod(B,product_prod(E,F4)))),fun(set(fun(A,option(product_prod(B,product_prod(E,F4))))),fun(A,option(product_prod(B,product_prod(E,F4))))),partial_fun_lub(option(product_prod(B,product_prod(E,F4))),option(product_prod(B,product_prod(E,F4))),A),partial_flat_lub(option(product_prod(B,product_prod(E,F4))),none(product_prod(B,product_prod(E,F4))))),aa(fun(option(product_prod(B,product_prod(E,F4))),fun(option(product_prod(B,product_prod(E,F4))),$o)),fun(fun(A,option(product_prod(B,product_prod(E,F4)))),fun(fun(A,option(product_prod(B,product_prod(E,F4)))),$o)),partial_fun_ord(option(product_prod(B,product_prod(E,F4))),option(product_prod(B,product_prod(E,F4))),A),partial_flat_ord(option(product_prod(B,product_prod(E,F4))),none(product_prod(B,product_prod(E,F4))))),aa(G2,fun(fun(A,option(product_prod(B,product_prod(E,F4)))),$o),aTP_Lamp_abx(fun(G2,fun(A,fun(E,fun(B,fun(F4,$o))))),fun(G2,fun(fun(A,option(product_prod(B,product_prod(E,F4)))),$o)),Pa),Xa)) ).

% heap_step_admissible
tff(fact_7442_mult__one__div__unit__factor,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A,B3: A] : ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),divide_divide(A,one_one(A),unit_f5069060285200089521factor(A,B3))) = divide_divide(A,A3,unit_f5069060285200089521factor(A,B3)) ) ) ).

% mult_one_div_unit_factor
tff(fact_7443_unit__factor__idem,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A] : ( unit_f5069060285200089521factor(A,unit_f5069060285200089521factor(A,A3)) = unit_f5069060285200089521factor(A,A3) ) ) ).

% unit_factor_idem
tff(fact_7444_unit__factor__eq__0__iff,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A] :
          ( ( unit_f5069060285200089521factor(A,A3) = zero_zero(A) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% unit_factor_eq_0_iff
tff(fact_7445_unit__factor__0,axiom,
    ! [A: $tType] :
      ( semido2269285787275462019factor(A)
     => ( unit_f5069060285200089521factor(A,zero_zero(A)) = zero_zero(A) ) ) ).

% unit_factor_0
tff(fact_7446_unit__factor__1,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ( unit_f5069060285200089521factor(A,one_one(A)) = one_one(A) ) ) ).

% unit_factor_1
tff(fact_7447_inv__unit__factor__eq__0__iff,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A] :
          ( ( divide_divide(A,one_one(A),unit_f5069060285200089521factor(A,A3)) = zero_zero(A) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% inv_unit_factor_eq_0_iff
tff(fact_7448_unit__factor__mult__unit__left,axiom,
    ! [A: $tType] :
      ( semido2269285787275462019factor(A)
     => ! [A3: A,B3: A] :
          ( dvd_dvd(A,A3,one_one(A))
         => ( unit_f5069060285200089521factor(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),unit_f5069060285200089521factor(A,B3)) ) ) ) ).

% unit_factor_mult_unit_left
tff(fact_7449_unit__factor__mult__unit__right,axiom,
    ! [A: $tType] :
      ( semido2269285787275462019factor(A)
     => ! [A3: A,B3: A] :
          ( dvd_dvd(A,A3,one_one(A))
         => ( unit_f5069060285200089521factor(A,aa(A,A,aa(A,fun(A,A),times_times(A),B3),A3)) = aa(A,A,aa(A,fun(A,A),times_times(A),unit_f5069060285200089521factor(A,B3)),A3) ) ) ) ).

% unit_factor_mult_unit_right
tff(fact_7450_gcd__mult__distrib,axiom,
    ! [A: $tType] :
      ( semiri6843258321239162965malize(A)
     => ! [K3: A,A3: A,B3: A] : ( aa(A,A,aa(A,fun(A,A),times_times(A),K3),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B3)) = 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),K3),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),K3),B3))),unit_f5069060285200089521factor(A,K3)) ) ) ).

% gcd_mult_distrib
tff(fact_7451_mult__gcd__right,axiom,
    ! [A: $tType] :
      ( semiri6843258321239162965malize(A)
     => ! [A3: A,B3: 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),B3)),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),B3),C2))),unit_f5069060285200089521factor(A,C2)) ) ) ).

% mult_gcd_right
tff(fact_7452_mult__gcd__left,axiom,
    ! [A: $tType] :
      ( semiri6843258321239162965malize(A)
     => ! [C2: A,A3: A,B3: 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),B3)) = 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),B3))) ) ) ).

% mult_gcd_left
tff(fact_7453_unit__factor__self,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A] : dvd_dvd(A,unit_f5069060285200089521factor(A,A3),A3) ) ).

% unit_factor_self
tff(fact_7454_unit__factor__mult,axiom,
    ! [A: $tType] :
      ( normal6328177297339901930cative(A)
     => ! [A3: A,B3: A] : ( unit_f5069060285200089521factor(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)) = aa(A,A,aa(A,fun(A,A),times_times(A),unit_f5069060285200089521factor(A,A3)),unit_f5069060285200089521factor(A,B3)) ) ) ).

% unit_factor_mult
tff(fact_7455_is__unit__unit__factor,axiom,
    ! [A: $tType] :
      ( semido2269285787275462019factor(A)
     => ! [A3: A] :
          ( dvd_dvd(A,A3,one_one(A))
         => ( unit_f5069060285200089521factor(A,A3) = A3 ) ) ) ).

% is_unit_unit_factor
tff(fact_7456_unit__factor__dvd,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A,B3: A] :
          ( ( A3 != zero_zero(A) )
         => dvd_dvd(A,unit_f5069060285200089521factor(A,A3),B3) ) ) ).

% unit_factor_dvd
tff(fact_7457_dvd__unit__factor__div,axiom,
    ! [A: $tType] :
      ( normal6328177297339901930cative(A)
     => ! [B3: A,A3: A] :
          ( dvd_dvd(A,B3,A3)
         => ( unit_f5069060285200089521factor(A,divide_divide(A,A3,B3)) = divide_divide(A,unit_f5069060285200089521factor(A,A3),unit_f5069060285200089521factor(A,B3)) ) ) ) ).

% dvd_unit_factor_div
tff(fact_7458_unit__factor__is__unit,axiom,
    ! [A: $tType] :
      ( semido2269285787275462019factor(A)
     => ! [A3: A] :
          ( ( A3 != zero_zero(A) )
         => dvd_dvd(A,unit_f5069060285200089521factor(A,A3),one_one(A)) ) ) ).

% unit_factor_is_unit
tff(fact_7459_coprime__crossproduct_H,axiom,
    ! [A: $tType] :
      ( semiri6843258321239162965malize(A)
     => ! [B3: A,D3: A,A3: A,C2: A] :
          ( ( B3 != zero_zero(A) )
         => ( ( unit_f5069060285200089521factor(A,B3) = unit_f5069060285200089521factor(A,D3) )
           => ( algebr8660921524188924756oprime(A,A3,B3)
             => ( 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),B3),C2) )
                <=> ( ( A3 = C2 )
                    & ( B3 = D3 ) ) ) ) ) ) ) ) ).

% coprime_crossproduct'
tff(fact_7460_Lcm__no__units,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [Aa2: set(A)] : ( gcd_Lcm(A,Aa2) = gcd_Lcm(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_aby(A,$o)))) ) ) ).

% Lcm_no_units
tff(fact_7461_is__pure__assn__pure,axiom,
    ! [Pa: $o] : is_pure_assn(pure_assn((Pa))) ).

% is_pure_assn_pure
tff(fact_7462_is__pure__assn__starI,axiom,
    ! [A3: assn,B3: assn] :
      ( is_pure_assn(A3)
     => ( is_pure_assn(B3)
       => is_pure_assn(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),A3),B3)) ) ) ).

% is_pure_assn_starI
tff(fact_7463_Lcm__empty,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ( gcd_Lcm(A,bot_bot(set(A))) = one_one(A) ) ) ).

% Lcm_empty
tff(fact_7464_Lcm__nat__empty,axiom,
    gcd_Lcm(nat,bot_bot(set(nat))) = one_one(nat) ).

% Lcm_nat_empty
tff(fact_7465_is__pure__assn__basic__simps_I2_J,axiom,
    is_pure_assn(one_one(assn)) ).

% is_pure_assn_basic_simps(2)
tff(fact_7466_is__pure__assn__def,axiom,
    ! [A3: assn] :
      ( is_pure_assn(A3)
    <=> ? [P6: $o] : ( A3 = pure_assn((P6)) ) ) ).

% is_pure_assn_def
tff(fact_7467_is__pure__assnE,axiom,
    ! [A3: assn] :
      ( is_pure_assn(A3)
     => ~ ! [P3: $o] : ( A3 != pure_assn((P3)) ) ) ).

% is_pure_assnE
tff(fact_7468_is__pure__assn__basic__simps_I1_J,axiom,
    is_pure_assn(bot_bot(assn)) ).

% is_pure_assn_basic_simps(1)
tff(fact_7469_Lcm__coprime,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [Aa2: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( ( Aa2 != bot_bot(set(A)) )
           => ( ! [A4: A,B4: A] :
                  ( member(A,A4,Aa2)
                 => ( member(A,B4,Aa2)
                   => ( ( A4 != B4 )
                     => algebr8660921524188924756oprime(A,A4,B4) ) ) )
             => ( gcd_Lcm(A,Aa2) = normal6383669964737779283malize(A,aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7121269368397514597t_prod(A,A),aTP_Lamp_abz(A,A)),Aa2)) ) ) ) ) ) ).

% Lcm_coprime
tff(fact_7470_Lcm__eq__Max__nat,axiom,
    ! [M5: set(nat)] :
      ( aa(set(nat),$o,finite_finite2(nat),M5)
     => ( ( M5 != bot_bot(set(nat)) )
       => ( ~ member(nat,zero_zero(nat),M5)
         => ( ! [M6: nat,N5: nat] :
                ( member(nat,M6,M5)
               => ( member(nat,N5,M5)
                 => member(nat,aa(nat,nat,aa(nat,fun(nat,nat),gcd_lcm(nat),M6),N5),M5) ) )
           => ( gcd_Lcm(nat,M5) = lattic643756798349783984er_Max(nat,M5) ) ) ) ) ) ).

% Lcm_eq_Max_nat
tff(fact_7471_normalize__idem,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A] : ( normal6383669964737779283malize(A,normal6383669964737779283malize(A,A3)) = normal6383669964737779283malize(A,A3) ) ) ).

% normalize_idem
tff(fact_7472_normalize__eq__0__iff,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A] :
          ( ( normal6383669964737779283malize(A,A3) = zero_zero(A) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% normalize_eq_0_iff
tff(fact_7473_normalize__0,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ( normal6383669964737779283malize(A,zero_zero(A)) = zero_zero(A) ) ) ).

% normalize_0
tff(fact_7474_normalize__mult__normalize__right,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A,B3: A] : ( normal6383669964737779283malize(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),normal6383669964737779283malize(A,B3))) = normal6383669964737779283malize(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)) ) ) ).

% normalize_mult_normalize_right
tff(fact_7475_normalize__mult__normalize__left,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A,B3: A] : ( normal6383669964737779283malize(A,aa(A,A,aa(A,fun(A,A),times_times(A),normal6383669964737779283malize(A,A3)),B3)) = normal6383669964737779283malize(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)) ) ) ).

% normalize_mult_normalize_left
tff(fact_7476_normalize__1,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ( normal6383669964737779283malize(A,one_one(A)) = one_one(A) ) ) ).

% normalize_1
tff(fact_7477_normalize__dvd__iff,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A,B3: A] :
          ( dvd_dvd(A,normal6383669964737779283malize(A,A3),B3)
        <=> dvd_dvd(A,A3,B3) ) ) ).

% normalize_dvd_iff
tff(fact_7478_dvd__normalize__iff,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A,B3: A] :
          ( dvd_dvd(A,A3,normal6383669964737779283malize(A,B3))
        <=> dvd_dvd(A,A3,B3) ) ) ).

% dvd_normalize_iff
tff(fact_7479_coprime__normalize__left__iff,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A,B3: A] :
          ( algebr8660921524188924756oprime(A,normal6383669964737779283malize(A,A3),B3)
        <=> algebr8660921524188924756oprime(A,A3,B3) ) ) ).

% coprime_normalize_left_iff
tff(fact_7480_coprime__normalize__right__iff,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A,B3: A] :
          ( algebr8660921524188924756oprime(A,A3,normal6383669964737779283malize(A,B3))
        <=> algebr8660921524188924756oprime(A,A3,B3) ) ) ).

% coprime_normalize_right_iff
tff(fact_7481_Lcm__insert,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A3: A,Aa2: set(A)] : ( gcd_Lcm(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),Aa2)) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),gcd_Lcm(A,Aa2)) ) ) ).

% Lcm_insert
tff(fact_7482_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)),normal6383669964737779283malize(A,A3)) = A3 ) ) ).

% unit_factor_mult_normalize
tff(fact_7483_normalize__mult__unit__factor,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A] : ( aa(A,A,aa(A,fun(A,A),times_times(A),normal6383669964737779283malize(A,A3)),unit_f5069060285200089521factor(A,A3)) = A3 ) ) ).

% normalize_mult_unit_factor
tff(fact_7484_div__unit__factor,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A] : ( divide_divide(A,A3,unit_f5069060285200089521factor(A,A3)) = normal6383669964737779283malize(A,A3) ) ) ).

% div_unit_factor
tff(fact_7485_div__normalize,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A] : ( divide_divide(A,A3,normal6383669964737779283malize(A,A3)) = unit_f5069060285200089521factor(A,A3) ) ) ).

% div_normalize
tff(fact_7486_normalize__mult__unit__right,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [B3: A,A3: A] :
          ( dvd_dvd(A,B3,one_one(A))
         => ( normal6383669964737779283malize(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)) = normal6383669964737779283malize(A,A3) ) ) ) ).

% normalize_mult_unit_right
tff(fact_7487_normalize__mult__unit__left,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A,B3: A] :
          ( dvd_dvd(A,A3,one_one(A))
         => ( normal6383669964737779283malize(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)) = normal6383669964737779283malize(A,B3) ) ) ) ).

% normalize_mult_unit_left
tff(fact_7488_unit__factor__normalize,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A] :
          ( ( A3 != zero_zero(A) )
         => ( unit_f5069060285200089521factor(A,normal6383669964737779283malize(A,A3)) = one_one(A) ) ) ) ).

% unit_factor_normalize
tff(fact_7489_normalize__unit__factor,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A] :
          ( ( A3 != zero_zero(A) )
         => ( normal6383669964737779283malize(A,unit_f5069060285200089521factor(A,A3)) = one_one(A) ) ) ) ).

% normalize_unit_factor
tff(fact_7490_lcm__mult__gcd,axiom,
    ! [A: $tType] :
      ( semiri6843258321239162965malize(A)
     => ! [A3: A,B3: A] : ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B3)),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B3)) = aa(A,A,aa(A,fun(A,A),times_times(A),normal6383669964737779283malize(A,A3)),normal6383669964737779283malize(A,B3)) ) ) ).

% lcm_mult_gcd
tff(fact_7491_gcd__mult__lcm,axiom,
    ! [A: $tType] :
      ( semiri6843258321239162965malize(A)
     => ! [A3: A,B3: A] : ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B3)),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B3)) = aa(A,A,aa(A,fun(A,A),times_times(A),normal6383669964737779283malize(A,A3)),normal6383669964737779283malize(A,B3)) ) ) ).

% gcd_mult_lcm
tff(fact_7492_Lcm__2,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A3: A,B3: A] : ( gcd_Lcm(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),B3),bot_bot(set(A))))) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B3) ) ) ).

% Lcm_2
tff(fact_7493_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)),insert3(A),A3),bot_bot(set(A)))) = normal6383669964737779283malize(A,A3) ) ) ).

% Lcm_singleton
tff(fact_7494_normalize__div,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A] : ( divide_divide(A,normal6383669964737779283malize(A,A3),A3) = divide_divide(A,one_one(A),unit_f5069060285200089521factor(A,A3)) ) ) ).

% normalize_div
tff(fact_7495_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)),insert3(A),A3),bot_bot(set(A)))) = normal6383669964737779283malize(A,A3) ) ) ).

% Gcd_singleton
tff(fact_7496_Lcm__nat__insert,axiom,
    ! [N: nat,M5: set(nat)] : ( gcd_Lcm(nat,aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert3(nat),N),M5)) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_lcm(nat),N),gcd_Lcm(nat,M5)) ) ).

% Lcm_nat_insert
tff(fact_7497_Lcm__in__lcm__closed__set__nat,axiom,
    ! [M5: set(nat)] :
      ( aa(set(nat),$o,finite_finite2(nat),M5)
     => ( ( M5 != bot_bot(set(nat)) )
       => ( ! [M6: nat,N5: nat] :
              ( member(nat,M6,M5)
             => ( member(nat,N5,M5)
               => member(nat,aa(nat,nat,aa(nat,fun(nat,nat),gcd_lcm(nat),M6),N5),M5) ) )
         => member(nat,gcd_Lcm(nat,M5),M5) ) ) ) ).

% Lcm_in_lcm_closed_set_nat
tff(fact_7498_coprime__crossproduct,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,D3: A,B3: A,C2: A] :
          ( algebr8660921524188924756oprime(A,A3,D3)
         => ( algebr8660921524188924756oprime(A,B3,C2)
           => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),normal6383669964737779283malize(A,A3)),normal6383669964737779283malize(A,C2)) = aa(A,A,aa(A,fun(A,A),times_times(A),normal6383669964737779283malize(A,B3)),normal6383669964737779283malize(A,D3)) )
            <=> ( ( normal6383669964737779283malize(A,A3) = normal6383669964737779283malize(A,B3) )
                & ( normal6383669964737779283malize(A,C2) = normal6383669964737779283malize(A,D3) ) ) ) ) ) ) ).

% coprime_crossproduct
tff(fact_7499_lcm__coprime,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B3: A] :
          ( algebr8660921524188924756oprime(A,A3,B3)
         => ( aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B3) = normal6383669964737779283malize(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)) ) ) ) ).

% lcm_coprime
tff(fact_7500_gcd__lcm,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B3: A] :
          ( ( A3 != zero_zero(A) )
         => ( ( B3 != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B3) = normal6383669964737779283malize(A,divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B3))) ) ) ) ) ).

% gcd_lcm
tff(fact_7501_gcd__mult__distrib_H,axiom,
    ! [A: $tType] :
      ( semiri6843258321239162965malize(A)
     => ! [C2: A,A3: A,B3: A] : ( aa(A,A,aa(A,fun(A,A),times_times(A),normal6383669964737779283malize(A,C2)),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B3)) = 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),B3)) ) ) ).

% gcd_mult_distrib'
tff(fact_7502_lcm__gcd__prod,axiom,
    ! [A: $tType] :
      ( semiri6843258321239162965malize(A)
     => ! [A3: A,B3: A] : ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B3)),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B3)) = normal6383669964737779283malize(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)) ) ) ).

% lcm_gcd_prod
tff(fact_7503_gcd__mult__right,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,C2: A,B3: 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),B3),C2)) = normal6383669964737779283malize(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),B3),A3)),C2)) ) ) ).

% gcd_mult_right
tff(fact_7504_gcd__mult__left,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [C2: A,A3: A,B3: 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),B3)) = 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),B3))) ) ) ).

% gcd_mult_left
tff(fact_7505_lcm__gcd,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B3: A] : ( aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B3) = normal6383669964737779283malize(A,divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B3))) ) ) ).

% lcm_gcd
tff(fact_7506_lcm__mult__unit1,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B3: A,C2: A] :
          ( 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),B3),A3)),C2) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),B3),C2) ) ) ) ).

% lcm_mult_unit1
tff(fact_7507_lcm__mult__unit2,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B3: A,C2: A] :
          ( dvd_dvd(A,A3,one_one(A))
         => ( aa(A,A,aa(A,fun(A,A),gcd_lcm(A),B3),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),B3),C2) ) ) ) ).

% lcm_mult_unit2
tff(fact_7508_associated__iff__dvd,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A,B3: A] :
          ( ( normal6383669964737779283malize(A,A3) = normal6383669964737779283malize(A,B3) )
        <=> ( dvd_dvd(A,A3,B3)
            & dvd_dvd(A,B3,A3) ) ) ) ).

% associated_iff_dvd
tff(fact_7509_associated__eqI,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A,B3: A] :
          ( dvd_dvd(A,A3,B3)
         => ( dvd_dvd(A,B3,A3)
           => ( ( normal6383669964737779283malize(A,A3) = A3 )
             => ( ( normal6383669964737779283malize(A,B3) = B3 )
               => ( A3 = B3 ) ) ) ) ) ) ).

% associated_eqI
tff(fact_7510_associatedD2,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A,B3: A] :
          ( ( normal6383669964737779283malize(A,A3) = normal6383669964737779283malize(A,B3) )
         => dvd_dvd(A,B3,A3) ) ) ).

% associatedD2
tff(fact_7511_associatedD1,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A,B3: A] :
          ( ( normal6383669964737779283malize(A,A3) = normal6383669964737779283malize(A,B3) )
         => dvd_dvd(A,A3,B3) ) ) ).

% associatedD1
tff(fact_7512_associatedI,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A,B3: A] :
          ( dvd_dvd(A,A3,B3)
         => ( dvd_dvd(A,B3,A3)
           => ( normal6383669964737779283malize(A,A3) = normal6383669964737779283malize(A,B3) ) ) ) ) ).

% associatedI
tff(fact_7513_normalize__mult,axiom,
    ! [A: $tType] :
      ( normal6328177297339901930cative(A)
     => ! [A3: A,B3: A] : ( normal6383669964737779283malize(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)) = aa(A,A,aa(A,fun(A,A),times_times(A),normal6383669964737779283malize(A,A3)),normal6383669964737779283malize(A,B3)) ) ) ).

% normalize_mult
tff(fact_7514_lcm__mult__left,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [C2: A,A3: A,B3: 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),B3)) = 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),B3))) ) ) ).

% lcm_mult_left
tff(fact_7515_lcm__mult__right,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,C2: A,B3: 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),B3),C2)) = normal6383669964737779283malize(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),B3),A3)),C2)) ) ) ).

% lcm_mult_right
tff(fact_7516_lcm__mult__distrib_H,axiom,
    ! [A: $tType] :
      ( semiri6843258321239162965malize(A)
     => ! [C2: A,A3: A,B3: A] : ( aa(A,A,aa(A,fun(A,A),times_times(A),normal6383669964737779283malize(A,C2)),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B3)) = 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),B3)) ) ) ).

% lcm_mult_distrib'
tff(fact_7517_normalize__idem__imp__is__unit__iff,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A] :
          ( ( normal6383669964737779283malize(A,A3) = A3 )
         => ( dvd_dvd(A,A3,one_one(A))
          <=> ( A3 = one_one(A) ) ) ) ) ).

% normalize_idem_imp_is_unit_iff
tff(fact_7518_is__unit__normalize,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A] :
          ( dvd_dvd(A,A3,one_one(A))
         => ( normal6383669964737779283malize(A,A3) = one_one(A) ) ) ) ).

% is_unit_normalize
tff(fact_7519_normalize__1__iff,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A] :
          ( ( normal6383669964737779283malize(A,A3) = one_one(A) )
        <=> dvd_dvd(A,A3,one_one(A)) ) ) ).

% normalize_1_iff
tff(fact_7520_associated__unit,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A,B3: A] :
          ( ( normal6383669964737779283malize(A,A3) = normal6383669964737779283malize(A,B3) )
         => ( dvd_dvd(A,A3,one_one(A))
           => dvd_dvd(A,B3,one_one(A)) ) ) ) ).

% associated_unit
tff(fact_7521_dvd__normalize__div,axiom,
    ! [A: $tType] :
      ( normal6328177297339901930cative(A)
     => ! [B3: A,A3: A] :
          ( dvd_dvd(A,B3,A3)
         => ( normal6383669964737779283malize(A,divide_divide(A,A3,B3)) = divide_divide(A,normal6383669964737779283malize(A,A3),normal6383669964737779283malize(A,B3)) ) ) ) ).

% dvd_normalize_div
tff(fact_7522_Gcd__mult,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [C2: A,Aa2: set(A)] : ( gcd_Gcd(A,aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),times_times(A),C2)),Aa2)) = normal6383669964737779283malize(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),gcd_Gcd(A,Aa2))) ) ) ).

% Gcd_mult
tff(fact_7523_lcm__pos__nat,axiom,
    ! [M2: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),gcd_lcm(nat),M2),N)) ) ) ).

% lcm_pos_nat
tff(fact_7524_normalize__idem__imp__unit__factor__eq,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A] :
          ( ( normal6383669964737779283malize(A,A3) = A3 )
         => ( unit_f5069060285200089521factor(A,A3) = aa($o,A,zero_neq_one_of_bool(A),A3 != zero_zero(A)) ) ) ) ).

% normalize_idem_imp_unit_factor_eq
tff(fact_7525_lcm__mult__distrib,axiom,
    ! [A: $tType] :
      ( semiri6843258321239162965malize(A)
     => ! [K3: A,A3: A,B3: A] : ( aa(A,A,aa(A,fun(A,A),times_times(A),K3),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B3)) = 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),K3),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),K3),B3))),unit_f5069060285200089521factor(A,K3)) ) ) ).

% lcm_mult_distrib
tff(fact_7526_mult__lcm__right,axiom,
    ! [A: $tType] :
      ( semiri6843258321239162965malize(A)
     => ! [A3: A,B3: 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),B3)),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),B3),C2))),unit_f5069060285200089521factor(A,C2)) ) ) ).

% mult_lcm_right
tff(fact_7527_mult__lcm__left,axiom,
    ! [A: $tType] :
      ( semiri6843258321239162965malize(A)
     => ! [C2: A,A3: A,B3: 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),B3)) = 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),B3))) ) ) ).

% mult_lcm_left
tff(fact_7528_normalize__unit__factor__eqI,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A,B3: A] :
          ( ( normal6383669964737779283malize(A,A3) = normal6383669964737779283malize(A,B3) )
         => ( ( unit_f5069060285200089521factor(A,A3) = unit_f5069060285200089521factor(A,B3) )
           => ( A3 = B3 ) ) ) ) ).

% normalize_unit_factor_eqI
tff(fact_7529_Lcm__mult,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [Aa2: set(A),C2: A] :
          ( ( Aa2 != bot_bot(set(A)) )
         => ( gcd_Lcm(A,aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),times_times(A),C2)),Aa2)) = normal6383669964737779283malize(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),gcd_Lcm(A,Aa2))) ) ) ) ).

% Lcm_mult
tff(fact_7530_Gcd__fin__mult,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [Aa2: set(A),B3: A] :
          ( aa(set(A),$o,finite_finite2(A),Aa2)
         => ( semiring_gcd_Gcd_fin(A,aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),times_times(A),B3)),Aa2)) = normal6383669964737779283malize(A,aa(A,A,aa(A,fun(A,A),times_times(A),B3),semiring_gcd_Gcd_fin(A,Aa2))) ) ) ) ).

% Gcd_fin_mult
tff(fact_7531_Lcm__set__eq__fold,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [Xsa: list(A)] : ( gcd_Lcm(A,aa(list(A),set(A),set2(A),Xsa)) = aa(A,A,aa(list(A),fun(A,A),aa(fun(A,fun(A,A)),fun(list(A),fun(A,A)),fold(A,A),gcd_lcm(A)),Xsa),one_one(A)) ) ) ).

% Lcm_set_eq_fold
tff(fact_7532_Lcm__nat__set__eq__fold,axiom,
    ! [Xsa: list(nat)] : ( gcd_Lcm(nat,aa(list(nat),set(nat),set2(nat),Xsa)) = aa(nat,nat,aa(list(nat),fun(nat,nat),aa(fun(nat,fun(nat,nat)),fun(list(nat),fun(nat,nat)),fold(nat,nat),gcd_lcm(nat)),Xsa),one_one(nat)) ) ).

% Lcm_nat_set_eq_fold
tff(fact_7533_Lcm__fin_Oset__eq__fold,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [Xsa: list(A)] : ( semiring_gcd_Lcm_fin(A,aa(list(A),set(A),set2(A),Xsa)) = aa(A,A,aa(list(A),fun(A,A),aa(fun(A,fun(A,A)),fun(list(A),fun(A,A)),fold(A,A),gcd_lcm(A)),Xsa),one_one(A)) ) ) ).

% Lcm_fin.set_eq_fold
tff(fact_7534_Lcm__fin_Oinsert__remove,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,Aa2: set(A)] : ( semiring_gcd_Lcm_fin(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),Aa2)) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),semiring_gcd_Lcm_fin(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),bot_bot(set(A)))))) ) ) ).

% Lcm_fin.insert_remove
tff(fact_7535_Lcm__fin_Oempty,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ( semiring_gcd_Lcm_fin(A,bot_bot(set(A))) = one_one(A) ) ) ).

% Lcm_fin.empty
tff(fact_7536_Lcm__fin_Oinsert,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,Aa2: set(A)] : ( semiring_gcd_Lcm_fin(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),Aa2)) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),semiring_gcd_Lcm_fin(A,Aa2)) ) ) ).

% Lcm_fin.insert
tff(fact_7537_lcm__list__dvd__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [Xsa: list(A),B3: A] :
          ( dvd_dvd(A,semiring_gcd_Lcm_fin(A,aa(list(A),set(A),set2(A),Xsa)),B3)
        <=> ! [X6: A] :
              ( member(A,X6,aa(list(A),set(A),set2(A),Xsa))
             => dvd_dvd(A,X6,B3) ) ) ) ).

% lcm_list_dvd_iff
tff(fact_7538_lcm__list__least,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [Bs: list(A),A3: A] :
          ( ! [B4: A] :
              ( member(A,B4,aa(list(A),set(A),set2(A),Bs))
             => dvd_dvd(A,B4,A3) )
         => dvd_dvd(A,semiring_gcd_Lcm_fin(A,aa(list(A),set(A),set2(A),Bs)),A3) ) ) ).

% lcm_list_least
tff(fact_7539_Lcm__int__set__eq__fold,axiom,
    ! [Xsa: list(int)] : ( gcd_Lcm(int,aa(list(int),set(int),set2(int),Xsa)) = aa(int,int,aa(list(int),fun(int,int),aa(fun(int,fun(int,int)),fun(list(int),fun(int,int)),fold(int,int),gcd_lcm(int)),Xsa),one_one(int)) ) ).

% Lcm_int_set_eq_fold
tff(fact_7540_Lcm__fin__mult,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [Aa2: set(A),B3: A] :
          ( ( Aa2 != bot_bot(set(A)) )
         => ( semiring_gcd_Lcm_fin(A,aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),times_times(A),B3)),Aa2)) = normal6383669964737779283malize(A,aa(A,A,aa(A,fun(A,A),times_times(A),B3),semiring_gcd_Lcm_fin(A,Aa2))) ) ) ) ).

% Lcm_fin_mult
tff(fact_7541_Lcm__fin_Oremove,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,Aa2: set(A)] :
          ( member(A,A3,Aa2)
         => ( semiring_gcd_Lcm_fin(A,Aa2) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),semiring_gcd_Lcm_fin(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),bot_bot(set(A)))))) ) ) ) ).

% Lcm_fin.remove
tff(fact_7542_properI,axiom,
    ! [Pa: fun(product_prod(heap_ext(product_unit),set(nat)),$o)] :
      ( ! [As: set(nat),H: heap_ext(product_unit)] :
          ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,Pa,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),As))
         => aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As)) )
     => ( ! [As: set(nat),H: heap_ext(product_unit),H6: heap_ext(product_unit)] :
            ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,Pa,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),As))
           => ( relH(As,H,H6)
             => ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H6),As))
               => aa(product_prod(heap_ext(product_unit),set(nat)),$o,Pa,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),As)) ) ) )
       => aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o,proper,Pa) ) ) ).

% properI
tff(fact_7543_properD2,axiom,
    ! [Pa: fun(product_prod(heap_ext(product_unit),set(nat)),$o),H2: heap_ext(product_unit),As2: set(nat),H4: heap_ext(product_unit)] :
      ( aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o,proper,Pa)
     => ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,Pa,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))
       => ( relH(As2,H2,H4)
         => ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),As2))
           => aa(product_prod(heap_ext(product_unit),set(nat)),$o,Pa,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),As2)) ) ) ) ) ).

% properD2
tff(fact_7544_bool__assn__proper_I4_J,axiom,
    ! [Pa: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Qa: fun(product_prod(heap_ext(product_unit),set(nat)),$o)] :
      ( aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o,proper,Pa)
     => ( aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o,proper,Qa)
       => 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_aca(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)),Pa),Qa)) ) ) ).

% bool_assn_proper(4)
tff(fact_7545_bool__assn__proper_I3_J,axiom,
    ! [Pa: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Qa: fun(product_prod(heap_ext(product_unit),set(nat)),$o)] :
      ( aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o,proper,Pa)
     => ( aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o,proper,Qa)
       => 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_acb(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)),Pa),Qa)) ) ) ).

% bool_assn_proper(3)
tff(fact_7546_bool__assn__proper_I2_J,axiom,
    aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o,proper,aTP_Lamp_abf(product_prod(heap_ext(product_unit),set(nat)),$o)) ).

% bool_assn_proper(2)
tff(fact_7547_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_7548_ex__assn__proper,axiom,
    ! [A: $tType,Pa: fun(A,fun(product_prod(heap_ext(product_unit),set(nat)),$o))] :
      ( ! [X: A] : aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o,proper,aa(A,fun(product_prod(heap_ext(product_unit),set(nat)),$o),Pa,X))
     => aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o,proper,aTP_Lamp_acc(fun(A,fun(product_prod(heap_ext(product_unit),set(nat)),$o)),fun(product_prod(heap_ext(product_unit),set(nat)),$o),Pa)) ) ).

% ex_assn_proper
tff(fact_7549_snga__assn__proper,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [R2: array(A),Xa: list(A)] : aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o,proper,snga_assn_raw(A,R2,Xa)) ) ).

% snga_assn_proper
tff(fact_7550_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_7551_times__assn__proper,axiom,
    ! [Pa: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Qa: fun(product_prod(heap_ext(product_unit),set(nat)),$o)] :
      ( aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o,proper,Pa)
     => ( aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o,proper,Qa)
       => aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o,proper,times_assn_raw(Pa,Qa)) ) ) ).

% times_assn_proper
tff(fact_7552_wand__proper,axiom,
    ! [Pa: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Qa: fun(product_prod(heap_ext(product_unit),set(nat)),$o)] : aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o,proper,wand_raw(Pa,Qa)) ).

% wand_proper
tff(fact_7553_sngr__assn__proper,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [R2: ref(A),Xa: A] : aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o,proper,sngr_assn_raw(A,R2,Xa)) ) ).

% sngr_assn_proper
tff(fact_7554_bool__assn__proper_I5_J,axiom,
    ! [Pa: fun(product_prod(heap_ext(product_unit),set(nat)),$o)] :
      ( aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o,proper,Pa)
     => aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o,proper,aTP_Lamp_acd(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(heap_ext(product_unit),set(nat)),$o),Pa)) ) ).

% bool_assn_proper(5)
tff(fact_7555_pure__assn__proper,axiom,
    ! [B3: $o] : aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o,proper,pure_assn_raw(heap_ext(product_unit),nat,(B3))) ).

% pure_assn_proper
tff(fact_7556_Abs__assn__inverse,axiom,
    ! [Y: fun(product_prod(heap_ext(product_unit),set(nat)),$o)] :
      ( member(fun(product_prod(heap_ext(product_unit),set(nat)),$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_7557_Abs__assn__cases,axiom,
    ! [Xa: assn] :
      ~ ! [Y2: fun(product_prod(heap_ext(product_unit),set(nat)),$o)] :
          ( ( Xa = aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),assn,abs_assn,Y2) )
         => ~ 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_7558_Abs__assn__induct,axiom,
    ! [Pa: fun(assn,$o),Xa: assn] :
      ( ! [Y2: fun(product_prod(heap_ext(product_unit),set(nat)),$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,Pa,aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),assn,abs_assn,Y2)) )
     => aa(assn,$o,Pa,Xa) ) ).

% Abs_assn_induct
tff(fact_7559_Abs__assn__inject,axiom,
    ! [Xa: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Y: fun(product_prod(heap_ext(product_unit),set(nat)),$o)] :
      ( member(fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xa,aa(fun(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o),set(fun(product_prod(heap_ext(product_unit),set(nat)),$o)),collect(fun(product_prod(heap_ext(product_unit),set(nat)),$o)),proper))
     => ( member(fun(product_prod(heap_ext(product_unit),set(nat)),$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)),$o),assn,abs_assn,Xa) = aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),assn,abs_assn,Y) )
        <=> ( Xa = Y ) ) ) ) ).

% Abs_assn_inject
tff(fact_7560_Rep__assn__induct,axiom,
    ! [Y: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Pa: fun(fun(product_prod(heap_ext(product_unit),set(nat)),$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))
     => ( ! [X: assn] : aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o,Pa,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,X))
       => aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o,Pa,Y) ) ) ).

% Rep_assn_induct
tff(fact_7561_Rep__assn__cases,axiom,
    ! [Y: fun(product_prod(heap_ext(product_unit),set(nat)),$o)] :
      ( member(fun(product_prod(heap_ext(product_unit),set(nat)),$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))
     => ~ ! [X: assn] : ( Y != aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,X) ) ) ).

% Rep_assn_cases
tff(fact_7562_Rep__assn,axiom,
    ! [Xa: assn] : 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,Xa),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_7563_properD1,axiom,
    ! [Pa: fun(product_prod(heap_ext(product_unit),set(nat)),$o),H2: heap_ext(product_unit),As2: set(nat)] :
      ( aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o,proper,Pa)
     => ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,Pa,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))
       => 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),As2)) ) ) ).

% properD1
tff(fact_7564_proper__iff,axiom,
    ! [Pa: fun(product_prod(heap_ext(product_unit),set(nat)),$o),As2: set(nat),H2: heap_ext(product_unit),H4: heap_ext(product_unit)] :
      ( aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o,proper,Pa)
     => ( relH(As2,H2,H4)
       => ( aa(product_prod(heap_ext(product_unit),set(nat)),$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),As2))
         => ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,Pa,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As2))
          <=> aa(product_prod(heap_ext(product_unit),set(nat)),$o,Pa,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),As2)) ) ) ) ) ).

% proper_iff
tff(fact_7565_proper__def,axiom,
    ! [Pa: fun(product_prod(heap_ext(product_unit),set(nat)),$o)] :
      ( aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o,proper,Pa)
    <=> ! [H3: heap_ext(product_unit),H8: heap_ext(product_unit),As8: set(nat)] :
          ( ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,Pa,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),As8))
           => aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H3),As8)) )
          & ( ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,Pa,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),As8))
              & relH(As8,H3,H8)
              & aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H8),As8)) )
           => aa(product_prod(heap_ext(product_unit),set(nat)),$o,Pa,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)),H8),As8)) ) ) ) ).

% proper_def
tff(fact_7566_list__ex__iff__not__all__inverval__nat,axiom,
    ! [Pa: fun(nat,$o),Ia: nat,J: nat] :
      ( aa(list(nat),$o,aa(fun(nat,$o),fun(list(nat),$o),list_ex(nat),Pa),upt(Ia,J))
    <=> ~ all_interval_nat(comp($o,$o,nat,fNot,Pa),Ia,J) ) ).

% list_ex_iff_not_all_inverval_nat
tff(fact_7567_list__all__iff__all__interval__nat,axiom,
    ! [Pa: fun(nat,$o),Ia: nat,J: nat] :
      ( aa(list(nat),$o,aa(fun(nat,$o),fun(list(nat),$o),list_all(nat),Pa),upt(Ia,J))
    <=> all_interval_nat(Pa,Ia,J) ) ).

% list_all_iff_all_interval_nat
tff(fact_7568_all__interval__nat__def,axiom,
    ! [Pa: fun(nat,$o),Ia: nat,J: nat] :
      ( all_interval_nat(Pa,Ia,J)
    <=> ! [X6: nat] :
          ( member(nat,X6,set_or7035219750837199246ssThan(nat,Ia,J))
         => aa(nat,$o,Pa,X6) ) ) ).

% all_interval_nat_def
tff(fact_7569_of__rat__less__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R2: rat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),field_char_0_of_rat(A,R2)),one_one(A))
        <=> aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),R2),one_one(rat)) ) ) ).

% of_rat_less_1_iff
tff(fact_7570_one__less__of__rat__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R2: rat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),field_char_0_of_rat(A,R2))
        <=> aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),one_one(rat)),R2) ) ) ).

% one_less_of_rat_iff
tff(fact_7571_of__rat__less__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R2: rat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),field_char_0_of_rat(A,R2)),zero_zero(A))
        <=> aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),R2),zero_zero(rat)) ) ) ).

% of_rat_less_0_iff
tff(fact_7572_zero__less__of__rat__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R2: rat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),field_char_0_of_rat(A,R2))
        <=> aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),R2) ) ) ).

% zero_less_of_rat_iff
tff(fact_7573_of__rat__mult,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: rat,B3: rat] : ( field_char_0_of_rat(A,aa(rat,rat,aa(rat,fun(rat,rat),times_times(rat),A3),B3)) = aa(A,A,aa(A,fun(A,A),times_times(A),field_char_0_of_rat(A,A3)),field_char_0_of_rat(A,B3)) ) ) ).

% of_rat_mult
tff(fact_7574_of__rat__diff,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: rat,B3: rat] : ( field_char_0_of_rat(A,aa(rat,rat,aa(rat,fun(rat,rat),minus_minus(rat),A3),B3)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),field_char_0_of_rat(A,A3)),field_char_0_of_rat(A,B3)) ) ) ).

% of_rat_diff
tff(fact_7575_of__rat__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R2: rat,S2: rat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),field_char_0_of_rat(A,R2)),field_char_0_of_rat(A,S2))
        <=> aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),R2),S2) ) ) ).

% of_rat_less
tff(fact_7576_listsp__mono,axiom,
    ! [A: $tType,Aa2: fun(A,$o),B2: fun(A,$o)] :
      ( aa(fun(A,$o),$o,aa(fun(A,$o),fun(fun(A,$o),$o),ord_less_eq(fun(A,$o)),Aa2),B2)
     => aa(fun(list(A),$o),$o,aa(fun(list(A),$o),fun(fun(list(A),$o),$o),ord_less_eq(fun(list(A),$o)),listsp(A,Aa2)),listsp(A,B2)) ) ).

% listsp_mono
tff(fact_7577_partial__function__definitions_Ofixp__induct__uc,axiom,
    ! [B: $tType,A: $tType,E: $tType,Leq: fun(A,fun(A,$o)),Lub: fun(set(A),A),U2: fun(E,fun(B,A)),Fa: fun(E,E),C3: fun(fun(B,A),E),F2: E,Pa: fun(fun(B,A),$o)] :
      ( partia7178651479351089652itions(A,Leq,Lub)
     => ( ! [X: B] : aa(fun(fun(B,A),A),$o,aa(fun(A,fun(A,$o)),fun(fun(fun(B,A),A),$o),aa(fun(fun(B,A),fun(fun(B,A),$o)),fun(fun(A,fun(A,$o)),fun(fun(fun(B,A),A),$o)),comple7038119648293358887notone(fun(B,A),A),aa(fun(A,fun(A,$o)),fun(fun(B,A),fun(fun(B,A),$o)),partial_fun_ord(A,A,B),Leq)),Leq),aa(B,fun(fun(B,A),A),aa(fun(fun(B,A),E),fun(B,fun(fun(B,A),A)),aa(fun(E,E),fun(fun(fun(B,A),E),fun(B,fun(fun(B,A),A))),aTP_Lamp_ace(fun(E,fun(B,A)),fun(fun(E,E),fun(fun(fun(B,A),E),fun(B,fun(fun(B,A),A)))),U2),Fa),C3),X))
       => ( ( F2 = aa(fun(B,A),E,C3,comple187402453842119260l_fixp(fun(B,A),aa(fun(set(A),A),fun(set(fun(B,A)),fun(B,A)),partial_fun_lub(A,A,B),Lub),aa(fun(A,fun(A,$o)),fun(fun(B,A),fun(fun(B,A),$o)),partial_fun_ord(A,A,B),Leq),aa(fun(fun(B,A),E),fun(fun(B,A),fun(B,A)),aa(fun(E,E),fun(fun(fun(B,A),E),fun(fun(B,A),fun(B,A))),aTP_Lamp_acf(fun(E,fun(B,A)),fun(fun(E,E),fun(fun(fun(B,A),E),fun(fun(B,A),fun(B,A)))),U2),Fa),C3))) )
         => ( ! [F3: fun(B,A)] : ( aa(E,fun(B,A),U2,aa(fun(B,A),E,C3,F3)) = F3 )
           => ( comple1908693960933563346ssible(fun(B,A),aa(fun(set(A),A),fun(set(fun(B,A)),fun(B,A)),partial_fun_lub(A,A,B),Lub),aa(fun(A,fun(A,$o)),fun(fun(B,A),fun(fun(B,A),$o)),partial_fun_ord(A,A,B),Leq),Pa)
             => ( aa(fun(B,A),$o,Pa,aTP_Lamp_acg(fun(set(A),A),fun(B,A),Lub))
               => ( ! [F3: E] :
                      ( aa(fun(B,A),$o,Pa,aa(E,fun(B,A),U2,F3))
                     => aa(fun(B,A),$o,Pa,aa(E,fun(B,A),U2,aa(E,E,Fa,F3))) )
                 => aa(fun(B,A),$o,Pa,aa(E,fun(B,A),U2,F2)) ) ) ) ) ) ) ) ).

% partial_function_definitions.fixp_induct_uc
tff(fact_7578_listsp__conj__eq,axiom,
    ! [A: $tType,Aa2: fun(A,$o),B2: fun(A,$o),X4: list(A)] :
      ( aa(list(A),$o,listsp(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_bx(fun(A,$o),fun(fun(A,$o),fun(A,$o)),Aa2),B2)),X4)
    <=> ( aa(list(A),$o,listsp(A,Aa2),X4)
        & aa(list(A),$o,listsp(A,B2),X4) ) ) ).

% listsp_conj_eq
tff(fact_7579_listsp__simps_I1_J,axiom,
    ! [A: $tType,Aa2: fun(A,$o)] : aa(list(A),$o,listsp(A,Aa2),nil(A)) ).

% listsp_simps(1)
tff(fact_7580_in__listspI,axiom,
    ! [A: $tType,Xsa: list(A),Aa2: fun(A,$o)] :
      ( ! [X: A] :
          ( member(A,X,aa(list(A),set(A),set2(A),Xsa))
         => aa(A,$o,Aa2,X) )
     => aa(list(A),$o,listsp(A,Aa2),Xsa) ) ).

% in_listspI
tff(fact_7581_append__in__listsp__conv,axiom,
    ! [A: $tType,Aa2: fun(A,$o),Xsa: list(A),Ys: list(A)] :
      ( aa(list(A),$o,listsp(A,Aa2),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xsa),Ys))
    <=> ( aa(list(A),$o,listsp(A,Aa2),Xsa)
        & aa(list(A),$o,listsp(A,Aa2),Ys) ) ) ).

% append_in_listsp_conv
tff(fact_7582_listsp__inf__eq,axiom,
    ! [A: $tType,Aa2: fun(A,$o),B2: 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,$o)),Aa2),B2)) = aa(fun(list(A),$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,Aa2)),listsp(A,B2)) ) ).

% listsp_inf_eq
tff(fact_7583_listspE,axiom,
    ! [A: $tType,Aa2: fun(A,$o),Xa: A,L: list(A)] :
      ( aa(list(A),$o,listsp(A,Aa2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),L))
     => ~ ( aa(A,$o,Aa2,Xa)
         => ~ aa(list(A),$o,listsp(A,Aa2),L) ) ) ).

% listspE
tff(fact_7584_listsp_OCons,axiom,
    ! [A: $tType,Aa2: fun(A,$o),A3: A,L: list(A)] :
      ( aa(A,$o,Aa2,A3)
     => ( aa(list(A),$o,listsp(A,Aa2),L)
       => aa(list(A),$o,listsp(A,Aa2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),L)) ) ) ).

% listsp.Cons
tff(fact_7585_listsp__simps_I2_J,axiom,
    ! [A: $tType,Aa2: fun(A,$o),Xa: A,Xsa: list(A)] :
      ( aa(list(A),$o,listsp(A,Aa2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa))
    <=> ( aa(A,$o,Aa2,Xa)
        & aa(list(A),$o,listsp(A,Aa2),Xsa) ) ) ).

% listsp_simps(2)
tff(fact_7586_in__listspD,axiom,
    ! [A: $tType,Aa2: fun(A,$o),Xsa: list(A)] :
      ( aa(list(A),$o,listsp(A,Aa2),Xsa)
     => ! [X4: A] :
          ( member(A,X4,aa(list(A),set(A),set2(A),Xsa))
         => aa(A,$o,Aa2,X4) ) ) ).

% in_listspD
tff(fact_7587_in__listsp__conv__set,axiom,
    ! [A: $tType,Aa2: fun(A,$o),Xsa: list(A)] :
      ( aa(list(A),$o,listsp(A,Aa2),Xsa)
    <=> ! [X6: A] :
          ( member(A,X6,aa(list(A),set(A),set2(A),Xsa))
         => aa(A,$o,Aa2,X6) ) ) ).

% in_listsp_conv_set
tff(fact_7588_listsp_ONil,axiom,
    ! [A: $tType,Aa2: fun(A,$o)] : aa(list(A),$o,listsp(A,Aa2),nil(A)) ).

% listsp.Nil
tff(fact_7589_listsp_Ocases,axiom,
    ! [A: $tType,Aa2: fun(A,$o),A3: list(A)] :
      ( aa(list(A),$o,listsp(A,Aa2),A3)
     => ( ( A3 != nil(A) )
       => ~ ! [A4: A,L3: list(A)] :
              ( ( A3 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A4),L3) )
             => ( aa(A,$o,Aa2,A4)
               => ~ aa(list(A),$o,listsp(A,Aa2),L3) ) ) ) ) ).

% listsp.cases
tff(fact_7590_listsp_Osimps,axiom,
    ! [A: $tType,Aa2: fun(A,$o),A3: list(A)] :
      ( aa(list(A),$o,listsp(A,Aa2),A3)
    <=> ( ( A3 = nil(A) )
        | ? [A7: A,L4: list(A)] :
            ( ( A3 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A7),L4) )
            & aa(A,$o,Aa2,A7)
            & aa(list(A),$o,listsp(A,Aa2),L4) ) ) ) ).

% listsp.simps
tff(fact_7591_listsp__lists__eq,axiom,
    ! [A: $tType,Aa2: set(A),X4: list(A)] :
      ( aa(list(A),$o,listsp(A,aa(set(A),fun(A,$o),aTP_Lamp_a(set(A),fun(A,$o)),Aa2)),X4)
    <=> member(list(A),X4,aa(set(A),set(list(A)),lists(A),Aa2)) ) ).

% listsp_lists_eq
tff(fact_7592_lists__def,axiom,
    ! [A: $tType,X4: set(A)] : ( aa(set(A),set(list(A)),lists(A),X4) = aa(fun(list(A),$o),set(list(A)),collect(list(A)),listsp(A,aa(set(A),fun(A,$o),aTP_Lamp_a(set(A),fun(A,$o)),X4))) ) ).

% lists_def
tff(fact_7593_listsp__infI,axiom,
    ! [A: $tType,Aa2: fun(A,$o),L: list(A),B2: fun(A,$o)] :
      ( aa(list(A),$o,listsp(A,Aa2),L)
     => ( aa(list(A),$o,listsp(A,B2),L)
       => aa(list(A),$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)),Aa2),B2)),L) ) ) ).

% listsp_infI
tff(fact_7594_Gr__def,axiom,
    ! [B: $tType,A: $tType,Aa2: set(A),F2: fun(A,B)] : ( bNF_Gr(A,B,Aa2,F2) = aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,B),fun(product_prod(A,B),$o),aTP_Lamp_ach(set(A),fun(fun(A,B),fun(product_prod(A,B),$o)),Aa2),F2)) ) ).

% Gr_def
tff(fact_7595_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_7596_GrD2,axiom,
    ! [A: $tType,B: $tType,Xa: A,Fx: B,Aa2: set(A),F2: fun(A,B)] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa),Fx),bNF_Gr(A,B,Aa2,F2))
     => ( aa(A,B,F2,Xa) = Fx ) ) ).

% GrD2
tff(fact_7597_GrD1,axiom,
    ! [B: $tType,A: $tType,Xa: A,Fx: B,Aa2: set(A),F2: fun(A,B)] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa),Fx),bNF_Gr(A,B,Aa2,F2))
     => member(A,Xa,Aa2) ) ).

% GrD1
tff(fact_7598_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_7599_max_Osemilattice__order__axioms,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => semilattice_order(A,ord_max(A),aTP_Lamp_ao(A,fun(A,$o)),aTP_Lamp_ys(A,fun(A,$o))) ) ).

% max.semilattice_order_axioms
tff(fact_7600_semilattice__neutr__order_Oaxioms_I2_J,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z2: A,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o))] :
      ( semila1105856199041335345_order(A,F2,Z2,Less_eq,Less)
     => semilattice_order(A,F2,Less_eq,Less) ) ).

% semilattice_neutr_order.axioms(2)
tff(fact_7601_semilattice__order_Omono,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A3: A,C2: A,B3: A,D3: A] :
      ( semilattice_order(A,F2,Less_eq,Less)
     => ( aa(A,$o,aa(A,fun(A,$o),Less_eq,A3),C2)
       => ( aa(A,$o,aa(A,fun(A,$o),Less_eq,B3),D3)
         => aa(A,$o,aa(A,fun(A,$o),Less_eq,aa(A,A,aa(A,fun(A,A),F2,A3),B3)),aa(A,A,aa(A,fun(A,A),F2,C2),D3)) ) ) ) ).

% semilattice_order.mono
tff(fact_7602_semilattice__order_OorderE,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A3: A,B3: A] :
      ( semilattice_order(A,F2,Less_eq,Less)
     => ( aa(A,$o,aa(A,fun(A,$o),Less_eq,A3),B3)
       => ( A3 = aa(A,A,aa(A,fun(A,A),F2,A3),B3) ) ) ) ).

% semilattice_order.orderE
tff(fact_7603_semilattice__order_OorderI,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A3: A,B3: A] :
      ( semilattice_order(A,F2,Less_eq,Less)
     => ( ( A3 = aa(A,A,aa(A,fun(A,A),F2,A3),B3) )
       => aa(A,$o,aa(A,fun(A,$o),Less_eq,A3),B3) ) ) ).

% semilattice_order.orderI
tff(fact_7604_semilattice__order_Oabsorb1,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A3: A,B3: A] :
      ( semilattice_order(A,F2,Less_eq,Less)
     => ( aa(A,$o,aa(A,fun(A,$o),Less_eq,A3),B3)
       => ( aa(A,A,aa(A,fun(A,A),F2,A3),B3) = A3 ) ) ) ).

% semilattice_order.absorb1
tff(fact_7605_semilattice__order_Oabsorb2,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),B3: A,A3: A] :
      ( semilattice_order(A,F2,Less_eq,Less)
     => ( aa(A,$o,aa(A,fun(A,$o),Less_eq,B3),A3)
       => ( aa(A,A,aa(A,fun(A,A),F2,A3),B3) = B3 ) ) ) ).

% semilattice_order.absorb2
tff(fact_7606_semilattice__order_Oabsorb3,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A3: A,B3: A] :
      ( semilattice_order(A,F2,Less_eq,Less)
     => ( aa(A,$o,aa(A,fun(A,$o),Less,A3),B3)
       => ( aa(A,A,aa(A,fun(A,A),F2,A3),B3) = A3 ) ) ) ).

% semilattice_order.absorb3
tff(fact_7607_semilattice__order_Oabsorb4,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),B3: A,A3: A] :
      ( semilattice_order(A,F2,Less_eq,Less)
     => ( aa(A,$o,aa(A,fun(A,$o),Less,B3),A3)
       => ( aa(A,A,aa(A,fun(A,A),F2,A3),B3) = B3 ) ) ) ).

% semilattice_order.absorb4
tff(fact_7608_semilattice__order_OboundedE,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A3: A,B3: A,C2: A] :
      ( semilattice_order(A,F2,Less_eq,Less)
     => ( aa(A,$o,aa(A,fun(A,$o),Less_eq,A3),aa(A,A,aa(A,fun(A,A),F2,B3),C2))
       => ~ ( aa(A,$o,aa(A,fun(A,$o),Less_eq,A3),B3)
           => ~ aa(A,$o,aa(A,fun(A,$o),Less_eq,A3),C2) ) ) ) ).

% semilattice_order.boundedE
tff(fact_7609_semilattice__order_OboundedI,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A3: A,B3: A,C2: A] :
      ( semilattice_order(A,F2,Less_eq,Less)
     => ( aa(A,$o,aa(A,fun(A,$o),Less_eq,A3),B3)
       => ( aa(A,$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),F2,B3),C2)) ) ) ) ).

% semilattice_order.boundedI
tff(fact_7610_semilattice__order_Oorder__iff,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A3: A,B3: A] :
      ( semilattice_order(A,F2,Less_eq,Less)
     => ( aa(A,$o,aa(A,fun(A,$o),Less_eq,A3),B3)
      <=> ( A3 = aa(A,A,aa(A,fun(A,A),F2,A3),B3) ) ) ) ).

% semilattice_order.order_iff
tff(fact_7611_semilattice__order_Ocobounded1,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A3: A,B3: A] :
      ( semilattice_order(A,F2,Less_eq,Less)
     => aa(A,$o,aa(A,fun(A,$o),Less_eq,aa(A,A,aa(A,fun(A,A),F2,A3),B3)),A3) ) ).

% semilattice_order.cobounded1
tff(fact_7612_semilattice__order_Ocobounded2,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A3: A,B3: A] :
      ( semilattice_order(A,F2,Less_eq,Less)
     => aa(A,$o,aa(A,fun(A,$o),Less_eq,aa(A,A,aa(A,fun(A,A),F2,A3),B3)),B3) ) ).

% semilattice_order.cobounded2
tff(fact_7613_semilattice__order_Oabsorb__iff1,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A3: A,B3: A] :
      ( semilattice_order(A,F2,Less_eq,Less)
     => ( aa(A,$o,aa(A,fun(A,$o),Less_eq,A3),B3)
      <=> ( aa(A,A,aa(A,fun(A,A),F2,A3),B3) = A3 ) ) ) ).

% semilattice_order.absorb_iff1
tff(fact_7614_semilattice__order_Oabsorb__iff2,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),B3: A,A3: A] :
      ( semilattice_order(A,F2,Less_eq,Less)
     => ( aa(A,$o,aa(A,fun(A,$o),Less_eq,B3),A3)
      <=> ( aa(A,A,aa(A,fun(A,A),F2,A3),B3) = B3 ) ) ) ).

% semilattice_order.absorb_iff2
tff(fact_7615_semilattice__order_Obounded__iff,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A3: A,B3: A,C2: A] :
      ( semilattice_order(A,F2,Less_eq,Less)
     => ( aa(A,$o,aa(A,fun(A,$o),Less_eq,A3),aa(A,A,aa(A,fun(A,A),F2,B3),C2))
      <=> ( aa(A,$o,aa(A,fun(A,$o),Less_eq,A3),B3)
          & aa(A,$o,aa(A,fun(A,$o),Less_eq,A3),C2) ) ) ) ).

% semilattice_order.bounded_iff
tff(fact_7616_semilattice__order_OcoboundedI1,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A3: A,C2: A,B3: A] :
      ( semilattice_order(A,F2,Less_eq,Less)
     => ( aa(A,$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),F2,A3),B3)),C2) ) ) ).

% semilattice_order.coboundedI1
tff(fact_7617_semilattice__order_OcoboundedI2,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),B3: A,C2: A,A3: A] :
      ( semilattice_order(A,F2,Less_eq,Less)
     => ( aa(A,$o,aa(A,fun(A,$o),Less_eq,B3),C2)
       => aa(A,$o,aa(A,fun(A,$o),Less_eq,aa(A,A,aa(A,fun(A,A),F2,A3),B3)),C2) ) ) ).

% semilattice_order.coboundedI2
tff(fact_7618_semilattice__order_Ostrict__boundedE,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A3: A,B3: A,C2: A] :
      ( semilattice_order(A,F2,Less_eq,Less)
     => ( aa(A,$o,aa(A,fun(A,$o),Less,A3),aa(A,A,aa(A,fun(A,A),F2,B3),C2))
       => ~ ( aa(A,$o,aa(A,fun(A,$o),Less,A3),B3)
           => ~ aa(A,$o,aa(A,fun(A,$o),Less,A3),C2) ) ) ) ).

% semilattice_order.strict_boundedE
tff(fact_7619_semilattice__order_Ostrict__order__iff,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A3: A,B3: A] :
      ( semilattice_order(A,F2,Less_eq,Less)
     => ( aa(A,$o,aa(A,fun(A,$o),Less,A3),B3)
      <=> ( ( A3 = aa(A,A,aa(A,fun(A,A),F2,A3),B3) )
          & ( A3 != B3 ) ) ) ) ).

% semilattice_order.strict_order_iff
tff(fact_7620_semilattice__order_Ostrict__coboundedI1,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A3: A,C2: A,B3: A] :
      ( semilattice_order(A,F2,Less_eq,Less)
     => ( aa(A,$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),F2,A3),B3)),C2) ) ) ).

% semilattice_order.strict_coboundedI1
tff(fact_7621_semilattice__order_Ostrict__coboundedI2,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),B3: A,C2: A,A3: A] :
      ( semilattice_order(A,F2,Less_eq,Less)
     => ( aa(A,$o,aa(A,fun(A,$o),Less,B3),C2)
       => aa(A,$o,aa(A,fun(A,$o),Less,aa(A,A,aa(A,fun(A,A),F2,A3),B3)),C2) ) ) ).

% semilattice_order.strict_coboundedI2
tff(fact_7622_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_7623_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_7624_sup_Osemilattice__order__axioms,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => semilattice_order(A,sup_sup(A),aTP_Lamp_yu(A,fun(A,$o)),aTP_Lamp_yv(A,fun(A,$o))) ) ).

% sup.semilattice_order_axioms
tff(fact_7625_comm__monoid__list__set_Oset__conv__list,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,fun(A,A)),Z2: A,G: fun(B,A),Xsa: list(B)] :
      ( groups4802862169904069756st_set(A,F2,Z2)
     => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F2,Z2),G),aa(list(B),set(B),set2(B),Xsa)) = aa(list(A),A,groups_monoid_F(A,F2,Z2),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),G),aa(list(B),list(B),remdups(B),Xsa))) ) ) ).

% comm_monoid_list_set.set_conv_list
tff(fact_7626_comm__monoid__list__set_Odistinct__set__conv__list,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,fun(A,A)),Z2: A,Xsa: list(B),G: fun(B,A)] :
      ( groups4802862169904069756st_set(A,F2,Z2)
     => ( aa(list(B),$o,distinct(B),Xsa)
       => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F2,Z2),G),aa(list(B),set(B),set2(B),Xsa)) = aa(list(A),A,groups_monoid_F(A,F2,Z2),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),G),Xsa)) ) ) ) ).

% comm_monoid_list_set.distinct_set_conv_list
tff(fact_7627_comm__monoid__mult__class_Oprod__def,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ( groups7121269368397514597t_prod(A,B) = groups_comm_monoid_F(B,A,times_times(B),one_one(B)) ) ) ).

% comm_monoid_mult_class.prod_def
tff(fact_7628_comm__monoid__set_Otriangle__reindex,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z2: A,G: fun(nat,fun(nat,A)),N: nat] :
      ( groups778175481326437816id_set(A,F2,Z2)
     => ( aa(set(product_prod(nat,nat)),A,aa(fun(product_prod(nat,nat),A),fun(set(product_prod(nat,nat)),A),groups_comm_monoid_F(A,product_prod(nat,nat),F2,Z2),aa(fun(nat,fun(nat,A)),fun(product_prod(nat,nat),A),product_case_prod(nat,nat,A),G)),aa(fun(product_prod(nat,nat),$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,$o),aTP_Lamp_hr(nat,fun(nat,fun(nat,$o)),N)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z2),aa(fun(nat,fun(nat,A)),fun(nat,A),aa(A,fun(fun(nat,fun(nat,A)),fun(nat,A)),aTP_Lamp_acj(fun(A,fun(A,A)),fun(A,fun(fun(nat,fun(nat,A)),fun(nat,A))),F2),Z2),G)),aa(nat,set(nat),set_ord_lessThan(nat),N)) ) ) ).

% comm_monoid_set.triangle_reindex
tff(fact_7629_comm__monoid__set_Ozero__middle,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z2: A,P2: nat,K3: nat,G: fun(nat,A),H2: fun(nat,A)] :
      ( groups778175481326437816id_set(A,F2,Z2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),P2)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K3),P2)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z2),aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aa(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),aTP_Lamp_ack(A,fun(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A)))),Z2),K3),G),H2)),aa(nat,set(nat),set_ord_atMost(nat),P2)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z2),aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_acl(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K3),G),H2)),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),P2),aa(nat,nat,suc,zero_zero(nat))))) ) ) ) ) ).

% comm_monoid_set.zero_middle
tff(fact_7630_comm__monoid__set_Osetdiff__irrelevant,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,fun(A,A)),Z2: A,Aa2: set(B),G: fun(B,A)] :
      ( groups778175481326437816id_set(A,F2,Z2)
     => ( aa(set(B),$o,finite_finite2(B),Aa2)
       => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F2,Z2),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),Aa2),aa(fun(B,$o),set(B),collect(B),aa(fun(B,A),fun(B,$o),aTP_Lamp_acm(A,fun(fun(B,A),fun(B,$o)),Z2),G)))) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F2,Z2),G),Aa2) ) ) ) ).

% comm_monoid_set.setdiff_irrelevant
tff(fact_7631_comm__monoid__set_Oempty,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,fun(A,A)),Z2: A,G: fun(B,A)] :
      ( groups778175481326437816id_set(A,F2,Z2)
     => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F2,Z2),G),bot_bot(set(B))) = Z2 ) ) ).

% comm_monoid_set.empty
tff(fact_7632_comm__monoid__set_Oinsert__if,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,fun(A,A)),Z2: A,Aa2: set(B),G: fun(B,A),Xa: B] :
      ( groups778175481326437816id_set(A,F2,Z2)
     => ( aa(set(B),$o,finite_finite2(B),Aa2)
       => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F2,Z2),G),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert3(B),Xa),Aa2)) = $ite(member(B,Xa,Aa2),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F2,Z2),G),Aa2),aa(A,A,aa(A,fun(A,A),F2,aa(B,A,G,Xa)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F2,Z2),G),Aa2))) ) ) ) ).

% comm_monoid_set.insert_if
tff(fact_7633_comm__monoid__set_Oinsert,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,fun(A,A)),Z2: A,Aa2: set(B),Xa: B,G: fun(B,A)] :
      ( groups778175481326437816id_set(A,F2,Z2)
     => ( aa(set(B),$o,finite_finite2(B),Aa2)
       => ( ~ member(B,Xa,Aa2)
         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F2,Z2),G),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert3(B),Xa),Aa2)) = aa(A,A,aa(A,fun(A,A),F2,aa(B,A,G,Xa)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F2,Z2),G),Aa2)) ) ) ) ) ).

% comm_monoid_set.insert
tff(fact_7634_comm__monoid__set_Oreindex__bij__witness__not__neutral,axiom,
    ! [B: $tType,A: $tType,E: $tType,F2: fun(A,fun(A,A)),Z2: A,S3: set(B),T6: set(E),S: set(B),Ia: fun(E,B),J: fun(B,E),T4: set(E),G: fun(B,A),H2: fun(E,A)] :
      ( groups778175481326437816id_set(A,F2,Z2)
     => ( aa(set(B),$o,finite_finite2(B),S3)
       => ( aa(set(E),$o,finite_finite2(E),T6)
         => ( ! [A4: B] :
                ( member(B,A4,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S),S3))
               => ( aa(E,B,Ia,aa(B,E,J,A4)) = A4 ) )
           => ( ! [A4: B] :
                  ( member(B,A4,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S),S3))
                 => member(E,aa(B,E,J,A4),aa(set(E),set(E),aa(set(E),fun(set(E),set(E)),minus_minus(set(E)),T4),T6)) )
             => ( ! [B4: E] :
                    ( member(E,B4,aa(set(E),set(E),aa(set(E),fun(set(E),set(E)),minus_minus(set(E)),T4),T6))
                   => ( aa(B,E,J,aa(E,B,Ia,B4)) = B4 ) )
               => ( ! [B4: E] :
                      ( member(E,B4,aa(set(E),set(E),aa(set(E),fun(set(E),set(E)),minus_minus(set(E)),T4),T6))
                     => member(B,aa(E,B,Ia,B4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S),S3)) )
                 => ( ! [A4: B] :
                        ( member(B,A4,S3)
                       => ( aa(B,A,G,A4) = Z2 ) )
                   => ( ! [B4: E] :
                          ( member(E,B4,T6)
                         => ( aa(E,A,H2,B4) = Z2 ) )
                     => ( ! [A4: B] :
                            ( member(B,A4,S)
                           => ( aa(E,A,H2,aa(B,E,J,A4)) = aa(B,A,G,A4) ) )
                       => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F2,Z2),G),S) = aa(set(E),A,aa(fun(E,A),fun(set(E),A),groups_comm_monoid_F(A,E,F2,Z2),H2),T4) ) ) ) ) ) ) ) ) ) ) ) ).

% comm_monoid_set.reindex_bij_witness_not_neutral
tff(fact_7635_comm__monoid__set_Ohead__if,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z2: A,G: fun(nat,A),M2: nat,N: nat] :
      ( groups778175481326437816id_set(A,F2,Z2)
     => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z2),G),set_or1337092689740270186AtMost(nat,M2,N)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),M2),Z2,aa(A,A,aa(A,fun(A,A),F2,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z2),G),set_or7035219750837199246ssThan(nat,M2,N))),aa(nat,A,G,N))) ) ) ).

% comm_monoid_set.head_if
tff(fact_7636_comm__monoid__set_Oop__ivl__Suc,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z2: A,G: fun(nat,A),M2: nat,N: nat] :
      ( groups778175481326437816id_set(A,F2,Z2)
     => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z2),G),set_or7035219750837199246ssThan(nat,M2,aa(nat,nat,suc,N))) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),M2),Z2,aa(A,A,aa(A,fun(A,A),F2,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z2),G),set_or7035219750837199246ssThan(nat,M2,N))),aa(nat,A,G,N))) ) ) ).

% comm_monoid_set.op_ivl_Suc
tff(fact_7637_comm__monoid__set_OatLeast__Suc__lessThan,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z2: A,M2: nat,N: nat,G: fun(nat,A)] :
      ( groups778175481326437816id_set(A,F2,Z2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),N)
       => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z2),G),set_or7035219750837199246ssThan(nat,M2,N)) = aa(A,A,aa(A,fun(A,A),F2,aa(nat,A,G,M2)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z2),G),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M2),N))) ) ) ) ).

% comm_monoid_set.atLeast_Suc_lessThan
tff(fact_7638_comm__monoid__set_Ocl__ivl__Suc,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z2: A,G: fun(nat,A),M2: nat,N: nat] :
      ( groups778175481326437816id_set(A,F2,Z2)
     => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z2),G),set_or1337092689740270186AtMost(nat,M2,aa(nat,nat,suc,N))) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,N)),M2),Z2,aa(A,A,aa(A,fun(A,A),F2,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z2),G),set_or1337092689740270186AtMost(nat,M2,N))),aa(nat,A,G,aa(nat,nat,suc,N)))) ) ) ).

% comm_monoid_set.cl_ivl_Suc
tff(fact_7639_comm__monoid__set_Oinsert_H,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,fun(A,A)),Z2: A,Ia2: set(B),P2: fun(B,A),Ia: B] :
      ( groups778175481326437816id_set(A,F2,Z2)
     => ( aa(set(B),$o,finite_finite2(B),aa(fun(B,$o),set(B),collect(B),aa(fun(B,A),fun(B,$o),aa(set(B),fun(fun(B,A),fun(B,$o)),aTP_Lamp_acn(A,fun(set(B),fun(fun(B,A),fun(B,$o))),Z2),Ia2),P2)))
       => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_G(A,B,F2,Z2),P2),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert3(B),Ia),Ia2)) = $ite(member(B,Ia,Ia2),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_G(A,B,F2,Z2),P2),Ia2),aa(A,A,aa(A,fun(A,A),F2,aa(B,A,P2,Ia)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_G(A,B,F2,Z2),P2),Ia2))) ) ) ) ).

% comm_monoid_set.insert'
tff(fact_7640_comm__monoid__set_Omono__neutral__cong__right_H,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,fun(A,A)),Z2: A,S: set(B),T4: set(B),G: fun(B,A),H2: fun(B,A)] :
      ( groups778175481326437816id_set(A,F2,Z2)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),S),T4)
       => ( ! [X: B] :
              ( member(B,X,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T4),S))
             => ( aa(B,A,G,X) = Z2 ) )
         => ( ! [X: B] :
                ( member(B,X,S)
               => ( aa(B,A,G,X) = aa(B,A,H2,X) ) )
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_G(A,B,F2,Z2),G),T4) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_G(A,B,F2,Z2),H2),S) ) ) ) ) ) ).

% comm_monoid_set.mono_neutral_cong_right'
tff(fact_7641_comm__monoid__set_Omono__neutral__cong__left_H,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,fun(A,A)),Z2: A,S: set(B),T4: set(B),H2: fun(B,A),G: fun(B,A)] :
      ( groups778175481326437816id_set(A,F2,Z2)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),S),T4)
       => ( ! [I3: B] :
              ( member(B,I3,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T4),S))
             => ( aa(B,A,H2,I3) = Z2 ) )
         => ( ! [X: B] :
                ( member(B,X,S)
               => ( aa(B,A,G,X) = aa(B,A,H2,X) ) )
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_G(A,B,F2,Z2),G),S) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_G(A,B,F2,Z2),H2),T4) ) ) ) ) ) ).

% comm_monoid_set.mono_neutral_cong_left'
tff(fact_7642_comm__monoid__set_Omono__neutral__right_H,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,fun(A,A)),Z2: A,S: set(B),T4: set(B),G: fun(B,A)] :
      ( groups778175481326437816id_set(A,F2,Z2)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),S),T4)
       => ( ! [X: B] :
              ( member(B,X,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T4),S))
             => ( aa(B,A,G,X) = Z2 ) )
         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_G(A,B,F2,Z2),G),T4) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_G(A,B,F2,Z2),G),S) ) ) ) ) ).

% comm_monoid_set.mono_neutral_right'
tff(fact_7643_comm__monoid__set_Omono__neutral__left_H,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,fun(A,A)),Z2: A,S: set(B),T4: set(B),G: fun(B,A)] :
      ( groups778175481326437816id_set(A,F2,Z2)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),S),T4)
       => ( ! [X: B] :
              ( member(B,X,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T4),S))
             => ( aa(B,A,G,X) = Z2 ) )
         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_G(A,B,F2,Z2),G),S) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_G(A,B,F2,Z2),G),T4) ) ) ) ) ).

% comm_monoid_set.mono_neutral_left'
tff(fact_7644_comm__monoid__set_Oempty_H,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,fun(A,A)),Z2: A,P2: fun(B,A)] :
      ( groups778175481326437816id_set(A,F2,Z2)
     => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_G(A,B,F2,Z2),P2),bot_bot(set(B))) = Z2 ) ) ).

% comm_monoid_set.empty'
tff(fact_7645_prod_Ocomm__monoid__set__axioms,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => groups778175481326437816id_set(A,times_times(A),one_one(A)) ) ).

% prod.comm_monoid_set_axioms
tff(fact_7646_comm__monoid__set_Oivl__cong,axiom,
    ! [A: $tType,B: $tType] :
      ( ord(B)
     => ! [F2: fun(A,fun(A,A)),Z2: A,A3: B,C2: B,B3: B,D3: B,G: fun(B,A),H2: fun(B,A)] :
          ( groups778175481326437816id_set(A,F2,Z2)
         => ( ( A3 = C2 )
           => ( ( B3 = D3 )
             => ( ! [X: B] :
                    ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),C2),X)
                   => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X),D3)
                     => ( aa(B,A,G,X) = aa(B,A,H2,X) ) ) )
               => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F2,Z2),G),set_or7035219750837199246ssThan(B,A3,B3)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F2,Z2),H2),set_or7035219750837199246ssThan(B,C2,D3)) ) ) ) ) ) ) ).

% comm_monoid_set.ivl_cong
tff(fact_7647_comm__monoid__set_Osubset__diff,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,fun(A,A)),Z2: A,B2: set(B),Aa2: set(B),G: fun(B,A)] :
      ( groups778175481326437816id_set(A,F2,Z2)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),B2),Aa2)
       => ( aa(set(B),$o,finite_finite2(B),Aa2)
         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F2,Z2),G),Aa2) = aa(A,A,aa(A,fun(A,A),F2,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F2,Z2),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),Aa2),B2))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F2,Z2),G),B2)) ) ) ) ) ).

% comm_monoid_set.subset_diff
tff(fact_7648_comm__monoid__set_Osame__carrier,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,fun(A,A)),Z2: A,C3: set(B),Aa2: set(B),B2: set(B),G: fun(B,A),H2: fun(B,A)] :
      ( groups778175481326437816id_set(A,F2,Z2)
     => ( aa(set(B),$o,finite_finite2(B),C3)
       => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),Aa2),C3)
         => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),B2),C3)
           => ( ! [A4: B] :
                  ( member(B,A4,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),C3),Aa2))
                 => ( aa(B,A,G,A4) = Z2 ) )
             => ( ! [B4: B] :
                    ( member(B,B4,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),C3),B2))
                   => ( aa(B,A,H2,B4) = Z2 ) )
               => ( ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F2,Z2),G),Aa2) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F2,Z2),H2),B2) )
                <=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F2,Z2),G),C3) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F2,Z2),H2),C3) ) ) ) ) ) ) ) ) ).

% comm_monoid_set.same_carrier
tff(fact_7649_comm__monoid__set_Osame__carrierI,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,fun(A,A)),Z2: A,C3: set(B),Aa2: set(B),B2: set(B),G: fun(B,A),H2: fun(B,A)] :
      ( groups778175481326437816id_set(A,F2,Z2)
     => ( aa(set(B),$o,finite_finite2(B),C3)
       => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),Aa2),C3)
         => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),B2),C3)
           => ( ! [A4: B] :
                  ( member(B,A4,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),C3),Aa2))
                 => ( aa(B,A,G,A4) = Z2 ) )
             => ( ! [B4: B] :
                    ( member(B,B4,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),C3),B2))
                   => ( aa(B,A,H2,B4) = Z2 ) )
               => ( ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F2,Z2),G),C3) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F2,Z2),H2),C3) )
                 => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F2,Z2),G),Aa2) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F2,Z2),H2),B2) ) ) ) ) ) ) ) ) ).

% comm_monoid_set.same_carrierI
tff(fact_7650_comm__monoid__set_Omono__neutral__left,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,fun(A,A)),Z2: A,T4: set(B),S: set(B),G: fun(B,A)] :
      ( groups778175481326437816id_set(A,F2,Z2)
     => ( aa(set(B),$o,finite_finite2(B),T4)
       => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),S),T4)
         => ( ! [X: B] :
                ( member(B,X,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T4),S))
               => ( aa(B,A,G,X) = Z2 ) )
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F2,Z2),G),S) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F2,Z2),G),T4) ) ) ) ) ) ).

% comm_monoid_set.mono_neutral_left
tff(fact_7651_comm__monoid__set_Omono__neutral__right,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,fun(A,A)),Z2: A,T4: set(B),S: set(B),G: fun(B,A)] :
      ( groups778175481326437816id_set(A,F2,Z2)
     => ( aa(set(B),$o,finite_finite2(B),T4)
       => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),S),T4)
         => ( ! [X: B] :
                ( member(B,X,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T4),S))
               => ( aa(B,A,G,X) = Z2 ) )
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F2,Z2),G),T4) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F2,Z2),G),S) ) ) ) ) ) ).

% comm_monoid_set.mono_neutral_right
tff(fact_7652_comm__monoid__set_Omono__neutral__cong__left,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,fun(A,A)),Z2: A,T4: set(B),S: set(B),H2: fun(B,A),G: fun(B,A)] :
      ( groups778175481326437816id_set(A,F2,Z2)
     => ( aa(set(B),$o,finite_finite2(B),T4)
       => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),S),T4)
         => ( ! [X: B] :
                ( member(B,X,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T4),S))
               => ( aa(B,A,H2,X) = Z2 ) )
           => ( ! [X: B] :
                  ( member(B,X,S)
                 => ( aa(B,A,G,X) = aa(B,A,H2,X) ) )
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F2,Z2),G),S) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F2,Z2),H2),T4) ) ) ) ) ) ) ).

% comm_monoid_set.mono_neutral_cong_left
tff(fact_7653_comm__monoid__set_Omono__neutral__cong__right,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,fun(A,A)),Z2: A,T4: set(B),S: set(B),G: fun(B,A),H2: fun(B,A)] :
      ( groups778175481326437816id_set(A,F2,Z2)
     => ( aa(set(B),$o,finite_finite2(B),T4)
       => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),S),T4)
         => ( ! [X: B] :
                ( member(B,X,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T4),S))
               => ( aa(B,A,G,X) = Z2 ) )
           => ( ! [X: B] :
                  ( member(B,X,S)
                 => ( aa(B,A,G,X) = aa(B,A,H2,X) ) )
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F2,Z2),G),T4) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F2,Z2),H2),S) ) ) ) ) ) ) ).

% comm_monoid_set.mono_neutral_cong_right
tff(fact_7654_comm__monoid__set_OInt__Diff,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,fun(A,A)),Z2: A,Aa2: set(B),G: fun(B,A),B2: set(B)] :
      ( groups778175481326437816id_set(A,F2,Z2)
     => ( aa(set(B),$o,finite_finite2(B),Aa2)
       => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F2,Z2),G),Aa2) = aa(A,A,aa(A,fun(A,A),F2,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F2,Z2),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),Aa2),B2))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F2,Z2),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),Aa2),B2))) ) ) ) ).

% comm_monoid_set.Int_Diff
tff(fact_7655_comm__monoid__set_Omono__neutral__cong,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,fun(A,A)),Z2: A,T4: set(B),S: set(B),H2: fun(B,A),G: fun(B,A)] :
      ( groups778175481326437816id_set(A,F2,Z2)
     => ( aa(set(B),$o,finite_finite2(B),T4)
       => ( aa(set(B),$o,finite_finite2(B),S)
         => ( ! [I3: B] :
                ( member(B,I3,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T4),S))
               => ( aa(B,A,H2,I3) = Z2 ) )
           => ( ! [I3: B] :
                  ( member(B,I3,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S),T4))
                 => ( aa(B,A,G,I3) = Z2 ) )
             => ( ! [X: B] :
                    ( member(B,X,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),S),T4))
                   => ( aa(B,A,G,X) = aa(B,A,H2,X) ) )
               => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F2,Z2),G),S) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F2,Z2),H2),T4) ) ) ) ) ) ) ) ).

% comm_monoid_set.mono_neutral_cong
tff(fact_7656_comm__monoid__set_Oreindex__bij__betw__not__neutral,axiom,
    ! [B: $tType,A: $tType,E: $tType,F2: fun(A,fun(A,A)),Z2: A,S3: set(B),T6: set(E),H2: fun(B,E),S: set(B),T4: set(E),G: fun(E,A)] :
      ( groups778175481326437816id_set(A,F2,Z2)
     => ( aa(set(B),$o,finite_finite2(B),S3)
       => ( aa(set(E),$o,finite_finite2(E),T6)
         => ( bij_betw(B,E,H2,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S),S3),aa(set(E),set(E),aa(set(E),fun(set(E),set(E)),minus_minus(set(E)),T4),T6))
           => ( ! [A4: B] :
                  ( member(B,A4,S3)
                 => ( aa(E,A,G,aa(B,E,H2,A4)) = Z2 ) )
             => ( ! [B4: E] :
                    ( member(E,B4,T6)
                   => ( aa(E,A,G,B4) = Z2 ) )
               => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F2,Z2),aa(fun(E,A),fun(B,A),aTP_Lamp_aco(fun(B,E),fun(fun(E,A),fun(B,A)),H2),G)),S) = aa(set(E),A,aa(fun(E,A),fun(set(E),A),groups_comm_monoid_F(A,E,F2,Z2),G),T4) ) ) ) ) ) ) ) ).

% comm_monoid_set.reindex_bij_betw_not_neutral
tff(fact_7657_comm__monoid__set_Oremove,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,fun(A,A)),Z2: A,Aa2: set(B),Xa: B,G: fun(B,A)] :
      ( groups778175481326437816id_set(A,F2,Z2)
     => ( aa(set(B),$o,finite_finite2(B),Aa2)
       => ( member(B,Xa,Aa2)
         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F2,Z2),G),Aa2) = aa(A,A,aa(A,fun(A,A),F2,aa(B,A,G,Xa)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F2,Z2),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),Aa2),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert3(B),Xa),bot_bot(set(B)))))) ) ) ) ) ).

% comm_monoid_set.remove
tff(fact_7658_comm__monoid__set_Oinsert__remove,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,fun(A,A)),Z2: A,Aa2: set(B),G: fun(B,A),Xa: B] :
      ( groups778175481326437816id_set(A,F2,Z2)
     => ( aa(set(B),$o,finite_finite2(B),Aa2)
       => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F2,Z2),G),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert3(B),Xa),Aa2)) = aa(A,A,aa(A,fun(A,A),F2,aa(B,A,G,Xa)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F2,Z2),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),Aa2),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert3(B),Xa),bot_bot(set(B)))))) ) ) ) ).

% comm_monoid_set.insert_remove
tff(fact_7659_comm__monoid__set_Ounion__disjoint,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,fun(A,A)),Z2: A,Aa2: set(B),B2: set(B),G: fun(B,A)] :
      ( groups778175481326437816id_set(A,F2,Z2)
     => ( aa(set(B),$o,finite_finite2(B),Aa2)
       => ( aa(set(B),$o,finite_finite2(B),B2)
         => ( ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),Aa2),B2) = bot_bot(set(B)) )
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F2,Z2),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),Aa2),B2)) = aa(A,A,aa(A,fun(A,A),F2,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F2,Z2),G),Aa2)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F2,Z2),G),B2)) ) ) ) ) ) ).

% comm_monoid_set.union_disjoint
tff(fact_7660_comm__monoid__set_Ounion__diff2,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,fun(A,A)),Z2: A,Aa2: set(B),B2: set(B),G: fun(B,A)] :
      ( groups778175481326437816id_set(A,F2,Z2)
     => ( aa(set(B),$o,finite_finite2(B),Aa2)
       => ( aa(set(B),$o,finite_finite2(B),B2)
         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F2,Z2),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),Aa2),B2)) = aa(A,A,aa(A,fun(A,A),F2,aa(A,A,aa(A,fun(A,A),F2,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F2,Z2),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),Aa2),B2))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F2,Z2),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),B2),Aa2)))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F2,Z2),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),Aa2),B2))) ) ) ) ) ).

% comm_monoid_set.union_diff2
tff(fact_7661_comm__monoid__set_Odelta__remove,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,fun(A,A)),Z2: A,S: set(B),A3: B,B3: fun(B,A),C2: fun(B,A)] :
      ( groups778175481326437816id_set(A,F2,Z2)
     => ( aa(set(B),$o,finite_finite2(B),S)
       => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F2,Z2),aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_acp(B,fun(fun(B,A),fun(fun(B,A),fun(B,A))),A3),B3),C2)),S) = $ite(member(B,A3,S),aa(A,A,aa(A,fun(A,A),F2,aa(B,A,B3,A3)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F2,Z2),C2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert3(B),A3),bot_bot(set(B)))))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F2,Z2),C2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert3(B),A3),bot_bot(set(B)))))) ) ) ) ).

% comm_monoid_set.delta_remove
tff(fact_7662_comm__monoid__set_OUNION__disjoint,axiom,
    ! [E: $tType,A: $tType,B: $tType,F2: fun(A,fun(A,A)),Z2: A,Ia2: set(B),Aa2: fun(B,set(E)),G: fun(E,A)] :
      ( groups778175481326437816id_set(A,F2,Z2)
     => ( aa(set(B),$o,finite_finite2(B),Ia2)
       => ( ! [X: B] :
              ( member(B,X,Ia2)
             => aa(set(E),$o,finite_finite2(E),aa(B,set(E),Aa2,X)) )
         => ( ! [X: B] :
                ( member(B,X,Ia2)
               => ! [Xa5: B] :
                    ( member(B,Xa5,Ia2)
                   => ( ( X != Xa5 )
                     => ( aa(set(E),set(E),aa(set(E),fun(set(E),set(E)),inf_inf(set(E)),aa(B,set(E),Aa2,X)),aa(B,set(E),Aa2,Xa5)) = bot_bot(set(E)) ) ) ) )
           => ( aa(set(E),A,aa(fun(E,A),fun(set(E),A),groups_comm_monoid_F(A,E,F2,Z2),G),aa(set(set(E)),set(E),complete_Sup_Sup(set(E)),aa(set(B),set(set(E)),image2(B,set(E),Aa2),Ia2))) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F2,Z2),aa(fun(E,A),fun(B,A),aa(fun(B,set(E)),fun(fun(E,A),fun(B,A)),aa(A,fun(fun(B,set(E)),fun(fun(E,A),fun(B,A))),aTP_Lamp_acq(fun(A,fun(A,A)),fun(A,fun(fun(B,set(E)),fun(fun(E,A),fun(B,A)))),F2),Z2),Aa2),G)),Ia2) ) ) ) ) ) ).

% comm_monoid_set.UNION_disjoint
tff(fact_7663_comm__monoid__set_OUnion__disjoint,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,fun(A,A)),Z2: A,C3: set(set(B)),G: fun(B,A)] :
      ( groups778175481326437816id_set(A,F2,Z2)
     => ( ! [X: set(B)] :
            ( member(set(B),X,C3)
           => aa(set(B),$o,finite_finite2(B),X) )
       => ( ! [X: set(B)] :
              ( member(set(B),X,C3)
             => ! [Xa5: set(B)] :
                  ( member(set(B),Xa5,C3)
                 => ( ( X != Xa5 )
                   => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),X),Xa5) = bot_bot(set(B)) ) ) ) )
         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F2,Z2),G),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),C3)) = aa(set(set(B)),A,aa(fun(B,A),fun(set(set(B)),A),comp(fun(set(B),A),fun(set(set(B)),A),fun(B,A),groups_comm_monoid_F(A,set(B),F2,Z2),groups_comm_monoid_F(A,B,F2,Z2)),G),C3) ) ) ) ) ).

% comm_monoid_set.Union_disjoint
tff(fact_7664_semilattice__neutr__order_Ointro,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z2: A,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o))] :
      ( semilattice_neutr(A,F2,Z2)
     => ( semilattice_order(A,F2,Less_eq,Less)
       => semila1105856199041335345_order(A,F2,Z2,Less_eq,Less) ) ) ).

% semilattice_neutr_order.intro
tff(fact_7665_semilattice__neutr__order__def,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z2: A,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o))] :
      ( semila1105856199041335345_order(A,F2,Z2,Less_eq,Less)
    <=> ( semilattice_neutr(A,F2,Z2)
        & semilattice_order(A,F2,Less_eq,Less) ) ) ).

% semilattice_neutr_order_def
tff(fact_7666_semilattice__neutr__order_Oaxioms_I1_J,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z2: A,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o))] :
      ( semila1105856199041335345_order(A,F2,Z2,Less_eq,Less)
     => semilattice_neutr(A,F2,Z2) ) ).

% semilattice_neutr_order.axioms(1)
tff(fact_7667_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_7668_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_7669_subset__eq__mset__impl_Opelims,axiom,
    ! [A: $tType,Xa: list(A),Xaa: list(A),Y: option($o)] :
      ( ( subset_eq_mset_impl(A,Xa,Xaa) = 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)),Xa),Xaa))
       => ( ( ( Xa = nil(A) )
           => ( ( Y = aa($o,option($o),some($o),Xaa != nil(A)) )
             => ~ accp(product_prod(list(A),list(A)),subset751672762298770561pl_rel(A),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),Xaa)) ) )
         => ~ ! [X: A,Xs: list(A)] :
                ( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) )
               => ( ( Y = case_option(option($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_acs(list(A),fun(list(A),fun(product_prod(A,list(A)),option($o))),Xs)),extract(A,aa(A,fun(A,$o),fequal(A),X),Xaa)) )
                 => ~ 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),X),Xs)),Xaa)) ) ) ) ) ) ).

% subset_eq_mset_impl.pelims
tff(fact_7670_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_7671_lexordp__eq_Ocases,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A1: list(A),A22: list(A)] :
          ( ord_lexordp_eq(A,A1,A22)
         => ( ( A1 != nil(A) )
           => ( ! [X: A] :
                  ( ? [Xs: list(A)] : ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) )
                 => ! [Y2: A] :
                      ( ? [Ys5: list(A)] : ( A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys5) )
                     => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y2) ) )
             => ~ ! [X: A,Y2: A,Xs: list(A)] :
                    ( ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) )
                   => ! [Ys5: list(A)] :
                        ( ( A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys5) )
                       => ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y2)
                         => ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y2),X)
                           => ~ ord_lexordp_eq(A,Xs,Ys5) ) ) ) ) ) ) ) ) ).

% lexordp_eq.cases
tff(fact_7672_lexordp__eq_Osimps,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A1: list(A),A22: list(A)] :
          ( ord_lexordp_eq(A,A1,A22)
        <=> ( ? [Ys4: list(A)] :
                ( ( A1 = nil(A) )
                & ( A22 = Ys4 ) )
            | ? [X6: A,Y4: A,Xs2: list(A),Ys4: list(A)] :
                ( ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X6),Xs2) )
                & ( A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys4) )
                & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X6),Y4) )
            | ? [X6: A,Y4: A,Xs2: list(A),Ys4: list(A)] :
                ( ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X6),Xs2) )
                & ( A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys4) )
                & ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X6),Y4)
                & ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y4),X6)
                & ord_lexordp_eq(A,Xs2,Ys4) ) ) ) ) ).

% lexordp_eq.simps
tff(fact_7673_lexordp__eq__simps_I2_J,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Xsa: list(A)] :
          ( ord_lexordp_eq(A,Xsa,nil(A))
        <=> ( Xsa = nil(A) ) ) ) ).

% lexordp_eq_simps(2)
tff(fact_7674_lexordp__eq__simps_I1_J,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Ys: list(A)] : ord_lexordp_eq(A,nil(A),Ys) ) ).

% lexordp_eq_simps(1)
tff(fact_7675_lexordp__eq__simps_I4_J,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Xa: A,Xsa: list(A),Y: A,Ys: list(A)] :
          ( ord_lexordp_eq(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
            | ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xa)
              & ord_lexordp_eq(A,Xsa,Ys) ) ) ) ) ).

% lexordp_eq_simps(4)
tff(fact_7676_lexordp__eq__simps_I3_J,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Xa: A,Xsa: list(A)] : ~ ord_lexordp_eq(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa),nil(A)) ) ).

% lexordp_eq_simps(3)
tff(fact_7677_lexordp__eq_ONil,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Ys: list(A)] : ord_lexordp_eq(A,nil(A),Ys) ) ).

% lexordp_eq.Nil
tff(fact_7678_lexordp__eq__pref,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [U: list(A),V2: list(A)] : ord_lexordp_eq(A,U,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),U),V2)) ) ).

% lexordp_eq_pref
tff(fact_7679_lexordp__eq__antisym,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xsa: list(A),Ys: list(A)] :
          ( ord_lexordp_eq(A,Xsa,Ys)
         => ( ord_lexordp_eq(A,Ys,Xsa)
           => ( Xsa = Ys ) ) ) ) ).

% lexordp_eq_antisym
tff(fact_7680_lexordp__eq__linear,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xsa: list(A),Ys: list(A)] :
          ( ord_lexordp_eq(A,Xsa,Ys)
          | ord_lexordp_eq(A,Ys,Xsa) ) ) ).

% lexordp_eq_linear
tff(fact_7681_lexordp__eq__trans,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xsa: list(A),Ys: list(A),Zs: list(A)] :
          ( ord_lexordp_eq(A,Xsa,Ys)
         => ( ord_lexordp_eq(A,Ys,Zs)
           => ord_lexordp_eq(A,Xsa,Zs) ) ) ) ).

% lexordp_eq_trans
tff(fact_7682_lexordp__eq__refl,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Xsa: list(A)] : ord_lexordp_eq(A,Xsa,Xsa) ) ).

% lexordp_eq_refl
tff(fact_7683_lexordp__into__lexordp__eq,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Xsa: list(A),Ys: list(A)] :
          ( aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),Xsa),Ys)
         => ord_lexordp_eq(A,Xsa,Ys) ) ) ).

% lexordp_into_lexordp_eq
tff(fact_7684_lexordp__eq__conv__lexord,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xsa: list(A),Ys: list(A)] :
          ( ord_lexordp_eq(A,Xsa,Ys)
        <=> ( ( Xsa = Ys )
            | aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),Xsa),Ys) ) ) ) ).

% lexordp_eq_conv_lexord
tff(fact_7685_lexordp__conv__lexordp__eq,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xsa: list(A),Ys: list(A)] :
          ( aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),Xsa),Ys)
        <=> ( ord_lexordp_eq(A,Xsa,Ys)
            & ~ ord_lexordp_eq(A,Ys,Xsa) ) ) ) ).

% lexordp_conv_lexordp_eq
tff(fact_7686_lexordp__eq_OCons,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Xa: A,Y: A,Xsa: list(A),Ys: list(A)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
         => ord_lexordp_eq(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)) ) ) ).

% lexordp_eq.Cons
tff(fact_7687_lexordp__eq_OCons__eq,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Xa: A,Y: A,Xsa: list(A),Ys: list(A)] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
         => ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xa)
           => ( ord_lexordp_eq(A,Xsa,Ys)
             => ord_lexordp_eq(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)) ) ) ) ) ).

% lexordp_eq.Cons_eq
tff(fact_7688_num__of__nat__inverse,axiom,
    ! [N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
     => ( nat_of_num(num_of_nat(N)) = N ) ) ).

% num_of_nat_inverse
tff(fact_7689_numeral__sqr,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [K3: num] : ( numeral_numeral(A,sqr(K3)) = aa(A,A,aa(A,fun(A,A),times_times(A),numeral_numeral(A,K3)),numeral_numeral(A,K3)) ) ) ).

% numeral_sqr
tff(fact_7690_nat__of__num__pos,axiom,
    ! [Xa: num] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),nat_of_num(Xa)) ).

% nat_of_num_pos
tff(fact_7691_less__num__def,axiom,
    ! [M2: num,N: num] :
      ( aa(num,$o,aa(num,fun(num,$o),ord_less(num),M2),N)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),nat_of_num(M2)),nat_of_num(N)) ) ).

% less_num_def
tff(fact_7692_bdd__below_Opreordering__bdd__axioms,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => condit622319405099724424ng_bdd(A,aTP_Lamp_aav(A,fun(A,$o)),aTP_Lamp_ij(A,fun(A,$o))) ) ).

% bdd_below.preordering_bdd_axioms
tff(fact_7693_natural__decr,axiom,
    ! [N: code_natural] :
      ( ( N != zero_zero(code_natural) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),code_nat_of_natural(N)),aa(nat,nat,suc,zero_zero(nat)))),code_nat_of_natural(N)) ) ).

% natural_decr
tff(fact_7694_less__natural_Orep__eq,axiom,
    ! [Xa: code_natural,Xaa: code_natural] :
      ( aa(code_natural,$o,aa(code_natural,fun(code_natural,$o),ord_less(code_natural),Xa),Xaa)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),code_nat_of_natural(Xa)),code_nat_of_natural(Xaa)) ) ).

% less_natural.rep_eq
tff(fact_7695_bdd__above_Opreordering__bdd__axioms,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => condit622319405099724424ng_bdd(A,ord_less_eq(A),ord_less(A)) ) ).

% bdd_above.preordering_bdd_axioms
tff(fact_7696_sndOp__def,axiom,
    ! [A: $tType,B: $tType,E: $tType,Pa: fun(E,fun(A,$o)),Qa: fun(A,fun(B,$o)),Ac: product_prod(E,B)] : ( bNF_sndOp(E,A,B,Pa,Qa,Ac) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),bNF_pick_middlep(E,A,B,Pa,Qa,aa(product_prod(E,B),E,product_fst(E,B),Ac),aa(product_prod(E,B),B,product_snd(E,B),Ac))),aa(product_prod(E,B),B,product_snd(E,B),Ac)) ) ).

% sndOp_def
tff(fact_7697_fstOp__def,axiom,
    ! [B: $tType,E: $tType,A: $tType,Pa: fun(A,fun(B,$o)),Qa: fun(B,fun(E,$o)),Ac: product_prod(A,E)] : ( bNF_fstOp(A,B,E,Pa,Qa,Ac) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(product_prod(A,E),A,product_fst(A,E),Ac)),bNF_pick_middlep(A,B,E,Pa,Qa,aa(product_prod(A,E),A,product_fst(A,E),Ac),aa(product_prod(A,E),E,product_snd(A,E),Ac))) ) ).

% fstOp_def
tff(fact_7698_CODE__ABORT__def,axiom,
    ! [A: $tType,F2: fun(product_unit,A)] : ( cODE_ABORT(A,F2) = aa(product_unit,A,F2,product_Unity) ) ).

% CODE_ABORT_def
tff(fact_7699_abstract__filter__def,axiom,
    ! [A: $tType,F2: fun(product_unit,filter(A))] : ( abstract_filter(A,F2) = aa(product_unit,filter(A),F2,product_Unity) ) ).

% abstract_filter_def
tff(fact_7700_next_Osimps,axiom,
    ! [V2: 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),V2),W2)) = $let(
          v: code_natural,
          v:= minus_shift(numeral_numeral(code_natural,bit1(bit1(bit0(bit1(bit0(bit1(bit0(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(one2))))))))))))))))))))))))))))))),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),times_times(code_natural),modulo_modulo(code_natural,V2,numeral_numeral(code_natural,bit0(bit0(bit1(bit0(bit0(bit1(bit0(bit1(bit1(bit0(bit0(bit0(bit1(bit0(bit1(one2)))))))))))))))))),numeral_numeral(code_natural,bit0(bit1(bit1(bit1(bit0(bit0(bit1(bit0(bit0(bit0(bit1(bit1(bit1(bit0(bit0(one2))))))))))))))))),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),times_times(code_natural),divide_divide(code_natural,V2,numeral_numeral(code_natural,bit0(bit0(bit1(bit0(bit0(bit1(bit0(bit1(bit1(bit0(bit0(bit0(bit1(bit0(bit1(one2)))))))))))))))))),numeral_numeral(code_natural,bit1(bit1(bit0(bit0(bit1(bit1(bit0(bit1(bit1(bit1(bit1(bit1(bit0(one2)))))))))))))))),
          $let(
            w2: code_natural,
            w2:= minus_shift(numeral_numeral(code_natural,bit1(bit1(bit1(bit0(bit0(bit0(bit0(bit0(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(one2))))))))))))))))))))))))))))))),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),times_times(code_natural),modulo_modulo(code_natural,W2,numeral_numeral(code_natural,bit0(bit1(bit1(bit0(bit0(bit1(bit0(bit0(bit0(bit1(bit1(bit1(bit0(bit0(bit1(one2)))))))))))))))))),numeral_numeral(code_natural,bit0(bit0(bit1(bit0(bit1(bit1(bit1(bit1(bit0(bit1(bit1(bit1(bit1(bit0(bit0(one2))))))))))))))))),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),times_times(code_natural),divide_divide(code_natural,W2,numeral_numeral(code_natural,bit0(bit1(bit1(bit0(bit0(bit1(bit0(bit0(bit0(bit1(bit1(bit1(bit0(bit0(bit1(one2)))))))))))))))))),numeral_numeral(code_natural,bit1(bit1(bit1(bit1(bit0(bit0(bit1(bit1(bit0(bit1(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(numeral_numeral(code_natural,bit0(bit1(bit0(bit1(bit0(bit1(bit0(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(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_7701_select__weight__member,axiom,
    ! [A: $tType,Xsa: list(product_prod(code_natural,A)),S2: product_prod(code_natural,code_natural)] :
      ( aa(code_natural,$o,aa(code_natural,fun(code_natural,$o),ord_less(code_natural),zero_zero(code_natural)),groups8242544230860333062m_list(code_natural,aa(list(product_prod(code_natural,A)),list(code_natural),aa(fun(product_prod(code_natural,A),code_natural),fun(list(product_prod(code_natural,A)),list(code_natural)),map(product_prod(code_natural,A),code_natural),product_fst(code_natural,A)),Xsa)))
     => member(A,aa(product_prod(A,product_prod(code_natural,code_natural)),A,product_fst(A,product_prod(code_natural,code_natural)),aa(product_prod(code_natural,code_natural),product_prod(A,product_prod(code_natural,code_natural)),select_weight(A,Xsa),S2)),aa(list(A),set(A),set2(A),aa(list(product_prod(code_natural,A)),list(A),aa(fun(product_prod(code_natural,A),A),fun(list(product_prod(code_natural,A)),list(A)),map(product_prod(code_natural,A),A),product_snd(code_natural,A)),Xsa))) ) ).

% select_weight_member
tff(fact_7702_full__exhaustive__natural_H_Ocases,axiom,
    ! [Xa: product_prod(fun(product_prod(code_natural,fun(product_unit,code_term)),option(product_prod($o,list(code_term)))),product_prod(code_natural,code_natural))] :
      ~ ! [F3: fun(product_prod(code_natural,fun(product_unit,code_term)),option(product_prod($o,list(code_term)))),D4: code_natural,I3: code_natural] : ( Xa != 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)),F3),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),D4),I3)) ) ).

% full_exhaustive_natural'.cases
tff(fact_7703_full__exhaustive__fun_H_Ocases,axiom,
    ! [A: $tType,B: $tType] :
      ( ( quickc3360725361186068524ustive(B)
        & cl_HOL_Oequal(A)
        & quickc3360725361186068524ustive(A) )
     => ! [Xa: product_prod(fun(product_prod(fun(A,B),fun(product_unit,code_term)),option(product_prod($o,list(code_term)))),product_prod(code_natural,code_natural))] :
          ~ ! [F3: fun(product_prod(fun(A,B),fun(product_unit,code_term)),option(product_prod($o,list(code_term)))),I3: code_natural,D4: code_natural] : ( Xa != aa(product_prod(code_natural,code_natural),product_prod(fun(product_prod(fun(A,B),fun(product_unit,code_term)),option(product_prod($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)),F3),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),I3),D4)) ) ) ).

% full_exhaustive_fun'.cases
tff(fact_7704_exhaustive__natural_H_Ocases,axiom,
    ! [Xa: product_prod(fun(code_natural,option(product_prod($o,list(code_term)))),product_prod(code_natural,code_natural))] :
      ~ ! [F3: fun(code_natural,option(product_prod($o,list(code_term)))),D4: code_natural,I3: code_natural] : ( Xa != 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)),F3),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),D4),I3)) ) ).

% exhaustive_natural'.cases
tff(fact_7705_exhaustive__fun_H_Ocases,axiom,
    ! [A: $tType,B: $tType] :
      ( ( quickc658316121487927005ustive(B)
        & cl_HOL_Oequal(A)
        & quickc658316121487927005ustive(A) )
     => ! [Xa: product_prod(fun(fun(A,B),option(product_prod($o,list(code_term)))),product_prod(code_natural,code_natural))] :
          ~ ! [F3: fun(fun(A,B),option(product_prod($o,list(code_term)))),I3: code_natural,D4: code_natural] : ( Xa != aa(product_prod(code_natural,code_natural),product_prod(fun(fun(A,B),option(product_prod($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)),F3),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),I3),D4)) ) ) ).

% exhaustive_fun'.cases
tff(fact_7706_iter_Ocases,axiom,
    ! [A: $tType,Xa: 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)] : ( Xa != 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_7707_log_Ocases,axiom,
    ! [Xa: product_prod(code_natural,code_natural)] :
      ~ ! [B4: code_natural,I3: code_natural] : ( Xa != 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),B4),I3) ) ).

% log.cases
tff(fact_7708_select__weight__cons__zero,axiom,
    ! [A: $tType,Xa: A,Xsa: 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)),Xa)),Xsa)) = select_weight(A,Xsa) ) ).

% select_weight_cons_zero
tff(fact_7709_pick__member,axiom,
    ! [A: $tType,Ia: code_natural,Xsa: list(product_prod(code_natural,A))] :
      ( aa(code_natural,$o,aa(code_natural,fun(code_natural,$o),ord_less(code_natural),Ia),groups8242544230860333062m_list(code_natural,aa(list(product_prod(code_natural,A)),list(code_natural),aa(fun(product_prod(code_natural,A),code_natural),fun(list(product_prod(code_natural,A)),list(code_natural)),map(product_prod(code_natural,A),code_natural),product_fst(code_natural,A)),Xsa)))
     => member(A,pick(A,Xsa,Ia),aa(list(A),set(A),set2(A),aa(list(product_prod(code_natural,A)),list(A),aa(fun(product_prod(code_natural,A),A),fun(list(product_prod(code_natural,A)),list(A)),map(product_prod(code_natural,A),A),product_snd(code_natural,A)),Xsa))) ) ).

% pick_member
tff(fact_7710_select__weight__select,axiom,
    ! [A: $tType,Xsa: list(A)] :
      ( ( Xsa != nil(A) )
     => ( select_weight(A,aa(list(A),list(product_prod(code_natural,A)),aa(fun(A,product_prod(code_natural,A)),fun(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))),Xsa)) = select(A,Xsa) ) ) ).

% select_weight_select
tff(fact_7711_select,axiom,
    ! [A: $tType,Xsa: list(A),S2: product_prod(code_natural,code_natural)] :
      ( ( Xsa != nil(A) )
     => member(A,aa(product_prod(A,product_prod(code_natural,code_natural)),A,product_fst(A,product_prod(code_natural,code_natural)),aa(product_prod(code_natural,code_natural),product_prod(A,product_prod(code_natural,code_natural)),select(A,Xsa),S2)),aa(list(A),set(A),set2(A),Xsa)) ) ).

% select
tff(fact_7712_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_acu(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_7713_Lazy__Sequence_Oiterate__upto_Ocases,axiom,
    ! [A: $tType,Xa: product_prod(fun(code_natural,A),product_prod(code_natural,code_natural))] :
      ~ ! [F3: fun(code_natural,A),N5: code_natural,M6: code_natural] : ( Xa != 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)),F3),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),N5),M6)) ) ).

% Lazy_Sequence.iterate_upto.cases
tff(fact_7714_small__lazy_H_Ocases,axiom,
    ! [Xa: product_prod(int,int)] :
      ~ ! [D4: int,I3: int] : ( Xa != aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),D4),I3) ) ).

% small_lazy'.cases
tff(fact_7715_select__weight__def,axiom,
    ! [A: $tType,Xsa: list(product_prod(code_natural,A))] : ( select_weight(A,Xsa) = 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),aa(fun(product_prod(code_natural,A),code_natural),fun(list(product_prod(code_natural,A)),list(code_natural)),map(product_prod(code_natural,A),code_natural),product_fst(code_natural,A)),Xsa))),aTP_Lamp_acv(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)))),Xsa)) ) ).

% select_weight_def
tff(fact_7716_Predicate_Oiterate__upto_Opinduct,axiom,
    ! [A: $tType,A0: fun(code_natural,A),A1: code_natural,A22: code_natural,Pa: fun(fun(code_natural,A),fun(code_natural,fun(code_natural,$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)))
     => ( ! [F3: fun(code_natural,A),N5: code_natural,M6: 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)),F3),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),N5),M6)))
           => ( ! [X4: product_unit] :
                  ( ~ aa(code_natural,$o,aa(code_natural,fun(code_natural,$o),ord_less(code_natural),M6),N5)
                 => aa(code_natural,$o,aa(code_natural,fun(code_natural,$o),aa(fun(code_natural,A),fun(code_natural,fun(code_natural,$o)),Pa,F3),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),plus_plus(code_natural),N5),one_one(code_natural))),M6) )
             => aa(code_natural,$o,aa(code_natural,fun(code_natural,$o),aa(fun(code_natural,A),fun(code_natural,fun(code_natural,$o)),Pa,F3),N5),M6) ) )
       => aa(code_natural,$o,aa(code_natural,fun(code_natural,$o),aa(fun(code_natural,A),fun(code_natural,fun(code_natural,$o)),Pa,A0),A1),A22) ) ) ).

% Predicate.iterate_upto.pinduct
tff(fact_7717_scomp__Pair,axiom,
    ! [E: $tType,B: $tType,A: $tType,Xa: fun(A,product_prod(B,E))] : ( product_scomp(A,B,E,product_prod(B,E),Xa,product_Pair(B,E)) = Xa ) ).

% scomp_Pair
tff(fact_7718_Pair__scomp,axiom,
    ! [A: $tType,B: $tType,E: $tType,Xa: E,F2: fun(E,fun(A,B))] : ( product_scomp(A,E,A,B,aa(E,fun(A,product_prod(E,A)),product_Pair(E,A),Xa),F2) = aa(E,fun(A,B),F2,Xa) ) ).

% Pair_scomp
tff(fact_7719_Random_Orange__def,axiom,
    ! [K3: code_natural] : ( range(K3) = 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(numeral_numeral(code_natural,bit1(bit0(bit0(bit1(bit0(bit1(bit0(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(one2))))))))))))))))))))))))))))))),K3),aTP_Lamp_acx(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_acy(code_natural,fun(code_natural,fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural)))),K3)) ) ).

% Random.range_def
tff(fact_7720_iterate_Oelims,axiom,
    ! [A: $tType,B: $tType,Xa: code_natural,Xaa: 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,Xa,Xaa),Xb) = Y )
     => ( Y = $ite(Xa = 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)),Xaa,Xb),iterate(B,A,aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),minus_minus(code_natural),Xa),one_one(code_natural)),Xaa))) ) ) ).

% iterate.elims
tff(fact_7721_iterate_Osimps,axiom,
    ! [A: $tType,B: $tType,K3: code_natural,F2: fun(B,fun(A,product_prod(B,A))),Xa: B] :
      ( aa(B,fun(A,product_prod(B,A)),iterate(B,A,K3,F2),Xa) = $ite(K3 = zero_zero(code_natural),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Xa),product_scomp(A,B,A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),F2,Xa),iterate(B,A,aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),minus_minus(code_natural),K3),one_one(code_natural)),F2))) ) ).

% iterate.simps
tff(fact_7722_select__def,axiom,
    ! [A: $tType,Xsa: list(A)] : ( select(A,Xsa) = 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(code_natural_of_nat(aa(list(A),nat,size_size(list(A)),Xsa))),aTP_Lamp_acz(list(A),fun(code_natural,fun(product_prod(code_natural,code_natural),product_prod(A,product_prod(code_natural,code_natural)))),Xsa)) ) ).

% select_def
tff(fact_7723_pick__same,axiom,
    ! [A: $tType,L: nat,Xsa: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),L),aa(list(A),nat,size_size(list(A)),Xsa))
     => ( pick(A,aa(list(A),list(product_prod(code_natural,A)),aa(fun(A,product_prod(code_natural,A)),fun(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))),Xsa),code_natural_of_nat(L)) = aa(nat,A,nth(A,Xsa),L) ) ) ).

% pick_same
tff(fact_7724_less__natural_Oabs__eq,axiom,
    ! [Xa: nat,Xb: nat] :
      ( aa(code_natural,$o,aa(code_natural,fun(code_natural,$o),ord_less(code_natural),code_natural_of_nat(Xa)),code_natural_of_nat(Xb))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Xb) ) ).

% less_natural.abs_eq
tff(fact_7725_log_Opelims,axiom,
    ! [Xa: code_natural,Xaa: code_natural,Y: code_natural] :
      ( ( log(Xa,Xaa) = 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),Xa),Xaa))
       => ~ ( ( Y = $ite(
                  ( aa(code_natural,$o,aa(code_natural,fun(code_natural,$o),ord_less_eq(code_natural),Xa),one_one(code_natural))
                  | aa(code_natural,$o,aa(code_natural,fun(code_natural,$o),ord_less(code_natural),Xaa),Xa) ),
                  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(Xa,divide_divide(code_natural,Xaa,Xa))) ) )
           => ~ 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),Xa),Xaa)) ) ) ) ).

% log.pelims
tff(fact_7726_iterate_Opelims,axiom,
    ! [A: $tType,B: $tType,Xa: code_natural,Xaa: 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,Xa,Xaa),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)),Xa),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),Xaa),Xb)))
       => ~ ( ( Y = $ite(Xa = 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)),Xaa,Xb),iterate(B,A,aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),minus_minus(code_natural),Xa),one_one(code_natural)),Xaa))) )
           => ~ 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)),Xa),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),Xaa),Xb))) ) ) ) ).

% iterate.pelims
tff(fact_7727_iterate_Ocases,axiom,
    ! [B: $tType,A: $tType,Xa: product_prod(code_natural,product_prod(fun(A,fun(B,product_prod(A,B))),A))] :
      ~ ! [K: code_natural,F3: fun(A,fun(B,product_prod(A,B))),X: A] : ( Xa != 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)),K),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),F3),X)) ) ).

% iterate.cases
tff(fact_7728_iter_H_Ocases,axiom,
    ! [A: $tType] :
      ( quickcheck_random(A)
     => ! [Xa: 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)] : ( Xa != 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_7729_in__range_Osimps,axiom,
    ! [H2: heap_ext(product_unit),As2: set(nat)] :
      ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As2))
    <=> ! [X6: nat] :
          ( member(nat,X6,As2)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X6),lim(product_unit,H2)) ) ) ).

% in_range.simps
tff(fact_7730_in__range_Oelims_I3_J,axiom,
    ! [Xa: product_prod(heap_ext(product_unit),set(nat))] :
      ( ~ aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,Xa)
     => ~ ! [H: heap_ext(product_unit),As: set(nat)] :
            ( ( Xa = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As) )
           => ! [X: nat] :
                ( member(nat,X,As)
               => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),lim(product_unit,H)) ) ) ) ).

% in_range.elims(3)
tff(fact_7731_in__range_Oelims_I2_J,axiom,
    ! [Xa: product_prod(heap_ext(product_unit),set(nat))] :
      ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,Xa)
     => ~ ! [H: heap_ext(product_unit),As: set(nat)] :
            ( ( Xa = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As) )
           => ~ ! [X4: nat] :
                  ( member(nat,X4,As)
                 => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X4),lim(product_unit,H)) ) ) ) ).

% in_range.elims(2)
tff(fact_7732_in__range_Oelims_I1_J,axiom,
    ! [Xa: product_prod(heap_ext(product_unit),set(nat)),Y: $o] :
      ( ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,Xa)
      <=> (Y) )
     => ~ ! [H: heap_ext(product_unit),As: 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),As) )
           => ( (Y)
            <=> ~ ! [X6: nat] :
                    ( member(nat,X6,As)
                   => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X6),lim(product_unit,H)) ) ) ) ) ).

% in_range.elims(1)
tff(fact_7733_new__addrs__def,axiom,
    ! [H2: heap_ext(product_unit),As2: set(nat),H4: heap_ext(product_unit)] : ( hoare_new_addrs(H2,As2,H4) = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As2),aa(fun(nat,$o),set(nat),collect(nat),aa(heap_ext(product_unit),fun(nat,$o),aTP_Lamp_ada(heap_ext(product_unit),fun(heap_ext(product_unit),fun(nat,$o)),H2),H4))) ) ).

% new_addrs_def
tff(fact_7734_sngr__assn__raw_Osimps,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [R2: ref(A),Xa: A,H2: heap_ext(product_unit),As2: set(nat)] :
          ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,sngr_assn_raw(A,R2,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)),H2),As2))
        <=> ( ( ref_get(A,H2,R2) = Xa )
            & ( As2 = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert3(nat),addr_of_ref(A,R2)),bot_bot(set(nat))) )
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),addr_of_ref(A,R2)),lim(product_unit,H2)) ) ) ) ).

% sngr_assn_raw.simps
tff(fact_7735_relH__ref,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [As2: set(nat),H2: heap_ext(product_unit),H4: heap_ext(product_unit),R2: ref(A)] :
          ( relH(As2,H2,H4)
         => ( member(nat,addr_of_ref(A,R2),As2)
           => ( ref_get(A,H2,R2) = ref_get(A,H4,R2) ) ) ) ) ).

% relH_ref
tff(fact_7736_sngr__assn__raw_Oelims_I3_J,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Xa: ref(A),Xaa: A,Xb: product_prod(heap_ext(product_unit),set(nat))] :
          ( ~ aa(product_prod(heap_ext(product_unit),set(nat)),$o,sngr_assn_raw(A,Xa,Xaa),Xb)
         => ~ ! [H: heap_ext(product_unit),As: 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),As) )
               => ( ( ref_get(A,H,Xa) = Xaa )
                  & ( As = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert3(nat),addr_of_ref(A,Xa)),bot_bot(set(nat))) )
                  & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),addr_of_ref(A,Xa)),lim(product_unit,H)) ) ) ) ) ).

% sngr_assn_raw.elims(3)
tff(fact_7737_sngr__assn__raw_Oelims_I2_J,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Xa: ref(A),Xaa: A,Xb: product_prod(heap_ext(product_unit),set(nat))] :
          ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,sngr_assn_raw(A,Xa,Xaa),Xb)
         => ~ ! [H: heap_ext(product_unit),As: 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),As) )
               => ~ ( ( ref_get(A,H,Xa) = Xaa )
                    & ( As = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert3(nat),addr_of_ref(A,Xa)),bot_bot(set(nat))) )
                    & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),addr_of_ref(A,Xa)),lim(product_unit,H)) ) ) ) ) ).

% sngr_assn_raw.elims(2)
tff(fact_7738_sngr__assn__raw_Oelims_I1_J,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Xa: ref(A),Xaa: A,Xb: product_prod(heap_ext(product_unit),set(nat)),Y: $o] :
          ( ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,sngr_assn_raw(A,Xa,Xaa),Xb)
          <=> (Y) )
         => ~ ! [H: heap_ext(product_unit),As: 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),As) )
               => ( (Y)
                <=> ~ ( ( ref_get(A,H,Xa) = Xaa )
                      & ( As = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert3(nat),addr_of_ref(A,Xa)),bot_bot(set(nat))) )
                      & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),addr_of_ref(A,Xa)),lim(product_unit,H)) ) ) ) ) ) ).

% sngr_assn_raw.elims(1)
tff(fact_7739_sngr__assn__raw_Opelims_I3_J,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Xa: ref(A),Xaa: A,Xb: product_prod(heap_ext(product_unit),set(nat))] :
          ( ~ aa(product_prod(heap_ext(product_unit),set(nat)),$o,sngr_assn_raw(A,Xa,Xaa),Xb)
         => ( accp(product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),sngr_assn_raw_rel(A),aa(product_prod(A,product_prod(heap_ext(product_unit),set(nat))),product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),aa(ref(A),fun(product_prod(A,product_prod(heap_ext(product_unit),set(nat))),product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat))))),product_Pair(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),Xa),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(A,product_prod(heap_ext(product_unit),set(nat))),aa(A,fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),product_Pair(A,product_prod(heap_ext(product_unit),set(nat))),Xaa),Xb)))
           => ~ ! [H: heap_ext(product_unit),As: 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),As) )
                 => ( accp(product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),sngr_assn_raw_rel(A),aa(product_prod(A,product_prod(heap_ext(product_unit),set(nat))),product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),aa(ref(A),fun(product_prod(A,product_prod(heap_ext(product_unit),set(nat))),product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat))))),product_Pair(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),Xa),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(A,product_prod(heap_ext(product_unit),set(nat))),aa(A,fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),product_Pair(A,product_prod(heap_ext(product_unit),set(nat))),Xaa),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),As))))
                   => ( ( ref_get(A,H,Xa) = Xaa )
                      & ( As = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert3(nat),addr_of_ref(A,Xa)),bot_bot(set(nat))) )
                      & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),addr_of_ref(A,Xa)),lim(product_unit,H)) ) ) ) ) ) ) ).

% sngr_assn_raw.pelims(3)
tff(fact_7740_sngr__assn__raw_Opelims_I2_J,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Xa: ref(A),Xaa: A,Xb: product_prod(heap_ext(product_unit),set(nat))] :
          ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,sngr_assn_raw(A,Xa,Xaa),Xb)
         => ( accp(product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),sngr_assn_raw_rel(A),aa(product_prod(A,product_prod(heap_ext(product_unit),set(nat))),product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),aa(ref(A),fun(product_prod(A,product_prod(heap_ext(product_unit),set(nat))),product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat))))),product_Pair(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),Xa),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(A,product_prod(heap_ext(product_unit),set(nat))),aa(A,fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),product_Pair(A,product_prod(heap_ext(product_unit),set(nat))),Xaa),Xb)))
           => ~ ! [H: heap_ext(product_unit),As: 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),As) )
                 => ( accp(product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),sngr_assn_raw_rel(A),aa(product_prod(A,product_prod(heap_ext(product_unit),set(nat))),product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),aa(ref(A),fun(product_prod(A,product_prod(heap_ext(product_unit),set(nat))),product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat))))),product_Pair(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),Xa),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(A,product_prod(heap_ext(product_unit),set(nat))),aa(A,fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),product_Pair(A,product_prod(heap_ext(product_unit),set(nat))),Xaa),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),As))))
                   => ~ ( ( ref_get(A,H,Xa) = Xaa )
                        & ( As = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert3(nat),addr_of_ref(A,Xa)),bot_bot(set(nat))) )
                        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),addr_of_ref(A,Xa)),lim(product_unit,H)) ) ) ) ) ) ) ).

% sngr_assn_raw.pelims(2)
tff(fact_7741_sngr__assn__raw_Opelims_I1_J,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Xa: ref(A),Xaa: A,Xb: product_prod(heap_ext(product_unit),set(nat)),Y: $o] :
          ( ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,sngr_assn_raw(A,Xa,Xaa),Xb)
          <=> (Y) )
         => ( accp(product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),sngr_assn_raw_rel(A),aa(product_prod(A,product_prod(heap_ext(product_unit),set(nat))),product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),aa(ref(A),fun(product_prod(A,product_prod(heap_ext(product_unit),set(nat))),product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat))))),product_Pair(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),Xa),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(A,product_prod(heap_ext(product_unit),set(nat))),aa(A,fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),product_Pair(A,product_prod(heap_ext(product_unit),set(nat))),Xaa),Xb)))
           => ~ ! [H: heap_ext(product_unit),As: 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),As) )
                 => ( ( (Y)
                    <=> ( ( ref_get(A,H,Xa) = Xaa )
                        & ( As = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert3(nat),addr_of_ref(A,Xa)),bot_bot(set(nat))) )
                        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),addr_of_ref(A,Xa)),lim(product_unit,H)) ) )
                   => ~ accp(product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),sngr_assn_raw_rel(A),aa(product_prod(A,product_prod(heap_ext(product_unit),set(nat))),product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),aa(ref(A),fun(product_prod(A,product_prod(heap_ext(product_unit),set(nat))),product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat))))),product_Pair(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),Xa),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(A,product_prod(heap_ext(product_unit),set(nat))),aa(A,fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),product_Pair(A,product_prod(heap_ext(product_unit),set(nat))),Xaa),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),As)))) ) ) ) ) ) ).

% sngr_assn_raw.pelims(1)
tff(fact_7742_relH__set__ref,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [R2: ref(A),As2: set(nat),H2: heap_ext(product_unit),Xa: A] :
          ( ~ member(nat,addr_of_ref(A,R2),As2)
         => ( aa(product_prod(heap_ext(product_unit),set(nat)),$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),As2))
           => relH(As2,H2,ref_set(A,R2,Xa,H2)) ) ) ) ).

% relH_set_ref
tff(fact_7743_nth__rule,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Ia: nat,Xsa: list(A),A3: array(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),aa(list(A),nat,size_size(list(A)),Xsa))
         => hoare_hoare_triple(A,snga_assn(A,A3,Xsa),array_nth(A,A3,Ia),aa(array(A),fun(A,assn),aa(list(A),fun(array(A),fun(A,assn)),aTP_Lamp_adb(nat,fun(list(A),fun(array(A),fun(A,assn))),Ia),Xsa),A3)) ) ) ).

% nth_rule
tff(fact_7744_false__rule,axiom,
    ! [A: $tType,C2: heap_Time_Heap(A),Qa: fun(A,assn)] : hoare_hoare_triple(A,bot_bot(assn),C2,Qa) ).

% false_rule
tff(fact_7745_norm__pre__pure__iff,axiom,
    ! [A: $tType,Pa: assn,B3: $o,F2: heap_Time_Heap(A),Qa: fun(A,assn)] :
      ( hoare_hoare_triple(A,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Pa),pure_assn((B3))),F2,Qa)
    <=> ( (B3)
       => hoare_hoare_triple(A,Pa,F2,Qa) ) ) ).

% norm_pre_pure_iff
tff(fact_7746_norm__pre__pure__iff__sng,axiom,
    ! [A: $tType,B3: $o,F2: heap_Time_Heap(A),Qa: fun(A,assn)] :
      ( hoare_hoare_triple(A,pure_assn((B3)),F2,Qa)
    <=> ( (B3)
       => hoare_hoare_triple(A,one_one(assn),F2,Qa) ) ) ).

% norm_pre_pure_iff_sng
tff(fact_7747_is__hoare__triple,axiom,
    ! [A: $tType,Pa: assn,C2: heap_Time_Heap(A),Qa: fun(A,assn)] :
      ( hoare_hoare_triple(A,Pa,C2,Qa)
     => hoare_hoare_triple(A,Pa,C2,Qa) ) ).

% is_hoare_triple
tff(fact_7748_frame__rule__left,axiom,
    ! [A: $tType,Pa: assn,C2: heap_Time_Heap(A),Qa: fun(A,assn),R: assn] :
      ( hoare_hoare_triple(A,Pa,C2,Qa)
     => hoare_hoare_triple(A,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),R),Pa),C2,aa(assn,fun(A,assn),aTP_Lamp_adc(fun(A,assn),fun(assn,fun(A,assn)),Qa),R)) ) ).

% frame_rule_left
tff(fact_7749_fi__rule,axiom,
    ! [A: $tType,Pa: assn,C2: heap_Time_Heap(A),Qa: fun(A,assn),Ps: assn,Fa: assn] :
      ( hoare_hoare_triple(A,Pa,C2,Qa)
     => ( aa(assn,$o,aa(assn,fun(assn,$o),entails,Ps),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Pa),Fa))
       => hoare_hoare_triple(A,Ps,C2,aa(assn,fun(A,assn),aTP_Lamp_zp(fun(A,assn),fun(assn,fun(A,assn)),Qa),Fa)) ) ) ).

% fi_rule
tff(fact_7750_frame__rule,axiom,
    ! [A: $tType,Pa: assn,C2: heap_Time_Heap(A),Qa: fun(A,assn),R: assn] :
      ( hoare_hoare_triple(A,Pa,C2,Qa)
     => hoare_hoare_triple(A,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Pa),R),C2,aa(assn,fun(A,assn),aTP_Lamp_zp(fun(A,assn),fun(assn,fun(A,assn)),Qa),R)) ) ).

% frame_rule
tff(fact_7751_norm__pre__pure__rule1,axiom,
    ! [A: $tType,B3: $o,Pa: assn,F2: heap_Time_Heap(A),Qa: fun(A,assn)] :
      ( ( (B3)
       => hoare_hoare_triple(A,Pa,F2,Qa) )
     => hoare_hoare_triple(A,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Pa),pure_assn((B3))),F2,Qa) ) ).

% norm_pre_pure_rule1
tff(fact_7752_cons__post__rule,axiom,
    ! [A: $tType,Pa: assn,C2: heap_Time_Heap(A),Qa: fun(A,assn),Q: fun(A,assn)] :
      ( hoare_hoare_triple(A,Pa,C2,Qa)
     => ( ! [X: A] : aa(assn,$o,aa(assn,fun(assn,$o),entails,aa(A,assn,Qa,X)),aa(A,assn,Q,X))
       => hoare_hoare_triple(A,Pa,C2,Q) ) ) ).

% cons_post_rule
tff(fact_7753_cons__rule,axiom,
    ! [A: $tType,Pa: assn,P: assn,Qa: fun(A,assn),Q: fun(A,assn),C2: heap_Time_Heap(A)] :
      ( aa(assn,$o,aa(assn,fun(assn,$o),entails,Pa),P)
     => ( ! [X: A] : aa(assn,$o,aa(assn,fun(assn,$o),entails,aa(A,assn,Qa,X)),aa(A,assn,Q,X))
       => ( hoare_hoare_triple(A,P,C2,Qa)
         => hoare_hoare_triple(A,Pa,C2,Q) ) ) ) ).

% cons_rule
tff(fact_7754_hoare__triple__preI,axiom,
    ! [A: $tType,Pa: assn,C2: heap_Time_Heap(A),Qa: fun(A,assn)] :
      ( ! [H: product_prod(heap_ext(product_unit),set(nat))] :
          ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,Pa),H)
         => hoare_hoare_triple(A,Pa,C2,Qa) )
     => hoare_hoare_triple(A,Pa,C2,Qa) ) ).

% hoare_triple_preI
tff(fact_7755_cons__pre__rule,axiom,
    ! [A: $tType,Pa: assn,P: assn,C2: heap_Time_Heap(A),Qa: fun(A,assn)] :
      ( aa(assn,$o,aa(assn,fun(assn,$o),entails,Pa),P)
     => ( hoare_hoare_triple(A,P,C2,Qa)
       => hoare_hoare_triple(A,Pa,C2,Qa) ) ) ).

% cons_pre_rule
tff(fact_7756_norm__pre__pure__rule2,axiom,
    ! [A: $tType,B3: $o,F2: heap_Time_Heap(A),Qa: fun(A,assn)] :
      ( ( (B3)
       => hoare_hoare_triple(A,one_one(assn),F2,Qa) )
     => hoare_hoare_triple(A,pure_assn((B3)),F2,Qa) ) ).

% norm_pre_pure_rule2
tff(fact_7757_case__prod__rule,axiom,
    ! [A: $tType,B: $tType,E: $tType,Xa: product_prod(A,B),Pa: assn,F2: fun(A,fun(B,heap_Time_Heap(E))),Qa: fun(E,assn)] :
      ( ! [A4: A,B4: B] :
          ( ( Xa = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B4) )
         => hoare_hoare_triple(E,Pa,aa(B,heap_Time_Heap(E),aa(A,fun(B,heap_Time_Heap(E)),F2,A4),B4),Qa) )
     => hoare_hoare_triple(E,Pa,aa(product_prod(A,B),heap_Time_Heap(E),aa(fun(A,fun(B,heap_Time_Heap(E))),fun(product_prod(A,B),heap_Time_Heap(E)),product_case_prod(A,B,heap_Time_Heap(E)),F2),Xa),Qa) ) ).

% case_prod_rule
tff(fact_7758_cons__rulet,axiom,
    ! [A: $tType,Pa: assn,P: assn,Qa: fun(A,assn),Q: fun(A,assn),C2: heap_Time_Heap(A)] :
      ( entailst(Pa,P)
     => ( ! [X: A] : entailst(aa(A,assn,Qa,X),aa(A,assn,Q,X))
       => ( hoare_hoare_triple(A,P,C2,aTP_Lamp_add(fun(A,assn),fun(A,assn),Qa))
         => hoare_hoare_triple(A,Pa,C2,aTP_Lamp_add(fun(A,assn),fun(A,assn),Q)) ) ) ) ).

% cons_rulet
tff(fact_7759_cons__pre__rulet,axiom,
    ! [A: $tType,Pa: assn,P: assn,C2: heap_Time_Heap(A),Qa: fun(A,assn)] :
      ( entailst(Pa,P)
     => ( hoare_hoare_triple(A,P,C2,aTP_Lamp_add(fun(A,assn),fun(A,assn),Qa))
       => hoare_hoare_triple(A,Pa,C2,aTP_Lamp_add(fun(A,assn),fun(A,assn),Qa)) ) ) ).

% cons_pre_rulet
tff(fact_7760_cons__post__rulet,axiom,
    ! [A: $tType,Pa: assn,C2: heap_Time_Heap(A),Qa: fun(A,assn),Q: fun(A,assn)] :
      ( hoare_hoare_triple(A,Pa,C2,aTP_Lamp_add(fun(A,assn),fun(A,assn),Qa))
     => ( ! [X: A] : entailst(aa(A,assn,Qa,X),aa(A,assn,Q,X))
       => hoare_hoare_triple(A,Pa,C2,aTP_Lamp_add(fun(A,assn),fun(A,assn),Q)) ) ) ).

% cons_post_rulet
tff(fact_7761_if__rule__split,axiom,
    ! [A: $tType,B3: $o,Pa: assn,F2: heap_Time_Heap(A),Q1: fun(A,assn),G: heap_Time_Heap(A),Q22: fun(A,assn),Qa: fun(A,assn)] :
      ( ( (B3)
       => hoare_hoare_triple(A,Pa,F2,Q1) )
     => ( ( ~ (B3)
         => hoare_hoare_triple(A,Pa,G,Q22) )
       => ( ! [X: A] : aa(assn,$o,aa(assn,fun(assn,$o),entails,aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(A,assn,Q1,X)),pure_assn((B3)))),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(A,assn,Q22,X)),pure_assn(~ (B3))))),aa(A,assn,Qa,X))
         => hoare_hoare_triple(A,Pa,
              $ite((B3),F2,G),
              Qa) ) ) ) ).

% if_rule_split
tff(fact_7762_upd__rule,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Ia: nat,Xsa: list(A),A3: array(A),Xa: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),aa(list(A),nat,size_size(list(A)),Xsa))
         => hoare_hoare_triple(array(A),snga_assn(A,A3,Xsa),array_upd(A,Ia,Xa,A3),aa(A,fun(array(A),assn),aa(array(A),fun(A,fun(array(A),assn)),aa(list(A),fun(array(A),fun(A,fun(array(A),assn))),aTP_Lamp_ade(nat,fun(list(A),fun(array(A),fun(A,fun(array(A),assn)))),Ia),Xsa),A3),Xa)) ) ) ).

% upd_rule
tff(fact_7763_make__rule,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [N: nat,F2: fun(nat,A)] : hoare_hoare_triple(array(A),one_one(assn),array_make(A,N,F2),aa(fun(nat,A),fun(array(A),assn),aTP_Lamp_adf(nat,fun(fun(nat,A),fun(array(A),assn)),N),F2)) ) ).

% make_rule
tff(fact_7764_new__rule,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [N: nat,Xa: A] : hoare_hoare_triple(array(A),one_one(assn),array_new(A,N,Xa),aa(A,fun(array(A),assn),aTP_Lamp_adg(nat,fun(A,fun(array(A),assn)),N),Xa)) ) ).

% new_rule
tff(fact_7765_length__rule,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [A3: array(A),Xsa: list(A)] : hoare_hoare_triple(nat,snga_assn(A,A3,Xsa),array_len(A,A3),aa(list(A),fun(nat,assn),aTP_Lamp_adh(array(A),fun(list(A),fun(nat,assn)),A3),Xsa)) ) ).

% length_rule
tff(fact_7766_update__wp__rule,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [R2: ref(A),Y: A,Xa: A,Qa: fun(product_unit,assn)] : hoare_hoare_triple(product_unit,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),sngr_assn(A,R2,Y)),wand_assn(sngr_assn(A,R2,Xa),aa(product_unit,assn,Qa,product_Unity))),ref_update(A,R2,Xa),Qa) ) ).

% update_wp_rule
tff(fact_7767_lookup__rule,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [P2: ref(A),Xa: A] : hoare_hoare_triple(A,sngr_assn(A,P2,Xa),ref_lookup(A,P2),aa(A,fun(A,assn),aTP_Lamp_adi(ref(A),fun(A,fun(A,assn)),P2),Xa)) ) ).

% lookup_rule
tff(fact_7768_update__rule,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [P2: ref(A),Y: A,Xa: A] : hoare_hoare_triple(product_unit,sngr_assn(A,P2,Y),ref_update(A,P2,Xa),aa(A,fun(product_unit,assn),aTP_Lamp_adj(ref(A),fun(A,fun(product_unit,assn)),P2),Xa)) ) ).

% update_rule
tff(fact_7769_hoare__triple__def,axiom,
    ! [A: $tType,Pa: assn,C2: heap_Time_Heap(A),Qa: fun(A,assn)] :
      ( hoare_hoare_triple(A,Pa,C2,Qa)
    <=> ! [H3: heap_ext(product_unit),As8: set(nat)] :
          ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,Pa),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H3),As8))
         => ? [R6: A,H8: heap_ext(product_unit)] :
              ( ? [T3: nat] : ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,C2),H3) = aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),R6),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),H8),T3))) )
              & aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,aa(A,assn,Qa,R6)),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H8),hoare_new_addrs(H3,As8,H8)))
              & relH(aa(fun(nat,$o),set(nat),collect(nat),aa(set(nat),fun(nat,$o),aTP_Lamp_adk(heap_ext(product_unit),fun(set(nat),fun(nat,$o)),H3),As8)),H3,H8)
              & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),lim(product_unit,H3)),lim(product_unit,H8)) ) ) ) ).

% hoare_triple_def
tff(fact_7770_hoare__tripleI,axiom,
    ! [A: $tType,Pa: assn,C2: heap_Time_Heap(A),Qa: fun(A,assn)] :
      ( ! [H: heap_ext(product_unit),As: set(nat)] :
          ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,Pa),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As))
         => ? [R14: A,H9: heap_ext(product_unit),T7: nat] :
              ( ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,C2),H) = aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),R14),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),H9),T7))) )
              & aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,aa(A,assn,Qa,R14)),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H9),hoare_new_addrs(H,As,H9)))
              & relH(aa(fun(nat,$o),set(nat),collect(nat),aa(set(nat),fun(nat,$o),aTP_Lamp_adk(heap_ext(product_unit),fun(set(nat),fun(nat,$o)),H),As)),H,H9)
              & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),lim(product_unit,H)),lim(product_unit,H9)) ) )
     => hoare_hoare_triple(A,Pa,C2,Qa) ) ).

% hoare_tripleI
tff(fact_7771_Heap__cases,axiom,
    ! [A: $tType,F2: heap_Time_Heap(A),H2: heap_ext(product_unit)] :
      ( ! [X: A,H6: product_prod(heap_ext(product_unit),nat)] : ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,F2),H2) != aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),X),H6)) )
     => ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,F2),H2) = none(product_prod(A,product_prod(heap_ext(product_unit),nat))) ) ) ).

% Heap_cases
tff(fact_7772_timeFrame_Ocases,axiom,
    ! [A: $tType,Xa: product_prod(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat))))] :
      ( ! [N5: nat,R3: A,H: heap_ext(product_unit),N4: nat] : ( Xa != aa(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),product_prod(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),aa(nat,fun(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),product_prod(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat))))),product_Pair(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),N5),aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),R3),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),H),N4)))) )
     => ~ ! [N5: nat] : ( Xa != aa(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),product_prod(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),aa(nat,fun(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),product_prod(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat))))),product_Pair(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),N5),none(product_prod(A,product_prod(heap_ext(product_unit),nat)))) ) ) ).

% timeFrame.cases
tff(fact_7773_hoare__tripleE,axiom,
    ! [A: $tType,Pa: assn,C2: heap_Time_Heap(A),Qa: fun(A,assn),H2: heap_ext(product_unit),As2: set(nat)] :
      ( hoare_hoare_triple(A,Pa,C2,Qa)
     => ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,Pa),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As2))
       => ~ ! [R3: A,H6: heap_ext(product_unit)] :
              ( ? [T8: nat] : ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,C2),H2) = aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),R3),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),H6),T8))) )
             => ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,aa(A,assn,Qa,R3)),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H6),hoare_new_addrs(H2,As2,H6)))
               => ( relH(aa(fun(nat,$o),set(nat),collect(nat),aa(set(nat),fun(nat,$o),aTP_Lamp_adk(heap_ext(product_unit),fun(set(nat),fun(nat,$o)),H2),As2)),H2,H6)
                 => ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),lim(product_unit,H2)),lim(product_unit,H6)) ) ) ) ) ) ).

% hoare_tripleE
tff(fact_7774_execute__lookup,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [R2: ref(A),H2: heap_ext(product_unit)] : ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,ref_lookup(A,R2)),H2) = aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),ref_get(A,H2,R2)),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),H2),one_one(nat)))) ) ) ).

% execute_lookup
tff(fact_7775_execute__update,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [R2: ref(A),V2: A,H2: heap_ext(product_unit)] : ( aa(heap_ext(product_unit),option(product_prod(product_unit,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(product_unit,ref_update(A,R2,V2)),H2) = aa(product_prod(product_unit,product_prod(heap_ext(product_unit),nat)),option(product_prod(product_unit,product_prod(heap_ext(product_unit),nat))),some(product_prod(product_unit,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(product_unit,product_prod(heap_ext(product_unit),nat)),aa(product_unit,fun(product_prod(heap_ext(product_unit),nat),product_prod(product_unit,product_prod(heap_ext(product_unit),nat))),product_Pair(product_unit,product_prod(heap_ext(product_unit),nat)),product_Unity),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),ref_set(A,R2,V2,H2)),one_one(nat)))) ) ) ).

% execute_update
tff(fact_7776_execute__change,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [F2: fun(A,A),R2: ref(A),H2: heap_ext(product_unit)] : ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,ref_change(A,F2,R2)),H2) = aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),aa(A,A,F2,ref_get(A,H2,R2))),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),ref_set(A,R2,aa(A,A,F2,ref_get(A,H2,R2)),H2)),numeral_numeral(nat,bit1(one2))))) ) ) ).

% execute_change
tff(fact_7777_Ref__Time_Oupdate__def,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [R2: ref(A),V2: A] : ( ref_update(A,R2,V2) = heap_Time_heap(product_unit,aa(A,fun(heap_ext(product_unit),product_prod(product_unit,product_prod(heap_ext(product_unit),nat))),aTP_Lamp_adl(ref(A),fun(A,fun(heap_ext(product_unit),product_prod(product_unit,product_prod(heap_ext(product_unit),nat)))),R2),V2)) ) ) ).

% Ref_Time.update_def
tff(fact_7778_execute__tap,axiom,
    ! [A: $tType,F2: fun(heap_ext(product_unit),A),H2: heap_ext(product_unit)] : ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,heap_Time_tap(A,F2)),H2) = aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),aa(heap_ext(product_unit),A,F2,H2)),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),H2),one_one(nat)))) ) ).

% execute_tap
tff(fact_7779_execute__assert_H,axiom,
    ! [Pa: $o,H2: heap_ext(product_unit)] :
      ( aa(heap_ext(product_unit),option(product_prod(product_unit,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(product_unit,refine_Imp_assert((Pa))),H2) = $ite((Pa),aa(product_prod(product_unit,product_prod(heap_ext(product_unit),nat)),option(product_prod(product_unit,product_prod(heap_ext(product_unit),nat))),some(product_prod(product_unit,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(product_unit,product_prod(heap_ext(product_unit),nat)),aa(product_unit,fun(product_prod(heap_ext(product_unit),nat),product_prod(product_unit,product_prod(heap_ext(product_unit),nat))),product_Pair(product_unit,product_prod(heap_ext(product_unit),nat)),product_Unity),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),H2),zero_zero(nat)))),none(product_prod(product_unit,product_prod(heap_ext(product_unit),nat)))) ) ).

% execute_assert'
tff(fact_7780_assert_H__rule,axiom,
    ! [Pa: assn,Phi2: $o] :
      ( ! [H: product_prod(heap_ext(product_unit),set(nat))] :
          ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,Pa),H)
         => (Phi2) )
     => hoare_hoare_triple(product_unit,Pa,refine_Imp_assert((Phi2)),aTP_Lamp_adm(assn,fun(product_unit,assn),Pa)) ) ).

% assert'_rule
tff(fact_7781_execute__ureturn,axiom,
    ! [A: $tType,Xa: A] : ( heap_Time_execute(A,heap_Time_ureturn(A,Xa)) = comp(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_ext(product_unit),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aTP_Lamp_adn(A,fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),Xa)) ) ).

% execute_ureturn
tff(fact_7782_execute__assert_I1_J,axiom,
    ! [A: $tType,Pa: fun(A,$o),Xa: A,H2: heap_ext(product_unit)] :
      ( aa(A,$o,Pa,Xa)
     => ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,heap_Time_assert(A,Pa,Xa)),H2) = aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),Xa),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),H2),one_one(nat)))) ) ) ).

% execute_assert(1)
tff(fact_7783_ureturn__def,axiom,
    ! [A: $tType,Xa: A] : ( heap_Time_ureturn(A,Xa) = heap_Time_heap(A,aTP_Lamp_adn(A,fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),Xa)) ) ).

% ureturn_def
tff(fact_7784_execute__return,axiom,
    ! [A: $tType,Xa: A] : ( heap_Time_execute(A,heap_Time_return(A,Xa)) = comp(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_ext(product_unit),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aTP_Lamp_ado(A,fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),Xa)) ) ).

% execute_return
tff(fact_7785_of__list__rule,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Xsa: list(A)] : hoare_hoare_triple(array(A),one_one(assn),array_of_list(A,Xsa),aa(list(A),fun(array(A),assn),aTP_Lamp_ci(list(A),fun(array(A),assn)),Xsa)) ) ).

% of_list_rule
tff(fact_7786_return__cons__rule,axiom,
    ! [A: $tType,Pa: assn,Qa: fun(A,assn),Xa: A] :
      ( aa(assn,$o,aa(assn,fun(assn,$o),entails,Pa),aa(A,assn,Qa,Xa))
     => hoare_hoare_triple(A,Pa,heap_Time_return(A,Xa),Qa) ) ).

% return_cons_rule
tff(fact_7787_return__sp__rule,axiom,
    ! [A: $tType,Pa: assn,Xa: A] : hoare_hoare_triple(A,Pa,heap_Time_return(A,Xa),aa(A,fun(A,assn),aTP_Lamp_adp(assn,fun(A,fun(A,assn)),Pa),Xa)) ).

% return_sp_rule
tff(fact_7788_return__def,axiom,
    ! [A: $tType,Xa: A] : ( heap_Time_return(A,Xa) = heap_Time_heap(A,aTP_Lamp_ado(A,fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),Xa)) ) ).

% return_def
tff(fact_7789_array__of__list__make,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Xsa: list(A)] : ( array_of_list(A,Xsa) = array_make(A,aa(list(A),nat,size_size(list(A)),Xsa),nth(A,Xsa)) ) ) ).

% array_of_list_make
tff(fact_7790_timeFrame_Oelims,axiom,
    ! [A: $tType,Xa: nat,Xaa: option(product_prod(A,product_prod(heap_ext(product_unit),nat))),Y: option(product_prod(A,product_prod(heap_ext(product_unit),nat)))] :
      ( ( heap_Time_timeFrame(A,Xa,Xaa) = Y )
     => ( ! [R3: A,H: heap_ext(product_unit),N4: nat] :
            ( ( Xaa = aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),R3),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),H),N4))) )
           => ( Y != aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),R3),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),H),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Xa),N4)))) ) )
       => ~ ( ( Xaa = none(product_prod(A,product_prod(heap_ext(product_unit),nat))) )
           => ( Y != none(product_prod(A,product_prod(heap_ext(product_unit),nat))) ) ) ) ) ).

% timeFrame.elims
tff(fact_7791_timeFrame_Osimps_I1_J,axiom,
    ! [A: $tType,N: nat,R2: A,H2: heap_ext(product_unit),N6: nat] : ( heap_Time_timeFrame(A,N,aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),R2),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),H2),N6)))) = aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),R2),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),H2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),N6)))) ) ).

% timeFrame.simps(1)
tff(fact_7792_execute__make,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [N: nat,F2: fun(nat,A),H2: heap_ext(product_unit)] : ( aa(heap_ext(product_unit),option(product_prod(array(A),product_prod(heap_ext(product_unit),nat))),heap_Time_execute(array(A),array_make(A,N,F2)),H2) = aa(product_prod(array(A),product_prod(heap_ext(product_unit),nat)),option(product_prod(array(A),product_prod(heap_ext(product_unit),nat))),some(product_prod(array(A),product_prod(heap_ext(product_unit),nat))),aa(product_prod(array(A),heap_ext(product_unit)),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),aa(fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),fun(product_prod(array(A),heap_ext(product_unit)),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),product_case_prod(array(A),heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),aTP_Lamp_adq(nat,fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),N)),array_alloc(A,aa(list(nat),list(A),aa(fun(nat,A),fun(list(nat),list(A)),map(nat,A),F2),upt(zero_zero(nat),N)),H2))) ) ) ).

% execute_make
tff(fact_7793_Array__Time_Omake__def,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [N: nat,F2: fun(nat,A)] : ( array_make(A,N,F2) = heap_Time_heap(array(A),aa(fun(nat,A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),aTP_Lamp_adr(nat,fun(fun(nat,A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),N),F2)) ) ) ).

% Array_Time.make_def
tff(fact_7794_of__list__def,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Xsa: list(A)] : ( array_of_list(A,Xsa) = heap_Time_heap(array(A),aTP_Lamp_adt(list(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),Xsa)) ) ) ).

% of_list_def
tff(fact_7795_execute__of__list,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Xsa: list(A),H2: heap_ext(product_unit)] : ( aa(heap_ext(product_unit),option(product_prod(array(A),product_prod(heap_ext(product_unit),nat))),heap_Time_execute(array(A),array_of_list(A,Xsa)),H2) = aa(product_prod(array(A),product_prod(heap_ext(product_unit),nat)),option(product_prod(array(A),product_prod(heap_ext(product_unit),nat))),some(product_prod(array(A),product_prod(heap_ext(product_unit),nat))),aa(product_prod(array(A),heap_ext(product_unit)),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),aa(fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),fun(product_prod(array(A),heap_ext(product_unit)),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),product_case_prod(array(A),heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),aTP_Lamp_ads(list(A),fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),Xsa)),array_alloc(A,Xsa,H2))) ) ) ).

% execute_of_list
tff(fact_7796_new__def,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [N: nat,Xa: A] : ( array_new(A,N,Xa) = heap_Time_heap(array(A),aa(A,fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),aTP_Lamp_adu(nat,fun(A,fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),N),Xa)) ) ) ).

% new_def
tff(fact_7797_execute__new,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [N: nat,Xa: A,H2: heap_ext(product_unit)] : ( aa(heap_ext(product_unit),option(product_prod(array(A),product_prod(heap_ext(product_unit),nat))),heap_Time_execute(array(A),array_new(A,N,Xa)),H2) = aa(product_prod(array(A),product_prod(heap_ext(product_unit),nat)),option(product_prod(array(A),product_prod(heap_ext(product_unit),nat))),some(product_prod(array(A),product_prod(heap_ext(product_unit),nat))),aa(product_prod(array(A),heap_ext(product_unit)),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),aa(fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),fun(product_prod(array(A),heap_ext(product_unit)),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),product_case_prod(array(A),heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),aTP_Lamp_adq(nat,fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),N)),array_alloc(A,aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),Xa),H2))) ) ) ).

% execute_new
tff(fact_7798_freeze__rule,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [A3: array(A),Xsa: list(A)] : hoare_hoare_triple(list(A),snga_assn(A,A3,Xsa),array_freeze(A,A3),aa(list(A),fun(list(A),assn),aTP_Lamp_adv(array(A),fun(list(A),fun(list(A),assn)),A3),Xsa)) ) ).

% freeze_rule
tff(fact_7799_effect__makeI,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [A3: array(A),H4: heap_ext(product_unit),F2: fun(nat,A),N: nat,H2: heap_ext(product_unit)] :
          ( ( aa(heap_ext(product_unit),product_prod(array(A),heap_ext(product_unit)),aa(array(A),fun(heap_ext(product_unit),product_prod(array(A),heap_ext(product_unit))),product_Pair(array(A),heap_ext(product_unit)),A3),H4) = array_alloc(A,aa(list(nat),list(A),aa(fun(nat,A),fun(list(nat),list(A)),map(nat,A),F2),upt(zero_zero(nat),N)),H2) )
         => heap_Time_effect(array(A),array_make(A,N,F2),H2,H4,A3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))) ) ) ).

% effect_makeI
tff(fact_7800_effectI,axiom,
    ! [A: $tType,C2: heap_Time_Heap(A),H2: heap_ext(product_unit),R2: A,H4: heap_ext(product_unit),N: nat] :
      ( ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,C2),H2) = aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),R2),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),H4),N))) )
     => heap_Time_effect(A,C2,H2,H4,R2,N) ) ).

% effectI
tff(fact_7801_effect__def,axiom,
    ! [A: $tType,C2: heap_Time_Heap(A),H2: heap_ext(product_unit),H4: heap_ext(product_unit),R2: A,N: nat] :
      ( heap_Time_effect(A,C2,H2,H4,R2,N)
    <=> ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,C2),H2) = aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),R2),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),H4),N))) ) ) ).

% effect_def
tff(fact_7802_hoare__triple__effect,axiom,
    ! [A: $tType,Pa: assn,C2: heap_Time_Heap(A),Qa: fun(A,assn),H2: heap_ext(product_unit),As2: set(nat)] :
      ( hoare_hoare_triple(A,Pa,C2,Qa)
     => ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,Pa),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As2))
       => ? [H6: heap_ext(product_unit),R3: A,T8: nat] :
            ( heap_Time_effect(A,C2,H2,H6,R3,T8)
            & aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,aa(A,assn,Qa,R3)),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),hoare_new_addrs(H2,As2,H6))) ) ) ) ).

% hoare_triple_effect
tff(fact_7803_effect__of__listI,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [A3: array(A),H4: heap_ext(product_unit),Xsa: list(A),H2: heap_ext(product_unit)] :
          ( ( aa(heap_ext(product_unit),product_prod(array(A),heap_ext(product_unit)),aa(array(A),fun(heap_ext(product_unit),product_prod(array(A),heap_ext(product_unit))),product_Pair(array(A),heap_ext(product_unit)),A3),H4) = array_alloc(A,Xsa,H2) )
         => heap_Time_effect(array(A),array_of_list(A,Xsa),H2,H4,A3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(list(A),nat,size_size(list(A)),Xsa))) ) ) ).

% effect_of_listI
tff(fact_7804_effect__newI,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [A3: array(A),H4: heap_ext(product_unit),N: nat,Xa: A,H2: heap_ext(product_unit)] :
          ( ( aa(heap_ext(product_unit),product_prod(array(A),heap_ext(product_unit)),aa(array(A),fun(heap_ext(product_unit),product_prod(array(A),heap_ext(product_unit))),product_Pair(array(A),heap_ext(product_unit)),A3),H4) = array_alloc(A,aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),Xa),H2) )
         => heap_Time_effect(array(A),array_new(A,N,Xa),H2,H4,A3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))) ) ) ).

% effect_newI
tff(fact_7805_tap__def,axiom,
    ! [A: $tType,F2: fun(heap_ext(product_unit),A)] : ( heap_Time_tap(A,F2) = heap_Time_Heap2(A,aTP_Lamp_adw(fun(heap_ext(product_unit),A),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),F2)) ) ).

% tap_def
tff(fact_7806_timeFrame_Opelims,axiom,
    ! [A: $tType,Xa: nat,Xaa: option(product_prod(A,product_prod(heap_ext(product_unit),nat))),Y: option(product_prod(A,product_prod(heap_ext(product_unit),nat)))] :
      ( ( heap_Time_timeFrame(A,Xa,Xaa) = Y )
     => ( accp(product_prod(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_T5500966940807335491me_rel(A),aa(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),product_prod(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),aa(nat,fun(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),product_prod(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat))))),product_Pair(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),Xa),Xaa))
       => ( ! [R3: A,H: heap_ext(product_unit),N4: nat] :
              ( ( Xaa = aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),R3),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),H),N4))) )
             => ( ( Y = aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),R3),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),H),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Xa),N4)))) )
               => ~ accp(product_prod(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_T5500966940807335491me_rel(A),aa(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),product_prod(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),aa(nat,fun(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),product_prod(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat))))),product_Pair(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),Xa),aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),R3),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),H),N4))))) ) )
         => ~ ( ( Xaa = none(product_prod(A,product_prod(heap_ext(product_unit),nat))) )
             => ( ( Y = none(product_prod(A,product_prod(heap_ext(product_unit),nat))) )
               => ~ accp(product_prod(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_T5500966940807335491me_rel(A),aa(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),product_prod(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),aa(nat,fun(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),product_prod(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat))))),product_Pair(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),Xa),none(product_prod(A,product_prod(heap_ext(product_unit),nat))))) ) ) ) ) ) ).

% timeFrame.pelims
tff(fact_7807_wait__def,axiom,
    ! [N: nat] : ( heap_Time_wait(N) = heap_Time_Heap2(product_unit,aTP_Lamp_adx(nat,fun(heap_ext(product_unit),option(product_prod(product_unit,product_prod(heap_ext(product_unit),nat)))),N)) ) ).

% wait_def
tff(fact_7808_execute__bind_I1_J,axiom,
    ! [A: $tType,B: $tType,F2: heap_Time_Heap(A),H2: heap_ext(product_unit),Xa: A,H4: heap_ext(product_unit),N: nat,G: fun(A,heap_Time_Heap(B))] :
      ( ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,F2),H2) = aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),Xa),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),H4),N))) )
     => ( aa(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(B,heap_Time_bind(A,B,F2,G)),H2) = heap_Time_timeFrame(B,N,aa(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(B,aa(A,heap_Time_Heap(B),G,Xa)),H4)) ) ) ).

% execute_bind(1)
tff(fact_7809_bind__rule_H,axiom,
    ! [A: $tType,B: $tType,E: $tType,Pa: assn,F2: heap_Time_Heap(B),R: fun(B,assn),G: fun(B,heap_Time_Heap(E)),Qa: fun(E,assn)] :
      ( ! [R3: A] : hoare_hoare_triple(B,Pa,F2,aa(fun(B,assn),fun(B,assn),aTP_Lamp_ady(assn,fun(fun(B,assn),fun(B,assn)),Pa),R))
     => ( ! [X: B] : hoare_hoare_triple(E,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,R,X)),Pa),aa(B,heap_Time_Heap(E),G,X),Qa)
       => hoare_hoare_triple(E,Pa,heap_Time_bind(B,E,F2,G),Qa) ) ) ).

% bind_rule'
tff(fact_7810_assert_H__bind__rule,axiom,
    ! [A: $tType,Pa: assn,Phi2: $o,C2: heap_Time_Heap(A),Qa: fun(A,assn)] :
      ( ! [H: product_prod(heap_ext(product_unit),set(nat))] :
          ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,Pa),H)
         => (Phi2) )
     => ( ( (Phi2)
         => hoare_hoare_triple(A,Pa,C2,Qa) )
       => hoare_hoare_triple(A,Pa,heap_Time_bind(product_unit,A,refine_Imp_assert((Phi2)),aTP_Lamp_adz(heap_Time_Heap(A),fun(product_unit,heap_Time_Heap(A)),C2)),Qa) ) ) ).

% assert'_bind_rule
tff(fact_7811_execute__bind__eq__SomeI,axiom,
    ! [A: $tType,B: $tType,F2: heap_Time_Heap(A),H2: heap_ext(product_unit),Xa: A,H4: heap_ext(product_unit),N: nat,G: fun(A,heap_Time_Heap(B)),Y: B,H10: heap_ext(product_unit),N6: nat] :
      ( ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,F2),H2) = aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),Xa),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),H4),N))) )
     => ( ( aa(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(B,aa(A,heap_Time_Heap(B),G,Xa)),H4) = aa(product_prod(B,product_prod(heap_ext(product_unit),nat)),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),some(product_prod(B,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(B,product_prod(heap_ext(product_unit),nat)),aa(B,fun(product_prod(heap_ext(product_unit),nat),product_prod(B,product_prod(heap_ext(product_unit),nat))),product_Pair(B,product_prod(heap_ext(product_unit),nat)),Y),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),H10),N6))) )
       => ( aa(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(B,heap_Time_bind(A,B,F2,G)),H2) = aa(product_prod(B,product_prod(heap_ext(product_unit),nat)),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),some(product_prod(B,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(B,product_prod(heap_ext(product_unit),nat)),aa(B,fun(product_prod(heap_ext(product_unit),nat),product_prod(B,product_prod(heap_ext(product_unit),nat))),product_Pair(B,product_prod(heap_ext(product_unit),nat)),Y),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),H10),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),N6)))) ) ) ) ).

% execute_bind_eq_SomeI
tff(fact_7812_success__bind__executeI,axiom,
    ! [A: $tType,B: $tType,F2: heap_Time_Heap(A),H2: heap_ext(product_unit),Xa: A,H4: heap_ext(product_unit),N: nat,G: fun(A,heap_Time_Heap(B))] :
      ( ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,F2),H2) = aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),Xa),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),H4),N))) )
     => ( heap_Time_success(B,aa(A,heap_Time_Heap(B),G,Xa),H4)
       => heap_Time_success(B,heap_Time_bind(A,B,F2,G),H2) ) ) ).

% success_bind_executeI
tff(fact_7813_ref__rule,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Xa: A] : hoare_hoare_triple(ref(A),one_one(assn),ref_ref(A,Xa),aa(A,fun(ref(A),assn),aTP_Lamp_ch(A,fun(ref(A),assn)),Xa)) ) ).

% ref_rule
tff(fact_7814_hoare__triple__success,axiom,
    ! [A: $tType,Pa: assn,C2: heap_Time_Heap(A),Qa: fun(A,assn),H2: heap_ext(product_unit),As2: set(nat)] :
      ( hoare_hoare_triple(A,Pa,C2,Qa)
     => ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,Pa),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))
       => heap_Time_success(A,C2,H2) ) ) ).

% hoare_triple_success
tff(fact_7815_successE,axiom,
    ! [A: $tType,F2: heap_Time_Heap(A),H2: heap_ext(product_unit)] :
      ( heap_Time_success(A,F2,H2)
     => ~ ! [R3: A,H6: product_prod(heap_ext(product_unit),nat)] : ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,F2),H2) != aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),R3),H6)) ) ) ).

% successE
tff(fact_7816_execute__ref,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [V2: A,H2: heap_ext(product_unit)] : ( aa(heap_ext(product_unit),option(product_prod(ref(A),product_prod(heap_ext(product_unit),nat))),heap_Time_execute(ref(A),ref_ref(A,V2)),H2) = aa(product_prod(ref(A),product_prod(heap_ext(product_unit),nat)),option(product_prod(ref(A),product_prod(heap_ext(product_unit),nat))),some(product_prod(ref(A),product_prod(heap_ext(product_unit),nat))),aa(product_prod(ref(A),heap_ext(product_unit)),product_prod(ref(A),product_prod(heap_ext(product_unit),nat)),aa(fun(ref(A),fun(heap_ext(product_unit),product_prod(ref(A),product_prod(heap_ext(product_unit),nat)))),fun(product_prod(ref(A),heap_ext(product_unit)),product_prod(ref(A),product_prod(heap_ext(product_unit),nat))),product_case_prod(ref(A),heap_ext(product_unit),product_prod(ref(A),product_prod(heap_ext(product_unit),nat))),aTP_Lamp_aea(ref(A),fun(heap_ext(product_unit),product_prod(ref(A),product_prod(heap_ext(product_unit),nat))))),ref_alloc(A,V2,H2))) ) ) ).

% execute_ref
tff(fact_7817_ref__def,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [V2: A] : ( ref_ref(A,V2) = heap_Time_heap(ref(A),aTP_Lamp_aeb(A,fun(heap_ext(product_unit),product_prod(ref(A),product_prod(heap_ext(product_unit),nat))),V2)) ) ) ).

% ref_def
tff(fact_7818_effect__refI,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [R2: ref(A),H4: heap_ext(product_unit),V2: A,H2: heap_ext(product_unit),N: nat] :
          ( ( aa(heap_ext(product_unit),product_prod(ref(A),heap_ext(product_unit)),aa(ref(A),fun(heap_ext(product_unit),product_prod(ref(A),heap_ext(product_unit))),product_Pair(ref(A),heap_ext(product_unit)),R2),H4) = ref_alloc(A,V2,H2) )
         => ( ( N = one_one(nat) )
           => heap_Time_effect(ref(A),ref_ref(A,V2),H2,H4,R2,N) ) ) ) ).

% effect_refI
tff(fact_7819_Ref__Time_Opresent__def,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [H2: heap_ext(product_unit),R2: ref(A)] :
          ( ref_present(A,H2,R2)
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),addr_of_ref(A,R2)),lim(product_unit,H2)) ) ) ).

% Ref_Time.present_def
tff(fact_7820_execute__upd_I1_J,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Ia: nat,H2: heap_ext(product_unit),A3: array(A),Xa: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),array_length(A,H2,A3))
         => ( aa(heap_ext(product_unit),option(product_prod(array(A),product_prod(heap_ext(product_unit),nat))),heap_Time_execute(array(A),array_upd(A,Ia,Xa,A3)),H2) = aa(product_prod(array(A),product_prod(heap_ext(product_unit),nat)),option(product_prod(array(A),product_prod(heap_ext(product_unit),nat))),some(product_prod(array(A),product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),aa(array(A),fun(product_prod(heap_ext(product_unit),nat),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),product_Pair(array(A),product_prod(heap_ext(product_unit),nat)),A3),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),array_update(A,A3,Ia,Xa,H2)),one_one(nat)))) ) ) ) ).

% execute_upd(1)
tff(fact_7821_next__present,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [R2: ref(A),H4: heap_ext(product_unit),Xa: A,H2: heap_ext(product_unit)] :
          ( ( aa(heap_ext(product_unit),product_prod(ref(A),heap_ext(product_unit)),aa(ref(A),fun(heap_ext(product_unit),product_prod(ref(A),heap_ext(product_unit))),product_Pair(ref(A),heap_ext(product_unit)),R2),H4) = ref_alloc(A,Xa,H2) )
         => ref_present(A,H4,R2) ) ) ).

% next_present
tff(fact_7822_next__fresh,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [R2: ref(A),H4: heap_ext(product_unit),Xa: A,H2: heap_ext(product_unit)] :
          ( ( aa(heap_ext(product_unit),product_prod(ref(A),heap_ext(product_unit)),aa(ref(A),fun(heap_ext(product_unit),product_prod(ref(A),heap_ext(product_unit))),product_Pair(ref(A),heap_ext(product_unit)),R2),H4) = ref_alloc(A,Xa,H2) )
         => ~ ref_present(A,H2,R2) ) ) ).

% next_fresh
tff(fact_7823_Array__Time_Olength__alloc,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Xsa: list(A),H2: heap_ext(product_unit),Ys: list(A)] : ( array_length(A,aa(product_prod(array(A),heap_ext(product_unit)),heap_ext(product_unit),product_snd(array(A),heap_ext(product_unit)),array_alloc(A,Xsa,H2)),aa(product_prod(array(A),heap_ext(product_unit)),array(A),product_fst(array(A),heap_ext(product_unit)),array_alloc(A,Ys,H2))) = aa(list(A),nat,size_size(list(A)),Xsa) ) ) ).

% Array_Time.length_alloc
tff(fact_7824_effect__updI,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Ia: nat,H2: heap_ext(product_unit),A3: array(A),H4: heap_ext(product_unit),V2: A,N: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),array_length(A,H2,A3))
         => ( ( H4 = array_update(A,A3,Ia,V2,H2) )
           => ( ( N = one_one(nat) )
             => heap_Time_effect(array(A),array_upd(A,Ia,V2,A3),H2,H4,A3,N) ) ) ) ) ).

% effect_updI
tff(fact_7825_effect__updE,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Ia: nat,V2: A,A3: array(A),H2: heap_ext(product_unit),H4: heap_ext(product_unit),R2: array(A),N: nat] :
          ( heap_Time_effect(array(A),array_upd(A,Ia,V2,A3),H2,H4,R2,N)
         => ~ ( ( R2 = A3 )
             => ( ( H4 = array_update(A,A3,Ia,V2,H2) )
               => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),array_length(A,H2,A3))
                 => ( N != one_one(nat) ) ) ) ) ) ) ).

% effect_updE
tff(fact_7826_success__updI,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Ia: nat,H2: heap_ext(product_unit),A3: array(A),Xa: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),array_length(A,H2,A3))
         => heap_Time_success(array(A),array_upd(A,Ia,Xa,A3),H2) ) ) ).

% success_updI
tff(fact_7827_success__nthI,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Ia: nat,H2: heap_ext(product_unit),A3: array(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),array_length(A,H2,A3))
         => heap_Time_success(A,array_nth(A,A3,Ia),H2) ) ) ).

% success_nthI
tff(fact_7828_execute__len,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [A3: array(A),H2: heap_ext(product_unit)] : ( aa(heap_ext(product_unit),option(product_prod(nat,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(nat,array_len(A,A3)),H2) = aa(product_prod(nat,product_prod(heap_ext(product_unit),nat)),option(product_prod(nat,product_prod(heap_ext(product_unit),nat))),some(product_prod(nat,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(nat,product_prod(heap_ext(product_unit),nat)),aa(nat,fun(product_prod(heap_ext(product_unit),nat),product_prod(nat,product_prod(heap_ext(product_unit),nat))),product_Pair(nat,product_prod(heap_ext(product_unit),nat)),array_length(A,H2,A3)),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),H2),one_one(nat)))) ) ) ).

% execute_len
tff(fact_7829_upd__def,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Ia: nat,Xa: A,A3: array(A)] : ( array_upd(A,Ia,Xa,A3) = heap_Time_guard(array(A),aa(array(A),fun(heap_ext(product_unit),$o),aTP_Lamp_aec(nat,fun(array(A),fun(heap_ext(product_unit),$o)),Ia),A3),aa(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),aa(A,fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),aTP_Lamp_aed(nat,fun(A,fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat))))),Ia),Xa),A3)) ) ) ).

% upd_def
tff(fact_7830_execute__nth_I1_J,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Ia: nat,H2: heap_ext(product_unit),A3: array(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),array_length(A,H2,A3))
         => ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,array_nth(A,A3,Ia)),H2) = aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),aa(nat,A,nth(A,array_get(A,H2,A3)),Ia)),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),H2),one_one(nat)))) ) ) ) ).

% execute_nth(1)
tff(fact_7831_get__update__elem__neqIndex,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Ia: nat,J: nat,A3: array(A),V2: A,H2: heap_ext(product_unit)] :
          ( ( Ia != J )
         => ( aa(nat,A,nth(A,array_get(A,array_update(A,A3,J,V2,H2),A3)),Ia) = aa(nat,A,nth(A,array_get(A,H2,A3)),Ia) ) ) ) ).

% get_update_elem_neqIndex
tff(fact_7832_get__update__eq,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [A3: array(A),Ia: nat,V2: A,H2: heap_ext(product_unit)] : ( array_get(A,array_update(A,A3,Ia,V2,H2),A3) = aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),array_get(A,H2,A3)),Ia),V2) ) ) ).

% get_update_eq
tff(fact_7833_length__def,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [H2: heap_ext(product_unit),A3: array(A)] : ( array_length(A,H2,A3) = aa(list(A),nat,size_size(list(A)),array_get(A,H2,A3)) ) ) ).

% length_def
tff(fact_7834_Array__Time_Onth__def,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [A3: array(A),Ia: nat] : ( array_nth(A,A3,Ia) = heap_Time_guard(A,aa(nat,fun(heap_ext(product_unit),$o),aTP_Lamp_aee(array(A),fun(nat,fun(heap_ext(product_unit),$o)),A3),Ia),aa(nat,fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),aTP_Lamp_aef(array(A),fun(nat,fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)))),A3),Ia)) ) ) ).

% Array_Time.nth_def
tff(fact_7835_execute__freeze,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [A3: array(A),H2: heap_ext(product_unit)] : ( aa(heap_ext(product_unit),option(product_prod(list(A),product_prod(heap_ext(product_unit),nat))),heap_Time_execute(list(A),array_freeze(A,A3)),H2) = aa(product_prod(list(A),product_prod(heap_ext(product_unit),nat)),option(product_prod(list(A),product_prod(heap_ext(product_unit),nat))),some(product_prod(list(A),product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(list(A),product_prod(heap_ext(product_unit),nat)),aa(list(A),fun(product_prod(heap_ext(product_unit),nat),product_prod(list(A),product_prod(heap_ext(product_unit),nat))),product_Pair(list(A),product_prod(heap_ext(product_unit),nat)),array_get(A,H2,A3)),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),H2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),array_length(A,H2,A3))))) ) ) ).

% execute_freeze
tff(fact_7836_freeze__def,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [A3: array(A)] : ( array_freeze(A,A3) = heap_Time_heap(list(A),aTP_Lamp_aeg(array(A),fun(heap_ext(product_unit),product_prod(list(A),product_prod(heap_ext(product_unit),nat))),A3)) ) ) ).

% freeze_def
tff(fact_7837_effect__nthI,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Ia: nat,H2: heap_ext(product_unit),A3: array(A),H4: heap_ext(product_unit),R2: A,N: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),array_length(A,H2,A3))
         => ( ( H4 = H2 )
           => ( ( R2 = aa(nat,A,nth(A,array_get(A,H2,A3)),Ia) )
             => ( ( N = one_one(nat) )
               => heap_Time_effect(A,array_nth(A,A3,Ia),H2,H4,R2,N) ) ) ) ) ) ).

% effect_nthI
tff(fact_7838_effect__nthE,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [A3: array(A),Ia: nat,H2: heap_ext(product_unit),H4: heap_ext(product_unit),R2: A,N: nat] :
          ( heap_Time_effect(A,array_nth(A,A3,Ia),H2,H4,R2,N)
         => ~ ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),array_length(A,H2,A3))
             => ( ( R2 = aa(nat,A,nth(A,array_get(A,H2,A3)),Ia) )
               => ( ( H4 = H2 )
                 => ( N != one_one(nat) ) ) ) ) ) ) ).

% effect_nthE
tff(fact_7839_execute__map__entry_I1_J,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Ia: nat,H2: heap_ext(product_unit),A3: array(A),F2: fun(A,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),array_length(A,H2,A3))
         => ( aa(heap_ext(product_unit),option(product_prod(array(A),product_prod(heap_ext(product_unit),nat))),heap_Time_execute(array(A),array_map_entry(A,Ia,F2,A3)),H2) = aa(product_prod(array(A),product_prod(heap_ext(product_unit),nat)),option(product_prod(array(A),product_prod(heap_ext(product_unit),nat))),some(product_prod(array(A),product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),aa(array(A),fun(product_prod(heap_ext(product_unit),nat),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),product_Pair(array(A),product_prod(heap_ext(product_unit),nat)),A3),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),array_update(A,A3,Ia,aa(A,A,F2,aa(nat,A,nth(A,array_get(A,H2,A3)),Ia)),H2)),numeral_numeral(nat,bit0(one2))))) ) ) ) ).

% execute_map_entry(1)
tff(fact_7840_execute__swap_I1_J,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Ia: nat,H2: heap_ext(product_unit),A3: array(A),Xa: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),array_length(A,H2,A3))
         => ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,array_swap(A,Ia,Xa,A3)),H2) = aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),aa(nat,A,nth(A,array_get(A,H2,A3)),Ia)),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),array_update(A,A3,Ia,Xa,H2)),numeral_numeral(nat,bit0(one2))))) ) ) ) ).

% execute_swap(1)
tff(fact_7841_success__swapI,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Ia: nat,H2: heap_ext(product_unit),A3: array(A),Xa: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),array_length(A,H2,A3))
         => heap_Time_success(A,array_swap(A,Ia,Xa,A3),H2) ) ) ).

% success_swapI
tff(fact_7842_success__map__entryI,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Ia: nat,H2: heap_ext(product_unit),A3: array(A),F2: fun(A,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),array_length(A,H2,A3))
         => heap_Time_success(array(A),array_map_entry(A,Ia,F2,A3),H2) ) ) ).

% success_map_entryI
tff(fact_7843_effect__map__entryI,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Ia: nat,H2: heap_ext(product_unit),A3: array(A),H4: heap_ext(product_unit),F2: fun(A,A),R2: array(A),N: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),array_length(A,H2,A3))
         => ( ( H4 = array_update(A,A3,Ia,aa(A,A,F2,aa(nat,A,nth(A,array_get(A,H2,A3)),Ia)),H2) )
           => ( ( R2 = A3 )
             => ( ( N = numeral_numeral(nat,bit0(one2)) )
               => heap_Time_effect(array(A),array_map_entry(A,Ia,F2,A3),H2,H4,R2,N) ) ) ) ) ) ).

% effect_map_entryI
tff(fact_7844_effect__map__entryE,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Ia: nat,F2: fun(A,A),A3: array(A),H2: heap_ext(product_unit),H4: heap_ext(product_unit),R2: array(A),N: nat] :
          ( heap_Time_effect(array(A),array_map_entry(A,Ia,F2,A3),H2,H4,R2,N)
         => ~ ( ( R2 = A3 )
             => ( ( H4 = array_update(A,A3,Ia,aa(A,A,F2,aa(nat,A,nth(A,array_get(A,H2,A3)),Ia)),H2) )
               => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),array_length(A,H2,A3))
                 => ( N != numeral_numeral(nat,bit0(one2)) ) ) ) ) ) ) ).

% effect_map_entryE
tff(fact_7845_effect__swapI,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Ia: nat,H2: heap_ext(product_unit),A3: array(A),H4: heap_ext(product_unit),Xa: A,R2: A,N: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),array_length(A,H2,A3))
         => ( ( H4 = array_update(A,A3,Ia,Xa,H2) )
           => ( ( R2 = aa(nat,A,nth(A,array_get(A,H2,A3)),Ia) )
             => ( ( N = numeral_numeral(nat,bit0(one2)) )
               => heap_Time_effect(A,array_swap(A,Ia,Xa,A3),H2,H4,R2,N) ) ) ) ) ) ).

% effect_swapI
tff(fact_7846_effect__swapE,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Ia: nat,Xa: A,A3: array(A),H2: heap_ext(product_unit),H4: heap_ext(product_unit),R2: A,N: nat] :
          ( heap_Time_effect(A,array_swap(A,Ia,Xa,A3),H2,H4,R2,N)
         => ~ ( ( R2 = aa(nat,A,nth(A,array_get(A,H2,A3)),Ia) )
             => ( ( H4 = array_update(A,A3,Ia,Xa,H2) )
               => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),array_length(A,H2,A3))
                 => ( N != numeral_numeral(nat,bit0(one2)) ) ) ) ) ) ) ).

% effect_swapE
tff(fact_7847_swap__def,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Ia: nat,Xa: A,A3: array(A)] : ( array_swap(A,Ia,Xa,A3) = heap_Time_guard(A,aa(array(A),fun(heap_ext(product_unit),$o),aTP_Lamp_aec(nat,fun(array(A),fun(heap_ext(product_unit),$o)),Ia),A3),aa(array(A),fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(A,fun(array(A),fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)))),aTP_Lamp_aeh(nat,fun(A,fun(array(A),fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))))),Ia),Xa),A3)) ) ) ).

% swap_def
tff(fact_7848_map__entry__def,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Ia: nat,F2: fun(A,A),A3: array(A)] : ( array_map_entry(A,Ia,F2,A3) = heap_Time_guard(array(A),aa(array(A),fun(heap_ext(product_unit),$o),aTP_Lamp_aec(nat,fun(array(A),fun(heap_ext(product_unit),$o)),Ia),A3),aa(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),aa(fun(A,A),fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),aTP_Lamp_aei(nat,fun(fun(A,A),fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat))))),Ia),F2),A3)) ) ) ).

% map_entry_def
tff(fact_7849_effect__of__listE,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Xsa: list(A),H2: heap_ext(product_unit),H4: heap_ext(product_unit),R2: array(A),N6: nat] :
          ( heap_Time_effect(array(A),array_of_list(A,Xsa),H2,H4,R2,N6)
         => ~ ( ( R2 = aa(product_prod(array(A),heap_ext(product_unit)),array(A),product_fst(array(A),heap_ext(product_unit)),array_alloc(A,Xsa,H2)) )
             => ( ( H4 = aa(product_prod(array(A),heap_ext(product_unit)),heap_ext(product_unit),product_snd(array(A),heap_ext(product_unit)),array_alloc(A,Xsa,H2)) )
               => ( ( array_get(A,H4,R2) = Xsa )
                 => ( array_present(A,H4,R2)
                   => ( ~ array_present(A,H2,R2)
                     => ( N6 != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(list(A),nat,size_size(list(A)),Xsa)) ) ) ) ) ) ) ) ) ).

% effect_of_listE
tff(fact_7850_snga__assn__raw_Oelims_I3_J,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Xa: array(A),Xaa: list(A),Xb: product_prod(heap_ext(product_unit),set(nat))] :
          ( ~ aa(product_prod(heap_ext(product_unit),set(nat)),$o,snga_assn_raw(A,Xa,Xaa),Xb)
         => ~ ! [H: heap_ext(product_unit),As: 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),As) )
               => ( ( array_get(A,H,Xa) = Xaa )
                  & ( As = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert3(nat),addr_of_array(A,Xa)),bot_bot(set(nat))) )
                  & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),addr_of_array(A,Xa)),lim(product_unit,H)) ) ) ) ) ).

% snga_assn_raw.elims(3)
tff(fact_7851_Array__Time_Opresent__def,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [H2: heap_ext(product_unit),A3: array(A)] :
          ( array_present(A,H2,A3)
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),addr_of_array(A,A3)),lim(product_unit,H2)) ) ) ).

% Array_Time.present_def
tff(fact_7852_relH__array,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [As2: set(nat),H2: heap_ext(product_unit),H4: heap_ext(product_unit),R2: array(A)] :
          ( relH(As2,H2,H4)
         => ( member(nat,addr_of_array(A,R2),As2)
           => ( array_get(A,H2,R2) = array_get(A,H4,R2) ) ) ) ) ).

% relH_array
tff(fact_7853_snga__assn__raw_Osimps,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [R2: array(A),Xa: list(A),H2: heap_ext(product_unit),As2: set(nat)] :
          ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,snga_assn_raw(A,R2,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)),H2),As2))
        <=> ( ( array_get(A,H2,R2) = Xa )
            & ( As2 = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert3(nat),addr_of_array(A,R2)),bot_bot(set(nat))) )
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),addr_of_array(A,R2)),lim(product_unit,H2)) ) ) ) ).

% snga_assn_raw.simps
tff(fact_7854_snga__assn__raw_Oelims_I1_J,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Xa: array(A),Xaa: list(A),Xb: product_prod(heap_ext(product_unit),set(nat)),Y: $o] :
          ( ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,snga_assn_raw(A,Xa,Xaa),Xb)
          <=> (Y) )
         => ~ ! [H: heap_ext(product_unit),As: 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),As) )
               => ( (Y)
                <=> ~ ( ( array_get(A,H,Xa) = Xaa )
                      & ( As = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert3(nat),addr_of_array(A,Xa)),bot_bot(set(nat))) )
                      & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),addr_of_array(A,Xa)),lim(product_unit,H)) ) ) ) ) ) ).

% snga_assn_raw.elims(1)
tff(fact_7855_snga__assn__raw_Oelims_I2_J,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Xa: array(A),Xaa: list(A),Xb: product_prod(heap_ext(product_unit),set(nat))] :
          ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,snga_assn_raw(A,Xa,Xaa),Xb)
         => ~ ! [H: heap_ext(product_unit),As: 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),As) )
               => ~ ( ( array_get(A,H,Xa) = Xaa )
                    & ( As = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert3(nat),addr_of_array(A,Xa)),bot_bot(set(nat))) )
                    & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),addr_of_array(A,Xa)),lim(product_unit,H)) ) ) ) ) ).

% snga_assn_raw.elims(2)
tff(fact_7856_snga__assn__raw_Opelims_I3_J,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Xa: array(A),Xaa: list(A),Xb: product_prod(heap_ext(product_unit),set(nat))] :
          ( ~ aa(product_prod(heap_ext(product_unit),set(nat)),$o,snga_assn_raw(A,Xa,Xaa),Xb)
         => ( accp(product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),snga_assn_raw_rel(A),aa(product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),aa(array(A),fun(product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),Xa),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),aa(list(A),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(list(A),product_prod(heap_ext(product_unit),set(nat))),Xaa),Xb)))
           => ~ ! [H: heap_ext(product_unit),As: 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),As) )
                 => ( accp(product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),snga_assn_raw_rel(A),aa(product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),aa(array(A),fun(product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),Xa),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),aa(list(A),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(list(A),product_prod(heap_ext(product_unit),set(nat))),Xaa),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),As))))
                   => ( ( array_get(A,H,Xa) = Xaa )
                      & ( As = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert3(nat),addr_of_array(A,Xa)),bot_bot(set(nat))) )
                      & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),addr_of_array(A,Xa)),lim(product_unit,H)) ) ) ) ) ) ) ).

% snga_assn_raw.pelims(3)
tff(fact_7857_snga__assn__raw_Opelims_I2_J,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Xa: array(A),Xaa: list(A),Xb: product_prod(heap_ext(product_unit),set(nat))] :
          ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,snga_assn_raw(A,Xa,Xaa),Xb)
         => ( accp(product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),snga_assn_raw_rel(A),aa(product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),aa(array(A),fun(product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),Xa),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),aa(list(A),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(list(A),product_prod(heap_ext(product_unit),set(nat))),Xaa),Xb)))
           => ~ ! [H: heap_ext(product_unit),As: 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),As) )
                 => ( accp(product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),snga_assn_raw_rel(A),aa(product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),aa(array(A),fun(product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),Xa),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),aa(list(A),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(list(A),product_prod(heap_ext(product_unit),set(nat))),Xaa),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),As))))
                   => ~ ( ( array_get(A,H,Xa) = Xaa )
                        & ( As = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert3(nat),addr_of_array(A,Xa)),bot_bot(set(nat))) )
                        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),addr_of_array(A,Xa)),lim(product_unit,H)) ) ) ) ) ) ) ).

% snga_assn_raw.pelims(2)
tff(fact_7858_snga__assn__raw_Opelims_I1_J,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Xa: array(A),Xaa: list(A),Xb: product_prod(heap_ext(product_unit),set(nat)),Y: $o] :
          ( ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,snga_assn_raw(A,Xa,Xaa),Xb)
          <=> (Y) )
         => ( accp(product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),snga_assn_raw_rel(A),aa(product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),aa(array(A),fun(product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),Xa),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),aa(list(A),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(list(A),product_prod(heap_ext(product_unit),set(nat))),Xaa),Xb)))
           => ~ ! [H: heap_ext(product_unit),As: 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),As) )
                 => ( ( (Y)
                    <=> ( ( array_get(A,H,Xa) = Xaa )
                        & ( As = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert3(nat),addr_of_array(A,Xa)),bot_bot(set(nat))) )
                        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),addr_of_array(A,Xa)),lim(product_unit,H)) ) )
                   => ~ accp(product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),snga_assn_raw_rel(A),aa(product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),aa(array(A),fun(product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),Xa),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),aa(list(A),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(list(A),product_prod(heap_ext(product_unit),set(nat))),Xaa),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),As)))) ) ) ) ) ) ).

% snga_assn_raw.pelims(1)
tff(fact_7859_nth__update__neq,axiom,
    ! [B: $tType,A: $tType] :
      ( ( heap(A)
        & heap(B) )
     => ! [A3: array(A),B3: array(B),J: nat,V2: B,H2: heap_ext(product_unit),Ia: nat] :
          ( array_noteq(A,B,A3,B3)
         => ( aa(nat,A,nth(A,array_get(A,array_update(B,B3,J,V2,H2),A3)),Ia) = aa(nat,A,nth(A,array_get(A,H2,A3)),Ia) ) ) ) ).

% nth_update_neq
tff(fact_7860_relH__set__array,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [R2: array(A),As2: set(nat),H2: heap_ext(product_unit),Xa: list(A)] :
          ( ~ member(nat,addr_of_array(A,R2),As2)
         => ( aa(product_prod(heap_ext(product_unit),set(nat)),$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),As2))
           => relH(As2,H2,array_set(A,R2,Xa,H2)) ) ) ) ).

% relH_set_array
tff(fact_7861_Array__Time_Oupdate__def,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [A3: array(A),Ia: nat,Xa: A,H2: heap_ext(product_unit)] : ( array_update(A,A3,Ia,Xa,H2) = array_set(A,A3,aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),array_get(A,H2,A3)),Ia),Xa),H2) ) ) ).

% Array_Time.update_def
tff(fact_7862_Array__Time_Oalloc__def,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Xsa: list(A),H2: heap_ext(product_unit)] :
          ( array_alloc(A,Xsa,H2) = $let(
              l: nat,
              l:= lim(product_unit,H2),
              $let(
                r: array(A),
                r:= array2(A,l),
                aa(heap_ext(product_unit),product_prod(array(A),heap_ext(product_unit)),aa(array(A),fun(heap_ext(product_unit),product_prod(array(A),heap_ext(product_unit))),product_Pair(array(A),heap_ext(product_unit)),r),array_set(A,r,Xsa,lim_update(product_unit,aTP_Lamp_aej(nat,fun(nat,nat),l),H2))) ) ) ) ) ).

% Array_Time.alloc_def
tff(fact_7863_Ref__Time_Oalloc__def,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Xa: A,H2: heap_ext(product_unit)] :
          ( ref_alloc(A,Xa,H2) = $let(
              l: nat,
              l:= lim(product_unit,H2),
              $let(
                r: ref(A),
                r:= ref2(A,l),
                aa(heap_ext(product_unit),product_prod(ref(A),heap_ext(product_unit)),aa(ref(A),fun(heap_ext(product_unit),product_prod(ref(A),heap_ext(product_unit))),product_Pair(ref(A),heap_ext(product_unit)),r),ref_set(A,r,Xa,lim_update(product_unit,aTP_Lamp_aej(nat,fun(nat,nat),l),H2))) ) ) ) ) ).

% Ref_Time.alloc_def
tff(fact_7864_raise__iff,axiom,
    ! [A: $tType,Pa: assn,S2: list(char),Qa: fun(A,assn)] :
      ( hoare_hoare_triple(A,Pa,heap_Time_raise(A,S2),Qa)
    <=> ( Pa = bot_bot(assn) ) ) ).

% raise_iff
tff(fact_7865_raise__rule,axiom,
    ! [A: $tType,S2: list(char),Qa: fun(A,assn)] : hoare_hoare_triple(A,bot_bot(assn),heap_Time_raise(A,S2),Qa) ).

% raise_rule
tff(fact_7866_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))),X4: product_prod(code_natural,code_natural)] : ( random_union(A,R1,R22,X4) = 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_ael(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,X4)) ) ).

% Random_Pred.union_def
tff(fact_7867_Nitpick_Otranclp__unfold,axiom,
    ! [A: $tType,R2: fun(A,fun(A,$o)),A3: A,B3: A] :
      ( aa(A,$o,aa(A,fun(A,$o),transitive_tranclp(A,R2),A3),B3)
    <=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B3),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_7868_less__nat__rel,axiom,
    ord_less(nat) = transitive_tranclp(nat,aTP_Lamp_aem(nat,fun(nat,$o))) ).

% less_nat_rel
tff(fact_7869_tranclp__power,axiom,
    ! [A: $tType,Pa: fun(A,fun(A,$o)),Xa: A,Y: A] :
      ( aa(A,$o,aa(A,fun(A,$o),transitive_tranclp(A,Pa),Xa),Y)
    <=> ? [N2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N2)
          & aa(A,$o,aa(A,fun(A,$o),compow(fun(A,fun(A,$o)),N2,Pa),Xa),Y) ) ) ).

% tranclp_power
tff(fact_7870_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,Pa: fun(A,fun(B,$o))] :
      ( aa(product_prod(A,B),$o,aa(product_prod(A,B),fun(product_prod(A,B),$o),transitive_tranclp(product_prod(A,B),R2),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Bx),By))
     => ( ! [A4: A,B4: B] :
            ( aa(product_prod(A,B),$o,aa(product_prod(A,B),fun(product_prod(A,B),$o),R2,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B4))
           => aa(B,$o,aa(A,fun(B,$o),Pa,A4),B4) )
       => ( ! [A4: A,B4: B,Aa3: A,Ba: B] :
              ( aa(product_prod(A,B),$o,aa(product_prod(A,B),fun(product_prod(A,B),$o),transitive_tranclp(product_prod(A,B),R2),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B4))
             => ( aa(product_prod(A,B),$o,aa(product_prod(A,B),fun(product_prod(A,B),$o),R2,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B4)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa3),Ba))
               => ( aa(B,$o,aa(A,fun(B,$o),Pa,A4),B4)
                 => aa(B,$o,aa(A,fun(B,$o),Pa,Aa3),Ba) ) ) )
         => aa(B,$o,aa(A,fun(B,$o),Pa,Bx),By) ) ) ) ).

% tranclp_induct2
tff(fact_7871_tranclp__trancl__eq,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),X4: A,Xa4: A] :
      ( aa(A,$o,aa(A,fun(A,$o),transitive_tranclp(A,aTP_Lamp_cz(set(product_prod(A,A)),fun(A,fun(A,$o)),R2)),X4),Xa4)
    <=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Xa4),transitive_trancl(A,R2)) ) ).

% tranclp_trancl_eq
tff(fact_7872_trancl__def,axiom,
    ! [A: $tType,X4: set(product_prod(A,A))] : ( transitive_trancl(A,X4) = aa(fun(product_prod(A,A),$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,$o),transitive_tranclp(A,aTP_Lamp_cz(set(product_prod(A,A)),fun(A,fun(A,$o)),X4)))) ) ).

% trancl_def
tff(fact_7873_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_7874_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))),F2: fun(B,fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural)))),X4: product_prod(code_natural,code_natural)] : ( random_bind(B,A,R,F2,X4) = 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_aep(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)))),F2)),aa(product_prod(code_natural,code_natural),product_prod(pred(B),product_prod(code_natural,code_natural)),R,X4)) ) ).

% Random_Pred.bind_def
tff(fact_7875_reflp__refl__eq,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( reflp(A,aTP_Lamp_cz(set(product_prod(A,A)),fun(A,fun(A,$o)),R2))
    <=> refl_on(A,top_top(set(A)),R2) ) ).

% reflp_refl_eq
tff(fact_7876_list_Orel__compp__Grp,axiom,
    ! [B: $tType,A: $tType,R: fun(A,fun(B,$o))] : ( aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),R) = 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)),$o),set(list(product_prod(A,B))),collect(list(product_prod(A,B))),aTP_Lamp_rq(fun(A,fun(B,$o)),fun(list(product_prod(A,B)),$o),R)),aa(fun(product_prod(A,B),A),fun(list(product_prod(A,B)),list(A)),map(product_prod(A,B),A),product_fst(A,B)))),bNF_Grp(list(product_prod(A,B)),list(B),aa(fun(list(product_prod(A,B)),$o),set(list(product_prod(A,B))),collect(list(product_prod(A,B))),aTP_Lamp_rq(fun(A,fun(B,$o)),fun(list(product_prod(A,B)),$o),R)),aa(fun(product_prod(A,B),B),fun(list(product_prod(A,B)),list(B)),map(product_prod(A,B),B),product_snd(A,B)))) ) ).

% list.rel_compp_Grp
tff(fact_7877_relcompp__distrib2,axiom,
    ! [A: $tType,B: $tType,E: $tType,S: fun(A,fun(E,$o)),T4: fun(A,fun(E,$o)),R: fun(E,fun(B,$o))] : ( relcompp(A,E,B,aa(fun(A,fun(E,$o)),fun(A,fun(E,$o)),aa(fun(A,fun(E,$o)),fun(fun(A,fun(E,$o)),fun(A,fun(E,$o))),sup_sup(fun(A,fun(E,$o))),S),T4),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))),relcompp(A,E,B,S,R)),relcompp(A,E,B,T4,R)) ) ).

% relcompp_distrib2
tff(fact_7878_relcompp__distrib,axiom,
    ! [A: $tType,B: $tType,E: $tType,R: fun(A,fun(E,$o)),S: fun(E,fun(B,$o)),T4: fun(E,fun(B,$o))] : ( relcompp(A,E,B,R,aa(fun(E,fun(B,$o)),fun(E,fun(B,$o)),aa(fun(E,fun(B,$o)),fun(fun(E,fun(B,$o)),fun(E,fun(B,$o))),sup_sup(fun(E,fun(B,$o))),S),T4)) = 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))),relcompp(A,E,B,R,S)),relcompp(A,E,B,R,T4)) ) ).

% relcompp_distrib
tff(fact_7879_relcompp__bot1,axiom,
    ! [E: $tType,B: $tType,A: $tType,R: fun(E,fun(B,$o))] : ( relcompp(A,E,B,bot_bot(fun(A,fun(E,$o))),R) = bot_bot(fun(A,fun(B,$o))) ) ).

% relcompp_bot1
tff(fact_7880_relcompp__bot2,axiom,
    ! [E: $tType,B: $tType,A: $tType,R: fun(A,fun(E,$o))] : ( relcompp(A,E,B,R,bot_bot(fun(E,fun(B,$o)))) = bot_bot(fun(A,fun(B,$o))) ) ).

% relcompp_bot2
tff(fact_7881_relcomp__def,axiom,
    ! [A: $tType,E: $tType,B: $tType,X4: set(product_prod(A,B)),Xa4: set(product_prod(B,E))] : ( relcomp(A,B,E,X4,Xa4) = aa(fun(product_prod(A,E),$o),set(product_prod(A,E)),collect(product_prod(A,E)),aa(fun(A,fun(E,$o)),fun(product_prod(A,E),$o),product_case_prod(A,E,$o),relcompp(A,B,E,aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_bi(set(product_prod(A,B)),fun(A,fun(B,$o))),X4),aTP_Lamp_aeq(set(product_prod(B,E)),fun(B,fun(E,$o)),Xa4)))) ) ).

% relcomp_def
tff(fact_7882_list_Orel__Grp,axiom,
    ! [B: $tType,A: $tType,Aa2: set(A),F2: fun(A,B)] : ( aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),bNF_Grp(A,B,Aa2,F2)) = bNF_Grp(list(A),list(B),aa(fun(list(A),$o),set(list(A)),collect(list(A)),aTP_Lamp_cy(set(A),fun(list(A),$o),Aa2)),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2)) ) ).

% list.rel_Grp
tff(fact_7883_transp__relcompp__less__eq,axiom,
    ! [A: $tType,R2: fun(A,fun(A,$o))] :
      ( transp(A,R2)
     => 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))),relcompp(A,A,A,R2,R2)),R2) ) ).

% transp_relcompp_less_eq
tff(fact_7884_relcompp__mono,axiom,
    ! [A: $tType,E: $tType,B: $tType,R5: fun(A,fun(B,$o)),R2: fun(A,fun(B,$o)),S4: fun(B,fun(E,$o)),S2: fun(B,fun(E,$o))] :
      ( aa(fun(A,fun(B,$o)),$o,aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),$o),ord_less_eq(fun(A,fun(B,$o))),R5),R2)
     => ( aa(fun(B,fun(E,$o)),$o,aa(fun(B,fun(E,$o)),fun(fun(B,fun(E,$o)),$o),ord_less_eq(fun(B,fun(E,$o))),S4),S2)
       => aa(fun(A,fun(E,$o)),$o,aa(fun(A,fun(E,$o)),fun(fun(A,fun(E,$o)),$o),ord_less_eq(fun(A,fun(E,$o))),relcompp(A,B,E,R5,S4)),relcompp(A,B,E,R2,S2)) ) ) ).

% relcompp_mono
tff(fact_7885_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_7886_converse__relcompp,axiom,
    ! [A: $tType,E: $tType,B: $tType,R2: fun(B,fun(E,$o)),S2: fun(E,fun(A,$o))] : ( conversep(B,A,relcompp(B,E,A,R2,S2)) = relcompp(A,E,B,conversep(E,A,S2),conversep(B,E,R2)) ) ).

% converse_relcompp
tff(fact_7887_relcompp_Ocases,axiom,
    ! [A: $tType,B: $tType,E: $tType,R2: fun(A,fun(B,$o)),S2: fun(B,fun(E,$o)),A1: A,A22: E] :
      ( aa(E,$o,aa(A,fun(E,$o),relcompp(A,B,E,R2,S2),A1),A22)
     => ~ ! [B4: B] :
            ( aa(B,$o,aa(A,fun(B,$o),R2,A1),B4)
           => ~ aa(E,$o,aa(B,fun(E,$o),S2,B4),A22) ) ) ).

% relcompp.cases
tff(fact_7888_relcompp_Osimps,axiom,
    ! [A: $tType,B: $tType,E: $tType,R2: fun(A,fun(B,$o)),S2: fun(B,fun(E,$o)),A1: A,A22: E] :
      ( aa(E,$o,aa(A,fun(E,$o),relcompp(A,B,E,R2,S2),A1),A22)
    <=> ? [A7: A,B8: B,C6: E] :
          ( ( A1 = A7 )
          & ( A22 = C6 )
          & aa(B,$o,aa(A,fun(B,$o),R2,A7),B8)
          & aa(E,$o,aa(B,fun(E,$o),S2,B8),C6) ) ) ).

% relcompp.simps
tff(fact_7889_OO__eq,axiom,
    ! [B: $tType,A: $tType,R: fun(A,fun(B,$o))] : ( relcompp(A,B,B,R,fequal(B)) = R ) ).

% OO_eq
tff(fact_7890_eq__OO,axiom,
    ! [B: $tType,A: $tType,R: fun(A,fun(B,$o))] : ( relcompp(A,A,B,fequal(A),R) = R ) ).

% eq_OO
tff(fact_7891_reflpD,axiom,
    ! [A: $tType,R2: fun(A,fun(A,$o)),Xa: A] :
      ( reflp(A,R2)
     => aa(A,$o,aa(A,fun(A,$o),R2,Xa),Xa) ) ).

% reflpD
tff(fact_7892_reflpE,axiom,
    ! [A: $tType,R2: fun(A,fun(A,$o)),Xa: A] :
      ( reflp(A,R2)
     => aa(A,$o,aa(A,fun(A,$o),R2,Xa),Xa) ) ).

% reflpE
tff(fact_7893_reflpI,axiom,
    ! [A: $tType,R2: fun(A,fun(A,$o))] :
      ( ! [X: A] : aa(A,$o,aa(A,fun(A,$o),R2,X),X)
     => reflp(A,R2) ) ).

% reflpI
tff(fact_7894_reflp__def,axiom,
    ! [A: $tType,R2: fun(A,fun(A,$o))] :
      ( reflp(A,R2)
    <=> ! [X6: A] : aa(A,$o,aa(A,fun(A,$o),R2,X6),X6) ) ).

% reflp_def
tff(fact_7895_relcomppE,axiom,
    ! [A: $tType,B: $tType,E: $tType,R2: fun(A,fun(B,$o)),S2: fun(B,fun(E,$o)),A3: A,C2: E] :
      ( aa(E,$o,aa(A,fun(E,$o),relcompp(A,B,E,R2,S2),A3),C2)
     => ~ ! [B4: B] :
            ( aa(B,$o,aa(A,fun(B,$o),R2,A3),B4)
           => ~ aa(E,$o,aa(B,fun(E,$o),S2,B4),C2) ) ) ).

% relcomppE
tff(fact_7896_relcomppI,axiom,
    ! [A: $tType,B: $tType,E: $tType,R2: fun(A,fun(B,$o)),A3: A,B3: B,S2: fun(B,fun(E,$o)),C2: E] :
      ( aa(B,$o,aa(A,fun(B,$o),R2,A3),B3)
     => ( aa(E,$o,aa(B,fun(E,$o),S2,B3),C2)
       => aa(E,$o,aa(A,fun(E,$o),relcompp(A,B,E,R2,S2),A3),C2) ) ) ).

% relcomppI
tff(fact_7897_reflp__mono,axiom,
    ! [A: $tType,R: fun(A,fun(A,$o)),Qa: fun(A,fun(A,$o))] :
      ( reflp(A,R)
     => ( ! [X: A,Y2: A] :
            ( aa(A,$o,aa(A,fun(A,$o),R,X),Y2)
           => aa(A,$o,aa(A,fun(A,$o),Qa,X),Y2) )
       => reflp(A,Qa) ) ) ).

% reflp_mono
tff(fact_7898_reflp__equality,axiom,
    ! [A: $tType] : reflp(A,fequal(A)) ).

% reflp_equality
tff(fact_7899_relcompp__apply,axiom,
    ! [A: $tType,B: $tType,E: $tType,R: fun(A,fun(B,$o)),S: fun(B,fun(E,$o)),A3: A,C2: E] :
      ( aa(E,$o,aa(A,fun(E,$o),relcompp(A,B,E,R,S),A3),C2)
    <=> ? [B8: B] :
          ( aa(B,$o,aa(A,fun(B,$o),R,A3),B8)
          & aa(E,$o,aa(B,fun(E,$o),S,B8),C2) ) ) ).

% relcompp_apply
tff(fact_7900_relcompp__assoc,axiom,
    ! [A: $tType,F4: $tType,B: $tType,E: $tType,R2: fun(A,fun(F4,$o)),S2: fun(F4,fun(E,$o)),T2: fun(E,fun(B,$o))] : ( relcompp(A,E,B,relcompp(A,F4,E,R2,S2),T2) = relcompp(A,F4,B,R2,relcompp(F4,E,B,S2,T2)) ) ).

% relcompp_assoc
tff(fact_7901_list_Orel__reflp,axiom,
    ! [A: $tType,R: fun(A,fun(A,$o))] :
      ( reflp(A,R)
     => reflp(list(A),aa(fun(A,fun(A,$o)),fun(list(A),fun(list(A),$o)),list_all2(A,A),R)) ) ).

% list.rel_reflp
tff(fact_7902_list_Orel__compp,axiom,
    ! [A: $tType,B: $tType,E: $tType,R: fun(A,fun(E,$o)),S: fun(E,fun(B,$o))] : ( aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),relcompp(A,E,B,R,S)) = relcompp(list(A),list(E),list(B),aa(fun(A,fun(E,$o)),fun(list(A),fun(list(E),$o)),list_all2(A,E),R),aa(fun(E,fun(B,$o)),fun(list(E),fun(list(B),$o)),list_all2(E,B),S)) ) ).

% list.rel_compp
tff(fact_7903_rel__filter__distr,axiom,
    ! [A: $tType,B: $tType,E: $tType,Aa2: fun(A,fun(E,$o)),B2: fun(E,fun(B,$o))] : ( relcompp(filter(A),filter(E),filter(B),rel_filter(A,E,Aa2),rel_filter(E,B,B2)) = rel_filter(A,B,relcompp(A,E,B,Aa2,B2)) ) ).

% rel_filter_distr
tff(fact_7904_rel__filter__Grp,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B)] : ( rel_filter(A,B,bNF_Grp(A,B,top_top(set(A)),F2)) = bNF_Grp(filter(A),filter(B),top_top(set(filter(A))),aa(fun(A,B),fun(filter(A),filter(B)),filtermap(A,B),F2)) ) ).

% rel_filter_Grp
tff(fact_7905_relcompp__SUP__distrib2,axiom,
    ! [E: $tType,B: $tType,A: $tType,F4: $tType,R2: fun(F4,fun(A,fun(E,$o))),Ia2: set(F4),S2: fun(E,fun(B,$o))] : ( relcompp(A,E,B,aa(set(fun(A,fun(E,$o))),fun(A,fun(E,$o)),complete_Sup_Sup(fun(A,fun(E,$o))),aa(set(F4),set(fun(A,fun(E,$o))),image2(F4,fun(A,fun(E,$o)),R2),Ia2)),S2) = aa(set(fun(A,fun(B,$o))),fun(A,fun(B,$o)),complete_Sup_Sup(fun(A,fun(B,$o))),aa(set(F4),set(fun(A,fun(B,$o))),image2(F4,fun(A,fun(B,$o)),aa(fun(E,fun(B,$o)),fun(F4,fun(A,fun(B,$o))),aTP_Lamp_aer(fun(F4,fun(A,fun(E,$o))),fun(fun(E,fun(B,$o)),fun(F4,fun(A,fun(B,$o)))),R2),S2)),Ia2)) ) ).

% relcompp_SUP_distrib2
tff(fact_7906_relcompp__SUP__distrib,axiom,
    ! [E: $tType,B: $tType,A: $tType,F4: $tType,S2: fun(A,fun(E,$o)),R2: fun(F4,fun(E,fun(B,$o))),Ia2: set(F4)] : ( relcompp(A,E,B,S2,aa(set(fun(E,fun(B,$o))),fun(E,fun(B,$o)),complete_Sup_Sup(fun(E,fun(B,$o))),aa(set(F4),set(fun(E,fun(B,$o))),image2(F4,fun(E,fun(B,$o)),R2),Ia2))) = aa(set(fun(A,fun(B,$o))),fun(A,fun(B,$o)),complete_Sup_Sup(fun(A,fun(B,$o))),aa(set(F4),set(fun(A,fun(B,$o))),image2(F4,fun(A,fun(B,$o)),aa(fun(F4,fun(E,fun(B,$o))),fun(F4,fun(A,fun(B,$o))),aTP_Lamp_aes(fun(A,fun(E,$o)),fun(fun(F4,fun(E,fun(B,$o))),fun(F4,fun(A,fun(B,$o)))),S2),R2)),Ia2)) ) ).

% relcompp_SUP_distrib
tff(fact_7907_relcompp__relcomp__eq,axiom,
    ! [A: $tType,B: $tType,E: $tType,R2: set(product_prod(A,E)),S2: set(product_prod(E,B)),X4: A,Xa4: B] :
      ( aa(B,$o,aa(A,fun(B,$o),relcompp(A,E,B,aTP_Lamp_aet(set(product_prod(A,E)),fun(A,fun(E,$o)),R2),aTP_Lamp_aeu(set(product_prod(E,B)),fun(E,fun(B,$o)),S2)),X4),Xa4)
    <=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Xa4),relcomp(A,E,B,R2,S2)) ) ).

% relcompp_relcomp_eq
tff(fact_7908_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_7909_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_7910_ord_Olexordp__eq__simps_I3_J,axiom,
    ! [A: $tType,Less: fun(A,fun(A,$o)),Xa: A,Xsa: list(A)] : ~ aa(list(A),$o,aa(list(A),fun(list(A),$o),lexordp_eq(A,Less),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa)),nil(A)) ).

% ord.lexordp_eq_simps(3)
tff(fact_7911_ord_Olexordp__eq__simps_I4_J,axiom,
    ! [A: $tType,Less: fun(A,fun(A,$o)),Xa: A,Xsa: list(A),Y: A,Ys: list(A)] :
      ( aa(list(A),$o,aa(list(A),fun(list(A),$o),lexordp_eq(A,Less),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys))
    <=> ( aa(A,$o,aa(A,fun(A,$o),Less,Xa),Y)
        | ( ~ aa(A,$o,aa(A,fun(A,$o),Less,Y),Xa)
          & aa(list(A),$o,aa(list(A),fun(list(A),$o),lexordp_eq(A,Less),Xsa),Ys) ) ) ) ).

% ord.lexordp_eq_simps(4)
tff(fact_7912_ord_Olexordp__eq__simps_I2_J,axiom,
    ! [A: $tType,Less: fun(A,fun(A,$o)),Xsa: list(A)] :
      ( aa(list(A),$o,aa(list(A),fun(list(A),$o),lexordp_eq(A,Less),Xsa),nil(A))
    <=> ( Xsa = nil(A) ) ) ).

% ord.lexordp_eq_simps(2)
tff(fact_7913_ord_Olexordp__eq__simps_I1_J,axiom,
    ! [A: $tType,Less: fun(A,fun(A,$o)),Ys: list(A)] : aa(list(A),$o,aa(list(A),fun(list(A),$o),lexordp_eq(A,Less),nil(A)),Ys) ).

% ord.lexordp_eq_simps(1)
tff(fact_7914_ord_Olexordp__eq_OCons__eq,axiom,
    ! [A: $tType,Less: fun(A,fun(A,$o)),Xa: A,Y: A,Xsa: list(A),Ys: list(A)] :
      ( ~ aa(A,$o,aa(A,fun(A,$o),Less,Xa),Y)
     => ( ~ aa(A,$o,aa(A,fun(A,$o),Less,Y),Xa)
       => ( aa(list(A),$o,aa(list(A),fun(list(A),$o),lexordp_eq(A,Less),Xsa),Ys)
         => aa(list(A),$o,aa(list(A),fun(list(A),$o),lexordp_eq(A,Less),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)) ) ) ) ).

% ord.lexordp_eq.Cons_eq
tff(fact_7915_ord_Olexordp__eq_OCons,axiom,
    ! [A: $tType,Less: fun(A,fun(A,$o)),Xa: A,Y: A,Xsa: list(A),Ys: list(A)] :
      ( aa(A,$o,aa(A,fun(A,$o),Less,Xa),Y)
     => aa(list(A),$o,aa(list(A),fun(list(A),$o),lexordp_eq(A,Less),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)) ) ).

% ord.lexordp_eq.Cons
tff(fact_7916_ord_Olexordp__eq_Osimps,axiom,
    ! [A: $tType,Less: fun(A,fun(A,$o)),A1: list(A),A22: list(A)] :
      ( aa(list(A),$o,aa(list(A),fun(list(A),$o),lexordp_eq(A,Less),A1),A22)
    <=> ( ? [Ys4: list(A)] :
            ( ( A1 = nil(A) )
            & ( A22 = Ys4 ) )
        | ? [X6: A,Y4: A,Xs2: list(A),Ys4: list(A)] :
            ( ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X6),Xs2) )
            & ( A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys4) )
            & aa(A,$o,aa(A,fun(A,$o),Less,X6),Y4) )
        | ? [X6: A,Y4: A,Xs2: list(A),Ys4: list(A)] :
            ( ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X6),Xs2) )
            & ( A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys4) )
            & ~ aa(A,$o,aa(A,fun(A,$o),Less,X6),Y4)
            & ~ aa(A,$o,aa(A,fun(A,$o),Less,Y4),X6)
            & aa(list(A),$o,aa(list(A),fun(list(A),$o),lexordp_eq(A,Less),Xs2),Ys4) ) ) ) ).

% ord.lexordp_eq.simps
tff(fact_7917_ord_Olexordp__eq_Ocases,axiom,
    ! [A: $tType,Less: fun(A,fun(A,$o)),A1: list(A),A22: list(A)] :
      ( aa(list(A),$o,aa(list(A),fun(list(A),$o),lexordp_eq(A,Less),A1),A22)
     => ( ( A1 != nil(A) )
       => ( ! [X: A] :
              ( ? [Xs: list(A)] : ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) )
             => ! [Y2: A] :
                  ( ? [Ys5: list(A)] : ( A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys5) )
                 => ~ aa(A,$o,aa(A,fun(A,$o),Less,X),Y2) ) )
         => ~ ! [X: A,Y2: A,Xs: list(A)] :
                ( ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) )
               => ! [Ys5: list(A)] :
                    ( ( A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys5) )
                   => ( ~ aa(A,$o,aa(A,fun(A,$o),Less,X),Y2)
                     => ( ~ aa(A,$o,aa(A,fun(A,$o),Less,Y2),X)
                       => ~ aa(list(A),$o,aa(list(A),fun(list(A),$o),lexordp_eq(A,Less),Xs),Ys5) ) ) ) ) ) ) ) ).

% ord.lexordp_eq.cases
tff(fact_7918_ord_Olexordp__into__lexordp__eq,axiom,
    ! [A: $tType,Less: fun(A,fun(A,$o)),Xsa: list(A),Ys: list(A)] :
      ( aa(list(A),$o,aa(list(A),fun(list(A),$o),lexordp2(A,Less),Xsa),Ys)
     => aa(list(A),$o,aa(list(A),fun(list(A),$o),lexordp_eq(A,Less),Xsa),Ys) ) ).

% ord.lexordp_into_lexordp_eq
tff(fact_7919_ord_Olexordp__eq_ONil,axiom,
    ! [A: $tType,Less: fun(A,fun(A,$o)),Ys: list(A)] : aa(list(A),$o,aa(list(A),fun(list(A),$o),lexordp_eq(A,Less),nil(A)),Ys) ).

% ord.lexordp_eq.Nil
tff(fact_7920_ord_Olexordp__eq__pref,axiom,
    ! [A: $tType,Less: fun(A,fun(A,$o)),U: list(A),V2: list(A)] : aa(list(A),$o,aa(list(A),fun(list(A),$o),lexordp_eq(A,Less),U),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),U),V2)) ).

% ord.lexordp_eq_pref
tff(fact_7921_ord_Olexordp__eq__refl,axiom,
    ! [A: $tType,Less: fun(A,fun(A,$o)),Xsa: list(A)] : aa(list(A),$o,aa(list(A),fun(list(A),$o),lexordp_eq(A,Less),Xsa),Xsa) ).

% ord.lexordp_eq_refl
tff(fact_7922_ord_Olexordp__eq_Ocong,axiom,
    ! [A: $tType,Less: fun(A,fun(A,$o))] : ( lexordp_eq(A,Less) = lexordp_eq(A,Less) ) ).

% ord.lexordp_eq.cong
tff(fact_7923_pure__assn__raw_Opelims_I3_J,axiom,
    ! [B: $tType,A: $tType,Xa: $o,Xaa: product_prod(A,set(B))] :
      ( ~ aa(product_prod(A,set(B)),$o,pure_assn_raw(A,B,(Xa)),Xaa)
     => ( accp(product_prod($o,product_prod(A,set(B))),pure_assn_raw_rel(A,B),aa(product_prod(A,set(B)),product_prod($o,product_prod(A,set(B))),aa($o,fun(product_prod(A,set(B)),product_prod($o,product_prod(A,set(B)))),product_Pair($o,product_prod(A,set(B))),(Xa)),Xaa))
       => ~ ! [H: A,As: set(B)] :
              ( ( Xaa = aa(set(B),product_prod(A,set(B)),aa(A,fun(set(B),product_prod(A,set(B))),product_Pair(A,set(B)),H),As) )
             => ( accp(product_prod($o,product_prod(A,set(B))),pure_assn_raw_rel(A,B),aa(product_prod(A,set(B)),product_prod($o,product_prod(A,set(B))),aa($o,fun(product_prod(A,set(B)),product_prod($o,product_prod(A,set(B)))),product_Pair($o,product_prod(A,set(B))),(Xa)),aa(set(B),product_prod(A,set(B)),aa(A,fun(set(B),product_prod(A,set(B))),product_Pair(A,set(B)),H),As)))
               => ( ( As = bot_bot(set(B)) )
                  & (Xa) ) ) ) ) ) ).

% pure_assn_raw.pelims(3)
tff(fact_7924_pure__assn__raw_Opelims_I2_J,axiom,
    ! [B: $tType,A: $tType,Xa: $o,Xaa: product_prod(A,set(B))] :
      ( aa(product_prod(A,set(B)),$o,pure_assn_raw(A,B,(Xa)),Xaa)
     => ( accp(product_prod($o,product_prod(A,set(B))),pure_assn_raw_rel(A,B),aa(product_prod(A,set(B)),product_prod($o,product_prod(A,set(B))),aa($o,fun(product_prod(A,set(B)),product_prod($o,product_prod(A,set(B)))),product_Pair($o,product_prod(A,set(B))),(Xa)),Xaa))
       => ~ ! [H: A,As: set(B)] :
              ( ( Xaa = aa(set(B),product_prod(A,set(B)),aa(A,fun(set(B),product_prod(A,set(B))),product_Pair(A,set(B)),H),As) )
             => ( accp(product_prod($o,product_prod(A,set(B))),pure_assn_raw_rel(A,B),aa(product_prod(A,set(B)),product_prod($o,product_prod(A,set(B))),aa($o,fun(product_prod(A,set(B)),product_prod($o,product_prod(A,set(B)))),product_Pair($o,product_prod(A,set(B))),(Xa)),aa(set(B),product_prod(A,set(B)),aa(A,fun(set(B),product_prod(A,set(B))),product_Pair(A,set(B)),H),As)))
               => ~ ( ( As = bot_bot(set(B)) )
                    & (Xa) ) ) ) ) ) ).

% pure_assn_raw.pelims(2)
tff(fact_7925_pure__assn__raw_Opelims_I1_J,axiom,
    ! [A: $tType,B: $tType,Xa: $o,Xaa: product_prod(A,set(B)),Y: $o] :
      ( ( aa(product_prod(A,set(B)),$o,pure_assn_raw(A,B,(Xa)),Xaa)
      <=> (Y) )
     => ( accp(product_prod($o,product_prod(A,set(B))),pure_assn_raw_rel(A,B),aa(product_prod(A,set(B)),product_prod($o,product_prod(A,set(B))),aa($o,fun(product_prod(A,set(B)),product_prod($o,product_prod(A,set(B)))),product_Pair($o,product_prod(A,set(B))),(Xa)),Xaa))
       => ~ ! [H: A,As: set(B)] :
              ( ( Xaa = aa(set(B),product_prod(A,set(B)),aa(A,fun(set(B),product_prod(A,set(B))),product_Pair(A,set(B)),H),As) )
             => ( ( (Y)
                <=> ( ( As = bot_bot(set(B)) )
                    & (Xa) ) )
               => ~ accp(product_prod($o,product_prod(A,set(B))),pure_assn_raw_rel(A,B),aa(product_prod(A,set(B)),product_prod($o,product_prod(A,set(B))),aa($o,fun(product_prod(A,set(B)),product_prod($o,product_prod(A,set(B)))),product_Pair($o,product_prod(A,set(B))),(Xa)),aa(set(B),product_prod(A,set(B)),aa(A,fun(set(B),product_prod(A,set(B))),product_Pair(A,set(B)),H),As))) ) ) ) ) ).

% pure_assn_raw.pelims(1)
tff(fact_7926_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_7927_is__arg__max__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F2: fun(A,B),Pa: fun(A,$o),Xa: A] :
          ( lattic501386751176901750rg_max(A,B,F2,Pa,Xa)
        <=> ( aa(A,$o,Pa,Xa)
            & ~ ? [Y4: A] :
                  ( aa(A,$o,Pa,Y4)
                  & aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,Xa)),aa(A,B,F2,Y4)) ) ) ) ) ).

% is_arg_max_def
tff(fact_7928_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)
     => condit16957441358409770ng_bdd(A,Less_eq,bot_bot(set(A))) ) ).

% preordering_bdd.empty
tff(fact_7929_lazI,axiom,
    ! [A: $tType,B: $tType,A3: list(A),B3: list(B),Pa: fun(A,fun(B,$o))] :
      ( ( aa(list(A),nat,size_size(list(A)),A3) = aa(list(B),nat,size_size(list(B)),B3) )
     => ( ! [I3: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(list(B),nat,size_size(list(B)),B3))
           => aa(B,$o,aa(A,fun(B,$o),Pa,aa(nat,A,nth(A,A3),I3)),aa(nat,B,nth(B,B3),I3)) )
       => list_all_zip(A,B,Pa,A3,B3) ) ) ).

% lazI
tff(fact_7930_laz__conj,axiom,
    ! [A: $tType,B: $tType,Pa: fun(A,fun(B,$o)),Qa: fun(A,fun(B,$o)),A3: list(A),B3: list(B)] :
      ( list_all_zip(A,B,aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aTP_Lamp_aev(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),Pa),Qa),A3,B3)
    <=> ( list_all_zip(A,B,Pa,A3,B3)
        & list_all_zip(A,B,Qa,A3,B3) ) ) ).

% laz_conj
tff(fact_7931_laz__weak__Pa,axiom,
    ! [B: $tType,A: $tType,Pa: fun(A,$o),Aa2: list(A),B2: list(B)] :
      ( list_all_zip(A,B,aTP_Lamp_aew(fun(A,$o),fun(A,fun(B,$o)),Pa),Aa2,B2)
    <=> ( ( aa(list(A),nat,size_size(list(A)),Aa2) = aa(list(B),nat,size_size(list(B)),B2) )
        & ! [X6: A] :
            ( member(A,X6,aa(list(A),set(A),set2(A),Aa2))
           => aa(A,$o,Pa,X6) ) ) ) ).

% laz_weak_Pa
tff(fact_7932_laz__weak__Pb,axiom,
    ! [A: $tType,B: $tType,Pa: fun(B,$o),Aa2: list(A),B2: list(B)] :
      ( list_all_zip(A,B,aTP_Lamp_ym(fun(B,$o),fun(A,fun(B,$o)),Pa),Aa2,B2)
    <=> ( ( aa(list(A),nat,size_size(list(A)),Aa2) = aa(list(B),nat,size_size(list(B)),B2) )
        & ! [X6: B] :
            ( member(B,X6,aa(list(B),set(B),set2(B),B2))
           => aa(B,$o,Pa,X6) ) ) ) ).

% laz_weak_Pb
tff(fact_7933_laz__len,axiom,
    ! [A: $tType,B: $tType,Pa: fun(A,fun(B,$o)),A3: list(A),B3: list(B)] :
      ( list_all_zip(A,B,Pa,A3,B3)
     => ( aa(list(A),nat,size_size(list(A)),A3) = aa(list(B),nat,size_size(list(B)),B3) ) ) ).

% laz_len
tff(fact_7934_laz__eq,axiom,
    ! [A: $tType,A3: list(A),B3: list(A)] :
      ( list_all_zip(A,A,fequal(A),A3,B3)
    <=> ( A3 = B3 ) ) ).

% laz_eq
tff(fact_7935_list__all__zip_Osimps_I1_J,axiom,
    ! [A: $tType,B: $tType,Pa: fun(A,fun(B,$o))] : list_all_zip(A,B,Pa,nil(A),nil(B)) ).

% list_all_zip.simps(1)
tff(fact_7936_laz__swap__ex,axiom,
    ! [B: $tType,A: $tType,E: $tType,Pa: fun(A,fun(B,fun(E,$o))),Aa2: list(A),B2: list(B)] :
      ( list_all_zip(A,B,aTP_Lamp_aex(fun(A,fun(B,fun(E,$o))),fun(A,fun(B,$o)),Pa),Aa2,B2)
     => ~ ! [C9: list(E)] :
            ( list_all_zip(A,E,aTP_Lamp_aey(fun(A,fun(B,fun(E,$o))),fun(A,fun(E,$o)),Pa),Aa2,C9)
           => ~ list_all_zip(B,E,aTP_Lamp_aez(fun(A,fun(B,fun(E,$o))),fun(B,fun(E,$o)),Pa),B2,C9) ) ) ).

% laz_swap_ex
tff(fact_7937_list__all__zip_Osimps_I2_J,axiom,
    ! [A: $tType,B: $tType,Pa: fun(A,fun(B,$o)),A3: A,As2: list(A),B3: B,Bs: list(B)] :
      ( list_all_zip(A,B,Pa,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),As2),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B3),Bs))
    <=> ( aa(B,$o,aa(A,fun(B,$o),Pa,A3),B3)
        & list_all_zip(A,B,Pa,As2,Bs) ) ) ).

% list_all_zip.simps(2)
tff(fact_7938_list__all__zip_Oelims_I3_J,axiom,
    ! [A: $tType,B: $tType,Xa: fun(A,fun(B,$o)),Xaa: list(A),Xb: list(B)] :
      ( ~ list_all_zip(A,B,Xa,Xaa,Xb)
     => ( ! [A4: A,As: list(A)] :
            ( ( Xaa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A4),As) )
           => ! [B4: B,Bs2: list(B)] :
                ( ( Xb = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B4),Bs2) )
               => ( aa(B,$o,aa(A,fun(B,$o),Xa,A4),B4)
                  & list_all_zip(A,B,Xa,As,Bs2) ) ) )
       => ( ( ? [V3: A,Va: list(A)] : ( Xaa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V3),Va) )
           => ( Xb != nil(B) ) )
         => ~ ( ( Xaa = nil(A) )
             => ! [V3: B,Va: list(B)] : ( Xb != aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),V3),Va) ) ) ) ) ) ).

% list_all_zip.elims(3)
tff(fact_7939_list__all__zip_Oelims_I2_J,axiom,
    ! [A: $tType,B: $tType,Xa: fun(A,fun(B,$o)),Xaa: list(A),Xb: list(B)] :
      ( list_all_zip(A,B,Xa,Xaa,Xb)
     => ( ( ( Xaa = nil(A) )
         => ( Xb != nil(B) ) )
       => ~ ! [A4: A,As: list(A)] :
              ( ( Xaa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A4),As) )
             => ! [B4: B,Bs2: list(B)] :
                  ( ( Xb = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B4),Bs2) )
                 => ~ ( aa(B,$o,aa(A,fun(B,$o),Xa,A4),B4)
                      & list_all_zip(A,B,Xa,As,Bs2) ) ) ) ) ) ).

% list_all_zip.elims(2)
tff(fact_7940_list__all__zip_Oelims_I1_J,axiom,
    ! [B: $tType,A: $tType,Xa: fun(A,fun(B,$o)),Xaa: list(A),Xb: list(B),Y: $o] :
      ( ( list_all_zip(A,B,Xa,Xaa,Xb)
      <=> (Y) )
     => ( ( ( Xaa = nil(A) )
         => ( ( Xb = nil(B) )
           => ~ (Y) ) )
       => ( ! [A4: A,As: list(A)] :
              ( ( Xaa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A4),As) )
             => ! [B4: B,Bs2: list(B)] :
                  ( ( Xb = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B4),Bs2) )
                 => ( (Y)
                  <=> ~ ( aa(B,$o,aa(A,fun(B,$o),Xa,A4),B4)
                        & list_all_zip(A,B,Xa,As,Bs2) ) ) ) )
         => ( ( ? [V3: A,Va: list(A)] : ( Xaa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V3),Va) )
             => ( ( Xb = nil(B) )
               => (Y) ) )
           => ~ ( ( Xaa = nil(A) )
               => ( ? [V3: B,Va: list(B)] : ( Xb = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),V3),Va) )
                 => (Y) ) ) ) ) ) ) ).

% list_all_zip.elims(1)
tff(fact_7941_list__all__zip_Osimps_I4_J,axiom,
    ! [A: $tType,B: $tType,Pa: fun(A,fun(B,$o)),V2: B,Va2: list(B)] : ~ list_all_zip(A,B,Pa,nil(A),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),V2),Va2)) ).

% list_all_zip.simps(4)
tff(fact_7942_list__all__zip_Osimps_I3_J,axiom,
    ! [A: $tType,B: $tType,Pa: fun(A,fun(B,$o)),V2: A,Va2: list(A)] : ~ list_all_zip(A,B,Pa,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V2),Va2),nil(B)) ).

% list_all_zip.simps(3)
tff(fact_7943_list__all__zip__map1,axiom,
    ! [E: $tType,A: $tType,B: $tType,Pa: fun(A,fun(B,$o)),F2: fun(E,A),As2: list(E),Bs: list(B)] :
      ( list_all_zip(A,B,Pa,aa(list(E),list(A),aa(fun(E,A),fun(list(E),list(A)),map(E,A),F2),As2),Bs)
    <=> list_all_zip(E,B,aa(fun(E,A),fun(E,fun(B,$o)),aTP_Lamp_rt(fun(A,fun(B,$o)),fun(fun(E,A),fun(E,fun(B,$o))),Pa),F2),As2,Bs) ) ).

% list_all_zip_map1
tff(fact_7944_list__all__zip__map2,axiom,
    ! [A: $tType,B: $tType,E: $tType,Pa: fun(A,fun(B,$o)),As2: list(A),F2: fun(E,B),Bs: list(E)] :
      ( list_all_zip(A,B,Pa,As2,aa(list(E),list(B),aa(fun(E,B),fun(list(E),list(B)),map(E,B),F2),Bs))
    <=> list_all_zip(A,E,aa(fun(E,B),fun(A,fun(E,$o)),aTP_Lamp_rs(fun(A,fun(B,$o)),fun(fun(E,B),fun(A,fun(E,$o))),Pa),F2),As2,Bs) ) ).

% list_all_zip_map2
tff(fact_7945_list__all__zip__alt,axiom,
    ! [A: $tType,B: $tType,Pa: fun(A,fun(B,$o)),As2: list(A),Bs: list(B)] :
      ( list_all_zip(A,B,Pa,As2,Bs)
    <=> ( ( aa(list(A),nat,size_size(list(A)),As2) = aa(list(B),nat,size_size(list(B)),Bs) )
        & ! [I2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),As2))
           => aa(B,$o,aa(A,fun(B,$o),Pa,aa(nat,A,nth(A,As2),I2)),aa(nat,B,nth(B,Bs),I2)) ) ) ) ).

% list_all_zip_alt
tff(fact_7946_list__all__zip_Opelims_I3_J,axiom,
    ! [A: $tType,B: $tType,Xa: fun(A,fun(B,$o)),Xaa: list(A),Xb: list(B)] :
      ( ~ list_all_zip(A,B,Xa,Xaa,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,$o)),product_prod(list(A),list(B))),Xa),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xaa),Xb)))
       => ( ! [A4: A,As: list(A)] :
              ( ( Xaa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A4),As) )
             => ! [B4: B,Bs2: list(B)] :
                  ( ( Xb = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B4),Bs2) )
                 => ( accp(product_prod(fun(A,fun(B,$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))),Xa),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A4),As)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B4),Bs2))))
                   => ( aa(B,$o,aa(A,fun(B,$o),Xa,A4),B4)
                      & list_all_zip(A,B,Xa,As,Bs2) ) ) ) )
         => ( ! [V3: A,Va: list(A)] :
                ( ( Xaa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V3),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,$o)),product_prod(list(A),list(B))),Xa),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V3),Va)),nil(B)))) ) )
           => ~ ( ( Xaa = nil(A) )
               => ! [V3: B,Va: list(B)] :
                    ( ( Xb = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),V3),Va) )
                   => ~ accp(product_prod(fun(A,fun(B,$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))),Xa),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),V3),Va)))) ) ) ) ) ) ) ).

% list_all_zip.pelims(3)
tff(fact_7947_list__all__zip_Opelims_I2_J,axiom,
    ! [A: $tType,B: $tType,Xa: fun(A,fun(B,$o)),Xaa: list(A),Xb: list(B)] :
      ( list_all_zip(A,B,Xa,Xaa,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,$o)),product_prod(list(A),list(B))),Xa),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xaa),Xb)))
       => ( ( ( Xaa = nil(A) )
           => ( ( Xb = nil(B) )
             => ~ accp(product_prod(fun(A,fun(B,$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))),Xa),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)))) ) )
         => ~ ! [A4: A,As: list(A)] :
                ( ( Xaa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A4),As) )
               => ! [B4: B,Bs2: list(B)] :
                    ( ( Xb = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B4),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,$o)),product_prod(list(A),list(B))),Xa),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A4),As)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B4),Bs2))))
                     => ~ ( aa(B,$o,aa(A,fun(B,$o),Xa,A4),B4)
                          & list_all_zip(A,B,Xa,As,Bs2) ) ) ) ) ) ) ) ).

% list_all_zip.pelims(2)
tff(fact_7948_list__all__zip_Opelims_I1_J,axiom,
    ! [A: $tType,B: $tType,Xa: fun(A,fun(B,$o)),Xaa: list(A),Xb: list(B),Y: $o] :
      ( ( list_all_zip(A,B,Xa,Xaa,Xb)
      <=> (Y) )
     => ( accp(product_prod(fun(A,fun(B,$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))),Xa),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)),Xaa),Xb)))
       => ( ( ( Xaa = 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,$o)),product_prod(list(A),list(B))),Xa),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),nil(A)),nil(B)))) ) ) )
         => ( ! [A4: A,As: list(A)] :
                ( ( Xaa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A4),As) )
               => ! [B4: B,Bs2: list(B)] :
                    ( ( Xb = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B4),Bs2) )
                   => ( ( (Y)
                      <=> ( aa(B,$o,aa(A,fun(B,$o),Xa,A4),B4)
                          & list_all_zip(A,B,Xa,As,Bs2) ) )
                     => ~ accp(product_prod(fun(A,fun(B,$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))),Xa),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A4),As)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B4),Bs2)))) ) ) )
           => ( ! [V3: A,Va: list(A)] :
                  ( ( Xaa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V3),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,$o)),product_prod(list(A),list(B))),Xa),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V3),Va)),nil(B)))) ) ) )
             => ~ ( ( Xaa = nil(A) )
                 => ! [V3: B,Va: list(B)] :
                      ( ( Xb = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),V3),Va) )
                     => ( ~ (Y)
                       => ~ accp(product_prod(fun(A,fun(B,$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))),Xa),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),V3),Va)))) ) ) ) ) ) ) ) ) ).

% list_all_zip.pelims(1)
tff(fact_7949_less__option__def,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xa: option(A),Y: option(A)] :
          ( aa(option(A),$o,aa(option(A),fun(option(A),$o),ord_less(option(A)),Xa),Y)
        <=> case_option($o,A,$false,aTP_Lamp_afa(option(A),fun(A,$o),Xa),Y) ) ) ).

% less_option_def
tff(fact_7950_Pair__transfer,axiom,
    ! [A: $tType,E: $tType,F4: $tType,B: $tType,Aa2: fun(A,fun(B,$o)),B2: fun(E,fun(F4,$o))] : aa(fun(B,fun(F4,product_prod(B,F4))),$o,aa(fun(A,fun(E,product_prod(A,E))),fun(fun(B,fun(F4,product_prod(B,F4))),$o),bNF_rel_fun(A,B,fun(E,product_prod(A,E)),fun(F4,product_prod(B,F4)),Aa2,bNF_rel_fun(E,F4,product_prod(A,E),product_prod(B,F4),B2,basic_rel_prod(A,B,E,F4,Aa2,B2))),product_Pair(A,E)),product_Pair(B,F4)) ).

% Pair_transfer
tff(fact_7951_rel__prod__inject,axiom,
    ! [B: $tType,A: $tType,E: $tType,F4: $tType,R1: fun(A,fun(B,$o)),R22: fun(E,fun(F4,$o)),A3: A,B3: E,C2: B,D3: F4] :
      ( aa(product_prod(B,F4),$o,aa(product_prod(A,E),fun(product_prod(B,F4),$o),basic_rel_prod(A,B,E,F4,R1,R22),aa(E,product_prod(A,E),aa(A,fun(E,product_prod(A,E)),product_Pair(A,E),A3),B3)),aa(F4,product_prod(B,F4),aa(B,fun(F4,product_prod(B,F4)),product_Pair(B,F4),C2),D3))
    <=> ( aa(B,$o,aa(A,fun(B,$o),R1,A3),C2)
        & aa(F4,$o,aa(E,fun(F4,$o),R22,B3),D3) ) ) ).

% rel_prod_inject
tff(fact_7952_prod__filter__parametric,axiom,
    ! [A: $tType,E: $tType,F4: $tType,B: $tType,R: fun(A,fun(B,$o)),S: fun(E,fun(F4,$o))] : aa(fun(filter(B),fun(filter(F4),filter(product_prod(B,F4)))),$o,aa(fun(filter(A),fun(filter(E),filter(product_prod(A,E)))),fun(fun(filter(B),fun(filter(F4),filter(product_prod(B,F4)))),$o),bNF_rel_fun(filter(A),filter(B),fun(filter(E),filter(product_prod(A,E))),fun(filter(F4),filter(product_prod(B,F4))),rel_filter(A,B,R),bNF_rel_fun(filter(E),filter(F4),filter(product_prod(A,E)),filter(product_prod(B,F4)),rel_filter(E,F4,S),rel_filter(product_prod(A,E),product_prod(B,F4),basic_rel_prod(A,B,E,F4,R,S)))),prod_filter(A,E)),prod_filter(B,F4)) ).

% prod_filter_parametric
tff(fact_7953_product__transfer,axiom,
    ! [A: $tType,E: $tType,F4: $tType,B: $tType,Aa2: fun(A,fun(B,$o)),B2: fun(E,fun(F4,$o))] : aa(fun(list(B),fun(list(F4),list(product_prod(B,F4)))),$o,aa(fun(list(A),fun(list(E),list(product_prod(A,E)))),fun(fun(list(B),fun(list(F4),list(product_prod(B,F4)))),$o),bNF_rel_fun(list(A),list(B),fun(list(E),list(product_prod(A,E))),fun(list(F4),list(product_prod(B,F4))),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Aa2),bNF_rel_fun(list(E),list(F4),list(product_prod(A,E)),list(product_prod(B,F4)),aa(fun(E,fun(F4,$o)),fun(list(E),fun(list(F4),$o)),list_all2(E,F4),B2),aa(fun(product_prod(A,E),fun(product_prod(B,F4),$o)),fun(list(product_prod(A,E)),fun(list(product_prod(B,F4)),$o)),list_all2(product_prod(A,E),product_prod(B,F4)),basic_rel_prod(A,B,E,F4,Aa2,B2)))),product(A,E)),product(B,F4)) ).

% product_transfer
tff(fact_7954_zip__transfer,axiom,
    ! [A: $tType,E: $tType,F4: $tType,B: $tType,Aa2: fun(A,fun(B,$o)),B2: fun(E,fun(F4,$o))] : aa(fun(list(B),fun(list(F4),list(product_prod(B,F4)))),$o,aa(fun(list(A),fun(list(E),list(product_prod(A,E)))),fun(fun(list(B),fun(list(F4),list(product_prod(B,F4)))),$o),bNF_rel_fun(list(A),list(B),fun(list(E),list(product_prod(A,E))),fun(list(F4),list(product_prod(B,F4))),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Aa2),bNF_rel_fun(list(E),list(F4),list(product_prod(A,E)),list(product_prod(B,F4)),aa(fun(E,fun(F4,$o)),fun(list(E),fun(list(F4),$o)),list_all2(E,F4),B2),aa(fun(product_prod(A,E),fun(product_prod(B,F4),$o)),fun(list(product_prod(A,E)),fun(list(product_prod(B,F4)),$o)),list_all2(product_prod(A,E),product_prod(B,F4)),basic_rel_prod(A,B,E,F4,Aa2,B2)))),zip(A,E)),zip(B,F4)) ).

% zip_transfer
tff(fact_7955_rtrancl__parametric,axiom,
    ! [A: $tType,B: $tType,Aa2: fun(A,fun(B,$o))] :
      ( bi_unique(A,B,Aa2)
     => ( bi_total(A,B,Aa2)
       => aa(fun(set(product_prod(B,B)),set(product_prod(B,B))),$o,aa(fun(set(product_prod(A,A)),set(product_prod(A,A))),fun(fun(set(product_prod(B,B)),set(product_prod(B,B))),$o),bNF_rel_fun(set(product_prod(A,A)),set(product_prod(B,B)),set(product_prod(A,A)),set(product_prod(B,B)),bNF_rel_set(product_prod(A,A),product_prod(B,B),basic_rel_prod(A,B,A,B,Aa2,Aa2)),bNF_rel_set(product_prod(A,A),product_prod(B,B),basic_rel_prod(A,B,A,B,Aa2,Aa2))),transitive_rtrancl(A)),transitive_rtrancl(B)) ) ) ).

% rtrancl_parametric
tff(fact_7956_partition__transfer,axiom,
    ! [A: $tType,B: $tType,Aa2: fun(A,fun(B,$o))] : aa(fun(fun(B,$o),fun(list(B),product_prod(list(B),list(B)))),$o,aa(fun(fun(A,$o),fun(list(A),product_prod(list(A),list(A)))),fun(fun(fun(B,$o),fun(list(B),product_prod(list(B),list(B)))),$o),bNF_rel_fun(fun(A,$o),fun(B,$o),fun(list(A),product_prod(list(A),list(A))),fun(list(B),product_prod(list(B),list(B))),bNF_rel_fun(A,B,$o,$o,Aa2,fequal($o)),bNF_rel_fun(list(A),list(B),product_prod(list(A),list(A)),product_prod(list(B),list(B)),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Aa2),basic_rel_prod(list(A),list(B),list(A),list(B),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Aa2),aa(fun(A,fun(B,$o)),fun(list(A),fun(list(B),$o)),list_all2(A,B),Aa2)))),partition(A)),partition(B)) ).

% partition_transfer
tff(fact_7957_rel__prod_Ocases,axiom,
    ! [B: $tType,A: $tType,E: $tType,F4: $tType,R1: fun(A,fun(B,$o)),R22: fun(E,fun(F4,$o)),A1: product_prod(A,E),A22: product_prod(B,F4)] :
      ( aa(product_prod(B,F4),$o,aa(product_prod(A,E),fun(product_prod(B,F4),$o),basic_rel_prod(A,B,E,F4,R1,R22),A1),A22)
     => ~ ! [A4: A,B4: B,C5: E] :
            ( ( A1 = aa(E,product_prod(A,E),aa(A,fun(E,product_prod(A,E)),product_Pair(A,E),A4),C5) )
           => ! [D4: F4] :
                ( ( A22 = aa(F4,product_prod(B,F4),aa(B,fun(F4,product_prod(B,F4)),product_Pair(B,F4),B4),D4) )
               => ( aa(B,$o,aa(A,fun(B,$o),R1,A4),B4)
                 => ~ aa(F4,$o,aa(E,fun(F4,$o),R22,C5),D4) ) ) ) ) ).

% rel_prod.cases
tff(fact_7958_rel__prod_Osimps,axiom,
    ! [B: $tType,A: $tType,E: $tType,F4: $tType,R1: fun(A,fun(B,$o)),R22: fun(E,fun(F4,$o)),A1: product_prod(A,E),A22: product_prod(B,F4)] :
      ( aa(product_prod(B,F4),$o,aa(product_prod(A,E),fun(product_prod(B,F4),$o),basic_rel_prod(A,B,E,F4,R1,R22),A1),A22)
    <=> ? [A7: A,B8: B,C6: E,D6: F4] :
          ( ( A1 = aa(E,product_prod(A,E),aa(A,fun(E,product_prod(A,E)),product_Pair(A,E),A7),C6) )
          & ( A22 = aa(F4,product_prod(B,F4),aa(B,fun(F4,product_prod(B,F4)),product_Pair(B,F4),B8),D6) )
          & aa(B,$o,aa(A,fun(B,$o),R1,A7),B8)
          & aa(F4,$o,aa(E,fun(F4,$o),R22,C6),D6) ) ) ).

% rel_prod.simps
tff(fact_7959_rel__prod_Ointros,axiom,
    ! [E: $tType,A: $tType,B: $tType,F4: $tType,R1: fun(A,fun(B,$o)),A3: A,B3: B,R22: fun(E,fun(F4,$o)),C2: E,D3: F4] :
      ( aa(B,$o,aa(A,fun(B,$o),R1,A3),B3)
     => ( aa(F4,$o,aa(E,fun(F4,$o),R22,C2),D3)
       => aa(product_prod(B,F4),$o,aa(product_prod(A,E),fun(product_prod(B,F4),$o),basic_rel_prod(A,B,E,F4,R1,R22),aa(E,product_prod(A,E),aa(A,fun(E,product_prod(A,E)),product_Pair(A,E),A3),C2)),aa(F4,product_prod(B,F4),aa(B,fun(F4,product_prod(B,F4)),product_Pair(B,F4),B3),D3)) ) ) ).

% rel_prod.intros
tff(fact_7960_refines__def,axiom,
    ! [A: $tType,P2: heap_Time_Heap(A),Q2: heap_Time_Heap(A)] :
      ( refine_Imp_refines(A,P2,Q2)
    <=> ! [H3: heap_ext(product_unit)] : case_option($o,product_prod(A,product_prod(heap_ext(product_unit),nat)),$true,aa(fun(A,fun(product_prod(heap_ext(product_unit),nat),$o)),fun(product_prod(A,product_prod(heap_ext(product_unit),nat)),$o),product_case_prod(A,product_prod(heap_ext(product_unit),nat),$o),aa(heap_ext(product_unit),fun(A,fun(product_prod(heap_ext(product_unit),nat),$o)),aTP_Lamp_afc(heap_Time_Heap(A),fun(heap_ext(product_unit),fun(A,fun(product_prod(heap_ext(product_unit),nat),$o))),P2),H3)),aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,Q2),H3)) ) ).

% refines_def
tff(fact_7961_group_Oaxioms_I2_J,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A)] :
      ( group(A,F2,Z2,Inverse)
     => group_axioms(A,F2,Z2,Inverse) ) ).

% group.axioms(2)
tff(fact_7962_group__axioms__def,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A)] :
      ( group_axioms(A,F2,Z2,Inverse)
    <=> ( ! [A7: A] : ( aa(A,A,aa(A,fun(A,A),F2,Z2),A7) = A7 )
        & ! [A7: A] : ( aa(A,A,aa(A,fun(A,A),F2,aa(A,A,Inverse,A7)),A7) = Z2 ) ) ) ).

% group_axioms_def
tff(fact_7963_group__axioms_Ointro,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A)] :
      ( ! [A4: A] : ( aa(A,A,aa(A,fun(A,A),F2,Z2),A4) = A4 )
     => ( ! [A4: A] : ( aa(A,A,aa(A,fun(A,A),F2,aa(A,A,Inverse,A4)),A4) = Z2 )
       => group_axioms(A,F2,Z2,Inverse) ) ) ).

% group_axioms.intro
tff(fact_7964_group_Ointro,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A)] :
      ( semigroup(A,F2)
     => ( group_axioms(A,F2,Z2,Inverse)
       => group(A,F2,Z2,Inverse) ) ) ).

% group.intro
tff(fact_7965_group__def,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A)] :
      ( group(A,F2,Z2,Inverse)
    <=> ( semigroup(A,F2)
        & group_axioms(A,F2,Z2,Inverse) ) ) ).

% group_def
tff(fact_7966_add_Osemigroup__axioms,axiom,
    ! [A: $tType] :
      ( semigroup_add(A)
     => semigroup(A,plus_plus(A)) ) ).

% add.semigroup_axioms
tff(fact_7967_mult_Osemigroup__axioms,axiom,
    ! [A: $tType] :
      ( semigroup_mult(A)
     => semigroup(A,times_times(A)) ) ).

% mult.semigroup_axioms
tff(fact_7968_inf_Osemigroup__axioms,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => semigroup(A,inf_inf(A)) ) ).

% inf.semigroup_axioms
tff(fact_7969_max_Osemigroup__axioms,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => semigroup(A,ord_max(A)) ) ).

% max.semigroup_axioms
tff(fact_7970_append_Osemigroup__axioms,axiom,
    ! [A: $tType] : semigroup(list(A),append(A)) ).

% append.semigroup_axioms
tff(fact_7971_semigroup_Oassoc,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),A3: A,B3: A,C2: A] :
      ( semigroup(A,F2)
     => ( aa(A,A,aa(A,fun(A,A),F2,aa(A,A,aa(A,fun(A,A),F2,A3),B3)),C2) = aa(A,A,aa(A,fun(A,A),F2,A3),aa(A,A,aa(A,fun(A,A),F2,B3),C2)) ) ) ).

% semigroup.assoc
tff(fact_7972_semigroup_Ointro,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A))] :
      ( ! [A4: A,B4: A,C5: A] : ( aa(A,A,aa(A,fun(A,A),F2,aa(A,A,aa(A,fun(A,A),F2,A4),B4)),C5) = aa(A,A,aa(A,fun(A,A),F2,A4),aa(A,A,aa(A,fun(A,A),F2,B4),C5)) )
     => semigroup(A,F2) ) ).

% semigroup.intro
tff(fact_7973_semigroup__def,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A))] :
      ( semigroup(A,F2)
    <=> ! [A7: A,B8: A,C6: A] : ( aa(A,A,aa(A,fun(A,A),F2,aa(A,A,aa(A,fun(A,A),F2,A7),B8)),C6) = aa(A,A,aa(A,fun(A,A),F2,A7),aa(A,A,aa(A,fun(A,A),F2,B8),C6)) ) ) ).

% semigroup_def
tff(fact_7974_min_Osemigroup__axioms,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => semigroup(A,ord_min(A)) ) ).

% min.semigroup_axioms
tff(fact_7975_group_Oaxioms_I1_J,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A)] :
      ( group(A,F2,Z2,Inverse)
     => semigroup(A,F2) ) ).

% group.axioms(1)
tff(fact_7976_sup_Osemigroup__axioms,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => semigroup(A,sup_sup(A)) ) ).

% sup.semigroup_axioms
tff(fact_7977_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_7978_bot__pred__def,axiom,
    ! [A: $tType] : ( bot_bot(pred(A)) = pred2(A,bot_bot(fun(A,$o))) ) ).

% bot_pred_def
tff(fact_7979_ordering__top__axioms__def,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,$o)),Top: A] :
      ( ordering_top_axioms(A,Less_eq,Top)
    <=> ! [A7: A] : aa(A,$o,aa(A,fun(A,$o),Less_eq,A7),Top) ) ).

% ordering_top_axioms_def
tff(fact_7980_ordering__top__axioms_Ointro,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,$o)),Top: A] :
      ( ! [A4: A] : aa(A,$o,aa(A,fun(A,$o),Less_eq,A4),Top)
     => ordering_top_axioms(A,Less_eq,Top) ) ).

% ordering_top_axioms.intro
tff(fact_7981_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_7982_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_7983_dual__order_Oordering__axioms,axiom,
    ! [A: $tType] :
      ( order(A)
     => ordering(A,aTP_Lamp_afd(A,fun(A,$o)),aTP_Lamp_afe(A,fun(A,$o))) ) ).

% dual_order.ordering_axioms
tff(fact_7984_order_Oordering__axioms,axiom,
    ! [A: $tType] :
      ( order(A)
     => ordering(A,ord_less_eq(A),ord_less(A)) ) ).

% order.ordering_axioms
tff(fact_7985_ordering_Oeq__iff,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A3: A,B3: A] :
      ( ordering(A,Less_eq,Less)
     => ( ( A3 = B3 )
      <=> ( aa(A,$o,aa(A,fun(A,$o),Less_eq,A3),B3)
          & aa(A,$o,aa(A,fun(A,$o),Less_eq,B3),A3) ) ) ) ).

% ordering.eq_iff
tff(fact_7986_ordering_Oantisym,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A3: A,B3: A] :
      ( ordering(A,Less_eq,Less)
     => ( aa(A,$o,aa(A,fun(A,$o),Less_eq,A3),B3)
       => ( aa(A,$o,aa(A,fun(A,$o),Less_eq,B3),A3)
         => ( A3 = B3 ) ) ) ) ).

% ordering.antisym
tff(fact_7987_ordering_Oorder__iff__strict,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A3: A,B3: A] :
      ( ordering(A,Less_eq,Less)
     => ( aa(A,$o,aa(A,fun(A,$o),Less_eq,A3),B3)
      <=> ( aa(A,$o,aa(A,fun(A,$o),Less,A3),B3)
          | ( A3 = B3 ) ) ) ) ).

% ordering.order_iff_strict
tff(fact_7988_ordering_Ostrict__iff__order,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A3: A,B3: A] :
      ( ordering(A,Less_eq,Less)
     => ( aa(A,$o,aa(A,fun(A,$o),Less,A3),B3)
      <=> ( aa(A,$o,aa(A,fun(A,$o),Less_eq,A3),B3)
          & ( A3 != B3 ) ) ) ) ).

% ordering.strict_iff_order
tff(fact_7989_ordering_Ostrict__implies__not__eq,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A3: A,B3: A] :
      ( ordering(A,Less_eq,Less)
     => ( aa(A,$o,aa(A,fun(A,$o),Less,A3),B3)
       => ( A3 != B3 ) ) ) ).

% ordering.strict_implies_not_eq
tff(fact_7990_ordering_Onot__eq__order__implies__strict,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A3: A,B3: A] :
      ( ordering(A,Less_eq,Less)
     => ( ( A3 != B3 )
       => ( aa(A,$o,aa(A,fun(A,$o),Less_eq,A3),B3)
         => aa(A,$o,aa(A,fun(A,$o),Less,A3),B3) ) ) ) ).

% ordering.not_eq_order_implies_strict
tff(fact_7991_ordering__strictI,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o))] :
      ( ! [A4: A,B4: A] :
          ( aa(A,$o,aa(A,fun(A,$o),Less_eq,A4),B4)
        <=> ( aa(A,$o,aa(A,fun(A,$o),Less,A4),B4)
            | ( A4 = B4 ) ) )
     => ( ! [A4: A,B4: A] :
            ( aa(A,$o,aa(A,fun(A,$o),Less,A4),B4)
           => ~ aa(A,$o,aa(A,fun(A,$o),Less,B4),A4) )
       => ( ! [A4: A] : ~ aa(A,$o,aa(A,fun(A,$o),Less,A4),A4)
         => ( ! [A4: A,B4: A,C5: A] :
                ( aa(A,$o,aa(A,fun(A,$o),Less,A4),B4)
               => ( aa(A,$o,aa(A,fun(A,$o),Less,B4),C5)
                 => aa(A,$o,aa(A,fun(A,$o),Less,A4),C5) ) )
           => ordering(A,Less_eq,Less) ) ) ) ) ).

% ordering_strictI
tff(fact_7992_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_7993_ordering__dualI,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o))] :
      ( ordering(A,aTP_Lamp_cn(fun(A,fun(A,$o)),fun(A,fun(A,$o)),Less_eq),aTP_Lamp_cn(fun(A,fun(A,$o)),fun(A,fun(A,$o)),Less))
     => ordering(A,Less_eq,Less) ) ).

% ordering_dualI
tff(fact_7994_ID_Opred__set,axiom,
    ! [A: $tType,Pa: fun(A,$o),X4: A] :
      ( aa(A,$o,bNF_id_bnf(fun(A,$o),Pa),X4)
    <=> ! [Xa2: A] :
          ( member(A,Xa2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),X4),bot_bot(set(A))))
         => aa(A,$o,Pa,Xa2) ) ) ).

% ID.pred_set
tff(fact_7995_Predicate_Oiterate__upto_Opelims,axiom,
    ! [A: $tType,Xa: fun(code_natural,A),Xaa: code_natural,Xb: code_natural,Y: pred(A)] :
      ( ( iterate_upto(A,Xa,Xaa,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)),Xa),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),Xaa),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_aff(fun(code_natural,A),fun(code_natural,fun(code_natural,fun(product_unit,seq(A)))),Xa),Xaa),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)),Xa),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),Xaa),Xb))) ) ) ) ).

% Predicate.iterate_upto.pelims
tff(fact_7996_ID_Opred__cong,axiom,
    ! [A: $tType,Xa: A,Ya: A,Pa: fun(A,$o),Paa: fun(A,$o)] :
      ( ( Xa = Ya )
     => ( ! [Z3: A] :
            ( member(A,Z3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Ya),bot_bot(set(A))))
           => ( aa(A,$o,Pa,Z3)
            <=> aa(A,$o,Paa,Z3) ) )
       => ( aa(A,$o,bNF_id_bnf(fun(A,$o),Pa),Xa)
        <=> aa(A,$o,bNF_id_bnf(fun(A,$o),Paa),Ya) ) ) ) ).

% ID.pred_cong
tff(fact_7997_ID_Opred__mono__strong,axiom,
    ! [A: $tType,Pa: fun(A,$o),Xa: A,Paa: fun(A,$o)] :
      ( aa(A,$o,bNF_id_bnf(fun(A,$o),Pa),Xa)
     => ( ! [Z3: A] :
            ( member(A,Z3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),bot_bot(set(A))))
           => ( aa(A,$o,Pa,Z3)
             => aa(A,$o,Paa,Z3) ) )
       => aa(A,$o,bNF_id_bnf(fun(A,$o),Paa),Xa) ) ) ).

% ID.pred_mono_strong
tff(fact_7998_Predicate_Oiterate__upto_Opsimps,axiom,
    ! [A: $tType,F2: fun(code_natural,A),N: code_natural,M2: 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),N),M2)))
     => ( iterate_upto(A,F2,N,M2) = seq2(A,aa(code_natural,fun(product_unit,seq(A)),aa(code_natural,fun(code_natural,fun(product_unit,seq(A))),aTP_Lamp_aff(fun(code_natural,A),fun(code_natural,fun(code_natural,fun(product_unit,seq(A)))),F2),N),M2)) ) ) ).

% Predicate.iterate_upto.psimps
tff(fact_7999_Random__Pred_Oiterate__upto__def,axiom,
    ! [A: $tType,F2: fun(code_natural,A),N: code_natural,M2: code_natural] : ( random_iterate_upto(A,F2,N,M2) = 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,F2,N,M2)) ) ).

% Random_Pred.iterate_upto_def
tff(fact_8000_of__seq__code_I1_J,axiom,
    ! [A: $tType] : ( set_of_seq(A,empty(A)) = bot_bot(set(A)) ) ).

% of_seq_code(1)
tff(fact_8001_of__seq__code_I2_J,axiom,
    ! [A: $tType,Xa: A,Pa: pred(A)] : ( set_of_seq(A,insert2(A,Xa,Pa)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Xa),set_of_pred(A,Pa)) ) ).

% of_seq_code(2)
tff(fact_8002_of__pred__code,axiom,
    ! [A: $tType,F2: fun(product_unit,seq(A))] : ( set_of_pred(A,seq2(A,F2)) = case_seq(set(A),A,bot_bot(set(A)),aTP_Lamp_afg(A,fun(pred(A),set(A))),aTP_Lamp_afh(pred(A),fun(seq(A),set(A))),aa(product_unit,seq(A),F2,product_Unity)) ) ).

% of_pred_code
tff(fact_8003_ATP_Olambda__1,axiom,
    ! [Uu: product_prod(int,int)] :
      ( aa(product_prod(int,int),product_prod(int,int),aTP_Lamp_aah(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_8004_ATP_Olambda__2,axiom,
    ! [A: $tType,Uu: A] : ( aa(A,set(product_prod(A,A)),aTP_Lamp_kz(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))),insert3(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_2
tff(fact_8005_ATP_Olambda__3,axiom,
    ! [Uu: product_prod(int,int)] : ( aa(product_prod(int,int),product_prod(int,int),aTP_Lamp_aag(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_3
tff(fact_8006_ATP_Olambda__4,axiom,
    ! [A: $tType,Uu: product_prod(A,A)] :
      ( aa(product_prod(A,A),$o,aTP_Lamp_ox(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_4
tff(fact_8007_ATP_Olambda__5,axiom,
    ! [A: $tType,Uu: list(A)] : ( aa(list(A),product_prod(nat,list(A)),aTP_Lamp_xs(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_5
tff(fact_8008_ATP_Olambda__6,axiom,
    ! [Uu: nat] : ( aa(nat,nat,aTP_Lamp_kf(nat,nat),Uu) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu),aa(nat,nat,suc,zero_zero(nat))) ) ).

% ATP.lambda_6
tff(fact_8009_ATP_Olambda__7,axiom,
    ! [B: $tType,Uu: B] : ( aa(B,product_prod(B,B),aTP_Lamp_ow(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_7
tff(fact_8010_ATP_Olambda__8,axiom,
    ! [A: $tType,Uu: A] : ( aa(A,product_prod(A,A),aTP_Lamp_ii(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_8
tff(fact_8011_ATP_Olambda__9,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [Uu: A] :
          ( aa(A,$o,aTP_Lamp_aby(A,$o),Uu)
        <=> dvd_dvd(A,Uu,one_one(A)) ) ) ).

% ATP.lambda_9
tff(fact_8012_ATP_Olambda__10,axiom,
    ! [Uu: nat] : ( aa(nat,product_prod(nat,nat),aTP_Lamp_tg(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_10
tff(fact_8013_ATP_Olambda__11,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A] : ( aa(A,list(A),aTP_Lamp_ir(A,list(A)),Uu) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uu),nil(A)) ) ) ).

% ATP.lambda_11
tff(fact_8014_ATP_Olambda__12,axiom,
    ! [A: $tType,Uu: A] : ( aa(A,list(A),aTP_Lamp_eu(A,list(A)),Uu) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uu),nil(A)) ) ).

% ATP.lambda_12
tff(fact_8015_ATP_Olambda__13,axiom,
    ! [A: $tType,Uu: A] : ( aa(A,set(A),aTP_Lamp_at(A,set(A)),Uu) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Uu),bot_bot(set(A))) ) ).

% ATP.lambda_13
tff(fact_8016_ATP_Olambda__14,axiom,
    ! [B: $tType,Uu: list(B)] :
      ( aa(list(B),$o,aTP_Lamp_td(list(B),$o),Uu)
    <=> ( Uu = nil(B) ) ) ).

% ATP.lambda_14
tff(fact_8017_ATP_Olambda__15,axiom,
    ! [A: $tType,Uu: list(A)] :
      ( aa(list(A),$o,aTP_Lamp_tc(list(A),$o),Uu)
    <=> ( Uu = nil(A) ) ) ).

% ATP.lambda_15
tff(fact_8018_ATP_Olambda__16,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_acx(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_acw(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_16
tff(fact_8019_ATP_Olambda__17,axiom,
    ! [B: $tType,Uu: list(B)] : ( aa(list(B),fun(nat,nat),aTP_Lamp_nu(list(B),fun(nat,nat)),Uu) = aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(B),nat,size_size(list(B)),Uu)),aa(nat,nat,suc,zero_zero(nat)))) ) ).

% ATP.lambda_17
tff(fact_8020_ATP_Olambda__18,axiom,
    ! [A: $tType,Uu: A] : ( aa(A,fun(set(product_prod(A,A)),set(product_prod(A,A))),aTP_Lamp_ky(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))),insert3(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_18
tff(fact_8021_ATP_Olambda__19,axiom,
    ! [B: $tType,Uu: list(B)] :
      ( aa(list(B),$o,aTP_Lamp_nv(list(B),$o),Uu)
    <=> ( Uu != nil(B) ) ) ).

% ATP.lambda_19
tff(fact_8022_ATP_Olambda__20,axiom,
    ! [A: $tType,Uu: list(A)] :
      ( aa(list(A),$o,aTP_Lamp_jg(list(A),$o),Uu)
    <=> ( Uu != nil(A) ) ) ).

% ATP.lambda_20
tff(fact_8023_ATP_Olambda__21,axiom,
    ! [B: $tType,E: $tType,A: $tType,Uu: A] : ( aa(A,fun(product_prod(B,E),product_prod(product_prod(A,B),E)),aTP_Lamp_yf(A,fun(product_prod(B,E),product_prod(product_prod(A,B),E))),Uu) = aa(fun(B,fun(E,product_prod(product_prod(A,B),E))),fun(product_prod(B,E),product_prod(product_prod(A,B),E)),product_case_prod(B,E,product_prod(product_prod(A,B),E)),aTP_Lamp_ye(A,fun(B,fun(E,product_prod(product_prod(A,B),E))),Uu)) ) ).

% ATP.lambda_21
tff(fact_8024_ATP_Olambda__22,axiom,
    ! [A: $tType,E: $tType,B: $tType,Uu: B] : ( aa(B,fun(product_prod(A,E),product_prod(A,product_prod(B,E))),aTP_Lamp_im(B,fun(product_prod(A,E),product_prod(A,product_prod(B,E)))),Uu) = aa(fun(A,fun(E,product_prod(A,product_prod(B,E)))),fun(product_prod(A,E),product_prod(A,product_prod(B,E))),product_case_prod(A,E,product_prod(A,product_prod(B,E))),aTP_Lamp_il(B,fun(A,fun(E,product_prod(A,product_prod(B,E)))),Uu)) ) ).

% ATP.lambda_22
tff(fact_8025_ATP_Olambda__23,axiom,
    ! [A: $tType,Uu: multiset(A)] : ( aa(multiset(A),set(A),aTP_Lamp_sk(multiset(A),set(A)),Uu) = aa(fun(A,$o),set(A),collect(A),aTP_Lamp_sh(multiset(A),fun(A,$o),Uu)) ) ).

% ATP.lambda_23
tff(fact_8026_ATP_Olambda__24,axiom,
    ! [Uu: nat] : ( aa(nat,set(nat),aTP_Lamp_mn(nat,set(nat)),Uu) = aa(fun(nat,$o),set(nat),collect(nat),aTP_Lamp_di(nat,fun(nat,$o),Uu)) ) ).

% ATP.lambda_24
tff(fact_8027_ATP_Olambda__25,axiom,
    ! [B: $tType,Uu: fun(B,nat)] :
      ( aa(fun(B,nat),$o,aTP_Lamp_abv(fun(B,nat),$o),Uu)
    <=> aa(set(B),$o,finite_finite2(B),aa(fun(B,$o),set(B),collect(B),aTP_Lamp_abu(fun(B,nat),fun(B,$o),Uu))) ) ).

% ATP.lambda_25
tff(fact_8028_ATP_Olambda__26,axiom,
    ! [A: $tType,Uu: fun(A,$o)] :
      ( aa(fun(A,$o),$o,aTP_Lamp_abt(fun(A,$o),$o),Uu)
    <=> aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_ce(fun(A,$o),fun(A,$o),Uu))) ) ).

% ATP.lambda_26
tff(fact_8029_ATP_Olambda__27,axiom,
    ! [A: $tType,Uu: fun(A,nat)] :
      ( aa(fun(A,nat),$o,aTP_Lamp_sj(fun(A,nat),$o),Uu)
    <=> aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_be(fun(A,nat),fun(A,$o),Uu))) ) ).

% ATP.lambda_27
tff(fact_8030_ATP_Olambda__28,axiom,
    ! [B: $tType,Uu: list(B)] : ( aa(list(B),fun(nat,nat),aTP_Lamp_nt(list(B),fun(nat,nat)),Uu) = aa(nat,fun(nat,nat),ord_max(nat),aa(list(B),nat,size_size(list(B)),Uu)) ) ).

% ATP.lambda_28
tff(fact_8031_ATP_Olambda__29,axiom,
    ! [A: $tType,Uu: list(A)] : ( aa(list(A),fun(nat,nat),aTP_Lamp_nq(list(A),fun(nat,nat)),Uu) = aa(nat,fun(nat,nat),ord_max(nat),aa(list(A),nat,size_size(list(A)),Uu)) ) ).

% ATP.lambda_29
tff(fact_8032_ATP_Olambda__30,axiom,
    ! [Uu: int] : ( aa(int,fun(int,product_prod(int,int)),aTP_Lamp_aak(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_30
tff(fact_8033_ATP_Olambda__31,axiom,
    ! [Uu: int] : ( aa(int,fun(int,product_prod(int,int)),aTP_Lamp_aaj(int,fun(int,product_prod(int,int))),Uu) = aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),abs_abs(int,Uu)) ) ).

% ATP.lambda_31
tff(fact_8034_ATP_Olambda__32,axiom,
    ! [Uu: nat] : ( aa(nat,fun(nat,product_prod(nat,nat)),aTP_Lamp_gv(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_32
tff(fact_8035_ATP_Olambda__33,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A] : ( aa(A,filter(A),aTP_Lamp_np(A,filter(A)),Uu) = aa(set(A),filter(A),principal(A),set_ord_atLeast(A,Uu)) ) ) ).

% ATP.lambda_33
tff(fact_8036_ATP_Olambda__34,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Uu: A] : ( aa(A,filter(A),aTP_Lamp_nn(A,filter(A)),Uu) = aa(set(A),filter(A),principal(A),set_ord_atLeast(A,Uu)) ) ) ).

% ATP.lambda_34
tff(fact_8037_ATP_Olambda__35,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A] : ( aa(A,filter(A),aTP_Lamp_no(A,filter(A)),Uu) = aa(set(A),filter(A),principal(A),aa(A,set(A),set_ord_atMost(A),Uu)) ) ) ).

% ATP.lambda_35
tff(fact_8038_ATP_Olambda__36,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Uu: A] : ( aa(A,filter(A),aTP_Lamp_nm(A,filter(A)),Uu) = aa(set(A),filter(A),principal(A),aa(A,set(A),set_ord_atMost(A),Uu)) ) ) ).

% ATP.lambda_36
tff(fact_8039_ATP_Olambda__37,axiom,
    ! [A: $tType,Uu: product_prod(A,A)] :
      ( aa(product_prod(A,A),$o,aTP_Lamp_ll(product_prod(A,A),$o),Uu)
    <=> ? [X6: A] : ( Uu = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X6),X6) ) ) ).

% ATP.lambda_37
tff(fact_8040_ATP_Olambda__38,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_lu(product_prod(product_prod($o,A),product_prod($o,B)),$o),Uu)
    <=> ? [X6: A,Y4: B] : ( Uu = aa(product_prod($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),$true),X6)),aa(B,product_prod($o,B),aa($o,fun(B,product_prod($o,B)),product_Pair($o,B),$false),Y4)) ) ) ).

% ATP.lambda_38
tff(fact_8041_ATP_Olambda__39,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: nat,Uua: nat] :
          ( aa(nat,A,aTP_Lamp_ew(nat,fun(nat,A),Uu),Uua) = $ite(dvd_dvd(nat,numeral_numeral(nat,bit0(one2)),Uua),aa(nat,A,semiring_1_of_nat(A),binomial(Uu,Uua)),zero_zero(A)) ) ) ).

% ATP.lambda_39
tff(fact_8042_ATP_Olambda__40,axiom,
    ! [E: $tType,B: $tType,A: $tType,Uu: product_prod(A,E),Uua: product_prod(E,B)] :
      ( aa(product_prod(E,B),list(product_prod(A,B)),aTP_Lamp_pe(product_prod(A,E),fun(product_prod(E,B),list(product_prod(A,B))),Uu),Uua) = $ite(aa(product_prod(A,E),E,product_snd(A,E),Uu) = aa(product_prod(E,B),E,product_fst(E,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,E),A,product_fst(A,E),Uu)),aa(product_prod(E,B),B,product_snd(E,B),Uua))),nil(product_prod(A,B))),nil(product_prod(A,B))) ) ).

% ATP.lambda_40
tff(fact_8043_ATP_Olambda__41,axiom,
    ! [A: $tType,Uu: list(A),Uua: nat] :
      ( aa(nat,option(A),aTP_Lamp_up(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_41
tff(fact_8044_ATP_Olambda__42,axiom,
    ! [Uu: int,Uua: int] :
      ( aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_aai(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)),abs_abs(int,Uu))) ) ).

% ATP.lambda_42
tff(fact_8045_ATP_Olambda__43,axiom,
    ! [Uu: int,Uua: int] :
      ( aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_aau(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_43
tff(fact_8046_ATP_Olambda__44,axiom,
    ! [A: $tType,Uu: set(fun(A,nat)),Uua: A] :
      ( aa(A,nat,aa(set(fun(A,nat)),fun(A,nat),aTP_Lamp_sz(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_sy(A,fun(fun(A,nat),nat),Uua)),Uu))) ) ).

% ATP.lambda_44
tff(fact_8047_ATP_Olambda__45,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: nat,Uua: nat] :
          ( aa(nat,A,aTP_Lamp_ex(nat,fun(nat,A),Uu),Uua) = $ite(~ dvd_dvd(nat,numeral_numeral(nat,bit0(one2)),Uua),aa(nat,A,semiring_1_of_nat(A),binomial(Uu,Uua)),zero_zero(A)) ) ) ).

% ATP.lambda_45
tff(fact_8048_ATP_Olambda__46,axiom,
    ! [A: $tType] :
      ( ( lattice(A)
        & order_bot(A) )
     => ! [Uu: fun(A,A),Uua: nat] : ( aa(nat,A,aTP_Lamp_ny(fun(A,A),fun(nat,A),Uu),Uua) = aa(A,A,compow(fun(A,A),Uua,Uu),bot_bot(A)) ) ) ).

% ATP.lambda_46
tff(fact_8049_ATP_Olambda__47,axiom,
    ! [A: $tType,Uu: list(list(A)),Uua: list(A)] :
      ( aa(list(A),$o,aTP_Lamp_ry(list(list(A)),fun(list(A),$o),Uu),Uua)
    <=> aa(list(list(A)),$o,aa(list(A),fun(list(list(A)),$o),aa(fun(A,fun(list(A),$o)),fun(list(A),fun(list(list(A)),$o)),list_all2(A,list(A)),aTP_Lamp_rx(A,fun(list(A),$o))),Uua),Uu) ) ).

% ATP.lambda_47
tff(fact_8050_ATP_Olambda__48,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Uu: option(A),Uua: A] :
          ( aa(A,$o,aTP_Lamp_afa(option(A),fun(A,$o),Uu),Uua)
        <=> case_option($o,A,$true,aa(A,fun(A,$o),aTP_Lamp_ij(A,fun(A,$o)),Uua),Uu) ) ) ).

% ATP.lambda_48
tff(fact_8051_ATP_Olambda__49,axiom,
    ! [A: $tType,Uu: list(list(A)),Uua: nat] : ( aa(nat,list(A),aTP_Lamp_jz(list(list(A)),fun(nat,list(A)),Uu),Uua) = aa(list(nat),list(A),aa(fun(nat,A),fun(list(nat),list(A)),map(nat,A),aa(nat,fun(nat,A),aTP_Lamp_jy(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_49
tff(fact_8052_ATP_Olambda__50,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : ( aa(nat,A,aTP_Lamp_hv(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_hu(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua)),aa(nat,set(nat),set_ord_atMost(nat),Uua)) ) ) ).

% ATP.lambda_50
tff(fact_8053_ATP_Olambda__51,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : ( aa(nat,A,aTP_Lamp_ht(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_hs(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua)),aa(nat,set(nat),set_ord_atMost(nat),Uua)) ) ) ).

% ATP.lambda_51
tff(fact_8054_ATP_Olambda__52,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_zy(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_52
tff(fact_8055_ATP_Olambda__53,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: nat,Uua: nat] : ( aa(nat,A,aTP_Lamp_gi(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),binomial(Uu,Uua))) ) ) ).

% ATP.lambda_53
tff(fact_8056_ATP_Olambda__54,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_aaf(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_54
tff(fact_8057_ATP_Olambda__55,axiom,
    ! [A: $tType,Uu: set(A),Uua: list(A)] :
      ( aa(list(A),$o,aTP_Lamp_dr(set(A),fun(list(A),$o),Uu),Uua)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uua)),Uu)
        & aa(list(A),$o,distinct(A),Uua) ) ) ).

% ATP.lambda_55
tff(fact_8058_ATP_Olambda__56,axiom,
    ! [A: $tType,Uu: set(A),Uua: list(A)] :
      ( aa(list(A),$o,aTP_Lamp_dc(set(A),fun(list(A),$o),Uu),Uua)
    <=> ( ( aa(list(A),set(A),set2(A),Uua) = Uu )
        & aa(list(A),$o,distinct(A),Uua) ) ) ).

% ATP.lambda_56
tff(fact_8059_ATP_Olambda__57,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: nat,Uua: nat] : ( aa(nat,A,aTP_Lamp_gk(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),binomial(Uu,Uua))) ) ) ).

% ATP.lambda_57
tff(fact_8060_ATP_Olambda__58,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: A] : ( aa(A,set(set(A)),aTP_Lamp_ug(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))),insert3(set(A)),image(A,A,Uu,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Uua),bot_bot(set(A))))),bot_bot(set(set(A)))) ) ).

% ATP.lambda_58
tff(fact_8061_ATP_Olambda__59,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : ( aa(nat,A,aTP_Lamp_gl(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,Uu),Uua)),aa(A,A,aa(A,fun(A,A),minus_minus(A),divide_divide(A,Uu,numeral_numeral(A,bit0(one2)))),aa(nat,A,semiring_1_of_nat(A),Uua))) ) ) ).

% ATP.lambda_59
tff(fact_8062_ATP_Olambda__60,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : ( aa(nat,A,aTP_Lamp_fy(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_60
tff(fact_8063_ATP_Olambda__61,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Uu: array(A),Uua: heap_ext(product_unit)] : ( aa(heap_ext(product_unit),product_prod(list(A),product_prod(heap_ext(product_unit),nat)),aTP_Lamp_aeg(array(A),fun(heap_ext(product_unit),product_prod(list(A),product_prod(heap_ext(product_unit),nat))),Uu),Uua) = aa(product_prod(heap_ext(product_unit),nat),product_prod(list(A),product_prod(heap_ext(product_unit),nat)),aa(list(A),fun(product_prod(heap_ext(product_unit),nat),product_prod(list(A),product_prod(heap_ext(product_unit),nat))),product_Pair(list(A),product_prod(heap_ext(product_unit),nat)),array_get(A,Uua,Uu)),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),Uua),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),array_length(A,Uua,Uu)))) ) ) ).

% ATP.lambda_61
tff(fact_8064_ATP_Olambda__62,axiom,
    ! [A: $tType,Uu: set(set(A)),Uua: set(A)] :
      ( aa(set(A),$o,aTP_Lamp_mt(set(set(A)),fun(set(A),$o),Uu),Uua)
    <=> ( member(set(A),Uua,Uu)
        & ! [X6: set(A)] :
            ( member(set(A),X6,Uu)
           => ~ aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),Uua),X6) ) ) ) ).

% ATP.lambda_62
tff(fact_8065_ATP_Olambda__63,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)),$o),aTP_Lamp_uj(set(product_prod(A,A)),fun(set(product_prod(A,A)),$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)
        & ! [A7: A,B8: A,C6: A] :
            ( ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A7),B8),Uua)
              & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B8),C6),Uu) )
           => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A7),B8),Uu) ) ) ) ).

% ATP.lambda_63
tff(fact_8066_ATP_Olambda__64,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),aTP_Lamp_hy(set(A),fun(set(A),$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_64
tff(fact_8067_ATP_Olambda__65,axiom,
    ! [A: $tType,B: $tType,Uu: B,Uua: A] : ( aa(A,set(product_prod(B,A)),aTP_Lamp_it(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))),insert3(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_65
tff(fact_8068_ATP_Olambda__66,axiom,
    ! [B: $tType,A: $tType,Uu: A,Uua: B] : ( aa(B,set(product_prod(A,B)),aTP_Lamp_qk(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))),insert3(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_66
tff(fact_8069_ATP_Olambda__67,axiom,
    ! [A: $tType,Uu: fun(set(A),$o),Uua: set(A)] :
      ( aa(set(A),$o,aa(fun(set(A),$o),fun(set(A),$o),aTP_Lamp_oh(fun(set(A),$o),fun(set(A),$o)),Uu),Uua)
    <=> ( ( Uua = bot_bot(set(A)) )
        | ? [A11: set(A),A7: A] :
            ( ( Uua = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A7),A11) )
            & aa(set(A),$o,Uu,A11) ) ) ) ).

% ATP.lambda_67
tff(fact_8070_ATP_Olambda__68,axiom,
    ! [A: $tType,B: $tType,Uu: set(B),Uua: fun(A,B)] :
      ( aa(fun(A,B),$o,aTP_Lamp_za(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_68
tff(fact_8071_ATP_Olambda__69,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: A] : ( aa(A,set(set(A)),aTP_Lamp_uf(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)),insert3(A),Uua),bot_bot(set(A))),Uu) ) ).

% ATP.lambda_69
tff(fact_8072_ATP_Olambda__70,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: set(A),Uua: list(A)] :
          ( aa(list(A),$o,aTP_Lamp_yw(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_70
tff(fact_8073_ATP_Olambda__71,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_gn(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),finite_card(product_prod(A,A),Uu)) ) ) ).

% ATP.lambda_71
tff(fact_8074_ATP_Olambda__72,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_gy(nat,fun(nat,$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_72
tff(fact_8075_ATP_Olambda__73,axiom,
    ! [A: $tType,B: $tType,E: $tType,Uu: fun(E,product_prod(A,B)),Uua: A] : ( aa(A,set(B),aTP_Lamp_ps(fun(E,product_prod(A,B)),fun(A,set(B)),Uu),Uua) = aa(set(E),set(B),image2(E,B,comp(product_prod(A,B),B,E,product_snd(A,B),Uu)),top_top(set(E))) ) ).

% ATP.lambda_73
tff(fact_8076_ATP_Olambda__74,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,A),Uua: nat] : ( aa(nat,A,aTP_Lamp_ek(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),numeral_numeral(nat,bit0(one2))),Uua))),aa(nat,A,Uu,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),numeral_numeral(nat,bit0(one2))),Uua)))) ) ) ).

% ATP.lambda_74
tff(fact_8077_ATP_Olambda__75,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : ( aa(nat,A,aTP_Lamp_fl(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_75
tff(fact_8078_ATP_Olambda__76,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Uu: fun(nat,A),Uua: nat] : ( aa(nat,A,aTP_Lamp_fb(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_76
tff(fact_8079_ATP_Olambda__77,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : ( aa(nat,A,aTP_Lamp_fw(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_77
tff(fact_8080_ATP_Olambda__78,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : ( aa(nat,A,aTP_Lamp_ft(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_78
tff(fact_8081_ATP_Olambda__79,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : ( aa(nat,A,aTP_Lamp_fm(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,Uu,Uua)),aa(nat,A,Uu,aa(nat,nat,suc,Uua))) ) ) ).

% ATP.lambda_79
tff(fact_8082_ATP_Olambda__80,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu: fun(A,$o),Uua: A] :
          ( aa(A,$o,aTP_Lamp_ub(fun(A,$o),fun(A,$o),Uu),Uua)
        <=> ( aa(A,$o,Uu,Uua)
            & ! [Y4: A] :
                ( aa(A,$o,Uu,Y4)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),Y4) ) ) ) ) ).

% ATP.lambda_80
tff(fact_8083_ATP_Olambda__81,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Uu: fun(A,$o),Uua: A] :
          ( aa(A,$o,aTP_Lamp_ud(fun(A,$o),fun(A,$o),Uu),Uua)
        <=> ( aa(A,$o,Uu,Uua)
            & ! [Y4: A] :
                ( aa(A,$o,Uu,Y4)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y4),Uua) ) ) ) ) ).

% ATP.lambda_81
tff(fact_8084_ATP_Olambda__82,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: A] : ( aa(A,product_prod(nat,A),aTP_Lamp_xu(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_82
tff(fact_8085_ATP_Olambda__83,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: A] : ( aa(A,product_prod(B,A),aTP_Lamp_tm(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_83
tff(fact_8086_ATP_Olambda__84,axiom,
    ! [A: $tType,Uu: fun(A,assn),Uua: A] : ( aa(A,assn,aTP_Lamp_add(fun(A,assn),fun(A,assn),Uu),Uua) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(A,assn,Uu,Uua)),top_top(assn)) ) ).

% ATP.lambda_84
tff(fact_8087_ATP_Olambda__85,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,A),Uua: B] : ( aa(B,list(A),aTP_Lamp_hc(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_85
tff(fact_8088_ATP_Olambda__86,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,A),Uua: B] : ( aa(B,set(A),aTP_Lamp_dk(fun(B,A),fun(B,set(A)),Uu),Uua) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),aa(B,A,Uu,Uua)),bot_bot(set(A))) ) ).

% ATP.lambda_86
tff(fact_8089_ATP_Olambda__87,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: fun(A,B),Uua: A] :
          ( aa(A,$o,aTP_Lamp_fi(fun(A,B),fun(A,$o),Uu),Uua)
        <=> ( aa(A,B,Uu,Uua) = zero_zero(B) ) ) ) ).

% ATP.lambda_87
tff(fact_8090_ATP_Olambda__88,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: fun(A,B),Uua: A] :
          ( aa(A,$o,aTP_Lamp_dh(fun(A,B),fun(A,$o),Uu),Uua)
        <=> ( aa(A,B,Uu,Uua) = one_one(B) ) ) ) ).

% ATP.lambda_88
tff(fact_8091_ATP_Olambda__89,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Uu: list(A),Uua: heap_ext(product_unit)] : ( aa(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),aTP_Lamp_adt(list(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),Uu),Uua) = aa(product_prod(array(A),heap_ext(product_unit)),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),aa(fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),fun(product_prod(array(A),heap_ext(product_unit)),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),product_case_prod(array(A),heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),aTP_Lamp_ads(list(A),fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),Uu)),array_alloc(A,Uu,Uua)) ) ) ).

% ATP.lambda_89
tff(fact_8092_ATP_Olambda__90,axiom,
    ! [A: $tType,Uu: list(fun(A,nat)),Uua: A] : ( aa(A,list(nat),aTP_Lamp_xt(list(fun(A,nat)),fun(A,list(nat)),Uu),Uua) = aa(list(fun(A,nat)),list(nat),aa(fun(fun(A,nat),nat),fun(list(fun(A,nat)),list(nat)),map(fun(A,nat),nat),aTP_Lamp_sy(A,fun(fun(A,nat),nat),Uua)),Uu) ) ).

% ATP.lambda_90
tff(fact_8093_ATP_Olambda__91,axiom,
    ! [A: $tType,Uu: list(A),Uua: list(A)] : ( aa(list(A),list(list(A)),aTP_Lamp_ig(list(A),fun(list(A),list(list(A))),Uu),Uua) = aa(list(A),list(list(A)),aa(fun(A,list(A)),fun(list(A),list(list(A))),map(A,list(A)),aa(list(A),fun(A,list(A)),aTP_Lamp_if(list(A),fun(A,list(A))),Uua)),Uu) ) ).

% ATP.lambda_91
tff(fact_8094_ATP_Olambda__92,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A)] :
      ( aa(set(A),$o,aTP_Lamp_ok(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_92
tff(fact_8095_ATP_Olambda__93,axiom,
    ! [Uu: assn,Uua: product_prod(heap_ext(product_unit),set(nat))] :
      ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aTP_Lamp_abi(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$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_93
tff(fact_8096_ATP_Olambda__94,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_acd(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_94
tff(fact_8097_ATP_Olambda__95,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,fun(B,$o)),Uua: list(product_prod(A,B))] :
      ( aa(list(product_prod(A,B)),$o,aTP_Lamp_rq(fun(A,fun(B,$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_95
tff(fact_8098_ATP_Olambda__96,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),$o),aTP_Lamp_oo(product_prod(A,B),fun(product_prod(A,B),$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_96
tff(fact_8099_ATP_Olambda__97,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),$o),aTP_Lamp_op(product_prod(A,B),fun(product_prod(A,B),$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_97
tff(fact_8100_ATP_Olambda__98,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Uu: nat,Uua: nat] : ( aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_my(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_98
tff(fact_8101_ATP_Olambda__99,axiom,
    ! [A: $tType,Uu: pred(A),Uua: seq(A)] : ( aa(seq(A),set(A),aa(pred(A),fun(seq(A),set(A)),aTP_Lamp_afh(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_99
tff(fact_8102_ATP_Olambda__100,axiom,
    ! [A: $tType,Uu: list(list(A)),Uua: A] : ( aa(A,list(list(A)),aTP_Lamp_ih(list(list(A)),fun(A,list(list(A))),Uu),Uua) = aa(list(list(A)),list(list(A)),aa(fun(list(A),list(A)),fun(list(list(A)),list(list(A))),map(list(A),list(A)),aa(A,fun(list(A),list(A)),cons(A),Uua)),aa(list(list(A)),list(list(A)),product_lists(A),Uu)) ) ).

% ATP.lambda_100
tff(fact_8103_ATP_Olambda__101,axiom,
    ! [A: $tType,Uu: list(A),Uua: list(A)] :
      ( aa(list(A),$o,aa(list(A),fun(list(A),$o),aTP_Lamp_mj(list(A),fun(list(A),$o)),Uu),Uua)
    <=> ( aa(list(A),nat,size_size(list(A)),Uu) = aa(list(A),nat,size_size(list(A)),Uua) ) ) ).

% ATP.lambda_101
tff(fact_8104_ATP_Olambda__102,axiom,
    ! [A: $tType,Uu: set(A),Uua: list(A)] :
      ( aa(list(A),$o,aTP_Lamp_cy(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_102
tff(fact_8105_ATP_Olambda__103,axiom,
    ! [A: $tType,B: $tType,Uu: list(B),Uua: A] : ( aa(A,list(product_prod(A,B)),aTP_Lamp_ie(list(B),fun(A,list(product_prod(A,B))),Uu),Uua) = aa(list(B),list(product_prod(A,B)),aa(fun(B,product_prod(A,B)),fun(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_103
tff(fact_8106_ATP_Olambda__104,axiom,
    ! [A: $tType,Uu: set(nat),Uua: product_prod(A,nat)] :
      ( aa(product_prod(A,nat),$o,aTP_Lamp_pc(set(nat),fun(product_prod(A,nat),$o),Uu),Uua)
    <=> member(nat,aa(product_prod(A,nat),nat,product_snd(A,nat),Uua),Uu) ) ).

% ATP.lambda_104
tff(fact_8107_ATP_Olambda__105,axiom,
    ! [Uu: set(nat),Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_oc(set(nat),fun(nat,$o),Uu),Uua)
    <=> member(nat,aa(nat,nat,suc,Uua),Uu) ) ).

% ATP.lambda_105
tff(fact_8108_ATP_Olambda__106,axiom,
    ! [A: $tType,Uu: nat,Uua: list(A)] :
      ( aa(list(A),$o,aTP_Lamp_ea(nat,fun(list(A),$o),Uu),Uua)
    <=> ( aa(list(A),nat,size_size(list(A)),Uua) = Uu ) ) ).

% ATP.lambda_106
tff(fact_8109_ATP_Olambda__107,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_go(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),numeral_numeral(A,bit0(one2))),Uua)),one_one(A))) ) ) ).

% ATP.lambda_107
tff(fact_8110_ATP_Olambda__108,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_gp(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),numeral_numeral(A,bit0(one2))),Uua)) ) ) ).

% ATP.lambda_108
tff(fact_8111_ATP_Olambda__109,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : ( aa(nat,A,aTP_Lamp_aa(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),divide_divide(A,aa(nat,A,semiring_1_of_nat(A),Uua),numeral_numeral(A,bit0(one2)))) ) ) ).

% ATP.lambda_109
tff(fact_8112_ATP_Olambda__110,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_ha(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)),insert3(A),Uu)),Uua)) ) ).

% ATP.lambda_110
tff(fact_8113_ATP_Olambda__111,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: list(A),Uua: A] :
          ( aa(A,$o,aTP_Lamp_jl(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),divide_divide(nat,aa(list(A),nat,size_size(list(A)),Uu),numeral_numeral(nat,bit0(one2))))) ) ) ).

% ATP.lambda_111
tff(fact_8114_ATP_Olambda__112,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: list(A),Uua: A] :
          ( aa(A,$o,aTP_Lamp_jm(list(A),fun(A,$o),Uu),Uua)
        <=> ( Uua = aa(nat,A,nth(A,Uu),divide_divide(nat,aa(list(A),nat,size_size(list(A)),Uu),numeral_numeral(nat,bit0(one2)))) ) ) ) ).

% ATP.lambda_112
tff(fact_8115_ATP_Olambda__113,axiom,
    ! [A: $tType,Uu: A,Uua: heap_ext(product_unit)] : ( aa(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)),aTP_Lamp_adn(A,fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),Uu),Uua) = aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),Uu),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),Uua),zero_zero(nat))) ) ).

% ATP.lambda_113
tff(fact_8116_ATP_Olambda__114,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Uu: ref(A),Uua: heap_ext(product_unit)] : ( aa(heap_ext(product_unit),product_prod(ref(A),product_prod(heap_ext(product_unit),nat)),aa(ref(A),fun(heap_ext(product_unit),product_prod(ref(A),product_prod(heap_ext(product_unit),nat))),aTP_Lamp_aea(ref(A),fun(heap_ext(product_unit),product_prod(ref(A),product_prod(heap_ext(product_unit),nat)))),Uu),Uua) = aa(product_prod(heap_ext(product_unit),nat),product_prod(ref(A),product_prod(heap_ext(product_unit),nat)),aa(ref(A),fun(product_prod(heap_ext(product_unit),nat),product_prod(ref(A),product_prod(heap_ext(product_unit),nat))),product_Pair(ref(A),product_prod(heap_ext(product_unit),nat)),Uu),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),Uua),one_one(nat))) ) ) ).

% ATP.lambda_114
tff(fact_8117_ATP_Olambda__115,axiom,
    ! [A: $tType,Uu: A,Uua: heap_ext(product_unit)] : ( aa(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)),aTP_Lamp_ado(A,fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),Uu),Uua) = aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),Uu),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),Uua),one_one(nat))) ) ).

% ATP.lambda_115
tff(fact_8118_ATP_Olambda__116,axiom,
    ! [A: $tType,B: $tType,Uu: set(product_prod(B,A)),Uua: B] : ( aa(B,set(A),aTP_Lamp_rz(set(product_prod(B,A)),fun(B,set(A)),Uu),Uua) = image(B,A,Uu,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert3(B),Uua),bot_bot(set(B)))) ) ).

% ATP.lambda_116
tff(fact_8119_ATP_Olambda__117,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,B),Uua: B] : ( aa(B,set(A),aTP_Lamp_qz(fun(A,B),fun(B,set(A)),Uu),Uua) = vimage(A,B,Uu,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert3(B),Uua),bot_bot(set(B)))) ) ).

% ATP.lambda_117
tff(fact_8120_ATP_Olambda__118,axiom,
    ! [A: $tType,Uu: set(A),Uua: A] :
      ( aa(A,$o,aTP_Lamp_uc(set(A),fun(A,$o),Uu),Uua)
    <=> ( Uu = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Uua),bot_bot(set(A))) ) ) ).

% ATP.lambda_118
tff(fact_8121_ATP_Olambda__119,axiom,
    ! [A: $tType,Uu: fun(nat,A),Uua: nat] : ( aa(nat,product_prod(nat,A),aTP_Lamp_kg(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_119
tff(fact_8122_ATP_Olambda__120,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: A] : ( aa(A,product_prod(A,B),aTP_Lamp_tn(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_120
tff(fact_8123_ATP_Olambda__121,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,option(B)),Uua: A] : ( aa(A,product_prod(A,B),aTP_Lamp_vi(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_121
tff(fact_8124_ATP_Olambda__122,axiom,
    ! [A: $tType,Uu: nat,Uua: list(A)] :
      ( aa(list(A),$o,aTP_Lamp_ja(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_122
tff(fact_8125_ATP_Olambda__123,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [Uu: A,Uua: nat] : ( aa(nat,A,aTP_Lamp_en(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_123
tff(fact_8126_ATP_Olambda__124,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : ( aa(nat,A,aTP_Lamp_em(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),Uu),aa(nat,A,semiring_1_of_nat(A),Uua)) ) ) ).

% ATP.lambda_124
tff(fact_8127_ATP_Olambda__125,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_vf(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_125
tff(fact_8128_ATP_Olambda__126,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] :
      ( aa(list(A),$o,aa(A,fun(list(A),$o),aTP_Lamp_rx(A,fun(list(A),$o)),Uu),Uua)
    <=> member(A,Uu,aa(list(A),set(A),set2(A),Uua)) ) ).

% ATP.lambda_126
tff(fact_8129_ATP_Olambda__127,axiom,
    ! [A: $tType,Uu: list(set(A)),Uua: set(A)] :
      ( aa(set(A),$o,aTP_Lamp_ec(list(set(A)),fun(set(A),$o),Uu),Uua)
    <=> member(set(A),Uua,aa(list(set(A)),set(set(A)),set2(set(A)),Uu)) ) ).

% ATP.lambda_127
tff(fact_8130_ATP_Olambda__128,axiom,
    ! [A: $tType,Uu: list(A),Uua: A] :
      ( aa(A,$o,aTP_Lamp_ji(list(A),fun(A,$o),Uu),Uua)
    <=> member(A,Uua,aa(list(A),set(A),set2(A),Uu)) ) ).

% ATP.lambda_128
tff(fact_8131_ATP_Olambda__129,axiom,
    ! [A: $tType,Uu: A,Uua: pred(A)] : ( aa(pred(A),set(A),aa(A,fun(pred(A),set(A)),aTP_Lamp_afg(A,fun(pred(A),set(A))),Uu),Uua) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Uu),set_of_pred(A,Uua)) ) ).

% ATP.lambda_129
tff(fact_8132_ATP_Olambda__130,axiom,
    ! [A: $tType,Uu: nat,Uua: list(A)] :
      ( aa(list(A),$o,aTP_Lamp_eb(nat,fun(list(A),$o),Uu),Uua)
    <=> ( Uu = aa(list(A),nat,size_size(list(A)),Uua) ) ) ).

% ATP.lambda_130
tff(fact_8133_ATP_Olambda__131,axiom,
    ! [A: $tType,Uu: list(A),Uua: A] :
      ( aa(A,$o,aTP_Lamp_sp(list(A),fun(A,$o),Uu),Uua)
    <=> ( Uua = aa(list(A),A,hd(A),Uu) ) ) ).

% ATP.lambda_131
tff(fact_8134_ATP_Olambda__132,axiom,
    ! [A: $tType,Uu: list(A),Uua: A] :
      ( aa(A,$o,aTP_Lamp_so(list(A),fun(A,$o),Uu),Uua)
    <=> ( Uua = last(A,Uu) ) ) ).

% ATP.lambda_132
tff(fact_8135_ATP_Olambda__133,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_aem(nat,fun(nat,$o)),Uu),Uua)
    <=> ( Uua = aa(nat,nat,suc,Uu) ) ) ).

% ATP.lambda_133
tff(fact_8136_ATP_Olambda__134,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A)] :
      ( aa(set(A),$o,aTP_Lamp_by(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_134
tff(fact_8137_ATP_Olambda__135,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_nr(nat,fun(nat,$o)),Uu),Uua)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Uua),Uu) ) ).

% ATP.lambda_135
tff(fact_8138_ATP_Olambda__136,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_dn(A,fun(A,$o)),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),Uu) ) ) ).

% ATP.lambda_136
tff(fact_8139_ATP_Olambda__137,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_yu(A,fun(A,$o)),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),Uu) ) ) ).

% ATP.lambda_137
tff(fact_8140_ATP_Olambda__138,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_xc(A,fun(A,$o)),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),Uu) ) ) ).

% ATP.lambda_138
tff(fact_8141_ATP_Olambda__139,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_aav(A,fun(A,$o)),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),Uu) ) ) ).

% ATP.lambda_139
tff(fact_8142_ATP_Olambda__140,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_ao(A,fun(A,$o)),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),Uu) ) ) ).

% ATP.lambda_140
tff(fact_8143_ATP_Olambda__141,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_afd(A,fun(A,$o)),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),Uu) ) ) ).

% ATP.lambda_141
tff(fact_8144_ATP_Olambda__142,axiom,
    ! [Uu: nat,Uua: nat] : ( aa(nat,nat,aTP_Lamp_dt(nat,fun(nat,nat),Uu),Uua) = modulo_modulo(nat,Uua,Uu) ) ).

% ATP.lambda_142
tff(fact_8145_ATP_Olambda__143,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A] : ( aa(A,A,aTP_Lamp_ay(A,fun(A,A),Uu),Uua) = divide_divide(A,Uua,Uu) ) ) ).

% ATP.lambda_143
tff(fact_8146_ATP_Olambda__144,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_ac(nat,fun(nat,$o)),Uu),Uua)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uua),Uu) ) ).

% ATP.lambda_144
tff(fact_8147_ATP_Olambda__145,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_do(A,fun(A,$o)),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Uua),Uu) ) ) ).

% ATP.lambda_145
tff(fact_8148_ATP_Olambda__146,axiom,
    ! [A: $tType] :
      ( unboun7993243217541854897norder(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_wj(A,fun(A,$o),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Uua),Uu) ) ) ).

% ATP.lambda_146
tff(fact_8149_ATP_Olambda__147,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_yv(A,fun(A,$o)),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Uua),Uu) ) ) ).

% ATP.lambda_147
tff(fact_8150_ATP_Olambda__148,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_xd(A,fun(A,$o)),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Uua),Uu) ) ) ).

% ATP.lambda_148
tff(fact_8151_ATP_Olambda__149,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_ij(A,fun(A,$o)),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Uua),Uu) ) ) ).

% ATP.lambda_149
tff(fact_8152_ATP_Olambda__150,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_ys(A,fun(A,$o)),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Uua),Uu) ) ) ).

% ATP.lambda_150
tff(fact_8153_ATP_Olambda__151,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_afe(A,fun(A,$o)),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Uua),Uu) ) ) ).

% ATP.lambda_151
tff(fact_8154_ATP_Olambda__152,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_cu(A,fun(A,$o),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Uua),Uu) ) ) ).

% ATP.lambda_152
tff(fact_8155_ATP_Olambda__153,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A] : ( aa(A,A,aTP_Lamp_ed(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),Uua),Uu) ) ) ).

% ATP.lambda_153
tff(fact_8156_ATP_Olambda__154,axiom,
    ! [Uu: nat,Uua: nat] : ( aa(nat,nat,aTP_Lamp_ej(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),Uu) ) ).

% ATP.lambda_154
tff(fact_8157_ATP_Olambda__155,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Uu: A,Uua: A] : ( aa(A,A,aTP_Lamp_ak(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),Uua),Uu) ) ) ).

% ATP.lambda_155
tff(fact_8158_ATP_Olambda__156,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu: A,Uua: A] : ( aa(A,A,aTP_Lamp_af(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),Uua),Uu) ) ) ).

% ATP.lambda_156
tff(fact_8159_ATP_Olambda__157,axiom,
    ! [A: $tType,Uu: A,Uua: multiset(A)] : ( aa(multiset(A),nat,aTP_Lamp_sf(A,fun(multiset(A),nat),Uu),Uua) = aa(A,nat,aa(multiset(A),fun(A,nat),count(A),Uua),Uu) ) ).

% ATP.lambda_157
tff(fact_8160_ATP_Olambda__158,axiom,
    ! [Uu: code_integer,Uua: code_integer] : ( aa(code_integer,code_integer,aTP_Lamp_la(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_158
tff(fact_8161_ATP_Olambda__159,axiom,
    ! [Uu: nat,Uua: nat] : ( aa(nat,nat,aTP_Lamp_kc(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uu) ) ).

% ATP.lambda_159
tff(fact_8162_ATP_Olambda__160,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Uu: A,Uua: ref(A)] : ( aa(ref(A),assn,aa(A,fun(ref(A),assn),aTP_Lamp_ch(A,fun(ref(A),assn)),Uu),Uua) = sngr_assn(A,Uua,Uu) ) ) ).

% ATP.lambda_160
tff(fact_8163_ATP_Olambda__161,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Uu: list(A),Uua: array(A)] : ( aa(array(A),assn,aa(list(A),fun(array(A),assn),aTP_Lamp_ci(list(A),fun(array(A),assn)),Uu),Uua) = snga_assn(A,Uua,Uu) ) ) ).

% ATP.lambda_161
tff(fact_8164_ATP_Olambda__162,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_di(nat,fun(nat,$o),Uu),Uua)
    <=> dvd_dvd(nat,Uua,Uu) ) ).

% ATP.lambda_162
tff(fact_8165_ATP_Olambda__163,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_bs(A,fun(A,$o),Uu),Uua)
        <=> dvd_dvd(A,Uua,Uu) ) ) ).

% ATP.lambda_163
tff(fact_8166_ATP_Olambda__164,axiom,
    ! [Uu: nat,Uua: nat] : ( aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),aTP_Lamp_mx(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_164
tff(fact_8167_ATP_Olambda__165,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_io(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_165
tff(fact_8168_ATP_Olambda__166,axiom,
    ! [A: $tType,Uu: A,Uua: nat] : ( aa(nat,product_prod(nat,A),aTP_Lamp_od(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_166
tff(fact_8169_ATP_Olambda__167,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_ot(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_167
tff(fact_8170_ATP_Olambda__168,axiom,
    ! [A: $tType,Uu: list(A),Uua: A] : ( aa(A,list(A),aa(list(A),fun(A,list(A)),aTP_Lamp_if(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_168
tff(fact_8171_ATP_Olambda__169,axiom,
    ! [B: $tType,Uu: set(B),Uua: B] :
      ( aa(B,$o,aTP_Lamp_wu(set(B),fun(B,$o),Uu),Uua)
    <=> member(B,Uua,Uu) ) ).

% ATP.lambda_169
tff(fact_8172_ATP_Olambda__170,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [Uu: set(A),Uua: A] :
          ( aa(A,$o,aTP_Lamp_rk(set(A),fun(A,$o),Uu),Uua)
        <=> member(A,Uua,Uu) ) ) ).

% ATP.lambda_170
tff(fact_8173_ATP_Olambda__171,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Uu: set(A),Uua: A] :
          ( aa(A,$o,aTP_Lamp_rn(set(A),fun(A,$o),Uu),Uua)
        <=> member(A,Uua,Uu) ) ) ).

% ATP.lambda_171
tff(fact_8174_ATP_Olambda__172,axiom,
    ! [A: $tType,Uu: set(A),Uua: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_a(set(A),fun(A,$o)),Uu),Uua)
    <=> member(A,Uua,Uu) ) ).

% ATP.lambda_172
tff(fact_8175_ATP_Olambda__173,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: nat] : ( aa(nat,set(product_prod(A,A)),aTP_Lamp_gm(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),Uu),Uua) = compow(set(product_prod(A,A)),Uua,Uu) ) ).

% ATP.lambda_173
tff(fact_8176_ATP_Olambda__174,axiom,
    ! [A: $tType,Uu: list(A),Uua: nat] : ( aa(nat,list(A),aTP_Lamp_zw(list(A),fun(nat,list(A)),Uu),Uua) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),Uua),Uu) ) ).

% ATP.lambda_174
tff(fact_8177_ATP_Olambda__175,axiom,
    ! [A: $tType,Uu: nat,Uua: list(A)] : ( aa(list(A),A,aTP_Lamp_jb(nat,fun(list(A),A),Uu),Uua) = aa(nat,A,nth(A,Uua),Uu) ) ).

% ATP.lambda_175
tff(fact_8178_ATP_Olambda__176,axiom,
    ! [A: $tType,Uu: A,Uua: A] :
      ( aa(A,$o,aTP_Lamp_ad(A,fun(A,$o),Uu),Uua)
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_176
tff(fact_8179_ATP_Olambda__177,axiom,
    ! [Uu: nat,Uua: nat] : ( aa(nat,nat,aTP_Lamp_aej(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu),one_one(nat)) ) ).

% ATP.lambda_177
tff(fact_8180_ATP_Olambda__178,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] : ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),aTP_Lamp_iy(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_178
tff(fact_8181_ATP_Olambda__179,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] : ( aa(list(A),list(list(A)),aa(A,fun(list(A),list(list(A))),aTP_Lamp_iz(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_179
tff(fact_8182_ATP_Olambda__180,axiom,
    ! [A: $tType,Uu: A,Uua: A] : ( aa(A,set(A),aTP_Lamp_qs(A,fun(A,set(A)),Uu),Uua) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Uu),bot_bot(set(A))) ) ).

% ATP.lambda_180
tff(fact_8183_ATP_Olambda__181,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Uu: A,Uua: heap_ext(product_unit)] : ( aa(heap_ext(product_unit),product_prod(ref(A),product_prod(heap_ext(product_unit),nat)),aTP_Lamp_aeb(A,fun(heap_ext(product_unit),product_prod(ref(A),product_prod(heap_ext(product_unit),nat))),Uu),Uua) = aa(product_prod(ref(A),heap_ext(product_unit)),product_prod(ref(A),product_prod(heap_ext(product_unit),nat)),aa(fun(ref(A),fun(heap_ext(product_unit),product_prod(ref(A),product_prod(heap_ext(product_unit),nat)))),fun(product_prod(ref(A),heap_ext(product_unit)),product_prod(ref(A),product_prod(heap_ext(product_unit),nat))),product_case_prod(ref(A),heap_ext(product_unit),product_prod(ref(A),product_prod(heap_ext(product_unit),nat))),aTP_Lamp_aea(ref(A),fun(heap_ext(product_unit),product_prod(ref(A),product_prod(heap_ext(product_unit),nat))))),ref_alloc(A,Uu,Uua)) ) ) ).

% ATP.lambda_181
tff(fact_8184_ATP_Olambda__182,axiom,
    ! [A: $tType,Uu: multiset(A),Uua: A] :
      ( aa(A,$o,aTP_Lamp_sh(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_182
tff(fact_8185_ATP_Olambda__183,axiom,
    ! [B: $tType,Uu: fun(B,nat),Uua: B] :
      ( aa(B,$o,aTP_Lamp_abu(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_183
tff(fact_8186_ATP_Olambda__184,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: A] :
      ( aa(A,$o,aTP_Lamp_be(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_184
tff(fact_8187_ATP_Olambda__185,axiom,
    ! [A: $tType,Uu: set(multiset(A)),Uua: A] :
      ( aa(A,$o,aTP_Lamp_sg(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_sf(A,fun(multiset(A),nat),Uua)),Uu))) ) ).

% ATP.lambda_185
tff(fact_8188_ATP_Olambda__186,axiom,
    ! [Uu: nat,Uua: nat] : ( aa(nat,set(nat),aTP_Lamp_zk(nat,fun(nat,set(nat)),Uu),Uua) = order_underS(nat,bNF_Ca8665028551170535155natLeq,Uu) ) ).

% ATP.lambda_186
tff(fact_8189_ATP_Olambda__187,axiom,
    ! [A: $tType,Uu: fun(nat,$o),Uua: product_prod(A,nat)] :
      ( aa(product_prod(A,nat),$o,aTP_Lamp_pa(fun(nat,$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_187
tff(fact_8190_ATP_Olambda__188,axiom,
    ! [A: $tType,Uu: fun(nat,$o),Uua: product_prod(A,nat)] :
      ( aa(product_prod(A,nat),$o,aTP_Lamp_pb(fun(nat,$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_188
tff(fact_8191_ATP_Olambda__189,axiom,
    ! [E: $tType,B: $tType,A: $tType,Uu: fun(B,E),Uua: product_prod(A,B)] : ( aa(product_prod(A,B),E,aTP_Lamp_ou(fun(B,E),fun(product_prod(A,B),E),Uu),Uua) = aa(B,E,Uu,aa(product_prod(A,B),B,product_snd(A,B),Uua)) ) ).

% ATP.lambda_189
tff(fact_8192_ATP_Olambda__190,axiom,
    ! [E: $tType,B: $tType,A: $tType,Uu: fun(A,E),Uua: product_prod(A,B)] : ( aa(product_prod(A,B),E,aTP_Lamp_oq(fun(A,E),fun(product_prod(A,B),E),Uu),Uua) = aa(A,E,Uu,aa(product_prod(A,B),A,product_fst(A,B),Uua)) ) ).

% ATP.lambda_190
tff(fact_8193_ATP_Olambda__191,axiom,
    ! [Uu: fun(nat,$o),Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_vt(fun(nat,$o),fun(nat,$o),Uu),Uua)
    <=> aa(nat,$o,Uu,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_191
tff(fact_8194_ATP_Olambda__192,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,A),Uua: nat] : ( aa(nat,A,aTP_Lamp_dl(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ) ) ).

% ATP.lambda_192
tff(fact_8195_ATP_Olambda__193,axiom,
    ! [A: $tType,Uu: fun(nat,A),Uua: nat] : ( aa(nat,A,aTP_Lamp_kh(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_193
tff(fact_8196_ATP_Olambda__194,axiom,
    ! [A: $tType,E: $tType,Uu: E,Uua: fun(E,set(set(A)))] : ( aa(fun(E,set(set(A))),set(set(A)),aTP_Lamp_zd(E,fun(fun(E,set(set(A))),set(set(A))),Uu),Uua) = aa(E,set(set(A)),Uua,Uu) ) ).

% ATP.lambda_194
tff(fact_8197_ATP_Olambda__195,axiom,
    ! [A: $tType,Uu: A,Uua: fun(A,nat)] : ( aa(fun(A,nat),nat,aTP_Lamp_sy(A,fun(fun(A,nat),nat),Uu),Uua) = aa(A,nat,Uua,Uu) ) ).

% ATP.lambda_195
tff(fact_8198_ATP_Olambda__196,axiom,
    ! [B: $tType,A: $tType,Uu: fun(set(A),A),Uua: B] : ( aa(B,A,aTP_Lamp_acg(fun(set(A),A),fun(B,A),Uu),Uua) = aa(set(A),A,Uu,bot_bot(set(A))) ) ).

% ATP.lambda_196
tff(fact_8199_ATP_Olambda__197,axiom,
    ! [A: $tType,Uu: list(A),Uua: list(A)] : ( aa(list(A),fun(product_prod(A,list(A)),option($o)),aTP_Lamp_acs(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($o))),aTP_Lamp_acr(list(A),fun(list(A),fun(A,fun(list(A),option($o)))),Uu),Uua)) ) ).

% ATP.lambda_197
tff(fact_8200_ATP_Olambda__198,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_vb(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_va(A,fun(list(A),fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A))))))),Uu),Uua)) ) ).

% ATP.lambda_198
tff(fact_8201_ATP_Olambda__199,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_ne(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_nd(nat,fun(nat,fun(nat,fun(nat,product_prod(nat,nat)))),Uu),Uua)) ) ).

% ATP.lambda_199
tff(fact_8202_ATP_Olambda__200,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_nc(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_nb(nat,fun(nat,fun(nat,fun(nat,product_prod(nat,nat)))),Uu),Uua)) ) ).

% ATP.lambda_200
tff(fact_8203_ATP_Olambda__201,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_na(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,$o)),aTP_Lamp_mz(nat,fun(nat,fun(nat,fun(nat,$o))),Uu),Uua)) ) ).

% ATP.lambda_201
tff(fact_8204_ATP_Olambda__202,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_mv(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_mu(nat,fun(nat,fun(nat,fun(nat,product_prod(nat,nat)))),Uu),Uua)) ) ).

% ATP.lambda_202
tff(fact_8205_ATP_Olambda__203,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_ju(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_jt(fun(A,B),fun(list(A),fun(list(A),fun(list(A),list(A)))),Uu),Uua)) ) ) ).

% ATP.lambda_203
tff(fact_8206_ATP_Olambda__204,axiom,
    ! [A: $tType,B: $tType,E: $tType,Uu: fun(E,fun(A,E)),Uua: E] : ( aa(E,fun(product_prod(B,A),E),aTP_Lamp_gt(fun(E,fun(A,E)),fun(E,fun(product_prod(B,A),E)),Uu),Uua) = aa(fun(B,fun(A,E)),fun(product_prod(B,A),E),product_case_prod(B,A,E),aa(E,fun(B,fun(A,E)),aTP_Lamp_gs(fun(E,fun(A,E)),fun(E,fun(B,fun(A,E))),Uu),Uua)) ) ).

% ATP.lambda_204
tff(fact_8207_ATP_Olambda__205,axiom,
    ! [A: $tType,Uu: fun(heap_ext(product_unit),A),Uua: heap_ext(product_unit)] : ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),aTP_Lamp_adw(fun(heap_ext(product_unit),A),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),Uu),Uua) = aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),aa(heap_ext(product_unit),A,Uu,Uua)),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),Uua),one_one(nat)))) ) ).

% ATP.lambda_205
tff(fact_8208_ATP_Olambda__206,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,option(A)),Uua: B] :
      ( aa(B,$o,aTP_Lamp_un(fun(B,option(A)),fun(B,$o),Uu),Uua)
    <=> ( aa(B,option(A),Uu,Uua) != none(A) ) ) ).

% ATP.lambda_206
tff(fact_8209_ATP_Olambda__207,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,set(B)),Uua: A] : ( aa(A,set(product_prod(A,B)),aTP_Lamp_ql(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_qk(A,fun(B,set(product_prod(A,B))),Uua)),aa(A,set(B),Uu,Uua))) ) ).

% ATP.lambda_207
tff(fact_8210_ATP_Olambda__208,axiom,
    ! [A: $tType,B: $tType,E: $tType,Uu: list(product_prod(E,B)),Uua: product_prod(A,E)] : ( aa(product_prod(A,E),list(product_prod(A,B)),aTP_Lamp_pf(list(product_prod(E,B)),fun(product_prod(A,E),list(product_prod(A,B))),Uu),Uua) = aa(list(list(product_prod(A,B))),list(product_prod(A,B)),concat(product_prod(A,B)),aa(list(product_prod(E,B)),list(list(product_prod(A,B))),aa(fun(product_prod(E,B),list(product_prod(A,B))),fun(list(product_prod(E,B)),list(list(product_prod(A,B)))),map(product_prod(E,B),list(product_prod(A,B))),aTP_Lamp_pe(product_prod(A,E),fun(product_prod(E,B),list(product_prod(A,B))),Uua)),Uu)) ) ).

% ATP.lambda_208
tff(fact_8211_ATP_Olambda__209,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_acw(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),numeral_numeral(code_natural,bit1(bit0(bit0(bit1(bit0(bit1(bit0(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(one2)))))))))))))))))))))))))))))))))) ) ).

% ATP.lambda_209
tff(fact_8212_ATP_Olambda__210,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : ( aa(nat,fun(A,A),aTP_Lamp_er(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_210
tff(fact_8213_ATP_Olambda__211,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: A,Uua: nat] : ( aa(nat,fun(A,A),aTP_Lamp_eq(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_211
tff(fact_8214_ATP_Olambda__212,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_acz(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),code_nat_of_natural(Uua))) ) ).

% ATP.lambda_212
tff(fact_8215_ATP_Olambda__213,axiom,
    ! [A: $tType,Uu: list(A),Uua: A] :
      ( aa(A,$o,aTP_Lamp_jj(list(A),fun(A,$o),Uu),Uua)
    <=> ~ member(A,Uua,aa(list(A),set(A),set2(A),Uu)) ) ).

% ATP.lambda_213
tff(fact_8216_ATP_Olambda__214,axiom,
    ! [A: $tType,Uu: list(A),Uua: A] :
      ( aa(A,$o,aTP_Lamp_ko(list(A),fun(A,$o),Uu),Uua)
    <=> ( Uua != last(A,Uu) ) ) ).

% ATP.lambda_214
tff(fact_8217_ATP_Olambda__215,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_acy(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_215
tff(fact_8218_ATP_Olambda__216,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_aek(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_216
tff(fact_8219_ATP_Olambda__217,axiom,
    ! [E: $tType,A: $tType,B: $tType,Uu: A,Uua: B] : ( aa(B,fun(E,product_prod(product_prod(A,B),E)),aTP_Lamp_ye(A,fun(B,fun(E,product_prod(product_prod(A,B),E))),Uu),Uua) = aa(product_prod(A,B),fun(E,product_prod(product_prod(A,B),E)),product_Pair(product_prod(A,B),E),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uu),Uua)) ) ).

% ATP.lambda_217
tff(fact_8220_ATP_Olambda__218,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_acv(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)),pick(A,Uu,Uua)) ) ).

% ATP.lambda_218
tff(fact_8221_ATP_Olambda__219,axiom,
    ! [B: $tType,A: $tType,Uu: list(product_prod(A,B)),Uua: A] : ( aa(A,B,aTP_Lamp_uk(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_219
tff(fact_8222_ATP_Olambda__220,axiom,
    ! [B: $tType,A: $tType,Uu: B,Uua: A] : ( aa(A,fun(set(product_prod(B,A)),set(product_prod(B,A))),aTP_Lamp_iu(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))),insert3(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_220
tff(fact_8223_ATP_Olambda__221,axiom,
    ! [A: $tType,B: $tType,Uu: A,Uua: B] : ( aa(B,fun(set(product_prod(A,B)),set(product_prod(A,B))),aTP_Lamp_qm(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))),insert3(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_221
tff(fact_8224_ATP_Olambda__222,axiom,
    ! [A: $tType,Uu: set(A),Uua: A] :
      ( aa(A,$o,aTP_Lamp_cd(set(A),fun(A,$o),Uu),Uua)
    <=> ~ member(A,Uua,Uu) ) ).

% ATP.lambda_222
tff(fact_8225_ATP_Olambda__223,axiom,
    ! [A: $tType,Uu: A,Uua: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_je(A,fun(A,$o)),Uu),Uua)
    <=> ( Uu != Uua ) ) ).

% ATP.lambda_223
tff(fact_8226_ATP_Olambda__224,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_top(A) )
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_wg(A,fun(A,$o),Uu),Uua)
        <=> ( Uua != Uu ) ) ) ).

% ATP.lambda_224
tff(fact_8227_ATP_Olambda__225,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_bot(A) )
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_wa(A,fun(A,$o),Uu),Uua)
        <=> ( Uua != Uu ) ) ) ).

% ATP.lambda_225
tff(fact_8228_ATP_Olambda__226,axiom,
    ! [A: $tType,Uu: A,Uua: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_cg(A,fun(A,$o)),Uu),Uua)
    <=> ( Uua != Uu ) ) ).

% ATP.lambda_226
tff(fact_8229_ATP_Olambda__227,axiom,
    ! [A: $tType,Uu: A,Uua: A] : ( aa(A,set(A),aTP_Lamp_qq(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)),insert3(A),Uu),bot_bot(set(A)))) ) ).

% ATP.lambda_227
tff(fact_8230_ATP_Olambda__228,axiom,
    ! [Uu: nat,Uua: heap_ext(product_unit)] : ( aa(heap_ext(product_unit),option(product_prod(product_unit,product_prod(heap_ext(product_unit),nat))),aTP_Lamp_adx(nat,fun(heap_ext(product_unit),option(product_prod(product_unit,product_prod(heap_ext(product_unit),nat)))),Uu),Uua) = aa(product_prod(product_unit,product_prod(heap_ext(product_unit),nat)),option(product_prod(product_unit,product_prod(heap_ext(product_unit),nat))),some(product_prod(product_unit,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(product_unit,product_prod(heap_ext(product_unit),nat)),aa(product_unit,fun(product_prod(heap_ext(product_unit),nat),product_prod(product_unit,product_prod(heap_ext(product_unit),nat))),product_Pair(product_unit,product_prod(heap_ext(product_unit),nat)),product_Unity),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),Uua),Uu))) ) ).

% ATP.lambda_228
tff(fact_8231_ATP_Olambda__229,axiom,
    ! [E: $tType,B: $tType,Uu: fun(B,set(E)),Uua: B] : ( aa(B,set(product_prod(E,E)),aTP_Lamp_zg(fun(B,set(E)),fun(B,set(product_prod(E,E))),Uu),Uua) = bNF_Ca6860139660246222851ard_of(E,aa(B,set(E),Uu,Uua)) ) ).

% ATP.lambda_229
tff(fact_8232_ATP_Olambda__230,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,set(B)),Uua: A] : ( aa(A,set(product_prod(B,B)),aTP_Lamp_zh(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_230
tff(fact_8233_ATP_Olambda__231,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,set(product_prod(B,B))),Uua: A] : ( aa(A,set(product_prod(B,B)),aTP_Lamp_ev(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_231
tff(fact_8234_ATP_Olambda__232,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,A),Uua: nat] : ( aa(nat,fun(A,A),aTP_Lamp_ds(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_232
tff(fact_8235_ATP_Olambda__233,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [Uu: fun(A,B),Uua: A] : ( aa(A,option(B),aTP_Lamp_uy(fun(A,B),fun(A,option(B)),Uu),Uua) = aa(B,option(B),some(B),aa(A,B,Uu,Uua)) ) ) ).

% ATP.lambda_233
tff(fact_8236_ATP_Olambda__234,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: A] : ( aa(A,option(B),aTP_Lamp_vh(fun(A,B),fun(A,option(B)),Uu),Uua) = aa(B,option(B),some(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_234
tff(fact_8237_ATP_Olambda__235,axiom,
    ! [B: $tType,A: $tType,E: $tType,Uu: fun(E,set(product_prod(B,A))),Uua: E] : ( aa(E,set(product_prod(A,B)),aTP_Lamp_se(fun(E,set(product_prod(B,A))),fun(E,set(product_prod(A,B))),Uu),Uua) = converse(B,A,aa(E,set(product_prod(B,A)),Uu,Uua)) ) ).

% ATP.lambda_235
tff(fact_8238_ATP_Olambda__236,axiom,
    ! [B: $tType,A: $tType,E: $tType,Uu: fun(E,A),Uua: E] : ( aa(E,fun(B,product_prod(A,B)),aTP_Lamp_hn(fun(E,A),fun(E,fun(B,product_prod(A,B))),Uu),Uua) = aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(E,A,Uu,Uua)) ) ).

% ATP.lambda_236
tff(fact_8239_ATP_Olambda__237,axiom,
    ! [E: $tType,B: $tType,A: $tType,Uu: fun(A,B),Uua: A] : ( aa(A,fun(E,product_prod(B,E)),aTP_Lamp_uw(fun(A,B),fun(A,fun(E,product_prod(B,E))),Uu),Uua) = aa(B,fun(E,product_prod(B,E)),product_Pair(B,E),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_237
tff(fact_8240_ATP_Olambda__238,axiom,
    ! [E: $tType,F4: $tType,Uu: fun(F4,set(E)),Uua: F4] : ( aa(F4,filter(E),aTP_Lamp_tl(fun(F4,set(E)),fun(F4,filter(E)),Uu),Uua) = aa(set(E),filter(E),principal(E),aa(F4,set(E),Uu,Uua)) ) ).

% ATP.lambda_238
tff(fact_8241_ATP_Olambda__239,axiom,
    ! [F4: $tType,E: $tType,Uu: fun(E,set(F4)),Uua: E] : ( aa(E,filter(F4),aTP_Lamp_tt(fun(E,set(F4)),fun(E,filter(F4)),Uu),Uua) = aa(set(F4),filter(F4),principal(F4),aa(E,set(F4),Uu,Uua)) ) ).

% ATP.lambda_239
tff(fact_8242_ATP_Olambda__240,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,set(A)),Uua: B] : ( aa(B,filter(A),aTP_Lamp_nj(fun(B,set(A)),fun(B,filter(A)),Uu),Uua) = aa(set(A),filter(A),principal(A),aa(B,set(A),Uu,Uua)) ) ).

% ATP.lambda_240
tff(fact_8243_ATP_Olambda__241,axiom,
    ! [G2: $tType,A: $tType,Uu: fun(A,set(G2)),Uua: A] : ( aa(A,filter(G2),aTP_Lamp_ts(fun(A,set(G2)),fun(A,filter(G2)),Uu),Uua) = aa(set(G2),filter(G2),principal(G2),aa(A,set(G2),Uu,Uua)) ) ).

% ATP.lambda_241
tff(fact_8244_ATP_Olambda__242,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,set(B)),Uua: A] : ( aa(A,filter(B),aTP_Lamp_nl(fun(A,set(B)),fun(A,filter(B)),Uu),Uua) = aa(set(B),filter(B),principal(B),aa(A,set(B),Uu,Uua)) ) ).

% ATP.lambda_242
tff(fact_8245_ATP_Olambda__243,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,set(B)),Uua: A] : ( aa(A,nat,aTP_Lamp_gu(fun(A,set(B)),fun(A,nat),Uu),Uua) = finite_card(B,aa(A,set(B),Uu,Uua)) ) ).

% ATP.lambda_243
tff(fact_8246_ATP_Olambda__244,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,list(A)),Uua: B] : ( aa(B,set(A),aTP_Lamp_dj(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_244
tff(fact_8247_ATP_Olambda__245,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: A] : ( aa(A,fun(set(B),set(B)),aTP_Lamp_hz(fun(A,B),fun(A,fun(set(B),set(B))),Uu),Uua) = aa(B,fun(set(B),set(B)),insert3(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_245
tff(fact_8248_ATP_Olambda__246,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: A] : ( aa(A,nat,aTP_Lamp_ik(fun(A,nat),fun(A,nat),Uu),Uua) = aa(nat,nat,suc,aa(A,nat,Uu,Uua)) ) ).

% ATP.lambda_246
tff(fact_8249_ATP_Olambda__247,axiom,
    ! [B: $tType,Uu: fun(B,$o),Uua: B] :
      ( aa(B,$o,aTP_Lamp_am(fun(B,$o),fun(B,$o),Uu),Uua)
    <=> ~ aa(B,$o,Uu,Uua) ) ).

% ATP.lambda_247
tff(fact_8250_ATP_Olambda__248,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: A] :
      ( aa(A,$o,aTP_Lamp_ce(fun(A,$o),fun(A,$o),Uu),Uua)
    <=> ~ aa(A,$o,Uu,Uua) ) ).

% ATP.lambda_248
tff(fact_8251_ATP_Olambda__249,axiom,
    ! [B: $tType,A: $tType] :
      ( finite_finite(A)
     => ! [Uu: fun(B,fun(A,$o)),Uua: B] :
          ( aa(B,$o,aTP_Lamp_vx(fun(B,fun(A,$o)),fun(B,$o),Uu),Uua)
        <=> ! [X_12: A] : aa(A,$o,aa(B,fun(A,$o),Uu,Uua),X_12) ) ) ).

% ATP.lambda_249
tff(fact_8252_ATP_Olambda__250,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,fun(B,$o)),Uua: A] :
      ( aa(A,$o,aTP_Lamp_aac(fun(A,fun(B,$o)),fun(A,$o),Uu),Uua)
    <=> ! [X_12: B] : aa(B,$o,aa(A,fun(B,$o),Uu,Uua),X_12) ) ).

% ATP.lambda_250
tff(fact_8253_ATP_Olambda__251,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,fun(B,$o)),Uua: A] :
      ( aa(A,$o,aTP_Lamp_vy(fun(A,fun(B,$o)),fun(A,$o),Uu),Uua)
    <=> ? [X_12: B] : aa(B,$o,aa(A,fun(B,$o),Uu,Uua),X_12) ) ).

% ATP.lambda_251
tff(fact_8254_ATP_Olambda__252,axiom,
    ! [Uu: nat,Uua: nat] : ( aa(nat,set(nat),aTP_Lamp_zl(nat,fun(nat,set(nat)),Uu),Uua) = aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_ac(nat,fun(nat,$o)),Uu)) ) ).

% ATP.lambda_252
tff(fact_8255_ATP_Olambda__253,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A)] : ( aa(set(A),filter(set(A)),aTP_Lamp_oj(set(A),fun(set(A),filter(set(A))),Uu),Uua) = aa(set(set(A)),filter(set(A)),principal(set(A)),aa(fun(set(A),$o),set(set(A)),collect(set(A)),aa(set(A),fun(set(A),$o),aTP_Lamp_oi(set(A),fun(set(A),fun(set(A),$o)),Uu),Uua))) ) ).

% ATP.lambda_253
tff(fact_8256_ATP_Olambda__254,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_yj(fun(A,$o),fun(fun(B,$o),filter(product_prod(A,B)))),Uu),Uua) = aa(set(product_prod(A,B)),filter(product_prod(A,B)),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_yi(fun(A,$o),fun(fun(B,$o),fun(A,fun(B,$o))),Uu),Uua)))) ) ).

% ATP.lambda_254
tff(fact_8257_ATP_Olambda__255,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,fun(A,$o)),Uua: A] :
      ( aa(A,$o,aTP_Lamp_abj(fun(B,fun(A,$o)),fun(A,$o),Uu),Uua)
    <=> ? [X6: B] : aa(A,$o,aa(B,fun(A,$o),Uu,X6),Uua) ) ).

% ATP.lambda_255
tff(fact_8258_ATP_Olambda__256,axiom,
    ! [A: $tType,Uu: fun(A,fun(product_prod(heap_ext(product_unit),set(nat)),$o)),Uua: product_prod(heap_ext(product_unit),set(nat))] :
      ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aTP_Lamp_acc(fun(A,fun(product_prod(heap_ext(product_unit),set(nat)),$o)),fun(product_prod(heap_ext(product_unit),set(nat)),$o),Uu),Uua)
    <=> ? [X6: A] : aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(A,fun(product_prod(heap_ext(product_unit),set(nat)),$o),Uu,X6),Uua) ) ).

% ATP.lambda_256
tff(fact_8259_ATP_Olambda__257,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,fun(B,$o)),Uua: B] :
      ( aa(B,$o,aTP_Lamp_aaz(fun(A,fun(B,$o)),fun(B,$o),Uu),Uua)
    <=> ? [A7: A] : aa(B,$o,aa(A,fun(B,$o),Uu,A7),Uua) ) ).

% ATP.lambda_257
tff(fact_8260_ATP_Olambda__258,axiom,
    ! [A: $tType,Uu: list(A),Uua: A] :
      ( aa(A,$o,aTP_Lamp_ls(list(A),fun(A,$o),Uu),Uua)
    <=> ? [I2: nat] :
          ( ( Uua = aa(nat,A,nth(A,Uu),I2) )
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Uu)) ) ) ).

% ATP.lambda_258
tff(fact_8261_ATP_Olambda__259,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A)] :
      ( aa(set(A),$o,aTP_Lamp_ly(set(A),fun(set(A),$o),Uu),Uua)
    <=> ? [B12: set(A)] :
          ( ( Uua = aa(set(A),set(A),uminus_uminus(set(A)),B12) )
          & member(set(A),Uu,pow(A,B12)) ) ) ).

% ATP.lambda_259
tff(fact_8262_ATP_Olambda__260,axiom,
    ! [A: $tType,Uu: set(filter(A)),Uua: filter(A)] :
      ( aa(filter(A),$o,aTP_Lamp_ng(set(filter(A)),fun(filter(A),$o),Uu),Uua)
    <=> ! [X6: filter(A)] :
          ( member(filter(A),X6,Uu)
         => aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),Uua),X6) ) ) ).

% ATP.lambda_260
tff(fact_8263_ATP_Olambda__261,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Uu: set(A),Uua: A] :
          ( aa(A,$o,aTP_Lamp_xx(set(A),fun(A,$o),Uu),Uua)
        <=> ! [X6: A] :
              ( member(A,X6,Uu)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),X6) ) ) ) ).

% ATP.lambda_261
tff(fact_8264_ATP_Olambda__262,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Uu: set(A),Uua: A] :
          ( aa(A,$o,aTP_Lamp_xp(set(A),fun(A,$o),Uu),Uua)
        <=> ! [X6: A] :
              ( member(A,X6,Uu)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X6),Uua) ) ) ) ).

% ATP.lambda_262
tff(fact_8265_ATP_Olambda__263,axiom,
    ! [A: $tType,Uu: set(filter(A)),Uua: fun(A,$o)] :
      ( aa(fun(A,$o),$o,aTP_Lamp_wv(set(filter(A)),fun(fun(A,$o),$o),Uu),Uua)
    <=> ! [X6: filter(A)] :
          ( member(filter(A),X6,Uu)
         => aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Uua),X6) ) ) ).

% ATP.lambda_263
tff(fact_8266_ATP_Olambda__264,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: fun(A,$o),Uua: A] :
          ( aa(A,$o,aTP_Lamp_wl(fun(A,$o),fun(A,$o),Uu),Uua)
        <=> ! [Y4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),Y4)
             => aa(A,$o,Uu,Y4) ) ) ) ).

% ATP.lambda_264
tff(fact_8267_ATP_Olambda__265,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: set(A)] :
      ( aa(set(A),$o,aTP_Lamp_vj(set(product_prod(A,A)),fun(set(A),$o),Uu),Uua)
    <=> ! [X6: A] :
          ( member(A,X6,Uua)
         => ! [Xa2: A] :
              ( member(A,Xa2,Uua)
             => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X6),Xa2),Uu)
                | member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa2),X6),Uu) ) ) ) ) ).

% ATP.lambda_265
tff(fact_8268_ATP_Olambda__266,axiom,
    ! [B: $tType,A: $tType,Uu: set(product_prod(A,B)),Uua: A] :
      ( aa(A,$o,aTP_Lamp_abk(set(product_prod(A,B)),fun(A,$o),Uu),Uua)
    <=> ? [Y4: B] : member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),Y4),Uu) ) ).

% ATP.lambda_266
tff(fact_8269_ATP_Olambda__267,axiom,
    ! [A: $tType,Uu: fun(A,assn),Uua: product_prod(heap_ext(product_unit),set(nat))] :
      ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aTP_Lamp_abd(fun(A,assn),fun(product_prod(heap_ext(product_unit),set(nat)),$o),Uu),Uua)
    <=> ? [X6: A] : aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,aa(A,assn,Uu,X6)),Uua) ) ).

% ATP.lambda_267
tff(fact_8270_ATP_Olambda__268,axiom,
    ! [A: $tType,B: $tType,Uu: A,Uua: product_prod(A,B)] :
      ( aa(product_prod(A,B),$o,aTP_Lamp_uq(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_268
tff(fact_8271_ATP_Olambda__269,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: A] :
      ( aa(A,$o,aTP_Lamp_lk(fun(B,A),fun(A,$o),Uu),Uua)
    <=> ? [X6: B] : ( Uua = aa(B,A,Uu,X6) ) ) ).

% ATP.lambda_269
tff(fact_8272_ATP_Olambda__270,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_lt(set(product_prod(A,B)),fun(product_prod(product_prod($o,A),product_prod($o,B)),$o),Uu),Uua)
    <=> ? [X6: A,Y4: B] :
          ( ( Uua = aa(product_prod($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),$false),X6)),aa(B,product_prod($o,B),aa($o,fun(B,product_prod($o,B)),product_Pair($o,B),$false),Y4)) )
          & member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X6),Y4),Uu) ) ) ).

% ATP.lambda_270
tff(fact_8273_ATP_Olambda__271,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,option(B)),Uua: product_prod(A,B)] :
      ( aa(product_prod(A,B),$o,aTP_Lamp_vd(fun(A,option(B)),fun(product_prod(A,B),$o),Uu),Uua)
    <=> ? [A7: A,B8: B] :
          ( ( Uua = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A7),B8) )
          & ( aa(A,option(B),Uu,A7) = aa(B,option(B),some(B),B8) ) ) ) ).

% ATP.lambda_271
tff(fact_8274_ATP_Olambda__272,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_rd(set(product_prod(A,A)),fun(product_prod(set(A),set(A)),$o),Uu),Uua)
    <=> ? [X9: 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)),X9),Y8) )
          & ( X9 != bot_bot(set(A)) )
          & ! [X6: A] :
              ( member(A,X6,Y8)
             => ? [Xa2: A] :
                  ( member(A,Xa2,X9)
                  & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa2),X6),Uu) ) ) ) ) ).

% ATP.lambda_272
tff(fact_8275_ATP_Olambda__273,axiom,
    ! [A: $tType,Uu: set(A),Uua: A] : ( aa(A,set(A),aTP_Lamp_rb(set(A),fun(A,set(A)),Uu),Uua) = aa(set(A),set(A),uminus_uminus(set(A)),Uu) ) ).

% ATP.lambda_273
tff(fact_8276_ATP_Olambda__274,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] : ( aa(list(A),option(A),aa(A,fun(list(A),option(A)),aTP_Lamp_vc(A,fun(list(A),option(A))),Uu),Uua) = aa(A,option(A),some(A),Uu) ) ).

% ATP.lambda_274
tff(fact_8277_ATP_Olambda__275,axiom,
    ! [E: $tType,B: $tType,Uu: set(product_prod(B,B)),Uua: E] : ( aa(E,set(B),aTP_Lamp_zc(set(product_prod(B,B)),fun(E,set(B)),Uu),Uua) = aa(set(product_prod(B,B)),set(B),field2(B),Uu) ) ).

% ATP.lambda_275
tff(fact_8278_ATP_Olambda__276,axiom,
    ! [A: $tType,B: $tType,Uu: set(product_prod(B,B)),Uua: A] : ( aa(A,set(B),aTP_Lamp_zf(set(product_prod(B,B)),fun(A,set(B)),Uu),Uua) = aa(set(product_prod(B,B)),set(B),field2(B),Uu) ) ).

% ATP.lambda_276
tff(fact_8279_ATP_Olambda__277,axiom,
    ! [E: $tType,A: $tType,Uu: set(product_prod(A,A)),Uua: E] : ( aa(E,set(A),aTP_Lamp_zb(set(product_prod(A,A)),fun(E,set(A)),Uu),Uua) = aa(set(product_prod(A,A)),set(A),field2(A),Uu) ) ).

% ATP.lambda_277
tff(fact_8280_ATP_Olambda__278,axiom,
    ! [B: $tType,A: $tType,Uu: set(product_prod(A,A)),Uua: B] : ( aa(B,set(A),aTP_Lamp_ze(set(product_prod(A,A)),fun(B,set(A)),Uu),Uua) = aa(set(product_prod(A,A)),set(A),field2(A),Uu) ) ).

% ATP.lambda_278
tff(fact_8281_ATP_Olambda__279,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: A] : ( aa(A,set(A),aTP_Lamp_qc(set(product_prod(A,A)),fun(A,set(A)),Uu),Uua) = aa(set(product_prod(A,A)),set(A),field2(A),Uu) ) ).

% ATP.lambda_279
tff(fact_8282_ATP_Olambda__280,axiom,
    ! [A: $tType,B: $tType,Uu: list(B),Uua: A] : ( aa(A,set(B),aTP_Lamp_px(list(B),fun(A,set(B)),Uu),Uua) = aa(list(B),set(B),set2(B),Uu) ) ).

% ATP.lambda_280
tff(fact_8283_ATP_Olambda__281,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_ti(A,fun(list(A),fun(set(A),set(A)))),Uu),Uua) = aa(A,fun(set(A),set(A)),insert3(A),Uu) ) ).

% ATP.lambda_281
tff(fact_8284_ATP_Olambda__282,axiom,
    ! [A: $tType,Uu: set(A),Uua: list(A)] : ( aa(list(A),set(list(A)),aTP_Lamp_qd(set(A),fun(list(A),set(list(A))),Uu),Uua) = aa(set(A),set(list(A)),lists(A),Uu) ) ).

% ATP.lambda_282
tff(fact_8285_ATP_Olambda__283,axiom,
    ! [A: $tType,Uu: nat,Uua: A] : ( aa(A,nat,aa(nat,fun(A,nat),aTP_Lamp_aj(nat,fun(A,nat)),Uu),Uua) = aa(nat,nat,suc,Uu) ) ).

% ATP.lambda_283
tff(fact_8286_ATP_Olambda__284,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_kx(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),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),numeral_numeral(code_integer,bit0(one2))),Uua)),one_one(code_integer))),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),Uub),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),numeral_numeral(code_integer,bit0(one2))),Uua)),Uub)) ) ).

% ATP.lambda_284
tff(fact_8287_ATP_Olambda__285,axiom,
    ! [Uu: num,Uua: nat,Uub: nat] :
      ( aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),aTP_Lamp_gw(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),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),numeral_numeral(nat,bit0(one2))),Uua)),one_one(nat))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),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),numeral_numeral(nat,bit0(one2))),Uua)),Uub)) ) ).

% ATP.lambda_285
tff(fact_8288_ATP_Olambda__286,axiom,
    ! [Uu: num,Uua: int,Uub: int] :
      ( aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_gx(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),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),numeral_numeral(int,bit0(one2))),Uua)),one_one(int))),aa(int,int,aa(int,fun(int,int),minus_minus(int),Uub),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),numeral_numeral(int,bit0(one2))),Uua)),Uub)) ) ).

% ATP.lambda_286
tff(fact_8289_ATP_Olambda__287,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_gz(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),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),numeral_numeral(A,bit0(one2))),Uua)),one_one(A))),aa(A,A,aa(A,fun(A,A),minus_minus(A),Uub),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),numeral_numeral(A,bit0(one2))),Uua)),Uub)) ) ) ).

% ATP.lambda_287
tff(fact_8290_ATP_Olambda__288,axiom,
    ! [A: $tType,Uu: A,Uua: A,Uub: list(A)] :
      ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),aTP_Lamp_of(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_288
tff(fact_8291_ATP_Olambda__289,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_te(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_289
tff(fact_8292_ATP_Olambda__290,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: A,Uub: A] :
      ( aa(A,nat,aa(A,fun(A,nat),aTP_Lamp_sv(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_290
tff(fact_8293_ATP_Olambda__291,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(A)
     => ! [Uu: B,Uua: A,Uub: B] :
          ( aa(B,A,aa(A,fun(B,A),aTP_Lamp_sq(B,fun(A,fun(B,A)),Uu),Uua),Uub) = $ite(Uu = Uub,Uua,zero_zero(A)) ) ) ).

% ATP.lambda_291
tff(fact_8294_ATP_Olambda__292,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(A)
     => ! [Uu: B,Uua: A,Uub: B] :
          ( aa(B,A,aa(A,fun(B,A),aTP_Lamp_sr(B,fun(A,fun(B,A)),Uu),Uua),Uub) = $ite(Uub = Uu,Uua,zero_zero(A)) ) ) ).

% ATP.lambda_292
tff(fact_8295_ATP_Olambda__293,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_md(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_293
tff(fact_8296_ATP_Olambda__294,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_nf(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),abs_abs(code_integer,Uu)),Uub))) ) ).

% ATP.lambda_294
tff(fact_8297_ATP_Olambda__295,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_mc(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_295
tff(fact_8298_ATP_Olambda__296,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_lc(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_296
tff(fact_8299_ATP_Olambda__297,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_km(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)),insert3(A),Uua),Uub),Uub) ) ).

% ATP.lambda_297
tff(fact_8300_ATP_Olambda__298,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,$o),fun(fun(A,nat),fun(A,nat)),aTP_Lamp_tb(fun(A,$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_298
tff(fact_8301_ATP_Olambda__299,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: fun(A,$o),Uub: A] :
      ( aa(A,nat,aa(fun(A,$o),fun(A,nat),aTP_Lamp_ta(fun(A,nat),fun(fun(A,$o),fun(A,nat)),Uu),Uua),Uub) = $ite(aa(A,$o,Uua,Uub),aa(A,nat,Uu,Uub),zero_zero(nat)) ) ).

% ATP.lambda_299
tff(fact_8302_ATP_Olambda__300,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,$o),fun(B,fun(A,A)),aTP_Lamp_on(fun(B,fun(A,A)),fun(fun(B,$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_300
tff(fact_8303_ATP_Olambda__301,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,A),Uua: fun(B,$o),Uub: B] :
      ( aa(B,option(A),aa(fun(B,$o),fun(B,option(A)),aTP_Lamp_ur(fun(B,A),fun(fun(B,$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_301
tff(fact_8304_ATP_Olambda__302,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_qn(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_qm(A,fun(B,fun(set(product_prod(A,B)),set(product_prod(A,B)))),Uua),Uub,Uu) ) ).

% ATP.lambda_302
tff(fact_8305_ATP_Olambda__303,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_pr(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_iu(B,fun(A,fun(set(product_prod(B,A)),set(product_prod(B,A)))),Uua),Uub,Uu) ) ).

% ATP.lambda_303
tff(fact_8306_ATP_Olambda__304,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_pm(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_304
tff(fact_8307_ATP_Olambda__305,axiom,
    ! [A: $tType,Uu: fun(A,A),Uua: A,Uub: nat] : ( aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_tv(fun(A,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,compow(fun(A,A),Uub,Uu),Uua) ) ).

% ATP.lambda_305
tff(fact_8308_ATP_Olambda__306,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_hu(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_306
tff(fact_8309_ATP_Olambda__307,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_hs(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_307
tff(fact_8310_ATP_Olambda__308,axiom,
    ! [A: $tType,Uu: fun(nat,fun(nat,A)),Uua: nat,Uub: nat] : ( aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_aci(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_308
tff(fact_8311_ATP_Olambda__309,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_vz(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_309
tff(fact_8312_ATP_Olambda__310,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_vw(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_310
tff(fact_8313_ATP_Olambda__311,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_sc(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_311
tff(fact_8314_ATP_Olambda__312,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_ku(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_312
tff(fact_8315_ATP_Olambda__313,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_pp(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_313
tff(fact_8316_ATP_Olambda__314,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_kv(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_314
tff(fact_8317_ATP_Olambda__315,axiom,
    ! [A: $tType,Uu: fun(A,fun(A,$o)),Uua: A,Uub: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_cn(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_315
tff(fact_8318_ATP_Olambda__316,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_ma(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_316
tff(fact_8319_ATP_Olambda__317,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)),$o),aTP_Lamp_abm(set(set(A)),fun(set(set(A)),fun(set(set(A)),$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_317
tff(fact_8320_ATP_Olambda__318,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_acu(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_act(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_318
tff(fact_8321_ATP_Olambda__319,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_aep(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_aeo(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_319
tff(fact_8322_ATP_Olambda__320,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : ( aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_aas(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_aar(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uua),Uub)),quotient_of(Uu)) ) ).

% ATP.lambda_320
tff(fact_8323_ATP_Olambda__321,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : ( aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_aaq(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_aap(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uua),Uub)),quotient_of(Uu)) ) ).

% ATP.lambda_321
tff(fact_8324_ATP_Olambda__322,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : ( aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_aao(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_aan(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uua),Uub)),quotient_of(Uu)) ) ).

% ATP.lambda_322
tff(fact_8325_ATP_Olambda__323,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : ( aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_aam(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_aal(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uua),Uub)),quotient_of(Uu)) ) ).

% ATP.lambda_323
tff(fact_8326_ATP_Olambda__324,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_yb(fun(B,A),fun(filter(B),fun(fun(A,$o),$o)),Uu),Uua),Uub)
    <=> aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),eventually(B),aa(fun(A,$o),fun(B,$o),aTP_Lamp_ya(fun(B,A),fun(fun(A,$o),fun(B,$o)),Uu),Uub)),Uua) ) ).

% ATP.lambda_324
tff(fact_8327_ATP_Olambda__325,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Uu: array(A),Uua: nat,Uub: heap_ext(product_unit)] : ( aa(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(nat,fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),aTP_Lamp_aef(array(A),fun(nat,fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)))),Uu),Uua),Uub) = aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),aa(nat,A,nth(A,array_get(A,Uub,Uu)),Uua)),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),Uub),one_one(nat))) ) ) ).

% ATP.lambda_325
tff(fact_8328_ATP_Olambda__326,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_fc(fun(nat,A),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),Uub)),aa(nat,A,Uua,Uub)) ) ) ).

% ATP.lambda_326
tff(fact_8329_ATP_Olambda__327,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),$o),aTP_Lamp_ic(set(product_prod(A,A)),fun(list(A),fun(list(A),$o)),Uu),Uua),Uub)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),Uua)),aa(list(A),nat,size_size(list(A)),Uub))
        | ( ( aa(list(A),nat,size_size(list(A)),Uua) = aa(list(A),nat,size_size(list(A)),Uub) )
          & member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Uua),Uub),lex(A,Uu)) ) ) ) ).

% ATP.lambda_327
tff(fact_8330_ATP_Olambda__328,axiom,
    ! [A: $tType,Uu: nat,Uua: list(A),Uub: list(A)] :
      ( aa(list(A),$o,aa(list(A),fun(list(A),$o),aTP_Lamp_le(nat,fun(list(A),fun(list(A),$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_328
tff(fact_8331_ATP_Olambda__329,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),$o),aTP_Lamp_lq(set(product_prod(A,A)),fun(list(A),fun(list(A),$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),X6: A,Y4: A,Xs5: list(A),Ys6: list(A)] :
            ( ( Uua = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xys2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X6),Xs5)) )
            & ( Uub = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xys2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys6)) )
            & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X6),Y4),Uu) ) ) ) ).

% ATP.lambda_329
tff(fact_8332_ATP_Olambda__330,axiom,
    ! [A: $tType,Uu: nat,Uua: set(A),Uub: list(A)] :
      ( aa(list(A),$o,aa(set(A),fun(list(A),$o),aTP_Lamp_es(nat,fun(set(A),fun(list(A),$o)),Uu),Uua),Uub)
    <=> ( ( aa(list(A),nat,size_size(list(A)),Uub) = Uu )
        & aa(list(A),$o,distinct(A),Uub)
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uub)),Uua) ) ) ).

% ATP.lambda_330
tff(fact_8333_ATP_Olambda__331,axiom,
    ! [A: $tType,Uu: set(A),Uua: nat,Uub: list(A)] :
      ( aa(list(A),$o,aa(nat,fun(list(A),$o),aTP_Lamp_az(set(A),fun(nat,fun(list(A),$o)),Uu),Uua),Uub)
    <=> ( ( aa(list(A),nat,size_size(list(A)),Uub) = Uua )
        & aa(list(A),$o,distinct(A),Uub)
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uub)),Uu) ) ) ).

% ATP.lambda_331
tff(fact_8334_ATP_Olambda__332,axiom,
    ! [A: $tType,Uu: set(A),Uua: nat,Uub: list(A)] :
      ( aa(list(A),$o,aa(nat,fun(list(A),$o),aTP_Lamp_qo(set(A),fun(nat,fun(list(A),$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_332
tff(fact_8335_ATP_Olambda__333,axiom,
    ! [A: $tType,Uu: nat,Uua: list(A),Uub: list(A)] :
      ( aa(list(A),$o,aa(list(A),fun(list(A),$o),aTP_Lamp_dw(nat,fun(list(A),fun(list(A),$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_333
tff(fact_8336_ATP_Olambda__334,axiom,
    ! [A: $tType,Uu: set(A),Uua: nat,Uub: list(A)] :
      ( aa(list(A),$o,aa(nat,fun(list(A),$o),aTP_Lamp_dz(set(A),fun(nat,fun(list(A),$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_334
tff(fact_8337_ATP_Olambda__335,axiom,
    ! [A: $tType,Uu: set(A),Uua: nat,Uub: list(A)] :
      ( aa(list(A),$o,aa(nat,fun(list(A),$o),aTP_Lamp_dp(set(A),fun(nat,fun(list(A),$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_335
tff(fact_8338_ATP_Olambda__336,axiom,
    ! [Uu: heap_ext(product_unit),Uua: heap_ext(product_unit),Uub: nat] :
      ( aa(nat,$o,aa(heap_ext(product_unit),fun(nat,$o),aTP_Lamp_ada(heap_ext(product_unit),fun(heap_ext(product_unit),fun(nat,$o)),Uu),Uua),Uub)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),lim(product_unit,Uu)),Uub)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uub),lim(product_unit,Uua)) ) ) ).

% ATP.lambda_336
tff(fact_8339_ATP_Olambda__337,axiom,
    ! [Uu: set(nat),Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_bg(set(nat),fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> ( member(nat,aa(nat,nat,suc,Uub),Uu)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uub),Uua) ) ) ).

% ATP.lambda_337
tff(fact_8340_ATP_Olambda__338,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),$o),aTP_Lamp_pd(set(nat),fun(nat,fun(product_prod(A,nat),$o)),Uu),Uua),Uub)
    <=> member(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(product_prod(A,nat),nat,product_snd(A,nat),Uub)),Uua),Uu) ) ).

% ATP.lambda_338
tff(fact_8341_ATP_Olambda__339,axiom,
    ! [A: $tType,Uu: list(A),Uua: set(nat),Uub: nat] :
      ( aa(nat,$o,aa(set(nat),fun(nat,$o),aTP_Lamp_ob(list(A),fun(set(nat),fun(nat,$o)),Uu),Uua),Uub)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uub),aa(list(A),nat,size_size(list(A)),Uu))
        & member(nat,Uub,Uua) ) ) ).

% ATP.lambda_339
tff(fact_8342_ATP_Olambda__340,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: list(A),Uub: nat] :
      ( aa(nat,$o,aa(list(A),fun(nat,$o),aTP_Lamp_jk(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_340
tff(fact_8343_ATP_Olambda__341,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: list(A),Uua: fun(A,$o),Uub: A] :
          ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ve(list(A),fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
        <=> ( member(A,Uub,aa(list(A),set(A),set2(A),Uu))
            & aa(A,$o,Uua,Uub) ) ) ) ).

% ATP.lambda_341
tff(fact_8344_ATP_Olambda__342,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: list(A),Uub: A] :
      ( aa(A,$o,aa(list(A),fun(A,$o),aTP_Lamp_jd(fun(A,$o),fun(list(A),fun(A,$o)),Uu),Uua),Uub)
    <=> ( member(A,Uub,aa(list(A),set(A),set2(A),Uua))
        & aa(A,$o,Uu,Uub) ) ) ).

% ATP.lambda_342
tff(fact_8345_ATP_Olambda__343,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat,Uub: nat] : ( aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_el(A,fun(nat,fun(nat,A)),Uu),Uua),Uub) = divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Uu),aa(nat,A,semiring_1_of_nat(A),Uub)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),Uub))) ) ) ).

% ATP.lambda_343
tff(fact_8346_ATP_Olambda__344,axiom,
    ! [Uu: heap_ext(product_unit),Uua: set(nat),Uub: nat] :
      ( aa(nat,$o,aa(set(nat),fun(nat,$o),aTP_Lamp_adk(heap_ext(product_unit),fun(set(nat),fun(nat,$o)),Uu),Uua),Uub)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uub),lim(product_unit,Uu))
        & ~ member(nat,Uub,Uua) ) ) ).

% ATP.lambda_344
tff(fact_8347_ATP_Olambda__345,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: set(A),Uub: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_mk(set(product_prod(A,A)),fun(set(A),fun(A,$o)),Uu),Uua),Uub)
    <=> ( member(A,Uub,aa(set(product_prod(A,A)),set(A),field2(A),Uu))
        & ! [X6: A] :
            ( member(A,X6,Uua)
           => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),X6),Uu) ) ) ) ).

% ATP.lambda_345
tff(fact_8348_ATP_Olambda__346,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: set(A),Uub: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_mm(set(product_prod(A,A)),fun(set(A),fun(A,$o)),Uu),Uua),Uub)
    <=> ( member(A,Uub,aa(set(product_prod(A,A)),set(A),field2(A),Uu))
        & ! [X6: A] :
            ( member(A,X6,Uua)
           => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X6),Uub),Uu) ) ) ) ).

% ATP.lambda_346
tff(fact_8349_ATP_Olambda__347,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: set(A),Uub: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_ml(set(product_prod(A,A)),fun(set(A),fun(A,$o)),Uu),Uua),Uub)
    <=> ( member(A,Uub,aa(set(product_prod(A,A)),set(A),field2(A),Uu))
        & ! [X6: A] :
            ( member(A,X6,Uua)
           => ( ( Uub != X6 )
              & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),X6),Uu) ) ) ) ) ).

% ATP.lambda_347
tff(fact_8350_ATP_Olambda__348,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: set(A),Uub: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_xi(set(product_prod(A,A)),fun(set(A),fun(A,$o)),Uu),Uua),Uub)
    <=> ( member(A,Uub,aa(set(product_prod(A,A)),set(A),field2(A),Uu))
        & ! [X6: A] :
            ( member(A,X6,Uua)
           => ( ( Uub != X6 )
              & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X6),Uub),Uu) ) ) ) ) ).

% ATP.lambda_348
tff(fact_8351_ATP_Olambda__349,axiom,
    ! [A: $tType,Uu: list(A),Uua: set(nat),Uub: nat] :
      ( aa(nat,$o,aa(set(nat),fun(nat,$o),aTP_Lamp_nz(list(A),fun(set(nat),fun(nat,$o)),Uu),Uua),Uub)
    <=> member(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),aa(list(A),nat,size_size(list(A)),Uu)),Uua) ) ).

% ATP.lambda_349
tff(fact_8352_ATP_Olambda__350,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_ix(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_350
tff(fact_8353_ATP_Olambda__351,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [Uu: set(A),Uua: nat,Uub: A] :
          ( aa(A,$o,aa(nat,fun(A,$o),aTP_Lamp_rj(set(A),fun(nat,fun(A,$o)),Uu),Uua),Uub)
        <=> ( member(A,Uub,Uu)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),infini527867602293511546merate(A,Uu,Uua)),Uub) ) ) ) ).

% ATP.lambda_351
tff(fact_8354_ATP_Olambda__352,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),$o),aTP_Lamp_xj(set(product_prod(A,A)),fun(set(A),fun(set(A),$o)),Uu),Uua),Uub)
    <=> ( order_ofilter(A,Uu,Uua)
        & ( Uua != aa(set(product_prod(A,A)),set(A),field2(A),Uu) )
        & order_ofilter(A,Uu,Uub)
        & ( Uub != aa(set(product_prod(A,A)),set(A),field2(A),Uu) )
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),Uua),Uub) ) ) ).

% ATP.lambda_352
tff(fact_8355_ATP_Olambda__353,axiom,
    ! [Uu: set(nat),Uua: set(nat),Uub: nat] :
      ( aa(nat,$o,aa(set(nat),fun(nat,$o),aTP_Lamp_ri(set(nat),fun(set(nat),fun(nat,$o)),Uu),Uua),Uub)
    <=> ( member(nat,Uub,Uu)
        & member(nat,finite_card(nat,aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_rh(set(nat),fun(nat,fun(nat,$o)),Uu),Uub))),Uua) ) ) ).

% ATP.lambda_353
tff(fact_8356_ATP_Olambda__354,axiom,
    ! [Uu: set(nat),Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_bh(set(nat),fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> ( member(nat,Uub,Uu)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uub),aa(nat,nat,suc,Uua)) ) ) ).

% ATP.lambda_354
tff(fact_8357_ATP_Olambda__355,axiom,
    ! [A: $tType,Uu: multiset(A),Uua: multiset(A),Uub: A] :
      ( aa(A,$o,aa(multiset(A),fun(A,$o),aTP_Lamp_si(multiset(A),fun(multiset(A),fun(A,$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_355
tff(fact_8358_ATP_Olambda__356,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_xr(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_356
tff(fact_8359_ATP_Olambda__357,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_abg(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_357
tff(fact_8360_ATP_Olambda__358,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A),Uub: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_bu(set(A),fun(set(A),fun(A,$o)),Uu),Uua),Uub)
    <=> ( member(A,Uub,Uu)
        | member(A,Uub,Uua) ) ) ).

% ATP.lambda_358
tff(fact_8361_ATP_Olambda__359,axiom,
    ! [A: $tType,Uu: A,Uua: set(A),Uub: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_cp(A,fun(set(A),fun(A,$o)),Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        | member(A,Uub,Uua) ) ) ).

% ATP.lambda_359
tff(fact_8362_ATP_Olambda__360,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A),Uub: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),aTP_Lamp_hw(set(A),fun(set(A),fun(set(A),$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_360
tff(fact_8363_ATP_Olambda__361,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A),Uub: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),aTP_Lamp_hx(set(A),fun(set(A),fun(set(A),$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_361
tff(fact_8364_ATP_Olambda__362,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_abh(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_362
tff(fact_8365_ATP_Olambda__363,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_aat(nat,fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> ( dvd_dvd(nat,Uub,Uua)
        & dvd_dvd(nat,Uub,Uu) ) ) ).

% ATP.lambda_363
tff(fact_8366_ATP_Olambda__364,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_xa(filter(A),fun(filter(A),fun(fun(A,$o),$o)),Uu),Uua),Uub)
    <=> ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Uub),Uu)
        & aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Uub),Uua) ) ) ).

% ATP.lambda_364
tff(fact_8367_ATP_Olambda__365,axiom,
    ! [A: $tType,Uu: set(A),Uua: A,Uub: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_mi(set(A),fun(A,fun(A,$o)),Uu),Uua),Uub)
    <=> ( member(A,Uua,Uu)
        & member(A,Uub,Uu) ) ) ).

% ATP.lambda_365
tff(fact_8368_ATP_Olambda__366,axiom,
    ! [Uu: set(nat),Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_rh(set(nat),fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> ( member(nat,Uub,Uu)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uub),Uua) ) ) ).

% ATP.lambda_366
tff(fact_8369_ATP_Olambda__367,axiom,
    ! [A: $tType,Uu: set(set(A)),Uua: A,Uub: set(A)] :
      ( aa(set(A),$o,aa(A,fun(set(A),$o),aTP_Lamp_yt(set(set(A)),fun(A,fun(set(A),$o)),Uu),Uua),Uub)
    <=> ( member(set(A),Uub,Uu)
        & member(A,Uua,Uub) ) ) ).

% ATP.lambda_367
tff(fact_8370_ATP_Olambda__368,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A),Uub: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_bw(set(A),fun(set(A),fun(A,$o)),Uu),Uua),Uub)
    <=> ( member(A,Uub,Uu)
        & member(A,Uub,Uua) ) ) ).

% ATP.lambda_368
tff(fact_8371_ATP_Olambda__369,axiom,
    ! [A: $tType,Uu: A,Uua: A,Uub: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_aax(A,fun(A,fun(A,$o)),Uu),Uua),Uub)
    <=> ( ( Uua = Uu )
        & ( Uub = Uu ) ) ) ).

% ATP.lambda_369
tff(fact_8372_ATP_Olambda__370,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A),Uub: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),aTP_Lamp_da(set(A),fun(set(A),fun(set(A),$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_370
tff(fact_8373_ATP_Olambda__371,axiom,
    ! [A: $tType,Uu: list(A),Uua: fun(A,nat),Uub: A] : ( aa(A,nat,aa(fun(A,nat),fun(A,nat),aTP_Lamp_is(list(A),fun(fun(A,nat),fun(A,nat)),Uu),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(A,nat,count_list(A,Uu),Uub)),aa(A,nat,Uua,Uub)) ) ).

% ATP.lambda_371
tff(fact_8374_ATP_Olambda__372,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: list(A),Uub: A] : ( aa(A,nat,aa(list(A),fun(A,nat),aTP_Lamp_iq(fun(A,nat),fun(list(A),fun(A,nat)),Uu),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(A,nat,count_list(A,Uua),Uub)),aa(A,nat,Uu,Uub)) ) ).

% ATP.lambda_372
tff(fact_8375_ATP_Olambda__373,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: set(A),Uub: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_wk(fun(A,$o),fun(set(A),fun(A,$o)),Uu),Uua),Uub)
    <=> ( member(A,Uub,Uua)
       => aa(A,$o,Uu,Uub) ) ) ).

% ATP.lambda_373
tff(fact_8376_ATP_Olambda__374,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu: set(A),Uua: fun(A,$o),Uub: A] :
          ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_rl(set(A),fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
        <=> ( member(A,Uub,Uu)
            & aa(A,$o,Uua,Uub) ) ) ) ).

% ATP.lambda_374
tff(fact_8377_ATP_Olambda__375,axiom,
    ! [A: $tType,Uu: set(A),Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_br(set(A),fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> ( member(A,Uub,Uu)
        & aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_375
tff(fact_8378_ATP_Olambda__376,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: set(A),Uub: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_kn(fun(A,$o),fun(set(A),fun(A,$o)),Uu),Uua),Uub)
    <=> ( member(A,Uub,Uua)
        & aa(A,$o,Uu,Uub) ) ) ).

% ATP.lambda_376
tff(fact_8379_ATP_Olambda__377,axiom,
    ! [A: $tType,Uu: A,Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_cs(A,fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> ( ( Uu = Uub )
        & aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_377
tff(fact_8380_ATP_Olambda__378,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: A,Uub: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_lb(fun(A,$o),fun(A,fun(A,$o)),Uu),Uua),Uub)
    <=> ( ( Uua = Uub )
        & aa(A,$o,Uu,Uua) ) ) ).

% ATP.lambda_378
tff(fact_8381_ATP_Olambda__379,axiom,
    ! [A: $tType,Uu: A,Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_cr(A,fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        & aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_379
tff(fact_8382_ATP_Olambda__380,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,$o),aTP_Lamp_qh(set(A),fun(fun(A,set(B)),fun(A,$o)),Uu),Uua),Uub)
    <=> ( member(A,Uub,Uu)
        & ( aa(A,set(B),Uua,Uub) != bot_bot(set(B)) ) ) ) ).

% ATP.lambda_380
tff(fact_8383_ATP_Olambda__381,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,$o),aTP_Lamp_ks(set(A),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> ( member(A,Uub,Uu)
            & ( aa(A,B,Uua,Uub) != zero_zero(B) ) ) ) ) ).

% ATP.lambda_381
tff(fact_8384_ATP_Olambda__382,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,$o),aTP_Lamp_kq(set(A),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> ( member(A,Uub,Uu)
            & ( aa(A,B,Uua,Uub) != zero_zero(B) ) ) ) ) ).

% ATP.lambda_382
tff(fact_8385_ATP_Olambda__383,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,$o),aTP_Lamp_bb(set(A),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> ( member(A,Uub,Uu)
            & ( aa(A,B,Uua,Uub) != one_one(B) ) ) ) ) ).

% ATP.lambda_383
tff(fact_8386_ATP_Olambda__384,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Uu: array(A),Uua: list(A),Uub: nat] : ( aa(nat,assn,aa(list(A),fun(nat,assn),aTP_Lamp_adh(array(A),fun(list(A),fun(nat,assn)),Uu),Uua),Uub) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),snga_assn(A,Uu,Uua)),pure_assn(Uub = aa(list(A),nat,size_size(list(A)),Uua))) ) ) ).

% ATP.lambda_384
tff(fact_8387_ATP_Olambda__385,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Uu: ref(A),Uua: A,Uub: A] : ( aa(A,assn,aa(A,fun(A,assn),aTP_Lamp_adi(ref(A),fun(A,fun(A,assn)),Uu),Uua),Uub) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),sngr_assn(A,Uu,Uua)),pure_assn(Uub = Uua)) ) ) ).

% ATP.lambda_385
tff(fact_8388_ATP_Olambda__386,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Uu: array(A),Uua: list(A),Uub: list(A)] : ( aa(list(A),assn,aa(list(A),fun(list(A),assn),aTP_Lamp_adv(array(A),fun(list(A),fun(list(A),assn)),Uu),Uua),Uub) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),snga_assn(A,Uu,Uua)),pure_assn(Uub = Uua)) ) ) ).

% ATP.lambda_386
tff(fact_8389_ATP_Olambda__387,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A),Uub: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_cf(set(A),fun(set(A),fun(A,$o)),Uu),Uua),Uub)
    <=> ( member(A,Uub,Uu)
        & ~ member(A,Uub,Uua) ) ) ).

% ATP.lambda_387
tff(fact_8390_ATP_Olambda__388,axiom,
    ! [A: $tType,B: $tType,Uu: list(product_prod(A,B)),Uua: A,Uub: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_uv(list(product_prod(A,B)),fun(A,fun(B,$o)),Uu),Uua),Uub)
    <=> ( aa(A,option(B),map_of(A,B,Uu),Uua) = aa(B,option(B),some(B),Uub) ) ) ).

% ATP.lambda_388
tff(fact_8391_ATP_Olambda__389,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_hr(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_389
tff(fact_8392_ATP_Olambda__390,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A,Uub: A] : ( aa(A,A,aa(A,fun(A,A),aTP_Lamp_eh(A,fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),minus_minus(A),divide_divide(A,Uub,Uu)),Uua) ) ) ).

% ATP.lambda_390
tff(fact_8393_ATP_Olambda__391,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A,Uub: A] : ( aa(A,A,aa(A,fun(A,A),aTP_Lamp_ef(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_391
tff(fact_8394_ATP_Olambda__392,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A,Uub: A] : ( aa(A,A,aa(A,fun(A,A),aTP_Lamp_eg(A,fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,Uub,Uu)),Uua) ) ) ).

% ATP.lambda_392
tff(fact_8395_ATP_Olambda__393,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A,Uub: A] : ( aa(A,A,aa(A,fun(A,A),aTP_Lamp_ee(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_393
tff(fact_8396_ATP_Olambda__394,axiom,
    ! [B: $tType,E: $tType,Uu: set(product_prod(E,B)),Uua: E,Uub: B] :
      ( aa(B,$o,aa(E,fun(B,$o),aTP_Lamp_aeu(set(product_prod(E,B)),fun(E,fun(B,$o)),Uu),Uua),Uub)
    <=> member(product_prod(E,B),aa(B,product_prod(E,B),aa(E,fun(B,product_prod(E,B)),product_Pair(E,B),Uua),Uub),Uu) ) ).

% ATP.lambda_394
tff(fact_8397_ATP_Olambda__395,axiom,
    ! [E: $tType,B: $tType,Uu: set(product_prod(B,E)),Uua: B,Uub: E] :
      ( aa(E,$o,aa(B,fun(E,$o),aTP_Lamp_aeq(set(product_prod(B,E)),fun(B,fun(E,$o)),Uu),Uua),Uub)
    <=> member(product_prod(B,E),aa(E,product_prod(B,E),aa(B,fun(E,product_prod(B,E)),product_Pair(B,E),Uua),Uub),Uu) ) ).

% ATP.lambda_395
tff(fact_8398_ATP_Olambda__396,axiom,
    ! [A: $tType,B: $tType,Uu: set(product_prod(B,A)),Uua: B,Uub: A] :
      ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_rw(set(product_prod(B,A)),fun(B,fun(A,$o)),Uu),Uua),Uub)
    <=> member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Uua),Uub),Uu) ) ).

% ATP.lambda_396
tff(fact_8399_ATP_Olambda__397,axiom,
    ! [E: $tType,A: $tType,Uu: set(product_prod(A,E)),Uua: A,Uub: E] :
      ( aa(E,$o,aa(A,fun(E,$o),aTP_Lamp_aet(set(product_prod(A,E)),fun(A,fun(E,$o)),Uu),Uua),Uub)
    <=> member(product_prod(A,E),aa(E,product_prod(A,E),aa(A,fun(E,product_prod(A,E)),product_Pair(A,E),Uua),Uub),Uu) ) ).

% ATP.lambda_397
tff(fact_8400_ATP_Olambda__398,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_bi(set(product_prod(A,B)),fun(A,fun(B,$o))),Uu),Uua),Uub)
    <=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),Uub),Uu) ) ).

% ATP.lambda_398
tff(fact_8401_ATP_Olambda__399,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_cz(set(product_prod(A,A)),fun(A,fun(A,$o)),Uu),Uua),Uub)
    <=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uua),Uub),Uu) ) ).

% ATP.lambda_399
tff(fact_8402_ATP_Olambda__400,axiom,
    ! [A: $tType,B: $tType,Uu: set(product_prod(B,A)),Uua: A,Uub: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_sd(set(product_prod(B,A)),fun(A,fun(B,$o)),Uu),Uua),Uub)
    <=> member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Uub),Uua),Uu) ) ).

% ATP.lambda_400
tff(fact_8403_ATP_Olambda__401,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_dq(set(product_prod(A,A)),fun(A,fun(A,$o)),Uu),Uua),Uub)
    <=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),Uua),Uu) ) ).

% ATP.lambda_401
tff(fact_8404_ATP_Olambda__402,axiom,
    ! [A: $tType,Uu: list(list(A)),Uua: nat,Uub: nat] : ( aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_jy(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_402
tff(fact_8405_ATP_Olambda__403,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Uu: fun(A,B),Uua: list(A),Uub: A] :
          ( aa(A,$o,aa(list(A),fun(A,$o),aTP_Lamp_js(fun(A,B),fun(list(A),fun(A,$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),divide_divide(nat,aa(list(A),nat,size_size(list(A)),Uua),numeral_numeral(nat,bit0(one2)))))),aa(A,B,Uu,Uub)) ) ) ).

% ATP.lambda_403
tff(fact_8406_ATP_Olambda__404,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: fun(B,A),Uub: A] :
      ( aa(A,$o,aa(fun(B,A),fun(A,$o),aTP_Lamp_xy(fun(A,B),fun(fun(B,A),fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(B,A,Uua,aa(A,B,Uu,Uub)) = Uub ) ) ).

% ATP.lambda_404
tff(fact_8407_ATP_Olambda__405,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_yl(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_405
tff(fact_8408_ATP_Olambda__406,axiom,
    ! [Uu: fun(nat,$o),Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_ct(fun(nat,$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_406
tff(fact_8409_ATP_Olambda__407,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Uu: fun(A,B),Uua: list(A),Uub: A] :
          ( aa(A,$o,aa(list(A),fun(A,$o),aTP_Lamp_jq(fun(A,B),fun(list(A),fun(A,$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),divide_divide(nat,aa(list(A),nat,size_size(list(A)),Uua),numeral_numeral(nat,bit0(one2)))))) ) ) ).

% ATP.lambda_407
tff(fact_8410_ATP_Olambda__408,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Uu: fun(A,B),Uua: list(A),Uub: A] :
          ( aa(A,$o,aa(list(A),fun(A,$o),aTP_Lamp_jr(fun(A,B),fun(list(A),fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(A,B,Uu,Uub) = aa(A,B,Uu,aa(nat,A,nth(A,Uua),divide_divide(nat,aa(list(A),nat,size_size(list(A)),Uua),numeral_numeral(nat,bit0(one2))))) ) ) ) ).

% ATP.lambda_408
tff(fact_8411_ATP_Olambda__409,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [Uu: fun(B,A),Uua: B,Uub: B] :
          ( aa(B,$o,aa(B,fun(B,$o),aTP_Lamp_hp(fun(B,A),fun(B,fun(B,$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_409
tff(fact_8412_ATP_Olambda__410,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,$o),aTP_Lamp_wm(fun(A,B),fun(fun(A,B),fun(A,$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_410
tff(fact_8413_ATP_Olambda__411,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_ba(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_411
tff(fact_8414_ATP_Olambda__412,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_nk(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_412
tff(fact_8415_ATP_Olambda__413,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_fp(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_413
tff(fact_8416_ATP_Olambda__414,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_fh(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_414
tff(fact_8417_ATP_Olambda__415,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_qg(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_415
tff(fact_8418_ATP_Olambda__416,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_ss(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_416
tff(fact_8419_ATP_Olambda__417,axiom,
    ! [A: $tType,Uu: fun(A,assn),Uua: fun(A,assn),Uub: A] : ( aa(A,assn,aa(fun(A,assn),fun(A,assn),aTP_Lamp_zv(fun(A,assn),fun(fun(A,assn),fun(A,assn)),Uu),Uua),Uub) = aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),aa(A,assn,Uu,Uub)),aa(A,assn,Uua,Uub)) ) ).

% ATP.lambda_417
tff(fact_8420_ATP_Olambda__418,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_xo(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_418
tff(fact_8421_ATP_Olambda__419,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_xw(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_419
tff(fact_8422_ATP_Olambda__420,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_sw(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_420
tff(fact_8423_ATP_Olambda__421,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_sl(fun(A,multiset(B)),fun(fun(A,multiset(B)),fun(A,multiset(B))),Uu),Uua),Uub) = union_mset(B,aa(A,multiset(B),Uu,Uub),aa(A,multiset(B),Uua,Uub)) ) ).

% ATP.lambda_421
tff(fact_8424_ATP_Olambda__422,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_ro(fun(A,multiset(B)),fun(fun(A,multiset(B)),fun(A,multiset(B))),Uu),Uua),Uub) = inter_mset(B,aa(A,multiset(B),Uu,Uub),aa(A,multiset(B),Uua,Uub)) ) ).

% ATP.lambda_422
tff(fact_8425_ATP_Olambda__423,axiom,
    ! [A: $tType,B: $tType,E: $tType,Uu: fun(E,A),Uua: fun(E,B),Uub: E] : ( aa(E,product_prod(A,B),aa(fun(E,B),fun(E,product_prod(A,B)),aTP_Lamp_qt(fun(E,A),fun(fun(E,B),fun(E,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(E,A,Uu,Uub)),aa(E,B,Uua,Uub)) ) ).

% ATP.lambda_423
tff(fact_8426_ATP_Olambda__424,axiom,
    ! [B: $tType,E: $tType,A: $tType,Uu: fun(A,B),Uua: fun(A,E),Uub: A] : ( aa(A,product_prod(B,E),aa(fun(A,E),fun(A,product_prod(B,E)),aTP_Lamp_yh(fun(A,B),fun(fun(A,E),fun(A,product_prod(B,E))),Uu),Uua),Uub) = aa(E,product_prod(B,E),aa(B,fun(E,product_prod(B,E)),product_Pair(B,E),aa(A,B,Uu,Uub)),aa(A,E,Uua,Uub)) ) ).

% ATP.lambda_424
tff(fact_8427_ATP_Olambda__425,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_bt(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_425
tff(fact_8428_ATP_Olambda__426,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)),$o),aTP_Lamp_acb(fun(product_prod(heap_ext(product_unit),set(nat)),$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_426
tff(fact_8429_ATP_Olambda__427,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_bv(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_427
tff(fact_8430_ATP_Olambda__428,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)),$o),aTP_Lamp_aca(fun(product_prod(heap_ext(product_unit),set(nat)),$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_428
tff(fact_8431_ATP_Olambda__429,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_bx(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_429
tff(fact_8432_ATP_Olambda__430,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_jc(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_430
tff(fact_8433_ATP_Olambda__431,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Uu: fun(A,B),Uua: A,Uub: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_jp(fun(A,B),fun(A,fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(A,B,Uu,Uua) = aa(A,B,Uu,Uub) ) ) ) ).

% ATP.lambda_431
tff(fact_8434_ATP_Olambda__432,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: A,Uub: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_tr(fun(A,B),fun(A,fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(A,B,Uu,Uua) = aa(A,B,Uu,Uub) ) ) ).

% ATP.lambda_432
tff(fact_8435_ATP_Olambda__433,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: fun(B,A),Uub: B] :
      ( aa(B,$o,aa(fun(B,A),fun(B,$o),aTP_Lamp_wi(fun(B,A),fun(fun(B,A),fun(B,$o)),Uu),Uua),Uub)
    <=> ( aa(B,A,Uu,Uub) = aa(B,A,Uua,Uub) ) ) ).

% ATP.lambda_433
tff(fact_8436_ATP_Olambda__434,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_wb(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_434
tff(fact_8437_ATP_Olambda__435,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: fun(A,B),Uub: A] :
      ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_uh(fun(A,B),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(A,B,Uu,Uub) = aa(A,B,Uua,Uub) ) ) ).

% ATP.lambda_435
tff(fact_8438_ATP_Olambda__436,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Uu: A,Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_ld(A,fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(A,B,Uua,Uu) = aa(A,B,Uua,Uub) ) ) ) ).

% ATP.lambda_436
tff(fact_8439_ATP_Olambda__437,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(B)
     => ! [Uu: fun(A,B),Uua: fun(A,$o),Uub: A] : ( aa(A,B,aa(fun(A,$o),fun(A,B),aTP_Lamp_fa(fun(A,B),fun(fun(A,$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_437
tff(fact_8440_ATP_Olambda__438,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_ue(fun(A,fun(A,$o)),fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(A,$o,Uua,Uub)
        & ! [Y4: A] :
            ( aa(A,$o,Uua,Y4)
           => aa(A,$o,aa(A,fun(A,$o),Uu,Uub),Y4) ) ) ) ).

% ATP.lambda_438
tff(fact_8441_ATP_Olambda__439,axiom,
    ! [A: $tType,Uu: fun(A,assn),Uua: fun(A,assn),Uub: A] : ( aa(A,assn,aa(fun(A,assn),fun(A,assn),aTP_Lamp_zu(fun(A,assn),fun(fun(A,assn),fun(A,assn)),Uu),Uua),Uub) = aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),aa(A,assn,Uu,Uub)),ex_assn(A,Uua)) ) ).

% ATP.lambda_439
tff(fact_8442_ATP_Olambda__440,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,option(B)),Uua: A,Uub: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_uu(fun(A,option(B)),fun(A,fun(B,$o)),Uu),Uua),Uub)
    <=> ( aa(A,option(B),Uu,Uua) = aa(B,option(B),some(B),Uub) ) ) ).

% ATP.lambda_440
tff(fact_8443_ATP_Olambda__441,axiom,
    ! [A: $tType,B: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_ws(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,Uu,Uub)),Uua) ) ) ).

% ATP.lambda_441
tff(fact_8444_ATP_Olambda__442,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_wp(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,Uu,Uub)),Uua) ) ) ).

% ATP.lambda_442
tff(fact_8445_ATP_Olambda__443,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,$o),aTP_Lamp_wq(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,Uu,Uub)),Uua) ) ) ).

% ATP.lambda_443
tff(fact_8446_ATP_Olambda__444,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_kk(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,Uu,Uub)),Uua) ) ) ).

% ATP.lambda_444
tff(fact_8447_ATP_Olambda__445,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_fe(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_445
tff(fact_8448_ATP_Olambda__446,axiom,
    ! [A: $tType,Uu: fun(A,assn),Uua: assn,Uub: A] : ( aa(A,assn,aa(assn,fun(A,assn),aTP_Lamp_zp(fun(A,assn),fun(assn,fun(A,assn)),Uu),Uua),Uub) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(A,assn,Uu,Uub)),Uua) ) ).

% ATP.lambda_446
tff(fact_8449_ATP_Olambda__447,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_cx(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_447
tff(fact_8450_ATP_Olambda__448,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_aw(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_448
tff(fact_8451_ATP_Olambda__449,axiom,
    ! [A: $tType,Uu: fun(A,assn),Uua: assn,Uub: A] : ( aa(A,assn,aa(assn,fun(A,assn),aTP_Lamp_zq(fun(A,assn),fun(assn,fun(A,assn)),Uu),Uua),Uub) = aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),aa(A,assn,Uu,Uub)),Uua) ) ).

% ATP.lambda_449
tff(fact_8452_ATP_Olambda__450,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_mp(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_450
tff(fact_8453_ATP_Olambda__451,axiom,
    ! [A: $tType,Uu: fun(A,assn),Uua: assn,Uub: A] : ( aa(A,assn,aa(assn,fun(A,assn),aTP_Lamp_zs(fun(A,assn),fun(assn,fun(A,assn)),Uu),Uua),Uub) = aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),aa(A,assn,Uu,Uub)),Uua) ) ).

% ATP.lambda_451
tff(fact_8454_ATP_Olambda__452,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_mr(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_452
tff(fact_8455_ATP_Olambda__453,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_bj(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_453
tff(fact_8456_ATP_Olambda__454,axiom,
    ! [A: $tType,B: $tType,E: $tType,Uu: fun(A,filter(B)),Uua: filter(E),Uub: A] : ( aa(A,filter(product_prod(B,E)),aa(filter(E),fun(A,filter(product_prod(B,E))),aTP_Lamp_yp(fun(A,filter(B)),fun(filter(E),fun(A,filter(product_prod(B,E)))),Uu),Uua),Uub) = aa(filter(E),filter(product_prod(B,E)),aa(filter(B),fun(filter(E),filter(product_prod(B,E))),prod_filter(B,E),aa(A,filter(B),Uu,Uub)),Uua) ) ).

% ATP.lambda_454
tff(fact_8457_ATP_Olambda__455,axiom,
    ! [F4: $tType,A: $tType,B: $tType,E: $tType,Uu: fun(F4,fun(A,fun(E,$o))),Uua: fun(E,fun(B,$o)),Uub: F4] : ( aa(F4,fun(A,fun(B,$o)),aa(fun(E,fun(B,$o)),fun(F4,fun(A,fun(B,$o))),aTP_Lamp_aer(fun(F4,fun(A,fun(E,$o))),fun(fun(E,fun(B,$o)),fun(F4,fun(A,fun(B,$o)))),Uu),Uua),Uub) = relcompp(A,E,B,aa(F4,fun(A,fun(E,$o)),Uu,Uub),Uua) ) ).

% ATP.lambda_455
tff(fact_8458_ATP_Olambda__456,axiom,
    ! [F4: $tType,A: $tType,B: $tType,E: $tType,Uu: fun(F4,set(product_prod(A,E))),Uua: set(product_prod(E,B)),Uub: F4] : ( aa(F4,set(product_prod(A,B)),aa(set(product_prod(E,B)),fun(F4,set(product_prod(A,B))),aTP_Lamp_ph(fun(F4,set(product_prod(A,E))),fun(set(product_prod(E,B)),fun(F4,set(product_prod(A,B)))),Uu),Uua),Uub) = relcomp(A,E,B,aa(F4,set(product_prod(A,E)),Uu,Uub),Uua) ) ).

% ATP.lambda_456
tff(fact_8459_ATP_Olambda__457,axiom,
    ! [E: $tType,A: $tType,B: $tType,Uu: fun(E,set(product_prod(B,A))),Uua: set(B),Uub: E] : ( aa(E,set(A),aa(set(B),fun(E,set(A)),aTP_Lamp_sa(fun(E,set(product_prod(B,A))),fun(set(B),fun(E,set(A))),Uu),Uua),Uub) = image(B,A,aa(E,set(product_prod(B,A)),Uu,Uub),Uua) ) ).

% ATP.lambda_457
tff(fact_8460_ATP_Olambda__458,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: $o,Uub: A] :
      ( aa(A,$o,aa($o,fun(A,$o),aTP_Lamp_aad(fun(A,$o),fun($o,fun(A,$o)),Uu),(Uua)),Uub)
    <=> ( aa(A,$o,Uu,Uub)
       => (Uua) ) ) ).

% ATP.lambda_458
tff(fact_8461_ATP_Olambda__459,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,B),Uua: set(B),Uub: A] :
      ( aa(A,$o,aa(set(B),fun(A,$o),aTP_Lamp_qx(fun(A,B),fun(set(B),fun(A,$o)),Uu),Uua),Uub)
    <=> member(B,aa(A,B,Uu,Uub),Uua) ) ).

% ATP.lambda_459
tff(fact_8462_ATP_Olambda__460,axiom,
    ! [B: $tType,A: $tType,Uu: set(A),Uua: fun(B,A),Uub: B] :
      ( aa(B,$o,aa(fun(B,A),fun(B,$o),aTP_Lamp_tp(set(A),fun(fun(B,A),fun(B,$o)),Uu),Uua),Uub)
    <=> member(A,aa(B,A,Uua,Uub),Uu) ) ).

% ATP.lambda_460
tff(fact_8463_ATP_Olambda__461,axiom,
    ! [E: $tType,A: $tType,B: $tType,Uu: fun(E,set(B)),Uua: fun(B,set(A)),Uub: E] : ( aa(E,set(A),aa(fun(B,set(A)),fun(E,set(A)),aTP_Lamp_abo(fun(E,set(B)),fun(fun(B,set(A)),fun(E,set(A))),Uu),Uua),Uub) = bind3(B,A,aa(E,set(B),Uu,Uub),Uua) ) ).

% ATP.lambda_461
tff(fact_8464_ATP_Olambda__462,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: $o,Uub: A] :
      ( aa(A,$o,aa($o,fun(A,$o),aTP_Lamp_wc(fun(A,$o),fun($o,fun(A,$o)),Uu),(Uua)),Uub)
    <=> ( aa(A,$o,Uu,Uub)
        | (Uua) ) ) ).

% ATP.lambda_462
tff(fact_8465_ATP_Olambda__463,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: $o,Uub: A] :
      ( aa(A,$o,aa($o,fun(A,$o),aTP_Lamp_aab(fun(A,$o),fun($o,fun(A,$o)),Uu),(Uua)),Uub)
    <=> ( aa(A,$o,Uu,Uub)
        & (Uua) ) ) ).

% ATP.lambda_463
tff(fact_8466_ATP_Olambda__464,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_jo(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(A,B,Uu,Uub) = Uua ) ) ) ).

% ATP.lambda_464
tff(fact_8467_ATP_Olambda__465,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,B),Uua: B,Uub: A] :
      ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_tz(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(A,B,Uu,Uub) = Uua ) ) ).

% ATP.lambda_465
tff(fact_8468_ATP_Olambda__466,axiom,
    ! [B: $tType,A: $tType,Uu: A,Uua: fun(B,A),Uub: B] :
      ( aa(B,$o,aa(fun(B,A),fun(B,$o),aTP_Lamp_acm(A,fun(fun(B,A),fun(B,$o)),Uu),Uua),Uub)
    <=> ( aa(B,A,Uua,Uub) = Uu ) ) ).

% ATP.lambda_466
tff(fact_8469_ATP_Olambda__467,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Uu: nat,Uua: fun(nat,A),Uub: heap_ext(product_unit)] : ( aa(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),aa(fun(nat,A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),aTP_Lamp_adr(nat,fun(fun(nat,A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),Uu),Uua),Uub) = aa(product_prod(array(A),heap_ext(product_unit)),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),aa(fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),fun(product_prod(array(A),heap_ext(product_unit)),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),product_case_prod(array(A),heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),aTP_Lamp_adq(nat,fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),Uu)),array_alloc(A,aa(list(nat),list(A),aa(fun(nat,A),fun(list(nat),list(A)),map(nat,A),Uua),upt(zero_zero(nat),Uu)),Uub)) ) ) ).

% ATP.lambda_467
tff(fact_8470_ATP_Olambda__468,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Uu: nat,Uua: A,Uub: heap_ext(product_unit)] : ( aa(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),aa(A,fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),aTP_Lamp_adu(nat,fun(A,fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),Uu),Uua),Uub) = aa(product_prod(array(A),heap_ext(product_unit)),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),aa(fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),fun(product_prod(array(A),heap_ext(product_unit)),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),product_case_prod(array(A),heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),aTP_Lamp_adq(nat,fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),Uu)),array_alloc(A,aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),Uu),Uua),Uub)) ) ) ).

% ATP.lambda_468
tff(fact_8471_ATP_Olambda__469,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_ael(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_aek(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_469
tff(fact_8472_ATP_Olambda__470,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_abc(set(product_prod(A,A)),fun(A,fun(A,$o)),Uu),Uua),Uub)
    <=> ( ( Uub != Uua )
        & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uua),Uub),Uu) ) ) ).

% ATP.lambda_470
tff(fact_8473_ATP_Olambda__471,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_yd(set(product_prod(A,A)),fun(A,fun(A,$o)),Uu),Uua),Uub)
    <=> ( ( Uub != Uua )
        & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),Uua),Uu) ) ) ).

% ATP.lambda_471
tff(fact_8474_ATP_Olambda__472,axiom,
    ! [A: $tType,Uu: A,Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_co(A,fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> ( ( Uub != Uu )
       => aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_472
tff(fact_8475_ATP_Olambda__473,axiom,
    ! [A: $tType,Uu: A,Uua: A,Uub: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_mf(A,fun(A,fun(A,$o)),Uu),Uua),Uub)
    <=> ( ( Uua != Uu )
        & ( Uub != Uu ) ) ) ).

% ATP.lambda_473
tff(fact_8476_ATP_Olambda__474,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: A,Uub: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_ui(fun(A,$o),fun(A,fun(A,$o)),Uu),Uua),Uub)
    <=> ( ~ aa(A,$o,Uu,Uub)
        | ( Uua = Uub ) ) ) ).

% ATP.lambda_474
tff(fact_8477_ATP_Olambda__475,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_ez(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_475
tff(fact_8478_ATP_Olambda__476,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_aae(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_476
tff(fact_8479_ATP_Olambda__477,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_ol(fun(list(A),fun(list(A),$o)),fun(list(A),fun(list(A),$o))),Uu),Uua),Uub)
        <=> ( ? [Y4: A,Ys4: list(A)] :
                ( ( Uua = nil(A) )
                & ( Uub = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys4) ) )
            | ? [X6: A,Y4: A,Xs2: list(A),Ys4: list(A)] :
                ( ( Uua = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X6),Xs2) )
                & ( Uub = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys4) )
                & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X6),Y4) )
            | ? [X6: A,Y4: A,Xs2: list(A),Ys4: list(A)] :
                ( ( Uua = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X6),Xs2) )
                & ( Uub = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys4) )
                & ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X6),Y4)
                & ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y4),X6)
                & aa(list(A),$o,aa(list(A),fun(list(A),$o),Uu,Xs2),Ys4) ) ) ) ) ).

% ATP.lambda_477
tff(fact_8480_ATP_Olambda__478,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A),Uub: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),aTP_Lamp_oi(set(A),fun(set(A),fun(set(A),$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_478
tff(fact_8481_ATP_Olambda__479,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),$o),aTP_Lamp_rg(set(product_prod(A,A)),fun(set(A),fun(set(A),$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)) )
        & ! [X6: A] :
            ( member(A,X6,Uua)
           => ? [Xa2: A] :
                ( member(A,Xa2,Uub)
                & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X6),Xa2),Uu) ) ) ) ) ).

% ATP.lambda_479
tff(fact_8482_ATP_Olambda__480,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_sb(set(product_prod(A,A)),fun(A,fun(A,set(A))),Uu),Uua),Uub) = image(A,A,converse(A,A,Uu),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Uua),bot_bot(set(A)))) ) ).

% ATP.lambda_480
tff(fact_8483_ATP_Olambda__481,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_jx(fun(A,B),fun(list(A),fun(A,fun(list(A),list(A)))),Uu),Uua),Uub) = aa(fun(A,fun(list(A),list(A))),fun(list(A),list(A)),aa(list(A),fun(fun(A,fun(list(A),list(A))),fun(list(A),list(A))),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_jw(fun(A,B),fun(list(A),fun(A,fun(A,fun(list(A),list(A))))),Uu),Uua),Uub)) ) ) ).

% ATP.lambda_481
tff(fact_8484_ATP_Olambda__482,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_ib(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_482
tff(fact_8485_ATP_Olambda__483,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Uu: A,Uua: A,Uub: nat] : ( aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_gj(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_483
tff(fact_8486_ATP_Olambda__484,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: A,Uua: A,Uub: nat] : ( aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_gg(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_484
tff(fact_8487_ATP_Olambda__485,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Uu: nat,Uua: array(A),Uub: heap_ext(product_unit)] : ( aa(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),aa(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),aTP_Lamp_adq(nat,fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),Uu),Uua),Uub) = aa(product_prod(heap_ext(product_unit),nat),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),aa(array(A),fun(product_prod(heap_ext(product_unit),nat),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),product_Pair(array(A),product_prod(heap_ext(product_unit),nat)),Uua),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),Uub),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu),one_one(nat)))) ) ) ).

% ATP.lambda_485
tff(fact_8488_ATP_Olambda__486,axiom,
    ! [A: $tType,Uu: A,Uua: set(A),Uub: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_vo(A,fun(set(A),fun(A,$o)),Uu),Uua),Uub)
    <=> member(A,Uub,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Uua),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Uu),bot_bot(set(A))))) ) ).

% ATP.lambda_486
tff(fact_8489_ATP_Olambda__487,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Uu: list(A),Uua: array(A),Uub: heap_ext(product_unit)] : ( aa(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),aa(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),aTP_Lamp_ads(list(A),fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),Uu),Uua),Uub) = aa(product_prod(heap_ext(product_unit),nat),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),aa(array(A),fun(product_prod(heap_ext(product_unit),nat),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),product_Pair(array(A),product_prod(heap_ext(product_unit),nat)),Uua),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),Uub),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(list(A),nat,size_size(list(A)),Uu)))) ) ) ).

% ATP.lambda_487
tff(fact_8490_ATP_Olambda__488,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Uu: nat,Uua: fun(nat,A),Uub: array(A)] : ( aa(array(A),assn,aa(fun(nat,A),fun(array(A),assn),aTP_Lamp_adf(nat,fun(fun(nat,A),fun(array(A),assn)),Uu),Uua),Uub) = snga_assn(A,Uub,aa(list(nat),list(A),aa(fun(nat,A),fun(list(nat),list(A)),map(nat,A),Uua),upt(zero_zero(nat),Uu))) ) ) ).

% ATP.lambda_488
tff(fact_8491_ATP_Olambda__489,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_fx(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_489
tff(fact_8492_ATP_Olambda__490,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,option(B)),Uua: set(A),Uub: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_vk(fun(A,option(B)),fun(set(A),fun(A,$o)),Uu),Uua),Uub)
    <=> member(A,Uub,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Uua),dom(A,B,Uu))) ) ).

% ATP.lambda_490
tff(fact_8493_ATP_Olambda__491,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,option(B)),Uua: set(A),Uub: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_vl(fun(A,option(B)),fun(set(A),fun(A,$o)),Uu),Uua),Uub)
    <=> member(A,Uub,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Uua),dom(A,B,Uu))) ) ).

% ATP.lambda_491
tff(fact_8494_ATP_Olambda__492,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Uu: nat,Uua: array(A),Uub: heap_ext(product_unit)] :
          ( aa(heap_ext(product_unit),$o,aa(array(A),fun(heap_ext(product_unit),$o),aTP_Lamp_aec(nat,fun(array(A),fun(heap_ext(product_unit),$o)),Uu),Uua),Uub)
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uu),array_length(A,Uub,Uua)) ) ) ).

% ATP.lambda_492
tff(fact_8495_ATP_Olambda__493,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Uu: array(A),Uua: nat,Uub: heap_ext(product_unit)] :
          ( aa(heap_ext(product_unit),$o,aa(nat,fun(heap_ext(product_unit),$o),aTP_Lamp_aee(array(A),fun(nat,fun(heap_ext(product_unit),$o)),Uu),Uua),Uub)
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uua),array_length(A,Uub,Uu)) ) ) ).

% ATP.lambda_493
tff(fact_8496_ATP_Olambda__494,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_fj(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_494
tff(fact_8497_ATP_Olambda__495,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Uu: nat,Uua: A,Uub: array(A)] : ( aa(array(A),assn,aa(A,fun(array(A),assn),aTP_Lamp_adg(nat,fun(A,fun(array(A),assn)),Uu),Uua),Uub) = snga_assn(A,Uub,aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),Uu),Uua)) ) ) ).

% ATP.lambda_495
tff(fact_8498_ATP_Olambda__496,axiom,
    ! [A: $tType,B: $tType,E: $tType,Uu: A,Uua: B,Uub: E] : ( aa(E,product_prod(A,product_prod(B,E)),aa(B,fun(E,product_prod(A,product_prod(B,E))),aa(A,fun(B,fun(E,product_prod(A,product_prod(B,E)))),aTP_Lamp_in(A,fun(B,fun(E,product_prod(A,product_prod(B,E))))),Uu),Uua),Uub) = aa(product_prod(B,E),product_prod(A,product_prod(B,E)),aa(A,fun(product_prod(B,E),product_prod(A,product_prod(B,E))),product_Pair(A,product_prod(B,E)),Uu),aa(E,product_prod(B,E),aa(B,fun(E,product_prod(B,E)),product_Pair(B,E),Uua),Uub)) ) ).

% ATP.lambda_496
tff(fact_8499_ATP_Olambda__497,axiom,
    ! [A: $tType,B: $tType,E: $tType,Uu: B,Uua: A,Uub: E] : ( aa(E,product_prod(A,product_prod(B,E)),aa(A,fun(E,product_prod(A,product_prod(B,E))),aTP_Lamp_il(B,fun(A,fun(E,product_prod(A,product_prod(B,E)))),Uu),Uua),Uub) = aa(product_prod(B,E),product_prod(A,product_prod(B,E)),aa(A,fun(product_prod(B,E),product_prod(A,product_prod(B,E))),product_Pair(A,product_prod(B,E)),Uua),aa(E,product_prod(B,E),aa(B,fun(E,product_prod(B,E)),product_Pair(B,E),Uu),Uub)) ) ).

% ATP.lambda_497
tff(fact_8500_ATP_Olambda__498,axiom,
    ! [A: $tType,Uu: A,Uua: list(A),Uub: nat] : ( aa(nat,list(A),aa(list(A),fun(nat,list(A)),aTP_Lamp_zm(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),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),Uub),Uua)) ) ).

% ATP.lambda_498
tff(fact_8501_ATP_Olambda__499,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_abw(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_499
tff(fact_8502_ATP_Olambda__500,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,$o),aTP_Lamp_rm(fun(A,B),fun(set(A),fun(B,$o)),Uu),Uua),Uub)
        <=> member(B,Uub,aa(set(A),set(B),image2(A,B,Uu),Uua)) ) ) ).

% ATP.lambda_500
tff(fact_8503_ATP_Olambda__501,axiom,
    ! [E: $tType,B: $tType,A: $tType] :
      ( comm_monoid_mult(E)
     => ! [Uu: fun(A,set(B)),Uua: fun(B,E),Uub: A] : ( aa(A,E,aa(fun(B,E),fun(A,E),aTP_Lamp_dy(fun(A,set(B)),fun(fun(B,E),fun(A,E)),Uu),Uua),Uub) = aa(set(B),E,aa(fun(B,E),fun(set(B),E),groups7121269368397514597t_prod(B,E),Uua),aa(A,set(B),Uu,Uub)) ) ) ).

% ATP.lambda_501
tff(fact_8504_ATP_Olambda__502,axiom,
    ! [E: $tType,B: $tType,A: $tType] :
      ( comm_monoid_add(E)
     => ! [Uu: fun(A,set(B)),Uua: fun(B,E),Uub: A] : ( aa(A,E,aa(fun(B,E),fun(A,E),aTP_Lamp_fo(fun(A,set(B)),fun(fun(B,E),fun(A,E)),Uu),Uua),Uub) = aa(set(B),E,aa(fun(B,E),fun(set(B),E),groups7311177749621191930dd_sum(B,E),Uua),aa(A,set(B),Uu,Uub)) ) ) ).

% ATP.lambda_502
tff(fact_8505_ATP_Olambda__503,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_wr(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Uua),aa(A,B,Uu,Uub)) ) ) ).

% ATP.lambda_503
tff(fact_8506_ATP_Olambda__504,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_wn(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Uua),aa(A,B,Uu,Uub)) ) ) ).

% ATP.lambda_504
tff(fact_8507_ATP_Olambda__505,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] :
          ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_kp(fun(B,A),fun(A,fun(B,$o)),Uu),Uua),Uub)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Uua),aa(B,A,Uu,Uub)) ) ) ).

% ATP.lambda_505
tff(fact_8508_ATP_Olambda__506,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_wo(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),Uua),aa(A,B,Uu,Uub)) ) ) ).

% ATP.lambda_506
tff(fact_8509_ATP_Olambda__507,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_kl(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),Uua),aa(A,B,Uu,Uub)) ) ) ).

% ATP.lambda_507
tff(fact_8510_ATP_Olambda__508,axiom,
    ! [B: $tType,Uu: assn,Uua: fun(B,assn),Uub: B] : ( aa(B,assn,aa(fun(B,assn),fun(B,assn),aTP_Lamp_ady(assn,fun(fun(B,assn),fun(B,assn)),Uu),Uua),Uub) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Uu),aa(B,assn,Uua,Uub)) ) ).

% ATP.lambda_508
tff(fact_8511_ATP_Olambda__509,axiom,
    ! [A: $tType,Uu: assn,Uua: fun(A,assn),Uub: A] : ( aa(A,assn,aa(fun(A,assn),fun(A,assn),aTP_Lamp_zo(assn,fun(fun(A,assn),fun(A,assn)),Uu),Uua),Uub) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Uu),aa(A,assn,Uua,Uub)) ) ).

% ATP.lambda_509
tff(fact_8512_ATP_Olambda__510,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_tf(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_510
tff(fact_8513_ATP_Olambda__511,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_fd(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_511
tff(fact_8514_ATP_Olambda__512,axiom,
    ! [A: $tType,Uu: fun(A,assn),Uua: assn,Uub: A] : ( aa(A,assn,aa(assn,fun(A,assn),aTP_Lamp_adc(fun(A,assn),fun(assn,fun(A,assn)),Uu),Uua),Uub) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Uua),aa(A,assn,Uu,Uub)) ) ).

% ATP.lambda_512
tff(fact_8515_ATP_Olambda__513,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: nat,Uub: A] : ( aa(A,nat,aa(nat,fun(A,nat),aTP_Lamp_sx(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_513
tff(fact_8516_ATP_Olambda__514,axiom,
    ! [A: $tType,Uu: fun(A,assn),Uua: A,Uub: assn] : ( aa(assn,assn,aa(A,fun(assn,assn),aTP_Lamp_qb(fun(A,assn),fun(A,fun(assn,assn)),Uu),Uua),Uub) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Uub),aa(A,assn,Uu,Uua)) ) ).

% ATP.lambda_514
tff(fact_8517_ATP_Olambda__515,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_mq(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_515
tff(fact_8518_ATP_Olambda__516,axiom,
    ! [A: $tType,Uu: assn,Uua: fun(A,assn),Uub: A] : ( aa(A,assn,aa(fun(A,assn),fun(A,assn),aTP_Lamp_zr(assn,fun(fun(A,assn),fun(A,assn)),Uu),Uua),Uub) = aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),Uu),aa(A,assn,Uua,Uub)) ) ).

% ATP.lambda_516
tff(fact_8519_ATP_Olambda__517,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_av(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_517
tff(fact_8520_ATP_Olambda__518,axiom,
    ! [A: $tType,Uu: assn,Uua: fun(A,assn),Uub: A] : ( aa(A,assn,aa(fun(A,assn),fun(A,assn),aTP_Lamp_zt(assn,fun(fun(A,assn),fun(A,assn)),Uu),Uua),Uub) = aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),Uu),aa(A,assn,Uua,Uub)) ) ).

% ATP.lambda_518
tff(fact_8521_ATP_Olambda__519,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_mo(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_519
tff(fact_8522_ATP_Olambda__520,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_ms(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_520
tff(fact_8523_ATP_Olambda__521,axiom,
    ! [B: $tType,E: $tType,A: $tType,Uu: filter(B),Uua: fun(A,filter(E)),Uub: A] : ( aa(A,filter(product_prod(B,E)),aa(fun(A,filter(E)),fun(A,filter(product_prod(B,E))),aTP_Lamp_yo(filter(B),fun(fun(A,filter(E)),fun(A,filter(product_prod(B,E)))),Uu),Uua),Uub) = aa(filter(E),filter(product_prod(B,E)),aa(filter(B),fun(filter(E),filter(product_prod(B,E))),prod_filter(B,E),Uu),aa(A,filter(E),Uua,Uub)) ) ).

% ATP.lambda_521
tff(fact_8524_ATP_Olambda__522,axiom,
    ! [A: $tType,E: $tType,B: $tType,Uu: fun(A,E),Uua: fun(B,filter(E)),Uub: B] : ( aa(B,filter(A),aa(fun(B,filter(E)),fun(B,filter(A)),aTP_Lamp_xh(fun(A,E),fun(fun(B,filter(E)),fun(B,filter(A))),Uu),Uua),Uub) = filtercomap(A,E,Uu,aa(B,filter(E),Uua,Uub)) ) ).

% ATP.lambda_522
tff(fact_8525_ATP_Olambda__523,axiom,
    ! [A: $tType,B: $tType,E: $tType,Uu: fun(A,B),Uua: fun(E,filter(B)),Uub: E] : ( aa(E,filter(A),aa(fun(E,filter(B)),fun(E,filter(A)),aTP_Lamp_xg(fun(A,B),fun(fun(E,filter(B)),fun(E,filter(A))),Uu),Uua),Uub) = filtercomap(A,B,Uu,aa(E,filter(B),Uua,Uub)) ) ).

% ATP.lambda_523
tff(fact_8526_ATP_Olambda__524,axiom,
    ! [A: $tType,E: $tType,B: $tType,F4: $tType,Uu: fun(A,fun(E,$o)),Uua: fun(F4,fun(E,fun(B,$o))),Uub: F4] : ( aa(F4,fun(A,fun(B,$o)),aa(fun(F4,fun(E,fun(B,$o))),fun(F4,fun(A,fun(B,$o))),aTP_Lamp_aes(fun(A,fun(E,$o)),fun(fun(F4,fun(E,fun(B,$o))),fun(F4,fun(A,fun(B,$o)))),Uu),Uua),Uub) = relcompp(A,E,B,Uu,aa(F4,fun(E,fun(B,$o)),Uua,Uub)) ) ).

% ATP.lambda_524
tff(fact_8527_ATP_Olambda__525,axiom,
    ! [A: $tType,B: $tType,E: $tType,Uu: fun(E,B),Uua: A,Uub: E] : ( aa(E,product_prod(A,B),aa(A,fun(E,product_prod(A,B)),aTP_Lamp_hm(fun(E,B),fun(A,fun(E,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(E,B,Uu,Uub)) ) ).

% ATP.lambda_525
tff(fact_8528_ATP_Olambda__526,axiom,
    ! [A: $tType,E: $tType,B: $tType,F4: $tType,Uu: set(product_prod(A,E)),Uua: fun(F4,set(product_prod(E,B))),Uub: F4] : ( aa(F4,set(product_prod(A,B)),aa(fun(F4,set(product_prod(E,B))),fun(F4,set(product_prod(A,B))),aTP_Lamp_pi(set(product_prod(A,E)),fun(fun(F4,set(product_prod(E,B))),fun(F4,set(product_prod(A,B)))),Uu),Uua),Uub) = relcomp(A,E,B,Uu,aa(F4,set(product_prod(E,B)),Uua,Uub)) ) ).

% ATP.lambda_526
tff(fact_8529_ATP_Olambda__527,axiom,
    ! [A: $tType,B: $tType,E: $tType,Uu: fun(B,A),Uua: fun(E,filter(B)),Uub: E] : ( aa(E,filter(A),aa(fun(E,filter(B)),fun(E,filter(A)),aTP_Lamp_xz(fun(B,A),fun(fun(E,filter(B)),fun(E,filter(A))),Uu),Uua),Uub) = aa(filter(B),filter(A),aa(fun(B,A),fun(filter(B),filter(A)),filtermap(B,A),Uu),aa(E,filter(B),Uua,Uub)) ) ).

% ATP.lambda_527
tff(fact_8530_ATP_Olambda__528,axiom,
    ! [A: $tType,B: $tType,E: $tType,Uu: set(product_prod(B,A)),Uua: fun(E,set(B)),Uub: E] : ( aa(E,set(A),aa(fun(E,set(B)),fun(E,set(A)),aTP_Lamp_rv(set(product_prod(B,A)),fun(fun(E,set(B)),fun(E,set(A))),Uu),Uua),Uub) = image(B,A,Uu,aa(E,set(B),Uua,Uub)) ) ).

% ATP.lambda_528
tff(fact_8531_ATP_Olambda__529,axiom,
    ! [A: $tType,Uu: $o,Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_we($o,fun(fun(A,$o),fun(A,$o)),(Uu)),Uua),Uub)
    <=> ( (Uu)
       => aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_529
tff(fact_8532_ATP_Olambda__530,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,set(A)),Uua: B,Uub: A] :
      ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_dd(fun(B,set(A)),fun(B,fun(A,$o)),Uu),Uua),Uub)
    <=> member(A,Uub,aa(B,set(A),Uu,Uua)) ) ).

% ATP.lambda_530
tff(fact_8533_ATP_Olambda__531,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_cw(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)),insert3(B),Uu),aa(A,set(B),Uua,Uub)) ) ).

% ATP.lambda_531
tff(fact_8534_ATP_Olambda__532,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_au(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)),insert3(A),Uu),aa(B,set(A),Uua,Uub)) ) ).

% ATP.lambda_532
tff(fact_8535_ATP_Olambda__533,axiom,
    ! [E: $tType,B: $tType,A: $tType] :
      ( condit1219197933456340205attice(E)
     => ! [Uu: fun(A,set(B)),Uua: fun(B,E),Uub: A] : ( aa(A,set(E),aa(fun(B,E),fun(A,set(E)),aTP_Lamp_xl(fun(A,set(B)),fun(fun(B,E),fun(A,set(E))),Uu),Uua),Uub) = aa(set(B),set(E),image2(B,E,Uua),aa(A,set(B),Uu,Uub)) ) ) ).

% ATP.lambda_533
tff(fact_8536_ATP_Olambda__534,axiom,
    ! [A: $tType,Uu: $o,Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_wd($o,fun(fun(A,$o),fun(A,$o)),(Uu)),Uua),Uub)
    <=> ( (Uu)
        | aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_534
tff(fact_8537_ATP_Olambda__535,axiom,
    ! [A: $tType,Uu: $o,Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_aaa($o,fun(fun(A,$o),fun(A,$o)),(Uu)),Uua),Uub)
    <=> ( (Uu)
        & aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_535
tff(fact_8538_ATP_Olambda__536,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: fun(list(A),A),Uua: list(A),Uub: A] :
          ( aa(A,$o,aa(list(A),fun(A,$o),aTP_Lamp_jf(fun(list(A),A),fun(list(A),fun(A,$o)),Uu),Uua),Uub)
        <=> ( Uub = aa(list(A),A,Uu,Uua) ) ) ) ).

% ATP.lambda_536
tff(fact_8539_ATP_Olambda__537,axiom,
    ! [A: $tType,Uu: assn,Uua: A,Uub: A] : ( aa(A,assn,aa(A,fun(A,assn),aTP_Lamp_adp(assn,fun(A,fun(A,assn)),Uu),Uua),Uub) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Uu),pure_assn(Uub = Uua)) ) ).

% ATP.lambda_537
tff(fact_8540_ATP_Olambda__538,axiom,
    ! [A: $tType,Uu: list(A),Uua: set(nat),Uub: A] :
      ( aa(A,$o,aa(set(nat),fun(A,$o),aTP_Lamp_oa(list(A),fun(set(nat),fun(A,$o)),Uu),Uua),Uub)
    <=> member(A,Uub,aa(list(A),set(A),set2(A),aa(set(nat),list(A),aa(list(A),fun(set(nat),list(A)),nths(A),Uu),Uua))) ) ).

% ATP.lambda_538
tff(fact_8541_ATP_Olambda__539,axiom,
    ! [B: $tType,A: $tType,E: $tType,Uu: set(A),Uua: set(E),Uub: fun(A,B)] : ( aa(fun(A,B),set(fun(A,E)),aa(set(E),fun(fun(A,B),set(fun(A,E))),aTP_Lamp_yx(set(A),fun(set(E),fun(fun(A,B),set(fun(A,E)))),Uu),Uua),Uub) = bNF_Wellorder_Func(A,E,Uu,Uua) ) ).

% ATP.lambda_539
tff(fact_8542_ATP_Olambda__540,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A),Uub: A] : ( aa(A,set(A),aa(set(A),fun(A,set(A)),aTP_Lamp_ra(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_540
tff(fact_8543_ATP_Olambda__541,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_yc(set(product_prod(A,A)),fun(A,fun(A,set(A))),Uu),Uua),Uub) = order_underS(A,Uu,Uua) ) ).

% ATP.lambda_541
tff(fact_8544_ATP_Olambda__542,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Uu: ref(A),Uua: A,Uub: product_unit] : ( aa(product_unit,assn,aa(A,fun(product_unit,assn),aTP_Lamp_adj(ref(A),fun(A,fun(product_unit,assn)),Uu),Uua),Uub) = sngr_assn(A,Uu,Uua) ) ) ).

% ATP.lambda_542
tff(fact_8545_ATP_Olambda__543,axiom,
    ! [A: $tType,B: $tType,E: $tType,Uu: set(B),Uua: set(E),Uub: A] : ( aa(A,set(sum_sum(B,E)),aa(set(E),fun(A,set(sum_sum(B,E))),aTP_Lamp_zj(set(B),fun(set(E),fun(A,set(sum_sum(B,E)))),Uu),Uua),Uub) = sum_Plus(B,E,Uu,Uua) ) ).

% ATP.lambda_543
tff(fact_8546_ATP_Olambda__544,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_bq(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_544
tff(fact_8547_ATP_Olambda__545,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_qa(B,fun(set(B),fun(A,set(B))),Uu),Uua),Uub) = aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert3(B),Uu),Uua) ) ).

% ATP.lambda_545
tff(fact_8548_ATP_Olambda__546,axiom,
    ! [A: $tType,B: $tType,F4: $tType,Uu: fun(F4,B),Uua: set(F4),Uub: A] : ( aa(A,set(B),aa(set(F4),fun(A,set(B)),aTP_Lamp_qj(fun(F4,B),fun(set(F4),fun(A,set(B))),Uu),Uua),Uub) = aa(set(F4),set(B),image2(F4,B,Uu),Uua) ) ).

% ATP.lambda_546
tff(fact_8549_ATP_Olambda__547,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_pz(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_547
tff(fact_8550_ATP_Olambda__548,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: A,Uub: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_df(fun(A,nat),fun(A,fun(A,$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_548
tff(fact_8551_ATP_Olambda__549,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_bf(fun(A,nat),fun(fun(A,$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_549
tff(fact_8552_ATP_Olambda__550,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Uu: ref(A),Uua: A,Uub: heap_ext(product_unit)] : ( aa(heap_ext(product_unit),product_prod(product_unit,product_prod(heap_ext(product_unit),nat)),aa(A,fun(heap_ext(product_unit),product_prod(product_unit,product_prod(heap_ext(product_unit),nat))),aTP_Lamp_adl(ref(A),fun(A,fun(heap_ext(product_unit),product_prod(product_unit,product_prod(heap_ext(product_unit),nat)))),Uu),Uua),Uub) = aa(product_prod(heap_ext(product_unit),nat),product_prod(product_unit,product_prod(heap_ext(product_unit),nat)),aa(product_unit,fun(product_prod(heap_ext(product_unit),nat),product_prod(product_unit,product_prod(heap_ext(product_unit),nat))),product_Pair(product_unit,product_prod(heap_ext(product_unit),nat)),product_Unity),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),ref_set(A,Uu,Uua,Uub)),one_one(nat))) ) ) ).

% ATP.lambda_550
tff(fact_8553_ATP_Olambda__551,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: fun(A,nat),Uub: A] :
      ( aa(A,$o,aa(fun(A,nat),fun(A,$o),aTP_Lamp_dg(fun(A,nat),fun(fun(A,nat),fun(A,$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_551
tff(fact_8554_ATP_Olambda__552,axiom,
    ! [Uu: fun(nat,$o),Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_vu(fun(nat,$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_552
tff(fact_8555_ATP_Olambda__553,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_aay(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_553
tff(fact_8556_ATP_Olambda__554,axiom,
    ! [E: $tType,A: $tType,B: $tType,Uu: fun(product_prod(A,B),E),Uua: A,Uub: B] : ( aa(B,E,aa(A,fun(B,E),aTP_Lamp_gr(fun(product_prod(A,B),E),fun(A,fun(B,E)),Uu),Uua),Uub) = aa(product_prod(A,B),E,Uu,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),Uub)) ) ).

% ATP.lambda_554
tff(fact_8557_ATP_Olambda__555,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: list(A),Uub: nat] :
      ( aa(nat,$o,aa(list(A),fun(nat,$o),aTP_Lamp_ke(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_555
tff(fact_8558_ATP_Olambda__556,axiom,
    ! [B: $tType,A: $tType,E: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [Uu: fun(multiset(A),B),Uua: fun(E,multiset(A)),Uub: E] : ( aa(E,B,aa(fun(E,multiset(A)),fun(E,B),aTP_Lamp_rp(fun(multiset(A),B),fun(fun(E,multiset(A)),fun(E,B)),Uu),Uua),Uub) = aa(multiset(A),B,Uu,aa(E,multiset(A),Uua,Uub)) ) ) ).

% ATP.lambda_556
tff(fact_8559_ATP_Olambda__557,axiom,
    ! [A: $tType,E: $tType,B: $tType,Uu: fun(E,A),Uua: fun(B,E),Uub: B] : ( aa(B,A,aa(fun(B,E),fun(B,A),aTP_Lamp_bo(fun(E,A),fun(fun(B,E),fun(B,A)),Uu),Uua),Uub) = aa(E,A,Uu,aa(B,E,Uua,Uub)) ) ).

% ATP.lambda_557
tff(fact_8560_ATP_Olambda__558,axiom,
    ! [A: $tType,B: $tType,E: $tType,Uu: fun(B,A),Uua: fun(E,B),Uub: E] : ( aa(E,A,aa(fun(E,B),fun(E,A),aTP_Lamp_bn(fun(B,A),fun(fun(E,B),fun(E,A)),Uu),Uua),Uub) = aa(B,A,Uu,aa(E,B,Uua,Uub)) ) ).

% ATP.lambda_558
tff(fact_8561_ATP_Olambda__559,axiom,
    ! [B: $tType,A: $tType,E: $tType] :
      ( ( condit1219197933456340205attice(A)
        & condit1219197933456340205attice(B) )
     => ! [Uu: fun(A,B),Uua: fun(E,A),Uub: E] : ( aa(E,B,aa(fun(E,A),fun(E,B),aTP_Lamp_xq(fun(A,B),fun(fun(E,A),fun(E,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(E,A,Uua,Uub)) ) ) ).

% ATP.lambda_559
tff(fact_8562_ATP_Olambda__560,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_wh(fun(A,$o),fun(fun(B,A),fun(B,$o)),Uu),Uua),Uub)
    <=> aa(A,$o,Uu,aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_560
tff(fact_8563_ATP_Olambda__561,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_rt(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_561
tff(fact_8564_ATP_Olambda__562,axiom,
    ! [B: $tType,A: $tType,E: $tType,Uu: fun(A,B),Uua: fun(E,A),Uub: E] : ( aa(E,B,aa(fun(E,A),fun(E,B),aTP_Lamp_to(fun(A,B),fun(fun(E,A),fun(E,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(E,A,Uua,Uub)) ) ).

% ATP.lambda_562
tff(fact_8565_ATP_Olambda__563,axiom,
    ! [A: $tType,E: $tType,B: $tType,Uu: fun(B,E),Uua: fun(E,A),Uub: B] : ( aa(B,A,aa(fun(E,A),fun(B,A),aTP_Lamp_aco(fun(B,E),fun(fun(E,A),fun(B,A)),Uu),Uua),Uub) = aa(E,A,Uua,aa(B,E,Uu,Uub)) ) ).

% ATP.lambda_563
tff(fact_8566_ATP_Olambda__564,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: fun(A,$o),Uub: B] :
      ( aa(B,$o,aa(fun(A,$o),fun(B,$o),aTP_Lamp_ya(fun(B,A),fun(fun(A,$o),fun(B,$o)),Uu),Uua),Uub)
    <=> aa(A,$o,Uua,aa(B,A,Uu,Uub)) ) ).

% ATP.lambda_564
tff(fact_8567_ATP_Olambda__565,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: fun(B,$o),Uub: A] :
      ( aa(A,$o,aa(fun(B,$o),fun(A,$o),aTP_Lamp_qv(fun(A,B),fun(fun(B,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> aa(B,$o,Uua,aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_565
tff(fact_8568_ATP_Olambda__566,axiom,
    ! [E: $tType,B: $tType,A: $tType] :
      ( comm_monoid_mult(E)
     => ! [Uu: fun(A,B),Uua: fun(B,E),Uub: A] : ( aa(A,E,aa(fun(B,E),fun(A,E),aTP_Lamp_abq(fun(A,B),fun(fun(B,E),fun(A,E)),Uu),Uua),Uub) = aa(B,E,Uua,aa(A,B,Uu,Uub)) ) ) ).

% ATP.lambda_566
tff(fact_8569_ATP_Olambda__567,axiom,
    ! [E: $tType,B: $tType,A: $tType] :
      ( comm_monoid_add(E)
     => ! [Uu: fun(A,B),Uua: fun(B,E),Uub: A] : ( aa(A,E,aa(fun(B,E),fun(A,E),aTP_Lamp_abp(fun(A,B),fun(fun(B,E),fun(A,E)),Uu),Uua),Uub) = aa(B,E,Uua,aa(A,B,Uu,Uub)) ) ) ).

% ATP.lambda_567
tff(fact_8570_ATP_Olambda__568,axiom,
    ! [E: $tType,B: $tType,A: $tType] :
      ( semiring_1(E)
     => ! [Uu: fun(A,B),Uua: fun(B,E),Uub: A] : ( aa(A,E,aa(fun(B,E),fun(A,E),aTP_Lamp_fq(fun(A,B),fun(fun(B,E),fun(A,E)),Uu),Uua),Uub) = aa(B,E,Uua,aa(A,B,Uu,Uub)) ) ) ).

% ATP.lambda_568
tff(fact_8571_ATP_Olambda__569,axiom,
    ! [E: $tType,B: $tType,A: $tType,Uu: fun(A,B),Uua: fun(B,E),Uub: A] : ( aa(A,E,aa(fun(B,E),fun(A,E),aTP_Lamp_xf(fun(A,B),fun(fun(B,E),fun(A,E)),Uu),Uua),Uub) = aa(B,E,Uua,aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_569
tff(fact_8572_ATP_Olambda__570,axiom,
    ! [B: $tType,A: $tType,E: $tType,Uu: fun(E,fun(A,E)),Uua: E,Uub: B] : ( aa(B,fun(A,E),aa(E,fun(B,fun(A,E)),aTP_Lamp_gs(fun(E,fun(A,E)),fun(E,fun(B,fun(A,E))),Uu),Uua),Uub) = aa(E,fun(A,E),Uu,Uua) ) ).

% ATP.lambda_570
tff(fact_8573_ATP_Olambda__571,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,$o),Uua: B,Uub: A] :
      ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_yn(fun(B,$o),fun(B,fun(A,$o)),Uu),Uua),Uub)
    <=> aa(B,$o,Uu,Uua) ) ).

% ATP.lambda_571
tff(fact_8574_ATP_Olambda__572,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,$o),Uua: A,Uub: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_aew(fun(A,$o),fun(A,fun(B,$o)),Uu),Uua),Uub)
    <=> aa(A,$o,Uu,Uua) ) ).

% ATP.lambda_572
tff(fact_8575_ATP_Olambda__573,axiom,
    ! [B: $tType,E: $tType,A: $tType,Uu: fun(A,E),Uua: A,Uub: B] : ( aa(B,E,aa(A,fun(B,E),aTP_Lamp_or(fun(A,E),fun(A,fun(B,E)),Uu),Uua),Uub) = aa(A,E,Uu,Uua) ) ).

% ATP.lambda_573
tff(fact_8576_ATP_Olambda__574,axiom,
    ! [A: $tType,Uu: heap_Time_Heap(A),Uua: heap_ext(product_unit),Uub: A] : ( aa(A,fun(product_prod(heap_ext(product_unit),nat),$o),aa(heap_ext(product_unit),fun(A,fun(product_prod(heap_ext(product_unit),nat),$o)),aTP_Lamp_afc(heap_Time_Heap(A),fun(heap_ext(product_unit),fun(A,fun(product_prod(heap_ext(product_unit),nat),$o))),Uu),Uua),Uub) = aa(fun(heap_ext(product_unit),fun(nat,$o)),fun(product_prod(heap_ext(product_unit),nat),$o),product_case_prod(heap_ext(product_unit),nat,$o),aa(A,fun(heap_ext(product_unit),fun(nat,$o)),aa(heap_ext(product_unit),fun(A,fun(heap_ext(product_unit),fun(nat,$o))),aTP_Lamp_afb(heap_Time_Heap(A),fun(heap_ext(product_unit),fun(A,fun(heap_ext(product_unit),fun(nat,$o)))),Uu),Uua),Uub)) ) ).

% ATP.lambda_574
tff(fact_8577_ATP_Olambda__575,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),$o)),aTP_Lamp_iw(set(product_prod(A,A)),fun(A,fun(A,fun(product_prod(A,A),$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,$o))),aTP_Lamp_iv(set(product_prod(A,A)),fun(A,fun(A,fun(A,fun(A,$o)))),Uu),Uua),Uub)) ) ).

% ATP.lambda_575
tff(fact_8578_ATP_Olambda__576,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_aen(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_576
tff(fact_8579_ATP_Olambda__577,axiom,
    ! [B: $tType,A: $tType,E: $tType,Uu: fun(A,fun(B,fun(E,$o))),Uua: A,Uub: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_aex(fun(A,fun(B,fun(E,$o))),fun(A,fun(B,$o)),Uu),Uua),Uub)
    <=> ? [X_12: E] : aa(E,$o,aa(B,fun(E,$o),aa(A,fun(B,fun(E,$o)),Uu,Uua),Uub),X_12) ) ).

% ATP.lambda_577
tff(fact_8580_ATP_Olambda__578,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_aeo(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_aen(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_578
tff(fact_8581_ATP_Olambda__579,axiom,
    ! [E: $tType,B: $tType,A: $tType] :
      ( condit1219197933456340205attice(E)
     => ! [Uu: fun(A,set(B)),Uua: fun(B,E),Uub: A] : ( aa(A,E,aa(fun(B,E),fun(A,E),aTP_Lamp_xm(fun(A,set(B)),fun(fun(B,E),fun(A,E)),Uu),Uua),Uub) = aa(set(E),E,complete_Sup_Sup(E),aa(set(B),set(E),image2(B,E,Uua),aa(A,set(B),Uu,Uub))) ) ) ).

% ATP.lambda_579
tff(fact_8582_ATP_Olambda__580,axiom,
    ! [E: $tType,B: $tType,A: $tType] :
      ( condit1219197933456340205attice(E)
     => ! [Uu: fun(A,set(B)),Uua: fun(B,E),Uub: A] : ( aa(A,E,aa(fun(B,E),fun(A,E),aTP_Lamp_xv(fun(A,set(B)),fun(fun(B,E),fun(A,E)),Uu),Uua),Uub) = aa(set(E),E,complete_Inf_Inf(E),aa(set(B),set(E),image2(B,E,Uua),aa(A,set(B),Uu,Uub))) ) ) ).

% ATP.lambda_580
tff(fact_8583_ATP_Olambda__581,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_tj(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_581
tff(fact_8584_ATP_Olambda__582,axiom,
    ! [B: $tType,A: $tType,E: $tType,Uu: fun(A,fun(B,fun(E,$o))),Uua: A,Uub: E] :
      ( aa(E,$o,aa(A,fun(E,$o),aTP_Lamp_aey(fun(A,fun(B,fun(E,$o))),fun(A,fun(E,$o)),Uu),Uua),Uub)
    <=> ? [B8: B] : aa(E,$o,aa(B,fun(E,$o),aa(A,fun(B,fun(E,$o)),Uu,Uua),B8),Uub) ) ).

% ATP.lambda_582
tff(fact_8585_ATP_Olambda__583,axiom,
    ! [B: $tType,A: $tType,E: $tType,Uu: fun(A,fun(B,fun(E,$o))),Uua: B,Uub: E] :
      ( aa(E,$o,aa(B,fun(E,$o),aTP_Lamp_aez(fun(A,fun(B,fun(E,$o))),fun(B,fun(E,$o)),Uu),Uua),Uub)
    <=> ? [A7: A] : aa(E,$o,aa(B,fun(E,$o),aa(A,fun(B,fun(E,$o)),Uu,A7),Uua),Uub) ) ).

% ATP.lambda_583
tff(fact_8586_ATP_Olambda__584,axiom,
    ! [B: $tType,A: $tType,E: $tType,Uu: fun(A,fun(B,E)),Uua: set(E),Uub: A] :
      ( aa(A,$o,aa(set(E),fun(A,$o),aTP_Lamp_oz(fun(A,fun(B,E)),fun(set(E),fun(A,$o)),Uu),Uua),Uub)
    <=> ? [B8: B] : member(E,aa(B,E,aa(A,fun(B,E),Uu,Uub),B8),Uua) ) ).

% ATP.lambda_584
tff(fact_8587_ATP_Olambda__585,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),$o),aTP_Lamp_lf(list(A),fun(list(B),fun(product_prod(A,B),$o)),Uu),Uua),Uub)
    <=> ? [I2: 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),I2)),aa(nat,B,nth(B,Uua),I2)) )
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),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_585
tff(fact_8588_ATP_Olambda__586,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),$o),aTP_Lamp_ach(set(A),fun(fun(A,B),fun(product_prod(A,B),$o)),Uu),Uua),Uub)
    <=> ? [A7: A] :
          ( ( Uub = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A7),aa(A,B,Uua,A7)) )
          & member(A,A7,Uu) ) ) ).

% ATP.lambda_586
tff(fact_8589_ATP_Olambda__587,axiom,
    ! [A: $tType,Uu: list(A),Uua: set(nat),Uub: A] :
      ( aa(A,$o,aa(set(nat),fun(A,$o),aTP_Lamp_nx(list(A),fun(set(nat),fun(A,$o)),Uu),Uua),Uub)
    <=> ? [I2: nat] :
          ( ( Uub = aa(nat,A,nth(A,Uu),I2) )
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Uu))
          & member(nat,I2,Uua) ) ) ).

% ATP.lambda_587
tff(fact_8590_ATP_Olambda__588,axiom,
    ! [A: $tType,Uu: nat,Uua: list(A),Uub: A] :
      ( aa(A,$o,aa(list(A),fun(A,$o),aTP_Lamp_lv(nat,fun(list(A),fun(A,$o)),Uu),Uua),Uub)
    <=> ? [I2: nat] :
          ( ( Uub = aa(nat,A,nth(A,Uua),I2) )
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Uu),I2)
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Uua)) ) ) ).

% ATP.lambda_588
tff(fact_8591_ATP_Olambda__589,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,$o),aTP_Lamp_abn(set(B),fun(fun(B,set(A)),fun(A,$o)),Uu),Uua),Uub)
    <=> ? [X6: set(A)] :
          ( member(set(A),X6,aa(set(B),set(set(A)),image2(B,set(A),Uua),Uu))
          & member(A,Uub,X6) ) ) ).

% ATP.lambda_589
tff(fact_8592_ATP_Olambda__590,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [Uu: set(A),Uua: A,Uub: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_vr(set(A),fun(A,fun(A,$o)),Uu),Uua),Uub)
        <=> ? [A7: A] :
              ( ( Uub = aa(A,A,aa(A,fun(A,A),sup_sup(A),Uua),A7) )
              & member(A,A7,Uu) ) ) ) ).

% ATP.lambda_590
tff(fact_8593_ATP_Olambda__591,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [Uu: set(A),Uua: A,Uub: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_vp(set(A),fun(A,fun(A,$o)),Uu),Uua),Uub)
        <=> ? [A7: A] :
              ( ( Uub = aa(A,A,aa(A,fun(A,A),inf_inf(A),Uua),A7) )
              & member(A,A7,Uu) ) ) ) ).

% ATP.lambda_591
tff(fact_8594_ATP_Olambda__592,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),$o),aTP_Lamp_lx(fun(B,set(A)),fun(list(B),fun(set(A),$o)),Uu),Uua),Uub)
    <=> ? [I2: nat] :
          ( ( Uub = aa(B,set(A),Uu,aa(nat,B,nth(B,Uua),I2)) )
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(B),nat,size_size(list(B)),Uua)) ) ) ).

% ATP.lambda_592
tff(fact_8595_ATP_Olambda__593,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),$o),aTP_Lamp_lz(fun(B,set(A)),fun(list(B),fun(set(A),$o)),Uu),Uua),Uub)
    <=> ? [A7: B] :
          ( ( Uub = aa(B,set(A),Uu,A7) )
          & member(B,A7,aa(list(B),set(B),set2(B),Uua)) ) ) ).

% ATP.lambda_593
tff(fact_8596_ATP_Olambda__594,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: set(B),Uub: A] :
      ( aa(A,$o,aa(set(B),fun(A,$o),aTP_Lamp_lh(fun(B,A),fun(set(B),fun(A,$o)),Uu),Uua),Uub)
    <=> ? [L4: B] :
          ( ( Uub = aa(B,A,Uu,L4) )
          & member(B,L4,Uua) ) ) ).

% ATP.lambda_594
tff(fact_8597_ATP_Olambda__595,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: fun(B,$o),Uub: A] :
      ( aa(A,$o,aa(fun(B,$o),fun(A,$o),aTP_Lamp_li(fun(B,A),fun(fun(B,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> ? [X6: B] :
          ( ( Uub = aa(B,A,Uu,X6) )
          & aa(B,$o,Uua,X6) ) ) ).

% ATP.lambda_595
tff(fact_8598_ATP_Olambda__596,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,option(A)),Uua: list(B),Uub: A] :
      ( aa(A,$o,aa(list(B),fun(A,$o),aTP_Lamp_ut(fun(B,option(A)),fun(list(B),fun(A,$o)),Uu),Uua),Uub)
    <=> ? [X6: B] :
          ( member(B,X6,aa(list(B),set(B),set2(B),Uua))
          & ( aa(B,option(A),Uu,X6) = aa(A,option(A),some(A),Uub) ) ) ) ).

% ATP.lambda_596
tff(fact_8599_ATP_Olambda__597,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,$o),aTP_Lamp_vv(set(A),fun(fun(B,fun(A,$o)),fun(B,$o)),Uu),Uua),Uub)
    <=> ! [X6: A] :
          ( member(A,X6,Uu)
         => aa(A,$o,aa(B,fun(A,$o),Uua,Uub),X6) ) ) ).

% ATP.lambda_597
tff(fact_8600_ATP_Olambda__598,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,$o),aTP_Lamp_zz(set(A),fun(fun(B,fun(A,$o)),fun(B,$o)),Uu),Uua),Uub)
    <=> ? [X6: A] :
          ( member(A,X6,Uu)
          & aa(A,$o,aa(B,fun(A,$o),Uua,Uub),X6) ) ) ).

% ATP.lambda_598
tff(fact_8601_ATP_Olambda__599,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_rr(fun(B,fun(A,$o)),fun(set(B),fun(A,$o)),Uu),Uua),Uub)
    <=> ? [X6: B] :
          ( member(B,X6,Uua)
          & aa(A,$o,aa(B,fun(A,$o),Uu,X6),Uub) ) ) ).

% ATP.lambda_599
tff(fact_8602_ATP_Olambda__600,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_mh(set(product_prod(A,A)),fun(A,fun(A,$o)),Uu),Uua),Uub)
    <=> ! [X6: product_prod(A,A)] :
          ( member(product_prod(A,A),X6,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_mg(set(product_prod(A,A)),fun(A,fun(A,fun(A,fun(A,$o)))),Uu),Uua),Uub)),X6) ) ) ).

% ATP.lambda_600
tff(fact_8603_ATP_Olambda__601,axiom,
    ! [A: $tType,B: $tType,E: $tType,Uu: fun(A,fun(B,E)),Uua: set(E),Uub: product_prod(A,B)] :
      ( aa(product_prod(A,B),$o,aa(set(E),fun(product_prod(A,B),$o),aTP_Lamp_oy(fun(A,fun(B,E)),fun(set(E),fun(product_prod(A,B),$o)),Uu),Uua),Uub)
    <=> ? [P4: product_prod(A,B)] :
          ( ( Uub = P4 )
          & member(E,aa(B,E,aa(A,fun(B,E),Uu,aa(product_prod(A,B),A,product_fst(A,B),P4)),aa(product_prod(A,B),B,product_snd(A,B),P4)),Uua) ) ) ).

% ATP.lambda_601
tff(fact_8604_ATP_Olambda__602,axiom,
    ! [A: $tType,B: $tType,Uu: set(product_prod(B,A)),Uua: set(B),Uub: A] :
      ( aa(A,$o,aa(set(B),fun(A,$o),aTP_Lamp_ru(set(product_prod(B,A)),fun(set(B),fun(A,$o)),Uu),Uua),Uub)
    <=> ? [X6: B] :
          ( member(B,X6,Uua)
          & member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X6),Uub),Uu) ) ) ).

% ATP.lambda_602
tff(fact_8605_ATP_Olambda__603,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: set(A),Uub: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_rf(fun(A,B),fun(set(A),fun(A,$o)),Uu),Uua),Uub)
    <=> ? [X6: A] :
          ( member(A,X6,Uua)
          & ( aa(A,B,Uu,X6) = aa(A,B,Uu,Uub) ) ) ) ).

% ATP.lambda_603
tff(fact_8606_ATP_Olambda__604,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,option(A)),Uua: set(B),Uub: A] :
      ( aa(A,$o,aa(set(B),fun(A,$o),aTP_Lamp_ux(fun(B,option(A)),fun(set(B),fun(A,$o)),Uu),Uua),Uub)
    <=> ? [X6: B] :
          ( member(B,X6,Uua)
          & ( aa(B,option(A),Uu,X6) = aa(A,option(A),some(A),Uub) ) ) ) ).

% ATP.lambda_604
tff(fact_8607_ATP_Olambda__605,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: set(B),Uub: A] :
      ( aa(A,$o,aa(set(B),fun(A,$o),aTP_Lamp_re(fun(B,A),fun(set(B),fun(A,$o)),Uu),Uua),Uub)
    <=> ? [X6: B] :
          ( member(B,X6,Uua)
          & ( Uub = aa(B,A,Uu,X6) ) ) ) ).

% ATP.lambda_605
tff(fact_8608_ATP_Olambda__606,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_xe(fun(A,B),fun(filter(B),fun(fun(A,$o),$o)),Uu),Uua),Uub)
    <=> ? [Q7: fun(B,$o)] :
          ( aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),eventually(B),Q7),Uua)
          & ! [X6: A] :
              ( aa(B,$o,Q7,aa(A,B,Uu,X6))
             => aa(A,$o,Uub,X6) ) ) ) ).

% ATP.lambda_606
tff(fact_8609_ATP_Olambda__607,axiom,
    ! [A: $tType,Uu: fun(A,A),Uua: A,Uub: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_tk(fun(A,A),fun(A,fun(A,$o)),Uu),Uua),Uub)
    <=> ? [N2: nat] : ( Uub = aa(A,A,compow(fun(A,A),N2,Uu),Uua) ) ) ).

% ATP.lambda_607
tff(fact_8610_ATP_Olambda__608,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),$o),aTP_Lamp_lw(fun(B,A),fun(fun(B,fun(B,$o)),fun(product_prod(A,A),$o)),Uu),Uua),Uub)
    <=> ? [A7: B,B8: B] :
          ( ( Uub = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(B,A,Uu,A7)),aa(B,A,Uu,B8)) )
          & aa(B,$o,aa(B,fun(B,$o),Uua,A7),B8) ) ) ).

% ATP.lambda_608
tff(fact_8611_ATP_Olambda__609,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),$o),aTP_Lamp_yg(set(product_prod(B,B)),fun(fun(B,A),fun(product_prod(A,A),$o)),Uu),Uua),Uub)
    <=> ? [A7: B,B8: B] :
          ( ( Uub = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(B,A,Uua,A7)),aa(B,A,Uua,B8)) )
          & member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),A7),B8),Uu) ) ) ).

% ATP.lambda_609
tff(fact_8612_ATP_Olambda__610,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),$o),aTP_Lamp_lo(set(product_prod(A,A)),fun(list(A),fun(list(A),$o)),Uu),Uua),Uub)
    <=> ? [A7: 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),A7),V4)) )
          | ? [U4: list(A),Aa5: A,B8: A,Va3: list(A),W3: list(A)] :
              ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Aa5),B8),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),Aa5),Va3)) )
              & ( 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),B8),W3)) ) ) ) ) ).

% ATP.lambda_610
tff(fact_8613_ATP_Olambda__611,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [Uu: set(A),Uua: set(A),Uub: A] :
          ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_vs(set(A),fun(set(A),fun(A,$o)),Uu),Uua),Uub)
        <=> ? [A7: A,B8: A] :
              ( ( Uub = aa(A,A,aa(A,fun(A,A),sup_sup(A),A7),B8) )
              & member(A,A7,Uu)
              & member(A,B8,Uua) ) ) ) ).

% ATP.lambda_611
tff(fact_8614_ATP_Olambda__612,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [Uu: set(A),Uua: set(A),Uub: A] :
          ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_vq(set(A),fun(set(A),fun(A,$o)),Uu),Uua),Uub)
        <=> ? [A7: A,B8: A] :
              ( ( Uub = aa(A,A,aa(A,fun(A,A),inf_inf(A),A7),B8) )
              & member(A,A7,Uu)
              & member(A,B8,Uua) ) ) ) ).

% ATP.lambda_612
tff(fact_8615_ATP_Olambda__613,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(list(A)),Uub: list(A)] :
      ( aa(list(A),$o,aa(set(list(A)),fun(list(A),$o),aTP_Lamp_lj(set(A),fun(set(list(A)),fun(list(A),$o)),Uu),Uua),Uub)
    <=> ? [X6: A,Xs2: list(A)] :
          ( ( Uub = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X6),Xs2) )
          & member(A,X6,Uu)
          & member(list(A),Xs2,Uua) ) ) ).

% ATP.lambda_613
tff(fact_8616_ATP_Olambda__614,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_xb(filter(A),fun(filter(A),fun(fun(A,$o),$o)),Uu),Uua),Uub)
    <=> ? [Q7: fun(A,$o),R7: fun(A,$o)] :
          ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Q7),Uu)
          & aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),R7),Uua)
          & ! [X6: A] :
              ( ( aa(A,$o,Q7,X6)
                & aa(A,$o,R7,X6) )
             => aa(A,$o,Uub,X6) ) ) ) ).

% ATP.lambda_614
tff(fact_8617_ATP_Olambda__615,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),$o),aTP_Lamp_rc(set(product_prod(A,A)),fun(multiset(A),fun(multiset(A),$o)),Uu),Uua),Uub)
    <=> ? [A7: A,M03: multiset(A),K8: multiset(A)] :
          ( ( Uub = add_mset(A,A7,M03) )
          & ( Uua = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M03),K8) )
          & ! [B8: A] :
              ( member(A,B8,aa(multiset(A),set(A),set_mset(A),K8))
             => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B8),A7),Uu) ) ) ) ).

% ATP.lambda_615
tff(fact_8618_ATP_Olambda__616,axiom,
    ! [B: $tType,A: $tType,G2: $tType,E: $tType,F4: $tType,Uu: fun(G2,fun(A,fun(E,fun(B,fun(F4,$o))))),Uua: G2,Uub: fun(A,option(product_prod(B,product_prod(E,F4))))] :
      ( aa(fun(A,option(product_prod(B,product_prod(E,F4)))),$o,aa(G2,fun(fun(A,option(product_prod(B,product_prod(E,F4)))),$o),aTP_Lamp_abx(fun(G2,fun(A,fun(E,fun(B,fun(F4,$o))))),fun(G2,fun(fun(A,option(product_prod(B,product_prod(E,F4)))),$o)),Uu),Uua),Uub)
    <=> ! [H3: A,H8: E,R6: B,N2: F4] :
          ( ( aa(A,option(product_prod(B,product_prod(E,F4))),Uub,H3) = aa(product_prod(B,product_prod(E,F4)),option(product_prod(B,product_prod(E,F4))),some(product_prod(B,product_prod(E,F4))),aa(product_prod(E,F4),product_prod(B,product_prod(E,F4)),aa(B,fun(product_prod(E,F4),product_prod(B,product_prod(E,F4))),product_Pair(B,product_prod(E,F4)),R6),aa(F4,product_prod(E,F4),aa(E,fun(F4,product_prod(E,F4)),product_Pair(E,F4),H8),N2))) )
         => aa(F4,$o,aa(B,fun(F4,$o),aa(E,fun(B,fun(F4,$o)),aa(A,fun(E,fun(B,fun(F4,$o))),aa(G2,fun(A,fun(E,fun(B,fun(F4,$o)))),Uu,Uua),H3),H8),R6),N2) ) ) ).

% ATP.lambda_616
tff(fact_8619_ATP_Olambda__617,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),$o),aTP_Lamp_ln(set(product_prod(A,A)),fun(list(A),fun(list(A),$o)),Uu),Uua),Uub)
    <=> ? [Us2: list(A),Z5: A,Z10: A,Vs3: list(A)] :
          ( ( Uua = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Z5),Vs3)) )
          & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z5),Z10),Uu)
          & ( Uub = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Z10),Vs3)) ) ) ) ).

% ATP.lambda_617
tff(fact_8620_ATP_Olambda__618,axiom,
    ! [A: $tType,Uu: fun(A,fun(A,A)),Uua: A,Uub: fun(nat,fun(nat,A)),Uuc: nat] : ( aa(nat,A,aa(fun(nat,fun(nat,A)),fun(nat,A),aa(A,fun(fun(nat,fun(nat,A)),fun(nat,A)),aTP_Lamp_acj(fun(A,fun(A,A)),fun(A,fun(fun(nat,fun(nat,A)),fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,Uu,Uua),aa(nat,fun(nat,A),aTP_Lamp_aci(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uub),Uuc)),aa(nat,set(nat),set_ord_atMost(nat),Uuc)) ) ).

% ATP.lambda_618
tff(fact_8621_ATP_Olambda__619,axiom,
    ! [E: $tType,A: $tType,B: $tType,F4: $tType,Uu: fun(B,fun(E,fun(F4,A))),Uua: F4,Uub: B,Uuc: E] : ( aa(E,A,aa(B,fun(E,A),aa(F4,fun(B,fun(E,A)),aTP_Lamp_gq(fun(B,fun(E,fun(F4,A))),fun(F4,fun(B,fun(E,A))),Uu),Uua),Uub),Uuc) = aa(F4,A,aa(E,fun(F4,A),aa(B,fun(E,fun(F4,A)),Uu,Uub),Uuc),Uua) ) ).

% ATP.lambda_619
tff(fact_8622_ATP_Olambda__620,axiom,
    ! [A: $tType,B: $tType,E: $tType,Uu: product_prod(E,A),Uua: A,Uub: B,Uuc: set(product_prod(E,B))] :
      ( aa(set(product_prod(E,B)),set(product_prod(E,B)),aa(B,fun(set(product_prod(E,B)),set(product_prod(E,B))),aa(A,fun(B,fun(set(product_prod(E,B)),set(product_prod(E,B)))),aTP_Lamp_pg(product_prod(E,A),fun(A,fun(B,fun(set(product_prod(E,B)),set(product_prod(E,B))))),Uu),Uua),Uub),Uuc) = $ite(aa(product_prod(E,A),A,product_snd(E,A),Uu) = Uua,aa(set(product_prod(E,B)),set(product_prod(E,B)),aa(product_prod(E,B),fun(set(product_prod(E,B)),set(product_prod(E,B))),insert3(product_prod(E,B)),aa(B,product_prod(E,B),aa(E,fun(B,product_prod(E,B)),product_Pair(E,B),aa(product_prod(E,A),E,product_fst(E,A),Uu)),Uub)),Uuc),Uuc) ) ).

% ATP.lambda_620
tff(fact_8623_ATP_Olambda__621,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_vn(fun(A,B),fun(set(A),fun(A,fun(B,A))),Uu),Uua),Uub),Uuc) = $ite(member(B,Uuc,aa(set(A),set(B),image2(A,B,Uu),Uua)),the_inv_into(A,B,Uua,Uu,Uuc),Uub) ) ).

% ATP.lambda_621
tff(fact_8624_ATP_Olambda__622,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_pq(set(A),fun(A,fun(B,fun(set(B),set(B)))),Uu),Uua),Uub),Uuc) = $ite(member(A,Uua,Uu),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert3(B),Uub),Uuc),Uuc) ) ).

% ATP.lambda_622
tff(fact_8625_ATP_Olambda__623,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_gc(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_623
tff(fact_8626_ATP_Olambda__624,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_eo(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_624
tff(fact_8627_ATP_Olambda__625,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_ep(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_625
tff(fact_8628_ATP_Olambda__626,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_gd(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_626
tff(fact_8629_ATP_Olambda__627,axiom,
    ! [A: $tType,Uu: nat,Uua: fun(nat,A),Uub: fun(nat,A),Uuc: nat] :
      ( aa(nat,A,aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_acl(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_627
tff(fact_8630_ATP_Olambda__628,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_tu(fun(A,B),fun(set(A),fun(fun(A,B),fun(A,B))),Uu),Uua),Uub),Uuc) = $ite(member(A,Uuc,Uua),aa(A,B,Uu,Uuc),aa(A,B,Uub,Uuc)) ) ).

% ATP.lambda_628
tff(fact_8631_ATP_Olambda__629,axiom,
    ! [A: $tType,B: $tType,Uu: B,Uua: fun(B,A),Uub: fun(B,A),Uuc: B] :
      ( aa(B,A,aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_acp(B,fun(fun(B,A),fun(fun(B,A),fun(B,A))),Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa(B,A,Uua,Uuc),aa(B,A,Uub,Uuc)) ) ).

% ATP.lambda_629
tff(fact_8632_ATP_Olambda__630,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_dx(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_630
tff(fact_8633_ATP_Olambda__631,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_fn(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_631
tff(fact_8634_ATP_Olambda__632,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_ei(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_632
tff(fact_8635_ATP_Olambda__633,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_vg(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_633
tff(fact_8636_ATP_Olambda__634,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_uo(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_634
tff(fact_8637_ATP_Olambda__635,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_qw(set(A),fun(B,fun(B,fun(A,B))),Uu),Uua),Uub),Uuc) = $ite(member(A,Uuc,Uu),Uua,Uub) ) ).

% ATP.lambda_635
tff(fact_8638_ATP_Olambda__636,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_aff(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),insert2(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_636
tff(fact_8639_ATP_Olambda__637,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_ns(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_637
tff(fact_8640_ATP_Olambda__638,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_al(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_638
tff(fact_8641_ATP_Olambda__639,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_dm(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_639
tff(fact_8642_ATP_Olambda__640,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,$o),fun(fun(A,B),fun(A,B)),aTP_Lamp_tq(fun(A,B),fun(fun(A,$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_640
tff(fact_8643_ATP_Olambda__641,axiom,
    ! [E: $tType,B: $tType,A: $tType,Uu: set(product_prod(A,B)),Uua: E,Uub: A,Uuc: set(product_prod(E,B))] : ( aa(set(product_prod(E,B)),set(product_prod(E,B)),aa(A,fun(set(product_prod(E,B)),set(product_prod(E,B))),aa(E,fun(A,fun(set(product_prod(E,B)),set(product_prod(E,B)))),aTP_Lamp_po(set(product_prod(A,B)),fun(E,fun(A,fun(set(product_prod(E,B)),set(product_prod(E,B))))),Uu),Uua),Uub),Uuc) = finite_fold(product_prod(A,B),set(product_prod(E,B)),aa(fun(A,fun(B,fun(set(product_prod(E,B)),set(product_prod(E,B))))),fun(product_prod(A,B),fun(set(product_prod(E,B)),set(product_prod(E,B)))),product_case_prod(A,B,fun(set(product_prod(E,B)),set(product_prod(E,B)))),aa(A,fun(A,fun(B,fun(set(product_prod(E,B)),set(product_prod(E,B))))),aTP_Lamp_pn(E,fun(A,fun(A,fun(B,fun(set(product_prod(E,B)),set(product_prod(E,B)))))),Uua),Uub)),Uuc,Uu) ) ).

% ATP.lambda_641
tff(fact_8644_ATP_Olambda__642,axiom,
    ! [A: $tType,E: $tType,B: $tType,Uu: set(product_prod(B,E)),Uua: A,Uub: B,Uuc: set(product_prod(A,E))] : ( aa(set(product_prod(A,E)),set(product_prod(A,E)),aa(B,fun(set(product_prod(A,E)),set(product_prod(A,E))),aa(A,fun(B,fun(set(product_prod(A,E)),set(product_prod(A,E)))),aTP_Lamp_pl(set(product_prod(B,E)),fun(A,fun(B,fun(set(product_prod(A,E)),set(product_prod(A,E))))),Uu),Uua),Uub),Uuc) = finite_fold(product_prod(B,E),set(product_prod(A,E)),aa(fun(B,fun(E,fun(set(product_prod(A,E)),set(product_prod(A,E))))),fun(product_prod(B,E),fun(set(product_prod(A,E)),set(product_prod(A,E)))),product_case_prod(B,E,fun(set(product_prod(A,E)),set(product_prod(A,E)))),aa(B,fun(B,fun(E,fun(set(product_prod(A,E)),set(product_prod(A,E))))),aTP_Lamp_pk(A,fun(B,fun(B,fun(E,fun(set(product_prod(A,E)),set(product_prod(A,E)))))),Uua),Uub)),Uuc,Uu) ) ).

% ATP.lambda_642
tff(fact_8645_ATP_Olambda__643,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_me(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_643
tff(fact_8646_ATP_Olambda__644,axiom,
    ! [A: $tType,Uu: fun(A,fun(A,$o)),Uua: A,Uub: list(A),Uuc: $o] :
      ( aa($o,$o,aa(list(A),fun($o,$o),aa(A,fun(list(A),fun($o,$o)),aTP_Lamp_abb(fun(A,fun(A,$o)),fun(A,fun(list(A),fun($o,$o))),Uu),Uua),Uub),(Uuc))
    <=> aa(list(A),$o,aa(fun(A,fun(list(A),$o)),fun(list(A),$o),aa($o,fun(fun(A,fun(list(A),$o)),fun(list(A),$o)),case_list($o,A),$true),aa($o,fun(A,fun(list(A),$o)),aa(A,fun($o,fun(A,fun(list(A),$o))),aTP_Lamp_aba(fun(A,fun(A,$o)),fun(A,fun($o,fun(A,fun(list(A),$o)))),Uu),Uua),(Uuc))),Uub) ) ).

% ATP.lambda_644
tff(fact_8647_ATP_Olambda__645,axiom,
    ! [B: $tType,E: $tType,A: $tType,Uu: fun(E,fun(list(E),fun(B,B))),Uua: fun(A,E),Uub: A,Uuc: list(A)] : ( aa(list(A),fun(B,B),aa(A,fun(list(A),fun(B,B)),aa(fun(A,E),fun(A,fun(list(A),fun(B,B))),aTP_Lamp_th(fun(E,fun(list(E),fun(B,B))),fun(fun(A,E),fun(A,fun(list(A),fun(B,B)))),Uu),Uua),Uub),Uuc) = aa(list(E),fun(B,B),aa(E,fun(list(E),fun(B,B)),Uu,aa(A,E,Uua,Uub)),aa(list(A),list(E),aa(fun(A,E),fun(list(A),list(E)),map(A,E),Uua),Uuc)) ) ).

% ATP.lambda_645
tff(fact_8648_ATP_Olambda__646,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,$o)),aTP_Lamp_hi(fun(A,fun(A,$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_646
tff(fact_8649_ATP_Olambda__647,axiom,
    ! [B: $tType,A: $tType,E: $tType,F4: $tType,Uu: fun(B,fun(E,A)),Uua: fun(F4,B),Uub: fun(F4,E),Uuc: F4] : ( aa(F4,A,aa(fun(F4,E),fun(F4,A),aa(fun(F4,B),fun(fun(F4,E),fun(F4,A)),aTP_Lamp_hh(fun(B,fun(E,A)),fun(fun(F4,B),fun(fun(F4,E),fun(F4,A))),Uu),Uua),Uub),Uuc) = aa(E,A,aa(B,fun(E,A),Uu,aa(F4,B,Uua,Uuc)),aa(F4,E,Uub,Uuc)) ) ).

% ATP.lambda_647
tff(fact_8650_ATP_Olambda__648,axiom,
    ! [A: $tType,E: $tType,B: $tType,F4: $tType] :
      ( ( order(E)
        & order(A) )
     => ! [Uu: fun(A,fun(B,E)),Uua: fun(F4,B),Uub: A,Uuc: F4] : ( aa(F4,E,aa(A,fun(F4,E),aa(fun(F4,B),fun(A,fun(F4,E)),aTP_Lamp_om(fun(A,fun(B,E)),fun(fun(F4,B),fun(A,fun(F4,E))),Uu),Uua),Uub),Uuc) = aa(B,E,aa(A,fun(B,E),Uu,Uub),aa(F4,B,Uua,Uuc)) ) ) ).

% ATP.lambda_648
tff(fact_8651_ATP_Olambda__649,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,$o)),aTP_Lamp_rs(fun(A,fun(B,$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_649
tff(fact_8652_ATP_Olambda__650,axiom,
    ! [A: $tType,B: $tType,E: $tType,Uu: fun(A,fun(B,A)),Uua: fun(E,B),Uub: A,Uuc: E] : ( aa(E,A,aa(A,fun(E,A),aa(fun(E,B),fun(A,fun(E,A)),aTP_Lamp_hl(fun(A,fun(B,A)),fun(fun(E,B),fun(A,fun(E,A))),Uu),Uua),Uub),Uuc) = aa(B,A,aa(A,fun(B,A),Uu,Uub),aa(E,B,Uua,Uuc)) ) ).

% ATP.lambda_650
tff(fact_8653_ATP_Olambda__651,axiom,
    ! [B: $tType,A: $tType,E: $tType] :
      ( semiring_0(A)
     => ! [Uu: fun(B,A),Uua: fun(E,A),Uub: set(E),Uuc: B] : ( aa(B,A,aa(set(E),fun(B,A),aa(fun(E,A),fun(set(E),fun(B,A)),aTP_Lamp_fg(fun(B,A),fun(fun(E,A),fun(set(E),fun(B,A))),Uu),Uua),Uub),Uuc) = aa(set(E),A,aa(fun(E,A),fun(set(E),A),groups7311177749621191930dd_sum(E,A),aa(B,fun(E,A),aa(fun(E,A),fun(B,fun(E,A)),aTP_Lamp_ff(fun(B,A),fun(fun(E,A),fun(B,fun(E,A))),Uu),Uua),Uuc)),Uub) ) ) ).

% ATP.lambda_651
tff(fact_8654_ATP_Olambda__652,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: set(B),Uub: filter(B),Uuc: fun(A,$o)] :
      ( aa(fun(A,$o),$o,aa(filter(B),fun(fun(A,$o),$o),aa(set(B),fun(filter(B),fun(fun(A,$o),$o)),aTP_Lamp_abs(fun(B,A),fun(set(B),fun(filter(B),fun(fun(A,$o),$o))),Uu),Uua),Uub),Uuc)
    <=> aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),eventually(B),aa(fun(A,$o),fun(B,$o),aa(set(B),fun(fun(A,$o),fun(B,$o)),aTP_Lamp_abr(fun(B,A),fun(set(B),fun(fun(A,$o),fun(B,$o))),Uu),Uua),Uuc)),Uub) ) ).

% ATP.lambda_652
tff(fact_8655_ATP_Olambda__653,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_wx(set(B),fun(fun(B,A),fun(filter(B),fun(fun(A,$o),$o))),Uu),Uua),Uub),Uuc)
    <=> aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),eventually(B),aa(fun(A,$o),fun(B,$o),aa(fun(B,A),fun(fun(A,$o),fun(B,$o)),aTP_Lamp_ww(set(B),fun(fun(B,A),fun(fun(A,$o),fun(B,$o))),Uu),Uua),Uuc)),Uub) ) ).

% ATP.lambda_653
tff(fact_8656_ATP_Olambda__654,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_jw(fun(A,B),fun(list(A),fun(A,fun(A,fun(list(A),list(A))))),Uu),Uua),Uub),Uuc) = aa(fun(A,fun(list(A),list(A))),fun(list(A),list(A)),
              aa(list(A),fun(fun(A,fun(list(A),list(A))),fun(list(A),list(A))),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_jv(fun(A,B),fun(list(A),fun(A,fun(list(A),list(A)))),Uu),Uua)) ) ) ).

% ATP.lambda_654
tff(fact_8657_ATP_Olambda__655,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_abl(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_655
tff(fact_8658_ATP_Olambda__656,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,$o)),aTP_Lamp_aev(fun(A,fun(B,$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_656
tff(fact_8659_ATP_Olambda__657,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,fun(B,assn)),Uua: $o,Uub: A,Uuc: B] : ( aa(B,assn,aa(A,fun(B,assn),aa($o,fun(A,fun(B,assn)),aTP_Lamp_ar(fun(A,fun(B,assn)),fun($o,fun(A,fun(B,assn))),Uu),(Uua)),Uub),Uuc) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),Uu,Uub),Uuc)),pure_assn((Uua))) ) ).

% ATP.lambda_657
tff(fact_8660_ATP_Olambda__658,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_act(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(numeral_numeral(code_natural,bit0(bit1(bit0(bit1(bit0(bit1(bit0(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(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(numeral_numeral(code_natural,bit0(bit1(bit1(bit0(bit0(bit0(bit0(bit0(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(one2))))))))))))))))))))))))))))))),Uua))) ) ).

% ATP.lambda_658
tff(fact_8661_ATP_Olambda__659,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,$o)),aTP_Lamp_st(fun(A,fun(A,$o)),fun(set(A),fun(A,fun(A,$o))),Uu),Uua),Uub),Uuc)
    <=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),Uuc),product_Sigma(A,A,Uua,aTP_Lamp_pt(set(A),fun(A,set(A)),Uua)))
        & aa(A,$o,aa(A,fun(A,$o),Uu,Uub),Uuc) ) ) ).

% ATP.lambda_659
tff(fact_8662_ATP_Olambda__660,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_mu(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_660
tff(fact_8663_ATP_Olambda__661,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Uu: nat,Uua: A,Uub: array(A),Uuc: heap_ext(product_unit)] : ( aa(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(array(A),fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(A,fun(array(A),fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)))),aTP_Lamp_aeh(nat,fun(A,fun(array(A),fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))))),Uu),Uua),Uub),Uuc) = aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),aa(nat,A,nth(A,array_get(A,Uuc,Uub)),Uu)),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),array_update(A,Uub,Uu,Uua,Uuc)),numeral_numeral(nat,bit0(one2)))) ) ) ).

% ATP.lambda_661
tff(fact_8664_ATP_Olambda__662,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_qy(set(product_prod(A,A)),fun(set(A),fun(A,fun(A,$o))),Uu),Uua),Uub),Uuc)
    <=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),Uuc),Uu)
        & ~ member(A,Uub,Uua)
        & ~ member(A,Uuc,Uua) ) ) ).

% ATP.lambda_662
tff(fact_8665_ATP_Olambda__663,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_ka(fun(A,nat),fun(set(product_prod(A,A)),fun(A,fun(A,$o))),Uu),Uua),Uub),Uuc)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,Uu,Uub)),aa(A,nat,Uu,Uuc))
        | ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,Uu,Uub)),aa(A,nat,Uu,Uuc))
          & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),Uuc),Uua) ) ) ) ).

% ATP.lambda_663
tff(fact_8666_ATP_Olambda__664,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_xn(set(product_prod(B,B)),fun(fun(A,B),fun(A,fun(A,$o))),Uu),Uua),Uub),Uuc)
    <=> member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),aa(A,B,Uua,Uub)),aa(A,B,Uua,Uuc)),Uu) ) ).

% ATP.lambda_664
tff(fact_8667_ATP_Olambda__665,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_us(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_665
tff(fact_8668_ATP_Olambda__666,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_ga(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),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_666
tff(fact_8669_ATP_Olambda__667,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_fz(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),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_667
tff(fact_8670_ATP_Olambda__668,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_lr(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),X6: A,Y4: A,Xs5: list(A),Ys6: list(A)] :
            ( ( Uub = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xys2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X6),Xs5)) )
            & ( Uuc = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xys2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys6)) )
            & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X6),Y4),Uu) ) ) ) ).

% ATP.lambda_668
tff(fact_8671_ATP_Olambda__669,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_fv(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_669
tff(fact_8672_ATP_Olambda__670,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,$o)),aTP_Lamp_bl(fun(B,A),fun(set(B),fun(fun(A,$o),fun(A,$o))),Uu),Uua),Uub),Uuc)
    <=> ( member(A,Uuc,aa(set(B),set(A),image2(B,A,Uu),Uua))
        & aa(A,$o,Uub,Uuc) ) ) ).

% ATP.lambda_670
tff(fact_8673_ATP_Olambda__671,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_fr(set(A),fun(fun(A,B),fun(B,fun(A,$o))),Uu),Uua),Uub),Uuc)
    <=> ( member(A,Uuc,Uu)
        & ( aa(A,B,Uua,Uuc) = Uub ) ) ) ).

% ATP.lambda_671
tff(fact_8674_ATP_Olambda__672,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_ty(fun(A,B),fun(set(A),fun(B,fun(A,$o))),Uu),Uua),Uub),Uuc)
    <=> ( member(A,Uuc,Uua)
        & ( aa(A,B,Uu,Uuc) = Uub ) ) ) ).

% ATP.lambda_672
tff(fact_8675_ATP_Olambda__673,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_fu(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_673
tff(fact_8676_ATP_Olambda__674,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_jt(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),aa(fun(A,B),fun(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),aa(fun(A,B),fun(list(A),list(A)),linorder_sort_key(A,B),Uu),Uuc))) ) ) ).

% ATP.lambda_674
tff(fact_8677_ATP_Olambda__675,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aa(nat,fun(nat,fun(nat,$o)),aTP_Lamp_mz(nat,fun(nat,fun(nat,fun(nat,$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_675
tff(fact_8678_ATP_Olambda__676,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_nb(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_676
tff(fact_8679_ATP_Olambda__677,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_nd(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_677
tff(fact_8680_ATP_Olambda__678,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_dv(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)),aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),Uua),Uuc)) ) ).

% ATP.lambda_678
tff(fact_8681_ATP_Olambda__679,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_du(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)),aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),Uuc),Uua)) ) ).

% ATP.lambda_679
tff(fact_8682_ATP_Olambda__680,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),$o)),aTP_Lamp_yk(filter(A),fun(filter(B),fun(fun(A,$o),fun(fun(B,$o),$o))),Uu),Uua),Uub),Uuc)
    <=> ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Uub),Uu)
        & aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),eventually(B),Uuc),Uua) ) ) ).

% ATP.lambda_680
tff(fact_8683_ATP_Olambda__681,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_ua(A,fun(B,fun(A,fun(B,$o))),Uu),Uua),Uub),Uuc)
    <=> ( ( Uu = Uub )
        & ( Uua = Uuc ) ) ) ).

% ATP.lambda_681
tff(fact_8684_ATP_Olambda__682,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,$o)),aTP_Lamp_bm(fun(B,A),fun(set(B),fun(fun(A,$o),fun(B,$o))),Uu),Uua),Uub),Uuc)
    <=> ( member(B,Uuc,Uua)
        & aa(A,$o,Uub,aa(B,A,Uu,Uuc)) ) ) ).

% ATP.lambda_682
tff(fact_8685_ATP_Olambda__683,axiom,
    ! [A: $tType,B: $tType,E: $tType,Uu: fun(E,set(product_prod(A,B))),Uua: E,Uub: A,Uuc: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(E,fun(A,fun(B,$o)),aTP_Lamp_db(fun(E,set(product_prod(A,B))),fun(E,fun(A,fun(B,$o))),Uu),Uua),Uub),Uuc)
    <=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uub),Uuc),aa(E,set(product_prod(A,B)),Uu,Uua)) ) ).

% ATP.lambda_683
tff(fact_8686_ATP_Olambda__684,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_bc(set(A),fun(fun(A,B),fun(fun(A,B),fun(A,$o))),Uu),Uua),Uub),Uuc)
        <=> ( member(A,Uuc,Uu)
            & ( aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uua,Uuc)),aa(A,B,Uub,Uuc)) != one_one(B) ) ) ) ) ).

% ATP.lambda_684
tff(fact_8687_ATP_Olambda__685,axiom,
    ! [B: $tType,A: $tType,Uu: A,Uua: set(B),Uub: fun(B,A),Uuc: B] :
      ( aa(B,$o,aa(fun(B,A),fun(B,$o),aa(set(B),fun(fun(B,A),fun(B,$o)),aTP_Lamp_acn(A,fun(set(B),fun(fun(B,A),fun(B,$o))),Uu),Uua),Uub),Uuc)
    <=> ( member(B,Uuc,Uua)
        & ( aa(B,A,Uub,Uuc) != Uu ) ) ) ).

% ATP.lambda_685
tff(fact_8688_ATP_Olambda__686,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Uu: nat,Uua: list(A),Uub: array(A),Uuc: A] : ( aa(A,assn,aa(array(A),fun(A,assn),aa(list(A),fun(array(A),fun(A,assn)),aTP_Lamp_adb(nat,fun(list(A),fun(array(A),fun(A,assn))),Uu),Uua),Uub),Uuc) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),snga_assn(A,Uub,Uua)),pure_assn(Uuc = aa(nat,A,nth(A,Uua),Uu))) ) ) ).

% ATP.lambda_686
tff(fact_8689_ATP_Olambda__687,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_gb(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_687
tff(fact_8690_ATP_Olambda__688,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_wz(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))
        & member(A,Uuc,Uu) ) ) ).

% ATP.lambda_688
tff(fact_8691_ATP_Olambda__689,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,$o)),aTP_Lamp_abr(fun(B,A),fun(set(B),fun(fun(A,$o),fun(B,$o))),Uu),Uua),Uub),Uuc)
    <=> ( aa(A,$o,Uub,aa(B,A,Uu,Uuc))
        & member(B,Uuc,Uua) ) ) ).

% ATP.lambda_689
tff(fact_8692_ATP_Olambda__690,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,$o)),aTP_Lamp_ww(set(B),fun(fun(B,A),fun(fun(A,$o),fun(B,$o))),Uu),Uua),Uub),Uuc)
    <=> ( aa(A,$o,Uub,aa(B,A,Uua,Uuc))
        & member(B,Uuc,Uu) ) ) ).

% ATP.lambda_690
tff(fact_8693_ATP_Olambda__691,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,$o)),aTP_Lamp_hq(fun(A,$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_691
tff(fact_8694_ATP_Olambda__692,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_kw(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_692
tff(fact_8695_ATP_Olambda__693,axiom,
    ! [B: $tType,A: $tType,E: $tType] :
      ( ( semiring_0(A)
        & comm_monoid_add(B)
        & times(B) )
     => ! [Uu: fun(B,A),Uua: fun(E,A),Uub: B,Uuc: E] : ( aa(E,A,aa(B,fun(E,A),aa(fun(E,A),fun(B,fun(E,A)),aTP_Lamp_sm(fun(B,A),fun(fun(E,A),fun(B,fun(E,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,Uu,Uub)),aa(E,A,Uua,Uuc)) ) ) ).

% ATP.lambda_693
tff(fact_8696_ATP_Olambda__694,axiom,
    ! [B: $tType,A: $tType,E: $tType] :
      ( semiring_0(A)
     => ! [Uu: fun(B,A),Uua: fun(E,A),Uub: B,Uuc: E] : ( aa(E,A,aa(B,fun(E,A),aa(fun(E,A),fun(B,fun(E,A)),aTP_Lamp_ff(fun(B,A),fun(fun(E,A),fun(B,fun(E,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,Uu,Uub)),aa(E,A,Uua,Uuc)) ) ) ).

% ATP.lambda_694
tff(fact_8697_ATP_Olambda__695,axiom,
    ! [A: $tType,E: $tType,B: $tType] :
      ( semiring_0(E)
     => ! [Uu: fun(A,E),Uua: fun(B,E),Uub: A,Uuc: B] : ( aa(B,E,aa(A,fun(B,E),aa(fun(B,E),fun(A,fun(B,E)),aTP_Lamp_tw(fun(A,E),fun(fun(B,E),fun(A,fun(B,E))),Uu),Uua),Uub),Uuc) = aa(E,E,aa(E,fun(E,E),times_times(E),aa(A,E,Uu,Uub)),aa(B,E,Uua,Uuc)) ) ) ).

% ATP.lambda_695
tff(fact_8698_ATP_Olambda__696,axiom,
    ! [A: $tType,E: $tType,F4: $tType,B: $tType,Uu: fun(A,filter(E)),Uua: fun(B,filter(F4)),Uub: A,Uuc: B] : ( aa(B,filter(product_prod(E,F4)),aa(A,fun(B,filter(product_prod(E,F4))),aa(fun(B,filter(F4)),fun(A,fun(B,filter(product_prod(E,F4)))),aTP_Lamp_yq(fun(A,filter(E)),fun(fun(B,filter(F4)),fun(A,fun(B,filter(product_prod(E,F4))))),Uu),Uua),Uub),Uuc) = aa(filter(F4),filter(product_prod(E,F4)),aa(filter(E),fun(filter(F4),filter(product_prod(E,F4))),prod_filter(E,F4),aa(A,filter(E),Uu,Uub)),aa(B,filter(F4),Uua,Uuc)) ) ).

% ATP.lambda_696
tff(fact_8699_ATP_Olambda__697,axiom,
    ! [E: $tType,A: $tType,B: $tType,F4: $tType,Uu: fun(E,A),Uua: fun(F4,B),Uub: E,Uuc: F4] : ( aa(F4,product_prod(A,B),aa(E,fun(F4,product_prod(A,B)),aa(fun(F4,B),fun(E,fun(F4,product_prod(A,B))),aTP_Lamp_ho(fun(E,A),fun(fun(F4,B),fun(E,fun(F4,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(E,A,Uu,Uub)),aa(F4,B,Uua,Uuc)) ) ).

% ATP.lambda_697
tff(fact_8700_ATP_Olambda__698,axiom,
    ! [A: $tType,E: $tType,F4: $tType,B: $tType,Uu: fun(A,E),Uua: fun(B,F4),Uub: A,Uuc: B] : ( aa(B,product_prod(E,F4),aa(A,fun(B,product_prod(E,F4)),aa(fun(B,F4),fun(A,fun(B,product_prod(E,F4))),aTP_Lamp_lm(fun(A,E),fun(fun(B,F4),fun(A,fun(B,product_prod(E,F4)))),Uu),Uua),Uub),Uuc) = aa(F4,product_prod(E,F4),aa(E,fun(F4,product_prod(E,F4)),product_Pair(E,F4),aa(A,E,Uu,Uub)),aa(B,F4,Uua,Uuc)) ) ).

% ATP.lambda_698
tff(fact_8701_ATP_Olambda__699,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,$o)),aTP_Lamp_yi(fun(A,$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_699
tff(fact_8702_ATP_Olambda__700,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_jh(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_700
tff(fact_8703_ATP_Olambda__701,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_jv(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_ju(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),divide_divide(nat,aa(list(A),nat,size_size(list(A)),Uua),numeral_numeral(nat,bit0(one2))))),Uua)) ) ) ).

% ATP.lambda_701
tff(fact_8704_ATP_Olambda__702,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,$o)),aTP_Lamp_bk(fun(A,fun(A,$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_702
tff(fact_8705_ATP_Olambda__703,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_wt(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_703
tff(fact_8706_ATP_Olambda__704,axiom,
    ! [A: $tType,E: $tType,B: $tType] :
      ( semiring_1(E)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: fun(B,E),Uuc: B] : ( aa(B,E,aa(fun(B,E),fun(B,E),aa(fun(A,B),fun(fun(B,E),fun(B,E)),aTP_Lamp_fs(set(A),fun(fun(A,B),fun(fun(B,E),fun(B,E))),Uu),Uua),Uub),Uuc) = aa(E,E,aa(E,fun(E,E),times_times(E),aa(nat,E,semiring_1_of_nat(E),finite_card(A,aa(fun(A,$o),set(A),collect(A),aa(B,fun(A,$o),aa(fun(A,B),fun(B,fun(A,$o)),aTP_Lamp_fr(set(A),fun(fun(A,B),fun(B,fun(A,$o))),Uu),Uua),Uuc))))),aa(B,E,Uub,Uuc)) ) ) ).

% ATP.lambda_704
tff(fact_8707_ATP_Olambda__705,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),$o)),aTP_Lamp_og(fun(A,fun(A,$o)),fun(fun(list(A),fun(list(A),$o)),fun(list(A),fun(list(A),$o))),Uu),Uua),Uub),Uuc)
    <=> ( ? [Y4: A,Ys4: list(A)] :
            ( ( Uub = nil(A) )
            & ( Uuc = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys4) ) )
        | ? [X6: A,Y4: A,Xs2: list(A),Ys4: list(A)] :
            ( ( Uub = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X6),Xs2) )
            & ( Uuc = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys4) )
            & aa(A,$o,aa(A,fun(A,$o),Uu,X6),Y4) )
        | ? [X6: A,Y4: A,Xs2: list(A),Ys4: list(A)] :
            ( ( Uub = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X6),Xs2) )
            & ( Uuc = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys4) )
            & ~ aa(A,$o,aa(A,fun(A,$o),Uu,X6),Y4)
            & ~ aa(A,$o,aa(A,fun(A,$o),Uu,Y4),X6)
            & aa(list(A),$o,aa(list(A),fun(list(A),$o),Uua,Xs2),Ys4) ) ) ) ).

% ATP.lambda_705
tff(fact_8708_ATP_Olambda__706,axiom,
    ! [A: $tType,B: $tType,Uu: $o,Uua: fun(A,fun(B,assn)),Uub: A,Uuc: B] : ( aa(B,assn,aa(A,fun(B,assn),aa(fun(A,fun(B,assn)),fun(A,fun(B,assn)),aTP_Lamp_aq($o,fun(fun(A,fun(B,assn)),fun(A,fun(B,assn))),(Uu)),Uua),Uub),Uuc) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),pure_assn((Uu))),aa(B,assn,aa(A,fun(B,assn),Uua,Uub),Uuc)) ) ).

% ATP.lambda_706
tff(fact_8709_ATP_Olambda__707,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_zx(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),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Uua),Uuc),Uub)) ) ).

% ATP.lambda_707
tff(fact_8710_ATP_Olambda__708,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Uu: nat,Uua: fun(A,A),Uub: array(A),Uuc: heap_ext(product_unit)] : ( aa(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),aa(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),aa(fun(A,A),fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),aTP_Lamp_aei(nat,fun(fun(A,A),fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat))))),Uu),Uua),Uub),Uuc) = aa(product_prod(heap_ext(product_unit),nat),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),aa(array(A),fun(product_prod(heap_ext(product_unit),nat),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),product_Pair(array(A),product_prod(heap_ext(product_unit),nat)),Uub),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),array_update(A,Uub,Uu,aa(A,A,Uua,aa(nat,A,nth(A,array_get(A,Uuc,Uub)),Uu)),Uuc)),numeral_numeral(nat,bit0(one2)))) ) ) ).

% ATP.lambda_708
tff(fact_8711_ATP_Olambda__709,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Uu: nat,Uua: A,Uub: array(A),Uuc: heap_ext(product_unit)] : ( aa(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),aa(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),aa(A,fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),aTP_Lamp_aed(nat,fun(A,fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat))))),Uu),Uua),Uub),Uuc) = aa(product_prod(heap_ext(product_unit),nat),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),aa(array(A),fun(product_prod(heap_ext(product_unit),nat),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),product_Pair(array(A),product_prod(heap_ext(product_unit),nat)),Uub),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),array_update(A,Uub,Uu,Uua,Uuc)),one_one(nat))) ) ) ).

% ATP.lambda_709
tff(fact_8712_ATP_Olambda__710,axiom,
    ! [B: $tType,A: $tType,E: $tType,Uu: fun(E,A),Uua: fun(E,B),Uub: set(E),Uuc: A] : ( aa(A,set(B),aa(set(E),fun(A,set(B)),aa(fun(E,B),fun(set(E),fun(A,set(B))),aTP_Lamp_qu(fun(E,A),fun(fun(E,B),fun(set(E),fun(A,set(B)))),Uu),Uua),Uub),Uuc) = aa(set(E),set(B),image2(E,B,Uua),aa(set(E),set(E),aa(set(E),fun(set(E),set(E)),inf_inf(set(E)),vimage(E,A,Uu,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Uuc),bot_bot(set(A))))),Uub)) ) ).

% ATP.lambda_710
tff(fact_8713_ATP_Olambda__711,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_acr(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_711
tff(fact_8714_ATP_Olambda__712,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_bp(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_712
tff(fact_8715_ATP_Olambda__713,axiom,
    ! [A: $tType,B: $tType,E: $tType,Uu: fun(B,A),Uua: fun(E,fun(list(E),B)),Uub: E,Uuc: list(E)] : ( aa(list(E),A,aa(E,fun(list(E),A),aa(fun(E,fun(list(E),B)),fun(E,fun(list(E),A)),aTP_Lamp_cj(fun(B,A),fun(fun(E,fun(list(E),B)),fun(E,fun(list(E),A))),Uu),Uua),Uub),Uuc) = aa(B,A,Uu,aa(list(E),B,aa(E,fun(list(E),B),Uua,Uub),Uuc)) ) ).

% ATP.lambda_713
tff(fact_8716_ATP_Olambda__714,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,$o)),aTP_Lamp_kd(fun(A,$o),fun(nat,fun(list(A),fun(nat,$o))),Uu),Uua),Uub),Uuc)
    <=> aa(A,$o,Uu,aa(nat,A,nth(A,aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),Uua),Uub)),Uuc)) ) ).

% ATP.lambda_714
tff(fact_8717_ATP_Olambda__715,axiom,
    ! [A: $tType,F4: $tType,B: $tType,E: $tType,Uu: fun(product_prod(B,E),A),Uua: fun(F4,B),Uub: F4,Uuc: E] : ( aa(E,A,aa(F4,fun(E,A),aa(fun(F4,B),fun(F4,fun(E,A)),aTP_Lamp_hk(fun(product_prod(B,E),A),fun(fun(F4,B),fun(F4,fun(E,A))),Uu),Uua),Uub),Uuc) = aa(product_prod(B,E),A,Uu,aa(E,product_prod(B,E),aa(B,fun(E,product_prod(B,E)),product_Pair(B,E),aa(F4,B,Uua,Uub)),Uuc)) ) ).

% ATP.lambda_715
tff(fact_8718_ATP_Olambda__716,axiom,
    ! [A: $tType,B: $tType,E: $tType,F4: $tType,Uu: fun(product_prod(B,E),A),Uua: fun(F4,E),Uub: B,Uuc: F4] : ( aa(F4,A,aa(B,fun(F4,A),aa(fun(F4,E),fun(B,fun(F4,A)),aTP_Lamp_hj(fun(product_prod(B,E),A),fun(fun(F4,E),fun(B,fun(F4,A))),Uu),Uua),Uub),Uuc) = aa(product_prod(B,E),A,Uu,aa(E,product_prod(B,E),aa(B,fun(E,product_prod(B,E)),product_Pair(B,E),Uub),aa(F4,E,Uua,Uuc))) ) ).

% ATP.lambda_716
tff(fact_8719_ATP_Olambda__717,axiom,
    ! [E: $tType,A: $tType,B: $tType,Uu: fun(E,fun(B,A)),Uua: fun(E,E),Uub: fun(fun(B,A),E),Uuc: fun(B,A)] : ( aa(fun(B,A),fun(B,A),aa(fun(fun(B,A),E),fun(fun(B,A),fun(B,A)),aa(fun(E,E),fun(fun(fun(B,A),E),fun(fun(B,A),fun(B,A))),aTP_Lamp_acf(fun(E,fun(B,A)),fun(fun(E,E),fun(fun(fun(B,A),E),fun(fun(B,A),fun(B,A)))),Uu),Uua),Uub),Uuc) = aa(E,fun(B,A),Uu,aa(E,E,Uua,aa(fun(B,A),E,Uub,Uuc))) ) ).

% ATP.lambda_717
tff(fact_8720_ATP_Olambda__718,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_kj(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_ki(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_718
tff(fact_8721_ATP_Olambda__719,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_hg(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,$o)))),aTP_Lamp_hf(set(product_prod(A,A)),fun(set(product_prod(B,B)),fun(A,fun(B,fun(A,fun(B,$o))))),Uu),Uua),Uub),Uuc)) ) ).

% ATP.lambda_719
tff(fact_8722_ATP_Olambda__720,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),$o))),aTP_Lamp_he(fun(A,$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,$o)))),aTP_Lamp_hd(fun(A,$o),fun(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(A,fun(B,$o))))),Uu),Uua),Uub),Uuc)) ) ).

% ATP.lambda_720
tff(fact_8723_ATP_Olambda__721,axiom,
    ! [B: $tType,A: $tType,E: $tType] :
      ( ( semiring_0(A)
        & comm_monoid_add(B)
        & times(B) )
     => ! [Uu: fun(B,A),Uua: fun(E,A),Uub: multiset(E),Uuc: B] : ( aa(B,A,aa(multiset(E),fun(B,A),aa(fun(E,A),fun(multiset(E),fun(B,A)),aTP_Lamp_sn(fun(B,A),fun(fun(E,A),fun(multiset(E),fun(B,A))),Uu),Uua),Uub),Uuc) = comm_m7189776963980413722m_mset(A,image_mset(E,A,aa(B,fun(E,A),aa(fun(E,A),fun(B,fun(E,A)),aTP_Lamp_sm(fun(B,A),fun(fun(E,A),fun(B,fun(E,A))),Uu),Uua),Uuc),Uub)) ) ) ).

% ATP.lambda_721
tff(fact_8724_ATP_Olambda__722,axiom,
    ! [A: $tType,E: $tType,F4: $tType,B: $tType,Uu: set(B),Uua: fun(A,filter(E)),Uub: fun(B,filter(F4)),Uuc: A] : ( aa(A,filter(product_prod(E,F4)),aa(fun(B,filter(F4)),fun(A,filter(product_prod(E,F4))),aa(fun(A,filter(E)),fun(fun(B,filter(F4)),fun(A,filter(product_prod(E,F4)))),aTP_Lamp_yr(set(B),fun(fun(A,filter(E)),fun(fun(B,filter(F4)),fun(A,filter(product_prod(E,F4))))),Uu),Uua),Uub),Uuc) = aa(set(filter(product_prod(E,F4))),filter(product_prod(E,F4)),complete_Inf_Inf(filter(product_prod(E,F4))),aa(set(B),set(filter(product_prod(E,F4))),image2(B,filter(product_prod(E,F4)),aa(A,fun(B,filter(product_prod(E,F4))),aa(fun(B,filter(F4)),fun(A,fun(B,filter(product_prod(E,F4)))),aTP_Lamp_yq(fun(A,filter(E)),fun(fun(B,filter(F4)),fun(A,fun(B,filter(product_prod(E,F4))))),Uua),Uub),Uuc)),Uu)) ) ).

% ATP.lambda_722
tff(fact_8725_ATP_Olambda__723,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_aan(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_723
tff(fact_8726_ATP_Olambda__724,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_aar(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_724
tff(fact_8727_ATP_Olambda__725,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_uz(fun(A,$o),fun(list(A),fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A))))))),Uu),Uua),Uub),Uuc) = aa(product_prod(list(A),product_prod(A,list(A))),option(product_prod(list(A),product_prod(A,list(A)))),some(product_prod(list(A),product_prod(A,list(A)))),aa(product_prod(A,list(A)),product_prod(list(A),product_prod(A,list(A))),aa(list(A),fun(product_prod(A,list(A)),product_prod(list(A),product_prod(A,list(A)))),product_Pair(list(A),product_prod(A,list(A))),aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),takeWhile(A),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_725
tff(fact_8728_ATP_Olambda__726,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_va(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_726
tff(fact_8729_ATP_Olambda__727,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_aap(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_727
tff(fact_8730_ATP_Olambda__728,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_aal(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_728
tff(fact_8731_ATP_Olambda__729,axiom,
    ! [A: $tType,E: $tType,B: $tType,Uu: set(product_prod(B,B)),Uua: fun(A,fun(B,set(E))),Uub: B,Uuc: A] : ( aa(A,set(E),aa(B,fun(A,set(E)),aa(fun(A,fun(B,set(E))),fun(B,fun(A,set(E))),aTP_Lamp_xk(set(product_prod(B,B)),fun(fun(A,fun(B,set(E))),fun(B,fun(A,set(E)))),Uu),Uua),Uub),Uuc) = aa(set(set(E)),set(E),complete_Sup_Sup(set(E)),aa(set(B),set(set(E)),image2(B,set(E),aa(A,fun(B,set(E)),Uua,Uuc)),image(B,B,Uu,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert3(B),Uub),bot_bot(set(B)))))) ) ).

% ATP.lambda_729
tff(fact_8732_ATP_Olambda__730,axiom,
    ! [A: $tType,B: $tType,E: $tType,Uu: set(product_prod(A,E)),Uua: set(product_prod(E,B)),Uub: A,Uuc: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(set(product_prod(E,B)),fun(A,fun(B,$o)),aTP_Lamp_pj(set(product_prod(A,E)),fun(set(product_prod(E,B)),fun(A,fun(B,$o))),Uu),Uua),Uub),Uuc)
    <=> ? [Y4: E] :
          ( member(product_prod(A,E),aa(E,product_prod(A,E),aa(A,fun(E,product_prod(A,E)),product_Pair(A,E),Uub),Y4),Uu)
          & member(product_prod(E,B),aa(B,product_prod(E,B),aa(E,fun(B,product_prod(E,B)),product_Pair(E,B),Y4),Uuc),Uua) ) ) ).

% ATP.lambda_730
tff(fact_8733_ATP_Olambda__731,axiom,
    ! [B: $tType,A: $tType,E: $tType,Uu: set(E),Uua: fun(E,A),Uub: fun(E,B),Uuc: product_prod(A,B)] :
      ( aa(product_prod(A,B),$o,aa(fun(E,B),fun(product_prod(A,B),$o),aa(fun(E,A),fun(fun(E,B),fun(product_prod(A,B),$o)),aTP_Lamp_mb(set(E),fun(fun(E,A),fun(fun(E,B),fun(product_prod(A,B),$o))),Uu),Uua),Uub),Uuc)
    <=> ? [A7: E] :
          ( ( Uuc = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(E,A,Uua,A7)),aa(E,B,Uub,A7)) )
          & member(E,A7,Uu) ) ) ).

% ATP.lambda_731
tff(fact_8734_ATP_Olambda__732,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_lg(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 )
            | member(B,L4,Uub) ) ) ) ).

% ATP.lambda_732
tff(fact_8735_ATP_Olambda__733,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_lp(A,fun(A,fun(fun(A,B),fun(B,$o))),Uu),Uua),Uub),Uuc)
        <=> ? [I2: A] :
              ( ( Uuc = aa(A,B,Uub,I2) )
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uu),I2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),I2),Uua) ) ) ) ).

% ATP.lambda_733
tff(fact_8736_ATP_Olambda__734,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_abe(set(A),fun(set(product_prod(B,B)),fun(fun(A,B),fun(product_prod(A,A),$o))),Uu),Uua),Uub),Uuc)
    <=> ? [A16: A,A24: A] :
          ( ( Uuc = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A16),A24) )
          & member(A,A16,Uu)
          & member(A,A24,Uu)
          & 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,A16)),aa(A,B,Uub,A24)),Uua) ) ) ).

% ATP.lambda_734
tff(fact_8737_ATP_Olambda__735,axiom,
    ! [A: $tType,E: $tType,B: $tType,Uu: fun(A,fun(A,A)),Uua: A,Uub: fun(B,set(E)),Uuc: fun(E,A),Uud: B] : ( aa(B,A,aa(fun(E,A),fun(B,A),aa(fun(B,set(E)),fun(fun(E,A),fun(B,A)),aa(A,fun(fun(B,set(E)),fun(fun(E,A),fun(B,A))),aTP_Lamp_acq(fun(A,fun(A,A)),fun(A,fun(fun(B,set(E)),fun(fun(E,A),fun(B,A)))),Uu),Uua),Uub),Uuc),Uud) = aa(set(E),A,aa(fun(E,A),fun(set(E),A),groups_comm_monoid_F(A,E,Uu,Uua),Uuc),aa(B,set(E),Uub,Uud)) ) ).

% ATP.lambda_735
tff(fact_8738_ATP_Olambda__736,axiom,
    ! [A: $tType,B: $tType,E: $tType,Uu: E,Uua: A,Uub: A,Uuc: B,Uud: set(product_prod(E,B))] :
      ( aa(set(product_prod(E,B)),set(product_prod(E,B)),aa(B,fun(set(product_prod(E,B)),set(product_prod(E,B))),aa(A,fun(B,fun(set(product_prod(E,B)),set(product_prod(E,B)))),aa(A,fun(A,fun(B,fun(set(product_prod(E,B)),set(product_prod(E,B))))),aTP_Lamp_pn(E,fun(A,fun(A,fun(B,fun(set(product_prod(E,B)),set(product_prod(E,B)))))),Uu),Uua),Uub),Uuc),Uud) = $ite(Uua = Uub,aa(set(product_prod(E,B)),set(product_prod(E,B)),aa(product_prod(E,B),fun(set(product_prod(E,B)),set(product_prod(E,B))),insert3(product_prod(E,B)),aa(B,product_prod(E,B),aa(E,fun(B,product_prod(E,B)),product_Pair(E,B),Uu),Uuc)),Uud),Uud) ) ).

% ATP.lambda_736
tff(fact_8739_ATP_Olambda__737,axiom,
    ! [B: $tType,E: $tType,A: $tType,Uu: A,Uua: B,Uub: B,Uuc: E,Uud: set(product_prod(A,E))] :
      ( aa(set(product_prod(A,E)),set(product_prod(A,E)),aa(E,fun(set(product_prod(A,E)),set(product_prod(A,E))),aa(B,fun(E,fun(set(product_prod(A,E)),set(product_prod(A,E)))),aa(B,fun(B,fun(E,fun(set(product_prod(A,E)),set(product_prod(A,E))))),aTP_Lamp_pk(A,fun(B,fun(B,fun(E,fun(set(product_prod(A,E)),set(product_prod(A,E)))))),Uu),Uua),Uub),Uuc),Uud) = $ite(Uua = Uub,aa(set(product_prod(A,E)),set(product_prod(A,E)),aa(product_prod(A,E),fun(set(product_prod(A,E)),set(product_prod(A,E))),insert3(product_prod(A,E)),aa(E,product_prod(A,E),aa(A,fun(E,product_prod(A,E)),product_Pair(A,E),Uu),Uuc)),Uud),Uud) ) ).

% ATP.lambda_737
tff(fact_8740_ATP_Olambda__738,axiom,
    ! [A: $tType,Uu: A,Uua: nat,Uub: fun(nat,A),Uuc: fun(nat,A),Uud: nat] :
      ( aa(nat,A,aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aa(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),aTP_Lamp_ack(A,fun(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A)))),Uu),Uua),Uub),Uuc),Uud) = $ite(
          aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uud),Uua),
          aa(nat,A,Uub,Uud),
          $ite(Uud = Uua,Uu,aa(nat,A,Uuc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uud),aa(nat,nat,suc,zero_zero(nat))))) ) ) ).

% ATP.lambda_738
tff(fact_8741_ATP_Olambda__739,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_nw(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_739
tff(fact_8742_ATP_Olambda__740,axiom,
    ! [A: $tType,E: $tType,B: $tType,Uu: fun(E,fun(B,A)),Uua: fun(E,E),Uub: fun(fun(B,A),E),Uuc: B,Uud: fun(B,A)] : ( aa(fun(B,A),A,aa(B,fun(fun(B,A),A),aa(fun(fun(B,A),E),fun(B,fun(fun(B,A),A)),aa(fun(E,E),fun(fun(fun(B,A),E),fun(B,fun(fun(B,A),A))),aTP_Lamp_ace(fun(E,fun(B,A)),fun(fun(E,E),fun(fun(fun(B,A),E),fun(B,fun(fun(B,A),A)))),Uu),Uua),Uub),Uuc),Uud) = aa(B,A,aa(E,fun(B,A),Uu,aa(E,E,Uua,aa(fun(B,A),E,Uub,Uud))),Uuc) ) ).

% ATP.lambda_740
tff(fact_8743_ATP_Olambda__741,axiom,
    ! [B: $tType,A: $tType,E: $tType,Uu: fun(A,fun(E,A)),Uua: fun(A,fun(B,fun(E,B))),Uub: A,Uuc: B,Uud: E] : ( aa(E,product_prod(A,B),aa(B,fun(E,product_prod(A,B)),aa(A,fun(B,fun(E,product_prod(A,B))),aa(fun(A,fun(B,fun(E,B))),fun(A,fun(B,fun(E,product_prod(A,B)))),aTP_Lamp_os(fun(A,fun(E,A)),fun(fun(A,fun(B,fun(E,B))),fun(A,fun(B,fun(E,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(E,A,aa(A,fun(E,A),Uu,Uub),Uud)),aa(E,B,aa(B,fun(E,B),aa(A,fun(B,fun(E,B)),Uua,Uub),Uuc),Uud)) ) ).

% ATP.lambda_741
tff(fact_8744_ATP_Olambda__742,axiom,
    ! [A: $tType,Uu: fun(A,fun(A,$o)),Uua: A,Uub: $o,Uuc: A,Uud: list(A)] :
      ( aa(list(A),$o,aa(A,fun(list(A),$o),aa($o,fun(A,fun(list(A),$o)),aa(A,fun($o,fun(A,fun(list(A),$o))),aTP_Lamp_aba(fun(A,fun(A,$o)),fun(A,fun($o,fun(A,fun(list(A),$o)))),Uu),Uua),(Uub)),Uuc),Uud)
    <=> ( aa(A,$o,aa(A,fun(A,$o),Uu,Uua),Uuc)
        & (Uub) ) ) ).

% ATP.lambda_742
tff(fact_8745_ATP_Olambda__743,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_gh(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_743
tff(fact_8746_ATP_Olambda__744,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_ge(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_744
tff(fact_8747_ATP_Olambda__745,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,$o))),aTP_Lamp_ia(A,fun(A,fun(set(product_prod(A,A)),fun(A,fun(A,$o)))),Uu),Uua),Uub),Uuc),Uud)
    <=> ( ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uuc),Uu),transitive_trancl(A,Uub))
          | ( Uuc = Uu ) )
        & ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uua),Uud),transitive_trancl(A,Uub))
          | ( Uud = Uua ) ) ) ) ).

% ATP.lambda_745
tff(fact_8748_ATP_Olambda__746,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_gf(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_746
tff(fact_8749_ATP_Olambda__747,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,$o))),aTP_Lamp_iv(set(product_prod(A,A)),fun(A,fun(A,fun(A,fun(A,$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)),insert3(A),Uua),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Uub),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Uuc),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Uud),bot_bot(set(A))))))),aa(set(product_prod(A,A)),set(A),field2(A),Uu))
        & ( ( ( Uua = Uuc )
            & ( Uub = Uud ) )
          | member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),bNF_We1388413361240627857o_max2(A,Uu,Uua,Uub)),bNF_We1388413361240627857o_max2(A,Uu,Uuc,Uud)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),minus_minus(set(product_prod(A,A))),Uu),id2(A)))
          | ( ( bNF_We1388413361240627857o_max2(A,Uu,Uua,Uub) = bNF_We1388413361240627857o_max2(A,Uu,Uuc,Uud) )
            & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uua),Uuc),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),minus_minus(set(product_prod(A,A))),Uu),id2(A))) )
          | ( ( bNF_We1388413361240627857o_max2(A,Uu,Uua,Uub) = bNF_We1388413361240627857o_max2(A,Uu,Uuc,Uud) )
            & ( Uua = Uuc )
            & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),Uud),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),minus_minus(set(product_prod(A,A))),Uu),id2(A))) ) ) ) ) ).

% ATP.lambda_747
tff(fact_8750_ATP_Olambda__748,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,$o))),aTP_Lamp_qr(A,fun(A,fun(set(product_prod(A,A)),fun(A,fun(A,$o)))),Uu),Uua),Uub),Uuc),Uud)
    <=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uuc),Uu),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),Uub))
        & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uua),Uud),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),Uub)) ) ) ).

% ATP.lambda_748
tff(fact_8751_ATP_Olambda__749,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Uu: nat,Uua: list(A),Uub: array(A),Uuc: A,Uud: array(A)] : ( aa(array(A),assn,aa(A,fun(array(A),assn),aa(array(A),fun(A,fun(array(A),assn)),aa(list(A),fun(array(A),fun(A,fun(array(A),assn))),aTP_Lamp_ade(nat,fun(list(A),fun(array(A),fun(A,fun(array(A),assn)))),Uu),Uua),Uub),Uuc),Uud) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),snga_assn(A,Uub,aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Uua),Uu),Uuc))),pure_assn(Uud = Uub)) ) ) ).

% ATP.lambda_749
tff(fact_8752_ATP_Olambda__750,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,$o))),aTP_Lamp_mg(set(product_prod(A,A)),fun(A,fun(A,fun(A,fun(A,$o)))),Uu),Uua),Uub),Uuc),Uud)
    <=> ( ( Uub = Uuc )
       => 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_750
tff(fact_8753_ATP_Olambda__751,axiom,
    ! [A: $tType,Uu: heap_Time_Heap(A),Uua: heap_ext(product_unit),Uub: A,Uuc: heap_ext(product_unit),Uud: nat] :
      ( aa(nat,$o,aa(heap_ext(product_unit),fun(nat,$o),aa(A,fun(heap_ext(product_unit),fun(nat,$o)),aa(heap_ext(product_unit),fun(A,fun(heap_ext(product_unit),fun(nat,$o))),aTP_Lamp_afb(heap_Time_Heap(A),fun(heap_ext(product_unit),fun(A,fun(heap_ext(product_unit),fun(nat,$o)))),Uu),Uua),Uub),Uuc),Uud)
    <=> ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,Uu),Uua) = aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),Uub),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),Uuc),Uud))) ) ) ).

% ATP.lambda_751
tff(fact_8754_ATP_Olambda__752,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,fun(B,assn)),Uua: list(A),Uub: list(B),Uuc: nat,Uud: assn] : ( aa(assn,assn,aa(nat,fun(assn,assn),aa(list(B),fun(nat,fun(assn,assn)),aa(list(A),fun(list(B),fun(nat,fun(assn,assn))),aTP_Lamp_hb(fun(A,fun(B,assn)),fun(list(A),fun(list(B),fun(nat,fun(assn,assn)))),Uu),Uua),Uub),Uuc),Uud) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Uud),aa(B,assn,aa(A,fun(B,assn),Uu,aa(nat,A,nth(A,Uua),Uuc)),aa(nat,B,nth(B,Uub),Uuc))) ) ).

% ATP.lambda_752
tff(fact_8755_ATP_Olambda__753,axiom,
    ! [A: $tType,E: $tType,B: $tType] :
      ( semiring_0(E)
     => ! [Uu: fun(A,E),Uua: fun(B,E),Uub: set(A),Uuc: set(B),Uud: E] :
          ( aa(E,$o,aa(set(B),fun(E,$o),aa(set(A),fun(set(B),fun(E,$o)),aa(fun(B,E),fun(set(A),fun(set(B),fun(E,$o))),aTP_Lamp_tx(fun(A,E),fun(fun(B,E),fun(set(A),fun(set(B),fun(E,$o)))),Uu),Uua),Uub),Uuc),Uud)
        <=> ? [A7: A,B8: B] :
              ( ( Uud = aa(E,E,aa(E,fun(E,E),times_times(E),aa(A,E,Uu,A7)),aa(B,E,Uua,B8)) )
              & member(A,A7,Uub)
              & member(B,B8,Uuc) ) ) ) ).

% ATP.lambda_753
tff(fact_8756_ATP_Olambda__754,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_hf(set(product_prod(A,A)),fun(set(product_prod(B,B)),fun(A,fun(B,fun(A,fun(B,$o))))),Uu),Uua),Uub),Uuc),Uud),Uue)
    <=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),Uud),Uu)
        | ( ( Uub = Uud )
          & member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Uuc),Uue),Uua) ) ) ) ).

% ATP.lambda_754
tff(fact_8757_ATP_Olambda__755,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,$o)))),aTP_Lamp_hd(fun(A,$o),fun(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(A,fun(B,$o))))),Uu),Uua),Uub),Uuc),Uud),Uue)
    <=> ( ( Uub = Uud )
        & aa(A,$o,Uu,Uud)
        & member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Uuc),Uue),aa(A,set(product_prod(B,B)),Uua,Uud)) ) ) ).

% ATP.lambda_755
tff(fact_8758_ATP_Olambda__756,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_ki(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(
          x2: b,
          x2:= aa(a,b,Uu,Uub),
          $ite(
            aa(b,$o,aa(b,fun(b,$o),ord_less(b),x2),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),x2),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_756
tff(fact_8759_ATP_Olambda__757,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,fun(B,assn)),Uua: A,Uub: list(A),Uuc: B,Uud: list(B),Uue: nat,Uuf: assn] : ( aa(assn,assn,aa(nat,fun(assn,assn),aa(list(B),fun(nat,fun(assn,assn)),aa(B,fun(list(B),fun(nat,fun(assn,assn))),aa(list(A),fun(B,fun(list(B),fun(nat,fun(assn,assn)))),aa(A,fun(list(A),fun(B,fun(list(B),fun(nat,fun(assn,assn))))),aTP_Lamp_id(fun(A,fun(B,assn)),fun(A,fun(list(A),fun(B,fun(list(B),fun(nat,fun(assn,assn)))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Uuf),aa(B,assn,aa(A,fun(B,assn),Uu,aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uua),Uub)),Uue)),aa(nat,B,nth(B,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Uuc),Uud)),Uue))) ) ).

% ATP.lambda_757
tff(fact_8760_ATP_Olambda__758,axiom,
    ! [A: $tType,Uu: heap_Time_Heap(A),Uua: product_unit] : ( aa(product_unit,heap_Time_Heap(A),aTP_Lamp_adz(heap_Time_Heap(A),fun(product_unit,heap_Time_Heap(A)),Uu),Uua) = Uu ) ).

% ATP.lambda_758
tff(fact_8761_ATP_Olambda__759,axiom,
    ! [A: $tType,B: $tType,Uu: multiset(B),Uua: A] : ( aa(A,multiset(B),aTP_Lamp_ap(multiset(B),fun(A,multiset(B)),Uu),Uua) = Uu ) ).

% ATP.lambda_759
tff(fact_8762_ATP_Olambda__760,axiom,
    ! [Uu: assn,Uua: product_unit] : ( aa(product_unit,assn,aTP_Lamp_adm(assn,fun(product_unit,assn),Uu),Uua) = Uu ) ).

% ATP.lambda_760
tff(fact_8763_ATP_Olambda__761,axiom,
    ! [A: $tType,Uu: assn,Uua: A] : ( aa(A,assn,aTP_Lamp_zn(assn,fun(A,assn),Uu),Uua) = Uu ) ).

% ATP.lambda_761
tff(fact_8764_ATP_Olambda__762,axiom,
    ! [A: $tType,Uu: $o,Uua: A] :
      ( aa(A,$o,aTP_Lamp_ae($o,fun(A,$o),(Uu)),Uua)
    <=> (Uu) ) ).

% ATP.lambda_762
tff(fact_8765_ATP_Olambda__763,axiom,
    ! [E: $tType,F4: $tType,Uu: set(F4),Uua: E] : ( aa(E,set(F4),aTP_Lamp_qi(set(F4),fun(E,set(F4)),Uu),Uua) = Uu ) ).

% ATP.lambda_763
tff(fact_8766_ATP_Olambda__764,axiom,
    ! [B: $tType,E: $tType,Uu: set(E),Uua: B] : ( aa(B,set(E),aTP_Lamp_qe(set(E),fun(B,set(E)),Uu),Uua) = Uu ) ).

% ATP.lambda_764
tff(fact_8767_ATP_Olambda__765,axiom,
    ! [A: $tType,E: $tType,Uu: set(E),Uua: A] : ( aa(A,set(E),aTP_Lamp_qf(set(E),fun(A,set(E)),Uu),Uua) = Uu ) ).

% ATP.lambda_765
tff(fact_8768_ATP_Olambda__766,axiom,
    ! [E: $tType,B: $tType,Uu: set(B),Uua: E] : ( aa(E,set(B),aTP_Lamp_yz(set(B),fun(E,set(B)),Uu),Uua) = Uu ) ).

% ATP.lambda_766
tff(fact_8769_ATP_Olambda__767,axiom,
    ! [B: $tType,Uu: set(B),Uua: B] : ( aa(B,set(B),aTP_Lamp_py(set(B),fun(B,set(B)),Uu),Uua) = Uu ) ).

% ATP.lambda_767
tff(fact_8770_ATP_Olambda__768,axiom,
    ! [A: $tType,B: $tType,Uu: set(B),Uua: A] : ( aa(A,set(B),aTP_Lamp_pv(set(B),fun(A,set(B)),Uu),Uua) = Uu ) ).

% ATP.lambda_768
tff(fact_8771_ATP_Olambda__769,axiom,
    ! [A: $tType,Uu: set(A),Uua: list(A)] : ( aa(list(A),set(A),aTP_Lamp_qp(set(A),fun(list(A),set(A)),Uu),Uua) = Uu ) ).

% ATP.lambda_769
tff(fact_8772_ATP_Olambda__770,axiom,
    ! [E: $tType,A: $tType,Uu: set(A),Uua: E] : ( aa(E,set(A),aTP_Lamp_yy(set(A),fun(E,set(A)),Uu),Uua) = Uu ) ).

% ATP.lambda_770
tff(fact_8773_ATP_Olambda__771,axiom,
    ! [B: $tType,A: $tType,Uu: set(A),Uua: B] : ( aa(B,set(A),aTP_Lamp_an(set(A),fun(B,set(A)),Uu),Uua) = Uu ) ).

% ATP.lambda_771
tff(fact_8774_ATP_Olambda__772,axiom,
    ! [A: $tType,Uu: set(A),Uua: A] : ( aa(A,set(A),aTP_Lamp_pt(set(A),fun(A,set(A)),Uu),Uua) = Uu ) ).

% ATP.lambda_772
tff(fact_8775_ATP_Olambda__773,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,$o),Uua: A] : ( aa(A,fun(B,$o),aTP_Lamp_ym(fun(B,$o),fun(A,fun(B,$o)),Uu),Uua) = Uu ) ).

% ATP.lambda_773
tff(fact_8776_ATP_Olambda__774,axiom,
    ! [A: $tType,E: $tType,B: $tType,Uu: fun(B,E),Uua: A] : ( aa(A,fun(B,E),aTP_Lamp_ov(fun(B,E),fun(A,fun(B,E)),Uu),Uua) = Uu ) ).

% ATP.lambda_774
tff(fact_8777_ATP_Olambda__775,axiom,
    ! [B: $tType,E: $tType,A: $tType,Uu: fun(A,fun(E,E)),Uua: B] : ( aa(B,fun(A,fun(E,E)),aTP_Lamp_kt(fun(A,fun(E,E)),fun(B,fun(A,fun(E,E))),Uu),Uua) = Uu ) ).

% ATP.lambda_775
tff(fact_8778_ATP_Olambda__776,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [Uu: B,Uua: A] : ( aa(A,B,aTP_Lamp_ah(B,fun(A,B),Uu),Uua) = Uu ) ) ).

% ATP.lambda_776
tff(fact_8779_ATP_Olambda__777,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [Uu: B,Uua: A] : ( aa(A,B,aTP_Lamp_ai(B,fun(A,B),Uu),Uua) = Uu ) ) ).

% ATP.lambda_777
tff(fact_8780_ATP_Olambda__778,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(B)
     => ! [Uu: B,Uua: A] : ( aa(A,B,aTP_Lamp_ax(B,fun(A,B),Uu),Uua) = Uu ) ) ).

% ATP.lambda_778
tff(fact_8781_ATP_Olambda__779,axiom,
    ! [A: $tType,B: $tType,Uu: B,Uua: A] : ( aa(A,B,aTP_Lamp_bd(B,fun(A,B),Uu),Uua) = Uu ) ).

% ATP.lambda_779
tff(fact_8782_ATP_Olambda__780,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: A,Uua: B] : ( aa(B,A,aTP_Lamp_de(A,fun(B,A),Uu),Uua) = Uu ) ) ).

% ATP.lambda_780
tff(fact_8783_ATP_Olambda__781,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A,Uua: list(A)] : ( aa(list(A),A,aa(A,fun(list(A),A),aTP_Lamp_kb(A,fun(list(A),A)),Uu),Uua) = Uu ) ) ).

% ATP.lambda_781
tff(fact_8784_ATP_Olambda__782,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(A)
     => ! [Uu: A,Uua: B] : ( aa(B,A,aTP_Lamp_ey(A,fun(B,A),Uu),Uua) = Uu ) ) ).

% ATP.lambda_782
tff(fact_8785_ATP_Olambda__783,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] : ( aa(list(A),A,aa(A,fun(list(A),A),aTP_Lamp_mw(A,fun(list(A),A)),Uu),Uua) = Uu ) ).

% ATP.lambda_783
tff(fact_8786_ATP_Olambda__784,axiom,
    ! [A: $tType,Uu: A,Uua: nat] : ( aa(nat,A,aTP_Lamp_oe(A,fun(nat,A),Uu),Uua) = Uu ) ).

% ATP.lambda_784
tff(fact_8787_ATP_Olambda__785,axiom,
    ! [B: $tType,A: $tType,Uu: A,Uua: B] : ( aa(B,A,aTP_Lamp_as(A,fun(B,A),Uu),Uua) = Uu ) ).

% ATP.lambda_785
tff(fact_8788_ATP_Olambda__786,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] : ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),aTP_Lamp_cb(A,fun(list(A),list(A))),Uu),Uua) = Uua ) ).

% ATP.lambda_786
tff(fact_8789_ATP_Olambda__787,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] :
      ( aa(list(A),$o,aa(A,fun(list(A),$o),aTP_Lamp_ca(A,fun(list(A),$o)),Uu),Uua)
    <=> $false ) ).

% ATP.lambda_787
tff(fact_8790_ATP_Olambda__788,axiom,
    ! [A: $tType,Uu: A,Uua: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_aaw(A,fun(A,$o)),Uu),Uua)
    <=> $false ) ).

% ATP.lambda_788
tff(fact_8791_ATP_Olambda__789,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] :
      ( aa(list(A),$o,aa(A,fun(list(A),$o),aTP_Lamp_bz(A,fun(list(A),$o)),Uu),Uua)
    <=> $true ) ).

% ATP.lambda_789
tff(fact_8792_ATP_Olambda__790,axiom,
    ! [A: $tType,Uu: A,Uua: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_cm(A,fun(A,$o)),Uu),Uua)
    <=> $true ) ).

% ATP.lambda_790
tff(fact_8793_ATP_Olambda__791,axiom,
    ! [Uu: nat] : ( aa(nat,nat,aTP_Lamp_et(nat,nat),Uu) = Uu ) ).

% ATP.lambda_791
tff(fact_8794_ATP_Olambda__792,axiom,
    ! [Uu: int] : ( aa(int,int,aTP_Lamp_jn(int,int),Uu) = Uu ) ).

% ATP.lambda_792
tff(fact_8795_ATP_Olambda__793,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: A] : ( aa(A,A,aTP_Lamp_fk(A,A),Uu) = Uu ) ) ).

% ATP.lambda_793
tff(fact_8796_ATP_Olambda__794,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A] : ( aa(A,A,aTP_Lamp_ip(A,A),Uu) = Uu ) ) ).

% ATP.lambda_794
tff(fact_8797_ATP_Olambda__795,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Uu: A] : ( aa(A,A,aTP_Lamp_ck(A,A),Uu) = Uu ) ) ).

% ATP.lambda_795
tff(fact_8798_ATP_Olambda__796,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [Uu: A] : ( aa(A,A,aTP_Lamp_abz(A,A),Uu) = Uu ) ) ).

% ATP.lambda_796
tff(fact_8799_ATP_Olambda__797,axiom,
    ! [A: $tType,Uu: A] : ( aa(A,A,aTP_Lamp_ab(A,A),Uu) = Uu ) ).

% ATP.lambda_797
tff(fact_8800_ATP_Olambda__798,axiom,
    ! [A: $tType,Uu: A] : ( aa(A,set($o),aTP_Lamp_zi(A,set($o)),Uu) = top_top(set($o)) ) ).

% ATP.lambda_798
tff(fact_8801_ATP_Olambda__799,axiom,
    ! [A: $tType,B: $tType,Uu: A] : ( aa(A,set(B),aTP_Lamp_pw(A,set(B)),Uu) = top_top(set(B)) ) ).

% ATP.lambda_799
tff(fact_8802_ATP_Olambda__800,axiom,
    ! [E: $tType,B: $tType,Uu: E] : ( aa(E,set(B),aTP_Lamp_nh(E,set(B)),Uu) = bot_bot(set(B)) ) ).

% ATP.lambda_800
tff(fact_8803_ATP_Olambda__801,axiom,
    ! [B: $tType,A: $tType,Uu: B] : ( aa(B,set(A),aTP_Lamp_cv(B,set(A)),Uu) = bot_bot(set(A)) ) ).

% ATP.lambda_801
tff(fact_8804_ATP_Olambda__802,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: B] : ( aa(B,A,aTP_Lamp_ag(B,A),Uu) = bot_bot(A) ) ) ).

% ATP.lambda_802
tff(fact_8805_ATP_Olambda__803,axiom,
    ! [A: $tType,F4: $tType,Uu: A] : ( aa(A,set(F4),aTP_Lamp_ni(A,set(F4)),Uu) = bot_bot(set(F4)) ) ).

% ATP.lambda_803
tff(fact_8806_ATP_Olambda__804,axiom,
    ! [A: $tType,B: $tType,Uu: A] : ( aa(A,set(B),aTP_Lamp_pu(A,set(B)),Uu) = bot_bot(set(B)) ) ).

% ATP.lambda_804
tff(fact_8807_ATP_Olambda__805,axiom,
    ! [A: $tType] :
      ( mult_zero(A)
     => ! [Uu: A] : ( aa(A,A,aTP_Lamp_cl(A,A),Uu) = zero_zero(A) ) ) ).

% ATP.lambda_805
tff(fact_8808_ATP_Olambda__806,axiom,
    ! [A: $tType,Uu: A] : ( aa(A,nat,aTP_Lamp_su(A,nat),Uu) = zero_zero(nat) ) ).

% ATP.lambda_806
tff(fact_8809_ATP_Olambda__807,axiom,
    ! [B: $tType,A: $tType,Uu: B] : ( aa(B,option(A),aTP_Lamp_ul(B,option(A)),Uu) = none(A) ) ).

% ATP.lambda_807
tff(fact_8810_ATP_Olambda__808,axiom,
    ! [A: $tType,B: $tType,Uu: A] : ( aa(A,option(B),aTP_Lamp_um(A,option(B)),Uu) = none(B) ) ).

% ATP.lambda_808
tff(fact_8811_ATP_Olambda__809,axiom,
    ! [A: $tType,B: $tType,Uu: A] : ( aa(A,B,aTP_Lamp_vm(A,B),Uu) = undefined(B) ) ).

% ATP.lambda_809
tff(fact_8812_ATP_Olambda__810,axiom,
    ! [Uu: product_prod(heap_ext(product_unit),set(nat))] :
      ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aTP_Lamp_abf(product_prod(heap_ext(product_unit),set(nat)),$o),Uu)
    <=> $false ) ).

% ATP.lambda_810
tff(fact_8813_ATP_Olambda__811,axiom,
    ! [Uu: nat] :
      ( aa(nat,$o,aTP_Lamp_wf(nat,$o),Uu)
    <=> $false ) ).

% ATP.lambda_811
tff(fact_8814_ATP_Olambda__812,axiom,
    ! [A: $tType,Uu: A] :
      ( aa(A,$o,aTP_Lamp_cq(A,$o),Uu)
    <=> $false ) ).

% ATP.lambda_812
tff(fact_8815_ATP_Olambda__813,axiom,
    ! [A: $tType,Uu: fun(A,$o)] :
      ( aa(fun(A,$o),$o,aTP_Lamp_wy(fun(A,$o),$o),Uu)
    <=> $true ) ).

% ATP.lambda_813
tff(fact_8816_ATP_Olambda__814,axiom,
    ! [A: $tType,Uu: A] :
      ( aa(A,$o,aTP_Lamp_cc(A,$o),Uu)
    <=> $true ) ).

% ATP.lambda_814
tff(fact_8817_ATP_Olambda__815,axiom,
    ! [A: $tType,Uu: A] : ( aa(A,fun(nat,nat),aTP_Lamp_kr(A,fun(nat,nat)),Uu) = suc ) ).

% ATP.lambda_815

% Type constructors (831)
tff(tcon_Heap__Time__Monad_OHeap___Code__Evaluation_Oterm__of,axiom,
    ! [A17: $tType] :
      ( typerep(A17)
     => code_term_of(heap_Time_Heap(A17)) ) ).

tff(tcon_Heap__Time__Monad_OHeap___Typerep_Otyperep,axiom,
    ! [A17: $tType] :
      ( typerep(A17)
     => typerep(heap_Time_Heap(A17)) ) ).

tff(tcon_Code__Numeral_Onatural___Code__Evaluation_Oterm__of_1,axiom,
    code_term_of(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Typerep_Otyperep_2,axiom,
    typerep(code_natural) ).

tff(tcon_Code__Numeral_Ointeger___Code__Evaluation_Oterm__of_3,axiom,
    code_term_of(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Typerep_Otyperep_4,axiom,
    typerep(code_integer) ).

tff(tcon_Code__Evaluation_Oterm___Code__Evaluation_Oterm__of_5,axiom,
    code_term_of(code_term) ).

tff(tcon_Code__Evaluation_Oterm___Typerep_Otyperep_6,axiom,
    typerep(code_term) ).

tff(tcon_Heap_Oheap_Oheap__ext___Code__Evaluation_Oterm__of_7,axiom,
    ! [A17: $tType] :
      ( typerep(A17)
     => code_term_of(heap_ext(A17)) ) ).

tff(tcon_Heap_Oheap_Oheap__ext___Typerep_Otyperep_8,axiom,
    ! [A17: $tType] :
      ( typerep(A17)
     => typerep(heap_ext(A17)) ) ).

tff(tcon_Product__Type_Ounit___Code__Evaluation_Oterm__of_9,axiom,
    code_term_of(product_unit) ).

tff(tcon_Product__Type_Ounit___Enum_Oenum,axiom,
    enum(product_unit) ).

tff(tcon_Product__Type_Ounit___Typerep_Otyperep_10,axiom,
    typerep(product_unit) ).

tff(tcon_Product__Type_Oprod___Code__Evaluation_Oterm__of_11,axiom,
    ! [A17: $tType,A18: $tType] :
      ( ( typerep(A17)
        & typerep(A18) )
     => code_term_of(product_prod(A17,A18)) ) ).

tff(tcon_Product__Type_Oprod___Enum_Oenum_12,axiom,
    ! [A17: $tType,A18: $tType] :
      ( ( enum(A17)
        & enum(A18) )
     => enum(product_prod(A17,A18)) ) ).

tff(tcon_Product__Type_Oprod___Typerep_Otyperep_13,axiom,
    ! [A17: $tType,A18: $tType] :
      ( ( typerep(A17)
        & typerep(A18) )
     => typerep(product_prod(A17,A18)) ) ).

tff(tcon_Multiset_Omultiset___Code__Evaluation_Oterm__of_14,axiom,
    ! [A17: $tType] :
      ( typerep(A17)
     => code_term_of(multiset(A17)) ) ).

tff(tcon_Multiset_Omultiset___Typerep_Otyperep_15,axiom,
    ! [A17: $tType] :
      ( typerep(A17)
     => typerep(multiset(A17)) ) ).

tff(tcon_Assertions_Oassn___Typerep_Otyperep_16,axiom,
    typerep(assn) ).

tff(tcon_Predicate_Opred___Code__Evaluation_Oterm__of_17,axiom,
    ! [A17: $tType] :
      ( typerep(A17)
     => code_term_of(pred(A17)) ) ).

tff(tcon_Predicate_Opred___Typerep_Otyperep_18,axiom,
    ! [A17: $tType] :
      ( typerep(A17)
     => typerep(pred(A17)) ) ).

tff(tcon_Predicate_Oseq___Code__Evaluation_Oterm__of_19,axiom,
    ! [A17: $tType] :
      ( typerep(A17)
     => code_term_of(seq(A17)) ) ).

tff(tcon_Predicate_Oseq___Typerep_Otyperep_20,axiom,
    ! [A17: $tType] :
      ( typerep(A17)
     => typerep(seq(A17)) ) ).

tff(tcon_Option_Ooption___Code__Evaluation_Oterm__of_21,axiom,
    ! [A17: $tType] :
      ( typerep(A17)
     => code_term_of(option(A17)) ) ).

tff(tcon_Option_Ooption___Enum_Oenum_22,axiom,
    ! [A17: $tType] :
      ( enum(A17)
     => enum(option(A17)) ) ).

tff(tcon_Option_Ooption___Typerep_Otyperep_23,axiom,
    ! [A17: $tType] :
      ( typerep(A17)
     => typerep(option(A17)) ) ).

tff(tcon_Filter_Ofilter___Code__Evaluation_Oterm__of_24,axiom,
    ! [A17: $tType] :
      ( typerep(A17)
     => code_term_of(filter(A17)) ) ).

tff(tcon_Filter_Ofilter___Typerep_Otyperep_25,axiom,
    ! [A17: $tType] :
      ( typerep(A17)
     => typerep(filter(A17)) ) ).

tff(tcon_Sum__Type_Osum___Code__Evaluation_Oterm__of_26,axiom,
    ! [A17: $tType,A18: $tType] :
      ( ( typerep(A17)
        & typerep(A18) )
     => code_term_of(sum_sum(A17,A18)) ) ).

tff(tcon_Sum__Type_Osum___Enum_Oenum_27,axiom,
    ! [A17: $tType,A18: $tType] :
      ( ( enum(A17)
        & enum(A18) )
     => enum(sum_sum(A17,A18)) ) ).

tff(tcon_Sum__Type_Osum___Typerep_Otyperep_28,axiom,
    ! [A17: $tType,A18: $tType] :
      ( ( typerep(A17)
        & typerep(A18) )
     => typerep(sum_sum(A17,A18)) ) ).

tff(tcon_Heap_Oarray___Code__Evaluation_Oterm__of_29,axiom,
    ! [A17: $tType] :
      ( typerep(A17)
     => code_term_of(array(A17)) ) ).

tff(tcon_Heap_Oarray___Typerep_Otyperep_30,axiom,
    ! [A17: $tType] : typerep(array(A17)) ).

tff(tcon_List_Olist___Code__Evaluation_Oterm__of_31,axiom,
    ! [A17: $tType] :
      ( typerep(A17)
     => code_term_of(list(A17)) ) ).

tff(tcon_List_Olist___Typerep_Otyperep_32,axiom,
    ! [A17: $tType] :
      ( typerep(A17)
     => typerep(list(A17)) ) ).

tff(tcon_Heap_Oref___Code__Evaluation_Oterm__of_33,axiom,
    ! [A17: $tType] :
      ( typerep(A17)
     => code_term_of(ref(A17)) ) ).

tff(tcon_Heap_Oref___Typerep_Otyperep_34,axiom,
    ! [A17: $tType] : typerep(ref(A17)) ).

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,
    ! [A17: $tType] :
      ( typerep(A17)
     => code_term_of(set(A17)) ) ).

tff(tcon_Set_Oset___Enum_Oenum_39,axiom,
    ! [A17: $tType] :
      ( enum(A17)
     => enum(set(A17)) ) ).

tff(tcon_Set_Oset___Typerep_Otyperep_40,axiom,
    ! [A17: $tType] :
      ( typerep(A17)
     => typerep(set(A17)) ) ).

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,
    ! [A17: $tType] :
      ( typerep(A17)
     => code_term_of(itself(A17)) ) ).

tff(tcon_itself___Typerep_Otyperep_50,axiom,
    ! [A17: $tType] :
      ( typerep(A17)
     => typerep(itself(A17)) ) ).

tff(tcon_fun___Code__Evaluation_Oterm__of_51,axiom,
    ! [A17: $tType,A18: $tType] :
      ( ( typerep(A17)
        & typerep(A18) )
     => code_term_of(fun(A17,A18)) ) ).

tff(tcon_fun___Enum_Oenum_52,axiom,
    ! [A17: $tType,A18: $tType] :
      ( ( enum(A17)
        & enum(A18) )
     => enum(fun(A17,A18)) ) ).

tff(tcon_fun___Typerep_Otyperep_53,axiom,
    ! [A17: $tType,A18: $tType] :
      ( ( typerep(A17)
        & typerep(A18) )
     => typerep(fun(A17,A18)) ) ).

tff(tcon_fun___Conditionally__Complete__Lattices_Oconditionally__complete__lattice,axiom,
    ! [A17: $tType,A18: $tType] :
      ( comple6319245703460814977attice(A18)
     => condit1219197933456340205attice(fun(A17,A18)) ) ).

tff(tcon_fun___Complete__Lattices_Ocomplete__distrib__lattice,axiom,
    ! [A17: $tType,A18: $tType] :
      ( comple592849572758109894attice(A18)
     => comple592849572758109894attice(fun(A17,A18)) ) ).

tff(tcon_fun___Quickcheck__Exhaustive_Ofull__exhaustive,axiom,
    ! [A17: $tType,A18: $tType] :
      ( ( cl_HOL_Oequal(A17)
        & quickc3360725361186068524ustive(A17)
        & quickc3360725361186068524ustive(A18) )
     => quickc3360725361186068524ustive(fun(A17,A18)) ) ).

tff(tcon_fun___Lattices_Obounded__semilattice__sup__bot,axiom,
    ! [A17: $tType,A18: $tType] :
      ( bounded_lattice(A18)
     => bounde4967611905675639751up_bot(fun(A17,A18)) ) ).

tff(tcon_fun___Lattices_Obounded__semilattice__inf__top,axiom,
    ! [A17: $tType,A18: $tType] :
      ( bounded_lattice(A18)
     => bounde4346867609351753570nf_top(fun(A17,A18)) ) ).

tff(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
    ! [A17: $tType,A18: $tType] :
      ( comple6319245703460814977attice(A18)
     => comple6319245703460814977attice(fun(A17,A18)) ) ).

tff(tcon_fun___Quickcheck__Exhaustive_Oexhaustive,axiom,
    ! [A17: $tType,A18: $tType] :
      ( ( cl_HOL_Oequal(A17)
        & quickc658316121487927005ustive(A17)
        & quickc658316121487927005ustive(A18) )
     => quickc658316121487927005ustive(fun(A17,A18)) ) ).

tff(tcon_fun___Boolean__Algebras_Oboolean__algebra,axiom,
    ! [A17: $tType,A18: $tType] :
      ( boolea8198339166811842893lgebra(A18)
     => boolea8198339166811842893lgebra(fun(A17,A18)) ) ).

tff(tcon_fun___Lattices_Obounded__lattice__top,axiom,
    ! [A17: $tType,A18: $tType] :
      ( bounded_lattice(A18)
     => bounded_lattice_top(fun(A17,A18)) ) ).

tff(tcon_fun___Lattices_Obounded__lattice__bot,axiom,
    ! [A17: $tType,A18: $tType] :
      ( bounded_lattice(A18)
     => bounded_lattice_bot(fun(A17,A18)) ) ).

tff(tcon_fun___Complete__Partial__Order_Occpo,axiom,
    ! [A17: $tType,A18: $tType] :
      ( comple6319245703460814977attice(A18)
     => comple9053668089753744459l_ccpo(fun(A17,A18)) ) ).

tff(tcon_fun___Quickcheck__Random_Orandom,axiom,
    ! [A17: $tType,A18: $tType] :
      ( ( code_term_of(A17)
        & cl_HOL_Oequal(A17)
        & quickcheck_random(A18) )
     => quickcheck_random(fun(A17,A18)) ) ).

tff(tcon_fun___Lattices_Osemilattice__sup,axiom,
    ! [A17: $tType,A18: $tType] :
      ( semilattice_sup(A18)
     => semilattice_sup(fun(A17,A18)) ) ).

tff(tcon_fun___Lattices_Osemilattice__inf,axiom,
    ! [A17: $tType,A18: $tType] :
      ( semilattice_inf(A18)
     => semilattice_inf(fun(A17,A18)) ) ).

tff(tcon_fun___Lattices_Odistrib__lattice,axiom,
    ! [A17: $tType,A18: $tType] :
      ( distrib_lattice(A18)
     => distrib_lattice(fun(A17,A18)) ) ).

tff(tcon_fun___Lattices_Obounded__lattice,axiom,
    ! [A17: $tType,A18: $tType] :
      ( bounded_lattice(A18)
     => bounded_lattice(fun(A17,A18)) ) ).

tff(tcon_fun___Orderings_Oorder__top,axiom,
    ! [A17: $tType,A18: $tType] :
      ( order_top(A18)
     => order_top(fun(A17,A18)) ) ).

tff(tcon_fun___Orderings_Oorder__bot,axiom,
    ! [A17: $tType,A18: $tType] :
      ( order_bot(A18)
     => order_bot(fun(A17,A18)) ) ).

tff(tcon_fun___Orderings_Opreorder,axiom,
    ! [A17: $tType,A18: $tType] :
      ( preorder(A18)
     => preorder(fun(A17,A18)) ) ).

tff(tcon_fun___Finite__Set_Ofinite,axiom,
    ! [A17: $tType,A18: $tType] :
      ( ( finite_finite(A17)
        & finite_finite(A18) )
     => finite_finite(fun(A17,A18)) ) ).

tff(tcon_fun___Lattices_Olattice,axiom,
    ! [A17: $tType,A18: $tType] :
      ( lattice(A18)
     => lattice(fun(A17,A18)) ) ).

tff(tcon_fun___Orderings_Oorder,axiom,
    ! [A17: $tType,A18: $tType] :
      ( order(A18)
     => order(fun(A17,A18)) ) ).

tff(tcon_fun___Orderings_Otop,axiom,
    ! [A17: $tType,A18: $tType] :
      ( top(A18)
     => top(fun(A17,A18)) ) ).

tff(tcon_fun___Orderings_Oord,axiom,
    ! [A17: $tType,A18: $tType] :
      ( ord(A18)
     => ord(fun(A17,A18)) ) ).

tff(tcon_fun___Orderings_Obot,axiom,
    ! [A17: $tType,A18: $tType] :
      ( bot(A18)
     => bot(fun(A17,A18)) ) ).

tff(tcon_fun___Groups_Ouminus,axiom,
    ! [A17: $tType,A18: $tType] :
      ( uminus(A18)
     => uminus(fun(A17,A18)) ) ).

tff(tcon_fun___Groups_Ominus,axiom,
    ! [A17: $tType,A18: $tType] :
      ( minus(A18)
     => minus(fun(A17,A18)) ) ).

tff(tcon_fun___HOL_Oequal,axiom,
    ! [A17: $tType,A18: $tType] :
      ( ( enum(A17)
        & cl_HOL_Oequal(A18) )
     => cl_HOL_Oequal(fun(A17,A18)) ) ).

tff(tcon_itself___Quickcheck__Random_Orandom_54,axiom,
    ! [A17: $tType] :
      ( typerep(A17)
     => quickcheck_random(itself(A17)) ) ).

tff(tcon_itself___HOL_Oequal_55,axiom,
    ! [A17: $tType] : cl_HOL_Oequal(itself(A17)) ).

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___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_Osemiring__1__no__zero__divisors,axiom,
    semiri2026040879449505780visors(int) ).

tff(tcon_Int_Oint___Rings_Olinordered__nonzero__semiring,axiom,
    linord181362715937106298miring(int) ).

tff(tcon_Int_Oint___Euclidean__Division_Oeuclidean__ring,axiom,
    euclid5891614535332579305n_ring(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_Osemiring__1__cancel,axiom,
    semiring_1_cancel(int) ).

tff(tcon_Int_Oint___Rings_Oalgebraic__semidom,axiom,
    algebraic_semidom(int) ).

tff(tcon_Int_Oint___Groups_Ocomm__monoid__mult,axiom,
    comm_monoid_mult(int) ).

tff(tcon_Int_Oint___Groups_Oab__semigroup__add,axiom,
    ab_semigroup_add(int) ).

tff(tcon_Int_Oint___Rings_Oordered__semiring,axiom,
    ordered_semiring(int) ).

tff(tcon_Int_Oint___Rings_Oordered__ring__abs,axiom,
    ordered_ring_abs(int) ).

tff(tcon_Int_Oint___Parity_Osemiring__parity,axiom,
    semiring_parity(int) ).

tff(tcon_Int_Oint___Groups_Ocomm__monoid__add,axiom,
    comm_monoid_add(int) ).

tff(tcon_Int_Oint___Rings_Osemiring__modulo,axiom,
    semiring_modulo(int) ).

tff(tcon_Int_Oint___Rings_Olinordered__ring,axiom,
    linordered_ring(int) ).

tff(tcon_Int_Oint___Rings_Olinordered__idom,axiom,
    linordered_idom(int) ).

tff(tcon_Int_Oint___Rings_Ocomm__semiring__1,axiom,
    comm_semiring_1(int) ).

tff(tcon_Int_Oint___Rings_Ocomm__semiring__0,axiom,
    comm_semiring_0(int) ).

tff(tcon_Int_Oint___Groups_Osemigroup__mult,axiom,
    semigroup_mult(int) ).

tff(tcon_Int_Oint___Rings_Osemidom__modulo,axiom,
    semidom_modulo(int) ).

tff(tcon_Int_Oint___Rings_Osemidom__divide,axiom,
    semidom_divide(int) ).

tff(tcon_Int_Oint___Num_Osemiring__numeral,axiom,
    semiring_numeral(int) ).

tff(tcon_Int_Oint___Groups_Osemigroup__add,axiom,
    semigroup_add(int) ).

tff(tcon_Int_Oint___Rings_Ozero__less__one,axiom,
    zero_less_one(int) ).

tff(tcon_Int_Oint___Rings_Ocomm__semiring,axiom,
    comm_semiring(int) ).

tff(tcon_Int_Oint___Nat_Osemiring__char__0,axiom,
    semiring_char_0(int) ).

tff(tcon_Int_Oint___Groups_Oab__group__add,axiom,
    ab_group_add(int) ).

tff(tcon_Int_Oint___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_63,axiom,
    preorder(int) ).

tff(tcon_Int_Oint___Orderings_Olinorder,axiom,
    linorder(int) ).

tff(tcon_Int_Oint___Groups_Omonoid__mult,axiom,
    monoid_mult(int) ).

tff(tcon_Int_Oint___Rings_Oidom__modulo,axiom,
    idom_modulo(int) ).

tff(tcon_Int_Oint___Rings_Oidom__divide,axiom,
    idom_divide(int) ).

tff(tcon_Int_Oint___Rings_Ocomm__ring__1,axiom,
    comm_ring_1(int) ).

tff(tcon_Int_Oint___Groups_Omonoid__add,axiom,
    monoid_add(int) ).

tff(tcon_Int_Oint___Rings_Osemiring__1,axiom,
    semiring_1(int) ).

tff(tcon_Int_Oint___Rings_Osemiring__0,axiom,
    semiring_0(int) ).

tff(tcon_Int_Oint___Orderings_Ono__top,axiom,
    no_top(int) ).

tff(tcon_Int_Oint___Orderings_Ono__bot,axiom,
    no_bot(int) ).

tff(tcon_Int_Oint___Lattices_Olattice_64,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_65,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_66,axiom,
    ord(int) ).

tff(tcon_Int_Oint___Groups_Ouminus_67,axiom,
    uminus(int) ).

tff(tcon_Int_Oint___Rings_Oring__1,axiom,
    ring_1(int) ).

tff(tcon_Int_Oint___Rings_Oabs__if,axiom,
    abs_if(int) ).

tff(tcon_Int_Oint___Groups_Otimes,axiom,
    times(int) ).

tff(tcon_Int_Oint___Groups_Ominus_68,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___Groups_Oplus,axiom,
    plus(int) ).

tff(tcon_Int_Oint___Rings_Oring,axiom,
    ring(int) ).

tff(tcon_Int_Oint___Rings_Oidom,axiom,
    idom(int) ).

tff(tcon_Int_Oint___Groups_Oone,axiom,
    one(int) ).

tff(tcon_Int_Oint___Rings_Odvd,axiom,
    dvd(int) ).

tff(tcon_Int_Oint___Heap_Oheap,axiom,
    heap(int) ).

tff(tcon_Int_Oint___HOL_Oequal_69,axiom,
    cl_HOL_Oequal(int) ).

tff(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_70,axiom,
    condit6923001295902523014norder(nat) ).

tff(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_71,axiom,
    condit1219197933456340205attice(nat) ).

tff(tcon_Nat_Onat___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_72,axiom,
    bit_un5681908812861735899ations(nat) ).

tff(tcon_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_73,axiom,
    semiri1453513574482234551roduct(nat) ).

tff(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring__with__nat_74,axiom,
    euclid5411537665997757685th_nat(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_75,axiom,
    ordere1937475149494474687imp_le(nat) ).

tff(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring_76,axiom,
    euclid3128863361964157862miring(nat) ).

tff(tcon_Nat_Onat___Euclidean__Division_Oeuclidean__semiring__cancel_77,axiom,
    euclid4440199948858584721cancel(nat) ).

tff(tcon_Nat_Onat___Rings_Onormalization__semidom__multiplicative_78,axiom,
    normal6328177297339901930cative(nat) ).

tff(tcon_Nat_Onat___Divides_Ounique__euclidean__semiring__numeral_79,axiom,
    unique1627219031080169319umeral(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors__cancel_80,axiom,
    semiri6575147826004484403cancel(nat) ).

tff(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add_81,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_82,axiom,
    ordere580206878836729694up_add(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le_83,axiom,
    ordere2412721322843649153imp_le(nat) ).

tff(tcon_Nat_Onat___Bit__Operations_Osemiring__bit__operations_84,axiom,
    bit_se359711467146920520ations(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__comm__semiring__strict_85,axiom,
    linord2810124833399127020strict(nat) ).

tff(tcon_Nat_Onat___Quickcheck__Exhaustive_Ofull__exhaustive_86,axiom,
    quickc3360725361186068524ustive(nat) ).

tff(tcon_Nat_Onat___Groups_Ostrict__ordered__comm__monoid__add_87,axiom,
    strict7427464778891057005id_add(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__cancel__comm__monoid__add_88,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_89,axiom,
    euclid3725896446679973847miring(nat) ).

tff(tcon_Nat_Onat___Groups_Olinordered__ab__semigroup__add_90,axiom,
    linord4140545234300271783up_add(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__1__no__zero__divisors_91,axiom,
    semiri2026040879449505780visors(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring_92,axiom,
    linord181362715937106298miring(nat) ).

tff(tcon_Nat_Onat___Rings_Osemidom__divide__unit__factor_93,axiom,
    semido2269285787275462019factor(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__semiring__strict_94,axiom,
    linord8928482502909563296strict(nat) ).

tff(tcon_Nat_Onat___Quickcheck__Exhaustive_Oexhaustive_95,axiom,
    quickc658316121487927005ustive(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors_96,axiom,
    semiri3467727345109120633visors(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add_97,axiom,
    ordere6658533253407199908up_add(nat) ).

tff(tcon_Nat_Onat___GCD_Osemiring__gcd__mult__normalize_98,axiom,
    semiri6843258321239162965malize(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__comm__monoid__add_99,axiom,
    ordere6911136660526730532id_add(nat) ).

tff(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add_100,axiom,
    cancel2418104881723323429up_add(nat) ).

tff(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add_101,axiom,
    cancel1802427076303600483id_add(nat) ).

tff(tcon_Nat_Onat___Rings_Ocomm__semiring__1__cancel_102,axiom,
    comm_s4317794764714335236cancel(nat) ).

tff(tcon_Nat_Onat___Bit__Operations_Osemiring__bits_103,axiom,
    bit_semiring_bits(nat) ).

tff(tcon_Nat_Onat___Rings_Oordered__comm__semiring_104,axiom,
    ordere2520102378445227354miring(nat) ).

tff(tcon_Nat_Onat___Rings_Onormalization__semidom_105,axiom,
    normal8620421768224518004emidom(nat) ).

tff(tcon_Nat_Onat___Groups_Ocancel__semigroup__add_106,axiom,
    cancel_semigroup_add(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__semiring_107,axiom,
    linordered_semiring(nat) ).

tff(tcon_Nat_Onat___Rings_Oordered__semiring__0_108,axiom,
    ordered_semiring_0(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__semidom_109,axiom,
    linordered_semidom(nat) ).

tff(tcon_Nat_Onat___Quickcheck__Random_Orandom_110,axiom,
    quickcheck_random(nat) ).

tff(tcon_Nat_Onat___Lattices_Osemilattice__sup_111,axiom,
    semilattice_sup(nat) ).

tff(tcon_Nat_Onat___Lattices_Osemilattice__inf_112,axiom,
    semilattice_inf(nat) ).

tff(tcon_Nat_Onat___Lattices_Odistrib__lattice_113,axiom,
    distrib_lattice(nat) ).

tff(tcon_Nat_Onat___Groups_Oab__semigroup__mult_114,axiom,
    ab_semigroup_mult(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__1__cancel_115,axiom,
    semiring_1_cancel(nat) ).

tff(tcon_Nat_Onat___Rings_Oalgebraic__semidom_116,axiom,
    algebraic_semidom(nat) ).

tff(tcon_Nat_Onat___Groups_Ocomm__monoid__mult_117,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_118,axiom,
    ab_semigroup_add(nat) ).

tff(tcon_Nat_Onat___Rings_Oordered__semiring_119,axiom,
    ordered_semiring(nat) ).

tff(tcon_Nat_Onat___Parity_Osemiring__parity_120,axiom,
    semiring_parity(nat) ).

tff(tcon_Nat_Onat___Groups_Ocomm__monoid__add_121,axiom,
    comm_monoid_add(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__modulo_122,axiom,
    semiring_modulo(nat) ).

tff(tcon_Nat_Onat___Rings_Ocomm__semiring__1_123,axiom,
    comm_semiring_1(nat) ).

tff(tcon_Nat_Onat___Rings_Ocomm__semiring__0_124,axiom,
    comm_semiring_0(nat) ).

tff(tcon_Nat_Onat___Groups_Osemigroup__mult_125,axiom,
    semigroup_mult(nat) ).

tff(tcon_Nat_Onat___Rings_Osemidom__modulo_126,axiom,
    semidom_modulo(nat) ).

tff(tcon_Nat_Onat___Rings_Osemidom__divide_127,axiom,
    semidom_divide(nat) ).

tff(tcon_Nat_Onat___Num_Osemiring__numeral_128,axiom,
    semiring_numeral(nat) ).

tff(tcon_Nat_Onat___Groups_Osemigroup__add_129,axiom,
    semigroup_add(nat) ).

tff(tcon_Nat_Onat___Rings_Ozero__less__one_130,axiom,
    zero_less_one(nat) ).

tff(tcon_Nat_Onat___Rings_Ocomm__semiring_131,axiom,
    comm_semiring(nat) ).

tff(tcon_Nat_Onat___Orderings_Owellorder,axiom,
    wellorder(nat) ).

tff(tcon_Nat_Onat___Orderings_Oorder__bot_132,axiom,
    order_bot(nat) ).

tff(tcon_Nat_Onat___Nat_Osemiring__char__0_133,axiom,
    semiring_char_0(nat) ).

tff(tcon_Nat_Onat___Rings_Ozero__neq__one_134,axiom,
    zero_neq_one(nat) ).

tff(tcon_Nat_Onat___Orderings_Opreorder_135,axiom,
    preorder(nat) ).

tff(tcon_Nat_Onat___Orderings_Olinorder_136,axiom,
    linorder(nat) ).

tff(tcon_Nat_Onat___Groups_Omonoid__mult_137,axiom,
    monoid_mult(nat) ).

tff(tcon_Nat_Onat___Groups_Omonoid__add_138,axiom,
    monoid_add(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__1_139,axiom,
    semiring_1(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__0_140,axiom,
    semiring_0(nat) ).

tff(tcon_Nat_Onat___Orderings_Ono__top_141,axiom,
    no_top(nat) ).

tff(tcon_Nat_Onat___Lattices_Olattice_142,axiom,
    lattice(nat) ).

tff(tcon_Nat_Onat___GCD_Osemiring__gcd_143,axiom,
    semiring_gcd(nat) ).

tff(tcon_Nat_Onat___GCD_Osemiring__Gcd_144,axiom,
    semiring_Gcd(nat) ).

tff(tcon_Nat_Onat___Rings_Omult__zero_145,axiom,
    mult_zero(nat) ).

tff(tcon_Nat_Onat___Orderings_Oorder_146,axiom,
    order(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring_147,axiom,
    semiring(nat) ).

tff(tcon_Nat_Onat___Orderings_Oord_148,axiom,
    ord(nat) ).

tff(tcon_Nat_Onat___Orderings_Obot_149,axiom,
    bot(nat) ).

tff(tcon_Nat_Onat___Groups_Otimes_150,axiom,
    times(nat) ).

tff(tcon_Nat_Onat___Groups_Ominus_151,axiom,
    minus(nat) ).

tff(tcon_Nat_Onat___Power_Opower_152,axiom,
    power(nat) ).

tff(tcon_Nat_Onat___Num_Onumeral_153,axiom,
    numeral(nat) ).

tff(tcon_Nat_Onat___Groups_Ozero_154,axiom,
    zero(nat) ).

tff(tcon_Nat_Onat___Groups_Oplus_155,axiom,
    plus(nat) ).

tff(tcon_Nat_Onat___Groups_Oone_156,axiom,
    one(nat) ).

tff(tcon_Nat_Onat___Rings_Odvd_157,axiom,
    dvd(nat) ).

tff(tcon_Nat_Onat___Heap_Oheap_158,axiom,
    heap(nat) ).

tff(tcon_Nat_Onat___HOL_Oequal_159,axiom,
    cl_HOL_Oequal(nat) ).

tff(tcon_Nat_Onat___Nat_Osize,axiom,
    size(nat) ).

tff(tcon_Num_Onum___Quickcheck__Exhaustive_Ofull__exhaustive_160,axiom,
    quickc3360725361186068524ustive(num) ).

tff(tcon_Num_Onum___Quickcheck__Random_Orandom_161,axiom,
    quickcheck_random(num) ).

tff(tcon_Num_Onum___Orderings_Opreorder_162,axiom,
    preorder(num) ).

tff(tcon_Num_Onum___Orderings_Olinorder_163,axiom,
    linorder(num) ).

tff(tcon_Num_Onum___Orderings_Oorder_164,axiom,
    order(num) ).

tff(tcon_Num_Onum___Orderings_Oord_165,axiom,
    ord(num) ).

tff(tcon_Num_Onum___Groups_Otimes_166,axiom,
    times(num) ).

tff(tcon_Num_Onum___Groups_Oplus_167,axiom,
    plus(num) ).

tff(tcon_Num_Onum___HOL_Oequal_168,axiom,
    cl_HOL_Oequal(num) ).

tff(tcon_Num_Onum___Nat_Osize_169,axiom,
    size(num) ).

tff(tcon_Rat_Orat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_170,axiom,
    semiri1453513574482234551roduct(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_171,axiom,
    ordere1937475149494474687imp_le(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors__cancel_172,axiom,
    semiri6575147826004484403cancel(rat) ).

tff(tcon_Rat_Orat___Groups_Ostrict__ordered__ab__semigroup__add_173,axiom,
    strict9044650504122735259up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__cancel__ab__semigroup__add_174,axiom,
    ordere580206878836729694up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add__imp__le_175,axiom,
    ordere2412721322843649153imp_le(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__comm__semiring__strict_176,axiom,
    linord2810124833399127020strict(rat) ).

tff(tcon_Rat_Orat___Quickcheck__Exhaustive_Ofull__exhaustive_177,axiom,
    quickc3360725361186068524ustive(rat) ).

tff(tcon_Rat_Orat___Groups_Ostrict__ordered__comm__monoid__add_178,axiom,
    strict7427464778891057005id_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__cancel__comm__monoid__add_179,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_180,axiom,
    linord715952674999750819strict(rat) ).

tff(tcon_Rat_Orat___Orderings_Ounbounded__dense__linorder,axiom,
    unboun7993243217541854897norder(rat) ).

tff(tcon_Rat_Orat___Groups_Olinordered__ab__semigroup__add_181,axiom,
    linord4140545234300271783up_add(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__1__no__zero__divisors_182,axiom,
    semiri2026040879449505780visors(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__nonzero__semiring_183,axiom,
    linord181362715937106298miring(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semiring__strict_184,axiom,
    linord8928482502909563296strict(rat) ).

tff(tcon_Rat_Orat___Quickcheck__Exhaustive_Oexhaustive_185,axiom,
    quickc658316121487927005ustive(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors_186,axiom,
    semiri3467727345109120633visors(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add_187,axiom,
    ordere6658533253407199908up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__group__add__abs_188,axiom,
    ordere166539214618696060dd_abs(rat) ).

tff(tcon_Rat_Orat___Archimedean__Field_Ofloor__ceiling,axiom,
    archim2362893244070406136eiling(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__comm__monoid__add_189,axiom,
    ordere6911136660526730532id_add(rat) ).

tff(tcon_Rat_Orat___Groups_Olinordered__ab__group__add_190,axiom,
    linord5086331880401160121up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Ocancel__ab__semigroup__add_191,axiom,
    cancel2418104881723323429up_add(rat) ).

tff(tcon_Rat_Orat___Rings_Oring__1__no__zero__divisors_192,axiom,
    ring_15535105094025558882visors(rat) ).

tff(tcon_Rat_Orat___Groups_Ocancel__comm__monoid__add_193,axiom,
    cancel1802427076303600483id_add(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__ring__strict_194,axiom,
    linord4710134922213307826strict(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring__1__cancel_195,axiom,
    comm_s4317794764714335236cancel(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__comm__semiring_196,axiom,
    ordere2520102378445227354miring(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semiring__1_197,axiom,
    linord6961819062388156250ring_1(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__group__add_198,axiom,
    ordered_ab_group_add(rat) ).

tff(tcon_Rat_Orat___Groups_Ocancel__semigroup__add_199,axiom,
    cancel_semigroup_add(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semiring_200,axiom,
    linordered_semiring(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__semiring__0_201,axiom,
    ordered_semiring_0(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semidom_202,axiom,
    linordered_semidom(rat) ).

tff(tcon_Rat_Orat___Quickcheck__Random_Orandom_203,axiom,
    quickcheck_random(rat) ).

tff(tcon_Rat_Orat___Orderings_Odense__linorder,axiom,
    dense_linorder(rat) ).

tff(tcon_Rat_Orat___Lattices_Osemilattice__sup_204,axiom,
    semilattice_sup(rat) ).

tff(tcon_Rat_Orat___Lattices_Osemilattice__inf_205,axiom,
    semilattice_inf(rat) ).

tff(tcon_Rat_Orat___Lattices_Odistrib__lattice_206,axiom,
    distrib_lattice(rat) ).

tff(tcon_Rat_Orat___Groups_Oab__semigroup__mult_207,axiom,
    ab_semigroup_mult(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__1__cancel_208,axiom,
    semiring_1_cancel(rat) ).

tff(tcon_Rat_Orat___Groups_Ocomm__monoid__mult_209,axiom,
    comm_monoid_mult(rat) ).

tff(tcon_Rat_Orat___Groups_Oab__semigroup__add_210,axiom,
    ab_semigroup_add(rat) ).

tff(tcon_Rat_Orat___Fields_Olinordered__field,axiom,
    linordered_field(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__semiring_211,axiom,
    ordered_semiring(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__ring__abs_212,axiom,
    ordered_ring_abs(rat) ).

tff(tcon_Rat_Orat___Groups_Ocomm__monoid__add_213,axiom,
    comm_monoid_add(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__ring_214,axiom,
    linordered_ring(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__idom_215,axiom,
    linordered_idom(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring__1_216,axiom,
    comm_semiring_1(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring__0_217,axiom,
    comm_semiring_0(rat) ).

tff(tcon_Rat_Orat___Orderings_Odense__order,axiom,
    dense_order(rat) ).

tff(tcon_Rat_Orat___Groups_Osemigroup__mult_218,axiom,
    semigroup_mult(rat) ).

tff(tcon_Rat_Orat___Rings_Osemidom__divide_219,axiom,
    semidom_divide(rat) ).

tff(tcon_Rat_Orat___Num_Osemiring__numeral_220,axiom,
    semiring_numeral(rat) ).

tff(tcon_Rat_Orat___Groups_Osemigroup__add_221,axiom,
    semigroup_add(rat) ).

tff(tcon_Rat_Orat___Fields_Odivision__ring,axiom,
    division_ring(rat) ).

tff(tcon_Rat_Orat___Rings_Ozero__less__one_222,axiom,
    zero_less_one(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring_223,axiom,
    comm_semiring(rat) ).

tff(tcon_Rat_Orat___Nat_Osemiring__char__0_224,axiom,
    semiring_char_0(rat) ).

tff(tcon_Rat_Orat___Groups_Oab__group__add_225,axiom,
    ab_group_add(rat) ).

tff(tcon_Rat_Orat___Fields_Ofield__char__0,axiom,
    field_char_0(rat) ).

tff(tcon_Rat_Orat___Rings_Ozero__neq__one_226,axiom,
    zero_neq_one(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__ring_227,axiom,
    ordered_ring(rat) ).

tff(tcon_Rat_Orat___Rings_Oidom__abs__sgn_228,axiom,
    idom_abs_sgn(rat) ).

tff(tcon_Rat_Orat___Orderings_Opreorder_229,axiom,
    preorder(rat) ).

tff(tcon_Rat_Orat___Orderings_Olinorder_230,axiom,
    linorder(rat) ).

tff(tcon_Rat_Orat___Groups_Omonoid__mult_231,axiom,
    monoid_mult(rat) ).

tff(tcon_Rat_Orat___Rings_Oidom__divide_232,axiom,
    idom_divide(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__ring__1_233,axiom,
    comm_ring_1(rat) ).

tff(tcon_Rat_Orat___Groups_Omonoid__add_234,axiom,
    monoid_add(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__1_235,axiom,
    semiring_1(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__0_236,axiom,
    semiring_0(rat) ).

tff(tcon_Rat_Orat___Orderings_Ono__top_237,axiom,
    no_top(rat) ).

tff(tcon_Rat_Orat___Orderings_Ono__bot_238,axiom,
    no_bot(rat) ).

tff(tcon_Rat_Orat___Lattices_Olattice_239,axiom,
    lattice(rat) ).

tff(tcon_Rat_Orat___Groups_Ogroup__add_240,axiom,
    group_add(rat) ).

tff(tcon_Rat_Orat___Rings_Omult__zero_241,axiom,
    mult_zero(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__ring_242,axiom,
    comm_ring(rat) ).

tff(tcon_Rat_Orat___Orderings_Oorder_243,axiom,
    order(rat) ).

tff(tcon_Rat_Orat___Num_Oneg__numeral_244,axiom,
    neg_numeral(rat) ).

tff(tcon_Rat_Orat___Nat_Oring__char__0_245,axiom,
    ring_char_0(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring_246,axiom,
    semiring(rat) ).

tff(tcon_Rat_Orat___Orderings_Oord_247,axiom,
    ord(rat) ).

tff(tcon_Rat_Orat___Groups_Ouminus_248,axiom,
    uminus(rat) ).

tff(tcon_Rat_Orat___Rings_Oring__1_249,axiom,
    ring_1(rat) ).

tff(tcon_Rat_Orat___Rings_Oabs__if_250,axiom,
    abs_if(rat) ).

tff(tcon_Rat_Orat___Groups_Otimes_251,axiom,
    times(rat) ).

tff(tcon_Rat_Orat___Groups_Ominus_252,axiom,
    minus(rat) ).

tff(tcon_Rat_Orat___Fields_Ofield,axiom,
    field(rat) ).

tff(tcon_Rat_Orat___Power_Opower_253,axiom,
    power(rat) ).

tff(tcon_Rat_Orat___Num_Onumeral_254,axiom,
    numeral(rat) ).

tff(tcon_Rat_Orat___Groups_Ozero_255,axiom,
    zero(rat) ).

tff(tcon_Rat_Orat___Groups_Oplus_256,axiom,
    plus(rat) ).

tff(tcon_Rat_Orat___Rings_Oring_257,axiom,
    ring(rat) ).

tff(tcon_Rat_Orat___Rings_Oidom_258,axiom,
    idom(rat) ).

tff(tcon_Rat_Orat___Groups_Oone_259,axiom,
    one(rat) ).

tff(tcon_Rat_Orat___Rings_Odvd_260,axiom,
    dvd(rat) ).

tff(tcon_Rat_Orat___HOL_Oequal_261,axiom,
    cl_HOL_Oequal(rat) ).

tff(tcon_Set_Oset___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_262,axiom,
    ! [A17: $tType] : condit1219197933456340205attice(set(A17)) ).

tff(tcon_Set_Oset___Complete__Lattices_Ocomplete__distrib__lattice_263,axiom,
    ! [A17: $tType] : comple592849572758109894attice(set(A17)) ).

tff(tcon_Set_Oset___Quickcheck__Exhaustive_Ofull__exhaustive_264,axiom,
    ! [A17: $tType] :
      ( quickc3360725361186068524ustive(A17)
     => quickc3360725361186068524ustive(set(A17)) ) ).

tff(tcon_Set_Oset___Lattices_Obounded__semilattice__sup__bot_265,axiom,
    ! [A17: $tType] : bounde4967611905675639751up_bot(set(A17)) ).

tff(tcon_Set_Oset___Lattices_Obounded__semilattice__inf__top_266,axiom,
    ! [A17: $tType] : bounde4346867609351753570nf_top(set(A17)) ).

tff(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_267,axiom,
    ! [A17: $tType] : comple6319245703460814977attice(set(A17)) ).

tff(tcon_Set_Oset___Quickcheck__Exhaustive_Oexhaustive_268,axiom,
    ! [A17: $tType] :
      ( quickc658316121487927005ustive(A17)
     => quickc658316121487927005ustive(set(A17)) ) ).

tff(tcon_Set_Oset___Boolean__Algebras_Oboolean__algebra_269,axiom,
    ! [A17: $tType] : boolea8198339166811842893lgebra(set(A17)) ).

tff(tcon_Set_Oset___Lattices_Obounded__lattice__top_270,axiom,
    ! [A17: $tType] : bounded_lattice_top(set(A17)) ).

tff(tcon_Set_Oset___Lattices_Obounded__lattice__bot_271,axiom,
    ! [A17: $tType] : bounded_lattice_bot(set(A17)) ).

tff(tcon_Set_Oset___Complete__Partial__Order_Occpo_272,axiom,
    ! [A17: $tType] : comple9053668089753744459l_ccpo(set(A17)) ).

tff(tcon_Set_Oset___Quickcheck__Random_Orandom_273,axiom,
    ! [A17: $tType] :
      ( quickcheck_random(A17)
     => quickcheck_random(set(A17)) ) ).

tff(tcon_Set_Oset___Lattices_Osemilattice__sup_274,axiom,
    ! [A17: $tType] : semilattice_sup(set(A17)) ).

tff(tcon_Set_Oset___Lattices_Osemilattice__inf_275,axiom,
    ! [A17: $tType] : semilattice_inf(set(A17)) ).

tff(tcon_Set_Oset___Lattices_Odistrib__lattice_276,axiom,
    ! [A17: $tType] : distrib_lattice(set(A17)) ).

tff(tcon_Set_Oset___Lattices_Obounded__lattice_277,axiom,
    ! [A17: $tType] : bounded_lattice(set(A17)) ).

tff(tcon_Set_Oset___Orderings_Oorder__top_278,axiom,
    ! [A17: $tType] : order_top(set(A17)) ).

tff(tcon_Set_Oset___Orderings_Oorder__bot_279,axiom,
    ! [A17: $tType] : order_bot(set(A17)) ).

tff(tcon_Set_Oset___Orderings_Opreorder_280,axiom,
    ! [A17: $tType] : preorder(set(A17)) ).

tff(tcon_Set_Oset___Finite__Set_Ofinite_281,axiom,
    ! [A17: $tType] :
      ( finite_finite(A17)
     => finite_finite(set(A17)) ) ).

tff(tcon_Set_Oset___Lattices_Olattice_282,axiom,
    ! [A17: $tType] : lattice(set(A17)) ).

tff(tcon_Set_Oset___Orderings_Oorder_283,axiom,
    ! [A17: $tType] : order(set(A17)) ).

tff(tcon_Set_Oset___Orderings_Otop_284,axiom,
    ! [A17: $tType] : top(set(A17)) ).

tff(tcon_Set_Oset___Orderings_Oord_285,axiom,
    ! [A17: $tType] : ord(set(A17)) ).

tff(tcon_Set_Oset___Orderings_Obot_286,axiom,
    ! [A17: $tType] : bot(set(A17)) ).

tff(tcon_Set_Oset___Groups_Ouminus_287,axiom,
    ! [A17: $tType] : uminus(set(A17)) ).

tff(tcon_Set_Oset___Groups_Ominus_288,axiom,
    ! [A17: $tType] : minus(set(A17)) ).

tff(tcon_Set_Oset___HOL_Oequal_289,axiom,
    ! [A17: $tType] :
      ( cl_HOL_Oequal(A17)
     => cl_HOL_Oequal(set(A17)) ) ).

tff(tcon_HOL_Obool___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_290,axiom,
    condit1219197933456340205attice($o) ).

tff(tcon_HOL_Obool___Complete__Lattices_Ocomplete__distrib__lattice_291,axiom,
    comple592849572758109894attice($o) ).

tff(tcon_HOL_Obool___Quickcheck__Exhaustive_Ofull__exhaustive_292,axiom,
    quickc3360725361186068524ustive($o) ).

tff(tcon_HOL_Obool___Lattices_Obounded__semilattice__sup__bot_293,axiom,
    bounde4967611905675639751up_bot($o) ).

tff(tcon_HOL_Obool___Lattices_Obounded__semilattice__inf__top_294,axiom,
    bounde4346867609351753570nf_top($o) ).

tff(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_295,axiom,
    comple6319245703460814977attice($o) ).

tff(tcon_HOL_Obool___Boolean__Algebras_Oboolean__algebra_296,axiom,
    boolea8198339166811842893lgebra($o) ).

tff(tcon_HOL_Obool___Lattices_Obounded__lattice__top_297,axiom,
    bounded_lattice_top($o) ).

tff(tcon_HOL_Obool___Lattices_Obounded__lattice__bot_298,axiom,
    bounded_lattice_bot($o) ).

tff(tcon_HOL_Obool___Complete__Partial__Order_Occpo_299,axiom,
    comple9053668089753744459l_ccpo($o) ).

tff(tcon_HOL_Obool___Quickcheck__Random_Orandom_300,axiom,
    quickcheck_random($o) ).

tff(tcon_HOL_Obool___Lattices_Osemilattice__sup_301,axiom,
    semilattice_sup($o) ).

tff(tcon_HOL_Obool___Lattices_Osemilattice__inf_302,axiom,
    semilattice_inf($o) ).

tff(tcon_HOL_Obool___Lattices_Odistrib__lattice_303,axiom,
    distrib_lattice($o) ).

tff(tcon_HOL_Obool___Lattices_Obounded__lattice_304,axiom,
    bounded_lattice($o) ).

tff(tcon_HOL_Obool___Orderings_Oorder__top_305,axiom,
    order_top($o) ).

tff(tcon_HOL_Obool___Orderings_Oorder__bot_306,axiom,
    order_bot($o) ).

tff(tcon_HOL_Obool___Orderings_Opreorder_307,axiom,
    preorder($o) ).

tff(tcon_HOL_Obool___Orderings_Olinorder_308,axiom,
    linorder($o) ).

tff(tcon_HOL_Obool___Finite__Set_Ofinite_309,axiom,
    finite_finite($o) ).

tff(tcon_HOL_Obool___Lattices_Olattice_310,axiom,
    lattice($o) ).

tff(tcon_HOL_Obool___Orderings_Oorder_311,axiom,
    order($o) ).

tff(tcon_HOL_Obool___Orderings_Otop_312,axiom,
    top($o) ).

tff(tcon_HOL_Obool___Orderings_Oord_313,axiom,
    ord($o) ).

tff(tcon_HOL_Obool___Orderings_Obot_314,axiom,
    bot($o) ).

tff(tcon_HOL_Obool___Groups_Ouminus_315,axiom,
    uminus($o) ).

tff(tcon_HOL_Obool___Groups_Ominus_316,axiom,
    minus($o) ).

tff(tcon_HOL_Obool___Heap_Oheap_317,axiom,
    heap($o) ).

tff(tcon_HOL_Obool___HOL_Oequal_318,axiom,
    cl_HOL_Oequal($o) ).

tff(tcon_Heap_Oref___Quickcheck__Exhaustive_Ofull__exhaustive_319,axiom,
    ! [A17: $tType] :
      ( typerep(A17)
     => quickc3360725361186068524ustive(ref(A17)) ) ).

tff(tcon_Heap_Oref___Quickcheck__Random_Orandom_320,axiom,
    ! [A17: $tType] :
      ( typerep(A17)
     => quickcheck_random(ref(A17)) ) ).

tff(tcon_Heap_Oref___Heap_Oheap_321,axiom,
    ! [A17: $tType] : heap(ref(A17)) ).

tff(tcon_Heap_Oref___HOL_Oequal_322,axiom,
    ! [A17: $tType] : cl_HOL_Oequal(ref(A17)) ).

tff(tcon_Heap_Oref___Nat_Osize_323,axiom,
    ! [A17: $tType] : size(ref(A17)) ).

tff(tcon_List_Olist___Quickcheck__Exhaustive_Ofull__exhaustive_324,axiom,
    ! [A17: $tType] :
      ( quickc3360725361186068524ustive(A17)
     => quickc3360725361186068524ustive(list(A17)) ) ).

tff(tcon_List_Olist___Quickcheck__Random_Orandom_325,axiom,
    ! [A17: $tType] :
      ( quickcheck_random(A17)
     => quickcheck_random(list(A17)) ) ).

tff(tcon_List_Olist___Heap_Oheap_326,axiom,
    ! [A17: $tType] :
      ( heap(A17)
     => heap(list(A17)) ) ).

tff(tcon_List_Olist___HOL_Oequal_327,axiom,
    ! [A17: $tType] : cl_HOL_Oequal(list(A17)) ).

tff(tcon_List_Olist___Nat_Osize_328,axiom,
    ! [A17: $tType] : size(list(A17)) ).

tff(tcon_Heap_Oarray___Quickcheck__Exhaustive_Ofull__exhaustive_329,axiom,
    ! [A17: $tType] :
      ( typerep(A17)
     => quickc3360725361186068524ustive(array(A17)) ) ).

tff(tcon_Heap_Oarray___Quickcheck__Random_Orandom_330,axiom,
    ! [A17: $tType] :
      ( typerep(A17)
     => quickcheck_random(array(A17)) ) ).

tff(tcon_Heap_Oarray___Heap_Oheap_331,axiom,
    ! [A17: $tType] : heap(array(A17)) ).

tff(tcon_Heap_Oarray___HOL_Oequal_332,axiom,
    ! [A17: $tType] : cl_HOL_Oequal(array(A17)) ).

tff(tcon_Heap_Oarray___Nat_Osize_333,axiom,
    ! [A17: $tType] : size(array(A17)) ).

tff(tcon_Sum__Type_Osum___Quickcheck__Exhaustive_Ofull__exhaustive_334,axiom,
    ! [A17: $tType,A18: $tType] :
      ( ( quickc3360725361186068524ustive(A17)
        & quickc3360725361186068524ustive(A18) )
     => quickc3360725361186068524ustive(sum_sum(A17,A18)) ) ).

tff(tcon_Sum__Type_Osum___Quickcheck__Random_Orandom_335,axiom,
    ! [A17: $tType,A18: $tType] :
      ( ( quickcheck_random(A17)
        & quickcheck_random(A18) )
     => quickcheck_random(sum_sum(A17,A18)) ) ).

tff(tcon_Sum__Type_Osum___Finite__Set_Ofinite_336,axiom,
    ! [A17: $tType,A18: $tType] :
      ( ( finite_finite(A17)
        & finite_finite(A18) )
     => finite_finite(sum_sum(A17,A18)) ) ).

tff(tcon_Sum__Type_Osum___Heap_Oheap_337,axiom,
    ! [A17: $tType,A18: $tType] :
      ( ( heap(A17)
        & heap(A18) )
     => heap(sum_sum(A17,A18)) ) ).

tff(tcon_Sum__Type_Osum___HOL_Oequal_338,axiom,
    ! [A17: $tType,A18: $tType] : cl_HOL_Oequal(sum_sum(A17,A18)) ).

tff(tcon_Sum__Type_Osum___Nat_Osize_339,axiom,
    ! [A17: $tType,A18: $tType] : size(sum_sum(A17,A18)) ).

tff(tcon_Filter_Ofilter___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_340,axiom,
    ! [A17: $tType] : condit1219197933456340205attice(filter(A17)) ).

tff(tcon_Filter_Ofilter___Lattices_Obounded__semilattice__sup__bot_341,axiom,
    ! [A17: $tType] : bounde4967611905675639751up_bot(filter(A17)) ).

tff(tcon_Filter_Ofilter___Lattices_Obounded__semilattice__inf__top_342,axiom,
    ! [A17: $tType] : bounde4346867609351753570nf_top(filter(A17)) ).

tff(tcon_Filter_Ofilter___Complete__Lattices_Ocomplete__lattice_343,axiom,
    ! [A17: $tType] : comple6319245703460814977attice(filter(A17)) ).

tff(tcon_Filter_Ofilter___Lattices_Obounded__lattice__top_344,axiom,
    ! [A17: $tType] : bounded_lattice_top(filter(A17)) ).

tff(tcon_Filter_Ofilter___Lattices_Obounded__lattice__bot_345,axiom,
    ! [A17: $tType] : bounded_lattice_bot(filter(A17)) ).

tff(tcon_Filter_Ofilter___Complete__Partial__Order_Occpo_346,axiom,
    ! [A17: $tType] : comple9053668089753744459l_ccpo(filter(A17)) ).

tff(tcon_Filter_Ofilter___Lattices_Osemilattice__sup_347,axiom,
    ! [A17: $tType] : semilattice_sup(filter(A17)) ).

tff(tcon_Filter_Ofilter___Lattices_Osemilattice__inf_348,axiom,
    ! [A17: $tType] : semilattice_inf(filter(A17)) ).

tff(tcon_Filter_Ofilter___Lattices_Odistrib__lattice_349,axiom,
    ! [A17: $tType] : distrib_lattice(filter(A17)) ).

tff(tcon_Filter_Ofilter___Lattices_Obounded__lattice_350,axiom,
    ! [A17: $tType] : bounded_lattice(filter(A17)) ).

tff(tcon_Filter_Ofilter___Orderings_Oorder__top_351,axiom,
    ! [A17: $tType] : order_top(filter(A17)) ).

tff(tcon_Filter_Ofilter___Orderings_Oorder__bot_352,axiom,
    ! [A17: $tType] : order_bot(filter(A17)) ).

tff(tcon_Filter_Ofilter___Orderings_Opreorder_353,axiom,
    ! [A17: $tType] : preorder(filter(A17)) ).

tff(tcon_Filter_Ofilter___Lattices_Olattice_354,axiom,
    ! [A17: $tType] : lattice(filter(A17)) ).

tff(tcon_Filter_Ofilter___Orderings_Oorder_355,axiom,
    ! [A17: $tType] : order(filter(A17)) ).

tff(tcon_Filter_Ofilter___Orderings_Otop_356,axiom,
    ! [A17: $tType] : top(filter(A17)) ).

tff(tcon_Filter_Ofilter___Orderings_Oord_357,axiom,
    ! [A17: $tType] : ord(filter(A17)) ).

tff(tcon_Filter_Ofilter___Orderings_Obot_358,axiom,
    ! [A17: $tType] : bot(filter(A17)) ).

tff(tcon_Filter_Ofilter___HOL_Oequal_359,axiom,
    ! [A17: $tType] :
      ( cl_HOL_Oequal(A17)
     => cl_HOL_Oequal(filter(A17)) ) ).

tff(tcon_Option_Ooption___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_360,axiom,
    ! [A17: $tType] :
      ( comple5582772986160207858norder(A17)
     => condit6923001295902523014norder(option(A17)) ) ).

tff(tcon_Option_Ooption___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_361,axiom,
    ! [A17: $tType] :
      ( comple6319245703460814977attice(A17)
     => condit1219197933456340205attice(option(A17)) ) ).

tff(tcon_Option_Ooption___Complete__Lattices_Ocomplete__distrib__lattice_362,axiom,
    ! [A17: $tType] :
      ( comple592849572758109894attice(A17)
     => comple592849572758109894attice(option(A17)) ) ).

tff(tcon_Option_Ooption___Quickcheck__Exhaustive_Ofull__exhaustive_363,axiom,
    ! [A17: $tType] :
      ( quickc3360725361186068524ustive(A17)
     => quickc3360725361186068524ustive(option(A17)) ) ).

tff(tcon_Option_Ooption___Lattices_Obounded__semilattice__sup__bot_364,axiom,
    ! [A17: $tType] :
      ( lattice(A17)
     => bounde4967611905675639751up_bot(option(A17)) ) ).

tff(tcon_Option_Ooption___Lattices_Obounded__semilattice__inf__top_365,axiom,
    ! [A17: $tType] :
      ( bounded_lattice_top(A17)
     => bounde4346867609351753570nf_top(option(A17)) ) ).

tff(tcon_Option_Ooption___Complete__Lattices_Ocomplete__linorder,axiom,
    ! [A17: $tType] :
      ( comple5582772986160207858norder(A17)
     => comple5582772986160207858norder(option(A17)) ) ).

tff(tcon_Option_Ooption___Complete__Lattices_Ocomplete__lattice_366,axiom,
    ! [A17: $tType] :
      ( comple6319245703460814977attice(A17)
     => comple6319245703460814977attice(option(A17)) ) ).

tff(tcon_Option_Ooption___Lattices_Obounded__lattice__top_367,axiom,
    ! [A17: $tType] :
      ( bounded_lattice_top(A17)
     => bounded_lattice_top(option(A17)) ) ).

tff(tcon_Option_Ooption___Lattices_Obounded__lattice__bot_368,axiom,
    ! [A17: $tType] :
      ( lattice(A17)
     => bounded_lattice_bot(option(A17)) ) ).

tff(tcon_Option_Ooption___Complete__Partial__Order_Occpo_369,axiom,
    ! [A17: $tType] :
      ( comple6319245703460814977attice(A17)
     => comple9053668089753744459l_ccpo(option(A17)) ) ).

tff(tcon_Option_Ooption___Quickcheck__Random_Orandom_370,axiom,
    ! [A17: $tType] :
      ( quickcheck_random(A17)
     => quickcheck_random(option(A17)) ) ).

tff(tcon_Option_Ooption___Lattices_Osemilattice__sup_371,axiom,
    ! [A17: $tType] :
      ( semilattice_sup(A17)
     => semilattice_sup(option(A17)) ) ).

tff(tcon_Option_Ooption___Lattices_Osemilattice__inf_372,axiom,
    ! [A17: $tType] :
      ( semilattice_inf(A17)
     => semilattice_inf(option(A17)) ) ).

tff(tcon_Option_Ooption___Lattices_Odistrib__lattice_373,axiom,
    ! [A17: $tType] :
      ( distrib_lattice(A17)
     => distrib_lattice(option(A17)) ) ).

tff(tcon_Option_Ooption___Lattices_Obounded__lattice_374,axiom,
    ! [A17: $tType] :
      ( bounded_lattice_top(A17)
     => bounded_lattice(option(A17)) ) ).

tff(tcon_Option_Ooption___Orderings_Owellorder_375,axiom,
    ! [A17: $tType] :
      ( wellorder(A17)
     => wellorder(option(A17)) ) ).

tff(tcon_Option_Ooption___Orderings_Oorder__top_376,axiom,
    ! [A17: $tType] :
      ( order_top(A17)
     => order_top(option(A17)) ) ).

tff(tcon_Option_Ooption___Orderings_Oorder__bot_377,axiom,
    ! [A17: $tType] :
      ( order(A17)
     => order_bot(option(A17)) ) ).

tff(tcon_Option_Ooption___Orderings_Opreorder_378,axiom,
    ! [A17: $tType] :
      ( preorder(A17)
     => preorder(option(A17)) ) ).

tff(tcon_Option_Ooption___Orderings_Olinorder_379,axiom,
    ! [A17: $tType] :
      ( linorder(A17)
     => linorder(option(A17)) ) ).

tff(tcon_Option_Ooption___Finite__Set_Ofinite_380,axiom,
    ! [A17: $tType] :
      ( finite_finite(A17)
     => finite_finite(option(A17)) ) ).

tff(tcon_Option_Ooption___Lattices_Olattice_381,axiom,
    ! [A17: $tType] :
      ( lattice(A17)
     => lattice(option(A17)) ) ).

tff(tcon_Option_Ooption___Orderings_Oorder_382,axiom,
    ! [A17: $tType] :
      ( order(A17)
     => order(option(A17)) ) ).

tff(tcon_Option_Ooption___Orderings_Otop_383,axiom,
    ! [A17: $tType] :
      ( order_top(A17)
     => top(option(A17)) ) ).

tff(tcon_Option_Ooption___Orderings_Oord_384,axiom,
    ! [A17: $tType] :
      ( preorder(A17)
     => ord(option(A17)) ) ).

tff(tcon_Option_Ooption___Orderings_Obot_385,axiom,
    ! [A17: $tType] :
      ( order(A17)
     => bot(option(A17)) ) ).

tff(tcon_Option_Ooption___Heap_Oheap_386,axiom,
    ! [A17: $tType] :
      ( heap(A17)
     => heap(option(A17)) ) ).

tff(tcon_Option_Ooption___HOL_Oequal_387,axiom,
    ! [A17: $tType] : cl_HOL_Oequal(option(A17)) ).

tff(tcon_Option_Ooption___Nat_Osize_388,axiom,
    ! [A17: $tType] : size(option(A17)) ).

tff(tcon_Predicate_Oseq___HOL_Oequal_389,axiom,
    ! [A17: $tType] : cl_HOL_Oequal(seq(A17)) ).

tff(tcon_Predicate_Opred___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_390,axiom,
    ! [A17: $tType] : condit1219197933456340205attice(pred(A17)) ).

tff(tcon_Predicate_Opred___Complete__Lattices_Ocomplete__distrib__lattice_391,axiom,
    ! [A17: $tType] : comple592849572758109894attice(pred(A17)) ).

tff(tcon_Predicate_Opred___Lattices_Obounded__semilattice__sup__bot_392,axiom,
    ! [A17: $tType] : bounde4967611905675639751up_bot(pred(A17)) ).

tff(tcon_Predicate_Opred___Lattices_Obounded__semilattice__inf__top_393,axiom,
    ! [A17: $tType] : bounde4346867609351753570nf_top(pred(A17)) ).

tff(tcon_Predicate_Opred___Complete__Lattices_Ocomplete__lattice_394,axiom,
    ! [A17: $tType] : comple6319245703460814977attice(pred(A17)) ).

tff(tcon_Predicate_Opred___Boolean__Algebras_Oboolean__algebra_395,axiom,
    ! [A17: $tType] : boolea8198339166811842893lgebra(pred(A17)) ).

tff(tcon_Predicate_Opred___Lattices_Obounded__lattice__top_396,axiom,
    ! [A17: $tType] : bounded_lattice_top(pred(A17)) ).

tff(tcon_Predicate_Opred___Lattices_Obounded__lattice__bot_397,axiom,
    ! [A17: $tType] : bounded_lattice_bot(pred(A17)) ).

tff(tcon_Predicate_Opred___Complete__Partial__Order_Occpo_398,axiom,
    ! [A17: $tType] : comple9053668089753744459l_ccpo(pred(A17)) ).

tff(tcon_Predicate_Opred___Lattices_Osemilattice__sup_399,axiom,
    ! [A17: $tType] : semilattice_sup(pred(A17)) ).

tff(tcon_Predicate_Opred___Lattices_Osemilattice__inf_400,axiom,
    ! [A17: $tType] : semilattice_inf(pred(A17)) ).

tff(tcon_Predicate_Opred___Lattices_Odistrib__lattice_401,axiom,
    ! [A17: $tType] : distrib_lattice(pred(A17)) ).

tff(tcon_Predicate_Opred___Lattices_Obounded__lattice_402,axiom,
    ! [A17: $tType] : bounded_lattice(pred(A17)) ).

tff(tcon_Predicate_Opred___Orderings_Oorder__top_403,axiom,
    ! [A17: $tType] : order_top(pred(A17)) ).

tff(tcon_Predicate_Opred___Orderings_Oorder__bot_404,axiom,
    ! [A17: $tType] : order_bot(pred(A17)) ).

tff(tcon_Predicate_Opred___Orderings_Opreorder_405,axiom,
    ! [A17: $tType] : preorder(pred(A17)) ).

tff(tcon_Predicate_Opred___Lattices_Olattice_406,axiom,
    ! [A17: $tType] : lattice(pred(A17)) ).

tff(tcon_Predicate_Opred___Orderings_Oorder_407,axiom,
    ! [A17: $tType] : order(pred(A17)) ).

tff(tcon_Predicate_Opred___Orderings_Otop_408,axiom,
    ! [A17: $tType] : top(pred(A17)) ).

tff(tcon_Predicate_Opred___Orderings_Oord_409,axiom,
    ! [A17: $tType] : ord(pred(A17)) ).

tff(tcon_Predicate_Opred___Orderings_Obot_410,axiom,
    ! [A17: $tType] : bot(pred(A17)) ).

tff(tcon_Predicate_Opred___Groups_Ouminus_411,axiom,
    ! [A17: $tType] : uminus(pred(A17)) ).

tff(tcon_Predicate_Opred___Groups_Ominus_412,axiom,
    ! [A17: $tType] : minus(pred(A17)) ).

tff(tcon_Predicate_Opred___HOL_Oequal_413,axiom,
    ! [A17: $tType] : cl_HOL_Oequal(pred(A17)) ).

tff(tcon_Assertions_Oassn___Lattices_Obounded__semilattice__sup__bot_414,axiom,
    bounde4967611905675639751up_bot(assn) ).

tff(tcon_Assertions_Oassn___Lattices_Obounded__semilattice__inf__top_415,axiom,
    bounde4346867609351753570nf_top(assn) ).

tff(tcon_Assertions_Oassn___Boolean__Algebras_Oboolean__algebra_416,axiom,
    boolea8198339166811842893lgebra(assn) ).

tff(tcon_Assertions_Oassn___Lattices_Obounded__lattice__top_417,axiom,
    bounded_lattice_top(assn) ).

tff(tcon_Assertions_Oassn___Lattices_Obounded__lattice__bot_418,axiom,
    bounded_lattice_bot(assn) ).

tff(tcon_Assertions_Oassn___Lattices_Osemilattice__sup_419,axiom,
    semilattice_sup(assn) ).

tff(tcon_Assertions_Oassn___Lattices_Osemilattice__inf_420,axiom,
    semilattice_inf(assn) ).

tff(tcon_Assertions_Oassn___Lattices_Odistrib__lattice_421,axiom,
    distrib_lattice(assn) ).

tff(tcon_Assertions_Oassn___Lattices_Obounded__lattice_422,axiom,
    bounded_lattice(assn) ).

tff(tcon_Assertions_Oassn___Groups_Oab__semigroup__mult_423,axiom,
    ab_semigroup_mult(assn) ).

tff(tcon_Assertions_Oassn___Groups_Ocomm__monoid__mult_424,axiom,
    comm_monoid_mult(assn) ).

tff(tcon_Assertions_Oassn___Groups_Osemigroup__mult_425,axiom,
    semigroup_mult(assn) ).

tff(tcon_Assertions_Oassn___Orderings_Oorder__top_426,axiom,
    order_top(assn) ).

tff(tcon_Assertions_Oassn___Orderings_Oorder__bot_427,axiom,
    order_bot(assn) ).

tff(tcon_Assertions_Oassn___Orderings_Opreorder_428,axiom,
    preorder(assn) ).

tff(tcon_Assertions_Oassn___Groups_Omonoid__mult_429,axiom,
    monoid_mult(assn) ).

tff(tcon_Assertions_Oassn___Lattices_Olattice_430,axiom,
    lattice(assn) ).

tff(tcon_Assertions_Oassn___Orderings_Oorder_431,axiom,
    order(assn) ).

tff(tcon_Assertions_Oassn___Orderings_Otop_432,axiom,
    top(assn) ).

tff(tcon_Assertions_Oassn___Orderings_Oord_433,axiom,
    ord(assn) ).

tff(tcon_Assertions_Oassn___Orderings_Obot_434,axiom,
    bot(assn) ).

tff(tcon_Assertions_Oassn___Groups_Ouminus_435,axiom,
    uminus(assn) ).

tff(tcon_Assertions_Oassn___Groups_Otimes_436,axiom,
    times(assn) ).

tff(tcon_Assertions_Oassn___Groups_Ominus_437,axiom,
    minus(assn) ).

tff(tcon_Assertions_Oassn___Power_Opower_438,axiom,
    power(assn) ).

tff(tcon_Assertions_Oassn___Groups_Oone_439,axiom,
    one(assn) ).

tff(tcon_Assertions_Oassn___Rings_Odvd_440,axiom,
    dvd(assn) ).

tff(tcon_Multiset_Omultiset___Quickcheck__Exhaustive_Ofull__exhaustive_441,axiom,
    ! [A17: $tType] :
      ( quickc3360725361186068524ustive(A17)
     => quickc3360725361186068524ustive(multiset(A17)) ) ).

tff(tcon_Multiset_Omultiset___Groups_Oordered__ab__semigroup__add_442,axiom,
    ! [A17: $tType] :
      ( preorder(A17)
     => ordere6658533253407199908up_add(multiset(A17)) ) ).

tff(tcon_Multiset_Omultiset___Groups_Ocancel__ab__semigroup__add_443,axiom,
    ! [A17: $tType] : cancel2418104881723323429up_add(multiset(A17)) ).

tff(tcon_Multiset_Omultiset___Groups_Ocancel__comm__monoid__add_444,axiom,
    ! [A17: $tType] : cancel1802427076303600483id_add(multiset(A17)) ).

tff(tcon_Multiset_Omultiset___Groups_Ocancel__semigroup__add_445,axiom,
    ! [A17: $tType] : cancel_semigroup_add(multiset(A17)) ).

tff(tcon_Multiset_Omultiset___Quickcheck__Random_Orandom_446,axiom,
    ! [A17: $tType] :
      ( quickcheck_random(A17)
     => quickcheck_random(multiset(A17)) ) ).

tff(tcon_Multiset_Omultiset___Groups_Ocomm__monoid__diff_447,axiom,
    ! [A17: $tType] : comm_monoid_diff(multiset(A17)) ).

tff(tcon_Multiset_Omultiset___Groups_Oab__semigroup__add_448,axiom,
    ! [A17: $tType] : ab_semigroup_add(multiset(A17)) ).

tff(tcon_Multiset_Omultiset___Groups_Ocomm__monoid__add_449,axiom,
    ! [A17: $tType] : comm_monoid_add(multiset(A17)) ).

tff(tcon_Multiset_Omultiset___Groups_Osemigroup__add_450,axiom,
    ! [A17: $tType] : semigroup_add(multiset(A17)) ).

tff(tcon_Multiset_Omultiset___Orderings_Opreorder_451,axiom,
    ! [A17: $tType] :
      ( preorder(A17)
     => preorder(multiset(A17)) ) ).

tff(tcon_Multiset_Omultiset___Groups_Omonoid__add_452,axiom,
    ! [A17: $tType] : monoid_add(multiset(A17)) ).

tff(tcon_Multiset_Omultiset___Orderings_Oorder_453,axiom,
    ! [A17: $tType] :
      ( preorder(A17)
     => order(multiset(A17)) ) ).

tff(tcon_Multiset_Omultiset___Orderings_Oord_454,axiom,
    ! [A17: $tType] :
      ( preorder(A17)
     => ord(multiset(A17)) ) ).

tff(tcon_Multiset_Omultiset___Groups_Ominus_455,axiom,
    ! [A17: $tType] : minus(multiset(A17)) ).

tff(tcon_Multiset_Omultiset___Groups_Ozero_456,axiom,
    ! [A17: $tType] : zero(multiset(A17)) ).

tff(tcon_Multiset_Omultiset___Groups_Oplus_457,axiom,
    ! [A17: $tType] : plus(multiset(A17)) ).

tff(tcon_Multiset_Omultiset___HOL_Oequal_458,axiom,
    ! [A17: $tType] :
      ( cl_HOL_Oequal(A17)
     => cl_HOL_Oequal(multiset(A17)) ) ).

tff(tcon_Multiset_Omultiset___Nat_Osize_459,axiom,
    ! [A17: $tType] : size(multiset(A17)) ).

tff(tcon_Product__Type_Oprod___Quickcheck__Exhaustive_Ofull__exhaustive_460,axiom,
    ! [A17: $tType,A18: $tType] :
      ( ( quickc3360725361186068524ustive(A17)
        & quickc3360725361186068524ustive(A18) )
     => quickc3360725361186068524ustive(product_prod(A17,A18)) ) ).

tff(tcon_Product__Type_Oprod___Quickcheck__Exhaustive_Oexhaustive_461,axiom,
    ! [A17: $tType,A18: $tType] :
      ( ( quickc658316121487927005ustive(A17)
        & quickc658316121487927005ustive(A18) )
     => quickc658316121487927005ustive(product_prod(A17,A18)) ) ).

tff(tcon_Product__Type_Oprod___Quickcheck__Random_Orandom_462,axiom,
    ! [A17: $tType,A18: $tType] :
      ( ( quickcheck_random(A17)
        & quickcheck_random(A18) )
     => quickcheck_random(product_prod(A17,A18)) ) ).

tff(tcon_Product__Type_Oprod___Finite__Set_Ofinite_463,axiom,
    ! [A17: $tType,A18: $tType] :
      ( ( finite_finite(A17)
        & finite_finite(A18) )
     => finite_finite(product_prod(A17,A18)) ) ).

tff(tcon_Product__Type_Oprod___Heap_Oheap_464,axiom,
    ! [A17: $tType,A18: $tType] :
      ( ( heap(A17)
        & heap(A18) )
     => heap(product_prod(A17,A18)) ) ).

tff(tcon_Product__Type_Oprod___HOL_Oequal_465,axiom,
    ! [A17: $tType,A18: $tType] : cl_HOL_Oequal(product_prod(A17,A18)) ).

tff(tcon_Product__Type_Oprod___Nat_Osize_466,axiom,
    ! [A17: $tType,A18: $tType] : size(product_prod(A17,A18)) ).

tff(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_467,axiom,
    condit6923001295902523014norder(product_unit) ).

tff(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_468,axiom,
    condit1219197933456340205attice(product_unit) ).

tff(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__distrib__lattice_469,axiom,
    comple592849572758109894attice(product_unit) ).

tff(tcon_Product__Type_Ounit___Quickcheck__Exhaustive_Ofull__exhaustive_470,axiom,
    quickc3360725361186068524ustive(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Obounded__semilattice__sup__bot_471,axiom,
    bounde4967611905675639751up_bot(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Obounded__semilattice__inf__top_472,axiom,
    bounde4346867609351753570nf_top(product_unit) ).

tff(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__linorder_473,axiom,
    comple5582772986160207858norder(product_unit) ).

tff(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__lattice_474,axiom,
    comple6319245703460814977attice(product_unit) ).

tff(tcon_Product__Type_Ounit___Boolean__Algebras_Oboolean__algebra_475,axiom,
    boolea8198339166811842893lgebra(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Obounded__lattice__top_476,axiom,
    bounded_lattice_top(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Obounded__lattice__bot_477,axiom,
    bounded_lattice_bot(product_unit) ).

tff(tcon_Product__Type_Ounit___Complete__Partial__Order_Occpo_478,axiom,
    comple9053668089753744459l_ccpo(product_unit) ).

tff(tcon_Product__Type_Ounit___Quickcheck__Random_Orandom_479,axiom,
    quickcheck_random(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Osemilattice__sup_480,axiom,
    semilattice_sup(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Osemilattice__inf_481,axiom,
    semilattice_inf(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Odistrib__lattice_482,axiom,
    distrib_lattice(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Obounded__lattice_483,axiom,
    bounded_lattice(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Owellorder_484,axiom,
    wellorder(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Oorder__top_485,axiom,
    order_top(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Oorder__bot_486,axiom,
    order_bot(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Opreorder_487,axiom,
    preorder(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Olinorder_488,axiom,
    linorder(product_unit) ).

tff(tcon_Product__Type_Ounit___Finite__Set_Ofinite_489,axiom,
    finite_finite(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Olattice_490,axiom,
    lattice(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Oorder_491,axiom,
    order(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Otop_492,axiom,
    top(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Oord_493,axiom,
    ord(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Obot_494,axiom,
    bot(product_unit) ).

tff(tcon_Product__Type_Ounit___Groups_Ouminus_495,axiom,
    uminus(product_unit) ).

tff(tcon_Product__Type_Ounit___Groups_Ominus_496,axiom,
    minus(product_unit) ).

tff(tcon_Product__Type_Ounit___Heap_Oheap_497,axiom,
    heap(product_unit) ).

tff(tcon_Product__Type_Ounit___HOL_Oequal_498,axiom,
    cl_HOL_Oequal(product_unit) ).

tff(tcon_Heap_Oheap_Oheap__ext___Quickcheck__Random_Orandom_499,axiom,
    ! [A17: $tType] :
      ( quickcheck_random(A17)
     => quickcheck_random(heap_ext(A17)) ) ).

tff(tcon_Heap_Oheap_Oheap__ext___HOL_Oequal_500,axiom,
    ! [A17: $tType] : cl_HOL_Oequal(heap_ext(A17)) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_501,axiom,
    bit_un5681908812861735899ations(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_502,axiom,
    semiri1453513574482234551roduct(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring__with__nat_503,axiom,
    euclid5411537665997757685th_nat(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__monoid__add__imp__le_504,axiom,
    ordere1937475149494474687imp_le(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring_505,axiom,
    euclid3128863361964157862miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring__cancel_506,axiom,
    euclid4440199948858584721cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Divides_Ounique__euclidean__semiring__numeral_507,axiom,
    unique1627219031080169319umeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__ring__cancel_508,axiom,
    euclid8851590272496341667cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors__cancel_509,axiom,
    semiri6575147826004484403cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__ab__semigroup__add_510,axiom,
    strict9044650504122735259up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__ab__semigroup__add_511,axiom,
    ordere580206878836729694up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add__imp__le_512,axiom,
    ordere2412721322843649153imp_le(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bit__operations_513,axiom,
    bit_se359711467146920520ations(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__comm__semiring__strict_514,axiom,
    linord2810124833399127020strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Quickcheck__Exhaustive_Ofull__exhaustive_515,axiom,
    quickc3360725361186068524ustive(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__comm__monoid__add_516,axiom,
    strict7427464778891057005id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__comm__monoid__add_517,axiom,
    ordere8940638589300402666id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring_518,axiom,
    euclid3725896446679973847miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1__strict_519,axiom,
    linord715952674999750819strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__semigroup__add_520,axiom,
    linord4140545234300271783up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Oring__bit__operations_521,axiom,
    bit_ri3973907225187159222ations(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1__no__zero__divisors_522,axiom,
    semiri2026040879449505780visors(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__nonzero__semiring_523,axiom,
    linord181362715937106298miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__ring_524,axiom,
    euclid5891614535332579305n_ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__strict_525,axiom,
    linord8928482502909563296strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Quickcheck__Exhaustive_Oexhaustive_526,axiom,
    quickc658316121487927005ustive(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors_527,axiom,
    semiri3467727345109120633visors(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add_528,axiom,
    ordere6658533253407199908up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add__abs_529,axiom,
    ordere166539214618696060dd_abs(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__comm__monoid__add_530,axiom,
    ordere6911136660526730532id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__group__add_531,axiom,
    linord5086331880401160121up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocancel__ab__semigroup__add_532,axiom,
    cancel2418104881723323429up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oring__1__no__zero__divisors_533,axiom,
    ring_15535105094025558882visors(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocancel__comm__monoid__add_534,axiom,
    cancel1802427076303600483id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring__strict_535,axiom,
    linord4710134922213307826strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1__cancel_536,axiom,
    comm_s4317794764714335236cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bits_537,axiom,
    bit_semiring_bits(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__comm__semiring_538,axiom,
    ordere2520102378445227354miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1_539,axiom,
    linord6961819062388156250ring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add_540,axiom,
    ordered_ab_group_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocancel__semigroup__add_541,axiom,
    cancel_semigroup_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring_542,axiom,
    linordered_semiring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring__0_543,axiom,
    ordered_semiring_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semidom_544,axiom,
    linordered_semidom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Quickcheck__Random_Orandom_545,axiom,
    quickcheck_random(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__mult_546,axiom,
    ab_semigroup_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1__cancel_547,axiom,
    semiring_1_cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oalgebraic__semidom_548,axiom,
    algebraic_semidom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__mult_549,axiom,
    comm_monoid_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__add_550,axiom,
    ab_semigroup_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring_551,axiom,
    ordered_semiring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring__abs_552,axiom,
    ordered_ring_abs(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Parity_Osemiring__parity_553,axiom,
    semiring_parity(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__add_554,axiom,
    comm_monoid_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__modulo_555,axiom,
    semiring_modulo(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring_556,axiom,
    linordered_ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__idom_557,axiom,
    linordered_idom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1_558,axiom,
    comm_semiring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__0_559,axiom,
    comm_semiring_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__mult_560,axiom,
    semigroup_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemidom__modulo_561,axiom,
    semidom_modulo(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemidom__divide_562,axiom,
    semidom_divide(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Num_Osemiring__numeral_563,axiom,
    semiring_numeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__add_564,axiom,
    semigroup_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ozero__less__one_565,axiom,
    zero_less_one(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring_566,axiom,
    comm_semiring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Nat_Osemiring__char__0_567,axiom,
    semiring_char_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oab__group__add_568,axiom,
    ab_group_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ozero__neq__one_569,axiom,
    zero_neq_one(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring_570,axiom,
    ordered_ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oidom__abs__sgn_571,axiom,
    idom_abs_sgn(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Parity_Oring__parity_572,axiom,
    ring_parity(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Opreorder_573,axiom,
    preorder(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Olinorder_574,axiom,
    linorder(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Omonoid__mult_575,axiom,
    monoid_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oidom__modulo_576,axiom,
    idom_modulo(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oidom__divide_577,axiom,
    idom_divide(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring__1_578,axiom,
    comm_ring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Omonoid__add_579,axiom,
    monoid_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1_580,axiom,
    semiring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__0_581,axiom,
    semiring_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ogroup__add_582,axiom,
    group_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Omult__zero_583,axiom,
    mult_zero(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring_584,axiom,
    comm_ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Oorder_585,axiom,
    order(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Num_Oneg__numeral_586,axiom,
    neg_numeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Nat_Oring__char__0_587,axiom,
    ring_char_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring_588,axiom,
    semiring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Oord_589,axiom,
    ord(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ouminus_590,axiom,
    uminus(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oring__1_591,axiom,
    ring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oabs__if_592,axiom,
    abs_if(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Otimes_593,axiom,
    times(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ominus_594,axiom,
    minus(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Power_Opower_595,axiom,
    power(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Num_Onumeral_596,axiom,
    numeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ozero_597,axiom,
    zero(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oplus_598,axiom,
    plus(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oring_599,axiom,
    ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oidom_600,axiom,
    idom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oone_601,axiom,
    one(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Odvd_602,axiom,
    dvd(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___HOL_Oequal_603,axiom,
    cl_HOL_Oequal(code_integer) ).

tff(tcon_Code__Numeral_Onatural___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_604,axiom,
    bit_un5681908812861735899ations(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Euclidean__Division_Ounique__euclidean__semiring__with__nat_605,axiom,
    euclid5411537665997757685th_nat(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__monoid__add__imp__le_606,axiom,
    ordere1937475149494474687imp_le(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Euclidean__Division_Ounique__euclidean__semiring_607,axiom,
    euclid3128863361964157862miring(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Euclidean__Division_Oeuclidean__semiring__cancel_608,axiom,
    euclid4440199948858584721cancel(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Osemiring__no__zero__divisors__cancel_609,axiom,
    semiri6575147826004484403cancel(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Ostrict__ordered__ab__semigroup__add_610,axiom,
    strict9044650504122735259up_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Oordered__cancel__ab__semigroup__add_611,axiom,
    ordere580206878836729694up_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__add__imp__le_612,axiom,
    ordere2412721322843649153imp_le(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Bit__Operations_Osemiring__bit__operations_613,axiom,
    bit_se359711467146920520ations(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Olinordered__comm__semiring__strict_614,axiom,
    linord2810124833399127020strict(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Quickcheck__Exhaustive_Ofull__exhaustive_615,axiom,
    quickc3360725361186068524ustive(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Ostrict__ordered__comm__monoid__add_616,axiom,
    strict7427464778891057005id_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Oordered__cancel__comm__monoid__add_617,axiom,
    ordere8940638589300402666id_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Euclidean__Division_Oeuclidean__semiring_618,axiom,
    euclid3725896446679973847miring(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Olinordered__ab__semigroup__add_619,axiom,
    linord4140545234300271783up_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Osemiring__1__no__zero__divisors_620,axiom,
    semiri2026040879449505780visors(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Olinordered__nonzero__semiring_621,axiom,
    linord181362715937106298miring(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Olinordered__semiring__strict_622,axiom,
    linord8928482502909563296strict(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Quickcheck__Exhaustive_Oexhaustive_623,axiom,
    quickc658316121487927005ustive(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Osemiring__no__zero__divisors_624,axiom,
    semiri3467727345109120633visors(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__add_625,axiom,
    ordere6658533253407199908up_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Oordered__comm__monoid__add_626,axiom,
    ordere6911136660526730532id_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Ocancel__ab__semigroup__add_627,axiom,
    cancel2418104881723323429up_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Ocancel__comm__monoid__add_628,axiom,
    cancel1802427076303600483id_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__1__cancel_629,axiom,
    comm_s4317794764714335236cancel(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Bit__Operations_Osemiring__bits_630,axiom,
    bit_semiring_bits(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Oordered__comm__semiring_631,axiom,
    ordere2520102378445227354miring(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Ocancel__semigroup__add_632,axiom,
    cancel_semigroup_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Olinordered__semiring_633,axiom,
    linordered_semiring(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Oordered__semiring__0_634,axiom,
    ordered_semiring_0(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Olinordered__semidom_635,axiom,
    linordered_semidom(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Quickcheck__Random_Orandom_636,axiom,
    quickcheck_random(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Oab__semigroup__mult_637,axiom,
    ab_semigroup_mult(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Osemiring__1__cancel_638,axiom,
    semiring_1_cancel(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Oalgebraic__semidom_639,axiom,
    algebraic_semidom(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__mult_640,axiom,
    comm_monoid_mult(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__diff_641,axiom,
    comm_monoid_diff(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Oab__semigroup__add_642,axiom,
    ab_semigroup_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Oordered__semiring_643,axiom,
    ordered_semiring(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Parity_Osemiring__parity_644,axiom,
    semiring_parity(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__add_645,axiom,
    comm_monoid_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Osemiring__modulo_646,axiom,
    semiring_modulo(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__1_647,axiom,
    comm_semiring_1(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__0_648,axiom,
    comm_semiring_0(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Osemigroup__mult_649,axiom,
    semigroup_mult(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Osemidom__modulo_650,axiom,
    semidom_modulo(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Osemidom__divide_651,axiom,
    semidom_divide(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Num_Osemiring__numeral_652,axiom,
    semiring_numeral(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Osemigroup__add_653,axiom,
    semigroup_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Ozero__less__one_654,axiom,
    zero_less_one(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring_655,axiom,
    comm_semiring(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Nat_Osemiring__char__0_656,axiom,
    semiring_char_0(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Ozero__neq__one_657,axiom,
    zero_neq_one(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Orderings_Opreorder_658,axiom,
    preorder(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Orderings_Olinorder_659,axiom,
    linorder(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Omonoid__mult_660,axiom,
    monoid_mult(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Omonoid__add_661,axiom,
    monoid_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Osemiring__1_662,axiom,
    semiring_1(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Osemiring__0_663,axiom,
    semiring_0(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Omult__zero_664,axiom,
    mult_zero(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Orderings_Oorder_665,axiom,
    order(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Osemiring_666,axiom,
    semiring(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Orderings_Oord_667,axiom,
    ord(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Otimes_668,axiom,
    times(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Ominus_669,axiom,
    minus(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Power_Opower_670,axiom,
    power(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Num_Onumeral_671,axiom,
    numeral(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Ozero_672,axiom,
    zero(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Oplus_673,axiom,
    plus(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Oone_674,axiom,
    one(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Odvd_675,axiom,
    dvd(code_natural) ).

tff(tcon_Code__Numeral_Onatural___HOL_Oequal_676,axiom,
    cl_HOL_Oequal(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Nat_Osize_677,axiom,
    size(code_natural) ).

tff(tcon_Heap__Time__Monad_OHeap___Quickcheck__Random_Orandom_678,axiom,
    ! [A17: $tType] :
      ( quickcheck_random(A17)
     => quickcheck_random(heap_Time_Heap(A17)) ) ).

tff(tcon_Heap__Time__Monad_OHeap___HOL_Oequal_679,axiom,
    ! [A17: $tType] : cl_HOL_Oequal(heap_Time_Heap(A17)) ).

tff(tcon_Heap__Time__Monad_OHeap___Nat_Osize_680,axiom,
    ! [A17: $tType] : size(heap_Time_Heap(A17)) ).

% Helper facts (6)
tff(help_fAll_1_1_U,axiom,
    ! [A: $tType,P8: fun(A,$o),X10: A] :
      ( ~ aa(fun(A,$o),$o,fAll(A),P8)
      | aa(A,$o,P8,X10) ) ).

tff(help_fNot_2_1_U,axiom,
    ! [P8: $o] :
      ( (P8)
      | aa($o,$o,fNot,(P8)) ) ).

tff(help_fNot_1_1_U,axiom,
    ! [P8: $o] :
      ( ~ aa($o,$o,fNot,(P8))
      | ~ (P8) ) ).

tff(help_fequal_2_1_T,axiom,
    ! [A: $tType,X10: A,Y: A] :
      ( ( X10 != Y )
      | aa(A,$o,aa(A,fun(A,$o),fequal(A),X10),Y) ) ).

tff(help_fequal_1_1_T,axiom,
    ! [A: $tType,X10: A,Y: A] :
      ( ~ aa(A,$o,aa(A,fun(A,$o),fequal(A),X10),Y)
      | ( X10 = Y ) ) ).

tff(help_fChoice_1_1_T,axiom,
    ! [A: $tType,P8: fun(A,$o)] :
      ( aa(A,$o,P8,fChoice(A,P8))
      = ( ? [X9: A] : aa(A,$o,P8,X9) ) ) ).

% Free types (1)
tff(tfree_0,hypothesis,
    linorder(b) ).

% Conjectures (1)
tff(conj_0,conjecture,
    aa(assn,$o,aa(assn,fun(assn,$o),entails,p),q) ).

%------------------------------------------------------------------------------